Fast Multipole Boundary Element Method for Solving Two-dimensional Acoustic Wave Problems PDF Download
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Author: Milind Shrikant Bapat
Publisher:
ISBN:
Category :
Languages : en
Pages : 85
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Book Description
The boundary element method (BEM) is a numerical method for solving boundary value problems. The boundary element method has a clear advantage over other techniques like finite element method (FEM) in problems involving infinite domains. Hence the boundary element method has found many applications in the field of acoustics which often exist in infinite domains. The traditional approach for finding solutions to acoustic problems using the boundary element method has a computational complexity of the order O(N 2). This makes the computation very slow as the number of nodes increase. A new technique called fast multipole method (FMM) has emerged in the last decade. Replacing the normal matrix-vector multiplication with the fast multipole method reduces the computational time to order O(N). In this thesis the fast multipole method has been used to accelerate the boundary element method for 2-D acoustic wave problems. The relevant formulae are derived. It is shown that the computational time is of the order O(N) for this formulation. It is also observed that the memory required is much lesser and hence larger models can be solved. The formulation is a very basic one and gives good results as shown by the numerical examples. Use of higher-order elements and hypersingular formulation will result in a very capable and robust solver in the future.
Author: Milind Shrikant Bapat
Publisher:
ISBN:
Category :
Languages : en
Pages : 85
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Book Description
The boundary element method (BEM) is a numerical method for solving boundary value problems. The boundary element method has a clear advantage over other techniques like finite element method (FEM) in problems involving infinite domains. Hence the boundary element method has found many applications in the field of acoustics which often exist in infinite domains. The traditional approach for finding solutions to acoustic problems using the boundary element method has a computational complexity of the order O(N 2). This makes the computation very slow as the number of nodes increase. A new technique called fast multipole method (FMM) has emerged in the last decade. Replacing the normal matrix-vector multiplication with the fast multipole method reduces the computational time to order O(N). In this thesis the fast multipole method has been used to accelerate the boundary element method for 2-D acoustic wave problems. The relevant formulae are derived. It is shown that the computational time is of the order O(N) for this formulation. It is also observed that the memory required is much lesser and hence larger models can be solved. The formulation is a very basic one and gives good results as shown by the numerical examples. Use of higher-order elements and hypersingular formulation will result in a very capable and robust solver in the future.
Author: Yijun Liu
Publisher: Cambridge University Press
ISBN: 113947944X
Category : Technology & Engineering
Languages : en
Pages : 255
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Book Description
The fast multipole method is one of the most important algorithms in computing developed in the 20th century. Along with the fast multipole method, the boundary element method (BEM) has also emerged as a powerful method for modeling large-scale problems. BEM models with millions of unknowns on the boundary can now be solved on desktop computers using the fast multipole BEM. This is the first book on the fast multipole BEM, which brings together the classical theories in BEM formulations and the recent development of the fast multipole method. Two- and three-dimensional potential, elastostatic, Stokes flow, and acoustic wave problems are covered, supplemented with exercise problems and computer source codes. Applications in modeling nanocomposite materials, bio-materials, fuel cells, acoustic waves, and image-based simulations are demonstrated to show the potential of the fast multipole BEM. Enables students, researchers, and engineers to learn the BEM and fast multipole method from a single source.
Author: Liang Shen
Publisher:
ISBN:
Category :
Languages : en
Pages : 122
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Book Description
As a numerical method used in the simulations of many potential and acoustic problems, the boundary element method (BEM) has suffered from high solution cost for quite some time, although it has the advantage in the modeling or meshing stage. One way to improve the solution efficiency of the BEM is to use the fast multipole method (FMM). The reduction of the computing cost with the FMM is achieved by using multilevel clustering of the boundary elements, the use of multipole expansions of the fundamental solutions and adaptive fast multipole algorithms. In combination with iterative solvers, the fast multipole boundary element method (FMBEM) is capable of solving many large-scale 3-D problems on desktop PCs. In this dissertation, 3-D adaptive fast multipole boundary element methods for solving large-scale potential (e.g., thermal and electrostatic) and acoustic wave problems are developed. For large-scale potential problems, an adaptive fast multipole algorithm is developed in the FMBEM implementation. The conventional boundary integral equation (CBIE), hyper-singular boundary integral equation (HBIE) and their combination, dual boundary integral equation (CHBIE), are adopted and can be selectively chosen to solve different models. Both the conventional and the new fast multipole method with diagonal translations are implemented and their performances are compared. Implementation issues related to reusing the pre-conditioner and storing the coefficients to further improve the efficiency are addressed. Numerical examples, ranging from simple block models to heat sink and large-scale models of micro-electro-mechanical-systems are tested and presented. For large-scale acoustic problems, a modified version of adaptive fast multipole algorithm is developed for full-space problems first. The Burton-Miller formulation using a linear combination of the CBIE and HBIE is used to overcome the non-uniqueness difficulties in the BIEs for exterior problems. Several large-scale radiation and scattering problems, including scattering and radiating spheres and an engine model are tested. Then, the full-space algorithm is further modified and extended to solving half-space problems. Instead of using a tree structure that contains both real domain and its mirror image, the same tree structure that has been used in the full-space domain is used in the half-pace domain, which greatly simplifies the implementation of half-space FMBEM and reduces the memory storage size. Several examples including spheres sitting on the ground and sound barriers are tested. All the numerical examples of the potential and acoustic problems presented in this dissertation clearly demonstrate the effectiveness and efficiency of the developed adaptive fast multipole boundary element methods. The adaptive FMBEM code for potential problems and the adaptive FMEBM code for acoustic problems have been integrated in a single software package, which is well structured, modularized and extendable to handling other types of problems. Three journal papers have been published based on the work reported in this dissertation, and one journal paper on the half-space problem is in preparation. This dissertation research has significantly advanced the FMBEM for solving large-scale 3-D potential and acoustic problems. The developed adaptive fast multipole algorithms can be easily extended to the FMBEM for 3-D single-domain elasticity, Stokes flow, and multi-domain potential, acoustic, elasticity and Stokes problems for applications in large-scale modeling of composites, functionally-graded materials, micro-electro-mechanical-systems, and biological materials and fluids.
