stabilized finite element formulation for elastic plastic finite deformations abstract this paper presents a stabilized finite element formulation for nearly incompressible finite deformations in hyperelastic plastic solids such as metals an updated lagrangian finite element formulation is developed where mesh dependent terms are added to enhance the stability of the mixed finite element formulation this formulation circumvents the restriction on the displacement and pressure fields due to the babuska brezzi condition and provides freedom in choosing interpolation functions in the incompressible or nearly incompressible limit typical in metal forming applications moreover it facilitates the use of low order simplex elements i e p1 p1 reducing the degrees of freedom required for the solution in the incompressible limit when stable elements are necessary linearization of the weak form is derived for implementation into a finite element code numerical experiments with p1 p1 elements show that the method is effective in incompressible conditions and can be advantageous in metal forming analysis

hit and run algorithms for the identification of nonredundant linear inequalities two probabilistic hit and run algorithms are presented to detect nonredundant constraints in a full dimensional system of linear inequalities the algorithms proceed by generating a random sequence of interior points whose limiting distribution is uniform and by searching for a nonredundant constraint in the direction of a random vector from each point in the sequence in the hypersphere directions algorithm the direction vector is drawn from a uniform distribution on a hypersphere in the computationally superior coordinate directions algorithm a search is carried out along one of the coordinate vectors the algorithms are terminated through the use of a bayesian stopping rule computational experience with the algorithms and the stopping rule will be reported

a finite element method for the simulation of strong and weak discontinuities in solid mechanics in this paper we introduce and analyze a finite element method for elasticity problems with interfaces the method allows for discontinuities internal to the elements in the approximation across the interface we propose a general approach that can handle both perfectly and imperfectly bonded interfaces without modifications of the code for the case of linear elasticity we show that optimal order of convergence holds without restrictions on the location of the interface relative to the mesh we present numerical examples for the linear case as well as for contact and crack propagation model problems

on the complexity of computing the volume of a polyhedron we show that computing the volume of a polyhedron given either as a list of facets or as a list of vertices is as hard as computing the permanent of a matrix

variational and projection methods for the volume constraint in finite deformation elasto plasticity this paper focuses on the treatment of volume constraints which in the context of elasto plasticity typically arise as a result of assuming volume preserving plastic flow projection methods based on the modification of the discrete gradient operator b often proposed on an ad hoc basis are systematically obtained in the variational context furnished by a three field hu washizu principle the fully nonlinear formulation proposed here is based on a local multiplicative split of the deformation gradient into volume preserving and dilatational parts without relying on rate forms of the weak form of momentum balance this approach fits naturally in a formulation of plasticity based on the multiplicative decomposition of the deformation gradient and enables one to exactly enforce the condition of volume preserving plastic flow within the framework proposed in this paper rate forms and incrementally objective algorithms are entirely bypassed

stress analysis around crack tips in finite strain problems using the extended finite element method fracture of rubber like materials is still an open problem indeed it deals with modelling issues crack growth law bulk behaviour and computational issues robust crack growth in 2d and 3d incompressibility the present study focuses on the application of the extended finite element method x fem to large strain fracture mechanics for plane stress problems two important issues are investigated the choice of the formulation used to solve the problem and the determination of suitable enrichment functions it is demonstrated that the results obtained with the method are in good agreement with previously published works

a finite element method for the computational modelling of cohesive cracks the present contribution is concerned with the computational modelling of cohesive cracks in quasi brittle materials whereby the discontinuity is not limited to interelement boundaries but is allowed to propagate freely through the elements in the elements which are intersected by the discontinuity additional displacement degrees of freedom are introduced at the existing nodes therefore two independent copies of the standard basis functions are used one set is put to zero on one side of the discontinuity while it takes its usual values on the opposite side and vice versa for the other set to model inelastic material behaviour a discrete damage type constitutive model is applied formulated in terms of displacements and tractions at the surface some details on the numerical implementation are given concerning the failure criterion the determination of the direction of the discontinuity and the integration scheme finally numerical examples show the performance of the method

hit and run mixes fast it is shown that the hit and run algorithm for sampling from a convex body k introduced by r l smith mixes in time o n2r2 r2 where r and r are the radii of the inscribed and circumscribed balls of k thus after appropriate preprocessing hit and run produces an approximately uniformly distributed sample point in time o n3 which matches the best known bound for other sampling algorithms we show that the bound is best possible in terms of r r and n

hit and run from a corner we show that the hit and run random walk mixes rapidly starting from any interior point of a convex body this is the first random walk known to have this property in contrast the ball walk can take exponentially many steps from some starting points the proof extends to sampling an exponential density over a convex body

