splines and a three body separable expansion for scattering problems abstract this paper investigates an energy dependant finite rank approximation to the three body amplitude at scattering energies both below and above the breakup threshold the expansion method is one proposed by adhikari and sloan as an approximation method for solving few body integral equations cubic spline functions are used to evaluate the expansion terms without contour rotation numerical results are obtained for a system of three identical bosons and the utility of this expansion method in a numerical treatment of the integral equations for four body scattering is briefly discussed due to logarithmic singularities in the off shell three body amplitude at positive energies no pointwise agreement is found between the separable expansion and the exact three body amplitude at energies above the breakup threshold

a three dimensional finite element method with arbitrary polyhedral elements the variable element topology finite element method vetfem is a finite element like galerkin approximation method in which the elements may take arbitrary polyhedral form a complete development of the vetfem is given here for both two and three dimensions a kinematic enhancement of the displacement based formulation is also given which effectively treats the case of near incompressibility convergence of the method is discussed and then illustrated by way of a 2d problem in elastostatics also the vetfem s performance is compared to that of the conventional fem with eight node hex elements in a 3d finite deformation elastic plastic problem the main attraction of the new method is its freedom from the strict rules of construction of conventional finite element meshes making automatic mesh generation on complex domains a significantly simpler matter copyright 2006 john wiley sons ltd

topology optimization using polytopes abstract meshing complex engineering domains is a challenging task arbitrary polyhedral meshes can provide the much needed flexibility in automated discretization of such domains the geometric property of polyhedral meshes such as its unstructured nature and the connectivity of faces between elements makes them specially attractive for topology optimization applications numerical anomalies in designs such as the single node connections and checkerboard pattern can be naturally circumvented with polyhedrons in the current work we solve the governing three dimensional elasticity state equation using the virtual element method vem approach the main characteristic difference between vem and standard finite element methods fem is that in vem the canonical basis functions are not constructed explicitly rather the stiffness matrix is computed directly utilizing a projection map which extracts the linear component of the deformation such a construction guarantees the satisfaction of the patch test used by engineers as an indicator of optimal convergence of numerical solutions under mesh refinement finally the computations reduce to the evaluation of matrices which contain purely geometric surface facet quantities the present work focuses on the first order vem in which the degrees of freedom are associated with the vertices the features of the current optimization approach are demonstrated using numerical examples for compliance minimization and compliant mechanism problems

a stabilized conforming nodal integration for galerkin mesh free methods domain integration by gauss quadrature in the galerkin mesh free methods adds considerable complexity to solution procedures direct nodal integration on the other hand leads to a numerical instability due to under integration and vanishing derivatives of shape functions at the nodes a strain smoothing stabilization for nodal integration is proposed to eliminate spatial instability in nodal integration for convergence an integration constraint ic is introduced as a necessary condition for a linear exactness in the mesh free galerkin approximation the gradient matrix of strain smoothing is shown to satisfy ic using a divergence theorem no numerical control parameter is involved in the proposed strain smoothing stabilization the numerical results show that the accuracy and convergent rates in the mesh free method with a direct nodal integration are improved considerably by the proposed stabilized conforming nodal integration method it is also demonstrated that the gauss integration method fails to meet ic in mesh free discretization for this reason the proposed method provides even better accuracy than gauss integration for galerkin mesh free method as presented in several numerical examples copyright 2001 john wiley sons ltd

efficient isoparametric trimmed hexahedral elements with explicit shape functions abstract in this study we develop isoparametric trimmed hexahedral th elements with explicit shape functions for finite element analysis using trimmed meshes th elements are placed at the boundary of a trimmed mesh generated by cutting a regular hexahedral mesh with a geometric surface master th elements are defined based on the marching cubes algorithm for an isoparametric mapping from the reference coordinates to the physical coordinates shape function values at the support points of master th elements are precomputed by solving laplace s equation with appropriate boundary conditions explicit shape functions for master th elements are constructed by interpolating the precomputed shape function values at the support points of master th elements the present explicit shape functions satisfy the required properties of fem and significantly reduce the computational cost compared with the previously developed polyhedral shape functions

quadratic serendipity finite elements over convex polyhedra summary the conventional approach to construct quadratic elements for an n sided polygon will yield n n 1 2 shape functions which increases the computational effort it is well known that the serendipity elements based on isoparametric formulation suffers from mesh distortion floater and lai proposed a systematic way to construct higher order approximations over arbitrary polygons using the generalized barycentric coordinates and triangular coordinates this approach ensures 2n shape functions with nodes only on the boundary of the polygon in this paper we extend the polygonal splines approach to three dimensions and construct serendipity shape functions over hexahedra and convex polyhedra this is done by expressing the shape functions using the barycentric coordinates and the local tetrahedral coordinates the quadratic shape functions possess kronecker delta property and satisfy constant linear and quadratic precision the accuracy and the convergence properties of the quadratic serendipity shape elements are demonstrated with a series of standard patch tests the numerical results show that the quadratic serendipity elements pass the patch test yield optimal convergence rate and can tolerate extreme mesh distortion this article is protected by copyright all rights reserved

an embedded dirichlet formulation for 3d continua this paper presents a new approach for imposing dirichlet conditions weakly on non fitting finite element meshes such conditions also called embedded dirichlet conditions are typically but not exclusively encountered when prescribing dirichlet conditions in the context of the extended finite element method xfem the methods key idea is the use of an additional stress field as the constraining lagrange multiplier function the resulting mixed hybrid formulation is applicable to 1d 2d and 3d problems the method does not require stabilization for the lagrange multiplier unknowns and allows the complete condensation of these unknowns on the element level furthermore only non zero diagonal terms are present in the tangent stiffness which allows the straight forward application of state of the art iterative solvers like algebraic multigrid amg techniques within this paper the method is applied to the linear momentum equation of an elastic continuum and to the transient incompressible navier stokes equations steady and unsteady benchmark computations show excellent agreement with reference values the general formulation presented in this paper can also be applied to other continuous field problems

