isogeometric structural shape optimization an old dilemma in structural shape optimization is the needed tight link between design model or geometric description and analysis model the intention of this paper is to show that isogeometric analysis offers a potential and promising way out of this dilemma to this end we show a structural shape optimization framework based on the isogeometric analysis approach with the discretization based on non uniform rational b splines nurbs the analysis model represents the structural geometry exactly furthermore nurbs enable efficient geometry control together with smooth boundaries they are the de facto standard in cad systems but are also widely used in a shape optimal design context to define the geometry representation and the design variables with the presented isogeometric approach to shape optimization the analysis model is inherently merged with the design model omitting the typically involved interplay between both we derive analytical sensitivities for nurbs discretizations which allow application of efficient gradient based optimization algorithms the present contribution is restricted to two dimensional problems of linear elasticity but the extension to three dimensions and other problem classes is straightforward some representative examples demonstrate and validate the methodology further the potential of boundary continuity control within isogeometric structural shape optimization is explored to trigger smooth or less smooth angular designs

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

a hierarchical approach to adaptive local refinement in isogeometric analysis abstract adaptive local refinement is one of the key issues in isogeometric analysis in this article we present an adaptive local refinement technique for isogeometric analysis based on extensions of hierarchical b splines we investigate the theoretical properties of the spline space to ensure fundamental properties like linear independence and partition of unity furthermore we use concepts well established in finite element analysis to fully integrate hierarchical spline spaces into the isogeometric setting this also allows us to access a posteriori error estimation techniques numerical results for several different examples are given and they turn out to be very promising

rotation free isogeometric thin shell analysis using pht splines this paper presents a novel approach for isogeometric analysis of thin shells using polynomial splines over hierarchical t meshes pht splines the method exploits the flexibility of t meshes for local refinement the main advantage of the pht splines in the context of thin shell theory is that it achieves c1 continuity so the kirchhoff love theory can be used in pristine form no rotational degrees of freedom are needed numerical results show the excellent performance of the present method

functional approach to the error control in adaptive iga schemes for elliptic boundary value problems abstract this work presents a numerical study of functional type a posteriori error estimates for iga approximation schemes in the context of elliptic boundary value problems along with the detailed discussion of the most crucial properties of such estimates we present the algorithm of a reliable solution approximation together with the scheme of an efficient a posteriori error bound generation in this approach we take advantage of b thb spline s high smoothness for the auxiliary vector function reconstruction which at the same time allows to use much coarser meshes and decrease the number of unknowns substantially the most representative numerical results obtained during a systematic testing of error estimates are presented in the second part of the paper the efficiency of the obtained error bounds is analysed from both the error estimation indication and the computational expenses points of view several examples illustrate that functional error estimates alternatively referred to as the majorants and minorants of deviation from an exact solution perform a much sharper error control than for instance residual based error estimates simultaneously assembling and solving routines for an auxiliary variables reconstruction which generate the majorant or minorant of an error can be executed several times faster than the routines for a primal unknown

on the classification of g spheres i equivariant transversality this paper is the first in a series of three stated in geometric terms the papers examine locally linear group actions on spheres for odd order groups g in essentially equivalent homotopy theoretic terms the papers study the homotopy types of the spaces plc v and tope v of equivariant pl homeomorphisms and homeomorphisms of a linear representation v in fact it is the homogeneous spaces f6 v plg v and top6 v pl6 v we study where fc v is the space of proper equivariant homotopy equivalences our results generalize theorems of haefliger kirby siebenmann sullivan and wall and others the link between classification of manifolds and classification of homotopy types is transversality and this is the subject of the present first paper the second paper generalizes wall s classification of fake lens spaces to the classification of g spheres which are equivariantly homotopy equivalent to a given linear action this involves the determination of the equivariant homotopy type of the g space f pl and a complete calculation of the pl equivariant surgery sequence for a linear g sphere as a result we show that the homotopy groups of pl v u plc v vanish in dimensions less than dim v g in the third paper we study g triangulation theory in particular we study the homotopy groups of top6 v u topc v in a range the homotopy groups of the stiefel spaces plc v u plc v and topc v u top6 v are in turn needed for equivariant transversality so the three papers are locked together in an inductive fashion the reader is referred to w 4 below and to the individual parts ii and iii for more information we now give a brief discussion of the content of the present paper let g denote an arbitrary finite group we consider the g transversality question in the locally linear

