Numerical homogenization of N‐component composites including stochastic interface defects

The purpose of this paper is to present a mathematical formulation and numerical analysis for a homogenization problem of random elastic composites with stochastic interface defects. The homogenization of composites so defined is carried out in two steps: (i) probabilistic averaging of stochastic discontinuities in the interphase region, (ii) probabilistic homogenization by extending the effective modules method to media random in the micro‐scale. To obtain such an approach the classical mathematical homogenization method is formulated for n‐component composite with random elastic components and implemented in the FEM‐based computer program. The article contains also numerous computational experiments illustrating stochastic sensitivity of the model to interface defects parameters and verifying statistical convergence of probabilistic simulation procedure. Copyright © 2000 John Wiley & Sons, Ltd.     

Multiscale thermomechanical contact: Computational homogenization with isogeometric analysis

A computational homogenization framework is developed in the context of the thermomechanical contact of two boundary layers with microscopically rough surfaces. The major goal is to accurately capture the temperature jump across the macroscopic interface in the finite deformation regime with finite deviations from the equilibrium temperature. Motivated by the limit of scale separation, a two‐phase thermomechanically decoupled methodology is introduced, wherein a purely mechanical contact problem is followed by a purely thermal one. In order to correctly take into account finite size effects that are inherent to the problem, this algorithmically consistent two‐phase framework is cast within a self‐consistent iterative scheme that acts as a first‐order corrector. For a comparison with alternative coupled homogenization frameworks as well as for numerical validation, a mortar‐based thermomechanical contact algorithm is introduced. This algorithm is uniformly applicable to all orders of isogeometric discretizations through non‐uniform rational B‐spline basis functions. Overall, the two‐phase approach combined with the mortar contact algorithm delivers a computational framework of optimal efficiency that can accurately represent the geometry of smooth surface textures. Copyright © 2013 John Wiley & Sons, Ltd.     

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.     

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.     

Variational multiscale enrichment for modeling coupled mechano‐diffusion problems

In this paper, a new multiscale–multiphysics computational methodology is devised for the analysis of coupled diffusion–deformation problems. The proposed methodology is based on the variational multiscale principles. The basic premise of the approach is accurate fine‐scale representation at a small subdomain where it is known a priori that important physical phenomena are likely to occur. The response within the remainder of the problem domain is idealized on the basis of coarse‐scale representation. We apply this idea to evaluate a coupled mechano‐diffusion problem that idealizes the response of titanium structures subjected to a thermo–chemo–mechanical environment. The proposed methodology is used to devise a multiscale model in which the transport of oxygen into titanium is modeled as a diffusion process, whereas the mechanical response is idealized using concentration‐dependent elasticity equations. A coupled solution strategy based on operator split is formulated to evaluate the coupled multiphysics and multiscale problem. Numerical experiments are conducted to assess the accuracy and computational performance of the proposed methodology. Numerical simulations indicate that the variational multiscale enrichment has reasonable accuracy and is computationally efficient in modeling the coupled mechano‐diffusion response. Copyright © 2011 John Wiley & Sons, Ltd.     

Approximation of Cahn–Hilliard diffuse interface models using parallel adaptive mesh refinement and coarsening with C1 elements

A variational formulation and C¹ finite element scheme with adaptive mesh refinement and coarsening are developed for phase‐separation processes described by the Cahn–Hilliard diffuse interface model of transport in a mixture or alloy. The adaptive scheme is guided by a Laplacian jump indicator based on the corresponding term arising from the weak formulation of the fourth‐order non‐linear problem, and is implemented in a parallel solution framework. It is then applied to resolve complex evolving interfacial solution behavior for 2D and 3D simulations of the classic spinodal decomposition problem from a random initial mixture and to other phase‐transformation applications of interest. Simulation results and adaptive performance are discussed. The scheme permits efficient, robust multiscale resolution and interface characterization. Copyright © 2008 John Wiley & Sons, Ltd.     

