mixed variational principles and robust finite element implementations of gradient plasticity at small strains summary this work outlines a theoretical and computational framework of gradient plasticity based on a rigorous exploitation of mixed variational principles in contrast to classical local approaches to plasticity based on locally evolving internal variables order parameter fields are taken into account governed by additional balance type pdes including micro structural boundary conditions this incorporates non local plastic effects based on length scales which reflect properties of the material micro structure we develop a unified variational framework based on mixed saddle point principles for the evolution problem of gradient plasticity which is outlined for the simple model problem of von mises plasticity with gradient extended hardening softening response the mixed variational structure includes the hardening softening variable itself as well as its dual driving force the numerical implementation exploits the underlying variational structure yielding a canonical symmetric structure of the monolithic problem it results in a novel finite element fe design of the coupled problem incorporating a long range hardening softening parameter and its dual driving force this allows a straightforward local definition of plastic loading unloading driven by the long range fields providing very robust fe implementations of gradient plasticity this includes a rational method for the definition of elastic plastic boundaries in gradient plasticity along with a post processor that defines the plastic variables in the elastic range we discuss alternative mixed fe designs of the coupled problem including a local global solution strategy of short range and long range fields this includes several new aspects such as extended q1p0 type and mini type finite elements for gradient plasticity all methods are derived in a rigorous format from variational principles numerical benchmarks address advantages and disadvantages of alternative fe designs and provide a guide for the evaluation of simple and robust schemes for variational gradient plasticity copyright 2013 john wiley sons ltd

isogeometric analysis of the cahn hilliard equation a convergence study herein we present a numerical convergence study of the cahn hilliard phase field model within an isogeometric finite element analysis framework using a manufactured solution a mixed formulation of the cahn hilliard equation and the direct discretisation of the weak form which requires a c 1 continuous approximation are compared in terms of convergence rates for approximations that are higher than second order in space the direct discretisation is found to be superior suboptimal convergence rates occur when splines of order p 2 are used this is validated with a priori error estimates for linear problems the convergence analysis is completed with an investigation of the temporal discretisation second order accuracy is found for the generalised method this ensures the functionality of an adaptive time stepping scheme which is required for the efficient numerical solution of the cahn hilliard equation the isogeometric finite element framework is eventually validated by two numerical examples of spinodal decomposition

corrigendum to attractors for non dissipative irrotational von karman plates with boundary damping j differential equations 253 12 2012 3568 3609

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

a variational multiscale discontinuous galerkin formulation for both implicit and explicit dynamic modeling of interfacial fracture abstract a stabilized discontinuous galerkin method is presented for modeling interfacial debonding under dynamic conditions the method s distinctive feature is its treatment of the inelastic gap at the interface as an internal variable that is evolved through a traction separation constitutive model enabling initially perfect adhesion to be captured numerical stability is derived using variational multiscale ideas and comparisons are drawn with both the intrinsic cohesive zone method and hybrid discontinuous galerkin cohesive zone method to expose subtle differences the proposed formulation is applied to model the nonlinear response of a composite unit cell under impact loading both explicit and implicit time integration schemes provide similar accuracy and stability and the critical time step for explicit analyses is only slightly reduced below the value for the continuous galerkin mesh without an interface further studies of mesh refinement and adjustments to the initial elastic interface stiffness in the reference cohesive zone model reveal the improved accuracy of the variational multiscale discontinuous galerkin method on coarser grids

isogeometric analysis and hierarchical refinement for higher order phase field models abstract while the interest in higher order models in physics and mechanics grows their numerical simulation still poses a challenge especially for arbitrary shaped three dimensional domains this contribution presents the mathematical framework as well as the application to different problems in the field of material science fracture mechanics and diffusion problems all models under consideration require at least c 1 continuity which prevents the application of standard finite element analysis and local mesh refinements introducing isogeometric analysis iga for the discretization in a finite element framework enables us to deal with these requirements moreover a general hierarchical refinement scheme based on a subdivision projection is presented here for one two and three dimensional simulations this technique allows to enhance the approximation space using finer splines on each level but preserves the partition of unity as well as the continuity properties of the original discretization using this mathematical framework the improved convergence of a kuramoto sivashinsky model a mesh adapted thermal diffusion simulation and computations of a priori unknown crack propagation in different fracture modes underline the versatility of the presented hierarchical refinement scheme

