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1
NOVELINO, LARISSA SIMOES. "APPLICATION OF FAST MULTIPOLE TECHNIQUES IN THE BOUNDARY ELEMENT METHODS." PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO, 2015. http://www.maxwell.vrac.puc-rio.br/Busca_etds.php?strSecao=resultado&nrSeq=37003@1.
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PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIROCOORDENAÇÃO DE APERFEIÇOAMENTO DO PESSOAL DE ENSINO SUPERIOR
PROGRAMA DE EXCELENCIA ACADEMICA
Este trabalho visa à implementação de um programa de elementos de contorno para problemas com milhões de graus de liberdade. Isto é obtido com a implementação do Método Fast Multipole (FMM), que pode reduzir o número de operações, para a solução de um problema com N graus de liberdade, de O(N(2)) para O(NlogN) ou O(N). O uso de memória também é reduzido, por não haver o armazenamento de matrizes de grandes dimensões como no caso de outros métodos numéricos. A implementação proposta é baseada em um desenvolvimento consistente do convencional, Método de colocação dos elementos de contorno (BEM) – com conceitos provenientes do Hibrido BEM – para problemas de potencial e elasticidade de larga escala em 2D e 3D. A formulação é especialmente vantajosa para problemas de topologia complicada ou que requerem soluções fundamentais complicadas. A implementação apresentada, usa um esquema para expansões de soluções fundamentais genéricas em torno de níveis hierárquicos de polos campo e fonte, tornando o FMM diretamente aplicável para diferentes soluções fundamentais. A árvore hierárquica dos polos é construída a partir de um conceito topológico de superelementos dentro de superelementos. A formulação é inicialmente acessada e validada em termos de um problema de potencial 2D. Como resolvedores iterativos não são necessários neste estágio inicial de simulação numérica, podese acessar a eficiência relativa à implementação do FMM.
This work aims to present an implementation of a boundary element solver for problems with millions of degrees of freedom. This is achieved through a Fast Multipole Method (FMM) implementation, which can lower the number of operations for solving a problem, with N degrees of freedom, from O(N(2)) to O(NlogN) or O(N). The memory usage is also very small, as there is no need to store large matrixes such as required by other numerical methods. The proposed implementations are based on a consistent development of the conventional, collocation boundary element method (BEM) - with concepts taken from the variationally-based hybrid BEM - for large-scale 2D and 3D problems of potential and elasticity. The formulation is especially advantageous for problems of complicated topology or requiring complicated fundamental solutions. The FMM implementation presented in this work uses a scheme for expansions of a generic fundamental solution about hierarchical levels of source and field poles. This makes the FMM directly applicable to different kinds of fundamental solutions. The hierarchical tree of poles is built upon a topological concept of superelements inside superelements. The formulation is initially assessed and validated in terms of a simple 2D potential problem. Since iterative solvers are not required in this first step of numerical simulations, an isolated efficiency assessment of the implemented fast multipole technique is possible.
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Bapat, Milind S. "New Developments in Fast Boundary Element Method." University of Cincinnati / OhioLINK, 2012. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1331296947.
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Ding, Jian. "Fast boundary element method solutions for three dimensional large scale problems." Available online, Georgia Institute of Technology, 2005, 2004. http://etd.gatech.edu/theses/available/etd-01102005-174227/unrestricted/ding%5Fjian%5F200505%5Fphd.pdf.
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Thesis (Ph. D.)--Mechanical Engineering, Georgia Institute of Technology, 2005.Mucha, Peter, Committee Member ; Qu, Jianmin, Committee Member ; Ye, Wenjing, Committee Chair ; Hesketh, Peter, Committee Member ; Gray, Leonard J., Committee Member. Vita. Includes bibliographical references.
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Bagur, Laura. "Modeling fluid injection effects in dynamic fault rupture using Fast Boundary Element Methods." Electronic Thesis or Diss., Institut polytechnique de Paris, 2024. http://www.theses.fr/2024IPPAE010.
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Les tremblements de terre d'origine naturelle ou anthropique provoquent d'importants dégâts humains et matériels. Dans les deux cas, la présence de fluides interstitiels influe sur le déclenchement des instabilités sismiques. Une nouvelle question d'actualité dans la communauté est de montrer que l'instabilité sismique peut être atténuée par un contrôle actif de la pression des fluides. Dans ce travail, nous étudions la capacité des méthodes d'éléments de frontière rapides (Fast BEMs) à fournir un solveur robuste multi-physique à grande échelle nécessaire à la modélisation des processus sismiques, de la sismicité induite et de leur atténuation.Dans une première partie, un solveur BEM rapide avec différents algorithmes d'intégration temporelle est utilisé. Nous évaluons les performances de diverses méthodes à pas de temps adaptatif sur la base de problèmes de cycles sismiques 2D usuels pour les failles planes.Nous proposons une solution asismique analytique pour effectuer des études de convergence et fournir une comparaison rigoureuse des capacités des différentes méthodes en plus des problèmes de cycles sismiques de référence testés.Nous montrons qu'une méthode hybride prédiction-correction / Runge-Kutta à pas de temps adaptatif permet non seulement une résolution précise mais aussi d'incorporer à la fois les effets inertiels et les couplages hydro-mécaniques dans les simulations de rupture dynamique de faille.Dans une deuxième partie, une fois les outils numériques développés pour des configurations standards, notre objectif est de prendre en compte les effets de l'injection de fluide sur le glissement sismique. Nous choisissons le cadre poroélastodynamique pour incorporer les effets de l'injection sur l'instabilité sismique. Un modèle poroélastodynamique complet nécessiterait des coûts de calcul ou des approximations non négligeables. Nous justifions rigoureusement quels effets fluides prédominants sont en jeu lors d'un tremblement de Terre ou d'un cycle sismique. Pour cela, nous effectuons une analyse dimensionnelle des équations, et illustrons les résultats en utilisant un problème de poroelastodynamique 1D simplifié. Plus précisément, nous montrons qu'à l'échelle de temps de l'instabilité sismique, les effets inertiels sont prédominants alors qu'une combinaison de la diffusion du fluide et de la déformation élastique de la matrice solide due à la variation de la pression interstitielle devrait être privilégiée à l'échelle de temps du cycle sismique, au lieu du modèle de diffusion principalement utilisé dans la littératureEarthquakes due to either natural or anthropogenic sources cause important human and material damage. In both cases, the presence of pore fluids influences the triggering of seismic instabilities.A new and timely question in the community is to show that the earthquake instability could be mitigated by active control of the fluid pressure. In this work, we study the ability of Fast Boundary Element Methods (Fast BEMs) to provide a multi-physic large-scale robust solver required for modeling earthquake processes, human induced seismicity and their mitigation.In a first part, a Fast BEM solver with different temporal integration algorithms is used. We assess the performances of various possible adaptive time-step methods on the basis of 2D seismic cycle benchmarks available for planar faults. We design an analytical aseismic solution to perform convergence studies and provide a rigorous comparison of the capacities of the different solving methods in addition to the seismic cycles benchmarks tested. We show that a hybrid prediction-correction / adaptive time-step Runge-Kutta method allows not only for an accurate solving but also to incorporate both inertial effects and hydro-mechanical couplings in dynamic fault rupture simulations.In a second part, once the numerical tools are developed for standard fault configurations, our objective is to take into account fluid injection effects on the seismic slip. We choose the poroelastodynamic framework to incorporate injection effects on the earthquake instability. A complete poroelastodynamic model would require non-negligible computational costs or approximations. We justify rigorously which predominant fluid effects are at stake during an earthquake or a seismic cycle. To this aim, we perform a dimensional analysis of the equations, and illustrate the results using a simplified 1D poroelastodynamic problem. We formally show that at the timescale of the earthquake instability, inertial effects are predominant whereas a combination of diffusion and elastic deformation due to pore pressure change should be privileged at the timescale of the seismic cycle, instead of the diffusion model mainly used in the literature
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SHEN, LIANG. "ADAPTIVE FAST MULTIPOLE BOUNDARY ELEMENT METHODS FOR THREE-DIMENSIONAL POTENTIAL AND ACOUSTIC WAVE PROBLEMS." University of Cincinnati / OhioLINK, 2007. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1193706024.
