Point-wise Discretization Errors in Boundary Element Method for Elasticity Problem Bart F. Zalewski...

Point-wise Discretization Errors inBoundary Element Method for Elasticity

Problem

Bart F. ZalewskiCase Western Reserve University

Robert L. MullenCase Western Reserve University

Reliability in EngineeringComputing

For most engineering systems, exact solutions to partial differential equations cannot be obtained.

Numerical methods have been developed to approximate the true solutions by discretizing the governing partial differential equation.

Due to the increase in computing power, many experiments are replaced with numerical simulations.

Thus, there is a growing need for reliability in engineering computing.

Reliability in EngineeringComputing (Cont.)

To achieve reliable solutions, the following causes of uncertainty must be addressed:

Uncertainty in the parameters of the system (i.e. material properties)

Uncertainty in boundary conditions

Errors in numerical integration

Errors in solving the resulting linear system of equations

Discretization errors

Why Intervals?

Interval approach is one potential mechanism for handling errors and uncertainties in an integrated and elegant fashion.

Intervals have been used to treat truncation errors, integration errors, and uncertain parameters (p-boxes).

Interval treatment of discretization error for Elasticity problem?

Why Intervals? (Cont.)

Interval finite element analysis has been developed to address:

Material, loading, and geometric uncertainty in static problems (Muhanna and Mullen 2001, Neumaier and Pownuk 2007, Modares and Mullen 2008)

Material uncertainty in dynamic problems (Modares and Mullen 2004)

Geometrical instability (Modares et al. 2005)

Why Intervals? (Cont.)

Interval boundary element analysis addresses:

Uncertainty in boundary conditions in static problems (Zalewski et al. 2007)

Truncation and integration errors (Zalewski et al. 2007)

Local discretization errors for Laplace equation (Zalewski and Mullen 2007)

Local discretization errors for Elasticity problem (Zalewski and Mullen 2008)

Objective

To explore the applications of interval concepts to quantifythe discretization error in numerical methods

Procedure

To use boundary element method

as an exemplar of integrated interval treatment

of uncertainties and discretization errors

Presentation Outline

Conventional BEA for Elasticity problem (Brebbia 1978)

Local Discretization Error in BEA- Interval Solver- Interval Kernel Splitting Technique- Parametric Interval Solver

Examples

Conclusion

Boundary Element Analysis

Boundary Element Analysis (BEA) is a method for obtaining approximate solutions of partial differential equations.

BEA reduces the dimension of the problem by transforming domain variables to variables on the boundary of the domain using Green’s functions.

The transformed boundary integral equations are solved using collocation methods in which the weighted residual exists only on the boundary of the system.

Source points are located sequentially at all boundary nodes that map domain variables such that they coincide to their nodal values.

The Elasticity problem is:

is the domain of the system

is the boundary of the system

, are the values at the boundary

Boundary Element Analysisof the Elasticity Problem

)(

)(

)ˆ(u

0

ˆ,ˆ

0

2

1

2

1

21

,

i

ii

i

ijij

and

onttonuu

inb

)ˆ(t

The boundary element formulation for the Elasticity problem can be derived starting from Betti’s reciprocal theorem:

The equilibrium condition is substituted into the above equation resulting in:

Boundary Element Analysisof the Elasticity Problem (Cont.)

dubdutdubdut iiiiiiii****

**, ijij b

dtudbudutdu iiiiiiijij****

,

The boundary element formulation requires that the weighted residual exists only on the boundary of the domain. This condition is satisfied if the weighted residual function is chosen as the Green’s function which is obtained by applying a concentrated force in direction at a source point as:

is the field point at which the response to the concentrated force is measured. The resulting fundamental solution is:

Boundary Element Analysisof the Elasticity Problem (Cont.)

)(

ijij ax )(*,

)(x

)( ia

jjii auu ** jjii att **

and are components of displacement and traction, respectively, due to the applied concentrated force in direction ( j ). These two kernel functions are given as:

Boundary Element Analysisof the Elasticity Problem (Cont.)

)( *jiu )( *

jit )(i

r

jx

r

ixr

Gu ijij

ln34)1(8

1*

xy

ij

ij

nr

jxn

r

ix

r

nx

r

jx

r

ix

rq

21

221

)1(4

1*

Boundary Element Analysisof the Elasticity Problem (Cont.)

Substituting the fundamental solution into the integral equation results in:

The indices are exchanged in the integral terms and the constant coefficients are cancelled out resulting in:

For simplicity the body force is neglected:

,)( *** dtaudbauduatau ijjiijjiijjiii

,)( *** dtudbudutu jijjijjiji

,)( ** dtudutu jijjiji

)( ia

The boundary integral equation is integrated such that the source point is enclosed by the half-circular boundary of radius , as :

Boundary Element Analysisof the Elasticity Problem (Cont.)

