Comparison of three accelerated FFT‐based schemes for computing the mechanical response of composite materials

Since the early 1990s when an iterative method based on Fourier transforms was introduced to compute the mechanical properties of heterogeneous materials, several algorithms have been proposed to increase the convergence rate of the initial scheme. This paper is devoted to the comparison of three of these accelerated schemes. It shows that two of them are special cases of the third, corresponding to particular choices of parameters of the method. An upper bound of the spectral radius of the schemes is determined, from which sufficient conditions of convergence of the schemes are derived. Conditions are found for minimizing this upper bound. In particular, the accelerated scheme, which minimizes this upper bound, is exhibited. The paper discusses the choice of the convergence test used in the schemes. Copyright © 2013 John Wiley & Sons, Ltd.     

A computational approach for multi‐scale analysis of heterogeneous elasto‐plastic media subjected to short duration loads

The asymptotic expansion homogenization (AEH) approach has found wide acceptance for the study of heterogeneous structures due to its ability to account for multi‐scale features. The emphasis of the present study is to develop consistent AEH numerical formulations to address elasto‐plastic material response of structures subjected to short‐duration transient loading. A second‐order accurate velocity‐based explicit time integration method, in conjunction with the AEH approach, is currently developed that accounts for large deformation non‐linear material response. The approach is verified under degenerate homogeneous conditions using existing experimental data in the literature and its ability to account for heterogeneous conditions is demonstrated for a number of test problems. Copyright © 2004 John Wiley & Sons, Ltd.     

Stability of an explicit high‐order spectral element method for acoustics in heterogeneous media based on local element stability criteria

This paper considers the stability of an explicit leapfrog time marching scheme for the simulation of acoustic wave propagation in heterogeneous media with high‐order spectral elements. The global stability criterion is taken as a minimum over local element stability criteria, obtained through the solution of element‐borne eigenvalue problems. First, an explicit stability criterion is obtained for the particular case of a strongly heterogeneous and/or rapidly fluctuating medium using asymptotic analysis. This criterion is only dependent upon the maximum velocity at the vertices of the mesh elements, and not on the velocity at the interior nodes of the high‐order elements. Second, in a more general setting, bounds are derived using statistics of the coefficients of the elemental dispersion matrices. Different bounds are presented, discussed, and compared. Several numerical experiments show the accuracy of the proposed criteria in one‐dimensional test cases as well as in more realistic large‐scale three‐dimensional problems.     

Improved guaranteed computable bounds on homogenized properties of periodic media by the Fourier–Galerkin method with exact integration

Moulinec and Suquet introduced FFT‐based homogenization in 1994, and 20years later, their approach is still effective for evaluating the homogenized properties arising from the periodic cell problem. This paper builds on the author's (2013) variational reformulation approximated by trigonometric polynomials establishing two numerical schemes: Galerkin approximation (Ga) and a version with numerical integration (GaNi). The latter approach, fully equivalent to the original Moulinec–Suquet algorithm, was used to evaluate guaranteed upper–lower bounds on homogenized coefficients incorporating a closed‐form double‐grid quadrature. Here, these concepts, based on the primal and dual formulations, are employed for the Ga scheme. For the same computational effort, the Ga outperforms the GaNi with more accurate guaranteed bounds and more predictable numerical behaviors. The quadrature technique leading to block‐sparse linear systems is extended here to materials defined via high‐resolution images in a way that allows for effective treatment using the FFT. Memory demands are reduced by a reformulation of the double‐grid scheme to the original grid scheme using FFT shifts. Minimization of the bounds during iterations of conjugate gradients is effective, particularly when incorporating a solution from a coarser grid. The methodology presented here for the scalar linear elliptic problem could be extended to more complex frameworks. Copyright © 2015 John Wiley & Sons, Ltd.     

Non‐local dispersive model for wave propagation in heterogeneous media: multi‐dimensional case

Three non‐dispersive models in multi‐dimensions have been developed. The first model consists of a leading‐order homogenized equation of motion subjected to the secularity constraints imposing uniform validity of asymptotic expansions. The second, non‐local model, contains a fourth‐order spatial derivative and thus requires C¹ continuous finite element formulation. The third model, which is limited to the constant mass density and a macroscopically orthotropic heterogeneous medium, requires C⁰ continuity only and its finite element formulation is almost identical to the classical local approach with the exception of the mass matrix. The modified mass matrix consists of the classical mass matrix (lumped or consistent) perturbed with a stiffness matrix whose constitutive matrix depends on the unit cell solution. Numerical results are presented to validate the present formulations. Copyright © 2002 John Wiley & Sons, Ltd.     

