Spectral element method is well known as high-order method, and has potential better parallel feature as compared with low order methods. In this paper, a parallel preconditioned conjugate gradient iterative method is...Spectral element method is well known as high-order method, and has potential better parallel feature as compared with low order methods. In this paper, a parallel preconditioned conjugate gradient iterative method is proposed to solving the spectral element approximation of the Helmholtz equation. The parallel algorithm is shown to have good performance as compared to non parallel cases, especially when the stiffness matrix is not memorized. A series of numerical experiments in one dimensional case is carried out to demonstrate the efficiency of the proposed method.展开更多
In this paper, we consider solving the Helmholtz equation in the Cartesian domain , subject to homogeneous Dirichlet boundary condition, discretized with the Chebyshev pseudo-spectral method. The main purpose of this ...In this paper, we consider solving the Helmholtz equation in the Cartesian domain , subject to homogeneous Dirichlet boundary condition, discretized with the Chebyshev pseudo-spectral method. The main purpose of this paper is to present the formulation of a two-level decomposition scheme for decoupling the linear system obtained from the discretization into independent subsystems. This scheme takes advantage of the homogeneity property of the physical problem along one direction to reduce a 2D problem to several 1D problems via a block diagonalization approach and the reflexivity property along the second direction to decompose each of the 1D problems to two independent subproblems using a reflexive decomposition, effectively doubling the number of subproblems. Based on the special structure of the coefficient matrix of the linear system derived from the discretization and a reflexivity property of the second-order Chebyshev differentiation matrix, we show that the decomposed submatrices exhibits a similar property, enabling the system to be decomposed using reflexive decompositions. Explicit forms of the decomposed submatrices are derived. The decomposition not only yields more efficient algorithm but introduces coarse-grain parallelism. Furthermore, it preserves all eigenvalues of the original matrix.展开更多
针对声学有限元分析中四节点等参单元计算精度低,对网格质量敏感的问题,将无网格径向插值技术引入到标准有限元中,构造径向插值形函数,推导径向插值有限元法(Radial interpolation finite element method,RIFEM)的二维声学数值计算公式...针对声学有限元分析中四节点等参单元计算精度低,对网格质量敏感的问题,将无网格径向插值技术引入到标准有限元中,构造径向插值形函数,推导径向插值有限元法(Radial interpolation finite element method,RIFEM)的二维声学数值计算公式。二维声学RIFEM采用标准有限元法形函数构造系统离散方程的声学刚度矩阵和边界积分矢量,保证了声压梯度和边界条件在区域边界的积分精度;采用径向插值形函数构造系统离散方程的质量矩阵,提高了声压数值近似函数的插值精度。对管道二维声腔模型和某轿车二维声腔模型的数值分析结果表明,与标准有限元法和SFEM相比,RIFEM的计算精度更高,对波数、单元尺寸和网格扭曲程度的灵敏度更低。因此RIFEM可以很好地应用于二维声学数值分析,具有广阔的工程应用前景。展开更多
The 2008 Wenchuan earthquake occurred in an active earthquake zone, i.e., Longmenshan tectonic zone. Seismic waves triggered by this earthquake can be used to explore the characteristics of the fault rupture process a...The 2008 Wenchuan earthquake occurred in an active earthquake zone, i.e., Longmenshan tectonic zone. Seismic waves triggered by this earthquake can be used to explore the characteristics of the fault rupture process and the hierarchical structure of the Earth's interior. We employ spectral element method incorporated with large-scale parallel computing technology, to investigate the characteristics of seismic wave propagation excited by Wenchuan earthquake. We calculate synthetic seismograms with one-point source model and three-point source model respectively. The AK135 model is employed as a prototype of our numerical global Earth model. The Earth's ellipticity, Earth’s medium attenuation, and topography data are taken into consideration. These wave propagation processes are simulated by solving three-dimensional elastic wave governing equations. Three-dimensional visualization of our numerical results displays the profile of the seismic wave propagation. The three-point source, which is proposed from the latest investigations through field observation and reverse estimation, can better demonstrate the spatial and temporal characteristics of the source rupture process than the one-point source. We take comparison of synthetic seismograms with observational data recorded at 16 observatory stations. Primary results show that the synthetic seismograms calculated from three-point source agree well with the observations. This can further reveal that the source rupture process of Wenchuan earthquake is a multi-rupture process, which is composed by at least three or more stages of rupture processes.展开更多
This paper introduce a cascadic multigrid method for solving semilinear elliptic equations based on a multilevel correction method.Instead of the common costly way of directly solving semilinear elliptic equation on a...This paper introduce a cascadic multigrid method for solving semilinear elliptic equations based on a multilevel correction method.Instead of the common costly way of directly solving semilinear elliptic equation on a very fine space,the new method contains some smoothing steps on a series of multilevel finite element spaces and some solving steps to semilinear elliptic equations on a very coarse space.To prove the efficiency of the new method,we derive two results,one of the optimal convergence rate by choosing the appro- priate sequence of finite element spaces and the number of smoothing steps,and the other of the optimal computational work by applying the parallel computing technique.Moreover,the requirement of bounded second order derivatives of nonlinear term in the existing multigrid methods is reduced to a bounded first order derivative in the new method.Some numerical experiments are presented to validate our theoretical analysis.展开更多
Semi-implicit spectral element schemes for 2-D shallow water equation are given, and numerical techniques are discussed. The EBE (element by element) idea is generalized to unsymmetric caes. We design mass-matrix diag...Semi-implicit spectral element schemes for 2-D shallow water equation are given, and numerical techniques are discussed. The EBE (element by element) idea is generalized to unsymmetric caes. We design mass-matrix diagonal pre-conditioned conjugate gradient method. The parallel computing is covered, and implemented on PC cluster. The research shows that spectral element has high precision and good scalability for shallow water simulation, and fits on the high-latency PC cluster perfectly.展开更多
基金This work was supported by Natural Science Foundation of Pujian under Grant A0310002 the Excellent Young Teachers Program (EYTP) of MOE of China.
