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Numerical Heat Transfer, Part B: FundamentalsAn International Journal of Computation and Methodology

ISSN: 1040-7790 (Print) 1521-0626 (Online) Journal homepage: https://www.tandfonline.com/loi/unhb20

A new meshless “fragile points method” and alocal variational iteration method for generaltransient heat conduction in anisotropicnonhomogeneous media. Part I: Theory andimplementation

Yue Guan, Rade Grujicic, Xuechuan Wang, Leiting Dong & Satya N. Atluri

To cite this article: Yue Guan, Rade Grujicic, Xuechuan Wang, Leiting Dong & SatyaN. Atluri (2020): A new meshless “fragile points method” and a local variational iterationmethod for general transient heat conduction in anisotropic nonhomogeneous media.Part I: Theory and implementation, Numerical Heat Transfer, Part B: Fundamentals, DOI:10.1080/10407790.2020.1747278

To link to this article: https://doi.org/10.1080/10407790.2020.1747278

Published online: 07 Apr 2020.

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A new meshless “fragile points method” and a local variationaliteration method for general transient heat conductionin anisotropic nonhomogeneous media. Part I: Theoryand implementation

Yue Guana, Rade Grujicicb, Xuechuan Wangc, Leiting Dongd , and Satya N. Atluria

aDepartment of Mechanical Engineering, Texas Tech University, Lubbock, USA; bFaculty of MechanicalEngineering, University of Montenegro, Podgorica, Montenegro; cSchool of Astronautics, NorthwesternPolytechnical University, Xi’an, P.R. China; dSchool of Aeronautic Science and Engineering, Beihang University,Beijing, P.R. China

ABSTRACTA new and effective computational approach is presented for analyzingtransient heat conduction problems. The approach consists of a meshlessFragile Points Method (FPM) being utilized for spatial discretization, and aLocal Variational Iteration (LVI) scheme for time discretization. Anisotropyand nonhomogeneity do not give rise to any difficulties in the presentimplementation. The meshless FPM is based on a Galerkin weak-form for-mulation and thus leads to symmetric matrices. Local, very simple, polyno-mial and discontinuous trial and test functions are employed. In themeshless FPM, Interior Penalty Numerical Fluxes are introduced to ensurethe consistency of the method. The LVIM in the time domain is a combin-ation of the Variational Iteration Method (VIM) and a collocation method ineach finitely large time interval. The present methodology represents aconsiderable improvement to the current state of science in computationaltransient heat conduction in anisotropic nonhomogeneous media. In thisfirst part of the two-paper series, we focus on the theoretical formulationand implementation of the proposed methodology. Numerical results andvalidation are then presented in Part II.

ARTICLE HISTORYReceived 10 January 2020Accepted 19 March 2020

1. Introduction

Transient heat conduction analysis in nonhomogeneous media is of great interest in research andengineering applications [1–3]. Typical nonhomogeneous media include conventional compositematerials and functionally graded materials (FGMs), etc. Widely used composite materials such asCarbon-Fiber Reinforced Composites (CFRPs) mostly have anisotropic material properties [4, 5].FGMs, which have found prospective applications in aerospace industry, computer circuit indus-try, etc. [6], have material properties varying gradually in space [5, 7]. Therefore, there is anincreasing demand for a reliable, accurate and efficient numerical approach for heat conductionproblems in nonhomogeneous and anisotropic materials.

Numerical methods for solving transient heat conduction problems usually have two discret-ization stages [8, 9]. First, spatial discretization is employed. This step reduces the original partialdifferential equation (PDE) to a set of ordinary differential equations (ODEs) in time. These

CONTACT Yue Guan yuguan@ttu.edu Department of Mechanical Engineering, Texas Tech University, Lubbock, TX79415 USA; Leiting Dong ltdong@buaa.edu.cn School of Aeronautic Science and Engineering, Beihang University,Beijing 100191, P.R. China.� 2020 Taylor & Francis Group, LLC

NUMERICAL HEAT TRANSFER, PART B: FUNDAMENTALShttps://doi.org/10.1080/10407790.2020.1747278

ODEs, known as “semi-discrete equations”, are then discretized in time using ODE solvers. Themost commonly used discretization methods are the Finite Element Method (FEM) [10] in thespatial domain and the finite difference schemes [11] in the time domain. FEM codes usuallydemand a high-quality mesh, otherwise the accuracy of solutions can be significantly affected.FEM also has difficulties in analyzing problems with rupture and fragmentation, e.g., crack devel-opment in thermally shocked brittle materials. Finite difference methods in the time domain, onthe other hand, may have stability and accuracy problems while using large time steps. In general,the current commercial codes have disadvantages and limitations.

