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arXiv · 2506.10894

Numerical approximation of a PDE-constrained Optimization problem that appears in Data-Driven Computational Mechanics

Pedro B. Bazon·Cristian G. Gebhardt·Gustavo C. Buscaglia·Roberto F. Ausas
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Abstract

We investigate an optimization problem that arises when working within the paradigm of Data-Driven Computational Mechanics. In the context of the diffusion-reaction problem, such an optimization problem seeks for the continuous primal fields (gradient and flux) that are closest to some predefined discrete fields taken from a material data set. The optimization is performed over primal fields that satisfy the physical conservation law and the geometrical compatibility. We consider a reaction term in the conservation law, which has the effect of coupling all the optimality conditions. We first establish the well-posedness in the continuous setting. Then, we propose stable finite element discretizations that consistently approximate the continuous formulation, preserving its saddle-point structure and allowing for equal-order interpolation of all fields. Finally, we demonstrate the effectiveness of the proposed methods through a set of numerical examples.

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Pedro B. Bazon, Cristian G. Gebhardt, Gustavo C. Buscaglia, Roberto F. Ausas. 2025-06-12. Numerical approximation of a PDE-constrained Optimization problem that appears in Data-Driven Computational Mechanics. https://arxiv.org/abs/2506.10894

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