A deterministic approach for shape and topology optimization under uncertain loads

International audienceThe present work aims at handling uncertain loads in shape and topology optimization. More specifically, we minimize objective functions combining mean values and variances of standard cost functions and assume that uncertainties are small and generated by a finite number N of random variables. A deterministic approach that relies on a second-order Taylor expansion of the cost function has been proposed by Allaire & Dapogny. 1 That method requires a computational effort comparable to the one for an N-load problem. This work presents a general framework to handle uncertainties on arbitrary static load cases where perturbations on both surface forces and body forces are considered. We demonstrate the effectiveness of the approach in the context of level-set-based topology optimization for the robust compliance minimization on three-dimensional test cases

The GNAT method for nonlinear model reduction: effective implementation and application to computational fluid dynamics and turbulent flows

The Gauss--Newton with approximated tensors (GNAT) method is a nonlinear model reduction method that operates on fully discretized computational models. It achieves dimension reduction by a Petrov--Galerkin projection associated with residual minimization; it delivers computational efficency by a hyper-reduction procedure based on the `gappy POD' technique. Originally presented in Ref. [1], where it was applied to implicit nonlinear structural-dynamics models, this method is further developed here and applied to the solution of a benchmark turbulent viscous flow problem. To begin, this paper develops global state-space error bounds that justify the method's design and highlight its advantages in terms of minimizing components of these error bounds. Next, the paper introduces a `sample mesh' concept that enables a distributed, computationally efficient implementation of the GNAT method in finite-volume-based computational-fluid-dynamics (CFD) codes. The suitability of GNAT for parameterized problems is highlighted with the solution of an academic problem featuring moving discontinuities. Finally, the capability of this method to reduce by orders of magnitude the core-hours required for large-scale CFD computations, while preserving accuracy, is demonstrated with the simulation of turbulent flow over the Ahmed body. For an instance of this benchmark problem with over 17 million degrees of freedom, GNAT outperforms several other nonlinear model-reduction methods, reduces the required computational resources by more than two orders of magnitude, and delivers a solution that differs by less than 1% from its high-dimensional counterpart

Approximation par trains de tenseurs de lois de comportement élasto-viscoplastiques paramétrées

This work presents a novel approach to construct surrogate models of parametric Differential Algebraic Equations based on a tensor representation of the solutions. The procedure consists in building simultaneously, for every output of the reference model, an approximation given in tensor-train format. A parsimonious exploration of the parameter space coupled with a compact data representation allows to alleviate the curse of dimensionality. The approach is thus appropriate when many parameters with large domains of variation are involved. The numerical results obtained for a nonlinear elasto-viscoplastic constitutive law show that the constructed surrogate model is sufficiently accurate to enable parametric studies such as the calibration of material coefficients.Ce travail présente une nouvelle approche pour construire des modèles de substitution pour les équations différentielles algébriques dépendant de paramètres qui s’appuient sur une représentation tensorielle des solutions. La procédure consiste à construire de manière simultanée, pour chaque sortie d’intérêt du modèle de référence, une approximation donnée sous la forme d’un train de tenseurs. Une exploration parcimonieuse du domaine des paramètres combinée à un format de données compact permet d’atténuer la malédiction de la dimension. Cette approche est donc appropriée en présence de nombreux paramètres avec de larges domaines de variations. Les résultats numériques obtenus pour une loi de comportement élasto-viscoplastique non-linéaire montrent que le modèle de substitution ainsi construit est suffisamment précis pour être utilisé dans le cadre d’études paramétriques telles que l’identification de coefficients matériaux

Multiple Tensor Train Approximation of Parametric Constitutive Equations in Elasto-Viscoplasticity

This work presents a novel approach to construct surrogate models of parametric differential algebraic equations based on a tensor representation of the solutions. The procedure consists of building simultaneously an approximation given in tensor-train format, for every output of the reference model. A parsimonious exploration of the parameter space coupled with a compact data representation allows alleviating the curse of dimensionality. The approach is thus appropriate when many parameters with large domains of variation are involved. The numerical results obtained for a nonlinear elasto-viscoplastic constitutive law show that the constructed surrogate model is sufficiently accurate to enable parametric studies such as the calibration of material coefficients