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

new finite elements with embedded strong discontinuities for the modeling of failure in electromechanical coupled solids this paper presents new finite elements with embedded strong discontinuities in the form of jumps in the mechanical displacement field as well as jumps in the electric potential for the modeling of failure in electromechanical coupled solids such materials which are commonly used in smart systems in the form of actuators or sensors are prone to defects in the form of cracks represented by such strong discontinuities in this work a discrete account of the individual jumps is made with the possibility of their propagation through the individual finite elements without the need of remeshing or refinement strategies originated for purely mechanical based materials and commonly referred as the strong discontinuity approach an extension to account for the electromechanical coupling and the additional jumps in the electric potential is made in this work the decomposition of that methodology into a global problem representing here the electromechanical boundary value problem and a local problem through which the strong discontinuities are introduced serves convenient when developing the corresponding finite elements based on the requirement to avoid stress locking phenomena this idea is extended in this work to develop new finite elements based on the incorporation of electrical separation modes directly into the discrete formulation keeping its local and computational efficient property with regard to the storage and possible static condensation of the parameters used to describe the mechanical and electrical discontinuities on a constitutive level the mechanical traction separation law is extended to account for a softening type behavior in the relation between the normal component of the electric displacement field and the surface charge density this is accomplished through the derivation of a localized electromechanical damage model based on the principle of maximum damage dissipation the performance of the resulting new finite elements is illustrated with several numerical simulations including a compact tension test and a three point bending test of piezoelectric ceramics and compared with experimental results available in the literature

formulation and numerical exploitation of mixed variational principles for coupled problems of cahn hilliard type and standard diffusion in elastic solids summary this work develops variational principles for the coupled problem of standard and extended cahn hilliard type species diffusion in solids undergoing finite elastic deformations it shows that the coupled problem of diffusion in deforming solids accounting for phenomena like swelling diffusion induced stress generation and possible phase segregation caused by the diffusing species is related to an intrinsic mixed variational principle it determines the rates of deformation and concentration along with the chemical potential where the latter plays the role of a mixed variable the principle characterizes a canonically compact model structure where the three governing equations involved that is the mechanical equilibrium condition the mass balance for the species content and a microforce balance that determines the chemical potential appear as the euler equation of a variational statement the existence of the variational principle underlines an inherent symmetry of the coupled deformation diffusion problem this can be exploited in the numerical implementation by the construction of time discrete and space discrete incremental potentials which fully determine the update problems of typical time stepping procedures the mixed variational principles provide the most fundamental approach to the monolithic finite element solution of the coupled deformation diffusion problem based on low order basis functions they induce in a natural format the choice of symmetric solvers for newton type iterative updates providing a speedup and reduction of data storage when compared with non symmetric implementations this is a strong argument for the use of the developed variational principles in the computational design of deformation diffusion problems copyright 2014 john wiley sons ltd

conservative numerical schemes for the vlasov equation abstract a new scheme for solving the vlasov equation using a phase space grid is proposed the algorithm is based on the conservation of the flux of particles and the distribution function is reconstructed using various techniques that allow control of spurious oscillations or preservation of the positivity several numerical results are presented in two and four dimensional phase space and the scheme is compared with the semilagrangian method this method is almost as accurate as the semilagrangian one and the local reconstruction technique is well suited for parallel computation

evolution semigroups in supersonic flow plate interactions abstract we consider the well posedness of a model for a flow structure interaction this model describes the dynamics of an elastic flexible plate with clamped boundary conditions immersed in a supersonic flow a perturbed wave equation describes the flow potential the plate s out of plane displacement can be modeled by various nonlinear plate equations including von karman and berger supersonic regimes corresponding to the flow provide for new mathematical challenge that is related to the loss of ellipticity in a stationary dynamics this difficulty is present also in the linear model we show that the linearized model is well posed on the state space as given by finite energy considerations and generates a strongly continuous semigroup we make use of these results along with sharp regularity of airy s stress function obtained by compensated compactness method to conclude global in time well posedness for the fully nonlinear model the proof of generation has two novel features namely 1 we introduce a new flow potential velocity type variable which makes it possible to cover both subsonic and supersonic cases and to split the dynamics generating operator into a skew adjoint component and a perturbation acting outside of the state space performing semigroup analysis also requires a nontrivial approximation of the domain of the generator the latter is due to the loss of ellipticity and 2 we make critical use of hidden trace regularity for the flow component of the model in the abstract setup for the semigroup problem which allows us to develop a fixed point argument and eventually conclude well posedness this well posedness result for supersonic flows in the absence of regularizing rotational inertia has been hereto open the use of semigroup methods to obtain well posedness opens this model to long time behavior considerations

