Showing papers in "International Journal for Numerical Methods in Engineering in 2011"

Filters in topology optimization based on Helmholtz‐type differential equations

Abstract: The aim of this paper is to apply a Helmholtz-type partial differential equation as an alternative to standard density filtering in topology optimization problems. Previously, this approach has been successfully applied as a sensitivity filter. The usual filtering techniques in topology optimization require information about the neighbor cells, which is difficult to obtain for fine meshes or complex domains and geometries. The complexity of the problem increases further in parallel computing, when the design domain is decomposed into multiple non-overlapping partitions. Obtaining information from the neighbor subdomains is an expensive operation. The proposed filter technique requires only mesh information necessary for the finite element discretization of the problem. The main idea is to define the filtered variable implicitly as a solution of a Helmholtz-type differential equation with homogeneous Neumann boundary conditions. The properties of the filter are demonstrated for various 2D and 3D topology optimization problems in linear elasticity, solved on serial and parallel computers. Copyright © 2010 John Wiley & Sons, Ltd.

Efficient non-linear model reduction via a least-squares Petrov–Galerkin projection and compressive tensor approximations

Abstract: The first author acknowledges the partial support by a National Science Foundation Graduate Fellowship and the partial support by a National Defense Science and Engineering Graduate Fellowship. The second and third authors acknowledge the partial support by the Motor Sports Division of the Toyota Motor Corporation under Agreement Number 48737, and the partial support by a research grant from the Academic Excellence Alliance program between King Abdullah University of Science and Technology (KAUST) and Stanford University. All authors also acknowledge the constructive comments received during the review process.

Isogeometric finite element data structures based on Bézier extraction of T-splines

Abstract: We develop finite element data structures for T-splines based on Bezier extraction generalizing our previous work for NURBS. As in traditional finite element analysis, the extracted Bezier elements are defined in terms of a fixed set of polynomial basis functions, the so-called Bernstein basis. The Bezier elements may be processed in the same way as in a standard finite element computer program, utilizing exactly the same data processing arrays. In fact, only the shape function subroutine needs to be modified while all other aspects of a finite element program remain the same. A byproduct of the extraction process is the element extraction operator. This operator localizes the topological and global smoothness information to the element level, and represents a canonical treatment of T-junctions, referred to as ‘hanging nodes’ in finite element analysis and a fundamental feature of T-splines. A detailed example is presented to illustrate the ideas. Copyright © 2011 John Wiley & Sons, Ltd.

A convected particle domain interpolation technique to extend applicability of the material point method for problems involving massive deformations

Abstract: SUMMARY A new algorithm is developed to improve the accuracy and efficiency of the material point method for problems involving extremely large tensile deformations and rotations. In the proposed procedure, particle domains are convected with the material motion more accurately than in the generalized interpolation material point method. This feature is crucial to eliminate instability in extension, which is a common shortcoming of most particle methods. Also, a novel alternative set of grid basis functions is proposed for efficiently calculating nodal force and consistent mass integrals on the grid. Specifically, by taking advantage of initially parallelogram-shaped particle domains, and treating the deformation gradient as constant over the particle domain, the convected particle domain is a reshaped parallelogram in the deformed configuration. Accordingly, an alternative grid basis function over the particle domain is constructed by a standard 4-node finite element interpolation on the parallelogram. Effectiveness of the proposed modifications is demonstrated using several large deformation solid mechanics problems. Copyright 2011 John Wiley & Sons, Ltd.

X-FEM in isogeometric analysis for linear fracture mechanics

Abstract: The extended finite element method (X-FEM) has proven to be an accurate, robust method for solving problems in fracture mechanics. X-FEM has typically been used with elements using linear basis functions, although some work has been performed using quadratics. In the current work, the X-FEM formulation is incorporated into isogeometric analysis to obtain solutions with higher order convergence rates for problems in linear fracture mechanics. In comparison with X-FEM with conventional finite elements of equal degree, the NURBS-based isogeometric analysis gives equal asymptotic convergence rates and equal accuracy with fewer degrees of freedom (DOF). Results for linear through quartic NURBS basis functions are presented for a multiplicity of one or a multiplicity equal the degree.