Apart from the FEM, other mesh-based numerical methods such as the Finite Volume Method(FVM) [12] and Boundary Element Method (BEM) [13, 14] can also be utilized in spatial discret-ization in heat transfer problems. In BEM, the non-availability of fundamental solutions in non-homogeneous anisotropic media is a serious limitation which is often insurmountable. The sameas the FEM, the FVM has drawbacks in solving problems with crack developments, and its accur-acy is also threatened by mesh distortion.

Another category of methods, known as “meshless methods”, is partly or completely free ofmesh discretization. As a result, the human and computer cost in generating a high-quality con-tiguous mesh can be eliminated or reduced. This is a great improvement especially in 3D prob-lems involving complex geometries. The Smoothed Particle Hydrodynamics (SPH) proposed byRandles and Libersky [15] is one of the earliest meshless methods. Though its original formula-tion has a problem of stability and particle deficiency on and near the boundaries, a number ofimproved methods based on the SPH have been carried out, including the modified SPH methodby Randles and Libersky [15], the Reproducing Kernel Particle Method (RKPM) by Liu et al. [16]and the Corrective Smoothed Particle Method (CSPM) by Chen et al. [17], etc. These methodsare extensively used in thermal analysis and fluid/solid mechanics. The SPH and its improvedmethods are based on a strong form, making it difficult to strictly prove their stability and con-vergence. Yet numerical tests have implied that the SPH method may turn unstable under ran-dom point distributions [18].

There are also meshless methods which are based on weak-forms or variational principles. TheDiffuse Element Method (DEM) [19] was initially introduced as a generalization of the FEM byremoving some limitations related to the trial functions and mesh generations. The original for-mulation fails in passing the patch test. Nevertheless, Krongauz and Belytschko [20] establishedan improved DEM based on Petrov-Galerkin formulation (PG DEM) which satisfies the patchtest but resulting in asymmetric matrices. Based on the DEM, the Element-Free Galerkin (EFG)method was carried out by Belytschko et al. [21] in which the shape functions are developed byMoving Least Squares (MLS) or Radial Basis Function (RBF) approximations. The MLS approxi-mation does not have delta-function properties and hence imposition of essential boundary con-ditions necessaires special treatments [22]. In addition, the integration of the Galerkin functionalin EFG requires back-ground meshes. Furthermore, the EFG is not necessarily objective when theback-ground meshes are rotated. After that, a Local Boundary Integral Equation (LBIE) Methodis introduced by Zhu et al. [23]. While analogous to the BEM, the LBIE method circumvents theproblem of global fundamental solutions in nonhomogeneous anisotropic materials in BEM. Ituses local fundamental solutions (assuming locally homogeneous material properties) orHeaviside functions as test functions. Finally, Atluri and Zhu [24] proposed the Meshless LocalPetrov-Galerkin (MLPG) approach. Compared with the EFG method, the MLPG approachemploys the local Petrov-Galerkin weak formulation instead of the global Galerkin weak formula-tion. The MLPG method is a truly meshless method and has shown its capability and accuracy in2D and 3D transient heat conduction analysis involving anisotropy, nonhomogeneity and tem-perature-dependent material properties [5, 25]. Yet the MLPG method still has its limitations: thematrices are asymmetric; the numerical integration in the Petrov-Galerkin weak-form over localsubdomains is expensive, as a result of the complex shape functions; and (the same as the EFG

2 Y. GUAN ET AL.

method) the essential boundary conditions cannot be imposed directly if MLS approximation isused as trial functions. A modified collocation method has to be applied to enforce the essentialboundary conditions [22].

In contrast to the complex, global continuous MLS approximated shape function used in theEFG and MLPG, local, simple, polynomial, piecewise-continuous trial functions are applied ingenerating the Fragile Points Method (FPM) in [26]. Nevertheless, the method would becomeinconsistent if the Galerkin weak-form is employed directly with these discontinuous trial andtest functions. The Numerical Flux Correction, which is widely used in Discontinuous Galerkin(DG) methods [27, 28], is introduced to remedy the problem. Whereas on the other hand, theinherent discontinuity can also be a benefit. Since it is convenient to relax the continuity require-ment between neighboring points, the FPM has a great potential in analyzing systems involvingcracks, ruptures and fragmentations, such as in problems of thermal shock damage in brittle sol-ids. The method has already shown its stability, accuracy and efficiency in solving 1D and 2DPoisson equations [26] and elasticity problems [29]. In the current work, the method is extendedto 2D and 3D heat transfer analysis for spatial discretization for the first time. The Galerkin func-tional in the FPM can be integrated quite simply, and finally leads to sparse and symmetricmatrices. Thus, the FPM based on a Galerkin weak-form is far more efficient than either the EFGor the MLPG method.

