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artificial boundary method FOR THE THREE-DIMENSIONAL EXTERIOR PROBLEM OF ELASTICITY

Journal of Computational Mathematics 2005年第6期23卷 603-618页

作者： Hou-de Han Chun-xiong Zheng Department of Mathematical Sciences Tsinghua University Beijing 100084 China Department of Mathematics University of Science and Technology of China Hefei 230026 China

The exact boundary condition on a spherical artificial boundary is derived for the three-dimensional exterior problem of linear elasticity in this paper. After this boundary condition is imposed on the artificial boundary, a reduced problem only defined in a bounded domain is obtained. A series of approximate problems with increasing accuracy can be derived if one truncates the series term in the variational formulation, which is equivalent to the reduced problem. An error estimate is presented to show how the error depends on the finite element discretization and the accuracy of the approximate problem. In the end, a numerical example is given to demonstrate the performance of the proposed method.

关键词： artificial boundary method unbounded domains elasticity

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artificial boundary method for the Zakharov-Rubenchik equations

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NUMERICAL ALGORITHMS 2024年第3期97卷 1017-1037页

作者： Li, Hongwei Zhang, Xiangyu Shandong Normal Univ Sch Math & Stat Jinan 250358 Peoples R China

The paper aims to study the numerical solution of the Zakharov-Rubenchik equations on unbounded domains, which model the propagation of two kinds of waves in plasma physics, by applying the artificial boundary method. Based on ideas of the operator splitting method and artificial boundary method of hyperbolic system and nonlinear Schrodinger equation, the artificial boundary conditions of the Zakharov-Rubenchik equations are designed on the introduced artificial boundaries to reduce the original problem defined on an unbounded domain into an initial boundary value problem on the bounded computational domain, which can be efficiently solved by finite difference method. A series of auxiliary variables is introduced to rigorously analyze the stability of the reduced initial boundary value problem through energy estimation. Numerical examples are presented to demonstrate the effectiveness of the proposed artificial boundary conditions, validate the theoretical analysis, and simulate the physical phenomena.

关键词： Zakharov-Rubenchik equations artificial boundary method Stability

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artificial boundary method for Anisotropic Problems in Semi-infinite Strips

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ENGINEERING LETTERS 2018年第4期26卷 473-477页

作者： Chen, Yajun Du, Qikui Nanjing Normal Univ Sch Math Sci Nanjing 210023 Jiangsu Peoples R China Shanghai Maritime Univ Dept Math Shanghai 200136 Peoples R China

In this paper, the artificial boundary method for anisotropic problems in semi-infinite strips is investigated. The exact and approximate boundary conditions on a segment artificial boundary are given. Finite element approximations are applied to the problem in a bounded computational domain and error estimates are obtained. Finally, some numerical examples show the effectiveness of this method.

关键词： artificial boundary method anisotropic problem error estimate

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THE ELLIPSOID artificial boundary method FOR THREE-DIMENSIONAL UNBOUNDED DOMAINS

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Journal of Computational Mathematics 2009年第2期27卷 196-214页

作者： Hongying Huang Dehao Yu Information College Zhejiang Ocean University Zhoushan 316004 China LSEC ICMSEC Academy of Mathematics and System Sciences Chinese Academy of SciencesBeijing 100080 China

The artificial boundary method is applied to solve three-dimensional exterior problems. Two kind of rotating ellipsoids are chosen as the artificial boundaries and the exact artificial boundary conditions are derived explicitly in terms of an infinite series. Then the well-posedness of the coupled variational problem is obtained. It is found that error estimates derived depend on the mesh size, truncation term and the location of the artificial boundary. Three numerical examples are presented to demonstrate the effectiveness and accuracy of the proposed method.

关键词： artificial boundary method Exterior harmonic problem Finite element method,Natural boundary reduction Oblate ellipsoid Prolate ellipsoid.