a numerical method for solving the vlasov poisson equation based on the conservative ido scheme we have applied the conservative form of the interpolated differential operator ido cf scheme in order to solve the vlasov poisson equation which is one of the multi moment schemes through numerical tests of the nonlinear landau damping and two stream instability we compared the present scheme with other schemes such as the spline and cip ones we mainly investigated the conservation property of the l1 norm energy entropy and phase space area for each scheme and demonstrated that the ido cf scheme is capable of performing stable long time scale simulation while maintaining high accuracy the scheme is based on an eulerian approach and it can thus be directly used for fokker planck high dimensional vlasov poisson and also guiding center drift simulations aiming at particular problems of plasma physics the benchmark tests for such simulations have shown that the ido cf scheme is superior in keeping the conservation properties without causing serious phase error

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

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

a two level implicit scheme for the numerical solution of the linearized vorticity equation a numerical scheme is developed to study the stability of the linearized vorticity equations the linear growth rates of kelvin helmholtz instabilities resulting from a velocity shear are calculated numerically the theory of neutrally stable waves in two dimensions is extended to include the cases where the stream function is zero or periodic at the boundary and is verified numerically for the case of a velocity profile having the shape of a cosine function the instability growth rates of the oscillations in the neighbourhood of the neutrally stable waves are calculated numerically and are shown to be in very good agreement with the theory the results are also of interest in relation to studies on the instabilities of a two dimensional guiding centre plasma and also in the study of the diocotron instability in electronics

weak and strong solutions of a nonlinear subsonic flow structure interaction semigroup approach abstract we consider a non rotational subsonic flow structure interaction describing the flow of gas above a flexible plate a perturbed wave equation describes the flow and a second order nonlinear plate equation describes the plate s displacement it is shown that the linearization of the model generates a strongly continuous semigroup with respect to the topology generated by finite energy considerations an interesting feature of the problem is that linear perturbed flow structure interaction is not monotone with respect to the standard norm describing the finite energy space the main tool used in overcoming this difficulty is the construction of a suitable inner product on the finite energy space which allows the application of maximal monotone operator theory the obtained result allows us to employ suitable perturbation theory in order to discuss well posedness of weak and strong solutions corresponding to several classes of nonlinear dynamics including the full flow structure interaction with von karman berger s and other semilinear plate equations

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

an adaptive meshfree method for phase field models of biomembranes part i we present an adaptive meshfree method for phase field models of biomembranes max ent smooth approximants can deal with the high order phase field equations the method resolves adaptively the models at an affordable computational cost convergence to sharp interface limit is shown with small regularization parameters we present an adaptive meshfree method to approximate phase field models of biomembranes in such models the helfrich curvature elastic energy the surface area and the enclosed volume of a vesicle are written as functionals of a continuous phase field which describes the interface in a smeared manner such functionals involve up to second order spatial derivatives of the phase field leading to fourth order euler lagrange partial differential equations pde the solutions develop sharp internal layers in the vicinity of the putative interface and are nearly constant elsewhere thanks to the smoothness of the local maximum entropy max ent meshfree basis functions we approximate numerically this high order phase field model with a direct ritz galerkin method the flexibility of the meshfree method allows us to easily adapt the grid to resolve the sharp features of the solutions thus the proposed approach is more efficient than common tensor product methods e g finite differences or spectral methods and simpler than unstructured c 0 finite element methods applicable by reformulating the model as a system of second order pde the proposed method implemented here under the assumption of axisymmetry allows us to show numerical evidence of convergence of the phase field solutions to the sharp interface limit as the regularization parameter approaches zero in a companion paper we present a lagrangian method based on the approximants analyzed here to study the dynamics of vesicles embedded in a viscous fluid

