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.     

Stochastic multiscale analysis in hydrodynamic lubrication

A stochastic multiscale analysis framework is developed for hydrodynamic lubrication problems with random surface roughness. The approach is based on a multi‐resolution computational strategy wherein the deterministic solution of the multiscale problem for each random surface realization is achieved through a coarse‐scale analysis with a local upscaling that is achieved through homogenization theory. The stochastic nature of this solution because of the underlying randomness is then characterized through local and global quantities of interest, accompanied by a detailed discussion regarding suitable choices of the numerical parameters in order to achieve a desired stochastic predictive capability while ensuring numerical efficiency. Finally, models of the stochastic interface response are constructed, and their performance is demonstrated for representative problem settings. Overall, the developed approach offers a computational framework, which can essentially predict the significant influence of interface heterogeneity in the absence of a strict scale separation. Copyright © 2017 John Wiley & Sons, Ltd.     

Smart element method I. The Zienkiewicz–Zhu feedback

A new error control finite element formulation is developed and implemented based on the variational multiscale method, the inclusion theory in homogenization, and the Zienkiewicz–Zhu error estimator. By synthesizing variational multiscale method in computational mechanics, the equivalent eigenstrain principle in micromechanics, and the Zienkiewicz–Zhu error estimator in the finite element method (FEM), the new finite element formulation can automatically detect and subsequently homogenize its own discretization errors in a self‐adaptive and a self‐adjusting manner. It is the first finite element formulation that combines an optimal feedback mechanism and a precisely defined homogenization procedure to reduce its own discretization errors and hence to control numerical pollutions. The paper focuses on the following two issues: (1) how to combine a multiscale method with the existing finite element error estimate criterion through a feedback mechanism, and (2) convergence study. It has been shown that by combining the proposed variational multiscale homogenization method with the Zienkiewicz–Zhu error estimator a clear improvement can be made on the coarse scale computation. It is also shown that when the finite element mesh is refined, the solution obtained by the variational eigenstrain multiscale method will converge to the exact solution. Copyright © 2004 John Wiley & Sons, Ltd.     

Multiscale fatigue life prediction model for heterogeneous materials

A multiscale fatigue life prediction model is developed for heterogeneous materials. The proposed model combines a two‐scale asymptotic homogenization approach in time with a ‘block cycle jump’ technique into a unified temporal multiscale framework that can be effectively utilized for arbitrary material architectures and constitutive equations of microphases. The unified temporal multiscale approach in combination with a spatial multiscale approach based on the reduced order homogenization is characterized for high temperature ceramic matrix composites. Copyright © 2012 John Wiley & Sons, Ltd.     

The finite element square reduced (FE2R) method with GPU acceleration: towards three‐dimensional two‐scale simulations

The FE² method is a renown computational multiscale simulation technique for solid materials with fine‐scale microstructure. It allows for the accurate prediction of the mechanical behavior of structures made of heterogeneous materials with nonlinear material behavior. However, the FE² method leads to excessive CPU time and storage requirements, even for academic two‐dimensional problems. In order to allow for realistic three‐dimensional two‐scale simulations, a significant reduction of the CPU and memory usage is required. For this purpose, the authors have recently proposed a reduced basis homogenization scheme based on a mixed incremental variational principle. The approach exploits the potential structure of generalized standard materials. Thereby, important speed‐ups and memory savings can be achieved. Using high‐performance GPUs, the reduced‐basis method can be further accelerated. In the present contribution, our previous works are combined and extended to form the FE²‐reduced method: the FE^2R. The FE^2R can be used to simulate three‐dimensional structural problems with consideration of the nonlinearity and microstructure of the underlying material at acceptable computational cost. Thereby, it allows for a new level of complexity in nonlinear multiscale simulations. Numerical examples illustrate the capabilities of the chosen approach. Copyright © 2016 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.     

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.     

