a mesh based partition of unity method for discontinuity modeling this study explores a numerical analysis scheme the manifold method within the framework of the partition of unity method the manifold method is rooted in the discrete element method and has been used in solving discrete continuum interaction problems at its core two meshes are employed in an analysis the mathematical mesh provides the nodes for building a finite covering of the solution domain and the partition of unity functions while the physical mesh provides the domain of integration after providing a geometric interpretation and giving a construct overview the working of the method is illustrated with a number of examples

cutfem topology optimization of 3d laminar incompressible flow problems abstract this paper studies the characteristics and applicability of the cutfem approach of burman et al 2014 as the core of a robust topology optimization framework for 3d laminar incompressible flow and species transport problems at low reynolds numbers r e 200 cutfem is a methodology for discretizing partial differential equations on complex geometries by immersed boundary techniques in this study the geometry of the fluid domain is described by an explicit level set method where the parameters of a discretized level set function are defined as functions of the optimization variables the fluid behavior is modeled by the incompressible navier stokes equations species transport is modeled by an advection diffusion equation the governing equations are discretized in space by a generalized extended finite element method face oriented ghost penalty terms are added for stability reasons and to improve the conditioning of the system the boundary conditions are enforced weakly via nitsche s method the emergence of isolated volumes of fluid surrounded by solid during the optimization process leads to a singular analysis problem an auxiliary indicator field is modeled to identify these volumes and to impose a constraint on the average pressure numerical results for 3d steady state and transient problems demonstrate that the cutfem analyses are sufficiently accurate and the optimized designs agree well with results from prior studies solved in 2d or by density approaches

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

finite cover method for linear and non linear analyses of heterogeneous solids we introduce the finite cover method fcm as a generalization of the finite element method fem and extend it to analyse the linear and non linear mechanical behaviour of heterogeneous solids and structures the name fcm is actually an alias for the manifold method mm and the basic idea of the method has already been established for linear analyses of structures with homogeneous materials after reviewing the concept of physical and mathematical covers for approximating functions in the fcm we present the formulation for the static equilibrium state of a structure with arbitrary physical boundaries including material interfaces the problem essentially involves the discontinuities in strains and possibly has the discontinuities in displacement caused by interfacial debonding or rupture of material interfaces we simulate such non linear mechanical behaviour after presenting simple numerical examples that demonstrate the equivalence between the approximation capabilities of the fcm and those of the fem copyright 2003 john wiley sons ltd

adjoint based optimal control using meshfree discretizations the paper at hand presents a combination of optimal control approaches for pdes with meshless discretizations applying a classical lagrangian type particle method to optimization problems with hyperbolic constraints several adjoint based strategies differing in the sequential order of optimization and discretization of the lagrangian or eulerian problem formulation are proposed and compared the numerical results confirm the theoretically predicted independence principle of the optimization approaches and show the expected convergence behavior moreover they exemplify the superiority of meshless methods over the conventional mesh based approaches for the numerical handling and optimization of problems with time dependent geometries and freely moving boundaries

renormalization and blow up for charge one equivariant critical wave maps we prove the existence of equivariant finite time blow up solutions for the wave map problem from 2 1 s 2 of the form u t r q lambda t r mathcal r t r where u is the polar angle on the sphere q r 2 arctan r is the ground state harmonic map t t 1 and mathcal r t r is a radiative error with local energy going to zero as t 0 the number nu frac 1 2 can be prescribed arbitrarily this is accomplished by first renormalizing the blow up profile followed by a perturbative analysis

generating binary trees using rotations a new algorithm that for the first time exploits the rotational geometry of binary trees is developed in order to allow for the lexicographic generation of computer representations of these trees in average time o 1 per tree rotation codewords for these trees in average time o 1 per tree are also generated it is shown how these codewords relate to lattice paths and using this relationship that n n 1 n 2 is the average number of rotations needed to generate a binary tree on n nodes finally a necessary and sufficient condition that a codeword represent a full binary tree each node has 0 or 2 sons on n 2 m 1 nodes is given and how to contract this codeword to obtain the codeword for the binary tree on m nodes for which this full tree is the extended binary tree is shown

asymptotic stability of large energy harmonic maps under the wave map from 2d hyperbolic spaces to 2d hyperbolic spaces abstract in this paper we prove that the large energy harmonic maps from h 2 to h 2 are asymptotically stable under the wave map equation this work is the last paper in a series of works 46 45 44 which together prove the asymptotic stability for all admissible harmonic maps of arbitrary size without any symmetry assumptions

on the formation of singularities in the critical o 3 sigma model we study the phenomena of energy concentration for the critical o 3 sigma model also known as the wave map flow from 2 1 minkowski space into the sphere s 2 we establish rigorously and constructively existence of a set of smooth initial data resulting in a dynamic finite time formation of singularities the construction and analysis are done in the context of the k equivariant symmetry reduction and we restrict to maps with homotopy class k 4 the concentration mechanism we uncover is essentially due to a resonant self focusing shrinking of a corresponding harmonic map we show that the phenomenon is generic e g in certain sobolev spaces in that it persists under small perturbations of initial data while the resulting blowup is bounded by a log modified self similar asymptotic

the generation of binary trees as a numerical problem the problem of generating random uniformly distributed binarytrees is considered a closed formula that counts the number of treeshaving a left subtee with k 1 nodes k 1 2 n is found by inverting the formula random trees with n nodes are generated according to the appropriateprobability distribution determining the number of nodes in the leftand right subtrees that can be generated recursively the procedure isshown to run in time o n occupying an extra space in the order of o n