After the spatial discretization is carried out, a semi-discrete system is obtained. The computa-tional accuracy and efficiency are also significantly affected by the ODE solver employed in thetime domain. Though nonlinearity is not emphasized in this paper, here we focus on generalizedtime discretization methods that can deal with linear as well as nonlinear ODEs. These methodscan roughly be divided into two categories: 1). The finite difference method, which is simple andthe most widely used, and 2). weak-form method based on weighted residual approximations[30]. The most well-known finite difference schemes include: the central, forward and backwarddifference schemes [31], Runge-Kutta method [32], Newmark-b method [33] and Hilber-Hughes-Taylor (HHT)-a method [34]. The Houbolt’s method [35] based on a third-order interpolation isalso famous in dynamic analysis. Ode45, the most popular, highly optimized built-in ODE solverin MATLAB, is based on an explicit Runge-Kutta formula [36, 37]. However, the efficiency of theclassic finite difference methods may not be sufficient for solving the high-dimensional semi-dis-crete system generated with the 2D or 3D FPM. Moreover, many finite difference methods mayencounter stability problems when considering nonlinearity or under large time steps [33, 34, 38].

On the other hand, the weak-form methods, though somewhat hard to implement, have apotential in analyzing high dimensional and nonlinear systems more efficiently. For periodic sys-tems, the Harmonic Balance (HB) method [39] and the Spectral Time Domain Collocation(TDC) method [40] were developed. Considering more general transient solutions, a series ofanalytical or semi-analytical asymptotic methods are introduced. He [41] proposed theVariational Iteration Method (VIM) which can be seen as an extension of the Newton-Raphsonmethod to nonlinear algebraic equations (NAEs). The Adomian Decomposition Method (ADM)is then developed by Adomian [42], in which the initial guess is corrected step by step by addingcomponents of an Adomian polynomial. Following that, the Picard Iteration Method (PIM) iscarried out by Fukushima [43] and modified by Woollands et al. [44]. The method is also basedon an initial guess and correctional iterative formula. The formula is relatively concise, yet com-puting the integral of nonlinear terms in each time step is a challenge. Though developed inde-pendently, the previous VIM, ADM and PIM approaches can be unified using a generalizedLagrange multiplier [45]. Based on that, Wang et al. [46] employed the VIM over a finitely largetime interval, along with a collocation method and numerical discretization, leading to the LocalVariational Iteration Method (LVIM). Unlike the VIM, the LVIM is a numerical method, applic-able to digital computation and has the potential in being implemented with parallel processing.The initial guess can be constructed in a relatively simple form. The method possesses excellent

NUMERICAL HEAT TRANSFER, PART B: FUNDAMENTALS 3

efficiency in solving nonlinear ODEs in fluid mechanics, structural mechanics, and astrophysics,etc [46]. Several approximate algorithms can be generated to further improve the computing effi-ciency. In the current work, in order to maintain the best stability, only the classic LVIM basedon the first kind of Chebyshev polynomials is considered. This method is also named asChebyshev Local Iterative Collocation-1 (CLIC-1) algorithm in [45].

In this paper, we focus on transient heat conduction problems in anisotropic nonhomogeneousmedia. The system is discretized by using the FPM in space, and the LVIM in the time domain.Section 2 presents the governing equations for heat conduction in anisotropic nonhomogeneousmedia, and the formulation of the FPM. The LVIM and its numerical implementations are thenintroduced in Section 3. Numerical results and validation of the overall methodology, as well as aparametric discussion, are given in Part II of this series.

