Fast Algorithms for Composite Materials

1995 ◽  
Vol 408 ◽  
Author(s):  
L. Greengard
Keyword(s):  
Composite Materials ◽  
Large Scale ◽  
Fast Multipole Method ◽  
Fast Algorithms ◽  
Boundary Integral ◽  
Cpu Time ◽  
Electrostatic Fields ◽  
Background Material ◽  
Uniform Background ◽  
Time Required

AbstractWe briefly review recently developed fast algorithms for the evaluation of electrostatic fields in composite materials consisting of a collection of piecewise homogeneous inclusions embedded in a uniform background. These algorithms are based on combining a suitable boundary integral equation with the fast multipole method and a conjugate gradient-like iterative method. The CPU time required grows linearly with the number of points in the discretization of the interface between the inclusions and the background material, bringing large-scale calculations within practical reach.

On the numerical evaluation of electrostatic fields in composite materials

Acta Numerica ◽  
1994 ◽  
Vol 3 ◽  
pp. 379-410 ◽  
Author(s):  
Leslie Greengard ◽  
Monique Moura
Keyword(s):  
Composite Material ◽  
Effective Properties ◽  
Fast Multipole Method ◽  
Classical Problem ◽  
Boundary Integral ◽  
Electrostatic Fields ◽  
Background Material ◽  
Uniform Background ◽  
Time Required

A classical problem in electrostatics is the determination of the effective electrical conductivity in a composite material consisting of a collection of piecewise homogeneous inclusions embedded in a uniform background. We discuss recently developed fast algorithms for the evaluation of the potential and electrostatic fields induced in multiphase composites by an applied potential, from which the desired effective properties may be easily obtained. The schemes are based on combining a suitable boundary integral equation with the Fast Multipole Method and the GMRES iterative method; the CPU time required grows linearly with the number of points in the discretization of the interface between the inclusions and the background material.A variety of other questions in electrostatics, magnetostatics and diffusion can be formulated in terms of interface problems. These include the evaluation of electrostatic fields in the presence of dielectric inclusions, the determination of magnetostatic fields in media with variable magnetic permeability, and the calculation of the effective thermal conductivity of a composite material. The methods presented here apply with minor modification to these other situations as well.

scholarly journals Statistical and machine learning models for optimizing energy in parallel applications

2019 ◽  
Vol 33 (6) ◽  
pp. 1079-1097 ◽  
Author(s):  
Mark Endrei ◽  
Chao Jin ◽  
Minh Ngoc Dinh ◽  
David Abramson ◽  
Heidi Poxon ◽  
...  
Keyword(s):  
Machine Learning ◽  
Energy Efficiency ◽  
High Performance ◽  
Large Scale ◽  
Energy Use ◽  
Parallel Applications ◽  
Learning Models ◽  
Trade Off ◽  
Time Required ◽  
Machine Learning Models

Rising power costs and constraints are driving a growing focus on the energy efficiency of high performance computing systems. The unique characteristics of a particular system and workload and their effect on performance and energy efficiency are typically difficult for application users to assess and to control. Settings for optimum performance and energy efficiency can also diverge, so we need to identify trade-off options that guide a suitable balance between energy use and performance. We present statistical and machine learning models that only require a small number of runs to make accurate Pareto-optimal trade-off predictions using parameters that users can control. We study model training and validation using several parallel kernels and more complex workloads, including Algebraic Multigrid (AMG), Large-scale Atomic Molecular Massively Parallel Simulator, and Livermore Unstructured Lagrangian Explicit Shock Hydrodynamics. We demonstrate that we can train the models using as few as 12 runs, with prediction error of less than 10%. Our AMG results identify trade-off options that provide up to 45% improvement in energy efficiency for around 10% performance loss. We reduce the sample measurement time required for AMG by 90%, from 13 h to 74 min.

scholarly journals Geometric series expansion of the Neumann–Poincaré operator: Application to composite materials

2021 ◽  
pp. 1-26
Author(s):  
ELENA CHERKAEV ◽  
MINWOO KIM ◽  
MIKYOUNG LIM
Keyword(s):  
Composite Materials ◽  
Series Expansion ◽  
Function Theory ◽  
Effective Conductivity ◽  
Two Dimensions ◽  
Boundary Integral ◽  
Geometric Function Theory ◽  
Boundary Integral Formulation ◽  
Geometric Function ◽  
Series Expression

The Neumann–Poincaré (NP) operator, a singular integral operator on the boundary of a domain, naturally appears when one solves a conductivity transmission problem via the boundary integral formulation. Recently, a series expression of the NP operator was developed in two dimensions based on geometric function theory [34]. In this paper, we investigate geometric properties of composite materials using this series expansion. In particular, we obtain explicit formulas for the polarisation tensor and the effective conductivity for an inclusion or a periodic array of inclusions of arbitrary shape with extremal conductivity, in terms of the associated exterior conformal mapping. Also, we observe by numerical computations that the spectrum of the NP operator has a monotonic behaviour with respect to the shape deformation of the inclusion. Additionally, we derive inequality relations of the coefficients of the Riemann mapping of an arbitrary Lipschitz domain using the properties of the polarisation tensor corresponding to the domain.