decay for the wave and schrodinger evolutions on manifolds with conical ends part ii let c n be a compact imbedded riemannian manifold of dimension d 1 and define the d 1 dimensional riemannian manifold m x r x w x with r 0 and smooth and the natural metric ds 2 1 r x 2 dx 2 r 2 x ds 2 we require that m has conical ends r x x o x 1 as x the hamiltonian flow on such manifolds always exhibits trapping dispersive estimates for the schrodinger evolution e it m and the wave evolution e it m are obtained for data of the form f x y n u x where y n are eigenfunctions of with eigenvalues u 2 n in this paper we discuss all cases d n 1 if n 0 there is the following accelerated local decay estimate with 0 1 e it m y n f l m c n m t d 1 2 w 1 f l 1 m where w x x and similarly for the wave evolution our method combines two main ingredients a a detailed scattering analysis of schrodinger operators of the form 2 v 2 1 4 2 u on the line where u is real valued and smooth with u l 0 3 l for all l 0 as and v 0 in particular we introduce the notion of a zero energy resonance for this class and derive an asymptotic expansion of the wronskian between the outgoing jost solutions as the energy tends to zero in particular the division into part i and part ii can be explained by the former being resonant at zero energy where the present paper deals with the nonresonant case b estimation of oscillatory integrals by non stationary phase

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

a hybrid extended finite element level set method for modeling phase transformations a hybrid numerical method for modelling the evolution of sharp phase interfaces on fixed grids is presented we focus attention on two dimensional solidification problems where the temperature field evolves according to classical heat conduction in two subdomains separated by a moving freezing front the enrichment strategies of the extended finite element method x fem are employed to represent the jump in the temperature gradient that governs the velocity of the phase boundary a new approach with the x fem is suggested for this class of problems whereby the partition of unity is constructed with c1 polynomials and enriched with a c0 function this approach leads to jumps in temperature gradient occurring only at the phase boundary and is shown to significantly improve estimates for the front velocity temporal derivatives of the temperature field in the vicinity of the phase front are obtained with a projection that employs discontinuous enrichment in conjunction with a finer finite difference grid the level set method is used to represent the evolution of the phase interface an iterative procedure is adopted to satisfy the constraints on the temperature field on the phase boundary the robustness and utility of the method is demonstrated with several benchmark problems of phase transformation copyright 2002 john wiley sons ltd

effects of time integration schemes on discontinuous deformation simulations using the numerical manifold method numerical stability by using certain time integration scheme is a critical issue for accurate simulation of discontinuous deformations of solids to investigate the effects of the time integration schemes on the numerical stability of the numerical manifold method the implicit time integration schemes ie the newmark the hht alpha and the wbz alpha methods and the explicit time integration algorithms ie the central difference the zhai s and chung lee methods are implemented their performance is examined by conducting transient response analysis of an elastic strip subjected to constant loading impact analysis of an elastic rod with an initial velocity and excavation analysis of jointed rock masses respectively parametric studies using different time steps are conducted for different time integration algorithms and the convergence efficiency of the open close iterations for the contact problems is also investigated it is proved that the hilber hughes taylor alpha hht alpha wood bossak zienkiewicz alpha wbz alpha zhai s and chung lee methods are more attractive in solving discontinuous deformation problems involving nonlinear contacts it is also found that the examined explicit algorithms showed higher computational efficiency compared to those implicit algorithms within acceptable computational accuracy

a rigorous and unified mass lumping scheme for higher order elements abstract in dynamic analysis with explicit time integration schemes a lumped mass matrix lmm is preferable because lmm can avoid solving the large scale simultaneous algebraic equations mathematically rigorous mass lumping schemes such as the mass lumping by nodal quadrature and the row sum technique are applicable to only linear or bilinear elements for higher order elements such as 8 node serendipity elements the diagonal scaling procedure is the only lumping method that can be recommended to generate positive definite diagonal element mass matrices unfortunately there is no mathematical theory in support of this approach this study proposes a general mass lumping scheme applicable to higher order elements where the virtual work of initial force is integrated over the problem domain that is viewed as the manifold covered by the finite element patches by a series of numerical experiments both free and forced vibration problems it is suggested that even in the implicit time integration scheme the consistent mass matrix cmm can be superseded by the proposed lmm furthermore the proposed lmm has much stronger adaptability to distorted meshes

scattering for wave maps exterior to a ball abstract we consider 1 equivariant wave maps from r t r x 3 b s 3 where b is a ball centered at 0 and b gets mapped to a fixed point on s 3 we show that 1 equivariant maps of degree zero scatter to zero irrespective of their energy for positive degrees we prove asymptotic stability of the unique harmonic maps in the energy class determined by the degree

an integrated procedure for three dimensional structural analysis with the finite cover method in this paper an integrated procedure for three dimensional 3d structural analyses with the finite cover method fcm is introduced in the pre process of this procedure the geometry of a structure is modelled by 3d cad followed by digitization to have the corresponding voxel model and then the structure is covered by a union of mathematical covers namely a mathematical mesh independently generated for approximation purposes since the mesh topology in the fcm does not need to conform to the physical boundaries of the structure the mesh can be regular and structured thus the numerical analysis procedure is free from the difficulties mesh generation typically poses and in this sense enables us to realize the mesh free analysis after formulating the fcm with interface elements for the static equilibrium state of a structure we detail the procedure of the finite cover modelling including the geometry modelling with 3d cad and the identification of the geometry covered by a regular mesh for numerical integration prior to full 3d modelling and analysis we present a simple numerical example to confirm the equivalence of the performance of the fcm and that of the standard finite element method fem finally representative numerical examples are presented to demonstrate the capabilities of the proposed analysis procedure

