192 Scientific Paper Structural Engineering International 22014

Peer-reviewed by international ex-perts and accepted for publication by SEI Editorial Board

Paper received July 15 2013Paper accepted January 24 2014

Wetting and Drying of Concrete Modelling and Finite Element Formulation for Stable Convergence Kaustav Sarkar Research Scholar Bishwajit Bhattacharjee Prof Department of Civil Engineering Indian Institute of

Technology Delhi IIT Delhi New Delhi Delhi India Contact srkrkaustavgmailcom

DOI 102749101686614X13830790993401

Abstract

The simulation of moisture movement in concrete is vital for estimating its durability performance under the given conditions of service A robust analy-sis of the nonlinear phenomenon relies on the implementation of an efficient numerical algorithm This entails a comprehension of the underlying mathe-matical models the numerical analysis scheme and the related issues of stabil-ity and convergence This paper describes a computational scheme to facilitate the estimation of moisture distribution in concrete subjected to wettingndashdrying exposure due to intermittent rains an exposure type which severely impairs the durability performance of structural concrete under tropical climatic con-ditions Beginning with a brief review of the conventional moisture transport models for concrete this paper proposes a modified model using dimension-less parameters which enhances the computational efficiency of the delineated scheme This paper also describes the formulation of a one-dimensional non-linear finite element scheme for the modified model and discusses the issue of numerical oscillations concomitant to the numerical analysis of the problem The suitability of the first-order element in providing stable convergence when implemented in conjunction with a lumped-mass scheme has been mathemati-cally interpreted The validity of the delineated proposition has been illustrated using a test problem

Keywords wettingndashdrying exposure moisture transport models FE analysis numerical oscillation concrete wetting-drying exposure

of its performance while under service The pertinence is especially critical for exposure conditions that strongly influ-ence the state of moisture in concrete and hence expedite the impending degradation

In this context the effect of alternat-ing wet and dry conditions is known to be very severe in causing corrosion of steel reinforcement embedded in structural concrete In a tropical cli-mate the exposure is predominantly due to the occurrence of intermittent rainfall events While exposure to rain causes a conspicuous ingress of mois-ture in concrete subsequent drying renders the medium unsaturated and the embedded reinforcement suscepti-ble to corrosion owing to the presence of both air and moisture in the pores Estimating the possible extent of ensu-ing degradation requires appropriate modelling of moisture transport and implementation of an efficient numeri-cal scheme to address the highly non-linear character of the problem

The modelling of moisture movement in hydrated cement systems has been

conventionally carried out using two distinct approaches The more rigorous of the two starts with the formulation of mass and energy balance equations at a microlevel followed by volume averaging to obtain the macroscopic description The method is particularly suitable for modelling the hygrother-mal behaviour of concrete at early ages4 and the deformations occurring due to the effect of shrinkage and creep5 This approach however relies on the experimental estimation of sev-eral material properties and is thus less viable for durability studies The alter-native approach is phenomenological and starts directly from the macro-scopic level with the effect of various influencing factors lumped into a few model parameters to be determined experimentally This approach culmi-nates into a diffusion model and which because of its relative simplicity has been used extensively to describe the transfer of moisture and heat in porous building materials6ndash10 Most of these studies have implemented numerical methods to handle the mathematical nonlinearity of diffusion model but have not elaborated adequately the associated issues of numerical stabil-ity and oscillation-free convergence In contrast several studies pertaining to the domain of water-infiltration mod-elling in unsaturated soil medium have evaluated the efficiency of numerical schemes and tested their stability in relation to element order mass type and time-step size11ndash18 the delineated conclusions however remain to be tes-tified and adapted for the analysis of moisture movement in cement-based materials

This paper provides an exhaustive description of the outlined prob-lem reviews the associated aspects of moisture transport modelling and elaborates the constitution of a robust finite element (FE) scheme for effi-cient analysis of the problem At the outset the paper briefly describes the formulation of conventional equations governing the phenomena of moisture transport in unsaturated concrete The

Introduction

Structural concrete during service con-stantly remains exposed to the action of temperature and humidity cycles oper-ating at diurnal and seasonal scales Under tropical climatic conditions it is also subjected to intermittent spells of rainfall These conditions induce flow of moisture in liquid and vapour phases through the porous matrix of concrete often rendering it unsaturated1ndash3 The subsequent interaction of concrete with gases and other invading species carried by the imbibed liquid phase alters its physical and chemical consti-tution resulting in gradual degradation and consequent disruption of durabil-ity The degradation mechanisms being critically moisture dependent necessi-tate the study of moisture movement in concrete to enable a reliable prediction

Structural Engineering International 22014 Scientific Paper 193

model is subsequently restated using a set of nondimensional terms to mini-mize the range of orders involved in the underlying calculations This aids in reducing the numerical errors con-comitant to computer arithmetic and in easily adapting the simulated results to predict the behaviour of hydrauli-cally similar materials Further using the modified model this paper elu-cidates the formulation of a one-dimensional nonlinear FE scheme to determine the evolution of moisture distribution in concrete subjected to wettingndashdrying exposure due to inter-mittent rains The paper also discusses the issue of numerical oscillations typi-cally related to the FE-based analysis of nonlinear diffusion models A pru-dent evaluation of the system of non-linear algebraic equations generated in the FE analysis has been carried out to ascertain the suitability of the mass type to be implemented with a linear element for obtaining an oscillation-free convergence The study leads to a set of criteria governing the maximum permissible value of time-step size for stable convergence The validity of the derived criteria has been tested for a typical concrete subjected to rainfall exposure

The thrust of the present work is therefore on establishing the condi-tions that enable a robust one-dimen-sional transient nonlinear FE analysis of moisture flow in concrete described using a phenomenological model The developed scheme can be imple-mented to examine the impact of various influencing parameters on the state of moisture distribution in con-crete without resorting to extensive experimental programmes The pres-ent work thus bears a special signifi-cance in the context of model-based service life analysis of concrete struc-tures in particular for tropics where the occurrence of rainfall in intermit-tent spells is a characteristic climatic phenomenon

Brief Review of Phenomenological Moisture Transport Models

Moisture Ingress

When a concrete surface is exposed to rain flux the concomitant ingress of moisture is caused due to the action of capillary forces and the resulting dis-tribution of moisture is governed by the gradient of capillary pressure It is thus rational to represent the ensuing

movement of water using convection models that consider the gradient of pressure as the primary driving force19 Darcyrsquos law the first convection model was proposed in the form of an empir-ical equation to describe the flow of water through saturated porous media20 Mathematically Darcyrsquos law is stated as

Q = ndashksA DpL (1a)

ie u = ndashks DpL (1b)

In a local sense Eq (1b) can be expressed as

u = ndashksnablap (1c)

where Q is the volume rate of flow (m3 sminus1) ks is the Darcean permeabil-ity (kgminus1 m3 s) A is the cross-sectional area of specimen (m2) p is the hydro-static head (kg mminus1 sminus2) L is the thick-ness of the medium (m) and u is the average velocity of flow (msminus1)

By expressing the hydrostatic pres-sure in terms of pressure potential the equation further reduces to

u = ndashKsnablap (1d)

where Ks = ksrwg is the permeability of the saturated medium (m sminus1) P = prwg is the pressure potential (m) rw is the density of water (kg mminus3) and g is the acceleration due to gravity (msminus2)

When applied to the analysis of unsat-urated flow the permeability term in Darcyrsquos relation is replaced by a gen-eralized transport property called the capillary conductivity K(ql)(msminus1) which is a strong function of the liq-uid water content and the pressure potential is replaced by the capillary potential y (m)21 Thus Eq (1d) gets modified to

u = ndashK(ql)nablay (1e)

The above relation can be applied directly to describe the steady-state unsaturated flow However to incor-porate the effect of unsteady condi-tions Eq (1e) for apparent velocity has to be combined with the mass bal-ance equation

icircrvicirc t + nabla(jl) = 0 (2a)

where rv is the partial moisture den-sity (kg mminus3) t is the time variable (s) and jl is the liquid flux (kg mminus2 sminus1)Equation (2a) can be restated in terms of volumetric liquid water content ql (m3m3) as

icirc (rwql) icirc t + nabla(rwu) = 0 (2b)

The combination of Eqs (1e) and (2b) results in

icircqlicirc t = nabla(K(ql)nablay) (2c)

Equation (2c) can be expressed in terms of q rather than y as

icircqlicirc t = nabla(D(ql)nablaql) (2d)

with D(ql) = K(ql)icircy icircql = K(ql)(icircqlicircy )= K(ql)c(ql) where c(ql) is specific water capacity (mminus1)

The extended Darcyrsquos law represented by Eq (2d) governs the flow of water in unsaturated porous medium and is also referred to as Richardrsquos equa-tion Mathematically it is analogous to Fickrsquos law of diffusion

Moisture Egress

The evaporative drying of moisture imbibed in concrete starts soon after the end of a rainfall event and involves flow in both liquid and vapour phases This multiphase movement of water can be described using the diffusion models that assume concentration gradient to be the primary driving force Krischerrsquos model in particular considers the transport of water and water vapour as caused sepa-rately due to the respective gradients of liquid moisture content and partial pres-sure of vapour in air19Thus the liquid and vapour fluxes are expressed as

jl = ndashrwDl(ql)nablaql (3a)

jv = ndash(DvmRvT)nablapv (3b)

The corresponding mass balance equa-tions are given by

rw icircql ___ icirc t

= rwnabla(Dl(ql)nablaql) (4a)

(4b)

Here Dl(ql) is the diffusivity of liquid water (m2sminus1) Rv is the gas constant of water vapour (J kgminus1 Kminus1) T is the tem-perature (K) pv is the partial vapour pressure (Nmminus2) Dv is the diffusion coefficient of water vapour in air (m2 sminus1) and m is the resistance factor of the porous medium to vapour diffusion

The total mass balance of moisture after accounting for the effects of the Stefan diffusion and the concentration of water in the pores can be obtained from Eqs (4a) and (4b) as

(4c)

194 Scientific Paper Structural Engineering International 22014

where f is the open porosity (m3m3) and pt is the pressure of vapourndashair mixture (Nmminus2)

Equation (4c) represents Krischerrsquos model for combined liquid and vapour transport The expression can be fur-ther simplified by considering the aspects of capillary pressure and rela-tive humidity that prevail within the porous matrix

For a given moisture content the capil-lary pressure within the porous matrix is constant and the relative humidity of the associated vapour phase is

h = pv ps (5a)

where ps is the saturated vapour pres-sure (Nmminus2) Now using Kelvinrsquos equa-tion pv can be stated as

(5b)

where pc is the capillary pressure (Nmminus2)

From the above expression it becomes evident that under isothermal con-ditions the partial vapour pressure depends only on capillary pressure and hence varies as a function of the mois-ture content Thus from Eq (5b)

nablapv = ps(icirchicircql)T nablaql (5c)

The vapour flux in Eq (3b) can now be expressed as

(6a)

Thus as in Eq (4c) the total mass bal-ance of moisture can now be expressed as

(6b)

that is

icirc ___ icirct (ql + qv) = nabla(Dl(ql)nablaql) + nabla(Dv(ql)nablaql)

which is of the form

icircq ___ icirc t = nabla(D(ql)nablaql) (6c)

Here qv is the volumetric water vapour content (m3m3) Dv(ql) is the diffusiv-ity of water vapour in porous medium (m2 sminus1) q = ql + qv is the total volumet-ric moisture content and D(ql) = Dl(ql) + Dv(ql) is the isothermal moisture

diffusivity (m2 sminus1) The derived equa-tion governing the combined flow of liquid and vapour phases of water takes up the form of Fickrsquos law of diffusion

Models for Diffusivity Function

Diffusivity of a porous material is a measure of its tendency to transmit fluid through its interconnected pore space while in a state of unsatura-tion21 When the fluid being trans-mitted is water it is referred to as the hydraulic diffusivity The hydrau-lic diffusivity is a continuum level property that takes into account the combined effect of pore structure geometry state of saturation of pore space and the transport mechanisms driving the moisture movement phe-nomenon22 In a phenomenological model the hydraulic diffusivity is the only decisive parameter that governs the evolution of moisture distribution in a medium Considering the pore structure to remain stable the strong dependence of hydraulic diffusivity on moisture content is generally repre-sented using monotonically increasing functions The trend follows from the underlying mechanisms of moisture transport which enhance the rate of mass transfer with increasing levels of pore saturation initiating with the slow gaseous diffusion of vapour in a near-dry state the rate compounds through the movement of water in adsorbed and condensed layers to the phase of bulk flow in pore space under the rising levels of saturation2324 Furthermore the empiric modelling of hydraulic diffusivity leads to typical functional forms for wetting and dry-ing conditions characterizing the asso-ciated moisture transport mechanisms For liquid water ingress which owes to the effect of capillarity the hydraulic diffusivity has been successfully repre-sented by an exponential function of the form21ndash28

Dr (qr) = Dd_wetexp(nqr) (7a)

with

qr = (q ndash qo)(qs ndash qo)

where qs is the saturation moisture content (m3m3) qo is the moisture content corresponding to the capillary dry state (m3m3) Dr(qr) is the hydrau-lic diffusivity as a function of nondi-mensional moisture content (m2 sminus1) and Dd_wet is the wetting diffusivity of concrete in the capillary dry state (m2 sminus1) The value of n has been suggested

to be in the range of 6ndash8 for building materials29 A precise investigation carried out using nuclear magnetic resonance has suggested a value of n = 6 for an initially dry concrete28

The drying behaviour of concrete on the other hand has been described by an S-shaped diffusivity function of the form30ndash33

(7b)

where Dw_dry is the drying diffusivity of concrete in completely wet state (m2 sminus1) ao is the ratio of minimum to maximum diffusivity and the param-eters n and qc characterize respec-tively the spread and the location of drop of the curve

The empirical diffusivity models broadly describe the hygroscopic behaviour of concrete under wetting and drying conditions The application of these models for a given concrete is however dependent on the experi-mental estimation of the function parameters Comprehensive studies to establish the dependence of these parameters on temperature initial saturation level and parameters gov-erning the pore structure properties of concrete would enable a direct imple-mentation of these models for durabil-ity studies

FE Formulation

Governing Equation

As has been described in the Brief Review of Phenomenological Moisture Transport Models section the phe-nomenon of one-dimensional isother-mal moisture transport in a porous medium can be represented using the equation

icircq

___ icirc t = icirc ___ icircx ( D(q )

icircq ___ icircx ) (8a)

Eq (8a) is a nonlinear parabolic partial differential equation The nonlinearity is caused by the term D(q ) a strong function of moisture content that fol-lows distinct characteristic trends for wetting and for drying conditions respectively Owing to its highly non-linear nature Eq (8a) is not amenable to analytical solution and requires the implementation of numerical proce-dures The equation being first order in time and second order in space the

Structural Engineering International 22014 Scientific Paper 195

solution of the problem relies on the specification of an initial condition and two boundary conditions The bound-ary conditions may either be provided as the value of moisture content q (Dirichletessential boundary condi-tion) or as the value of moisture flux stated as D(q )icircqicircx = ndashVo (Neumannnatural boundary condition) where Vo is the orthogonal rain flux incident on the exposed surface (msminus1) The initial moisture content across the physi-cal domain of analysis is generally described by an initial moisture profile q (xt = 0) = qini(x)

Since the parameters constituting the given problem range over several orders of magnitude restating Eq (8a) using the following dimensionless terms

Reduced moisture q r = (q ndash qo)(qs ndash qo)

(8b)

Reduced distance xr = (VoDd_wet)x (8c)

Reduced distance tr = ( V o 2 Dd_wet)t (8d)

aids in minimizing the computing errors

Restating Eq (8a) in terms of non-dimensional parameters gives

(8e)

A scrutiny of these terms reveals their appropriateness in reducing the range of orders involved in computation the most prominent aspect in the modi-fied model represented by Eq (8e) is the normalization of the diffusivity function Dr(qr)

with the term Dd_wet Moreover expressing the model in terms of these parameters enables adaptation of results to determine the behaviour of all hydraulically similar materials The idea will be clarified fur-ther in the following section

Considering the following simplified representations for the hydraulic dif-fusivity function

for wetting Dr (qr) = Dd_wet fw(q r) (8f)

for drying Dr (qr) = Dw_dry fd(q r) (8g)

where fw(q r) and fd(q r) are functions of reduced moisture content in wet-ting and drying diffusivity functions respectively and subsequently equat-ing the boundary flux term

to wetting and drying fluxes yields respectively the following conditions

for wetting fw (qr) icircqr ___ icircxr

= ndash1(q s ndash q o)

(8h)

for drying

(8i)

where Jv is drying boundary flux acting in the direction of outward normal to the surface (msminus1)

It is worth mentioning that the adap-tation of Eq (8a) most widely used in previous studies implements only the reduced moisture content as in Eq (8b) and has the form

(8j)

Formulation of Element-Level Governing Equation

Using the governing differential equa-tion stated in Eq (8e) the FE formula-tion can be carried out using Galerkinrsquos weighted residual technique34 For the given problem the weighted residual statement is given as

(9a)

where wk is the weight function (k = 1 to number of nodes in an element) and l is the reduced size of element consti-tuting the FE mesh

Integration of Eq (9a) by parts leads to

(9b)

Now for a linear element with two nodes it is known that

(9c)

Thus

(9d)

where H is the interpolation matrix B is the gradient matrix and de is the vector of elemental degrees of freedom

Following Galerkinrsquos method the weights in Eq (9b) are taken equal to the interpolation functions

Thus for w1 = 1 ndash (xrl) Eq (9b) yields

(9e)

And for w2 = (xrl)

(9f)

Combining Eqs (9e) and (9f) the gen-eral element-level governing equation can be obtained as

(9g)

Substituting Eqs (8f) and (8g) in Eq (9g) and using Eqs (8h) and (8i) in tandem the governing equation for the boundary element respective to the cases of wetting and drying can be obtained as

(9h)

(9i)

196 Scientific Paper Structural Engineering International 22014

An inspection of Eqs (9h) and (9i) reveals their dependence on the functions fw(qr) and fd(qr) which are known to follow characteristic expo-nential and S-shaped forms for the respective cases of wetting and drying of concrete Also as the terms (Dw_dry Dd_wet) and (1 qs ndash qo) are relative in nature the derived expressions can be considered to represent the behaviour of all hydraulically similar materials

Equations (9h) and (9i) are semi-dis-crete and can be represented in matrix form as

[m]de + [k]de = q (9j)

where m and k are the element-level mass and the diffusivity matrices respectively de is the vector of time derivatives of elemental degrees of freedom and q is the vector of ele-mental nodal fluxes

In order to obtain a fully discretized system of equations the time deriva-tives of the field variable in these equa-tions are to be further approximated using the method of finite difference Adopting the CrankndashNicolson scheme a completely discrete system of equa-tions is obtained as

([m] + 05Δtr[k]n+1)den+1

= ([m] + 05Δtr[k]n)den

+ 05Δtr(qn+1 + qn) (9k)

where the superscripts n and (n+1) denote the previous and present time levels respectively in the course of transient analysis

Numerical Stability of the FE Scheme

The highly nonlinear nature of the problem necessitates the implementa-tion of an extremely refined mesh and a small time-step size to facilitate the stable convergence of simulated results However earlier studies based on FE analysis of parabolic partial differen-tial equations reported that the mini-mization of the time-step size beyond a certain threshold value induces numerical oscillations in the simulated results3536 The issue was addressed in the context of heat diffusion problems by determining the minimum time-step sizes for some typical one- and two-dimensional elements16 These crite-ria16 that had actually been developed for strictly constant material properties were tested for unsaturated seepage problems involving mildly nonlinear material properties and were found to

be equally useful17 In another study18 it was demonstrated that the use of lumped-mass scheme with linear ele-ments offers stable convergence and consistent results for FE-based solu-tion of Richardrsquos equation The obser-vation was however empirical and was not deduced mathematically

