Lippmann-Schwinger solvers for the computational homogenization of materials with pores

We show that under suitable hypotheses on the nonporous material law and a geometric regularity condition on the pore space, Moulinec-Suquet's basic solution scheme converges linearly. We also discuss for which derived solvers a (super)linear convergence behavior may be obtained, and for which such results do not hold, in general. The key technical argument relies on a specific subspace on which the homogenization problem is nondegenerate, and which is preserved by iterations of the basic scheme. Our line of argument is based in the nondiscretized setting, and we draw conclusions on the convergence behavior for discretized solution schemes in FFT-based computational homogenization. Also, we see how the geometry of the pores' interface enters the convergence estimates. We provide computational experiments underlining our claims.     

Computational homogenization of nonlinear elastic materials using neural networks

In this work, a decoupled computational homogenization method for nonlinear elastic materials is proposed using neural networks. In this method, the effective potential is represented as a response surface parameterized by the macroscopic strains and some microstructural parameters. The discrete values of the effective potential are computed by finite element method through random sampling in the parameter space, and neural networks are used to approximate the surface response and to derive the macroscopic stress and tangent tensor components. We show through several numerical convergence analyses that smooth functions can be efficiently evaluated in parameter spaces with dimension up to 10, allowing to consider three‐dimensional representative volume elements and an explicit dependence of the effective behavior on microstructural parameters like volume fraction. We present several applications of this technique to the homogenization of nonlinear elastic composites, involving a two‐scale example of heterogeneous structure with graded nonlinear properties. Copyright © 2015 John Wiley & Sons, Ltd.     

Many-scale finite strain computational homogenization via Concentric Interpolation

A method for efficient computational homogenization of hyperelastic materials under finite strains is proposed. Multiple spatial scales are homogenized in a recursive procedure: starting on the smallest scale, few high fidelity FE computations are performed. The resulting fields of deformation gradient fluctuations are processed by a snapshot POD resulting in a reduced basis (RB) model. By means of the computationally efficient RB model, a large set of samples of the homogenized material response is created. This data set serves as the support for the Concentric Interpolation (CI) scheme, interpolating the effective stress and stiffness. Then, the same procedure is invoked on the next larger scale with this CI surrogating the homogenized material law. A three-scale homogenization process is completed within few hours on a standard workstation. The resulting model is evaluated within minutes on a laptop computer in order to generate fourth-scale results. Open source code is provided.     

An algorithmically consistent macroscopic tangent operator for FFT‐based computational homogenization

The present work addresses a multiscale framework for fast‐Fourier‐transform–based computational homogenization. The framework considers the scale bridging between microscopic and macroscopic scales. While the macroscopic problem is discretized with finite elements, the microscopic problems are solved by means of fast‐Fourier‐transforms (FFTs) on periodic representative volume elements (RVEs). In such multiscale scenario, the computation of the effective properties of the microstructure is crucial. While effective quantities in terms of stresses and deformations can be computed from surface integrals along the boundary of the RVE, the computation of the associated moduli is not straightforward. The key contribution of the present paper is the derivation and implementation of an algorithmically consistent macroscopic tangent operator which directly resembles the effective moduli of the microstructure. The macroscopic tangent is derived by means of the classical Lippmann‐Schwinger equation and can be computed from a simple system of linear equations. This is performed through an efficient FFT‐based approach along with a conjugate gradient solver. The viability and efficiency of the method is demonstrated for a number of two‐ and three‐dimensional boundary value problems incorporating linear and nonlinear elasticity as well as viscoelastic material response.     

