FEM for Singularities - [PDF Document] (2024)

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JOURNAL OF COMPUTATIONAL PHYS ICS %, 391410 (1991)

An Efficient Finite Element Method for

Treating Singularities in Laplaces Equation

LORRAINE

G.

OLSON, GEORGIOS

C.

GEORGIOU,* AND WILLIAM

W.

SCHULTZ

Department of Mechanical Engineering and Applie d Mechanics,

University of Michigan, Ann Arbor, Michigan 48109

Received June 16, 1989; revised February 2, 1990

We present a new f in ite element method for solving part ia ldi f ferent ial equations with

singulari t ies caused by abrupt changes in boundary condit ionsor sudden changes in boundary

shape. Term s from the local solut ion supplement the ordinarybasis funct ions in the f in ite

eleme nt solution. All singular con tributions reduce to boundary integrals a fter a double

appl icat ion of the divergence theorem to the Galerkinintegrals, and the essential boundary

condit ions are weak ly enforced using Lagrange mult ip l iers.The proposed m ethod el iminates

the need for high-order integrat ion, improves the overallaccura cy, and yields very accurate

est imates for the singular coeff tcients. I t a lso acceleratesthe convergence with regular mesh

ref inemen t and converges rapidly with the number of singularfunct ions. Al though here we

solve the Laplace equation in two dimensions, the method is applicable to a more general

class of problems.

0 1991 Academic Press, Inc.

1. INTRODUCTION

Singularities often occur in models of engineering problems dueto discontinuities

in the boundary conditions or abrupt changes in the boundaryshape. Two well-

known examples are the crack-tip problem in fracture mechanics[l] and the

sudden-expansion problem in fluid mechanics [2].

When using numerical methods to solve problems withsingularities, one must

pay special attention to the singular regions. In both thefinite difference and the

finite element methods, local refinement is often employed nearthe singularity to

achieve reasonable accuracy. However, the accuracy achieved andthe rate of

convergence are generally not uniform nor satisfactory [1,3].

Incorporating the form of the singularity in a numerical schemeis generally more

effective than mesh refinement. This idea has been successfullyadopted in a variety

of methods such as finite differences [4, 51, finite elements [1, 6133, global

elements [14, 151, and boundary elements [16].

* Presently at Unite de Mecanique Appl iquee, Universi t tCathol ique de Louvain, 2 Place du Levant,

1348 Louvain-la-Neuve, Belgium.

391

0021-9991/91 3.00

581/96/2-l 1

Copyright 0 1991 by Academic Press, Inc.

Al l rights of reproduction in any form reserved.

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392

OLSON, GEORGIOU, AND SCHULTZ

The form of the singularity is obtained by a local analysis. Theasymptotic

(r + 0) solution for Laplaces equation in two dimensions isgiven by

where u is the dependent variable, (r, 0) are the polarcoordinates centered at the

singular point, cl i are the singular coefficients, ;li are thesingularity exponents, and

f,(0) represent the 0 dependence of the eigensolution. Theasymptotic solution

satisfies the governing equation in the domain and the boundaryconditions along

the boundary segments adjacent to the singular point.

In this work, we focus on the finite element method, which isvery popular in

many engineering fields for solving problems with complexgeometries. The local

solution can be incorporated into a finite element scheme in twobasic ways:

1.

Singular finite element approach.

Special elements are used in a small

region around the singularity, while ordinary elements are usedin the rest of the

domain. The shape functions defined on the special elements takeinto account

the known form of the singularity. This approach has onefundamental drawback:

the size of the singular elements, and thus the region overwhich the singularity is

given special attention, is reduced as the mesh is refined.

2. Singular basis function approach. A set of supplementaryfunctions that

reproduce the functional form of the leading terms of thesingularity solution is

added to the ordinary finite element expansion. In thisapproach, the singular

representation is independent of the mesh refinement, and thesingular coefficients

are directly calculated.

Accurate estimates of the leading singular coefficients areoften desirable, as in frac-

ture mechanics [13] (the first coefficient is the stressintensity factor, a measure of

the stress a body can endure before fracture occurs). Forreviews on singular finite

element approaches, see Fix [13], Gallagher [17], and Georgiou[18].