Méthode de décomposition tensorielle pour la calibration de lois de comportement en sciences des matériaux

International audienceOn souhaite construire un métamodèle à représentation tensorielle sur un vaste espace paramétrique de lois de comportement élastoviscoplastiques pour en faciliter la calibration. Un train de tenseurs est construit pour chaque variable mécanique à l'aide de la méthode TT-cross. Pour avoir un échantillonnage parcimonieux du domaine paramétrique, la méthode est complétée par une approximation des fibres multidimensionnelles par la Gappy POD sur des bases réduites issues de la snapshot POD. Les performances (en ligne et hors ligne) de la méthode sont illustrées sur un modèle à 6 paramètres

Optimization of oriented and parametric cellular structures by the homogenization method

International audienceWe present here a topology optimization method based on a homogenization approach to design oriented and parametrized cellular structures. The present work deals with 2-D square cells featuring a rectangular hole, because their structure is close to that of rank-2 sequential laminates, which are optimal for compliance optimization. For several cells, the value and the parametric sensitivities of their effective elastic tensor can easily be computed, by the resolution of a cell problem. The obtained results can be used to build a surrogate model for the homogenized constitutive law. Moreover, we add the local orientation of the cells to our problem. Then, an optimal composite shape is computed thanks to an alternate directions algorithm. The crucial ingredient of the methodology is the extraction of a quasi-periodic and additive manufacturable structure from the previously obtained composite shape, based on the introduction of a space transformation

Parametric Damage Mechanics Empowering Structural Health Monitoring of 3D Woven Composites

This paper presents a data-driven structural health monitoring (SHM) method by the use of so-called reduced-order models relying on an offline training/online use for unidirectional fiber and matrix failure detection in a 3D woven composite plate. During the offline phase (or learning) a dataset of possible damage localization, fiber and matrix failure ratios is generated through high-fidelity simulations (ABAQUS software). Then, a reduced model in a lower-dimensional approximation subspace based on the so-called sparse proper generalized decomposition (sPGD) is constructed. The parametrized approach of the sPGD method reduces the computational burden associated with a high-fidelity solver and allows a faster evaluation of all possible failure configurations. However, during the testing phase, it turns out that classical sPGD fails to capture the influence of the damage localization on the solution. To alleviate the just-referred difficulties, the present work proposes an adaptive sPGD. First, a change of variable is carried out to place all the damage areas on the same reference region, where an adapted interpolation can be done. During the online use, an optimization algorithm is employed with numerical experiments to evaluate the damage localization and damage ratio which allow us to define the health state of the structure

A R&D software platform for shape and topology optimization using body-fitted meshes

International audienceTopology optimization is devoted to the optimal design of structures: It aims at finding the best material distribution inside a working domain while fulfilling mechanical, geometrical and manufacturing specifications. Conceptually different from parametric or size optimization, topology optimization relies on a freeform approach enabling to search for the optimal design in a larger space of configurations and promoting disruptive design. The need for lighter and efficient structural solutions has made topology optimization a vigorous research field in both academic and industrial structural engineering communities. This contribution presents a Research and Development software platform for shape and topology optimization where the computational process is carried out in a level set framework combined with a body-fitted approach

Shape Optimization of a Coupled Thermal Fluid-Structure Problem in a Level Set Mesh Evolution Framework

International audienceHadamard's method of shape differentiation is applied to topology optimization of a weakly coupled three physics problem. The coupling is weak because the equations involved are solved consecutively, namely the steady state Navier-Stokes equations for the fluid domain, first, the convection diffusion equation for the whole domain, second, and the linear thermo-elasticity system in the solid domain, third. Shape sensitivities are derived in a fully Lagrangian setting which allows us to obtain shape derivatives of general objective functions. An emphasis is given on the derivation of the adjoint interface condition dual to the one of equality of the normal stresses at the fluid solid interface. The arguments allowing to obtain this surprising condition are specifically detailed on a simplified scalar problem. Numerical test cases are presented using the level set mesh evolution framework of [4]. It is demonstrated how the implementation enables to treat a variety of shape optimization problems. keywords. Topology and shape optimization, adjoint methods, fluid structure interaction, convective heat transfer, adaptive remeshing