Configurational-force-based structural updates and adaptive remeshing strategies

The application of configurational forces in h -adaptive strategies for fracture mechanics and inelasticity is investigated. Starting from a global Clausius-Planck inequality, dual equilibrium conditions are derived by means of a Coleman-type exploitation method. The remaining reduced dissipation inequality is used for the derivation of evolution equations for the internal variables. In fracture mechanics, crack loading conditions as well as a normality rule for the crack propagation are obtained. In the discrete setting, the crack propagation is governed by a configurational-force-driven update of the geometry model. The material balance equation is used to set up a h -adaptive refinement indicator. A relative global criterion is defined used for the decision on mesh refinement. In addition, a criterion on the element level is evaluated controlling the local refinement procedure. The capability of the proposed procedures is demonstrated by means of numerical examples. (© 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)     

Strategies for the Computation of Configurational Forces in Dissipative Media

Configurational forces can be interpreted as driving forces on material inhomogeneities such as crack tips. In dissipative media the total configurational force on an inhomogeneity consists of an elastic contribution and a contribution due to the dissipative processes in the material. For the computation of discrete configurational forces acting at the nodes of a finite element mesh, the elastic and dissipative contributions must be evaluated at integration point level. While the evaluation of the elastic contribution is straightforward, the evaluation of the dissipative part is faced with certain difficulties. This is because gradients of internal variables are necessary in order to compute the dissipative part of the configurational force. For the sake of efficiency, these internal variables are usually treated as local history data at integration point level in finite element (FE) implementations. Thus, the history data needs to be projected to the nodes of the FE mesh in order to compute the gradients by means of shape function interpolations of nodal data as it is standard practice. However, this is a rather cumbersome method which does not easily integrate into standard finite element frameworks. An alternative approach which facilitates the computation of gradients of local history data is investigated in this work. This approach is based on the definition of subelements within the elements of the FE mesh and allows for a straightforward integration of the configurational force computation into standard finite element software. The suitability and the numerical accuracy of different projection approaches and the subelement technique are discussed and analyzed exemplarily within the context of a crystal plasticity model. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Configurational-Force-Based Adaptive FE Solver for a Phase Field Model of Fracture

The computational modeling of failure mechanisms in solids due to fracture based on sharp crack discontinuities suffers in situations with complex crack topologies. This can be overcome by diffusive crack modeling, based on the introduction of a crack phase field as outlined in [1, 2]. Following these formulations, we outline a thermodynamically consistent framework for phase field models of crack propagation in elastic solids, develop incremental variational principles and, as an extension to [1, 2], consider their numerical implementations by an efficient h-adaptive finite element method. A key problem of the phase field formulation is the mesh density, which is required for the resolution of the diffusive crack patterns. To this end, we embed the computational framework into an adaptive mesh refinement strategy that resolves the fracture process zones. We construct a configurational-force-based framework for h-adaptive finite element discretizations of the gradient-type diffusive fracture model. We develop a staggered computational scheme for the solution of the coupled balances in physical and material space. The balance in the material space is then used to set up indicators for the quality of the finite element mesh and accounts for a subsequent h-type mesh refinement. The capability of the proposed method is demonstrated by means of a numerical example. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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)     

Exploitation of Configurational Forces in Extended Nonlocal Continua with Microstructure

We investigate aspects of the application of configurational forces in extended nonlocal continua with microstructure. Focussing on multifield approaches to gradient–type inelastic solids, the coupled problem is governed by the macroscopic deformation field, while nonlocal inelastic effects on the microstructure are described by a family of order parameter fields. The dual macro– and micro–field equations are derived within an incremental variational framework. Using an incremental principle, due to the variation with respect to the material position, an additional balance in the material space appears with the dual macro–micro–balances in the physical space. In view of the numerical implementation of this coupled problem by a finite element method, the incremental variational framework is recast into a discrete format in terms of discrete macro– and micro–physical nodal forces and configurational nodal forces. Applying a staggered solution scheme, the configurational branch is used as a postprocessing procedure with all the ingredients known from the solution of the coupled macro–micro–problem. The procedure is implemented for a nonlocal, viscous damage model. The consequences with regard to the configurational nodal forces are assessed by means of a numerical example. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A computational framework of three dimensional configurational-force-driven crack propagation

