0045-7949(95)00148-4

~ompuru~ & Smcrurcs Vol. 58. No. 2. pp. 289-297. 1996 Copyright c, 1995 Elsevier Sctence Ltd

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ERROR ANALYSIS IN APPLYING A BOUNDARY

ELEMENT METHOD FOR THREE-DIMENSIONAL

STEADY STATE POTENTIAL PROBLEMS

0. P. CuptatS and S. S. Babu§

tDepartment of Mechanical Engineering, Indian Institute of Technology, Kharagpur 721 302, India

$Department of Mechanical Engineering, Bapatla Engineering College, Bapatla 522 101, India

(Received 7 July 1994)

Abstract-The boundary element method (BEM) has been applied to the case of three-dimensional steady state heat flow with no internal heat generation. The edge and corner effects and the near surface difficulties due to singular integral kernels of BEM are considered. Certain existing methods to overcome these difficulties have been implemented and tested. An algorithm is proposed which considers floating temporary nodes and local refinements of the mesh for interior points close to the boundary. The effects of the different methods of reducing the error are studied in the case of temperature distribution in a cube. The severity of the various errors increases with the coarsening of the mesh. The proposed method, however, is found to solve the near surface problems very effectively, even in the case of a very coarse mesh with negligibly small excess processing time.

1. INTRODUCTION

The boundary element method (BEM) has gained much popularity in recent years due to the many inherent advantages when compared to the finite element method. The main features of BEM include less computational time and less computer memory requirement, as the process involves discretization only on the boundary of the domain. These features enable BEM to be used for certain problems requir- ing three-dimensional analysis which otherwise would have been almost impossible to solve using FEM.

The method provides a fairly accurate estimate of the parameter within the domain, even if the bound- ary surface is divided into a very coarse mesh with large elements. The estimation of potential (tempera- ture in the case of thermal analysis) at an internal point in the domain is nearly independent of the nature of the shape function within the surface elements, because it is calculated using the basic integral equation for the whole surface enclosing the domain.

However, this estimate at an interior point tends to become inaccurate as it approaches the corner or an edge of the boundary surface or the surface itself. Some times these errors are very high. This paper addresses the problem of eliminating or reducing these errors.

The general belief that reducing the size of the surface elements will reduce the error does not seem to prove satisfactory for this type of error. Also, the

$ To whom all correspondence should be addressed.

full unbanded matrices resulting in BEM would put a restriction on the maximum number of nodes that can be used in a problem, as the computational time becomes very high and the very purpose for which the method has been employed could be lost.

Several methods have been suggested in the litera- ture which are found to be quite effective. The edge and corner difficulties are resolved by considering multinodes [I] or by employing discontinuous bound- ary elements [2,3]. On the other hand, the near surface problems have been attempted by the use of external collocation points [l, 41 or by altering the integration scheme, employing a coordinate trans- formation method [5,6].

In this paper the problem of three-dimensional steady state temperature distribution in a cube with- out any internal heat generation, is considered for the purpose of illustrating the nature of distribution of the error in the calculated value of the functional at points near the boundary. A method of reducing the error by considering a temporary extra node on the surface nearest to the source point has been investi- gated, along with some other methods from the literature. Linear triangular elements have been used for discretizing the surface of the domain. The mag- nitude of the error in any case has been calculated as the difference between the value obtained by solving the boundary integral equation and that obtained from the analytical solution.

2. BASIC FORMULATION

The basic equation for three-dimensional steady state heat flow without any internal

289

290

heat generation is

V>(U) = 0.

The boundary conditions are

u = L? on surface

and

0. P. Gupta and S. S. Babu

(1)

SI

q = q on surface s2,

where u is the temperature and q = (&j&t) is the slope (temperature gradient) in the direction of the outward drawn normal at any point on the boundary.

