A sparse basis for equilibrium stress fields with application for direct data-driven mechanics

We present a new class of solvers for direct data-driven mechanical problems based on a sparse basis representation of equilibrium stress fields. Our first contribution is an efficient algorithm for computing the required sparse null-space basis on tetrahedral meshes. Only a single QR decomposition is needed to compute a small remaining set of dense basis vectors associated with boundary conditions and topological holes which can be handled efficiently via a partitioned Cholesky factorization. Building on this, we demonstrate how standard iterative solvers-such as the Newton-Raphson method-can be applied to direct data-driven formulations. The proposed approach is particularly valuable for challenging problems with complex data distributions requiring systematic exploration of the space of equilibrium stress fields. To this end, we introduce an algorithm that constructs a hierarchical solution set through an eigenvalue decomposition in the joint space of equilibrium stress and compatible strain fields. We demonstrate the proposed methodology with a numerical example involving brittle fracture with probabilistic tensile strength. The resulting family of failure patterns offers valuable insights for uncertainty quantification and design decision-making.