modified moving least squares with polynomial bases for scattered data approximation one common problem encountered in many fields is the generation of surfaces based on values at irregularly distributed nodes to tackle such problems we present a modified robust moving least squares mls method for scattered data smoothing and approximation the error functional used in the derivation of the classical mls approximation is augmented with additional terms based on the coefficients of the polynomial base functions this allows quadratic base functions to be used with the same size of the support domain as linear base functions resulting in better approximation capability the increased robustness of the modified mls method to irregular nodal distributions makes it suitable for use across many fields the analysis is supported by several univariate and bivariate examples

s r decomposition based numerical manifold method the numerical manifold method nmm surmounting the mesh dependence has successfully solved very complicated problems involving small deformation and large movement but had few applications to large deformation and large rotation problems because the false volume expansion and other issues exist in this study it is shown that the false volume expansion in nmm can be excellently resolved by using the s r strain rotation decomposition theorem which can precisely reflect complex physical behaviors occurring in the process of large rotation and large deformation the numerical methods based on the s r decomposition theorem have been limited to the static analysis of large deformations to remove this limitation a new formulation taking into account dynamical features is proposed based on the weak form of momentum conservation law under the framework of nmm the generalized a method is employed to discretize the temporal variables the updates of variables are described using the updated co moving coordinate system thus a new method named s r d based nmm is established the new formulation can be implemented in any other partition of unity based methods as well so as to improve the performances of such methods in the analysis of dynamic large deformations c 2016 elsevier b v all rights reserved

generalized nodes and high performance elements the paper concerns the development of robust and high accuracy finite elements with only corner nodes using a partition of unity based finite element approximation construction of the partition of unity based approximation is accomplished by a physically defined local function of displacements a 4 node quadratic tetrahedral element and a 3 node quadratic triangular element are developed eigenvalue analysis shows that linear dependencies in the partition of unity based finite element approximation constructed for the new elements are eliminable numerical calculations demonstrate that the new elements are robust insensitive to mesh distortion and offer quadratic accuracy while also keeping mesh generation extremely simple copyright 2005 john wiley sons ltd

a coupled element free galerkin boundary element method for stress analysis of two dimensional solids element free galerkin efg method is a newly developed meshless method for solving partial differential equations using moving least squares interpolants it is however computationally expensive for many problems a coupled efg boundary element be method is proposed in this paper to improve the solution efficiency a procedure is developed for the coupled efg be method so that the continuity and compatibility are preserved on the interface of the two domains where the efg and be methods are applied the present coupled efg be method has been coded in fortran the validity and efficiency of the efg be method are demonstrated through a number of examples it is found that the present method can take the full advantages of both efg and be methods it is very easy to implement and very flexible for computing displacements and stresses of desired accuracy in solids with or without infinite domains

frictional contact analysis of functionally graded materials with lagrange finite block method summary based on the one dimensional differential matrix derived from the lagrange series expansion the finite block method was recently developed to solve both the elasticity and transient heat conduction problems of anisotropic and functionally graded materials in this paper the formulation of the lagrange finite block method with boundary type in the strong form is presented and applied to non conforming contact problems for the functionally graded materials subjected to either static or dynamic loads the first order partial differential matrices are only needed both in the governing equations and in the neumann boundary condition by introducing the mapping technique a block of quadratic type is transformed from the cartesian coordinate of global system to the normalized coordinate with eight seeds time dependent partial differential equations are analyzed in the laplace transformed domain and the durbin s inversion method is applied to determine all the physical values in the time domain conforming and non conforming contacts are investigated by using the iterative algorithm with full load technique illustrative numerical examples are given and comparisons have been made with analytical solutions copyright 2015 john wiley sons ltd

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

a perturbation method for stochastic meshless analysis in elastostatics a stochastic meshless method is presented for solving boundary value problems in linear elasticity that involves random material properties the material property was modelled as a homogeneous random field a meshless formulation was developed to predict stochastic structural response unlike the finite element method the meshless method requires no structured mesh since only a scattered set of nodal points is required in the domain of interest there is no need for fixed connectivities between nodes in conjunction with the meshless equations classical perturbation expansions were derived to predict second moment characteristics of response numerical examples based on one and two dimensional problems are presented to examine the accuracy and convergence of the stochastic meshless method a good agreement is obtained between the results of the proposed method and monte carlo simulation since mesh generation of complex structures can be a far more time consuming and costly effort than the solution of a discrete set of equations the meshless method provides an attractive alternative to finite element method for solving stochastic mechanics problems copyright 2001 john wiley sons ltd

gap eigenvalues and asymptotic dynamics of geometric wave equations on hyperbolic space in this paper we study k equivariant wave maps from the hyperbolic plane into the 2 sphere as well as the energy critical equivariant su 2 s u 2 yang mills problem on 4 dimensional hyperbolic space the latter problem bears many similarities to a 2 equivariant wave map into a surface of revolution as in the case of 1 equivariant wave maps considered in 9 both problems admit a family of stationary solutions indexed by a parameter that determines how far the image of the map wraps around the target manifold here we show that if the image of a stationary solution is contained in a geodesically convex subset of the target then it is asymptotically stable in the energy space however for a stationary solution that covers a large enough portion of the target we prove that the schrodinger operator obtained by linearizing about such a harmonic map admits a simple positive eigenvalue in the spectral gap as there is no a priori nonlinear obstruction to asymptotic stability this gives evidence for the existence of metastable states i e solutions with anomalously slow decay rates in these simple geometric models