guaranteed error bounds and local indicators for adaptive solvers using stabilised space time iga approximations to parabolic problems abstract the paper is concerned with space time iga approximations to parabolic initial boundary value problems we deduce guaranteed and fully computable error bounds adapted to special features of such type of approximations and investigate their efficiency the derivation of error estimates is based on the analysis of the corresponding integral identity and exploits purely functional arguments in the maximal parabolic regularity setting the estimates are valid for any approximation from the admissible energy class and do not contain mesh dependent constants they provide computable and fully guaranteed error bounds for the norms arising in stabilised space time approximations furthermore a posterior error estimates yield efficient error indicators enhancing the performance of adaptive solvers and generate very successful mesh refinement procedures theoretical results are verified with a series of numerical examples in which approximate solutions and the corresponding fluxes are recovered by iga techniques the numerical results confirm the high efficiency of the method in the context of the two main goals of a posteriori error analysis estimation of global errors and mesh adaptation

topological classification of systems of bilinear and sesquilinear forms abstract let a and b be two systems consisting of the same vector spaces c n 1 c n t and bilinear or sesquilinear forms a i b i c n k i c n l i c for i 1 s we prove that a is transformed to b by homeomorphisms within c n 1 c n t if and only if a is transformed to b by linear bijections within c n 1 c n t

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

integration by interpolation and look up for galerkin based isogeometric analysis abstract even though isogeometric analysis has a clear advantage regarding the number of degrees of freedom needed to achieve a certain level of accuracy the time needed for matrix assembly by means of numerical integration of stiffness or mass matrix entries constitutes a severe bottleneck in the process in order to overcome this difficulty we propose the new approach of integration by interpolation and lookup iil firstly applying spline interpolation to the common factors in the occurring integrals approximately transforms them into integrals of piecewise polynomial functions whose integrands are expressed in tensor product b spline form the common factors represent the influence of the geometry mapping i e the nurbs domain parameterization and the contributions of possibly non constant material coefficients secondly these integrals are evaluated exactly using pre computed look up tables for integrals of tri products of univariate b splines and their derivatives for the model case of elliptic problems we perform a theoretical analysis to demonstrate that the iil method maintains the overall approximation order of the galerkin discretization provided that the spline interpolation is sufficiently accurate moreover we provide a comparison of the computational complexity of our method with that of a standard gauss quadrature method finally we present experimental results to illustrate the performance of the iil method and to support our theoretical results

nonlinear similarity begins in dimension six the real representations pi and p2 of a finite group g are topologically similar written pi t p2 if there is a homeomorphism h v p1 v pa where v pi denotes the vector space of the representation pi such that h p1 g v p2 g h v for v e v p1 and g e g i e the representation spaces are equivariantly homeomorphic de rham dr conjectured that topological similarity implies the linear equivalence of the two representations the first two authors showed that de rham s conjecture is true for representations of dimension less than six cs6 but false for representations of dimension greater than or equal to nine cs2 here we give examples of nonlinear similarity i e topological similarity between linearly inequivalent representations in dimension six which is therefore the minimal dimension in which nonlinear similarity may occur since linear equivalence is detected by cyclic subgroups a minimal counterexample to de rham s conjecture will occur for g cyclic moreover by hp and mr nonlinear similarity may only occur for groups whose order is divisible by four thus we shall restrict attention to g z4q the cyclic group of order 4q we shall construct six dimensional nonlinear similarities of z4q for every q greater than two moreover we shall show in a later paper cssww2 joint with weinberger that the examples here represent all six dimensional nonlinear similarities of cyclic groups actually we shall give the complete topological classification of representations of the form p 6 c where p is free i e z4q acts freely away from the origin in the representation space of p and 6 and e are the nontrivial and trivial one dimensional representations respectively representations of the form p 6 were studied in cs2 suppose that pi 6 c 1t p2 6 e with p1 and p2 free let lp be the orbit space of the unit