polyhedral elements with strain smoothing for coupling hexahedral meshes at arbitrary nonmatching interfaces abstract managing nonmatching interfaces between dissimilar hexahedral meshes is complicated because the interfaces which are arbitrarily formed by the intersection of element edges are composed of various types of polygons in this paper a simple and efficient scheme is proposed for dealing with nonmatching interfaces using polyhedral elements because polyhedral elements can have arbitrary numbers of polygonal faces and nodes they can be used as transition elements for coupling nonmatching meshes a strain smoothing technique in the cell based smoothed finite element method is introduced to resolve critical difficulties in defining shape functions and in conducting numerical integration for the polyhedral elements as well as to further enhance the accuracy and efficiency of finite element analyses the effectiveness of the proposed scheme is demonstrated through several numerical examples involving nonmatching interfaces the effects of decomposition types of smoothing cells in the strain smoothing technique are also discussed in terms of numerical stability and accuracy

a new polyhedral element for the analysis of hexahedral dominant finite element models and its application to nonlinear solid mechanics problems abstract a hexahedral dominant finite element mesh can be easily constructed by cutting regular hexahedral elements in a simple block with cad surfaces representing outer surfaces of a geometric model polyhedral elements with straight edges but possibly non planar faces are generated at the domain boundaries while regular hexahedral elements remain in the interior region shape functions for polyhedral elements are derived from moving least square approximation based on a tetrahedral subdivision of polyhedral domains by a centroid based subdivision technique the polyhedral shape functions in this study have similar properties to conventional finite element shape functions in terms of continuity and completeness within elements compatibility across inter element boundaries and the kronecker delta property furthermore the present approach using hexahedral dominant meshes with polyhedral elements at domain boundaries is successfully applied to solve large deformation problems of hyperelastic and elastic plastic materials

hp generalized fem and crack surface representation for non planar 3 d cracks a high order generalized finite element method gfem for non planar three dimensional crack surfaces is presented discontinuous p hierarchical enrichment functions are applied to strongly graded tetrahedral meshes automatically created around crack fronts the gfem is able to model a crack arbitrarily located within a finite element fe mesh and thus the proposed method allows fully automated fracture analysis using an existing fe discretization without cracks we also propose a crack surface representation that is independent of the underlying gfem discretization and controlled only by the physics of the problem the representation preserves continuity of the crack surface while being able to represent non planar non smooth crack surfaces inside of elements of any size the proposed representation also provides support for the implementation of accurate robust and computationally efficient numerical integration of the weak form over elements cut by the crack surface numerical simulations using the proposed gfem show high convergence rates of extracted stress intensity factors along non planar curved crack fronts and the robustness of the method copyright 2008 john wiley sons ltd

integration of singular enrichment functions in the generalized extended finite element method for three dimensional problems a mapping method is developed to integrate weak singularities which result from enrichment functions in the generalized extended finite element method the integration scheme is applicable to 2d and 3d problems including arbitrarily shaped triangles and tetrahedra implementation of the proposed scheme in existing codes is straightforward numerical examples for 2d and 3d problems demonstrate the accuracy and convergence properties of the technique copyright 2008 john wiley sons ltd

virtual element methods for hyperbolic problems on polygonal meshes in the present paper we develop the virtual element method for hyperbolic problems on polygonal meshes considering the linear wave equations as our model problem after presenting the semi discrete scheme we derive the convergence estimates in h 1 semi norm and l 2 norm moreover we develop a theoretical analysis on the stability for the fully discrete problem by comparing the newmark method and the bathe method finally we show the practical behaviour of the proposed method through a large set of numerical tests

spline galerkin solution of integral equations for three body scattering above break up abstract we investigate a galerkin method for solving the integral equations for three body scattering at energies above the break up threshold the scattering equations that we consider are one dimensional integral equations that arise from a separable potential model cubic spline approximants with multiple knots are used to construct a non smooth solution function numerical results are obtained both for a system of spin 0 boson and spin 1 2 fermion particles interacting via separable two body potentials the results demonstrate that our numerical treatment of this problem is both robust and accurate with a small number of basis functions

quadratic maximum entropy serendipity shape functions for arbitrary planar polygons in this paper we present the development of quadratic serendipity shape functions on planar convex and nonconvex polygons drawing on the work of bompadre et al 2012 1 and hormann and sukumar 2008 2 we adopt a relative entropy measure for signed positive or negative shape functions with nodal prior weight functions that have the appropriate zero set on the boundary of the polygon we maximize the objective functional subject to the constraints for quadratic completeness proposed by rand et al 2013 3 along an edge of a polygon the approximation is identical to univariate bernstein polynomials the choice of the nodal prior weight function ensures that the shape functions satisfy a weak kronecker delta property on each edge the shape functions are well defined for arbitrary planar polygons without self intersections on using a modified numerical integration scheme we show that the quadratic patch test is passed on polygonal meshes with convex and nonconvex elements numerical tests for the poisson equation on self similar trapezoidal meshes and quasiuniform polygonal meshes are presented which reveal the sound accuracy of the method and optimal rates of convergence in the l2l2 norm and the h1h1 seminorm are established

virtual element method for a nonlocal elliptic problem of kirchhoff type on polygonal meshes abstract we consider the discretization of the nonlocal elliptic problem of kirchhoff type using the virtual element method vem over polygonal meshes the nonlocal diffusion coefficient is approximated by using the l 2 projection operator which is directly computable from the degrees of freedom however the presence of the nonlocal term reduces the sparsity of the jacobian matrix which would increase the computational burden to avoid this and to retain the sparsity of the jacobian the nonlinear system is replaced by an equivalent system inspired by the work on the fem by gudi 2012 the numerical results show that the proposed methodology yields optimal convergence rate in the l 2 norm theoretical estimates are derived for h 1 norms and are verified by solving a numerical example

estimation of the effect of numerical integration in finite element eigenvalue approximation finite element approximations of the eigenpairs of differential operators are computed as eigenpairs of matrices whose elements involve integrals which must be evaluated by numerical integration the effect of this numerical integration on the eigenvalue and eigenfunction error is estimated specifically for 2nd order selfadjoint eigenvalue problems we show that finite element approximations with quadrature satisfy the well known estimates for approximations without quadrature provided the quadrature rules have appropriate degrees of precision