stabilized mixed continuous enriched galerkin formulations for locally mass conservative poromechanics abstract local element wise mass conservation is often highly desired for numerical solutions to coupled poromechanical problems as an efficient numerical method featuring this property mixed continuous galerkin cg enriched galerkin eg finite elements have recently been proposed in which piecewise constant functions are enriched to the pore pressure interpolation functions of the conventional mixed cg cg elements while this enrichment of the pressure space provides local mass conservation it unavoidably alters the stability condition for mixed finite elements because no stabilization method has been available for the new stability condition high order displacement interpolation has been required for mixed cg eg elements if undrained condition is expected to circumvent this requirement here we develop stabilized formulations for the mixed cg eg elements that permit equal order interpolation functions even in the undrained limit we begin by identifying the inf sup condition for mixed cg eg elements by phrasing an enriched poromechanical problem as a twofold saddle point problem we then derive two types of stabilized formulations one based on the polynomial pressure projection ppp method and the other based on the fluid pressure laplacian fpl method a key finding of this work is that both methods lead to stabilization terms that should be augmented only to the cg part of the pore pressure field not to the enrichment part the two stabilized formulations are verified and investigated through numerical examples that involve various conditions spanning 1d to 3d isotropy to anisotropy and homogeneous to heterogeneous domains the methodology presented in this work may also help stabilize other types of mixed finite elements in which the constraint field is enriched by additional functions

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

localizing gradient damage model with decreasing interactions summary nonlocal integral and or gradient enhancements are widely used to resolve the mesh dependency issue with standard continuum damage models however it is reported that whereas the structural response is mesh independent a spurious damage growth is observed accordingly a class of modified nonlocal enhancements is developed in literature where the interaction domain increases with damage in this contribution we adopt a contrary view that the interaction domain decreases with damage this is motivated by the fact that the fracture of quasi brittle materials typically starts as a diffuse network of microcracks before localizing into a macroscopic crack to ensure thermodynamics consistency the micromorphic theory is adopted in the model development the ensuing microforce balance resembles closely the helmholtz expression in a conventional gradient damage model the superior performance of the localizing gradient damage model is demonstrated through a one dimensional problem as well as mode i and ii failure in plane deformation for all three cases a localized deformation band at material failure is obtained this article is protected by copyright all rights reserved

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

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 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

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

smooth attractors of finite dimension for von karman evolutions with nonlinear frictional damping localized in a boundary layer abstract in this paper dynamic von karman equations with localized interior damping supported in a boundary collar are considered hadamard well posedness for von karman plates with various types of nonlinear damping are well known and the long time behavior of nonlinear plates has been a topic of recent interest since the von karman plate system is of hyperbolic type with critical nonlinearity noncompact with respect to the phase space this latter topic is particularly challenging in the case of geometrically constrained nonlinear damping in this paper we first show the existence of a compact global attractor for finite energy solutions and we then prove that the attractor is both smooth and finite dimensional thus the hyperbolic like flow is stabilized asymptotically to a smooth and finite dimensional set

a linearized viscous compressible flow plate interaction with non dissipative coupling abstract we address semigroup well posedness for a linear compressible viscous fluid interacting at its boundary with an elastic plate we derive the model by linearizing the compressible navier stokes equations about an arbitrary flow state so the fluid pde includes an ambient flow profile u in contrast to model in 6 we track the effect of this term at the flow structure interface yielding a velocity matching condition involving the material derivative of the structure this destroys the dissipative nature of the coupling of the dynamics we adopt here a lumer phillips approach with a view of associating fluid structure solutions with a c 0 semigroup e a t t 0 on a chosen finite energy space of data given this approach the challenge becomes establishing the maximal dissipativity of an operator a yielding the flow structure dynamics

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

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

an algorithm based on incompatible modes for the global tracking of strong discontinuities in shear localization analyses abstract numerical methods for predicting localized shear failure in elasto plastic solids have experienced considerable advancements in the last decades among these approaches the so called embedded strong discontinuity esd method is often successfully used to accurately simulate the post localization response with negligible dependence on the finite element discretization however it was observed that the employed discontinuity tracking strategy plays a crucial role in the successful localization analysis in this contribution we propose a novel strategy for the global tracking of discontinuity surfaces it is based on exploiting information obtained from the enhanced parameters employed in enhanced assumed strain eas formulations it is well known that enhanced strain element formulations are able to better capture localized shear deformations as compared to standard finite elements this can be explained as a consequence of the improved performance in bending we observed that the approximation of the strain jumps delimiting the shear band is connected with a deformation field characterized by opposite bending curvatures across these two discontinuities hence in view of the relations existing between the kinematics of strong and weak discontinuities we formulate a proper scalar function of the enhanced parameters to identify potential strong discontinuity surfaces which are evaluated in each step of the analysis with negligible computational cost this proposed approach has a global character as it is based upon evaluating discontinuity surfaces defined in the complete analysis domain that are by construction continuous across elements we demonstrate that the tracking algorithm correctly identifies the potential strong discontinuity surface already in early loading stages even before a localization condition is fulfilled in those elements which are crossed by the potential failure surface and which also satisfy the localization condition the kinematics of embedded strong discontinuities is activated to capture the shear failure surface the performance of the new tracking algorithm is demonstrated by means of several numerical shear localization analyses using associative and non associative drucker prager elastoplastic models to simulate 2 d and 3 d benchmarkanalyses

