duality based two level error estimation for time dependent pdes application to linear and nonlinear parabolic equations we introduce a duality based two level error estimator for linear and nonlinear time dependent problems the error measure can be a space time norm energy norm final time error or other error related functional the general methodology is developed for an abstract nonlinear parabolic pde and subsequently applied to linear heat and nonlinear cahn hilliard equations the error due to finite element approximations is estimated with a residual weighted approximate dual solution which is computed with two primal approximations at nested levels we prove that the exact error is estimated by our estimator up to higher order remainder terms numerical experiments confirm the theory regarding consistency of the dual based two level estimator we also present a novel space time adaptive strategy to control errors based on the new estimator

an isogeometric collocation approach for bernoulli euler beams and kirchhoff plates abstract in this paper iga collocation methods are for the first time introduced for the solution of thin structural problems described by the bernoulli euler beam and kirchhoff plate models in particular a precise description of the proposed methods of the relevant implementation details and of the strategy to efficiently deal with different combinations of boundary conditions is given finally several numerical experiments confirm that the proposed formulations represent an efficient and geometrically flexible tool for the simulation of thin structures

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

local refinement of analysis suitable t splines we develop a local refinement algorithm for analysis suitable t splines which does not produce excessive propagation of control points we then demonstrate its use as an adaptive framework for isogeometric analysis analysis suitable t splines are a class of t splines which are linearly independent and form a partition of unity these properties coupled with local refinement make this class of t splines appealing as a basis for isogeometric analysis

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

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

stable time step estimation for multi material eulerian hydrocodes the time step estimates used in explicit multi material eulerian hydrocodes are based on a series of upper bound approximations that can lead to unnecessarily small time steps since the cost of the calculation is proportional to the number of time steps the underestimation of the time step size has a direct adverse effect on the cost an accurate method for calculating the stable time step size is presented with example calculations

hierarchically refined and coarsened splines for moving interface problems with particular application to phase field models of prostate tumor growth abstract moving interface problems are ubiquitous in science and engineering to develop an accurate and efficient methodology for this class of problems we present algorithms for local h adaptivity of hierarchical b splines to be utilized in isogeometric analysis we extend bezier projection an efficient quadrature free local projection technique to the hierarchical setting in this case extraction operators may not be invertible to address this issue we develop a multi level reconstruction operator which maintains the locality properties of the projection we also introduce a balance parameter to control the overlap of hierarchical functions leading to improved numerical conditioning we apply our algorithms to the simulation of localized prostate cancer growth we model this disease using the phase field method and a set of diffusion reaction equations to account for the dynamics of nutrients and a key biomarker termed prostate specific antigen our results include examples on simple 2d and 3d domains and a more compelling tissue scale patient specific simulation which is run over a prostate anatomy extracted from medical images our methods for local h adaptivity efficiently capture the evolving interface between the tumor and the neighboring healthy tissue with remarkable accuracy in all cases

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 3d isogeometric be fe analysis with dynamic remeshing for the simulation of a deformable particle in shear flows abstract a three dimensional isogeometric coupled boundary element and finite element approach based on analysis suitable t splines is developed for the simulation of deformable capsules suspended in shear flows boundary element analysis is used to solve the fluid stokes equation whereas the hydrodynamic membrane load is computed via isogeometric analysis under the assumption that the membrane is a hyper elastic thin shell with negligible bending resistance the smoothness of the t spline basis functions accommodate large deformations of the capsule without the need for additional smoothing techniques and can be used to accurately compute the membrane load a balanced distribution of membrane elements can be constructed using an unstructured locally refined mesh these properties are coupled with an adaptive temporal implicit integration scheme several benchmark examples are solved to illustrate the accuracy and potential of the method the approach is then applied to simulate the dynamics of a capsule in a real geometry of a brain capillary

tests for a change point summary the problem considered is that of testing a sequence of independent normal random variables with constant known or unknown variance for no change in mean versus alternatives with a single change point various tests such as those based on the likelihood ratio and recursive residuals are studied power approximations are developed by integrating approximations for conditional boundary crossing probabilities a comparison of several tests is made