In this paper we follow the singular basis function approach.The supplementary

basis functions W, take the general form

Wi = Qrlf, (f3),

(2)

where Q is an optional blending function. At least three kindsof blending functions

appear in the literature:

1. Functions with two-zone blending [ 1, 71. For example,

{

1,

O

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FINITE ELEMENT METHOD FOR SINGULARITIES 393

Pi smoothly forces the singular functions to zero at r = R,. Thefunctional form

near the singular point is not affected by the blending, and noextra boundary terms

appear in the finite element formulation.

2. Functions with one-zone blending [6, 191. Here, Q modifiesthe singular

terms even near the singular point. One choice is simply tolet

Ri go

to 0 in (3).

Another choice is to conform the blending to the underlyingmesh, e.g.,

for a rectangular grid in Cartesian coordinates with thesingularity at the origin.

Again, no extra boundary terms appear in the formulation if theboundaries

are located at x2, y2 >

H,

since the singular contributions are zero along the

boundaries.

3. Exact function (no blending) [6]. In this case, Q = 1.Additional

boundary terms appear in the finite element formulation, sincethe singular

functions are not zero along parts of the boundary. Also, theessential boundary

conditions must be enforced separately.

Many researchers have used the singular basis function approachto solve a

variety of problems such as the cracked-beam problem [13], theL-shaped mem-

brane vibration problem [ 191, and reentrant-corner problems[20]. The two-zone

and one-zone blending methods have two main advantages overother singular

treatments: they are easy to program; and they are easilyextended to nonlinear

problems. However, the blending function

introduces additional arbitrary

parameters, contaminates the singular functions, and generallycauses inaccurate

estimates in the second and higher singular coefficients. Inaddition, a high-

accuracy quadrature rule must be used to integrate the blendedsingular functions

in the neighborhood of the singular point [ 11. The unblendedmethod avoids the

singular function contamination problem, but requires separateenforcement of the

essential boundary conditions and accurate quadrature rules toevaluate the

integrals.

In this paper we present a new method with no blending function:the integrated

singular basis function method (ISBFM). In the ISBFM, theanalytical form of the

asymptotic terms are used as the singular functions, and thevolume integrals with

singular contributions are reduced to boundary integrals bymeans of a double

application of the divergence theorem. This method avoids thereduced accuracy

associated with the blending function and eliminates the need toevaluate singular

integrands.

We demonstrate the ISBFM on two Laplace-equation problems thathave been

studied extensively: the Motz problem [4] and the cracked-beamproblem [19],

described below.

The Motz problem.

Figure 1 shows the geometry, governing equations, and bound-

ary conditions for the Motz problem [4] as modified by Wait andMitchell [7];

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394

OLSON, GEORGIOU, AND SCHULTZ

y=l

SF-0

a2 -

u = 500

y=o

Sl

Y e

s2

z=-1

OX

h-0

==I

u=o

av -

FIG. 1. The Motz proble m.

this is considered a benchmark problem for testing the varioussingular approaches

proposed in the literature. A singularity arises at x= y=O,where the boundary

condition suddenly changes from u=O to au/+ = 0. The localsolution is given

by WI

u(y, 0) = f c@- V2

i= 1

cos[(+ij.

(5)

Whiteman [S] employed special finite difference methods to solvethe problem,

while Hendry and Delves [lS] and Kermode et al. [14] used theglobal element

method to determine the singular coefficients cli. Wait andMitchell [7] used a

combination of two-zone blended singular basis functions andmesh refinement and

obtained disappointing results for the singular coefficients.The zones they

employed were very small, and some of the singular functionswere inappropriate

for the Motz problem. Morley [6] applied both one-zone blendingand no blending

functions. While the singular treatments improved the solution,the actual values of

the singular coefficients were not satisfactory.

