geometrically accurate efficient and flexible quadrature techniques for the tetrahedral finite cell method abstract we illustrate the importance of geometrically accurate volume quadrature for obtaining optimal accuracy with non boundary fitted finite element discretizations when the problem domain is defined by sharp boundaries we consider the tetrahedral finite cell method tetfcm and replace its recursive subdivision based integration approach with geometrically accurate quadrature rules that emanate from higher order geometric parametrizations of cut tetrahedral elements the element wise parametrization procedure relies on the identification of the intersection topology and a series of higher order mappings based on lagrange polynomials we demonstrate with several 3d examples that geometrically faithful local parametrization ensures optimal accuracy while significantly reducing the number of quadrature points with respect to recursive subdivision on the other hand we highlight the strength of subdivision quadrature in the context of a patient specific workflow for the simulation based performance analysis of coupled bone implant configurations in particular we show that accuracy flexibility and computational efficiency of the tetfcm critically depend on flexibly applying the two different quadrature variants for fuzzy imaging data and sharp boundary representations respectively

optimal orlicz domains in sobolev embeddings into marcinkiewicz spaces abstract in this paper we present a method for determining whether there exists a largest orlicz space l a satisfying the sobolev embedding w m l a y where y stands for an arbitrary so called marcinkiewicz endpoint space the tool developed in this work enables us to investigate the optimality of orlicz domain spaces in sobolev embeddings and also in sobolev trace embeddings on domains in r n with various regularity

a primal dual active set method and predictor corrector mesh adaptivity for computing fracture propagation using a phase field approach abstract in this paper we consider phase field based fracture propagation in elastic media the main purpose is the development of a robust and efficient numerical scheme to enforce crack irreversibility as a constraint we use a primal dual active set strategy which can be identified as a semi smooth newton method the active set iteration is merged with the newton iteration for solving the fully coupled nonlinear partial differential equation discretized using finite elements resulting in a single rapidly converging nonlinear scheme it is well known that phase field models require fine meshes to accurately capture the propagation dynamics of the crack because traditional estimators based on adaptive mesh refinement schemes are not appropriate we develop a predictor corrector scheme for local mesh adaptivity to reduce the computational cost this method is both robust and efficient and allows us to treat temporal and spatial refinements and to study the influence of model regularization parameters finally our proposed approach is substantiated with different numerical tests for crack propagation in elastic media and pressurized fracture propagation in homogeneous and heterogeneous media

computational modeling of localized failure in solids xfem vs pf czm abstract this work addressesnumerical comparison between the extended finite element method xfem and the phase field regularized cohesive zone model pf czm wu 2017 2018a for the modeling of cohesive fracture induced localized failure in solids a novel implementation of the pf czm is proposed similarly to the xfem in which the added degrees of freedom characterizing the crack behavior are associated only with those nodes within a small sub domain representative benchmark examples show that both methods are independent of the mesh discretization for problems with the crack path known a priori both methods give almost coincident numerical results though the xfem is advantageous due to its high coarse mesh resolution for complex problems with arbitrary crack propagation even though some discrepancies are observed the numerical results given by both methods are also quantitatively similar if the predicted crack trajectories are close to each other despite the higher computational cost in this case the pf czm is much more favored due to its intrinsic capability of modeling complex crack configurations e g nucleation propagation merging and branching etc with no need of any extra strategy the above facts validate our previous theoretical analysis that the pf czm approaches asymptotically to a discontinuous czm for a vanishing length scale as it avoids most challenging issues exhibited by other approaches the pf czm is not only theoretically elegant but also numerically appealing for the modeling of localized failure in solids

a phase field multirate scheme with stabilized iterative coupling for pressure driven fracture propagation in porous media abstract phase field methods have the potential to simulate large scale evolution of networks of fractures in porous media without the need to explicitly track interfaces practical field simulations require however that robust and efficient decoupling techniques can be applied for solving these complex systems in this work we focus on the mechanics step that involves the coupling of elasticity and the phase field variable we develop a multirate scheme in which a coarser time grid is employed for the mechanics equation i e the displacements and a finer time grid is taken for the phase field problem the performance of this algorithm is demonstrated for two test cases

cracking node method for dynamic fracture with finite elements a new method for modeling discrete cracks based on the extended finite element method is described in the method the growth of the actual crack is tracked and approximated with contiguous discrete crack segments that lie on finite element nodes and span only two adjacent elements the method can deal with complicated fracture patterns because it needs no explicit representation of the topology of the actual crack path a set of effective rules for injection of crack segments is presented so that fracture behavior beginning from arbitrary crack nucleations to macroscopic crack propagation is seamlessly modeled the effectiveness of the method is demonstrated with several dynamic fracture problems that involve complicated crack patterns such as fragmentation and crack branching copyright 2008 john wiley sons ltd

anisotropic mesh adaptation for the generalized ambrosio tortorelli functional with application to brittle fracture abstract quasi static crack propagation in brittle materials is modeled via the ambrosio tortorelli approximation the crack is modeled by a smooth phase field defined on the whole computational domain since the crack is confined to a thin layer the employment of anisotropic adapted grids is shown to be a really effective tool in containing computational costs we extend the anisotropic error analysis applied to the classical ambrosio tortorelli approximation by artina et al to the generalized ambrosio tortorelli functional where a unified framework for several elasticity laws is dealt with as well as a non convex fracture energy can be accommodated after deriving an anisotropic a posteriori error estimator we devise an algorithm which alternates optimization and mesh adaptation both anti plane and plane strain configurations are numerically checked

thermodynamically consistent phase field models of fracture variational principles and multi field fe implementations the computational modeling of failure mechanisms in solids due to fracture based on sharp crack discontinuities suffers in situations with complex crack topologies this can be overcome by a diffusive crack modeling based on the introduction of a crack phase field in this paper we outline a thermodynamically consistent framework for phase field models of crack propagation in elastic solids develop incremental variational principles and consider their numerical implementations by multi field finite element methods we start our investigation with an intuitive and descriptive derivation of a regularized crack surface functional that converges for vanishing length scale parameter to a sharp crack topology functional this functional provides the basis for the definition of suitable convex dissipation functions that govern the evolution of the crack phase field here we propose alternative rate independent and viscous over force models that ensure the local growth of the phase field next we define an energy storage function whose positive tensile part degrades with increasing phase field with these constitutive functionals at hand we derive the coupled balances of quasi static stress equilibrium and gradient type phase field evolution in the solid from the argument of virtual power here we consider a canonical two field setting for rate independent response and a time regularized three field formulation with viscous over force response it is then shown that these balances follow as the euler equations of incremental variational principles that govern the multi field problems these principles make the proposed formulation extremely compact and provide a perfect base for the finite element implementation including features such as the symmetry of the monolithic tangent matrices we demonstrate the performance of the proposed phase field formulations of fracture by means of representative numerical examples copyright 2010 john wiley sons ltd