semilinear schrodinger flows on hyperbolic spaces scattering in h 1 we prove global well posedness and scattering in h1 for the defocusing nonlinear schrodinger equations left begin array ll i partial t delta g u u u 2 sigma u 0 phi end array right on the hyperbolic spaces mathbb h d d 2 for exponents sigma in 0 2 d 2 the main unexpected conclusion is scattering to linear solutions in the case of small exponents for comparison on euclidean spaces scattering in h1 is not known for any exponent sigma in 1 d 2 d and is known to fail for sigma in 0 1 d our main ingredients are certain noneuclidean global in time strichartz estimates and noneuclidean morawetz inequalities

generating random binary trees a survey this paper surveys algorithms for generating unbiased random binary trees there exist several linear time algorithms the best algorithms use only integers of size o n to generate binary trees on n nodes

on generation of lumped mass matrices in partition of unity based methods summary the partition of unity based methods such as the extended finite element method and the numerical manifold method are able to construct global functions that accurately reflect local behaviors through introducing locally defined basis functions beyond polynomials in the dynamic analysis of cracked bodies using an explicit time integration algorithm as a result huge difficulties arise in deriving lumped mass matrices because of the presence of those physically meaningless degrees of freedom associated with those locally defined functions observing no spatial derivatives of trial or test functions exist in the virtual work of inertia force we approximate the virtual work of inertia force in a coarser manner than the virtual work of stresses where we inversely utilize the from local to global skill the proposed lumped mass matrix is strictly diagonal and can yield the results in agreement with the consistent mass matrix but has more excellent dynamic property than the latter meanwhile the critical time step of the numerical manifold method equipped with an explicit time integration scheme and the proposed mass lumping scheme does not decrease even if the crack in study approaches the mesh nodes a very excellent dynamic property copyright 2017 john wiley sons ltd

treatment of material discontinuity in the element free galerkin method abstract a method is presented for application of the element free galerkin method efg to solid mechanics problems containing material discontinuities by using the efg method the trial and test functions for the weak form are constructed with moving least square interpolants in each material domain additional constraints are imposed at the variational level to ensure satisfaction of interface conditions as a result only nodal data no elemental connectivity is needed given c 1 continuity of the interpolant weighting function the dependent variable and its gradient are continuous in each material domain two linear elastic bi material examples are presented to illustrate the effectiveness of the method

nonlinear waves on 3d hyperbolic space in this article global in time dispersive estimates and strichartz estimates are explored for the wave equation on three dimensional hyperbolic space due to the negative curvature extra dispersion is noted as compared to the euclidean case and a wider range of strichartz estimates is proved using these small data global existence to semilinear wave equations is shown for a range of powers that is broader than that known for euclidean space

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 linear time algorithm for binary tree sequences transformation using left arm and right arm rotations in this paper we consider a transformation on binary trees using new types of rotations each of the newly proposed rotations is permitted only at nodes on the left arm or the fight arm of a tree consequently we develop a linear time algorithm with at most n 1 rotations for converting weight sequences between any two binary trees in particular from an analysis of aggregate method for a sequence of rotations each rotation of the proposed algorithm can be performed in a constant amortized time next we show that a specific directed rooted tree called rotation tree can be constructed using one of the new type rotations as a by product a naive algorithm for enumerating weight sequences of binary trees in lexicographic order can be implemented by traversing the rotation tree

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

new strategies for some issues of numerical manifold method in simulation of crack propagation summary aiming to solve in a unified way continuous and discontinuous problems in geotechnical engineering the numerical manifold method introduces two covers namely the mathematical cover and the physical cover in order to reach the goal some issues in the simulation of crack propagation have to be solved among which are the four issues to be treated in this study 1 to reduce the rank deficiency induced by high degree polynomials as local approximation a new variational principle is formulated which suppresses the gradient dependent dofs 2 to evaluate the integrals with singularity of 1 r a new numerical quadrature scheme is developed which is simpler but more efficient than the existing duffy transformation 3 to analyze kinked cracks a sign convention for argument in the polar system at the crack tip is specified which leads to more accurate results in a simpler way than the existing mapping technique and 4 to demonstrate the mesh independency of numerical manifold method in handling strong singularity a mesh deployment scheme is advised which can reproduce all singular locations of the crack with regard to the mesh corresponding to the four issues typical examples are given to demonstrate the effectiveness of the proposed schemes copyright 2013 john wiley sons ltd

efg approximation with discontinuous derivatives a technique for incorporating discontinuities in derivatives into meshless methods is presented the technique enriches the approximation by adding special shape functions that contain discontinuities in derivatives the special shape functions have compact support which results in banded matrix equations the technique is described in element free galerkin context but is easily applicable to other meshless methods and projections numerical results for problems in one and two dimensions are reported 1998 john wiley sons ltd

