on the kohn sham equations with periodic background potentials we study the question of existence and uniqueness for the finite temperature kohn sham equations for finite volumes a unique soluion is shown to exists if the effective potential satisfies a set of general conditions and the coupling constant is smaller than a certain value for periodic background potentials this value is proven to be volume independent in this case the finite volume solutions are shown to converge as the thermodynamic limit is considered the local density approximation is shown to satisfy the general conditions mentioned above

viscoplasticity using peridynamics peridynamics is a continuum reformulation of the standard theory of solid mechanics unlike the partial differential equations of the standard theory the basic equations of peridynamics are applicable even when cracks and other singularities appear in the deformation field the assumptions in the original peridynamic theory resulted in severe restrictions on the types of material response that could be modeled including a limitation on the poisson ratio recent theoretical developments have shown promise for overcoming these limitations but have not previously incorporated rate dependence and have not been demonstrated in realistic applications in this paper a new method for implementing a rate dependent plastic material within a peridynamic numerical model is proposed and demonstrated the resulting material model implementation is fitted to rate dependent test data on 6061 t6 aluminum alloy it is shown that with this material model the peridynamic method accurately reproduces the experimental results for taylor impact tests over a wide range of impact velocities the resulting model retains the advantages of the peridynamic formulation regarding discontinuities while allowing greater generality in material response than was previously possible copyright 2009 john wiley sons ltd

a discussion of stress rates in finite deformation problems abstract it has recently been shown that the finite elastic plastic solution of the simple shear problem exhibits an oscillatory stress response when kinematic hardening is employed while the solution for isotropic hardening gives a monotonically increasing stress this paper analyzes this response on the basis of continuum mechanical descriptions of the problem three objective stress rates are recalled and spatial descriptions of plasticity at finite deformation are reviewed for the usual generalization of the infinitesimal theory as well as a theory based on an invariant measure of true stress in light of the equations for the evolution of the yield surface the hypoelastic solution to the simple shear problem for each of the three stress rates is presented it is shown that the use of the jaumann rate in the generalization of the infinitesimal theory leads to an oscillation in the evolution of the yield surface in simple shear which is explained on the basis of the hypoelastic solution an alternative theory which makes use of the polar decomposition predicts a monotonically increasing shear stress

an axisymmetric ordinary state based peridynamic model for linear elastic solids abstract in this study a new axisymmetric ordinary state based peridynamic pd model for axisymmetric problems of linear elastic solids is presented a fracture criterion based on the pd bond energy density is proposed adaptive dynamic relaxation adr method is adopted to obtain equilibrium solutions and a viable fictitious density of the model is derived and proven to be valid for implementation in the adr method performance and validity of the proposed axisymmetric pd model are demonstrated by three kinds of numerical problems i e compression tests pull out deformation and indentation fracture in the compression tests with focus on constant strain deformations the displacements predicted by the present model are compared with classical analytical solutions and the results show good agreements both m convergence and convergence behaviors under four influence functions are investigated and discussed based on a thorough error analysis in different compression cases the recovery of the poisson s ratios in the model is tested in detail as well the capability of capturing general non uniform axisymmetric deformation by the proposed model is verified in the pull out analysis the equilibrium displacement fields predicted by the present model agree very well with those by the finite element method the peridynamic evaluation of strains and stresses also show good match with the finite element ones the proposed fracture criterion is validated by the third example of indentation cracking which is compared with the available experimental data the model developed can effectively be used to analyze axisymmetric problems of linear elastic solids in the framework of the ordinary state based peridynamics

non ordinary state based peridynamic analysis of stationary crack problems an implicit implementation of the non ordinary state based peridynamics formulation for quasi static linearly elastic solids is presented emphasis is placed on assessing the accuracy of the numerical scheme in the vicinity of the crack front and other sources of stress concentration we also present a comparative study of methods used to control the zero energy modes inherent in the nonlocal definition of the strain tensor and reduce the spurious oscillations present particularly in regions of high strain gradients the accuracy of the peridynamics scheme including the impact of the lattice spacing and configuration is assessed by performing an analysis of the near tip stress and displacement fields k fields for 2d problems the manuscript also summarizes a verification study based on the classical 3d penny shaped crack problem and a validation study of a 3d notched fracture specimen

