Fast multipole method applied to the coupling of elastostatic BEM with FEM

BEM–FEM coupling is desirable for three-dimensional problems involving specific features such as (i) large or unbounded media with linear constitutive properties, (ii) cracks, (iii) critical parts of complex geometry requiring accurate stress analyses. However, for cases with a BEM discretization involving a large number NBEM of degrees of freedom, setting up the BEM contribution to the coupled problem using conventional techniques is an expensive $O(NBEM^{2})$ task. Moreover, the fully-populated BEM block entails a $O(NBEM^{2})$ storage requirement and a $O(NBEM^{3})$ contribution to the solution time via usual direct solvers. To overcome these pitfalls, the BEM contribution is formulated using the fast multipole method (FMM) and the coupled equations are solved by means of an iterative GMRES solver. Both the storage requirements and the solution times are found to be close to O(NBEM). A preconditioner based on the sparse approximate inverse of the BEM block is shown to improve the convergence of the GMRES solver. Numerical examples involving NBEM = O(10^5 − 10^6) unknowns, run on a PC computer, are presented; they include the Eshelby inclusion (as a validation example), a many-inclusion configuration, and a dam structure

Fast multipole method applied to Symmetric Galerkin boundary element method for 3D elasticity and fracture problems

International audienceThe solution of three-dimensional elastostatic problems using the Symmetric Galerkin Boundary Element Method (SGBEM) gives rise to fully-populated (albeit symmetric) matrix equations, entailing high solution times for large models. This article is concerned with the formulation and implementation of a multi-level fast multipole SGBEM (FM-SGBEM) for elastic solid with cracks. Arbitrary geometries and boundary conditions may be considered. Numerical results on test problems involving a cube, single or multiple cracks in an unbounded medium, and a cracked cylindrical solid are presented. BEM models involving up to $10^{6}$ BEM unknowns are considered, and the desirable predicted trends of the elastostatic FM-SGBEM, such as a $O(N)$ complexity per iteration, are verified

Sensitivity Analysis for Coupled Structural-Acoustic System with Absorbing Material Using FEM/BEM

Since the acoustic impedance in water cannot be neglected with respect to the mechanical impedance, the acoustic radiation caused by the vibration of structures in the compressible fluid would react to the structure. Therefore, the fluid-structure interaction needs to be considered. The finite element method is used for structure vibration analysis and the boundary element method for acoustic analysis. Sound absorption materials are used to reduce the scattering sound field in the reference region. The sensitivity analysis of a fully coupled structural-acoustic system is proposed. Numerical tests verify the correctness of the proposed algorithm

The fast multipole method at exascale

This thesis presents a top to bottom analysis on designing and implementing fast algorithms for current and future systems. We present new analysis, algorithmic techniques, and implementations of the Fast Multipole Method (FMM) for solving N- body problems. We target the FMM because it is broadly applicable to a variety of scientific particle simulations used to study electromagnetic, fluid, and gravitational phenomena, among others. Importantly, the FMM has asymptotically optimal time complexity with guaranteed approximation accuracy. As such, it is among the most attractive solutions for scalable particle simulation on future extreme scale systems. We specifically address two key challenges. The first challenge is how to engineer fast code for today’s platforms. We present the first in-depth study of multicore op- timizations and tuning for FMM, along with a systematic approach for transforming a conventionally-parallelized FMM into a highly-tuned one. We introduce novel opti- mizations that significantly improve the within-node scalability of the FMM, thereby enabling high-performance in the face of multicore and manycore systems. The second challenge is how to understand scalability on future systems. We present a new algorithmic complexity analysis of the FMM that considers both intra- and inter- node communication costs. Using these models, we present results for choosing the optimal algorithmic tuning parameter. This analysis also yields the surprising prediction that although the FMM is largely compute-bound today, and therefore highly scalable on current systems, the trajectory of processor architecture designs, if there are no significant changes could cause it to become communication-bound as early as the year 2015. This prediction suggests the utility of our analysis approach, which directly relates algorithmic and architectural characteristics, for enabling a new kind of highlevel algorithm-architecture co-design. To demonstrate the scientific significance of FMM, we present two applications namely, direct simulation of blood which is a multi-scale multi-physics problem and large-scale biomolecular electrostatics. MoBo (Moving Boundaries) is the infrastruc- ture for the direct numerical simulation of blood. It comprises of two key algorithmic components of which FMM is one. We were able to simulate blood flow using Stoke- sian dynamics on 200,000 cores of Jaguar, a peta-flop system and achieve a sustained performance of 0.7 Petaflop/s. The second application we propose as future work in this thesis is biomolecular electrostatics where we solve for the electrical potential using the boundary-integral formulation discretized with boundary element methods (BEM). The computational kernel in solving the large linear system is dense matrix vector multiply which we propose can be calculated using our scalable FMM. We propose to begin with the two dielectric problem where the electrostatic field is cal- culated using two continuum dielectric medium, the solvent and the molecule. This is only a first step to solving biologically challenging problems which have more than two dielectric medium, ion-exclusion layers, and solvent filled cavities. Finally, given the difficulty in producing high-performance scalable code, productivity is a key concern. Recently, numerical algorithms are being redesigned to take advantage of the architectural features of emerging multicore processors. These new classes of algorithms express fine-grained asynchronous parallelism and hence reduce the cost of synchronization. We performed the first extensive performance study of a recently proposed parallel programming model, called Concurrent Collections (CnC). In CnC, the programmer expresses her computation in terms of application-specific operations, partially-ordered by semantic scheduling constraints. The CnC model is well-suited to expressing asynchronous-parallel algorithms, so we evaluate CnC using two dense linear algebra algorithms in this style for execution on state-of-the-art mul- ticore systems. Our implementations in CnC was able to match and in some cases even exceed competing vendor-tuned and domain specific library codes. We combine these two distinct research efforts by expressing FMM in CnC, our approach tries to marry performance with productivity that will be critical on future systems. Looking forward, we would like to extend this to distributed memory machines, specifically implement FMM in the new distributed CnC, distCnC to express fine-grained paral- lelism which would require significant effort in alternative models.Ph.D

