A DPG method for the circular arch problem

TL;DR

This paper introduces a DPG method with optimal test functions for modeling a circular arch, achieving predicted convergence rates and demonstrating improved accuracy through test space norm scaling.

Contribution

It develops a novel ultra-weak variational formulation and DPG approximation for the circular arch problem, incorporating membrane, shear, and bending effects.

Findings

Theoretical analysis predicts optimal convergence rates.

Numerical experiments confirm theoretical predictions.

Accuracy improves with scaled test space norm.

Abstract

We consider an elastic model for a circular arch that incorporates membrane, transverse shear, and bending effects. The central line of the arch is partitioned into elements, and an ultra-weak variational formulation is developed alongside a discontinuous Petrov-Galerkin (DPG) approximation procedure based on so-called optimal test functions. The formulation uses discontinuous stress and displacement interpolations on the element mesh, with corresponding interface variables defined at the nodes. Theoretical analysis predicts optimal convergence rates for all quantities of interest, while also revealing potential error amplification influenced by the curvature of the arch and the imposed boundary conditions. The method is tested on examples with different support configurations. The numerical experiments confirm the theoretical predictions and further demonstrate that the accuracy of the DPG method can be improved by employing a suitably scaled test space norm.