Wigley [21] obtained very accurate estimates for the leadingsingular coefficients

with an inherently iterative approach. He first generated anapproximate numerical

solution using finite differences and then estimated the firstsingular coefficient from

this solution. Next, he modified the original problem bysubtracting out the first

singular term and again generated an approximate numericalsolution to this

modified problem, which he used to estimate the second singularcoefficient, and so

on. He obtained excellent results for the first several singularcoefficients in the

Motz problem as well as the cracked-beam problem.

The Cracked-Beam Problem. The second problem we examine is thecracked-

beam problem [ 13, 211 illustrated in Fig. 2. (In the originalproblem, V2u = - 1 and

u = 0 along y = 4. The transformation u = u + y2/2 leads to theproblem considered

here.) Clearly, the form of the local solution is the same asthat of the Motz

problem.

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FINITE E LEMENT METHOD FOR SINGULARITIES

u = 0.125

Ti

a-0

a2 -

v2u = 0

395

FIG. 2. The cracked-beam proble m.

Fix et al. [ 193 used blended singular basis function approachesfor this problem.

They found that their method was efficient and moderatelyaccurate. Fix [ 131 later

reviewed singular basis function approaches and discussed thecracked-beam

problem. As mentioned above, Wigley [21] obtained very accuratecoefficients for

the cracked-beam using an iterative method.

In the remainder of this paper, we discuss the solution of thetwo problems above

with

1. Ordinary finite elements,

2. One- and two-zone blending functions (we refer to this methodas the

blended singular basis function method, BSBFM), and

3. The ISBFM.

In Section 2 we present the finite element formulation for allthree methods. The

results for the Motz problem, in Section 3, indicate that theaccuracy of both the

BSBFM and ISBFM is substantially better than that of theordinary elements.

With the BSBFM, we examine three different blendings and studythe effects of the

order of integration around the singular point and the size ofthe region over which

the singular functions are defined. As mentioned above,high-order integration is

not required with the ISBFM, and the singular functions aredefined over the entire

domain. The estimates of the leading coefficients with the ISBFMare more accurate

than those with the BSBFM. The ISBFM yields an algebraic rate ofconvergence

with mesh refinement, in agreement with the theoretical errorestimates. It also

converges rapidly with the number of singular functions. InSection 4 we solve the

cracked-beam problem and confirm the accuracy and rate ofconvergence of the

ISBFM. Finally, in Section 5 we summarize the conclusions.

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396 OLSON, GEORGIOU, AND SCHULTZ

2. FINITE ELEMENT FORMULATION

We now present the Galerkin finite element formulation for theMotz problem.

The formulation for the cracked-beam problem is similar; thus,it is not included

in this paper.

2.1. Ordinary Finite Elements

The unknown u is expanded in terms of biquadratic basisfunctions @:

u= 2 UicDi,

i= I

(6)

where N, is the number of unknowns, and ui are the nodalvalues.

We apply Galerkins method by weighting the governing equationwith the basis

functions @ and then using the divergence theorem:

s

z@dS-j. Vu.V@dV=O,

dn

i=

1, 2, . N,.

V

Here, V is the physical domain, S denotes its boundary(consisting of five different

part as shown in Fig. l), and n is the outward normal from S.The boundary terms

can be omitted because there are only essential and hom*ogeneousnatural boundary

conditions, and (7) is simplified to

- Vu.V@dV=O,

s

i= 1, 2, . N,.

(8)

V

The equations for the essential boundary conditions along S, orS, replace the

corresponding equations in (8). Equation (8) constitutes asymmetric and banded

linear system of equations that is solved using standardsubroutines.

2.2. The Blended Singular Basis Function Method

To the ordinary finite element expansion, we add the singularbasis functions w,

u= z .i@+Nyriwi,

(9)

i= 1

i=l

where N,,, is the number of singular functions, and ai are theunknown singular

coefficients.

For comparison, we construct three different sets of blendedsingular functions:

f

r(2i- I)/2 cos

2i- 1

- 0

2

O

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FINITE ELEMENT METHOD FOR SINGULARITIES

1 2i- 1

BSBF2: W=

7

R

v- yr - R)2(2r + R) cos 2 8,

0,

397

OR

elsewhere.

