University of Hertfordshire Department of Mathematics Study on the Dual Reciprocity Boundary Element Method Wattana Toutip Technical Report 3 July 999 Preface The boundary Element method (BEM) is now recognised as a well-established numerical technique for solving problems engineering and applied science. The main advantage of the BEM is its unique ability to provide a complete solution in terms of boundary value only. An initial restriction of the BEM is that the fundamental solution to the original partial differential equation is required in order to obtain an equivalent boundary integral equation. Another is that non-homogeneous terms are included in the formulation by means of domain integrals, thus making the technique lose the attraction of its boundary only character.the resolution of these problems has been the subect of considerable research over the past decade and several methods have been suggested. It is our opinion that the most successful so far is the dual reciprocity method (DRM) by means of accuracy and programming point of view.
. Introduction There are three classical methods for solving problems in engineering and applied science. The first approach is the finite difference method. This technique approximates the derivatives in the differential equation which govern each problem using some type of truncated Taylor expansion. The second one is the finite element method (FEM). This method involves the approximation of the variables over small parts of the domain, called elements, in term of polynomial interpolation function. The disadvantages of FEM are that large quantities of data are required to discretize the full domain. The third one is the boundary element method (BEM). This approach is developed as a response to that problem. The method requires discretization of the boundary only thus reducing the quantity of data necessary to run a program. However, there are some difficulties of extending the technique to several applications such as non-homogeneous, non-linear and time-dependent problems for examples. The main drawback in these case is the need to discretize the domain into a series of internal cells to deal with the terms not taken to the boundary by application of the fundamental solution. This additional dicretization destroys some of the attraction of the method in terms of the data required to run the program and the complexity of the extra operations involved. It was then realised that a new approach was needed to deal with domain integrals in boundary elements. Several methods have been proposed by different authors. The most important of them are:. Analytic Integration of the Domain Integrals.. The Use of Fourier Expansion. 3. The Galerkin Vector Technique. 4. The Multiple Reciprocity Method. 5. The Dual Reciprocity Method The Dual Reciprocity Method (DRM) is essentially a generalised way of constructing particular solutions that can be used to solve non-linear and time-dependent problems as well as to represent any internal source distribution. This work is intended to study this method.
. The boundary Element Method for the equation u 0 and u b. Laplace equation The Laplace equation satisfied by the potential u in a domain with a boundary as shown in Figure can be converted into the well known integral equation (Brebbia, 978) cu uq d qu d (.) u where and we assume that u u on and q q on. n denotes n the unit outward normal to and u is the fundamental solution of the Laplace equation; c is a real number which is described in the following paragraphs. n u u u 0 u q q n Figure. : Laplace equation posed on the domain with the boundary Let P be a point in the domain and assume that P is an internal point. Then equation (.) becomes (Paris and Cañas, 997) u( P) qu d uq d (.) u where u is a fundamental solution and q. n On the other hand, in the case when P is a boundary point, equation (.) is of the form (Paris and Cañas, 997) where c( P) u( P) uq d qu d (.3) ( P) c( P), (P) is the internal angle at the boundary point P. 3
For a two-dimensional domain, the fundamental solution is (Gilbert and Howard, 990) u ln( ) r (.4) On the other hand, in a three-dimensional domain (Gilbert and Howard, 990) u 4r where r is the distance between the source point and the field point, see Figure.. Field point r Source point Figure. : Distance r between the source point and the field point The boundary integral equation (.3) is discretized by partitioning the boundary into elements. The element lies between node and node + as shown in Figure.3. This is equivalent to replacing the boundary curve by a polygon Target element + [] r i 3 Base node Figure.3 : Discretization of the boundary into elements 4