the l theory of laurent extensions and genus 0 function fields the stable classification of quadratic forms over a field is given by the witt group topology via surgery theory has embedded the witt groups in a general theory of forms over any ring with involution in this paper we use geometrically inspired methods to make computations general results on the l theory of a laurent polynomial extension are used to study the witt groups of genus 0 function fields

goal oriented error estimation based on equilibrated flux reconstruction for finite element approximations of elliptic problems abstract we propose an approach for goal oriented error estimation in finite element approximations of second order elliptic problems that combines the dual weighted residual method and equilibrated flux reconstruction methods for the primal and dual problems the objective is to be able to consider discretization schemes for the dual solution that may be different from those used for the primal solution it is only assumed here that the discretization methods come with a priori error estimates and an equilibrated flux reconstruction algorithm a high order discontinuous galerkin dg method is actually the preferred choice for the approximation of the dual solution thanks to its flexibility and straightforward construction of equilibrated fluxes one contribution of the paper is to show how the order of the dg method for asymptotic exactness of the proposed estimator can be chosen in the cases where a conforming finite element method a dg method or a mixed raviart thomas method is used for the solution of the primal problem numerical experiments are also presented to illustrate the performance and convergence of the error estimates in quantities of interest with respect to the mesh size

a subdivision based implementation of the hierarchical b spline finite element method abstract a novel technique is presented to facilitate the implementation of hierarchical b splines and their interfacing with conventional finite element implementations the discrete interpretation of the two scale relation as common in subdivision schemes is used to establish algebraic relations between the basis functions and their coefficients on different levels of the hierarchical b spline basis the subdivision projection technique introduced allows us first to compute all element matrices and vectors using a fixed number of same level basis functions their subsequent multiplication with subdivision matrices projects them during the assembly stage to the correct levels of the hierarchical b spline basis the proposed technique is applied to convergence studies of linear and geometrically nonlinear problems in one two and three space dimensions

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

on the use of adaptive refinement in isogeometric digital image correlation a novel global digital image correlation method was developed using adaptive refinement of isogeometric shape functions non uniform rational b spline shape functions are used because of their flexibility and versatility which enable them to capture a wide range of kinematics the goal of this work was to explore the full potential of isogeometric shape functions for digital image correlation dic this is reached by combining a global dic method with an adaptive refinement algorithm adaptive isogeometric gdic the shape functions are automatically adjusted to be able to describe the kinematics of the sought displacement field with an optimized number of degrees of freedom this results in an accurate method without the need of making problem specific choices regarding the structure of the shape functions which makes the method less user input dependent than regular global dic methods while keeping the number of degrees of freedom limited to realize optimum regularization of the ill posed dic problem the method s accuracy is demonstrated by a virtual experiment with a predefined highly localized displacement field real experiments with a complex sample geometry demonstrate the effectiveness in practice

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

isogeometric analysis using lr b splines abstract the recently proposed locally refined b splines denoted lr b splines by dokken et al 2013 6 may have the potential to be a framework for isogeometric analysis to enable future interoperable computer aided design and finite element analysis in this paper we propose local refinement strategies for adaptive isogeometric analysis using lr b splines and investigate its performance by doing numerical tests on well known benchmark cases the theory behind lr b spline is not presented in full details but the main conceptual ingredients are explained and illustrated by a number of examples

the role of bezier extraction in adaptive isogeometric analysis local refinement and hierarchical refinement summary we present two adaptive refinement techniques namely adaptive local refinement and adaptive hierarchical refinement for isogeometric analysis an element wise point of view is adopted exploiting bezier extraction which facilitates the implementation of adaptive refinement in isogeometric analysis locally refined t splines lr t splines and hierarchical t splines are employed for the description of the geometry as well as for the approximation of the solution space in the analysis the refinement is carried out with the aid of a subdivision operator which is computed by again exploiting the bezier extraction operator the concept and algorithm of an element based adaptive isogeometric analysis are illustrated numerical examples are given to examine the accuracy the convergence and the condition number this article is protected by copyright all rights reserved

guaranteed and sharp a posteriori error estimates in isogeometric analysis we present functional type a posteriori error estimates in isogeometric analysis these estimates derived on functional grounds provide guaranteed and sharp upper bounds of the exact error in the energy norm moreover since these estimates do not contain any unknown generic constants they are fully computable and thus provide quantitative information on the error by exploiting the properties of non uniform rational b splines we present efficient computation of these error estimates the numerical realization and the quality of the computed error distribution are addressed the potential and the limitations of the proposed approach are illustrated using several computational examples