Non‐local dispersive model for wave propagation in heterogeneous media: one‐dimensional case

Non‐local dispersive model for wave propagation in heterogeneous media is derived from the higher‐order mathematical homogenization theory with multiple spatial and temporal scales. In addition to the usual space–time co‐ordinates, a fast spatial scale and a slow temporal scale are introduced to account for rapid spatial fluctuations of material properties as well as to capture the long‐term behaviour of the homogenized solution. By combining various order homogenized equations of motion the slow time dependence is eliminated giving rise to the fourth‐order differential equation, also known as a ‘bad’ Boussinesq problem. Regularization procedures are then introduced to construct the so‐called ‘good’ Boussinesq problem, where the need for C¹ continuity is eliminated. Numerical examples are presented to validate the present formulation. Copyright © 2002 John Wiley & Sons, Ltd.     

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.     

FFT‐based homogenization for microstructures discretized by linear hexahedral elements

The FFT‐based homogenization method of Moulinec–Suquet has recently emerged as a powerful tool for computing the macroscopic response of complex microstructures for elastic and inelastic problems. In this work, we generalize the method to problems discretized by trilinear hexahedral elements on Cartesian grids and physically nonlinear elasticity problems. We present an implementation of the basic scheme that reduces the memory requirements by a factor of four and of the conjugate gradient scheme that reduces the storage necessary by a factor of nine compared with a naive implementation. For benchmark problems in linear elasticity, the solver exhibits mesh‐ and contrast‐independent convergence behavior and enables the computational homogenization of complex structures, for instance, arising from computed tomography computed tomography (CT) imaging techniques. There exist 3D microstructures involving pores and defects, for which the original FFT‐based homogenization scheme does not converge. In contrast, for the proposed scheme, convergence is ensured. Also, the solution fields are devoid of the spurious oscillations and checkerboarding artifacts associated to conventional schemes. We demonstrate the power of the approach by computing the elasto‐plastic response of a long‐fiber reinforced thermoplastic material with 172 × 10⁶ (displacement) degrees of freedom. Copyright © 2016 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 Fourier‐series‐based virtual fields method for the identification of 2‐D stiffness distributions

The virtual fields method (VFM) is a powerful technique for the calculation of spatial distributions of material properties from experimentally determined displacement fields. A Fourier‐series‐based extension to the VFM (the F‐VFM) is presented here, in which the unknown stiffness distribution is parameterised in the spatial frequency domain rather than in the spatial domain as used in the classical VFM. We present in this paper the theory of the F‐VFM for the case of elastic isotropic thin structures with known boundary conditions. An efficient numerical algorithm based on the two‐dimensional Fast Fourier Transform (FFT) is presented, which reduces the computation time by three to four orders of magnitude compared with a direct implementation of the F‐VFM for typical experimental dataset sizes. Artefacts specific to the F‐VFM (ringing at the highest spatial frequency near to modulus discontinuities) can be largely removed through the use of appropriate filtering strategies. Reconstruction of stiffness distributions with the F‐VFM has been validated on three stiffness distribution scenarios under varying levels of noise in the input displacement fields. Robust reconstructions are achieved even when the displacement noise is higher than in typical experimental fields.Copyright © 2014 John Wiley & Sons, Ltd.     

A polarization‐based FFT iterative scheme for computing the effective properties of elastic composites with arbitrary contrast

It is recognized that the convergence of FFT‐based iterative schemes used for computing the effective properties of elastic composite materials drastically depends on the contrast between the phases. Particularly, the rate of convergence of the strain‐based iterative scheme strongly decreases when the composites contain very stiff inclusions and the method diverges in the case of rigid inclusions. Reversely, the stress‐based iterative scheme converges rapidly in the case of composites with very stiff or rigid inclusions but leads to low convergence rates when soft inclusions are considered and to divergence for composites containing voids. It follows that the computation of effective properties is costly when the heterogeneous medium contains simultaneously soft and stiff phases. Particularly, the problem of composites containing voids and rigid inclusions cannot be solved by the strain or the stress‐based approaches. In this paper, we propose a new polarization‐based iterative scheme for computing the macroscopic properties of elastic composites with an arbitrary contrast which is nearly as simple as the basic schemes (strain and stress‐based) but which has the ability to compute the overall properties of multiphase composites with arbitrary elastic moduli, as illustrated through several examples. Copyright © 2011 John Wiley & Sons, Ltd.     