After discretizing and writing the abave equation in the boundary integral form, it becomes

c,u;=J:qu*dr-[;q*di. (2)

where U* = 1147~ is the fundamental solution and q* = &*/an is the gradient of U* [7]. This yields the matrix equation,

Out of the 2n unknown values of u and 4, n values are specified on the boundary as known temperature or gradient and only n unknown terms are ultimately left. By separating the known and the unknown terms, the matrix equation can be rewritten as

L4w = {FL (4)

where {X} and (F} are the vectors of unknown and known quantities, respectively. The vector of un- known quantities can be found by solving the set of

equations represented by eqn (4). Once the solution on the boundary is obtained, the potential at any point within the domain can be found as

where ci is equal to 1 for interior points. The numerical integration scheme used considers

13 integration points on the linear triangular el- ements. The solution for the integral Jfq* ds is ob- tained by writing it in the following form:

= s “x!?&= 1 S[ 1 -- s dr an J 4nr2

cos 9 ds,

where cos 8 = &/an, 0 being the angle between the

line joining the source point and the observation point and the outward drawn normal to the surface, at the observation point. This angle is calculated at every integration point in an element, by using the

vectorial method.

3. ERROR ANALYSIS

The algorithm explained above has been used to calculate the temperature distribution at different points within a cube of side 40 mm. The temperatures at the two opposite faces are maintained at 100 and 10°C respectively, while the other faces are con- sidered as insulated. This represents the simple case of steady state heat flow through an infinite wall, for which the temperature is known to vary linearly from the high temperature face to the one at low temperature.

The exact values of temperatures being known at any point, the error could be determined easily. The mesh configurations with 386 and 26 nodes are

shown in Figs 1 and 2, respectively. The various error-reducing algorithms are discussed here.

3.1. Sharp corners and sharp variation in the boundary conditions

The ambiguity in the normal direction at sharp corners, or insufficiently defined boundary conditions at points where sharp variations in boundary con-

ditions exist, cause errors in the calculated gradient of the potential.

Mitra and Ingber [I] developed a modified multi- node concept for solving these difficulties. In their method, they placed two nodes at the same geometri-

cal location and an extra collocation point outside the domain for each pair of such nodes. This allows for the obtaining of an additional equation from each of such collocation points, without introducing any further unknowns.

Patterson and Sheikh [2] have used the concept of non-conforming boundary elements for handling the

Fig. 1. Mesh configuration with 3% nodes.

Error analysis in applying a boundary element method

Fig. 2. Mesh configuration with 26 nodes.

modelling difficulties at singular boundary conditions and concluded that the results are generally accept- able.

Xu and Brebbia [3], while emphasizing the advan- tages of discontinuous boundary elements, have car- ried out numerical investigations and concluded that the optimum positions for the nodes are those corre- sponding to the Gauss integration points.

Figure 3 shows the distribution of the error in the calculated slopes on the face 1 of the cube with 386 nodes. In spite of the fineness of the mesh used, it can be observed that the error is quite considerable and it becomes rather unpresentable for the other case with 26 nodes.

The multinode concept as suggested in Ref. [l] has been implemented and new node numbers are given on the faces 1 and 2 in both the cases. After this, all the points where the boundary condition changes from temperature to slope have double nodes. Both the nodes are used as collocation points so that no additional points are needed. The linear elements used

Fig. 3. Absolute error contours of slope (face 1,386 nodes).

291

here and the different types of boundary conditions specified at the double nodes avoid the problem of getting singular equations from points at the same geometric location.

The error contours of the calculated gradients of temperature on face 1 of the cube with 42 nodes, as shown in Fig. 4, clearly indicate the improvement obtained. It can be seen that the error has been brought down by a large extent, even with the coarse mesh case. When one of the collocation points for each pair of coincident nodes is placed outside the domain, the result has not shown any further im- provement than that obtained above.

3.2. Errors at points close to the boundary

The singular integral kernels lead to errors in the calculated functional when the source points are close to the boundary. For problems where near surface details are important, these errors can give rise to many difficulties.

