level set topology optimization of structural problems with interface cohesion summary this paper presents a finite element topology optimization framework for the design of two phase structural systems considering contact and cohesion phenomena along the interface the geometry of the material interface is described by an explicit level set method and the structural response is predicted by the extended finite element method in this work the interface condition is described by a bilinear cohesive zone model on the basis of the traction separation constitutive relation the non penetration condition in the presence of compressive interface forces is enforced by a stabilized lagrange multiplier method the mechanical model assumes a linear elastic isotropic material infinitesimal strain theory and a quasi static response the optimization problem is solved by a nonlinear programming method and the design sensitivities are computed by the adjoint method the performance of the presented method is evaluated by 2d and 3d numerical examples the results obtained from topology optimization reveal distinct design characteristics for the various interface phenomena considered in addition 3d examples demonstrate optimal geometries that cannot be fully captured by reduced dimensionality the optimization framework presented is limited to two phase structural systems where the material interface is coincident in the undeformed configuration and to structural responses that remain valid considering small strain kinematics copyright 2017 john wiley sons ltd

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

boundary element formulations for the numerical solution of two dimensional diffusion problems with variable coefficients this paper presents new formulations of the radial integration boundary integral equation ribie and the radial integration boundary integro differential equation ribide methods for the numerical solution of two dimensional diffusion problems with variable coefficients the methods use either a specially constructed parametrix levi function or the standard fundamental solution for the laplace equation to reduce the boundary value problem bvp to a boundary domain integral equation bdie or boundary domain integro differential equation bdide the radial integration method rim is then employed to convert the domain integrals arising in both bdie and bdide methods into equivalent boundary integrals the resulting formulations lead to pure boundary integral and integro differential equations with no domain integrals furthermore a subdomain decomposition technique sdbdie is proposed which leads to a sparse system of linear equations thus avoiding the need to calculate a large number of domain integrals numerical examples are presented for several simple problems for which exact solutions are available to demonstrate the efficiency of the proposed approaches

extra dof free and linearly independent enrichments in gfem abstract a gfem without extra dof is developed without the extra dof the resulting linear system becomes independent in size of the order of local approximation the resulting pu approximation becomes free of linear dependence and the resulting stiffness matrix becomes good conditioned numerical studies show the new gfem s excellent convergence properties and excellent stability

numerical simulation of premixed combustion using an enriched finite element method in this paper we present a novel discretization technique for the simulation of premixed combustion based on a locally enriched finite element method fem use is made of the g function approach to premixed combustion in which the domain is divided into two parts one part containing the burned and another containing the unburned gases a level set or g function is used to define the flame interface separating burned from unburned gases the extended finite element method x fem is employed which allows for velocity and pressure fields that are discontinuous across the flame interface lagrange multipliers are used to enforce the correct essential interface conditions in the form of jump conditions across the embedded flame interface a persisting problem with the use of lagrange multipliers in x fem has been the discretization of the lagrange multipliers in this paper the distributed lagrange multiplier technique is adopted we will provide results from a spatial convergence analysis showing good convergence however a small modification of the interface is required to ensure a unique solution finally results are presented from the application of the method to the problems of moving flame fronts the darrieus landau instability and a piloted bunsen burner flame

surfaces generated by moving least squares methods an analysis of moving least squares m l s methods for smoothing and interpolating scattered data is presented in particular theorems are proved concerning the smoothness of interpolants and the description of m l s processes as projection methods some properties of compositions of the m l s projector with projectors associated with finiteelement schemes are also considered the analysis is accompanied by examples of univariate and bivariate problems

the intrinsic xfem a method for arbitrary discontinuities without additional unknowns a new method for treating arbitrary discontinuities in a finite element fe context is presented unlike the standard extended fe method xfem no additional unknowns are introduced at the nodes whose supports are crossed by discontinuities the method constructs an approximation space consisting of mesh based enriched moving least squares mls functions near discontinuities and standard fe shape functions elsewhere there is only one shape function per node and these functions are able to represent known characteristics of the solution such as discontinuities singularities etc the mls method constructs shape functions based on an intrinsic basis by minimizing a weighted error functional thereby weight functions are involved and special mesh based weight functions are proposed in this work the enrichment is achieved through the intrinsic basis the method is illustrated for linear elastic examples involving strong and weak discontinuities and matches optimal rates of convergence even for crack tip applications copyright 2006 john wiley sons ltd

