Isabelle Ramière

Marc Josien, Anas El Hachimi, Isabelle Ramière

In this article, we design an original solver based on Quantized Tensor Trains (QTT) for linear elliptic equations with heterogeneous coefficient field, that allows for extremely fine meshes. It can achieve full-field simulations in dimensions $d=2$ and $d=3$ with a number of Degrees of Freedom (DoFs) up to $20$ orders of magnitude beyond the classical solvers, recovering accurately the solution as well as its gradient in the $\LL^2$ norm. For treating such an enormous amount of data, the solver crucially relies on the exponential compression properties of QTTs. This significantly improves upon the existing literature. The main ingredient of the proposed solver consists in the introduction of a penalization term involving the Helmholtz--Leray projector in the equation governing the gradient unknown. For practical reasons related to the expression of the Helmholtz--Leray projector, the penalized equation is solved in Fourier space. The primal solution is then obtained from the gradient via the Green operator. A core property of the solver is that it is unconditionally stable with respect to the mesh size. Based on numerical evidence supported by mathematical analysis, we show that reliable gradients and solutions can be obtained, and guaranteed by the proposed a posteriori error estimator. As an illustration, we successfully solve an elliptic equation in a microstructured material with up to $10^{37}$ virtual degrees of freedom in dimension $d=3$.

Daria Koliesnikova, Isabelle Ramière

We propose a unified iterative framework for the solution of frictionless mechanical contact problems, which relies exclusively on the solution of standard stiffness systems. The framework is built upon a two-step fixed-point algorithm: first, the displacement is computed for given contact forces; second, the contact forces are updated based on the displacement solution. The choice of the dual update scheme depends on the numerical contact formulation under consideration. Specifically, the Uzawa iterative scheme is obtained for the Lagrange multiplier formulation, while a penalty-based operator-splitting strategy is proposed for the penalty contact formulation. The main interest of such displacement-force splitting strategy is to involve only standard rigidity matrices in the solving step: no saddle-point or penalized ill-conditionned coefficient matrices have to be handled. Moreover only the right-hand side of the system is updated throughout the iterations, which enables matrix factorization reuse or efficient iterative solvers initialization. The main limitation of such splitting iterative strategies lies in the inherently slow convergence of the underlying fixed-point iterations. Moreover, convergence is guaranteed only within a narrow range of numerical parameter values (i.e., the augmentation or penalty parameter). This work addresses both issues by applying the Crossed-Secant fixed-point acceleration strategy, which substantially improves the convergence rate and renders the iterative schemes effectively parameter-unconstrained. To the best of our knowledge, this contribution provides the first computational demonstration of efficient, parameter-unbounded convergence for such contact formulations. The substantial practical benefits of the proposed approach are illustrated through representative three-dimensional academic and industrial frictionless contact problems.