A level set based model for damage growth: The thick level set approach

Abstract: In this paper, we introduce a new way to model damage growth in solids. A level set is used to separate the undamaged zone from the damaged zone. In the damaged zone, the damage variable is an explicit function of the level set. This function is a parameter of the model. Beyond a critical length, we assume the material to be totally damaged, thus allowing a straightforward transition to fracture. The damage growth is expressed as a level set propagation. The configurational force driving the damage front is non-local in the sense that it averages information over the thickness in the wake of the front. The computational and theoretical advantages of the new damage model are stressed. Numerical examples demonstrate the capability of the new model to initiate cracks and propagate them even in complex topological patterns (branching and merging for instance).

A large deformation frictional contact formulation using NURBS‐based isogeometric analysis

Abstract: This paper focuses on the application of NURBS-based isogeometric analysis to Coulomb frictional contact problems between deformable bodies, in the context of large deformations. A mortar-based approach is presented to treat the contact constraints, whereby the discretization of the continuum is performed with arbitrary order NURBS, as well as C0-continuous Lagrange polynomial elements for comparison purposes. The numerical examples show that the proposed contact formulation in conjunction with the NURBS discretization delivers accurate and robust predictions. Results of lower quality are obtained from the Lagrange discretization, as well as from a different contact formulation based on the enforcement of the contact constraints at every integration point on the contact surface. Copyright © 2011 John Wiley & Sons, Ltd.

Truly monolithic algebraic multigrid for fluid–structure interaction

Abstract: The coupling of flexible structures to incompressible fluids draws a lot attention during the last decade. Many different solution schemes have been proposed. In this contribution we concentrate on strong coupling fluid-structure interaction by means of monolithic solution schemes. Therein, a Newton-Krylov method is applied to the monolithic set of nonlinear equations. Such schemes require good preconditioning to be efficient. We propose two preconditioners that apply algebraic multigrid techniques to the entire fluid-structure interaction system of equations. The first is based on a standard block Gauss-Seidel approach where approximate inverses of the individual field blocks are based on a algebraic multigrid hierarchy tailored for the type of the underlying physical problem. The second is based on a monolithic coarsening scheme for the coupled system that makes use of prolongation and restriction projections constructed for the individual fields. The resulting nonsymmetric monolithic algebraic multigrid method therefore involves coupling of the fields on coarse approximations to the problem yielding significantly enhanced performance.

Mesh‐independent matrix cracking and delamination modeling in laminated composites

Abstract: The initiation and evolution of transverse matrix cracks and delaminations are predicted within a mesh-independent cracking (MIC) framework MIC is a regularized extended finite element method (x-FEM) that allows the insertion of cracks in directions that are independent of the mesh orientation The Heaviside step function that is typically used to introduce a displacement discontinuity across a crack surface is replaced by a continuous function approximated by using the original displacement shape functions Such regularization allows the preservation of the Gaussian integration schema regardless of the enrichment required to model cracking in an arbitrary direction The interaction between plies is anchored on the integration point distribution, which remains constant through the entire simulation Initiation and propagation of delaminations between plies as well as intra-ply MIC opening is implemented by using a mixed-mode cohesive formulation in a fully three-dimensional model that includes residual thermal stresses The validity of the proposed methodology was tested against a variety of problems ranging from simple evolution of delamination from existing transverse cracks to strength predictions of complex laminates withouttextita priori knowledge of damage location or initiation Good agreement with conventional numerical solutions and/or experimental data was observed in all the problems considered Published 2011 This article is a US Government work and is in the public domain in the USA

Implicit Large Eddy Simulation of transition to turbulence at low Reynolds numbers using a Discontinuous Galerkin method

Abstract: The present work predicts the formation of laminar separation bubbles at low Reynolds numbers and the related transition to turbulence by means of Implicit Large Eddy Simulations with a high-order Discontinuous Galerkin method. The flow around an SD7003 infinite wing at an angle of attack of 4° is considered at Reynolds numbers of 10 000, 22 000, and 60 000 in order to gain insight into the characteristics of the laminar and turbulent regimes. At the lowest Reynolds number studied, the flow remains laminar and two dimensional over the wing surface, with a periodic vortex shedding. For higher Reynolds numbers, the flow is unsteady over the upper wing surface and exhibits a separation bubble along which the flow transitions to turbulence. Tollmien–Schlichting (TS) waves are observed in the boundary layer, and transition is found to be caused by unstable TS modes as revealed by the growth of the stream-wise amplification factor. Copyright © 2010 John Wiley & Sons, Ltd.