a kohn sham system at zero temperature a one dimensional kohn sham system for spin particles is considered which effectively describes semiconductor nanostructures and which is investigated at zero temperature we prove the existence of solutions and derive a priori estimates for this purpose we find estimates for eigenvalues of the schrodinger operator with effective kohn sham potential and obtain w1 2 bounds of the associated particle density operator afterwards compactness and continuity results allow us to apply schauder s fixed point theorem in the case of vanishing exchange correlation potential uniqueness is shown by monotonicity arguments finally we investigate the behavior of the system if the temperature approaches zero

a simple level set method for solving stefan problems a simple level set method for solving stefan problems is presented this method can be applied to problems involving dendritic solidification our method consists of an implicit finite difference scheme for solving the heat equation and a level set approach for capturing the front between solid and liquid phases of a pure substance our method is accurate with respect to some exact solutions of the stefan problem results indicate that this method can handle topology changes and complicated interfacial shapes and that it can numerically simulate many of the physical features of dendritic solidification

a mesh grading material point method and its parallelization for problems with localized extreme deformation abstract as a kind of meshless method material point method mpm applies an eulerian background grid served as a finite element mesh in each time step and therefore its accuracy and efficiency are mainly dependent on the cell size setting of background grid however the conventional mpm commonly uses a regular background grid with uniform cells which is not apposite for localized extreme deformation problems from the view point of computation efficiency where in fact a local refined background grid is preferable hence a mesh grading material point method mgmpm is proposed here for such problems to supply mpm with the ability for local refinement simulation the edge displacement continuity associated with mesh grading is embedded in the nodal shape functions besides the truss element is incorporated into mgmpm to model the steel reinforcement bars in reinforced concrete impacting problems based on our previous work furthermore the proposed method is parallelized using openmp open multi processing to take advantage of pc power with multi core and hyper threading technologies for large scale engineering problems where both loop level parallelism and code block parallelism are used several numerical examples including stress wave propagation taylor bar impact and penetration problems are studied which show that the efficiency of mgmpm is much higher than that of conventional mpm and with lower memory requirement

constant mean curvature graphs on exterior domains of the hyperbolic plane we prove an existence result for non rotational constant mean curvature ends in mathbb h 2 times mathbb r where mathbb h 2 is the hyperbolic real plane the value of the curvature is h in big 0 frac 1 2 big we use schauder theory and a continuity method for solutions of the prescribed mean curvature equation on exterior domains of mathbb h 2 we also prove a fine property of the asymptotic behavior of the rotational ends introduced by sa earp and toubiana

numerical experiments on the convergence properties of state based peridynamic laws and influence functions in two dimensional problems abstract peridynamics is widely used as the theoretical basis for numerical studies of fracture evolution propagation and behavior while the theory has been shown to converge to continuum mechanics in the theoretical limit its behavior as a discrete numerical approximation with respect to classic problems has not been shown in this study we use standard analytical solutions to thoroughly test the numerical accuracy and rate of convergence of the spatial discretization obtained by peridynamics we analyze the accuracy and rate of convergence of three different peridynamic constitutive responses of these two involve a state based dilation and the third is based on the estimation of the deformation gradient additionally we study the choice of the peridynamic influence function in each of the constitutive responses the peridynamic materials are solved in the linear elastic regime by solving a linear system of equations obtained by symbolically differentiating the force states we test the methods against standard constant strain solutions for uniaxial compression isotropic compression and simple shear we also apply the methods to a finite material with a pressurized thin crack using the westergaard s solution method to obtain an analytical displacement field for comparison the two dilation based peridynamic constitutive responses are found to only converge to one of the constant strain solutions while the deformation gradient based law converges in all cases with an appropriate choice of the influence functions we show that a cubic influence function is the best choice of those considered in all methods only the deformation gradient based model converges for all three linear deformation problems but is less accurate than the dilation based models for the thin crack problem because of instabilities we demonstrate an ad hoc smoothing technique based on the influence function that is able to alleviate these instabilities and improve the accuracy of the deformation gradient based model