a cardinal function algorithm for computing multivariate quadrature points we present a new algorithm for numerically computing quadrature formulas for arbitrary domains which exactly integrate a given polynomial space an effective method for constructing quadrature formulas has been to numerically solve a nonlinear set of equations for the quadrature points and their associated weights symmetry conditions are often used to reduce the number of equations and unknowns our algorithm instead relies on the construction of cardinal functions and thus requires that the number of quadrature points n be equal to the dimension of a prescribed lower dimensional polynomial space the cardinal functions allow us to treat the quadrature weights as dependent variables and remove them as well as an equivalent number of equations from the numerical optimization procedure we give results for the triangle where for all degrees d le 25 we find quadrature formulas of this form which have positive weights and contain no points outside the triangle seven of these quadrature formulas improve on previously known results

solving three body integral equations with blending functions this paper investigates a numerical method for solving three body integral equations the method is a finite element approach similar to that of hammerlin et al which utilizes an approximating surface in three dimensional space commonly referred to as coons patches or blending functions however unlike hammerlin et al who use smoothest splines the present paper uses a basis of c piecewise polynomial functions numerical accuracy of the approximation is tested in a single variable three boson model at bound energies and scattering threshold it is shown that the method converges to exact three body solutions and that the rate of convergence is much faster than a usual tensor product of c piecewise polynomial functions the blending function approximation can be used to construct finite rank separable approximations of the three body amplitude which should be of use in four body calculations

hp theory on a smooth domain in rn and elliptic boundary value problems in this paper we answer the following two questions question 1 let be an appropriate domain inrn what are the possible natural notions of hp that generalize the usual hardy spaces hp rn we shall see that several versions are possible we concentrate on the two most relevant here which may be considered to be the largest version hpr and the smallest version hpz for the second question consider the dirichlet and neumann problems for the former consists in finding the u g which solves u in and u 0 the latter consists in finding the u g which solves u in and u n 0 question 2 in the context of the relevant hp can one obtain the boundedness of 2g xi xj and g xi xj

a paradigm for higher order polygonal elements in finite elasticity using a gradient correction scheme abstract recent studies have demonstrated that polygonal elements possess great potential in the study of nonlinear elastic materials under finite deformations on the one hand these elements are well suited to model complex microstructures e g particulate microstructures and microstructures involving different length scales and incorporating periodic boundary conditions on the other hand polygonal elements are found to be more tolerant to large localized deformations than the standard finite elements and to produce more accurate results in bending and shear with mixed formulations lower order mixed polygonal elements are also shown to be numerically stable on voronoi type meshes without any additional stabilization treatment however polygonal elements generally suffer from persistent consistency errors under mesh refinement with the commonly used numerical integration schemes as a result non convergent finite element results typically occur which severely limit their applications in this work a general gradient correction scheme is adopted that restores the polynomial consistency by adding a minimal perturbation to the gradient of the displacement field with the correction scheme the recovery of optimal convergence for solutions of displacement based and mixed formulations with both linear and quadratic displacement interpolants is confirmed by numerical studies of several boundary value problems in finite elasticity in addition for mixed polygonal elements the various choices of the pressure field approximations are discussed and their performance on stability and accuracy are numerically investigated we present applications of those elements in physically based examples including a study of filled elastomers with interphasial effect and a qualitative comparison with cavitation experiments for fiber reinforced elastomers

a numerical algorithm for the construction of efficient quadrature rules in two and higher dimensions we present a numerical algorithm for the construction of efficient high order quadratures in two and higher dimensions quadrature rules constructed via this algorithm possess positive weights and interior nodes resembling the gaussian quadratures in one dimension in addition rules can be generated with varying degrees of symmetry adaptable to individual domains we illustrate the performance of our method with numerical examples and report quadrature rules for polynomials on triangles squares and cubes up to degree 50 these formulae are near optimal in the number of nodes used and many of them appear to be new

a stress displacement virtual element method for plane elasticity problems abstract the numerical approximation of 2d elasticity problems is considered in the framework of the small strain theory and in connection with the mixed hellinger reissner variational formulation a low order virtual element method vem with a priori symmetric stresses is proposed several numerical tests are provided along with a rigorous stability and convergence analysis

on the virtual element method for three dimensional linear elasticity problems on arbitrary polyhedral meshes abstract we explore the recently proposed virtual element method vem for the numerical solution of boundary value problems on arbitrary polyhedral meshes more specifically we focus on the linear elasticity equations in three dimensions and elaborate upon the key concepts underlying the first order vem while the point of departure is a conforming galerkin framework the distinguishing feature of vem is that it does not require an explicit computation of the trial and test spaces thereby circumventing a barrier to standard finite element discretizations on arbitrary grids at the heart of the method is a particular kinematic decomposition of element deformation states which in turn leads to a corresponding decomposition of strain energy by capturing the energy of linear deformations exactly one can guarantee satisfaction of the patch test and optimal convergence of numerical solutions the decomposition itself is enabled by local projection maps that appropriately extract the rigid body motion and constant strain components of the deformation as we show computing these projection maps and subsequently the local stiffness matrices in practice reduces to the computation of purely geometric quantities in addition to discussing aspects of implementation of the method we present several numerical studies in order to verify convergence of the vem and evaluate its performance for various types of meshes

polyhedral elements by means of node edge based smoothed finite element method summary the node or edge based smoothed finite element method is extended to develop polyhedral elements that are allowed to have an arbitrary number of nodes or faces and so retain a good geometric adaptability the strain smoothing technique and implicit shape functions based on the linear point interpolation makes the element formulation simple and straightforward the resulting polyhedral elements are free from the excessive zero energy modes and yield a robust solution very much insensitive to mesh distortion several numerical examples within the framework of linear elasticity demonstrate the accuracy and convergence behavior the smoothed finite element methods based polyhedral elements in general yield solutions of better accuracy and faster convergence rate than those of the conventional finite element methods this article is protected by copyright all rights reserved

