Multiscale aggregating discontinuities method for micro–macro failure of composites

The application of a multiscale method, called the multiscale aggregating discontinuities (MAD) method, to the failure analysis of composites is described. Two distinct features of the MAD method are the use of perforated unit cells, and the extraction of coarse-grained failure information. In the perforated unit cell, all subdomains of the unit cell that are not strictly elliptic are excluded, which enables the decomposition of its stable and unstable material. By means of these concepts, it is possible to compute an equivalent discontinuity at the macroscale, including both the direction and the magnitude of the discontinuity. This equivalent discontinuity is then passed to the macroscale along with the computed stress from the unit cell. The macroscale discontinuity is injected into the macro model by the extended finite element method (XFEM) procedure. In this paper, the method is improved by adding hourglass modes to the unit cell deformations, which better model growing cracks. Several examples comparing the MAD method with direct numerical simulations are presented.     

Coarse‐graining of multiscale crack propagation

A method for coarse graining of microcrack growth to the macroscale through the multiscale aggregating discontinuity (MAD) method is further developed. Three new features are: (1) methods for treating nucleating cracks, (2) the linking of the micro unit cell with the macroelement by the hourglass mode, and (3) methods for recovering macrocracks with variable crack opening. Unlike in the original MAD method, ellipticity is not retained at the macroscale in the bulk material, but we show that the element stiffness of the bulk material is positive definite. Several examples with comparisons with direct numerical simulations are given to demonstrate the effectiveness of the method. Copyright © 2009 John Wiley & Sons, Ltd.     

A generalized phase field multiscale finite element method for brittle fracture

A generalized multiscale finite element method is introduced to address the computationally taxing problem of elastic fracture across scales. Crack propagation is accounted for at the microscale utilizing phase field theory. Both the displacement-based equilibrium equations and phase field state equations at the microscale are mapped on a coarser scale. The latter is defined by a set of multinode coarse elements, where solution of the governing equations is performed. Mapping is achieved by employing a set of numerically derived multiscale shape functions. A set of representative benchmark tests is used to verify the proposed procedure and assess its performance in terms of accuracy and efficiency compared with the standard phase field finite element implementation.     

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 modelling of particle debonding in reinforced elastomers subjected to finite deformations

Interfacial damage nucleation and evolution in reinforced elastomers subjected to finite strains is modelled using the mathematical theory of homogenization based on the asymptotic expansion of unknown variables. The microscale is characterized by a periodic unit cell, which contains particles dispersed in a blend and the particle matrix interface is characterized by a cohesive law. A novel numerical framework based on the perturbed Petrov–Galerkin method for the treatment of nearly incompressible behaviour is employed to solve the resulting boundary value problem on the microscale and the deformation path of a macroscale particle is predefined as in the micro‐history recovery procedure. A fully implicit and efficient finite element formulation, including consistent linearization, is presented. The proposed multiscale framework is capable of predicting the non‐homogeneous micro‐fields and damage nucleation and propagation along the particle matrix interface, as well as the macroscopic response and mechanical properties of the damaged continuum. Examples are considered involving simple unit cells in order to illustrate the multiscale algorithm and demonstrate the complexity of the underlying physical processes. Copyright © 2005 John Wiley & Sons, Ltd.     

Elastoplasticity with linear tetrahedral elements: A variational multiscale method

We present a computational framework for the simulation of J₂‐elastic/plastic materials in complex geometries based on simple piecewise linear finite elements on tetrahedral grids. We avoid spurious numerical instabilities by means of a specific stabilization method of the variational multiscale kind. Specifically, we introduce the concept of subgrid‐scale displacements, velocities, and pressures, approximated as functions of the governing equation residuals. The subgrid‐scale displacements/velocities are scaled using an effective (tangent) elastoplastic shear modulus, and we demonstrate the beneficial effects of introducing a subgrid‐scale pressure in the plastic regime. We provide proofs of stability and convergence of the proposed algorithms. These methods are initially presented in the context of static computations and then extended to the case of dynamics, where we demonstrate that, in general, naïve extensions of stabilized methods developed initially for static computations seem not effective. We conclude by proposing a dynamic version of the stabilizing mechanisms, which obviates this problematic issue. In its final form, the proposed approach is simple and efficient, as it requires only minimal additional computational and storage cost with respect to a standard finite element relying on a piecewise linear approximation of the displacement field.     