2. Fragile points method (FPM) based on a Galerkin weak-form and point“stiffness” matrices

2.1. The heat conduction problem and governing equation

Consider a transient heat conduction problem in a continuously anisotropic nonhomogeneousmedium, which is governed by the following partial differential equation [5, 47]:

qðxÞcðxÞ @u@t

ðx, tÞ ¼ r � kruðx, tÞ½ � þ Qðx, tÞ, x 2 X, t 2 0,T½ �, (1)

where X is the entire 2D or 3D domain under study, x ¼ xy½ � in 2D or x ¼ xyz½ � in 3D, uðx, tÞ isthe temperature field, and Qðx, tÞ is the density of heat sources. r is the gradient operator. Thethermal conductivity tensor kðxÞ, mass density qðxÞ and specific heat capacity cðxÞ are dependenton the spatial coordinates in nonhomogeneous media. The thermal conductivity tensor compo-nents kij can also be directionally dependent for anisotropic materials.

Three main kinds of boundary conditions are considered:

1Þ: Dirichlet bc : uðx, tÞ ¼ euDðx, tÞ, on CD,2Þ: Neumann bc : qðx, tÞ :¼ kruðx, tÞ � n ¼ eqNðx, tÞ, on CN ,3Þ: Robin ðconvectiveÞ bc : qðx, tÞ ¼ hðxÞ euRðxÞ � uðx, tÞ½ �, on CR,

(2)

where the global boundary @X ¼ CD [ CN [ CR, n is the unit outward normal of @X, hðxÞ is theheat transfer coefficient, and euRðxÞ is the temperature of the medium outside the convect-ive boundary.

The initial condition is assumed as:

uðx, tÞ t¼0 ¼ uðx, 0Þ, in X [ @X:j (3)

2.2. Local, polynomial, point-based discontinuous trial and test functions

In the domain X, a set of random points are introduced. The global domain can then be parti-tioned into several confirming and nonoverlapping subdomains, with only one point in each sub-domain (as shown in Figure 1a–c). The subdomains could be of arbitrary geometric shapes. Andthe partition is not unique. For example, the Voronoi Diagram partition [48] can be applied.Figure 1d shows the traditional triangular elements used in FEM as a comparison. Unlike theFEM, the trial and test functions in the present Fragile Points Method (FPM) are not element-based interpolations, but are based on distribution of the random points. The trial functions areassumed to be simple polynomials in each subdomain. As a result, the temperature field could bediscontinuous between subdomains. In FEM on the other hand, the trial and test functions areelement-based, and are continuous at the inter-element boundaries.

4 Y. GUAN ET AL.

In each subdomain, we define the simple, local, polynomial trial function in terms of tempera-ture u and its gradient at the internal point. For instance, the trial function uh in subdomain E0which contains an internal point P0 can be written as:

uhðxÞ ¼ u0 þ ðx � x0Þ � rujP0 , x 2 E0 (4)

where u0 is the value of uh at P0, and x0 is the location vector of P0.The gradient of temperature ru at point P0 remains unknown. In the current work, we employ

the Generalized Finite Difference (GFD) method [49] to relaterujP0 to the value uh at several neigh-boring points of P0. Unlike the common definition of the support of P0 which includes all the pointswithin a distance r (as shown in Figure 2a), in the present work, the support of P0 is defined toinvolve all the nearest neighboring points of P0 in subdomains sharing boundaries with E0 in thepartition (shown in Figure 2b). The points are named as P1, P2, � � � , Pm.

In order to relate the gradient rujP0 to uh at neighboring points, we minimize the followingweighted discrete L2 norm J:

J ¼Xmi¼1

ðxi � x0Þ � rujP0 � ðui � u0Þ� �2wi, (5)

where xi donates the coordinates of Pi, ui is the value of uh at Pi, and wi is the value of the weightfunction at Pi (i ¼ 1, 2, :::,m). For convenience, we assume constant weight functions in the cur-rent work. Hence, we have:

Figure 1. The domain X and its partitions. (a) 2D domain with points distributed inside it (P 2 X). (b) 2D domain with pointsdistributed inside it and on its boundary (P 2 X [ @X). (c) 3D domain with points distributed inside it (P 2 X). (d) 2D domainwith traditional triangular elements and nodes (e.g., used in the FEM).

NUMERICAL HEAT TRANSFER, PART B: FUNDAMENTALS 5

rujP0 ¼ BuE, (6)

where

uE ¼ u0 u1 u2 � � � um� �T

,B ¼ ðATAÞ�1AT I1 I2

� �,

I1 ¼ �1 �1 � � � �1� �

1�m

� �T,

I2 ¼

1 0 � � � 0

0 1 . .. ..

.

..

. . .. . .

