A DPG method for the circular arch problem
For computational mechanics researchers, this provides a robust numerical method for arch problems, but it is an incremental extension of DPG to a specific geometry.
The paper develops a DPG method for circular arch elasticity, achieving optimal convergence rates in numerical experiments and showing that accuracy improves with a scaled test space norm.
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.