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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.

The finite element method (FEM) [

Among the main advantages of the SEM over the FEM is the high accuracy of approximation of the solution at a substantially smaller number of mesh elements required. The error of the numerical solution of SEM decreases exponentially with the order of elements [

The industrial application of the SEM is hindered by the problem of mesh generation [

This article discusses the variant of implementation of spectral element method on hybrid curvilinear meshes for three-dimensional problems of elasticity theory and its industrial application in CAE Fidesys [

Let the domain

Each reference point is determined by the index

Generally, the Jacobian [

The class of quadrilaterals includes the following types of finite elements, named by number of reference points: four-node, QUAD4

The shape functions for the element

The approximation of solution of the problem

GLL-weights are calculated as follows:

The class of triangles contains the following types of finite elements named by the number of reference points: three-node, TRI3; six-node, TRI6. The reference element

Shape functions for the element

The class of hexahedrons contains the following types of finite elements named by the number of reference points: eight-node, HEX8; twenty-node, HEX20; twenty-seven-node, HEX27. The reference element

The shape functions for the element

The approximation of solution of the problem

Coordinates of the GLL-nodes and GLL-weights (indexes

The class of tetrahedrons contains the following types of finite elements named by the number of reference points: four-node, TETRA4; ten-node, TETRA10. The reference element

The shape functions for the element

The class of pyramids contains the following types of finite elements named by the number of reference points: five-node, PYRAMID5; thirteen-node, PYRAMID13. The reference element

The class of prisms contains the following types of finite elements named by the number of reference points: six-node: WEDGE6; fifteen-node, WEDGE15. The reference element

The nodes of a prismatic spectral element are the points

The shape functions for the element

The main steps of application of the spectral element method to the problems of the theory of elasticity are similar to the steps of problems solving using the finite element method, such as discretization of equilibrium equations in the integral form; selection of quadrature for calculation of integrals; building of local matrices of stiffness, mass, and damping for each element; assembling global matrices of stiffness, mass, and damping. At present, most of mesh generators for geometric models build only the finite element meshes of the first and second order, which forces us to rebuild the mesh model for calculation using the spectral element method. The most long-lasting operation at this step is to construct a graph of mesh connectivity, so one can use the connectivity graph of the initial finite element mesh model in order to speed up the algorithm. Increasing the speed of the algorithm is associated with an increase in memory consumption. In particular, for high order elements it is necessary to store the points of integration, quadrature weights, and computed values of derivatives of functions at these points. An important feature of the algorithm is the ability to use the spectral element method on any hybrid finite element mesh of the first and second order. In fact, uncoated classes of reference mesh elements currently are beam and shell elements.

This approach to the implementation of spectral element method on hybrid curvilinear meshes for elasticity problems was industrially implemented in CAE Fidesys. Let us give the spectral element solution of the problem of stress analysis for a structural element and the results of spectral element analysis of wave propagation [

Numerical experiments clearly demonstrate that the mesh convergence in the model is achieved much faster when using the spectral element method as compared with the finite element method.

The wave propagation in unbounded elastic medium was analysed using the spectral element method, and the results were compared with the solution obtained via analytical calculation of displacement vector components, depending on time at fixed points of the body [

The calculation was performed for the finite elements of the first and second orders, as well as for the spectral elements of the orders from 1 to 4. An error of vector norm of maximum displacement of the bridge span was estimated. As a reference value for the assessment of the accuracy of the obtained solution, a value corresponding to the model mesh convergence was taken. Results are provided for the following cases: case 1, 10 thousand elements (the character element size is 0.8); case 2, 28 thousand elements (the character element size is 0.4); case 3, 104 thousand elements (the character element size is 0.2); case 4, 849 thousand elements (the character element size is 0.1). The case number is laid off as abscissa.

As it can be seen in Figure

This article discusses the variant of spectral element method implementation on the hybrid curvilinear meshes for problems of elasticity theory and its industrial application in CAE Fidesys. The comparison with the finite element method was conducted, which allowed us to draw a conclusion about the high accuracy of the method and the correctness of algorithms and the program developed. In the future, it is planned to expand the implementation of the method for the cases of shell and beam elements within the static and dynamic elasticity problems.

The authors declare that they have no conflicts of interest.

The research for this article was performed in FIDESYS LLC and was financially supported by Russian Ministry of Education and Science (Project no. 14.579.21.0112, Project ID RFMEFI57915X0112).