Here the radius R and the length H define the size of thesingular domain. BSBFl

are functions with two-zone blending as suggested by Strang andFix [ 11. The coef-

ficients a and b are determined by demanding continuity of w andits first

derivative at Y= R/2. BSBF2 and BSBF3 are functions withone-zone blending. As

r goes to zero, they all converge to the asymptoticsolution.

Applying Galerkins principle, we weight the governing equationby @ and

W.

After using the divergence theorem we obtain

Vu.V@dV=O,

i =

1, 2, . N,,

V

(10)

and

s

e

WdS-j

dn

Vu.VWdV=O,

i =

1, 2, . NSBF,

(11)

V

Note that the total number of unknowns is now N, + N,,,. Theboundary terms

in (10) and (11) can be omitted on those parts of the boundarywhere a

hom*ogeneous natural boundary condition is applied. Theboundary

are ignored in (10) because we have essential conditions for Uiand in

w are zero. Therefore,

- Vu.V@dV=O,

I

i=

1, 2, . N,,

V

and

- Vu.VWdV=O,

s

i= 1, 2, . NseF.

V

terms on S,

( 11) because

(12)

(13)

Again, the equations for the essential boundary conditions alongS, and S3 replace

the corresponding equations in (12). High-order integration isrequired for the

terms involving singular functions in (13). Equations (12) and(13) constitute a

symmetric linear system.

Using the singular functions destroys the banded structure ofthe stiffness matrix.

The additional equations attach full rows and columns to thematrix that is

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398 OLSON, GEORGIOU, AND SCHULTZ

otherwise banded about the main diagonal, resulting in anarrow-shaped matrix

structure. Matrices of this structure are very often encounteredwhen a basic

boundary value problem is augmented with scalar constraints oris solved simulta-

neously with densely coupled algebraic equations [ 1, 221.Skyline solvers or

extensions of other standard algorithms may be used in invertingthe stiffness

matrix to avoid extra operations during elimination as well asextra storage

requirements. Strang and Fix [l] proposed a modified Choleskisfactorization

algorithm for arrow-shaped matrices, while Thomas and Brown [22]developed a

modified LU-decomposition subroutine.

2.3. The Integrated Singular Basis Function Method

In the ISBFM we subtract the asymptotic terms directly from thegoverning

equation. The singular functions are now identical to the formof the corresponding

asymptotic expansion terms:

(141

Let us be the singular part of u,

NSBF

us= c cqw,

(15)

i=l

and u* be the part of the solution approximated by the ordinaryfinite element

expansion,

u*=u-u3

(16)

Note that us satisfies the governing equation and the boundaryconditions along

y = 0, and the original problem is transformed to that shown inFig. 3.

a u u - =

ar at

V2u = 0 u* + u = 500

FIG. 3. The modified Motz problem

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FINITE ELEMENT METHOD FOR SINGULARITIES

399

Again we use the Galerkin method (and the divergence theorem) toobtain

s

f&d&[

s an

Vu* .V@dV=O, i= 1, 2, . N,,

(17)

V

and

I

WdS-j- Vu*.VWdV=O,

i = 1, 2, . NSBF.

(18)

V

To reduce the singular volume integrals of (18) to boundaryintegrals, we apply the

divergence theorem once more:

s

Vu* .VWdV=

s

u*zdS-1 u*V2WidV.

(19)

V S

V

The volume integral on the right-hand side of (19) is zero,since the W satisfy

Laplaces equation, and (18) becomes

dS=O,

i = 1, 2, . NSBF.

(20)

The boundary terms are not ignored, since &*/an =

-&P/an.

To impose the

originally essential boundary condition on S,, we employLagrange multipliers n:

expanded in terms of quadratic basis functions M:

(21)

NY is the number of nodes on S3. (Bertsekas [23] and Babuska[24] discuss the

general use of Lagrange multipliers with constrainedoptimization problems and

finite elements.) The final equations are

js,(I.W+u~)dy-j+( W+u*~)dx-js~(~W+u*~)dy

= 500 fs, z dy, i = 1,2, . NSBF,

(23)

and

fs, (u* + u) M dy = 500 ss, M dy,

i= 1, 2, . N,,.