isogeometric analysis of free surface flow abstract this paper presents the first application of isogeometric analysis a new computational technology built on higher order and higher continuity basis functions employed in computer aided design and computer graphics to the computation of free surface phenomena described using the level set approach the method is based on the variational framework that is suitable for discretization by standard finite elements as well as the basis functions employed in isogeometric analysis the underlying numerical formulation globally conserves mass and preserves a sharp air water interface for the entire length of the simulation the numerical tests indicate that the proposed methodology gives an accurate description of the free surface behavior in both quasi steady and dynamic regimes furthermore very good per degree of freedom accuracy is obtained when higher order and higher continuity isogeometric discretizations are employed in free surface computations

on the completeness of hierarchical tensor product b splines abstract given a grid in r d consisting of d bi infinite sequences of hyperplanes possibly with multiplicities orthogonal to the d axes of the coordinate system we consider the spaces of tensor product spline functions of a given degree on a multi cell domain such a domain consists of finite set of cells which are defined by the grid a piecewise polynomial function belongs to the spline space if its polynomial pieces on adjacent cells have a contact according to the multiplicity of the hyperplanes in the grid we prove that the connected components of the associated set of tensor product b splines whose support intersects the multi cell domain form a basis of this spline space more precisely if the intersection of the support of a tensor product b spline with the multi cell domain consists of several connected components then each of these components contributes one basis function in order to establish the connection to earlier results we also present further details relating to the three dimensional case with single knots only a hierarchical b spline basis is defined by specifying nested hierarchies of spline spaces and multi cell domains we adapt the techniques from giannelli and juttler 2013 to the more general setting and prove the completeness of this basis in the sense that its span contains all piecewise polynomial functions on the hierarchical grid with the smoothness specified by the grid and the degrees under certain assumptions on the domain hierarchy finally we introduce a decoupled version of the hierarchical spline basis that allows to relax the assumptions on the domain hierarchy in certain situations such as quadratic tensor product splines the decoupled basis provides the completeness property for any choice of the domain hierarchy

the k theoretic farrell jones conjecture for hyperbolic groups we prove the k theoretic farrell jones conjecture for hyperbolic groups with twisted coefficients in any associative ring with unit

regularizing decompositions for matrix pencils and a topological classification of pairs of linear mappings abstract by kronecker s theorem each matrix pencil a b over a field f is strictly equivalent to its regularizing decomposition i e a direct sum i r d m 1 n 1 m t n t where d is an r r nonsingular matrix and each m i n i is of the form i k j k 0 j k 0 i k l k r k or l k t r k t in which l k and r k are obtained from i k by deleting its last or respectively first row and j k 0 is a singular jordan block we give a method for constructing a regularizing decomposition of an m n matrix pencil a b which is formulated in terms of the linear mappings a b f n f m two m n pencils a b and a b over f r or c are said to be topologically equivalent if the pairs of linear mappings a b f n f m and a b f n f m coincide up to homeomorphisms of the spaces f n and f m we prove that two pencils are topologically equivalent if and only if their regularizing decompositions coincide up to permutation of summands and replacement of d by a nonsingular matrix d such that the linear operators d d f r f r coincide up to a homeomorphism of f r

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

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

adaptive finite element methods for elliptic equations over hierarchical t meshes isogeometric analysis using nurbs non uniform rational b splines as basis functions gives accurate representation of the geometry and the solution but it is not well suited for local refinement in this paper we use the polynomial splines over hierarchical t meshes pht splines to construct basis functions which not only share the nice smoothness properties as the b splines but also allow us to effectively refine meshes locally we develop a residual based a posteriori error estimate for the finite element discretization of elliptic equations using pht splines basis functions and study their approximation properties in addition we conduct numerical experiments to verify the theory and to demonstrate the effectiveness of the error estimate and the high order approximations provided by the numerical solution

least supported bases and local linear independence we introduce the concept of least supported basis which is very useful for numerical purposes we prove that this concept is equivalent to the local linear independence of the basis for any given locally linearly independent basis we characterize all the bases of the space sharing the same property several examples for spline spaces are given