a hybrid variationally consistent homogenization approach based on ritz s method summary multiscale approaches based on homogenization theory provide a suitable framework to incorporate information associated with a small scale microscale problem into the considered large scale macroscopic problem in this connection the present paper proposes a novel computationally efficient hybrid homogenization method its backbone is a variationally consistent fe2 approach in which every aspect is governed by energy minimization in particular scale bridging is realized by the canonical principle of energy equivalence as a direct implementation of the aforementioned variationally consistent fe2 approach is numerically extensive an efficient approximation based on ritz s method is advocated by doing so the material parameters defining an effective macroscopic material model capturing the underlying microstructure can be efficiently computed furthermore the variational scale bridging principle provides some guidance to choose a suitable family of macroscopic material models comparisons between the results predicted by the novel hybrid homogenization method and full field finite element simulations show that the novel method is indeed very promising for multiscale analyses copyright 2013 john wiley sons ltd

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

a mass conservative scheme for solving the vlasov poisson equation using characteristic curve abstract in this paper we introduce a mass conservative scheme for solving the vlasov poisson equation this scheme is based on an eulerian approach and is constructed using an interpolation scheme with limiters in order to preserve the mass the difference in the values for numerical flux functions on each cell is used for this the flux functions are constructed by preserving both the solution along a characteristics and the mass in each cell we mainly investigate the conservation of l 1 and l 2 norms of the distribution function total energy entropy and minimum value in addition we show that this scheme is bounded on the total variation to demonstrate the efficiency of the proposed scheme this scheme is compared with the flux balance scheme positive and flux conservative scheme umeda s scheme and fifth order weno reconstruction finite volume scheme

high order boundary and finite elements for 3d fracture propagation in brittle materials abstract the quasi static propagation of fracture in brittle materials was studied in several recent publications a variational formulation salvadori 2008 salvadori and carini 2011 salvadori and fantoni 2013 led to three dimensional crack tracking strategies salvadori and fantoni 2014 2 3 salvadori and fantoni 2016 one of the complexities of this new class of algorithms is the evaluation of high order terms of the expansion of the crack opening and sliding in this paper new types of finite and boundary elements are formulated that capture the near crack front asymptotical displacement behavior up to the order 3 2 the use of these elements with the quasi static crack propagation algorithms of the above references is demonstrated for a simple crack configuration

computational homogenization of mesoscale gradient viscoplasticity variationally consistent selective homogenization is applied to a class of gradient enhanced dissipative materials which is adopted for the mesoscale modeling the adopted standard first order homogenization assumption results in the classical equilibrium equation for a local continuum on the macroscale while the internal variables live on the mesoscale only among the issues considered in the paper we note i the variationally consistent setting of homogenization of gradient theory on the mesoscale ii the variational basis of the sve problem sve statistical volume element in the time incremental setting the sve problem is formulated for the classical boundary conditions dirichlet and neumann pertinent to the standard momentum balance as well as the micro momentum balance the weak format of the sve problem constitutes the stationarity condition of an incremental sve potential which represents an extension of the situation for a single phase continuum model the macroscale stress in a given time increment is derivable from an incremental macroscale pseudo elastic strain energy moreover the saddle point properties of the sve potential are shown to form the basis for establishing upper and lower bounds on the pseudo elastic strain energy bingham viscoplasticity with gradient enhanced hardening is chosen as the prototype model problem for the numerical evaluation the computed stress strain response relations confirm the theoretical predictions of the influence of different combinations of boundary conditions on the sve