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MITRA, KAUSIK PRADIP. "APPLICATION OF MULTIPOLE EXPANSIONS TO BOUNDARY ELEMENT METHOD." University of Cincinnati / OhioLINK, 2002. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1026411773.
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Rahman, Mizanur. "Fast boundary element methods for integral equations on infinite domains and scattering by unbounded surfaces." Thesis, Brunel University, 2000. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.324648.
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Ding, Jian. "Fast Boundary Element Method Solutions For Three Dimensional Large Scale Problems." Diss., Georgia Institute of Technology, 2005. http://hdl.handle.net/1853/6830.
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Efficiency is one of the key issues in numerical simulation of large-scale problems with complex 3-D geometry. Traditional domain based methods, such as finite element methods, may not be suitable for these problems due to, for example, the complexity of mesh generation. The Boundary Element Method (BEM), based on boundary integral formulations (BIE), offers one possible solution to this issue by discretizing only the surface of the domain. However, to date, successful applications of the BEM are mostly limited to linear and continuum problems. The challenges in the extension of the BEM to nonlinear problems or problems with non-continuum boundary conditions (BC) include, but are not limited to, the lack of appropriate BIE and the difficulties in the treatment of the volume integrals that result from the nonlinear terms. In this thesis work, new approaches and techniques based on the BEM have been developed for 3-D nonlinear problems and Stokes problems with slip BC.
For nonlinear problems, a major difficulty in applying the BEM is the treatment of the volume integrals in the BIE. An efficient approach, based on the precorrected-FFT technique, is developed to evaluate the volume integrals. In this approach, the 3-D uniform grid constructed initially to accelerate surface integration is used as the baseline mesh to evaluate volume integrals. The cubes enclosing part of the boundary are partitioned using surface panels. No volume discretization of the interior cubes is necessary. This grid is also used to accelerate volume integration. Based on this approach, accelerated BEM solvers for non-homogeneous and nonlinear problems are developed and tested. Good agreement is achieved between simulation results and analytical results. Qualitative comparison is made with current approaches.
Stokes problems with slip BC are of particular importance in micro gas flows such as those encountered in MEMS devices. An efficient approach based on the BEM combined with the precorrected-FFT technique has been proposed and various techniques have been developed to solve these problems. As the applications of the developed method, drag forces on oscillating objects immersed in an unbounded slip flow are calculated and validated with either analytic solutions or experimental results.
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Grasso, Eva. "Modelling visco-elastic seismic wave propagation : a fast-multipole boundary element method and its coupling with finite elements." Phd thesis, Université Paris-Est, 2012. http://tel.archives-ouvertes.fr/tel-00730752.
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The numerical simulation of elastic wave propagation in unbounded media is a topical issue. This need arises in a variety of real life engineering problems, from the modelling of railway- or machinery-induced vibrations to the analysis of seismic wave propagation and soil-structure interaction problems. Due to the complexity of the involved geometries and materials behavior, modelling such situations requires sophisticated numerical methods. The Boundary Element method (BEM) is a very effective approach for dynamical problems in spatially-extended regions (idealized as unbounded), especially since the advent of fast BEMs such as the Fast Multipole Method (FMM) used in this work. The BEM is based on a boundary integral formulation which requires the discretization of the only domain boundary (i.e. a surface in 3-D) and accounts implicitly for the radiation conditions at infinity. As a main disadvantage, the BEM leads a priori to a fully-populated and (using the collocation approach) non-symmetrical coefficient matrix, which make the traditional implementation of this method prohibitive for large problems (say O(106) boundary DoFs). Applied to the BEM, the Multi-Level Fast Multipole Method (ML-FMM) strongly lowers the complexity in computational work and memory that hinder the classical formulation, making the ML-FMBEM very competitive in modelling elastic wave propagation. The elastodynamic version of the Fast Multipole BEM (FMBEM), in a form enabling piecewise-homogeneous media, has for instance been successfully used to solve seismic wave propagation problems in a previous work (thesis dissertation of S. Chaillat, ENPC, 2008). This thesis aims at extending the capabilities of the existing frequency-domain elastodynamic FMBEM in two directions. Firstly, the time-harmonic elastodynamic ML-FMBEM formulation has been extended to the case of weakly dissipative viscoelastic media. Secondly, the FMBEM and the Finite Element Method (FEM) have been coupled to take advantage of the versatility of the FEM to model complex geometries and non-linearities while the FM-BEM accounts for wave propagation in the surrounding unbounded medium. In this thesis, we consider two strategies for coupling the FMBEM and the FEM to solve three-dimensional time-harmonic wave propagation problems in unbounded domains. The main idea is to separate one or more bounded subdomains (modelled by the FEM) from the complementary semi-infinite viscoelastic propagation medium (modelled by the FMBEM) through a non-overlapping domain decomposition. Two coupling strategies have been implemented and their performances assessed and compared on several examples
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BAPAT, MILIND SHRIKANT. "FAST MULTIPOLE BOUNDARY ELEMENT METHOD FOR SOLVING TWO-DIMENSIONAL ACOUSTIC WAVE PROBLEMS." University of Cincinnati / OhioLINK, 2006. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1163773308.
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Li, Yuxiang. "A Fast Multipole Boundary Element Method for Solving Two-dimensional Thermoelasticity Problems." University of Cincinnati / OhioLINK, 2014. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1397477834.
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Huang, Shuo. "A Fast Multipole Boundary Element Method for the Thin Plate Bending Problem." University of Cincinnati / OhioLINK, 2013. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1368026582.
