Showing papers in "International Journal for Numerical Methods in Engineering in 2007"

A meshfree thin shell method for non‐linear dynamic fracture

Abstract: A meshfree method for thin shells with finite strains and arbitrary evolving cracks is described. The C 1 displacement continuity requirement is met by the approximation, so no special treatments for fulfilling the Kirchhoff condition are necessary. Membrane locking is eliminated by the use of a cubic or quartic polynomial basis. The shell is tested for several elastic and elasto-plastic examples and shows good results. The shell is subsequently extended to modelling cracks. Since no discretization of the director field is needed, the incorporation of discontinuities is easy to implement and straightforward.

Theoretical aspects of the smoothed finite element method (SFEM)

Abstract: This paper examines the theoretical bases for the smoothed finite element method (SFEM), which was formulated by incorporating cell-wise strain smoothing operation into standard compatible finite element method (FEM) The weak form of SFEM can be derived from the Hu–Washizu three-field variational principle For elastic problems, it is proved that 1D linear element and 2D linear triangle element in SFEM are identical to their counterparts in FEM, while 2D bilinear quadrilateral elements in SFEM are different from that of FEM: when the number of smoothing cells (SCs) of the elements equals 1, the SFEM solution is proved to be ‘variationally consistent’ and has the same properties with those of FEM using reduced integration; when SC approaches infinity, the SFEM solution will approach the solution of the standard displacement compatible FEM model; when SC is a finite number larger than 1, the SFEM solutions are not ‘variationally consistent’ but ‘energy consistent’, and will change monotonously from the solution of SFEM (SC = 1) to that of SFEM (SC → ∞) It is suggested that there exists an optimal number of SC such that the SFEM solution is closest to the exact solution The properties of SFEM are confirmed by numerical examples Copyright © 2006 John Wiley & Sons, Ltd

A projection semi-implicit scheme for the coupling of an elastic structure with an incompressible fluid

Abstract: We address the numerical simulation of fluid-structure systems involving an incompressible viscous fluid. This issue is particularly difficult to face when the fluid added-mass acting on the structure is strong, as it happens in hemodynamics for example. Indeed, several works have shown that, in such situations, implicit coupling seems to be necessary in order to avoid numerical instabilities. Although significant improvements have been achieved during the last years, solving implicit coupling often exhibits a prohibitive computational cost. In this work, we introduce a semi-implicit coupling scheme which remains stable for a reasonable range of the discretization parameters. The first idea consists in treating implicitly the added-mass effect, whereas the other contributions (geometrical non-linearities, viscous and convective effects) are treated explicitly. The second idea, relies on the fact that this kind of explicit-implicit splitting can be naturally performed using a Chorin-Temam projection scheme in the fluid. We prove (conditional) stability of the scheme for a fully discrete formulation. Several numerical experiments point out the efficiency of the present scheme compared to several implicit approaches.

An extended finite element library

Abstract: SUMMARY This paper presents and exercises a general structure for an object-oriented enriched finite element code. The programming environment provides a robust tool for extended finite element (XFEM) computations and a modular and extensible system. The program structure has been designed to meet all natural requirements for modularity, extensibility, and robustness. To facilitate meshgeometry interactions with hundreds of enrichment items, a mesh generator and mesh database are included. The salient features of the program are: flexibility in the integration schemes (subtriangles, subquadrilaterals, independent near-tip and discontinuous quadrature rules); domain integral methods for homogeneous and bi-material interface cracks arbitrarily oriented with respect to the mesh; geometry is described and updated by level sets, vector level sets or a standard method; standard and enriched approximations are independent; enrichment detection schemes: topological, geometrical, narrow-band, etc.; multi-material problem with an arbitrary number of interfaces and slip-interfaces; non-linear material models such as J2 plasticity with linear, isotropic and kinematic hardening. To illustrate the possible applications of our paradigm, we present two-dimensional linear elastic fracture mechanics for hundreds of cracks with local near-tip refinement, and crack propagation in two dimensions as well as complex three-dimensional industrial problems. Copyright c 2006 John Wiley & Sons, Ltd.