isogeometric analysis of the cahn hilliard phase field model abstract the cahn hilliard equation involves fourth order spatial derivatives finite element solutions are not common because primal variational formulations of fourth order operators are only well defined and integrable if the finite element basis functions are piecewise smooth and globally c 1 continuous there are a very limited number of two dimensional finite elements possessing c 1 continuity applicable to complex geometries but none in three dimensions we propose isogeometric analysis as a technology that possesses a unique combination of attributes for complex problems involving higher order differential operators namely higher order accuracy robustness two and three dimensional geometric flexibility compact support and most importantly the possibility of c 1 and higher order continuity a nurbs based variational formulation for the cahn hilliard equation was tested on two and three dimensional problems we present steady state solutions in two dimensions and for the first time in three dimensions to achieve these results an adaptive time stepping method is introduced we also present a technique for desensitizing calculations to dependence on mesh refinement this enables the calculation of topologically correct solutions on coarse meshes opening the way to practical engineering applications of phase field methodology

a collocated c0 finite element method reduced quadrature perspective cost comparison with standard finite elements and explicit structural dynamics summary we demonstrate the potential of collocation methods for efficient higher order analysis on standard nodal finite element meshes we focus on a collocation method that is variationally consistent and geometrically flexible converges optimally embraces concepts of reduced quadrature and leads to symmetric stiffness and diagonal consistent mass matrices at the same time it minimizes the evaluation cost per quadrature point thus reducing formation and assembly effort significantly with respect to standard galerkin finite element methods we provide a detailed review of all components of the technology in the context of elastodynamics that is weighted residual formulation nodal basis functions on gauss lobatto quadrature points and symmetrization by averaging with the ultra weak formulation we quantify potential gains by comparing the computational efficiency of collocated and standard finite elements in terms of basic operation counts and timings our results show that collocation is significantly less expensive for problems dominated by the formation and assembly effort such as higher order elastostatic analysis furthermore we illustrate the potential of collocation for efficient higher order explicit dynamics throughout this work we advocate a straightforward implementation based on simple modifications of standard finite element codes we also point out the close connection to spectral element methods where many of the key ideas are already established copyright 2014 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

worst case multi objective error estimation and adaptivity this paper introduces a new computational methodology for determining a posteriori multi objective error estimates for finite element approximations and for constructing corresponding quasi optimal adaptive refinements of finite element spaces as opposed to the classical goal oriented approaches which consider only a single objective functional the presented methodology applies to general closed convex subsets of the dual space and constructs a worst case error estimate of the finite element approximation error this worst case multi objective error estimate conforms to a dual weighted residual in which the dual solution is associated with an approximate supporting functional of the objective set at the approximation error we regard both standard approximation errors and data incompatibility errors associated with incompatibility of boundary data with the trace of the finite element space numerical experiments are presented to demonstrate the efficacy of applying the proposed worst case multi objective error estimate in adaptive refinement procedures

a large deformation frictional contact formulation using nurbs based isogeometric analysis this paper focuses on the application of nurbs based isogeometric analysis to coulomb frictional contact problems between deformable bodies in the context of large deformations a mortar based approach is presented to treat the contact constraints whereby the discretization of the continuum is performed with arbitrary order nurbs as well as c0 continuous lagrange polynomial elements for comparison purposes the numerical examples show that the proposed contact formulation in conjunction with the nurbs discretization delivers accurate and robust predictions results of lower quality are obtained from the lagrange discretization as well as from a different contact formulation based on the enforcement of the contact constraints at every integration point on the contact surface copyright 2011 john wiley sons ltd

subdivision stabilised immersed b spline finite elements for moving boundary flows an immersed finite element method is presented to compute flows with complex moving boundaries on a fixed cartesian grid the viscous incompressible fluid flow equations are discretized with b spline basis functions the two scale relation for b splines is used to implement an elegant and efficient technique to satisfy the lbb condition on non grid aligned fluid domains and at moving boundaries the boundary conditions are enforced with a consistent penalty method as originally proposed by nitsche in addition a special extrapolation technique is employed to prevent the loss of numerical stability in presence of arbitrarily small cut cells the versatility and accuracy of the proposed approach is demonstrated by means of convergence studies and comparisons with previous experimental and computational investigations