Three‐dimensional finite elements with embedded strong discontinuities to model material failure in the infinitesimal range

This paper presents new three‐dimensional finite elements with embedded strong discontinuities in the small strain infinitesimal range. The goal is to model localized surfaces of failure in solids, such as cracks at fracture, through enhancements of the finite elements that capture the propagating discontinuities of the displacement field in the element interiors. In this way, such surfaces of discontinuity can be sharply resolved in general meshes not necessarily related to the detailed geometry of the surface, unknown a priori. An important issue is also the consideration of general finite element formulations in the developments (e.g., basic displacement‐based, mixed or enhanced assumed strain finite element formulations), as needed to optimally resolve the continuum problem in the bulk. The actual modeling of the discontinuity effects, including the incorporation of the cohesive law defining the discontinuity constitutive response, is carried out at the element level with the proper enhancement of the discrete strain field of the element. The added elemental degrees of freedom approximate the displacement jumps associated with the discontinuity and are defined independently from element to element, thus allowing their static condensation at the element level without affecting the global mechanical problem in terms of the number and topology of the global degrees of freedom. In fact, this global‐local structure of the finite element methods developed in this work arises naturally from a multi‐scale characterization of these localized solutions, with the discontinuities understood to appear in the small scales, thus leading directly to these computationally efficient numerical methods for their numerical resolution, easily incorporated to an existing finite element code. The focus in this paper is on the development of finite elements incorporating a linear interpolation of the displacement jumps in the general three‐dimensional setting. These interpolations are shown to be necessary for hexahedral elements to avoid the so‐called stress locking that occurs with simpler constant approximations of the jumps (namely, a spurious transfer of stresses across the discontinuity not allowing its full release and, hence, resulting in an overstiff or locked numerical solution). The design of the new finite elements is accomplished in this work by a direct identification of the separation modes to be incorporated in the discrete strain field of the element, rather than from an assumed discontinuous interpolation of the displacements, assuring with this approach their locking‐free response by design. An additional issue addressed in the paper is the geometric characterization and propagation of the discontinuity surfaces in the general three‐dimensional setting of interest here. The paper includes a series of numerical simulations illustrating and evaluating the properties of the new finite elements. Copyright © 2012 John Wiley & Sons, Ltd.     

Multiscale domain decomposition analysis of quasi‐brittle heterogeneous materials

A hybrid multiscale framework is presented, which processes the material scales in a concurrent manner, borrowing features from hierarchical multiscale methods. The framework is used for the analysis of non‐linear heterogeneous materials and is capable of tackling strain localization and failure phenomena. Domain decomposition techniques, such as the ?nite element tearing and interconnecting method, are used to partition the material in a number of non‐overlapping domains and adaptive re?nement is performed at those domains that are affected by damage processes. This re?nement is performed in terms of material scale and ?nite element size. It is veri?ed that the results are independent of the chosen domain decomposition. Moreover, the multiscale analyses are validated with reference solutions obtained with a full ?ne‐scale solution procedure. Copyright © 2011 John Wiley & Sons, Ltd.     