dynamic crack propagation based on loss of hyperbolicity and a new discontinuous enrichment summary a methodology is developed for switching from a continuum to a discrete discontinuity where the governing partial dierential equation loses hyperbolicity the approach is limited to rate independent materials so that the transition occurs on a set of measure zero the discrete discontinuity is treated by the extendednite element method xfem whereby arbitrary discontinuities can be incorporated in the model without remeshing loss of hyperbolicity is tracked by a hyperbolicity indicator that enables both the crack speed and crack direction to be determined for a given material model a new method was developed for the case when the discontinuity ends within an element it facilitates the modelling of crack tips that occur within an element in a dynamic setting the method is applied to several dynamic crack growth problems including the branching of cracks copyright 2003 john wiley sons ltd

on nonnegatively curved hypersurfaces in hyperbolic space author s bonini v ma s qing j abstract 2018 springer verlag gmbh germany part of springer nature in this paper we prove a conjecture of alexander and currier that states except for covering maps of equidistant surfaces in hyperbolic 3 space a complete nonnegatively curved immersed hypersurface in hyperbolic space is necessarily properly embedded

higher order xfem for curved strong and weak discontinuities the extended finite element method xfem enables the accurate approximation of solutions with jumps or kinks within elements optimal convergence rates have frequently been achieved for linear elements and piecewise planar interfaces higher order convergence for arbitrary curved interfaces relies on two major issues i an accurate quadrature of the galerkin weak form for the cut elements and ii a careful formulation of the enrichment which should preclude any problems in the blending elements for i we employ a strategy of subdividing the elements into subcells with only one curved side reference elements that are higher order on only one side are then used to map the integration points to the real element for ii we find that enrichments for strong discontinuities are easily extended to higher order accuracy in contrast problems in blending elements may hinder optimal convergence for weak discontinuities different formulations are investigated including the corrected xfem numerical results for several test cases involving strong or weak curved discontinuities are presented quadratic and cubic approximations are investigated optimal convergence rates are achieved using the standard xfem for the case of a strong discontinuity close to optimal convergence rates for the case of a weak discontinuity are achieved using the corrected xfem copyright 2009 john wiley sons ltd

reproducing kernel triangular b spline based fem for solving pdes abstract we propose a reproducing kernel triangular b spline based finite element method fem as an improvement to the conventional triangular b spline element for solving partial differential equations pdes in the latter unexpected errors can occur throughout the analysis domain mainly due to the excessive flexibilities in defining the b spline the performance therefore becomes unstable and cannot be controlled in a desirable way to address this issue the proposed improvement adopts the reproducing kernel approximation in the calculation of b spline kernel function three types of pde problems are tested to validate the present element and compare against the conventional triangular b spline it has been shown that the improved triangular b spline satisfies the partition of unity condition even for extreme conditions including corners and holes

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 nonlocal operator method for solving partial differential equations abstract a nonlocal operator method is proposed which is generally applicable for solving partial differential equations pdes of mechanical problems the nonlocal operator can be regarded as the integral form equivalent to the differential form in the sense of a nonlocal interaction model for solving the unknown field the variation of a nonlocal operator plays an equivalent role as the derivatives of the shape functions in the meshless methods or those of the finite element method thus it circumvents many difficulties in the calculation of shape functions and their derivatives the nonlocal operator method can consistently applied with common procedure leading to the weak forms i e the variational principle and the weighted residual method based on these the residual and the tangent stiffness matrix can be obtained with ease the nonlocal operator method is enhanced here also with an operator energy functional to satisfy the linear consistency of the field higher order nonlocal operators and higher order operator energy functional are hereby generalized a highlighted of the present method is the functional derived based on the nonlocal operator can convert the construction of residual and stiffness matrix into a series of matrix multiplications using the predefined nonlocal operators the nonlocal strong forms of different functionals can be obtained easily via the concept of support and dual support the two basic elements introduced in the paper several numerical examples of different types of pdes are presented in the end to show the effectiveness of the present method and also serve for validation

