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Meta 2010 & FEM 2010 – Rome, 13-15 December 2010 A Comparison between Hybrid Methods: FEM-BEM versus FEM-DBCI G. Aiello, S. Alfonzetti, S. A. Rizzo, and N. Salerno Università di Catania, Dipartimento di Ingegneria Elettrica, Elettronica e dei Sistemi (DIEES), Catania, Italy – E-mail: alfo@diees.unict.it In the literature several methods have been devised to enable the Finite Element Method (FEM) to solve static and quasi-static electromagnetic field problems in openboundary domains. Among these there are the hybrid FEM/BEM (Boundary Element Method) method [1], and the hybrid FEM-DBCI (Dirichlet Boundary Condition Iteration) method proposed by the authors [2]. This paper compares these two hybrid methods by referring to simple electrostatic field problems. Consider an electrostatic system made of voltaged conductors, dielectric objects and charge distributions embedded in air. In the FEM-BEM method a truncation boundary �F enclosing the system is introduced. On �F an unknown Neumann condition �r�v/�n=q is assumed. The FEM-BEM leads to the global system [1]: � A A F 0 � � v � �b 0 � � t � � � � � � � A F A FF C � � v F � � 0 � (1) �� 0 H � G�� q �� 0 � F � where: v and vF are the vectors of the unknown values of the potential v in the nodes inside the domain and on �F, respectively, A, AF and AFF are sparse matrices of coefficients, b0 is the known term vector due to the conductor potentials and sources, C is a sparse matrix of coefficients, and qF is the vector of the unknown values of q, evaluated in nodes other than those of v [1]. In the FEM-DBCI method a Dirichlet boundary condition is assumed on �F. The global system is [2]: � A A F � � v � �b0 � � � � � � � � (2) �� G' H' � �v F � � 0 � Comparing the two methods the following considerations can be made. First, by analyzing the dimensions of the various matrices, it can be shown that FEM-DBCI requires less memory than FEM-BEM. Moreover, the greater complexity of (1) with respect to (2) makes FEM-BEM more time-consuming than FEM-DBCI. From the point of view of accuracy, it can be noted that in FEM-DBCI a numerical derivative of the potential is performed on the integration curve, whereas this is not necessary in FEM-BEM. FEM-BEM can therefore be expected to give more accurate results than FEM-DBCI. These considerations have been verified by means of a set of examples, exhibiting analytical solutions. References [1] S. Alfonzetti, N. Salerno, “A non-standard family of boundary elements for the hybrid FEM-BEM method,” IEEE Trans. Magn., 45, 1312-1315, 2009. [2] G. Aiello, S. Alfonzetti, “Charge iteration: a procedure for the finite-element computation of unbounded electrical fields,” Int. J. Num. Meth. Engng, 37, 4147-4166, 1994. 44
Meta 2010 & FEM 2010 – Rome, 13-15 December 2010 A Comparison of Direct Methods for the Solution of Finite Element Systems on Shared Memory Computers Giuseppe Borzì University of Messina, Department of Civil Engineering Messina, Italy – E-mail: gborzi@ieee.org The numerical solution of electromagnetic problems by means of the finite element method involves the construction of a linear algebraic system and its solution. When the order of the system is very high, the solution is achieved by means of iterative solvers such as those of the Lanczos family [1], Multigrid [2] or Algebraic Multigrid [3]. The convergence characteristics of iterative solvers are not well understood, except for a few special cases, like Symmetric Positive Definite matrices for Lanczos based solvers. Moreover, for complex linear systems the convergence theory is even less understood than for real linear systems. So, for small and medium size linear systems, direct solvers for sparse matrices can be more suitable. Unlike iterative solvers, direct ones can be used as ‘black boxes’, that is to say, the user does not need to give parameters such as the convergence tolerance, or preconditioning parameters. Most of these parameters are chosen heuristically, and an unwise choice may lead wrong results or a breakdown of the iteration. The downside of direct methods is their higher memory and CPU usage when compared with iterative solvers, but with the commercial availability of multi core/threaded CPUs with huge memory this is no more a serious drawback. In this paper, some public available direct solvers for sparse matrices, such as those included in the suitesparse package [4], superlu and superlu_mt [5-6] and spooles [7] are compared on some test matrices resulting from the finite element discretization of electromagnetic problems. References [1] G. H. Golub and C. F. van Loan, Matrix Computations, 3rd Ed., The John Hopkins University Press, Baltimore, USA, 1996 [2] W. L. Briggs, A Multigrid Tutorial, SIAM Books, Philadelphia, USA, 1987 [3] K. Stueben, “Algebraic multigrid (AMG): An introduction with applications,” GMD - Forschungszentrum Informationstechnik GmbH, Tech. Rep. 53, Sankt Augustin, Germany, Mar. 1999 [4] T. A. Davis, Direct Methods for Sparse Linear Systems, SIAM Books, Philadelphia, USA, 2006 [5] J. W. Demmel, S. C. Eisenstat, J. R. Gilbert, X. S. Li, and J W. H. Liu, “A supernodal approach to sparse partial pivoting,” SIAM J. Matrix Analysis and Applications, 20, 720-755, 1999 [6] J. W. Demmel, J. R. Gilbert, and X. S. Li, “An Asynchronous Parallel Supernodal Algorithm for Sparse Gaussian Elimination,” SIAM J. Matrix Analysis and Applications, 20, 915-952, 1999 [7] C. Ashcraft, and R. Grimes, “SPOOLES: An object-oriented sparse matrix library,” Proceedings of the Ninth SIAM Conference on Parallel Processing, San Antonio, Texas, USA, March 22-24, 1999. 45
Page 1 and 2: Meta 2010 & FEM 2010 - Rome, 13-15
Page 3 and 4: Meta 2010 - Committees Chairmen Met
Page 7 and 8: Meta 2010 Foreword Ladies and Gentl
Page 9 and 10: FEM 2010 Foreword Ladies and Gentle
Page 25 and 26: Session FEM-1 Magnetic device model
Page 43: Session FEM-3 Methods and solvers M
Page 47 and 48: Session FEM-4 Design and applicatio
Page 67 and 68: Author Index Meta 2010 & FEM 2010 -

Abstracts - Dipartimento di Elettronica Applicata

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