In the following section an attempt has been made to mathematically interpret the issue of numerical oscilla-tions and to ascertain the suitability of a lumped-mass type implemented with linear elements in providing a stable convergence In addition a set of cri-teria has been developed for the modi-fied model to regulate the variation of time-step size as a function of element length and hydraulic diffusivity for stable convergence of the FE scheme

General Conditions for Stability

To determine the nodal values of the field variable at a particular time-step all the element-level equations are assembled to obtain a set of nonlinear algebraic equations in the form [A]X = b with

[A] = [M] + 05Δtr[K]n+1 (10a)

b = ([M] ndash 05Δtr[K]n)Xn

+ Δtr(Qn+1 + Qn) (10b)

where [M] and [K] are respectively the global consistent mass and diffu-sivity matrices and X and Q are the global vectors for degrees of freedom and nodal fluxes respectively

It is evident that spatial oscillations in the simulated results are caused by the numerical characteristics of matrix [A] and vector b From basic reasoning it can be perceived that the occurrence of numerical oscillations can be elimi-nated if

1 [A] is strictly diagonally dominant with positive terms in the principal diagonal and negative terms else-where with magnitudes lesser than the smallest element of the principal diagonal

2 All the elements of b are posi-tive which necessarily depends on the diagonal elements of ([M] ndash 05Δtr[K]n)Xn to remain positive

Depending on the method to be imple-mented for solving the system [A]X = b the implications of the stated conditions would be as follows

(a) The constitution of [A] would be such that its manipulation using any of the elimination methods37

bull will not produce negative ele-ments on its principal diagonal

bull will not require row subtractions during the elimination phase thus the elements in b shall always remain positive provided condition (2) is already met

(b) If [A] is to be inverted for X = [A]ndash1b then none of the elements in the inverted matrix would be negative it would otherwise fetch negative values in the resulting vector

(c) The diagonal dominance of [A] is a suffi cient condition for the itera-tive methods to converge for any initial solution vector37

The specified conditions restrict the generation of negative values at nodes which generally manifest in an alter-nating pattern when a wetting front invades a dry medium They also arrest the oscillations causing a saw-tooth pattern in simulated moisture profile Fulfilling the proposed con-ditions thus eliminates both forms of oscillations that are reported to influ-ence the quality of results simulated under a FE scheme17 Furthermore it is to be recognized that the numerical oscillations are primarily caused due to the composition of matrix [A] that is in relation to the violation of condi-tion (1) The perturbations caused due to the multiplication of the negative diagonal elements of [M] ndash 05Δtr[K]n with the elements of vector [X]n are largely counterbalanced by the contri-bution of the off-diagonal elements in the respective rows which are always positive and have the same order of magnitude

The mathematical criteria to be adopted to enforce the proposed con-ditions depend on the type of element being implemented in the FE model This study considers only the instance of linear element in the following section

Stability Criteria

For the FE formulation presented in the Formulation of Element Level Governing Equation section the global characteristic matrices for the phase of wetting will be tri-diagonal and are defined as

(11a)

Structural Engineering International 22014 Scientific Paper 197

(11b)

where M ij L is an element of the ith row

and jth column of lumped-mass matrix

The revised criteria for satisfying con-ditions (1) and (2) would now be

which reduces to

Δtr lt l2[ fw(qrmax) ndash fw(qrmin)] (12b)

and

which reduces to

Δtr lt l2fw(qrmax) (12c)

Equations (12b) and (12c) establish that adopting the lumped-mass scheme in conjunction with a two-noded linear element allows the minimization of time-step size to achieve convergence without the need of simultaneously refining the mesh density It is impor-tant here to note that although the condition in Eq (12c) imposes a more stringent limit on Δtr in comparison to Eq (12b) the two conditions converge within a value of qrmax = 02 for fw(qr) = exp (6qr) qrmin = 0 and l =05 as is depicted in Fig 1 In fact the conver-gence occurs at a steeper rate for finer meshes and remains within the stated threshold for the coarsest meshes prac-

ticable for the problem The delineated criteria are thus particularly stringent for near-dry conditions and can be relaxed at higher levels of moisture content using average value of mois-ture content that is qravg in place of qrmax

Similarly for the phase of drying the governing criteria for time-step size corresponding to conditions (1) and (2) would be

(13a)

and (13b)

Implementing the Time-Step Size Criteria

To illustrate an application of the delineated time-stepping criteria and the influence of mass type on numeri-cal oscillations we consider a one-dimensional moisture ingress problem involving an initially dry (ie qr = 0 at tr = 0) concrete medium of length L = 010 m subjected to a rainfall of inten-sity Vo = 10 mmh until the attainment of surface saturation (ie qr = 1 at xr = 0) The typical material parameters opted for the analysis are qo = 0 m3m3 qs = 01811 m3m3 Dw_dry = 945 times 10ndash10m2s and n = 6 The FE mesh has been constituted using linear elements of reduced size l = 05 which equals 017 mm for the adopted parameter values

The FE analysis of the considered problem governed by Eq 8(e) has been carried out by four distinct approaches

(i) using a consistent mass type and varying the time-step size in accordance to Eq (11c)

(ii) with a lumped-mass type and varying the time-step size as per Eqs (12b) and (12c)

(iii) as in (i) but relaxing the time-stepping constraints in Eq (11c) by replacing qrmax by qravg

(iv) as in (ii) but relaxing the time-step-ping constraints in Eqs (12b) and (12c) by replacing qrmax by qravg

The time-stepping in FE analysis has been implemented using the following scheme

1 At any instant tr calculate the pres-ent time-step size (Δtrpresent) if tr = 0 take Δtrpresent = 001l2fw(qr = 0) and for tr gt 0 take Δtrpresent = Δtrnext as determined in the previous instant of analysis

where m is the order of the matri-ces which equals the total number of unknowns Mij and Kij represent an element of the ith row and jth column of the global consistent mass and dif-fusivity matrices respectively

To fulfil conditions (1) and (2) at any time level the following criteria are to be satisfied

which reduces to

l23fw(qrmax) lt Δtr

lt 2l23[ fw(qrmax) ndash fw(qrmax)] (11c)

and

l __ 6 ndash 05Δtr 05 ___

l [2fw(qrmax)] gt 0

which reduces to

Δtr lt l23fw(qrmax) (11d)

Here qrmax and qrmin represent the maximum and minimum values respectively of the reduced moisture content at any time level

As is evident Eqs (11c) and (11d) con-tradict each other and cannot be satisfied simultaneously However as has been pointed out earlier and will be dem-onstrated in what follows violating the constraint imposed by condition (2) that is Eq (11d) results only in minor oscilla-tions that gradually damp out Equation (11c) however imposes a more severe constraint and explains the occurrence of oscillations that set in when the time-step size (Δtr) is reduced without simul-taneously refining the mesh while using a consistent mass scheme with linear ele-ments18 The identified constraints can however be circumvented by replac-ing the consistent mass matrix with a lumped-mass scheme defined as

(12a)

0100

1

2

Max

imum

Δt r

3

4

5

02 03 04Maximum moisture content ( tmax)

05 06 07 08 09 1

12b12c

Fig 1 Convergence pattern of the maxi-mum time limit criteria delineated in Eqs (12b) and (12c)

198 Scientific Paper Structural Engineering International 22014

02

0

005

ndash005

0

0 500 1000 1500

Time step number2000 2500 2 4

Distance fromexposed face (mm)

6 8 1000

02

04

06

08

1

0 500 1000 1500

Time step number

2000 2500

04

Red

uced

moi

stur

eR

educ

ed m

oist

ure

06

08

1(a) (c)

(b)

Fig 2 Approach (i) (a) evolution of nodal values (b) oscillating emergence of nodal val-ues from initial dry state (c) final moisture distribution across the medium

(a) (c)

(b)

02

0

005

ndash005

0

0 200 400 600 800 1000

Time step number14001200 1600 2 4

Distance fromexposed face (mm)

6 8 1000

02

04

06

08

1

0 400200 800600 1000 1200

Time step number

1400 1500

04

Red

uced

moi

stur

eR

educ

ed m

oist

ure

06

08

1

Fig 3 Approach (ii) (a) evolution of nodal values (b) oscillation-free emergence of nodal values from initial dry state (c) final moisture distribution across the medium

(a) (c)

(b)

02

0

005

ndash005

0

0 5 10 15 20 25

Time step number3530 4540 2 4

Distance fromexposed face (mm)

6 8 1000

02

04

06

08

1

0 105 2015 25 30

Time step number

35 40 45

04

Red

uced

moi

stur

eR

educ

ed m

oist

ure

06

08

1

Fig 4 Approach (iii) (a) evolution of nodal values (b) oscillating emergence of nodal values from initial dry state (c) final moisture distribution across the medium

2 Compute moisture distribution at the present instant tr

3 Calculate the limiting-time-step size (Δtrlimit) using appropriate criteria

4 Calculate time-step size for next instant to be simulated (Δtrnext) if (Δtrlimit gt 2 Δtrpresent) then Δtrnext = 2 Δtrpresent else Δtrnext = 095 Δtrlimit

5 Increment time tr =tr + Δtrpresent

Figures 2ndash5 present the simulated results corresponding to each of the analysis approaches These illustra-tions describe the evolution of reduced moisture content at nodes in succes-sive time-steps and the distribution of moisture across the medium at the attainment of surface saturation The observations clearly depict that the FE analysis based on consistent mass approach induces numerical oscillations in the simulated results An inspection of Figs 2 and 4 further reveals that the oscillations strongly influence the initial stages of analysis and then gradually reduce to a rela-tively less intense level with a consis-tent effect on the nodes in a near-dry state at the toe of the wetting front The analysis based on lumped-mass scheme on the other hand ensures a stable con-vergence of nodal values as is depicted in Figs 3 and 4 and also leads to con-vergence in relatively fewer time-steps thereby economizing the computation time Table 1 provides a summary of the time-steps computed and the simu-lated parameters Relaxing the time-stepping constraints as in approaches (iii) and (iv) causes some perturbations in the initial evolution of nodal values as depicted in Figs 4 and 5 However as summarized in Table 1 the analysis leads to satisfactory results with a sig-nificantly reduced computational time These findings thus establish the advan-tage of implementing a lumped-mass scheme in conjunction with the relaxed time-stepping criteria delineated in the study for achieving computational effi-ciency and stability in the FE analysis of moisture transport in concrete

The conventional model stated in Eq (8j) which has been widely adopted in several previous studies was also implemented to analyse the consid-ered test problem The observed trends of numerical stability were found to be similar to the ones obtained with the model proposed in this study However the computational expense for analys-ing the conventional model was found to be higher in relation to the proposed model A comparison of the results has been summarized in Table 1

Structural Engineering International 22014 Scientific Paper 199

Discussion and Conclusions

A rational study of moisture move-ment in concrete rests on appropriate modelling of associated mass transport phenomena and on the efficacy of the numerical analysis of the constituted mathematical model A sound compre-hension of these facets is essential for a plausible treatment of the problem The present work has addressed these aspects in the context of a rain-induced wettingndashdrying scenario and in its dis-course has emphasized the following ideas

(a) The parabolic partial differential equation provides a phenomeno-logical description of the fl ow of moisture in concrete under wet-ting and drying conditions albeit with distinct diffusivity functions

(b) The presently available empirical models for hydraulic diffusivity incorporate parameters which are known to depend on temperature

and the pore structure proper-ties of concrete This dependence however remains to be estab-lished mathematically in terms of the generic concrete proper-ties Future studies towards the refi nement of hydraulic diffusiv-ity models would enable a direct application of the delineated phe-nomenological approach in ana-lysing the moisture transport in structural concrete

(c) The restatement of the governing equation using the set of dimen-sionless terms qr = (q ndashqo)(qs ndashqo) xr = (VoDd_wet)x and tr = ( V o 2 Dd_wet)t (i) reduces the range of orders involved and hence minimizes the errors inherent in numerical computations (ii) pro-vides a representation character-istic of all hydraulically similar materials and (iii) economizes the computation time

(d) Galerkinrsquos method can be effec-tively implemented to formulate

a FE scheme for numerical anal-ysis of the moisture transport problem Its application has been elaborated using the modifi ed model to describe the constitu-tion of the characteristic matrices in detail

(e) The occurrence of numerical oscil-lations inherent to the FE analysis of diffusion models relates to the nature of the matrices involved in the analysis and can be effectively controlled by imposing the condi-tions outlined in the study

(f) The effi ciency of lumped-mass scheme implemented in conjunc-tion with a linear element for one-dimensional analysis has been mathematically interpreted A concomitant time-stepping crite-rion for stable convergence has been derived and its viability has been demonstrated through a test problem

The deliberations presented in this paper provide an insight into the math-ematical framework underlying the study of moisture transport phenom-ena in concrete and facilitate a robust FE analysis of moisture distribution in concrete subjected to wettingndashdrying exposure

References

[1] Andrade C Sarria J Alonso C Relative humidity in the interior of concrete exposed to natural and artificial weathering Cement Concrete Res 1999 29(8) 1249ndash1259

[2] Ryu DW Ko JW Noguchi T Effect of simu-lated environmental conditions on the internal relative humidity and relative moisture con-tent distribution of exposed concrete Cement Concrete Comp 2011 33(1) 142ndash153

[3] Zhang J Gao Y Han Y Interior humidity of concrete under dry-wet cycles J Mater Civil Eng 2012 24(3) 289ndash298

[4] Gawin D Pesavento F Schrefler BA Hygro-thermo-chemo-mechanical modelling of con-crete at early ages and beyond Part I Hydration and hygro-thermal phenomena Int J Numer Meth Eng 2006 67(3) 299ndash331

[5] Gawin D Pesavento F Schrefler BA Hygro-thermo-chemo-mechanical modelling of con-crete at early ages and beyond Part II Shrinkage and Creep of Concrete Int J Numer Meth Eng 2006 67(3) 332ndash363

[6] Saetta AV Schrefler BA Vitaliani RV The carbonation of concrete and the mechanism of moisture heat and carbon dioxide flow through porous materials Cement Concrete Res 1993 23(4) 761ndash772

[7] Saetta AV Schrefler BA Vitaliani RV 2-D model for carbonation and moistureheat flow in porous materials Cement Concrete Res 1995 25(8) 1703ndash1712

Approach No of time-steps computed

Time taken for surface saturation

(s)

Area under fi nal moisture profi le

(m3)

Length penetrated

(mm)

Eq (8e)

i 2629 24830 68988endash004 510ii 1762 24834 68998endash004 527iii 47 24887 69146endash004 510iv 42 25100 69734endash004 527

Eq (8j)

i 2652 24830 68989endash004 510ii 1785 24835 69000endash004 527iii 70 24862 69076endash004 510iv 65 25088 69703endash004 527

Table 1 Comparison of computational expense and simulated parameters for the proposed and conventional models

Fig 5 Approach (iv) (a) evolution of nodal values (b) oscillation-free emergence of nodal values from initial dry state (c) final moistur e distribution across the medium

(a) (c)

(b)

02

0

005

ndash005

0

0 5 10 15 20 25

Time step number3530 40 2 4

Distance fromexposed face (mm)

6 8 1000

02

04

06

08

1

0 105 2015 25 30

Time step number

35 40

04

Red

uced

moi

stur

eR

educ

ed m

oist

ure

06

08

1

Structural Engineering International 22014 Scientific Paper 193

model is subsequently restated using a set of nondimensional terms to mini-mize the range of orders involved in the underlying calculations This aids in reducing the numerical errors con-comitant to computer arithmetic and in easily adapting the simulated results to predict the behaviour of hydrauli-cally similar materials Further using the modified model this paper elu-cidates the formulation of a one-dimensional nonlinear FE scheme to determine the evolution of moisture distribution in concrete subjected to wettingndashdrying exposure due to inter-mittent rains The paper also discusses the issue of numerical oscillations typi-cally related to the FE-based analysis of nonlinear diffusion models A pru-dent evaluation of the system of non-linear algebraic equations generated in the FE analysis has been carried out to ascertain the suitability of the mass type to be implemented with a linear element for obtaining an oscillation-free convergence The study leads to a set of criteria governing the maximum permissible value of time-step size for stable convergence The validity of the derived criteria has been tested for a typical concrete subjected to rainfall exposure

The thrust of the present work is therefore on establishing the condi-tions that enable a robust one-dimen-sional transient nonlinear FE analysis of moisture flow in concrete described using a phenomenological model The developed scheme can be imple-mented to examine the impact of various influencing parameters on the state of moisture distribution in con-crete without resorting to extensive experimental programmes The pres-ent work thus bears a special signifi-cance in the context of model-based service life analysis of concrete struc-tures in particular for tropics where the occurrence of rainfall in intermit-tent spells is a characteristic climatic phenomenon

Brief Review of Phenomenological Moisture Transport Models

Moisture Ingress

When a concrete surface is exposed to rain flux the concomitant ingress of moisture is caused due to the action of capillary forces and the resulting dis-tribution of moisture is governed by the gradient of capillary pressure It is thus rational to represent the ensuing

movement of water using convection models that consider the gradient of pressure as the primary driving force19 Darcyrsquos law the first convection model was proposed in the form of an empir-ical equation to describe the flow of water through saturated porous media20 Mathematically Darcyrsquos law is stated as

Q = ndashksA DpL (1a)

ie u = ndashks DpL (1b)

In a local sense Eq (1b) can be expressed as

u = ndashksnablap (1c)

where Q is the volume rate of flow (m3 sminus1) ks is the Darcean permeabil-ity (kgminus1 m3 s) A is the cross-sectional area of specimen (m2) p is the hydro-static head (kg mminus1 sminus2) L is the thick-ness of the medium (m) and u is the average velocity of flow (msminus1)

By expressing the hydrostatic pres-sure in terms of pressure potential the equation further reduces to

u = ndashKsnablap (1d)

where Ks = ksrwg is the permeability of the saturated medium (m sminus1) P = prwg is the pressure potential (m) rw is the density of water (kg mminus3) and g is the acceleration due to gravity (msminus2)

When applied to the analysis of unsat-urated flow the permeability term in Darcyrsquos relation is replaced by a gen-eralized transport property called the capillary conductivity K(ql)(msminus1) which is a strong function of the liq-uid water content and the pressure potential is replaced by the capillary potential y (m)21 Thus Eq (1d) gets modified to

u = ndashK(ql)nablay (1e)

The above relation can be applied directly to describe the steady-state unsaturated flow However to incor-porate the effect of unsteady condi-tions Eq (1e) for apparent velocity has to be combined with the mass bal-ance equation

icircrvicirc t + nabla(jl) = 0 (2a)

where rv is the partial moisture den-sity (kg mminus3) t is the time variable (s) and jl is the liquid flux (kg mminus2 sminus1)Equation (2a) can be restated in terms of volumetric liquid water content ql (m3m3) as

icirc (rwql) icirc t + nabla(rwu) = 0 (2b)

The combination of Eqs (1e) and (2b) results in

icircqlicirc t = nabla(K(ql)nablay) (2c)

Equation (2c) can be expressed in terms of q rather than y as

icircqlicirc t = nabla(D(ql)nablaql) (2d)

with D(ql) = K(ql)icircy icircql = K(ql)(icircqlicircy )= K(ql)c(ql) where c(ql) is specific water capacity (mminus1)

The extended Darcyrsquos law represented by Eq (2d) governs the flow of water in unsaturated porous medium and is also referred to as Richardrsquos equa-tion Mathematically it is analogous to Fickrsquos law of diffusion

Moisture Egress

The evaporative drying of moisture imbibed in concrete starts soon after the end of a rainfall event and involves flow in both liquid and vapour phases This multiphase movement of water can be described using the diffusion models that assume concentration gradient to be the primary driving force Krischerrsquos model in particular considers the transport of water and water vapour as caused sepa-rately due to the respective gradients of liquid moisture content and partial pres-sure of vapour in air19Thus the liquid and vapour fluxes are expressed as

jl = ndashrwDl(ql)nablaql (3a)

jv = ndash(DvmRvT)nablapv (3b)