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

a higher order nonlocal operator method for solving partial differential equations abstract a higher order nonlocal operator method for the solution of boundary value problems is developed the proposed higher order nonlocal operator brings several advantages as compared to the original nonlocal operator method ren et al 2020 which only ensures first order convergence furthermore it can be applied to directly and efficiently obtain all partial derivatives of higher orders simultaneously without the need of using shape functions only the functionals based on the nonlocal operators termed as operator functional are needed to obtain the final discrete system of equations which significantly facilitates the implementation several numerical examples are presented to show the effectiveness and accuracy of the proposed higher order nonlocal operator method including the solution of the poisson equation in 2 5 dimensional space kirchhoff and von karman plate problems incompressible elastic materials as well as phase field modeling of fracture

a generalized ordinary finite deformation constitutive correspondence model for peridynamics abstract a finite deformation constitutive framework for peridynamics that uses generalized kirchhoff stress measures as intermediate quantities in computing the peridynamic material point interactions is proposed it is shown to be a generalization of a previous finite deformation constitutive correspondence theory and has additional useful properties the new model is shown to be an ordinary material model and therefore conserves angular momentum without additional restrictions additionally it does not suffer from well known surface effects present in some peridynamic material models has improved material stability properties and allows for inclusion of elasto plastic behavior in a straightforward manner

a morphing approach to couple state based peridynamics with classical continuum mechanics abstract a local nonlocal coupling technique called the morphing method is developed to couple classical continuum mechanics with state based peridynamics state based peridynamics which enables the description of cracks that appear and propagate spontaneously is applied to the key domain of a structure where damage and fracture are considered to have non negligible effects in the rest of the structure classical continuum mechanics is used to reduce computational costs and to simultaneously satisfy solution accuracy and boundary conditions both models are glued by the proposed morphing method in the transition region the morphing method creates a balance between the stiffness tensors of classical continuum mechanics and the weighted coefficients of state based peridynamics through the equivalent energy density of both models linearization of state based peridynamics is derived by taylor approximations based on vector operations the discrete formulation of coupled models is also described two dimensional numerical examples illustrate the validity and accuracy of the proposed technique it is shown that the morphing method originally developed for bond based peridynamics can be successfully extended to state based peridynamics through the original developments presented here

linearized state based peridynamics for 2 d problems summary in this work the authors formulate a 2 d linearized ordinary state based peridynamic model of elastic deformations and compute the stiffness matrix for 2 d plane stress strain conditions this model is then verified by testing the recovery of elastic properties for given poisson s ratios in the range 0 1 0 45 the convergence behavior of peridynamic solutions in terms of the size of the nonlocal region by comparison with the classical local mechanics model is also discussed the degree to which the peridynamic surface effect influences the recovery of elastic properties is examined and stress strain recovery values are found to have a definite influence on the results the technique used here can provide the basis for applying 2 d peridynamic models to the study of fatigue failure and quasi static fracture problems copyright 2016 john wiley sons ltd

finite propagation speed for limited flux diffusion equations we prove that the support of solutions of a limited flux diffusion equation known as a relativistic heat equation evolves at a constant speed identified as the speed of light c for that we construct entropy sub and super solutions which are fronts evolving at speed c and prove the corresponding comparison principle between entropy solutions and sub and super solutions respectively this enables us to prove the existence of discontinuity fronts moving at light s speed