(24)

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400 OLSON, GEORGIOU, AND SCHULTZ

The total number of unknowns is now N, + NssF + NY. The linearsystem of equa-

tions defined by (22)-(24) is symmetric. Since the singularfunctions satisfy the

boundary conditions near the singular point, all of the surfaceintegrands are non-

singular and are evaluated with an ordinary Gauss-Legendrequadrature. As with

the BSBFM, the banded structure of the stiffness matrix isdestroyed in the ISBFM.

This, along with the increase in the number of unknowns, resultsin some additional

computational costs for a given underlying mesh. However, sincethe accuracy is

high with the ISBFM, mesh refinement may not be necessary.

Neither of the singular function techniques (BSBFM or ISBFM)require a

rectangular domain.

Convergence estimates. For convergence studies on the ISBFM, wecalculate the

variationa/ indicator 17. For Laplaces equation, I7 may bewritten as [25]

lI=;, Vu.VudV+~dS.

V

The error measure (n, - 17,,) is of the same order as the energyerror and

satisfies the inequality

(fl,h-17,Y2G II~,,--hIIl. (26)

The subscript h denotes quantities calculated with a mesh sizeh, the subscript ex

indicates the exact solution, and the norm l]~ll~ has thestandard definition [26].

It is well known that the energy error satisfies

II ex ~111G Ch II~,,II, (27)

with

p=min(k, m- l},

(28)

where C is a constant, p is the convergence rate, k is the orderof the finite element

polynomial approximation (2, in this work), and m is the orderof the generalized

Sobolev space of the exact solution [3]. The order m is ameasure of the regularity

of the solution, or an inverse measure of the singularitystrength, given by

m=l+A,,

(29)

where 1, is the exponent in the first term of the local solution(1). For example,

m = for the Motz problem and the cracked-beam problem and,according to (28),

if no singular functions are used the convergence rate is p =0.5, independent of the

order of the finite element approximation k. Subtracting thefirst term of the local

solution changes m to and the rate of convergence wil l be p =1.5. Subtracting

two or more singular functions results in quadratic convergence(for k = 2).

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FINITE ELEMEN T METHOD FOR SINGULARITIES

401

3. RFSULTS FOR THE MOTZ PROBLE M

In order to compare the three approaches for the Motz problem(i.e., ordinary

finite elements, blended singular basis functions, andintegrated singular basis func-

tions), we first establish the best integration order andsingular domain for the

BSBFM. This question is addressed in Section 3.1. Section 3.2examines the con-

vergence of the ISBFM with mesh refinement and NsBF. Section 3.3compares the

actual solutions for ordinary finite elements, BSBFM, and ISBFM,and examines

the singular coefficients for BSBFM and ISBFM.

3.1. A Comparison of Blended Singular Basis Function Methods

Unlike the ISBFM, the BSBFM requires high-order integration ifconverged

results are to be obtained. To study the effect of the order ofthe Gauss-Legendre

quadrature, we first use a uniform mesh with 16 x 8 elements,set R and H (the size

of the singular domain) equal to the mesh size, and use NBsF =1. The computed

values of ~1, are plotted in Fig. 4. BSBFl and BSBF2 give poorestimates for ai,

whereas BSBF3 appears to converge to the exact value as theorder of integration

increases. We have found satisfactory results when high-orderintegration is

employed only over the two elements sharing the singular point.For all the

BSBFM results hereafter, each element adjacent to the singularpoint is divided into

64 rectangles over which a 15 x 15 Gauss quadrature is used.

The effect of mesh refinement on the first coefficient a, isshown in Fig. 5; here

BSBF 2

-I

0.0 20.0 40.0 60.0

Order of integration, M

FIG. 4. The coefficient a, as a function of the order ofintegrat ion m (16 x 8 mes h; R and H equal

to mes hsize; -- -- : analyt ical value).

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402

*1

OLSON, GEORGIOU, AND SCHULTZ

-I

OO. 2. 4. 6. 8. 10.