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PEIXOTO, HELVIO DE FARIAS COSTA. "A STUDY OF THE FAST MULTIPOLE METHOD APPLIED TO BOUNDARY ELEMENT PROBLEMS." PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO, 2014. http://www.maxwell.vrac.puc-rio.br/Busca_etds.php?strSecao=resultado&nrSeq=24364@1.
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PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIROCONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICO
FUNDAÇÃO DE APOIO À PESQUISA DO ESTADO DO RIO DE JANEIRO
BOLSA NOTA 10
Este trabalho faz parte de um projeto para a implementação de um programa que possa simular problemas com milhões de graus de liberdade em um computador pessoal. Para isto, combina-se o Método Fast Multipole (FMM) com o Método Expedito dos Elementos de Contorno (EBEM), além de serem utilizados resolvedores iterativos de sistemas de equações. O EBEM é especialmente vantajoso em problemas de complicada topologia, ou que usem funções fundamentais muito complexas. Neste trabalho apresenta-se uma formulação para o Método Fast Multipole (FMM) que pode ser usada para, virtualmente, qualquer função e também para contornos curvos, o que parece ser uma contribuição original. Esta formulação apresenta um formato mais compacto do que as já existentes na literatura, e também pode ser diretamente aplicada a diversos tipos de problemas praticamente sem modificação de sua estrutura básica. É apresentada a validação numérica da formulação proposta. Sua utilização em um contexto do EBEM permite que um programa prescinda de integrações sobre segmentos – mesmo curvos – do contorno quando estes estão distantes do ponto fonte.
This is part of a larger project that aims to develop a program able to simulate problems with millions of degrees of freedom on a personal computer. The Fast Multipole Method (FMM) is combined with the Expedite Boundary Element Method (EBEM) for integration, in the project s final version, with iterative equations solvers. The EBEM is especially advantageous when applied to problems with complicated topology as well as in the case of highly complex fundamental solutions. In this work, a FMM formulation is proposed for the use with virtually any type of fundamental solution and considering curved boundaries, which seems to be an original contribution. This formulation presents a more compact format than the ones shown in the technical literature, and can be directly applied to different kinds of problems without the need of manipulation of its basic structure, being numerically validated for a few applications. Its application in the context of the EBEM leads to the straightforward implementation of higher-order elements for generally curved boundaries that dispenses integration when the boundary segment is relatively far from the source point.
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Harbrecht, Helmut, and Reinhold Schneider. "Wavelets for the fast solution of boundary integral equations." Universitätsbibliothek Chemnitz, 2006. http://nbn-resolving.de/urn:nbn:de:swb:ch1-200600540.
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This paper presents a wavelet Galerkin scheme for the fast solution of boundary integral equations. Wavelet Galerkin schemes employ appropriate wavelet bases for the discretization of boundary integral operators. This yields quasi-sparse system matrices which can be compressed to O(N_J) relevant matrix entries without compromising the accuracy of the underlying Galerkin scheme. Herein, O(N_J) denotes the number of unknowns. The assembly of the compressed system matrix can be performed in O(N_J) operations. Therefore, we arrive at an algorithm which solves boundary integral equations within optimal complexity. By numerical experiments we provide results which corroborate the theory.
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Kachanovska, Maryna. "Fast, Parallel Techniques for Time-Domain Boundary Integral Equations." Doctoral thesis, Universitätsbibliothek Leipzig, 2014. http://nbn-resolving.de/urn:nbn:de:bsz:15-qucosa-132183.
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This work addresses the question of the efficient numerical solution of time-domain boundary integral equations with retarded potentials arising in the problems of acoustic and electromagnetic scattering. The convolutional form of the time-domain boundary operators allows to discretize them with the help of Runge-Kutta convolution quadrature. This method combines Laplace-transform and time-stepping approaches and requires the explicit form of the fundamental solution only in the Laplace domain to be known. Recent numerical and analytical studies revealed excellent properties of Runge-Kutta convolution quadrature, e.g. high convergence order, stability, low dissipation and dispersion.
As a model problem, we consider the wave scattering in three dimensions. The convolution quadrature discretization of the indirect formulation for the three-dimensional wave equation leads to the lower triangular Toeplitz system of equations. Each entry of this system is a boundary integral operator with a kernel defined by convolution quadrature. In this work we develop an efficient method of almost linear complexity for the solution of this system based on the existing recursive algorithm. The latter requires the construction of many discretizations of the Helmholtz boundary single layer operator for a wide range of complex wavenumbers. This leads to two main problems:
the need to construct many dense matrices and to evaluate many singular and near-singular integrals.
The first problem is overcome by the use of data-sparse techniques, namely, the high-frequency fast multipole method (HF FMM) and H-matrices. The applicability of both techniques for the discretization of the Helmholtz boundary single-layer operators with complex wavenumbers is analyzed. It is shown that the presence of decay can favorably affect the length of the fast multipole expansions and thus reduce the matrix-vector multiplication times. The performance of H-matrices and the HF FMM is compared for a range of complex wavenumbers, and the strategy to choose between two techniques is suggested.
The second problem, namely, the assembly of many singular and nearly-singular integrals, is solved by the use of the Huygens principle. In this work we prove that kernels of the boundary integral operators $w_n^h(d)$ ($h$ is the time step and $t_n=nh$ is the time) exhibit exponential decay outside of the neighborhood of $d=nh$ (this is the consequence of the Huygens principle). The size of the support of these kernels for fixed $h$ increases with $n$ as $n^a,a<1$, where $a$ depends on the order of the Runge-Kutta method and is (typically) smaller for Runge-Kutta methods of higher order. Numerical experiments demonstrate that theoretically predicted values of $a$ are quite close to optimal.
In the work it is shown how this property can be used in the recursive algorithm to construct only a few matrices with the near-field, while for the rest of the matrices the far-field only is assembled. The resulting method allows to solve the three-dimensional wave scattering problem with asymptotically almost linear complexity. The efficiency of the approach is confirmed by extensive numerical experiments.
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Brunner, Dominik. "Fast boundary element methods for large-scale simulations of the vibro-acoustic behavior of ship-like structures." Tönning Lübeck Marburg Der Andere Verl, 2009. http://d-nb.info/99703128X/04.
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Fischer, Matthias. "The fast multipole boundary element method and its application to structure acoustic field interaction." [S.l. : s.n.], 2004. http://www.bsz-bw.de/cgi-bin/xvms.cgi?SWB11380456.
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Huang, Shuo. "A New Multidomain Approach and Fast Direct Solver for the Boundary Element Method." University of Cincinnati / OhioLINK, 2017. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1505125721346283.
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Kachanovska, Maryna. "Fast, Parallel Techniques for Time-Domain Boundary Integral Equations." Doctoral thesis, Max-Planck-Institut für Mathematik in den Naturwissenschaften, 2013. https://ul.qucosa.de/id/qucosa%3A12278.
