Copyright 009 by ABCM January 04-08, 010, Foz do Iguaçu, PR, Brazil ADVANCES AT TELLES METHOD APPLIED IN SCIENTIFIC VISUALIZATION Carlos Andrés Reyna Vera-Tudela, candres@ufrrj.br Universidade Federal Rural do Rio de Janeiro, Departamento de Matemática, Caixa Postal 74517, CEP 3890-971, Seropédica, RJ, Brasil Marlucio Barbosa, marlucio@coc.ufrj.br Edivaldo Figueiredo Fontes Junior, fontesjunior@coc.ufrj.br José Claudio de Faria Telles, telles@coc.ufrj.br COPPE/Universidade Federal do Rio de Janeiro, Programa de Engenharia Civil, Caixa Postal 68506, CEP 1945-970, Rio de Janeiro, RJ, Brasil Abstract. Accurate numerical integration of line integrals is of fundamental importance for any reliable implementation of the boundary element method. The Telles transform introduces an efficient means of computing singular or nearly singular integrals currently found in two-dimensional, axisymmetric and three-dimensional applications. Recently, the authors started to develop a dedicated software for scientific visualization applications. The initial procedure was writen in Java, employing the Visualization Toolkit (VTK. VTK is an open-source, freely available software for 3D computer graphics, image processing, and visualization; this consists of a C++ class library and several interpreted interface layers including Tcl/Tk, Java, and Python. However, problems can occur if abrupt changes of stress values are to be accomodated since this needs special treatment when it comes to scientific visualization and numerical accuracy. The Telles transform is a good alternative to solve the problem of numerical accuracy. Recent research suggests that the use of the transformation for internal points near the boundary and in regions with severe stress gradients can improve the look and consistency of scientific visualization representing the solution to the problem. Thus, the present work discusses some advances in the Telles transformation for efficient use in scientific visualization. Examples are included to illustrate details in the implementation and the efficiency of the method. Keywords: Boundary Element Method, Telles transform, Scientific Visualization 1. INTRODUCTION The evolution of computational tools is fundamental for the development of computer software for new applications dealing with more complex problems. Traditional algorithms must be revised and, sometimes, modified to use in high performance applications. For example, parallel processing is growing fast and practically there ere no limitations in the use of computer memory. Scientific visualization is important for science and engineering since the observation of physical behavior is fundamental for proper analysis of the problem. The authors are working in the development of software to solve, initially, D elastostatic problems with the Boundary Element Method (BEM (Brebbia, et al., 1984. The software MEMEC was developed in its original version in Fortran, but for an efficient visualization is necessary to use Object Oriented Language. Hence, the choice was Java with the Visualization Toolkit (VTK (Schroeder, et al., 004. The traditional algorithms were modified to the new requirements of computational efficiency and visualization. New algorithms had to be implemented; e.g., mesh generator and/ or interaction BEM-FEM. Initial tests indicated how efficient and friendly the software had become, relative error decreased and visualization of the problem could be accomplished. The original physical problem, comprising loads and boundary conditions could also be shown. An independent visualization of stresses and displacements had to be produced, including final deformations too. It is now possible to observe 3D visualizations and move the body in all directions. Some algorithms, however, need to be analyzed in more detail, such as the case of the Telles Transform. Thus, this paper studies the effect of the singular integrals in this context, the numerical efficiency and their influence in scientific visualization. A classical example is presented and the results are compared with a traditional BEM result, the analytical solution and Finite Element result.. THE BOUNDARY ELEMENT METHOD The Boundary Element Method (BEM (Brebbia, et al., 1984 is a numerical method for solving linear partial differential equations expressed as integral equations. In the case of elastostatic problems, without body forces, the direct formulation leads to equation of the form: C ( ( x p ( x, ξ dγ p ( x u ( x, ξ dγ = 0 ξ, x Γ ui ( ξ + u j ξ (1 Γ Γ j