an isogeometric design through analysis methodology based on adaptive hierarchical refinement of nurbs immersed boundary methods and t spline cad surfaces we explore hierarchical refinement of nurbs as a basis for adaptive isogeometric and immersed boundary analysis we use the principle of b spline subdivision to derive a local refinement procedure which combines full analysis suitability of the basis with straightforward implementation in tree data structures and simple generalization to higher dimensions we test hierarchical refinement of nurbs for some elementary fluid and structural analysis problems in two and three dimensions and attain good results in all cases using the b spline version of the finite cell method we illustrate the potential of immersed boundary methods as a seamless isogeometric design through analysis procedure for complex engineering parts defined by t spline cad surfaces specifically a ship propeller and an automobile wheel we show that hierarchical refinement considerably increases the flexibility of this approach by adaptively resolving local features

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

three dimensional simulation of unstable gravity driven infiltration of water into a porous medium infiltration of water in dry porous media is subject to a powerful gravity driven instability although the phenomenon of unstable infiltration is well known its description using continuum mathematical models has posed a significant challenge for several decades the classical model of water flow in the unsaturated flow the richards equation is unable to reproduce the instability here we present a computational study of a model of unsaturated flow in porous media that extends the richards equation and is capable of predicting the instability and captures the key features of gravity fingering quantitatively the extended model is based on a phase field formulation and is fourth order in space the new model poses a set of challenges for numerical discretizations such as resolution of evolving interfaces stiffness in space and time treatment of singularly perturbed equations and discretization of higher order spatial partial differential operators we develop a numerical algorithm based on isogeometric analysis a generalization of the finite element method that permits the use of globally smooth basis functions leading to a simple and efficient discretization of higher order spatial operators in variational form we illustrate the accuracy efficiency and robustness of our method with several examples in two and three dimensions in both homogeneous and strongly heterogeneous media we simulate for the first time unstable gravity driven infiltration in three dimensions and confirm that the new theory reproduces the fundamental features of water infiltration into a porous medium our results are consistent with classical experimental observations that demonstrate a transition from stable to unstable fronts depending on the infiltration flux

duality and unified analysis of discrete approximations in structural dynamics and wave propagation comparison of p method finite elements with k method nurbs we study the discretization behavior of classical finite element and nurbs approximations on problems of structural vibrations and wave propagation we find that on the basis of equal numbers of degrees of freedom and bandwidth nurbs have superior approximation properties in fact we observe that the high mode behavior of classical finite elements is divergent with the order of approximation a surprisingly negative result on the other hand nurbs offer almost spectral approximation properties and all modes converge with increasing order of approximation

a galerkin characteristic unified finite element method for moving thermal fronts in porous media abstract we investigate the performance of a unified finite element method for the numerical solution of moving fronts in porous media under non isothermal flow conditions the governing equations consist of coupling the darcy equation for the pressure to two convection diffusion reaction equations for the temperature and depth of conversion the aim is to develop a non oscillatory unified galerkin characteristic method for efficient simulation of moving fronts in porous media the method is based on combining the modified method of characteristics with a galerkin finite element discretization of the governing equations the main feature of the proposed unified finite element method is that the same finite element space is used for all solutions to the problem including the pressure velocity temperature and concentration analysis of convergence and stability is also presented in this study and error estimates in the l 2 norm are established for the numerical solutions in addition due to the lagrangian treatment of convection terms the standard courant friedrichs lewy condition is relaxed and the time truncation errors are reduced in the diffusion reaction part we verify the method for the benchmark problem of moving fronts around an array of cylinders the numerical results obtained demonstrate the ability of the proposed method to capture the main flow features

an isogeometric analysis approach to gradient damage models continuum damage formulations are commonly used for the simulation of diffuse fracture processes implicit gradient damage models are employed to avoid the spurious mesh dependencies associated with local continuum damage models the c 0 continuity of traditional finite elements has hindered the study of higher order gradient damage approximations in this contribution we use isogeometric finite elements which allow for the construction of higher order continuous basis functions on complex domains we study the suitability of isogeometric finite elements for the discretization of higher order gradient damage approximations

an adaptive finite element approximation of a variational model of brittle fracture the energy of the francfort marigo model of brittle fracture can be approximated in the sense of gamma convergence by the ambrosio tortorelli functional in this work we formulate and analyze two adaptive finite element algorithms for the computation of its local minimizers for each algorithm we combine a newton type method with residual driven adaptive mesh refinement we present two theoretical results which demonstrate convergence of our algorithms to local minimizers of the ambrosio tortorelli functional