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This work addresses the question of the efficient numerical solution of time-domain boundary integral equations with retarded potentials arising in the problems of acoustic and electromagnetic scattering. The convolutional form of the time-domain boundary operators allows to discretize them with the help of Runge-Kutta convolution quadrature. This method combines Laplace-transform and time-stepping approaches and requires the explicit form of the fundamental solution only in the Laplace domain to be known. Recent numerical and analytical studies revealed excellent properties of Runge-Kutta convolution quadrature, e.g. high convergence order, stability, low dissipation and dispersion.
As a model problem, we consider the wave scattering in three dimensions. The convolution quadrature discretization of the indirect formulation for the three-dimensional wave equation leads to the lower triangular Toeplitz system of equations. Each entry of this system is a boundary integral operator with a kernel defined by convolution quadrature. In this work we develop an efficient method of almost linear complexity for the solution of this system based on the existing recursive algorithm. The latter requires the construction of many discretizations of the Helmholtz boundary single layer operator for a wide range of complex wavenumbers. This leads to two main problems:
the need to construct many dense matrices and to evaluate many singular and near-singular integrals.
The first problem is overcome by the use of data-sparse techniques, namely, the high-frequency fast multipole method (HF FMM) and H-matrices. The applicability of both techniques for the discretization of the Helmholtz boundary single-layer operators with complex wavenumbers is analyzed. It is shown that the presence of decay can favorably affect the length of the fast multipole expansions and thus reduce the matrix-vector multiplication times. The performance of H-matrices and the HF FMM is compared for a range of complex wavenumbers, and the strategy to choose between two techniques is suggested.
The second problem, namely, the assembly of many singular and nearly-singular integrals, is solved by the use of the Huygens principle. In this work we prove that kernels of the boundary integral operators $w_n^h(d)$ ($h$ is the time step and $t_n=nh$ is the time) exhibit exponential decay outside of the neighborhood of $d=nh$ (this is the consequence of the Huygens principle). The size of the support of these kernels for fixed $h$ increases with $n$ as $n^a,a<1$, where $a$ depends on the order of the Runge-Kutta method and is (typically) smaller for Runge-Kutta methods of higher order. Numerical experiments demonstrate that theoretically predicted values of $a$ are quite close to optimal.
In the work it is shown how this property can be used in the recursive algorithm to construct only a few matrices with the near-field, while for the rest of the matrices the far-field only is assembled. The resulting method allows to solve the three-dimensional wave scattering problem with asymptotically almost linear complexity. The efficiency of the approach is confirmed by extensive numerical experiments.
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Ballani, Jonas. "Fast Evaluation of Near-Field Boundary Integrals using Tensor Approximations." Doctoral thesis, Universitätsbibliothek Leipzig, 2012. http://nbn-resolving.de/urn:nbn:de:bsz:15-qucosa-97317.
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In this dissertation, we introduce and analyse a scheme for the fast evaluation of integrals stemming from boundary element methods including discretisations of the classical single and double layer potential operators. Our method is based on the parametrisation of boundary elements in terms of a d-dimensional parameter tuple. We interpret the integral as a real-valued function f depending on d parameters and show that f is smooth in a d-dimensional box. A standard interpolation of f by polynomials leads to a d-dimensional tensor which is given by the values of f at the interpolation points. This tensor may be approximated in a low rank tensor format like the canonical format or the hierarchical format. The tensor approximation has to be done only once and allows us to evaluate interpolants in O(dr(m+1)) operations in the canonical format, or O(dk³ + dk(m + 1)) operations in the hierarchical format, where m denotes the interpolation order and the ranks r, k are small integers. In particular, we apply an efficient black box scheme in the hierarchical tensor format in order to adaptively approximate tensors even in high dimensions d with a prescribed (but heuristic) target accuracy. By means of detailed numerical experiments, we demonstrate that highly accurate integral values can be obtained at very moderate costs.
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Karban, Ugur. "Three-dimensional Flow Solutions For Non-lifting Flows Using Fast Multipole Boundary Element Method." Master's thesis, METU, 2012. http://etd.lib.metu.edu.tr/upload/12615042/index.pdf.
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Driving aim of this study was to develop a solver which is accurate enough to be used in analysis and fast enough to be used in optimization purposes. As a first step, a three-dimensional potential flow solver is developed using Fast Multipole Boundary Element (FMBEM) for calculating the pressure distributions in non-lifting flows.
It is a steady state solver which uses planar triangular unstructured mesh. After the geometry is introduced, the program creates a prescribed wake surface attached to the trailing edge(s), obtains a solution using panel elements on which the doublet and source strengths vary linearly. The reason for using FMBEM instead of classical BEM is the availability of solutions of systems having DOFs up to several millions within a few hours using a standard computer which is impossible to accomplish with classical BEM. Solutions obtained for different test cases are compared with the analytical solution (if applicable), the experimental data or the results obtained by JavaFoil.
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Brancati, Alessandro. "Boundary element method for fast solution of acoustic problems : active and passive noise control." Thesis, Imperial College London, 2010. http://hdl.handle.net/10044/1/6139.
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This thesis presents boundary element formulations for three-dimensional acoustic problems of active (ANC) and passive (PNC) noise control. A new boundary element strategy, referred to as RABEM (Rapid Acoustic Boundary Element Method), has been formulated and implemented for acoustic problems. The assembly time for both the system matrix and the right hand side vector is accelerated using a Hierarchical-matrix approach based on the Adaptive Cross Approximation (ACA). Two different H-matrix-GMRES solvers (one without preconditioners and one with a block diagonal preconditioner) are developed and tested for low and high frequency problems including noise emanated by aircraft approaching an airport. A new formulation for solving the ANC based on attenuating the unwanted sound in a control volume (CV) rather than cancelling it at a single point is presented. The noise attenuation is obtained by minimising the square modules of two acoustic quantities - the potential and one component of the particle velocity - within the CV. The two formulations presented include a single and a double secondary source, respectively. Several examples are presented to demonstrate the e fficiency of the proposed technique. A new approach, based on sensitivity analysis, for determining the optimum locations of the CV and the optimum location/orientation of the secondary source is presented. The optimisation procedure is based upon a first order method and minimises a suitable cost function by using its gradient. The procedure to calculate the cost function gradients is explained in detail. Finally, a PNC strategy applied to the interior of an aircraft cabin is investigated. A lower noise level is achieved through the introduction of a new textile with a higher noise absorbing coe cient than a conventional textile, especially at low frequencies. The so-called "bubble concept", which consists of adding cap insertions at the sides of the passenger head, is also investigated.
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Wu, Shu-Wei. "A fast, robust and accurate procedure for radiation and scattering analyses of submerged elastic axisymmetric bodies." Thesis, Imperial College London, 1990. http://hdl.handle.net/10044/1/46618.
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Lee, Jimin. "Earthquake site effect modeling in sedimentary basins using a 3-D indirect boundary element-fast multipole method." Diss., Online access via UMI:, 2007.
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Wilkes, Daniel. "The development of a fast multipole boundary element method for coupled acoustic and elastic problems." Thesis, Curtin University, 2014. http://hdl.handle.net/20.500.11937/122.