numerical approximations to a new phase field model for two phase flows of complex fluids abstract we derive a new phase field theory for immiscible mixtures of nematic liquid crystals and viscous fluids using the variational principle coupled with the generalized onsager principle a novel phase transition mechanism is implemented to couple the nematic liquid crystal phase with the viscous fluid phase to arrive at the dissipative hydrodynamic model for incompressible fluid mixtures through a delicate explicit implicit numerical discretization we develop a decoupled linear scheme for a simplified version of the phase field model as well as a coupled nonlinear scheme for the full model both schemes are shown as unconditionally energy stable with consistent discrete dissipative energy laws several numerical examples are presented to show the effectiveness of the new model and the new numerical schemes developed for it

linear and unconditionally energy stable schemes for the binary fluid surfactant phase field model abstract in this paper we consider the numerical solution of a binary fluid surfactant phase field model in which the free energy contains a nonlinear coupling entropy a ginzburg landau double well potential and a logarithmic flory huggins potential the resulting system consists of two nonlinearly coupled cahn hilliard type equations we develop a first and a second order time stepping schemes for this system using the invariant energy quadratization approach in particular the system is transformed into an equivalent one by introducing appropriate auxiliary variables and all nonlinear terms are then treated semi explicitly both schemes are linear and lead to symmetric positive definite systems in space at each time step thus they can be efficiently solved we further prove that these schemes are unconditionally energy stable in the discrete sense various 2d and 3d numerical experiments are performed to validate the accuracy and energy stability of the proposed schemes

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

efficient linear schemes with unconditional energy stability for the phase field elastic bending energy model abstract in this paper we study efficient numerical schemes of the classical phase field elastic bending energy model that has been widely used to describe the shape deformation of biological lipid vesicles in which the free energy of the system consists of an elastic bending energy a surface area constraint and a volume constraint one major challenge in solving such model numerically is how to design appropriate temporal discretizations in order to preserve energy stability with large time step sizes at the semi discrete level we develop a first order and a second order time stepping scheme for this highly nonlinear and stiff parabolic pde system based on the invariant energy quadratization approach in particular the resulted semi discretizations lead to linear systems in space with symmetric positive definite operators at each time step thus can be efficiently solved in addition the proposed schemes are rigorously proved to be unconditionally energy stable various numerical experiments in 2d and 3d are presented to demonstrate the stability and accuracy of the proposed schemes

optimal mean square error calibration of classifier error estimators under bayesian models a recently proposed bayesian modeling framework for classification facilitates both the analysis and optimization of error estimation performance the bayesian error estimator is then defined to have optimal mean square error performance but in many situations closed form representations are unavailable and approximations may not be feasible to address this we present a method to optimally calibrate arbitrary error estimators for minimum mean square error performance within a supposed bayesian framework assuming a fixed sample size classification rule and error estimation rule as well as a fixed bayesian model the calibration is done by first computing a calibration function that maps error estimates to their optimally calibrated values off line once found this calibration function may be easily applied to error estimates on the fly whenever the assumptions apply we demonstrate that calibrated error estimators offer significant improvement in performance relative to classical error estimators under bayesian models with both linear and non linear classification rules

comparison of high order finite element and discontinuous galerkin methods for phase field equations application to structural damage abstract phase field equations are used to model a wide range of multiphase problems such as separation of fluids solidification viscous fingering fracture and fatigue a wide variety of methods to numerically solve phase field equations can be found in the literature in particular high order methods are an effective option when accuracy improvement is desired in the first part of this work we analyze the accuracy and computational efficiency of the high order finite element method fem and discontinuous galerkin dg method applied to the second order allen cahn ac and fourth order cahn hilliard ch equations several schemes for time integration are used for these equations the explicit schemes are the forward euler classical fourth order runge kutta rk4 and the strong stability preserving ten stages fourth order runge kutta rkssp 10 4 described in gottlieb et al 2011 the backward euler and trapezoidal implicit methods are adopted in the full and semi implicit schemes as proposed in eyre unpublished manufactured solutions for one dimensional problems are used in order to evaluate the errors and to compare the different numerical methods by choosing an adequate discretization for ac equations resulting from the previous analysis of the first part of the work in the second part we propose a numerical semi implicit scheme to solve the damage and fracture model described in boldrini et al 2016 this procedure employs the fem for spatial discretization the newmark method for time integration of the kinematics equation and the backward euler for the damage phase field evolution finally results for 2d benchmark tests are presented for the fracture phase field model and the convergence to a sharp crack for a small width of the damage phase field layer is verified

sobolev inequalities for the symmetric gradient in arbitrary domains abstract a form of sobolev inequalities for the symmetric gradient of vector valued functions is proposed which allows for arbitrary ground domains in r n in the relevant inequalities boundary regularity of domains is replaced with information on boundary traces of trial functions the inequalities so obtained exhibit the same exponents as in classical inequalities for the full gradient of sobolev functions in regular domains furthermore they involve constants independent of the geometry of the domain and hence yield novel results yet for smooth domains our approach relies upon a pointwise estimate for the functions in question via a riesz potential of their symmetric gradient and an unconventional potential depending on their boundary trace

an efficient two grid method for the cahn hilliard equation with the concentration dependent mobility and the logarithmic flory huggins bulk potential abstract in this paper we consider the two grid method for solving cahn hilliard equation with the concentration dependent mobility and the logarithmic flory huggins bulk potential this two grid method consists of two steps first step solve the cahn hilliard equation with full implicit mixed finite element method and newton iteration method on a coarse grid second step solve two linear equations on a fine grid the stability and convergence of the proposed scheme are established finally some numerical experiments are presented