We choose a suitable set of basis function w ( s) :,,..., where s is the distance around the boundary,, and consider the boundary element approximation u ~ ( s) w ( s) U q ~ ( s) w ( s) Q where U and Q are the, approximate, values of u and q at node. Setting u u~ and q q~ in equation (.3) we collocate at the nodal points to obtain a system of algebraic equation (Davies and Crann, 996) c u i i ( w s) U q ds w ( s) Q u ds (.5) which we may write as c u i i i Hˆ U G Q (.6) i where Hˆ G i i w ( s) q ds w ( s) q ds (.7.) w ( s) u ds w ( s) u ds (.7.) We shall consider linear elements in which w (s) is the usual hat function based at node as shown in Figure.4. 5
w (s) - - + + Figure.4 : Hat function based on node Consider an arbitrary segment such as the one shown in Figure.5, odal value of u and q = - = ode + ode Figure.5 : Relation between local co-ordinate and dimensionless co-ordinate, ( s s ) l s s l Local co-ordinate s s l With the basis function w (s) being the hat function shown in Figure.4, we see that the values of u and q at any point of an element can be defined in terms of their nodal values and the linear interpolation functions and such that u( ) U q( ) Q U Q U U Q Q 6
where is the dimensionless co-ordinate and and are the usual Lagrange interpolation polynomials ( ( ) ) The integral along the element in equation (5) becomes, for the left hand side, U U w q ds q ds h h U (.8) U [ ] where for each element we have two components, h q ds [ ] h q ds [ ] Similarly, for the right hand side we obtain Q Q w u ds q ds g g (.9) Q Q [ ] where for each element we have two components, g u ds [ ] g u ds [ ] Hence, from equation (.6), for each collocation node i we have H ˆ h h (.0.) i ( ) and G i g g( ) (.0.) 7
In this development of the boundary element method, the equation c U i i i Hˆ U G Q (.) i has been obtained by collocating at the nodes defining the function values i.e. the collocation and functional nodes are the same. In equation (.), defines a functional node and i denotes a collocation node. Let us define H ˆ where i i H i H i H ˆ c where i i i then we can write equation () as H iu G i Q (.) The system of linear equations may be written in matrix form HU = GQ (.3) Applying the boundary conditions to identify the Dirichlet and eumann boundary regions we can partition the system (.3) in the form U U Q H H G G Q where U and Q comprise known boundary values and U and Q comprise unknown boundary values. The equations are then rearranged in the form AX = F Where A G H, Q X, and F [ GQ HU] U Once the values of U and Q on the whole boundary are known we can calculate the value of u at any internal point P k by using equation (.). We obtain 8
u k G k Q Hˆ U (.4) k The LIBEM code (Mushtag, 995), based on Brebbia (978), is implemented by using this mathematical algorithm and works well with smooth domain boundaries because there are no corners in such boundaries. However, the MULBEM program (Toutip, 999), can solve problems caused corners and discontinuous boundary condithions using the multiple node method (Subia and Ingber, 995).. Poisson equation Consider the Poisson equation as shown in Figure.6 u b in (.5) n u u q q Figure.6 : Geometric Definitions of the Problem where b is at present assumed to be a known function. In a similar way as done in the Laplace equation, we have u bu d q qu d u u q d (.6) which inegrated by parts twice to produce u ud bu d qu d qu d uq d uq d (.7) After substituting the fundamental solution u of the Laplace equation into (.7) and grouping all boundary terms together we obtain 9