A concise interface constitutive law for analysis of delamination and splitting in composite materials and its application to scaled notched tensile specimens

Abstract: A concise constitutive law for cohesive interfaces is proposed in this paper. A new state variable is introduced to track the extent of damage accumulated at the interface. The constitutive equations not only account for mixed-mode delamination propagation in composite materials, but also satisfactorily deal with mode ratio change during the debonding process. The interface model is implemented in the LS-DYNA explicit finite element code. The model has been applied to scaled open hole tension tests on laminated composite material. Comparison between numerical results and experiments shows good correlation for failure modes and strengths for a range of different specimen sizes. Copyright © 2006 John Wiley & Sons, Ltd.

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

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

Pseudospectra, MUSIC, and dynamic wavelet neural network for damage detection of highrise buildings

Abstract: A non-parametric system identification-based model is presented for damage detection of highrise building structures subjected to seismic excitations using the dynamic fuzzy wavelet neural network (WNN) model developed by the authors. The model does not require complete measurements of the dynamic responses of the whole structure. A large structure is divided into a series of sub-structures around a few pre-selected floors where sensors are placed and measurements are made. The new model balances the global and local influences of the training data and incorporates the imprecision existing in the sensor data effectively, thus resulting in fast training convergence and high accuracy. A new damage evaluation method is proposed based on a power density spectrum method, called pseudospectrum. The multiple signal classification (MUSIC) method is employed to compute the pseudospectrum from the structural response time series. The methodology is validated using the data obtained for a 38-storey concrete test model. The results demonstrate the effectiveness of the WNN model together with the pseudospectrum method for damage detection of highrise buildings based on a small amount of sensed data. Copyright © 2007 John Wiley & Sons, Ltd.

Large-scale topology optimization using preconditioned Krylov subspace methods with recycling

Abstract: SUMMARY The computational bottleneck of topology optimization is the solution of a large number of linear systems arising in the finite element analysis. We propose fast iterative solvers for large threedimensional topology optimization problems to address this problem. Since the linear systems in the sequence of optimization steps change slowly from one step to the next, we can significantly reduce the number of iterations and the runtime of the linear solver by recycling selected search spaces from previous linear systems. In addition, we introduce a MINRES (Minimum Residual method) version with recycling (and a short term recurrence) to make recycling more efficient for symmetric problems. Furthermore, we discuss preconditioning to ensure fast convergence. We show that a proper rescaling of the linear systems reduces the huge condition numbers that typically occur in topology optimization to roughly those arising for a problem with constant density. We demonstrate the effectiveness of our solvers by solving a topology optimization problem with more than a million unknowns on a fast PC.

A simplified mesh-free method for shear bands with cohesive surfaces

Abstract: A simple methodology to model shear bands as strong displacement discontinuities in a mesh-free particle method is presented. The shear band is represented as a set of sheared particles. A sheared particle is developed through enrichment by tangential displacement discontinuities. The representation of the shear band as set of cohesive segments provides a simple and versatile model of shear bands. The loss of material stability is used as the criterion for switching from a classical continuum description of the constitutive behaviour to a traction-separation law acting on the discontinuity surface. The method is implemented for two and three dimensions. Examples of shear band progression in rate-dependent and rate-independent materials are presented, including the Kalthoff problem, where the transition from brittle fracture to shear banding is studied. Copyright © 2006 John Wiley & Sons, Ltd.

Dynamic refinement and boundary contact forces in SPH with applications in fluid flow problems

Abstract: This paper presents a general method for dynamic particle refinement in smoothed particle hydrodynamics (SPH). Candidate particles are split into several ‘daughter’ particles according to a given refinement pattern centred about the original particle. Through the solution of a non-linear minimization problem the optimal mass distribution of the daughter particles is obtained so as to reduce the errors introduced to the underlying density field. This procedure necessarily conserves the mass of the system. Conservation of energy and momentum results are also discussed. Copyright © 2007 John Wiley & Sons, Ltd.