On the strong discontinuity approach in finite deformation settings

Taking the strong discontinuity approach as a framework for modelling displacement discontinuities and strain localization phenomena, this work extends previous results in infinitesimal strain settings to finite deformation scenarios. By means of the strong discontinuity analysis, and taking isotropic damage models as target continuum (stress–strain) constitutive equation, projected discrete (tractions–displacement jumps) constitutive models are derived, together with the strong discontinuity conditions that restrict the stress states at the discontinuous regime. A variable bandwidth model, to automatically induce those strong discontinuity conditions, and a discontinuous bifurcation procedure, to determine the initiation and propagation of the discontinuity, are briefly sketched. The large strain counterpart of a non‐symmetric finite element with embedded discontinuities, frequently considered in the strong discontinuity approach for infinitesimal strains, is then presented. Finally, some numerical experiments display the theoretical issues, and emphasize the role of the large strain kinematics in the obtained results. Copyright © 2003 John Wiley & Sons, Ltd.     

Continuum-molecular modelling of graphene

In the present contribution we address the modeling of graphene membranes using a hierarchical modeling strategy to bridge the scales required to describe and understand the material. Quantum Mechanical (QM) and optimized Molecular Mechanical (MM) models are used to describe details on the nanoscale, while a multiscale continuum mechanical method is used to model the graphene response at the device or micrometer scale. The complete method is obtained on the basis of the Cauchy Born Rule (CBR), where the continuum model is coupled to the atomic field via the CBR and a local discrete fluctuation field. The MM method, often used to model carbon structures, involves the Tersoff–Brenner (TB) potential; however, when applying this potential to graphene with standard parameters one obtains material stress behavior much weaker than experiments. On the other hand, the more fundamental Hartree Fock and Density Functional Theory (DFT) methods are computationally too expensive and very limited in terms of their applicability to model the geometric scale at the device level. In this contribution a simple calibration of some of the TB parameters is proposed in order to reproduce the results obtained from QM calculations. Subsequently, the fine-tuned TB-potential is used for the multiscale modeling of a nano indentation sample, where experimental data are available. Effects of the mechanical response after the calibration are demonstrated.     

Finite elements with embedded strong discontinuities for the modeling of failure in solids

This paper presents new finite elements that incorporate strong discontinuities with linear interpolations of the displacement jumps for the modeling of failure in solids. The cases of interest are characterized by a localized cohesive law along a propagating discontinuity (e.g. a crack), with this propagation occurring in a general finite element mesh without remeshing. Plane problems are considered in the infinitesimal deformation range. The new elements are constructed by enhancing the strains of existing finite elements (including general displacement based, mixed, assumed and enhanced strain elements) with a series of strain modes that depend on the proper enhanced parameters local to the element. These strain modes are designed by identifying the strain fields to be captured exactly, including the rigid body motions of the two parts of a splitting element for a fully softened discontinuity, and the relative stretching of these parts for a linear tangential sliding of the discontinuity. This procedure accounts for the discrete kinematics of the underlying finite element and assures the lack of stress locking in general quadrilateral elements for linearly separating discontinuities, that is, spurious transfers of stresses through the discontinuity are avoided. The equations for the enhanced parameters are constructed by imposing the local equilibrium between the stresses in the bulk of the element and the tractions driving the aforementioned cohesive law, with the proper equilibrium operators to account for the linear kinematics of the discontinuity. Given the locality of all these considerations, the enhanced parameters can be eliminated by their static condensation at the element level, resulting in an efficient implementation of the resulting methods and involving minor modifications of an existing finite element code. A series of numerical tests and more general representative numerical simulations are presented to illustrate the performance of the new elements. Copyright © 2007 John Wiley & Sons, Ltd.     

Microstructural Modeling and Multiscale Mechanical Properties Analysis of Cancellous Bone

This paper is devoted to the microstructure geometric modeling and mechanical properties computation of cancellous bone. The microstructure of the cancellous bone determines its mechanical properties and a precise geometric modeling of this structure is important to predict the material properties. Based on the microscopic observation, a new microstructural unit cell model is established by introducing the Schwarz surface in this paper. And this model is very close to the real microstructure and satisfies the main biological characteristics of cancellous bone. By using the unit cell model, the multiscale analysis method is newly applied to predict the mechanical properties of cancellous bone. The effective stiffness parameters are calculated by the up-scaling multi-scale analysis. And the distribution of microscopic stress in cancellous bone is determined through the down-scaling procedure. In addition, the effect of porosity on the stiffness parameters is also investigated. The predictive mechanical properties are in good agreement with the available experimental results, which verifies the applicability of the proposed unit cell model and the validness of the multiscale analysis method to predict the mechanical properties of cancellous bone.     