Koizumi and Utamura [5] developed and tested a method featuring the transformation of the Cartesian coordinates into polar coordinates, with the foot of the perpendicular from the point under consideration onto the nearest surface as the origin and utilizing the coefficient, c, of the boundary integral equation for correcting the error in numerical integration. It has been shown for the case of potential distribution in a cube that this method reduces the near surface errors to a large extent. Hayami and Brebbia [6] have presented a rectangular polar transformation method for solving the singular and nearly singular integrals in three-dimensional boundary element analysis.

The nature of distribution of the absolute error in the temperatures calculated on a section parallel to the Y-Z plane and at adistance of 1 mm from it, for a mesh configuration with 450 nodes, is shown in Fig. 5

Fig. 4. Absolute error contours of slope after using multi- nodes (face I, 42 nodes).

292

2.48

2.00

1.53

1.05

0.58

0.10

0. P. Gupta and S. S. Babu

(a)

10 0.58 1.05 1.53 2.00 2.48 2.95 3.43 3.90

'2.48

0.58

0.10 0.10 0.58 I.35 1.53 2.00 2.48 2.95 3 43 3.90

Max. 5.5%

Fig. 5. (a) Absolute error contours at X = I mm (450 nodes) with no special correction scheme; (b) three-dimensional distribution of the error.

No special correction schemes have been employed in the nodes of the basic mesh. This observation has led this case. It seems to be difficult to comprehend the the authors to use an extra temporary node at a point behaviour of the error in relation to the geometrical on the boundary, just under the source point, as features of the mesh on the nearest boundary, which described in the next sub-section. is superimposed on the contours. However, it can be As suggested by Koizumi and Utamura [5], a seen that the error is minimum at points just under simple correction of the coefficient, cir appearing in

Error analysis in applying a boundary element method

0.10 0.58 1.05 1.53 2.00 2.48 2.95 3.43 3.90 3.90 3.90

3.43 3.43

2.95 2.95

2.48 2.48

2.00 2.00

1.53 1.53

1.05 1.05

0.58 0.58

0.10 0.10 0.10 0 58 1.05 1.53 2.00 2.48 2.95 3 43 3.90

Max. 4.S%

Fig. 6. Absolute error contours with corrected c, at X = 0. I mm (42 nodes).

293

the integral equation will improve the result to a large extent. The correction involves calculation of the coefficient c,, as is the case with the surface points which otherwise would expectedly be equal to unity.

The error contours obtained after implementing the ci correction as shown in Fig. 6 show a marked improvement even with the 42 nodes mesh configur- ation. However, the error magnitude and the area over which it is distributed is still considerable.

3.3. The floating node algorithm

The algorithm developed considers a floating extra node at a point where the perpendicular drawn from the internal source point meets the nearest surface. This is followed by a local rearrangement of the element connectivity to accommodate the new node. Different locations for the new nodes necessitate different varieties of adjustments in the connectivity, as shown in Fig. 7.

Figure 7a shows the type of adjustment used in the basic mesh when a temporary node falls very close to one of the nodal points of the basic mesh. Instead of forming new elements which will be highly dispropor- tionate, the modification shifts the existing node to the location of the temporary node. The element connectivity remains unaltered, as do also the total number of nodes and elements.

New sub-elements are formed when the new node falls well inside an element, as in Fig. 7b. The total number of nodes is increased by one and the elements

by two. This is expectedly the most effective modifi- cation, as long as the point falls well inside an element.

However, if the node falls very near to one of the sides of an element, this modification might lead to errors as one of the newly formed elements becomes very small. For such cases, or for the case when the point is falling on one of the sides of an element, the type of modification employed is as shown in Fig. 7c. Four sub-elements are formed around the central node in the place of two adjacent elements. The total number of elements and nodes in this case also increases by two and one, respectively. The tempera- ture and slope at any temporary node are calculated by interpolation using the nodal values of the element in which it falls.

