Configurational–Force–Based Adaptive Methods in Nonlocal Gradient-Type Inelastic Solids

We propose a configurational-force-based framework for h-adaptive finite element discretizations of solids with nonlocal, gradient-type constitutive response. Typical applications are related to gradient-type damage mechanics, strain gradient plasticity and regularized brittle fracture. On the theoretical side, we outline a general incremental variational framework for the multifield problem of gradient-type dissipative solids, where generalized internal variable fields account for the current state of evolving microstructures. The Euler equations of the multifield variational principle define the macroscopic balance of momentum along with balance-type evolution equations for the generalized internal variables in the physical space as well as the balance of configurational forces in the material space. We propose a staggered computational scheme for satisfying those balances in both the physical as well as the material space. The coupled micro- and macro-structural balances of momentum and internal variables provide a solution in the physical space for a given finite element mesh. The balance in the material space is then used to provide an indicator for the quality of the finite element mesh and accounts for a subsequent h-type mesh refinement. Such a configurational-force-based approach provides in a natural and unified format mesh refinement indicators for a broad class of complex nonlocal problems. This framework is applied to damage-type regularized brittle fracture. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On Configurational-Force-Driven Crack Propagation and Adaptive Mesh Refinement in Finite Inelasticity

We introduce a consistent variational framework for inelasticity at finite strains, yielding dual balances in physical and material space as the Euler equations. The formulation is employed for the simultaneous usage of configurational forces as both driving forces for crack propagation as well as h-adaptive mesh refinement. The theoretical basis builds upon a global balance of internal and external power, where the mechanical response is exclusively governed by two scalar functions, the free energy function and a dissipation potential. The resulting variational structure is exploited in the context of fracture mechanics and yields evolution equations for internal variables. In the discrete setting, we present a geometry model fully separated from the finite element mesh structure that represents structural changes of the material configuration due to crack propagation. Advanced meshing algorithms provide an optimal discretization at the crack tip. Local and global criteria are obtained via error estimators based on configurational forces being interpreted as indicators of an energetic misfit due to an insufficient discretization. The numerical handling is decomposed into a staggered algorithm scheme for the dual set of equilibrium equations in material and physical space and efficient mesh generation tools. Exemplary numerical examples are considered to illustrate the method and to underline the effects of inelastic material behaviour in the presented context. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Primal‐dual active set methods for Allen–Cahn variational inequalities with nonlocal constraints

We propose and analyze a primal‐dual active set method for discretized versions of the local and nonlocal Allen–Cahn variational inequalities. An existence result for the nonlocal variational inequality is shown in a formulation involving Lagrange multipliers for local and nonlocal constraints. Local convergence of the discrete method is shown by interpreting the approach as a semismooth Newton method. Properties of the method are discussed and several numerical simulations demonstrate its efficiency. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

The virtual element method for a nonhomogeneous double obstacle problem of Kirchhoff plate

This paper is intended to propose and analyze the lowest order conforming and nonconforming virtual element methods (VEMs) for a nonhomogeneous double obstacle problem in fourth-order variational inequality. We first present an abstract framework for the error analysis of an abstract discrete method. Then, we develop the conforming and fully nonconforming VEMs for the previous problem, with the optimal error estimates obtained by using the previous abstract framework combined with two modified interpolation operators and the related estimates. The discrete problem is solved by the primal–dual active set algorithm and the numerical results are in good agreement with theoretical findings.     

Macroscopic finite element analyses of discretematerials based on directly evaluated micro‐macro transitions

We consider a homogenized macro‐continuum with locally attached microstructure of granules and derive specific micromacro transitions by a consistent transfer of discrete micro‐variables to field variables on a continuous macrostructure. Displacements and rotational constraints are imposed on the granules on the defined boundary frame of the microstructure. The constraints for linear displacements and uniform tractions on the surface yield upper and lower bound characteristics for periodic boundary conditions with regard to the aggregate stiffness. Secondly, we perform two‐scale analyses where we link simulations on the macro‐ and the microscales. Therein, coupled boundary‐value problems are solved on both scales. The macroscopic homogeneous problem is solved by a finite element method where the material model is implemented using the directly evaluated micro‐macro transitions on the basis of the discrete microstructures. Finally, a model problem is investigated to clarify the proposed method. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Variational–Based Formulation of Regularized Brittle Fracture