weak coupling for isogeometric analysis of non matching and trimmed multi patch geometries nitsche s method can be used as a coupling tool for non matching discretizations by weakly enforcing interface constraints we explore the use of weak coupling based on nitsche s method in the context of higher order and higher continuity b splines and nurbs we demonstrate that weakly coupled spline discretizations do not compromise the accuracy of isogeometric analysis we show that the combination of weak coupling with the finite cell method opens the door for a truly isogeometric treatment of trimmed b spline and nurbs geometries that eliminates the need for costly reparameterization procedures we test our methodology for several relevant technical problems in two and three dimensions such as gluing together trimmed multi patches and connecting non matching meshes that contain b spline basis functions and standard triangular finite elements the results demonstrate that the concept of nitsche based weak coupling in conjunction with the finite cell method has the potential to considerably increase the flexibility of the design through analysis process in isogeometric analysis

goal oriented adaptivity using unconventional error representations for the 1d helmholtz equation in this work the error of a given output functional is represented using bilinear forms that are different from those given by the adjoint problem these representations can be employed to design novel h p and h p energy norm and goal oriented adaptive algorithms numerical results in 1d show that for wave propagation problems the advantages of this new representation are notorious when selecting the laplace equation as the dual problem specifically the computed upper bounds of the new error representation are sharper than the classical ones used in both energy norm and goal oriented adaptive methods especially when the dispersion pollution error is significant

topological classification of sesquilinear forms reduction to the nonsingular case abstract two sesquilinear forms c m c m c and c n c n c are called topologically equivalent if there exists a homeomorphism c m c n i e a continuous bijection whose inverse is also a continuous bijection such that x y x y for all x y c m r a horn and v v sergeichuk in 2006 constructed a regularizing decomposition of a square complex matrix a that is a direct sum s a s r j n 1 j n p in which s and r are nonsingular and each j n i is the n i by n i singular jordan block in this paper we prove that and are topologically equivalent if and only if the regularizing decompositions of their matrices coincide up to permutation of the singular summands j n i and replacement of r c r r by a nonsingular matrix r c r r such that r and r are the matrices of topologically equivalent forms c r c r c analogous results for bilinear forms over c and over r are also obtained

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

adaptive mesh refinement strategies in isogeometric analysis a computational comparison abstract we explain four variants of an adaptive finite element method with cubic splines and compare their performance in simple elliptic model problems the methods in comparison are truncated hierarchical b splines with two different refinement strategies t splines with the refinement strategy introduced by scott et al in 2012 and t splines with an alternative refinement strategy introduced by some of the authors in four examples including singular and non singular problems of linear elasticity and the poisson problem the h1 errors of the discrete solutions the number of degrees of freedom as well as sparsity patterns and condition numbers of the discretized problem are compared the quantitative results of this paper can help the reader to choose the most appropriate refinement method for the application

enhancing isogeometric analysis by a finite element based local refinement strategy abstract while isogeometric analysis has the potential to close the gap between computer aided design and finite element methods the underlying structure of nurbs non uniform rational b splines is a weakness when it comes to local refinement we propose a hybrid method that combines a globally c 1 continuous piecewise polynomial finite element basis with rational nurbs mappings in such a way that an isoparametric setting and exact geometry representation are preserved we define this basis over t meshes with a hierarchical structure that allows locally restricted refinement combined with a state of the art a posteriori error estimator we present an adaptive refinement procedure this concept is successfully demonstrated with the laplace equation advection diffusion problems and linear elasticity problems

configuration of separability and tests for multipartite entanglement in bell type experiments we derive tight quadratic inequalities for all kinds of hybrid separable inseparable n particle density operators on an arbitrary dimensional space this methodology enables us to derive a tight quadratic inequality as tests for full n partite entanglement in various bell type correlation experiments on the systems that may not be identified as a collection of qubits e g those involving photons measured by incomplete detectors it is also proved that when the two measured observables are assumed to precisely anticommute a stronger quadratic inequality can be used as a witness of full n partite entanglement