Locality constraints within multiscale model for non‐linear material behaviour

This paper presents a two‐scale approach for the mechanical and numerical modelling of materials with microstructure‐like concrete or fibre‐reinforced concrete in the non‐linear regime. It addresses applications, where the assumption of scale separation as the basis for classical homogenization methods does not hold. This occurs when the resolution of micro and macro scale does not differ ab initio or when evolving fluctuations in the macro‐fields are in the order of the micro scale during the loading progress. Typical examples are localization phenomena. The objective of the present study is to develop an efficient solution method exploiting the physically existing multiscale character of the problem. The proposed method belongs to the superposition‐based methods with local enrichment of the large‐scale solution ū by a small‐scale part u ′. The main focus of the present formulation is to allow for locality of the small‐scale solution within the large‐scale elements to achieve an efficient solution strategy. At the same time the small‐scale information exchange over the large‐scale element boundaries is facilitated while maintaining the accuracy of a refined complete solution. Thus, the emphasis lies on finding appropriate locality constraints for u ′. To illustrate the method the formulation is applied to a damage mechanics based material model for concrete‐like materials. Copyright © 2006 John Wiley & Sons, Ltd.     

An efficient coarse‐grained parallel algorithm for global–local multiscale computations on massively parallel systems

The existing global–local multiscale computational methods, using finite element discretization at both the macro‐scale and micro‐scale, are intensive both in terms of computational time and memory requirements and their parallelization using domain decomposition methods incur substantial communication overhead, limiting their application. We are interested in a class of explicit global–local multiscale methods whose architecture significantly reduces this communication overhead on massively parallel machines. However, a naïve task decomposition based on distributing individual macro‐scale integration points to a single group of processors is not optimal and leads to communication overheads and idling of processors. To overcome this problem, we have developed a novel coarse‐grained parallel algorithm in which groups of macro‐scale integration points are distributed to a layer of processors. Each processor in this layer communicates locally with a group of processors that are responsible for the micro‐scale computations. The overlapping groups of processors are shown to achieve optimal concurrency at significantly reduced communication overhead. Several example problems are presented to demonstrate the efficiency of the proposed algorithm. Copyright © 2009 John Wiley & Sons, Ltd.     

Computational continua for linear elastic heterogeneous solids on unstructured finite element meshes

The computational continua framework, which is a variant of higher‐order computational homogenization theories that is free of scale separation, does not require higher‐order finite element continuity, and is free of higher‐order boundary conditions, has been generalized to unstructured meshes. The salient features of the proposed generalization are (i) a nonlocal quadrature scheme for distorted elements that accounts for unit cell distortion in the parent element domain and (ii) an approximate variant of the nonlocal quadrature that eliminates the cost of computing positions of the quadrature points in the preprocessing stage. The performance of the computational continua framework on unstructured meshes has been compared to the first‐order homogenization theory and the direct numerical simulation.     

X-FEM implementation of VAMUCH: Application to active structural fiber multi-functional composite materials

In multiscale analysis of composite materials, there is usually a need to solve microstructures problems with complex geometries. The variational asymptotic method for unit cell homogenization (VAMUCH) is a recently developed variant of the asymptotic homogenization approach. In contrast to conventional asymptotic methods, VAMUCH carries out an asymptotic analysis of the variational statement, synthesizing the merits of both variational methods and asymptotic methods. This work gives an outline of the Extended Finite Element Method (X-FEM) implementation of VAMUCH for complex multi-material structures. The X-FEM allows one to use meshes not necessarily matching the physical surface of the problem while retaining the accuracy of the classical finite element approach. For material interfaces, this is achieved by introducing an enrichment strategy. The X-FEM/VAMUCH approach is applied successfully to many examples reported in the VAMUCH literature. Numerical experiments on the periodic homogenization of complex unit cells demonstrate the accuracy and simplicity of the X-FEM/VAMUCH approach.     

Contact mechanics at the nanoscale,a 3D multiscale approach

We present a multiscale coupling method to address contact problems. The components of the model are a molecular dynamics engine, a finite element program and a coupling scheme. We validate the approach, first on Hertzian contact and then with a rough surface contacting a rigid body plane. Various measures are provided to highlight limitations and new opportunities in conducting large‐scale simulations of contact brought by the proposed multiscale approach. Copyright © 2009 John Wiley & Sons, Ltd.     

