The diffuse Nitsche method: Dirichlet constraints on phase‐field boundaries

We explore diffuse formulations of Nitsche's method for consistently imposing Dirichlet boundary conditions on phase‐field approximations of sharp domains. Leveraging the properties of the phase‐field gradient, we derive the variational formulation of the diffuse Nitsche method by transferring all integrals associated with the Dirichlet boundary from a geometrically sharp surface format in the standard Nitsche method to a geometrically diffuse volumetric format. We also derive conditions for the stability of the discrete system and formulate a diffuse local eigenvalue problem, from which the stabilization parameter can be estimated automatically in each element. We advertise metastable phase‐field solutions of the Allen‐Cahn problem for transferring complex imaging data into diffuse geometric models. In particular, we discuss the use of mixed meshes, that is, an adaptively refined mesh for the phase‐field in the diffuse boundary region and a uniform mesh for the representation of the physics‐based solution fields. We illustrate accuracy and convergence properties of the diffuse Nitsche method and demonstrate its advantages over diffuse penalty‐type methods. In the context of imaging‐based analysis, we show that the diffuse Nitsche method achieves the same accuracy as the standard Nitsche method with sharp surfaces, if the inherent length scales, ie, the interface width of the phase‐field, the voxel spacing, and the mesh size, are properly related. We demonstrate the flexibility of the new method by analyzing stresses in a human vertebral body.     

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.     

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.     

Mesoscale models of interface mechanics in crystalline solids: a review

Theoretical and computational methods for representing mechanical behaviors of crystalline materials in the vicinity of planar interfaces are examined and compared. Emphasis is on continuum-type resolutions of microstructures at the nanometer and micrometer levels, i.e., mesoscale models. Grain boundary interfaces are considered first, with classes of models encompassing sharp interface, continuum defect (i.e., dislocation and disclination), and diffuse interface types. Twin boundaries are reviewed next, considering sharp interface and diffuse interface (e.g., phase field) models as well as pseudo-slip crystal plasticity approaches to deformation twinning. Several classes of models for evolving failure interfaces, i.e., fracture surfaces, in single crystals and polycrystals are then critically summarized, including cohesive zone approaches, continuum damage theories, and diffuse interface models. Important characteristics of compared classes of models for a given physical behavior include complexity, generality/flexibility, and predictive capability versus number of free or calibrated parameters.     

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.     

A front remeshing technique for a Lagrangian description of moving interfaces in two‐fluid flows

The numerical analysis of two‐fluid flows involves the treatment of a discontinuity that appears at the separating interface. Classical Lagrangian schemes applied to update the front position between two immiscible incompressible fluids have been long recognized to provide a sharp representation of the interface. However, the main drawback of these approaches is the progressive distortion in the distribution of the markers used to identify the material front. To avoid this problem, an interface remeshing algorithm based on the diffuse approximation of the interface curvature is proposed in this work. In addition, the remeshed front is enforced to preserve the global volume. These new aspects are incorporated in an existing fluid dynamics formulation for the analysis of two‐fluid flows problems. The resulting formulation is called in this work as the moving Lagrangian interface remeshing technique (MLIRT). Copyright © 2005 John Wiley & Sons, Ltd.     

On strategies for enforcing interfacial constraints and evaluating jump conditions with the extended finite element method

We consider a problem stemming from recent models of phase transitions in stimulus‐responsive hydrogels, wherein a sharp interface separates swelled and collapsed phases. Extended finite element methods that approximate the local solution with an enriched basis such that the mesh need not explicitly ‘fit’ the interface geometry are emphasized. Attention is focused on the weak enforcement of the normal configurational force balance and various options for evaluating the jump in the normal component of the solute flux at the interface. We show that as the reciprocal interfacial mobility vanishes, it plays the role of a penalty parameter enforcing a pure Dirichlet constraint, eventually triggering oscillations in the interfacial velocity. We also examine alternative formulations employing a Lagrange multiplier to enforce this constraint. It is shown that the most convenient choice of basis for the Lagrange multiplier results in oscillations in the multiplier field and a decrease in accuracy and rate of convergence in local error norms, suggesting a lack of stability in the discrete formulation. Under such conditions, neither the direct evaluation of the gradient of the approximation at the phase interface nor the interpretation of the Lagrange multiplier field provide a robust means to obtain the jump in the normal component of solute flux. Fortunately, the adaptation and use of local, domain‐integral methodologies considerably improves the flux evaluations. Several example problems are presented to compare and contrast the various techniques and methods. Copyright © 2004 John Wiley & Sons, Ltd.     

Port reduction in parametrized component static condensation: approximation and a posteriori error estimation

We introduce a port (interface) approximation and a posteriori error bound framework for a general component‐based static condensation method in the context of parameter‐dependent linear elliptic partial differential equations. The key ingredients are as follows: (i) efficient empirical port approximation spaces—the dimensions of these spaces may be chosen small to reduce the computational cost associated with formation and solution of the static condensation system; and (ii) a computationally tractable a posteriori error bound realized through a non‐conforming approximation and associated conditioner—the error in the global system approximation, or in a scalar output quantity, may be bounded relatively sharply with respect to the underlying finite element discretization. Our approximation and a posteriori error bound framework is of particular computational relevance for the static condensation reduced basis element (SCRBE) method. We provide several numerical examples within the SCRBE context, which serve to demonstrate the convergence rate of our port approximation procedure as well as the efficacy of our port reduction error bounds. Copyright © 2013 John Wiley & Sons, Ltd.     