An FFT-based method for computing weighted minimal surfaces in microstructures with applications to the computational homogenization of brittle fracture

Cell formulae for the effective crack resistance of a heterogeneous medium obeying Francfort-Marigo's formulation of linear elastic fracture mechanics have been proved recently, both in the context of periodic and stochastic homogenization. This work proposes a numerical strategy for computing the effective, possibly anisotropic, crack resistance of voxelized microstructures using the fast Fourier transform (FFT). Based on Strang's continuous minimum cut—maximum flow duality, we explore a primal-dual hybrid gradient method for computing the effective crack resistance, which may be readily integrated into an existing FFT-based code for homogenizing thermal conductivity. We close with demonstrative numerical experiments.     

A posteriori error estimation for numerical model reduction in computational homogenization of porous media

Numerical model reduction is adopted for solving the microscale problem that arizes from computational homogenization of a model problem of porous media with displacement and pressure as unknown fields. A reduced basis is obtained for the pressure field using (i) spectral decomposition (SD) and (ii) proper orthogonal decomposition (POD). This strategy has been used in previous work—the main contribution of this article is the extension with an a posteriori estimator for assessing the error in (i) energy norm and in (ii) a given quantity of interest. The error estimator builds on previous work by the authors; the novelty presented in this article is the generalization of the estimator to a coupled problem, and, more importantly, to accommodate the estimator for a POD basis rather than the SD basis. Guaranteed, fully computable and low-cost bounds are derived and the performance of the error estimates is demonstrated via numerical results.     

A computational homogenization approach for limit analysis of heterogeneous materials

The macroscopic strength domain and failure mode of heterogeneous or composite materials can be quantitatively determined by solving an auxiliary limit analysis problem formulated on a periodic representative volume element. In this paper, a novel numerical procedure based on kinematic theorem and homogenization theory for limit analysis of periodic composites is developed. The total velocity fields, instead of fluctuating (periodic) velocity, at microscopic level are approximated by finite elements, ensuring that the resulting optimization problem is similar to the limit analysis problem formulated for structures, except for additional constraints, which are periodic boundary conditions and averaging relations. The optimization problem is then formulated in the form of a standard second‐order cone programming problem to be solved by highly efficient solvers. Effects of loading condition, representative volume element architecture, and fiber/air void volume fraction on the macroscopic strength of perforated and fiber reinforced composites are studied in numerical examples. Copyright © 2017 John Wiley & Sons, Ltd.     

Lippmann-Schwinger solvers for the explicit jump discretization for thermal computational homogenization problems

We present a variational formulation and a Lippmann-Schwinger equation for the explicit jump discretization of thermal computational homogenization problems, together with fast and memory-efficient matrix-free solvers based on the fast Fourier transform (FFT). Wiegmann and Zemitis introduced the explicit jump discretization for volumetric image-based computational homogenization of thermal conduction. In contrast to Fourier and finite difference-based discretization methods classically used in FFT-based homogenization, the explicit jump discretization is devoid of ringing and checkerboarding artifacts. Originally, the explicit jump discretization was formulated as the discrete equivalent of a boundary integral equation for the jump in the temperature gradient. The resulting equations are not symmetric positive definite, and thus solved by the BiCGSTAB method. Still, the numerical scheme exhibits stable convergence behavior, also in the presence of pores. In this work, we exploit a reformulation of the explicit jump system in terms of harmonically averaged conductivities. The resulting system is intrinsically symmetric positive definite and admits a Lippmann-Schwinger formulation. A seamless integration into existing FFT-based software packages is ensured. We demonstrate our improvements by numerical experiments.     

A framework for implementation of RVE‐based multiscale models in computational homogenization using isogeometric analysis

This study presents an isogeometric framework for incorporating representative volume element–based multiscale models into computational homogenization. First‐order finite deformation homogenization theory is derived within the framework of the method of multiscale virtual power, and Lagrange multipliers are used to illustrate the effects of considering different kinematical constraints. Using a Lagrange multiplier approach in the numerical implementation of the discrete system naturally leads to a consolidated treatment of the commonly employed representative volume element boundary conditions. Implementation of finite deformation computational strain‐driven, stress‐driven, and mixed homogenization is detailed in the context of isogeometric analysis (IGA), and performance is compared to standard finite element analysis. As finite deformations are considered, a numerical multiscale stability analysis procedure is also detailed for use with IGA. Unique implementation aspects that arise when computational homogenization is performed using IGA are discussed, and the developed framework is applied to a complex curved microstructure representing an architectured material.     

