A Recurrent Neural Network-based Surrogate Model for History-Dependent Multi-scale Simulations of Composite Materials

In order to make Computational homogenization affordable, pre-off-line finite element simulations are conducted on the mesoscale problem in order to build a synthetic database that can, in turn, be used to train surrogate models, which can be used as a constitutive law on a classical finite element simulation, speeding up the multi-scale process by several orders. Artificial neural networks (NNWs) offer the possibility to serve as a surrogate model, but a difficulty arises for elasto-plasticity because of its history-dependency. This difficulty can be solved by considering a Recurrent Neural Network (RNN), which uses sequential information [1]. Nevertheless, in order to be accurate under multi-dimensional non-proportional loading conditions, a sufficiently wide database is required in order to perform the training. To this end, a sequential training synthetic database is obtained from finite element simulations on an elasto-plastic RVE subjected to random loading paths. The RNN predictions are thus found to be in agreement with the FE2 simulations, while reducing the computational cost by 4 orders. Nevertheless, such a paradigm is essentially used as a mapping between the macro-stress and macro-strain tensors of the micro-scale boundary value response and the micro-structure information could not be recovered in a so-called localization step. We thus also develop Recurrent Neural Networks (RNNs)-based surrogate of the local micro-structure state variables for complex loading scenarios [2]. In order to address the curse of dimensionality arising because of the large amount of internal state variables in the micro-structure, we enrich the RNN with PCA dimensionality reduction and dimensionality break down, i.e. the use of several RNNs instead of a single one. The sequential training strategy is optimized to allow for GPU usage

Homogenization of fibre reinforced composite with gradient enhanced damage model

Classical finite element simulations face the problems of losing uniqueness and strain localization when the strain softening of materials is involved. Thus, when using continuum damage model or plasticity softening model, numerical convergence will not be obtained with the refinement of the finite element discretization when strain localization occurs. Gradient-enhanced softening and non-local continua models have been proposed by several researchers in order to solve this problem. In such approaches, high-order spatial gradients of state variables are incorporated in the macroscopic constitutive equations. However, when dealing with complex heterogeneous materials, a direct simulation of the macroscopic structures is unreachable, motivating the development of non-local homogenization schemes. In this work, a non-local homogenization procedure is proposed for fiber reinforced materials. In this approach, the fiber is assumed to remain linear elastic while the matrix material is modeled as elasto-plastic coupled with a damage law described by a non-local constitutive model. Toward this end, the mean-field homogenization is based on the knowledge of the macroscopic deformation tensors, internal variables and their gradients, which are applied to a micro- structural representative volume element (RVE). Macro-stress is then obtained from a homogenization process

Full discontinuous Galerkin formulation of shells in large deformations with parallel and fracture mechanics applications

Fracture mechanical problems can be solved by coupling the finite elements with a cohesive approach. Unfortunately, the classical cohesive methods suffer from severe limitations. Indeed, on one hand, the intrinsic approach, which inserts the cohesive elements at the beginning, has to model the prefracture stage. This requires an initial slope in the traction separation law that should tend toward infinity to avoid lack of consistency leading to obvious numerical problems. On the other hand, the extrinsic cohesive method inserts the cohesive elements during the simulation when a fracture criterion is reached. This insertion requires topological mesh modifications and therefore a very complicated implementation, especially in a parallel code. To overcome these limitations, new methods were developed and in particular, an approach based on discontinuous Galerkin formulation (DG) has been pioneered by R. Radovitzky (Radovitzky cmame2011). The use of the DG principle allows to formulate the problem with discontinuous elements and the continuity between them is ensured weakly by terms integrated on the elements interface . These interface elements can be easily replaced by a cohesive element during the simulation. We have recently developed this approach for shells (Becker cmame2011) to obtain a full DG method. Moreover, a new cohesive law based on the reduced stresses of the thin bodies formulation is developed to propagate a fracture through the thickness. This cohesive model dissipates the right amount of energy during crack phenomena. These developments are implemented in parallel and validated by the study the blast of a notched cylinder, for which experimental and numerical (by XFEM method) data are reported in the literature by R. Larsson (Larsson ijnme2011). Finally, as thin structures are often made of ductile materials, which show large deformations before fracture, the formulation is extended to the non linear case with hyperelastic material law. This one can take into account the damage and a criterion based on the work of Huespe (Huespe plasticity2009) is developed to localize the damage leading to the apparition and propagation of cracks