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This thesis presents a dual fast multipole boundary element method (FMBEM) for modelling 3D acoustic coupled fluid-structure interaction problems in the frequency domain. Boundary integral representations are used to represent both the exterior fluid and interior elastic solid domains and the fast multipole method is employed to accelerate the calculations in both domains. The dual FMBEM yields a similar solution accuracy to the conventional models, while its solution times and memory requirements are substantially reduced.
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Ozyazicioglu, Mehmet H. "A Boundary Element Formulation For Axi-symmetric Problems In Poro-elasticity." Phd thesis, METU, 2006. http://etd.lib.metu.edu.tr/upload/3/12607351/index.pdf.
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A formulation is proposed for the boundary element analysis of poro-elastic media with axi-symmetric geometry. The boundary integral equation is reduced to a set of line integral equations in the generating plane for each of the Fourier coefficients, through complex Fourier series expansion of boundary quantities in circumferential direction. The method is implemented into a computer program, where the fundamental solutions are integrated by Gaussian Quadrature along the generator, while Fast Fourier Transform algorithm is employed for integrations in circumferential direction. The strongly singular integrands in boundary element equations are regularized by a special technique. The Fourier transform solution is then inverted in to Rθ
z space via inverse FFT. The success of the method is assessed by problems with analytical solutions. A good fit is observed in each case, which indicates effectiveness and reliability of the present method.
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Pester, M., and S. Rjasanow. "A parallel version of the preconditioned conjugate gradient method for boundary element equations." Universitätsbibliothek Chemnitz, 1998. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-199800455.
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The parallel version of precondition techniques is developed for
matrices arising from the Galerkin boundary element method for
two-dimensional domains with Dirichlet boundary conditions.
Results were obtained for implementations on a transputer network
as well as on an nCUBE-2 parallel computer showing that iterative
solution methods are very well suited for a MIMD computer. A
comparison of numerical results for iterative and direct solution
methods is presented and underlines the superiority of iterative
methods for large systems.
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Vuylsteke, Xavier. "Development of a reference method based on the fast multipole boundary element method for sound propagation problems in urban environments : formalism, improvements & applications." Thesis, Paris Est, 2014. http://www.theses.fr/2014PEST1174/document.
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Décrit comme l'un des algorithmes les plus prometteurs du 20ème siècle, le formalisme multipolaire appliqué à la méthode des éléments de frontière, permet de nos jours de traiter de larges problèmes encore inconcevables il y a quelques années. La motivation de ce travail de thèse est d'évaluer la capacité, ainsi que les avantages concernant les ressources numériques, de ce formalisme pour apporter une solution de référence aux problèmes de propagation sonore tri-dimensionnels en environnement urbain, dans l'objectif d'améliorer les algorithmes plus rapides déjà existants. Nous présentons la théorie nécessaire à l'obtention de l'équation intégrale de frontière pour la résolution de problèmes non bornés. Nous discutons également de l'équation intégrale de frontière conventionnelle et hyper-singulière pour traiter les artefacts numériques liés aux fréquences fictives, lorsque l'on résout des problèmes extérieurs. Nous présentons par la suite un bref aperçu historique et technique du formalisme multipolaire rapide et des outils mathématiques requis pour représenter la solution élémentaire de l'équation de Helmholtz. Nous décrivons les principales étapes, d'un point de vue numérique, du calcul multipolaire. Un problème de propagation sonore dans un quartier, composé de 5 bâtiments, nous a permis de mettre en évidence des problèmes d'instabilités dans le calcul par récursion des matrices de translations, se traduisant par des discontinuités sur le champs de pression de surface et une non convergence du solveur. Ceci nous a conduits à considérer le travail très récent de Gumerov et Duraiswamy en lien avec un processus récursif stable pour le calcul des coefficients des matrices de rotation. Cette version améliorée a ensuite été testée avec succès sur un cas de multi diffraction jusqu'à une taille dimensionnelle de problème de 207 longueur d'ondes. Nous effectuons finalement une comparaison entre un algorithme d'élément de frontière, Micado3D, un algorithme multipolaire et un algorithme basé sur le tir de rayons, Icare, pour le calcul de niveaux de pression moyennés dans une cour ouverte et fermée. L'algorithme multipolaire permet de valider les résultats obtenus par tir de rayons dans la cour ouverte jusqu'à 300 Hz (i.e. 100 longueur d'ondes), tandis que concernant la cour fermée, zone très sensible par l'absence de contribution directes ou réfléchies, des études complémentaires sur le préconditionnement de la matrice semblent requises afin de s'assurer de la pertinence des résultats obtenus à l'aide de solveurs itératifsDescribed as one of the best ten algorithms of the 20th century, the fast multipole formalism applied to the boundary element method allows to handle large problems which were inconceivable only a few years ago. Thus, the motivation of the present work is to assess the ability, as well as the benefits in term of computational resources provided by the application of this formalism to the boundary element method, for solving sound propagation problems and providing reference solutions, in three dimensional dense urban environments, in the aim of assessing or improving fast engineering tools. We first introduce the mathematical background required for the derivation of the boundary integral equation, for solving sound propagation problems in unbounded domains. We discuss the conventional and hyper-singular boundary integral equation to overcome the numerical artifact of fictitious eigen-frequencies, when solving exterior problems. We then make a brief historical and technical overview of the fast multipole principle and introduce the mathematical tools required to expand the elementary solution of the Helmholtz equation and describe the main steps, from a numerical viewpoint, of fast multipole calculations. A sound propagation problem in a city block made of 5 buildings allows us to highlight instabilities in the recursive computation of translation matrices, resulting in discontinuities of the surface pressure and a no convergence of the iterative solver. This observation leads us to consider the very recent work of Gumerov & Duraiswamy, related to a ``stable'' recursive computation of rotation matrices coefficients in the RCR decomposition. This new improved algorithm has been subsequently assessed successfully on a multi scattering problem up to a dimensionless domain size equal to 207 wavelengths. We finally performed comparisons between a BEM algorithm, extit{Micado3D}, the FMBEM algorithm and a ray tracing algorithm, Icare, for the calculation of averaged pressure levels in an opened and closed court yards. The fast multipole algorithm allowed to validate the results computed with Icare in the opened court yard up to 300 Hz corresponding, (i.e. 100 wavelengths), while in the closed court yard, a very sensitive area without direct or reflective fields, further investigations related to the preconditioning seem required to ensure reliable solutions provided by iterative solver based algorithms
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Fischer, Matthias [Verfasser]. "The fast multipole boundary element method and its application to structure-acoustic field interaction / Institut A für Mechanik der Universität Stuttgart. Matthias Fischer." Stuttgart : Inst. A für Mechanik, 2004. http://d-nb.info/972310819/34.
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PEIXOTO, HELVIO DE FARIAS COSTA. "A FAST MULTIPOLE METHOD FOR HIGH ORDER BOUNDARY ELEMENTS." PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO, 2018. http://www.maxwell.vrac.puc-rio.br/Busca_etds.php?strSecao=resultado&nrSeq=34740@1.