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The artificial boundary method for a Nonlinear Interface Problem on Unbounded Domain

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CMES-COMPUTER MODELING IN ENGINEERING & SCIENCES 2008年第3期35卷 227-252页

作者： Yu, De-hao Huang, Hong-ying Chinese Acad Sci ICMSEC LSEC Acad Math & Syst Sci Beijing 100190 Peoples R China Zhejiang Ocean Univ Sch Math Phys & Informat Sci Zhoushan 316004 Zhejiang Peoples R China

In this paper, we apply the artificial boundary method to solve a three-dimensional nonlinear interface problem on an unbounded domain. A spherical or ellipsoidal surface as the artificial boundary is introduced. The exact artificial boundary conditions are derived explicitly in terms of an infinite series and then the well-posedness of the coupled weak formulation in a bounded domain, which is equivalent to the original problem in the unbounded domain, is obtained. The error estimate depends on the mesh size, the term after truncating the infinite series and the location of the artificial boundary. Some numerical examples are presented to demonstrate the effectiveness and accuracy of this method.

关键词： artificial boundary method Dirichlet to Neumann mapping Finite element method Natural integral operator Nonlinear interface problem

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An artificial boundary method for Burgers' Equation in the Unbounded Domain

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CMES-COMPUTER MODELING IN ENGINEERING & SCIENCES 2014年第6期100卷 445-461页

作者： Zheng, Quan Fan, Lei Li, Xuezheng North China Univ Technol Coll Sci Beijing 100144 Peoples R China

In this paper, we construct a numerical method for one-dimensional Burgers' equation in the unbounded domain by using artificial boundary conditions. The original problem is converted by Hopf-Cole transformation to the heat equation in the unbounded domain, the latter is reduced to an equivalent problem in a bounded computational domain by using two artificial integral boundary conditions, a finite difference method with discrete artificial boundary conditions is established by using the method of reduction of order for the last problem, and thereupon the numerical solution of Burgers' equation is obtained. This artificial boundary method is proved and verified to be uniquely solvable, unconditionally stable and convergent with the order 2 in space and the order 3/2 in time for solving. Burgers' equation on the computational domain.

关键词： Burgers' equation Unbounded domain Hopf-Cole transformation artificial boundary method Finite difference method

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A survey on artificial boundary method Dedicated to Professor Shi Zhong-Ci on the Occasion of his 80th Birthday

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Science China Mathematics 2013年第12期56卷 2439-2488页

作者： HAN HouDe WU XiaoNan Department of Mathematical Sciences Tsinghua University Department of Mathematics Hong Kong Baptist University

The artificial boundary method is one of the most popular and effective numerical methods tor solving partial differential equations on unbounded domains, with more than thirty years development. The artificiM boundary method has reached maturity in recent years. It has been applied to various problems in scientific and engineering computations, and the theoretical issues such as the convergence and error estimates of the artificial boundary method have been solved gradually. This paper reviews the development and discusses different forms of the artificial boundary method.

关键词： artificial boundary method global artificial boundary condition local artificial boundary condi-tion discrete artificial boundary condition implicit artificial boundary condition nonlinear artificial boundarycondition

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Scaled boundary Perfectly Matched Layer (SBPML): A novel 3D time-domain artificial boundary method for wave problem in general-shaped and heterogeneous infinite domain

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COMPUTER methodS IN APPLIED MECHANICS AND ENGINEERING 2023年第PartB期403卷

作者： Zhang, Guoliang Zhao, Mi Zhang, Junqi Du, Xiuli Beijing Univ Technol Key Lab Urban Secur & Disaster Engn Minist Educ Beijing 100124 Peoples R China Tsinghua Univ Dept Hydraul Engn Beijing 100084 Peoples R China

