Strict error bounds for linear and nonlinear solid mechanics problems using a patch-based flux-free method

We discuss, in this paper, a common ux-free method for the computation of strict error bounds for linear and nonlinear Finite Element computations. In the linear case, the error bounds are on the energy norm of the error, while, in the nonlinear case, the concept of error in constitutive relation is used. In both cases, the error bounds are strict in the sense that they re- fer to the exact solution of the continuous equations, rather than to some FE computation over a refined mesh. For both linear and nonlinear solid mechanics, this method is based on the computation of a statically admissible stress field, which is performed as a series of local problems on patches of elements. There is no requirement to solve a previous problem of ux equilibration globally, as happens with other methods.Postprint (published version

Certification, Adaptivity, and Reduction in Non-Intrusive Model Couplings

The non-intrusive coupling method is a local analysis method that aims to separate the macroscopic and local scales both from mesh, operator, and loading viewpoints. This method has demonstrated that it can be used for a large set of problems such as those with local plasticity, and it has been implemented in an industrial code. In order to reduce the complexity of a given problem, a coarse model is defined on the total domain with its properties (geometry, connectivity, operator, solver) and a separated fine model is set locally with its different properties. The solution is obtained iteratively by exchanging data on the interface between the global and local models. However, the issue of certification and optimal driving of such a coupling method has been addressed in very few works until now. In the present work, we tackle this subject using the concept of constitutive relation error (CRE) which is a generic tool able to certify the quality of a numerical solution. The main goal is to control the solution of the non-intrusive global-local algorithm, in terms of global error or error in specific quantities of interest, and to optimally drive the coupling algorithm according to quantitative error indicators on individual error sources (discretization, modelling, iterative scheme). In this context, we will also study the coupling between a global solution raised from the well-known Finite Element Method and a local solution evaluated on-line from a virtual chart [4] constructed offline from a reduced order model in the local area of interest. Such a virtual chart is based on the PGD technique and integrates as variables (i.e. extra-parameters) some features of the local model such as boundary conditions, geometry, or material behavior. Complementing the non-intrusive global-local coupling with local ROM leads to a more flexible exchange between interface quantities, and higher performance in terms of computational efficiency and stability. The talk will start with an overview of the methodology of non-intrusive coupling methods, before presenting the error estimation procedure using the CRE concept and its decomposition into various source contributions for adaptive control. Eventually, the construction and the performance of ROM in the non-intrusive coupling method will be studied and discussed

Guaranteed control of switched control systems using model order reduction and state-space bisection

This paper considers discrete-time linear systems controlled by a quantized law, i.e., a piecewise constant time function taking a finite set of values. We show how to generate the control by, first, applying model reduction to the original system, then using a "state-space bisection" method for synthesizing a control at the reduced-order level, and finally computing an upper bound to the deviations between the controlled output trajectories of the reduced-order model and those of the original model. The effectiveness of our approach is illustrated on several examples of the literature

A posteriori error estimation and adaptive strategy for PGD model reduction applied to parametrized linear parabolic problems

We define an a posteriori verification procedure that enables to control and certify PGD-based model reduction techniques applied to parametrized linear elliptic or parabolic problems. Using the concept of constitutive relation error, it provides guaranteed and fully computable global/goal-oriented error estimates taking both discretization and PGD truncation errors into account. Splitting the error sources, it also leads to a natural greedy adaptive strategy which can be driven in order to optimize the accuracy of PGD approximations. The focus of the paper is on two technical points: (i) construction of equilibrated fields required to compute guaranteed error bounds; (ii) error splitting and adaptive process when performing PGD-based model reduction. Performances of the proposed verification and adaptation tools are shown on several multi-parameter mechanical problems