Copyright 009 by ABCM January 04-08, 010, Foz do Iguaçu, PR, Brazil where the first integral is in Cauchy principal value sense. In this equation, Γ is the boundary of the body, ξ is the source point, x is a field point, C is a coefficient which depends on the boundary geometry at ξ, u and p are the displacement and traction components, and u and p are the fundamental displacements and tractions at x, in j direction, due to a unit load at ξ in i direction. The discretization of the body, restricted only to its boundary, is accomplished with the aid of quadratic boundary elements. Thus, Eq. (1 can be written in discrete form, after a point collocation procedure. In matrix form and amenable to prescribe the boundary conditions, the resulting system of equations is written as, Hu = Gp ( 3. THE TELLES TRANSFORMATION The boundary element method requires integration of singular kernels, present in the fundamental solution and its derivatives. Depending on the actual order and boundary dimension (i.e.; 3-D or -D, there are three distinguishable singularities: (a weak singularity (integrable in the ordinary Riemann sense, (b strong singularity (integrable in Cauchy principal value sense, (c hypersingularity (integrable in Hadamard finite part sense. If the source point is outside the element, the integrals are actually non-singular. Nevertheless, from the point of view of numerical integration, one should devote great attention also to the evaluation of nearly singular integrals, which are potentially cumbersome with the source point located close to the element being integrated. This is so because the integrands produce gradient profiles varying in strong non-uniform manner over the elements. Hence, standard integration quadrature fails to produce acceptable results. (Sladek, et al., 001 In Telles (1987 this problem is discussed in detail and efficient means to compute singular or nearly singular integrals are introduced. Initially is analyzed the use of a second-degree polynomial transformation and is verified that the use of a third-degree polynomial transformation can be used in preference to the second-degree transformation. Special attention should be given to the parametric optimization presented in this work (e.g.; Telles and Oliveira, 1994. Consider the integral = 1 1 f ( d I (3 in which f ( is singular at a point. One can always choose a third-degree relation of the form: ( (4 3 γ = a γ + bγ + cγ + d so that, in order to define a, b, c and d, the conditions below apply: (1 = 1 ( 1 = 1 d ( γ = dγ d dγ = 0 = r J (5 where r is a function of D (distance of the source point to the axis. The functional dependence of r with respect to D can be obtained to produce a minimum quadrature error. Thus, the transformation is dependent on r : j j
Copyright 009 by ABCM January 04-08, 010, Foz do Iguaçu, PR, Brazil J = 3a γ + bγ + c a = b = c = q = p = 3 ( 1 r / Q 3( 1 r ( r + 3γ d = b Q = 1+ 3γ γ = 3 3 [ q + ( q + p ] + q ( q + p 3 3 1 ( 1+ r 1 ( 1+ r γ / Q / Q ( 3 r [ ] 3 1 1 + r 1+ r [ 4r ( 1 r + 3( 1 ] + 1+ r (6 For general usage, the first step is the computation of the normalized distance D as a function of distance from singular point to element. The following formula should be used: D R l R MIN (minimum MIN = (7 in which l is the real space distance between the two end points of the integration domain. Finally, the following expressions were found to determine r : r r r = 0.85 + 0.4ln ( D, = 0.893 + 0.083 ln ( D, = 1, 0.05 D 1.3 1.3 D 3.618 3.618 D (8 4. NUMERICAL EXAMPLES As a case study, the intrinsic influence of the numerical technique adopted for scientific visualization is analyzed. The original algorithm in Fortran, was reviewed and, in many cases, modified to adapt to other languages, such as Java, that is oriented to object. Thus, special considerations had be made, principally at singular points where a high gradient of colors is found. The Telles Transformation allows for a study of the influence of this singularity in the visualization of the problem. The actual problem is a rectangular plate with a circular hole (Timoshenko and Goodier, 1970, the geometry and the material properties are shown in Fig. (1.
Copyright 009 by ABCM January 04-08, 010, Foz do Iguaçu, PR, Brazil Figure 1. Rectangular plate with a circular hole This problem is solved with the MEMEC software (Barbosa, et al., 008; the boundary discretization was defined with 00 linear elements and 104 internal points. It is important to note that a high number of internal points is necessary for efficient scientific visualization. However, the processing of such points is fastest due to effective processing techniques. First, a solution was obtained with a BEM implementation without advanced techniques of integration. The visualization of results for stress in x direction is shown in Fig. (; as can be observed, some of the internal points are near boundary points P1 and P (defined in Fig. (1 and the final results indicate errors when compared with the analytical result (Timoshenko and Goodier, 1970. This error leads to misinterpretations since the maximum stresses are located in incorrect positions. To solve this problem, the Telles Transformation was implemented in the BEM algorithm and the test has been repeated. The visualization of the results is presented in Fig. (3 for x direction stresses. When results are compared with the analytical stresses, one can observe a relative error of 0.33% at the P1 and P points. These points present the highest gradient of stress in x direction. Figure. Stress in x direction without the Telles Transformation procedure.