phase field based nonlocal anisotropic damage mechanics model abstract a nonlocal anisotropic damage theory is developed in this work through the phase field method to address the anisotropic damage evolution in materials the anisotropic damage is discussed and appropriate nonconserved order parameters in three mutually perpendicular directions are defined to find the growth of the components of a second order diagonal damage tensor corresponding to the principal directions of a general second order damage tensor in contrast to the previous models two new tensors are proposed to act as interpolation and potential functions along with the allen cahn equation in order to obtain the evolution of the order parameters which is the basis of the definition of the damage rate the tensor formulation of the growth of the components of the damage tensor is proposed for the first time it is shown that by introducing a set of material parameters including a length scale parameter due to damage there is a robust and simplified way to model the nonlocal behavior of damage and predict the corresponding material behavior as components of a second order diagonal damage tensor

variational gradient plasticity at finite strains part i mixed potentials for the evolution and update problems of gradient extended dissipative solids this work outlines a theoretical and computational framework of finite inelasticity with length scales based on a rigorous exploitation of mixed variational principles in contrast to classical local approaches to inelasticity based on locally evolving internal variables order parameter fields are taken into account governed by additional balance type partial differential equations including micro structural boundary conditions this incorporates non local dissipative effects based on length scales which reflect properties of the material micro structure typical examples are phase field theories gradient damage and strain gradient plasticity we outline unified minimization and saddle point principles for the evolution problem of first order gradient type standard dissipative solids particular emphasis is put on mixed multi field representations where both the microstructural variable itself as well as its dual driving force are present these settings are needed for models with threshold functions formulated in the space of the driving forces in particular for finite gradient plasticity the central aim is to define constitutive rate type and algorithmic incremental potentials whose variational derivatives govern the coupled macro and micro balances in both the continuous as well as the time discrete setting their existence underlines the inherent symmetry of standard dissipative solids we demonstrate geometrically consistent constructions of these potentials for important classes of finite inelasticity providing a fresh look on models of multiplicative and additive gradient plasticity of single crystals and amorphous materials the potentials provide maximum compact representations of those complex material models are the cornerstones of a subsequent mixed finite element design and should be considered as the primary objects for the theoretical and computational modeling

a phase field model of fracture with frictionless contact and random fracture properties application to thin film fracture and soil desiccation abstract we present a new derivation for a phase field model of cohesive fracture that allows for fully damaged surfaces to properly transmit tractions under frictionless contact conditions the model is derived from an energy minimization standpoint and the governing equations are presented in a general allen cahn form unifying both brittle and cohesive fracture models a novel elastic energy split is proposed to enforce frictionless contact conditions along the regularized crack set a fixed point iterative algorithm is used to solve the system of equations and an active set strategy is incorporated to enforce the crack set irreversibility the model is then applied to study the mechanical fracture coupling in the fracture of thin films and soil desiccation problems in which intricate crack networks form we present results for one dimensional two dimensional and fully three dimensional configurations and examine the relationship between the generalized driving energies for fracture and the resulting fracture patterns we also investigate the impact of stochastic spatial variations in material parameters on fragment statistics and fracture network morphology to this end we develop a random field model for fracture properties and address a forward propagation problem the versatility offered by the probabilistic framework is finally highlighted by considering a statistical inverse problem based on physical experiments it is shown in particular that the formulation allows key characteristics of observed crack patterns to be reproduced with reasonable accuracy

the diffuse nitsche method dirichlet constraints on phase field boundaries summary we explore diffuse formulations of nitsche s method for consistently imposing dirichlet boundary conditions on phase field approximations of sharp domains leveraging the properties of the phase field gradient we derive the variational formulation of the diffuse nitsche method by transferring all integrals associated with the dirichlet boundary from a geometrically sharp surface format in the standard nitsche method to a geometrically diffuse volumetric format we also derive conditions for the stability of the discrete system and formulate a diffuse local eigenvalue problem from which the stabilization parameter can be estimated automatically in each element we advertise metastable phase field solutions of the allen cahn problem for transferring complex imaging data into diffuse geometric models in particular we discuss the use of mixed meshes that is an adaptively refined mesh for the phase field in the diffuse boundary region and a uniform mesh for the representation of the physics based solution fields we illustrate accuracy and convergence properties of the diffuse nitsche method and demonstrate its advantages over diffuse penalty type methods in the context of imaging based analysis we show that the diffuse nitsche method achieves the same accuracy as the standard nitsche method with sharp surfaces if the inherent length scales i e the interface width of the phase field the voxel spacing and the mesh size are properly related we demonstrate the flexibility of the new method by analyzing stresses in a human vertebral body this article is protected by copyright all rights reserved

a computational framework of three dimensional configurational force driven brittle crack propagation abstract we consider a variational formulation of quasi static brittle fracture and develop a new finite element based computational framework for propagation of cracks in three dimensional bodies we outline a consistent thermodynamical framework for crack propagation in elastic solids and show that both the elastic equilibrium response as well as the local crack evolution follow in a natural format by exploitation of a global clausius planck inequality in the sense of coleman s method consequently the crack propagation direction associated with the classical griffith criterion is identified by the material configurational force which maximizes the local dissipation at the crack front the variational formulation is realized numerically by a standard spatial discretization with finite elements which yields a discrete formulation of the global dissipation in terms configurational nodal forces therefore the constitutive setting of crack propagation in the space discretized finite element context is naturally related to discrete nodes of a typical finite element mesh in a consistent way with the node based setting the discretization of the evolving crack discontinuity is performed by the doubling of critical nodes and interface facets of the mesh the crucial step for the success of this procedure is its embedding into an r adaptive crack facet reorientation procedure based on configurational force based indicators in conjunction with crack front constraints we propose a staggered solution procedure that results in a sequence of positive definite discrete subproblems with successively decreasing overall stiffness providing a robust algorithmic setting in the postcritical range the predictive capabilitiy of the proposed formulation is demonstrated by means of representative numerical simulations