Number of elements, N

FIG. 5. Effect of mesh ref inement on a, (R or H are equal tothe me shs ize; ------: analytical value).

we use different uniform meshes and keep

R

(or H) equal to the mesh size. Each

mesh consists of 2N x N square elements, where N is the numberof elements in the

y direction. The results appear to approach the analytical valuewith mesh retine-

ment in all cases. Among the blended singular functions used,BSBF3 again appears

to give the best results.

The singular domain in the foregoing analysis includes only thetwo elements

sharing the singular point, or parts of them. It seems thatlarger values of R (or H)

would allow the ordinary finite element expansion coefficientsto adjust more effec-

tively to the addition of one or more singular terms. Toinvestigate this, we use a

16 x 8 element mesh, varying R from up to 1. The computedcoefficients are

plotted in Fig. 6. We observe that the results with BSBF2 andBSBF3 converge very

close to the exact value. The results with BSBFl exhibitanalogous behavior but

higher values of R (or more elements within the singular domain)are required to

reach a plateau.

We conclude that BSBF3 with H equal to the size of the domain isthe best of

the blended singular basis functions examined. Although thesetests were conducted

on the Motz problem, it seems reasonable to suppose that BSBF3would also prove

superior in other applications. All the results with the BSBFMhereafter are

obtained with the BSBF3.

3.2. ISBFM Convergence

Figure 7 shows the convergence rate achieved for the ISBFM:p=OSO when no

singular functions are used, p = 1.54 when N,,, = 1, and p = 1.9when Nse, = 2.

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FIN ITE ELEME NT METHOD FOR SINGULARITIES

403

ff l

Effect of the

-: analytical

0.0

0.2 0.4 0.6 0.8 1.0

R or H

the singular dom ain (R or H) on the f i rst coeffkient aI(16x8

uniform

IG. 6.

size of

me sh; --- value).

(Recall that Eq. (28) indicates that the rate of convergence of(ITh - Z7ex)12should

be p = 0.5 when no singular functions are used; when onesingular function is used,

p should increase to 1.5; and with two or more singularfunctions, p = 2.0.) For the

problems considered in this paper, no analytical expression forZZ is available;

therefore, we use the estimate from a very tine mesh (32 x 16elements, 20 singular

FIG. 7. Convergence with mesh ref inement for the Mo tz problem(0 : no singular funct ions; 0 : one

singular funct ion, 0 : two singular funct ions).

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404

OLSON, GEORGIOU, AND SCHULTZ

0.5 1 10 50

N

FIG. 8. Convergence of singular coefficients with mesh refinement for the Motz proble m (0 : a,

one singular function, 0: a, with two singular functions, 0: a2with two singular functions).

with

functions) as the exact value of I7. The slight discrepanciesfrom the theoretical

convergence rates may be partially due to this approximation forIl. As indicated

by (28), higher convergence rates can be achieved usinghigher-order elements.

Figure 8 shows the rate of convergence of the singularcoefficients as the mesh is

refined. With one singular basis function, CI~convergesessentially cubically. When

two singular basis functions are used, the rate of convergenceof a, is approximately

linear, while the rate of convergence of ~1~s essentiallyunchanged.

Figure 9 shows the rapid (roughly exponential) convergence of(I7, - Z7sX)2

0 2 4 6 6 10

NSBF

FIG. 9. Convergence with the numbe r of singular functions forthe Motz proble m (2 x 1 mesh).

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FINITE ELEMENT METHOD FOR SINGULARITIES

405

TABLE I

Lead ing Coefficients for the Motz Prob lem (Nsep=20,

16 x 8 Mesh)

i ISBFM

Wigley [21] Exact [27]

1 401.1625 401.163

401.1625

2 87.6559 87.655 87.65592

3 17.2379

17.238 17.23792

4 -8.0712 -8.071 -8.0712

5 1.4403 1.440

6 0.3310 0.331

7 0.2754 0.275

8 - 0.0869 - 0.087

9 0.0336 0.0336

10 0.0154 0.0154

11 0.0073 0.0073

12 - 0.0032 -0.0032

13 0.0012 0.0012

14 0.0005 OS005

versus the number of singular terms for the ISBFM. Theconvergence with the

number of singular functions is considerably faster than theconvergence with mesh

refinement.