3.4. Eflects of theJfoating node

Figures 8-i 1 show the error contours obtained after implementing the floating node concept along with the corrected ci. These figures explain the be- haviour of the algorithm proposed in controlling the magnitude and distribution of the error, in the calcu- lated temperatures at different sections in the cube. Figures 8 and 9 show the distribution of the error at sections parallel to the Y-Z plane and at distances of 0.1 and I mm, respectively, from face 1 of the cube. Figures 10 and I1 show error contours on similar sections near the low temperature face of the cube. The boundary mesh considered has 42 nodes.

294 0. P. Gupta and S. S. Babu

(a) Floating point close to one of the nodes ( Type 1 1

(b) Floating point well inside

an element (Type 2 )

(c)

2.95

2.48

0.58

, I’

/’ /’ rsJ ,, 6’

,/’ I’

I

/’

Point falling on one of (d)Point falling very close to the sides one of the sides

(Type 3)

(Dotted lines represent modified mesh )

Fig. 7. Different local modifications in element connectivity.

2.95

/’ ’ ’ , ‘I,, , , ‘J

1.05

0.58

0.10 .I0 0.58 1.05 1.53 2.00 2.48 2.95 3.43 3.90

Min. 0.70 Max. 1.50% Fig. 8. Error contours with floating node and c, correction, at X = 0. I mm (42 nodes).

Error analysis in applying a boundary element method 295

in n 58 1 n5 1 5.3 2.00 2.48 2.95 3.43 3.90

3.43

2.95

2.48 2.48

2.00

1.53

1.05

4 0.58 0.58

0.10 0. 10 0.58 1.05 1.53 2.00 2.48 2.95 3.43 3.90

Max. 1.1 %

Fig. 9. Error contours with floating node and c, correction at X = I mm (42 nodes).

3.90

1

3.90

3.43

0.10 0.58 1.05 1.53 2.00 2.48 2.95 3.43

3.90 \, , , [ / ,I I I I I /IIIlIl 11 1

_I 2.95

2.48

2.00

j 2.48

1.53

1.05

0.58 L 0.10 0. 10 0.58 1.05 1.53 2.00 2.48 2.95 3 43 3.90

Max. 1.5 Fig. 10. Error contours with floating node and c, correction at X = 39.9 mm (42 nodes)

296 0. P. Gupta and S. S. Babu

0.10 0.58 1.05 1.53 2.00 2 48 2.95 3.43 3.90

2.48

2.00

1.53

1.05

0.58

0.10 L

2.48

2.00

1.53

1.05

0 58

0.10 0 10 0.58 1 05 1.53 2.00 2.48 2.95 3 43 3.90

Max. -1.1

Fig. 1 I. Error contours with floating node and c, correction at X = 39 mm (42 nodes).

It can be seen that not only the magnitude of the maximum error has been reduced, but the areas over which the error is present have been confined to some small islands. These islands show the areas of maxi- mum error, in the ranges obtained in different cases. The magnitude of the error in general is increasing as the source point moves towards the boundary surface.

For points close to the nodes, the error is already less, for being in the vicinity of a node. Additionally, the slight shifting of a node, when it is not on one of the edges, so as to be just under the source point (type 1 modification of the mesh configuration as in Fig. 7) improved the result still further. This effect can be

seen in all of the Figs 8-l 1, in the areas around the central node, where the error is almost zero.

The central portion of any element is expectedly

the most advantageous position for the extra node to be placed as it involves formation of sub-elements of more or less equal sizes. Consequently, as can be seen in Figs 8-l 1, areas central to the elements, where the second type of modification (Fig. 7) in the mesh configuration is done, show less magnitude of error.

