Analysis of the Discontinuous Petrov-Galerkin Method with Optimal Test Functions for the Reissner-Mindlin Plate Bending Model

TL;DR

This paper analyzes the stability and convergence of the DPG method with optimal test functions for the Reissner-Mindlin plate bending model, providing theoretical error estimates and numerical validation.

Contribution

It establishes the well-posedness of the DPG formulation for the Reissner-Mindlin model and develops a stable numerical scheme with polynomial approximation of optimal test functions.

Findings

The DPG method is stable with a thickness-dependent constant.

Optimal test functions approximated by degree p+3 polynomials ensure stability.

Numerical experiments confirm convergence on various mesh types.

Abstract

We analyze the discontinuous Petrov-Galerkin (DPG) method with optimal test functions when applied to solve the Reissner-Mindlin model of plate bending. We prove that the hybrid variational formulation underlying the DPG method is well-posed (stable) with a thickness-dependent constant in a norm encompassing the $L_2$-norms of the bending moment, the shear force, the transverse deflection and the rotation vector. We then construct a numerical solution scheme based on quadrilateral scalar and vector finite elements of degree $p$. We show that for affine meshes the discretization inherits the stability of the continuous formulation provided that the optimal test functions are approximated by polynomials of degree $p+3$. We prove a theoretical error estimate in terms of the mesh size $h$ and polynomial degree $p$ and demonstrate numerical convergence on affine as well as non-affine mesh sequences.