A spectral element method has been recently developed for solving elastodynamic problems. The numerical solutions are obtained by using the weak formulation of the elastodynamic equation for heterogeneous media, based...A spectral element method has been recently developed for solving elastodynamic problems. The numerical solutions are obtained by using the weak formulation of the elastodynamic equation for heterogeneous media, based on the Galerkin approach applied to a partition, in small subdomains, of the original physical domain. In this work, some mathematical aspects of the method and the associated algorithm implementation are systematically investigated. Two kinds of orthogonal basis functions, constructed with Legendre and Chebyshev polynomials, and their related Gauss-Lobatto collocation points are introduced. The related integration formulas are obtained. The standard error estimations and expansion convergence are discussed. An element-by-element pre-conditioned conjugate gradient linear solver in the space domain and a staggered predictor/multi-corrector algorithm in the time integration are used for strong heterogeneous elastic media. As a consequence, neither the global matrices nor the effective force vector is assembled. When analytical formulas are used for the element quadrature, there is even no need for forming element matrix in order to further save memory without losing much in computational efficiency. The element-by-element algorithm uses an optimal tensor product scheme which makes this method much more efficient than finite-element methods from the point of view of both memory storage and computational time requirements. This work is divided into two parts. The first part mainly focuses on theoretical studies with a simple numerical result for the Che-byshev spectral element, and the second part, mainly with the Legendre spectral element, will give the algorithm implementation, numerical accuracy and efficiency analyses, and then the detailed modeling example comparisons of the proposed spectral element method with a pseudo-spectral method, which will be seen in another work by Lin, Wang and Zhang.展开更多
发展了一个可分辨大气临近空间且采用谱元数值计算技术的大气数值模式SEMANS(Spectral Element Model with Atmospheric Near Space resolved),并对模式模拟能力进行检验.数值模式采用立方体球面投影坐标,每个投影面分解为81个局地元,...发展了一个可分辨大气临近空间且采用谱元数值计算技术的大气数值模式SEMANS(Spectral Element Model with Atmospheric Near Space resolved),并对模式模拟能力进行检验.数值模式采用立方体球面投影坐标,每个投影面分解为81个局地元,在元内利用8次Gauss-Lobatto-Legendre插值多项式对变量进行谱离散;模式大气在铅直方向分为66层,大气顶气压取为4.5×10-6h Pa.进行了10年积分试验,利用ECMWF(European Center for Medium-Range Weather Forecasts)ERA-Interim再分析数据集和COSPAR(Committee on Space Research)国际参考大气1986对SEMANS模拟结果进行初步检验.结果表明,模式模拟出30 hPa等压面上北半球纬向2波特征及南半球纬向1波特征;模式能模拟出100 hPa和0.001 hPa等压面附近的低温区及1 hPa等压面附近和0.0001 hPa等压面高度之上的高温区;模式还模拟出0.001 hPa等压面高度以下1月及7月纬向平均纬向风随高度分布的主要特征.展开更多
基金the Abdus Salam International Centre for Theoretical Physics of UNESCOthe International Science Link Program by Department of Education,Science and Technology of Australiathe Hundred Talent Program of Chinese Academy of Sciences
文摘A spectral element method has been recently developed for solving elastodynamic problems. The numerical solutions are obtained by using the weak formulation of the elastodynamic equation for heterogeneous media, based on the Galerkin approach applied to a partition, in small subdomains, of the original physical domain. In this work, some mathematical aspects of the method and the associated algorithm implementation are systematically investigated. Two kinds of orthogonal basis functions, constructed with Legendre and Chebyshev polynomials, and their related Gauss-Lobatto collocation points are introduced. The related integration formulas are obtained. The standard error estimations and expansion convergence are discussed. An element-by-element pre-conditioned conjugate gradient linear solver in the space domain and a staggered predictor/multi-corrector algorithm in the time integration are used for strong heterogeneous elastic media. As a consequence, neither the global matrices nor the effective force vector is assembled. When analytical formulas are used for the element quadrature, there is even no need for forming element matrix in order to further save memory without losing much in computational efficiency. The element-by-element algorithm uses an optimal tensor product scheme which makes this method much more efficient than finite-element methods from the point of view of both memory storage and computational time requirements. This work is divided into two parts. The first part mainly focuses on theoretical studies with a simple numerical result for the Che-byshev spectral element, and the second part, mainly with the Legendre spectral element, will give the algorithm implementation, numerical accuracy and efficiency analyses, and then the detailed modeling example comparisons of the proposed spectral element method with a pseudo-spectral method, which will be seen in another work by Lin, Wang and Zhang.
基金Supported by National Natural Science Foundation of China(41276190)
文摘发展了一个可分辨大气临近空间且采用谱元数值计算技术的大气数值模式SEMANS(Spectral Element Model with Atmospheric Near Space resolved),并对模式模拟能力进行检验.数值模式采用立方体球面投影坐标,每个投影面分解为81个局地元,在元内利用8次Gauss-Lobatto-Legendre插值多项式对变量进行谱离散;模式大气在铅直方向分为66层,大气顶气压取为4.5×10-6h Pa.进行了10年积分试验,利用ECMWF(European Center for Medium-Range Weather Forecasts)ERA-Interim再分析数据集和COSPAR(Committee on Space Research)国际参考大气1986对SEMANS模拟结果进行初步检验.结果表明,模式模拟出30 hPa等压面上北半球纬向2波特征及南半球纬向1波特征;模式能模拟出100 hPa和0.001 hPa等压面附近的低温区及1 hPa等压面附近和0.0001 hPa等压面高度之上的高温区;模式还模拟出0.001 hPa等压面高度以下1月及7月纬向平均纬向风随高度分布的主要特征.