On the application of cohesive crack modeling in cementitious materials

Cohesive crack models—in particular the Fictitious Crack Model – are applied routinely in the analysis of crack propagation in concrete and mortar. Bridged crack models—where cohesive stresses are assumed to exist together with a stress singularity at the crack tip—on the other hand, are used typically for multi scale problems such as crack propagation in fiber reinforced composites. Mortar and concrete, however, are multi-scale materials and the question naturally arises, if bridged crack models in fact are more suitable for concrete and mortar as well? In trying to answer this question a model for a centrally cracked sheet is established applying semi-analytical, bridged and fictitious crack modeling. The semi-analytical crack model is compared with a FEM analysis and it is demonstrated, that the standard fictitious crack implementation in FEM packages (in this case DIANA) provides a good approximation. Further, a quantitative condition is established indicating when a bridged crack model can be approximated with a cohesive crack model with smooth crack closure in terms of the ratio between the energy dissipation associated with the crack tip and the process zone.     

Phase‐Field Modeling of Microstructural Evolution by Freeze‐Casting

Freeze‐casting has attracted great attention as a potential method for manufacturing bioinspired materials with excellent flexibility in microstructure control. The solidification of ice crystals in ceramic colloidal suspensions plays an important role during the dynamic process of freeze‐casting. During solidification, the formation of a microstructure results in a dendritic pattern within the ice‐template crystals, which determines the macroscopic properties of materials. In this paper, the authors propose a phase‐field model that describes the crystallization in an ice template and the evolution of particles during anisotropic solidification. Under the assumption that ceramic particles represent mass flow, namely a concentration field, the authors derive a sharp‐interface model and then transform the model into a continuous initial boundary value problem via the phase‐field method. The adaptive finite‐element technique and generalized single‐step single‐solve (GSSSS) time‐integration method are employed to reduce computational cost and reconstruct microstructure details. The numerical results are compared with experimental results, which demonstrate good agreement. Finally, a microstructural morphology map is constructed to demonstrate the effect of different concentration fields and input cooling rates. The authors observe that at particle concentrations ranging between 25 and 30% and cooling rate lower than ?5° min^?1 generates the optimal dendrite structure in freeze casting process.

     

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.     

Synthesis of shape and topology of multi-material structures with a phase-field method

In this paper, we present a phase-field method to the problem of shape and topology synthesis of structures with three materials. A single phase model is developed based on the classical phase-transition theory in the fields of mechanics and material sciences. The multi-material synthesis is formulated as a continuous optimization problem within a fixed reference domain. As a single parameter, the phase-field model represents regions made of any of the three distinct material phases and the interface between the regions. The Van der Waals–Cahn-Hilliard theory is applied to define a dynamic process of phase transition. The Γ-convergence theory is used for an approximate numerical solution to this free-discontinuity problem without any explicit tracking of the interface. Within this variational framework, we show that the phase-transition theory leads to a well-posed problem formulation with the effects of “domain regularization” and “region segmentation” incorporated naturally. The proposed phase-field method is illustrated with several 2D examples that have been extensively used in the recent literature of topology optimization, especially in the homogenization based methods. It is further suggested that such a phase-field approach may represent a promising alternative to the widely-used homogenization models for the design of heterogeneous materials and solids, with a possible extension to a general model of multiple material phases.     

Numerical differentiation for non‐trivial consistent tangent matrices: an application to the MRS‐Lade model

In a companion paper Pérez‐Foguet, A., Rodríguez‐Ferran, A. and Huerta, A. ‘Numerical differentiation for local and global tangent operators in computational plasticity’. Computer Methods in Applied Mechanics and Engineering, 2000, in press, the authors have shown that numerical differentiation is a competitive alternative to analytical derivatives for the computation of consistent tangent matrices. Relatively simple models were treated in that reference. The approach is extended here to a complex model: the MRS‐Lade model. This plastic model has a cone‐cap yield surface and exhibits strong coupling between the flow vector and the hardening moduli. Because of this, differentiating these quantities with respect to stresses and internal variables—the crucial step in obtaining consistent tangent matrices—is rather involved. Numerical differentiation is used here to approximate these derivatives. The approximated derivatives are then used to (1) compute consistent tangent matrices (global problem) and (2) integrate the constitutive equation at each Gauss point (local problem) with the Newton–Raphson method. The choice of the stepsize (i.e. the perturbation in the approximation schemes), based on the concept of relative stepsize, poses no difficulties. In contrast to previous approaches for the MRS‐Lade model, quadratic convergence is achieved, for both the local and the global problems. The computational efficiency (CPU time) and robustness of the proposed approach is illustrated by means of several numerical examples, where the major relevant topics are discussed in detail. Copyright © 2000 John Wiley & Sons, Ltd.     