effects of the smoothness of partitions of unity on the quality of representation of singular enrichments for gfem xfem stress approximations around brittle cracks abstract the convergence rates of the conventional generalized extended finite element method gfem xfem in crack modeling are similar to the convergence rates of the finite element method fem laborde et al 2005 and bechet et al 2005 unless the crack tip enrichment functions are applied in a subdomain with fixed dimension independently of the mesh parameter h and also demanding some special care for blending along the transition zone chahine et al 2006 and taracon et al 2009 thus to improve convergence rates more degrees of freedom dof are generated due to the larger quantity of enriched nodes this work seeks to identify and understand the advantages of better capturing the information provided by singular enrichments over mesh based smooth partitions of unity pou such pou with higher regularity can be built through the so called c k gfem framework following duarte et al 2006 based on a moving least square of degree zero and considering mesh based smooth weighting functions associated with arbitrary polygonal clouds the purpose herein is to investigate some possible advantages of mesh based smooth pou for modeling discontinuities and singularities in two dimensional problems of linear elastic fracture mechanics in such a fashion that the discretization error associated to stress discontinuities inherent in standard c 0 continuous gfem xfem approximations is eliminated the procedure shares features similar to the standard fem regarding domain partition and numerical integration but as neither the pou nor the enrichment functions are defined in natural domains the integrations are performed using only global coordinates the approximation capabilities of c k gfem discretizations with different patterns of singular enrichment distribution are investigated by analyzing the convergence rates of the h and p versions considering global measures in terms of strain energy and l 2 norm of displacements the effects on stability are also verified by analyzing the evolution of the condition number the effect of smoothness on conditioning is investigated and the eigenvalues distributions are used to identify the several aspects involved the smoothness of the pou the different types of enrichment functions and the pattern of enrichment the performance of the smooth approximations is compared to the c 0 counterparts built using conventional c 0 fem based pou it is shown that smoothness in the presence of extrinsically applied singular enrichments is important as the information provided by such enrichment is better captured in addition there are no stress jumps around singularities reducing error propagation beyond the neighborhood of the singularity

a locally extended finite element method for the simulation of multi fluid flows using the particle level set method abstract the simulation of immiscible two phase flows on eulerian meshes requires the use of special techniques to guarantee a sharp definition of the evolving fluid interface this work describes the combination of two distinct technologies with the goal of improving the accuracy of the target simulations first of all a spatial enrichment is employed to improve the approximation properties of the eulerian mesh this is done by injecting into the solution space new features to make it able to correctly resolve the solution in the vicinity of the moving interface then the lagrangian particle level set pls method is employed to keep trace of the evolving solution and to improve the mass conservation properties of the resulting method while the local enrichment can be understood in the general context of the xfem we employ an element local variant which allows preserving the matrix graph and hence highly improving the computational efficiency

enrichment and coupling of the finite element and meshless methods a mixed hierarchical approximation based on finite elements and meshless methods is presented two cases are considered the first one couples regions where finite elements or meshless methods are used to interpolate continuity and consistency is preserved the second one enriches a finite element mesh with particles thus there is no need to remesh in adaptive refinement processes in both cases the same formulation is used convergence is studied and examples are shown copyright 2000 john wiley sons ltd

an interface fitted mesh generator and virtual element methods for elliptic interface problems abstract a simple and efficient interface fitted mesh generation algorithm which can produce a semi structured interface fitted mesh in two and three dimensions quickly is developed in this paper elements in such interface fitted meshes are not restricted to simplices but can be polygons or polyhedra especially in 3d the polyhedra instead of tetrahedra can avoid slivers virtual element methods are applied to solve elliptic interface problems with solutions and flux jump conditions algebraic multigrid solvers are used to solve the resulting linear algebraic system numerical results are presented to illustrate the effectiveness of our method

a second order virtual node method for elliptic problems with interfaces and irregular domains in three dimensions we present a numerical method for the variable coefficient poisson equation in three dimensional irregular domains and with interfacial discontinuities the discretization embeds the domain and interface into a uniform cartesian grid augmented with virtual degrees of freedom to provide accurate treatment of jump and boundary conditions the matrix associated with the discretization is symmetric positive definite and equal to the standard 7 point finite difference poisson stencil away from embedded interfaces and boundaries numerical evidence suggests second order accuracy in the l norm our approach improves the treatment of dirichlet and jump constraints in the recent work of bedrossian et al 1 and introduces innovations necessary in three dimensions specifically we construct new constraint based lagrange multiplier spaces that significantly improve the conditioning of the associated linear system of equations we provide a method for cell local polyhedral approximation to the zero isocontour surface of a level set needed for three dimensional embedding and we show that the new lagrange multiplier spaces naturally lead to a class of easy to implement multigrid methods that achieve near optimal efficiency as shown by numerical examples for the specific case of a continuous poisson coefficient in interface problems we provide an expansive treatment of the construction of a particular solution that satisfies the value jump and flux jump constraints as in 1 this is used in a discontinuity removal technique that yields the standard 7 point stencil across the interface and only requires a modification to the right hand side of the linear system

