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 boun...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.展开更多
We propose a finite-difference ghost-point approach for the numerical solution of Cauchy-Navier equations in linear elasticity problems on arbitrary unbounded domains.The technique is based on a smooth coordinate tran...We propose a finite-difference ghost-point approach for the numerical solution of Cauchy-Navier equations in linear elasticity problems on arbitrary unbounded domains.The technique is based on a smooth coordinate transformation,which maps an unbounded domain into a unit square.Arbitrary geometries are defined by suitable level-set functions.The equations are discretized by classical nine-point stencil on interior points,while boundary conditions and high order reconstructions are used to define the field variables at ghost-points,which are grid nodes external to the domain with a neighbor inside the domain.The linear system arising from such discretization is solved by a multigrid strategy.The approach is then applied to solve elasticity problems in volcanology for computing the displacement caused by pressure sources.The method is suitable to treat problems in which the geometry of the source often changes(explore the effects of different scenarios,or solve inverse problems in which the geometry itself is part of the unknown),since it does not require complex re-meshing when the geometry is modified.Several numerical tests are successfully performed,which asses the effectiveness of the present approach.展开更多
With the development of numerical methods the numerical computations require higher and higher accuracy. This paper is devoted to the high-order local absorbing boundary conditions （ABCs） for heat equation. We prove...With the development of numerical methods the numerical computations require higher and higher accuracy. This paper is devoted to the high-order local absorbing boundary conditions （ABCs） for heat equation. We proved that the coupled system yields a stable problem between the obtained high-order local ABCs and the partial differential equation in the computational domain. This method has been used widely in wave propagation models only recently. We extend the spirit of the methodology to parabolic ones, which will become a basis to design the local ABCs for a class of nonlinear PDEs. Some numerical tests show that the new treatment is very efficient and tractable.展开更多
This paper discusses the numerical solution of Burgers＇ equation on unbounded domains. Two artificial boundaries are introduced and boundary conditions are obtained on the artificial boundaries, which are in nonlinea...This paper discusses the numerical solution of Burgers＇ equation on unbounded domains. Two artificial boundaries are introduced and boundary conditions are obtained on the artificial boundaries, which are in nonlinear forms. Then the original problem is reduced to an equivalent problem on a bounded domain. Finite difference method is applied to the reduced problem, and some numerical examples are given to show the effectiveness of the new approach.展开更多
In framework of the fictitious domain methods with immersed interfaces for the elasticity problem,the present contribution is to study and numerically validate the jump-integrated boundary conditions method with sharp...In framework of the fictitious domain methods with immersed interfaces for the elasticity problem,the present contribution is to study and numerically validate the jump-integrated boundary conditions method with sharp interface for the vector elasticity system discretized by a proposed finite volume method.The main idea of the fictitious domain approach consists in embedding the original domain of study into a geometrically larger and simpler one called the fictitious domain.Here,we present a cell-centered finite volume method to discretize the fictitious domain problem.The proposed method is numerically validated for different test cases.This work can be considered as a first step before more challenging problems such as fluid-structure interactions or moving interface problems.展开更多
We propose an artificial boundary method for solving the deterministic Kardar-Parisi-Zhang equation in one-,two-and three dimensional unbounded domains.The exact artificial boundary conditions are obtained on the arti...We propose an artificial boundary method for solving the deterministic Kardar-Parisi-Zhang equation in one-,two-and three dimensional unbounded domains.The exact artificial boundary conditions are obtained on the artificial boundaries.Then the original problems are reduced to equivalent problems in bounded domains.A fi-nite difference method is applied to solve the reduced problems,and some numerical examples are provided to show the effectiveness of the method.展开更多