Assembly of finite element methods on graphics processors

Abstract: Recently, graphics processing units (GPUs) have had great success in accelerating many numerical computations. We present their application to computations on unstructured meshes such as those in finite element methods. Multiple approaches in assembling and solving sparse linear systems with NVIDIA GPUs and the Compute Unified Device Architecture (CUDA) are created and analyzed. Multiple strategies for efficient use of global, shared, and local memory, methods to achieve memory coalescing, and optimal choice of parameters are introduced. We find that with appropriate preprocessing and arrangement of support data, the GPU coprocessor using single-precision arithmetic achieves speedups of 30 or more in comparison to a well optimized double-precision single core implementation. We also find that the optimal assembly strategy depends on the order of polynomials used in the finite element discretization. Copyright © 2010 John Wiley & Sons, Ltd.

An isogeometric analysis approach to gradient damage models

Abstract: Continuum damage formulations are commonly used for the simulation of diffuse fracture processes. Implicit gradient damage models are employed to avoid the spurious mesh dependencies associated with local continuum damage models. The C 0 -continuity of traditional finite elements has hindered the study of higher order gradient damage approximations. In this contribution we use isogeometric finite elements, which allow for the construction of higher order continuous basis functions on complex domains. We study the suitability of isogeometric finite elements for the discretization of higher order gradient damage approximations.

An isogeometric approach to cohesive zone modeling

Abstract: The possibility of enhancing a B-spline basis with discontinuities by means of knot insertion makes isogeometric finite elements a suitable candidate for modeling discrete cracks. In this contribution we use isogeometric finite elements to discretize the cohesive zone formulation for failure in materials. In the case of a pre-defined interface, non-uniform rational B-splines are used to obtain an efficient discretization. In the case that propagating cracks are considered, T-splines are found to be more suitable, due to their ability to generate localized discontinuities. Various numerical simulations demonstrate the suitability of the isogeometric approach to cohesive zone modeling.

On the performance of strain smoothing for quadratic and enriched finite element approximations (XFEM/GFEM/PUFEM)

Abstract: By using the strain smoothing technique proposed by Chen et al. (Comput. Mech. 2000; 25: 137-156) for meshless methods in the context of the finite element method (FEM), Liu et al. (Comput. Mech. 2007; 39(6): 859-877) developed the Smoothed FEM (SFEM). Although the SFEM is not yet well understood mathematically, numerical experiments point to potentially useful features of this particularly simple modification of the FEM. To date, the SFEM has only been investigated for bilinear and Wachspress approximations and is limited to linear reproducing conditions. The goal of this paper is to extend the strain smoothing to higher order elements and to investigate numerically in which condition strain smoothing is beneficial to accuracy and convergence of enriched finite element approximations. We focus on three widely used enrichment schemes, namely: (a) weak discontinuities; (b) strong discontinuities; (c) near-tip linear elastic fracture mechanics functions. The main conclusion is that strain smoothing in enriched approximation is only beneficial when the enrichment functions are polynomial (cases (a) and (b)), but that non-polynomial enrichment of type (c) lead to inferior methods compared to the standard enriched FEM (e.g. XFEM). Copyright (C) 2011 John Wiley & Sons, Ltd.

