The solution of boundary element problems using symbolic computation A. Almeida, H.L. Pina Institute Superior Tecnico, Lisbon, Portugal

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1 The solution of boundary element problems using symbolic computation A. Almeida, H.L. Pina Institute Superior Tecnico, Lisbon, Portugal Abstract The paper explores the potential of using Symbolic Computation in BEM problems. The work focuses on the following aspects: the development of analytic integration formulae to compute the element matrices with either regular or singular kernels, the study of how collocation node placement influences the results and the possibility of automatic FORTRAN code generation via Symbolic Computation Packages. 1 Introduction Let us consider the following boundary value problem (b.v.p.). C^(fi) (in order to be a solution in the classical sense) such that: Find u G v^(x) = o, Vxenc]R\ (i) Q is a open bounded subset of IR^, and F its Lipschitz-continuous boundary. The boundary conditions are taken as w(x) = iz(x), x 6 To (2) flu q(x)= = q(x), X l\ (3) where g(x) and %T(x) are given functions and n is the outer unit normal and r = TO u r\ (4) The previous b.v.p. can be solved by use of the Green function associated with the b.v.p. Let us introduce the following partial differential equation:

2 29 Boundary Element Technology W(x) + f(x-xi) = %'n,, (5) where u~ is the fundamental solution of Eq. (1) and 6 is the Dirac mass concentrated at point x. The fundamental solution for a two-dimensional space is given by (see [1]): «"(r) = ^log(^), (6) where r is the Euclidean distance in IR? from the field point x to the source point x;. Now we introduce the following Green's formula: Lemma 1 (Green's Formula) Let fl be a bounded open set and T its Lip shitz- continuous boundary, then the following formula holds: f (uv*v - v/n r an an Existence and uniqueness of the weak solution of Eq. (1) and Eq. (5) can easily be established. Both weak solutions are H* regular and so it is possible to apply Lemma 1 with v = u*. We obtain after some algebraic manipulation the following integral representation for u in fi, where z; H and u,u~ H*( l). Eq. (7) involves knowing the values of both u and du/dn on the boundary F. But by the problem definition (Eqs. (l)-(3)) we only know u in FQ and du/du in FI. In order to determine the value of u in FI and du/dn. in FQ instead of an integral representation in fi, we need now an integral representation on the boundary F. By a limiting process the interior point Xj can be taken to the boundary thus obtaining the integral representation for u on the boundary F. = - /r i^l> + /r where the fundamental solution u* has been substituted by Eq. (6) and C is a constant that is given by the fraction of solid angle subtended by F at x. For a regular boundary (that is piecewise smooth C* boundary) C = 1/2. We are now able to compute the values of u in any point of fi ( 1 is the closure of Q) by using Eq. (8) and/or Eq. (7).

3 Boundary Element Technology Problem discretization We can discretize the previous problem by replacing the unknown functions u and du/dn in the boundary (in TO and I\ respectively) by a finite element approximation on the boundary. Adopting a master element description of the discretized problem we will use the notation u and q for for the approximations of u and q = du/dn respectively in the master boundary finite element F. The elementary shape functions will be denoted by ^, where i = 1... N<, is the node number in F. As usual the master element coordinate is denoted by and 1 <(< 1. Thus we have: ( = <Mfc, (9) 1 = 1 «( = *4 (1) 1=1 where U{ and qi are the nodal values of u and g, respectively, in the master element. The discrete equation corresponding to Eq. (8) is thus: Z / %k, ir ^ = E / %^(xi j=l^3 OH y.' j=l ^J where x' is a node point and XQ is the collocation point; u^. and q^ are the restrictions of the approximation of u and q to the boundary element FJ; TV is the number of nodes. We can write Eq. (11) in matrix form: where ds. (12) (13) and N. i = E L (14) 3=1 *> The coordinate transformation between a generic boundary element Fj and the master element F is given by the parametric transformation: M X = t=l M (15) y =