The paper performs a comparative study of variational-based brittle fracture with its gradient-type regularization, and outlines aspects of the numerical implementations of both approaches. The latter smoothes out sharp displacement discontinuities of cracks. On the side of discrete crack modeling, we propose a variational framework of configurational-force-driven crack propagation. The latter provides the basis for the computation of material nodal forces and drives the crack propagation in our proposed finite element framework with adaptive nodal doubling. Such a formulation is of limited applicability for the modeling of crack inititation in homogeneous bodies without defects and in situations with complex crack branching. This can be overcome by a regularized crack modeling. Here, an elliptic approximation of the crack surface term yields a regularized two field functional, where an additional scalar field approximates the set of discontinuities. We provide a simple interpretation of such a transition from the sharp crack to the regularized setting. It results in a smooth continuum-damage-type theory with a specific gradient-damage and hardening terms, depending on a length scale that represents the width of a zone that surrounds the crack. Such a variational framework is implemented by a coupled two-field finite element framework in a staightforward manner. We compare representative numerical results obtained by both methods for selected crack patterns and highlight the pro and contra of both meshes. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A variational formulation of nonlocalmaterials withmicrostructure based on dual macro- and micro-balances

We investigate a variational setting of nonlocal materials with microstructure and outline aspects of its numerical implementation. Thereby, the current state of the evolving microstructure is described by independent global degrees in addition to the macroscopic displacement field, so-called order parameters. Focussing on standard-dissipative materials, the constitutive response is governed by two fundamental functions for the energy storage and the dissipation. Based on these functions, a global dissipation postulate is introduced. Its exploitation constitutes a global variation formulation of nonlocal materials, which can be related to a minimization principle. Following this methodology, we end up with coupled macro- and microscopic field equations and corresponding boundary conditions. On the numerical side, we consider the weak counterpart of these coupled field equations and obtain after linearization a fully coupled system for increments of the displacement and the order parameters. Due to the underlying variational structure, this system of equations is symmetric. In order to show the capability of the proposed setting, we specify the above outlined scenario to a model problem of isotropic damage mechanics. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Material-Force-Based Refinement Indicators in Adaptive Strategies

A material-force-based refinement indicator for adaptive finite element strategies for finite elasto-plasticity is proposed. Starting from the local format of the spatial balance of linear momentum, a dual material counterpart in terms of Eshelby's energy-momentum tensor is derived. For inelastic problems, this material balance law depends on the material gradient of the internal variables. In a global format the material balance equation coincides with an equilibrium condition of material forces. For a homogeneous body, this condition corresponds to vanishing discrete material nodal forces. However, due to insufficient discretization, spurious material forces occur at the interior nodes of the finite element mesh. These nodal forces are used as an indicator for mesh refinement. Assigning the ideas of elasticity, where material forces have a clear energetic meaning, the magnitude of the discrete nodal forces is used to define a relative global criterion governing the decision on mesh refinement. Following the same reasoning, in a second step a criterion on the element level is computed which governs the local h-adaptive refinement procedure. The mesh refinement is documented for a representative numerical example of finite elasto-plasticity. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Theory and Numerics of Rate-Dependent Incremental Variational Formulations in Ferroelectricity

This paper is concerned with macroscopic continuous and discrete variational formulations for domain switching effects at small strains, which occur in ferroelectric ceramics. The developed new three–dimensional model is thermodynamically–consistent and determined by two scalar–valued functions: the energy storage function (Helmholtz free energy) and the dissipation function, which is in particular rate–dependent. The constitutive model successfully reproduces the ferroelastic and the ferroelectric hysteresis as well as the butterfly hysteresis for ferroelectric ceramics. The rate–dependent character of the dissipation function allows us also to reproduce the experimentally observed rate dependency of the above mentioned hysteresis phenomena. An important aspect is the numerical implementation of the coupled problem. The discretization of the two–field problem appears, as a consequence of the proposed incremental variational principle, in a symmetric format. The performance of the proposed methods is demonstrated by means of a benchmark problem. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Configurational Forces in Crystal Plasticity

The surface morphology of micro machined surfaces depends on the heterogeneous microstructure. A crystal plasticity model is used to describe the plastic deformation in cp-titanium with its hcp crystal structure. Therefore the basal and prismatic slip systems are taken into account. Furthermore, self and latent hardening are considered. The rate dependency is motivated by a visco plastic evolution law. The cutting process of cp-titanium is modeled within the concept of configurational forces for a standard dissipative media. This framework is implemented into the finite element method. An example illustrates the effects of the microstructure on plastic deformation and configurational forces. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

一类单边约束问题的数值方法

本文分别基于原始变分形式与对偶混合变分形式，对一类单边约束问题进行了数值求解，提出了求解离散对偶混合变分问题的Uzawa型算法，并用数值例子验证了算法的有效性．     

Finite Element Solution of Conical Diffraction Problems

This paper is devoted to the numerical study of diffraction by periodic structures of plane waves under oblique incidence. For this situation Maxwell's equations can be reduced to a system of two Helmholtz equations in R ² coupled via quasiperiodic transmission conditions on the piecewise smooth interfaces between different materials. The numerical analysis is based on a strongly elliptic variational formulation of the differential problem in a bounded periodic cell involving nonlocal boundary operators. We obtain existence and uniqueness results for discrete solutions and provide the corresponding error analysis.     