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

space time meshfree collocation method methodology and application to initial boundary value problems a novel space time meshfree collocation method stmcm for solving systems of non linear ordinary and partial differential equations by a consistent discretization in both space and time is proposed as an alternative to established mesh based methods the stmcm belongs to the class of truly meshfree methods i e the methods that do not have any underlying mesh but work on a set of nodes only without any a priori node to node connectivity instead the neighbouring information is established on the fly the stmcm is constructed using the interpolating moving least squares technique which allows a simplified implementation of boundary conditions due to fulfillment of the kronecker delta property by the kernel functions which is not the case for the major part of other meshfree methods the method is validated by several examples ranging from interpolation problems to the solution of pdes whereas the stmcm solutions are compared with either analytical or reference ones

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

a spline based enrichment function for arbitrary inclusions in extended finite element method with applications to finite deformations summary a novel enrichment function which can model arbitrarily shaped inclusions within the framework of the extended finite element method is proposed the internal boundary of an arbitrary shaped inclusion is first discretized and a numerical enrichment function is constructed on the fly using spline interpolation we consider a piecewise cubic spline which is constructed from seven localized discrete boundary points the enrichment function is then determined by solving numerically a nonlinear equation which determines the distance from any point to the spline curve parametric convergence studies are carried out to show the accuracy of this approach compared with pointwise and linear segmentation of points for the construction of the enrichment function in the case of simple inclusions and arbitrarily shaped inclusions in linear elasticity moreover the viability of this approach is illustrated on a neo hookean hyperelastic material with a hole undergoing large deformation in this case the enrichment is able to adapt to the deformation and effectively capture the correct response without remeshing copyright 2013 john wiley sons ltd

generalized gaussian quadrature rules for discontinuities and crack singularities in the extended finite element method new gaussian integration schemes are presented for the efficient and accurate evaluation of weak form integrals in the extended finite element method for discontinuous functions we construct gauss like quadrature rules over arbitrarily shaped elements in two dimensions without the need for partitioning the finite element a point elimination algorithm is used in the construction of the quadratures which ensures that the final quadratures have minimal number of gauss points for weakly singular integrands we apply a polar transformation that eliminates the singularity so that the integration can be performed efficiently and accurately numerical examples in elastic fracture using the extended finite element method are presented to illustrate the performance of the new integration techniques

the design and analysis of the generalized finite element method abstract in this paper we introduce the generalized finite element method gfem as a combination of the classical finite element method fem and the partition of unity method pum the standard finite element spaces are augmented by adding special functions which reflect the known information about the boundary value problem and the input data e g the singular functions obtained from the local asymptotic expansion of the exact solution in the neighborhood of a corner point etc the special functions are multiplied with the partition of unity corresponding to the standard linear vertex shape functions and pasted together with the existing finite element basis to construct an augmented conforming finite element space in this way the local approximability afforded by the special functions is included in the approximation while maintaining the existing infrastructure of finite element codes the major features of the gfem are 1 the essential boundary conditions can be imposed exactly as in the standard fem unlike other partition of unity based methods where this is a major issue 2 the accuracy of the numerical integration of the entries of the stiffness matrix and load vector is controlled adaptively so that the errors in integration of the special functions do not affect the accuracy of the constructed approximation this issue also has not been sufficiently addressed in other implementations of partition of unity or meshless methods and 3 linear dependencies in the system of equations are resolved by employing an easy modification of the direct linear solver the power of the gfem for solving problems in domains with complex geometry with less error and less computer resources than the standard fem is illustrated by numerical examples

improved xfem an extra dof free well conditioning and interpolating xfem abstract although its unprecedentedly success in approximating the singular field around a crack tip independent of mesh discretization xfem still has been long hindered in engineering applications by its ill conditioning and the difficulty in mass lumping corresponding to extra dofs an improved xfem i xfem is developed to overcome those difficulties from three aspects 1 to eliminate the linear dependence and the ill conditioning issues of the standard and the corrected xfems 2 to get rid of extra dofs in crack tip enrichment to facilitate optimal mass lumping in dynamic analyses 3 to be interpolating at enriched nodes to enable direct essential contact boundary treatments the core of i xfem is a globally interpolating approximation based on local least squares fitting the new method is numerically examined with comprehensive comparisons with the standard xfem and the corrected xfem fries 2008 the new method s excellent performance in terms of accuracy convergence and conditioning stability are demonstrated