A fast algorithm for multifield representation and multiscale simulation of high‐quality 3D stochastic aggregate microstructures by concurrent coupling of stationary Gaussian and fractional Brownian random fields

This study presents a multiscale computational framework for the representation and generation of concrete aggregate microstructures on the basis of the multifield theory, which couples the stationary Gaussian random field with the fractional Brownian random field. Specifically, the stationary Gaussian field is utilized to simulate the morphological shape of an aggregate on the coarse scale, whereas the surface topography of the aggregate on the fine scale is represented by the fractional Brownian field. To bridge the 2 scales, a concurrent coupling formula is proposed. This coupled technique allows for smooth transition between the coarse and fine scales and permits the rapid generation of highly realistic concrete aggregates that can be tailored to the desired quality and requirements, making the algorithm computationally appealing. In the generation of the random fields on the 2 scales, the Fourier representation of block circulant covariance matrices with circulant blocks is exploited, which yields substantial efficiency advantages over the conventional Cholesky decomposition approach in factorizing covariance matrices as well as simulating random fields. Meanwhile, a microsurface postprocessing and reconstruction procedure is also developed to convert the generated random fields into realistic 3D shapes. The numerical methodology proposed in this study offers tremendous potential for a plethora of applications in cement‐based materials.     

Thermal contact conductance characterization via computational contact homogenization: A finite deformation theory framework

In order to predict the macroscopic thermal response of contact interfaces between rough surface topographies, a computational contact homogenization technique is developed at the finite deformation regime. The overall homogenization framework transfers macroscopic contact variables, such as surfacial stretch, pressure and heat flux, as boundary conditions on a test sample within a micromechanical interface testing procedure. An analysis of the thermal dissipation within the test sample reveals a thermodynamically consistent identification for the macroscopic thermal contact conductance parameter that enables the solution of a homogenized thermomechanical contact boundary value problem based on standard computational approaches. The homogenized contact response effectively predicts a temperature jump across the macroscale contact interface. The strong dependence of this homogenized response on macroscale solution variables of interest is demonstrated via representative three‐dimensional numerical investigations. The proposed contact homogenization framework is suitable for the analysis of similar energy transport phenomena across heterogeneous contact interfaces where the investigation of the sources for energy dissipation is of concern. Copyright © 2010 John Wiley & Sons, Ltd.     

FEM×DEM multiscale modeling: Model performance enhancement from Newton strategy to element loop parallelization

This paper presents a multiscale model based on a FEM×DEM approach, a method that couples discrete elements at the microscale and finite elements at the macroscale. FEM×DEM has proven to be an effective way to treat real‐scale engineering problems by embedding constitutive laws numerically obtained using discrete elements into a standard finite element framework. This proposed paper focuses on some numerical open issues of the method. Given the nonlinearity of the problem, Newton's method is required. The standard full Newton method is modified by adopting operators different from the consistent tangent matrix and by developing ad‐hoc solution strategies. The efficiency of several existing operators is compared, and a new and original strategy is proposed, which is shown to be numerically more efficient than the existing propositions. Furthermore, a shared memory parallelization framework using OpenMP directives is introduced. The combination of these enhancements allows to overcome the FEM×DEM computational limitations, thus making the approach competitive with classical FEM in terms of stability and computational cost.     

A multilevel projection‐based model order reduction framework for nonlinear dynamic multiscale problems in structural and solid mechanics

A reduction/hyper reduction framework is presented for dramatically accelerating the solution of nonlinear dynamic multiscale problems in structural and solid mechanics. At each scale, the dimensionality of the governing equations is reduced using the method of snapshots for proper orthogonal decomposition, and computational efficiency is achieved for the evaluation of the nonlinear reduced‐order terms using a carefully designed configuration of the energy conserving sampling and weighting method. Periodic boundary conditions at the microscales are treated as linear multipoint constraints and reduced via projection onto the span of a basis formed from the singular value decomposition of Lagrange multiplier snapshots. Most importantly, information is efficiently transmitted between the scales without incurring high‐dimensional operations. In this proposed proper orthogonal decomposition–energy conserving sampling and weighting nonlinear model reduction framework, training is performed in two steps. First, a microscale hyper reduced‐order model is constructed in situ, or using a mesh coarsening strategy, in order to achieve significant speedups even in non‐parametric settings. Next, a classical offline–online training approach is performed to build a parametric hyper reduced‐order macroscale model, which completes the construction of a fully hyper reduced‐order parametric multiscale model capable of fast and accurate multiscale simulations. A notable feature of this computational framework is the minimization, at the macroscale level, of the cost of the offline training using the in situ or coarsely trained hyper reduced‐order microscale model to accelerate snapshot acquisition. The effectiveness of the proposed hyper reduction framework at accelerating the solution of nonlinear dynamic multiscale problems is demonstrated for two problems in structural and solid mechanics. Speedup factors as high as five orders of magnitude are shown to be achievable. Copyright © 2017 John Wiley & Sons, Ltd.     