boundary conforming free surface flow computations interface tracking for linear higher order and isogeometric finite elements abstract the simulation of certain flow problems requires a means for modeling a free fluid surface examples being viscoelastic die swell or fluid sloshing in tanks in a finite element context this type of problem can among many other options be dealt with using an interface tracking approach with the deforming spatial domain stabilized space time dsd sst formulation a difficult issue that is connected with this type of approach is the determination of a suitable coupling mechanism between the fluid velocity at the boundary and the displacement of the boundary mesh nodes in order to avoid large mesh distortions one goal is to keep the nodal movements as small as possible but of course still compliant with the no penetration boundary condition standard displacement techniques are full velocity velocity in a specific coordinate direction and velocity in normal direction in this work we investigate how the interface tracking approach can be combined with isogeometric analysis for the spatial discretization if nurbs basis functions of sufficient order are used for both the geometry and the solution both a continuous normal vector as well as the velocity are available on the entire boundary this circumstance allows the weak imposition of the no penetration boundary condition we compare this option with an alternative that relies on strong imposition at discrete points furthermore we examine several coupling methods between the fluid equations boundary conditions and equations for the adjustment of interior control point positions

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

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

a 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

explicit time integration algorithms for structural dynamics with optimal numerical dissipation abstract a new predictor corrector explicit time integration algorithm is presented for solving structural dynamics problems the basis of the algorithm is the implicit generalized method developed by the authors like its implicit parent the explicit generalized method is a one parameter family of algorithms in which the parameter defines the high frequency numerical dissipation for a given value of high frequency dissipation the explicit generalized method minimizes low frequency dissipation the algorithm can be utilized effectively for structural dynamics calculations in which numerical dissipation is needed to reduce spurious oscillations parameter values are given that enable physical damping to be treated explicitly while maintaining second order accuracy without the need for an expensive second corrector pass

isogeometric spline forests in this paper we present isogeometric spline forests an isogeometric spline forest is a hierarchical spline representation capable of representing surfaces or volumes of arbitrarily complex geometry and topological genus spline forests can accomodate arbitrary degree and smoothness in the underlying hierarchical basis as well as non uniform knot interval configurations we describe adaptive h refinement and coarsening algorithms for isogeometric spline forests and develop a bezier extraction framework which provides a simple and efficient single level finite element description of the complex multi level unstructured hierarchical spline basis we then demonstrate the potential of spline forests as a basis for analysis in the context of transient advection diffusion problems where fully integrated adaptivity is demonstrated for the first time in an isogeometric simulation in all cases the adaptive process remains local even in the case of moving fronts and preserves exact geometry at the coarsest level of the discretization the accuracy and robustness of the approach is demonstrated in all cases

goal adaptive isogeometric analysis with hierarchical splines abstract in this work a method of goal adaptive isogeometric analysis is proposed we combine goal oriented error estimation and adaptivity with hierarchical b splines for local h refinement the goal oriented error estimator is computed with a p refined discrete dual space which is adaptively refined alongside the primal space this discrete dual space is proven to be a strict superset of the primal space hierarchical refinements are introduced in marked regions that are formed as the union of chosen coarse level spline supports from the primal basis we present two ways of extracting localized refinement indicators suitable for the hierarchical refinement procedure one based on a partitioning of the dual weighted residual into contributions of basis function supports and one based on the combination of element indicators within a basis function support the proposed goal oriented adaptive strategy is exemplified for the poisson problem and a free surface flow problem numerical experiments on these problems show convergence of the adaptive method with optimal rates furthermore the corresponding goal oriented error estimators are shown to be accurate with effectivity indices in the range of 0 7 1 1

