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

Computer Methods in Applied Mechanics and Engineering, vol. 453, pp. 118803

Abstract

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.

Authors 4

1. Erik Prume corresponding Aachen Institute of Applied Mechanics

   RWTH Aachen University

   Affiliation as printed

   Institute of Applied Mechanics, RWTH Aachen University, Mies-van-der-Rohe-Str. 1, Aachen, D-52074, Germany

2. Chenyi Ji Aachen Institute of Applied Mechanics Institutes of Applied Medical Engineering

   RWTH Aachen University

   Affiliation as printed

   Institute of Applied Mechanics, RWTH Aachen University, Mies-van-der-Rohe-Str. 1, Aachen, D-52074, Germany

   Institutes of Applied Medical Engineering, RWTH Aachen University, Paulwelsstr. 20, Aachen, D-52074, Germany

3. Stefanie Reese Aachen Institute of Applied Mechanics

   University of Siegen · RWTH Aachen University

   Affiliation as printed

   Institute of Applied Mechanics, RWTH Aachen University, Mies-van-der-Rohe-Str. 1, Aachen, D-52074, Germany

   University of Siegen, Adolf-Reichwein-Str. 2a, Siegen, D-57076, Germany

4. Michael Ortiz

   California Institute of Technology · International Center for Numerical Methods in Engineering · Universitat Politècnica de Catalunya

   Affiliation as printed

   Centre Internacional de Mètodes Numèrics en Enginyeria (CIMNE), Universitat Politècnica de Catalunya, Jordi Girona 1, Barcelona, 08034, Spain

   Division of Engineering and Applied Science, California Institute of Technology, 1200 E. California Blvd., Pasadena, 91125, CA, USA