phase field modeling of ductile fracture at finite strains a robust variational based numerical implementation of a gradient extended theory by micromorphic regularization summary this work provides a robust variational based numerical implementation of a phase field model of ductile fracture in elastic plastic solids undergoing large strains this covers a computationally efficient micromorphic regularization of the coupled gradient plasticity damage formulation the phase field approach regularizes sharp crack surfaces within a pure continuum setting by a specific gradient damage modeling with geometric features rooted in fracture mechanics it has proven immensely successful with regard to the analysis of complex crack topologies without the need for fracture specific computational structures such as finite element design of crack discontinuities or intricate crack tracking algorithms the proposed gradient extended plasticity damage formulation includes two independent length scales that regularize both the plastic response as well as the crack discontinuities this ensures that the damage zones of ductile fracture are inside of plastic zones or vice versa and guarantees on the computational side a mesh objectivity in post critical ranges the proposed setting is rooted in a canonical variational principle the coupling of gradient plasticity to gradient damage is realized by a constitutive work density function that includes the stored elastic energy and the dissipated work due to plasticity and fracture the latter represents a coupled resistance to plasticity and damage depending on the gradient extended internal variables that enter plastic yield functions and fracture threshold functions with this viewpoint on the generalized internal variables at hand the thermodynamic formulation is outlined for gradient extended dissipative solids with generalized internal variables that are passive in nature it is specified for a conceptual model of von mises type elasto plasticity at finite strains coupled with fracture the canonical theory proposed is shown to be governed by a rate type minimization principle which fully determines the coupled multi field evolution problem this is exploited on the numerical side by a fully symmetric monolithic finite element implementation an important aspect of this work is the regularization towards a micromorphic gradient plasticity damage setting by taking into account additional internal variable fields linked to the original ones by penalty terms this enhances the robustness of the finite element implementation in particular on the side of gradient plasticity the performance of the formulation is demonstrated by means of some representative examples copyright 2016 john wiley sons ltd

a higher order phase field model for brittle fracture formulation and analysis within the isogeometric analysis framework phase field models based on the variational formulation for brittle fracture have recently been gaining popularity these models have proven capable of accurately and robustly predicting complex crack behavior in both two and three dimensions in this work we propose a fourth order model for the phase field approximation of the variational formulation for brittle fracture we derive the thermodynamically consistent governing equations for the fourth order phase field model by way of a variational principle based on energy balance assumptions the resulting model leads to higher regularity in the exact phase field solution which can be exploited by the smooth spline function spaces utilized in isogeometric analysis this increased regularity improves the convergence rate of the numerical solution and opens the door to higher order convergence rates for fracture problems we present an analysis of our proposed theory and numerical examples that support this claim we also demonstrate the robustness of the model in capturing complex three dimensional crack behavior

the extended finite element method with new crack tip enrichment functions for an interface crack between two dissimilar piezoelectric materials summary this paper studies the static fracture problems of an interface crack in linear piezoelectric bimaterial by means of the extended finite element method x fem with new crack tip enrichment functions in the x fem crack modeling is facilitated by adding a discontinuous function and crack tip asymptotic functions to the classical finite element approximation within the framework of the partition of unity in this work the coupled effects of an elastic field and an electric field in piezoelectricity are considered corresponding to the two classes of singularities of the aforementioned interface crack problem namely class and class two classes of crack tip enrichment functions are newly derived and the former that exhibits oscillating feature at the crack tip is numerically investigated computation of the fracture parameter i e the j integral using the domain form of the contour integral is presented excellent accuracy of the proposed formulation is demonstrated on benchmark interface crack problems through comparisons with analytical solutions and numerical results obtained by the classical fem moreover it is shown that the geometrical enrichment combining the mesh with local refinement is substantially better in terms of accuracy and efficiency copyright 2015 john wiley sons ltd

an optimal interpolation theorem of marcinkiewicz type in orlicz spaces abstract an extension of the marcinkiewicz interpolation theorem is proved yielding a necessary and sufficient condition for every quasilinear operator satisfying given endpoint estimates of weak type to be bounded from an orlicz space into another

iterative coupling of flow geomechanics and adaptive phase field fracture including level set crack width approaches in this work we present numerical studies of fixed stress iterative coupling for solving flow and geomechanics with propagating fractures in a porous medium specifically fracture propagations are described by employing a phase field approach the extension to fixed stress splitting to propagating phase field fractures and systematic investigation of its properties are important enhancements to existing studies moreover we provide an accurate computation of the fracture opening using level set approaches and a subsequent finite element interpolation of the width the latter enters as fracture permeability into the pressure diffraction problem which is crucial for fluid filled fractures our developments are substantiated with several numerical tests that include comparisons of computational cost for iterative coupling and nonlinear and linear iterations as well as convergence studies in space and time fixed stress formulations for fluid filled phase field fractures accurate fracture width finite element computations using level sets numerical examples in 2d and 3d with spatial and temporal convergence studies

fast provably unconditionally energy stable and second order accurate algorithms for the anisotropic cahn hilliard model abstract in this paper we consider numerical approximations for solving the anisotropic cahn hilliard model we combine the scalar auxiliary variable sav approach with the stabilization technique to arrive at a novel stabilized sav approach where three linear stabilization terms which are shown to be crucial to remove the oscillations caused by the anisotropic coefficient are added to enhance the stability while keeping the required accuracy the schemes are very easy to implement and fast in the sense that all nonlinear terms are treated in a semi explicit way and one only needs to solve three decoupled linear equations with constant coefficients at each time step we further prove the unconditional energy stabilities rigorously and present numerous 2d and 3d numerical simulations to demonstrate the stability and accuracy

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

efficient non iterative and second order accurate numerical algorithms for the anisotropic allen cahn equation with precise nonlocal mass conservation abstract we propose two efficient non iterative and second order accurate algorithms to solving the anisotropic allen cahn equation with the nonlocal mass conservation the first scheme is the stabilized sav approach which combines the sav approach with the stabilization technique in which three linear stabilization terms are added to enhance the stability and keep the required accuracy while using large time steps the scheme not only can conserve the mass precisely but also is very easy to implement and non iterative where one only needs to solve three decoupled fourth order biharmonic equations with constant coefficients at each time step we further prove the scheme is unconditionally energy stable rigorously the second scheme is based on the linear stabilization approach where all nonlinear terms are treated in an explicit way therefore the scheme is quite efficient and stable that allows for large time steps in computations for both schemes we present a number of 2d and 3d numerical simulations to show stability and accuracy