A variational formulation of quasi-static brittle fracture is considered and a new finite-element-based computational framework is developed for propagation of cracks in three-dimensional bodies. We outline a consistent thermodynamical framework for crack propagation in elastic solids and show that the crack propagation direction associated with the classical Griffith criterion is identified by the material configurational force which maximizes the local dissipation at the crack front. The evolving crack discontinuity is realized by the doubling of critical nodes and triangular interface facets of the tetrahedral mesh. The crucial step for the success of the procedure is its embedding into an r-adaptive crack-facet reorientation procedure based on configurational-force-based indicators in conjunction with crack front constraints. We further propose a staggered algorithm which minimizes the stored energy at frozen crack state followed by the successive crack releases at frozen deformation. This constitutes a sequence of positive definite subproblems with successively decreasing overall stiffness, providing a very robust algorithmic setting in the postcritical range. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Computation of Material Forces in Thermoplasticity Based on Alternative Smoothing Algorithms

Different approaches to the computation of material forces in inelastic structures are investigated. Dissipative effects in inelastic materials are described by internal variables. The formulation of balance equations in the material space requires the computation of gradients of these internal variables. The computational evaluation of these gradients in the context of finite element simulations needs a global representation of the internal variable fields. On the one side, this request can be carried out by a global formulation that discretizes the internal variable fields in terms of nodal degrees additional to the displacements. A numerically more effective approach applies smoothing algorithms which project the internal variables of a typical local formulation from the integration points onto the nodal points. In detail, the implementation of two smoothing algorithms for the computation of material forces is dicussed. The L2–projection necessiates the solution of a system of equations on the global level. A patch recovery yields a smoothed solution from an element patch surrounding the nodal point of interest. The performance of both algorithms is compared for the material force computation in finite thermoplasticity. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Two grid finite element discretization method for semi‐linear hyperbolic integro‐differential equations

In this paper, we will investigate a two grid finite element discretization method for the semi‐linear hyperbolic integro‐differential equations by piecewise continuous finite element method. In order to deal with the semi‐linearity of the model, we use the two grid technique and derive that once the coarse and fine mesh sizes H, h satisfy the relation h = H² for the two‐step two grid discretization method, the two grid method achieves the same convergence accuracy as the ordinary finite element method. Both theoretical analysis and numerical experiments are given to verify the results.     

Computation of material forces for the thermo-mechanical response of dynamically loaded elastomers

Elastomers are widely used in today's life. The material is characterized by large deformability upon failure, elastic and time dependent as well as non-time dependent effects which can be also a function of temperature. In addition, cyclically loaded components show heat build-up which is due to dissipation. As a result, the temperature evolution of an elastomeric component can strongly influence the material properties and durability characteristics. Representing best the real thermo-mechanical behaviour of an elastomeric component in its design process is one motivation for the use of sophisticated, coupled material approaches within numerical simulations. In order to assess the durability characteristics, for example regarding crack propagation, material forces (configurational forces) are one possible approach to be applied. In the present contribution, the implementation of material forces for a thermo-mechanically coupled material model including a continuum mechanical damage (CMD) approach is demonstrated in the context of the Finite Element Method (FEM). Special emphasis is given to material forces resulting from internal variables (viscosity and damage variables), temperature field evolution and dynamic loading. Using the example of an elastomeric component, for which the material model parameters have been previously identified by a uniaxial extension test, material forces are evaluated quantitatively. The influence of each contribution (internal variables, temperature field and dynamics) is illustrated and compared to the overall material force response. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On a numerical scheme for curved crack propagation based on configurational forces and maximum dissipation

A numerical scheme is presented to predict crack trajectories in two dimensional components. First a relation between the curvature in mixed–mode crack propagation and the corresponding configurational forces is derived, based on the principle of maximum dissipation. With the help of this, a numerical scheme is presented which is based on a predictor–corrector method using the configurational forces acting on the crack together with their derivatives along real and test paths. With the help of this scheme it is possible to take bigger than usual propagation steps, represented by splines. Essential for this approach is the correct numerical determination of the configurational forces acting on the crack tip. The methods used by other authors are shortly reviewed and an approach valid for arbitrary non–homogenous and non–linear materials with mixed–mode cracks is presented. Numerical examples show, that the method is a able to predict the crack paths in components with holes, stiffeners etc. with good accuracy. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Formulation and Application of the Initially Rigid Cohesive Finite Element Method

The presented procedure for cohesive crack propagation is based on an adaptive finite element (FE) implementation, which enables the introduction of cohesive surfaces in dependence on the current crack state. In contrast to already existing formulations, the focus of the present model lies on failure processes that can be described at quasi-static conditions within an implicit framework. Furthermore, an extension for mesh independent crack propagation in terms of an additional mesh adaptive formulation is presented. By the evaluation of the failure criterion considering the preferred crack direction, a new crack tip coordinate is computed and the discretization is accordingly adjusted. The remaining mesh is modified for the new boundary representation. The application of the proposed method is shown by the numerical investigation of a concrete fracture specimen from an experimental research project. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A note on the least squares solution of the groundwater flow equation