A fully variational algorithmic formulation for wrinkling at finite strains

This contribution is concerned with an efficient novel algorithmic formulation for wrinkling at finite strains. In contrast to previously published numerical implementations, the advocated method is fully variational. More precisely, the parameters describing wrinkles or slacks, together with the unknown deformation mapping, are computed jointly by minimizing the potential energy of the considered mechanical system. Furthermore, the wrinkling criteria are naturally included within the presented variational framework. The presented approach allows to employ three–dimensional constitutive models directly, i.e., plane stress conditions characterizing membranes are variationally enforced by minimizing the potential energy with respect to the transversal strains. Since the proposed formulation for wrinkling in membranes is fully variational, it can be conveniently combined with other variational methods (based on energy minimization). As an example, a variationally consistent framework for finite strain plasticity theory is considered. More precisely, the minimization principle characterizing wrinkling in elastic membranes and that describing plasticity in inelastic solids are coupled leading to a novel variational approach for inelastic membranes. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Rate-dependent incremental variational formulation of ferroelectricity

The paper presents continuous and discrete variational formulations for the treatment of the non-linear response of piezoceramics under electrical loading. The point of departure is a general internal variable formulation that determines the hysteretic response of the material as a generalized standard medium in terms of an energy storage and a rate–dependent dissipation function. Consistent with this type of standard dissipative continua, we develop an incremental variational formulation of the coupled electromechanical boundary value problem. We specify the variational formulation for a setting based on a smooth rate–dependent dissipation function which governs the hysteretic response. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Primal Dual Methods for Wasserstein Gradient Flows

Combining the classical theory of optimal transport with modern operator splitting techniques, we develop a new numerical method for nonlinear, nonlocal partial differential equations, arising in models of porous media, materials science, and biological swarming. Our method proceeds as follows: first, we discretize in time, either via the classical JKO scheme or via a novel Crank–Nicolson-type method we introduce. Next, we use the Benamou–Brenier dynamical characterization of the Wasserstein distance to reduce computing the solution of the discrete time equations to solving fully discrete minimization problems, with strictly convex objective functions and linear constraints. Third, we compute the minimizers by applying a recently introduced, provably convergent primal dual splitting scheme for three operators (Yan in J Sci Comput 1–20, 2018). By leveraging the PDEs’ underlying variational structure, our method overcomes stability issues present in previous numerical work built on explicit time discretizations, which suffer due to the equations’ strong nonlinearities and degeneracies. Our method is also naturally positivity and mass preserving and, in the case of the JKO scheme, energy decreasing. We prove that minimizers of the fully discrete problem converge to minimizers of the spatially continuous, discrete time problem as the spatial discretization is refined. We conclude with simulations of nonlinear PDEs and Wasserstein geodesics in one and two dimensions that illustrate the key properties of our approach, including higher-order convergence our novel Crank–Nicolson-type method, when compared to the classical JKO method.

     

Multiscale modelling and simulation of micro machining of titanium

The microscale morphology of micro machined component surfaces is directly connected to the heterogeneous microstructure. The deformation depends on the crystal structure, in case of the considered cp-titanium, the hcp crystal structure. In a first approach the crystal plastic deformation is modeled with isotropic hardening. A visco-plastic evolution law accounts for the rate dependency. The concept of configurational forces is used with the framework of crystal plasticity to model the cutting process of cp-titanium. The setting is implemented into the finite element method. The examples show the effect of the material heterogeneity on the deforamtion behavior and on the related configurational forces. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Characterization of energy minimizers in micromagnetics

We present a new characterization of minimizing sequences and possible minimizers (all called the minimizing magnetizations) for a nonlocal micromagnetic-like energy (without the exchange energy). Our method is to replace the nonlocal energy functional and its relaxation with certain local integral functionals on divergence-free fields obtained by a two-step minimization of some auxiliary augmented functionals. Through this procedure, the minimization problem becomes equivalent to the minimization of a new local variational functional, called the dual variational functional, which has a unique minimizer. We then precisely characterize the minimizing magnetizations of original nonlocal functionals in terms of the unique minimizer of the dual variational functional. Finally, we give some remarks and ideas on solving the dual minimization problem.     

An Approach for Micromechanical Modelling of Damage Mechanisms

In the framework of numerical simulation of damaged materials, softening behaviour represents an important topic. Thereby, the decrease of stiffness is mainly caused by the evolution of microvoids. In contrast to the established phenomenological damage approaches, the explicit consideration of effects on the micro scale can lead to an improved approximation quality. In this work, we discuss an approach to describe microstructural evolution. Based on a two phase micro model representing the macroscopical material behaviour, the structural evolution on the micro scale will be modelled based on configurational forces. Besides some theoretical basics on configurational forces at two phase systems and the definition of suitable evolution laws, we present an application of this approach on void growth process in rubberlike material. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)