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

liquid vapor transformations with surfactants phase field model and isogeometric analysis surfactants are compounds that find energetically favorable to be located at the boundaries between fluids they are able to modify the properties of those interfaces for example reducing surface tension here we propose a new model for liquid vapor flows with surfactants which captures the dynamics of the surfactant and accounts for phase transformations in the fluid the aforementioned model is derived from a free energy functional by using a coleman noll approach the proposed theory emanates from the isothermal navier stokes korteweg equations which describe single component two phase flow and naturally allow for phase transformations we believe that our model has significant potential to study the influence of surfactants in vaporization and condensation processes from a numerical point of view the proposed model poses significant challenges to existing discretization methods including stiffness in space and time internal and boundary layers as well as higher order partial differential operators to overcome these challenges we propose algorithms based on isogeometric analysis which permit an accurate and efficient discretization finally we illustrate the viability of the theoretical framework and the effectiveness of our algorithms by solving several numerical problems in two and three dimensions

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

isogeometric finite element data structures based on bezier extraction of t splines we develop finite element data structures for t splines based on bezier extraction generalizing our previous work for nurbs as in traditional finite element analysis the extracted bezier elements are defined in terms of a fixed set of polynomial basis functions the so called bernstein basis the bezier elements may be processed in the same way as in a standard finite element computer program utilizing exactly the same data processing arrays in fact only the shape function subroutine needs to be modified while all other aspects of a finite element program remain the same a byproduct of the extraction process is the element extraction operator this operator localizes the topological and global smoothness information to the element level and represents a canonical treatment of t junctions referred to as hanging nodes in finite element analysis and a fundamental feature of t splines a detailed example is presented to illustrate the ideas copyright 2011 john wiley sons ltd

hierarchical t splines analysis suitability bezier extraction and application as an adaptive basis for isogeometric analysis abstract in this paper hierarchical analysis suitable t splines hasts are developed the resulting spaces are a superset of both analysis suitable t splines and hierarchical b splines the additional flexibility provided by the hierarchy of t spline spaces results in simple highly localized refinement algorithms which can be utilized in a design or analysis context a detailed theoretical formulation is presented bezier extraction is extended to hasts simplifying the implementation of hasts in existing finite element codes the behavior of a simple hasts refinement algorithm is compared to the local refinement algorithm for analysis suitable t splines demonstrating the superior efficiency and locality of the hasts algorithm finally hasts are utilized as a basis for adaptive isogeometric analysis

multi level bezier extraction for hierarchical local refinement of isogeometric analysis abstract one of the main topics of research on isogeometric analysis is local refinement among the various techniques currently studied and developed one of the most appealing referred to as hierarchical b splines consists of defining a suitable set of basis functions on different hierarchical levels this strategy can also be improved for example to recover partition of unity resorting to a truncation operation giving rise to the so called truncated hierarchical b splines despite its conceptual simplicity implementing the hierarchical definition of shape functions into an existing code can be rather involved in this work we present a simple way to bring the hierarchical isogeometric concept closer to a standard finite element formulation practically speaking the hierarchy of functions and knot spans is flattened into a sequence of elements being equipped with a standard single level basis in fact the proposed multi level extraction is a generalization of the classical bezier extraction and analogously offers a standard element structure to the hierarchical overlay of functions moreover this approach is suitable for an extension to non linear problems and for a parallel implementation the multi level extraction is presented as a general concept that can be applied to different kinds of refinements and basis functions finally few basic algorithms to compute the local multi level extraction operator for knot insertion on spline spaces are outlined and compared and some numerical examples are presented

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

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

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

a phase field 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

an interactive geometry modeling and parametric design platform for isogeometric analysis in this paper an interactive parametric design through analysis platform is proposed to help design engineers and analysts make more effective use of isogeometric analysis iga to improve their product design and performance we develop several rhinoceros rhino plug ins to take input design parameters through a user friendly interface generate appropriate surface and or volumetric models perform mechanical analysis and visualize the solution fields all within the same computer aided design cad program as part of this effort we propose and implement graphical generative algorithms for iga model creation and visualization based on grasshopper a visual programming interface to rhino the developed platform is demonstrated on two structural mechanics examples an actual wind turbine blade and a model of an integrally bladed rotor ibr in the latter example we demonstrate how the rhino functionality may be utilized to create conforming volumetric models for iga