The corresponding mass balance equa-tions are given by

rw icircql ___ icirc t

= rwnabla(Dl(ql)nablaql) (4a)

(4b)

Here Dl(ql) is the diffusivity of liquid water (m2sminus1) Rv is the gas constant of water vapour (J kgminus1 Kminus1) T is the tem-perature (K) pv is the partial vapour pressure (Nmminus2) Dv is the diffusion coefficient of water vapour in air (m2 sminus1) and m is the resistance factor of the porous medium to vapour diffusion

The total mass balance of moisture after accounting for the effects of the Stefan diffusion and the concentration of water in the pores can be obtained from Eqs (4a) and (4b) as

(4c)

194 Scientific Paper Structural Engineering International 22014

where f is the open porosity (m3m3) and pt is the pressure of vapourndashair mixture (Nmminus2)

Equation (4c) represents Krischerrsquos model for combined liquid and vapour transport The expression can be fur-ther simplified by considering the aspects of capillary pressure and rela-tive humidity that prevail within the porous matrix

For a given moisture content the capil-lary pressure within the porous matrix is constant and the relative humidity of the associated vapour phase is

h = pv ps (5a)

where ps is the saturated vapour pres-sure (Nmminus2) Now using Kelvinrsquos equa-tion pv can be stated as

(5b)

where pc is the capillary pressure (Nmminus2)

From the above expression it becomes evident that under isothermal con-ditions the partial vapour pressure depends only on capillary pressure and hence varies as a function of the mois-ture content Thus from Eq (5b)

nablapv = ps(icirchicircql)T nablaql (5c)

The vapour flux in Eq (3b) can now be expressed as

(6a)

Thus as in Eq (4c) the total mass bal-ance of moisture can now be expressed as

(6b)

that is

icirc ___ icirct (ql + qv) = nabla(Dl(ql)nablaql) + nabla(Dv(ql)nablaql)

which is of the form

icircq ___ icirc t = nabla(D(ql)nablaql) (6c)

Here qv is the volumetric water vapour content (m3m3) Dv(ql) is the diffusiv-ity of water vapour in porous medium (m2 sminus1) q = ql + qv is the total volumet-ric moisture content and D(ql) = Dl(ql) + Dv(ql) is the isothermal moisture

diffusivity (m2 sminus1) The derived equa-tion governing the combined flow of liquid and vapour phases of water takes up the form of Fickrsquos law of diffusion

Models for Diffusivity Function

Diffusivity of a porous material is a measure of its tendency to transmit fluid through its interconnected pore space while in a state of unsatura-tion21 When the fluid being trans-mitted is water it is referred to as the hydraulic diffusivity The hydrau-lic diffusivity is a continuum level property that takes into account the combined effect of pore structure geometry state of saturation of pore space and the transport mechanisms driving the moisture movement phe-nomenon22 In a phenomenological model the hydraulic diffusivity is the only decisive parameter that governs the evolution of moisture distribution in a medium Considering the pore structure to remain stable the strong dependence of hydraulic diffusivity on moisture content is generally repre-sented using monotonically increasing functions The trend follows from the underlying mechanisms of moisture transport which enhance the rate of mass transfer with increasing levels of pore saturation initiating with the slow gaseous diffusion of vapour in a near-dry state the rate compounds through the movement of water in adsorbed and condensed layers to the phase of bulk flow in pore space under the rising levels of saturation2324 Furthermore the empiric modelling of hydraulic diffusivity leads to typical functional forms for wetting and dry-ing conditions characterizing the asso-ciated moisture transport mechanisms For liquid water ingress which owes to the effect of capillarity the hydraulic diffusivity has been successfully repre-sented by an exponential function of the form21ndash28

Dr (qr) = Dd_wetexp(nqr) (7a)

with

qr = (q ndash qo)(qs ndash qo)

where qs is the saturation moisture content (m3m3) qo is the moisture content corresponding to the capillary dry state (m3m3) Dr(qr) is the hydrau-lic diffusivity as a function of nondi-mensional moisture content (m2 sminus1) and Dd_wet is the wetting diffusivity of concrete in the capillary dry state (m2 sminus1) The value of n has been suggested

to be in the range of 6ndash8 for building materials29 A precise investigation carried out using nuclear magnetic resonance has suggested a value of n = 6 for an initially dry concrete28

The drying behaviour of concrete on the other hand has been described by an S-shaped diffusivity function of the form30ndash33

(7b)

where Dw_dry is the drying diffusivity of concrete in completely wet state (m2 sminus1) ao is the ratio of minimum to maximum diffusivity and the param-eters n and qc characterize respec-tively the spread and the location of drop of the curve

The empirical diffusivity models broadly describe the hygroscopic behaviour of concrete under wetting and drying conditions The application of these models for a given concrete is however dependent on the experi-mental estimation of the function parameters Comprehensive studies to establish the dependence of these parameters on temperature initial saturation level and parameters gov-erning the pore structure properties of concrete would enable a direct imple-mentation of these models for durabil-ity studies

FE Formulation

Governing Equation

As has been described in the Brief Review of Phenomenological Moisture Transport Models section the phe-nomenon of one-dimensional isother-mal moisture transport in a porous medium can be represented using the equation

icircq

___ icirc t = icirc ___ icircx ( D(q )

icircq ___ icircx ) (8a)

Eq (8a) is a nonlinear parabolic partial differential equation The nonlinearity is caused by the term D(q ) a strong function of moisture content that fol-lows distinct characteristic trends for wetting and for drying conditions respectively Owing to its highly non-linear nature Eq (8a) is not amenable to analytical solution and requires the implementation of numerical proce-dures The equation being first order in time and second order in space the

Structural Engineering International 22014 Scientific Paper 195

solution of the problem relies on the specification of an initial condition and two boundary conditions The bound-ary conditions may either be provided as the value of moisture content q (Dirichletessential boundary condi-tion) or as the value of moisture flux stated as D(q )icircqicircx = ndashVo (Neumannnatural boundary condition) where Vo is the orthogonal rain flux incident on the exposed surface (msminus1) The initial moisture content across the physi-cal domain of analysis is generally described by an initial moisture profile q (xt = 0) = qini(x)

Since the parameters constituting the given problem range over several orders of magnitude restating Eq (8a) using the following dimensionless terms

Reduced moisture q r = (q ndash qo)(qs ndash qo)

(8b)

Reduced distance xr = (VoDd_wet)x (8c)

Reduced distance tr = ( V o 2 Dd_wet)t (8d)

aids in minimizing the computing errors

Restating Eq (8a) in terms of non-dimensional parameters gives

(8e)

A scrutiny of these terms reveals their appropriateness in reducing the range of orders involved in computation the most prominent aspect in the modi-fied model represented by Eq (8e) is the normalization of the diffusivity function Dr(qr)

with the term Dd_wet Moreover expressing the model in terms of these parameters enables adaptation of results to determine the behaviour of all hydraulically similar materials The idea will be clarified fur-ther in the following section

Considering the following simplified representations for the hydraulic dif-fusivity function

for wetting Dr (qr) = Dd_wet fw(q r) (8f)

for drying Dr (qr) = Dw_dry fd(q r) (8g)

where fw(q r) and fd(q r) are functions of reduced moisture content in wet-ting and drying diffusivity functions respectively and subsequently equat-ing the boundary flux term

to wetting and drying fluxes yields respectively the following conditions

for wetting fw (qr) icircqr ___ icircxr

= ndash1(q s ndash q o)

(8h)

for drying

(8i)

where Jv is drying boundary flux acting in the direction of outward normal to the surface (msminus1)

It is worth mentioning that the adap-tation of Eq (8a) most widely used in previous studies implements only the reduced moisture content as in Eq (8b) and has the form

(8j)

Formulation of Element-Level Governing Equation

Using the governing differential equa-tion stated in Eq (8e) the FE formula-tion can be carried out using Galerkinrsquos weighted residual technique34 For the given problem the weighted residual statement is given as

(9a)

where wk is the weight function (k = 1 to number of nodes in an element) and l is the reduced size of element consti-tuting the FE mesh

Integration of Eq (9a) by parts leads to

(9b)

Now for a linear element with two nodes it is known that

(9c)

Thus

(9d)

where H is the interpolation matrix B is the gradient matrix and de is the vector of elemental degrees of freedom

Following Galerkinrsquos method the weights in Eq (9b) are taken equal to the interpolation functions

Thus for w1 = 1 ndash (xrl) Eq (9b) yields

(9e)

And for w2 = (xrl)

(9f)

Combining Eqs (9e) and (9f) the gen-eral element-level governing equation can be obtained as

(9g)

Substituting Eqs (8f) and (8g) in Eq (9g) and using Eqs (8h) and (8i) in tandem the governing equation for the boundary element respective to the cases of wetting and drying can be obtained as

(9h)

(9i)

196 Scientific Paper Structural Engineering International 22014

An inspection of Eqs (9h) and (9i) reveals their dependence on the functions fw(qr) and fd(qr) which are known to follow characteristic expo-nential and S-shaped forms for the respective cases of wetting and drying of concrete Also as the terms (Dw_dry Dd_wet) and (1 qs ndash qo) are relative in nature the derived expressions can be considered to represent the behaviour of all hydraulically similar materials

Equations (9h) and (9i) are semi-dis-crete and can be represented in matrix form as

[m]de + [k]de = q (9j)

where m and k are the element-level mass and the diffusivity matrices respectively de is the vector of time derivatives of elemental degrees of freedom and q is the vector of ele-mental nodal fluxes

In order to obtain a fully discretized system of equations the time deriva-tives of the field variable in these equa-tions are to be further approximated using the method of finite difference Adopting the CrankndashNicolson scheme a completely discrete system of equa-tions is obtained as

([m] + 05Δtr[k]n+1)den+1

= ([m] + 05Δtr[k]n)den

+ 05Δtr(qn+1 + qn) (9k)

where the superscripts n and (n+1) denote the previous and present time levels respectively in the course of transient analysis

Numerical Stability of the FE Scheme

The highly nonlinear nature of the problem necessitates the implementa-tion of an extremely refined mesh and a small time-step size to facilitate the stable convergence of simulated results However earlier studies based on FE analysis of parabolic partial differen-tial equations reported that the mini-mization of the time-step size beyond a certain threshold value induces numerical oscillations in the simulated results3536 The issue was addressed in the context of heat diffusion problems by determining the minimum time-step sizes for some typical one- and two-dimensional elements16 These crite-ria16 that had actually been developed for strictly constant material properties were tested for unsaturated seepage problems involving mildly nonlinear material properties and were found to

be equally useful17 In another study18 it was demonstrated that the use of lumped-mass scheme with linear ele-ments offers stable convergence and consistent results for FE-based solu-tion of Richardrsquos equation The obser-vation was however empirical and was not deduced mathematically

In the following section an attempt has been made to mathematically interpret the issue of numerical oscilla-tions and to ascertain the suitability of a lumped-mass type implemented with linear elements in providing a stable convergence In addition a set of cri-teria has been developed for the modi-fied model to regulate the variation of time-step size as a function of element length and hydraulic diffusivity for stable convergence of the FE scheme

General Conditions for Stability

To determine the nodal values of the field variable at a particular time-step all the element-level equations are assembled to obtain a set of nonlinear algebraic equations in the form [A]X = b with

[A] = [M] + 05Δtr[K]n+1 (10a)

b = ([M] ndash 05Δtr[K]n)Xn

+ Δtr(Qn+1 + Qn) (10b)

where [M] and [K] are respectively the global consistent mass and diffu-sivity matrices and X and Q are the global vectors for degrees of freedom and nodal fluxes respectively

It is evident that spatial oscillations in the simulated results are caused by the numerical characteristics of matrix [A] and vector b From basic reasoning it can be perceived that the occurrence of numerical oscillations can be elimi-nated if

1 [A] is strictly diagonally dominant with positive terms in the principal diagonal and negative terms else-where with magnitudes lesser than the smallest element of the principal diagonal

2 All the elements of b are posi-tive which necessarily depends on the diagonal elements of ([M] ndash 05Δtr[K]n)Xn to remain positive

Depending on the method to be imple-mented for solving the system [A]X = b the implications of the stated conditions would be as follows

(a) The constitution of [A] would be such that its manipulation using any of the elimination methods37

bull will not produce negative ele-ments on its principal diagonal

bull will not require row subtractions during the elimination phase thus the elements in b shall always remain positive provided condition (2) is already met

(b) If [A] is to be inverted for X = [A]ndash1b then none of the elements in the inverted matrix would be negative it would otherwise fetch negative values in the resulting vector

(c) The diagonal dominance of [A] is a suffi cient condition for the itera-tive methods to converge for any initial solution vector37

The specified conditions restrict the generation of negative values at nodes which generally manifest in an alter-nating pattern when a wetting front invades a dry medium They also arrest the oscillations causing a saw-tooth pattern in simulated moisture profile Fulfilling the proposed con-ditions thus eliminates both forms of oscillations that are reported to influ-ence the quality of results simulated under a FE scheme17 Furthermore it is to be recognized that the numerical oscillations are primarily caused due to the composition of matrix [A] that is in relation to the violation of condi-tion (1) The perturbations caused due to the multiplication of the negative diagonal elements of [M] ndash 05Δtr[K]n with the elements of vector [X]n are largely counterbalanced by the contri-bution of the off-diagonal elements in the respective rows which are always positive and have the same order of magnitude

The mathematical criteria to be adopted to enforce the proposed con-ditions depend on the type of element being implemented in the FE model This study considers only the instance of linear element in the following section

Stability Criteria

For the FE formulation presented in the Formulation of Element Level Governing Equation section the global characteristic matrices for the phase of wetting will be tri-diagonal and are defined as

(11a)

Structural Engineering International 22014 Scientific Paper 197

(11b)

where M ij L is an element of the ith row

and jth column of lumped-mass matrix

The revised criteria for satisfying con-ditions (1) and (2) would now be

which reduces to

Δtr lt l2[ fw(qrmax) ndash fw(qrmin)] (12b)

and

which reduces to

Δtr lt l2fw(qrmax) (12c)

Equations (12b) and (12c) establish that adopting the lumped-mass scheme in conjunction with a two-noded linear element allows the minimization of time-step size to achieve convergence without the need of simultaneously refining the mesh density It is impor-tant here to note that although the condition in Eq (12c) imposes a more stringent limit on Δtr in comparison to Eq (12b) the two conditions converge within a value of qrmax = 02 for fw(qr) = exp (6qr) qrmin = 0 and l =05 as is depicted in Fig 1 In fact the conver-gence occurs at a steeper rate for finer meshes and remains within the stated threshold for the coarsest meshes prac-

ticable for the problem The delineated criteria are thus particularly stringent for near-dry conditions and can be relaxed at higher levels of moisture content using average value of mois-ture content that is qravg in place of qrmax

Similarly for the phase of drying the governing criteria for time-step size corresponding to conditions (1) and (2) would be

(13a)

and (13b)

Implementing the Time-Step Size Criteria

To illustrate an application of the delineated time-stepping criteria and the influence of mass type on numeri-cal oscillations we consider a one-dimensional moisture ingress problem involving an initially dry (ie qr = 0 at tr = 0) concrete medium of length L = 010 m subjected to a rainfall of inten-sity Vo = 10 mmh until the attainment of surface saturation (ie qr = 1 at xr = 0) The typical material parameters opted for the analysis are qo = 0 m3m3 qs = 01811 m3m3 Dw_dry = 945 times 10ndash10m2s and n = 6 The FE mesh has been constituted using linear elements of reduced size l = 05 which equals 017 mm for the adopted parameter values

The FE analysis of the considered problem governed by Eq 8(e) has been carried out by four distinct approaches

(i) using a consistent mass type and varying the time-step size in accordance to Eq (11c)

(ii) with a lumped-mass type and varying the time-step size as per Eqs (12b) and (12c)

(iii) as in (i) but relaxing the time-stepping constraints in Eq (11c) by replacing qrmax by qravg

(iv) as in (ii) but relaxing the time-step-ping constraints in Eqs (12b) and (12c) by replacing qrmax by qravg

The time-stepping in FE analysis has been implemented using the following scheme

1 At any instant tr calculate the pres-ent time-step size (Δtrpresent) if tr = 0 take Δtrpresent = 001l2fw(qr = 0) and for tr gt 0 take Δtrpresent = Δtrnext as determined in the previous instant of analysis

where m is the order of the matri-ces which equals the total number of unknowns Mij and Kij represent an element of the ith row and jth column of the global consistent mass and dif-fusivity matrices respectively

To fulfil conditions (1) and (2) at any time level the following criteria are to be satisfied

which reduces to

l23fw(qrmax) lt Δtr

lt 2l23[ fw(qrmax) ndash fw(qrmax)] (11c)

and

l __ 6 ndash 05Δtr 05 ___

l [2fw(qrmax)] gt 0

which reduces to

Δtr lt l23fw(qrmax) (11d)

Here qrmax and qrmin represent the maximum and minimum values respectively of the reduced moisture content at any time level

As is evident Eqs (11c) and (11d) con-tradict each other and cannot be satisfied simultaneously However as has been pointed out earlier and will be dem-onstrated in what follows violating the constraint imposed by condition (2) that is Eq (11d) results only in minor oscilla-tions that gradually damp out Equation (11c) however imposes a more severe constraint and explains the occurrence of oscillations that set in when the time-step size (Δtr) is reduced without simul-taneously refining the mesh while using a consistent mass scheme with linear ele-ments18 The identified constraints can however be circumvented by replac-ing the consistent mass matrix with a lumped-mass scheme defined as

(12a)

0100

1

2

Max

imum

Δt r

3

4

5

02 03 04Maximum moisture content ( tmax)

05 06 07 08 09 1

12b12c

Fig 1 Convergence pattern of the maxi-mum time limit criteria delineated in Eqs (12b) and (12c)

198 Scientific Paper Structural Engineering International 22014

02

0

005

ndash005

0

0 500 1000 1500

Time step number2000 2500 2 4

Distance fromexposed face (mm)

6 8 1000

02

04

06

08

1

0 500 1000 1500

Time step number

2000 2500

04

Red

uced

moi

stur

eR

educ

ed m

oist

ure

06

08

1(a) (c)

(b)

Fig 2 Approach (i) (a) evolution of nodal values (b) oscillating emergence of nodal val-ues from initial dry state (c) final moisture distribution across the medium

(a) (c)

(b)

02

0

005

ndash005

0

0 200 400 600 800 1000

Time step number14001200 1600 2 4

Distance fromexposed face (mm)

6 8 1000

02

04

06

08

1

0 400200 800600 1000 1200

Time step number

1400 1500

04

Red

uced

moi

stur

eR

educ

ed m

oist

ure

06

08

1

Fig 3 Approach (ii) (a) evolution of nodal values (b) oscillation-free emergence of nodal values from initial dry state (c) final moisture distribution across the medium

(a) (c)

(b)

02

0

005

ndash005

0

0 5 10 15 20 25

Time step number3530 4540 2 4

Distance fromexposed face (mm)

6 8 1000

02

04

06

08

1

0 105 2015 25 30

Time step number

35 40 45

04

Red

uced

moi

stur

eR

educ

ed m

oist

ure

06

08

1

Fig 4 Approach (iii) (a) evolution of nodal values (b) oscillating emergence of nodal values from initial dry state (c) final moisture distribution across the medium

2 Compute moisture distribution at the present instant tr

3 Calculate the limiting-time-step size (Δtrlimit) using appropriate criteria

4 Calculate time-step size for next instant to be simulated (Δtrnext) if (Δtrlimit gt 2 Δtrpresent) then Δtrnext = 2 Δtrpresent else Δtrnext = 095 Δtrlimit

