A fast isogeometric BEM for the three dimensional Laplace- and Helmholtz problems

We present an indirect higher order boundary element method utilising NURBS mappings for exact geometry representation and an interpolation-based fast multipole method for compression and reduction of computational complexity, to counteract the problems arising due to the dense matrices produced by boundary element methods. By solving Laplace and Helmholtz problems via a single layer approach we show, through a series of numerical examples suitable for easy comparison with other numerical schemes, that one can indeed achieve extremely high rates of convergence of the pointwise potential through the utilisation of higher order B-spline-based ansatz functions

O método rápido dos elementos de contorno com expansão em multipolos para problemas elástico- anisotrópicos em duas dimensões

Tese (doutorado)—Universidade de Brasília, Faculdade de Tecnologia, Departamento de Engenharia Mecânica, 2019.Este trabalho apresenta uma formulação do método dos elementos de contorno com expansão em multipolos rápidos (FMBEM) aplicado à análise de problemas elásticos anisotrópicos em duas dimensões. Equações integrais são obtidas usando a identidade de Somigliana. Soluções fundamentais de deslocamento e tração obtidas pelo formalismo de Lekhnitskii são usadas para transformar equações integrais de domínio em equações integrais de contorno. O contorno é discretizado em pequenos pedaços do contorno, chamados de elementos de contorno, considerando que deslocamentos e trações são constantes ao longo de cada elemento de contorno. As Integrais são divididas em campo próximo e campo distante. Campo próximo, quando os pontos fontes e os elementos de integração estão próximos, são tratados como no método do elemento de contorno padrão, ou seja, integrando ao longo do elemento e considerando a interação entre pontos fontes (nós) e os elementos. Por outro lado, no campo distante, quando os pontos fontes e os elementos de integração estão longes, o método dos multipolos rápidos é aplicado. Nesse caso, a solução fundamental é expandida em série de Laurent e a interação nó a nó é substituída por uma interação célula a célula. As células são geradas por uma decomposição hierárquica do domínio usando o algoritmo quad-tree. Diferentes operações de multipolos rápidos são usadas para tirar vantagem da decomposição hierárquica do domínio e das expansões das soluções fundamentais. As matrizes de influência nunca são explicitamente obtidas e o produto matriz-vetor pode ser realizado com complexidade linear. O sistema linear é resolvido por um método iterativo. Nesta tese o método dos resíduos mínimos generalizados (GMRES) foi escolhido com base em trabalhos anteriores. Uma matriz de precondicionamento é usada para reduzir o número de iterações para obter um resultado com a precisão especificada. A eficácia e eficiência na solução de problemas de larga escala são discutidas. A formulação apresentada nesta tese é baseada em uma representação de variáveis complexas dos integrandos, similar à formulação desenvolvida anteriormente para problemas potenciais (escalares). A validação é realizada através da comparação dos resultados obtidos pelas duas formulações: o método dos elementos de contorno padrão e o método dos elementos de contorno com expansão em multipolos rápidos. Analisa-se a influência do número de termos da expansão em séries no cálculo das soluções fundamentais e das matrizes de influência. O custo computacional de ambas as formulações é comparado. Exemplos numéricos são apresentados para demonstrar a eficiência, a precisão e os potencial do método dos elementos de contorno com expansão em multipolos rápidos para resolver problemas elásticos anisotrópicos de larga escala, ou seja, com dezenas de milhares de graus de liberdades.Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES).This work presents a formulation of the Fast Multipole Boundary Element Method (FMBEM) applied to the analysis of anisotropic elastic problems in two dimensions. Integral equations are obtained using the Somigliana identity. Displacement and traction fundamental solutions obtained by the Lekhnitskii formalism are used in order to transform domain integral equations into boundary integral equations. The boundary is discretized into small boundary pieces, called boundary elements, considering that displacements and tractions are constants along each boundary element. Integrals are divided into near and far field. Near field, when source points and integration elements are near, are treated as in standard boundary element method, i.e., integrating along the boundary and considering the interaction between source points (nodes) and elements. On the other hand, in far field, the fast multipole method is applied. In this case, the fundamental solution is expanded into Laurent series and the node to node interaction is substituted by a cell to cell interaction. Cells are generated by an hierarchical decomposition of the domain using the quad-tree algorithm. Different multipole operations are used in order to take advantage of the hierarchical decomposition of the domain and the expansions of fundamental solutions. Influence matrices are never explicitly obtained and the matrix-vector product can be carried out with linear complexity. The linear system is solved by an iterative method, in this problem the generalized minimum residue method (GMRES) was chosen based on previous work. A preconditioner matrix is used in order to reduce the number of iterations to obtain the specified accuracy. The effectiveness and efficiencies in solving large-scale problems are discussed. The formulation presented in this thesis is based on a complex-variable representation of the kernels, similar to the formulation developed earlier for potential (scalar) problems. Validation is carried out through comparison of results obtained by both formulations: the standard boundary element method and the fast multipole boundary element method. It is analyzed the influence of the number of terms in the series expansion on the computation of fundamental solution and influence matrices. The computational cost of both formulations are compared. Numerical examples are presented to further demonstrate the efficiency, accuracy and using the Somigliana identity. Displacement and traction fundamental solutions obtained by the Lekhnitskii formalism are used in order to transform domain integral equations into boundary integral equations. The boundary is discretized into small boundary pieces, called boundary elements, considering that displacements and tractions are constants along each boundary element. Integrals are divided into near and far field. Near field, when source points and integration elements are near, are treated as in standard boundary element method, i.e., integrating along the boundary and considering the interaction between source points (nodes) and elements. On the other hand, in far field, the fast multipole method is applied. In this case, the fundamental solution is expanded into Laurent series and the node to node interaction is substituted by a cell to cell interaction. Cells are generated by an hierarchical decomposition of the domain using the quad-tree algorithm. Different multipole operations are used in order to take advantage of the hierarchical decomposition of the domain and the expansions of fundamental solutions. Influence matrices are never explicitly obtained and the matrix-vector product can be carried out with linear complexity. The linear system is solved by an iterative method, in this problem the generalized minimum residue method (GMRES) was chosen based on previous work. A preconditioner matrix is used in order to reduce the number of iterations to obtain the specified accuracy. The effectiveness and efficiencies in solving large-scale problems are discussed. The formulation presented in this thesis is based on a complex-variable representation of the kernels, similar to the formulation developed earlier for potential (scalar) problems. Validation is carried out through comparison of results obtained by both formulations: the standard boundary element method and the fast multipole boundary element method. It is analyzed the influence of the number of terms in the series expansion on the computation of fundamental solution and influence matrices. The computational cost of both formulations are compared. Numerical examples are presented to further demonstrate the efficiency, accuracy and potentials of the fast multipole BEM for solving large-scale anisotropic elastic problems