Coupled lattice Boltzmann method and discrete element modelling of particle transport in turbulent fluid flows: Computational issues

Abstract: This paper presents essential numerical procedures in the context of the coupled lattice Boltzmann (LB) and discrete element (DE) solution strategy for the simulation of particle transport in turbulent fluid flows. Key computational issues involved are (1) the standard LB formulation for the solution of incompressible fluid flows, (2) the incorporation of large eddy simulation (LES)-based turbulence models in the LB equations for turbulent flows, (3) the computation of hydrodynamic interaction forces of the fluid and moving particles; and (4) the DE modelling of the interaction between solid particles. A complete list is provided for the conversion of relevant physical variables to lattice units to facilitate the understanding and implementation of the coupled methodology. Additional contributions made in this work include the application of the Smagorinsky turbulence model to moving particles and the proposal of a subcycling time integration scheme for the DE modelling to ensure an overall stable solution. A particle transport problem comprising 70 large particles and high Reynolds number (around 56 000) is provided to demonstrate the capability of the presented coupling strategy. Copyright © 2007 John Wiley & Sons, Ltd.

An enhanced multi-scale approach for masonry wall computations with localization of damage

Abstract: This contribution presents a multi-scale framework for the computational study of masonry structures. In order to overcome the need for excessively complex closed-form constitutive equations, a first-order computational homogenization framework is applied to infer the non-linear material behaviour of brick masonry in the presence of quasi-brittle damage. A localization analysis is carried out based on the macroscopic homogenized tangent stiffness. It is shown that localization is detected along preferential orientations, which are consistent with the underlying mesostructural failure patterns and with the applied loading. The macroscopic description is enhanced with a finite width damage band model in order to allow the treatment of macroscopic localization resulting from damage growth in the constituents. As a result of the use of homogenization techniques on finite volumes and the presence of quasi-brittle constituents, mesostructural snap-back may occur in the homogenized material response. A methodology to introduce this type of response in the multi-scale technique is proposed. The numerical implementation of the multi-scale solution scheme using a finite element method is outlined. The results obtained by the framework are illustrated by means of elementary examples, and by an example of a structural wall computation.

Numerical modelling of non-linear electroelasticity

Abstract: The numerical modelling of non-linear electroelasticity is presented in this work. Based on well-established basic equations of non-linear electroelasticity a variational formulation is built and the finite element method is employed to solve the non-linear electro-mechanical coupling problem. Numerical examples are presented to show the accuracy of the implemented formulation. Copyright © 2006 John Wiley & Sons, Ltd.

Topology optimization of acoustic-structure interaction problems using a mixed finite element formulation

Abstract: The paper presents a gradient-based topology optimization formulation that allows to solve acoustic–structure (vibro-acoustic) interaction problems without explicit boundary interface representation. In acoustic–structure interaction problems, the pressure and displacement fields are governed by Helmholtz equation and the elasticity equation, respectively. Normally, the two separate fields are coupled by surface-coupling integrals, however, such a formulation does not allow for free material re-distribution in connection with topology optimization schemes since the boundaries are not explicitly given during the optimization process. In this paper we circumvent the explicit boundary representation by using a mixed finite element formulation with displacements and pressure as primary variables (a u/p-formulation). The Helmholtz equation is obtained as a special case of the mixed formulation for the elastic shear modulus equating to zero. Hence, by spatial variation of the mass density, shear and bulk moduli we are able to solve the coupled problem by the mixed formulation. Using this modelling approach, the topology optimization procedure is simply implemented as a standard density approach. Several two-dimensional acoustic–structure problems are optimized in order to verify the proposed method. Copyright © 2006 John Wiley & Sons, Ltd.

