Frequency domain Bernstein-Bézier finite element solver for modelling short waves in elastodynamics

This work presents a high-order Bernstein-Bézier finite element (FE) discretisation to accurately solve time harmonic elastic wave problems on unstructured triangular mesh grids. Although high-order FEs possess many advantages over standard FEs, the computational cost of matrix assembly is a major issue in high-order computations. A key ingredient to address this drawback is to resort to low complexity procedures in building the local high order FE matrices. This is achieved in this work by exploiting the tensorial property of Bernstein polynomials and applying the sum factorisation method for curved elements. An efficient implementation of the analytical rules for affine elements is also proposed. Furthermore, element-level static condensation of the interior degrees of freedom is performed to reduce the memory requirements. Additionally, the applicability of the method with a variable polynomial order, based on a simple a priori indicator, is investigated.

The computational complexities of sum factorisation, analytical rules and standard quadrature are first evaluated, in terms of the CPU time against the polynomial order. The analysis shows that the achieved numerical complexities compare favourably to those expected theoretically. A significant runtime saving is also obtained by using analytical rules and sum factorisation. The performance of the Bernstein-Bézier FEs is then assessed on various benchmark tests, over a wide range of frequencies. Results from the elastic wave scattering problem demonstrate the effectiveness of this method in coping with the pollution error, and its accuracy in resolving high order evanescent wave modes. Additionally, a wave transmission problem with high wave speeds contrast and a curved interior interface is considered, where a simple a priori indicator is proposed to assign the variable polynomial order. The study provides evidence of the great benefit of a non uniform p-refinement in reducing the computational cost and enhancing accuracy.