A coupled molecular dynamics and extended finite element method for dynamic crack propagation

A multiscale method is presented which couples a molecular dynamics approach for describing fracture at the crack tip with an extended finite element method for discretizing the remainder of the domain. After recalling the basic equations of molecular dynamics and continuum mechanics, the discretization is discussed for the continuum subdomain where the partition‐of‐unity property of finite element shape functions is used, since in this fashion the crack in the wake of its tip is naturally modelled as a traction‐free discontinuity. Next, the zonal coupling method between the atomistic and continuum models is recapitulated. Finally, it is discussed how the stress has been computed in the atomic subdomain, and a two‐dimensional computation is presented of dynamic fracture using the coupled model. The result shows multiple branching, which is reminiscent of recent results from simulations on dynamic fracture using cohesive‐zone models. Copyright © 2009 John Wiley & Sons, Ltd.     

A Fourier transformation‐based method for gradient‐enhanced modeling of fatigue

A key limitation of the most constitutive models that reproduce a degradation of quasi‐brittle materials is that they generally do not address issues related to fatigue. One reason is the huge computational costs to resolve each load cycle on the structural level. The goal of this paper is the development of a temporal integration scheme, which significantly increases the computational efficiency of the finite element method in comparison to conventional temporal integrations. The essential constituent of the fatigue model is an implicit gradient‐enhanced formulation of the damage rate. The evolution of the field variables is computed as a multiscale Fourier series in time. On a microchronological scale attributed to single cycles, the initial boundary value problem is approximated by linear BVPs with respect to the Fourier coefficients. Using the adaptive cycle jump concept, the obtained damage rates are transferred to a coarser macrochronological scale associated with the duration of material deterioration. The performance of the developed method is hence improved due to an efficient numerical treatment of the microchronological problem in combination with the cycle jump technique on the macrochronological scale. Validation examples demonstrate the convergence of the obtained solutions to the reference simulations while significantly reducing the computational costs.     

Towards a taxonomy for multiscale methods in computational mechanics: building blocks of existing methods

Existing multiscale methods in computational mechanics are analyzed with respect to their computational building blocks, considering methods in both solid and fluid mechanics. From this analysis, a step towards a taxonomy for multiscale methods in computational mechanics is taken. The present article is not intended as a closed story; it is rather hoped that it may provide some basis for future discussions. Moreover, it might even provide a point of view to more clearly identify differences and similarities in the variety of multiscale methods currently existing or being developed in the future. The methods or their building blocks, respectively, are investigated with a view on their multiscale features regarding the underlying problem, spatial scale processing, and temporal scale processing. As expected, it turns out that the mechanics of the underlying problem strongly influences the necessary building blocks of an adequate multiscale method.     

Multiscale modeling of solid propellants: From particle packing to failure

We present a theoretical and computational framework for modeling the multiscale constitutive behavior of highly filled elastomers, such as solid propellants and other energetic materials. Special emphasis is placed on the effect of the particle debonding or dewetting process taking place at the microscale and on the macroscopic constitutive response. The microscale is characterized by a periodic unit cell, which contains a set of hard particles (such as ammonium perchlorate for AP-based propellants) dispersed in an elastomeric binder. The unit cell is created using a packing algorithm that treats the particles as spheres or discs, enabling us to generate packs which match the size distribution and volume fraction of actual propellants. A novel technique is introduced to characterize the pack geometry in a way suitable for meshing, allowing for the creation of high-quality periodic meshes with refinement zones in the regions of interest. The proposed numerical multiscale framework, based on the mathematical theory of homogenization, is capable of predicting the complex, heterogeneous stress and strain fields associated, at the microscale, with the nucleation and propagation of damage along the particle–matrix interface, as well as the macroscopic response and mechanical properties of the damaged continuum. Examples involving simple unit cells are presented to illustrate the multiscale algorithm and demonstrate the complexity of the underlying physical processes.     