A study of the representation of cracks with level sets

Abstract: The level set method has been used for a few years to represent cracks in fracture mechanics simulations instead of an explicit description of the cracks faces geometry. This paper studies in detail the shape of the level set functions around a crack in two dimensions that is propagating with a sharp kink, obtained both with level set update methods found in the literature and with several innovative update methods developed by the author. A criterion based on the computation of a J integral of a virtual displacement field obtained with the values of the level set functions is proposed in order to assess the quality of these update methods. With the help of this criterion, two optimal approaches are identified, which predict an accurate evolution of the crack with smooth and consistent level set functions. These methods are then applied in three dimensions to the case of an initially penny-shaped crack that propagates out of its plane. Copyright © 2006 John Wiley & Sons, Ltd.

Developing new enrichment functions for crack simulation in orthotropic media by the extended finite element method

Abstract: New enrichment functions are proposed for crack modelling in orthotropic media using the extended finite element method (XFEM). In this method, Heaviside and near-tip functions are utilized in the framework of the partition of unity method for modelling discontinuities in the classical finite element method. In this procedure, by using meshless based ideas, elements containing a crack are not required to conform to crack edges. Therefore, mesh generation is directly performed ignoring the existence of any crack while the method remains capable of extending the crack without any remeshing requirement. Furthermore, the type of elements around the crack-tip remains the same as other parts of the finite element model and the number of nodes and consequently degrees of freedom are reduced considerably in comparison to the classical finite element method. Mixed-mode stress intensity factors (SIFs) are evaluated to determine the fracture properties of domain and to compare the proposed approach with other available methods. In this paper, the interaction integral (M-integral) is adopted, which is considered as one of the most accurate numerical methods for calculating stress intensity factors.

A non‐stationary covariance‐based Kriging method for metamodelling in engineering design

Abstract: Metamodels are widely used to facilitate the analysis and optimization of engineering systems that involve computationally expensive simulations. Kriging is a metamodeling technique that is well known for its ability to build surrogate models of responses with nonlinear behavior. However, the assumption of a stationary covariance structure underlying Kriging does not hold in situations where the level of smoothness of a response varies significantly. Although nonstationary Gaussian process models have been studied for years in statistics and geostatistics communities, this has largely been for physical experimental data in relatively low dimensions. In this paper, the nonstationary covariance structure is incorporated into Kriging modeling for computer simulations. To represent the nonstationary covariance structure, we adopt a nonlinear mapping approach based on a parameterized density functions. To avoid over-parameterizing for the high dimension problems typical of engineering design, we propose a modified version of the nonlinear map approach, with a sparser, yet flexible, parameterization. The effectiveness of the proposed method is demonstrated through both mathematical and engineering examples. The robustness of the method is verified by testing multiple functions under various sampling settings. We also demonstrate that our method is effective in quantifying prediction uncertainty associated with the use of metamodels. Nomenclature

Discrete element method for modelling solid and particulate materials

Abstract: The discrete element method (DEM) is developed in this study as a general and robust technique for unified two-dimensional modelling of the mechanical behaviour of solid and particulate materials, including the transition from solid phase to particulate phase Inter-element parameters (contact stiffnesses and failure criteria) are theoretically established as functions of element size and commonly accepted material parameters including Young's modulus, Poisson's ratio, ultimate tensile strength, and fracture toughness A main feature of such an approach is that it promises to provide convergence with refinement of a DEM discretization Regarding contact failure, an energy criterion based on the material's ultimate tensile strength and fracture toughness is developed to limit the maximum contact forces and inter-element relative displacement This paper also addresses the issue of numerical stability in DEM computations and provides a theoretical method for the determination of a stable time-step The method developed herein is validated by modelling several test problems having analytic solutions and results show that indeed convergence is obtained Moreover, a very good agreement with the theoretical results is obtained in both elastic behaviour and fracture An example application of the method to high-speed penetration of a concrete beam is also given Copyright © 2006 John Wiley & Sons, Ltd