interior h1 estimates for second order elliptic equations with vanishing lmo coefficients an a priori interior h1 estimate in a bounded domain for a second order elliptic operator with vanishing lmo coefficients is proved via a corresponding estimate for the commutator of a singular integral with an lmo function the preceding estimate is of independent interest in harmonic analysis

polytope finite elements in this paper finite elements based on arbitrary convex and non convex polytopes are introduced polytopes in combination with natural element coordinates necs permit a uniform element formulation of interpolation functions that are independent of the dimension of space localization and the number of vertices necs based on the natural neighbor interpolation are restricted to the polytope and can be understood as an extension of the barycentric coordinates on simplexes the differentiation and integration of these interpolation functions on the basis of necs is essential for finite element approximations the accuracy of the finite element interpolation or approximation can be controlled by either applying the h version or by utilizing the p version of the finite element method fem advantages in the handling of hanging nodes are discussed furthermore we present construction methods for lagrangian as well as for hierarchical interpolation functions based on necs numerical experiments on different convex and non convex decompositions will show the usability accuracy and convergence of the developed polytope fem copyright 2007 john wiley sons ltd

a cartesian grid method for modeling multiple moving objects in 2d incompressible viscous flow we present an efficient method for solving 2d incompressible viscous flows around multiple moving objects our method employs an underlying regular cartesian grid to solve the system using a streamfunction vorticity formulation and with discontinuities representing the embedded objects the no penentration condition for the moving geometry is satisfied by superposing a homogenous solution to the poisson s equation for the streamfunction the no slip condition is satisfied by generating vorticity on the surfaces of the objects both the initial poisson solution and the evaluation of the homogenous solution require embedding irregular discontinuities in a fast poisson solver computation time is dictated by the time required to do a fast poisson solution plus solve an integral form of laplace s equation there is no significant increase in computational cost if the geometry of the embedded objects is variable and moving relative to the underlying grid we test the method against the canonical example of flow past a cylinder and obtained new results on the flow and forces of two cylinders moving relative to each other

fast integration and weight function blending in the extended finite element method two issues in the extended finite element method xfem are addressed efficient numerical integration of the weak form when the enrichment function is self equilibrating and blending of the enrichment the integration is based on transforming the domain integrals in the weak form into equivalent contour integrals it is shown that the contour form is computationally more efficient than the domain form especially when the enrichment function is singular and or discontinuous a method for alleviating the errors in the blending elements is also studied in this method the enrichment function is pre multiplied by a smooth weight function with compact support to allow for a completely smooth transition between enriched and unenriched subdomains a method for blending step function enrichment with singular enrichments is described it is also shown that if the enrichment is not shifted properly the weighted enrichment is equivalent to the standard enrichment an edge dislocation and a crack problem are used to benchmark the technique the influence of the variables that parameterize the weight function is analyzed the resulting method shows both improved accuracy and optimum convergence rates and is easily implemented into existing xfem codes

consistent pseudo derivatives in meshless methods a meshless petrov galerkin formulation is developed in which derivatives of the trial functions are obtained as a linear combination of derivatives of shepard functions a key contribution is the development of conditions on test functions and trial functions for nonintegrable pseudo derivatives for petrov galerkin method which pass the patch test numerical results show that the resulting method is substantially more accurate than the galerkin method with shepard approximants and exceeds the rate of convergence of linear finite elements

the effect of numerical quadrature in the p version of the finite element method we investigate the use of numerical quadrature in the p version of the finite element method we describe a set of minimal conditions that the quadrature rules should satisfy for various types of elements under sufficient assumptions of smoothness we establish optimality of the asymptotic rate of convergence some computational results are presented which illustrate under what conditions overintegration may be useful

on a finite element method with variable element topology a new finite element like approximation method for problems in solid mechanics here called the variable element topology finite element method vetfem is presented the displacement based variational basis of the conventional finite element method fem is retained in the vetfem as is the discretization of the problem into elements however vetfem elements are not subject to any of the geometric or topological restrictions of conventional elements they may contain any number of nodes in any arrangement the polynomial vetfem shape functions emerge from a constrained minimization process on each element instead of from an isoparametric transformation from a parent element as in the conventional fem all the powerful features normally associated with the conventional fem are exhibited by the new method in addition because of the absence of geometric and or topological restrictions on the elements automatic mesh generation is enormously simplified for this reason the vetfem is thought to be particularly useful for problems involving very complex geometry adaptive remeshing or crack extension

simulating curvilinear crack propagation in two dimensions with universal meshes summary we formulate a class of delicately controlled problems to model the kink free evolution of quasistatic cracks in brittle isotropic linearly elastic materials in two dimensions the evolving crack satisfies familiar principles griffith s criterion local symmetry and irreversibility a novel feature of the formulation is that in addition to the crack path the loading is also treated as an unknown specifically a scaling factor for prescribed dirichlet and neumann boundary conditions is computed as part of the solution to yield an always propagating and ostensibly kink free crack and a continuous loading history beyond the initial step a dimensionless statement of the problem depends only on the poisson s ratio of a homogeneous material and is in particular independent of its young s modulus and fracture toughness numerical resolution of the formulated problem relies on two new ideas the first is an algorithm to compute triangulations conforming to cracked domains by locally deforming a given background mesh in the vicinity of evolving cracks the algorithm is robust under mild assumptions on the sizes and angles of triangles near the crack and its smoothness hence a subset if not the entire family of cracked domains realized during the course of a simulation can be discretized with the same background mesh we term the latter a universal mesh for such a family of domains universal meshes facilitate adopting a discrete representation for the crack as splines in our examples preclude the need for local splitting retriangulation operations liberate the crack from following directions prescribed by the mesh and enable the adoption of standard fems to compute the elastic fields in the cracked solid second we employ a method specifically designed to approximate the stress intensity factors for curvilinear cracks we examine the performance of the resulting numerical method with detailed examples including comparisons with an exact solution of a crack propagating along a circular arc which we construct and comparisons with experimental fracture paths in all cases we observe convergence of computed paths their derivatives and loading histories with refinement of the universal mesh copyright 2014 john wiley sons ltd