对流-扩散-反应方程的变分多尺度解法

根据变分多尺度的思想求解了对流项和反应项占优的对流-扩散-反应方程.在变分多尺度思想的理论框架内,推导了附加于Galerkin变分弱形式的稳定化结构和具体的稳定化系数;阐述了这种稳定化结构和经典的SUPG稳定化结构之间的关系;数值算例表明,该稳定化系数可以适应均匀和非均匀的计算网格.通过网格的恰当加密,变分多尺度方法消除了算例中的数值伪振荡.     

A variational multiscale stabilized formulation for the incompressible Navier–Stokes equations

This paper presents a variational multiscale residual-based stabilized finite element method for the incompressible Navier–Stokes equations. Structure of the stabilization terms is derived based on the two level scale separation furnished by the variational multiscale framework. A significant feature of the new method is that the fine scales are solved in a direct nonlinear fashion, and a definition of the stabilization tensor τ is derived via the solution of the fine-scale problem. A computationally economic procedure is proposed to evaluate the advection part of the stabilization tensor. The new method circumvents the Babuska–Brezzi (inf–sup) condition and yields a stable formulation for high Reynolds number flows. A family of equal-order pressure-velocity elements comprising 4-and 10-node tetrahedral elements and 8- and 27-node hexahedral elements is developed. Convergence rates are reported and accuracy properties of the method are presented via the lid-driven cavity flow problem.     

Error controlled adaptive multiscale XFEM simulation of cracks

The accurate and efficient prediction of the interaction of microcracks with macrocracks has been a challenge for many years. In this paper a discretization error controlled adaptive multiscale technique for the accurate simulation of microstructural effects within a macroscopic component is presented. The simulation of cracks is achieved using the corrected XFEM. The error estimation procedure is based on the well known Zienkiewicz and Zhu method extended to the XFEM for cracks such that physically meaningful stress irregularities and non-smoothnesses are accurately reflected. The incorporation of microstructural features such as microcracks is achieved by means of the multiscale projection method. In this context an error controlled adaptive mesh refinement is performed on the fine scale where microstructural effects may lead to highly complex mechanical behavior. The presented method is applied to a few examples showing its validity and applicability to arbitrary problems within fracture mechanics.     

Topology optimization of multiscale elastoviscoplastic structures

This paper extends current concepts of topology optimization to the design of structures made of nonlinear microheterogeneous materials. The objective is to maximize the macroscopic structural stiffness for a prescribed material volume usage while accounting for the nonlinearity and the microstructure of the material. The resulting design problem considers two scales: the macroscopic scale at which the optimization is performed and the microscopic scale at which the material heterogeneities and the nonlinearities are observed. The topology optimization at the macroscopic scale is performed by means of the bi‐directional evolutionary structural optimization method. The solution of the macroscopic boundary value problem requires as inputs the effective constitutive response with full consideration of the microstructure. While computational homogenization methods such as the FE² method could be used to solve the nonlinear multiscale problem, the associated numerical expense (CPU time and memory) is highly unacceptable. In order to regain the computational feasibility of the computational scale transition, a recent model reduction technique of the authors is employed: the potential‐based reduced basis model order reduction with graphics processing unit acceleration. Numerical examples show the efficiency of the resulting nonlinear two‐scale designs. The impact of different load amplitudes on the design is examined. Copyright © 2015 John Wiley & Sons, Ltd.