Pablo Antolin Sanchez

DeepBND: A machine learning approach to enhance multiscale solid mechanics

Pablo Antolin Sanchez, Annalisa Buffa, Simone Deparis, Felipe Figueredo Rocha

Effective properties of materials with random heterogeneous structures are typically determined by homogenising the mechanical quantity of interest in a window of observation. The entire problem setting encompasses the solution of a local PDE and some averaging formula for the quantity of interest in such domain. There are relatively standard methods in the literature to completely determine the formulation except for two choices: i) the local domain itself and the ii) boundary conditions. Hence, the modelling errors are governed by the quality of these two choices. The choice i) relates to the degree of representativeness of a microscale sample, i.e., it is essentially a statistical characteristic. Naturally, its reliability is higher as the size of the observation window becomes larger and/or the number of samples increases. On the other hand, excepting few special cases there is no automatic guideline to handle ii). Although it is known that the overall effect of boundary condition becomes less important with the size of the microscale domain, the computational cost to simulate such large problem several times might be prohibitive even for relatively small accuracy requirements. Here we introduce a machine learning procedure to select most suitable boundary conditions for multiscale problems, particularly those arising in solid mechanics. We propose the combination Reduced-Order Models and Deep Neural Networks in an offline phase, whilst the online phase consists in the very same homogenisation procedure plus one (cheap) evaluation of the trained model for boundary conditions. Hence, the method allows an implementation with minimal changes in existing codes and the use of relatively small domains without losing accuracy, which reduces the computational cost by several orders of magnitude. A few test cases accounting for random circular and elliptical inclusions are reported aiming at proving the potentials of the DeepBND method.

2023

A localized reduced basis approach for unfitted domain methods on parameterized geometries

Pablo Antolin Sanchez, Annalisa Buffa, Margarita Chasapi

This work introduces a reduced order modeling (ROM) framework for the solution of parameterized second-order linear elliptic partial differential equations formulated on unfitted geometries. The goal is to construct efficient projection-based ROMs, which rely on techniques such as the reduced basis method and discrete empirical interpolation. The presence of geometrical parameters in unfitted domain discretizations entails challenges for the application of standard ROMs. Therefore, in this work we propose a methodology based on i) extension of snapshots on the background mesh and ii) localization strategies to decrease the number of reduced basis functions. The method we obtain is computationally efficient and accurate, while it is agnostic with respect to the underlying discretization choice. We test the applicability of the proposed framework with numerical experiments on two model problems, namely the Poisson and linear elasticity problems. In particular, we study several benchmarks formulated on two-dimensional, trimmed domains discretized with splines and we observe a significant reduction of the online computational cost compared to standard ROMs for the same level of accuracy. Moreover, we show the applicability of our methodology to a three-dimensional geometry of a linear elastic problem.

2022

Analysis-aware defeaturing of complex geometries

Pablo Antolin Sanchez, Ondine Gabrielle Chanon

Local modifications of a computational domain are often performed in order to simplify the meshing process and to reduce computational costs and memory requirements. However, removing geometrical features of a domain often introduces a non-negligible error in the solution of a differential problem in which it is defined. In this paper, we aim at generalizing the work from [1], in which an a posteriori estimator of the geometrical defeaturing error is derived for domains from which one geometrical feature is removed. More precisely, we study the case of domains containing an arbitrary number of distinct features, and we perform an analysis on Poisson's, linear elasticity, and Stokes' equations. We introduce a simple and computationally cheap a posteriori estimator of the geometrical defeaturing error, whose reliability and efficiency are rigorously proved, and we introduce a geometric refinement strategy that accounts for the defeaturing error: Starting from a fully defeatured geometry, the algorithm determines at each iteration step which features need to be added to the geometrical model to reduce the defeaturing error. These important features are then added to the (partially) defeatured geometrical model at the next iteration, until the solution attains a prescribed accuracy. A wide range of numerical experiments are finally reported to illustrate and validate this work.