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

a local solution approach for adaptive hierarchical refinement in isogeometric analysis abstract adaptively refining the state variables and fast solving the equations are two key issues in isogeometric analysis currently existing research more focuses on the first issue while the efficiency issue is not elaborated when the overall equations are solved at each refinement step this article develops an approach to promote the efficiency by locally solving the equations that govern the refined variables at each refinement step here hierarchical refinement of the state variables in nurbs is adopted and transformational matrix that transforms the basis functions from the current level to the refined level is utilized to handle the relations between the equations at different levels by dividing the coefficient matrix into blocks and estimating the error of state variables for different blocks we represent the global solution method in a hierarchical structure and separately construct and solve equations to obtain the solution for each level in the structure we validate the proposed method by some steady heat transfer and structural analysis problems in domains with different dimensionality we have compared the numerical result accuracy as well as the time cost of the proposed method with those of current available refinement approaches and it turns out to be very promising

on the similarities and differences between classical hierarchical truncated hierarchical and lr b splines abstract smooth spline functions such as b splines and nurbs are already an established technology in the field of computer aided design cad and have in recent years been given a lot of attention from the computer aided engineering cae community the advantages of local refinement are obvious for anyone working in either field and as such several approaches have been proposed among others we find the three strategies classical hierarchical b splines truncated hierarchical b splines and locally refined b splines we will in this paper present these three frameworks and highlight similarities and differences between them in particular we will look at the function space they span and the support of the basis functions we will then analyse the corresponding stiffness and mass matrices in terms of sparsity patterns and conditioning numbers we show that the basis functions in general do not span the same space and that conditioning numbers are comparable moreover we show that the weighting needed by the classical hierarchical basis to maintain partition of unity has significant implications on the conditioning numbers

accurate efficient and iso geometrically flexible collocation methods for phase field models we propose new collocation methods for phase field models our algorithms are based on isogeometric analysis a new technology that makes use of functions from computational geometry such as for example non uniform rational b splines nurbs nurbs exhibit excellent approximability and controllable global smoothness and can represent exactly most geometries encapsulated in computer aided design cad models these attributes permitted us to derive accurate efficient and geometrically flexible collocation methods for phase field models the performance of our method is demonstrated by several numerical examples of phase separation modeled by the cahn hilliard equation we feel that our method successfully combines the geometrical flexibility of finite elements with the accuracy and simplicity of pseudo spectral collocation methods and is a viable alternative to classical collocation methods

an energy based criterion of crack branching and its application on the multidimensional space method abstract in this paper a criterion of crack branching based on the energy release rate is proposed theory of multidimensional space method for crack branching is established in the framework of the extended finite element method xfem in which the formula of the boundary shift energy is deduced the boundary cracking energy is obtained because the boundary shift model is abstracted into the boundary cracking model the energy release rates of the crack branching in four quadrants are determined based on boundary cracking energy further the energy release rate of the crack branching is introduced into multidimensional space method to determine the crack branching if the crack tip branching the criteria of crack branching are applied to determine the direction of the branching cracks otherwise the criteria of maximum circumferential stress are applied to determine the direction of the cracks moreover a classical numerical example is investigated to verify the correctness of the criterion of crack branching in the framework of the multidimensional space method

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

a hybrid igafem igabem formulation for two dimensional stationary magnetic and magneto mechanical field problems abstract isogeometric analysis iga can bridge the gap between geometrical and numerical modelling to this end the same functions used in computer aided design are applied to represent geometry and approximate field variables in the numerical model the concept has already been implemented to solve field problems using finite element fem and boundary element methods bem but coupling of both methods has not been applied in the current work an isogeometric fem bem coupling is proposed and applied to two dimensional stationary magnetic field problems while fem is used to model magnetisable bodies allowing for heterogeneous structures and non linear constitutive behaviour the bem domain accounts for the surrounding free space both methods are coupled on the surface of the magnetisable body due to this hybrid igafem bem approach no meshing of free space is necessary and truncation errors are avoided for problems to be solved on open infinite or semi infinite domains once the solution for the magnetic problem is obtained using the hybrid method igafem is used to solve a magneto mechanical field problem with one sided coupling in a subdomain of the magnetic problem this one sided coupling is realised by a magnetic stress tensor computed from the solution of the stationary magnetic field problem from the comparison of error norms and convergence rates for nurbs and discretisations based on lagrangian polynomials smaller errors and similar convergence rates are found for the proposed method for the same polynomial order of the basis functions and a comparable mesh size