.0

0 � � � 0 1

266664377775m�m

, A ¼x1 � x0x2 � x0� � �xm � x0

26643775:

Therefore, the relation between uh and uE can be obtained:

uhðxÞ ¼ NuE, x 2 E0 (7)

where N is called the shape function of uh in E0:

N ¼ x� x0½ �Bþ 1 0 � � � 0� �

1�ðmþ1Þ (8)

The trial function and shape function uh can be derived in each subdomain Ei 2 X by thesame process. As no continuity requirement is imposed at the internal boundaries, the trial func-tion and shape function can be discontinuous across neighboring subdomains. Figure 3 shows thegraphs of shape functions in 2D and 3D domains respectively. And as an example, the trial func-tion simulating an exponential function ua ¼ e�10 x�xaj j is shown in Figure 4. As can be seen, thetrial function is a simple local polynomial in each subdomain, and it is piecewise-continuous inthe entire domain. The test function vh in the Galerkin weak-form in FPM is prescribed to havethe same piecewise-continuous shape as uh.

Unfortunately, the discontinuous trial and test functions will lead to inconsistent and inaccur-ate results with the traditional Galerkin weak-form. To resolve that problem, we introduceNumerical Flux Corrections to the present FPM.

Figure 2. Two kinds of support of a point in 3D.

6 Y. GUAN ET AL.

2.3. Numerical flux corrections

The Numerical Fluxes are widely used in Discontinuous Galerkin Methods [28] to ensure theconsistency and stability. In our work, the Interior Penalty (IP) Numerical Flux Correctionsare employed.

The governing equation Eq. (1) can be written in the local weak-form with test function v ineach subdomain E 2 X: After applying the Gauss divergence theorem, we can get:

ðEqcv

@u@t

dXþðErvTkrudX ¼

ðEQvdXþ

ð@EvnTkrudC, (9)

where @E is the boundary of the subdomain, and n is the unit vector outward to @E:Let C donate the set of all internal and external boundaries, i.e., C ¼ Ch þ @X ¼ Ch þ CD þ

CN þ CR, where Ch is the set of all internal boundaries. For each internal boundary e 2 Ch, thereshould be two neighboring subdomains. We define one as E1 and the other as E2 (the order doesnot affect the formulation of FPM). With nj being the unit vector normal to e and pointing out-ward from Ej (see Figure 5), we further define n so that ne ¼ n1 ¼ �n2 when e 2 Ch: When e 2@X, there is only one neighboring subdomain E1, and we define ne ¼ n1:

Figure 3. The shape functions. (a) 2D domain with 49 points. (b) 3D domain with 343 points (x, y, z � 0:5).

Figure 4. The trial functions for ua ¼ e�10jx�xa j: (a) 2D domain with 49 points, xa ¼ ½0:50:5�: (b) 3D domain with 343 points(xa ¼ ½0:50:5�), xa ¼ ½0:50:50:5�:

NUMERICAL HEAT TRANSFER, PART B: FUNDAMENTALS 7

We also define the jump operator ½� and average operator fg as (for 8w 2 R):

w½ � ¼ wjE1e � wjE2e e 2 Ch

wje e 2 @X, fwg ¼

12

wjE1e þ wjE2e� �

e 2 Ch

wje e 2 @X:

8<:8<:

Now we sum Eq. (9) over all subdomains and rewrite it considering vneTkru½ � ¼fvg neTkru½ � þ ½v� neTkruf g, the following equation is obtained:X

E2X

ðEqcv

@u@t

dXþXE2X

ðErvTkrudX ¼

ðXQvdX

þXe2Ch

ðefvg neTkru½ � þ ½v� neTkruf g

� �dCþ

Xe2CD[CN[CR

ðe½v� neTkruf gdC:

(10)

Substituting the boundary conditions (Eq. 2) into the last term in Eq. (10), we obtain:Xe2CN

ðe½v� neTkruf gdC ¼

Xe2CN

ðeveqNdC,X

e2CR

ðe½v� neTkruf gdC ¼

Xe2CR

ðehveuRdC�

Xe2CR

ðehvudC:

(11)

When u is the exact solution, we have neTkru½ � ¼ 0 and ½u� ¼ 0 for e 2 Ch due to continuityconditions. This leads to fvg neTkru½ � ¼ 0 and neTkrvf g½u� ¼ 0 for e 2 Ch: Hence, we canreplace the term fvg neTkru½ � in Eq. (10) by neTkrvf g½u� without influencing the consistency ofthe formula.