5 Increment time tr =tr + Δtrpresent

Figures 2ndash5 present the simulated results corresponding to each of the analysis approaches These illustra-tions describe the evolution of reduced moisture content at nodes in succes-sive time-steps and the distribution of moisture across the medium at the attainment of surface saturation The observations clearly depict that the FE analysis based on consistent mass approach induces numerical oscillations in the simulated results An inspection of Figs 2 and 4 further reveals that the oscillations strongly influence the initial stages of analysis and then gradually reduce to a rela-tively less intense level with a consis-tent effect on the nodes in a near-dry state at the toe of the wetting front The analysis based on lumped-mass scheme on the other hand ensures a stable con-vergence of nodal values as is depicted in Figs 3 and 4 and also leads to con-vergence in relatively fewer time-steps thereby economizing the computation time Table 1 provides a summary of the time-steps computed and the simu-lated parameters Relaxing the time-stepping constraints as in approaches (iii) and (iv) causes some perturbations in the initial evolution of nodal values as depicted in Figs 4 and 5 However as summarized in Table 1 the analysis leads to satisfactory results with a sig-nificantly reduced computational time These findings thus establish the advan-tage of implementing a lumped-mass scheme in conjunction with the relaxed time-stepping criteria delineated in the study for achieving computational effi-ciency and stability in the FE analysis of moisture transport in concrete

The conventional model stated in Eq (8j) which has been widely adopted in several previous studies was also implemented to analyse the consid-ered test problem The observed trends of numerical stability were found to be similar to the ones obtained with the model proposed in this study However the computational expense for analys-ing the conventional model was found to be higher in relation to the proposed model A comparison of the results has been summarized in Table 1

Structural Engineering International 22014 Scientific Paper 199

Discussion and Conclusions

A rational study of moisture move-ment in concrete rests on appropriate modelling of associated mass transport phenomena and on the efficacy of the numerical analysis of the constituted mathematical model A sound compre-hension of these facets is essential for a plausible treatment of the problem The present work has addressed these aspects in the context of a rain-induced wettingndashdrying scenario and in its dis-course has emphasized the following ideas

(a) The parabolic partial differential equation provides a phenomeno-logical description of the fl ow of moisture in concrete under wet-ting and drying conditions albeit with distinct diffusivity functions

(b) The presently available empirical models for hydraulic diffusivity incorporate parameters which are known to depend on temperature

and the pore structure proper-ties of concrete This dependence however remains to be estab-lished mathematically in terms of the generic concrete proper-ties Future studies towards the refi nement of hydraulic diffusiv-ity models would enable a direct application of the delineated phe-nomenological approach in ana-lysing the moisture transport in structural concrete

(c) The restatement of the governing equation using the set of dimen-sionless terms qr = (q ndashqo)(qs ndashqo) xr = (VoDd_wet)x and tr = ( V o 2 Dd_wet)t (i) reduces the range of orders involved and hence minimizes the errors inherent in numerical computations (ii) pro-vides a representation character-istic of all hydraulically similar materials and (iii) economizes the computation time

(d) Galerkinrsquos method can be effec-tively implemented to formulate

a FE scheme for numerical anal-ysis of the moisture transport problem Its application has been elaborated using the modifi ed model to describe the constitu-tion of the characteristic matrices in detail

(e) The occurrence of numerical oscil-lations inherent to the FE analysis of diffusion models relates to the nature of the matrices involved in the analysis and can be effectively controlled by imposing the condi-tions outlined in the study

(f) The effi ciency of lumped-mass scheme implemented in conjunc-tion with a linear element for one-dimensional analysis has been mathematically interpreted A concomitant time-stepping crite-rion for stable convergence has been derived and its viability has been demonstrated through a test problem

The deliberations presented in this paper provide an insight into the math-ematical framework underlying the study of moisture transport phenom-ena in concrete and facilitate a robust FE analysis of moisture distribution in concrete subjected to wettingndashdrying exposure

References

[1] Andrade C Sarria J Alonso C Relative humidity in the interior of concrete exposed to natural and artificial weathering Cement Concrete Res 1999 29(8) 1249ndash1259

[2] Ryu DW Ko JW Noguchi T Effect of simu-lated environmental conditions on the internal relative humidity and relative moisture con-tent distribution of exposed concrete Cement Concrete Comp 2011 33(1) 142ndash153

[3] Zhang J Gao Y Han Y Interior humidity of concrete under dry-wet cycles J Mater Civil Eng 2012 24(3) 289ndash298

[4] Gawin D Pesavento F Schrefler BA Hygro-thermo-chemo-mechanical modelling of con-crete at early ages and beyond Part I Hydration and hygro-thermal phenomena Int J Numer Meth Eng 2006 67(3) 299ndash331

[5] Gawin D Pesavento F Schrefler BA Hygro-thermo-chemo-mechanical modelling of con-crete at early ages and beyond Part II Shrinkage and Creep of Concrete Int J Numer Meth Eng 2006 67(3) 332ndash363

[6] Saetta AV Schrefler BA Vitaliani RV The carbonation of concrete and the mechanism of moisture heat and carbon dioxide flow through porous materials Cement Concrete Res 1993 23(4) 761ndash772

[7] Saetta AV Schrefler BA Vitaliani RV 2-D model for carbonation and moistureheat flow in porous materials Cement Concrete Res 1995 25(8) 1703ndash1712

Approach No of time-steps computed

Time taken for surface saturation

(s)

Area under fi nal moisture profi le

(m3)

Length penetrated

(mm)

Eq (8e)

i 2629 24830 68988endash004 510ii 1762 24834 68998endash004 527iii 47 24887 69146endash004 510iv 42 25100 69734endash004 527

Eq (8j)

i 2652 24830 68989endash004 510ii 1785 24835 69000endash004 527iii 70 24862 69076endash004 510iv 65 25088 69703endash004 527

Table 1 Comparison of computational expense and simulated parameters for the proposed and conventional models

Fig 5 Approach (iv) (a) evolution of nodal values (b) oscillation-free emergence of nodal values from initial dry state (c) final moistur e distribution across the medium

(a) (c)

(b)

02

0

005

ndash005

0

0 5 10 15 20 25

Time step number3530 40 2 4

Distance fromexposed face (mm)

6 8 1000

02

04

06

08

1

0 105 2015 25 30

Time step number

35 40

04

Red

uced

moi

stur

eR

educ

ed m

oist

ure

06

08

1

194 Scientific Paper Structural Engineering International 22014

where f is the open porosity (m3m3) and pt is the pressure of vapourndashair mixture (Nmminus2)

Equation (4c) represents Krischerrsquos model for combined liquid and vapour transport The expression can be fur-ther simplified by considering the aspects of capillary pressure and rela-tive humidity that prevail within the porous matrix

For a given moisture content the capil-lary pressure within the porous matrix is constant and the relative humidity of the associated vapour phase is

h = pv ps (5a)

where ps is the saturated vapour pres-sure (Nmminus2) Now using Kelvinrsquos equa-tion pv can be stated as

(5b)

where pc is the capillary pressure (Nmminus2)

From the above expression it becomes evident that under isothermal con-ditions the partial vapour pressure depends only on capillary pressure and hence varies as a function of the mois-ture content Thus from Eq (5b)

nablapv = ps(icirchicircql)T nablaql (5c)

The vapour flux in Eq (3b) can now be expressed as

(6a)

Thus as in Eq (4c) the total mass bal-ance of moisture can now be expressed as

(6b)

that is

icirc ___ icirct (ql + qv) = nabla(Dl(ql)nablaql) + nabla(Dv(ql)nablaql)

which is of the form

icircq ___ icirc t = nabla(D(ql)nablaql) (6c)

Here qv is the volumetric water vapour content (m3m3) Dv(ql) is the diffusiv-ity of water vapour in porous medium (m2 sminus1) q = ql + qv is the total volumet-ric moisture content and D(ql) = Dl(ql) + Dv(ql) is the isothermal moisture

diffusivity (m2 sminus1) The derived equa-tion governing the combined flow of liquid and vapour phases of water takes up the form of Fickrsquos law of diffusion

Models for Diffusivity Function

Diffusivity of a porous material is a measure of its tendency to transmit fluid through its interconnected pore space while in a state of unsatura-tion21 When the fluid being trans-mitted is water it is referred to as the hydraulic diffusivity The hydrau-lic diffusivity is a continuum level property that takes into account the combined effect of pore structure geometry state of saturation of pore space and the transport mechanisms driving the moisture movement phe-nomenon22 In a phenomenological model the hydraulic diffusivity is the only decisive parameter that governs the evolution of moisture distribution in a medium Considering the pore structure to remain stable the strong dependence of hydraulic diffusivity on moisture content is generally repre-sented using monotonically increasing functions The trend follows from the underlying mechanisms of moisture transport which enhance the rate of mass transfer with increasing levels of pore saturation initiating with the slow gaseous diffusion of vapour in a near-dry state the rate compounds through the movement of water in adsorbed and condensed layers to the phase of bulk flow in pore space under the rising levels of saturation2324 Furthermore the empiric modelling of hydraulic diffusivity leads to typical functional forms for wetting and dry-ing conditions characterizing the asso-ciated moisture transport mechanisms For liquid water ingress which owes to the effect of capillarity the hydraulic diffusivity has been successfully repre-sented by an exponential function of the form21ndash28

Dr (qr) = Dd_wetexp(nqr) (7a)

with

qr = (q ndash qo)(qs ndash qo)

where qs is the saturation moisture content (m3m3) qo is the moisture content corresponding to the capillary dry state (m3m3) Dr(qr) is the hydrau-lic diffusivity as a function of nondi-mensional moisture content (m2 sminus1) and Dd_wet is the wetting diffusivity of concrete in the capillary dry state (m2 sminus1) The value of n has been suggested

to be in the range of 6ndash8 for building materials29 A precise investigation carried out using nuclear magnetic resonance has suggested a value of n = 6 for an initially dry concrete28

The drying behaviour of concrete on the other hand has been described by an S-shaped diffusivity function of the form30ndash33

(7b)

where Dw_dry is the drying diffusivity of concrete in completely wet state (m2 sminus1) ao is the ratio of minimum to maximum diffusivity and the param-eters n and qc characterize respec-tively the spread and the location of drop of the curve

The empirical diffusivity models broadly describe the hygroscopic behaviour of concrete under wetting and drying conditions The application of these models for a given concrete is however dependent on the experi-mental estimation of the function parameters Comprehensive studies to establish the dependence of these parameters on temperature initial saturation level and parameters gov-erning the pore structure properties of concrete would enable a direct imple-mentation of these models for durabil-ity studies

FE Formulation

Governing Equation

As has been described in the Brief Review of Phenomenological Moisture Transport Models section the phe-nomenon of one-dimensional isother-mal moisture transport in a porous medium can be represented using the equation

icircq

___ icirc t = icirc ___ icircx ( D(q )

icircq ___ icircx ) (8a)

Eq (8a) is a nonlinear parabolic partial differential equation The nonlinearity is caused by the term D(q ) a strong function of moisture content that fol-lows distinct characteristic trends for wetting and for drying conditions respectively Owing to its highly non-linear nature Eq (8a) is not amenable to analytical solution and requires the implementation of numerical proce-dures The equation being first order in time and second order in space the

Structural Engineering International 22014 Scientific Paper 195

solution of the problem relies on the specification of an initial condition and two boundary conditions The bound-ary conditions may either be provided as the value of moisture content q (Dirichletessential boundary condi-tion) or as the value of moisture flux stated as D(q )icircqicircx = ndashVo (Neumannnatural boundary condition) where Vo is the orthogonal rain flux incident on the exposed surface (msminus1) The initial moisture content across the physi-cal domain of analysis is generally described by an initial moisture profile q (xt = 0) = qini(x)

Since the parameters constituting the given problem range over several orders of magnitude restating Eq (8a) using the following dimensionless terms

Reduced moisture q r = (q ndash qo)(qs ndash qo)

(8b)

Reduced distance xr = (VoDd_wet)x (8c)

Reduced distance tr = ( V o 2 Dd_wet)t (8d)

aids in minimizing the computing errors

Restating Eq (8a) in terms of non-dimensional parameters gives

(8e)

A scrutiny of these terms reveals their appropriateness in reducing the range of orders involved in computation the most prominent aspect in the modi-fied model represented by Eq (8e) is the normalization of the diffusivity function Dr(qr)

with the term Dd_wet Moreover expressing the model in terms of these parameters enables adaptation of results to determine the behaviour of all hydraulically similar materials The idea will be clarified fur-ther in the following section

Considering the following simplified representations for the hydraulic dif-fusivity function

for wetting Dr (qr) = Dd_wet fw(q r) (8f)

for drying Dr (qr) = Dw_dry fd(q r) (8g)

where fw(q r) and fd(q r) are functions of reduced moisture content in wet-ting and drying diffusivity functions respectively and subsequently equat-ing the boundary flux term

to wetting and drying fluxes yields respectively the following conditions

for wetting fw (qr) icircqr ___ icircxr

= ndash1(q s ndash q o)

(8h)

for drying

(8i)

where Jv is drying boundary flux acting in the direction of outward normal to the surface (msminus1)

It is worth mentioning that the adap-tation of Eq (8a) most widely used in previous studies implements only the reduced moisture content as in Eq (8b) and has the form

(8j)

Formulation of Element-Level Governing Equation

Using the governing differential equa-tion stated in Eq (8e) the FE formula-tion can be carried out using Galerkinrsquos weighted residual technique34 For the given problem the weighted residual statement is given as

(9a)

where wk is the weight function (k = 1 to number of nodes in an element) and l is the reduced size of element consti-tuting the FE mesh

Integration of Eq (9a) by parts leads to

(9b)

Now for a linear element with two nodes it is known that

(9c)

Thus

(9d)

where H is the interpolation matrix B is the gradient matrix and de is the vector of elemental degrees of freedom

Following Galerkinrsquos method the weights in Eq (9b) are taken equal to the interpolation functions

Thus for w1 = 1 ndash (xrl) Eq (9b) yields

(9e)

And for w2 = (xrl)

(9f)

Combining Eqs (9e) and (9f) the gen-eral element-level governing equation can be obtained as

(9g)

Substituting Eqs (8f) and (8g) in Eq (9g) and using Eqs (8h) and (8i) in tandem the governing equation for the boundary element respective to the cases of wetting and drying can be obtained as

(9h)

(9i)

196 Scientific Paper Structural Engineering International 22014

An inspection of Eqs (9h) and (9i) reveals their dependence on the functions fw(qr) and fd(qr) which are known to follow characteristic expo-nential and S-shaped forms for the respective cases of wetting and drying of concrete Also as the terms (Dw_dry Dd_wet) and (1 qs ndash qo) are relative in nature the derived expressions can be considered to represent the behaviour of all hydraulically similar materials

Equations (9h) and (9i) are semi-dis-crete and can be represented in matrix form as

[m]de + [k]de = q (9j)

where m and k are the element-level mass and the diffusivity matrices respectively de is the vector of time derivatives of elemental degrees of freedom and q is the vector of ele-mental nodal fluxes

In order to obtain a fully discretized system of equations the time deriva-tives of the field variable in these equa-tions are to be further approximated using the method of finite difference Adopting the CrankndashNicolson scheme a completely discrete system of equa-tions is obtained as

([m] + 05Δtr[k]n+1)den+1

= ([m] + 05Δtr[k]n)den

+ 05Δtr(qn+1 + qn) (9k)

where the superscripts n and (n+1) denote the previous and present time levels respectively in the course of transient analysis

Numerical Stability of the FE Scheme

The highly nonlinear nature of the problem necessitates the implementa-tion of an extremely refined mesh and a small time-step size to facilitate the stable convergence of simulated results However earlier studies based on FE analysis of parabolic partial differen-tial equations reported that the mini-mization of the time-step size beyond a certain threshold value induces numerical oscillations in the simulated results3536 The issue was addressed in the context of heat diffusion problems by determining the minimum time-step sizes for some typical one- and two-dimensional elements16 These crite-ria16 that had actually been developed for strictly constant material properties were tested for unsaturated seepage problems involving mildly nonlinear material properties and were found to

be equally useful17 In another study18 it was demonstrated that the use of lumped-mass scheme with linear ele-ments offers stable convergence and consistent results for FE-based solu-tion of Richardrsquos equation The obser-vation was however empirical and was not deduced mathematically

In the following section an attempt has been made to mathematically interpret the issue of numerical oscilla-tions and to ascertain the suitability of a lumped-mass type implemented with linear elements in providing a stable convergence In addition a set of cri-teria has been developed for the modi-fied model to regulate the variation of time-step size as a function of element length and hydraulic diffusivity for stable convergence of the FE scheme

General Conditions for Stability

To determine the nodal values of the field variable at a particular time-step all the element-level equations are assembled to obtain a set of nonlinear algebraic equations in the form [A]X = b with

[A] = [M] + 05Δtr[K]n+1 (10a)

b = ([M] ndash 05Δtr[K]n)Xn

+ Δtr(Qn+1 + Qn) (10b)

where [M] and [K] are respectively the global consistent mass and diffu-sivity matrices and X and Q are the global vectors for degrees of freedom and nodal fluxes respectively

It is evident that spatial oscillations in the simulated results are caused by the numerical characteristics of matrix [A] and vector b From basic reasoning it can be perceived that the occurrence of numerical oscillations can be elimi-nated if

1 [A] is strictly diagonally dominant with positive terms in the principal diagonal and negative terms else-where with magnitudes lesser than the smallest element of the principal diagonal

2 All the elements of b are posi-tive which necessarily depends on the diagonal elements of ([M] ndash 05Δtr[K]n)Xn to remain positive

Depending on the method to be imple-mented for solving the system [A]X = b the implications of the stated conditions would be as follows

(a) The constitution of [A] would be such that its manipulation using any of the elimination methods37

bull will not produce negative ele-ments on its principal diagonal

bull will not require row subtractions during the elimination phase thus the elements in b shall always remain positive provided condition (2) is already met

(b) If [A] is to be inverted for X = [A]ndash1b then none of the elements in the inverted matrix would be negative it would otherwise fetch negative values in the resulting vector

(c) The diagonal dominance of [A] is a suffi cient condition for the itera-tive methods to converge for any initial solution vector37

The specified conditions restrict the generation of negative values at nodes which generally manifest in an alter-nating pattern when a wetting front invades a dry medium They also arrest the oscillations causing a saw-tooth pattern in simulated moisture profile Fulfilling the proposed con-ditions thus eliminates both forms of oscillations that are reported to influ-ence the quality of results simulated under a FE scheme17 Furthermore it is to be recognized that the numerical oscillations are primarily caused due to the composition of matrix [A] that is in relation to the violation of condi-tion (1) The perturbations caused due to the multiplication of the negative diagonal elements of [M] ndash 05Δtr[K]n with the elements of vector [X]n are largely counterbalanced by the contri-bution of the off-diagonal elements in the respective rows which are always positive and have the same order of magnitude

The mathematical criteria to be adopted to enforce the proposed con-ditions depend on the type of element being implemented in the FE model This study considers only the instance of linear element in the following section

Stability Criteria

For the FE formulation presented in the Formulation of Element Level Governing Equation section the global characteristic matrices for the phase of wetting will be tri-diagonal and are defined as

(11a)

Structural Engineering International 22014 Scientific Paper 197

(11b)

where M ij L is an element of the ith row

and jth column of lumped-mass matrix

The revised criteria for satisfying con-ditions (1) and (2) would now be

which reduces to

Δtr lt l2[ fw(qrmax) ndash fw(qrmin)] (12b)

and

which reduces to

Δtr lt l2fw(qrmax) (12c)