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

explicit dynamics with a non local damage model using the thick level set approach summary in this paper we are interested in the dynamical response of a material body subjected to impact loadings such loadings are brutal and intense and may damage the material leading to strain localization and rupture before strain localization occurs computation of such problems is often accurate enough and very efficient when an explicit time integration scheme is applied however after strain localization occurs the mathematical relevance of a model is preserved only if non locality is introduced this is often resulting in a dramatic increase of computational costs we propose in this work to introduce non locality through the thick level set approach tls it is the first time the tls approach has been presented in a dynamical context in this approach additional computational efforts are limited in space to a domain slightly bigger than the strain localization region and the time discretization is explicit the non local computation is based on a new technique where basis functions are built on the damaged band the resulting function space has needed properties to compute non local fields copyright 2014 john wiley sons ltd

on the crack driving force of phase field models in linearized and finite elasticity abstract the phase field approach to fracture has been proven to be a mathematically sound and easy to implement method for computing crack propagation with arbitrary crack paths hereby crack growth is driven by energy minimization resulting in a variational crack driving force the definition of this force out of a tension related energy functional does however not always agree with the established failure criteria of fracture mechanics in this work different variational formulations for linear and finite elastic materials are discussed and ad hoc driving forces are presented which are motivated by general fracture mechanical considerations the superiority of the generalized approach is demonstrated by a series of numerical examples

a constrained shape optimization problem in orlicz sobolev spaces abstract in this manuscript we study the following optimization problem given a bounded and regular domain r n we look for an optimal shape for the w vanishing window on the boundary with prescribed measure over all admissible profiles in the framework of the orlicz sobolev spaces associated to constant for the sobolev trace embedding in this direction we establish existence of minimizer profiles and optimal sets as well as we obtain further properties for such extremals finally we also place special emphasis on analyzing the corresponding optimization problem involving an a vanishing hole inside the domain with volume constraint

numerical approximations for a phase field dendritic crystal growth model based on the invariant energy quadratization approach summary we present two accurate and efficient numerical schemes for a phase field dendritic crystal growth model which is derived from the variation of a free energy functional consisting of a temperature dependent bulk potential and a conformational entropy with a gradient dependent anisotropic coefficient we introduce a novel invariant energy quadratization approach to transform the free energy functional into a quadratic form by introducing new variables to substitute the nonlinear transformations based on the reformulated equivalent governing system we develop a first and a second order semi discretized scheme in time for the system in which all nonlinear terms are treated semi explicitly the resulting semi discretized equations consist of a linear elliptic equation system at each time step where the coefficient matrix operator is positive definite and thus the semi discrete system can be solved efficiently we further prove that the proposed schemes are unconditionally energy stable convergence test together with 2d and 3d numerical simulations for dendritic crystal growth are presented after the semi discrete schemes are fully discretized in space using the finite difference method to demonstrate the stability and the accuracy of the proposed schemes copyright 2016 john wiley sons ltd

an optimization based phase field method for continuous discontinuous crack propagation this is the peer reviewed version of the following article this article may be used for non commercial purposes in accordance with wiley terms and conditions for self archiving geelen r liu y dolbow j rodriguez ferran a an optimization based phase field method for continuous discontinuous crack propagation international journal for numerical methods in engineering 5 octubre 2018 vol 116 num 1 p 1 20 which has been published in final form at https onlinelibrary wiley com doi abs 10 1002 nme 5911

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

on penalization in variational phase field models of brittle fracture abstract irreversible evolution is one of the central concepts as well as implementation challenges of both the variational approach to fracture by francfort and marigo 1998 and its regularized counterpart by bourdin francfort and marigo 2000 2007 and 2008 which is commonly referred to as a phase field model of brittle fracture irreversibility of the crack phase field imposed to prevent fracture healing leads to a constrained minimization problem whose optimality condition is given by a variational inequality in our study the irreversibility is handled via penalization provided the penalty constant is well tuned the penalized formulation is a good approximation to the original one with the advantage that the induced equality based weak problem enables a much simpler algorithmic treatment we propose an analytical procedure for deriving the optimal penalty constant more precisely its lower bound which guarantees a sufficiently accurate enforcement of the crack phase field irreversibility our main tool is the notion of the optimal phase field profile as well as the convergence result it is shown that the explicit lower bound is a function of two formulation parameters the fracture toughness and the regularization length scale but is independent of the problem setup geometry boundary conditions etc the formulation ingredients degradation function tension compression split etc as well as the discretization mesh size the optimally penalized formulation is tested for three benchmark problems including one with available analytical solution we also compare our results with those obtained by using the alternative irreversibility technique based on the notion of the history field by miehe et al 2010

a phase field model for rate independent crack propagation robust algorithmic implementation based on operator splits the computational modeling of failure mechanisms in solids due to fracture based on sharp crack discontinuities suffers in situations with complex crack topologies this can be overcome by a diffusive crack modeling based on the introduction of a crack phase field following our recent work c miehe f welschinger m hofacker thermodynamically consistent phase field models of fracture variational principles and multi field fe implementations international journal for numerical methods in engineering doi 10 1002 nme 2861 on phase field type fracture we propose in this paper a new variational framework for rate independent diffusive fracture that bases on the introduction of a local history field it contains a maximum reference energy obtained in the deformation history which may be considered as a measure for the maximum tensile strain obtained in history it is shown that this local variable drives the evolution of the crack phase field the introduction of the history field provides a very transparent representation of the balance equation that governs the diffusive crack topology in particular it allows for the construction of a new algorithmic treatment of diffusive fracture here we propose an extremely robust operator split scheme that successively updates in a typical time step the history field the crack phase field and finally the displacement field a regularization based on a viscous crack resistance that even enhances the robustness of the algorithm may easily be added the proposed algorithm is considered to be the canonically simple scheme for the treatment of diffusive fracture in elastic solids we demonstrate the performance of the phase field formulation of fracture by means of representative numerical examples