)()( 0

,)(),()(),()(2

1 ** dxxtxudxxuxtu jijjiji

Any boundary can be discretized into boundary elements consisting of nodes, at which a value of either or is known, and assumed polynomial shape functions between nodes.

and are vectors of nodal values

is a vector of polynomial shape functions

Boundary ElementDiscretization

)( )( i)(u )(t

])][([)( iuxxu

])][([)( itxxt

][ iu ][ it

)]([ x

The discretized integral equation is written as:

or in matrix form:

Applying the boundary conditions, the system of linear equations is rearranged as:

and are fully populated non-symmetric matrices and matrix is singular.

Boundary ElementDiscretization (Cont.)

Elements

jijjElements

iji

xx

tdxxxuudxxxtu ][)]()[,(][)]()[,(][2

1 **

]][[]][[ tGuH

][]][[ bxA

][H][H ][G

Interval Arithmetic

In this work errors are treated as interval quantities

and interval solutions are shown to guarantee

the worst case bounds (interval enclosure) of the true solution.

The interval number is a closed set as (Moore 1966) and (Neumaier 1990):

Considering two interval numbers: and

Addition:

Subtraction:

Multiplication:

Division:

Subdistributive Property:

Interval Arithmetic (Cont.)

],[ dbca yx

],[ cbda yx

)],,,max(),,,,[min( bdbcadacbdbcadacyx

]),[0(],1

,1

[],[ dccd

ba y

x

zyzxzyx )(

],[ bax ],[ dcy

}|{],[ xxxxxx x

Interval boundary element analysis treats uncertain boundary conditions, truncation and integration errors (Zalewski et al. 2007):

The equation is rearranged as:

The system is solved using Newton-Krawczyk iteration (Krawczyk 1969).

Interval Boundary ElementFormulation

]][[]][[ tGuH

][]][[ bxA

Interval Equation Solver

The residual Krawczyk iteration is:

Substitution:

eeee xbAδ 1

eeeeeeeee xbAAxAbδδ 1

eeeeeee δAxAbδδ

Interval Equation Solver (Cont.)

Regrouping:

The iteration follows:

eeeeeee C δAxAbδδ

eeeeee CIC δAxAbδ

iii CIC δδAAxbδ 1

Interval Equation Solver (Cont.)

and are mid-point matrices of and

Interval Equation Solver (Cont.)

][][][ 10 bAx

][A ][b ][A ][b

][][][][ 1 AI AId

]][[][][][ 01 xA Abδ

][][ 1 δδ

The iteration follows as:

If

Interval Equation Solver (Cont.)

][][ 1δdel

]][[][][ 1 delIδδ d

][][ 1δdel

][][][ 10 δx x

The boundary is subdivided into elements. For each element, the interval values and are found that bound the functions and over an element such that:

Discretization Error

)(

)(k)(u )(t

)(u )(t

n

mmjmkij

n

mmjmkij

kjkkijkikkjkkij

k

kkkkkk

mmmmmm

n

mmjmkijkjkkij

n

mmjmkij

kjkkijkik

k

kkkkkk

mmmmmm

mm

kk

mkm

k

dxtxudxuxt

dxuxtudxtxu

tttFinduuuknownAlso

kmknownistttuuuAssumenk

Or

dxuxtdxtxudxtxu

dxuxtu

uuuFindtttknownAlso

kmknownistttuuuAssumenk

1

*

1

*

**

1

**

1

*

*

)(),()(),(

)(),()(2

1)(),(

.

.,,...,2,1

)(),()(),()(),(

)(),()(2

1

.

.,,...,2,1

Assuming constant elements results in constant interval bounds on the solution:

Assuming linear elements results in linear interval bounds:

Discretization Error (Cont.)

Kernel Splitting Technique

The kernel splitting technique has been used to bound Fredholm Equations of the First Kind in which the left hand side is deterministic (Dobner 2002). The boundary element integral equations have an interval left hand side and therefore an extended approach is developed.

dxxuxb )(),()( k

The integral of the product of two functions is bounded as:

The right hand side is expressed as a sum of the integrals:

or

Interval Kernel Splitting Technique

dxxdxxux uξaξaξb ),()(),()(

21

),(),(),( dxxdxxdxx uξauξauξa

0),( ξa x 0),( ξa x 1on

2on),(0 ξa x

The interval kernel is of the same sign on , thus can be taken out of the integral on :

cannot be taken out of the integral on due to subdistributive property.

Interval Kernel SplittingTechnique (Cont.)

21

),(),(),( dxxdxxdxx uξauξauξa

)( 1)(u)( 1

)(u )( 2

The interval kernel is bounded by its limits:

where

Interval Kernel SplittingTechnique (Cont.)

22

),( dxdxx auuξa

)}],(max{)},,([min{ ξεaξεaa xx

],[ ε

can be taken out of the integral and the integral equation becomes:

The kernels are bounded for all the elements resulting in the system of equations:

Interval Kernel SplittingTechnique (Cont.)

uauauaauauau

22

22

221

0001

0)(lim)(lim)(lim)(lim dxxxnxnxdx

n

ixxx

n

ix

uauξaξb

21

),()( dxdxx

][][]][[]][[ 21 bcuAuA

)(u

Transformation of the Interval Linear System of Equations

Considering system of equations:

Preconditioning the system:

Let , and

eeeee bxAxA 21

eeeee eeebAxAAxAA 1

21

11

111

Iee 1

11 AA eee 32

11 AAA

eee 11

1 bbA

eeee 13 bxAx

eee 1bxA

The interval bounds obtained by the solver are not sharp since the dependency of the location of the source point has not been considered.