4 292 Boundary Element Technology where (x;,2/{) is element Tj nodal coordinates and M is the number of nodes of transformation Tg. The basic assumptions that we will use throughout this paper are the following: Only constant and linear approximations of u and q in each element will be analyzed. Boundaries are C* piecewise smooth. Only symmetrical node placement w.r.t. = will be considered. In what concerns the continuity across elements we will deal only discontinuous approximations. As a direct consequence of the last assumption we will use different shape functions for geometry interpolation and u and q approximation. By Eq. (15) the coordinate transformation is: x = - -- f +, 2/2 + where Eq. (16) was obtained by adopting the Lagrange polynomials of first degree in the master element F for geometry interpolation, i.e.: (l-, (17) i(l +, (18) The subscript G indicates that these are shape functions for geometry interpolation. This assumption may seen, at first sight, strange. It was made in order to enable us to study the influence of the element collocation nodes position. We will return to this subject later on. 3 Formulae derivation using Maple We will enter now the specific subject of this paper which focus the use of a computer algebra system, in this case Maple [2, 3, 4], in the solution of problems by the Boundary Element Method (BEM). The first task for which we used Maple was the derivation of closed form expressions for the integrals that appear in Eqs. (13)-(14).

5 Boundary Element Technology 293 In the case of constant approximation of u and q in each element and by Eqs. (6), (13) we have the H and Q matrices entries: y = a, - E / ''fo where SH = - j- /' log(r)v'(*2-ai)* + (yj-»i)'#, (2) **" j=i-'-i r = (*' - ^ - 2i±Si)' + (,' - 2L^ - ^)>. (21) In what concerns linear approximation of u and q in each element Eqs. (16)-(18) still hold but the matrices H and Q entries are: ~ ~, \ > (22) (23) where r is given by Eq. (21). The shape functions are given by: ^ = o, v,,(( + 4-1), (24) 2 + &o + (%(, V"2 = ^ U + da ~ 1), (25) 2 + da 4- at where da and <f& are the distance from the element nodes (for function approximation) to the extreme nodes (for geometry interpolation ) as illustrated in Figure 1. For da. = db = the element nodes for geometry interpolation coincide with element nodes for function approximation. Only in this case continuity across elements is obtained. Note that for the constant element the shape function is ^i = 1 and the element node for function approximation is placed in the element center, corresponding to =. The integrals that appear in the expressions for computing %., and Qij are regular when the collocation point lies outside the domain of integration. Maple's int function can handle them quite easily. However if the collocation point x' lies inside the domain of integration the integrals for 7Y%j and Qij become singular.

6 294 Boundary Element Technology \ /2 da db node 1 node 2 Figure 1: Linear element and node location Obtaining closed form expressions for such integrals using Maple demands certain precautions in order to be able to compute such expressions successfully. We will compute the following integrals: Ik = (26) where fc =,1 and r is given by Eq. (21). Note that the nodal coordinates are given by: where a, b [,1] are such that: Xi = ax i + bxi Ui = &2/i + fy/2 (27) -6) (28) is the master element f length. In this case = 1 - (-1) = 2. Let us recall that d^ and d^ are the distance from the nearest element node to the left and right extremes ( = 1 and = 1) of the element respectively (see Eqs. (24) and (25)). Note that the integrals given by Eq. (26) are exactly the integrals contributing to Hij (Eqs. (2) and (21)). When x; belongs to the domain of integration, the integrals that appear in the expressions for computing Ti,ij are due to orthogonality between the distance r and the outer normal n. This very interesting result holds true only for straight elements. Therefore the computation of singular integrals reduces to computing closed form expressions for Eq. (26). Introducing the expression relative to Eq. (26) for k = in Maple and using the int function with -1 < < 1 we get a rather lengthy expression as