c u i uq d i bu d qu d (.8) otice that although the function b is known and consequently the integral in does not introduce any new unknowns, the problem has changed in character as we need now to carry out a domain integral as well as the boundary integral. The constant c i depends only on the boundary geometry at the point i under consideration. The simplest way of computing the domain integral term in equation (.8) is by subdividing the region into a series of internal cells, on each of which a numerical integration scheme such as Gauss quadrature can be applied. However, this technique loses the attraction of its boundary only character. It was then realised that a new approach was needed to deal with domain integrals in boundary elements. Several methods have been proposed by different authors. The most important of them are:. Analytic Integration of the Domain Integrals. This approach, although producing very accurate results, is only applicable to a limited number of cases for which the integrals can be evaluated analytically.. The Use of Fourier Expansion. The Fourier expansion method is not straightforward to apply in many cases as the calculation of the coefficients can be computationally cumbersome, although the method has been applied with some success to relatively simple cases. 3. The Galerkin Vector Technique. This approach uses a primitive, higherorder fundamental solution and Green s identity to transform certain types of domain integrals into equivalent boundary integrals. The main difficulty of the approach is that it can only solve comparatively simple cases. It has been extended to deal with other applications giving origin to the technique discussed in the paragraph. 4. The Multiple Reciprocity Method. This is an extension of the Galerkin vector technique which utilises as many higher-order fundamental solutions as required rather than using ust one. The main difficulty is that the method cannot be easily applied to general non-linear problems although it has been successfully used to solve some time-dependent problems. 5. The Dual Reciprocity Method. This is the subect of this work and constituted the only general technique other than cell integration. The Dual Reciprocity Method (DRM) is essentially a generalised way of constructing particular solutions that can be used to solve non-linear and time-dependent problems as well as to represent any internal source distribution. The method can be applied to define sources over the whole domain or only on part of it. The approach will be described in the next section. 0
3. The Dual Reciprocity Method for Equation of the Type u b x, y 3. Mathematical Development of the DRM for the Poisson Equation Consider the Poisson equation u b (3.) where b bx, y, that is, b is considered to be a known function of position. The solution to equation (3.) can be expressed as the sum of the solution of a homogenous and a particular solution as u uh uˆ where u h is the solution of the homogeneous equation and û is a particular solution of the Poisson equation such that uˆ b (3.) If there are boundary nodes and L internal nodes, as shown in Figure 3. : Boundary node : Internal node Figure 3. : Boundary and internal nodes there will be form L values of û and the approximation of b can be written in the
b L (3.3) f where the :,,..., L is a set of coefficients and the f are approximating functions. The particular solutions û, and the approximating, f are linked through the relation uˆ f (3.4) The function f in (3.3) can be compared with the usual interpolation function i in expansions such as which are used on the boundary elements themselves. Substituting equation (3.4) into (3.3) gives b u iu i (3.5) L uˆ (3.6) Equation (3.6) can be substituted into the original equation (3.) to give the following expression u L uˆ (3.7) Multiplying by the fundamental solution and integrating by parts over the domain, we obtain L u u d ( u) u d ˆ (3.8) ote that the same result may be obtained from equation ( u) u d bu d (3.9) Integrating the Laplacian terms by parts in (3.8), produces the following integral equation for each source node i, c u i i L q ud u qd c u q uˆ i i d ˆ u qˆ d (3.0)
The term qˆ in equation (3.0) is defined as outward normal to, and can be expanded to u qˆ ˆ, where n is the unit n uˆ x uˆ y qˆ (3.) x n y n ote that equation (3.0) involves no domain integrals. The next step is to write equation (3.0) in discretized form, with summations over the boundary elements replacing the integrals. This gives for a source node i the expression L ciui q ud u qd ciuˆ i q uˆ d u q ˆ d (3.) k k k k k k k k After introducing the interpolation function and integrating over each boundary element, the above equation can be written in terms of nodal values as c u i i k H ik u k k G ik q k L ciuˆ i H ikuˆ k Gik qˆ k (3.3) k k The index k is used for the boundary nodes which are the field points. After application to all boundary nodes using a collocation technique, equation (3.3) can be expressed in matrix form as Hu Gq L Hu Gˆ ˆ (3.4) q If each of the vectors û and qˆ is considered to be one column of the matrices Û and Qˆ respectively, then equation (3.4) may be written without the summation to produce Hu Gq Huˆ Gˆ (3.5) q Equation (3.5) is the basis for the application of the Dual Reciprocity Boundary Element Method and involves discretization of the boundary only. The process described forms the basis of the method and gives a process for extending to non-linear problems. However, in the linear case ( b = b(x, ), we can develop the method in an manner, such that we can easily adapt the existing MULBEM code, as follows: Consider the Poisson equation (3.) with the boundary conditions as shown in Figure.6. From equation (3.3) we have 3