linear smoothed polygonal and polyhedral finite elements the strain smoothing technique over higher order elements and arbitrary polytopes yields less accurate solutions than other techniques such as the conventional polygonal finite element method in this work we propose a linear strain smoothing scheme that improves the accuracy of linear and quadratic approximations over convex polytopes the main idea is to subdivide the polytope into simplicial subcells and use a linear smoothing function in each subcell to compute the strain this new strain is then used in the computation of the stiffness matrix the convergence properties and accuracy of the proposed scheme are discussed by solving a few benchmark problems numerical results show that the proposed linear strain smoothing scheme makes the approximation based on polytopes able to deliver the same optimal convergence rate as traditional quadrilateral and hexahedral approximations the accuracy is also improved and all the methods tested pass the patch test to machine precision

effect of numerical integration on meshless methods abstract in this paper we present the effect of numerical integration on meshless methods with shape functions that reproduce polynomials of degree k 1 the meshless method was used on a second order neumann problem and we derived an estimate for the energy norm of the error between the exact solution and the approximate solution from the meshless method under the presence of numerical integration this estimate was obtained under the assumption that the numerical integration scheme satisfied a form of green s formula we also indicated how to obtain numerical integration schemes satisfying this property

a displacement based finite element formulation for general polyhedra using harmonic shape functions summary a displacement based continuous galerkin finite element formulation for general polyhedra is presented for applications in nonlinear solid mechanics the polyhedra can have an arbitrary number of vertices or faces the faces of the polyhedra can have an arbitrary number of edges and can be nonplanar the polyhedra can be nonconvex with only the mild restriction of star convexity with respect to the vertex averaged centroid conforming shape functions are constructed using harmonic functions defined on the undeformed configuration thus requiring the use of a total lagrangian finite element formulation in large deformation applications for nonlinear applications with computationally intensive constitutive models it is important to minimize the number of element quadrature points for this reason an integration scheme is adopted in which the number of quadrature points is equal to the number of vertices as a first step toward verifying the element behavior in the general nonlinear setting several linear verification examples are presented using both random voronoi meshes and distorted hexahedral meshes for the hexahedral meshes results for the polyhedral formulation are compared to those of the standard trilinear hexahedral formulation the element behavior in the nearly incompressible regime is also examined copyright 2013 john wiley sons ltd

a new discretization methodology for diffusion problems on generalized polyhedral meshes we develop a family of inexpensive discretization schemes for diffusion problems on generalized polyhedral meshes with elements having non planar faces the material properties are described by a full tensor we also prove superconvergence for the scalar pressure variable under very general assumptions the theoretical results are confirmed with numerical experiments in the practically important case of logically cubic meshes with randomly perturbed nodes the mixed finite element with the lowest order raviart thomas elements does not converge while the proposed mimetic method has the optimal convergence rate 2007 elsevier b v all rights reserved

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

the effect of numerical integration on the finite element approximation of linear functionals in this paper we have studied the effect of numerical integration on the finite element method based on piecewise polynomials of degree k in the context of approximating linear functionals which are also known as quantities of interest we have obtained the optimal order of convergence mathcal o h 2k of the error in the computed functional when the integrals in the stiffness matrix and the load vector are computed with a quadrature rule of algebraic precision 2k 1 however this result was obtained under an increased regularity assumption on the data which is more than required to obtain the optimal order of convergence of the energy norm of the error in the finite element solution with quadrature we have obtained a lower bound of the error in the computed functional for a particular problem which indicates the necessity of the increased regularity requirement of the data numerical experiments have been presented indicating that over integration may be necessary to accurately approximate the functional when the data lack the increased regularity

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

an immersed interface method for viscous incompressible flows involving rigid and flexible boundaries we present an immersed interface method for the incompressible navier stokes equations capable of handling both rigid and flexible boundaries the immersed boundaries are represented by a number of lagrangian control points in order to ensure that the no slip condition on the rigid boundary is satisfied singular forces are applied on the fluid the forces are related to the jumps in pressure and the jumps in the derivatives of both pressure and velocity and are interpolated using cubic splines the strength of the singular forces at the rigid boundary is determined by solving a small system of equations at each timestep for flexible boundaries the forces that the boundary exerts on the fluid are computed from the constitutive relation of the flexible boundary and are applied to the fluid through the jump conditions the position of the flexible boundary is updated implicitly using a quasi newton method bfgs within each timestep the navier stokes equations are discretized on a staggered cartesian grid by a second order accurate projection method for pressure and velocity and the overall scheme is second order accurate

generalized gaussian quadrature rules on arbitrary polygons in this paper we present a numerical algorithm based on group theory and numerical optimization to compute efficient quadrature rules for integration of bivariate polynomials over arbitrary polygons these quadratures have desirable properties such as positivity of weights and interiority of nodes and can readily be used as software libraries where numerical integration within planar polygons is required we have used this algorithm for the construction of symmetric and non symmetric quadrature rules over convex and concave polygons while in the case of symmetric quadratures our results are comparable to available rules the proposed algorithm has the advantage of being flexible enough so that it can be applied to arbitrary planar regions for the integration of generalized classes of functions to demonstrate the efficiency of the new quadrature rules we have tested them for the integration of rational polygonal shape functions over a regular hexagon for a relative error of 10 8 in the computation of stiffness matrix entries one needs at least 198 evaluation points when the region is partitioned whereas 85 points suffice with our quadrature rule copyright 2009 john wiley sons ltd

a c 1 virtual element method for the cahn hilliard equation with polygonal meshes in this paper we develop an evolution of the c 1 virtual elements of minimal degree for the approximation of the cahn hilliard equation the proposed method has the advantage of being conforming in h 2 and making use of a very simple set of degrees of freedom namely 3 degrees of freedom per vertex of the mesh moreover although the present method is new also on triangles it can make use of general polygonal meshes as a theoretical and practical support we prove the convergence of the semidiscrete scheme and investigate the performance of the fully discrete scheme through a set of numerical tests