higher order conformal decomposition fem cdfem abstract a higher order accurate finite element method is proposed which uses automatically generated meshes based on implicit level set data for the description of boundaries and interfaces in two and three dimensions the method is an alternative for fictitious domain and extended finite element methods the domain of interest is immersed in a background mesh composed by higher order elements the zero level sets are identified and meshed followed by a decomposition of the cut background elements into conforming sub elements adaptivity is a crucial ingredient of the method to guarantee the success of the mesh generation it ensures the successful decomposition of cut elements and enables improved geometry descriptions and approximations it is confirmed that higher order accurate results with optimal convergence rates are achieved with the proposed conformal decomposition finite element method cdfem

on the construction of blending elements for local partition of unity enriched finite elements for computational efficiency partition of unity enrichments are preferably localized to the sub domains where they are needed it is shown that an appropriate construction of the elements in the blending area the regionwhere the enriched elements blend to unenriched elements is often crucial for good performance of local partition of unity enrichments an enhanced strain formulation is developed which leads to good performance the optimal rate of convergence is achieved for polynomial enrichments it is shown that a proper choice of the finite element shape functions and partition of unity shape functions also improves the accuracy and convergence the methods are illustrated by several examples the examples deal primarily with the signed distance function enrichment for treating discontinuous derivatives inside an element but other enrichments are also considered results show that both methods provide optimal rates of convergence

high order extended finite element method for cracked domains the aim of the paper is to study the capabilities of the extended finite element method xfem to achieve accurate computations in non smooth situations such as crack problems although the xfem method ensures a weaker error than classical nite element methods the rate of convergence is not improved when the mesh parameter h is going to zero because of the presence of a singularity the di culty can be overcome by modifying the enrichment of the nite element basis with the asymptotic crack tip displacement solutions as well as with the heaviside function numerical simulations show that the modi ed xfem method achieves an optimal rate of convergence i e like in a standard nite element method for a smooth problem

a new weight function enrichment in meshless methods for multiple cracks in linear elasticity summary a new enriched weight function for meshless methods is proposed for the numerical treatment of multiple arbitrary cracks in two dimensions the main novelty consists in modifying the weight function with an intrinsic enrichment which is discontinuous over the finite length of the crack represented by a segment but continuous all around the crack tips an analytical function is used to introduce discontinuities that are incorporated in the kernel in a simple multiplicative manner the resulting method allows a more straightforward implementation and simulation of the presence of multiple cracks crack branching and crack propagation in a meshless framework without using any of the existing algorithms such as visibility transparency and diffraction and without using additional unknowns and additional equations for the evolution of the level sets as in extrinsic partition of unity based methods stress intensity factors calculated using the j integral demonstrate excellent agreement with analytical solutions for classical fracture mechanics benchmarks copyright 2011 john wiley sons ltd

derivative recovery and a posteriori error estimate for extended finite elements this paper is the first attempt at error estimation for extended finite elements the goal of this work is to devise a simple and effective local a posteriori error estimate for partition of unity enriched finite element methods such as the extended finite element method xfem in each element the local estimator is the l2 norm of the difference between the raw xfem strain field and an enhanced strain field computed by extended moving least squares xmls derivative recovery obtained from the raw nodal xfem displacements the xmls construction is tailored to the nature of the solution the technique is applied to linear elastic fracture mechanics in which near tip asymptotic functions are added to the mls basis the xmls shape functions are constructed from weight functions following the diffraction criterion to represent the discontinuity the result is a very smooth enhanced strain solution including the singularity at the crack tip results are shown for two and three dimensional linear elastic fracture mechanics problems in mode i and mixed mode the effectivity index of the estimator is close to 1 and improves upon mesh refinement for the studied near tip problem it is also shown that for the linear elastic fracture mechanics problems treated the proposed estimator outperforms one of the superconvergent patch recovery technique of zienkiewicz and zhu which is only c0 parametric studies of the general performance of the estimator are also carried out