The third type of refinement in the mesh is also found to give good results when the new node is falling somewhere in the middle of the side of an element. However, at some locations, between the end and middle portions, particularly for the longer

Table 1. Results of the numerical investigation

Total Maximum error in no. of Location section plane the calculated

Aglorithm nodes where the error is estimated temperatures (X)

Implementing multi-nodes with no 450 Section parallel to the side face 1 and 5.5 other correction scheme. located at 1 mm from it. Multinodes along with c, correction. 42 Section parallel to the side face 1 and 4.58

located at 0.1 mm from it. Multinodes along with c, modification 42 Section parallel to the side face I and 1.50

and the floating node method. located at 0.1 mm from it. Multinodes along with c, modification 42 Section parallel to the side face I and 1.10

and the floating node method. located at 1 mm from it.

Table 1 shows conclusively that the floating node procedure suggested in this paper, in association with the existing technique of correcting the c, values is very effective in reducing the error at near surface points. This improvement could possibly be due to the balanced distribution of the integration points around a node. The formation and the solution of the basic equation around the floating node would be more unbiased and balanced.

Error analysis in applying a boundary element method 291

sides, both the second and the third type local adjustments in the mesh configuration seem to have

not been so effective. These are the locations for the islands of maximum error, shown in Figs 8-11. In a physical sense, the reason for this could be the disproportionately formed elements after read- justing.

4. DISCUSSION AND CONCLUSIONS

The boundary element method, though efficient, suffers from some drawbacks. The results could be erroneous when the confining boundary has sharp edges and corners or when the boundary conditions change sharply. Also, the singular integrals would lead to errors when a source point approaches close to one of the boundary surfaces. Various procedures, like the use of multiple coincident nodes, are suggested to reduce the effects of the first type of error.

However, the second type of error arising out of the proximity of the point of interest to the boundary surface has not been tackled sufficiently. except for the general modification of c, (eqn 2). This has some welcome features in reducing the error. How- ever, the floating node algorithm suggested in this paper greatly reduces the error when used in associ- ation with c, modification. The results of implement- ing these error reducing procedures have been discussed in the previous sections and are compiled in Table 1.

1.

2.

3.

4.

5.

6.

7.

REFERENCES

A. K. Mitra and M. S. Ingber, Resolving the difficulties caused by geometric corners and discontinuous bound- ary conditions. In: Boundary EIemenrs X, Vol. I, Math - ematical and Computafional Aspects (Edited by C. A Brebbia, W. L. Wendland and G. Kuhn), pp. 519-532. Springer, Berlin (1987). C. Patterson and M. A. Sheikh, Non-conforming boundary elements for stress analysis. In: Proc. 3rd Int. Seminar, Irvine, CA (Edited by C. A. Brebbia), pp. 137-152. Springer, Berlin (1987). J. M. Xu and C. A. Brebbia, Optimum positions for the nodes in discontinuous boundary elements. In: Proc. 8th Int. Cot~j., Tokyo, Japan, Vol. I1 (Edited by M. Tanaka and C. A. Brebbia), pp. 751-767. Springer, Berlin (1986). R. Yuuki, T. Matsumato and H. Kisu, The usefulness and the limit of direct regular method in boundary element elastostatic analysis. In: Boundary Elements VIII, Proceedings of the 8th International Conference, Tokyo, Japan, Vol. II, pp. 715-724. Springer, Berlin (1987). M. Koizumi and M. Utamura, A new approach to singular kernel integration for general curved elements. In: Proc. 8th Int. Co& Tokyo, Japan, Vol. II (Edited by M. Tanaka and C. A. Brebbia), pp. 665475. Springer, Berlin (1986). K. Hayami and C. A. Brebbia. A new coordinate transformation method for singular and nearly singular integrals over general curved boundary elements. In: Boundary Elements, IX, Vol. I, Mathemarical and Com- putational Aspects (Edited by C. A. Brebbia, W. L. Wendland and G. Kuhn), pp. 375-399. Springer, Berlin (1987). C. A. Brebbia and S. Walker, The Boundary Element Techniques in Engineering. Newnes-Butterworths. London ( 1980)