Copyright 009 by ABCM January 04-08, 010, Foz do Iguaçu, PR, Brazil Figure 3. Stress in x direction with the Telles Transformation procedure. Comparison of results obtained with the MEMEC software and the Finite Element Method (FEM: In order to validate the MEMEC software, results are compared with the solution obtained with a FEM implementation. The parameter calculated is the stress in x direction, Figs (3 and (4, obtained with MEMEC and FEM respectively. The discretization for the FEM implementation is shown in Fig. (5. Figures. (6 and (7 show displacements in x direction for finite elements and MEMEC respectively. The stress concentration factor (K at point P1 (Fig. 1 has been determined in order to compare with the analytical solution. Table (1 presents a summary of these results. Figure 4. Stress in x direction with FEM Figure 5. FEM grid with bi-linear quadrilateral elements Figure 6. FEM displacements in x direction Figure 7. BEM displacements in x direction with the Telles transformation.
Copyright 009 by ABCM January 04-08, 010, Foz do Iguaçu, PR, Brazil Table (1 shows that the error in the FEM implementation is 17.66%, whereas for the MEMEC software (BEM formulation this inaccuracy is equal to 0.33%. This result is even more evident when the actual CPU time is compared, since for the FEM implementation the computer effort has been greater than for the MEMEC software. Table 1. Stress concentration at points P1 and P Method Number Number of K Exact Solution Error (% of Nodal elements points BEM 04 00 3.01 3.0 0.3333 FEM 870 774.47 3.0 17.666 5. CONCLUSION This paper presents the Telles transformation for scientific visualization applications. It is verified that in conjunction with efficient computational algorithms, excellent results can be obtained. The comparison with a classical finite element solution and the analytical result is seen very important when error versus discretization is observed. The MEMEC software is a tool that the authors have been developing since the early stages and further applications are also to be included in the near future. 6. ACKNOWLEDGEMENTS This work has been carried out with the support of the FAPERJ and the Fundo Setorial de Infra-Estrutura (CT- INFRA with the MCT/CNPq Programa Primeiros Projetos and Apoio às Instituições de Ensino e Pesquisa Sediadas no Estado do Rio de Janeiro; the Universidade Federal Rural do Rio de Janeiro with Grupos Emergentes project. 7. REFERENCES Barbosa, M., Fontes Junior, E.F., Vera-Tudela, C.A.R. and Telles, J.C.F., 008. O Método dos Elementos de Contorno e a Visualização Científica à Resolução de Problemas da Mecânica Computacional, Proceedings of the XXXI Congresso Nacional de Matemática Aplicada e Computacional, Belém PA, Brasil. Brebbia, C.A., Telles, J.C.F. and Wrobel, L.C., 1984. Boundary Element Techniques: Theory and Applications in Engineering, Springer, Berlin. Schroeder, W., Ken, M. and Lorensen, B., 004. The Visualization Toolkit, Kitware, Inc. Publishers. Sladek, V., Sladek, J. and Tanaka, M., 001. Numerical Integration of Logarithmic and Nearly Logarithmic Singularity in BEMs, Applied Mathematical Modeling, Vol. 5, No. 11, pp. 901-9. Telles, J.C.F., 1987. A Self-Adaptive Co-ordinate Transformation for Efficient Numerical Evaluation of General Boundary Element Integrals, International Journal for Numerical Methods in Engineering, Vol. 4, pp. 959-973. Telles, J.C.F. and Oliveira, R.F., 1994. Third Degree Polynomial Transformation for Boundary Element Integrals: Further Improvements, Engineering Analysis with Boundary Elements, Vol. 13, pp. 135-141. Timoshenko, S.P. and Goodier, J.N., 1970. Theory of Elasticity, Mc Graw-Hill, New York, 567 p. 8. RESPONSIBILITY NOTICE The authors are the only responsible for the printed material included in this paper.