phase locking in weakly heterogeneous neuronal networks abstract we examine analytically the existence and stability of phase locked states in a weakly heterogeneous neuronal network we consider a model of n neurons with all to all synaptic coupling where the heterogeneity is in the intrinsic firing frequency of the individual neurons we consider both inhibitory and excitatory coupling we derive the conditions under which stable phase locking is possible in homogeneous networks many different periodic phase locked states are possible their stability depends on the dynamics of the neuron and the coupling for weak heterogeneity the phase locked states are perturbed from the homogeneous states and can remain stable if their homogeneous counterparts are stable for enough heterogeneity phase locked solutions either lose stability or are destroyed completely we analyze the possible states the network can take when phase locking is broken

a stable and optimally convergent latin cutfem algorithm for multiple unilateral contact problems in this paper we propose a novel unfitted finite element method for the simulation of multiple body contact the computational mesh is generated independently of the geometry of the interacting solids which can be arbitrarily complex the key novelty of the approach is the combination of elements of the cutfem technology namely the enrichment of the solution field via the definition of overlapping fictitious domains with a dedicated penalty type regularisation of discrete operators and the latin hybrid mixed formulation of complex interface conditions furthermore the novel p1 p1 discretisation scheme that we propose for the unfitted latin solver is shown to be stable robust and optimally convergent with mesh refinement finally the paper introduces a high performance 3d level set cutfem framework for the versatile and robust solution of contact problems involving multiple bodies of complex geometries with more than two bodies interacting at a single point

error estimation for crack simulations using the xfem summarythe extended nite element method xfem is by now well established for crack calculations in linear elas tic fracture mechanics an advantage of this method is its discretization independence for crack simulations nevertheless discretization errors occur when using the xfem in this paper a simple recovery based errorestimator for the discretization error in xfem calculations for cracks is presented the method is basedon the zienkiewicz and zhu error estimator enhanced smoothed stresses incorporating the discontinuitiesand singularities because of the cracks are recovered to enable the error estimation for arbitrary distributedcracks this approach also allows the consideration of materials with generally inelastic behaviour theenhanced stresses are computed by means of a least square t problem to assess the quality of the error esti mator global norms and the effectivity index for the global energy norm for examples with known analyticalsolutions are presented copyright 2012 john wiley sons ltd

a numerical method for solving variable coefficient elliptic equation with interfaces a new 2nd order accurate numerical method on non body fitting grids is proposed for solving the variable coefficient elliptic equation in disjoint subdomains w separated by interfaces c the variable coefficients the source term and hence the solution itself and its derivatives may be discontinuous across the interfaces jump conditions in solution and its co normal derivative at interface are prescribed instead of smooth the interfaces are only required to be lipschitz continuous as submanifold a weak formulation is developed the existence uniqueness and regularity of the solutions are studied the numerical method is derived by discretizing the weak formulation the method is different from traditional finite element methods extensive numerical experiments are presented and show that the method is 2nd order accurate in solution and 1st order accurate in its gradient in l norm if the interface is c 2 and solutions are c 2 on the closures of the subdomains the method can handle the problems when the solutions and or the interfaces are weaker than c 2 for example email protected h 2 w c is lipschitz continuous and their singularities coincide see example 18 in section 4 the accuracies of the method under various circumstances are listed in table 19

multiscale aggregating discontinuities a method for circumventing loss of material stability new methods for the analysis of failure by multiscale methods that invoke unit cells to obtain the subscale response are described these methods called multiscale aggregating discontinuities are based on the concept of perforated unit cells which exclude subdomains that are unstable i e exhibit loss of material stability using this concept it is possible to compute an equivalent discontinuity at the coarser scale including both the direction of the discontinuity and the magnitude of the jump these variables are then passed to the coarse scale model along with the stress in the unit cell the discontinuity is injected at the coarser scale by the extended finite element method analysis of the procedure shows that the method is consistent in power and yields a bulk stress strain response that is stable applications of this procedure to crack growth in heterogeneous materials are given copyright 2007 john wiley sons ltd

an effective xfem with equivalent eigenstrain for stress intensity factors of homogeneous plates abstract based on the concept of equivalent eigenstrain a low order accurate extended finite element method xfem is presented the goal is the determination of the stress intensity factors of cracked homogeneous specimens the proposed approach differs from the conspicuous amount of existing contributions on this topic the findings of the present paper highlight aspects so far neglected in the literature such as the mechanical meaning of additional fields and equations specific to the xfem approximation of the displacement field moreover based on the plane strain examples simulated in the present study the proposed xfem is generally computationally more robust and accurate than existing comparable xfems while keeping a minimal implementation effort