In Table I we list the singular coefficients calculated with theISBFM using 20

singular functions and a 16 x 8 mesh. They are in closeagreement with the exact

values available [27] and with Wigleys results for the highercoefficients [21].

At some point, we expect the condition number of the stiffnessmatrices will

become too large to permit an accurate solution of theequations. Table II shows

the condition number estimates, K, for a 2 x 1 mesh wih variousnumbers of singular

functions. With 35 coefficients, the condition number is quitepoor and the

TABLE II

Cond ition Numbers for the Motz

Problem (ISBFM, 2 x 1 mesh)

5 0.40 x 1o-5

10 0.60 x lo-

15 0.11 X 10-E

20 0.47 x 10-10

25 0.21 x 10-I

30 0.44 x lo-

35 0.12 x lo-l4

40 0.48 x lo-l6

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406

OLSON, GEORGIOU, AND SCHULTZ

numerical results begin to diverge. It is interesting to notethat when we use 30

singular functions, the ordinary contribution to the solution iszero to four decimal

places.

3.3. Comparison between Ordinary E lements, BSBFM, and ISBFM

Initially, we study the effect of increasing the number ofsingular functions for

both the BSBFM and the ISBFM with a 16 x 8-element mesh. Asshown in

Tables III and IV, increasing the number of functions for theISBFM improves the

accuracy of the leading coefficients, which appear to convergeto the analytical

values with increasing Ns,,. With the BSBFM, the firstcoefficient remains essen-

tially constant as we increase the number of singular functions,and the estimates

of the higher coefficients are poor.

The disappointing results with the BSBFM for the highercoefficients may be due

to contamination from the blending; extra higher-order termsthat do not satisfy

the governing equation are introduced with every W. For example,with the

BSBF3.

W = rj2( 1 - x2)( 1 - y2) cos i

= r112cos i - rs12cos i + r712cos i sin2 ecos2 8.

Only the rli2 cos(0/2) term should be associated with ai,although ~1,multiplies all

three terms in BSBF3. Therefore, one can expect a good estimateonly for the first

expansion coeffkient, as noted by Whiteman [20], Morley [6], andWait and

Mitchell [ 73.

Table V compares the values of the solution u at various pointsof the domain

for the ordinary finite element method (OFE), the BSBFM, and theISBFM. We

observe that both the BSBFM and the ISBFM yield improved resultscompared to

the ordinary elements. The ISBFM predictions, however, arecloser to the exact

values.

TABLE III

Values of the Leading Singular Coefticients for the Motz Problemfrom the BSBFM

(16 x 8 Uniform Mesh)

N

FISF

al

1 402.21518

2

402.21517

3 402.21516

4 402.21590

5 402.21758

Exact

401.1625

a2

10.867980

15.206817

15.206704

15.206599

87.65592

a3

-3.563013

- 3.562949

- 3.562890

17.23792

a4

- 5.609400

- 6.309 102

-8.0712

a5

-2.677194

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FINITE ELEMEN T METHOD FOR SINGULARITIES 407

TABLE IV

Values of the Leading Singular Coefficients for the Motz Problemfrom the ISBFM

(16 x 8 Uniform Mesh)

NSBF aI a2 a3 a4 a5

1 401.15943

2 401.15932 86.662605

3 401.15932 87.620499 14.603352

4 401.16197 87.620963 14.601024 - 7.47523 1

5 401.16224 87.620202 14.604841 - 7.470403 1.224633

Exact 401.1625 87.65592 17.23792 -8.0712

TABLE V

Solution of the Motz Problem at Various Points Compared withValues from the Literature

(l6x8Mesh, Nsar=l)

(Xi> i)

OFE BSBFM ISBFM Exact [27] Wigley [21]

(-617,617)

90.964 91.342 91.341

( - 217,217 )

78.053 78.560 78.559

(0,217)

140.477 141.562 141.560

(217,217)