The finite element method for the mechanically based model of non‐local continuum

In this paper the finite element method (FEM) for the mechanically based non‐local elastic continuum model is proposed. In such a model each volume element of the domain is considered mutually interacting with the others, beside classical interactions involved by the Cauchy stress field, by means of central body forces that are monotonically decreasing with their inter‐distance and proportional to the product of the interacting volume elements. The constitutive relations of the long‐range interactions involve the product of the relative displacement of the centroids of volume elements by a proper, distance‐decaying function, which accounts for the decrement of the long‐range interactions as long as distance increases. In this study, the elastic problem involving long‐range central interactions for isotropic elastic continuum will be solved with the aid of the FEM. The accuracy of the solution obtained with the proposed FEM code is compared with other solutions obtained with Galerkins' approximation as well as with finite difference method. Moreover, a parametric study regarding the effect of the material length scale in the mechanically based model and in the Kr”oner–Eringen non‐local elasticity has been investigated for a plane elasticity problem. Copyright © 2011 John Wiley & Sons, Ltd.     

A coupling extended multiscale finite element method for dynamic analysis of heterogeneous saturated porous media

A coupling extended multiscale finite element method (CEMsFEM) is developed for the dynamic analysis of heterogeneous saturated porous media. The coupling numerical base functions are constructed by a unified method with an equivalent stiffness matrix. To improve the computational accuracy, an additional coupling term that could reflect the interaction of the deformations among different directions is introduced into the numerical base functions. In addition, a kind of multi‐node coarse element is adopted to describe the complex high‐order deformation on the boundary of the coarse element for the two‐dimensional dynamic problem. The coarse element tests show that the coupling numerical base functions could not only take account of the interaction of the solid skeleton and the pore fluid but also consider the effect of the inertial force in the dynamic problems. On the other hand, based on the static balance condition of the coarse element, an improved downscaling technique is proposed to directly obtain the satisfying microscopic solutions in the CEMsFEM. Both one‐dimensional and two‐dimensional numerical examples of the heterogeneous saturated porous media are carried out, and the results verify the validity and the efficiency of the CEMsFEM by comparing with the conventional finite element method. Copyright © 2015 John Wiley & Sons, Ltd.     

Fourier‐series‐based virtual fields method combining with moiré interferometry for characterising elastic modulus distribution of laser repaired GH4169

Laser cladding is an effective technology for repairing damaged components with high efficiency and low cost. Characterisation for the mechanical properties of the repaired structure is of great importance to evaluate its reliability. In this study, the mechanical properties of laser repaired GH4169 are investigated by Fourier‐series‐based virtual fields method and moiré interferometry. The elastic modulus distribution of the whole structure is identified to understand the mechanical performance of the repaired zone and base material. A uniaxial tensile test is performed and the full‐field deformation is measured by moiré interferometry. With the measured strain field, the distribution of modulus in repaired structures can be calculated. In addition, simulation experiments of moiré interferometry are conducted to optimise the parameters of Fourier‐series‐based virtual fields method to improve the accuracy of the proposed method. On the basis of these results, experiments on the repaired GH4169 material are performed and the variation of modulus in the whole structure is identified. The results verify that the proposed method is effective for characterising modulus distribution of the laser repaired structures, and will have a good prospect for further application.     

Fourier‐based numerical approximation of the Weertman equation for moving dislocations

This work addresses the numerical approximation of solutions to a dimensionless form of the Weertman equation, which models a steadily moving dislocation and is an important extension (with advection term) of the celebrated Peierls‐Nabarro equation for a static dislocation. It belongs to the class of nonlinear reaction‐advection‐diffusion integro‐differential equations with Cauchy‐type kernel, thus involving an integration over an unbounded domain. In the Weertman problem, the unknowns are the shape of the core of the dislocation and the dislocation velocity. The proposed numerical method rests on a time‐dependent formulation that admits the Weertman equation as its long‐time limit. Key features are (1) time iterations are conducted by means of a new, robust, and inexpensive Preconditioned Collocation Scheme in the Fourier domain, which allows for explicit time evolution but amounts to implicit time integration, thus allowing for large time steps; (2) as the integration over the unbounded domain induces a solution with slowly decaying tails of important influence on the overall dislocation shape, the action of the operators at play is evaluated with exact asymptotic estimates of the tails, combined with discrete Fourier transform operations on a finite computational box of size L; (3) a specific device is developed to compute the moving solution in a comoving frame, to minimize the effects of the finite‐box approximation. Applications illustrate the efficiency of the approach for different types of nonlinearities, with systematic assessment of numerical errors. Converged numerical results are found insensitive to the time step, and scaling laws for the combined dependence of the numerical error with respect to L and to the spatial step size are obtained. The method proves fast and accurate and could be applied to a wide variety of equations with moving fronts as solutions; notably, Weertman‐type equations with the Cauchy‐type kernel replaced by a fractional Laplacian.     