Equations (12b) and (12c) establish that adopting the lumped-mass scheme in conjunction with a two-noded linear element allows the minimization of time-step size to achieve convergence without the need of simultaneously refining the mesh density It is impor-tant here to note that although the condition in Eq (12c) imposes a more stringent limit on Δtr in comparison to Eq (12b) the two conditions converge within a value of qrmax = 02 for fw(qr) = exp (6qr) qrmin = 0 and l =05 as is depicted in Fig 1 In fact the conver-gence occurs at a steeper rate for finer meshes and remains within the stated threshold for the coarsest meshes prac-

ticable for the problem The delineated criteria are thus particularly stringent for near-dry conditions and can be relaxed at higher levels of moisture content using average value of mois-ture content that is qravg in place of qrmax

Similarly for the phase of drying the governing criteria for time-step size corresponding to conditions (1) and (2) would be

(13a)

and (13b)

Implementing the Time-Step Size Criteria

To illustrate an application of the delineated time-stepping criteria and the influence of mass type on numeri-cal oscillations we consider a one-dimensional moisture ingress problem involving an initially dry (ie qr = 0 at tr = 0) concrete medium of length L = 010 m subjected to a rainfall of inten-sity Vo = 10 mmh until the attainment of surface saturation (ie qr = 1 at xr = 0) The typical material parameters opted for the analysis are qo = 0 m3m3 qs = 01811 m3m3 Dw_dry = 945 times 10ndash10m2s and n = 6 The FE mesh has been constituted using linear elements of reduced size l = 05 which equals 017 mm for the adopted parameter values

The FE analysis of the considered problem governed by Eq 8(e) has been carried out by four distinct approaches

(i) using a consistent mass type and varying the time-step size in accordance to Eq (11c)

(ii) with a lumped-mass type and varying the time-step size as per Eqs (12b) and (12c)

(iii) as in (i) but relaxing the time-stepping constraints in Eq (11c) by replacing qrmax by qravg

(iv) as in (ii) but relaxing the time-step-ping constraints in Eqs (12b) and (12c) by replacing qrmax by qravg

The time-stepping in FE analysis has been implemented using the following scheme

1 At any instant tr calculate the pres-ent time-step size (Δtrpresent) if tr = 0 take Δtrpresent = 001l2fw(qr = 0) and for tr gt 0 take Δtrpresent = Δtrnext as determined in the previous instant of analysis

where m is the order of the matri-ces which equals the total number of unknowns Mij and Kij represent an element of the ith row and jth column of the global consistent mass and dif-fusivity matrices respectively

To fulfil conditions (1) and (2) at any time level the following criteria are to be satisfied

which reduces to

l23fw(qrmax) lt Δtr

lt 2l23[ fw(qrmax) ndash fw(qrmax)] (11c)

and

l __ 6 ndash 05Δtr 05 ___

l [2fw(qrmax)] gt 0

which reduces to

Δtr lt l23fw(qrmax) (11d)

Here qrmax and qrmin represent the maximum and minimum values respectively of the reduced moisture content at any time level

As is evident Eqs (11c) and (11d) con-tradict each other and cannot be satisfied simultaneously However as has been pointed out earlier and will be dem-onstrated in what follows violating the constraint imposed by condition (2) that is Eq (11d) results only in minor oscilla-tions that gradually damp out Equation (11c) however imposes a more severe constraint and explains the occurrence of oscillations that set in when the time-step size (Δtr) is reduced without simul-taneously refining the mesh while using a consistent mass scheme with linear ele-ments18 The identified constraints can however be circumvented by replac-ing the consistent mass matrix with a lumped-mass scheme defined as

(12a)

0100

1

2

Max

imum

Δt r

3

4

5

02 03 04Maximum moisture content ( tmax)

05 06 07 08 09 1

12b12c

Fig 1 Convergence pattern of the maxi-mum time limit criteria delineated in Eqs (12b) and (12c)

198 Scientific Paper Structural Engineering International 22014

02

0

005

ndash005

0

0 500 1000 1500

Time step number2000 2500 2 4

Distance fromexposed face (mm)

6 8 1000

02

04

06

08

1

0 500 1000 1500

Time step number

2000 2500

04

Red

uced

moi

stur

eR

educ

ed m

oist

ure

06

08

1(a) (c)

(b)

Fig 2 Approach (i) (a) evolution of nodal values (b) oscillating emergence of nodal val-ues from initial dry state (c) final moisture distribution across the medium

(a) (c)

(b)

02

0

005

ndash005

0

0 200 400 600 800 1000

Time step number14001200 1600 2 4

Distance fromexposed face (mm)

6 8 1000

02

04

06

08

1

0 400200 800600 1000 1200

Time step number

1400 1500

04

Red

uced

moi

stur

eR

educ

ed m

oist

ure

06

08

1

Fig 3 Approach (ii) (a) evolution of nodal values (b) oscillation-free emergence of nodal values from initial dry state (c) final moisture distribution across the medium

(a) (c)

(b)

02

0

005

ndash005

0

0 5 10 15 20 25

Time step number3530 4540 2 4

Distance fromexposed face (mm)

6 8 1000

02

04

06

08

1

0 105 2015 25 30

Time step number

35 40 45

04

Red

uced

moi

stur

eR

educ

ed m

oist

ure

06

08

1

Fig 4 Approach (iii) (a) evolution of nodal values (b) oscillating emergence of nodal values from initial dry state (c) final moisture distribution across the medium

2 Compute moisture distribution at the present instant tr

3 Calculate the limiting-time-step size (Δtrlimit) using appropriate criteria

4 Calculate time-step size for next instant to be simulated (Δtrnext) if (Δtrlimit gt 2 Δtrpresent) then Δtrnext = 2 Δtrpresent else Δtrnext = 095 Δtrlimit

5 Increment time tr =tr + Δtrpresent

Figures 2ndash5 present the simulated results corresponding to each of the analysis approaches These illustra-tions describe the evolution of reduced moisture content at nodes in succes-sive time-steps and the distribution of moisture across the medium at the attainment of surface saturation The observations clearly depict that the FE analysis based on consistent mass approach induces numerical oscillations in the simulated results An inspection of Figs 2 and 4 further reveals that the oscillations strongly influence the initial stages of analysis and then gradually reduce to a rela-tively less intense level with a consis-tent effect on the nodes in a near-dry state at the toe of the wetting front The analysis based on lumped-mass scheme on the other hand ensures a stable con-vergence of nodal values as is depicted in Figs 3 and 4 and also leads to con-vergence in relatively fewer time-steps thereby economizing the computation time Table 1 provides a summary of the time-steps computed and the simu-lated parameters Relaxing the time-stepping constraints as in approaches (iii) and (iv) causes some perturbations in the initial evolution of nodal values as depicted in Figs 4 and 5 However as summarized in Table 1 the analysis leads to satisfactory results with a sig-nificantly reduced computational time These findings thus establish the advan-tage of implementing a lumped-mass scheme in conjunction with the relaxed time-stepping criteria delineated in the study for achieving computational effi-ciency and stability in the FE analysis of moisture transport in concrete

The conventional model stated in Eq (8j) which has been widely adopted in several previous studies was also implemented to analyse the consid-ered test problem The observed trends of numerical stability were found to be similar to the ones obtained with the model proposed in this study However the computational expense for analys-ing the conventional model was found to be higher in relation to the proposed model A comparison of the results has been summarized in Table 1

Structural Engineering International 22014 Scientific Paper 199

Discussion and Conclusions

A rational study of moisture move-ment in concrete rests on appropriate modelling of associated mass transport phenomena and on the efficacy of the numerical analysis of the constituted mathematical model A sound compre-hension of these facets is essential for a plausible treatment of the problem The present work has addressed these aspects in the context of a rain-induced wettingndashdrying scenario and in its dis-course has emphasized the following ideas

(a) The parabolic partial differential equation provides a phenomeno-logical description of the fl ow of moisture in concrete under wet-ting and drying conditions albeit with distinct diffusivity functions

(b) The presently available empirical models for hydraulic diffusivity incorporate parameters which are known to depend on temperature

and the pore structure proper-ties of concrete This dependence however remains to be estab-lished mathematically in terms of the generic concrete proper-ties Future studies towards the refi nement of hydraulic diffusiv-ity models would enable a direct application of the delineated phe-nomenological approach in ana-lysing the moisture transport in structural concrete

(c) The restatement of the governing equation using the set of dimen-sionless terms qr = (q ndashqo)(qs ndashqo) xr = (VoDd_wet)x and tr = ( V o 2 Dd_wet)t (i) reduces the range of orders involved and hence minimizes the errors inherent in numerical computations (ii) pro-vides a representation character-istic of all hydraulically similar materials and (iii) economizes the computation time

(d) Galerkinrsquos method can be effec-tively implemented to formulate

a FE scheme for numerical anal-ysis of the moisture transport problem Its application has been elaborated using the modifi ed model to describe the constitu-tion of the characteristic matrices in detail

(e) The occurrence of numerical oscil-lations inherent to the FE analysis of diffusion models relates to the nature of the matrices involved in the analysis and can be effectively controlled by imposing the condi-tions outlined in the study

(f) The effi ciency of lumped-mass scheme implemented in conjunc-tion with a linear element for one-dimensional analysis has been mathematically interpreted A concomitant time-stepping crite-rion for stable convergence has been derived and its viability has been demonstrated through a test problem

The deliberations presented in this paper provide an insight into the math-ematical framework underlying the study of moisture transport phenom-ena in concrete and facilitate a robust FE analysis of moisture distribution in concrete subjected to wettingndashdrying exposure

References

[1] Andrade C Sarria J Alonso C Relative humidity in the interior of concrete exposed to natural and artificial weathering Cement Concrete Res 1999 29(8) 1249ndash1259

[2] Ryu DW Ko JW Noguchi T Effect of simu-lated environmental conditions on the internal relative humidity and relative moisture con-tent distribution of exposed concrete Cement Concrete Comp 2011 33(1) 142ndash153

[3] Zhang J Gao Y Han Y Interior humidity of concrete under dry-wet cycles J Mater Civil Eng 2012 24(3) 289ndash298

[4] Gawin D Pesavento F Schrefler BA Hygro-thermo-chemo-mechanical modelling of con-crete at early ages and beyond Part I Hydration and hygro-thermal phenomena Int J Numer Meth Eng 2006 67(3) 299ndash331

[5] Gawin D Pesavento F Schrefler BA Hygro-thermo-chemo-mechanical modelling of con-crete at early ages and beyond Part II Shrinkage and Creep of Concrete Int J Numer Meth Eng 2006 67(3) 332ndash363

[6] Saetta AV Schrefler BA Vitaliani RV The carbonation of concrete and the mechanism of moisture heat and carbon dioxide flow through porous materials Cement Concrete Res 1993 23(4) 761ndash772

[7] Saetta AV Schrefler BA Vitaliani RV 2-D model for carbonation and moistureheat flow in porous materials Cement Concrete Res 1995 25(8) 1703ndash1712

Approach No of time-steps computed

Time taken for surface saturation

(s)

Area under fi nal moisture profi le

(m3)

Length penetrated

(mm)

Eq (8e)

i 2629 24830 68988endash004 510ii 1762 24834 68998endash004 527iii 47 24887 69146endash004 510iv 42 25100 69734endash004 527

Eq (8j)

i 2652 24830 68989endash004 510ii 1785 24835 69000endash004 527iii 70 24862 69076endash004 510iv 65 25088 69703endash004 527

Table 1 Comparison of computational expense and simulated parameters for the proposed and conventional models

Fig 5 Approach (iv) (a) evolution of nodal values (b) oscillation-free emergence of nodal values from initial dry state (c) final moistur e distribution across the medium

(a) (c)

(b)

02

0

005

ndash005

0

0 5 10 15 20 25

Time step number3530 40 2 4

Distance fromexposed face (mm)

6 8 1000

02

04

06

08

1

0 105 2015 25 30

Time step number

35 40

04

Red

uced

moi

stur

eR

educ

ed m

oist

ure

06

08

1

Structural Engineering International 22014 Scientific Paper 195

solution of the problem relies on the specification of an initial condition and two boundary conditions The bound-ary conditions may either be provided as the value of moisture content q (Dirichletessential boundary condi-tion) or as the value of moisture flux stated as D(q )icircqicircx = ndashVo (Neumannnatural boundary condition) where Vo is the orthogonal rain flux incident on the exposed surface (msminus1) The initial moisture content across the physi-cal domain of analysis is generally described by an initial moisture profile q (xt = 0) = qini(x)

Since the parameters constituting the given problem range over several orders of magnitude restating Eq (8a) using the following dimensionless terms

Reduced moisture q r = (q ndash qo)(qs ndash qo)

(8b)

Reduced distance xr = (VoDd_wet)x (8c)

Reduced distance tr = ( V o 2 Dd_wet)t (8d)

aids in minimizing the computing errors

Restating Eq (8a) in terms of non-dimensional parameters gives

(8e)

A scrutiny of these terms reveals their appropriateness in reducing the range of orders involved in computation the most prominent aspect in the modi-fied model represented by Eq (8e) is the normalization of the diffusivity function Dr(qr)

with the term Dd_wet Moreover expressing the model in terms of these parameters enables adaptation of results to determine the behaviour of all hydraulically similar materials The idea will be clarified fur-ther in the following section

Considering the following simplified representations for the hydraulic dif-fusivity function

for wetting Dr (qr) = Dd_wet fw(q r) (8f)

for drying Dr (qr) = Dw_dry fd(q r) (8g)

where fw(q r) and fd(q r) are functions of reduced moisture content in wet-ting and drying diffusivity functions respectively and subsequently equat-ing the boundary flux term

to wetting and drying fluxes yields respectively the following conditions

for wetting fw (qr) icircqr ___ icircxr

= ndash1(q s ndash q o)

(8h)

for drying

(8i)

where Jv is drying boundary flux acting in the direction of outward normal to the surface (msminus1)

It is worth mentioning that the adap-tation of Eq (8a) most widely used in previous studies implements only the reduced moisture content as in Eq (8b) and has the form

(8j)

Formulation of Element-Level Governing Equation

Using the governing differential equa-tion stated in Eq (8e) the FE formula-tion can be carried out using Galerkinrsquos weighted residual technique34 For the given problem the weighted residual statement is given as

(9a)

where wk is the weight function (k = 1 to number of nodes in an element) and l is the reduced size of element consti-tuting the FE mesh

Integration of Eq (9a) by parts leads to

(9b)

Now for a linear element with two nodes it is known that

(9c)

Thus

(9d)

where H is the interpolation matrix B is the gradient matrix and de is the vector of elemental degrees of freedom

Following Galerkinrsquos method the weights in Eq (9b) are taken equal to the interpolation functions

Thus for w1 = 1 ndash (xrl) Eq (9b) yields

(9e)

And for w2 = (xrl)

(9f)

Combining Eqs (9e) and (9f) the gen-eral element-level governing equation can be obtained as

(9g)

Substituting Eqs (8f) and (8g) in Eq (9g) and using Eqs (8h) and (8i) in tandem the governing equation for the boundary element respective to the cases of wetting and drying can be obtained as

(9h)

(9i)

196 Scientific Paper Structural Engineering International 22014

An inspection of Eqs (9h) and (9i) reveals their dependence on the functions fw(qr) and fd(qr) which are known to follow characteristic expo-nential and S-shaped forms for the respective cases of wetting and drying of concrete Also as the terms (Dw_dry Dd_wet) and (1 qs ndash qo) are relative in nature the derived expressions can be considered to represent the behaviour of all hydraulically similar materials

Equations (9h) and (9i) are semi-dis-crete and can be represented in matrix form as

[m]de + [k]de = q (9j)

where m and k are the element-level mass and the diffusivity matrices respectively de is the vector of time derivatives of elemental degrees of freedom and q is the vector of ele-mental nodal fluxes

In order to obtain a fully discretized system of equations the time deriva-tives of the field variable in these equa-tions are to be further approximated using the method of finite difference Adopting the CrankndashNicolson scheme a completely discrete system of equa-tions is obtained as

([m] + 05Δtr[k]n+1)den+1

= ([m] + 05Δtr[k]n)den

+ 05Δtr(qn+1 + qn) (9k)

where the superscripts n and (n+1) denote the previous and present time levels respectively in the course of transient analysis

Numerical Stability of the FE Scheme

The highly nonlinear nature of the problem necessitates the implementa-tion of an extremely refined mesh and a small time-step size to facilitate the stable convergence of simulated results However earlier studies based on FE analysis of parabolic partial differen-tial equations reported that the mini-mization of the time-step size beyond a certain threshold value induces numerical oscillations in the simulated results3536 The issue was addressed in the context of heat diffusion problems by determining the minimum time-step sizes for some typical one- and two-dimensional elements16 These crite-ria16 that had actually been developed for strictly constant material properties were tested for unsaturated seepage problems involving mildly nonlinear material properties and were found to

be equally useful17 In another study18 it was demonstrated that the use of lumped-mass scheme with linear ele-ments offers stable convergence and consistent results for FE-based solu-tion of Richardrsquos equation The obser-vation was however empirical and was not deduced mathematically

In the following section an attempt has been made to mathematically interpret the issue of numerical oscilla-tions and to ascertain the suitability of a lumped-mass type implemented with linear elements in providing a stable convergence In addition a set of cri-teria has been developed for the modi-fied model to regulate the variation of time-step size as a function of element length and hydraulic diffusivity for stable convergence of the FE scheme

General Conditions for Stability

To determine the nodal values of the field variable at a particular time-step all the element-level equations are assembled to obtain a set of nonlinear algebraic equations in the form [A]X = b with

[A] = [M] + 05Δtr[K]n+1 (10a)

b = ([M] ndash 05Δtr[K]n)Xn

+ Δtr(Qn+1 + Qn) (10b)

where [M] and [K] are respectively the global consistent mass and diffu-sivity matrices and X and Q are the global vectors for degrees of freedom and nodal fluxes respectively

It is evident that spatial oscillations in the simulated results are caused by the numerical characteristics of matrix [A] and vector b From basic reasoning it can be perceived that the occurrence of numerical oscillations can be elimi-nated if

1 [A] is strictly diagonally dominant with positive terms in the principal diagonal and negative terms else-where with magnitudes lesser than the smallest element of the principal diagonal

2 All the elements of b are posi-tive which necessarily depends on the diagonal elements of ([M] ndash 05Δtr[K]n)Xn to remain positive

Depending on the method to be imple-mented for solving the system [A]X = b the implications of the stated conditions would be as follows

(a) The constitution of [A] would be such that its manipulation using any of the elimination methods37

bull will not produce negative ele-ments on its principal diagonal

bull will not require row subtractions during the elimination phase thus the elements in b shall always remain positive provided condition (2) is already met

(b) If [A] is to be inverted for X = [A]ndash1b then none of the elements in the inverted matrix would be negative it would otherwise fetch negative values in the resulting vector

(c) The diagonal dominance of [A] is a suffi cient condition for the itera-tive methods to converge for any initial solution vector37

The specified conditions restrict the generation of negative values at nodes which generally manifest in an alter-nating pattern when a wetting front invades a dry medium They also arrest the oscillations causing a saw-tooth pattern in simulated moisture profile Fulfilling the proposed con-ditions thus eliminates both forms of oscillations that are reported to influ-ence the quality of results simulated under a FE scheme17 Furthermore it is to be recognized that the numerical oscillations are primarily caused due to the composition of matrix [A] that is in relation to the violation of condi-tion (1) The perturbations caused due to the multiplication of the negative diagonal elements of [M] ndash 05Δtr[K]n with the elements of vector [X]n are largely counterbalanced by the contri-bution of the off-diagonal elements in the respective rows which are always positive and have the same order of magnitude

The mathematical criteria to be adopted to enforce the proposed con-ditions depend on the type of element being implemented in the FE model This study considers only the instance of linear element in the following section

Stability Criteria

For the FE formulation presented in the Formulation of Element Level Governing Equation section the global characteristic matrices for the phase of wetting will be tri-diagonal and are defined as