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PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIROCONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICO
FUNDAÇÃO DE APOIO À PESQUISA DO ESTADO DO RIO DE JANEIRO
BOLSA NOTA 10
Desde a década de 1990, o Método Fast Multipole (FMM) tem sido usado em conjunto com o Métodos dos Elementos de Contorno (BEM) para a simulação de problemas de grande escala. Este método utiliza expansões em série de Taylor para aglomerar pontos da discretização do contorno, de forma a reduzir o tempo computacional necessário para completar a simulação. Ele se tornou uma ferramenta bastante importante para os BEMs, pois eles apresentam matrizes cheias e assimétricas, o que impossibilita a utilização de técnicas de otimização de solução de sistemas de equação. A aplicação do FMM ao BEM é bastante complexa e requer muita manipulação matemática. Este trabalho apresenta uma formulação do FMM que é independente da solução fundamental utilizada pelo BEM, o Método Fast Multipole Generalizado (GFMM), que se aplica a elementos de contorno curvos e de qualquer ordem. Esta característica é importante, já que os desenvolvimentos de fast multipole encontrados na literatura se restringem apenas a elementos constantes. Todos os aspectos são abordados neste trabalho, partindo da sua base matemática, passando por validação numérica, até a solução de problemas de potencial com muitos milhões de graus de liberdade. A aplicação do GFMM a problemas de potencial e elasticidade é discutida e validada, assim como os desenvolvimentos necessários para a utilização do GFMM com o Método Híbrido Simplificado de Elementos de Contorno (SHBEM). Vários resultados numéricos comprovam a eficiência e precisão do método apresentado. A literatura propõe que o FMM pode reduzir o tempo de execução do algoritmo do BEM de O(N2) para O(N), em que N é o número de graus de liberdade do problema. É demonstrado que esta redução é de fato possível no contexto do GFMM, sem a necessidade da utilização de qualquer técnica de otimização computacional.
The Fast Multipole Method (FMM) has been used since the 1990s with the Boundary Elements Method (BEM) for the simulation of large-scale problems. This method relies on Taylor series expansions of the underlying fundamental solutions to cluster the nodes on the discretised boundary of a domain, aiming to reduce the computational time required to carry out the simulation. It has become an important tool for the BEMs, as they present matrices that are full and nonsymmetric, so that the improvement of storage allocation and execution time is not a simple task. The application of the FMM to the BEM ends up with a very intricate code, and usually changing from one problem s fundamental solution to another is not a simple matter. This work presents a kernel-independent formulation of the FMM, here called the General Fast Multipole Method (GFMM), which is also able to deal with high order, curved boundary elements in a straightforward manner. This is an important feature, as the fast multipole implementations reported in the literature only apply to constant elements. All necessary aspects of this method are presented, starting with the mathematical basics of both FMM and BEM, carrying out some numerical assessments, and ending up with the solution of large potential problems. The application of the GFMM to both potential and elasticity problems is discussed and validated in the context of BEM. Furthermore, the formulation of the GFMM with the Simplified Hybrid Boundary Elements Method (SHBEM) is presented. Several numerical assessments show that the GFMM is highly efficient and may be as accurate as arbitrarily required, for problems with up to many millions of degrees of freedom. The literature proposes that the FMM is capable of reducing the time complexity of the BEM algorithms from O(N2) to O(N), where N is the number of degrees of freedom. In fact, it is shown that the GFMM is able to arrive at such time reduction without resorting to techniques of computational optimisation.
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Keuchel, Sören [Verfasser]. "Aufwandsreduzierungen in der Fast-Multipole-Boundary-Elemente-Methode / Sören Keuchel." Aachen : Shaker, 2017. http://d-nb.info/1138177822/34.
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Ostrowski, Jörg. "Boundary element methods for inductive hardening." [S.l. : s.n.], 2003. http://deposit.ddb.de/cgi-bin/dokserv?idn=973933941.
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Of, Günther, Gregory J. Rodin, Olaf Steinbach, and Matthias Taus. "Coupling Methods for Interior Penalty Discontinuous Galerkin Finite Element Methods and Boundary Element Methods." Universitätsbibliothek Chemnitz, 2012. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-qucosa-96885.
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This paper presents three new coupling methods for interior penalty discontinuous Galerkin finite element methods and boundary element methods. The new methods allow one to use discontinuous basis functions on the interface between the subdomains represented by the finite element and boundary element methods. This feature is particularly important when discontinuous Galerkin finite element methods are used. Error and stability analysis is presented for some of the methods. Numerical examples suggest that all three methods exhibit very similar convergence properties, consistent with available theoretical results.
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Onyango, Thomas Tonny Mboya. "Boundary element methods for solving inverse boundary conditions identification problems." Thesis, University of Leeds, 2008. http://etheses.whiterose.ac.uk/11283/.
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This thesis explores various features of the boundary element method (BEM) used in solving heat transfer boundary conditions identification problems. In particular, we present boundary integral equation (BIE) formulations and procedures of the numerical computation for the approximation of the boundary temperatures, heat fluxes and space, time or temperature dependent heat transfer coefficients. There are many practical heat transfer situations where such problems occur, for example in high temperature regions or hostile environments, such as in combustion chambers, steel cooling processes, etc., in which the actual method of heat transfer on the surface is unknown. In such situations the boundary condition relating the heat flux to the difference between the boundary temperature and that of the surrounding fluid is represented by an unknown function which may depend on space, time, or temperature. In these inverse heat conduction problems (IHCP), the BEM is formulated as a minimization of some functional that measures the discrepancy between the measured data, say the average temperature on a portion of the boundary or at an instant over the whole domain. The minimization provides solutions that are consistent with the data. This indicates that the BEM algorithms for the IRCP are robust, stable and predict reliable results. When the input data is noisy, we have used the truncated singular value decomposition and the Tikhonov regularisation methods to stabilise the solution of the IRCI' boundary conditions identification. Numerical approximations have been obtained and, where possible, the results obtained are compared to the analytical solutions.
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Shah, Nawazish A. "Boundary element methods for road vehicle aerodynamics." Thesis, Loughborough University, 1985. https://dspace.lboro.ac.uk/2134/26942.
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The technique of the boundary element method consists of subdividing the boundary of the field of a function into a series of discrete elements, over which the function can vary. This technique offers important advantages over domain type solutions such as finite elements and finite differences. One of the most important features of the method is the much smaller system of equations and the considerable reduction in data required to run a program. Furthermore, the method is well-suited to problems with an infinite domain. Boundary element methods can be formulated using two different approaches called the ‘direct' and the ‘indirect' methods.
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Leon, Ernesto Pineda. "Dual boundary element methods for creep fracture." Thesis, Queen Mary, University of London, 2006. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.435177.