optimal approximations by piecewise smooth functions and associated variational problems abstract this reprint will introduce and study the most basic properties of three new variational problems which are suggested by applications to computer vision in computer vision a fundamental problem is to appropriately decompose the domain r of a function g x y of two variables this problem starts by describing the physical situation which produces images assume that a three dimensional world is observed by an eye or camera from some point p and that g1 rho represents the intensity of the light in this world approaching the point sub 1 from a direction rho if one has a lens at p focusing this light on a retina or a film in both cases a plane domain r in which we may introduce coordinates x y then let g x y be the strength of the light signal striking r at a point with coordinates x y g x y is essentially the same as sub 1 rho possibly after a simple transformation given by the geometry of the imaging syste the function g x y defined on the plane domain r will be called an image what sort of function is g the light reflected off the surfaces si of various solid objects o sub i visible from p will strike the domain r in various open subsets r sub i when one object o1 is partially in front of another object o2 as seen from p but some of object o2 appears as the background to the sides of o1 then the open sets r1 and r2 will have a common boundary the edge of object o1 in the image defined on r and one usually expects the image g x y to be discontinuous along this boundary jhd

a phase field description of dynamic brittle fracture abstract in contrast to discrete descriptions of fracture phase field descriptions do not require numerical tracking of discontinuities in the displacement field this greatly reduces implementation complexity in this work we extend a phase field model for quasi static brittle fracture to the dynamic case we introduce a phase field approximation to the lagrangian for discrete fracture problems and derive the coupled system of equations that govern the motion of the body and evolution of the phase field we study the behavior of the model in one dimension and show how it influences material properties for the temporal discretization of the equations of motion we present both a monolithic and staggered time integration scheme we study the behavior of the dynamic model by performing a number of two and three dimensional numerical experiments we also introduce a local adaptive refinement strategy and study its performance in the context of locally refined t splines we show that the combination of the phase field model and local adaptive refinement provides an effective method for simulating fracture in three dimensions

an augmented lagrangian method for the phase field approach for pressurized fractures abstract in the modeling of pressurized fractures using phase field approaches the irreversibility of crack growth is enforced through an inequality constraint on the temporal derivative of the phase field function in comparison to the classical case in elasticity the presence of the pressure requires the additional constraint and makes the problem much harder to analyze after temporal discretization this induces a minimization problem in each time step over a solution dependent admissible set to avoid solving the resulting variational inequality corresponding to the first order necessary conditions a penalization approach is used commonly to remove the inequality constraint it is well known that for large penalty parameters the algorithm suffers from numerical instabilities in the solution process consequently to avoid such a drawback we propose an augmented lagrangian algorithm for the discrete in time and continuous in space phase field problems the final set of equations is solved in a decoupled fashion the proposed method is substantiated with several benchmark and prototype tests in two and three dimensions

the tetrahedral finite cell method higher order immersogeometric analysis on adaptive non boundary fitted meshes the finite cell method fcm is an immersed domain finite element method that combines higher order non boundary fitted meshes weak enforcement of dirichlet boundary conditions and adaptive quadrature based on recursive subdivision due to its ability to improve the geometric resolution of intersected elements it can be characterized as an immersogeometric method in this paper we extend the fcm so far only used with cartesian hexahedral elements to higher order non boundary fitted tetrahedral meshes based on a reformulation of the octree based subdivision algorithm for tetrahedral elements we show that the resulting tetfcm scheme is fully accurate in an immersogeometric sense that is the solution fields achieve optimal and exponential rates of convergence for h and p refinement if the immersed geometry is resolved with sufficient accuracy tetfcm can leverage the natural ability of tetrahedral elements for local mesh refinement in three dimensions its suitability for problems with sharp gradients and highly localized features is illustrated by the immersogeometric phase field fracture analysis of a human femur bone

sobolev embeddings in orlicz and lorentz spaces with measures abstract embedding theorems for orlicz sobolev spaces into orlicz or orlicz lorentz spaces in domains in r n endowed with a frostman measure are offered parallel embeddings for lorentz sobolev spaces into generalized lorentz spaces are also established the relevant embeddings yield an optimal target norm in the relevant classes whenever it does exist in particular various results available in the literature are improved by special choices of the function spaces and of the measure

a characterization of minimal orlicz sobolev norms in the affine class abstract the orlicz sobolev norm is invariant only under rigid motions but is not invariant under volume preserving affine transformations on r n in this paper we will consider the orlicz sobolev norm in the affine class and characterize its minimum

coupling fluid structure interaction with phase field fracture abstract in this work a concept for coupling fluid structure interaction with brittle fracture in elasticity is proposed the fluid structure interaction problem is modeled in terms of the arbitrary lagrangian eulerian technique and couples the isothermal incompressible navier stokes equations with nonlinear elastodynamics using the saint venant kirchhoff solid model the brittle fracture model is based on a phase field approach for cracks in elasticity and pressurized elastic solids in order to derive a common framework the phase field approach is re formulated in lagrangian coordinates to combine it with fluid structure interaction a crack irreversibility condition that is mathematically characterized as an inequality constraint in time is enforced with the help of an augmented lagrangian iteration the resulting problem is highly nonlinear and solved with a modified newton method e g error oriented that specifically allows for a temporary increase of the residuals the proposed framework is substantiated with several numerical tests in these examples computational stability in space and time is shown for several goal functionals which demonstrates reliability of numerical modeling and algorithmic techniques but also current limitations such as the necessity of using solid damping are addressed

isogeometric collocation for phase field fracture models abstract phase field models based on the variational formulation for brittle fracture have recently been shown capable of accurately and robustly predicting complex crack behavior their numerical implementation requires costly operations at the quadrature point level which may include finding eigenvalues and forming tensor projection operators we explore the application of isogeometric collocation methods for the discretization of second order and fourth order phase field fracture models we show that a switch from isogeometric galerkin to isogeometric collocation methods has the potential to significantly speed up phase field fracture computations due to a reduction of point evaluations we advocate a hybrid collocation galerkin formulation that provides a consistent way of weakly enforcing neumann boundary conditions and multi patch interface constraints is able to handle the multiple boundary integral terms that arise from the weighted residual formulation and offers the flexibility to adaptively improve the crack resolution in the fracture zone we present numerical examples in one and two dimensions that illustrate the advantages of our approach