artificial boundary method is widely used in the numerical modeling of unbounded wave problem. However, the accurate modeling of truncated infinite domain with general geometry and heterogeneous materials is still a challenging task, especially for direct time-domain analysis in three dimensions (3D). In this paper, a novel 3D time-domain artificial boundary method, called Scaled boundary Perfectly Matched Layer (SBPML), is proposed. This method is a generalization of the Perfectly Matched Layer (PML) based on a scaled boundary coordinates transformation inspired by the Scaled boundary Finite Element method (SBFEM), which is capable of using artificial boundary of general geometry (not necessarily convex) and considering plane physical surfaces and interfaces extending to infinity in the truncated infinite domain. Local scaled boundary coordinates are firstly introduced on the element-level into the truncated infinite domain to describe general geometry properties of the infinite domain. Then, a complex stretching function from PML is applied to radial direction of the local scaled boundary coordinates to map the physical space onto the complex space, resulting in a SBPML domain. The spatial discretization of the SBPML domain produces semi-discrete mixed displacement-stress unsplit-field formulations of third orders in time. The order of the obtained formulation can be reduced by one, enabling a seamless coupling with the standard displacementbased finite element formulation of the interior domain. The coupled system can be solved by an explicit time integration algorithm efficiently. The validation of the SBPML is demonstrated through several benchmark tests, including wave problems in unbounded domains with general geometries and heterogeneous material properties. Furthermore, the application of the SBPML in dynamic soil-structure interaction (SSI) is demonstrated using two engineering problems, including an impact analysis of soft rock-nuclear island system and a

关键词： artificial boundary method Unbounded domain Perfectly matched layer (PML) Scaled boundary finite element method (SBFEM) Explicit finite element method Dynamic soil-structure interaction

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Fast artificial boundary method for the heat equation on unbounded domains with strip tails

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2023年第1期425卷

作者： Zheng, Chunxiong Xie, Jiangming Xinjiang Univ Coll Math & Syst Sci Urumqi 830046 Peoples R China Tsinghua Univ Dept Math Sci Beijing 100084 Peoples R China

We propose a fast numerical algorithm for the multi-dimensional heat equation on unbounded domains with strips tails. The artificial boundary method is used to confine the computational domains. By applying BDF2 for the time discretization and performing the Z transform, we derive an exact semi-discrete artificial boundary condition which contains the discrete temporal convolution and the surface Laplacian operator. Based on the best relative Chebyshev approximation of the square-root function, we design a fast algorithm to approximate and localize the exact semi-discrete artificial boundary condition. The spatial discretization is realized by the Galerkin finite element method with a special boundary treatment. A complete error estimate for the fully discrete problems is performed, and numerical tests are presented to demonstrate the accuracy and efficiency of the proposed algorithm.(c) 2022 Elsevier B.V. All rights reserved.

关键词： Unbounded domains artificial boundary method Fast algorithm Error estimate

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Numerical solution of convected wave equation in free field using artificial boundary method

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NUMERICAL methodS FOR PARTIAL DIFFERENTIAL EQUATIONS 2024年第6期40卷

作者： Wang, Xin Wang, Jihong Di, Yana Zhang, Jiwei Wuhan Univ Sch Math & Stat Wuhan Peoples R China Zhejiang Lab Res Ctr Appl Math & Machine Intelligence Hangzhou Peoples R China Beijing Normal Univ Res Ctr Math Zhuhai Peoples R China BNU HKBU United Int Coll Div Sci & Technol Zhuhai Peoples R China Wuhan Univ Hubei Key Lab Computat Sci Wuhan Hubei Peoples R China

In this article, we propose two procedures focusing on the computation of the time-dependent convected wave equation in a free field with a uniform background flow. Both procedures are based on a framework, expended from Du et al. (SIAM J. Sci. Comput. 40 (2018), A1430-A1445.), of constructing the Dirichlet-to-Dirichlet (DtD)-type discrete absorbing boundary conditions (ABCs). The first procedure is dedicated to reducing the infinite problem into a finite problem by a direct application of the framework on the finite difference discretization of the convected wave equation. However, the presence of convection terms makes the stability analysis hard to implement, which motivates us to develop the second procedure. First, the convected wave equation is transformed into a standard wave equation by using the Prandtl-Glauert-Lorentz transformation. After that, we obtain the DtD-type ABC by using the above framework, and on this basis, derive an equivalent Dirichlet-to-Neumann-type ABCs, which makes stability and convergence analysis easy to be obtained by the classical energy method. The effectiveness and comparison of these two procedures are investigated through numerical experiments.

关键词： artificial boundary method convected wave equation DtD-type absorbing boundary conditions DtN-type absorbing boundary conditions free field Prandtl-Glauert-Lorentz transformation stability analysis

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