Variational framework for distance-minimizing method in data-driven computational mechanics

Abstract

A variational framework for the distance-minimizing data-driven computing method is proposed. We provide a mathematical view on data-driven boundary value problems as well as the associated mathematical tools. The notion of function spaces is introduced into a data-driven boundary value problem and thus it can be formulated as a continuous optimization problem. An explanation of a data-driven boundary value problem as a double-minimization problem is given. This problem is subsequently broken down into two single-minimization problems, one of which is a constrained optimization problem. The associated constraint is given in terms of differential equations and the method of choice is the method of Lagrange multipliers. For the constrained optimization problem, we seek solutions by solving a system of equations for nodal degrees of freedom. The second is an unconstrained optimization problem whose solution is sought by working with the values of the solution at quadrature points and the supplied material data.The proposed variational formulation renders the high-order polynomial interpolation to straightforward implementation. In particular, spectral element methods are employed to reduce the computational cost while assuring the high accuracy of the data-driven solution. Within the spirit of the variational formulation, diffusion and dynamic problems can be efficiently studied in the data-driven setting with only few necessary modifications. Some representative numerical examples for parabolic and hyperbolic equations are presented. Numerical convergence studies show the reliability and robustness of the proposed formulation.