three dimensional mortar based frictional contact treatment in isogeometric analysis with nurbs a three dimensional mortar based frictional contact treatment in isogeometric analysis with nurbs is presented in the finite deformation regime within a setting where the nurbs discretization of the contact surface is inherited directly from the nurbs discretization of the volume the contact integrals are evaluated through a mortar approach where the geometrical and frictional contact constraints are treated through a projection to control point quantities the formulation delivers a non negative pressure distribution and minimally oscillatory local contact interactions with respect to alternative lagrange discretizations independent of the discretization order these enable the achievement of improved smoothness in global contact forces and moments through higher order geometrical descriptions it is concluded that the presented mortar based approach serves as a common basis for treating isogeometric contact problems with varying orders of discretization throughout the contact surface and the volume

device independent witnesses of genuine multipartite entanglement we consider the problem of determining whether genuine multipartite entanglement was produced in an experiment without relying on a characterization of the systems observed or of the measurements performed we present an n partite inequality that is satisfied by all correlations produced by measurements on biseparable quantum states but which can be violated by n partite entangled states such as greenberger horne zeilinger states in contrast to traditional entanglement witnesses the violation of this inequality implies that the state is not biseparable independently of the hilbert space dimension and of the measured operators violation of this inequality does not imply however genuine multipartite nonlocality we show more generically how the problem of identifying genuine tripartite entanglement in a deviceindependent way can be addressed through semidefinite programming

large eddy simulation of turbulent taylor couette flow using isogeometric analysis and the residual based variational multiscale method we present an application of the residual based variational multiscale turbulence modeling rbvms methodology to the computation of turbulent taylor couette flow at high reynolds number we show that the rbvms formulation globally conserves angular momentum a feature that is felt to be important for flows dominated by rotation and that is not shared by standard stabilized formulations of fluid flow weak imposition of dirichlet boundary conditions is employed to enhance the accuracy of the rbvms framework in the presence of thin turbulent boundary layers near solid walls calculation of conservative boundary forces and torques is also presented for the case of weakly enforced boundary conditions nurbs based isogeometric analysis is employed for the spatial discretization and mesh refinement is performed to assess the convergence characteristics of the proposed methodology numerical tests show that very accurate results are obtained on relatively coarse grids to the best of the authors knowledge this paper is the first to report large eddy simulation computations of this challenging test case

adaptive isogeometric analysis by local h refinement with t splines isogeometric analysis based on non uniform rational b splines nurbs as basis functions preserves the exact geometry but suffers from the drawback of a rectangular grid of control points in the parameter space which renders a purely local refinement impossible this paper demonstrates how this difficulty can be overcome by using t splines instead t splines allow the introduction of so called t junctions which are related to hanging nodes in the standard fem obeying a few straightforward rules rectangular patches in the parameter space of the t splines can be subdivided and thus a local refinement becomes feasible while still preserving the exact geometry furthermore it is shown how state of the art a posteriori error estimation techniques can be combined with refinement by t splines numerical examples underline the potential of isogeometric analysis with t splines and give hints for further developments

isogeometric sizing and shape optimisation of beam structures based on the recently proposed framework of isogeometric analysis a study of structural sizing and shape optimisation of curved beam structures is presented the idea of the isogeometric approach is to analyse the structure using its intrinsic and hence exact geometric definition the structural response is approximated using the same basis functions describing the geometry the presented isogeometric framework employs non uniform rational b splines shape changes can be represented by altering both spatial location of control points and the corresponding weights towards the optimal design adaptation of identical geometric representations for both design and analysis resolves the problem of repetitive conversion between the computer aided and facet based geometries used in conventional finite element analysis sizing variables defined at the control points are interpolated using the same basis functions due to the fact that both analysis sizing and shape design share the same underlying description direct sensitivities can be easily evaluated analytically which improves the performance of the optimisation process while sizing should reflect local stress states shape design is preferably performed on a global level thus a multilevel design approach is proposed where shape design is carried on a coarser level projecting the shape design sensitivities bridges the gap between the different levels the proposed design formulation is applied to compliance minimisation problems numerical results indicate the applicability of the method

