A framework for data‐driven structural analysis in general elasticity based on nonlinear optimization: The dynamic case

Abstract

Summary In this article, we present an extension of the formulation recently developed by the authors to the structural dynamics setting. Inspired by a structure‐preserving family of variational integrators, our new formulation relies on a discrete balance equation that establishes the dynamic equilibrium. From this point of departure, we first derive an “exact” discrete‐continuous nonlinear optimization problem that works directly with data sets. We then develop this formulation further into an “approximate” nonlinear optimization problem that relies on a general constitutive model. This underlying model can be identified from a data set in an offline phase. To showcase the advantages of our framework, we specialize our methodology to the case of a geometrically exact beam formulation that makes use of all elements of our approach. We investigate three numerical examples of increasing difficulty that demonstrate the excellent computational behavior of the proposed framework and motivate future research in this direction.

Model-free Data-Driven Inference

“…We note that the paradigm is strictly data-driven and modelfree in the sense that solutions are obtained, or approximated, directly from the data set without recourse to any intervening modeling of the data. Extensions of the approach, applications and follow-up work have spawned a sizeable engineering literature to date (cf., e. g., [11,12,13,14,15,16,17] for a representative sample). 1.1.…”

Finite element solver for data-driven finite strain elasticity

“…Recently, several authors contributed to improve the DDCM numerical scheme: the entropy-maximizing solver developed in [18] is more robust to noisy material data sets with outliers; the issue of the high dimensionality of the phase space was addressed by [19] using tensor voting, a machine learning technique; [20] showed that the data-driven BVP is a well posed Mixed-Integer Quadratic Programming problem, allowing to reach the global minimizer, through (very expensive) branch-and-bound solvers. Hybrid methods have also been introduced to further improve the robustness of the data-driven approach: bridging the gap between inverse manifold reconstruction techniques [8] and direct DDCM solvers, several authors have incorporated the definition of a constitutive manifold into the datadriven distance minimizing problem [21,22,23,24]. For instance, [23,24] formulated an approximate nonlinear optimization problem and demonstrated its computational efficiency in the static and dynamic cases for specific configurations spaces of nonlinear kinematics.…”

“…Hybrid methods have also been introduced to further improve the robustness of the data-driven approach: bridging the gap between inverse manifold reconstruction techniques [8] and direct DDCM solvers, several authors have incorporated the definition of a constitutive manifold into the datadriven distance minimizing problem [21,22,23,24]. For instance, [23,24] formulated an approximate nonlinear optimization problem and demonstrated its computational efficiency in the static and dynamic cases for specific configurations spaces of nonlinear kinematics. Finally, the DDCM has been extended to other classes of problem, such as elasto-dynamics [25], diffusion problems [26], history-dependent behavior like visco-elasticity [27] or fracture mechanics [28].…”

Formulating and Heuristic Solving of Contact Problems in Hybrid Data Driven Computational Mechanics

“…An alternative inverse DDCM approach employs the data set provided to reconstruct a traditional material model that expresses the stresses as explicit functions of the strains by means of an energy functional [14,12,13]. In order to combine the strengths of the two approaches while mitigating their weaknesses, we have proposed a hybrid DDCM approach in [7,8]. This approach allows for non-traditional implicit material models given by a smooth constitutive manifold that has to be reconstructed from the data set in a first (off-line) step.…”