文摘Spectral element method is well known as high-order method, and has potential better parallel feature as compared with low order methods. In this paper, a parallel preconditioned conjugate gradient iterative method is proposed to solving the spectral element approximation of the Helmholtz equation. The parallel algorithm is shown to have good performance as compared to non parallel cases, especially when the stiffness matrix is not memorized. A series of numerical experiments in one dimensional case is carried out to demonstrate the efficiency of the proposed method.
文摘In this paper, we consider solving the Helmholtz equation in the Cartesian domain , subject to homogeneous Dirichlet boundary condition, discretized with the Chebyshev pseudo-spectral method. The main purpose of this paper is to present the formulation of a two-level decomposition scheme for decoupling the linear system obtained from the discretization into independent subsystems. This scheme takes advantage of the homogeneity property of the physical problem along one direction to reduce a 2D problem to several 1D problems via a block diagonalization approach and the reflexivity property along the second direction to decompose each of the 1D problems to two independent subproblems using a reflexive decomposition, effectively doubling the number of subproblems. Based on the special structure of the coefficient matrix of the linear system derived from the discretization and a reflexivity property of the second-order Chebyshev differentiation matrix, we show that the decomposed submatrices exhibits a similar property, enabling the system to be decomposed using reflexive decompositions. Explicit forms of the decomposed submatrices are derived. The decomposition not only yields more efficient algorithm but introduces coarse-grain parallelism. Furthermore, it preserves all eigenvalues of the original matrix.
文摘针对声学有限元分析中四节点等参单元计算精度低,对网格质量敏感的问题,将无网格径向插值技术引入到标准有限元中,构造径向插值形函数,推导径向插值有限元法(Radial interpolation finite element method,RIFEM)的二维声学数值计算公式。二维声学RIFEM采用标准有限元法形函数构造系统离散方程的声学刚度矩阵和边界积分矢量,保证了声压梯度和边界条件在区域边界的积分精度;采用径向插值形函数构造系统离散方程的质量矩阵,提高了声压数值近似函数的插值精度。对管道二维声腔模型和某轿车二维声腔模型的数值分析结果表明,与标准有限元法和SFEM相比,RIFEM的计算精度更高,对波数、单元尺寸和网格扭曲程度的灵敏度更低。因此RIFEM可以很好地应用于二维声学数值分析,具有广阔的工程应用前景。
基金National Basic Research Program of China (Grant No. 2004CB418406)National Natural Science Foundation of China (Grant Nos. 40774049 and 40474038)Computer Network Information Center, Chinese Academy of Sciences (Grant No. INF105-SCE-02-12)
文摘The 2008 Wenchuan earthquake occurred in an active earthquake zone, i.e., Longmenshan tectonic zone. Seismic waves triggered by this earthquake can be used to explore the characteristics of the fault rupture process and the hierarchical structure of the Earth's interior. We employ spectral element method incorporated with large-scale parallel computing technology, to investigate the characteristics of seismic wave propagation excited by Wenchuan earthquake. We calculate synthetic seismograms with one-point source model and three-point source model respectively. The AK135 model is employed as a prototype of our numerical global Earth model. The Earth's ellipticity, Earth’s medium attenuation, and topography data are taken into consideration. These wave propagation processes are simulated by solving three-dimensional elastic wave governing equations. Three-dimensional visualization of our numerical results displays the profile of the seismic wave propagation. The three-point source, which is proposed from the latest investigations through field observation and reverse estimation, can better demonstrate the spatial and temporal characteristics of the source rupture process than the one-point source. We take comparison of synthetic seismograms with observational data recorded at 16 observatory stations. Primary results show that the synthetic seismograms calculated from three-point source agree well with the observations. This can further reveal that the source rupture process of Wenchuan earthquake is a multi-rupture process, which is composed by at least three or more stages of rupture processes.
基金the National Science Foundation of China (NSFC 11401129).
文摘This paper introduce a cascadic multigrid method for solving semilinear elliptic equations based on a multilevel correction method.Instead of the common costly way of directly solving semilinear elliptic equation on a very fine space,the new method contains some smoothing steps on a series of multilevel finite element spaces and some solving steps to semilinear elliptic equations on a very coarse space.To prove the efficiency of the new method,we derive two results,one of the optimal convergence rate by choosing the appro- priate sequence of finite element spaces and the number of smoothing steps,and the other of the optimal computational work by applying the parallel computing technique.Moreover,the requirement of bounded second order derivatives of nonlinear term in the existing multigrid methods is reduced to a bounded first order derivative in the new method.Some numerical experiments are presented to validate our theoretical analysis.
文摘Semi-implicit spectral element schemes for 2-D shallow water equation are given, and numerical techniques are discussed. The EBE (element by element) idea is generalized to unsymmetric caes. We design mass-matrix diagonal pre-conditioned conjugate gradient method. The parallel computing is covered, and implemented on PC cluster. The research shows that spectral element has high precision and good scalability for shallow water simulation, and fits on the high-latency PC cluster perfectly.