b L f with uˆ f where f and û are known. Set L U u uˆ (3.6) Taking Laplacian operator both two sides we obtain U u L Substituting (3.) and (3.4) into equation (3.7) we have ( uˆ ) (3.7) U b L (3.8) f Finally, by substituting (3.3) into equation (3.8), we obtain a new Laplace eqaution U 0 (3.9) with boundary conditions U U L u uˆ on and Q Q q L qˆ on where Q is the normal derivative of U. After the Laplace eqution (3.9) has been solved, the values of U and Q are known and hence we also obtain the solution of the Poisson equation (3.) by the followings: L u U uˆ on boundary and inside (3.0) and q Q on boundary (3.) The MULBEM code is modified to the MULDRM by this manner. 4
Interior odes The definition of interior nodes is not normally a necessary condition to obtain a boundary solution, however, the solution will usually be more accurate if a number of such nodes is used. When interior nodes are defined, each one is independently placed, and they do not form part of any element or cell, thus the co-ordinates only are needed as input data. Hence these nodes may be defined in any order. The Vector The vector in equation (3.5) will now be considered. It was seen in equation (3.3) that b L (3.) f This may be expressed in matrix form as b F (3.3) where each column of F consists of a vector f containing the values of the function f at the ( L) DRM collocation points. In the case of the problems considered in this section, the function b in (3.) and (3.) is a known function of position. Thus equation (3.3) may be inverted to obtain, i.e. F b (3.4) The right-hand side of equation (3.5) is thus a known vector. Writing (3.5) as Hu Gq d (3.5) where d HUˆ GQˆ (3.6) Applying boundary conditions to (3.5), this equation reduces to the form Ax y (3.7) where x contains unknown boundary values of u and q. 5
Internal Solution After equation (3.7) is obtained using standard techniques, the values at any internal node can be calculated from equation (3.3), each one involving a separate multiplication of known vectors and matrices. In the case of internal nodes, as was explained in previous section, c and equation (3.3) becomes i u i k H ik u k k G ik q L k uˆ i H ikuˆ k Gikqˆ k (3.8) k k 3. Different f Expansions The particular solution, approximating functions û, its normal derivative, qˆ, and the corresponding f used in DRM analysis are not limited by formulation except that the resulting F matrix, equation (3.3), should be non-singular. In order to define these functions it is customary to propose an expansion for f and then compute û and qˆ using equations (3.4) and (3.), respectively. The originators of the method have proposed the following types of functions for f. Elements of the Pascal triangle. Trigonometric series 3. The distance function r use in the definition of the fundamental solution The r function was adopted first by ardini and Brebbia and then by most researchers as the simplest and most accurate alternative. The definition of r is that r r x r y (3.9) where r x and r y are the components of r in the direction of the x and y axes. If f r, it can easily be shown that the corresponding û function is dimensional case. The function qˆ will be given by r 9, in the two r qˆ rx cos( n, x) ry cos( n, (3.30) 3 6