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OLIVEIRA, MARIA FERNANDA FIGUEIREDO DE. "CONVENTIONAL, HYBRID AND SIMPLIFIED BOUNDARY ELEMENT METHODS." PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO, 2004. http://www.maxwell.vrac.puc-rio.br/Busca_etds.php?strSecao=resultado&nrSeq=5562@1.
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COORDENAÇÃO DE APERFEIÇOAMENTO DO PESSOAL DE ENSINO SUPERIORApresentam-se as formulações, consolidando a nomenclatura e os principais conceitos dos métodos de elementos de contorno: convencional (MCCEC), híbrido de tensões (MHTEC), híbrido de deslocamentos (MHDEC) e híbrido simplificado de tensões (MHSTEC). proposto o método híbrido simplificado de deslocamentos (MHSDEC), em contrapartida ao MHSTEC, baseando-se nas mesmas hipóteses de aproximação de tensões e deslocamentos do MHDEC e supondo que a solução fundamental em termos de tensões seja válida no contorno. Como decorrência do MHSTEC e do MHSDEC, é apresentado também o método híbrido de malha reduzida dos elementos de contorno (MHMREC), com aplicação computacionalmente vantajosa a problemas no domínio da freqüência ou envolvendo materiais não-homogêneos. A partir da investigação das equações matriciais desses métodos, são identificadas quatro novas relações matriciais, das quais uma verifica-se como válida para a obtenção dos elementos das matrizes de flexibilidade e de deslocamento que não podem ser determinados por integração ou avaliação direta. Também é proposta a correta consideração, ainda não muito bem explicada na literatura, de que forças de superfície devem ser interpoladas em função de atributos de superfície e não de atributos nodais. São apresentadas aplicações numéricas para problemas de potencial para cada método mencionado, em que é verificada a validade das novas relações matriciais.
A consolidated, unified formulation of the conventional (CCBEM), hybrid stress (HSBEM), hybrid displacement (HDBEM) and simplified hybrid stress (SHSBEM) boundary element methods is presented. As a counterpart of SHSBEM, the simplified hybrid displacement boundary element method (SHDBEM) is proposed on the basis of the same stress and displacement approximation hypotheses of the HDBEM and on the assumption that stress fundamental solutions are also valid on the boundary. A combination of the SHSBEM and the SHDBEM gives rise to a provisorily called mesh-reduced hybrid boundary element method (MRHBEM), which seems computationally advantageous when applied to frequency domain problems or non-homogeneous materials. Four new matrix relations are identified, one of which may be used to obtain the flexibility and displacement matrix coefficients that cannot be determined by integration or direct evaluation. It is also proposed the correct consideration, still not well explained in the technical literature, that traction forces should be interpolated as functions of surface and not of nodal attributes. Numerical examples of potential problems are presented for each method, in which the validity of the new matrix relations is verified.
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Ortiz, guzman John Erick. "Fast boundary element formulations for electromagnetic modelling in biological tissues." Thesis, Ecole nationale supérieure Mines-Télécom Atlantique Bretagne Pays de la Loire, 2017. http://www.theses.fr/2017IMTA0051/document.
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Cette thèse présente plusieurs nouvelles techniques pour la convergence rapide des solutions aux éléments de frontière de problèmes électromagnétiques. Une attention spéciale a été dédiée aux formulations pertinentes pour les solutions aux problèmes électromagnétiques dans les tissus biologiques à haute et basse fréquence. Pour les basses fréquences, de nouveaux schémas pour préconditionner et accélérer le problème direct de l'électroencéphalographie sont présentés dans cette thèse. La stratégie de régularisation repose sur une nouvelle formule de Calderon, obtenue dans cette thèse, alors que l'accélération exploite le paradigme d'approximation adaptive croisée (ACA). En ce qui concerne le régime haute fréquence, en vue d'applications de dosimétrie, l'attention de ce travail a été concentrée sur l'étude de la régularisation de l'équation intégrale de Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) à l'aide de techniques hiérarchiques. Le travail comprend une analyse complète de l'équation pour des géométries simplement et non-simplement connectées. Cela a permis de concevoir une nouvelle stratégie de régularisation avec une base hiérarchique permettant d'obtenir une équation pour les milieux pénétrable stable pour un large spectre de fréquence. Un cadre de travail propédeutique de discrétisation et une bibliothèque de calcul pour des thèmes de recherches sur les techniques de Calderon en 2D sont proposés en dernière partie de cette thèse. Cela permettra d'étendre nos recherches à l'imagerie par tomographieThis thesis presents several new techniques for rapidly converging boundary element solutions of electromagnetic problems. A special focus has been given to formulations that are relevant for electromagnetic solutions in biological tissues both at low and high frequencies. More specifically, as pertains the low-frequency regime, this thesis presents new schemes for preconditioning and accelerating the Forward Problem in Electroencephalography (EEG). The regularization strategy leveraged on a new Calderon formula, obtained in this thesis work, while the acceleration leveraged on an Adaptive-Cross-Approximation paradigm. As pertains the higher frequency regime, with electromagnetic dosimetry applications in mind, the attention of this work focused on the study and regularization of the Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) integral equation via hierarchical techniques. In this effort, a complete analysis of the equation for both simply and non-simply connected geometries has been obtained. This allowed to design a new hierarchical basis regularization strategy to obtain an equation for penetrable media which is stable in a wide spectrum of frequencies. A final part of this thesis work presents a propaedeutic discretization framework and associated computational library for 2D Calderon research which will enable our future investigations in tomographic imaging
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Hamina, M. (Martti). "Some boundary element methods for heat conduction problems." Doctoral thesis, University of Oulu, 2000. http://urn.fi/urn:isbn:951425614X.
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Abstract
This thesis summarizes certain boundary element methods
applied to some initial and boundary value problems.
Our model problem is the two-dimensional homogeneous heat conduction
problem with vanishing initial data. We use the heat potential
representation of the solution. The given boundary conditions,
as well as the choice of the representation formula,
yield various boundary integral equations. For the sake of simplicity,
we use the direct boundary integral approach, where
the unknown boundary density appearing in the boundary integral
equation is a quantity of physical meaning.
We consider two different sets of boundary conditions, the Dirichlet problem,
where the boundary temperature is given and the Neumann problem,
where the heat flux across the boundary is given.
Even a nonlinear Neumann condition satisfying certain monotonicity
and growth conditions is possible. The approach yields
a nonlinear boundary integral equation of the second kind.
In the stationary case, the model problem reduces to a potential
problem with a nonlinear Neumann condition. We use the spaces of smoothest
splines as trial functions. The nonlinearity is approximated by using the
L2-orthogonal projection. The resulting collocation scheme retains
the optimal L2-convergence. Numerical experiments are in
agreement with this result.
This approach generalizes to the time dependent case.
The trial functions are tensor products of piecewise linear
and piecewise constant splines. The proposed projection method
uses interpolation with respect to the space variable and the orthogonal
projection with respect to the time variable. Compared to the
Galerkin method, this approach simplifies the realization of the
discrete matrix equations.
In addition, the rate of the convergence is of optimal order.