An efficient framework for fluid–structure interaction using the lattice Boltzmann method and immersed moving boundaries

Abstract: This paper presents a serial computational framework that hydrodynamically couples the lattice Boltzmann method (LBM) and the discrete element method (DEM) for the solution of particle suspension problems in two and three dimensions. The single-relaxation-time Bhatnagar–Gross–Krook (LBGK) form of the lattice Boltzmann equation is employed with an immersed moving boundary method for the fluid–structure interaction. Similar algorithms have been previously reported in the literature, however, this work deliberately utilizes solution options that minimize the computational overheads of the framework to facilitate simulations of multibody structural fields in large fluid domains. In particular, mixed boundary conditions are employed which combine the simple bounce-back technique with the immersed moving boundary method, and the relatively inexpensive D3Q15 lattice is employed for 3D solutions. The fundamentals of the LBM are briefly discussed followed by a review of the coupling techniques available for FSI using the LBM. Options for mapping solid obstacles to the LBM grid are presented and an algorithm for automatic, dynamic subcycling of the two explicit solution schemes is outlined. The LBM–DEM framework is then validated and benchmarked against previously published LBM results, with comments made where appropriate on the comparative accuracy and convergence characteristics. Finally, a multi-particle suspension simulation is presented to qualitatively assess the performance of the framework when a large number of dynamic contacts exist.

A reduced integration solid-shell finite element based on the EAS and the ANS concept—Large deformation problems

Abstract: In this paper we address the extension of a recently proposed reduced integration eight-node solid-shell finite element to large deformations. The element requires only one integration point within the shell plane and at least two integration points over the thickness. The possibility to choose arbitrarily many Gauss points over the shell thickness enables a realistic and efficient modeling of the non-linear material behavior. Only one enhanced degree-of-freedom is needed to avoid volumetric and Poisson thickness locking. One key point of the formulation is the Taylor expansion of the inverse Jacobian matrix with respect to the element center leading to a very accurate modeling of arbitrary element shapes. The transverse shear and curvature thickness locking are cured by means of the assumed natural strain concept. Further crucial points are the Taylor expansion of the compatible cartesian strain with respect to the center of the element as well as the Taylor expansion of the second Piola–Kirchhoff stress tensor with respect to the normal through the center of the element. Copyright © 2010 John Wiley & Sons, Ltd.

Topology optimization for unsteady flow

Abstract: A computational methodology for optimizing the conceptual layout of unsteady flow problems at low Reynolds numbers is presented. The geometry of the design is described by the spatial distribution of a fictitious material with continuously varying porosity. The flow is predicted by a stabilized finite element formulation of the incompressible Navier–Stokes equations. A Brinkman penalization is used to enforce zero-velocities in solid material. The resulting parameter optimization problem is solved by a non-linear programming method. The paper studies the feasibility of the material interpolation approach for optimizing the topology of unsteady flow problems. The derivation of the governing equations and the adjoint sensitivity analysis are presented. A design-dependent stabilization scheme is introduced to mitigate numerical instabilities in porous material. The emergence of non-physical artifacts in the optimized material distribution is observed and linked to an insufficient resolution of the flow field and an improper representation of the pressure field within solid material by the Brinkman penalization. Two numerical examples demonstrate that the designs optimized for unsteady flow differ significantly from their steady-state counterparts. Copyright © 2011 John Wiley & Sons, Ltd.

A non-linear interface element for 3D mesoscale analysis of brick-masonry structures

Abstract: This paper presents a novel interface element for the geometric and material nonlinear analysis of unreinforced brick-masonry structures. In the proposed modelling approach, the blocks are modelled using 3D continuum solid elements, while the mortar and brick-mortar interfaces are modelled by means of the 2D nonlinear interface element. This enables the representation of any 3D arrangement for brick-masonry, accounting for the in-plane stacking mode and the through-thickness geometry, and importantly it allows the investigation of both the in-plane and the out-of-plane response of unreinforced masonry panels. A co-rotational approach is employed for the interface element, which shifts the treatment of geometric nonlinearity to the level of discrete entities, and enables the consideration of material nonlinearity within a simplified local framework employing first-order kinematics. In this respect, the internal interface forces are modelled by means of elasto-plastic material laws based on work-softening plasticity and employing multi-surface plasticity concepts. Following the presentation of the interface element formulation details, several experimentalnumerical comparisons are provided for the in-plane and out-of-plane static behaviour of brick-masonry panels. The favourable results achieved demonstrate the accuracy and the significant potential of using the developed interface element for the nonlinear analysis of brick-masonry structures under extreme loading conditions.