In the above, the direction cosine refer to the outward normal at the boundary with respect to the x and y axes. Formula (3.30) may be easily obtained suing (3.) and r rx r ry remembering that and. x r y r Furthermore, some recent works suggest that series f r is in fact one component of the f r r... r m (3.3) The û and qˆ functions corresponding to (3.3) are : uˆ r 4 3 r 9 m r... ( m ) (3.3) m x y r r qˆ rx ry... (3.33) n n 3 ( m ) In principal, any combination of terms may be selected from (3.3). To illustrate this, for Poisson-type equation, one case will be considered: f r The presence of the constant guarantees the completeness of the expansion and also implies that the leading diagonal of F is no longer zero. Equation (3.3) may be solved for using standard Gaussian elimination. This is the simplest alternative to program. It has already produced excellent results for a wide range of engineering problems. ote that in this case uˆ r 4 3 r 9 and qˆ r x x r n y y r n 3 We implement the DRBEM program first based on this manner. Furthermore, the program include the thin plate spline function for approximation. However this program is still separated the execution of solving the boundary solution and internal solution. Finally, the DRBEM is implemented to solve the whole system of equation in order to apply to solve non-linear case in the future work. We discuss computational results in the late section. 7
4. Computational Result of the MULDRM program Example : Square plate with internal heat generation (Gibson, 985) Consider the Poisson equation u (4.) The problem is to be solved over an isotropic square plate occupying the region 6 x 6 and 6 y 6. The symmetry of the region means we need consider only the negative quadrant as shown Figure 4.. (-6,6) 0 9 (0,6) 3 8 u 4 7 5 6 (-6,0) 6 3 4 5 (0,0) : Boundary node : Internal node : Multiple node Figure 4. : Discretization of the boundary into elements with 4 multiple nodes and 5 internal points Initially the boundary is modelled with boundary elements and 4 multiple nodes at the corners. The calculation of the potential is required at the five internal points shown in Figure 4..Boundary conditions are specified as zero flux on the line x 0, y 0and as zero temperature on the line x 6, y 6. The internal solutions obtained using MULDRM program are compared with those of the Monte Carlo method and the exact solution in Table 4.. 8
Error (%) DRBEM 3/8/09 Table 4. : Internal solutions of square plate with internal heat generation problem Internal Monte Calo method MULDRM Exact point 500 000 3000 Standard Multiple solution (-,) 8.985 8.543 8.537 8.48 8.690 8.690 (-4,) 5.80 5.645 5.736 5.58 5.77 5.748 (-3,3) 6.498 6.36 6.477 6.358 6.547 6.5 (-,4) 5.78 5.634 5.633 5.584 5.77 5.748 (-4,4) 3.96 3.987 3.98 3.889 3.987 3.98 Percentage errors of the potentials at the internal points are shown in Figure 4.. Percentage errors of potential 4 3 0 (-,) (-4,) (-3,3) (-,4) (-4,4) Monte Calo Standard Multiple node Figure 4. : Percentage errors of potential at the internal points We see from the figure that the percentage errors of the solution using the multiple node approach are less than the other methods. Example : The Torsion problem Consider the Poisson equation u (4.) on the elliptical domain as shown in Figure 4.3. 9
(0,) 5 4 3 6 0 (-,0) 9 (,0) 8 3 4 5 (0,-) 6 7 : Boundary node : Internal point : Multiple node Figure 4.3 : Discretization of the boundary into 6 elements 7 internal points and 4 multiple nodes in elliptical domain The elliptical section shown in Figure 3 has a semi-maor axis a = and a semiminor axis b =. The equation of the ellipse is x a b y (4.3) The boundary condition is the Dirichlet condition with u = 0 on the boundary. The exact solution is The normal derivative is 0 x y u.8 (4.4) a b q 0.( x 8y ) (4.5) We use the number of multiple nodes as 4, 8 and 6. The four multiple nodes are shown in Figure 4.3. The solutions are compared with the cell integration method and the exact solution as shown in Table 4.. 0
Error (%) DRBEM 3/8/09 Table 4. : Internal solutions of the Torsion problem Internal Cell MULDRM Program (o. of multiple nodes) Exact point Integration 0 4 8 6 solution (.5,0.0) 0.33 0.344 0.347 0.347 0.347 0.350 (.,-0.35) 0.40 0.40 0.49 0.49 0.48 0.44 (0.6,-0.45) 0.557 0.576 0.574 0.574 0.574 0.566 (0.0,-0.45) 0.69 0.648 0.646 0.646 0.646 0.638 (0.9,0.0) 0.66 0.646 0.644 0.644 0.643 0.638 (0.3,0.0) 0.77 0.793 0.790 0.790 0.790 0.78 (0.0,0.0) 0.79 0.80 0.808 0.808 0.808 0.800 Percentage errors of potential at the internal points are shown in Figure 4.4 Percentage errors of potential 6 5 4 3 0 (.5,0.0) (.,-0.35) (0.6,-0.45) (0.0,-0.45) (0.9,0.0) (0.3,0.0) (0.0,0.0) Cell Integration Standard Multiple node Figure 4.4 : Comparison percentage errors at the internal point We see that from Figure 4.4 that the multiple nodes do not seem to help much because the domain boundary is smooth. The boundary does not contain real corners. However, the solution using the method is quite better compared with the others.