Recent Development of the Fast Multipole Boundary Element Method for Modeling Acoustic Problems

2009 ◽  
Author(s):  
Yijun Liu ◽  
Milind Bapat
Keyword(s):  
Boundary Element Method ◽  
Acoustic Wave ◽  
Boundary Element ◽  
Large Scale ◽  
Boundary Integral ◽  
Practical Significance ◽  
Fast Multipole ◽  
Tree Structures ◽  
Wave Problems ◽  
Element Method

Some recent development of the fast multipole boundary element method (BEM) for modeling acoustic wave problems in both 2-D and 3-D domains are presented in this paper. First, the fast multipole BEM formulation for 2-D acoustic wave problems based on a dual boundary integral equation (BIE) formulation is presented. Second, some improvements on the adaptive fast multipole BEM for 3-D acoustic wave problems based on the earlier work are introduced. The improvements include adaptive tree structures, error estimates for determining the numbers of expansion terms, refined interaction lists, and others in the fast multipole BEM. Examples involving 2-D and 3-D radiation and scattering problems solved by the developed 2-D and 3-D fast multipole BEM codes, respectively, will be presented. The accuracy and efficiency of the fast multipole BEM results clearly demonstrate the potentials of the fast multipole BEM for solving large-scale acoustic wave problems that are of practical significance.

Fast Multipole Virtual Boundary Element - Least Square Method for Solving 2-D Elastic Problems

2011 ◽  
Vol 204-210 ◽  
pp. 2196-2201
Author(s):  
Yan Tao Jiang ◽  
Si Tian Chen ◽  
Cheng Hua Li
Keyword(s):  
Boundary Element ◽  
Large Scale ◽  
Fast Multipole Method ◽  
Main Idea ◽  
Least Square Method ◽  
Least Square ◽  
Fast Multipole ◽  
Large Scale Problems ◽  
Virtual Boundary ◽  
Virtual Boundary Element

In this paper, the fast multipole virtual boundary element - least square method (Fast Multipole VBE - LSM) is proposed and used to simulate 2-D elastic problems, which is based on the fast multipole method (FMM) and virtual boundary element - least square method (VBE - LSM).The main idea of the method is to change computational model by applying the FMM to conventional VBE - LSM. The memory and operations could be reduced to be of linear proportion to the degree of freedom (DOF) and large scale problems could be effectively solved on a common desktop with this method. Numerical results show that this method holds virtues of high feasibility, accuracy and efficiency. Moreover, the idea of this method can be generalized and extended in application.

2D Structural Acoustic Analysis Using the FEM/FMBEM with Different Coupled Element Types

2017 ◽  
Vol 42 (1) ◽  
pp. 37-48 ◽  
Author(s):  
Leilei Chen ◽  
Wenchang Zhao ◽  
Cheng Liu ◽  
Haibo Chen
Keyword(s):  
Acoustic Analysis ◽  
Interaction Analysis ◽  
Fast Multipole Method ◽  
Boundary Elements ◽  
Higher Order ◽  
Boundary Integral ◽  
The Matrix ◽  
Coupling Approach ◽  
Element Type ◽  
Relative Errors

Abstract A FEM-BEM coupling approach is used for acoustic fluid-structure interaction analysis. The FEM is used to model the structure and the BEM is used to model the exterior acoustic domain. The aim of this work is to improve the computational efficiency and accuracy of the conventional FEM-BEM coupling approach. The fast multipole method (FMM) is applied to accelerating the matrix-vector products in BEM. The Burton-Miller formulation is used to overcome the fictitious eigen-frequency problem when using a single Helmholtz boundary integral equation for exterior acoustic problems. The continuous higher order boundary elements and discontinuous higher order boundary elements for 2D problem are developed in this work to achieve higher accuracy in the coupling analysis. The performance for coupled element types is compared via a simple example with analytical solution, and the optimal element type is obtained. Numerical examples are presented to show the relative errors of different coupled element types.

scholarly journals Mathematical models and a constructive heuristic for finding minimum fundamental cycle bases

2005 ◽  
Vol 15 (1) ◽  
pp. 15-24 ◽  
Author(s):  
Leo Liberti ◽  
Edoardo Amaldi ◽  
Francesco Maffioli ◽  
Nelson Maculan
Keyword(s):  
Linear Programming ◽  
Total Cost ◽  
Fundamental Cycle ◽  
Cycle Basis ◽  
Cpu Time ◽  
Constructive Heuristic ◽  
Cycle Bases ◽  
Time Required ◽  
Application Fields

The problem of finding a fundamental cycle basis with minimum total cost in a graph arises in many application fields. In this paper we present some integer linear programming formulations and we compare their performances, in terms of instance size, CPU time required for the solution, and quality of the associated lower bound derived by solving the corresponding continuous relaxations. Since only very small instances can be solved to optimality with these formulations and very large instances occur in a number of applications, we present a new constructive heuristic and compare it with alternative heuristics.

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