(11a)

Structural Engineering International 22014 Scientific Paper 197

(11b)

where M ij L is an element of the ith row

and jth column of lumped-mass matrix

The revised criteria for satisfying con-ditions (1) and (2) would now be

which reduces to

Δtr lt l2[ fw(qrmax) ndash fw(qrmin)] (12b)

and

which reduces to

Δtr lt l2fw(qrmax) (12c)

Equations (12b) and (12c) establish that adopting the lumped-mass scheme in conjunction with a two-noded linear element allows the minimization of time-step size to achieve convergence without the need of simultaneously refining the mesh density It is impor-tant here to note that although the condition in Eq (12c) imposes a more stringent limit on Δtr in comparison to Eq (12b) the two conditions converge within a value of qrmax = 02 for fw(qr) = exp (6qr) qrmin = 0 and l =05 as is depicted in Fig 1 In fact the conver-gence occurs at a steeper rate for finer meshes and remains within the stated threshold for the coarsest meshes prac-

ticable for the problem The delineated criteria are thus particularly stringent for near-dry conditions and can be relaxed at higher levels of moisture content using average value of mois-ture content that is qravg in place of qrmax

Similarly for the phase of drying the governing criteria for time-step size corresponding to conditions (1) and (2) would be

(13a)

and (13b)

Implementing the Time-Step Size Criteria

To illustrate an application of the delineated time-stepping criteria and the influence of mass type on numeri-cal oscillations we consider a one-dimensional moisture ingress problem involving an initially dry (ie qr = 0 at tr = 0) concrete medium of length L = 010 m subjected to a rainfall of inten-sity Vo = 10 mmh until the attainment of surface saturation (ie qr = 1 at xr = 0) The typical material parameters opted for the analysis are qo = 0 m3m3 qs = 01811 m3m3 Dw_dry = 945 times 10ndash10m2s and n = 6 The FE mesh has been constituted using linear elements of reduced size l = 05 which equals 017 mm for the adopted parameter values

The FE analysis of the considered problem governed by Eq 8(e) has been carried out by four distinct approaches

(i) using a consistent mass type and varying the time-step size in accordance to Eq (11c)

(ii) with a lumped-mass type and varying the time-step size as per Eqs (12b) and (12c)

(iii) as in (i) but relaxing the time-stepping constraints in Eq (11c) by replacing qrmax by qravg

(iv) as in (ii) but relaxing the time-step-ping constraints in Eqs (12b) and (12c) by replacing qrmax by qravg

The time-stepping in FE analysis has been implemented using the following scheme

1 At any instant tr calculate the pres-ent time-step size (Δtrpresent) if tr = 0 take Δtrpresent = 001l2fw(qr = 0) and for tr gt 0 take Δtrpresent = Δtrnext as determined in the previous instant of analysis

where m is the order of the matri-ces which equals the total number of unknowns Mij and Kij represent an element of the ith row and jth column of the global consistent mass and dif-fusivity matrices respectively

To fulfil conditions (1) and (2) at any time level the following criteria are to be satisfied

which reduces to

l23fw(qrmax) lt Δtr

lt 2l23[ fw(qrmax) ndash fw(qrmax)] (11c)

and

l __ 6 ndash 05Δtr 05 ___

l [2fw(qrmax)] gt 0

which reduces to

Δtr lt l23fw(qrmax) (11d)

Here qrmax and qrmin represent the maximum and minimum values respectively of the reduced moisture content at any time level

As is evident Eqs (11c) and (11d) con-tradict each other and cannot be satisfied simultaneously However as has been pointed out earlier and will be dem-onstrated in what follows violating the constraint imposed by condition (2) that is Eq (11d) results only in minor oscilla-tions that gradually damp out Equation (11c) however imposes a more severe constraint and explains the occurrence of oscillations that set in when the time-step size (Δtr) is reduced without simul-taneously refining the mesh while using a consistent mass scheme with linear ele-ments18 The identified constraints can however be circumvented by replac-ing the consistent mass matrix with a lumped-mass scheme defined as

(12a)

0100

1

2

Max

imum

Δt r

3

4

5

02 03 04Maximum moisture content ( tmax)

05 06 07 08 09 1

12b12c

Fig 1 Convergence pattern of the maxi-mum time limit criteria delineated in Eqs (12b) and (12c)

198 Scientific Paper Structural Engineering International 22014

02

0

005

ndash005

0

0 500 1000 1500

Time step number2000 2500 2 4

Distance fromexposed face (mm)

6 8 1000

02

04

06

08

1

0 500 1000 1500

Time step number

2000 2500

04

Red

uced

moi

stur

eR

educ

ed m

oist

ure

06

08

1(a) (c)

(b)

Fig 2 Approach (i) (a) evolution of nodal values (b) oscillating emergence of nodal val-ues from initial dry state (c) final moisture distribution across the medium

(a) (c)

(b)

02

0

005

ndash005

0

0 200 400 600 800 1000

Time step number14001200 1600 2 4

Distance fromexposed face (mm)

6 8 1000

02

04

06

08

1

0 400200 800600 1000 1200

Time step number

1400 1500

04

Red

uced

moi

stur

eR

educ

ed m

oist

ure

06

08

1

Fig 3 Approach (ii) (a) evolution of nodal values (b) oscillation-free emergence of nodal values from initial dry state (c) final moisture distribution across the medium

(a) (c)

(b)

02

0

005

ndash005

0

0 5 10 15 20 25

Time step number3530 4540 2 4

Distance fromexposed face (mm)

6 8 1000

02

04

06

08

1

0 105 2015 25 30

Time step number

35 40 45

04

Red

uced

moi

stur

eR

educ

ed m

oist

ure

06

08

1

Fig 4 Approach (iii) (a) evolution of nodal values (b) oscillating emergence of nodal values from initial dry state (c) final moisture distribution across the medium

2 Compute moisture distribution at the present instant tr

3 Calculate the limiting-time-step size (Δtrlimit) using appropriate criteria

4 Calculate time-step size for next instant to be simulated (Δtrnext) if (Δtrlimit gt 2 Δtrpresent) then Δtrnext = 2 Δtrpresent else Δtrnext = 095 Δtrlimit

5 Increment time tr =tr + Δtrpresent

Figures 2ndash5 present the simulated results corresponding to each of the analysis approaches These illustra-tions describe the evolution of reduced moisture content at nodes in succes-sive time-steps and the distribution of moisture across the medium at the attainment of surface saturation The observations clearly depict that the FE analysis based on consistent mass approach induces numerical oscillations in the simulated results An inspection of Figs 2 and 4 further reveals that the oscillations strongly influence the initial stages of analysis and then gradually reduce to a rela-tively less intense level with a consis-tent effect on the nodes in a near-dry state at the toe of the wetting front The analysis based on lumped-mass scheme on the other hand ensures a stable con-vergence of nodal values as is depicted in Figs 3 and 4 and also leads to con-vergence in relatively fewer time-steps thereby economizing the computation time Table 1 provides a summary of the time-steps computed and the simu-lated parameters Relaxing the time-stepping constraints as in approaches (iii) and (iv) causes some perturbations in the initial evolution of nodal values as depicted in Figs 4 and 5 However as summarized in Table 1 the analysis leads to satisfactory results with a sig-nificantly reduced computational time These findings thus establish the advan-tage of implementing a lumped-mass scheme in conjunction with the relaxed time-stepping criteria delineated in the study for achieving computational effi-ciency and stability in the FE analysis of moisture transport in concrete

The conventional model stated in Eq (8j) which has been widely adopted in several previous studies was also implemented to analyse the consid-ered test problem The observed trends of numerical stability were found to be similar to the ones obtained with the model proposed in this study However the computational expense for analys-ing the conventional model was found to be higher in relation to the proposed model A comparison of the results has been summarized in Table 1

Structural Engineering International 22014 Scientific Paper 199

Discussion and Conclusions

A rational study of moisture move-ment in concrete rests on appropriate modelling of associated mass transport phenomena and on the efficacy of the numerical analysis of the constituted mathematical model A sound compre-hension of these facets is essential for a plausible treatment of the problem The present work has addressed these aspects in the context of a rain-induced wettingndashdrying scenario and in its dis-course has emphasized the following ideas

(a) The parabolic partial differential equation provides a phenomeno-logical description of the fl ow of moisture in concrete under wet-ting and drying conditions albeit with distinct diffusivity functions

(b) The presently available empirical models for hydraulic diffusivity incorporate parameters which are known to depend on temperature

and the pore structure proper-ties of concrete This dependence however remains to be estab-lished mathematically in terms of the generic concrete proper-ties Future studies towards the refi nement of hydraulic diffusiv-ity models would enable a direct application of the delineated phe-nomenological approach in ana-lysing the moisture transport in structural concrete

(c) The restatement of the governing equation using the set of dimen-sionless terms qr = (q ndashqo)(qs ndashqo) xr = (VoDd_wet)x and tr = ( V o 2 Dd_wet)t (i) reduces the range of orders involved and hence minimizes the errors inherent in numerical computations (ii) pro-vides a representation character-istic of all hydraulically similar materials and (iii) economizes the computation time

(d) Galerkinrsquos method can be effec-tively implemented to formulate

a FE scheme for numerical anal-ysis of the moisture transport problem Its application has been elaborated using the modifi ed model to describe the constitu-tion of the characteristic matrices in detail

(e) The occurrence of numerical oscil-lations inherent to the FE analysis of diffusion models relates to the nature of the matrices involved in the analysis and can be effectively controlled by imposing the condi-tions outlined in the study

(f) The effi ciency of lumped-mass scheme implemented in conjunc-tion with a linear element for one-dimensional analysis has been mathematically interpreted A concomitant time-stepping crite-rion for stable convergence has been derived and its viability has been demonstrated through a test problem

The deliberations presented in this paper provide an insight into the math-ematical framework underlying the study of moisture transport phenom-ena in concrete and facilitate a robust FE analysis of moisture distribution in concrete subjected to wettingndashdrying exposure

References

[1] Andrade C Sarria J Alonso C Relative humidity in the interior of concrete exposed to natural and artificial weathering Cement Concrete Res 1999 29(8) 1249ndash1259

[2] Ryu DW Ko JW Noguchi T Effect of simu-lated environmental conditions on the internal relative humidity and relative moisture con-tent distribution of exposed concrete Cement Concrete Comp 2011 33(1) 142ndash153

[3] Zhang J Gao Y Han Y Interior humidity of concrete under dry-wet cycles J Mater Civil Eng 2012 24(3) 289ndash298

[4] Gawin D Pesavento F Schrefler BA Hygro-thermo-chemo-mechanical modelling of con-crete at early ages and beyond Part I Hydration and hygro-thermal phenomena Int J Numer Meth Eng 2006 67(3) 299ndash331

[5] Gawin D Pesavento F Schrefler BA Hygro-thermo-chemo-mechanical modelling of con-crete at early ages and beyond Part II Shrinkage and Creep of Concrete Int J Numer Meth Eng 2006 67(3) 332ndash363

[6] Saetta AV Schrefler BA Vitaliani RV The carbonation of concrete and the mechanism of moisture heat and carbon dioxide flow through porous materials Cement Concrete Res 1993 23(4) 761ndash772

[7] Saetta AV Schrefler BA Vitaliani RV 2-D model for carbonation and moistureheat flow in porous materials Cement Concrete Res 1995 25(8) 1703ndash1712

Approach No of time-steps computed

Time taken for surface saturation

(s)

Area under fi nal moisture profi le

(m3)

Length penetrated

(mm)

Eq (8e)

i 2629 24830 68988endash004 510ii 1762 24834 68998endash004 527iii 47 24887 69146endash004 510iv 42 25100 69734endash004 527

Eq (8j)

i 2652 24830 68989endash004 510ii 1785 24835 69000endash004 527iii 70 24862 69076endash004 510iv 65 25088 69703endash004 527

Table 1 Comparison of computational expense and simulated parameters for the proposed and conventional models

Fig 5 Approach (iv) (a) evolution of nodal values (b) oscillation-free emergence of nodal values from initial dry state (c) final moistur e distribution across the medium

(a) (c)

(b)

02

0

005

ndash005

0

0 5 10 15 20 25

Time step number3530 40 2 4

Distance fromexposed face (mm)

6 8 1000

02

04

06

08

1

0 105 2015 25 30

Time step number

35 40

04

Red

uced

moi

stur

eR

educ

ed m

oist

ure

06

08

1

196 Scientific Paper Structural Engineering International 22014

An inspection of Eqs (9h) and (9i) reveals their dependence on the functions fw(qr) and fd(qr) which are known to follow characteristic expo-nential and S-shaped forms for the respective cases of wetting and drying of concrete Also as the terms (Dw_dry Dd_wet) and (1 qs ndash qo) are relative in nature the derived expressions can be considered to represent the behaviour of all hydraulically similar materials

Equations (9h) and (9i) are semi-dis-crete and can be represented in matrix form as

[m]de + [k]de = q (9j)

where m and k are the element-level mass and the diffusivity matrices respectively de is the vector of time derivatives of elemental degrees of freedom and q is the vector of ele-mental nodal fluxes

In order to obtain a fully discretized system of equations the time deriva-tives of the field variable in these equa-tions are to be further approximated using the method of finite difference Adopting the CrankndashNicolson scheme a completely discrete system of equa-tions is obtained as

([m] + 05Δtr[k]n+1)den+1

= ([m] + 05Δtr[k]n)den

+ 05Δtr(qn+1 + qn) (9k)

where the superscripts n and (n+1) denote the previous and present time levels respectively in the course of transient analysis

Numerical Stability of the FE Scheme

The highly nonlinear nature of the problem necessitates the implementa-tion of an extremely refined mesh and a small time-step size to facilitate the stable convergence of simulated results However earlier studies based on FE analysis of parabolic partial differen-tial equations reported that the mini-mization of the time-step size beyond a certain threshold value induces numerical oscillations in the simulated results3536 The issue was addressed in the context of heat diffusion problems by determining the minimum time-step sizes for some typical one- and two-dimensional elements16 These crite-ria16 that had actually been developed for strictly constant material properties were tested for unsaturated seepage problems involving mildly nonlinear material properties and were found to

be equally useful17 In another study18 it was demonstrated that the use of lumped-mass scheme with linear ele-ments offers stable convergence and consistent results for FE-based solu-tion of Richardrsquos equation The obser-vation was however empirical and was not deduced mathematically

In the following section an attempt has been made to mathematically interpret the issue of numerical oscilla-tions and to ascertain the suitability of a lumped-mass type implemented with linear elements in providing a stable convergence In addition a set of cri-teria has been developed for the modi-fied model to regulate the variation of time-step size as a function of element length and hydraulic diffusivity for stable convergence of the FE scheme

General Conditions for Stability

To determine the nodal values of the field variable at a particular time-step all the element-level equations are assembled to obtain a set of nonlinear algebraic equations in the form [A]X = b with

[A] = [M] + 05Δtr[K]n+1 (10a)

b = ([M] ndash 05Δtr[K]n)Xn

+ Δtr(Qn+1 + Qn) (10b)

where [M] and [K] are respectively the global consistent mass and diffu-sivity matrices and X and Q are the global vectors for degrees of freedom and nodal fluxes respectively

It is evident that spatial oscillations in the simulated results are caused by the numerical characteristics of matrix [A] and vector b From basic reasoning it can be perceived that the occurrence of numerical oscillations can be elimi-nated if

1 [A] is strictly diagonally dominant with positive terms in the principal diagonal and negative terms else-where with magnitudes lesser than the smallest element of the principal diagonal

2 All the elements of b are posi-tive which necessarily depends on the diagonal elements of ([M] ndash 05Δtr[K]n)Xn to remain positive

Depending on the method to be imple-mented for solving the system [A]X = b the implications of the stated conditions would be as follows

(a) The constitution of [A] would be such that its manipulation using any of the elimination methods37

bull will not produce negative ele-ments on its principal diagonal

bull will not require row subtractions during the elimination phase thus the elements in b shall always remain positive provided condition (2) is already met

(b) If [A] is to be inverted for X = [A]ndash1b then none of the elements in the inverted matrix would be negative it would otherwise fetch negative values in the resulting vector

(c) The diagonal dominance of [A] is a suffi cient condition for the itera-tive methods to converge for any initial solution vector37

The specified conditions restrict the generation of negative values at nodes which generally manifest in an alter-nating pattern when a wetting front invades a dry medium They also arrest the oscillations causing a saw-tooth pattern in simulated moisture profile Fulfilling the proposed con-ditions thus eliminates both forms of oscillations that are reported to influ-ence the quality of results simulated under a FE scheme17 Furthermore it is to be recognized that the numerical oscillations are primarily caused due to the composition of matrix [A] that is in relation to the violation of condi-tion (1) The perturbations caused due to the multiplication of the negative diagonal elements of [M] ndash 05Δtr[K]n with the elements of vector [X]n are largely counterbalanced by the contri-bution of the off-diagonal elements in the respective rows which are always positive and have the same order of magnitude

The mathematical criteria to be adopted to enforce the proposed con-ditions depend on the type of element being implemented in the FE model This study considers only the instance of linear element in the following section

Stability Criteria

For the FE formulation presented in the Formulation of Element Level Governing Equation section the global characteristic matrices for the phase of wetting will be tri-diagonal and are defined as

(11a)

Structural Engineering International 22014 Scientific Paper 197

(11b)

where M ij L is an element of the ith row

and jth column of lumped-mass matrix

The revised criteria for satisfying con-ditions (1) and (2) would now be

which reduces to

Δtr lt l2[ fw(qrmax) ndash fw(qrmin)] (12b)

and

which reduces to

Δtr lt l2fw(qrmax) (12c)

Equations (12b) and (12c) establish that adopting the lumped-mass scheme in conjunction with a two-noded linear element allows the minimization of time-step size to achieve convergence without the need of simultaneously refining the mesh density It is impor-tant here to note that although the condition in Eq (12c) imposes a more stringent limit on Δtr in comparison to Eq (12b) the two conditions converge within a value of qrmax = 02 for fw(qr) = exp (6qr) qrmin = 0 and l =05 as is depicted in Fig 1 In fact the conver-gence occurs at a steeper rate for finer meshes and remains within the stated threshold for the coarsest meshes prac-

ticable for the problem The delineated criteria are thus particularly stringent for near-dry conditions and can be relaxed at higher levels of moisture content using average value of mois-ture content that is qravg in place of qrmax

Similarly for the phase of drying the governing criteria for time-step size corresponding to conditions (1) and (2) would be

(13a)

and (13b)

Implementing the Time-Step Size Criteria

To illustrate an application of the delineated time-stepping criteria and the influence of mass type on numeri-cal oscillations we consider a one-dimensional moisture ingress problem involving an initially dry (ie qr = 0 at tr = 0) concrete medium of length L = 010 m subjected to a rainfall of inten-sity Vo = 10 mmh until the attainment of surface saturation (ie qr = 1 at xr = 0) The typical material parameters opted for the analysis are qo = 0 m3m3 qs = 01811 m3m3 Dw_dry = 945 times 10ndash10m2s and n = 6 The FE mesh has been constituted using linear elements of reduced size l = 05 which equals 017 mm for the adopted parameter values

The FE analysis of the considered problem governed by Eq 8(e) has been carried out by four distinct approaches

(i) using a consistent mass type and varying the time-step size in accordance to Eq (11c)

(ii) with a lumped-mass type and varying the time-step size as per Eqs (12b) and (12c)

(iii) as in (i) but relaxing the time-stepping constraints in Eq (11c) by replacing qrmax by qravg

(iv) as in (ii) but relaxing the time-step-ping constraints in Eqs (12b) and (12c) by replacing qrmax by qravg

