Nonlinear Optimization Methods for Data Driven Computational Mechanics
This talk explores nonlinear optimization methods in the context of Data Driven Computational Mechanics (DDCM), a paradigm introduced to replace traditional material models with experimental data. The DDCM approach reformulates Boundary Value Problems by directly incorporating strain-stress data, which eliminates the need for predefined material laws. We follow a hybrid DDCM approach, where we require that a smooth constitutive manifold is constructed to obtain a well-behaved nonlinear optimization problem (NLP) instead of a more challenging discrete-continuous NLP (DCNLP). Using this formulation, we include geometric inequality constraints to consider contact problems. To solve these contact problems, framed as a mathematical program with complementarity constraints (MPCC), we propose a heuristic quick-shot solution approach that solves up to four NLPs. In addition to this discrete optimization setting, we introduce theoretical considerations to determine the structural properties of DDCM problem formulations in function spaces. With respect to the function space regularity of the variables, we distinguish between a low-regularity formulation, where we establish initial existence results, and a high-regularity formulation, where we use a feedback control operator that enables solution strategies using optimal control methods. These theoretical foundations are complemented by a practical application, employing the Alternating Direction Method (ADM) using a promising initialization. Furthermore, technical investigations into the differentiability of the feedback control operator open up the potential for a successful application of optimal control techniques.