A robust preconditioning technique for the extended finite element method

Abstract: The extended finite element method enhances the approximation properties of the finite element space by using additional enrichment functions. But the resulting stiffness matrices can become ill-conditioned. In that case iterative solvers need a large number of iterations to obtain an acceptable solution. In this paper a procedure is described to obtain stiffness matrices whose condition number is close to the one of the finite element matrices without any enrichments. A domain decomposition is employed and the algorithm is very well suited for parallel computations. The method was tested in numerical experiments to show its effectiveness. The experiments have been conducted for structures containing cracks and material interfaces. We show that the corresponding enrichments can result in arbitrarily ill-conditioned matrices. The method proposed here, however, provides well-conditioned matrices and can be applied to any sort of enrichment. The complexity of this approach and its relation to the domain decomposition is discussed. Computation times have been measured for a structure containing multiple cracks. For this structure the computation times could be decreased by a factor of 2.

Contact algorithms for the material point method in impact and penetration simulation

Abstract: The inherent no-slip contact constraint in the standard material point method (MPM) creates a greater penetration resistance. Therefore, the standard MPM was not able to treat the problems involving impact and penetration very well. To overcome these deficiencies, two contact methods for MPM are presented and implemented in our 3D explicit MPM code, MPM3D. In MPM, the impenetrability condition may not satisfied on the redefined regular grid at the beginning of each time step, even if it has been imposed on the deformed grid at the end of last time step. The impenetrability condition between bodies is only imposed on the deformed grid in the first contact method, while it is imposed both on the deformed grid and redefined regular grid in the second contact method. Furthermore, three methods are proposed for impact and penetration simulation to determine the surface normal vectors that satisfy the collinearity conditions at the contact surface. The contact algorithms are verified by modeling the collision of two elastic rings and sphere rolling problems, and then applied to the simulation of penetration of steel ball and perforation of thick plate with a particle failure model. In the simulation of elastic ring collision, the first contact algorithm introduces significant disturbance into the total energy, but the second contact algorithm can obtain the stable solution by using much larger time step. It seems that both contact algorithms give good results for other problems, such as the sphere rolling and the projectile penetration. Copyright © 2010 John Wiley & Sons, Ltd.

A low‐cost, goal‐oriented ‘compact proper orthogonal decomposition’ basis for model reduction of static systems

Abstract: A novel model reduction technique for static systems is presented. The method is developed using a goal-oriented framework, and it extends the concept of snapshots for proper orthogonal decomposition (POD) to include (sensitivity) derivatives of the state with respect to system input parameters. The resulting reduced-order model generates accurate approximations due to its goal-oriented construction and the explicit 'training' of the model for parameter changes. The model is less computationally expensive to construct than typical POD approaches, since efficient multiple right-hand side solvers can be used to compute the sensitivity derivatives. The effectiveness of the method is demonstrated on a parameterized aerospace structure problem.

Accurate fracture modelling using meshless methods, the visibility criterion and level sets: Formulation and 2D modelling

Abstract: Fracture modelling using numerical methods is well-advanced in 2D using techniques such as the extended finite element method (XFEM). The use of meshless methods for these problems lags somewhat behind, but the potential benefits of no meshing (particularly in 3D) prompt continued research into their development. In methods where the crack face is not explicitly modelled (as the edge of an element for instance), two procedures are instead used to associate the displacement jump with the crack surface: the visibility criterion and the diffraction method. The visibility criterion is simple to implement and efficient to compute, especially with the help of level set coordinates. However, spurious discontinuities have been reported around crack tips using the visibility criterion, whereas implementing the diffraction method in 3D is much more complicated than the visibility criterion. In this paper, a tying procedure is proposed to remove the difficulty with the visibility criterion so that crack tip closure can be ensured while the advantages of the visibility criterion can be preserved. The formulation is based on the use of level set coordinates and the element-free Galerkin method, and is generally applicable for single or multiple crack problems in 2D or 3D. The paper explains the formulation and provides verification of the method against a number of 2D crack problems. Copyright © 2011 John Wiley & Sons, Ltd.