The time-stepping in FE analysis has been implemented using the following scheme

1 At any instant tr calculate the pres-ent time-step size (Δtrpresent) if tr = 0 take Δtrpresent = 001l2fw(qr = 0) and for tr gt 0 take Δtrpresent = Δtrnext as determined in the previous instant of analysis

where m is the order of the matri-ces which equals the total number of unknowns Mij and Kij represent an element of the ith row and jth column of the global consistent mass and dif-fusivity matrices respectively

To fulfil conditions (1) and (2) at any time level the following criteria are to be satisfied

which reduces to

l23fw(qrmax) lt Δtr

lt 2l23[ fw(qrmax) ndash fw(qrmax)] (11c)

and

l __ 6 ndash 05Δtr 05 ___

l [2fw(qrmax)] gt 0

which reduces to

Δtr lt l23fw(qrmax) (11d)

Here qrmax and qrmin represent the maximum and minimum values respectively of the reduced moisture content at any time level

As is evident Eqs (11c) and (11d) con-tradict each other and cannot be satisfied simultaneously However as has been pointed out earlier and will be dem-onstrated in what follows violating the constraint imposed by condition (2) that is Eq (11d) results only in minor oscilla-tions that gradually damp out Equation (11c) however imposes a more severe constraint and explains the occurrence of oscillations that set in when the time-step size (Δtr) is reduced without simul-taneously refining the mesh while using a consistent mass scheme with linear ele-ments18 The identified constraints can however be circumvented by replac-ing the consistent mass matrix with a lumped-mass scheme defined as

(12a)

0100

1

2

Max

imum

Δt r

3

4

5

02 03 04Maximum moisture content ( tmax)

05 06 07 08 09 1

12b12c

Fig 1 Convergence pattern of the maxi-mum time limit criteria delineated in Eqs (12b) and (12c)

198 Scientific Paper Structural Engineering International 22014

02

0

005

ndash005

0

0 500 1000 1500

Time step number2000 2500 2 4

Distance fromexposed face (mm)

6 8 1000

02

04

06

08

1

0 500 1000 1500

Time step number

2000 2500

04

Red

uced

moi

stur

eR

educ

ed m

oist

ure

06

08

1(a) (c)

(b)

Fig 2 Approach (i) (a) evolution of nodal values (b) oscillating emergence of nodal val-ues from initial dry state (c) final moisture distribution across the medium

(a) (c)

(b)

02

0

005

ndash005

0

0 200 400 600 800 1000

Time step number14001200 1600 2 4

Distance fromexposed face (mm)

6 8 1000

02

04

06

08

1

0 400200 800600 1000 1200

Time step number

1400 1500

04

Red

uced

moi

stur

eR

educ

ed m

oist

ure

06

08

1

Fig 3 Approach (ii) (a) evolution of nodal values (b) oscillation-free emergence of nodal values from initial dry state (c) final moisture distribution across the medium

(a) (c)

(b)

02

0

005

ndash005

0

0 5 10 15 20 25

Time step number3530 4540 2 4

Distance fromexposed face (mm)

6 8 1000

02

04

06

08

1

0 105 2015 25 30

Time step number

35 40 45

04

Red

uced

moi

stur

eR

educ

ed m

oist

ure

06

08

1

Fig 4 Approach (iii) (a) evolution of nodal values (b) oscillating emergence of nodal values from initial dry state (c) final moisture distribution across the medium

2 Compute moisture distribution at the present instant tr

3 Calculate the limiting-time-step size (Δtrlimit) using appropriate criteria

4 Calculate time-step size for next instant to be simulated (Δtrnext) if (Δtrlimit gt 2 Δtrpresent) then Δtrnext = 2 Δtrpresent else Δtrnext = 095 Δtrlimit

5 Increment time tr =tr + Δtrpresent

Figures 2ndash5 present the simulated results corresponding to each of the analysis approaches These illustra-tions describe the evolution of reduced moisture content at nodes in succes-sive time-steps and the distribution of moisture across the medium at the attainment of surface saturation The observations clearly depict that the FE analysis based on consistent mass approach induces numerical oscillations in the simulated results An inspection of Figs 2 and 4 further reveals that the oscillations strongly influence the initial stages of analysis and then gradually reduce to a rela-tively less intense level with a consis-tent effect on the nodes in a near-dry state at the toe of the wetting front The analysis based on lumped-mass scheme on the other hand ensures a stable con-vergence of nodal values as is depicted in Figs 3 and 4 and also leads to con-vergence in relatively fewer time-steps thereby economizing the computation time Table 1 provides a summary of the time-steps computed and the simu-lated parameters Relaxing the time-stepping constraints as in approaches (iii) and (iv) causes some perturbations in the initial evolution of nodal values as depicted in Figs 4 and 5 However as summarized in Table 1 the analysis leads to satisfactory results with a sig-nificantly reduced computational time These findings thus establish the advan-tage of implementing a lumped-mass scheme in conjunction with the relaxed time-stepping criteria delineated in the study for achieving computational effi-ciency and stability in the FE analysis of moisture transport in concrete

The conventional model stated in Eq (8j) which has been widely adopted in several previous studies was also implemented to analyse the consid-ered test problem The observed trends of numerical stability were found to be similar to the ones obtained with the model proposed in this study However the computational expense for analys-ing the conventional model was found to be higher in relation to the proposed model A comparison of the results has been summarized in Table 1

Structural Engineering International 22014 Scientific Paper 199

Discussion and Conclusions

A rational study of moisture move-ment in concrete rests on appropriate modelling of associated mass transport phenomena and on the efficacy of the numerical analysis of the constituted mathematical model A sound compre-hension of these facets is essential for a plausible treatment of the problem The present work has addressed these aspects in the context of a rain-induced wettingndashdrying scenario and in its dis-course has emphasized the following ideas

(a) The parabolic partial differential equation provides a phenomeno-logical description of the fl ow of moisture in concrete under wet-ting and drying conditions albeit with distinct diffusivity functions

(b) The presently available empirical models for hydraulic diffusivity incorporate parameters which are known to depend on temperature

and the pore structure proper-ties of concrete This dependence however remains to be estab-lished mathematically in terms of the generic concrete proper-ties Future studies towards the refi nement of hydraulic diffusiv-ity models would enable a direct application of the delineated phe-nomenological approach in ana-lysing the moisture transport in structural concrete

(c) The restatement of the governing equation using the set of dimen-sionless terms qr = (q ndashqo)(qs ndashqo) xr = (VoDd_wet)x and tr = ( V o 2 Dd_wet)t (i) reduces the range of orders involved and hence minimizes the errors inherent in numerical computations (ii) pro-vides a representation character-istic of all hydraulically similar materials and (iii) economizes the computation time

(d) Galerkinrsquos method can be effec-tively implemented to formulate

a FE scheme for numerical anal-ysis of the moisture transport problem Its application has been elaborated using the modifi ed model to describe the constitu-tion of the characteristic matrices in detail

(e) The occurrence of numerical oscil-lations inherent to the FE analysis of diffusion models relates to the nature of the matrices involved in the analysis and can be effectively controlled by imposing the condi-tions outlined in the study

(f) The effi ciency of lumped-mass scheme implemented in conjunc-tion with a linear element for one-dimensional analysis has been mathematically interpreted A concomitant time-stepping crite-rion for stable convergence has been derived and its viability has been demonstrated through a test problem

The deliberations presented in this paper provide an insight into the math-ematical framework underlying the study of moisture transport phenom-ena in concrete and facilitate a robust FE analysis of moisture distribution in concrete subjected to wettingndashdrying exposure

References

[1] Andrade C Sarria J Alonso C Relative humidity in the interior of concrete exposed to natural and artificial weathering Cement Concrete Res 1999 29(8) 1249ndash1259

[2] Ryu DW Ko JW Noguchi T Effect of simu-lated environmental conditions on the internal relative humidity and relative moisture con-tent distribution of exposed concrete Cement Concrete Comp 2011 33(1) 142ndash153

[3] Zhang J Gao Y Han Y Interior humidity of concrete under dry-wet cycles J Mater Civil Eng 2012 24(3) 289ndash298

[4] Gawin D Pesavento F Schrefler BA Hygro-thermo-chemo-mechanical modelling of con-crete at early ages and beyond Part I Hydration and hygro-thermal phenomena Int J Numer Meth Eng 2006 67(3) 299ndash331

[5] Gawin D Pesavento F Schrefler BA Hygro-thermo-chemo-mechanical modelling of con-crete at early ages and beyond Part II Shrinkage and Creep of Concrete Int J Numer Meth Eng 2006 67(3) 332ndash363

[6] Saetta AV Schrefler BA Vitaliani RV The carbonation of concrete and the mechanism of moisture heat and carbon dioxide flow through porous materials Cement Concrete Res 1993 23(4) 761ndash772

[7] Saetta AV Schrefler BA Vitaliani RV 2-D model for carbonation and moistureheat flow in porous materials Cement Concrete Res 1995 25(8) 1703ndash1712

Approach No of time-steps computed

Time taken for surface saturation

(s)

Area under fi nal moisture profi le

(m3)

Length penetrated

(mm)

Eq (8e)

i 2629 24830 68988endash004 510ii 1762 24834 68998endash004 527iii 47 24887 69146endash004 510iv 42 25100 69734endash004 527

Eq (8j)

i 2652 24830 68989endash004 510ii 1785 24835 69000endash004 527iii 70 24862 69076endash004 510iv 65 25088 69703endash004 527

Table 1 Comparison of computational expense and simulated parameters for the proposed and conventional models

Fig 5 Approach (iv) (a) evolution of nodal values (b) oscillation-free emergence of nodal values from initial dry state (c) final moistur e distribution across the medium

(a) (c)

(b)

02

0

005

ndash005

0

0 5 10 15 20 25

Time step number3530 40 2 4

Distance fromexposed face (mm)

6 8 1000

02

04

06

08

1

0 105 2015 25 30

Time step number

35 40

04

Red

uced

moi

stur

eR

educ

ed m

oist

ure

06

08

1

Structural Engineering International 22014 Scientific Paper 197

(11b)

where M ij L is an element of the ith row

and jth column of lumped-mass matrix

The revised criteria for satisfying con-ditions (1) and (2) would now be

which reduces to

Δtr lt l2[ fw(qrmax) ndash fw(qrmin)] (12b)

and

which reduces to

Δtr lt l2fw(qrmax) (12c)

Equations (12b) and (12c) establish that adopting the lumped-mass scheme in conjunction with a two-noded linear element allows the minimization of time-step size to achieve convergence without the need of simultaneously refining the mesh density It is impor-tant here to note that although the condition in Eq (12c) imposes a more stringent limit on Δtr in comparison to Eq (12b) the two conditions converge within a value of qrmax = 02 for fw(qr) = exp (6qr) qrmin = 0 and l =05 as is depicted in Fig 1 In fact the conver-gence occurs at a steeper rate for finer meshes and remains within the stated threshold for the coarsest meshes prac-

ticable for the problem The delineated criteria are thus particularly stringent for near-dry conditions and can be relaxed at higher levels of moisture content using average value of mois-ture content that is qravg in place of qrmax

Similarly for the phase of drying the governing criteria for time-step size corresponding to conditions (1) and (2) would be

(13a)

and (13b)

Implementing the Time-Step Size Criteria

To illustrate an application of the delineated time-stepping criteria and the influence of mass type on numeri-cal oscillations we consider a one-dimensional moisture ingress problem involving an initially dry (ie qr = 0 at tr = 0) concrete medium of length L = 010 m subjected to a rainfall of inten-sity Vo = 10 mmh until the attainment of surface saturation (ie qr = 1 at xr = 0) The typical material parameters opted for the analysis are qo = 0 m3m3 qs = 01811 m3m3 Dw_dry = 945 times 10ndash10m2s and n = 6 The FE mesh has been constituted using linear elements of reduced size l = 05 which equals 017 mm for the adopted parameter values

The FE analysis of the considered problem governed by Eq 8(e) has been carried out by four distinct approaches

(i) using a consistent mass type and varying the time-step size in accordance to Eq (11c)

(ii) with a lumped-mass type and varying the time-step size as per Eqs (12b) and (12c)

(iii) as in (i) but relaxing the time-stepping constraints in Eq (11c) by replacing qrmax by qravg

(iv) as in (ii) but relaxing the time-step-ping constraints in Eqs (12b) and (12c) by replacing qrmax by qravg

The time-stepping in FE analysis has been implemented using the following scheme

1 At any instant tr calculate the pres-ent time-step size (Δtrpresent) if tr = 0 take Δtrpresent = 001l2fw(qr = 0) and for tr gt 0 take Δtrpresent = Δtrnext as determined in the previous instant of analysis

where m is the order of the matri-ces which equals the total number of unknowns Mij and Kij represent an element of the ith row and jth column of the global consistent mass and dif-fusivity matrices respectively

To fulfil conditions (1) and (2) at any time level the following criteria are to be satisfied

which reduces to

l23fw(qrmax) lt Δtr

lt 2l23[ fw(qrmax) ndash fw(qrmax)] (11c)

and

l __ 6 ndash 05Δtr 05 ___

l [2fw(qrmax)] gt 0

which reduces to

Δtr lt l23fw(qrmax) (11d)

Here qrmax and qrmin represent the maximum and minimum values respectively of the reduced moisture content at any time level

As is evident Eqs (11c) and (11d) con-tradict each other and cannot be satisfied simultaneously However as has been pointed out earlier and will be dem-onstrated in what follows violating the constraint imposed by condition (2) that is Eq (11d) results only in minor oscilla-tions that gradually damp out Equation (11c) however imposes a more severe constraint and explains the occurrence of oscillations that set in when the time-step size (Δtr) is reduced without simul-taneously refining the mesh while using a consistent mass scheme with linear ele-ments18 The identified constraints can however be circumvented by replac-ing the consistent mass matrix with a lumped-mass scheme defined as

(12a)

0100

1

2

Max

imum

Δt r

3

4

5

02 03 04Maximum moisture content ( tmax)

05 06 07 08 09 1

12b12c

Fig 1 Convergence pattern of the maxi-mum time limit criteria delineated in Eqs (12b) and (12c)

198 Scientific Paper Structural Engineering International 22014

02

0

005

ndash005

0

0 500 1000 1500

Time step number2000 2500 2 4

Distance fromexposed face (mm)

6 8 1000

02

04

06

08

1

0 500 1000 1500

Time step number

2000 2500

04

Red

uced

moi

stur

eR

educ

ed m

oist

ure

06

08

1(a) (c)

(b)

Fig 2 Approach (i) (a) evolution of nodal values (b) oscillating emergence of nodal val-ues from initial dry state (c) final moisture distribution across the medium

(a) (c)

(b)

02

0

005

ndash005

0

0 200 400 600 800 1000

Time step number14001200 1600 2 4

Distance fromexposed face (mm)

6 8 1000

02

04

06

08

1

0 400200 800600 1000 1200

Time step number

1400 1500

04

Red

uced

moi

stur

eR

educ

ed m

oist

ure

06

08

1

Fig 3 Approach (ii) (a) evolution of nodal values (b) oscillation-free emergence of nodal values from initial dry state (c) final moisture distribution across the medium

(a) (c)

(b)

02

0

005

ndash005

0

0 5 10 15 20 25

Time step number3530 4540 2 4

Distance fromexposed face (mm)

6 8 1000

02

04

06

08

1

0 105 2015 25 30

Time step number

35 40 45

04

Red

uced

moi

stur

eR

educ

ed m

oist

ure

06

08

1

Fig 4 Approach (iii) (a) evolution of nodal values (b) oscillating emergence of nodal values from initial dry state (c) final moisture distribution across the medium

2 Compute moisture distribution at the present instant tr

3 Calculate the limiting-time-step size (Δtrlimit) using appropriate criteria

4 Calculate time-step size for next instant to be simulated (Δtrnext) if (Δtrlimit gt 2 Δtrpresent) then Δtrnext = 2 Δtrpresent else Δtrnext = 095 Δtrlimit

5 Increment time tr =tr + Δtrpresent

Figures 2ndash5 present the simulated results corresponding to each of the analysis approaches These illustra-tions describe the evolution of reduced moisture content at nodes in succes-sive time-steps and the distribution of moisture across the medium at the attainment of surface saturation The observations clearly depict that the FE analysis based on consistent mass approach induces numerical oscillations in the simulated results An inspection of Figs 2 and 4 further reveals that the oscillations strongly influence the initial stages of analysis and then gradually reduce to a rela-tively less intense level with a consis-tent effect on the nodes in a near-dry state at the toe of the wetting front The analysis based on lumped-mass scheme on the other hand ensures a stable con-vergence of nodal values as is depicted in Figs 3 and 4 and also leads to con-vergence in relatively fewer time-steps thereby economizing the computation time Table 1 provides a summary of the time-steps computed and the simu-lated parameters Relaxing the time-stepping constraints as in approaches (iii) and (iv) causes some perturbations in the initial evolution of nodal values as depicted in Figs 4 and 5 However as summarized in Table 1 the analysis leads to satisfactory results with a sig-nificantly reduced computational time These findings thus establish the advan-tage of implementing a lumped-mass scheme in conjunction with the relaxed time-stepping criteria delineated in the study for achieving computational effi-ciency and stability in the FE analysis of moisture transport in concrete

The conventional model stated in Eq (8j) which has been widely adopted in several previous studies was also implemented to analyse the consid-ered test problem The observed trends of numerical stability were found to be similar to the ones obtained with the model proposed in this study However the computational expense for analys-ing the conventional model was found to be higher in relation to the proposed model A comparison of the results has been summarized in Table 1

Structural Engineering International 22014 Scientific Paper 199

Discussion and Conclusions

A rational study of moisture move-ment in concrete rests on appropriate modelling of associated mass transport phenomena and on the efficacy of the numerical analysis of the constituted mathematical model A sound compre-hension of these facets is essential for a plausible treatment of the problem The present work has addressed these aspects in the context of a rain-induced wettingndashdrying scenario and in its dis-course has emphasized the following ideas

(a) The parabolic partial differential equation provides a phenomeno-logical description of the fl ow of moisture in concrete under wet-ting and drying conditions albeit with distinct diffusivity functions

(b) The presently available empirical models for hydraulic diffusivity incorporate parameters which are known to depend on temperature

and the pore structure proper-ties of concrete This dependence however remains to be estab-lished mathematically in terms of the generic concrete proper-ties Future studies towards the refi nement of hydraulic diffusiv-ity models would enable a direct application of the delineated phe-nomenological approach in ana-lysing the moisture transport in structural concrete

(c) The restatement of the governing equation using the set of dimen-sionless terms qr = (q ndashqo)(qs ndashqo) xr = (VoDd_wet)x and tr = ( V o 2 Dd_wet)t (i) reduces the range of orders involved and hence minimizes the errors inherent in numerical computations (ii) pro-vides a representation character-istic of all hydraulically similar materials and (iii) economizes the computation time

(d) Galerkinrsquos method can be effec-tively implemented to formulate

a FE scheme for numerical anal-ysis of the moisture transport problem Its application has been elaborated using the modifi ed model to describe the constitu-tion of the characteristic matrices in detail

(e) The occurrence of numerical oscil-lations inherent to the FE analysis of diffusion models relates to the nature of the matrices involved in the analysis and can be effectively controlled by imposing the condi-tions outlined in the study

(f) The effi ciency of lumped-mass scheme implemented in conjunc-tion with a linear element for one-dimensional analysis has been mathematically interpreted A concomitant time-stepping crite-rion for stable convergence has been derived and its viability has been demonstrated through a test problem

The deliberations presented in this paper provide an insight into the math-ematical framework underlying the study of moisture transport phenom-ena in concrete and facilitate a robust FE analysis of moisture distribution in concrete subjected to wettingndashdrying exposure

References

[1] Andrade C Sarria J Alonso C Relative humidity in the interior of concrete exposed to natural and artificial weathering Cement Concrete Res 1999 29(8) 1249ndash1259