2022

Quadrature-free immersed isogeometric analysis

Pablo Antolin Sanchez, Thibaut Hirschler

This paper presents a novel method for solving partial differential equations on three-dimensional CAD geometries by means of immersed isogeometric discretizations that do not require quadrature schemes. It relies on a newly developed technique for the evaluation of polynomial integrals over spline boundary representations that is exclusively based on analytical computations. First, through a consistent polynomial approximation step, the finite element operators of the Galerkin method are transformed into integrals involving only polynomial integrands. Then, by successive applications of the divergence theorem, those integrals over B-Reps are transformed into the first surface and then line integrals with polynomials integrands. Eventually, these line integrals are evaluated analytically with machine precision accuracy. The performance of the proposed method is demonstrated by means of numerical experiments in the context of 2D and 3D elliptic problems, retrieving optimal error convergence order in all cases. Finally, the methodology is illustrated for 3D CAD models with an industrial level of complexity.

SPRINGER2022

Reduced order modelling of nonaffine problems on parameterized NURBS multipatch geometries

Pablo Antolin Sanchez, Annalisa Buffa, Margarita Chasapi

This contribution explores the combined capabilities of reduced basis methods and IsoGeometric Analysis (IGA) in the context of parameterized partial differential equations. The introduction of IGA enables a unified simulation framework based on a single geometry representation for both design and analysis. The coupling of reduced basis methods with IGA has been motivated in particular by their combined capabilities for geometric design and solution of parameterized geometries. In most IGA applications, the geometry is modelled by multiple patches with different physical or geometrical parameters. In particular, we are interested in nonaffine problems characterized by a high-dimensional parameter space. We consider the Empirical Interpolation Method (EIM) to recover an affine parametric dependence and combine domain decomposition to reduce the dimensionality. We couple spline patches in a parameterized setting, where multiple evaluations are performed for a given set of geometrical parameters, and employ the Static Condensation Reduced Basis Element (SCRBE) method. At the common interface between adjacent patches a static condensation procedure is employed, whereas in the interior a reduced basis approximation enables an efficient offline/online decomposition. The full order model over which we setup the RB formulation is based on NURBS approximation, whereas the reduced basis construction relies on techniques such as the Greedy algorithm or proper orthogonal decomposition (POD). We demonstrate the developed procedure using an illustrative model problem on a three-dimensional geometry featuring a multi-dimensional geometrical parameterization. 18 pages, 7 figures, 1 table

2022

Robust numerical integration on curved polyhedra based on folded decompositions

Pablo Antolin Sanchez, Annalisa Buffa, Xiaodong Wei

We present a novel method to perform numerical integration over curved polyhedra enclosed by high-order parametric surfaces. Such a polyhedron is first decomposed into a set of triangular and/or rectangular pyramids, whose certain faces correspond to the given parametric surfaces. Each pyramid serves as an integration cell with a geometric mapping from a standard parent domain (e.g., a unit cube), where the tensor-product Gauss quadrature is adopted. As no constraint is imposed on the decomposition, certain resulting pyramids may intersect with themselves, and thus their geometric mappings may present negative Jacobian values. We call such cells folded cells and refer to the corresponding decomposition as a folded decomposition. We show that folded cells do not cause any issues in practice as they are only used to numerically compute certain integrals of interest. The same idea can be applied to planar curved polygons as well. We demonstrate both theoretically and numerically that folded cells can retain the same accuracy as the cells with strictly positive Jacobians. On the other hand, folded cells allow for a much easier and much more flexible decomposition for general curved polyhedra, on which one can robustly compute integrals. In the end, we show that folded cells can flexibly and robustly accommodate real-world complex geometries by presenting several examples in the context of immersed isogeometric analysis, where involved sharp features can be well respected in generating integration cells.(c) 2022 Elsevier B.V. All rights reserved.