A multiscale framework for computational nanomechanics: Application to the modeling of carbon nanotubes

A multiscale computational framework is presented that provides a coupled self‐consistent system of equations involving molecular mechanics at small scales and quasi‐continuum mechanics at large scales. The proposed method permits simultaneous resolution of quasi‐continuum and atomistic length scales and the associated displacement fields in a unified manner. Interatomic interactions are incorporated into the method through a set of analytical equations that contain nanoscale‐based material moduli. These material moduli are defined via internal variables that are functions of the local atomic configuration parameters. Point defects like vacancy defects in nanomaterials perturb the atomic structure locally and generate localized force fields. Formation energy of vacancy is evaluated via interatomic potentials and minimization of this energy leads to nanoscale force fields around defects. These nanoscale force fields are then employed in the multiscale method to solve for the localized displacement fields in the vicinity of vacancies and defects. The finite element method that is developed based on the hierarchical multiscale framework furnishes a two‐level statement of the problem. It concurrently feeds information at the molecular scale, formulated in terms of the nanoscale material moduli, into the quasi‐continuum equations. Representative numerical examples are shown to validate the model and demonstrate its range of applicability. Copyright © 2008 John Wiley & Sons, Ltd.     

Fully coupled hydromechanical multiscale model with microdynamic effects

In this paper, a multiscale finite element framework is developed based on the first‐order homogenization method for fully coupled saturated porous media using an extension of the Hill‐Mandel theory in the presence of microdynamic effects. The multiscale method is employed for the consolidation problem of a 2‐dimensional saturated soil medium generated from the periodic arrangement of circular particles embedded in a square matrix, which is compared with the direct numerical simulation method. The effects of various issues, including the boundary conditions, size effects, particle arrangements, and the integral domain constraints for the microscale boundary value problem, are numerically investigated to illustrate the performance of a representative volume element in the proposed computational homogenization method of fully coupled saturated porous media. This study is aimed to clarify the effect of scale separation and size dependence, and to introduce characteristics of a proper representative volume element in multiscale modeling of saturated porous media.     

Stochastic projection schemes for deterministic linear elliptic partial differential equations on random domains

We present stochastic projection schemes for approximating the solution of a class of deterministic linear elliptic partial differential equations defined on random domains. The key idea is to carry out spatial discretization using a combination of finite element methods and stochastic mesh representations. We prove a result to establish the conditions that the input uncertainty model must satisfy to ensure the validity of the stochastic mesh representation and hence the well posedness of the problem. Finite element spatial discretization of the governing equations using a stochastic mesh representation results in a linear random algebraic system of equations in a polynomial chaos basis whose coefficients of expansion can be non‐intrusively computed either at the element or the global level. The resulting randomly parametrized algebraic equations are solved using stochastic projection schemes to approximate the response statistics. The proposed approach is demonstrated for modeling diffusion in a square domain with a rough wall and heat transfer analysis of a three‐dimensional gas turbine blade model with uncertainty in the cooling core geometry. The numerical results are compared against Monte–Carlo simulations, and it is shown that the proposed approach provides high‐quality approximations for the first two statistical moments at modest computational effort. Copyright © 2010 John Wiley & Sons, Ltd.     

A multiscale virtual element method for the analysis of heterogeneous media

We introduce a novel heterogeneous multiscale method for the elastic analysis of two-dimensional domains with a complex microstructure. To this end, the multiscale finite element method is revisited and originally upgraded by introducing virtual element discretizations at the microscale, hence allowing for generalized polygonal and nonconvex elements. The microscale is upscaled through the numerical evaluation of a set of multiscale basis functions. The solution of the equilibrium equations is performed at the coarse scale at a reduced computational cost. We discuss the computation of the multiscale basis functions and corresponding virtual projection operators. The performance of the method in terms of accuracy and computational efficiency is evaluated through a set of numerical examples.     