An explicit finite element-finite difference method for analyzing the effects of two-dimensional visco-elastic localtopography on earthquake ground motion is prOPosed in this paper. In the method, at first, the finite...An explicit finite element-finite difference method for analyzing the effects of two-dimensional visco-elastic localtopography on earthquake ground motion is prOPosed in this paper. In the method, at first, the finite elementdiscrete model is formed by using the artificial boundary and finite element method, and the dynamic equationsof local nodes in the discrete model are obtained according to the theory of the special finite element method similar to the finite difference method, and then the explicit step-by-step integration formulas are presented by usingthe explicit difference method for solving the visco-elastic dynamic equation and Generalized Multi-transmittingBoundary. The method has the advantages of saving computing time and computer memory space, and it is suitable for any case of topography and has high computing accuracy and good computing stability.展开更多
This work presents a fast Cartesian grid-based integral equation method for unbounded interface problems with non-homogeneous source terms.The unbounded interface problem is solved with boundary integral equation meth...This work presents a fast Cartesian grid-based integral equation method for unbounded interface problems with non-homogeneous source terms.The unbounded interface problem is solved with boundary integral equation methods such that infinite boundary conditions are satisfied naturally.This work overcomes two difficulties.The first difficulty is the evaluation of singular integrals.Boundary and volume integrals are transformed into equivalent but much simpler bounded interface problems on rectangular domains,which are solved with FFT-based finite difference solvers.The second one is the expensive computational cost for volume integrals.Despite the use of efficient interface problem solvers,the evaluation for volume integrals is still expensive due to the evaluation of boundary conditions for the simple interface problem.The problem is alleviated by introducing an auxiliary circle as a bridge to indirectly evaluate boundary conditions.Since solving boundary integral equations on a circular boundary is so accurate,one only needs to select a fixed number of points for the discretization of the circle to reduce the computational cost.Numerical examples are presented to demonstrate the efficiency and the second-order accuracy of the proposed numerical method.展开更多
In this paper, some V-cycle multigrid algorithms are presented for the coupling system arising from the discretization of the Dirichlet exterior problem by coupling the natural boundary element method and finite eleme...In this paper, some V-cycle multigrid algorithms are presented for the coupling system arising from the discretization of the Dirichlet exterior problem by coupling the natural boundary element method and finite element method. The convergence of these multigrid algorithms is obtained even with only one smoothing on all levels. The rate of convergence is found uniformly bounded independent of the number of levels and the mesh sizes of all levels, which indicates that these multigrid algorithms are optimal. Some numerical results are also reported.展开更多
文摘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.
基金the OTRIONS project under the European Territorial Cooperation Programme Greece-Italy 2007-2013,and by PRIN 2009“Innovative numerical methods for hyperbolic problems with applications to fluid dynamics,kinetic theory and computational biology”.
文摘We propose a finite-difference ghost-point approach for the numerical solution of Cauchy-Navier equations in linear elasticity problems on arbitrary unbounded domains.The technique is based on a smooth coordinate transformation,which maps an unbounded domain into a unit square.Arbitrary geometries are defined by suitable level-set functions.The equations are discretized by classical nine-point stencil on interior points,while boundary conditions and high order reconstructions are used to define the field variables at ghost-points,which are grid nodes external to the domain with a neighbor inside the domain.The linear system arising from such discretization is solved by a multigrid strategy.The approach is then applied to solve elasticity problems in volcanology for computing the displacement caused by pressure sources.The method is suitable to treat problems in which the geometry of the source often changes(explore the effects of different scenarios,or solve inverse problems in which the geometry itself is part of the unknown),since it does not require complex re-meshing when the geometry is modified.Several numerical tests are successfully performed,which asses the effectiveness of the present approach.
文摘With the development of numerical methods the numerical computations require higher and higher accuracy. This paper is devoted to the high-order local absorbing boundary conditions （ABCs） for heat equation. We proved that the coupled system yields a stable problem between the obtained high-order local ABCs and the partial differential equation in the computational domain. This method has been used widely in wave propagation models only recently. We extend the spirit of the methodology to parabolic ones, which will become a basis to design the local ABCs for a class of nonlinear PDEs. Some numerical tests show that the new treatment is very efficient and tractable.
基金Research is supported in part by National Natural Science Foundation of China （No. 10471073） and RGC of Hong Kong and in part by RGC of Hong Kong and FRG of Hong Kong Baptist University.