Another two Numerical Flux terms are applied to Ch and CD in Eq. (10) with penalty parame-ters g1 and g2 respectively. The final formula of the FPM with IP Numerical Flux Corrections canbe then obtained:X

E2X

ðEqcv

@u@t

dXþXE2X

ðErvTkrudX�

Xe2Ch[CD

ðe

neTkruf g½v� þ neTkrvf g½u��

dC

þXe2CR

ðehvudCþ

Xe2Ch

g1he

ðe½u�½v�dCþ

Xe2CD

g2he

ðeuvdC

¼XE2X

ðEQvdXþ

Xe2CD

ðe

g2he

v� neTkrv� �euDdCþ

Xe2CN

ðeveqNdCþ

Xe2CR

ðehveuRdC,

(12)

where he is a boundary-dependent parameter with the unit of length. For instance, he can bedefined as the length of the boundary (in 2D), the square root of the boundary area (in 3D), orthe distance between the points in subdomains sharing the boundary. The penalty parameters g1,g2 are positive numbers having the same unit of k and independent of the boundary size. Themethod is only stable when the penalty parameters are large enough. However, on the otherhand, an excessively large penalty parameter is harmful for the accuracy and may cause a condi-tion number problem. A discussion on recommended values of these penalty parameters is pre-sented in Part II of this series.

It should be noted that, the IP Numerical Flux Correction terms on the Dirichlet boundary CD

vanish when the Dirichlet boundary conditions are satisfied a-priori. Thus, there are two ways toimpose the Dirichlet boundary conditions in practice. If there are no points distributed on theboundaries (as shown in Figure 1a), the Interior Penalty terms on CD are responsible for theboundary conditions. In this way, the Dirichlet boundary conditions are satisfied in a weak-sense.Alternatively, if boundary points are employed (as shown in Figure 1b), we can also impose u ¼euD strongly at the boundary points and thus the corresponding Internal Penalty terms on CD inEq. (12) are not applied.

8 Y. GUAN ET AL.

2.4. Numerical implementation

The formula of the FPM can be written in the matrix form finally:

C _u þ Ku ¼ q, (13)

where C and K are the global heat capacity and thermal conductivity matrices respectively, u isthe unknown vector with nodal temperatures, q is the heat flux vector.

Substituting the shape function N for uh and v, B for ru and rv, the point heat capacitymatrix CE, point thermal conductivity matrix KE and the boundary thermal conductivity matricesKh,KD,KR can be written as:

CE ¼ðEqcNTNdX, E 2 X,

KE ¼ðEBTkBdX, E 2 X,

Kh ¼ � 12

ðeNT

1nT1kB1 þ BT

1kTn1N1

� dCþ g1

he

ðeNT

1N1dC

� 12

ðeNT

2nT2kB2 þ BT

2kTn2N2

� dCþ g1

he

ðeNT

2N2dC

� 12

ðeNT

1nT1kB2 þ BT

1kTn2N2

� dC� g1

he

ðeNT

1N2dC

� 12

ðeNT

2nT2kB1 þ BT

2kTn1N1

� dC� g1

he

ðeNT

2N1dC, e 2 @E1 \ @E2,

KD ¼ �ðeNTnTkBþ BTkTnNð ÞdCþ g2

he

ðeNTNdC, e 2 CD,

KR ¼ðehNTNdC, e 2 CD,

(14)

The global heat capacity and thermal conductivity matrices can be established by assemblingall the submatrices. The process is the same as the FEM. For simplicity, in this work, we useGaussian quadrature rule and only one integration point is used in each subdomain. Similarly,the point and boundary heat flux vectors are developed:

qE ¼ðENTQdX, E 2 X,

qD ¼ �ðeBTkTneuDdCþ g2

he

ðeNTeuDdC, e 2 CD,

Figure 5. The inner boundary and normal vectors. (a) 2D case. (b) 3D case.

NUMERICAL HEAT TRANSFER, PART B: FUNDAMENTALS 9

qN ¼ðeNTeqNdC, e 2 CN ,

qR ¼ðehNTeuRdC, e 2 CR: (15)

The global heat flux vector is assembled in the same way. Eventually, a set of discretized ODEs(Eq. 13) with sparse and symmetric matrices are achieved.

3. Local variational iteration method (LVIM)

3.1. Functional recursive formula

Equation (13) can be rewritten as a system of standard first-order ODEs:

_u ¼ g u, tð Þ :¼ �C�1Kuþ C�1qðtÞ, t 2 0,T½ �: (16)

The unknown temperature vector u ¼ u1, u2, :::, uL½ �T, where L is the number of points used inthe FPM.