A multiscale projection method for macro/microcrack simulations

Abstract: We present a new multiscale method for crack simulations. This approach is based on a two-scale decomposition of the displacements and a projection to the coarse scale by using coarse scale test functions. The extended finite element method (XFEM) is used to take into account macrocracks as well as microcracks accurately. The transition of the field variables between the different scales and the role of the microfield in the coarse scale formulation are emphasized. The method is designed so that the fine scale computation can be done independently of the coarse scale computation, which is very efficient and ideal for parallelization. Several examples involving microcracks and macrocracks are given. It is shown that the effect of crack shielding and amplification for crack growth analyses can be captured efficiently. Copyright © 2007 John Wiley & Sons, Ltd.

An uncoupled directional damage model for fibred biological soft tissues. Formulation and computational aspects

Abstract: In this paper we present a fully three-dimensional finite-strain damage model for fibrous soft tissue. Continuum damage mechanics is used to describe the softening behaviour of soft tissues under large deformation. The structural model is formulated using the concept of internal variables that provides a very general description of materials involving irreversible effects. We considered the internal variables associated to damage to correspond to separated contributions of the matrix and fibres. In order to show clearly the performance of the constitutive model, we present 3D simulations of the behaviour of the human medial collateral ligament and of a coronary artery. Results show that the model is able to capture the typical stress–strain behaviour observed in fibrous soft tissues and seems to confirm the soundness of the proposed formulation. Copyright © 2006 John Wiley & Sons, Ltd.

A large deformation solid-shell concept based on reduced integration with hourglass stabilization

Abstract: In this paper a new eight-node (brick) solid-shell finite element formulation based on the concept of reduced integration with hourglass stabilization is presented. The work focuses on static problems. The starting point of the derivation is the three-field variational functional upon which meanwhile established 3D enhanced strain concepts are based. Important additional assumptions are made to transfer the approach into a powerful solid-shell. First of all, a Taylor expansion of the first Piola–Kirchhoff stress tensor with respect to the normal through the centre of the element is carried out. In this way the stress becomes a linear function of the shell surface co-ordinates whereas the dependence on the thickness co-ordinate remains non-linear. Secondly, the Jacobian matrix is replaced by its value in the centre of the element. These two assumptions lead to a computationally efficient shell element which requires only two Gauss points in the thickness direction (and one Gauss point in the plane of the shell element). Additionally three internal element degrees-of-freedom have to be determined to avoid thickness locking. One important advantage of the element is the fact that a fully three-dimensional stress state can be modelled without any modification of the constitutive law. The formulation has only displacement degrees-of-freedom and the geometry in the thickness direction is correctly displayed. Copyright © 2006 John Wiley & Sons, Ltd.

Shape optimization with topological changes and parametric control

Abstract: Recent advances in shape optimization rely on free-form implicit representations, such as level sets, to support boundary deformations and topological changes. By contrast, parametric shape optimization is formulated directly in terms of meaningful geometric design variables, but usually does not support free-form boundary and topological changes. We propose a novel approach to shape optimization that combines and retains the advantages of the earlier optimization techniques. The shapes in the design space are represented implicitly as level sets of a higher-dimensional function that is constructed using B-splines (to allow free-form deformations), and parameterized primitives combined with R-functions (to support desired parametric changes). Our approach to shape design and optimization offers great flexibility because it provides explicit parametric control of geometry and topology within a large space of free-form shapes. The resulting method is also general in that it subsumes most other types of shape optimization as special cases. We describe an implementation of the proposed technique with attractive numerical properties. The explicit construction of an implicit representation supports straightforward sensitivity analysis that can be used with most gradient-based optimization methods. Furthermore, our implementation does not require any error-prone polygonization or approximation of level sets (isocurves and isosurfaces). The effectiveness of the method is demonstrated by several numerical examples.