Results for different functions b b( x, The problem in Example will now be presented for different known function b( x, In all applications the same problem geometry will be used, that given in figure 4.3 with homogeneous condition u 0. ( a ) The case u x This case and another in this section will be modeled using the element geometry shown in Figure 4.3. The governing equation is The exact solution is given by u x (4.6) x x u y (4.7) 7 4 which satisfies the boundary condition u 0 on and produces x 3x q y (4.8) 4 Results for both the DRBEM program with varieties of multiple nodes and the cell integration are given in Table 4.3. Table 4.3 : Internal solution for the equation u x Internal Cell MULDRM Program (O. of multiple nodes) Exact point Integration 0 4 8 6 solution (.5, 0.00) 0.76 0.79 0.84 0.84 0.84 0.87 (., 0.35) 0.7 0.80 0.8 0.83 0.83 0.77 (0.6,-0.45) 0.8 0.3 0.5 0.5 0.5 0. (0.0,-0.45) 0.000 0.000 0.000 0.000 0.000 0.000 (0.9, 0.00) 0.00 0.05 0.09 0.09 0.09 0.05 (0.3, 0.00) 0.08 0.080 0.085 0.085 0.085 0.083 (0.0, 0.00) 0.000 0.000 0.000 0.000 0.000 0.000 Percentage errors of potential at the internal points are shown in Figure 4.5
error (%) DRBEM 3/8/09 Percetage errors of potential 7. 0 0 0 6. 0 0 0 5. 0 0 0 4. 0 0 0 3. 0 0 0. 0 0 0. 0 0 0 0. 0 0 0 (. 5, 0. 0 0 ) (., 0. 3 5 ) ( 0. 6, - 0. 4 5 ) ( 0. 9, 0. 0 0 ) ( 0. 3, 0. 0 0 ) C e ll in te g r a tio n S ta n d a r d M u ltip le n o d e s ( b ) The case u x In this case the governing equation is The exact solution is given by u x (4.9) x 50x 8y 33.6 y 4 u (4.0) 46 which again satisfies the boundary condition u 0 on and produces 3 x 50x 96xy 83.x 96x y 3y 83. yy 3 q (4.) 46 46 Results for the DRBEM program with varieties of multiple nodes and the exact solution are given in Table 4.4. 3
Error (%) DRBEM 3/8/09 Table 4.4 : Internal solution for the equation u x Internal MULDRM Program (o. of multiple nodes) Exact point 0 4 8 6 solution (.5,0.0) 0.649 0.65076 0.6504 0.6493 0.59 (.,-0.35) 0.8793 0.966 0.974 0.9798 0.0 (0.6,-0.45) 0.349 0.3553 0.35 0.35345 0.43 (0.0,-0.45) 0.0935 0.0904 0.09096 0.0900 0.03 (0.9,0.0) 0.3553 0.36444 0.3645 0.36486 0.40 (0.3,0.0) 0.40506 0.49 0.47 0.45 0.5 (0.0,0.0) 0.4800 0.659 0.6580 0.6689 0.36 Percentage errors of potential at the internal points are shown in Figure 4.6 Percentage errors of potential 0 8 6 4 0 (.5,0.0) (.,-0.35) (0.6,-0.45) (0.0,-0.45) (0.9,0.0) (0.3,0.0) (0.0,0.0) Standard Multiple node Figure 4.6 : Comparison percentage errors at the internal points We see from Figure 4.6 that the multiple nodes do not seem to help much in case of the smooth boundary as we mentioned in the first problem of Example. 4
5. Computational results of the DRBEM program We have implemented the program DRBEM to solved the Poisson equations in the form u u u p( x, p ( x, u p3( x, p4 ( x, x y (5.) using both of the radial basis function f r and f r log r with linear term. The previous DRBEM program is separated to solve the boundary solution and use the result to evaluate the internal solution. On the other hand this program solve the whole system of equations to obtain the boundary solution and the internal solution in the same time. The program is tested with some problems in 6 cases: Case : p p p p 0 3 4 p3 p4 p ( x, x x y p3 p4 p( x, 5e p3 p4 p ( x, p p4 p 3( x, p p 3( x,, p4 ( x, Case : p 0, Case 3: p 0, Case 4: p 0, Case 5: p 0, Case 6: p 0, We are going to present the result in each case. Case : p p p p 0 3 4 This case is a kind of Laplace equation. We test the program with a previous problem examined by LIBEM and MULBEM program. Consider the potential problem u 0 (5.) in a square plate 0 x 6, 0 y 6 with solution u 300 50x. We partition the boundary into elements without multiple nodes. The boundary values of function and normal derivative are shown in Figure 5.. 5