A recurrent neural network-accelerated multi-scale model for elasto-plastic heterogeneous materials subjected to random cyclic and non-proportional loading paths

his project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No 862015 for the project Multi-scale Optimisation for Additive Manufacturing of fatigue resistant shock-absorbing MetaMaterials (MOAMMM) of the H2020-EU.1.2.1. - FET Open Programm

Development of a Finite Strain Weakly Coupled Electro-Magneto-Thermo-Mechanical Model for Shape Memory Polymer Composites

Shape memory behaviour refers to the ability of the material to shift from a temporary configuration to the permanent/original configuration under the influence of an external stimulus, such as temperature, electromagnetism, or light. Triggering the shape change in polymer composites through contactless temperature control induced through the losses incurred by the alternating electromagnetic fields is the primary focus of this work. Shape memory polymer composite (SMPC) synthesized with electrically conductive fillers and magnetic particles in a polymer matrix are considered here. A coupled Electro-Thermo-Mechanical (E-TM) model for SMPC was recently developed in [1]. The present work extends the existing E-TM model by introducing the magnetic field, thus leading to a strongly coupled Electro-Magneto-Thermo-Mechanical (EM-TM) model. A finite strain phenomenological model for shape memory behaviour of semi-crystalline polymers under thermo-mechanical loading was developed in-house [2]. The viscoelasto-plastic constitutive model for the thermo-mechanics of SMP is extended considering i) the heat sources arising from the losses incurred by the alternating EM fields in the SMPC ii) the Maxwell stress contributions in the SMPC arising from the EM fields permeating in the SMPC and the surrounding air. First, a strong coupling EM-TM model with heating of the SMPC through EM sources in the on- linear finite deformations regime is demonstrated. Due to the considered frequency of the EM sources, the timescale of the EM problem is relatively small (ms or μs) compared to the heating and deformations observed in the SMPC (in s). Secondly, the large deformations in the elements belonging to the air domain surrounding the deforming SMPC lead to skewed elements which cause convergence issues. To circumvent these issues of the strong coupling, a novel multi-timescale scheme weakly coupling the dynamic EM and quasi-static TM problem was developed. The weak coupling employs a total Lagrangian formulation with only the SMPC domain for the TM problem. Whereas an updated Lagrangian formulation after a remeshing step, using the deformed SMPC and added inductor coil and air domains, is employed for the EM problem. The computational efficiency of the weak coupling is highlighted along with the validated results against the strong coupling. Finally, a test depicting the shape memory effects with shape recovery by temperature control in the SMPC will be demonstrated

Recurrent Neural Networks (RNNs) with dimension reduction and break down in the context of high dimensional localization step in multi-scale analysis

Artificial Neural Networks (NNWs) are appealing tools to serve as surrogate model of high dimensional and non-linear history-dependent problems in computational mechanics, in particular in multi-scale methods. Indeed, as shown in [1] once properly trained, they replace expensive micro-scale finite element resolutions allowing reducing the computational time by more than 4 orders of magnitude. Nevertheless, such a paradigm is essentially used as a mapping between the macro-stress and macro-strain tensors of the micro-scale boundary value response and the micro-structure information could not be recovered in a so-called localization step. In this talk, we present a Recurrent Neural Networks (RNNs)-based surrogate of the micro-scale boundary value problem, while being able to recover the evolution of the local micro-structure state variables for complex loading scenarios [2]. In order to address the curse of dimensionality arising because of the large amount of internal state variables in the micro-structure, we enrich the RNN with PCA dimension reduction and dimension break down, i.e. the use of several RNNs instead of a single one. Besides, the sequential training strategy is optimized in order to allow for GPU usage. [1] L. Wu, V.-D. Nguyen, N.G. Kilingar and L. Noels, Computer Methods in Applied Mechanics and Engineering, 360:113234, 2020. [2] L. Wu, L. Noels, Recurrent Neural Networks (RNNs) in computational mechanics for high dimensional problems: RNNs structure design through dimension reduction and break down; application to multi-scale localization step. Submitted.MOAMM

A stochastic Mean Field Homogenization model of Unidirectional composite materials

