Spectral Element Method

It is a unique feature of CAE Fidesys that, in addition to the of finite element method (FEM) used by default, it enables calculations by spectral element method (SEM). Spectral element method) is a FEM modification which often used in the explicit dynamic problems.

The main advantages of SEM in comparison to FEM:

High computational speed as there is no need to solve the system of linear algebraic equations due to diagonal form of mass matrix. The latter is obtained by specific quadrature formula for volume integration;
High precision of solution approximation at coarse meshing with low number of elements;
Ability of effective paralleling for OpenMP, MPI and CUDA.

SEM is most effective for the dynamic analysis using an explicit time scheme.

Summary and benefits of SEM

SEM is a FEM modification where piecewise functions are used as basic functions consisting of high degree polynomials.

The main advantages of SEM in comparison to FEM:

1) High computational speed as there is no need to solve the system of linear algebraic equations due to diagonal form of mass matrix;
2) High precision of solution approximation at coarse meshing (low number of elements). The solution error is estimated as.

where

then for SEM

Here C₁ and C₂ are constants, h is a characteristic element size, N is an element order, u_h is a numerical solution, [u]_h is an exact solution in mesh nodes.

3) Ability of effective paralleling for OpenMP, MPI and CUDA.

SEM is most effective for the dynamic analysis using an explicit time scheme.

Here are the results of classical problem of wave propagation in 2D plate (size 1x1).

To achieve the computational error 2% and less, it is necessary to generate one of the following meshes:

a) 3-noded triangular mesh of 6 390 197 elements (characteristic element size is 4e-4)

b) 4-noded quadrilateral mesh of 1 640 961 elements (characteristic element size is 3e-3)

c) coarse spectral element mesh with 4th element order (only 16 elements are required).

Fig.1. The distribution of the field of magnitude of displacement U across the plate at the time t1

Fig.2. The distribution of the field of magnitude of displacement U across the plate at the time t2

Examples of mesh with using spectral elements with different order:

2nd spectral element order

4th spectral element order

8th spectral element order

12th spectral element order

SEM Usage

In order to use the method of spectral elements instead of the finite element method to solve the problem, set the order of elements 3 and higher (exept spring and lumpmass):

Spectral elements used in CAE Fidesys.