when strictly locally convex hypersurfaces are embedded in this paper we will prove hadamard stoker type theorems in the following ambient spaces mathcal m n times mathbb r where mathcal m n is a 1 4 pinched manifold and certain killing submersions e g berger spheres and heisenberg spaces that is under the condition that the principal curvatures of an immersed hypersurface are greater than some non negative constant depending on the ambient space we prove that such a hypersurface is embedded and we also study its topology

a local strain based implementation strategy for the extended peridynamic model with bond rotation abstract the conventional grid displacements based implementation of the extended peridynamic model with bond rotation may give rise to some difficulty in predicting non uniform deformation field essentially due to the existence of local rigid rotation in solid this paper presents a novel alternative local strain based implementation technique for handling the numerical issue the key is to formulate the relative displacement between particles in terms of the strain tensor approximated locally rather than the direct use of displacements at peridynamic particles it is critically proved that bond stretch is independent of the rigid rotation tensor local strain is thus required only for approximating local shear deformation the model is currently implemented for static problems in a hybrid manner and in a finite element peridynamics coupling framework the latter allowing direct and correct imposition of boundary conditions to illustrate the performance of the proposed method in numerical accuracy and computational efficiency various examples are performed with varying poisson s ratio and comparisons with the results from finite element analysis are presented

a simple and robust three dimensional cracking particle method without enrichment abstract a new robust and efficient approach for modeling discrete cracks in meshfree methods is described the method is motivated by the cracking particle method rabczuk t belytschko t international journal for numerical methods in engineering 2004 where the crack is modeled by a set of cracked segments however in contrast to the above mentioned paper we do not introduce additional unknowns in the variational formulation to capture the displacement discontinuity instead the crack is modeled by splitting particles located on opposite sides of the associated crack segments and we make use of the visibility method in order to describe the crack kinematics we apply this method to several two and three dimensional problems in statics and dynamics and show through several numerical examples that the method does not show any mesh orientation bias

motion of curves constrained on surfaces using a level set approach the level set method has been successfully applied to a variety of problems that deal with curves in r2 or surfaces in r3 we present here a combination of these two cases creating a level set representation for curves constrained to lie on surfaces we study primarily geometrically based motions of these curves on stationary surfaces while allowing topological changes in the curves i e merging and breaking to occur applications include finding geodesic curves and shortest paths and curve shortening on surfaces further applications can be arrived at by extending those for curves moving in r2 to surfaces the problem of moving curves on surfaces can also be viewed as a simple constraint problem and may be useful in studying more difficult versions results show that our representation can accurately handle many geometrically based motions of curves on a wide variety of surfaces while automatically enforcing topological changes in the curves when they occur and automatically fixing the curves to lie on the surfaces the method can also be easily extended to higher dimensions

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

dual horizon peridynamics a stable solution to varying horizons abstract in this paper we present a dual horizon peridynamics formulation which allows for simulations with dual horizon with minimal spurious wave reflection we prove the general dual property for dual horizon peridynamics based on which the balance of momentum and angular momentum in pd are naturally satisfied we also analyze the crack pattern of random point distribution and the multiple materials issue in peridynamics for selected benchmark problems we show that dh pd is less sensitive to the spatial than the original pd formulation

improved one point quadrature algorithms for two dimensional peridynamic models based on analytical calculations the peridynamics theory is a reformulation of the classical theory of continuum mechanics based on long range interactions suitable for the description of material failure and damage integration plays a central role in this theory we study one point quadrature algorithms for the discretization of two dimensional integrals in peridynamics these algorithms are closely related to meshfree methods they assume an underlying reference lattice where each lattice point within a body is assigned a cell a main challenge in such algorithms is the accurate estimation of the area of the intersecting region between the neighborhood of a point and a neighbor cell we provide a classification of the different types of intersecting regions in square lattices and present analytical derivations for the exact calculation of their areas leading to improved integration accuracy to address convergence issues geometric centers of intersecting regions are taken as quadrature points replacing commonly used cell centers we present analytical derivations for the exact calculation of those geometric centers based on a decomposition of intersecting regions into subdomains of simple geometry by using exact values for the areas and geometric centers of intersecting regions we achieve an asymptotically monotonic convergence in numerical integration in contrast other algorithms from the literature exhibit a highly oscillatory behavior numerical results compare the proposed algorithms with others from the literature for different quantities of interest and demonstrate their improved accuracy and convergence error estimates for the quadrature algorithms are derived and extended and hybrid algorithms are discussed additional numerical studies using influence functions with a finite support and a controlled regularity suggest an alternative means to improve the convergence of discretization algorithms