Numerical study of magnetic processes: extending the Landau-Lifshitz-Gilbert approach from nanoscale to microscale

The micromagnetic theory describes the magnetic processes in magnetic materials on a microscopic time and space length. Therefore, micromagnetic models are since long employed in the design of for instants magnetic storage media as magnetic tapes and Random Access Memory elements, used in computers. The use of efficient numerical techniques and the availability of powerful computers now make it possible to apply the same micromagnetic models on larger and more complex material systems with the aim of increasing our insight in the experimentally observed magnetic phenomena. In this PhD research, an efficient numerical micromagnetic model is developed that enables the analysis of magnetic processes starting from the nanometer space scale up to the micrometer space scale. Therefore, efficient algorithms are presented on the one hand to simulate the ultra fast dynamics of the magnetic processes as described by the Landau-Lifshitz-Gilbert equation. On the other hand, powerful numerical techniques are developed to evaluate the magnetic fields, characteristic to the micromagnetic description, in a fast way. The developed micromagnetic model is validated extensively in comparative studies with other micromagnetic and macroscopic magnetic material models. Moreover, the model is successfully applied in different magnetic research domains: magnetic switching processes in classical samples with nanometer dimensions are analysed, magnetic domains are studied in structures with order micrometer dimensions and magnetic hysteresis properties are investigated

Surface-Only Simulation of Fluids

Surface-only simulation methods for fluid dynamics are those that perform computation only on a surface representation, without relying on any volumetric discretization. Such methods have superior asymptotic complexity in time and memory than the traditional volumetric discretization approaches, and thus are more tractable for simulation of complex fluid phenomena. Although for most computer graphics applications and many engineering applications, the interior flow inside the fluid phases is typically not of interest, the vast majority of existing numerical techniques still rely on discretization of the volumetric domain. My research first tackles the mesh-based surface tracking problem in the multimaterial setting, and then proposes surface-only simulation solutions for two scenarios: the soap-films and bubbles, and the general 3D liquids. Throughout these simulation approaches, all computation takes place on the surface, and volumetric discretization is entirely eliminated

PARTITION OF UNITY BOUNDARY ELEMENT AND FINITE ELEMENT METHOD: OVERCOMING NONUNIQUENESS AND COUPLING FOR ACOUSTIC SCATTERING IN HETEROGENEOUS MEDIA

The understanding of complex wave phenomenon, such as multiple scattering in heterogeneous media, is often hindered by lack of equations modelling the exact physics. Use of approximate numerical methods, such as Finite Element Method (FEM) and Boundary Element Method (BEM), is therefore needed to understand these complex wave problems. FEM is known for its ability to accurately model the physics of the problem but requires truncating the computational domain. On the other hand, BEM can accurately model waves in unbounded region but is suitable for homogeneous media only. Coupling FEM and BEM therefore is a natural way to solve problems involving a relatively small heterogeneity (to be modelled with FEM) surrounded by an unbounded homogeneous medium (to be modelled with BEM). The use of a classical FEM-BEM coupling can become computationally demanding due to high mesh density requirement at high frequencies. Secondly, BEM is an integral equation based technique and suffers from the problem of non-uniqueness. To overcome the requirement of high mesh density for high frequencies, a technique known as the ‘Partition of Unity’ (PU) method has been developed by previous researchers. The work presented in this thesis extends the concept of PU to BEM (PUBEM) while effectively treating the problem of non-uniqueness. Two of the well-known methods, namely CHIEF and Burton-Miller approaches, to overcome the non-uniqueness problem, are compared for PUBEM. It is shown that the CHIEF method is relatively easy to implement and results in at least one order of magnitude of improvement in the accuracy. A modified ‘PU’ concept is presented to solve the heterogeneous problems with the PU based FEM (PUFEM). It is shown that use of PUFEM results in close to two orders of magnitude improvement over FEM despite using a much coarser mesh. The two methods, namely PUBEM and PUFEM, are then coupled to solve the heterogeneous wave problems in unbounded media. Compared to PUFEM, the coupled PUFEM-PUBEM apporach is shown to result between 30-40% savings in the total degress of freedom required to achieve similar accuracy

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