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

computational design for long term numerical integration of the equations of fluid motion the integral constraints on quadratic quantities of physical importance such as conservation of mean kinetic energy and mean square vorticity will not be maintained in finite difference analogues of the equation of motion for two dimensional incompressible flow unless the finite difference jacobian expression for the advection term is restricted to a form which properly represents the interaction between grid points as derived in this paper it is shown that the derived form of the finite difference jacobian prevents nonlinear computational instability and thereby permits long term numerical integrations

γ convergence for high order phase field fracture continuum and isogeometric formulations abstract we consider high order phase field functionals introduced in borden et al 2014 and provide a rigorous proof that these functionals converge to a sharp crack brittle fracture energy we take into account three dimensional problems in linear elastic fracture mechanics and functionals defined both in sobolev spaces and in spaces of tensor product b splines in the latter convergence holds when the mesh size vanishes faster than the internal length of the phase field model on the theoretical level this condition is natural since the size of the phase field layer around the crack itself scales like the internal length on the numerical level it should be satisfied by local h refinement technically convergence holds in the sense of convergence with respect to the strong topology of l 1 while the sharp crack energy is defined in g s b d 2 the constraint on the phase field to take values in 0 1 is taken into account both in the sobolev setting and in the iso geometric setting in the latter it requires a special treatment since the projection operator on the space of tensor product b splines is not lagrangian i e interpolatory

a new lagrange multiplier approach for gradient flows abstract we propose a new lagrange multiplier approach to design unconditional energy stable schemes for gradient flows the new approach leads to unconditionally energy stable schemes that are as accurate and efficient as the recently proposed sav approach shen xu and yang 2018 but enjoys two additional advantages i schemes based on the new approach dissipate the original energy as opposed to a modified energy in the recently proposed sav approach shen xu and yang 2018 and ii they do not require the nonlinear part of the free energy to be bounded from below as is required in the sav approach the price we pay for these advantages is that a nonlinear algebraic equation has to be solved to determine the lagrange multiplier we present ample numerical results to validate the new approach and as a particular example of challenging applications we consider a block copolymer bcp coupled cahn hilliard model and carry out new and nontrivial simulations which are consistent with experiment results

mixed variational potentials and inherent symmetries of the cahn hilliard theory of diffusive phase separation this work shows that the cahn hilliard theory of diffusive phase separation is related to an intrinsic mixed variational principle that determines the rate of concentration and the chemical potential the principle characterizes a canonically compact model structure where the two balances involved for the species content and microforce appear as the euler equations of a variational statement the existence of the variational principle underlines an inherent symmetry in the two field representation of the cahn hilliard theory this can be exploited in the numerical implementation by the construction of time and space discrete incremental potentials which fully determine the update problems of typical time stepping procedures the mixed variational principles provide the most fundamental approach to the finite element solution of the cahn hilliard equation based on low order basis functions leading to monolithic symmetric algebraic systems of iterative update procedures based on a linearization of the nonlinear problem they induce in a natural format the choice of symmetric solvers for newton type iterative updates providing a speed up and reduction of data storage when compared with non symmetric implementations in this sense the potentials developed are believed to be fundamental ingredients to a deeper understanding of the cahn hilliard theory

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 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

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

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

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

a geometrically regularized gradient damage model with energetic equivalence abstract aiming to bridge the gap between damage and fracture mechanics for modeling localized failure in solids this paper addresses a novel geometrically regularized gradient damage model with energetic equivalence for cracking evolution with the free energy potential defined as usual in terms of the local strain and damage fields the constitutive relations are derived consistently from the standard framework of thermodynamics upon the sharp crack topology geometrically regularized by the damage localization band with a length scale the ensuing energetic equivalence naturally yields the damage evolution law of gradient type and the associated boundary condition of neumann type compared to other gradient damage models no extra assumptions like the nonlocal energy residual and insulation condition are introduced moreover the damage gradient physically accounting for microscopic nonlocal interactions is fully dissipative as expected in line with the unified phase field theory recently proposed by the author wu 2017 during failure processes the material behavior is uniquely characterized by two constitutive functions i e the degradation function defining the free energy potential of the bulk and the geometric function regularizing the sharp crack topology in particular optimal constitutive functions defining an equivalent cohesive zone model of general softening laws are postulated with the involved parameters calibrated from standard material properties the proposed model is numerically implemented into the multi field finite element method and applied to several benchmark tests of concrete under mode i and mixed mode failure it is found that the incorporated length scale can be regarded either a numerical parameter or a material property for the former considered in this work the length scale has negligible if not no effects on the global responses so long as the sharp crack topology and the damage field of high gradients within the localization band are well resolved furthermore the localization bandwidth does not exhibit spurious widening but rather it approaches to a finite value proportional to the length scale comparison between the numerical and experimental results regarding the curve of load versus displacement and crack path confirms validity of the proposed gradient damage model for characterizing localized failure in quasi brittle solids