Numerical plate testing for linear two‐scale analyses of composite plates with in‐plane periodicity

A method of numerical plate testing (NPT) for composite plates with in‐plane periodic heterogeneity is proposed. In the two‐scale boundary value problem, a thick plate model is employed at macroscale, while three‐dimensional solids are assumed at microscale. The NPT, which is nothing more or less than the homogenization analysis, is in fact a series of microscopic analyses on a unit cell that evaluates the macroscopic plate stiffnesses. The specific functional forms of microscopic displacements are originally presented so that the relationship between the macroscopic resultant stresses/moments and strains/curvatures to be consistent with the microscopic equilibrated state. In order to perform NPT by using general‐purpose FEM programs, we introduce control nodes to facilitate the multiple‐point constraints for in‐plane periodicity. Numerical examples are presented to verify that the proposed method of NPT reproduces the plate stiffnesses in classical plate and laminate theories. We also perform a series of homogenization, macroscopic, and localization analyses for an in‐plane heterogeneous composite plate to demonstrate the performance of the proposed method. Copyright © 2015 John Wiley & Sons, Ltd.     

A novel numerical method of high‐order accuracy for flow in unsaturated porous media

We construct finite volume schemes of very high order of accuracy in space and time for solving the nonlinear Richards equation (RE). The general scheme is based on a three‐stage predictor–corrector procedure. First, a high‐order weighted essentially non‐oscillatory (WENO) reconstruction procedure is applied to the cell averages at the current time level to guarantee monotonicity in the presence of steep gradients. Second, the temporal evolution of the WENO reconstruction polynomials is computed in a predictor stage by using a global weak form of the governing equations. A global space–time DG FEM is used to obtain a scheme without the parabolic time‐step restriction caused by the presence of the diffusion term in the RE. The resulting nonlinear algebraic system is solved by a Newton–Krylov method, where the generalized minimal residual method algorithm of Saad and Schulz is used to solve the linear subsystems. Finally, as a third step, the cell averages of the finite volume method are updated using a one‐step scheme, on the basis of the solution calculated previously in the space–time predictor stage. Our scheme is validated against analytical, experimental, and other numerical reference solutions in four test cases. A numerical convergence study performed allows us to show that the proposed novel scheme is high order accurate in space and time. Copyright © 2011 John Wiley & Sons, Ltd.     

A Bioinspired Interface Design for Improving the Strength and Electrical Conductivity of Graphene‐Based Fibers

Graphene‐based fibers (GBFs) are attractive for next‐generation wearable electronics due to their potentially high mechanical strength, superior flexibility, and excellent electrical and thermal conductivity. Many efforts have been devoted to improving these properties of GBFs in the past few years. However, fabricating GBFs with high strength and electrical conductivity simultaneously remains as a great challenge. Herein, inspired by nacre‐like multilevel structural design, an interface‐reinforced method is developed to improve both the mechanical property and electrical conductivity of the GBFs by introducing polydopamine‐derived N‐doped carbon species as resistance enhancers, binding agents, and conductive connection “bridges.” Remarkably, both the tensile strength and electrical conductivity of the obtained GBFs are significantly improved to ≈724 MPa and ≈6.6 × 10⁴ S m^?1, respectively, demonstrating great superiority compared to previously reported similar GBFs. These outstanding integrated performances of the GBFs provide it with great application potential in the fields of flexible and wearable microdevices such as sensors, actuators, supercapacitors, and batteries.     

Algorithms for automated meshing and unit cell analysis of periodic composites with hierarchical tri‐quadratic tetrahedral elements

Unit cell homogenization techniques together with the finite element method are very effective for computing equivalent mechanical properties of composites and heterogeneous materials systems. For systems with very complicated material arrangements, traditional, manual mesh generation can be a considerable obstacle to usage of these techniques. This problem is addressed here by developing automated meshing techniques that start from a hierarchical quad‐tree (in 2D) or oc‐tree (in 3D) mesh of pixel or voxel elements. From the pixel/voxel mesh, algorithms are presented for successive element splitting and nodal shifting to arrive at final meshes that accurately capture both material arrangements and constituent volume fractions, and the material‐scale stress and strain fields within the composite under different modalities of loading. The performance and associated convergence behaviour of the proposed techniques are demonstrated on both densely packed fibre and particulate composites, and on 3D textile‐reinforced composites. Copyright © 2003 John Wiley Sons, Ltd.