Error Sampling and Synthesis for High-Order Node Movement

The presented work focuses on the error sampling and synthesis procedure within an optimization framework for high-order, metric-based mesh adaptation in high-order, finite-element (FEM) discretization. This mesh optimization framework is designed to handle arbitrary FEM discretization order, geometry order, and element types. In performing a metric-based adaptation, the framework uses a high-order Riemannian metric field to encode the curvature, anisotropy, and global coupling between vertices and high-order geometry nodes. An error model and a cost model are employed to iteratively construct the desired Riemannian metric field and guide a series of globally coupled vertex (r-adaptation) and high-order geometry (q-adaptation) node movements. The resulting mesh is an optimal high-order (curved) mesh that conforms to the specified metric field. The error model requires an error sampling and synthesis procedure, which involves several steps, including element splitting, random sampling of high-order geometry node movements, and estimating the metric-based error kernel on each mesh element. This paper aims to: 1) discuss the theoretical underpinnings of a robust, a posteriori, metric-based error model for qr-adaptation and 2) provide a status update on the 1D HOMES algorithm, which is a native extension of the Mesh Optimization via Error Sampling and Synthesis (MOESS) algorithm to a higher order.

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Recommended citation: Devina Sanjaya, Ajay Rangarajan, Carl Gooch. (2025). "Error Sampling and Synthesis for High-Order Node Movement." AIAA SCITECH 2025 Forum, 2025.