(0,6) y q = 0 (6,6) u = 300 u = 0 (0,0) (6,0) q = 0 x Figure 5.: Boundary conditions for Problem The results are compared with those of using LIBEM and MULBEM program and shown in Table 5.. Table : The potential at the internal points Point x y LIBEM MULBEM DRBEM Exact f = +r f=tps solution.00.00 00.86 00.044 00.393 00.393 00.000 4.00 4.00 99.909 99.957 99.607 99.607 00.000 3.00 4.00 00.980 00.044 00.393 00.393 00.000 4 3.00 3.00 50.38 50.000 50.000 50.000 50.000 Case : p p p 0, 3 4 p ( x x, Consider the Poisson equation on the elliptical domain as shown in Figure 5. u x (5.3) 6
(0,) 5 4 3 6 0 (-,0) 9 (,0) 8 3 4 5 (0,-) 6 7 : Boundary node : Internal point Figure 5.: Discretization of the boundary into 6 elements and 7 internal points in elliptical domain The elliptical section shown in Figure has a semi-maor axis a = and a semiminor axis b =. The equation of the ellipse is x y (5.4) The boundary condition is the Dirichlet condition with u = 0 on the boundary. The exact solution is given by x 50x 8y 33.6 y 4 u (5.5) 46 which again satisfies the boundary condition u 0 on and produces 3 x 50x 96xy 83.x 96x y 3y 83. yy 3 q (5.6) 46 46 The potential solution at the internal nodes are shown in Table 5. and the normal derivatives at the boundary are shown in Table 5.3. 7
Table 5.: The potential at the internal nodes of the problem Point x y DRBEM DRBEM Exact f = +r f=tps f = +r f=tps solution.50 0.00 0.6345 0.780 0.6348 0.7800 0.60.0-0.35 0.7888 0.445 0.800 0.44460 0.0 3 0.60-0.45 0.348 0.76643 0.3574 0.7664 0.44 4 0.00-0.45 0.087549 0.380 0.087580 0.3808 0.04 5 0.00 0.00 0.45 0.7048 0.445 0.7050 0.37 6 0.30 0.00 0.3887 0.8456 0.3878 0.84566 0.5 7 0.90 0.00 0.33807 0.684 0.33798 0.6837 0.40 Table 5.3: The normal derivative at the boundary of the problem Point x y DRBEM DRBEM Exact f = +r f=tps f = +r f=tps solution - 0-0.90-0.853-0.90-0.853-0.950-0.8476-0.3868-0.986-0.93-0.986-0.93-0.947 3 -.44-0.707-0.855-0.8-0.855-0.8-0.790 4-0.76537-93880 -0.435-0.40-0.435-0.40-0.4 5 0 - -0.0-0.68-0.0-0.68-0.08 Case 3: p p p 0, 3 4 p( x, 5e x y Consider the Poisson equation u x y 5 e (5.7) on the quarter of a unit-circle x y. Discretization and boundary condition are shown in Figure 5.3. 8
9 (0,) 8 0 7 q e x y 6 u x y e (0,0) (,0) 3 4 5 q e x y Boundary node Internal node Figure 5.3: Discretization and boundary condition of the mixed problem The potential solutions at the internal nodes are shown in Table 5.4 and the normal derivatives at the boundary are shown in Table 5.5. Table 5.4: The potential at the internal nodes of the problem Point x y DRBEM DRBEB Exact f = +r f=tps f = +r f=tps solution 0.75 0.5 5.796 5.74004 5.790 5.7407 5.754 0.50 0.5 3.454453 3.458933 3.454437 3.458954 3.490 3 0.5 0.5.074674.06553.074635.06583.7 4 0.5 0.50.697.68564.6975.68580.78 5 0.50 0.50 4.44486 4.4549 4.444830 4.45407 4.48 6 0.75 0.50 7.350586 7.3607 7.35056 7.36695 7.389 7 0.50 0.75 5.737537 5.736434 5.73767 5.737045 5.755 9
Table 5.5: The normal derivative on the boundary of the problem Point x y DRBEM DRBEB Exact f = +r f=tps f = +r f=tps solution 0 4.8634 4.8000 4.86368 4.809980 4.778 0.9388 0.38683 0.747750 0.68550 0.74780 0.685450 0.75 3 0.70707 0.70707 7.858360 7.8670 7.858580 7.8560 7.696 4 0.38683 0.9388 9.06996 9.057966 9.06883 9.057836 9.48 5 0-0.90696-0.895357-0.90750-0.895443.78 Case 4: p p p 0, p ( x, 3 4 Consider the Poisson equation u u (5.8) on the boundary as shown in Case. Since homogeneous boundary condition will result in the trivial solution u q 0 at all nodes, a non-homogeneous condition has to be used, for example u sin x (5.9) The solutions of the program are also compared with those of the original program and the exact solutions and are shown in Table 5.6. Table 5.6: The potential at the internal nodes of the problem 30