[2] Ryu DW Ko JW Noguchi T Effect of simu-lated environmental conditions on the internal relative humidity and relative moisture con-tent distribution of exposed concrete Cement Concrete Comp 2011 33(1) 142ndash153

[3] Zhang J Gao Y Han Y Interior humidity of concrete under dry-wet cycles J Mater Civil Eng 2012 24(3) 289ndash298

[4] Gawin D Pesavento F Schrefler BA Hygro-thermo-chemo-mechanical modelling of con-crete at early ages and beyond Part I Hydration and hygro-thermal phenomena Int J Numer Meth Eng 2006 67(3) 299ndash331

[5] Gawin D Pesavento F Schrefler BA Hygro-thermo-chemo-mechanical modelling of con-crete at early ages and beyond Part II Shrinkage and Creep of Concrete Int J Numer Meth Eng 2006 67(3) 332ndash363

[6] Saetta AV Schrefler BA Vitaliani RV The carbonation of concrete and the mechanism of moisture heat and carbon dioxide flow through porous materials Cement Concrete Res 1993 23(4) 761ndash772

[7] Saetta AV Schrefler BA Vitaliani RV 2-D model for carbonation and moistureheat flow in porous materials Cement Concrete Res 1995 25(8) 1703ndash1712

Approach No of time-steps computed

Time taken for surface saturation

(s)

Area under fi nal moisture profi le

(m3)

Length penetrated

(mm)

Eq (8e)

i 2629 24830 68988endash004 510ii 1762 24834 68998endash004 527iii 47 24887 69146endash004 510iv 42 25100 69734endash004 527

Eq (8j)

i 2652 24830 68989endash004 510ii 1785 24835 69000endash004 527iii 70 24862 69076endash004 510iv 65 25088 69703endash004 527

Table 1 Comparison of computational expense and simulated parameters for the proposed and conventional models

Fig 5 Approach (iv) (a) evolution of nodal values (b) oscillation-free emergence of nodal values from initial dry state (c) final moistur e distribution across the medium

(a) (c)

(b)

02

0

005

ndash005

0

0 5 10 15 20 25

Time step number3530 40 2 4

Distance fromexposed face (mm)

6 8 1000

02

04

06

08

1

0 105 2015 25 30

Time step number

35 40

04

Red

uced

moi

stur

eR

educ

ed m

oist

ure

06

08

1

198 Scientific Paper Structural Engineering International 22014

02

0

005

ndash005

0

0 500 1000 1500

Time step number2000 2500 2 4

Distance fromexposed face (mm)

6 8 1000

02

04

06

08

1

0 500 1000 1500

Time step number

2000 2500

04

Red

uced

moi

stur

eR

educ

ed m

oist

ure

06

08

1(a) (c)

(b)

Fig 2 Approach (i) (a) evolution of nodal values (b) oscillating emergence of nodal val-ues from initial dry state (c) final moisture distribution across the medium

(a) (c)

(b)

02

0

005

ndash005

0

0 200 400 600 800 1000

Time step number14001200 1600 2 4

Distance fromexposed face (mm)

6 8 1000

02

04

06

08

1

0 400200 800600 1000 1200

Time step number

1400 1500

04

Red

uced

moi

stur

eR

educ

ed m

oist

ure

06

08

1

Fig 3 Approach (ii) (a) evolution of nodal values (b) oscillation-free emergence of nodal values from initial dry state (c) final moisture distribution across the medium

(a) (c)

(b)

02

0

005

ndash005

0

0 5 10 15 20 25

Time step number3530 4540 2 4

Distance fromexposed face (mm)

6 8 1000

02

04

06

08

1

0 105 2015 25 30

Time step number

35 40 45

04

Red

uced

moi

stur

eR

educ

ed m

oist

ure

06

08

1

Fig 4 Approach (iii) (a) evolution of nodal values (b) oscillating emergence of nodal values from initial dry state (c) final moisture distribution across the medium

2 Compute moisture distribution at the present instant tr

3 Calculate the limiting-time-step size (Δtrlimit) using appropriate criteria

4 Calculate time-step size for next instant to be simulated (Δtrnext) if (Δtrlimit gt 2 Δtrpresent) then Δtrnext = 2 Δtrpresent else Δtrnext = 095 Δtrlimit

5 Increment time tr =tr + Δtrpresent

Figures 2ndash5 present the simulated results corresponding to each of the analysis approaches These illustra-tions describe the evolution of reduced moisture content at nodes in succes-sive time-steps and the distribution of moisture across the medium at the attainment of surface saturation The observations clearly depict that the FE analysis based on consistent mass approach induces numerical oscillations in the simulated results An inspection of Figs 2 and 4 further reveals that the oscillations strongly influence the initial stages of analysis and then gradually reduce to a rela-tively less intense level with a consis-tent effect on the nodes in a near-dry state at the toe of the wetting front The analysis based on lumped-mass scheme on the other hand ensures a stable con-vergence of nodal values as is depicted in Figs 3 and 4 and also leads to con-vergence in relatively fewer time-steps thereby economizing the computation time Table 1 provides a summary of the time-steps computed and the simu-lated parameters Relaxing the time-stepping constraints as in approaches (iii) and (iv) causes some perturbations in the initial evolution of nodal values as depicted in Figs 4 and 5 However as summarized in Table 1 the analysis leads to satisfactory results with a sig-nificantly reduced computational time These findings thus establish the advan-tage of implementing a lumped-mass scheme in conjunction with the relaxed time-stepping criteria delineated in the study for achieving computational effi-ciency and stability in the FE analysis of moisture transport in concrete

The conventional model stated in Eq (8j) which has been widely adopted in several previous studies was also implemented to analyse the consid-ered test problem The observed trends of numerical stability were found to be similar to the ones obtained with the model proposed in this study However the computational expense for analys-ing the conventional model was found to be higher in relation to the proposed model A comparison of the results has been summarized in Table 1

Structural Engineering International 22014 Scientific Paper 199

Discussion and Conclusions

A rational study of moisture move-ment in concrete rests on appropriate modelling of associated mass transport phenomena and on the efficacy of the numerical analysis of the constituted mathematical model A sound compre-hension of these facets is essential for a plausible treatment of the problem The present work has addressed these aspects in the context of a rain-induced wettingndashdrying scenario and in its dis-course has emphasized the following ideas

(a) The parabolic partial differential equation provides a phenomeno-logical description of the fl ow of moisture in concrete under wet-ting and drying conditions albeit with distinct diffusivity functions

(b) The presently available empirical models for hydraulic diffusivity incorporate parameters which are known to depend on temperature

and the pore structure proper-ties of concrete This dependence however remains to be estab-lished mathematically in terms of the generic concrete proper-ties Future studies towards the refi nement of hydraulic diffusiv-ity models would enable a direct application of the delineated phe-nomenological approach in ana-lysing the moisture transport in structural concrete

(c) The restatement of the governing equation using the set of dimen-sionless terms qr = (q ndashqo)(qs ndashqo) xr = (VoDd_wet)x and tr = ( V o 2 Dd_wet)t (i) reduces the range of orders involved and hence minimizes the errors inherent in numerical computations (ii) pro-vides a representation character-istic of all hydraulically similar materials and (iii) economizes the computation time

(d) Galerkinrsquos method can be effec-tively implemented to formulate

a FE scheme for numerical anal-ysis of the moisture transport problem Its application has been elaborated using the modifi ed model to describe the constitu-tion of the characteristic matrices in detail

(e) The occurrence of numerical oscil-lations inherent to the FE analysis of diffusion models relates to the nature of the matrices involved in the analysis and can be effectively controlled by imposing the condi-tions outlined in the study

(f) The effi ciency of lumped-mass scheme implemented in conjunc-tion with a linear element for one-dimensional analysis has been mathematically interpreted A concomitant time-stepping crite-rion for stable convergence has been derived and its viability has been demonstrated through a test problem

The deliberations presented in this paper provide an insight into the math-ematical framework underlying the study of moisture transport phenom-ena in concrete and facilitate a robust FE analysis of moisture distribution in concrete subjected to wettingndashdrying exposure

References

[1] Andrade C Sarria J Alonso C Relative humidity in the interior of concrete exposed to natural and artificial weathering Cement Concrete Res 1999 29(8) 1249ndash1259

[2] Ryu DW Ko JW Noguchi T Effect of simu-lated environmental conditions on the internal relative humidity and relative moisture con-tent distribution of exposed concrete Cement Concrete Comp 2011 33(1) 142ndash153

[3] Zhang J Gao Y Han Y Interior humidity of concrete under dry-wet cycles J Mater Civil Eng 2012 24(3) 289ndash298

[4] Gawin D Pesavento F Schrefler BA Hygro-thermo-chemo-mechanical modelling of con-crete at early ages and beyond Part I Hydration and hygro-thermal phenomena Int J Numer Meth Eng 2006 67(3) 299ndash331

[5] Gawin D Pesavento F Schrefler BA Hygro-thermo-chemo-mechanical modelling of con-crete at early ages and beyond Part II Shrinkage and Creep of Concrete Int J Numer Meth Eng 2006 67(3) 332ndash363

[6] Saetta AV Schrefler BA Vitaliani RV The carbonation of concrete and the mechanism of moisture heat and carbon dioxide flow through porous materials Cement Concrete Res 1993 23(4) 761ndash772

[7] Saetta AV Schrefler BA Vitaliani RV 2-D model for carbonation and moistureheat flow in porous materials Cement Concrete Res 1995 25(8) 1703ndash1712

Approach No of time-steps computed

Time taken for surface saturation

(s)

Area under fi nal moisture profi le

(m3)

Length penetrated

(mm)

Eq (8e)

i 2629 24830 68988endash004 510ii 1762 24834 68998endash004 527iii 47 24887 69146endash004 510iv 42 25100 69734endash004 527

Eq (8j)

i 2652 24830 68989endash004 510ii 1785 24835 69000endash004 527iii 70 24862 69076endash004 510iv 65 25088 69703endash004 527

Table 1 Comparison of computational expense and simulated parameters for the proposed and conventional models

Fig 5 Approach (iv) (a) evolution of nodal values (b) oscillation-free emergence of nodal values from initial dry state (c) final moistur e distribution across the medium

(a) (c)

(b)

02

0

005

ndash005

0

0 5 10 15 20 25

Time step number3530 40 2 4

Distance fromexposed face (mm)

6 8 1000

02

04

06

08

1

0 105 2015 25 30

Time step number

35 40

04

Red

uced

moi

stur

eR

educ

ed m

oist

ure

06

08

1

Structural Engineering International 22014 Scientific Paper 199

Discussion and Conclusions

A rational study of moisture move-ment in concrete rests on appropriate modelling of associated mass transport phenomena and on the efficacy of the numerical analysis of the constituted mathematical model A sound compre-hension of these facets is essential for a plausible treatment of the problem The present work has addressed these aspects in the context of a rain-induced wettingndashdrying scenario and in its dis-course has emphasized the following ideas

(a) The parabolic partial differential equation provides a phenomeno-logical description of the fl ow of moisture in concrete under wet-ting and drying conditions albeit with distinct diffusivity functions

(b) The presently available empirical models for hydraulic diffusivity incorporate parameters which are known to depend on temperature

and the pore structure proper-ties of concrete This dependence however remains to be estab-lished mathematically in terms of the generic concrete proper-ties Future studies towards the refi nement of hydraulic diffusiv-ity models would enable a direct application of the delineated phe-nomenological approach in ana-lysing the moisture transport in structural concrete

(c) The restatement of the governing equation using the set of dimen-sionless terms qr = (q ndashqo)(qs ndashqo) xr = (VoDd_wet)x and tr = ( V o 2 Dd_wet)t (i) reduces the range of orders involved and hence minimizes the errors inherent in numerical computations (ii) pro-vides a representation character-istic of all hydraulically similar materials and (iii) economizes the computation time

(d) Galerkinrsquos method can be effec-tively implemented to formulate

a FE scheme for numerical anal-ysis of the moisture transport problem Its application has been elaborated using the modifi ed model to describe the constitu-tion of the characteristic matrices in detail

(e) The occurrence of numerical oscil-lations inherent to the FE analysis of diffusion models relates to the nature of the matrices involved in the analysis and can be effectively controlled by imposing the condi-tions outlined in the study

(f) The effi ciency of lumped-mass scheme implemented in conjunc-tion with a linear element for one-dimensional analysis has been mathematically interpreted A concomitant time-stepping crite-rion for stable convergence has been derived and its viability has been demonstrated through a test problem

The deliberations presented in this paper provide an insight into the math-ematical framework underlying the study of moisture transport phenom-ena in concrete and facilitate a robust FE analysis of moisture distribution in concrete subjected to wettingndashdrying exposure

References

[1] Andrade C Sarria J Alonso C Relative humidity in the interior of concrete exposed to natural and artificial weathering Cement Concrete Res 1999 29(8) 1249ndash1259

[2] Ryu DW Ko JW Noguchi T Effect of simu-lated environmental conditions on the internal relative humidity and relative moisture con-tent distribution of exposed concrete Cement Concrete Comp 2011 33(1) 142ndash153

[3] Zhang J Gao Y Han Y Interior humidity of concrete under dry-wet cycles J Mater Civil Eng 2012 24(3) 289ndash298

[4] Gawin D Pesavento F Schrefler BA Hygro-thermo-chemo-mechanical modelling of con-crete at early ages and beyond Part I Hydration and hygro-thermal phenomena Int J Numer Meth Eng 2006 67(3) 299ndash331

[5] Gawin D Pesavento F Schrefler BA Hygro-thermo-chemo-mechanical modelling of con-crete at early ages and beyond Part II Shrinkage and Creep of Concrete Int J Numer Meth Eng 2006 67(3) 332ndash363

[6] Saetta AV Schrefler BA Vitaliani RV The carbonation of concrete and the mechanism of moisture heat and carbon dioxide flow through porous materials Cement Concrete Res 1993 23(4) 761ndash772

[7] Saetta AV Schrefler BA Vitaliani RV 2-D model for carbonation and moistureheat flow in porous materials Cement Concrete Res 1995 25(8) 1703ndash1712

Approach No of time-steps computed

Time taken for surface saturation

(s)

Area under fi nal moisture profi le

(m3)

Length penetrated

(mm)

Eq (8e)

i 2629 24830 68988endash004 510ii 1762 24834 68998endash004 527iii 47 24887 69146endash004 510iv 42 25100 69734endash004 527

Eq (8j)

i 2652 24830 68989endash004 510ii 1785 24835 69000endash004 527iii 70 24862 69076endash004 510iv 65 25088 69703endash004 527

Table 1 Comparison of computational expense and simulated parameters for the proposed and conventional models

Fig 5 Approach (iv) (a) evolution of nodal values (b) oscillation-free emergence of nodal values from initial dry state (c) final moistur e distribution across the medium

(a) (c)

(b)

02

0

005

ndash005

0

0 5 10 15 20 25

Time step number3530 40 2 4

Distance fromexposed face (mm)

6 8 1000

02

04

06

08

1

0 105 2015 25 30

Time step number

35 40

04

Red

uced

moi

stur

eR

educ

ed m

oist

ure

06

08

1

200 Scientific Paper Structural Engineering International 22014

[8] Isgor OB Razaqpur AG Finite element modelling of coupled heat transfer moisture transport and carbonation processes in concrete structures Cement Concrete Comp 2004 26(1) 57ndash73

[9] Li C Li K Chen Z Numerical analysis of moisture influential depth in concrete and its application in durability design Tsinghua Sci Technol 2008 13(1) 7ndash12

[10] Li C Li K Chen Z Numerical analysis of moisture influential depth in concrete during drying-wetting cycles Tsinghua Sci Technol 2008 13(5) 696ndash701

[11] Vasconcellos CAB Amorim JCC Numerical simulation of unsaturated flow in porous media using a mass conservative model Paper pre-sented at COBEM 2001 Proceedings of 16th Brazilian Congress of Mechanical Engineering 2001 139ndash148

[12] Hao X Zhang R Kravchenko A A mass conservative switching method for simulating saturated-unsaturated flow J Hydrol 2005 311(1ndash4) 254ndash265

[13] Miller CT Abhishek C Farthing MW A spatially and temporarily adaptive solution for Richardrsquos equation Adv Water Resour 2006 29(4) 525ndash545

[14] Shahraiyni HT Ashtiani BA Comparison of finite difference schemes for water flow in unsaturated soils World Acad Sci Eng Technol 2008 40 21ndash25

[15] Shahraiyni HT Ashtiani BA Mathematical forms and numerical schemes for the solution of unsaturated flow equations J Irrig Drain E-ASCE 2012 138(1) 63ndash72

[16] Thomas HR Zhou Z Minimum time step size for diffusion problem in FEM analysis Int J Numer Meth Eng 1997 40(20) 3865ndash3880

[17] Karthikeyan M Tan TS Phoon KK Numerical analysis in seepage analysis of unsaturated soils Can Geotech J 2001 38(3) 639ndash651

[18] Ju SH Kung KJS Mass types element orders and solution schemes for the Richardrsquos equation Comput Geosci 1997 23(2) 175ndash187

[19] Cerny R Rovnanikova P Transport pro-cesses in concrete SPON Press London and New York 2002

[20] Freeze RA Henry Darcy and the fountains of Dijon Ground Water 1994 32(1) 23ndash30

[21] Hall C Hoff WD Water transport in brick stone and concrete Taylor amp Francis London and New York 2002

[22] Pradhan B Nagesh M Bhattacharjee B Prediction of hydraulic diffusivity from the pore size distribution of concrete Cement Concrete Res 2005 35(9) 1724ndash1733

[23] Maekawa K Chaube R Kishi T Modelling of concrete performance EampFN Spon London 1999

[24] Maekawa K Ishida T Kishi T Multi-scale Modelling of Structural Concrete Taylor and Francis New York 2009

[25] Lin SH Nonlinear water diffusion in unsat-urated porous solid materials Int J Eng Sci 1992 30(12) 1677ndash1682

[26] Pel L Moisture Transport in Porous Building Materials Technische Universiteit Eindhoven 1995

[27] Lockington D Parlange JY Dux P Sorptivity and the estimation of water penetra-tion into unsaturated concrete Mater Struct 1999 32(5) 342ndash347

[28] Leech C Lockington D Dux P Unsaturated diffusivity functions for concrete derived from NMR images Mater Struct 2003 36(6) 413ndash418

[29] Hall C Water sorptivity of mortars and concretes a review Mag Concrete Res 1989 41(147) 51ndash61

[30] Bazant ZP Najjar LJ Drying of concrete as a nonlinear diffusion problem Cement Concrete Res 1971 1(5) 461ndash473

[31] Bazant ZP Najjar LJ Nonlinear water dif-fusion in nonsaturated concrete Materiaux Et Construct 1972 5(25) 3ndash20

[32] Sakata K A study of moisture diffusion in drying and drying shrinkage of concrete Cement Concrete Res 1983 13(2) 216ndash224

[33] Wong SF Wee TH Swaddiwudhipong S Study of water movement in concrete Mag Concrete Res 2001 53(3) 205ndash220

[34] Reddy JN An Introduction to the Finite Element Method 3rd edn TMH New Delhi 2005

[35] Sandhu RS Liu H Singh KJ Numerical performance of some finite element schemes for analysis of seepage in porous media Int J Numer Anal Met 1977 1(2) 177ndash194

[36] Vermeer PA Verruijt A An accuracy condi-tion for consolidation by finite elements Int J Numer Anal Met 1981 5(1) 1ndash14

[37] Hoffman JD Numerical Methods for Engineers and Scientists 2nd edn Marcel Dekker Inc New York and Basel 2001

Elegance in StructuresIABSE Conference Nara Japan

Organised byThe Japanese Group of IABSE

May 13-15 2015

Call for Papers May 15 2014

wwwiabseorgnara2015

photo courtesy Nikken Sekkei Ltd