A nonlocal computational homogenization of softening quasi-brittle materials

In this paper, a computational counterpart of the experimental investigation is presented based on a nonlocal computational homogenization technique for extracting damage model parameters in quasi-brittle materials with softening behavior. The technique is illustrated by introducing the macroscopic nonlocal strain to eliminate the mesh sensitivity in the macroscale level as well as the size dependence of the representative volume element (RVE) in the first-order continuous homogenization. The macroscopic nonlocal strains are computed at each direction, and both the local and nonlocal strains are transferred to the microscale level. Two RVEs with similar geometries and material properties are introduced for each macroscopic Gauss point, in which the microscopic damage variables and the macroscale consistent tangent modulus and its derivatives are obtained by imposing the macroscopic nonlocal strain on the first RVE, and the macroscopic stress is computed by employing the microscopic damage variables and imposing the macroscopic local strain over the second RVE. Finally, numerical examples are solved to illustrate the performance of the proposed nonlocal computational homogenization technique for softening quasi-brittle materials.     

Computational nonlinear stochastic homogenization using a nonconcurrent multiscale approach for hyperelastic heterogeneous microstructures analysis

This paper is devoted to the computational nonlinear stochastic homogenization of a hyperelastic heterogeneous microstructure using a nonconcurrent multiscale approach. The geometry of the microstructure is random. The nonconcurrent multiscale approach for micro‐macro nonlinear mechanics is extended to the stochastic case. Because the nonconcurrent multiscale approach is based on the use of a tensorial decomposition, which is then submitted to the curse of dimensionality, we perform an analysis with respect to the stochastic dimension. The technique uses a database describing the strain energy density function (potential) in both the macroscopic Cauchy green strain space and the geometrical random parameters domain. Each value of the potential is numerically computed by means of the FEM on an elementary cell whose geometry is given by the random parameters and the corresponding macroscopic strains being prescribed as boundary conditions. An interpolation scheme is finally introduced to obtain a continuous explicit form of the potential, which, by derivation, allows to evaluate the macroscopic stress and elastic tangent tensors during the macroscopic structural computations. Two numerical examples are presented. Copyright © 2012 John Wiley & Sons, Ltd.     

Mesoscopic aspects of the computational homogenization with meshless modeling for masonry material

The multiscale homogenization scheme is becoming a diffused tool for the analysis of heterogeneous materials as masonry since it allows dealing with the complexity of formulating closed-form constitutive laws by retrieving the material response from the solution of a unit cell (UC) boundary value problem (BVP). The robustness of multiscale simulations depends on the robustness of the nested macroscopic and mesoscopic models. In this study, specific attention is paid to the meshless solution of the UC BVP under plane stress conditions, comparing performances related to the application of linear displacement or periodic boundary conditions (BCs). The effect of the geometry of the UC is also investigated since the BVP is formulated for the two simpliest UCs, according to a displacement-based variational formulation assuming the block indefinitely elastic and the mortar joints as zero-thickness elasto-plastic interfaces. It will be showed that the meshless discretization allows obtaining some advantages with respect to a standard FE mesh. The influence of the UC morphology as well as the BCs on the linear and nonlinear UC macroscopic response is discussed for pure modes of failure. The results can be constructive in view of performing a general Fe·Meshless or Meshless² analysis.     