isogeometric fluid structure interaction analysis with emphasis on non matching discretizations and with application to wind turbines abstract in this paper we develop a framework for fluid structure interaction fsi modeling and simulation with emphasis on isogeometric analysis iga and non matching fluid structure interface discretizations we take the augmented lagrangian approach to fsi as a point of departure here the lagrange multiplier field is defined on the fluid structure interface and is responsible for coupling of the two subsystems thus the fsi formulation does not rely on the continuity of the underlying function spaces across the fluid structure interface in order to produce the correct coupling conditions between the fluid and structural subdomains however in deriving the final fsi formulation the interface lagrange multiplier is formally eliminated and the formulation is written purely in terms of primal variables avoiding the use of lagrange multipliers adds efficiency to the proposed formulation as an added benefit the ability to employ non matching grids for multi physics simulations leads to significantly relaxed requirements that are placed on the geometry modeling and meshing tools for iga we show an application of the proposed fsi formulation to the simulation of the nrel 5 mw offshore wind turbine rotor where the aerodynamics domain is modeled using volumetric quadratic nurbs while the rotor structure is modeled using a cubic t spline based discretization of a rotation free kirchhoff love shell we conclude the article by showing fsi coupling of a t spline shell with a low order finite element method fem discretization of the aerodynamics equations this combined use of iga and fem is felt to be a good balance between speed robustness and accuracy of fsi simulations for this class of problems

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

efficient quadrature for nurbs based isogeometric analysis abstract we initiate the study of efficient quadrature rules for nurbs based isogeometric analysis a rule of thumb emerges the half point rule indicating that optimal rules involve a number of points roughly equal to half the number of degrees of freedom or equivalently half the number of basis functions of the space under consideration the half point rule is independent of the polynomial order of the basis efficient rules require taking into account the precise smoothness of basis functions across element boundaries several rules of practical interest are obtained and a numerical procedure for determining efficient rules is presented we compare the cost of quadrature for typical situations arising in structural mechanics and fluid dynamics the new rules represent improvements over those used previously in isogeometric analysis

a hybrid variational collocation immersed method for fluid structure interaction using unstructured t splines summary we present a hybrid variational collocation immersed and fully implicit formulation for fluid structure interaction fsi using unstructured t splines in our immersed methodology we define an eulerian mesh on the whole computational domain and a lagrangian mesh on the solid domain which moves arbitrarily on top of the eulerian mesh mathematically the problem reduces to solving three equations namely the linear momentum balance mass conservation and a condition of kinematic compatibility between the lagrangian displacement and the eulerian velocity we use a weighted residual approach for the linear momentum and mass conservation equations but we discretize directly the strong form of the kinematic relation deriving a hybrid variational collocation method we use t splines for both the spatial discretization and the information transfer between the eulerian mesh and the lagrangian mesh t splines offer us two main advantages against non uniform rational b splines they can be locally refined and they are unstructured the generalized method is used for the time discretization we validate our formulation with a common fsi benchmark problem achieving excellent agreement with the theoretical solution an example involving a partially immersed solid is also solved the numerical examples show how the use of t junctions and extraordinary nodes results in an accurate efficient and flexible method copyright 2015 john wiley sons ltd

isogeometric collocation cost comparison with galerkin methods and extension to adaptive hierarchical nurbs discretizations abstract we compare isogeometric collocation with isogeometric galerkin and standard c 0 finite element methods with respect to the cost of forming the matrix and residual vector the cost of direct and iterative solvers the accuracy versus degrees of freedom and the accuracy versus computing time on this basis we show that isogeometric collocation has the potential to increase the computational efficiency of isogeometric analysis and to outperform both isogeometric galerkin and standard c 0 finite element methods when a specified level of accuracy is to be achieved with minimum computational cost we then explore an adaptive isogeometric collocation method that is based on local hierarchical refinement of nurbs basis functions and collocation points derived from the corresponding multi level greville abscissae we introduce the concept of weighted collocation that can be consistently developed from the weighted residual form and the two scale relation of b splines using weighted collocation in the transition regions between hierarchical levels we are able to reliably handle coincident collocation points that naturally occur for multi level greville abscissae the resulting method combines the favorable properties of isogeometric collocation and hierarchical refinement in terms of computational efficiency local adaptivity robustness and straightforward implementation which we illustrate by numerical examples in one two and three dimensions

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