Homogenization-based multiscale approaches have been widely developed in order to account for micro-structural geometrical and material properties in an accurate way. However, most of the approaches assume the existence of a statistically Representative Volume Element (RVE), which does not always exist for composite materials due to the existing micro-structural uncertainties, in particular when studying the onset of failure. To address this lack of representativity, a stochastic multi-scale approach for unidirectional composite materials is developed with the aim of predicting scatter in the structural behavior. The first step consists in building Stochastic Volume Elements (SVE) [1] from experimental measurements. Toward this end, statistical functions of the fibers features are extracted from SEM images to generate statistical functions of the micro-structure. The dependent variables are then represented using the copula framework, allowing generating micro-structures respecting the statistical information using a fiber additive process [2]. Probabilistic meso-scale stochastic behaviors are then extracted from direct numerical simulations of the generated SVEs, defining random fields of homogenized properties [2]. Finally, in order to provide an efficient way of generating meso-scale random fields, while keeping information such as stress/strain fields at the micro-scale during the resolution of macro-scale stochastic finite element, a probabilistic Mean-Field-Homogenization (MFH) method is developed, first in the linear range [3] and then in the non-linear one. To this end, the phase parameters of the MFH are seen as random fields defined by inverse stochastic identification of the stochastic homogenized properties obtained through the stochastic direct simulations of the SVEs. The resulting micro-mechanics-based reduced order model allows studying composite failure in a probabilistic way. [1] M. Ostoja-Starzewski, X. Wang, Stochastic finite elements as a bridge between random material microstructure and global response, Computer Methods in Applied Mechanics and Engineering 168 (14) (1999) 35 - 49, [2] L. Wu, C.N. Chung, Z. Major, L. Adam, L. Noels. From SEM images to elastic responses: a stochastic multiscale analysis of UD fiber reinforced composites. Submitted to Composite Structures. [3] L. Wu, L. Adam, L. Noels, A micro-mechanics-based inverse study for stochastic order reduction of elastic UD-fiber reinforced composites analyzes, International Journal for Numerical Methods in Engineering (2018)The research has been funded by the Walloon Region under the agreement no 1410246 - STOMMMAC (CT-INT2013-03-28) in the context of the M-ERA.NET Joint Call 2014

A Recurrent Neural Network-based Surrogate Model for History-Dependent Multi-scale Simulations

Homogenization-based multi-scale analyses are widely used to account for the effect of material heterogeneity at a structural material point. Among the existing different homogenization methods, computational homogenization solves the meso-scale heterogeneous problems using a full field discretization of the micro-structure. When embedded in a multi-scale analyses, computational homogenization results in the so-called FE2 method, which is an accurate methodology but which yields prohibitive computational time. A more efficient approach is to conduct pre-off-line finite element simulations on the meso-scale problem in order to build a surrogate model by means of constructing mapping functions. Once this so-called training step is completed, the surrogate model can be used as the constitutive law of a single-scale simulation, leading to highly efficient simulations. Artificial neural networks (NNWs) offer the possibility to build such a mapping. However, one difficulty arises for history-dependent material behaviours, such as elasto-plasticity, since state variables are needed to account for the loading history. This difficulty can be solved by considering a Recurrent Neural Network (RNN), which uses sequential information. In [1] a RNN was designed using a Gated Recurrent Unit (GRU). In order to achieve accuracy under multi-dimensional non-proportional loading conditions, the sequential training data were obtained from finite element simulations on an elastoplastic composite RVE subjected to random loading paths. The RNN predictions were found to be in agreement with the finite elements simulations.This project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No. 862015 for the project “Multi-scale Optimisation for Additive Manufacturing of fatigue resistant shock-absorbing MetaMaterials (MOAMMM)” of the H2020-EU.1.2.1. - FET Open Programme

A Discontinuous Galerkin Formulation of Kirchhoff-Love Shells: From Linear Elasticity to Finite Deformations

Spatially-discontinuous Galerkin methods constitute a generalization of weak formulations, which allow for discontinuities of the problem unknowns in its domain interior [1]. When considering problems involving high-order derivatives, discontinuous Galerkin methods can also be seen as a means of enforcing higher-order continuity requirements in a weak manner [2,3]. Recently, the authors [4] have proposed a DG formulation for Kirchhoff-Love shell theory for which both the membrane and the bending response of the shell are considered. The proposed one-field formulation takes advantage of the weak enforcement in such a way that the displacements are the only discrete unknowns, while the C1 continuity is enforced weakly. The consistency, stability and rate of convergence of the numerical method are demonstrated for the case of a linear elastic material. In this work, this method is extended to shell problems involving finite displacements and finite deformations