Point x y Original DRBEM Exact f = +r f=+r+r^+r^3 f = +r f=tps solution.50 0.00 0.994 0.995 0.996 0.997 0.997.0-0.35 0.98 0.93 0.98 0.93 0.93 3 0.60-0.45 0.56 0.566 0.56 0.564 0.565 4 0.00-0.45 0.000 0.000 0.000 0.000 0.000 5 0.90 0.00 0.780 0.784 0.780 0.78 0.783 6 0.30 0.00 0.94 0.96 0.94 0.95 0.95 7 0.00 0.00 0.000 0.000 0.000 0.000 0.000 Case 5: : p p p 0, p ( x, 4 Consider the Poisson equation 3 u u (5.0) x on the boundary as shown in Case. Since homogeneous boundary condition will result in the trivial solution u q 0 at all nodes, a non-homogeneous condition has to be used, for example u x e (5.) The solutions of the program are also compared with those of the original program and the exact solutions and are shown in Table 5.7. Table 5.7: The potential at the internal nodes of the problem Point x y Original DRBEM Exact f = +r f=+r+r^+r^3 f = +r f=tps solution.50 0.00 0.9 0.4 0.9 0.5 0.3.0-0.35 0.307 0.74 0.307 0.305 0.30 3 0.60-0.45 0.555 0.53 0.555 0.553 0.549 4 0.00-0.45.003.006.003.005.000 5 0.90 0.00 0.4 0.363 0.4 0.40 0.406 6 0.30 0.00 0.745 0.75 0.745 0.745 0.74 7 0.00 0.00.00.00.00.005.000 3
Case 6: p p 0, p x,, p ( x, 3( 4 Consider the Poisson equation u u u (5.) x y on the boundary as shown in Case. Since homogeneous boundary condition will result in the trivial solution u q 0 at all nodes, a non-homogeneous condition has to be used, for example u x y e e (5.3) The solutions of the program are also compared with those of the original program and the exact solutions and are shown in Table 5.8. Table 5.8: The potential at the internal nodes of the problem Point x y Original DRBEM Exact f = +r f=+r+r^+r^3 f = +r f=tps solution.50 0.00.3.4.3.5.3.0-0.35.74.669.74.77.70 3 0.60-0.45.07.057.07.09.7 4 0.00-0.45.557.547.557.560.568 5 0.90 0.00.400.345.40.404.406 6 0.30 0.00.73.69.73.737.74 7 0.00 0.00.989.963.989.997.000 6. Conclusion We cannot distinguish the results computed by the standard linear elements and the multiple node approach in problems containing smooth boundary. The MULDRM program which transform Poisson equation to Laplace one works well but available only in case of the right hand side function is a position function. The DRBEM program which separates the execution of solving boundary and internal solution works as well as the DRBEM one which solves the whole solution in the same time. However the DRBEM is suitable to modify to solve the non-linear problem which is the future work. In problem containing corner domain, the normal derivative solution at the boundary is still poor. It is our purpose to modify the program to resolve this problem. For approximation of functions, the thin plate spline works better than the linear function f r for all cases. 3
References Golberg, M.A. (994) The theory of radial basis functions applied to the BEM for inhomogrneous partial differential equations. Boundary Elements Communications 5, 57-6. Golberg, M.A. (996) The method of fundamental solutions for Poisson's equation. Engineering Analysis with Boundary Elements 6, 05-3. Golberg, M.A., Chen, C.S., Bowman, H. and Power, H. (998) Some comments on the use of radial basis functions in the dual reciprocity method. Computational Mechanics, 6-69. Karur, S.R. and Ramachandran, P.A. (994) Radial basis function approximation in the dual reciprocity method. Mathematical and Computer Modelling 0, 59-70. Partridge, P.W. and Brebbia, C.A. (989) Computer implementation of the BEM dual reciprocity method for the solution of Poisson type equations. Software for Engineering Workstations 5, 99-06. Partridge, P.W. and Brebbia, C.A. (990) Computer implementation of the BEM dual reciprocity method for the solution of general field equations. Communications in Applied umerical Methods 6, 83-9. Partridge, P.W., Brebbia, C.A. and Wrobel, L.C. (99) The dual reciprocity boundary element method, Southampton: Computational Mechanics Publications. Powell, M.J.D. (994) The uniform covergence of thin plate spline interpolation in two dimension. umerical Mathematics 68, 07-8. ====================================== 33