a computational approach to handle complex microstructure geometries in multiscale analysis of components there is usually a need to solve microstructures with complex geometries in this paper we use the extended finite element method x fem to solve scales involving complex geometries the x fem allows one to use meshes not necessarily matching the physical surface of the problem while retaining the accuracy of the classical finite element approach for material interfaces this is achieved by introducing a new enrichment strategy although the mesh does not need to conform to the physical surfaces it needs to be fine enough to capture the geometry of these surfaces a simple algorithm is described to adaptively refine the mesh to meet this geometrical requirement numerical experiments on the periodic homogenization of two phase complex cells demonstrate the accuracy and simplicity of the x fem c 2003 elsevier b v all rights reserved

enrichment of enhanced assumed strain approximations for representing strong discontinuities addressing volumetric incompressibility and the discontinuous patch test summary we present a geometrically nonlinear assumed strain method that allows for the presence of arbitrary intra finite element discontinuities in the deformation map special attention is placed on the coarsemesh accuracy of these methods and their ability to avoid mesh locking in the incompressible limit given an underlying mesh and an arbitrary failure surface we first construct an enriched approximation for the deformation map with the nonlinear analogue of the extended finite element method x fem with regard to the richer space of functions spanned by the gradient of the enriched approximation we then adopt a broader interpretation of variational consistency for the construction of the enhanced strain in particular in those elements intersected by the failure surface we construct enhanced strain approximations which are orthogonal to piecewise constant stress fields contrast is drawn with existing strong discontinuity approaches where the enhanced strain variations in localized elements were constructed to be orthogonal to constant nominal stress fields importantly the present formulation gives rise to a symmetric tangent stiffness matrix even in localized elements the present modification also allows for the satisfaction of a discontinuous patch test wherein two different constant stress fields on each side of the failure surface lie in the solution space we demonstrate how the proposed modifications eliminate spurious stress oscillations along the failure surface particularly for nearly incompressible material response additional numerical examples are provided to illustrate the efficacy of the modified method for problems in hyperelastic fracture mechanics copyright c 2000 john wiley sons ltd

a generalized finite element method for polycrystals with discontinuous grain boundaries summary we present a generalized finite element method for the analysis of polycrystals with explicit treatment of grain boundaries grain boundaries and junctions understood as loci of possible displacement discontinuity are inserted into finite elements by exploiting the partition of unity property of finite element shape functions consequently the finite element mesh does not need to conform to the polycrystal topology the formulation is outlined and a numerical example is presented to demonstrate the potential and accuracy of the approach the proposed methodology can also be used for branched and intersecting cohesive cracks and comparisons are made to a related approach int j numer meth engng 2000 48 1741 copyright 2006 john wiley sons ltd

an unfitted hp adaptive finite element method based on hierarchical b splines for interface problems of complex geometry generating finite element discretizations with direct interface parameterizations constitutes a considerable computational expense in case of complex interface geometries the paper at hand introduces a b spline finite element method which circumvents parameterization of interfaces and offers fast and easy meshing irrespective of the geometric complexity involved its core idea is the adaptive approximation of discontinuities by hierarchical grid refinement which adds several levels of local basis functions in the close vicinity of interfaces but unfitted to their exact location so that a simple regular grid of knot span elements can be maintained numerical experiments show that an hp refinement strategy which simultaneously increases the polynomial degree of the b spline basis and the levels of refinement around interfaces achieves exponential rates of convergence despite the presence of discontinuities it is also demonstrated that the hierarchical b spline fem can be used to transfer the recently introduced finite cell concept to geometrically nonlinear problems its computational performance imposition of unfitted boundary conditions and fast hierarchical grid generation are illustrated for a set of benchmark problems in one two and three dimensions and the advantages of the regular grid approach for complex geometries are demonstrated by the geometrically nonlinear simulation of a voxel based foam composite

modeling holes and inclusions by level sets in the extended finite element method a methodology to model arbitrary holes and material interfaces inclusions without meshing the internal boundaries is proposed the numerical method couples the level set method s osher j a sethian j comput phys 79 1 1988 12 to the extended finite element method x fem n moes j dolbow t belytschko int j numer methods engrg 46 1 1999 131 in the x fem the finite element approximation is enriched by additional functions through the notion of partition of unity the level set method is used for representing the location of holes and material interfaces and in addition the level set function is used to develop the local enrichment for material interfaces numerical examples in two dimensional linear elastostatics are presented to demonstrate the accuracy and potential of the new technique