Multi‐scale constitutive modelling of heterogeneous materials with a gradient‐enhanced computational homogenization scheme

A gradient‐enhanced computational homogenization procedure, that allows for the modelling of microstructural size effects, is proposed within a general non‐linear framework. In this approach the macroscopic deformation gradient tensor and its gradient are imposed on a microstructural representative volume element (RVE). This enables us to incorporate the microstructural size and to account for non‐uniform macroscopic deformation fields within the microstructural cell. Every microstructural constituent is modelled as a classical continuum and the RVE problem is formulated in terms of standard equilibrium and boundary conditions. From the solution of the microstructural boundary value problem, the macroscopic stress tensor and the higher‐order stress tensor are derived based on an extension of the Hill–Mandel condition. This automatically delivers the microstructurally based constitutive response of the higher‐order macro continuum and deals with the microstructural size in a natural way. Several examples illustrate the approach, particularly the microstructural size effects. Copyright © 2002 John Wiley & Sons, Ltd.     

Formulation and implementation of stress‐driven and/or strain‐driven computational homogenization for finite strain

In this paper, we present a homogenization approach that can be used in the geometrically nonlinear regime for stress‐driven and strain‐driven homogenization and even a combination of both. Special attention is paid to the straightforward implementation in combination with the finite‐element method. The formulation follows directly from the principle of virtual work, the periodic boundary conditions, and the Hill–Mandel principle of macro‐homogeneity. The periodic boundary conditions are implemented using the Lagrange multiplier method to link macroscopic strain to the boundary displacements of the computational model of a representative volume element. We include the macroscopic strain as a set of additional degrees of freedom in the formulation. Via the Lagrange multipliers, the macroscopic stress naturally arises as the associated ‘forces’ that are conjugate to the macroscopic strain ‘displacements’. In contrast to most homogenization schemes, the second Piola–Kirchhoff stress and Green–Lagrange strain have been chosen for the macroscopic stress and strain measures in this formulation. The usage of other stress and strain measures such as the first Piola–Kirchhoff stress and the deformation gradient is discussed in the Appendix. Copyright © 2015 John Wiley & Sons, Ltd.     

A computational model for elasto-viscoplastic solids at finite strain with reference to thin shell applications

This work extends a previously developed methodology for computational plasticity at finite strains that is based on the exponential map and logarithmic stretches to the context of isotropic elasto-viscoplastic solids. A particular form of the strain-energy function, given in terms of its principal values is employed. It is noticeable that within the proposed framework, the small strain integration algorithms, and the corresponding consistent tangent operators, automatically extend to the finite strain regime. Central to the effort of this formulation is the derivation of the closed form of a tangent modulus obtained by linearization of incremental non-linear problem. This ensures asymptotically quadratic rates of convergence of the Newton–Raphson procedure in the implicit finite element solution. To illustrate the performance of the presented formulation, several numerical examples, involving failure by strain localization and finite deformations, are given. © 1998 John Wiley & Sons, Ltd.     

Estimation of composite damage model parameters using spectral finite element and neural network

A multi-layer perceptron (MLP) network using error back propagation algorithm is employed in this paper to estimate the damage parameters from broad-band spectral data as diagnostic signal. Various existing models of damage in laminated composite and the resulting stiffness degradation are discussed from comparative view-point. Degradation of ply properties can be considered to be one of the damage model parameters while monitoring transverse matrix cracks in cross-ply, splitting in longitudinal ply, and evolution of consecutive stages of damage, such as delaminations and fiber fracture. The stiffness degradation factor, the location and size of the damaged zone in laminated composite beam are considered as damage model parameters in the present paper. Fourier spectral data, which is typical to most of the diagnostic wave measurements, are used as input to the neural network. Since, training the neural network in such case involves many data sets and all of these data are difficult to generate using experiments, a spectral finite element model (SFEM) with embedded degraded zone in laminated composite beam is developed. Numerical simulation using this element is carried out, which shows the nature of temporal signal that are likely to be measured. Analytical studies on the performance of the neural network are presented based on numerically simulated data. Effect of measurement noise on the network performance is also reported.     