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 iterative polynomial chaos approach toward stochastic elastostatic structural analysis with non-Gaussian randomness

Stochastic analysis of structure with non-Gaussian material property and loading in the framework of polynomial chaos (PC) is considered. A new approach for the solution of stochastic mechanics problem with random coefficient is presented. The major focus of the method is to consider reduced size of expansion in an iterative manner to overcome the problem of large system matrix in conventional PC expansion. The iterative method is based on orthogonal expansion of stochastic responses and generation of an iterative PC based on the responses of the previous iteration. The polynomials are evaluated using Gram-Schmidt orthogonalization process. The numbers of random variables in PC expansion are reduced by considering only the dominant components of the response characteristics, which is evaluated using Karhunen-Loève (KL) expansion. In case of random material field problem, the KL expansion is used to discretize and simulate the non-Gaussian random field. Independent component analysis (ICA) is carried out on the non-Gaussian KL random variables to minimize statistical dependence. The usefulness of the proposed method in terms of accuracy and computational efficiency is examined. From the numerical analysis of three different types of structural mechanics problems, the proposed iterative method is observed to be computationally more efficient and accurate than conventional PC method for solution of linear elastostatic structural mechanics problems.     

A multiscale framework for localizing microstructures towards the onset of macroscopic discontinuity

This paper presents a multiscale computational homogenization model for the post localization behavior of a macroscale domain crossed by a cohesive discontinuity emanating from microstructural damage. The stress–strain and the cohesive macroscopic responses are obtained incorporating the underlying microstructure, in which the damage evolution results in the formation of a strain localization band. The macro structural kinematics entails a discontinuous displacement field and a non-uniform deformation field across the discontinuity. Novel scale transitions are formulated to provide a consistent coupling to the continuous microscale kinematics. From the solution of the micromechanical boundary value problem, the macroscale stress responses at both sides of the discontinuity are recovered, providing automatically the cohesive tractions at the interface. The effective displacement jump and deformation field discontinuity are derived from the same microscale analysis. This contribution focusses on scale transition relations and on the solution procedure at the microlevel; the highlights of the approach are demonstrated on microscale numerical examples. Coupled two-scale solution strategy will be presented in a subsequent paper.     

Multiscale design optimization of lithium ion batteries using adjoint sensitivity analysis

The capacity of lithium ion batteries can be improved through the use of functionally graded electrodes. Here, we present a computational framework for optimizing the layout of electrodes using a multiscale lithium ion battery cell model. The model accounts for nonlinear transient transport processes and mechanical deformations at multiple scales. A key component of the optimization methodology is the formulation of the adjoint sensitivity equations of the multiscale battery model. The efficient solution of the adjoint equations relies on the decomposition of the multiscale problem into multiple, computationally small problems associated with the individual realizations of the microscale model. This decomposition method is shown to significantly reduce the computational time needed for sensitivity analysis versus numerical finite differencing. The potential of the proposed optimization framework is illustrated with numerical problems involving both macroscale and microscale performance criteria and design variables. The usable capacity of a lithium ion battery cell is maximized while limiting the stress level in the electrode particles through manipulation of the local porosities and particle radii. The optimization results suggest that optimal functionally graded electrodes improve the performance of a battery cell over using uniform porosity and particle radius distributions. Copyright © 2012 John Wiley & Sons, Ltd.     

An extended finite element/level set method to study surface effects on the mechanical behavior and properties of nanomaterials

We present a new approach based on coupling the extended finite element method (XFEM) and level sets to study surface and interface effects on the mechanical behavior of nanostructures. The coupled XFEM‐level set approach enables a continuum solution to nanomechanical boundary value problems in which discontinuities in both strain and displacement due to surfaces and interfaces are easily handled, while simultaneously accounting for critical nanoscale surface effects, including surface energy, stress, elasticity and interface decohesion. We validate the proposed approach by studying the surface‐stress‐driven relaxation of homogeneous and bi‐layer nanoplates as well as the contribution from the surface elasticity to the effective stiffness of nanobeams. For each case, we compare the numerical results with new analytical solutions that we have derived for these simple problems; for the problem involving the surface‐stress‐driven relaxation of a homogeneous nanoplate, we further validate the proposed approach by comparing the results with those obtained from both fully atomistic simulations and previous multiscale calculations based upon the surface Cauchy–Born model. These numerical results show that the proposed method can be used to gain critical insights into how surface effects impact the mechanical behavior and properties of homogeneous and composite nanobeams under generalized mechanical deformation. Copyright © 2010 John Wiley & Sons, Ltd.