A Parallel Spectral Element Method for Acoustic Wave Modeling

1997 ◽  
Vol 05 (01) ◽  
pp. 53-69 ◽  
Author(s):  
Géza Seriani
Keyword(s):  
Large Scale ◽  
Spectral Element Method ◽  
Iterative Solution ◽  
Spectral Element ◽  
Solution Technique ◽  
Stiffness Matrices ◽  
The Matrix ◽  
Speed Performance ◽  
Element Approach ◽  
Element Method

The finite element method is well reputed for its great flexibility in solving problems with complex geometries and heterogeneous structures, but, in its classical form, it has fairly low accuracy and poor computational efficiency. This makes an adverse impact on a large-scale numerical simulation of acoustic wavefield propagation. It has been shown by the author and his co-workers that the spectral approach, based on the use of high-order orthogonal interpolating functions (the Spectral Element Method), yields high accuracy with almost no numerical artifacts; it also significantly reduces the simulation computing costs. In this paper, the method is presented in conjunction with an iterative solution technique in such a way that it fully exploits the currently available parallel computers. The underlying algorithm is based on the observations that the assembly of mass and stiffness matrices is not really needed, and that the matrix-vector product required for the iterations can be done concurrently via an element-by-element approach. Moreover, by using a tensor-product sum-factorization scheme, computational and storage requirements can be further reduced as neither element nor global matrices are ever formed. The method is tested on the Cray T3D MPP computer, and its parallel efficiency and speed performance are discussed.

scholarly journals The spectral element method on variable resolution grids: evaluating grid sensitivity and resolution-aware numerical viscosity

2014 ◽  
Vol 7 (3) ◽  
pp. 4081-4117 ◽  
Author(s):  
O. Guba ◽  
M. A. Taylor ◽  
P. A. Ullrich ◽  
J. R. Overfelt ◽  
M. N. Levy
Keyword(s):  
Large Scale ◽  
Spectral Element Method ◽  
Truncation Error ◽  
Spherical Geometry ◽  
Spectral Element ◽  
Metric Tensor ◽  
Variable Resolution ◽  
Lower Valence ◽  
Element Method

Abstract. We evaluate the performance of the Community Atmosphere Model's (CAM) spectral element method on variable resolution grids using the shallow water equations in spherical geometry. We configure the method as it is used in CAM, with dissipation of grid scale variance implemented using hyperviscosity. Hyperviscosity is highly scale selective and grid independent, but does require a resolution dependent coefficient. For the spectral element method with variable resolution grids and highly distorted elements, we obtain the best results if we introduce a tensor-based hyperviscosity with tensor coefficients tied to the eigenvalues of the local element metric tensor. The tensor hyperviscosity is constructed so that for regions of uniform resolution it matches the traditional constant coefficient hyperviscsosity. With the tensor hyperviscosity the large scale solution is almost completely unaffected by the presence of grid refinement. This later point is important for climate applications where long term climatological averages can be imprinted by stationary inhomogeneities in the truncation error. We also evaluate the robustness of the approach with respect to grid quality by considering unstructured conforming quadrilateral grids generated with a well-known grid-generating toolkit and grids generated by SQuadGen, a new open source alternative which produces lower valence nodes.

scholarly journals The spectral element method (SEM) on variable-resolution grids: evaluating grid sensitivity and resolution-aware numerical viscosity

2014 ◽  
Vol 7 (6) ◽  
pp. 2803-2816 ◽  
Author(s):  
O. Guba ◽  
M. A. Taylor ◽  
P. A. Ullrich ◽  
J. R. Overfelt ◽  
M. N. Levy
Keyword(s):  
Large Scale ◽  
Spectral Element Method ◽  
Truncation Error ◽  
Spherical Geometry ◽  
Spectral Element ◽  
Metric Tensor ◽  
Variable Resolution ◽  
Lower Valence ◽  
Element Method

Abstract. We evaluate the performance of the Community Atmosphere Model's (CAM) spectral element method on variable-resolution grids using the shallow-water equations in spherical geometry. We configure the method as it is used in CAM, with dissipation of grid scale variance, implemented using hyperviscosity. Hyperviscosity is highly scale selective and grid independent, but does require a resolution-dependent coefficient. For the spectral element method with variable-resolution grids and highly distorted elements, we obtain the best results if we introduce a tensor-based hyperviscosity with tensor coefficients tied to the eigenvalues of the local element metric tensor. The tensor hyperviscosity is constructed so that, for regions of uniform resolution, it matches the traditional constant-coefficient hyperviscosity. With the tensor hyperviscosity, the large-scale solution is almost completely unaffected by the presence of grid refinement. This later point is important for climate applications in which long term climatological averages can be imprinted by stationary inhomogeneities in the truncation error. We also evaluate the robustness of the approach with respect to grid quality by considering unstructured conforming quadrilateral grids generated with a well-known grid-generating toolkit and grids generated by SQuadGen, a new open source alternative which produces lower valence nodes.

