HDG methods for elastodynamics

Abstract

We derive and analyze a hybridizable discontinuous Galerkin (HDG) method for approximating weak solutions to the equations of time-harmonic linear elasticity on a bounded Lipschitz domain in three dimensions. The real symmetry of the stress tensor is strongly enforced and its coefficients as well as those of the displacement vector field are approximated simultaneously at optimal convergence with respect to the choice of approximating spaces, wavenumber, and mesh size. Sufficient conditions are given so that the system is indeed transferable onto a global hybrid variable that, for larger polynomial degrees, may be approximated via a smaller-dimensional space than the original variables. We construct several variants of this method and discuss their advantages and disadvantages, and give a systematic approach to the error analysis for these methods. We touch briefly on the application of this error analysis to the time-dependent problem, and finally, we examine two different implementations of the method over various polynomial degrees and numerically demonstrate the convergence properties proven herein.

New analytical tools for HDG in elasticity, with applications to elastodynamics

“…We show that the proof for optimal convergence can be easily obtained by using the new projection and some existing techniques in traditional HDG methods for evolutionary equations [6]. Moreover, we recover the error estimates for steady-state elasticity [19] and frequency domain elastodynamics [15] by using the projection-based analysis, and we show that the analysis can be simplified in both cases. Since the construction of the HDG+ projection involves first constructing a projection associated to an M-decomposition, it also shed some light upon the connections between these two kinds of methods.…”

“…Method and convergence estimates. In this section we give new proofs of error estimates to the second HDG+ method studied in [15] by using projection based analysis for the following time-harmonic linear elasticity problem…”

“…It uses only polynomial basis functions, achieves one order higher convergence rate for the displacement without post-processing, and its computational complexity is the same to the standard HDG method (order k for both stress and displacement) for their global systems. However, the existing error analysis of HDG+ methods are all based on using orthogonal projections [15,18,19,20,21], which make the analysis slightly more complicated (it requires a bootstrapping argument to prove convergence of all variables, as opposed to consecutive energy and duality proofs), and detached from the existing projection-based error analysis of HDG methods [3,6,7,8,9,10,11,14], where specifically constructed projections are used to make the analysis simple and concise. This motivates us to find a new kind of projection for HDG+ for elasticity.…”

“…In Section 4, we develop a systematic approach of changes of variables to obtain the projection on the physical element. In Section 5 and 6, we recover the error estimates in steady-state elasticity [19] and elastodynamics [15] using projection-based analysis. In Section 7, we present a semi-discrete HDG method for transient elastic waves and prove it is optimally convergent, uniformly in the time variable.…”

HDGlab: An Open-Source Implementation of the Hybridisable Discontinuous Galerkin Method in MATLAB

“…In the framework of elastodynamics, HDG with DIRK time integrators were introduced in [202], whereas in [256] an HDG spectral element method (HDG-SEM) is utilised to simulate wave propagation in coupled elastic-acoustic media. In the frequency-domain, HDG methods for elastodynamics are analysed and presented in [28,164].…”