a comment on the article a finite element method for simulation of strong and weak discontinuities in solid mechanics by a hansbo and p hansbo comput methods appl mech engrg 193 2004 3523 3540

the generalized finite element method this paper describes a pilot design and implementation of the generalized finite element method gfem as a direct extension of the standard finite element method sfem or fem which makes possible the accurate solution of engineering problems in complex domains which may be practically impossible to solve using the fem the development of the gfem is illustrated for the laplacian in two space dimensions in domains which may include several hundreds of voids and or cracks for which the construction of meshes used by the fem is practically impossible the two main capabilities are 1 it can construct the approximation using meshes which may overlap part or all of the domain boundary 2 it can incorporate into the approximation handbook functions which are known analytically or are generated numerically and approximate well the solution of the boundary value problem in the neighborhood of corner points voids cracks etc the main tool is a special integration algorithm which we call the fast remeshing approach which is robust and works for any domain with arbitrary complexity the incorporation of the handbook functions into the gfem is done by employing the partition of unity method pum the presented formulations and implementations can be easily extended to the multi material medium where the voids are replaced by inclusions of various shapes and sizes and to the case of the elasticity problem this work can also be understood as a pilot study for the feasibility and demonstration of the capabilities of the gfem which is needed before analogous implementations are attempted in the three dmensional and nonlinear cases which are the cases of main interest for future work

on the hardness of approximate reasoning abstract many ai problems when formalized reduce to evaluating the probability that a propositional expression is true in this paper we show that this problem is computationally intractable even in surprisingly restricted cases and even if we settle for an approximation to this probability we consider various methods used in approximate reasoning such as computing degree of belief and bayesian belief networks as well as reasoning techniques such as constraint satisfaction and knowledge compilation that use approximation to avoid computational difficulties and reduce them to model counting problems over a propositional domain we prove that counting satisfying assignments of propositional languages is intractable even for horn and monotone formulae and even when the size of clauses and number of occurrences of the variables are extremely limited this should be contrasted with the case of deductive reasoning where horn theories and theories with binary clauses are distinguished by the existence of linear time satisfiability algorithms what is even more surprising is that as we show even approximating the number of satisfying assignments i e approximating approximate reasoning is intractable for most of these restricted theories we also identify some restricted classes of propositional formulae for which efficient algorithms for counting satisfying assignments can be given

a generalized finite element method for the simulation of three dimensional dynamic crack propagation this paper is aimed at presenting a partition of unity method for the simulation of three dimensional dynamic crack propagation the method is a variation of the partition of unity finite element method and hp cloud method in the context of crack simulation this method allows for modeling of arbitrary dynamic crack propagation without any remeshing of the domain in the proposed method the approximation spaces are constructed using a partition of unity pu and local enrichment functions the pu is provided by a combination of shepard and finite element partitions of unity this combination of pus allows the inclusion of arbitrary crack geometry in a model without any modification of the initial discretization it also avoids the problems associated with the integration of moving least squares or conventional shepard partitions of unity used in several meshless methods the local enrichment functions can be polynomials or customized functions these functions can efficiently approximate the singular fields around crack fronts the crack propagation is modeled by modifying the partition of unity along the crack surface and does not require continuous remeshings or mappings of solutions between consecutive meshes as the crack propagates in contrast with the boundary element method the proposed method can be applied to any class of problems solvable by the classical finite element method in addition the proposed method can be implemented into most finite element data bases several numerical examples demonstrating the main features and computational efficiency of the proposed method for dynamic crack propagation are presented

piecewise affine state feedback for piecewise affine slab systems using convex optimization abstract this paper shows that lyapunov based state feedback controller synthesis for piecewise affine pwa slab systems can be cast as an optimization problem subject to a set of linear matrix inequalities lmis analytically parameterized by a vector furthermore it is shown that continuity of the control inputs at the switchings can be guaranteed by adding equality constraints to the problem without affecting its parameterization structure finally it is shown that piecewise affine state feedback controller synthesis for piecewise affine slab systems to maximize the decay rate of a quadratic control lyapunov function can be cast as a set of quasi concave optimization problems analytically parameterized by a vector before casting the synthesis in the format presented in this paper lyapunov based piecewise affine state feedback controller synthesis could only be formulated as a bi convex optimization program which is very expensive to solve globally thus the fundamental importance of the contributions of the paper relies on the fact that for the first time the piecewise affine state feedback synthesis problem has been formulated as a convex problem with a parameterized set of lmis that can be relaxed to a finite set of lmis and solved efficiently to a point near the global optimum using available software furthermore it is shown for the first time that in some situations the global can be exactly found by solving only one concave problem

