numerical modelling of crack propagation automatic remeshing and comparison of different criteria modelling of a crack propagating through a finite element mesh under mixed mode conditions is of prime importance in fracture mechanics three different crack growth criteria and the respective crack paths prediction for several test cases are compared the maximal circumferential stress criterion the strain energy density fracture criterion and the criterion of the maximal strain energy release rate are implemented using advanced finite element techniques a fully automatic remesher enables to deal with multiple boundaries and multiple materials the propagation of the crack is calculated with both remeshing and nodal relaxation several examples are presented to check for the robustness of the numerical techniques and to study specific features of each criterion

the mixed complementarity problem arising from non associative plasticity with non smooth yield surfaces abstract an elastic domain with multiple yield surfaces brings about huge troubles to numerical computations due to singularities while non associative plasticity leads further to the open issue of whether constitutive integration is well posed this study reduces the constitutive integration of non associative plasticity with multiple yield surfaces to a mixed complementarity problem represented by micp a special case of finite dimensional variational inequalities by means of the projection contraction algorithm for finite dimensional variational inequalities and the idea of the gauss seidel method a new projection contraction algorithm for micp is designed and denoted by gspc applying the monotonicity of the mapping of the micp gspc is proved convergent theoretically for associative plasticity for non associative plasticity the sufficient condition for gspc to be convergent is also established if the tension part of the mohr coulomb elastic domain is cut off typical examples are designed to illustrate gspc is highly efficient accurate and stable some of which cannot be solved by the conventional return mapping method in all the cases the computational efficiency of gspc is apparently above the mapping return method

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

dual form of discontinuous deformation analysis abstract discontinuous deformation analysis dda is a numerical method for analyzing dynamic behaviors of an assemblage of distinct blocks with the block displacements as the basic variables the contact conditions are approximately satisfied by the open close iteration which needs to fix or remove repeatedly the virtual springs between blocks in contact the results from dda are strongly dependent upon stiffness of these virtual springs excessively hard or soft springs all incur numerical problems this is believed to be the biggest obstacle to more extensive application of dda to avoid the introduction of virtual springs huge efforts have been made with little progress related to low efficiency in solution in this study the contact forces instead of the block displacements are taken as the basic variables stemming from the equations of momentum conservation of each block the block displacements can be expressed in terms of the contact forces acting on the block from the contact conditions a finite dimensional quasi variational inequality is derived with the contact forces as the independent variables on the basis of the projection contraction algorithm for the standard finite dimensional variational inequalities an iteration algorithm called the compatibility iteration is designed for the quasi variational inequality the main processes can be highly parallelized with no need to assemble the global stiffness matrix a number of numerical tests including those very challenging suggest that the proposed procedure has reached practical level in accuracy robustness and efficiency and the goal to abandon completely virtual springs has been reached

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

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

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

the best sobolev trace constant as limit of the usual sobolev constant for small strips near the boundary in this paper we prove that the best constant in the sobolev trace embedding h1 lq in a bounded smooth domain can be obtained as the limit as 0 of the best constant of the usual sobolev embedding h1 lq dx where x 2 dist x is a small neighborhood of the boundary we also analyze symmetry properties of extremals of this last embedding when is a ball

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

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 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

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

an efficient augmented finite element method for arbitrary cracking and crack interaction in solids summary this paper presents an augmentation method that enables bilinear finite elements to efficiently and accurately account for arbitrary multiple intra elemental discontinuities in 2d solids the augmented finite element method a fem employs four internal nodes to account for the crack displacements due to an intra elemental weak or strong discontinuity and it permits repeated elemental augmentation to include multiple interactive cracks it thus enables a unified treatment of damage evolution from a weak discontinuity to a strong discontinuity and to multiple interactive cohesive cracks all within a single bilinear element that employs standard external nodal dofs only a novel elemental condensation procedure has been developed to solve the internal nodal dofs as functions of the external nodal dofs for any irreversible piece wise linear cohesive laws it leads to a fully condensed elemental equilibrium equation with mathematical exactness eliminating the need for nonlinear equilibrium iterations at elemental level the new a fem s high fidelity simulation capabilities to interactive cohesive crack formation and propagation in homogeneous and heterogeneous solids have been demonstrated through several challenging numerical examples it is shown that the proposed a fem empowered by the new elemental condensation procedure is numerically very efficient accurate and robust copyright 2014 john wiley sons ltd

generalized bezout s theorem and its applications in coding theory this paper presents a generalized bezout theorem which can be used to determine a tighter lower bound of the number of distinct points of intersection of two or more plane curves a new approach to determine a lower bound on the minimum distance for algebraic geometric codes defined from a class of plane curves is introduced based on the generalized bezout theorem examples of more efficient linear codes are constructed using the generalized bezout theorem and the new approach for d 4 the linear codes constructed by the new construction are better than or equal to the known linear codes for d spl ges 5 these new codes are better than the known ag codes defined from whole spaces the klein codes 22 16 5 and 22 15 6 over gf 2 sup 3 and the improved hermitian code 64 56 6 over gf 2 sup 4 are also constructed

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