A hybrid variationally consistent homogenization approach based on Ritz's method

Multiscale approaches based on homogenization theory provide a suitable framework to incorporate information associated with a small‐scale (microscale) problem into the considered large‐scale (macroscopic) problem. In this connection, the present paper proposes a novel computationally efficient hybrid homogenization method. Its backbone is a variationally consistent FE² approach in which every aspect is governed by energy minimization. In particular, scale bridging is realized by the canonical principle of energy equivalence. As a direct implementation of the aforementioned variationally consistent FE² approach is numerically extensive, an efficient approximation based on Ritz's method is advocated. By doing so, the material parameters defining an effective macroscopic material model capturing the underlying microstructure can be efficiently computed. Furthermore, the variational scale bridging principle provides some guidance to choose a suitable family of macroscopic material models. Comparisons between the results predicted by the novel hybrid homogenization method and full field finite element simulations show that the novel method is indeed very promising for multiscale analyses.Copyright © 2013 John Wiley & Sons, Ltd.     

Stochastic homogenization of nano-thickness thin films including patterned holes using structural perturbation method

The aim of this paper is to investigate the statistical response of the homogenized mechanical behavior of nano-thickness thin films with circular holes. For this purpose, a stochastic multiscale framework is proposed. The proposed framework involves molecular dynamics simulation, surface modeling, asymptotic homogenization, moving-mesh technique, Monte-Carlo simulation, and a reduced computational scheme. The surface effect of thin-film material is predicted by the molecular dynamics (MD) approach. The volume fraction and location of each circular hole are considered as geometric uncertainties of a model. In order to investigate the statistical response of the homogenized mechanical behavior, Monte-Carlo simulation is performed to show the probability density distribution of the homogenized elastic modulus against geometric uncertainties. The reduced computational schematic based on the static reduction method and the structural perturbation method is proposed in order to overcome the issues of a cumbersome remeshing procedure and computational inefficiency of Monte-Carlo simulations involving a high number of repetitive trials. A guideline to minimize the coefficient of variation (CV) of the mechanical properties is suggested based on the parametric study.     

Multiscale modeling of impact on heterogeneous viscoelastic solids containing evolving microcracks

Multiscale computational techniques play a major role in solving problems related to viscoelastic composites due to the complexities inherent to these materials. In this paper, a numerical procedure for multiscale modeling of impact on heterogeneous viscoelastic solids containing evolving microcracks is proposed in which the (global scale) homogenized viscoelastic incremental constitutive equations have the same form as the local‐scale viscoelastic incremental constitutive equations, but the homogenized tangent constitutive tensor and the homogenized incremental history‐dependent stress tensor at the global scale depend on the amount of damage accumulated at the local scale. Furthermore, the developed technique allows the computation of the full anisotropic incremental constitutive tensor of viscoelastic solids containing evolving cracks (and other kinds of heterogeneities) by solving the micromechanical problem only once at each material point and each time step. The procedure is basically developed by relating the local‐scale displacement field to the global‐scale strain tensor and using first‐order homogenization techniques. The finite element formulation is developed and some example problems are presented in order to verify the approach and demonstrate the model capabilities. Copyright © 2009 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.     

Two‐level multiscale enrichment methodology for modeling of heterogeneous plates

A new two‐level multiscale enrichment methodology for analysis of heterogeneous plates is presented. The enrichments are applied in the displacement and strain levels: the displacement field of a Reissner–Mindlin plate is enriched using the multiscale enrichment functions based on the partition of unity principle; the strain field is enriched using the mathematical homogenization theory. The proposed methodology is implemented for linear and failure analysis of brittle heterogeneous plates. The eigendeformation‐based model reduction approach is employed to efficiently evaluate the non‐linear processes in case of failure. The capabilities of the proposed methodology are verified against direct three‐dimensional finite element models with full resolution of the microstructure. Copyright © 2009 John Wiley & Sons, Ltd.