A mass constraint formulation for structural topology optimization with multiphase materials

Abstract: This work is focused on the topology optimization of lightweight structures consisting of multiphase materials. Instead of adopting the common idea of using volume constraint, a new problem formulation with mass constraint is proposed. Meanwhile, recursive multiphase materials interpolation (RMMI) and uniform multiphase materials interpolation (UMMI) schemes are discussed and compared based on numerical tests and theoretical analysis. It is indicated that the nonlinearity of the mass constraint introduced by RMMI brings numerical difficulties to attain the global optimum of the optimization problem. On the contrary, the UMMI-2 scheme makes it possible to formulate the mass constraint in a linear form with separable design variables. One such formulation favors very much the problem resolution by means of mathematical programming approaches, especially the convex programming methods. Moreover, numerical analysis indicates that fully uniform initial weighting is beneficial to seek the global optimum when UMMI-2 scheme is used. Besides, the relationship between the volume constraint and mass constraint is theoretically revealed. The filtering technique is adapted to avoid the checkerboard pattern related to the problem with multiphase materials. Numerical examples show that the UMMI-2 scheme with fully uniform initial weighting is reliable and efficient to deal with the structural topology optimization with multiphase materials and mass constraint. Meanwhile, the mass constraint formulation is evidently more significant than the volume constraint formulation. Copyright © 2011 John Wiley & Sons, Ltd.

An XFEM method for modeling geometrically elaborate crack propagation in brittle materials

Abstract: We present a method for simulating quasistatic crack propagation in 2-D which combines the extended finite element method (XFEM) with a general algorithm for cutting triangulated domains, and introduce a simple yet general and flexible quadrature rule based on the same geometric algorithm. The combination of these methods gives several advantages. First, the cutting algorithm provides a flexible and systematic way of determining material connectivity, which is required by the XFEM enrichment functions. Also, our integration scheme is straightforward to implement and accurate, without requiring a triangulation that incorporates the new crack edges or the addition of new degrees of freedom to the system. The use of this cutting algorithm and integration rule allows for geometrically complicated domains and complex crack patterns. Copyright © 2011 John Wiley & Sons, Ltd.

Uncertainty quantification in homogenization of heterogeneous microstructures modeled by XFEM

Abstract: An extended finite element method (XFEM) coupled with a Monte Carlo approach is proposed to quantify the uncertainty in the homogenized effective elastic properties of multiphase materials. The methodology allows for an arbitrary number, aspect ratio, location and orientation of elliptic inclusions within a matrix, without the need for fine meshes in the vicinity of tightly packed inclusions and especially without the need to remesh for every different generated realization of the microstructure. Moreover, the number of degrees of freedom in the enriched elements is dynamically reallocated for each Monte Carlo sample run based on the given volume fraction. The main advantage of the proposed XFEM-based methodology is a major reduction in the computational effort in extensive Monte Carlo simulations compared with the standard FEM approach. Monte Carlo and XFEM appear to work extremely efficiently together. The Monte Carlo approach allows for the modeling of the size, aspect ratios, orientations, and spatial distribution of the elliptical inclusions as random variables with any prescribed probability distributions. Numerical results are presented and the uncertainty of the homogenized elastic properties is discussed. Copyright © 2011 John Wiley & Sons, Ltd.

An X-FEM and level set computational approach for image-based modelling: Application to homogenization

Abstract: The advances in material characterization by means of imaging techniques require powerful computational methods for numerical analysis. The present contribution focuses on highlighting the advantages of coupling the extended finite elements method and the level sets method, applied to solve microstructures with complex geometries. The process of obtaining the level set data starting from a digital image of a material structure and its input into an extended finite element framework is presented. The coupled method is validated using reference examples and applied to obtain homogenized properties for heterogeneous structures. Although the computational applications presented here are mainly two-dimensional, the method is equally applicable for three-dimensional problems. Copyright © 2010 John Wiley & Sons, Ltd.