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

stability of stationary equivariant wave maps from the hyperbolic plane in this paper we initiate the study of equivariant wave maps from 2d hyperbolic space bbb h 2 into rotationally symmetric surfaces this problem exhibits markedly different phenomena than its euclidean counterpart due to the exponential volume growth of concentric geodesic spheres on the domain in particular when the target is bbb s 2 we find a family of equivariant harmonic maps bbb h 2 to bbb s 2 indexed by a parameter that measures how far the image of each harmonic map wraps around the sphere these maps have energies taking all values between zero and the energy of the unique co rotational euclidean harmonic map q rm euc from bbb r 2 to bbb s 2 given by stereographic projection we prove that the harmonic maps are asymptotically stable for values of the parameter smaller than a threshold that is large enough to allow for maps that wrap more than halfway around the sphere indeed we prove strichartz estimates for the operator obtained by linearizing around such a harmonic map however for harmonic maps with energies approaching the euclidean energy of q rm euc asymptotic stability via a perturbative argument based on strichartz estimates is precluded by the existence of gap eigenvalues in the spectrum of the linearized operator when the target is bbb h 2 we find a continuous family of asymptotically stable equivariant harmonic maps bbb h 2 to bbb h 2 with arbitrarily small and arbitrarily large energies this stands in sharp contrast to the corresponding problem on euclidean space where all finite energy solutions scatter to zero as time tends to infinity

harmonic enrichment functions a unified treatment of multiple intersecting and branched cracks in the extended finite element method a unifying procedure to numerically compute enrichment functions for elastic fracture problems with the extended finite element method is presented within each element that is intersected by a crack the enrichment function for the crack is obtained via the solution of the laplace equation with dirichlet and vanishing neumann boundary conditions a single algorithm emanates for the enrichment field for multiple cracks as well as intersecting and branched cracks without recourse to special cases which provides flexibility over the existing approaches in which each case is treated separately numerical integration is rendered to be simple there is no need for partitioning of the finite elements into conforming subdivisions for the integration of discontinuous or weakly singular kernels stress intensity factor computations for different crack configurations are presented to demonstrate the accuracy and versatility of the proposed technique copyright 2010 john wiley sons ltd

a fictitious domain method for dirichlet problem and applications abstract in this article we discuss the solution of the dirichlet problem for a class of elliptic operators by a lagrange multiplier fictitious domain method this approach allows the use of regular grids and therefore of fast specialized solvers for problems on complicated geometries the resulting saddle point system can be solved by an uzawa conjugate gradient algorithm in the case of two dimensional problems a quasi optimal preconditioner has been found by fourier analysis and numerical experiments confirm its nice scaling properties the resulting methodology is applied to a nonlinear time dependent problem namely the flow of a viscous plastic medium in a cylindrical pipe showing the potential of this methodology for some classes of nonlinear problems

an immersed boundary method for fluids using the xfem and the hydrodynamic boltzmann transport equation abstract this paper presents a stabilized finite element formulation of the hydrodynamic boltzmann transport equation hbte to predict nearly incompressible fluid flow the hbte is discretized with hermite polynomials in the velocity variable and a streamline upwind petrov galerkin formulation is used to discretize the spatial variable a nonlinear stabilization scheme is presented from which a simple linear stabilization scheme is constructed in contrast to the navier stokes ns equations the hbte is a first order equation and allows for conveniently enforcing dirichlet conditions along immersed boundaries a simple and efficient formulation for enforcing dirichlet boundary conditions is presented and its accuracy is studied for immersed boundaries captured by the extended finite element method xfem numerical experiments indicate that both the linear and non linear stabilization methods are sufficiently accurate and stable but the linear formulation reduces the computational cost significantly the accuracy of enforcing boundary conditions is satisfactory and shows second order convergence as the mesh is refined augmenting the boundary condition formulation with a penalty term increases the accuracy of enforcing the boundary condition constraints but may degrade the accuracy of the global solution comparisons with results of a single relaxation time lattice boltzmann method show that the proposed finite element method features greater robustness and lesser dependence of the computational costs on the level of mesh refinement

a survey of combinatorial gray codes the term combinatorial gray code was introduced in 1980 to refer to any method for generating combinatorial objects so that successive objects differ in some prespecified small way this notion generalizes the classical binary reflected gray code scheme for listing n bit binary numbers so that successive numbers differ in exactly one bit position as well as work in the 1960s and 1970s on minimal change listings for other combinatorial families including permutations and combinations the area of combinatorial gray codes was popularized by herbert wilf in his invited address at the siam conference on discrete mathematics in 1988 and his subsequent siam monograph combinatorial algorithms an update 1989 in which he posed some open problems and variations on the theme this resulted in much recent activity in the area and most of the problems posed by wilf are now solved in this paper we survey the area of combinatorial gray codes describe recent results variations and trends and highlight some open problems

complementarity problem arising from static growth of multiple cracks and mls based numerical manifold method abstract the full response of a brittle structure containing multiple cracks to loading under the servo control is of vital importance in the evaluation of properties of the structure during crack growth the fracture toughness condition at all the crack tips as well as the equilibrium condition should be obeyed leading to a nonlinear complementarity problem ncp the vector valued function in the ncp depends implicitly on the cracking increments which in turn determine the stress field the stress field can be obtained through solving a mixed variational problem since the degrees of freedom in the discrete variational problem vary with cracking not only in magnitude but also in number the jacobian matrix of the ncp is hard to compute and it cannot be expected to be solved by the newton methods that involve the calculation of jacobian matrices therefore a well scaled projection contraction algorithm is designed the proposed procedure is able to simulate growth of multiple cracks in a natural way allowing the crack tips to stop anywhere with no sensitivity to node configuration or cracking increments through the analysis of some examples that have been widely tested many interesting and profound phenomena are found which have never been revealed

