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DOE OSTI · 3484530

Bridging mathematical rigor and data scalability: Data-driven finite element method

Cheung, Siu Wun [Lawrence Livermore National Laboratory]
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Cheung, Siu Wun [Lawrence Livermore National Laboratory]. 2026-08-25. Bridging mathematical rigor and data scalability: Data-driven finite element method. https://doi.org/10.2172/3484530

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High-order meshes are crucial for achieving optimal convergence rates in curvilinear domains, preserving symmetry, and aligning with key flow features in moving mesh simulations [1], but their quality is challenging to control. In prior work, we have developed techniques based on Target-Matrix Optimization Paradigm (TMOP) to adapt a given high-order mesh to the geometry and solution of the partial differential equation (PDE) [2, 3]. Here, we extend this framework to address two key gaps in the literature for highorder mesh 𝑟-adaptivity. First, we introduce tangential relaxation on curved surfaces using solely the discrete mesh representation, eliminating the need for access to underlying geometry (e.g., CAD model). Second, we ensure a continuously positive Jacobian determinant throughout the domain. This determinant positivity is essential for using the high-order mesh resulting from 𝑟-adaptivity with arbitrary quadrature schemes in simulations. The proposed approach is demonstrated to be robust using a variety of numerical experiments.

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