PDE-Constrained High-Order Mesh Optimization

2025-07-02

AI Summary

This study addresses the challenge of simultaneously optimizing mesh quality and numerical accuracy of PDE solutions in high-order mesh adaptation. We propose a PDE-constrained r-adaptation framework built upon Target-Matrix Optimization Paradigm (TMOP), which tightly couples finite-element discretization error estimation with adjoint-based sensitivity analysis and incorporates convolutional gradient regularization to enhance optimization stability. The framework is broadly applicable to arbitrary spatial dimensions, element types, and PDE systems. Numerical experiments on Poisson’s equation and linear elasticity demonstrate substantial improvements in mesh quality and up to one order-of-magnitude reduction in PDE solution error, confirming its effectiveness, robustness, and generalizability. The core contribution lies in the first systematic integration of error-driven adjoint gradients and regularization into high-order mesh optimization—enabling synergistic enhancement of both mesh quality and numerical solution accuracy.

Technology Category

• Machine Learning: Optimization
• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Constraint Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• User Modeling, Personalization and Recommendation: User modeling for targeted and personalized online advertising

Abstract

We present a novel framework for PDE-constrained $r$-adaptivity of high-order meshes. The proposed method formulates mesh movement as an optimization problem, with an objective function defined as a convex combination of a mesh quality metric and a measure of the accuracy of the PDE solution obtained via finite element discretization. The proposed formulation achieves optimized, well-defined high-order meshes by integrating mesh quality control, PDE solution accuracy, and robust gradient regularization. We adopt the Target-Matrix Optimization Paradigm to control geometric properties across the mesh, independent of the PDE of interest. To incorporate the accuracy of the PDE solution, we introduce error measures that control the finite element discretization error. The implicit dependence of these error measures on the mesh nodal positions is accurately captured by adjoint sensitivity analysis. Additionally, a convolution-based gradient regularization strategy is used to ensure stable and effective adaptation of high-order meshes. We demonstrate that the proposed framework can improve mesh quality and reduce the error by up to 10 times for the solution of Poisson and linear elasto-static problems. The approach is general with respect to the dimensionality, the order of the mesh, the types of mesh elements, and can be applied to any PDE that admits well-defined adjoint operators.

Problem

• Optimizing high-order mesh movement with PDE constraints
• Balancing mesh quality and PDE solution accuracy
• Ensuring stable adaptation via gradient regularization

Innovation

• PDE-constrained mesh optimization via convex combination
• Target-Matrix Optimization Paradigm for geometric control
• Adjoint sensitivity analysis for error measure integration