Finite Element Data Driven Computational approaches for Complex Materials and Microfluidics

• Grant number: 25/00975-9
• Support Opportunities: Regular Research Grants
• Start date: February 01, 2026
• End date: January 31, 2029
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Applied Mathematics
• Principal Investigator: Roberto Federico Ausas
• Grantee: Roberto Federico Ausas
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
• City of the host institution: São Carlos

Abstract

The primary motivation of this project is investigating a novel Data Driven approach for solving problems involving elliptic PDEs, whether the scalar Poisson problem or the elasticity problem, that incorporates experimental or synthetic constitutive data without relying on a smooth constitutive relation-between fluxes and gradients in the former case, or to stresses and deformations in the latter. This approach is known as the Data-Driven Computational Mechanics paradigm, which can be seen as a form of unsupervised machine learning applied to continuum mechanics. The ultimate goal of this work is to formulate a discrete-continuous optimization problem that seeks, under certain constraints, the fluxes-gradients (or stresses-deformations) that minimize the distance between the dataset and the subspace of corresponding compatible fields in equilibrium. Given the challenging nature of this optimization problem, we begin by analyzing a finite element formulation in which the data set is prescribed over the domain. Next, we develop strategies to initialize the data distribution, enabling the solution of a class of optimization subproblems. We then propose iterative strategies to approximate the global optimization problem. Additionally, we discuss other data-driven approaches of interest, along with potential extensions to parabolic problems and Stokes flows. These include applications in microfluidics, such as fluid-structure finite element formulations for simulating the behavior of microswimmers. (AU)