lipschitz regularity for a general class of quasilinear elliptic equations in convex domains abstract we establish a local lipschitz regularity near the boundary of weak solutions to a general class of homogeneous quasilinear elliptic equations with neumann boundary condition in bounded convex domains

modified newton methods for solving fully monolithic phase field quasi static brittle fracture propagation abstract our goal in this work is to develop and compare modified newton methods for fully monolithic quasi static brittle phase field fracture propagation in variational phase field fracture a smoothed phase field indicator variable denoting the crack path is coupled to elasticity moreover a crack irreversibility condition is incorporated to develop a fully monolithic scheme is an extremely challenging task since the underlying problem is non convex and the jacobian of newton s method is indefinite to split the problem using alternating minimization thus a partitioned approach is a possible resort however there are good reasons to consider monolithic approaches such as for example robustness and efficiency although an error oriented newton method can cope with a larger variety of configurations it appears that this method is not always robust and also not always efficient inspired by nonlinear flow problems as alternative we develop a modified newton scheme in which globalization is based on a dynamic modification of the jacobian matrix rather than utilizing line search or trust region strategies this variation switches smoothly between full newton and newton like steps in several 2d and 3d numerical examples all of them with different characteristic features our modified newton solver is compared to a backtracking line search newton method another line search method monitoring the global energy and allowing for negative curvatures and to already published results of an error oriented version these computations also include further modifications of newton s method and detailed discussions why certain schemes either work or fail revisiting all findings the main outcome of this paper is that the modified newton scheme with jacobian modification is currently the only method that works in a robust and efficient way for all provided examples whereas line search schemes or the error oriented scheme show deficiencies for certain configurations

linear first and second order unconditionally energy stable numerical schemes for the phase field model of homopolymer blends abstract in this paper we develop a series of efficient numerical schemes to solve the phase field model for homopolymer blends the governing system is derived from the energetic variational approach of a total free energy that consists of a nonlinear logarithmic flory huggins potential and a gradient entropy with a concentration dependent de gennes type coefficient the main challenging issue to solve this kind of models numerically is about the time marching problem i e how to develop suitable temporal discretizations for the nonlinear terms in order to preserve the energy stability at the discrete level we solve this issue in this paper by developing the first and second order temporal approximation schemes based on the invariant energy quadratization method where all nonlinear terms are treated semi explicitly consequently the resulting numerical schemes lead to a symmetric positive definite linear system to be solved at each time step the unconditional energy stabilities are further proved various numerical simulations of 2d and 3d are presented to demonstrate the stability and the accuracy of the proposed schemes

a phase field model for cohesive fracture in this paper a phase field model for cohesive fracture is developed after casting the cohesive zone approach in an energetic framework which is suitable for incorporation in phase field approaches the phase field approach to brittle fracture is recapitulated the approximation to the dirac function is discussed with particular emphasis on the dirichlet boundary conditions that arise in the phase field approximation the accuracy of the discretisation of the phase field including the sensitivity to the parameter that balances the field and the boundary contributions is assessed at the hand of a simple example the relation to gradient enhanced damage models is highlighted and some comments on the similarities and the differences between phase field approaches to fracture and gradient damage models are made a phase field representation for cohesive fracture is elaborated starting from the aforementioned energetic framework the strong as well as the weak formats are presented the latter being the starting point for the ensuing finite element discretisation which involves three fields the displacement field an auxiliary field that represents the jump in the displacement across the crack and the phase field compared to phase field approaches for brittle fracture the modelling of the jump of the displacement across the crack is a complication and the current work provides evidence that an additional constraint has to be provided in the sense that the auxiliary field must be constant in the direction orthogonal to the crack the sensitivity of the results with respect to the numerical parameter needed to enforce this constraint is investigated as well as how the results depend on the orders of the discretisation of the three fields finally examples are given that demonstrate grid insensitivity for adhesive and for cohesive failure the latter example being somewhat limited because only straight crack propagation is considered

orlicz sobolev boundary trace embeddings we prove a sharp trace embedding theorem for orlicz sobolev spaces into orlicz spaces on the boundary an optimal trace embedding into more general rearrangement invariant spaces on the boundary is also established

a model for the quasi static growth of brittle fractures existence and approximation results we give a precise mathematical formulation of a variational model for the irreversible quasi static evolution of brittle fractures proposed by g a francfort and j j marigo and based on griffith s theory of crack growth in the two dimensional case we prove an existence result for the quasi static evolution and show that the total energy is an absolutely continuous function of time although we cannot exclude the possibility that the bulk energy and the surface energy may present some jump discontinuities this existence result is proved by a time discretization process where at each step a global energy minimization is performed with the constraint that the new crack contains all cracks formed at the previous time steps this procedure provides an effective way to approximate the continuous time evolution

approximation of functional depending on jumps by elliptic functional via t convergence we show how it is possible to approximate the mumford shah see 29 image segmentation functional by elliptic functionals defined on sobolev spaces the heuristic idea is to consider functionals h u z with z ranging between 0 and 1 and related to the set k the minimizing zh are near to 1 in a neighborhood of the set k and far from the neighborhood they are very small the neighborhood shrinks as h for a similar approach to the problem compare kulkarni see 25 the approximation of h to takes place in a variational sense the de giorgi f convergence

an iterative staggered scheme for phase field brittle fracture propagation with stabilizing parameters abstract this paper concerns the analysis and implementation of a novel iterative staggered scheme for quasi static brittle fracture propagation models where the fracture evolution is tracked by a phase field variable the model we consider is a two field variational inequality system with the phase field function and the elastic displacements of the solid material as independent variables using a penalization strategy this variational inequality system is transformed into a variational equality system which is the formulation we take as the starting point for our algorithmic developments the proposed scheme involves a partitioning of this model into two subproblems phase field and mechanics with added stabilization terms to both subproblems for improved efficiency and robustness we analyze the convergence of the proposed scheme using a fixed point argument and find that under a natural condition the elastic mechanical energy remains bounded and if the diffusive zone around crack surfaces is sufficiently thick monotonic convergence is achieved finally the proposed scheme is validated numerically with several bench mark problems