7 Boundary Element Technology 295 a result. The expression by itself is of little interest. Instead we will explain some details of the procedure followed in its deduction. In the expression that Maple returns as a result of computing IQ two terms involving the arctan function appear. Those terms have a denominator (after factorization) involving the term 1 (a-t-6). Note that according to our assumptions both element nodes are inside the element (i.e. in [ 1, 1]) and we must have a = 1 6. If a susbstitution of this result is made in the expression of the integral using Maple subs function in the arctan terms, a division by zero error will be issued. This is not what we expected. We would like Maple to give us the value of the arctan terms that will be %/2 or?r/2 according to the sign of the numerator of those terms. However as Maple has a fully recursive evaluation mode for this kind of expressions (only procedures and tables are evaluated by name), Maple will first try to evaluate the arctan function argument which gives a division by zero, and thus signaling an error. The solution is to "isolate" the arctan terms from the remainder of the expression and then extracting the numerator and denominator of the arctan terms argument handling them separately. By doing so we are able to sucessfully compute the wanted expressions. The arctan terms will be either 7T/2 or?r/2 according to the sign of the factor 6 which appears in the denominator of one term and 6 1 for the other. Because b belongs to [, 1] we have: 1 < b<, -1 < 6-1 <, concluding that the arctan terms cancel each other for b G (, 1) because they have opposite signs and obviously ir/2 -f Tr/2 =. Accordingly we can now obtain the formula for IQ with b (, 1). - log 6-1 ) + 21og Zi)2 + (7/2 - y,)2))(z2-zi)2 + (2/2-3/1)'. (29) Putting a = 6 = 1/2 we obtain the formula for the integral contributing to Ga for the constant element which can be found in any introductory text to the BEM (e.g see [1]). /o = -(log((z2-zi)' + (3/2-3/i)')-2(log(2)-l))\(z2-21)2 + (3/2 - For /i (putting k = 1 in Eq. (26)) following a similar procedure we have -log 6- l\)b(b- 1) -26+ l)\(z:-zi)2 + (3,2-3,1)2, (31)

8 296 Boundary Element Technology Combining Eq. (29) and Eq. (31) we can obtain closed form expressions for linear elements with shape functions as those given by Eqs. (24) and (25). These expressions are valid for an arbitrary node placement in the element. The relations between the parameters a, 6 and d^ 4 are established by Eqs. (28). This enables us to study node placement influence in BEM solution behaviour. 4 Comparison between Maple's and classical BEM code solutions The model problem that we have solved consists on the following: V*u = in, (32) with n = [,5] x [,5] and boundary conditions and The solution to this b.v.p. is given by: fjf(«,) = (*,5) =, (33) %(,y) = A, %(5,3/) = B. (34) N B-A u(x,y) = - z + A. (35) o In this case we have taken A 1 and B = 2 thus u(x,y) 2x + 1. This extremely simple problem enables us to thoroughly analyze the several aspects concerning the influence of numerical integration in BEM solution behavior. Let us now compare the Q and H matrices obtained by use of Maple's int function (analytical integration) with those obtained by a two point Gauss-Legendre rule. We have solved the previous b.v.p. using a 16 element mesh with both constant and linear elements. Below we present a few typical entries of the Q and H matrices. The 7i matrix for constant elements (analytical integration)

9 Boundary Element Technology 297 The H matrix for constant elements with Gauss-Legendre quadrature The Q matrix for constant elements using Eq. (3) (analytical integration) The Q matrix for constant elements with Gauss-Legendre quadrature For linear elements the H matrices the results are similar. The diagonal terms are for the Gauss-Legendre quadrature and for analytical integration. The Q matrix for linear elements using Eq. (31) (analytical integration) The Q matrix for linear elements with Gauss-Legendre quadrature A comparison between the entries of the Q matrix for numerical and analytical integration reveals sensible differences in the entries corresponding to the singular integrals. By using numerical integration the singular behaviour of the integrand cannot be captured whereas through the use of analytical integration the integrand singular behaviour is properly taken in to account.