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

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

a finite element implementation of the nonlocal granular rheology summary inhomogeneous flows involving dense particulate media display clear size effects in which the particle length scale has an important effect on flow fields hence nonlocal constitutive relations must be used in order to predict these flows recently a class of nonlocal fluidity models has been developed for emulsions and subsequently adapted to granular materials these models have successfully provided a quantitative description of experimental flows in many different flow configurations in this work we present a finite element based numerical approach for solving the nonlocal constitutive equations for granular materials which involve an additional non standard nodal degree of freedom the granular fluidity which is a scalar state parameter describing the susceptibility of a granular element to flow our implementation is applied to three canonical inhomogeneous flow configurations 1 linear shear with gravity 2 annular shear flow without gravity and 3 annular shear flow with gravity we verify our implementation demonstrate convergence and show that our results are mesh independent copyright 2016 john wiley sons ltd

multi dimensional fully implicit spectral method for the vlasov maxwell equations with exact conservation laws in discrete form a spectral method for the numerical solution of the multi dimensional vlasov maxwell equations is presented the plasma distribution function is expanded in fourier for the spatial part and hermite for the velocity part basis functions leading to a truncated system of ordinary differential equations for the expansion coefficients moments that is discretized with an implicit second order accurate crank nicolson time discretization the discrete non linear system is solved with a preconditioned jacobian free newton krylov method it is shown analytically that the fourier hermite method features exact conservation laws for total mass momentum and energy in discrete form standard tests involving plasma waves and the whistler instability confirm the validity of the conservation laws numerically the whistler instability test also shows that we can step over the fastest time scale in the system without incurring in numerical instabilities some preconditioning strategies are presented showing that the number of linear iterations of the krylov solver can be drastically reduced and a significant gain in performance can be obtained

conservative form of interpolated differential operator scheme for compressible and incompressible fluid dynamics the proposed scheme which is a conservative form of the interpolated differential operator scheme ido cf can provide high accurate solutions for both compressible and incompressible fluid equations spatial discretizations with fourth order accuracy are derived from interpolation functions locally constructed by both cell integrated values and point values these values are coupled and time integrated by solving fluid equations in the flux forms for the cell integrated values and in the derivative forms for the point values the ido cf scheme exactly conserves mass momentum and energy retaining the high resolution more than the non conservative form of the ido scheme a direct numerical simulation of turbulence is carried out with comparable accuracy to that of spectral methods benchmark tests of riemann problems and lid driven cavity flows show that the ido cf scheme is immensely promising in compressible and incompressible fluid dynamics studies

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

stabilization of von karman plate in the presence of thermal effects in a subsonic potential flow of gas we discuss the problem of nonlinear oscillations of a clamped plate in the presence of thermal effects in a subsonic gas flow the dynamics of the plate is described by von karman system in the presence of thermal effects in which rotational inertia is accounted for to describe influence of the gas flow we apply the linearized theory of potential flows our main result states that each weak solution of the problem considered tends to the set of the stationary points of the problem

damage gradient of damage and principle of virtual power a theory of continuum damage mechanics is developed within the framework of the principle of virtual power because the damage in a solid results from microscopic movements we decided to include the power of these microscopic movements in the power of the internal forces the microscopic velocities are related to the damage rate the power of the internal forces we choose depends on the damage velocity and its gradient to take into account the interactions models issued from this theory are presented they are coherent from the mechanical and mathematical points of view the numerical computations show no mesh sensitivity they describe with good agreement the main experimental properties concrete is chosen as an example to illustrate the theory a model using two damage quantities is also presented it yields the correct description of the unilateral phenomenon observed in concrete finally an extension of the existing models is proposed to describe the fatigue damage behaviour