Author: Nail A Gumerov
Publisher: Elsevier
ISBN: 0080531598
Category : Mathematics
Languages : en
Pages : 551
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Book Description
This volume in the Elsevier Series in Electromagnetism presents a detailed, in-depth and self-contained treatment of the Fast Multipole Method and its applications to the solution of the Helmholtz equation in three dimensions. The Fast Multipole Method was pioneered by Rokhlin and Greengard in 1987 and has enjoyed a dramatic development and recognition during the past two decades. This method has been described as one of the best 10 algorithms of the 20th century. Thus, it is becoming increasingly important to give a detailed exposition of the Fast Multipole Method that will be accessible to a broad audience of researchers. This is exactly what the authors of this book have accomplished. For this reason, it will be a valuable reference for a broad audience of engineers, physicists and applied mathematicians. The Only book that provides comprehensive coverage of this topic in one location Presents a review of the basic theory of expansions of the Helmholtz equation solutions Comprehensive description of both mathematical and practical aspects of the fast multipole method and it's applications to issues described by the Helmholtz equation
Author: Yijun Liu
Publisher: Cambridge University Press
ISBN: 0521116597
Category : Mathematics
Languages : en
Pages : 255
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Book Description
First book on the fast multipole BEM, bringing together classical theory in BEM formulations and the fast multipole method.
Author: Stephen Kirkup
Publisher: Stephen Kirkup
ISBN: 9780953403103
Category : Acoustical engineering
Languages : en
Pages : 136
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Book Description
Author: Gernot Beer
Publisher: Springer Nature
ISBN: 3030233391
Category : Science
Languages : en
Pages : 335
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Book Description
This book discusses the introduction of isogeometric technology to the boundary element method (BEM) in order to establish an improved link between simulation and computer aided design (CAD) that does not require mesh generation. In the isogeometric BEM, non-uniform rational B-splines replace the Lagrange polynomials used in conventional BEM. This may seem a trivial exercise, but if implemented rigorously, it has profound implications for the programming, resulting in software that is extremely user friendly and efficient. The BEM is ideally suited for linking with CAD, as both rely on the definition of objects by boundary representation. The book shows how the isogeometric philosophy can be implemented and how its benefits can be maximised with a minimum of user effort. Using several examples, ranging from potential problems to elasticity, it demonstrates that the isogeometric approach results in a drastic reduction in the number of unknowns and an increase in the quality of the results. In some cases even exact solutions without refinement are possible. The book also presents a number of practical applications, demonstrating that the development is not only of academic interest. It then elegantly addresses heterogeneous and non-linear problems using isogeometric concepts, and tests them on several examples, including a severely non-linear problem in viscous flow. The book makes a significant contribution towards a seamless integration of CAD and simulation, which eliminates the need for tedious mesh generation and provides high-quality results with minimum user intervention and computing.
Author:
Publisher:
ISBN:
Category : Acoustical engineering
Languages : en
Pages : 274
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Book Description
Author: Kausik Mitra
Publisher:
ISBN:
Category :
Languages : en
Pages :
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Book Description
The Boundary Element Method (BEM) has long been considered to be a viable alternative to the Finite Element Method (FEM) for doing engineering analysis. The BEM reduces the dimensions of the problem by one and leads to smaller system of equations. One of the inherent limitations of the BEM has been the long time required for the solution of large problems. This makes the BEM prohibitively expensive to use while solving large problems involving crack propagation, elastodynamics, etc. This thesis is a successful attempt at reducing the solution time for the BEM. An iterative solver has been developed and the advantages it offers over the direct solver have been presented. The fast multipole method is a method used to reduce the number of computations while solving N body problems in astrophysics and molecular dynamics. A numerical formulation for accelerating the computation of boundary integrals based on the fast multipole method has been presented. An algorithm has been developed and it has been applied to the BEM for two-dimensional potential problems. It has been found that the use of this algorithm leads to savings in CPU time for large number of nodes. This method is very promising and future research can concentrate on improving the code so that more significant savings in time can be obtained.
Author: Bo Wang
Publisher:
ISBN:
Category : Boundary element methods
Languages : en
Pages : 74
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Book Description