extended finite element method for three dimensional crack modelling an extended finite element method x fem for three dimensional crack modelling is described a discontinuous function and the two dimensional asymptotic crack tip displacement fields are added to the finite element approximation to account for the crack using the notion of partition of unity this enables the domain to be modelled by finite elements with no explicit meshing of the crack surfaces computational geometry issues associated with the representation of the crack and the enrichment of the finite element approximation are discussed stress intensity factors sifs for planar three dimensional cracks are presented which are found to be in good agreement with benchmark solutions copyright 2000 john wiley sons ltd

gauss green cubature and moment computation over arbitrary geometries we have implemented in matlab a gauss like cubature formula over arbitrary bivariate domains with a piecewise regular boundary which is tracked by splines of maximum degree p spline curvilinear polygons the formula is exact for polynomials of degree at most 2n 1 using n cmn 2 nodes 1 c p m being the total number of points given on the boundary it does not need any decomposition of the domain but relies directly on univariate gauss legendre quadrature via green s integral formula several numerical tests are presented including computation of standard as well as orthogonal moments over a nonstandard planar region

improving the accuracy of xfem crack tip fields using higher order quadrature and statically admissible stress recovery this study is concerned with improving the accuracy of crack tip fields obtained using the extended generalized finite element method xfem first the numerical integration of the element stiffness matrices which guarantees convergence with quadrature of not only the regular nodal displacements but also additional degrees of freedom corresponding to the enrichment functions is studied as the accuracy of the stresses obtained by direct differentiation of the converged with quadrature regular nodal displacements and of the coefficients corresponding to enrichment functions is still not satisfactory a statically admissible stress recovery sar scheme is introduced sar uses basis functions which meet the equilibrium equations within the domain and the local traction conditions on the boundary and moving least squares mls to fit the stresses at sampling points e g quadrature points obtained by the xfem important parameters controlling the accuracy of crack tip fields using the xfem and sar namely the order of quadrature the number of retained terms in the crack tip asymptotic field the number of enriched layers and use of arbitrary branch functions a proper choice of the sampling points in the enriched element and the size of the domain of influence doi of mls are investigated copyright 2005 john wiley sons ltd

h text div h div and h mathbf curl h curl conforming virtual element methods in the present paper we construct virtual element spaces that are h mathrm div h div conforming and h mathbf curl h curl conforming on general polygonal and polyhedral elements these spaces can be interpreted as a generalization of well known finite elements we moreover present the basic tools needed to make use of these spaces in the approximation of partial differential equations finally we discuss the construction of exact sequences of vem spaces

on the elimination of quadrature subcells for discontinuous functions in the extended finite element method the introduction of discontinuous non differentiable functions in the extended finite element method allows to model discontinuities independent of the mesh structure however to compute the stiffness matrix of the elements intersected by the discontinuity a subdivision of the elements into quadrature subcells aligned with the discontinuity line is commonly adopted in the paper it is shown how standard gauss quadrature can be used in the elements containing the discontinuity without splitting the elements into subcells or introducing any additional approximation the technique is illustrated and developed in one two and three dimensions for crack and material discontinuity problems

quadrature schemes for arbitrary convex concave volumes and integration of weak form in enriched partition of unity methods abstract quadrature schemes are constructed based on moment fitting equations to integrate polynomials over arbitrary convex concave volumes that arise among others in enriched partition of unity finite element methods epum the building block of the scheme involves the divergence theorem of multivariable calculus which is used to integrate the base functions an efficient and robust point distribution method is proposed and the quadrature weights at the corresponding points are obtained by solving a least squares problem the method is applied initially to integrate given polynomial functions over complex volumes and further to simulate simple three dimensional fluid dynamic problems which involve very complex volumes when solved with epum accuracy of the present quadrature construction scheme is demonstrated by comparing the results with the available exact numerical solutions and efficiency of the method is proved by comparing the computational time with that of the widely used tessellation method

isogeometric analysis enhanced by the scaled boundary finite element method abstract by leveraging the information of a typical cad model in the analysis the intensive process of discretization can be circumvented this unification has led to the isogeometric analysis iga hughes et al 2005 however as the cad model provides information only of the boundary a 2d 3d stress analysis is still one major step away in this work the concepts of isogeometric analysis and the scaled boundary finite element method sbfem are combined the sbfem requires only the boundary information and hence provides a seamless integration with the cad modeling within the proposed framework the nurbs basis functions are used to discretize the unknown fields in the circumferential direction whilst analytical solution is sought in the radial direction we further extend the framework to problems with singularities and to dynamic analysis the accuracy and the convergence properties of the proposed method are demonstrated with benchmark problems in the context of linear elasticity and linear elastic fracture mechanics

symmetric gaussian quadrature formulae for tetrahedronal regions abstract quadrature formulae of degrees 2 to 6 are presented for the numerical integration of a function over tetrahedronal regions the formulae presented are of gaussian type and fully symmetric with respect to the four vertices of the tetrahedron

a cartesian grid method for solving the two dimensional streamfunction vorticity equations in irregular regions we describe a method for solving the two dimensional navier stokes equations in irregular physical domains our method is based on an underlying uniform cartesian grid and second order finite difference finite volume discretizations of the streamfunction vorticity equations geometry representing stationary solid obstacles in the flow domain is embedded in the cartesian grid and special discretizations near the embedded boundary ensure the accuracy of the solution in the cut cells along the embedded boundary we determine a distribution of vorticity sources needed to impose the no slip flow conditions this distribution appears as a right hand side term in the discretized fluid equations and so we can use fast solvers to solve the linear systems that arise to handle the advective terms we use the high resolution algorithms in clawpack we show that our stokes solver is second order accurate for steady state solutions and that our full navier stokes solver is between first and second order accurate and reproduces results from well studied benchmark problems in viscous fluid flow finally we demonstrate the robustness of our code on flow in a complex domain