ELSEVIER SCIENCE SA2022

Accurate equilibrium-based interlaminar stress recovery for isogeometric laminated composite Kirchhoff plates

Pablo Antolin Sanchez, Alessandro Reali

Despite the accelerated deployment of laminated composites in a wide variety of markets due to their peculiar engineering features, the design of those materials is often restrained by the lack of cost-efficient modeling techniques. In fact, the existing strategies allowing for cheap simulations usually fail to directly capture out of-plane through-the-thickness stresses, which prove to be typically responsible of delamination failure modes. In this paper, we introduce a fast and accurate stress recovery strategy to model the out-of-plane behavior of Kirchhoff laminated plates. The proposed technique can be regarded as a two-step approach: First, the classical composite plates theory, providing the lowest computational cost among known literature strategies, is applied to obtain a coarse displacement solution; afterwards, this solution is used to compute the necessary in-plane derivatives to recover the out-of-plane stresses directly imposing equilibrium in strong form. Since this a posteriori step relies on high-order in-plane continuity requirements, isogeometric analysis (IGA) represents a natural simulation framework given its accuracy and higher continuity properties. Both isogeometric Galerkin and collocation formulations are herein considered. The effectiveness of the proposed approach is proven by extensive numerical tests.

ELSEVIER SCI LTD2021

Damage-driven strain localisation in networks of fibres: A computational homogenisation approach

Pablo Antolin Sanchez, Felipe Figueredo Rocha

In many applications, such as textiles, fibreglass, paper and several kinds of biological fibrous tissues, the main load-bearing constituents at the micro-scale are arranged as a fibre network. In these materials, rupture is usually driven by micro-mechanical failure mechanisms, and strain localisation due to progressive damage evolution in the fibres is the main cause of macro-scale instability. We propose a strain-driven computational homogenisation formulationbased on Representative Volume Element (RVE), within a framework in which micro-scale fibre damage can lead to macro-scale localisation phenomena. The mechanical stiffness considered here for the fibrous structure system is due to: i) an intra-fibre mechanism in which each fibre is axially stretched, and as a result, it can suffer damage; ii) an inter-fibre mechanism in which the stiffness results from the variation of the relative angle between pairs of fibres. The homogenised tangent tensor, which comes from the contribution of these two mechanisms, is required to detect the so-called bifurcation point at the macro-scale, through the spectral analysis of the acoustic tensor. This analysis can precisely determine the instant at which the macro-scale problem becomes ill-posed. At such a point, the spectral analysis provides information about the macro-scale failure pattern (unit normal and crack-opening vectors). Special attention is devoted to present the theoretical fundamentals rigorously in the light of variational formulations for multi-scale models. Also, the impact of a recent derived more general boundary condition for fibre networks is assessed in the context of materials undergoing softening. Numerical examples showing the suitability of the present methodology are also shown and discussed.

2021

Efficient equilibrium-based stress recovery for isogeometric laminated curved structures

Pablo Antolin Sanchez, Alessandro Reali

This work extends the stress recovery for laminated composite solid plates, proposed in [1,2], to curved structures. Based on 3D Isogeometric Analysis (IGA) computations and equilibrium, this procedure uses a single element through the thickness in combination with a calibrated layer-by-layer integration rule or a homogenized approach, allowing for an inexpensive and accurate approximation in terms of in-plane stresses (and their derivatives), while through-the-thickness stress components are poorly approximated. Relying on the highorder continuity properties of IGA shape functions, an accurate out-of-plane stress state can also be recovered by means of direct integration of the equilibrium equations in strong form. The a posteriori step application, which is straightforward in the context of solid plates, is not trivial in the case of curved geometries. In fact, the notion of in-plane and out-of-plane directions is not clear when modeling this kind of structures in the global reference system, while adopting curvilinear coordinates to express the equilibrium gives rise to additional coupled terms that require an iterative process to resolve the balance of momentum equation. Therefore, we propose to apply the recovery locally, which, despite leading to more elaborated stress derivative terms because of the increasing geometry complexity, still allows for a direct reconstruction as the resolvent system is uncoupled. Several numerical results show the good performance of this approach particularly for composite stacks with significant radius-to-thickness ratio and number of plies.

ELSEVIER SCI LTD2021