10 298 Boundary Element Technology 5 Node placement influence in BEM solution behaviour When using analytical integration the node placement influence in solution behaviour can be analyzed. Although the problem presented here is extremely simple and consequently the node placement influence in the solution was less obvious than would be in a more elaborated problem, some variation in the solution was observed when the node placement is changed. Here we have considered only two positions for element node placement. This positions being = ±V3/3 (the Gauss-Legendre points) and = ±1/2. These positions are usually referred in the BEM literature as being the most appropriate for node placement, since they tend to contribute for a better performance of the BEM. The results we have obtained are presented in the table below. Nodes X y u f = W3/3 9= J %, u , f = ±1/2 <?* LI As we can observe, there are sizable diferences in the results particularly in the flux (derivatives) values. <7v 6 A brief comment on the Maple program The Maple program that was developed has basically seven different phases. The program layout is described in Figure 2. In all the phases full use of Maple's symbolic computation capability was made. Mesh generation was done by procedures that were developed specifically for the program. These procedures generate 3 type of curves: archs; parabolas and straight lines. The user specifies the number of points in the curve. The system of equations Tiu = gq was solved using Maple's built in function solve. This approach was preferred over a use of the linsolve function. In order to use the linsolve function the system should be in the form Ax = b. A somewhat complicated algorithm is needed for transforming Tiu = Qq in Ax = b. Using solve we can collect the unknown variables in to a set and give any set of equations (which is consistent) to solve that

11 Boundary Element Technology 299 Problem Data: V* = I Elementary Q Ji matrices I Mesh data reading] Q and H matrices assembly [Essential and natural B.C. imposition I Equation System solving I Computation of u and q in Figure 2: Program layout envolves those unknown variables. The disadvantage of following such an approach is that results become dependent on Maple's solving algorithm. However in the several examples that we have solved using this program the results obtained were always satisfactory. The computation of the diagonal terms of matrix Ji was done by considering a constant potential (or rigid body motions as in Mechanics). In such case the sum of all the off diagonal row elements is. Thus C is given by: N C = E%j. (36) The computation of the integrals that contribute to the Q and H matrices was done using the placeholder Int (for an explanation of the several Maple placeholders see [3]). Int is not an ordinary Maple function. Instead of trying to compute a value corresponding to the function invocation, a placeholder simply returns a string indicating the type of operation we would like to perform in a given expression. We can say that a placeholder "freezes" the evaluation of a Maple expression. The evaluation can be delayed until is strictly necessary. Maple's evaluator recognize placeholders and knows how to deal with such expressions in order to obtain (if possible) a numeric or algebraic expression. The placeholder concept enables Maple to simulate a stream, in the sense that when we use a placeholder an unevaluated expression and a way to evaluate it is being specified (albeit in an implicit fashion). In [3] a placeholder is designated also as a function inert form. In this paper we have prefered the first designation since, in our opinion, it describes more

12 3 Boundary Element Technology accurately how Maple deals with such objects. In the program that we have implemented the Q and *H matrices are assembled with their entries being placeholders for the int function. It is only after assembling the matrices that the numerical values of the entries are computed by using evalf. This approach avoids Maple from signaling divisions by zero while computing the integrals. Note that have we used int instead of Int and several division by zero errors would have been signaled when computing the integrals. A FORTRAN (C) output of Maple results can easily be obtained through the use of the f ortran (C) builtin function. This function invokes a parser to translate Maple algebraic expressions in to FORTRAN (C) code (see [3]). 7 Conclusions We have shown that the use of Symbolic Computation Packages can present advantages in the deduction of analytic (exact) integration formulae for computation of BEM matrices for the case of constant and linear shape functions. Uncertainties regarding numeric (approximate) integration are thus avoided and eventually more efficient computer programs are produced. Besides, the flexibility provided by these packages allow for easier code development and experimentation before one commits itself to a particular algorithm for production software as was exemplified with the collocation node localization study. Undoubtly we shall see in the near future mathematical software with fully integrated numerical and symbolic capabilities makeing knowledge of Symbolic Computation Packages indispensable. Although it is not yet possible to generate FORTRAN or C code directly from a symbolic manipulation language (only algebraic expressions are convertible to FORTRAN or C), compiled code is now obtainable (see [7]) making the symbolic approach to problem solving "almost" as efficient as a numerical approach coded in FOR- TRAN or C. Acknowledgements The present work was partially supported by JNICT, Junta Nacional de Investigate Cientifica, Programa CIENCIA, BM/2975/92-IB, and by Institute Superior Tecnico, Lisbon. References [1] C. A. Brebbia. The Boundary Element Method for Engineers. Pentech Press, 1978.

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