Nonlinear Elasticity for Mesh Deformation with High-Order Discontinuous Galerkin Methods for the Navier-Stokes Equations on Deforming Domains

We present a numerical framework for simulation of the compressible Navier-Stokes equations on problems with deforming domains where the boundary motion is prescribed by moving meshes. Our goal is a high-order accurate, efficient, robust, and general purpose simulation tool. To obtain this, we use a discontinuous Galerkin space discretization, diagonally implicit Runge-Kutta time integrators, and fully unstructured meshes of triangles and tetrahedra. To handle the moving boundaries, a mapping function is produced by first deforming the mesh using a neo-Hookean elasticity model and a high-order continuous Galerkin FEM method. The resulting nonlinear equations are solved using Newton’s method and a robust homotopy approach. From the deformed mesh, we compute grid velocities and deformations that are consistent with the time integration scheme. These are used in a mapping-based arbitrary Lagrangian-Eulerian formulation, with numerically computed mapping Jacobians which satisfy the geometric conservation law. We demonstrate our methods on a number of problems, ranging from model problems that confirm the high-order accuracy to the flow in domains with complex deformations.