A robust algorithm for configurational‐force‐driven brittle crack propagation with R‐adaptive mesh alignment

Abstract: The paper considers a variational formulation of brittle fracture in elastic solids and proposes a numerical implementation by a finite element method. On the theoretical side, we outline a consistent thermodynamic framework for crack propagation in an elastic solid. It is shown that both the elastic equilibrium response as well as the local crack evolution follow in a natural format by exploitation of a global Clausius–Planck inequality in the sense of Coleman's method. Here, the canonical direction of the crack propagation associated with the classical Griffith criterion is the direction of the material configurational force which maximizes the local dissipation at the crack tip and minimizes the incremental energy release. On the numerical side, we exploit this variational structure in terms of crack-driving configurational forces. First, a standard finite element discretization in space yields a discrete formulation of the global dissipation in terms configurational nodal forces. As a consequence, the constitutive setting of crack propagation in the space-discretized finite element context is naturally related to discrete nodes of a typical finite element mesh. Next, consistent with the node-based setting, the discretization of the evolving crack discontinuity is performed by the doubling of critical nodes and interface segments of the mesh. Critical for the success of this procedure is its embedding into an r-adaptive crack-segment reorientation procedure with configurational-force-based directional indicator. Here, successive crack releases appear in discrete steps associated with the given space discretization. These are performed by a staggered loading–release algorithm of energy minimization at frozen crack state followed by the successive crack releases at frozen deformation. This constitutes a sequence of positive-definite discrete subproblems with successively decreasing overall stiffness, providing an extremely robust algorithmic setting in the postcritical range. We demonstrate the performance of the formulation by means of representative numerical simulations. Copyright © 2007 John Wiley & Sons, Ltd.

An interior‐point algorithm for elastoplasticity

Abstract: The problem of small-deformation, rate-independent elastoplasticity is treated using convex programming theory and algorithms. A finite-step variational formulation is first derived after which the relevant potential is discretized in space and subsequently viewed as the Lagrangian associated with a convex mathematical program. Next, an algorithm, based on the classical primal–dual interior point method, is developed. Several key modifications to the conventional implementation of this algorithm are made to fully exploit the nature of the common elastoplastic boundary value problem. The resulting method is compared to state-of-the-art elastoplastic procedures for which both similarities and differences are found. Finally, a number of examples are solved, demonstrating the capabilities of the algorithm when applied to standard perfect plasticity, hardening multisurface plasticity, and problems involving softening. Copyright © 2006 John Wiley & Sons, Ltd.

Extrinsic cohesive modelling of dynamic fracture and microbranching instability in brittle materials

Abstract: Dynamic crack microbranching processes in brittle materials are investigated by means of a computational fracture mechanics approach using the finite element method with special interface elements and a topological data structure representation. Experiments indicate presence of a limiting crack speed for dynamic crack in brittle materials as well as increasing fracture resistance with crack speed. These phenomena are numerically investigated by means of a cohesive zone model (CZM) to characterize the fracture process. A critical evaluation of intrinsic versus extrinsic CZMs is briefly presented, which highlights the necessity of adopting an extrinsic approach in the current analysis. A novel topology-based data structure is employed to enable fast and robust manipulation of evolving mesh information when extrinsic cohesive elements are inserted adaptively. Compared to intrinsic CZMs, which include an initial hardening segment in the traction–separation curve, extrinsic CZMs involve additional issues both in implementing the procedure and in interpreting simulation results. These include time discontinuity in stress history, fracture pattern dependence on time step control, and numerical energy balance. These issues are investigated in detail through a ‘quasi-steady-state’ crack propagation problem in polymethylmethacrylate. The simulation results compare reasonably well with experimental observations both globally and locally, and demonstrate certain advantageous features of the extrinsic CZM with respect to the intrinsic CZM. Copyright © 2007 John Wiley & Sons, Ltd.