A computational model for the finite element analysis of thermoplasticity coupled with ductile damage at finite strains

An updated Lagrangian implicit FEM model for the analysis of large thermo‐mechanically coupled hyperelastic‐viscoplastic deformations of isotropic porous materials is considered. An appropriate framework for constitutive modelling is introduced that includes a stress‐free thermally expanded configuration and a plastically deformed unstressed damaged configuration. A two‐level iterative scheme is employed at each time increment to solve the field equations governing the conservation of momentum (mechanical step) and the conservation of energy (thermal step) for the coupled thermo‐mechanical problem. Exact linearizations for the calculation of the tangent stiffness are performed in each of these solution steps. A fully implicit, thermo‐mechanically coupled and incrementally objective Euler‐backward radial return based map is developed for the time integration of the constitutive equations. The present model is used to analyse a number of benchmark examples including metal forming processes wherein temperature and the accumulated damage play an important role in influencing the deformation mechanism and the nature of the deformed workpiece. Copyright © 1999 John Wiley & Sons, Ltd.     

A two‐scale failure model for heterogeneous materials: numerical implementation based on the finite element method

In the first part of this contribution, a brief theoretical revision of the mechanical and variational foundations of a Failure‐Oriented Multiscale Formulation devised for modeling failure in heterogeneous materials is described. The proposed model considers two well separated physical length scales, namely: (i) the macroscale where nucleation and evolution of a cohesive surface is considered as a medium to characterize the degradation phenomenon occurring at the lower length scale, and (ii) the microscale where some mechanical processes that lead to the material failure are taking place, such as strain localization, damage, shear band formation, and so on. These processes are modeled using the concept of Representative Volume Element (RVE). On the macroscale, the traction separation response, characterizing the mechanical behavior of the cohesive interface, is a result of the failure processes simulated in the microscale. The traction separation response is obtained by a particular homogenization technique applied on specific RVE sub‐domains. Standard, as well as, Non‐Standard boundary conditions are consistently derived in order to preserve objectivity of the homogenized response with respect to the micro‐cell size. In the second part of the paper, and as an original contribution, the detailed numerical implementation of the two‐scale model based on the finite element method is presented. Special attention is devoted to the topics, which are distinctive of the Failure‐Oriented Multiscale Formulation, such as: (i) the finite element technologies adopted in each scale along with their corresponding algorithmic expressions, (ii) the generalized treatment given to the kinematical boundary conditions in the RVE, and (iii) how these kinematical restrictions affect the capturing of macroscopic material instability modes and the posterior evolution of failure at the RVE level. Finally, a set of numerical simulations is performed in order to show the potentialities of the proposed methodology, as well as, to compare and validate the numerical solutions furnished by the two‐scale model with respect to a direct numerical simulation approach. Copyright © 2013 John Wiley & Sons, Ltd.     

A neural network constitutive model for hyperelasticity based on molecular dynamics simulations

Numerical analysis of the hyperelastic behavior of polymer materials has drawn significant interest from within the field of mechanical engineering. Currently, hyperelastic models based on the energy density function, such as the Neo-Hookean, Mooney-Rivlin, and Ogden models, are used to investigate the hyperelastic responses of materials. Conventionally, constants relating to materials were determined from experimental data by using global least-squares fitting. However, formulating a constitutive equation to capture the complex behavior of hyperelastic materials was difficult owing to the limitations of the analytical model and experimental data. This study addresses these limitations by using a system of neural networks (NNs) to design a data-driven surrogate model without a specific function formula, and employs molecular dynamics (MD) simulations to calculate the massive amount of combined loading data of hyperelastic materials. Thus, MD simulations were used to propose an NN constitutive model for hyperelasticity to derive the constitutive equation to model the complex hyperelastic response. In addition, the probability distributions of the numerical solutions of hyperelasticity are used to characterize the uncertainty of the MD models. These statistical finite element results not only present numerical results with reliability ranges but also scattered distributions of the solution obtained from the MD-based probability distributions.