242.783 243.814 243.812

(0, l/7)

102.056 103.772 103.768

(- l/28, l/28) 31.770 33.594 33.590

(0, l/28)

50.261 53.197 53.190

(l/28, l/28)

79.286 83.682 83.672

(W&O)

72.264 76.412 76.403

(3/28,0)

131.740 134.452 134.447

(l/7,0)

154.096 156.487 156.483

91.34

78.56

141.6

243.8

103.77

33.59

53.19

83.67

76.41

156.48

91.343

78.559

141.560

243.812

103.768

33.592

53.186

83.671

76.408

134.447

156.483

TABLE VI

Lea ding Coefficients for the Cracked-

Bea m Prob lem (NsaF = 20, 16 x 8 Mesh)

i ISBFM Wigley [21]

1 0.191119 0.19112

2 -0.118116 -0.11811

3 -0.OOOOOO o.ooooO

4 -O.OOOOOO O.OOOOO

5 -0.012547 -0.01256

6 -0.019033 -0.01905

7 O.OOOOOO

581/96/Z-12

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408 OLSON, GEORGIOU, AND SCHULTZ

TABLE VII

Comp uted Value s of c(, for the Cracked-

Beam Problem with Various Uniform

Meshes (2N x N Eleme nts, NSBF = 1)

N ISBFM BSBFM

1 0.19116 0.16486

2

0.19114 0.18843

4 0.19112 0.19123

8 0.19112 0.19140

10

0.19112 0.19139

20 0.19112 0.19135

4.

RESULTS OR THE CRACKED-BEAM PROBLEM

As for the Motz problem, the ISBFM accelerates the convergencein agreement

with (28) and gives very accurate estimates for the highercoefficients. The com-

puted rate of convergence of (Z7, - neJ1/* with mesh refinementfor the cracked-

beam problem (p = 0.5 when no singular functions are used; p =1.6 when N,,, = 1;

and p= 2.0 when NsBF = 2) is similar to that obtained for theMotz problem.

Table VI shows the first seven singular coefficients, whichagree well with Wigleys

iterative results [21]. Table VII compares the calculated valuesof c1i from the

ISBFM and the BSBFM as the underlying mesh is relined. The ISBFMagain con-

verges significantly faster than the BSBFM. Finally, the valuesof u obtained with

the ordinary finite element method, the BSBFM, and the ISBFM arecompared

with results from the literature in Table VIII. As in the Motzproblem, both the

BSBFM and the ISBFM yield improved results compared to theordinary element

method.

TABLE VIII

Solution of the Cracked-Beam Problem at Various Points Comparedwith Results from the Literature

(l6x8Mesh, N,a,=l)

(x,3

YJ

OFE BSBF M ISBFM Fix ef 01. [ 193 Wigley [21]

al124) 0.026192 0.02743 1 0.027429 0.027425 0.027428

( - 1 /24, l/4) 0.032847 0.032878 0.032879 0.032877 0.032878

(11124, l/4) 0.070657 0.070844

0.070844 0.070844 0.070844

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FINITE ELEMENT METHOD FOR SINGULARIT IES

409

5. CONCLUDING REMARKS

The proposed integrated singular basis function approach (ISBFM)eliminates

the need for high-order integration, improves the overallaccuracy, and yields very

accurate estimates for the singular coefficients. It alsoaccelerates the convergence

with mesh refinement, in agreement with theory; the same rate ofconvergence as

for regular problems is achieved by including a sufhcient numberof singular func-

tions. For a fixed mesh, convergence with the number of singularfunctions is very

rapid. Although we have demonstrated the method only on examplesinvolving

singularities in the 2D Laplaces equation in rectangulardomains, the approach is

quite general and can be applied to arbitrary geometries and awide range of

governing equations. The extension of the method to fluid flowproblems in

complex geometries is currently under investigation.

ACKNOWLEDGMENTS

This work was partially supported by The Ocean Engi neer ingDivision of the O&x of Naval

Research, Contract NOOO14-87-0509, and the Natio nal ScienceFound ation, Contract DMC-87 16766.

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