a diffuse interface method for the navier stokes darcy equations perfusion profile for a patient specific human liver based on mri scans abstract we present a diffuse interface method for coupling free and porous medium type flows modeled by the navier stokes and darcy equations its essential component is a diffuse geometry model generated from the phase field solution of a separate initial boundary value problem that is based on the allen cahn equation phase field approximations of the interface and its gradient are then employed to transfer all interface terms in the coupled variational flow formulation into volumetric terms this eliminates the need for an explicit interface parametrization between the two flow regimes we illustrate accuracy and convergence for a series of benchmark examples using standard low order stabilized finite element discretizations our diffuse interface method is particularly attractive for coupled flow analysis on imaging data with complex implicit interfaces where procedures for deriving explicit surface parametrizations constitute a significant bottleneck we demonstrate the potential of our method to establish seamless imaging through analysis workflows by computing a perfusion profile for a full scale 3d human liver based on mri scans

a marching cubes based failure surface propagation concept for three dimensional finite elements with non planar embedded strong discontinuities of higher order kinematics summary this paper presents an advanced failure surface propagation concept based on the marching cubes algorithm initially proposed in the field of computer graphics and applies it to the embedded finite element method when modeling three dimensional 3d solids at failure the propagation of the failure surface representing a crack or shear band should not exhibit a strong sensitivity to the details of the finite element discretization this results in the need for a propagation of the discrete failure zone through the individual finite elements which is possible for finite elements with embedded strong discontinuities whereas for two dimensional calculations the failure zone propagation location is easily predicted by the maximal principal stress direction more advanced strategies are needed to achieve a smooth failure surface in 3d simulations an example for such method is the global tracking algorithm which predicts the crack path by a scalar level set function computed on the basis of the solution of a simplified heat conduction like problem its prediction may though lead to various scenarios on how the failure surface may propagate through the individual finite elements in particular for a hexahedral eight node finite element 256 such cases exist to capture all those possibilities the marching cubes algorithm is combined with the global tracking algorithm and the finite elements with embedded strong discontinuities in this work in addition because many of the possible cases result in non planar failure surfaces within a single finite element and because the local quantities used to describe the kinematics of the embedded strong discontinuities are physically meaningful in a strict sense only for planar failure surfaces a remedy for such scenarios is proposed various 3d failure propagation simulations outline the performance of the proposed concept copyright 2013 john wiley sons ltd

weighted sobolev spaces and embedding theorems in the present paper we study embedding operators for weighted sobolev spaces whose weights satisfy the well known muckenhoupt a p condition sufficient conditions for boundedness and compactness of the embedding operators are obtained for smooth domains and domains with boundary singularities the proposed method is based on the concept of generalized quasiconformal homeomorphisms homeomorphisms with bounded mean distortion the choice of the homeomorphism type depends on the choice of the corresponding weighted sobolev space such classes of homeomorphisms induce bounded composition operators for weighted sobolev spaces with the help of these homeomorphism classes the embedding problem for non smooth domains is reduced to the corresponding classical embedding problem for smooth domains examples of domains with anisotropic holder singularities demonstrate the sharpness of our machinery comparatively with known results

fourth order phase field model for local max ent approximants applied to crack propagation we apply a fourth order phase field model for fracture based on local maximum entropy lme approximants the higher order continuity of the meshfree lme approximants allows to directly solve the fourth order phase field equations without splitting the fourth order differential equation into two second order differential equations we will first show that the crack surface can be captured more accurately in the fourth order model furthermore less nodes are needed for the fourth order model to resolve the crack path finally we demonstrate the performance of the proposed meshfree fourth order phase field formulation for 5 representative numerical examples computational results will be compared to analytical solutions within linear elastic fracture mechanics and experimental data for three dimensional crack propagation

bezier extraction and adaptive refinement of truncated hierarchical nurbs abstract this contribution presents bezier extraction of truncated hierarchical b splines and the application of the approach to adaptive isogeometric analysis the developed procedures allow for the implementation of hierarchical b splines and nurbs without the need for an explicit truncation of the basis moreover standard procedures of adaptive finite element analysis for error estimation and marking of elements are directly applicable due to the strict use of an element viewpoint starting from a multi level nested mesh that results from uniform h refinement standard bezier extraction is applied to active elements that contribute to the hierarchical approximation this results in a multi level system of equations without communication between individual hierarchy levels a hierarchical subdivision operator is developed to recover this communication by transforming the multi level system of equations into a hierarchical system of equations it is demonstrated that this approach implicitly defines the truncated hierarchical basis in terms of a simple matrix multiplication in this way the implementation effort is reduced to a minimum as shape function routines and bezier extraction procedures remain unchanged compared to standard isogeometric analysis the convergence and the computational efficiency of the approach are examined in three different demonstration problems of heat conduction linear elasticity and the phase field modelling of brittle fracture

nonlocal damage model using the phase field method theory and applications abstract a new nonlocal gradient based damage model is proposed for isotropic elastic damage using the phase field method in order to show the evolution of damage in brittle materials the general framework of the phase field model pfm is discussed and the order parameter is related to the damage variable in continuum damage mechanics cdm the time dependent ginzburg landau equation which is also termed the allen cahn equation is used to describe the damage evolution process specific length scale which addresses the interface region in which the process of changing undamaged solid to fully damaged material microcracks occurs is defined in order to capture the effect of the damaged localization zone a new implicit damage variable is proposed through the phase field theory details of the different aspects and regularization capabilities are illustrated by means of numerical examples and the validity and usefulness of the phase field modeling approach is demonstrated

strong discontinuity approaches an algorithm for robust performance and comparative assessment of accuracy abstract the strong discontinuity approach sda is a popular method to incorporate cracks as displacement discontinuities into finite elements in the first part of the paper different sda formulations denoted as sos kos and skon are assessed numerically based upon different error norms including a norm to evaluate the variational consistency of sda elements results are compared with analytical solutions as well as with results from interface elements and the element erosion technique for cohesionless cracks in the second part of the paper a new method to improve the computational robustness of sda analyses is proposed in addition to using arc length control to solve the nonlinear equation system a sequential un and reloading scheme is proposed with the crack state frozen during unloading to avoid that more than one new crack segment is activated within an increment the performance of the algorithm is demonstrated by numerical analyses of a tension test on a dog bone shaped specimen by means of different variants of the sda and for comparison also using interface elements