Fast integral equation methods for the modified Helmholtz equation

We present a collection of integral equation methods for the solution to the two-dimensional, modified Helmholtz equation, u(\x) - \alpha^2 \Delta u(\x) = 0, in bounded or unbounded multiply-connected domains. We consider both Dirichlet and Neumann problems. We derive well-conditioned Fredholm integral equations of the second kind, which are discretized using high-order, hybrid Gauss-trapezoid rules. Our fast multipole-based iterative solution procedure requires only O(N) or $O(N\log N)$ operations, where N is the number of nodes in the discretization of the boundary. We demonstrate the performance of the methods on several numerical examples.Comment: Published in Computers & Mathematics with Application

A SVD accelerated kernel-independent fast multipole method and its application to BEM

The kernel-independent fast multipole method (KIFMM) proposed in [1] is of almost linear complexity. In the original KIFMM the time-consuming M2L translations are accelerated by FFT. However, when more equivalent points are used to achieve higher accuracy, the efficiency of the FFT approach tends to be lower because more auxiliary volume grid points have to be added. In this paper, all the translations of the KIFMM are accelerated by using the singular value decomposition (SVD) based on the low-rank property of the translating matrices. The acceleration of M2L is realized by first transforming the associated translating matrices into more compact form, and then using low-rank approximations. By using the transform matrices for M2L, the orders of the translating matrices in upward and downward passes are also reduced. The improved KIFMM is then applied to accelerate BEM. The performance of the proposed algorithms are demonstrated by three examples. Numerical results show that, compared with the original KIFMM, the present method can reduce about 40% of the iterating time and 25% of the memory requirement.Comment: 19 pages, 4 figure