Semi‐analytic solution for multiple interacting three‐dimensional inhomogeneous inclusions of arbitrary shape in an infinite space

Abstract: This paper develops a semi-analytic solution for multiple arbitrarily shaped three-dimensional inhomogeneous inclusions embedded in an infinite isotropic matrix under external load. All interactions between the inhomogeneous inclusions are taken into account in this solution. The inhomogeneous inclusions are discretized into small cuboidal elements, each of which is treated as a cuboidal inclusion with initial eigenstrain plus unknown equivalent eigenstrain according to the Equivalent Inclusion Method. All the unknown equivalent eigenstrains are determined by solving a set of simultaneous constitutive equations established for each equivalent cuboidal inclusion. The final solution is obtained by summing up the closed-form solutions for each individual equivalent cuboidal inclusion in an infinite space. The solution evaluation is performed by application of the fast Fourier transform algorithm, which greatly increases the computational efficiency. Finally, the solution is validated by taking Eshelby's analytic solution of an ellipsoidal inhomogeneous inclusion as a benchmark and by the finite element analysis. A few sample results are also given to demonstrate the generality of the solution. The solution may have potentially significant applications in solving a wide range of inhomogeneity-related problems. Copyright © 2011 John Wiley & Sons, Ltd.

Hanging nodes and XFEM

Abstract: This paper investigates two approaches for the handling of hanging nodes in the framework of extended finite element methods (XFEM). Allowing for hanging nodes, locally refined meshes may be easily generated to improve the resolution of general, i.e. model-independent, steep gradients in the problem under consideration. Hence, a combination of these meshes with XFEM facilitates an appropriate modeling of jumps and kinks within elements that interact with steep gradients. Examples for such an interaction are, e.g. found in stress fields near crack fronts or in boundary layers near internal interfaces between two fluids. The two approaches for XFEM based on locally refined meshes with hanging nodes basically differ in whether (enriched) degrees of freedom are associated with the hanging nodes. Both approaches are applied to problems in linear elasticity and incompressible flows. Copyright © 2010 John Wiley & Sons, Ltd.

Variational principles in dissipative electro-magneto-mechanics: A framework for the macro-modeling of functional materials

Abstract: SUMMARY This paper presents a general framework for the macroscopic, continuum-based formulation and numerical implementation of dissipative functional materials with electro-magneto-mechanical couplings based on incremental variational principles. We focus on quasi-static problems, where mechanical inertia effects and time-dependent electro-magnetic couplings are a priori neglected and a time-dependence enters the formulation only through a possible rate-dependent dissipative material response. The underlying variational structure of non-reversible coupled processes is related to a canonical constitutive modeling approach, often addressed to so-called standard dissipative materials. It is shown to have enormous consequences with respect to all aspects of the continuum-based modeling in macroscopic electro-magnetomechanics. At first, the local constitutive modeling of the coupled dissipative response, i.e. stress, electric and magnetic fields versus strain, electric displacement and magnetic induction, is shown to be variational based, governed by incremental minimization and saddle-point principles. Next, the implications on the formulation of boundary-value problems are addressed, which appear in energy-based formulations as minimization principles and in enthalpy-based formulations in the form of saddle-point principles. Furthermore, the material stability of dissipative electro-magneto-mechanics on the macroscopic level is defined based on the convexity/concavity of incremental potentials. We provide a comprehensive outline of alternative variational structures and discuss details of their computational implementation, such as formulation of constitutive update algorithms and finite element solvers. From the viewpoint of constitutive modeling, including the understanding of the stability in coupled electro-magneto-mechanics, an energy-based formulation is shown to be the canonical setting. From the viewpoint of the computational convenience, an enthalpy-based formulation is the most convenient setting. A numerical investigation of a multiferroic composite demonstrates perspectives of the proposed framework with regard to the future design of new functional materials. Copyright 2011 John Wiley & Sons, Ltd. Received 1 October 2010; Revised 15 December 2010; Accepted 15 December 2010