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

the numerical manifold method for exterior problems abstract the numerical manifold method nmm a galerkin type numerical method has been successful in the solution of problems with finite definition domains yet it has never been applied to problems with unbounded domains or exterior problems this study aims to fill the big gap by constructing infinite patches together with the finite patches to cover the unbounded domain the local approximations of infinite patches can take the asymptotic estimations of the solutions at infinity which are available for all those well established boundary value problems compared with the infinite element methods in the finite element method fem the construction of the trial functions by nmm is more elegant in theory and more systematical in methodology resulting in more accurate solutions some typical examples in potential and half space elasticity problems are investigated to illustrate the applicability and accuracy of the proposed method

prediction of rank deficiency in partition of unity based methods with plane triangular or quadrilateral meshes in the partition of unity pu based methods the global approximation is built by multiplying a partition of unity by local approximations within this framework high order approximations are achieved by directly adopting high order polynomials as local approximations and therefore nodes along sides or inside elements which are usually adopted in the conventional finite element methods are no more required however the pu based approximation constructed in this way may suffer from rank deficiency due to the linear dependence of the global degrees of freedom in this paper the origin of the rank deficiency in the pu based approximation space is first dissected at an element level and then an approach to predict the rank deficiency for a mesh is proposed together with the principle of the increase of rank deficiency finally examples are investigated to validate the present approach the current work indicates such a fact that the rank deficiency is an unrelated issue to the nullity of the global matrix it can be resolved in its own manner

a new way to treat material discontinuities in the numerical manifold method abstract a new technique to describe the material discontinuities within the numerical manifold method framework is presented a new concept of weak discontinuous physical covers is introduced special local approximations which are continuous in the approximation but discontinuous in its first derivative are tailored for weak discontinuous physical covers this new method takes full advantage of the partition of unity several typical examples are presented results have shown the accuracy and convenience of the proposed method in modeling material interfaces especially composite materials with multiple holes inclusions

tensor based methods for numerical homogenization from high resolution images we present a complete numerical strategy based on tensor approximation techniques for the solution of numerical homogenization problems with geometrical data coming from high resolution images we first introduce specific numerical treatments for the translation of image based homogenization problems into a tensor framework it includes the tensor approximations in suitable tensor formats of fields of material properties or indicator functions of multiple material phases recovered from segmented images we then introduce some variants of proper generalized decomposition pgd methods for the construction of tensor decompositions in different tensor formats of the solution of boundary value problems a new definition of pgd is introduced which allows the progressive construction of a tucker decomposition of the solution this tensor format is well adapted to the present application and improves convergence properties of tensor decompositions finally we use a dual based error estimator on quantities of interest which was recently introduced in the context of pgd we exhibit its specificities when it is used for assessing the error on the homogenized properties of the heterogeneous material we also provide a complete goal oriented adaptive strategy for the progressive construction of tensor decompositions of primal and dual solutions yielding to predictions of homogenized quantities with a prescribed accuracy

on the smoothed finite element method recently liu et al proposed the smoothed finite element method by using the non mapped shape functions and then introducing the strain smoothing operator when evaluating the element stiffness in the framework of the finite element method however the theories and examples by liu et al are not sufficient for general quadrilateral elements this paper shows that the non mapped shape functions used in the smoothed finite element have disadvantages in existence linearity non negativity and patch test copyright 2008 john wiley sons ltd

non planar 3d crack growth by the extended finite element and level sets part i mechanical model a methodology for solving three dimensional crack problems with geometries that are independent of the mesh is described the method is based on the extended finite element method in which the crack discontinuity is introduced as a heaviside step function via a partition of unity in addition branch functions are introduced for all elements containing the crack front the branch functions include asymptotic near tip fields that improve the accuracy of the method the crack geometry is described by two signed distance functions which in turn can be defined by nodal values consequently no explicit representation of the crack is needed examples for three dimensional elastostatic problems are given and compared to analytic and benchmark solutions the method is readily extendable to inelastic fracture problems copyright 2002 john wiley sons ltd

resonances radiation damping and instabilitym in hamiltonian nonlinear wave equations we consider a class of nonlinear klein gordon equations which are hamiltonian and are perturbations of linear dispersive equations the unperturbed dynamical system has a bound state a spatially localized and time periodic solution we show that for ge neric nonlinear hamiltonian perturbations all small amplitude so lutions decay to zero as time tends to infinity at an anomalously slow rate in particular spatially localized and time periodic solutions of the linear problem are destroyed by generic nonlinear hamiltonian perturbations via slow radiation of energy to infinity these solutions can therefore be thought of as metastable states the main mecha nism is a nonlinear resonant interaction of bound states eigenfunc tions and radiation continuous spectral modes leading to energy transfer from the discrete to continuum modes this is in contrast to the kam theory in which appropriate nonresonance conditions im ply the persistence of invariant tori a hypothesis ensuring that such a resonance takes place is a nonlinear analogue of the fermi golden rule arising in the theory of resonances in quantum mechanics the techniques used involve i a time dependent method developed by the authors for the treatment of the quantum resonance problem and perturbations of embedded eigenvalues ii a generalization of the hamiltonian normal form appropriate for infinite dimensional dis persive systems and iii ideas from scattering theory the arguments are quite general and we expect them to apply to a large class of