On the other hand,
the Dirichlet problem, where the boundary temperature is given,
leads to a single layer heat operator equation of the first kind.
In the first approach, we use tensor products of piecewise linear splines
as trial functions with collocation at the nodal points.
Stability and suboptimal L2-convergence of the method were proved in the
case of a circular domain. Numerical experiments indicate the
expected quadratic L2-convergence.
Later, a Petrov-Galerkin approach was proposed, where the trial functions were
tensor products of piecewise linear and piecewise constant splines.
The resulting approximative scheme is stable and
convergent. The analysis has been carried out in the cases of
the single layer heat operator and the hypersingular heat operator.
The rate of the convergence with respect to the L2-norm
is also here of suboptimal order.
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Marin, Oana. "Boundary integral methods for Stokes flow : Quadrature techniques and fast Ewald methods." Doctoral thesis, KTH, Skolan för teknikvetenskap (SCI), 2012. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-105540.
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Fluid phenomena dominated by viscous effects can, in many cases, be modeled by the Stokes equations. The boundary integral form of the Stokes equations reduces the number of degrees of freedom in a numerical discretization by reformulating the three-dimensional problem to two-dimensional integral equations to be discretized over the boundaries of the domain. Hence for the study of objects immersed in a fluid, such as drops or elastic/solid particles, integral equations are to be discretized over the surfaces of these objects only. As outer boundaries or confinements are added these must also be included in the formulation. An inherent difficulty in the numerical treatment of boundary integrals for Stokes flow is the integration of the singular fundamental solution of the Stokes equations, e.g. the so called Stokeslet. To alleviate this problem we developed a set of high-order quadrature rules for the numerical integration of the Stokeslet over a flat surface. Such a quadrature rule was first designed for singularities of the type QC 20121122
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Nesemann, Leo [Verfasser]. "Finite element and boundary element methods for contact with adhesion / Leo Nesemann." Hannover : Technische Informationsbibliothek und Universitätsbibliothek Hannover (TIB), 2011. http://d-nb.info/1013365542/34.
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Peng, Xuan. "Isogeometric boundary element methods for linear elastic fracture mechanics." Thesis, Cardiff University, 2016. http://orca.cf.ac.uk/92543/.
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We develop in this work a procedure for obtaining the fatigue life of complex structures directly from Computer-Aided Design (CAD) data, without any mesh generation or regeneration as the cracks evolve. The method relies on a standard isogeometric boundary element method (IGABEM) where the same basis functions are used to both describe the geometry of the component and approximate the displacement and traction fields. The contributions of this work include: (1) Dual boundary integral equations have been applied to model 2D/3D fracture problems in the framework of IGA and that such simulations require no meshing or remeshing in the conventional sense; (2) Graded knot insertion and partition of unity enrichment have been used to capture the stress singularity around the crack tip. The contour-integral based methods and the virtual crack closure integral method are adopted to extract stress intensity factors in the framework of IGABEM; (3) Modifications on the singularity subtraction technique for (hyper-)singular integration are proposed to enhance the quadrature on distorted elements which commonly arise in IGA; (4)ANURBS-based geometry modification algorithm is developed to simulate fatigue crack growth in 2D/3D. smooth crack trajectory and crack front are obtained; (5) An implementation on trimmed NURBS is realized based on a localized double mapping method to perform the quadrature on trimmed elements. A phantom element method is subsequently proposed to model the surface crack (breaking crack) problem and the displacement discontinuity can be introduced without any reparametrization on the original patch.
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Vartiainen, Markku Juhani. "Singular boundary element methods for the hyperbolic wave equation." Thesis, University of Cambridge, 2000. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.621821.
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Suchivoraphanpong, Varanyu. "Fast integral equation methods for large acoustic scattering analyses." Thesis, Imperial College London, 2000. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.312269.
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Layton, Simon. "Fast multipole boundary element solutions with inexact Krylov iterations and relaxation strategies." Thesis, Boston University, 2013. https://hdl.handle.net/2144/11115.
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Thesis (Ph.D.)--Boston UniversityBoundary element methods (BEM) have been used for years to solve a multitude of engineering problems, ranging from Bioelectrostatics, to fluid flows over micro-electromechanical devices and deformations of cell membranes. Only requiring the discretization of a surface into panels rather than the entire domain, they effectively reduce the dimensionality of a problem by one. This reduction in dimensionality nevertheless comes at a cost. BEM requires the solution of a large, dense linear system with each matrix element formed of an integral between two panels, often performed used an iterative solver based on Krylov subspace methods. This requires the repeated calculation of a matrix vector product that can be approximated using a hierarchical approximation known as the fast multipole method (FMM). While adding complexity, this reduces order of the time-to-solution from O(cN^2) to OcN), where c is some function of the condition number of the dense matrix. This thesis obtains algorithmic speedups for the solutions of FMM-BEM systems by applying the mathematical theory behind inexact matrix-vector products to our solver, implementing a relaxation scheme to control the error incurred by the FMM in order to minimize the total time-to-solution. The theory is extensively verified for both Laplace equation and Stokes flow problems, with an investigation to determine how further problems may benefit from the addition of a relaxed solver. We also present experiments for the Stokes flow around both single and multiple red blood cells, an area of ongoing research, showing good speedups that would be applicable for any other code that chose to implement a similar relaxed solver. All of these results are obtained with an easy-to-use, extensible and open-source FMM-BEM code.
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雷哲翔 and Zhexiang Lei. "Time domain boundary element method & its applications." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 1993. http://hub.hku.hk/bib/B31233703.
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Lei, Zhexiang. "Time domain boundary element method & its applications /." [Hong Kong : University of Hong Kong], 1993. http://sunzi.lib.hku.hk/hkuto/record.jsp?B13570365.
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Harbrecht, Helmut, and Reinhold Schneider. "Wavelet based fast solution of boundary integral equations." Universitätsbibliothek Chemnitz, 2006. http://nbn-resolving.de/urn:nbn:de:swb:ch1-200600649.
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This paper presents a wavelet Galerkin scheme for the fast solution of boundary integral equations. Wavelet Galerkin schemes employ appropriate wavelet bases for the discretization of boundary integral operators which yields quasi-sparse system matrices. These matrices can be compressed such that the complexity for solving a boundary integral equation scales linearly with the number of unknowns without compromising the accuracy of the underlying Galerkin scheme. Based on the wavelet Galerkin scheme we present also an adaptive algorithm. By numerical experiments we provide results which demonstrate the performance of our algorithm.
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Tang, W. "A generalized approach for transforming domain integrals into boundary integrals in boundary element methods." Thesis, University of Southampton, 1988. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.378981.
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Weggler, Lucy Verfasser], and Sergej [Akademischer Betreuer] [Rjasanow. "High order boundary element methods / Lucy Weggler. Betreuer: Sergej Rjasanow." Saarbrücken : Saarländische Universitäts- und Landesbibliothek, 2012. http://d-nb.info/1051586801/34.
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