The uniqueness of the problem is not preserved since two source points are allowed to have the same location at one time resulting in rectangular matrices.

The parameterization considers each source point to have a unique location and allows for sharper interval bounds.

Parameterized IntervalEquation Solver

Parameterized IntervalEquation Solver (Cont.)

The system is parameterized such that . The system is solved by splitting the kernels for all subintervals such that:

This results in the system of equations for each

]1,0[ξ)( iξ

ξξ

i

n

i

1

)( iξ

])][([])][([])][([])][([ 2121 tξGtξGuξHuξH iiii

01

n

iiξ

The system of equations is rearranged:

Preconditioning and substitution as described before lead to:

The parameterization is incorporated into the solver:

is computed when

Parameterized IntervalEquation Solver (Cont.)

)]([])][([ 1 ii ξbxξA

)]([][][ 11

10 i

n

i

A ξbx

5.0][A

)]([])][([])][([ 21 iii ξbxξAxξA

The difference between the solution and the initial guess is computed and pre-multiplied by the preconditioning matrix :

The system is subjected to residual Krawczyk iteration.

Parameterized IntervalEquation Solver (Cont.)

][I

)]([][][][ 1

1i

n

id AI ξAI

])][([])][([)]([][ 02011

xξAxξAξbδ iii

n

i

The first example obtains the bounds on discretization error for the BEA of the Elasticity problem for the unit square boundary.

Boundary Conditions: ubottom=0, tsides=0, uy top=1, tx top=0

Examples

widthTrue

widthComputedindexyEffectivit

Examples (Cont.)

4 6 8 10 12 14 160.5

1

1.5

2Behavior of the Solution Width

Number of Elements

Sol

uti

on W

idth

4 6 8 10 12 14 161.3

1.4

1.5

1.6

1.7

1.8

1.9

2

2.1

2.2

2.3Behavior of the Effectivity Index

Number of Elements

Eff

ect

ivit

y In

dex

Examples (Cont.)

4 6 8 10 12 14 16-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4Bounds on the Solution

Number of Elements

Sol

uti

on

True SolutionInterval Bounds

Examples (Cont.)

0 0.2 0.4 0.6 0.8 1

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

4 Element Mesh

Sol

uti

on

0 0.2 0.4 0.6 0.8 1-0.2

0

0.2

0.4

0.6

0.8

1

1.2

8 Element Mesh

Sol

uti

on

0 0.2 0.4 0.6 0.8 1-0.2

0

0.2

0.4

0.6

0.8

1

1.2

16 Element Mesh

Sol

uti

on

0 0.2 0.4 0.6 0.8 1-0.2

0

0.2

0.4

0.6

0.8

1

1.2

12 Element Mesh

Sol

uti

on

Examples (Cont.)

2 3 4 5 6 7 8 9 100.26

0.28

0.3

0.32

0.34

0.36

0.38

0.4

0.42

0.44Effect of Parameterization on Interval Width

Number of Subintervals

Inte

rval

Wid

th

The second example shows the behavior of the solution for a hexagonal plate in tension. A symmetry model is considered to decrease computational time.

Examples (Cont.)

Examples (Cont.)

4 6 8 10 12 14 160.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2Behavior of the Solution Width

Number of Elements

Sol

uti

on W

idth

4 6 8 10 12 14 161.3

1.35

1.4

1.45

1.5

1.55

1.6

1.65

1.7

1.75

1.8Behavior of the Effectivity Index

Number of Elements

Eff

ect

ivit

y In

dex

Examples (Cont.)

4 6 8 10 12 14 16-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6Bounds on the Solution

Number of Elements

Sol

uti

on

True SolutionInterval Bounds

Examples (Cont.)

0 0.2 0.4 0.6 0.8 1

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

4 Element Mesh

Sol

uti

on

0 0.2 0.4 0.6 0.8 1-0.2

0

0.2

0.4

0.6

0.8

1

1.2

8 Element Mesh

Sol

uti

on

0 0.2 0.4 0.6 0.8 1-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

12 Element Mesh

Sol

uti

on

0 0.2 0.4 0.6 0.8 1-0.2

0

0.2

0.4

0.6

0.8

1

1.2

16 Element Mesh

Sol

uti

on

In this work, the point-wise discretization error is bounded using interval methods for the boundary element analysis of the elasticity problem.

The examples presented show the capability of the method to enclose the true solution with a desired accuracy.

The discretization error is shown to converge with the increasing number of elements, which is the expected behavior.

The interval width can be decreased with further parameterization for which the computational cost increases linearly.

Conclusions

Thank you.