The least squares finite element method is a member of the weighted residuals class of numerical methods for solving partial differential equations. The least squares finite element method is applied to the groundwater flow equation. Space is discretized with a C¹ continuous trial function and parameters are approximated with a C⁰ bilinear basis. Solutions for problems containing parameters with large localized spatial gradients are characterized by errors that are propagated throughout the entire domain. Second-order spatial convergence is observed, and extreme mesh refinement is required to match Galerkin and mixed least squares finite element results. Temporal discretization should be kept separate from the least squares spatial discretization. © 1994 John Wiley & Sons, Inc.     

Coherence of Structural Optimization and Configurational Mechanics

This contribution is concerned with the similarities of structural optimization and configurational mechanics. In structural optimization sensitivity analysis is used to obtain the sensitivity of continuum mechanical functions with respect to variations of the material body, i.e. the reference configuration. In the same manner in configurational mechanics we are interested in changes of the material body, e.g. crack propagation or phase transition problems. Consequently, variational design sensitivity analysis and the numerical techniques from structural optimization are applicable to problems fromconfigurational mechanics. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Discontinuous Galerkin finite element methods for variational inequalities of first and second kinds

We develop the error analysis for the h‐version of the discontinuous Galerkin finite element discretization for variational inequalities of first and second kinds. We establish an a priori error estimate for the method which is of optimal order in a mesh dependant as well as L²‐norm.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2007     

Two-level pressure projection finite element methods for Navier-Stokes equations with nonlinear slip boundary conditions

The two-level pressure projection stabilized finite element methods for Navier-Stokes equations with nonlinear slip boundary conditions are investigated in this paper, whose variational formulation is the Navier-Stokes type variational inequality problem of the second kind. Based on the P₁-P₁ triangular element and using the pressure projection stabilized finite element method, we solve a small Navier-Stokes type variational inequality problem on the coarse mesh with mesh size H and solve a large Stokes type variational inequality problem for simple iteration or a large Oseen type variational inequality problem for Oseen iteration on the fine mesh with mesh size h. The error analysis obtained in this paper shows that if h=O(H²), the two-level stabilized methods have the same convergence orders as the usual one-level stabilized finite element methods, which is only solving a large Navier-Stokes type variational inequality problem on the fine mesh. Finally, numerical results are given to verify the theoretical analysis.     

Passerelles volumes finis — éléments finis

The aim of this work is to introduce a new algorithm for the dicretization of second order elliptic operators in the context of finite volume schemes. The technique consists in matching to a finite volume discretization based on a given mesh, a finite element volume representation on the same given mesh. An inverse operator is also built. The results of numerical experiments concerning a system of two-dimensional, nonlinear partial differential equations on a unstructured mesh are presented.     

Simulation of brittle crack growth in functionally graded materials

We consider a thermodynamic consistent framework for crack propagation by applying a dissipation inequality to a time dependent migrating control volume. The direction of crack growth is obtained in terms of material forces as a result of the principle of maximum dissipation. In the numerical implementation a staggered algorithm – deformation update for fixed geometry followed by geometry update for fixed deformation – is employed within each time increment. The corresponding mesh is generated by combining Delaunay triangulation with local mesh refinement. A numerical example with inhomogeneous material properties illustrates the capability of the resulting algorithm. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A generalized cohesive element technique for arbitrary crack motion

A computational method for arbitrary crack motion through a finite element mesh, termed as the generalized cohesive element technique, is presented. In this method, an element with an internal discontinuity is replaced by two superimposed elements with a combination of original and imaginary nodes. Conventional cohesive zone modeling, limited to crack propagation along the edges of the elements, is extended to incorporate the intra-element mixed-mode crack propagation. Proposed numerical technique has been shown to be quite accurate, robust and mesh insensitive provided the cohesive zone ahead of the crack tip is resolved adequately. A series of numerical examples is presented to demonstrate the validity and applicability of the proposed method.     

A posteriori error control for finite element approximations of elliptic eigenvalue problems

We develop a new approach to a posteriori error estimation for Galerkin finite element approximations of symmetric and nonsymmetric elliptic eigenvalue problems. The idea is to embed the eigenvalue approximation into the general framework of Galerkin methods for nonlinear variational equations. In this context residual-based a posteriori error representations are available with explicitly given remainder terms. The careful evaluation of these error representations for the concrete situation of an eigenvalue problem results in a posteriori error estimates for the approximations of eigenvalues as well as eigenfunctions. These suggest local error indicators that are used in the mesh refinement process.