scholarly journals Electromechanical Impedance Analysis on Piezoelectric Smart Beam with a Crack Based on Spectral Element Method

2015 ◽  
Vol 2015 ◽  
pp. 1-13 ◽  
Author(s):  
Dansheng Wang ◽  
Hongyuan Song ◽  
Hongping Zhu
Keyword(s):  
Spectral Element Method ◽  
Crack Depth ◽  
Adhesive Layer ◽  
Impedance Analysis ◽  
Spectral Element ◽  
Peak Position ◽  
Electromechanical Impedance ◽  
Stiffness Matrices ◽  
Smart Beam ◽  
Element Method

An electromechanical impedance (EMI) analysis of a piezoelectric smart beam with a crack is implemented in this paper. Spectral element method (SEM) is used to analyze the EMI response of the piezoelectric smart beam. In this analysis, the spectral element stiffness matrices of different beam segments are derived in this paper. The crack is simulated using spring models, and the EMI signatures of piezoelectric smart beam with and without crack are calculated using SEM, respectively. From the analysis results, it is found that the peak position and amplitude of the EMI signatures have significant changes with the change in crack depth, especially in higher frequency ranges. Different vibration modes of the piezoelectric smart beam are analyzed, and the effect of thickness of the adhesive layer on the admittance is also researched. An experimental study is also implemented to verify the validity of the analysis results using SEM.

Spectral Element Approach for Coupled Radiative and Conductive Heat Transfer in Semitransparent Medium

2007 ◽  
Vol 129 (10) ◽  
pp. 1417-1424 ◽  
Author(s):  
J. M. Zhao ◽  
L. H. Liu
Keyword(s):  
Heat Transfer ◽  
Transfer Equation ◽  
Spectral Element Method ◽  
Radiative Transfer Equation ◽  
Spectral Element ◽  
Conductive Heat Transfer ◽  
Conductive Heat ◽  
Element Approach ◽  
Semitransparent Medium ◽  
Element Method

A spectral element method is presented to solve coupled radiative and conductive heat transfer problems in multidimensional semitransparent medium. The solution of radiative energy source is based on a second order radiative transfer equation. Both the second order radiative transfer equation and the heat diffusion equation are discretized by spectral element approach. Four various test problems are taken as examples to verify the performance of the spectral element method. The h-and the p-convergence characteristics of the spectral element method are studied. The convergence rate of p refinement for different values of Planck number follows the exponential law and is superior to that of h refinement. The spectral element method has good property to tolerate skewed meshes. The predicted dimensionless temperature distributions determined by the spectral element method agree well with the results in references. The presented method is very effective to solve coupled radiative and conductive heat transfer in semitransparent medium with complex configurations and demands little on the quality of mesh.

scholarly journals The Implementation of Spectral Element Method in a CAE System for the Solution of Elasticity Problems on Hybrid Curvilinear Meshes

2017 ◽  
Vol 2017 ◽  
pp. 1-7 ◽  
Author(s):  
Dmitriy Konovalov ◽  
Anatoly Vershinin ◽  
Konstantin Zingerman ◽  
Vladimir Levin
Keyword(s):  
Mathematical Simulation ◽  
High Performance ◽  
Spectral Element Method ◽  
New Technologies ◽  
Spectral Element ◽  
High Tech ◽  
Engineering Analysis ◽  
Elasticity Problems ◽  
Cae System ◽  
Element Method

Modern high-performance computing systems allow us to explore and implement new technologies and mathematical modeling algorithms into industrial software systems of engineering analysis. For a long time the finite element method (FEM) was considered as the basic approach to mathematical simulation of elasticity theory problems; it provided the problems solution within an engineering error. However, modern high-tech equipment allows us to implement design solutions with a high enough accuracy, which requires more sophisticated approaches within the mathematical simulation of elasticity problems in industrial packages of engineering analysis. One of such approaches is the spectral element method (SEM). The implementation of SEM in a CAE system for the solution of elasticity problems is considered. An important feature of the proposed variant of SEM implementation is a support of hybrid curvilinear meshes. The main advantages of SEM over the FEM are discussed. The shape functions for different classes of spectral elements are written. Some results of computations are given for model problems that have analytical solutions. The results show the better accuracy of SEM in comparison with FEM for the same meshes.

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