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

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

on the application of the method of difference potentials to linear elastic fracture mechanics the difference potential method dpm proved to be a very efficient tool for solving boundary value problems bvps in the case of complex geometries it allows bvps to be reduced to a boundary equation without the knowledge of green s functions the method has been successfully used for solving very different problems related to the solution of partial differential equations however it has mostly been considered in regular lipschitz domains in the current paper for the first time the method has been applied to a problem of linear elastic fracture mechanics this problem requires solving bvps in domains containing cracks for the first time dpm technology has been combined with the finite element method singular enrichment functions such as those used within the extended finite element formulations are introduced into the system in order to improve the approximation of the crack tip singularity near optimal convergence rates are achieved with the application of these enrichment functions for the dpm the reduction of the bvp to a boundary equation is based on generalised surface projections the projection is fully determined by the clear trace in the current paper for the first time the minimal clear trace for such problems has been numerically realised for a domain with a cut key words method of difference potentials extended finite element method fracture rate of convergence

a minimization principle for deformation diffusion processes in polymeric hydrogels constitutive modeling and fe implementation abstract this paper presents a recently developed innovative minimization principle for coupled deformation diffusion processes applied to hydrogels and compares this new structure with the classical saddle point formulation in both variational foundation and finite element implementation first balance equations and boundary conditions associated with dissipative fluid transport in solids undergoing large deformation are shown to be rooted in a canonical minimization formulation this two field principle determines the deformation rate and the fluid flux constitutively governed by the scalar free energy and the dissipation potential functions it can be used to derive the well known saddle point formulation by a legendre transformation of the dissipation potential next the variational potential is transformed to its incremental counterpart by means of a discretization in time which offers an intuitive and unconstrained discretization within the finite element method to this end vectorial raviart thomas shape functions are chosen for flux degrees of freedom in order to fulfill the required h d i v b conformity the need for this ansatz space can be interpreted as a counterpart to the lbb condition that arises within the saddle point principle and is usually addressed by mixed element formulations however we are able to demonstrate equivalent or superior performance of the minimization principle in several representative boundary value problems phenomena specific to hydrogels like diffusion induced large volume change with instability patterns in the presence of geometrical constraints are successfully modeled the proposed variational framework can thus be validated and its significance complementary to classical approaches is underlined with the inherent symmetry of the coupled problem as a key feature and consequence of the minimization formulation

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 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

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

accuracy study of the ido scheme by fourier analysis the numerical accuracy of the interpolated differential operator ido scheme is studied with fourier analysis for the solutions of partial differential equations pdes advection diffusion and poisson equations the ido scheme solves governing equations not only for physical variable but also for first order spatial derivative spatial discretizations are based on hermite interpolation functions with both of them in the fourier analysis for the ido scheme the fourier coefficients of the physical variable and the first order derivative are coupled by the equations derived from the governing equations the analysis shows the ido scheme resolves all the wavenumbers with higher accuracy than the fourth order finite difference fd and compact difference cd schemes for advection equation in particular for high wavenumbers the accuracy is superior to that of the sixth order combined compact difference ccd scheme the diffusion and poisson equations are also more accurately solved in comparison with the fd and cd schemes these results show that the ido scheme guarantees highly resolved solutions for all the terms of fluid flow equations

an energetic formulation of a gradient damage model for concrete and its numerical implementation abstract the energetic formulation of a rate independent system assumes that the evolution of the system is driven by two scalar valued functions the storage energy functional and the dissipation pseudo potential the evolution of the material system is then characterized by two energetic principles the global stability condition and the energy balance between stored and dissipated energies with the work of external loading rate independent generalized standard materials are endowed with such structure whereas the existence of such energetic structure is not so apparent in materials with non associated flow rule in this paper we consider a rate independent gradient damage model for concrete where the evolution of the damage does not follow the normality rule we will show that such model can be nevertheless derived by a dissipation potential at the expense of having a state variable dependent potential and therefore the energetic formulation can be obtained also in this case after introducing the incremental minimization problem consistent with such formulation we obtain a discrete version of the stability condition and establish lower and upper a priori energy bounds met by the energetic solution these are fundamental results for the analysis of the formulation the actual numerical solution of the incremental minimization problem is realized by considering first a variable splitting in order to treat the gradient of the damage field as an independent variable and then by applying the augmented lagrangian method to tackle with the resulting constrained optimization problem we solve the first order stationarity conditions of the augmented lagrangian functional by a path following newton s method based on the energy dissipation rate control we show that we are able to describe highly non linear responses of the material such as softening branches and snap back responses several numerical tests are performed to verify the objectivity of the formulation and of the proposed numerical method details of the numerical implementation are also given