solution of nearly incompressible field problems using a generalized finite element approach abstract it is well known that for problems approaching incompressible limits standard finite element approaches suffer from suboptimal convergence due to a phenomenon known as locking we analyze this phenomenon and present a robust generalized finite element formulation that enables optimal solution convergence this approach is explored for the two dimensional incompressible elasticity equations with a variable poisson s ratio as well as the two dimensional lid driven cavity stokes flow problem using a penalty pressure formulation

uncertainty quantification in homogenization of heterogeneous microstructures modeled by xfem 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

a random polynomial time algorithm for approximating the volume of convex bodies a randomized polynomial time algorithm for approximating the volume of a convex body k in n dimensional euclidean space is presented the proof of correctness of the algorithm relies on recent theory of rapidly mixing markov chains and isoperimetric inequalities to show that a certain random walk can be used to sample nearly uniformly from within k

efficient monte carlo procedures for generating points uniformly distributed over bounded regions we consider the monte carlo problem of generating points uniformly distributed within an arbitrary bounded measurable region the class of markovian methods considered generate points asymptotically uniformly distributed within the region computational experience suggests the methods are potentially superior to conventional rejection techniques for large dimensional regions

on probabilistic inference by weighted model counting a recent and effective approach to probabilistic inference calls for reducing the problem to one of weighted model counting wmc on a propositional knowledge base specifically the approach calls for encoding the probabilistic model typically a bayesian network as a propositional knowledge base in conjunctive normal form cnf with weights associated to each model according to the network parameters given this cnf computing the probability of some evidence becomes a matter of summing the weights of all cnf models consistent with the evidence a number of variations on this approach have appeared in the literature recently that vary across three orthogonal dimensions the first dimension concerns the specific encoding used to convert a bayesian network into a cnf the second dimensions relates to whether weighted model counting is performed using a search algorithm on the cnf or by compiling the cnf into a structure that renders wmc a polytime operation in the size of the compiled structure the third dimension deals with the specific properties of network parameters local structure which are captured in the cnf encoding in this paper we discuss recent work in this area across the above three dimensions and demonstrate empirically its practical importance in significantly expanding the reach of exact probabilistic inference we restrict our discussion to exact inference and model counting even though other proposals have been extended for approximate inference and approximate model counting

thermomechanical modeling of regressing heterogeneous solid propellants a numerical framework based on the generalized finite element method gfem is developed to capture the coupled effects of thermomechanical deformations and thermal gradients on the regression rate of a heterogeneous solid propellant the thermomechanical formulation is based on a multiplicative split of the deformation gradient and regression of the heterogeneous solid propellant is simulated using the level set method a spatial mesh convergence study is performed on a non regressing solid heterogeneous propellant system to examine the consistency of the coupled thermomechanical gfem solver the overall accuracy spatial and temporal of the coupled thermomechanical solver for regressing solid propellants is obtained from a periodic sandwich propellant configuration where the effects of thermomechanical deformations on its regression rate is investigated finally the effects of thermomechanical deformations in a regressing two dimensional heterogeneous propellant pack are studied and time average regression rates are reported

a high order generalized fem for through the thickness branched cracks this paper presents high order implementations of a generalized finite element method for through the thickness three dimensional branched cracks this approach can accurately represent discontinuities such as triple joints in polycrystalline materials and branched cracks independently of the background finite element mesh representative problems are investigated to illustrate the accuracy of the method in combination with various discretizations and refinement strategies the combination of local refinement at crack fronts and high order continuous and discontinuous enrichments proves to be an excellent combination which can deliver convergence rates close to that of problems with smooth solutions copyright 2007 john wiley sons ltd

interface material failure modeled by the extended finite element method and level sets multi phase materials are characterized by the fact that they possess a specific heterogeneous microstructure this feature often necessitates complex formulations in order to obtain realistic mechanical behavior with macroscopic material models furthermore such complex models cause substantial difficulties when the corresponding material parameters need to be identified and their physical meaning interpreted in multiscale analyses usually the microstructure of a material point is resolved directly thus the material behavior at the material point can be modeled in an elegant natural and simple way the present contribution aims at a detailed geometric modeling of multi phase materials as well as at a local mechanical modeling of material interfaces and interfacial failure the geometry of the material distribution within the microstructure is described through level set functions the mechanical modeling of material interfaces and interfacial cracks is accomplished by the extended finite element method x fem the combination of both the level set method and the x fem allows one to model such internal features of the microstructure without the adaptation of the mesh thus expensive meshing procedures can be avoided and considerable flexibility is obtained with respect to the variation of the material design in the x fem the finite element approximation is enriched by appropriate functions through the concept of partition of unity remarkably the level set method cannot only be applied to describe the geometry of the material interfaces and cracks respectively but also to support and even to develop the corresponding enrichment function