allman s triangle rotational dof and partition of unity a simplification of the 1984 allman triangle one of historically most important elements with rotational dofs is presented it is found that this old element takes a typical form of the partition of unity approximation the allman s rotation presented in the partition of unity form offers merits and convenience in formulation and practical applications the stiffness matrix of the 1984 allman triangle which is originally computed from the linear strain triangular element the 6 nodes quadratic triangle can be obtained instead in a cheaper way from that of the constant strain triangular element the constraint of the rotational terms during essential boundary treatment which remains equivocal and ambiguous is understood to be mandatory the partition of unity notion enables a straightforward extension of the allman s rotational dof to meshfree approximations in numerical examples we discuss suppression of spurious zero energy modes and patch tests standard benchmarks are carried out to assess performance of the newly formulated triangle and a meshfree approximation with the rotational dofs copyright 2006 john wiley sons ltd

two scale diffusion deformation coupling model for material deterioration involving micro crack propagation in this paper we introduce a two scale diffusion deformation coupled model that represents the aging material deterioration of two phase materials involving micro crack propagations the mathematical homogenization method is applied to relate the micro and macroscopic field variables and a weak coupling solution method is employed to solve the two way coupling phenomena between the diffusion of scalar fields and the deformation of quasi brittle solids the macroscopic mechanical behavior represented by the derived two scale two way coupled model reveals material nonlinearity due to micro scale cracking induced by the scalar field induced deformation which can be simulated by the finite cover method after verifying the fundamental validity of the proposed model and the analysis method we perform a simple numerical example to demonstrate their ability to predict aging material deterioration copyright 2010 john wiley sons ltd

numerical manifold space of hermitian form and application to kirchhoff s thin plate problems summary for second order problems where the behavior is described by second order partial differential equations the numerical manifold method nmm has gained great success because of difficulties in the construction of the h 2 regular lagrangian partition of unity subordinate to the finite element cover however few applications of the nmm have been found to fourth order problems such as kirchhoff s thin plate problems parallel to the finite element methods this study constructs the numerical manifold space of the hermitian form to solve fourth order problems from the minimum potential principle meanwhile the mixed primal formulation and the penalized formulation fitted to the nmm for kirchhoff s thin plate problems are derived the typical examples indicate that by the proposed procedures even those earliest developed elements in the finite element history such as zienkiewicz s plate element regain their vigor copyright 2013 john wiley sons ltd

development of contact algorithm for three dimensional numerical manifold method summary this paper customizes a contact detection and enforcing scheme to fit the three dimensional 3 d numerical manifold method nmm a hierarchical contact system is established for efficient contact detection the mathematical mesh a unique component in the nmm is utilized for global searching of possible contact blocks and elements followed by the local searching to identify primitive hierarchies all the potential contact pairs are then transformed into one of the two essential entrance modes point to plane and crossing lines modes among which real contact pairs are detected through a unified formula the penalty method is selected to enforce the contact constraints and a general contact solution procedure in the 3 d nmm is established because of the implicit framework an open close iteration is performed within each time step to determine the correct number of contact pairs among multi bodies and to achieve complete convergence of imposed contact force at corresponding position the proposed contact algorithm extensively utilizes most of the original components of the nmm namely the mathematical mesh cells and the manifold elements as well as the external components associated with contacts such as the contact body the contact facet and the contact vertex in particular the utilization of two mutually approaching mathematical cells is efficient in detecting contacting territory which makes this method particularly effective for both convex and non convex bodies the validity and accuracy of the proposed contact algorithm are verified and demonstrated through three benchmark problems copyright 2013 john wiley sons ltd

critical assessment of different mass lumping schemes for higher order serendipity finite elements abstract an efficient and robust finite element based transient analysis of structures is important in many engineering applications in this context a diagonal or lumped mass matrix is an essential prerequisite in the last decades several methods to construct such a mass matrix have been proposed and therefore a comprehensive and unbiased evaluation of these approaches is needed in the present article we consequently investigate established mass lumping schemes such as the row sum method and the diagonal scaling method as well as the recently proposed manifold based method furthermore the latter approach is rigorously extended to higher order serendipity finite elements in two and three dimensions note that the manifold based method and the diagonal scaling method are very general approaches in the sense that they are applicable to arbitrary nodal based shape functions with arbitrary polynomial orders while guaranteeing the positive definiteness of the lumped mass matrix with respect to the row sum method the positivity is not ensured and negative or zero diagonal components might be computed consequently a detailed analysis of the influence and performance of these mass lumping schemes on the numerical results is conducted to this end several dynamic benchmark models including modal harmonic and transient analyses are selected showing that only suboptimal rates of convergence are attained if serendipity finite elements based on a diagonal mass matrix are employed

non planar 3d crack growth by the extended finite element and level sets part ii level set update we present a level set method for treating the growth of non planar three dimensional cracks the crack is defined by two almost orthogonal level sets signed distance functions one of them describes the crack as a two dimensional surface in a three dimensional space and the second is used to describe the one dimensional crack front which is the intersection of the two level sets a hamilton jacobi equation is used to update the level sets a velocity extension is developed that preserves the old crack surface and can accurately generate the growing surface the technique is coupled with the extended finite element method which approximates the displacement field with a discontinuous partition of unity this displacement field is constructed directly in terms of the level sets so the discretization by finite elements requires no explicit representation of the crack surface numerical experiments show the robustness of the method both in accuracy and in treating cracks with significant changes in topology copyright 2002 john wiley sons ltd