A linearly conforming point interpolation method (LC-PIM) for three-dimensional elasticity problems

Abstract: Linearly conforming point interpolation method (LC-PIM) is formulated for three-dimensional elasticity problems. In this method, shape functions are generated using point interpolation method by adopting polynomial basis functions and local supporting nodes are selected based on the background cells. The shape functions so constructed have the Kronecker delta functions property and it allows straightforward imposition of point essential boundary conditions. Galerkin weak form is used for creating discretized system equations, and a nodal integration scheme with strain-smoothing operation is used to perform the numerical integration. The present LC-PIM can guarantee linear exactness and monotonic convergence for the numerical results. Numerical examples are used to examine the present method in terms of accuracy, convergence, and efficiency. Compared with the finite element method using linear elements, the LC-PIM can achieve better efficiency, and higher accuracy especially for stresses. Copyright © 2007 John Wiley & Sons, Ltd.

Level set X-FEM non matching meshes: application to dynamic crack propagation in elastic-plastic media

Abstract: This paper develops two aspects improving crack propagation modeling with the X-FEM method. On the one hand, it explains how one can use at the same time a regular structured mesh for a precise and ef ficient level set update and an unstructured irregular one for the mechani cal model. On the other hand, a new numerical scheme based on the X-FEM method is proposed for dynamic elastic-plastic situations. The simu lation results are compared with two experiments on PMMA for which crack speed and crack path are provided.

Overview and construction of meshfree basis functions: From moving least squares to entropy approximants

Abstract: In this paper, an overview of the construction of meshfree basis functions is presented, with particular emphasis on moving least-squares approximants, natural neighbour-based polygonal interpolants, and entropy approximants. The use of information-theoretic variational principles to derive approximation schemes is a recent development. In this setting, data approximation is viewed as an inductive inference problem, with the basis functions being synonymous with a discrete probability distribution and the polynomial reproducing conditions acting as the linear constraints. The maximization (minimization) of the Shannon–Jaynes entropy functional (relative entropy functional) is used to unify the construction of globally and locally supported convex approximation schemes. A JAVA applet is used to visualize the meshfree basis functions, and comparisons and links between different meshfree approximation schemes are presented. Copyright © 2006 John Wiley & Sons, Ltd.

On multiscale FE analyses of heterogeneous structures : From homogenization to multigrid solvers

Abstract: Heterogeneous structures like composites often need a fine-scale resolution of micro-effects which influence the macroscopic overall response. This is of particular relevance in the fully non-linear range of large strains and inelastic material response of the constituents. Suitable solution methods introduce a multifield scenario of hierarchically superimposed states on different length scales. For big differences of micro- and macro-scales, the argument of scale separation induces the application of homogenization methods. Such types of physical multiscale approaches can be treated by nested multilevel finite element analyses that discretize both the fine-scale micro-structure as well as the macroscopic boundary-value problem. In contrast, small-scale differences require full resolution of the heterogeneous structure. Effective solution methods for the resulting large-scale problems with strongly oscillating properties are suitably designed geometric multigrid techniques, which may be considered as numerical multiscale approaches. In both scenarios, a key ingredient is the suitable formulation of scale bridging algorithms that govern the transfer between different scales. The paper outlines new mesh-bridging techniques in a deformation-driven context for fully non-linear response, which exploit in a non-trivial manner weak constraints on the average deformation in typical finite element patches. The framework is based on an incremental variational structure of finite inelasticity. The proposed new formulations provide variational-based homogenization algorithms for physical multiscale scenarios and problem-dependent optimal finite element grid transfers for numerical multiscale scenarios of heterogeneous materials. Copyright © 2007 John Wiley & Sons, Ltd.

Transversely isotropic membrane shells with application to mitral valve mechanics. constitutive modelling and finite element implementation

Abstract: The present study addresses constitutive modelling and implementation of transversely isotropic hyperelastic material models for the analysis of the mitral valve. This valve separates the left atri ...