stable particle methods based on lagrangian kernels abstract a large deformation particle method based on the krongauz belytschko corrected gradient meshfree method with lagrangian kernels is developed in this form the gradient is corrected by a linear transformation so that linear completeness is satisfied for the test functions shepard functions are used this guarantees that the patch test is met lagrangian kernels are introduced to eliminate spurious distortions of the domain of material stability a mass allocation scheme is developed that captures correct reflection of waves without any explicit application of traction boundary conditions in addition the lagrangian kernel versions of various forms of smooth particle methods sph including the standard forms and the randles libersky modification are presented and studied results are obtained for a variety of problems that compare this method to standard forms of sph the randles libersky correction and large deformation versions of the element free galerkin method

the generalized finite element method an example of its implementation and illustration of its performance the generalized finite element method gfem was introduced in reference 1 as a combination of the standard fem and the partition of unity method the standard mapped polynomial finite element spaces are augmented by adding special functions which reflect the known information about the boundary value problem and the input data the geometry of the domain the loads and the boundary conditions the special functions are multiplied with the partition of unity corresponding to the standard linear vertex shape functions and are pasted to the existing finite element basis to construct a conforming approximation the essential boundary conditions can be imposed exactly as in the standard fem adaptive numerical quadrature is used to ensure that the errors in integration do not affect the accuracy of the approximation this paper gives an example of how the gfem can be developed for the laplacian in domains with multiple elliptical voids and illustrates implementation issues and the superior accuracy of the gfem versus the standard fem copyright 2000 john wiley sons ltd

numerical schemes for the hamilton jacobi and level set equations on triangulated domains borrowing from techniques developed for conservation law equations numerical schemes which discretize the hamilton jacobi h j level set and eikonal equations on triangulated domains are presented the first scheme is a provably monotone discretization for the h j equations unfortunately the basic scheme lacks lipschitz continuity of the numerical hamiltonian by employing a virtual edge flipping technique local lipschitz continuity of the numerical flux is restored on acute triangulations next schemes are introduced and developed based on the weaker concept of positive coefficient approximations for homogeneous hamiltonians these schemes possess a discrete maximum principle on arbitrary triangulations and exhibit local lipschitz continuity of the numerical hamiltonian finally a class of petrov galerkin approximations is considered these schemes are stabilized via a least squares bilinear form the petrov galerkin schemes do not possess a discrete maximum principle but generalize to high order accuracy discretization of the level set equation also requires the numerical approximation of a mean curvature term a simple mass lumped galerkin approximation is presented in section 6 and analyzed using maximum principle analysis the use of unstructured meshes permits several forms of mesh adaptation which have been incorporated into numerical examples these numerical examples include discretizations of convex and nonconvex forms of the h j equation the eikonal equation and the level set equation

motion of curves in three spatial dimensions using a level set approach the level set method was originally designed for problems dealing with codimension one objects where it has been extremely succesful especially when topological changes in the interface i e merging and breaking occur attempts have been made to modify it to handle objects of higher codimension such as vortex filaments while preserving the merging and breaking property we present numerical simulations of a level set based method for moving curves in r3 the model problem for higher codimension that allows for topological changes a vector valued level set function is used with the zero level set representing the curve our results show that this method can handle many types of curves moving under all types of geometrically based flows while automatically enforcing merging and breaking