A study on the prediction of the mechanical properties of nanoparticulate composites using the homogenization method with the effective interface concept

A sequential multi‐scale homogenization method combined with molecular dynamics (MD) simulation is developed for the mechanical characterization of nanoparticulate composites. In order to characterize the particle‐size effect of nanocomposites, the effective interface, which has been adopted in continuum micromechanics approaches, is considered as the characteristic phase. Owing to the existence of the interface and the size‐dependent elastic modulus that is observed from MD simulations, an analysis of the mechanical properties of nanocomposites with continuum micromechanics requires careful consideration of the particle‐concentration effect. Therefore, this study focuses on hierarchical information transfer from the molecular model to the continuum model through the homogenization method in lieu of an analytical micromechanics bridging method. Using the present multi‐scale homogenization method, the elastic properties of the effective interface are numerically evaluated and compared with the analytically obtained micromechanics solutions. In addition, the overall elastic modulus of nanocomposites is obtained from the present model and compared with the results of MD simulation, the micromechanics bridging model, and finite‐element analysis (FEA). Copyright © 2010 John Wiley & Sons, Ltd.     

Numerical approximation of a thermally driven interface using finite elements

A two‐dimensional finite element model for dendritic solidification has been developed that is based on the direct solution of the energy equation over a fixed mesh. The model tracks the position of the sharp solid–liquid interface using a set of marker points placed on the interface. The simulations require calculation of the temperature gradients on both sides of the interface in the direction normal to it; at the interface the heat flux is discontinuous due to the release of latent heat during the solidification (melting) process. Two ways to calculate the temperature gradients at the interface, evaluating their interpolants at Gauss points, were proposed. Using known one‐ and two‐dimensional solutions to stable solidification problems (the Stefan problem), it was shown that the method converges with second‐order accuracy. When applied to the unstable solidification of a crystal into an undercooled liquid, it was found that the numerical solution is extremely sensitive to the mesh size and the type of approximation used to calculate the temperature gradients at the interface, i.e. different approximations and different meshes can yield different solutions. The cause of these difficulties is examined, the effect of different types of interpolation on the simulations is investigated, and the necessary criteria to ensure converged solutions are established. Copyright © 2003 John Wiley & Sons, Ltd.     

A localized mixed‐hybrid method for imposing interfacial constraints in the extended finite element method (XFEM)

The paper proposes an approach for the imposition of constraints along moving or fixed immersed interfaces in the context of the extended finite element method. An enriched approximation space enables consistent representation of strong and weak discontinuities in the solution fields along arbitrarily‐shaped material interfaces using an unfitted background mesh. The use of Lagrange multipliers or penalty methods is circumvented by a localized mixed hybrid formulation of the model equations. In a defined region in the vicinity of the interface, the original problem is re‐stated in its auxiliary formulation. The availability of the auxiliary variable enables the consideration of a variety of interface constraints in the weak form. The contribution discusses the weak imposition of Dirichlet‐ and Neumann‐type interface conditions as well as continuity requirements not fulfilled a priori by the enriched approximation. The properties of the proposed approach applied to two‐dimensional linear scalar‐ and vector‐valued elliptic problems are investigated by studying the convergence behavior. Copyright © 2009 John Wiley & Sons, Ltd.     

Temporal homogenization of viscoelastic and viscoplastic solids subjected to locally periodic loading

 As a direct extension of the asymptotic spatial homogenization method we develop a temporal homogenization scheme for a class of homogeneous solids with an intrinsic time scale significantly longer than a period of prescribed loading. Two rate-dependent material models, the Maxwell viscoelastic model and the power-law viscoplastic model, are studied as an illustrative examples. Double scale asymptotic analysis in time domain is utilized to obtain a sequence of initial-boundary value problems with various orders of temporal scaling parameter. It is shown that various order initial-boundary value problems can be further decomposed into: (i) the global initial-boundary value problem with smooth loading for the entire loading history, and (ii) the local initial-boundary value problem with the remaining (oscillatory) portion of loading for a single load period. Large time increments can be used for integrating the global problem due to smooth loading, whereas the integration of the local initial-boundary value problem requires a significantly smaller time step, but only locally in a single load period. The present temporal homogenization approach has been found to be in good agreement with a closed-form analytical solution for one-dimensional case and with a numerical solution in multidimensional case obtained by using a sufficiently small time step required to resolve the load oscillations. Received: 22 November 2001 / Accepted: 21 May 2002     

Modeling of non-equilibrium solute diffusion upon rapid solidification: effective mobility versus kinetic energy

Abstract

The effective mobility approach is compared with the kinetic energy approach in terms of sharp interface modeling and phase-field modelling of non-equilibrium solute diffusion upon rapid solidification of binary alloys. The two approaches are equivalent for modelling of long range solute diffusion in bulk phases, but only the effective mobility approach can introduce the non-equilibrium solute diffusion effect to short range solute diffusion at a sharp interface or within a diffuse interface. Addition of the kinetic energy terms results in an unreasonable non-bilinear expression of the flux and thermodynamic driving force in the free energy production of interface migration or phase field propagation, whereas the effective mobility approach allows the thermodynamic extremal principle workable.     

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.