the extended finite element method xfem for solidification problems an enriched finite element method for the multi dimensional stefan problems is presented in this method the standard finite element basis is enriched with a discontinuity in the derivative of the temperature normal to the interface the approximation can then represent the phase interface and the associated discontinuity in the temperature gradient within an element the phase interface can be evolved without re meshing or the use of artificial heat capacity techniques the interface is described by a level set function that is updated by a stabilized finite element scheme several examples are solved by the proposed method to demonstrate the accuracy and robustness of the method copyright 2001 john wiley sons ltd

combination of the meshless finite difference approach with the monte carlo random walk technique for solution of elliptic problems abstract this paper proposes a stochastic approach for the fast and effective numerical analysis of the second order elliptic differential equations it is based upon the well known monte carlo mc method with a random walk rw technique carried out on the grid of points this method allows for accurate estimation of the solution of the differential equation at selected point s of the domain and or its boundary it extends the standard formulation of the monte carlo random walk mc rw approach by means of its appropriate combination with the meshless version of the finite difference method in this manner the proposed approach may deal with elliptic equations in more general non homogeneous form as well as boundary conditions of both essential and natural types moreover arbitrarily irregular clouds of nodes may be used with no a priori imposed nodes structure therefore the meshless mc rw approach may be applied to the significantly wider class of problems with more complex geometry this concept was examined on variety of 2d boundary value problems selected numerical results are presented and discussed a simple matlab code is included as well

time decay for solutions of schrodinger equations with rough and time dependent potentials in this paper we establish dispersive estimates for solutions to the linear schrodinger equation in three dimensions frac 1 i partial t psi triangle psi v psi 0 qquad psi s f 0 1 where v t x is a time dependent potential that satisfies the conditions sup t v t cdot l frac 3 2 mathbb r 3 sup x in mathbb r 3 int mathbb r 3 int infty infty frac v hat tau x x y d tau dy c 0 here c0 is some small constant and v hat tau x denotes the fourier transform with respect to the first variable we show that under these conditions 0 1 admits solutions l t l2x 3 l2t l6x 3 for any f l2 3 satisfying the dispersive inequality psi t infty le c t s frac32 f 1 text for all times t s 0 2 for the case of time independent potentials v x 0 2 remains true if int mathbb r 6 frac v x v y x y 2 dxdy 4 pi 2 text and v mathcal k sup x in mathbb r 3 int mathbb r 3 frac v y x y dy 4 pi we also establish the dispersive estimate with an e loss for large energies provided v mathcal k v 2 infty

a coupled interpolating meshfree method for computing sound radiation in infinite domain summary in this paper the coupling of the improved interpolating element free galerkin iiefg method and the variable order infinite acoustic wave envelope element wee method is studied a coupled iiefg wee method for computing sound radiation is proposed to make use of their advantages while evading their disadvantages the coupling is achieved by constructing the hybrid shape function of continuity and compatibility on the interface between the iiefg and wee domains in the iiefg domain the improved interpolating moving least squares iimls method is employed to form the shape functions satisfying the kronecker delta condition while nonsingular weight functions can be used the impacts of the size of the influence domain and the shape parameter on the performance of this coupled method are investigated the numerical results show that the coupled iiefg wee method can take full advantage of both the iiefg and wee methods and that it not only can achieve higher accuracy but also has a faster convergence speed than the conventional method of the finite element coupled with the wee the experimental results show that the method is very flexible for acoustic radiation prediction in the infinite domain this article is protected by copyright all rights reserved

a novel meshless scheme for solving surface integral equations with flat integral domains numerical solutions for electromagnetic em integral equations rely on the discretization of integral domains and the use of meshes for geometric description meshing geometries is very tedious especially for complicated structures with many details tiny parts and geometric discontinuities corners or edges and remeshing could be required in many scenarios to reduce the costs of generating quality meshes meshless or mesh free methods were developed and they have been extensively studied in mechanical engineering though there are less obvious interests in em community the meshless methods employ discrete nodes to replace meshes in the description of geometries but the background meshes for integrations are still needed traditionally in this work we first address the traditional meshless scheme for solving em integral equations based on the moving least square mls approximation for unknown currents and the use of background meshes for integrations and then develop a novel meshless scheme by applying the green s lemma to the em surface integral equations with flat domains the novel scheme transforms a surface integral over a flat domain into a line integral along its boundaries when excluding a singular patch in the domain since only the domain boundaries are discretized and no background meshes are needed the scheme is truly meshless numerical examples for em scattering by flat surface objects are presented to demonstrate the effectiveness of the novel scheme

interpolating element free galerkin method for the regularized long wave equation and its error analysis abstract combining the simplified interpolating moving least squares imls method with the galerkin weak form an interpolating element free galerkin iefg method is presented for the numerical study of the regularized long wave rlw equation the trial function in the imls method has one less unknown coefficient than in the traditional mls approximation and the shape functions satisfy the interpolating property at nodes then the iefg method can apply the essential boundary conditions naturally and directly and has high computational accuracy and then this paper studies the error estimates of the iefg method for rlw equation the error estimate of the semi discrete iefg method for the rlw equation on the influence domain radius is first studied and then the convergence rate of the full discrete scheme of the method on the influence domain radius and time step is studied the theoretical results show that the iefg method for rlw equation has good convergence rate for the purpose of demonstration three selected numerical examples are given to demonstrate the method and the mathematical theory of this paper