A kernel method for learning constitutive relation in data-driven computational elasticity

For numerical simulation of elastic structures, data-driven computational approaches attempt to use a data set of material responses, without resorting to conventional modeling of the material constitutive equation. In a material data set in the stress-strain space, the data points are considered to lie on or near a low-dimensional manifold, rather distribute ubiquitously in the space. This paper presents a kernel method for extracting this manifold. We formulate a regularized least-squares problem for learning a manifold, and show that its optimal solution corresponds to an eigenvector of a real symmetric matrix. Therefore, the method requires only simple computational task, and is easy to implement. We also give a description how to use the obtained solution in static equilibrium analysis of an elastic structure. Numerical experiments on two-dimensional continua are performed to demonstrate effectiveness and robustness of the proposed method.