Trust-Region Approaches for High-Order Node Movement

High-order node movement (q-adaptation) is a powerful technique for controlling solution and output errors in finite-element simulations without increasing the degrees of freedom. It is particularly effective when used in conjunction with high-order discretizations and vertex node movement (r-adaptation). This approach optimally positions high-order geometry nodes within mesh elements and can be interpreted as a mesh-curving strategy in which elements are curved to maximize accuracy and efficiency gains, while also providing accurate boundary and geometry representations. The High-Order Mesh Optimization via Error Sampling and Synthesis (High-Order MOESS, or HOMES) algorithm iteratively constructs a continuous mesh framework that encodes the shape, size, and anisotropy of high-order mesh elements in a Riemannian metric field. The high-order metric field is then used to strongly couple the movements of the vertices and the high-order geometry nodes. In this paper, we integrate a non-monotonic variant of a trust region with correction steps for non-smooth problems into the HOMES algorithm, replacing the original procedure used to update the step metric field at each adaptation iteration. A scaling matrix can be applied to the trust region if needed. Results show that HOMES with a trust region generally yields more optimally curved meshes and delivers more accuracy and efficiency gains than HOMES without a trust region.