文摘This paper discusses the numerical solution of Burgers＇ equation on unbounded domains. Two artificial boundaries are introduced and boundary conditions are obtained on the artificial boundaries, which are in nonlinear forms. Then the original problem is reduced to an equivalent problem on a bounded domain. Finite difference method is applied to the reduced problem, and some numerical examples are given to show the effectiveness of the new approach.
文摘In framework of the fictitious domain methods with immersed interfaces for the elasticity problem,the present contribution is to study and numerically validate the jump-integrated boundary conditions method with sharp interface for the vector elasticity system discretized by a proposed finite volume method.The main idea of the fictitious domain approach consists in embedding the original domain of study into a geometrically larger and simpler one called the fictitious domain.Here,we present a cell-centered finite volume method to discretize the fictitious domain problem.The proposed method is numerically validated for different test cases.This work can be considered as a first step before more challenging problems such as fluid-structure interactions or moving interface problems.
基金National Natural Science Foundation of China,Hong Kong Research Grants Council and FRG of Hong Kong Baptist University.
文摘We propose an artificial boundary method for solving the deterministic Kardar-Parisi-Zhang equation in one-,two-and three dimensional unbounded domains.The exact artificial boundary conditions are obtained on the artificial boundaries.Then the original problems are reduced to equivalent problems in bounded domains.A fi-nite difference method is applied to solve the reduced problems,and some numerical examples are provided to show the effectiveness of the method.
文摘An explicit finite element-finite difference method for analyzing the effects of two-dimensional visco-elastic localtopography on earthquake ground motion is prOPosed in this paper. In the method, at first, the finite elementdiscrete model is formed by using the artificial boundary and finite element method, and the dynamic equationsof local nodes in the discrete model are obtained according to the theory of the special finite element method similar to the finite difference method, and then the explicit step-by-step integration formulas are presented by usingthe explicit difference method for solving the visco-elastic dynamic equation and Generalized Multi-transmittingBoundary. The method has the advantages of saving computing time and computer memory space, and it is suitable for any case of topography and has high computing accuracy and good computing stability.
基金supported by the National Key R&D Program of China(Project No.2020YFA0712000)supported by the Shanghai Science and Technology Innovation Action Plan in Basic Research Area(Project No.22JC1401700)+1 种基金the Strategic Priority Research Program of Chinese Academy of Sciences(Grant No.XDA25010405)the National Natural Science Foundation of China(Grant No.DMS-11771290).
文摘This work presents a fast Cartesian grid-based integral equation method for unbounded interface problems with non-homogeneous source terms.The unbounded interface problem is solved with boundary integral equation methods such that infinite boundary conditions are satisfied naturally.This work overcomes two difficulties.The first difficulty is the evaluation of singular integrals.Boundary and volume integrals are transformed into equivalent but much simpler bounded interface problems on rectangular domains,which are solved with FFT-based finite difference solvers.The second one is the expensive computational cost for volume integrals.Despite the use of efficient interface problem solvers,the evaluation for volume integrals is still expensive due to the evaluation of boundary conditions for the simple interface problem.The problem is alleviated by introducing an auxiliary circle as a bridge to indirectly evaluate boundary conditions.Since solving boundary integral equations on a circular boundary is so accurate,one only needs to select a fixed number of points for the discretization of the circle to reduce the computational cost.Numerical examples are presented to demonstrate the efficiency and the second-order accuracy of the proposed numerical method.
基金This WOrk is supported by the National Basic Research Program of China under the grant 2005CB321701the National Natural Science Foundation of China under the grant 10531080 and 10601045the Research Starting Fund of Nankai University
文摘In this paper, some V-cycle multigrid algorithms are presented for the coupling system arising from the discretization of the Dirichlet exterior problem by coupling the natural boundary element method and finite element method. The convergence of these multigrid algorithms is obtained even with only one smoothing on all levels. The rate of convergence is found uniformly bounded independent of the number of levels and the mesh sizes of all levels, which indicates that these multigrid algorithms are optimal. Some numerical results are also reported.