generalized finite element enrichment functions for discontinuous gradient fields a general gfem xfem formulation is presented to solve two dimensional problems characterized by c0 continuity with gradient jumps along discrete lines such as those found in the thermal and structural analysis of heterogeneous materials or in line load problems in homogeneous media the new enrichment functions presented in this paper allow solving problems with multiple intersecting discontinuity lines such as those found at triple junctions in polycrystalline materials and in actively cooled microvascular materials with complex embedded networks we show how the introduction of enrichment functions yields accurate finite element solutions with meshes that do not conform to the geometry of the discontinuity lines the use of the proposed enrichments in both linear and quadratic approximations is investigated as well as their combination with interface enrichment functions available in the literature through a detailed convergence study we demonstrate that quadratic approximations do not require any correction to the method to recover optimal convergence rates and that they perform better than linear approximations for the same number of degrees of freedom in the solution of these types of problems in the linear case the effectiveness of correction functions proposed in the literature is also investigated copyright 2009 john wiley sons ltd

generalized finite element method for modeling nearly incompressible bimaterial hyperelastic solids an extension of the generalized finite element method to the class of mixed finite element methods is presented to tackle heterogeneous systems with nearly incompressible non linear hyperelastic material behavior in particular heterogeneous systems with large modulus mismatch across the material interface undergoing large strains are investigated using two formulations one based on a continuous deformation map the other on a discontinuous one a bimaterial patch test is formulated to assess the ability of the two formulations to reproduce constant stress fields while a mesh convergence study is used to examine the consistency of the formulations finally compression of a model heterogeneous propellant pack is simulated to demonstrate the robustness of the proposed discontinuous deformation map formulation

computing and estimating the volume of the solution space of smt la constraints abstract the satisfiability modulo theories smt problem is a decision problem i e deciding the satisfiability of logical formulas with respect to combinations of background theories like reals integers arrays bit vectors in this paper we study the counting version of smt with respect to linear arithmetic smt la which generalizes both model counting and volume computation of convex polytopes we describe a method for estimating the volume of convex polytopes based on the multiphase monte carlo method it employs a new technique to reutilize random points so that the number of random points can be significantly reduced we prove that the reutilization technique has no side effect on the error we also investigate a simplified version of hit and run method the coordinate directions method based on volume estimation method for polytopes we present an approach to estimating the volume of the solution space of smt la formulas it employs a heuristic strategy to accelerate the volume estimation procedure in addition we devise some specific techniques for instances that arise from program analysis

stabilized four node tetrahedron with nonlocal pressure for modeling hyperelastic materials summary non linear hyperelastic response of reinforced elastomers is modeled using a novel three dimensional mixed finite element method with a nonlocal pressure field the element is unconditionally convergent and free of spurious pressure modes nonlocal pressure is obtained by an implicit gradient technique and obeys the helmholtz equation physical motivation for this nonlocality is shown an implicit finite element scheme with consistent linearization is presented finally several hyperelastic examples are solved to demonstrate the computational algorithm including the inf sup and verifications tests copyright q 2008 john wiley sons ltd

hit and run algorithms for generating multivariate distributions we introduce a general class of hit and run algorithms for generating essentially arbitrary absolutely continuous distributions on rd they include the hypersphere directions algorithm and the coordinate directions algorithm that have been proposed for identifying nonredundant linear constraints and for generating uniform distributions over subsets of rd given a bounded open set s in rd an absolutely continuous probability distribution on s the target distribution and an arbitrary probability distribution i on the boundary of the d dimensional unit sphere centered at the origin the direction distribution the i hit and run algorithm produces a sequence of iteration points as follows given the nth iteration point x choose a direction i according to the distribution i and then choose the n 1st iteration point according to the conditionalization of the distribution along the line defined by x and x i under mild conditions we show that this sequence of points is a harris recurrent reversible markov chain converging in total variation to the target distribution

computational results of an o n4 volume algorithm abstract recently an o n4 volume algorithm has been presented for convex bodies by lovasz and vempala where n is the number of dimensions of the convex body essentially the algorithm is a series of monte carlo integrations in this paper we describe a computer implementation of the volume algorithm where we improved the computational aspects of the original algorithm by adding variance decreasing modifications a stratified sampling strategy double point integration and orthonormalised estimators formulas and methodology were developed so that the errors in each phase of the algorithm can be controlled some computational results for convex bodies in dimensions ranging from 2 to 10 are presented as well