an extended finite element method lagrange multiplier based approach for fluid structure interaction abstract this paper presents a new fixed grid fluid structure interaction scheme that can be applied to the interaction of most general structures with incompressible flow it is based on an extended finite element method xfem based strategy the extended eulerian fluid field and the lagrangian structural field are partitioned and iteratively coupled using lagrange multiplier techniques for non matching grids the approach allows the simulation of the interaction of thin and bulky structures exhibiting large deformations finally qualitative examples and a benchmark computation demonstrate key features and accuracy of the method

quadrature for meshless methods in this paper we discuss quadrature for meshless methods we present results of two analyses the first of these was developed in 1 and the second was developed in 2 using a simple example we show that a natural quadrature scheme that satisfies no special properties leads to severe inaccuracies in the approximate solution we then show in the first analysis that a simple correction to the stiffness matrix controls the effect of the quadrature in the second analysis we show that if the quadrature scheme satisfies green s formula then the effect of the quadrature is controlled

high order polygonal discontinuous petrov galerkin polydpg methods using ultraweak formulations abstract this work represents the first endeavor in using ultraweak formulations to implement high order polygonal finite element methods via the discontinuous petrov galerkin dpg methodology ultraweak variational formulations are nonstandard in that all the weight of the derivatives lies in the test space while most of the trial space can be chosen as copies of l 2 discretizations that have no need to be continuous across adjacent elements additionally the test spaces are broken along the mesh interfaces this allows one to construct conforming polygonal finite element methods termed here as polydpg methods by defining most spaces by restriction of a bounding triangle or box to the polygonal element the only variables that require nontrivial compatibility across elements are the so called interface or skeleton variables which can be defined directly on the element boundaries unlike other high order polygonal methods polydpg methods do not require ad hoc stabilization terms thanks to the crafted stability of the dpg methodology a proof of convergence of the form h p is provided and corroborated through several illustrative numerical examples these include polygonal meshes with n sided convex elements and with highly distorted concave elements as well as the modeling of discontinuous material properties along an arbitrary interface that cuts a uniform grid since polydpg methods have a natural a posteriori error estimator a polygonal adaptive strategy is developed and compared to standard adaptivity schemes based on constrained hanging nodes this work is also accompanied by an open source polydpg software supporting polygonal and conventional elements

hierarchical x fem for n phase flow n 2 the extended finite element method x fem has been successfully used in two phase flow problems involving a moving interface in order to simulate problems involving more than two phases the x fem has to be further extended the proposed approach is presented in the case of a quasi static stokes n phase flow and it is based on using an ordered collection of level set functions to describe the location of the phases a level set hierarchy allows describing triple junctions avoiding overlapping or voids between materials moreover an enriched solution accounting for several simultaneous phases inside one element is proposed the interpolation functions corresponding to the enriched degrees of freedom require redefining the associated ridge function accounting for all the level sets the computational implementation of this scheme involves calculating integrals in elements having several materials inside an adaptive quadrature accounting for the interfaces locations is proposed to accurately compute these integrals examples of the hierarchical x fem approach are given for a n phase stokes problem in 2 and 3 dimensions

quadrature rules for triangular and tetrahedral elements with generalized functions quadrature rules are developed for exactly integrating products of polynomials and generalized functions over triangular and tetrahedral domains these quadrature rules greatly simplify the implementation of finite element methods that involve integrals over volumes and interfaces that are not coincident with the element boundaries specifically the integrands considered here consist of a quadratic polynomial multiplied by a heaviside or dirac delta function operating on a linear polynomial this form allows for exact integration of expressions obtained from linear finite elements over domains and interfaces defined by a linear level set function exact quadrature rules are derived that involve fixed quadrature point locations with weights that depend continuously on the nodal level set values compared with methods involving explicit integration over subdomains the quadrature rules developed here accommodate degenerate interface geometries without any need for special consideration and provide analytical jacobian information describing the dependence of the integrals on the nodal level set values the accuracy of the method is demonstrated for a simple conduction problem with the neumann and robin type boundary conditions copyright 2007 john wiley sons ltd

some basic formulations of the virtual element method vem for finite deformations abstract we present a general virtual element method vem framework for finite elasticity which emphasizes two issues element level volume change volume average of the determinant of the deformation gradient and stabilization to address the former issue we provide exact evaluation of the average volume change in both 2d and 3d on properly constructed local displacement spaces for the later issue we provide a new stabilization scheme that is based on the trace of the material tangent modulus tensor which captures highly heterogeneous and localized deformations two vem formulations are presented a two field mixed and an equivalent displacement based which is free of volumetric locking convergence and accuracy of the vem formulations are verified by means of numerical examples and engineering applications are demonstrated

a new finite element approach for solving three dimensional problems using trimmed hexahedral elements summary in this paper a novel finite element approach is presented to solve three dimensional problems using trimmed hexahedral elements generated by cutting a simple block consisting of regular hexahedral elements with a computer aided design cad surface trimmed hexahedral elements which are polyhedral elements with curved faces are placed at the boundaries of finite element models and regular hexahedral elements remain in the interior regions shape functions for trimmed hexahedral elements are developed by using moving least square approximation with harmonic weight functions based on an extension of wachspress coordinates to curved faces a subdivision of polyhedral domains into tetrahedral sub domains is performed to construct shape functions for trimmed hexahedral elements and numerical integration of the weak form can be carried out consistently over the tetrahedral sub domains trimmed hexahedral elements have similar properties to conventional finite elements regarding the continuity the completeness the node element connectivity and the inter element compatibility numerical examples for three dimensional linear elastic problems with complex geometries show the efficiency and effectiveness of the present method copyright 2015 john wiley sons ltd

numerical integration over arbitrary polygonal domains based on schwarz christoffel conformal mapping this paper presents a new numerical integration technique oil arbitrary polygonal domains the polygonal domain is mapped conformally to the unit disk using schwarz christoffel mapping and a midpoint quadrature rule defined oil this unit disk is used this method eliminates the need for a two level isoparametric mapping usually required moreover the positivity of the jacobian is guaranteed numerical results presented for a few benchmark problems in the context of polygonal finite elements show that the proposed method yields accurate results

