Data-driven computational mechanics

Abstract: We develop a new computing paradigm, which we refer to as data-driven computing, according to which calculations are carried out directly from experimental material data and pertinent constraints and conservation laws, such as compatibility and equilibrium, thus bypassing the empirical material modeling step of conventional computing altogether. Data-driven solvers seek to assign to each material point the state from a prespecified data set that is closest to satisfying the conservation laws. Equivalently, data-driven solvers aim to find the state satisfying the conservation laws that is closest to the data set. The resulting data-driven problem thus consists of the minimization of a distance function to the data set in phase space subject to constraints introduced by the conservation laws. We motivate the data-driven paradigm and investigate the performance of data-driven solvers by means of two examples of application, namely, the static equilibrium of nonlinear three-dimensional trusses and linear elasticity. In these tests, the data-driven solvers exhibit good convergence properties both with respect to the number of data points and with regard to local data assignment. The variational structure of the data-driven problem also renders it amenable to analysis. We show that, as the data set approximates increasingly closely a classical material law in phase space, the data-driven solutions converge to the classical solution. We also illustrate the robustness of data-driven solvers with respect to spatial discretization. In particular, we show that the data-driven solutions of finite-element discretizations of linear elasticity converge jointly with respect to mesh size and approximation by the data set.

• The paper introduces data-driven computing by eliminating traditional empirical material models in favor of direct use of experimental data under physical constraints.
• It demonstrates applications on three-dimensional truss structures and linear elasticity, showcasing strong convergence with denser data sets and refined local assignments.
• The variational structure ensures that solutions converge to classical mechanics results, validating the approach through finite element discretizations and mesh convergence.

Data-driven Computational Mechanics

Introduction to Data-driven Computing

The paper "Data-driven computational mechanics" introduces an innovative computational paradigm known as data-driven computing. This approach fundamentally diverges from traditional methods by eliminating the empirical material modeling step typically inherent in computational mechanics. Instead, calculations are performed directly using experimental material data in conjunction with relevant constraints and conservation laws, such as compatibility and equilibrium. This involves using solvers that seek the optimal material state from a prescribed data set that best satisfies these conservation laws, effectively minimizing the distance to the data set in phase space subject to constraint conditions.

Application to Three-dimensional Truss Structures and Linear Elasticity

The practical implementation of data-driven computing is demonstrated through applications involving static equilibrium of nonlinear three-dimensional trusses and problems in linear elasticity. Within these examples, data-driven solvers exhibit commendable convergence properties in relation to both the density of data points and the local assignment of data. The performance evaluation in these scenarios demonstrates the viability of this methodology in solving complex mechanical systems with robustness toward spatial discretization intricacies.

Variational Structure and Convergence

A notable aspect of this approach is its variational structure which facilitates deeper mathematical analysis. The authors rigorously establish that solutions derived from data-driven solvers converge to classical solutions as the data sets increasingly approximate classical material laws in phase space. This convergence is pivotal since it bridges data-driven methodology with established theoretical frameworks, ensuring compatibility and accuracy in results.

Finite Element Discretization and Mesh Convergence

The robustness of data-driven computational mechanics is further highlighted by the convergence of solutions arising from finite-element discretizations of linear elasticity problems. The paper emphasizes the joint convergence of these solutions with respect to mesh size and approximation by the data sets. Therefore, the method effectively handles the discretization of spatial domains, a frequent challenge in classical computational mechanics.

Conclusion

The paper introduces significant advancements in computational mechanics through its data-driven approach, offering a viable alternative to the conventional empirical modeling processes. By focusing directly on experimental data and ensuring convergence to classical solutions, this paradigm enhances computational efficiency and accuracy. These developments portend potential expansions into broader applications in computational mechanics and material science, suggesting future integration with larger and more complex data sets as the availability and granularity of experimental material data continue to improve.