In a finitely large time interval ti, tiþ1½ � � 0,T½ �, with a given initial approximation u0ðsÞ, theLocal Variational Iteration Method (LVIM) approximates the exact solution u at any time t withthe following correctional iterative formula [46]:

unþ1ðtÞ ¼ unðtÞ þÐ ttikðsÞR un, sð Þds, (17)

where the error residual Rðun, sÞ is defined as:

R un, sð Þ ¼ _unðsÞ � g un, sð Þ, (18)

kðsÞ is a matrix of Lagrange multipliers for a given t, which are yet to be determined.Equation (17) can also be regarded as a correctional iteration based on an optimally weighted

error residual in time interval ti, t½ �, where kðsÞ is the set of optimal weighting functions.By making the right side of Eq. (17) stationary, we obtain the following constraints for kðsÞ :

Iþ kðsÞjs¼t ¼ 0_kðsÞ ¼ kðsÞJ un, sð Þ, s 2 ti, t½ � ,

((19)

where

J un, sð Þ ¼ @g un, sð Þ@un

¼ �C�1K (20)

is the Jacobian matrix. I is the unit matrix.Using the theory of Magnus series [50], we can prove that kðsÞ is the solution of kðtÞ in the

following equation for a given s [51]:

Iþ kðtÞjt¼s ¼ 0@kðtÞ@t

þ J un, tð ÞkðtÞ ¼ 0, t 2 s, tiþ1½ � :8<: (21)

Differentiating Eq. (17) and substituting Eq. (21) into it:

_unþ1ðtÞ ¼ _unðtÞ þ kðtÞR un, tð Þ þðtti

@kðsÞ@t

R un, sð Þds

¼ g un, tð Þ � J un, tð Þðtti

kðsÞR un, sð Þds¼ g un, tð Þ � J un, tð Þ unþ1ðtÞ � unðtÞ½ �:

(22)

10 Y. GUAN ET AL.

Therefore, we obtain the recursive formula:

_unþ1ðtÞ þ J un, tð Þunþ1ðtÞ ¼ g un, tð Þ þ J un, tð ÞunðtÞ, t 2 ti, tiþ1½ �: (23)

3.2. Collocation method and numerical discretization

Equation (23) can be written in the weak-form in the time interval ti, tiþ1½ � with a matrix of testfunctions vðtÞ :ðtiþ1

ti

vðtÞ _unþ1ðtÞ þ J un, tð Þunþ1ðtÞ� �

dt ¼ðtiþ1

ti

vðtÞ g un, tð Þ þ J un, tð ÞunðtÞ� �

dt: (24)

Let vðtÞ ¼ diagð v, v, :::, v½ �Þ, where v is the Dirac Delta function for each of the set of collocationnodes t1, t2, � � � , tM 2 ti, tiþ1½ �, that is:

v ¼ d t � tmð Þ, tm 2 ti, tiþ1½ �, m ¼ 1, 2, :::,M: (25)

The weak-form formula leads to:

_unþ1ðtmÞ þ J un, tmð Þunþ1ðtmÞ ¼ g un, tmð Þ þ J un, tmð ÞunðtmÞ,tm 2 ti, tiþ1½ �, m ¼ 1, 2, :::,M:

(26)

A set of orthogonal basis functions U ¼ f/0,/1, :::,/Ng are used to construct the trial functionue:

ueðtÞ ¼XNn¼0

ae, n/nðtÞ, (27)

where ueðtÞðe ¼ 1, 2, :::, LÞ are elements of the solution vector uðtÞ: A number of types of basisfunctions can be used in the collocation method, including harmonics, polynomials, Radial BasisFunctions (RBFs), etc. In the current work, we employed the first kind of Chebyshev polynomials[52] as an example. The collocation nodes are selected as Chebyshev-Gauss-Lobatto points. FromEq. (27), we can get:

Ue ¼ QAe, _Ue ¼ ðLQÞAe (28)

where

Ue ¼ ueðt1Þ, ueðt2Þ, :::, ueðtMÞ½ �T, Ae ¼ ae, 1, ae, 2, :::, ae,N½ �T,

Q ¼/0ðt1Þ /2ðt1Þ � � � /Nðt1Þ/0ðt2Þ /2ðt2Þ � � � /Nðt2Þ... ..