an adaptive hp version of the multilevel particle partition of unity method abstract this paper is concerned with the hp adaptive multilevel solution of second order elliptic partial differential equations using the meshfree particle partition of unity method the proposed refinement scheme automatically constructs new discretization points or particles the meshfree analogue of an adaptive h refinement and local approximation spaces with better local resolution a p refinement the refinement process is steered with the help of an a posteriori subdomain type error indicator we present results of numerical experiments in two and three space dimensions which demonstrated the overall efficiency of the proposed scheme

interface handling for three dimensional higher order xfem computations in fluid structure interaction three dimensional higher order extended finite element method xfem computations still pose challenging computational geometry problems especially for moving interfaces this paper provides a method for the localization of a higher order interface finite element fe mesh in an underlying three dimensional higher order fe mesh additionally it demonstrates how a subtetrahedralization of an intersected element can be obtained which preserves the possibly curved interface and allows therefore exact numerical integration the proposed interface algorithm collects initially a set of possibly intersecting elements by comparing their extended axis aligned bounding boxes the intersection method is applied to a highly reduced number of intersection candidates the resulting linearized interface is used as input for an elementwise constrained delaunay tetrahedralization which computes an appropriate subdivision for each intersected element the curved interface is recovered from the linearized interface in the last step the output comprises triangular integration cells representing the interface and tetrahedral integration cells for each intersected element application of the interface algorithm currently concentrates on fluid structure interaction problems on low order and higher order fe meshes which may be composed of any arbitrary element types such as hexahedra tetrahedra wedges etc nevertheless other xfem problems with explicitly given interfaces or discontinuities may be tackled in addition multiple structures and interfaces per intersected element can be handled without any additional difficulties several parallelization strategies exist depending on the desired domain decomposition approach numerical test cases including various geometrical exceptions demonstrate the accuracy robustness and efficiency of the interface handling

virtual and smoothed finite elements a connection and its application to polygonal polyhedral finite element methods summary we show both theoretically and numerically a connection between the smoothed finite element method sfem and the virtual element method and use this approach to derive stable cheap and optimally convergent polyhedral fem we show that the stiffness matrix computed with one subcell sfem is identical to the consistency term of the virtual element method irrespective of the topology of the element as long as the shape functions vary linearly on the boundary using this connection we propose a new stable approach to strain smoothing for polygonal polyhedral elements where instead of using sub triangulations we are able to use one single polygonal polyhedral subcell for each element while maintaining stability for a similar number of degrees of freedom the proposed approach is more accurate than the conventional sfem with triangular subcells the time to compute the stiffness matrix scales with the o dofs 1 1 in case of the conventional polygonal fem while it scales as o dofs 0 7 in the proposed approach the accuracy and the convergence properties of the sfem are studied with a few benchmark problems in 2d and 3d linear elasticity copyright 2015 john wiley sons ltd

the meshless finite element method a meshless method is presented which has the advantages of the good meshless methods concerning the ease of introduction of node connectivity in a bounded time of order n and the condition that the shape functions depend only on the node positions furthermore the method proposed also shares several of the advantages of the finite element method such as a the simplicity of the shape functions in a large part of the domain b c0 continuity between elements which allows the treatment of material discontinuities and c ease of introduction of the boundary conditions copyright 2003 john wiley sons ltd

the partitioned element method in computational solid mechanics abstract a finite element like approximation method is proposed for solid mechanics applications in which the elements can take essentially arbitrary polygonal form a distinguishing feature of the method herein called the partitioned element method is a partitioning of the elements into quadrature cells over which the shape functions are taken to be piecewise linear the gradient and constant values for each cell are determined by minimizing a quadratic function which represents a combined smoothness and compatibility measure linear completeness of the shape function formulation is proved robustness in the presence of element non convexity and geometric degeneracy e g nearly coincident nodes are particular goals of the method convergence for various 2d linear elasticity problems is demonstrated and results for a finite deformation elastic plastic problem are compared to those of the standard fem

virtual elements for linear elasticity problems we discuss the application of virtual elements to linear elasticity problems for both the compressible and the nearly incompressible case virtual elements are very close to mimetic finite differences see for linear elasticity l beirao da veiga m2an math model numer anal 44 2010 pp 231 250 and in particular to higher order mimetic finite differences as such they share the good features of being able to represent in an exact way certain physical properties conservation incompressibility etc and of being applicable in very general geometries the advantage of virtual elements is the ductility that easily allows high order accuracy and high order continuity

polynomial approximation of shape function gradients from element geometries a method is presented for the polynomial approximation of shape function gradients based solely on the geometry of finite element boundaries the method is founded on a least squares approach which leads to an integration scheme satisfying a necessary condition for convergence in its simplest form the method reduces to the well known uniform strain approach for finite elements the method is applicable to a broad class of problems such as connecting dissimilar meshes mesh adaptivity and transitioning and the construction of finite elements with variable topologies finite elements based on the polynomial approximations are shown to pass patch tests of various orders in contrast to standard elements higher order patch tests are passed without the need for nodes internal to element boundaries less sensitivity to volumetric locking under plane strain conditions is demonstrated through comparisons with a standard element formulation copyright 2001 john wiley sons ltd

gradient correction for polygonal and polyhedral finite elements summary previous studies have shown that the commonly used quadrature schemes for polygonal and polyhedral finite elements lead to consistency errors that persist under mesh refinement and subsequently render the approximations non convergent in this work we consider minimal perturbations to the gradient field at the element level in order to restore polynomial consistency and recover optimal convergence rates when the weak form integrals are evaluated using quadrature for finite elements of arbitrary order we state the accuracy requirements on the underlying volumetric and boundary quadrature rules and discuss the properties of the resulting corrected gradient operator we compare the proposed approach with the pseudo derivative method developed by belytschko and co workers and for linear elliptic problems with our previous remedy that involves splitting of polynomial and non polynomial of elemental energy bilinear form we present several numerical results for linear and nonlinear elliptic problems in two and three dimensions that not only confirm the recovery of optimal convergence rates but also suggest that the global error levels are close to those of approximations obtained from exact evaluation of the weak form integrals copyright 2015 john wiley sons ltd