. . .. ..

.

/0ðtMÞ /2ðtMÞ � � � /NðtMÞ

2666437775ðNþ1Þ�M

,

LQ ¼

_/0ðt1Þ _/2ðt1Þ � � � _/Nðt1Þ_/0ðt2Þ _/2ðt2Þ � � � _/Nðt2Þ... ..

. . .. ..

.

_/0ðtMÞ _/2ðtMÞ � � � _/NðtMÞ

266664377775ðNþ1Þ�M

:

Normally, we set M ¼ N þ 1: Thus, we obtain the relation between Ue and its derivative:

_Ue ¼ ðLQÞQ�1Ue: (29)

Finally, substituting the relation into Eq. (26) and rearranging the sequence of the collocationequations:

NUMERICAL HEAT TRANSFER, PART B: FUNDAMENTALS 11

eE þeJ� eUnþ1 ¼ eE þeJ� eUn � eR, (30)

where eU ¼ UT1 ,U

T2 , :::,U

TL

� �T, eE ¼ IL�L � ðLQÞQ�1

� �,eJ ¼ �CTKð Þ � IM�M, t ¼ t1, t2, :::, tM½ �T,eC ¼ C� IM�M, eK ¼ K� IM�M, eq ¼ qðtÞ,eR ¼ eE eUn þ eC�1eKeUn � eC�1eq,

here � denotes the Kronecker product.The LVIM can usually achieve good estimates with very simple initial guess functions, e.g.,

constant or linear functions. In the current approach, we simply assume eU0 ¼ uðtiÞ � 1, 1, :::, 1½ �Tas the initial guess. Moreover, to apply the initial condition in the time interval ti, tiþ1½ �, we havet1 ¼ ti, and we need to enforce that unþ1ðt1Þ ¼ uðtiÞ: However, this would make Eq. (30) overde-termined. To solve that problem, the collocation equations at t1 need to be eliminated. Thus, thefinal iteration formula in LVIM can be written as:

eUrnþ1 ¼ eUr

n � eEr þeJr� �1eRr, (31)

where ½�r stands for the reduced vector (or matrix) after eliminating the (pMþ 1)th rows (andcolumns) in Eq. (30), p ¼ 0, 1, :::, L� 1:

There are some other modifications of the LVIM in which the matrix of Lagrange multiplierskðsÞ is approximated in different ways [45]. Some of these modifications have potentials in fur-ther improving the computing efficiency by avoiding the inversion of the Jacobian matrix, espe-cially for systems dominated by a few eigenvalues. Yet in this work, we just concentrate on thebasic LVIM scheme shown in Eq. (31).

4. Conclusion

A new computational approach is developed for analyzing 2D and 3D transient heat conductionproblems in complex anisotropic nonhomogeneous media. The meshless Fragile Points Method(FPM) based on Galerkin weak-form formulation is employed for spatial discretization, while theLocal Variational Iteration (LVI) scheme is used to obtain the solution in the time domain. Themeshless FPM is a significant advancement over either the Element-Free Galerkin (EFG) Methodor the Meshless Local Petrov-Galerkin (MLPG) Method. The EFG is based on Global Galerkinweak-form and requires back-ground cells to integrate the weak-form terms. The integrationbecomes tedious while using the meshless Moving Least Squares (MLS) approximations. Alsowhen the mesh of back-ground cells is rotated, the EFG may not be an objective method. TheMLPG is a truly meshless method, based on a local Petrov-Galerkin weak-form, and the integra-tion of the weak-form is complicated when MLS approximations are used as trial functions.Although the FPM is also based on the Galerkin weak-form, it uses very simple polynomial trialand test functions, and thus the numerical integration involved in the weak-form is simple andaccurate. The FPM leads to sparse and symmetric matrices, which is good for modeling complexproblems with a large number of DoFs. And as discontinuous trial functions are used, it has alsoadvantages in modeling crack developments in solids. The time integration scheme LVIM is con-siderably superior to the finite difference methods. Thus, the FPMþ LVIM method for transientheat conduction in anisotropic nonhomogeneous solids presented in this paper is a superiormeshless method as compared to those in earlier literatures.

In this first part of the two-paper series, the theoretical formulation and implementation of theproposed methodology is given in detail. Numerical examples and validation are reported in PartII of the study.

12 Y. GUAN ET AL.

ORCID

Leiting Dong http://orcid.org/0000-0003-1460-1846

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