variational multiscale residual based turbulence modeling for large eddy simulation of incompressible flows we present an les type variational multiscale theory of turbulence our approach derives completely from the incompressible navier stokes equations and does not employ any ad hoc devices such as eddy viscosities we tested the formulation on forced homogeneous isotropic turbulence and turbulent channel flows in the calculations we employed linear quadratic and cubic nurbs a dispersion analysis of simple model problems revealed nurbs elements to be superior to classical finite elements in approximating advective and diffusive processes which play a significant role in turbulence computations the numerical results are very good and confirm the viability of the theoretical framework

adjoint based airfoil optimization with adaptive isogeometric discontinuous galerkin method abstract in this work an adjoint based airfoil shape optimization algorithm is developed based on the adaptive isogeometric discontinuous galerkin method for compressible euler equations to investigate the significance of each design variable of airfoil b spline parameterization we first parameterize the airfoil by b spline curve approximation with some control points viewed as design variables and build the b spline representation of the flow field with the curve to apply the goal oriented h adaptive isogeometric dg method for flow solution then we compute and employ the discrete adjoint solutions for both multi target error estimation in adaptive mesh refinement with the isogeometric nature not only all the geometrical cells but also the numerical basis functions can be analytically expressed by the design variables indicating that the numerical solutions and objective could be differentiable with respect to those variables consequently the gradient is totally computed in an accurate approach and the sensitivity analysis is thus improved by reducing the spatial discretization error and introducing the analytical expression of derivative to reveal the key parameters for optimization in an intuitive and efficient manner although the sqp optimization algorithm is adopted in the paper the given accurate gradient can be applied to any gradient based optimization algorithm the proposed algorithm is demonstrated on rae2822 airfoil with inviscid transonic flow where the shape is optimized to minimize the drag coefficient at a constrained lift and airfoil area the numerical results show that the drag is much more sensitive to the design variables near the tailing edge at the beginning but sensitivity is reduced when optimal

variational localizations of the dual weighted residual estimator the dual weighted residual method dwr and its localization for mesh adaptivity applied to elliptic partial differential equations are investigated the contribution of this paper is twofold first we introduce a novel localization technique based on the introduction of a partition of unity this new technique is very easy to apply as neither strong residuals nor jumps over element edges are required second we compare and analyze theoretically and numerically different localization techniques used for mesh adaptivity with respect to their effectivity here we focus on localizations in variational formulations that do not require the evaluation of the corresponding differential operator in the classical strong formulation in our mathematical analysis we show for different localization techniques established methods and our new approach that the local error indicators used for mesh adaptivity converge with proper order in the error functional several numerical tests substantiate our theoretical investigations

surgery with finite fundamental group the relationship between the topology of manifolds and homotopy theory has been analysed by surgery 1 18 28 31 however the effectiveness of this analysis depends on being able to compute surgery obstructions in this paper we introduce a new technique for evaluating the weakly simple surgery obstructions for degree 1 normal maps of closed manifolds and apply it to oriented manifolds with finite fundamental group our main result is theorem a below let x dx be a finite poincare pair of dimension n and a top bundle over x which is stably fibre homotopy equivalent to the spivak normal fibre space v let m dm be a compact n dimensional manifold and let

family of bell like inequalities as device independent witnesses for entanglement depth we present a simple family of bell inequalities applicable to a scenario involving arbitrarily many parties each of which performs two binary outcome measurements we show that these inequalities are members of the complete set of full correlation bell inequalities discovered by werner wolf ifmmode dot z else z fi ukowski brukner for scenarios involving a small number of parties we further verify that these inequalities are facet defining for the convex set of bell local correlations moreover we show that the amount of quantum violation of these inequalities naturally manifests the extent to which the underlying system is genuinely many body entangled in other words our bell inequalities when supplemented with the appropriate quantum bounds naturally serve as device independent witnesses for entanglement depth allowing one to certify genuine k partite entanglement in an arbitrary n ensuremath ge k partite scenario without relying on any assumption about the measurements being performed or the dimension of the underlying physical system a brief comparison is made between our witnesses and those based on some other bell inequalities as well as quantum fisher information a family of witnesses for genuine k partite nonlocality applicable to an arbitrary n ensuremath ge k partite scenario based on our bell inequalities is also presented

nurbs enhanced finite element method nefem the development of nurbs enhanced finite element method nefem is revisited this technique allows a seamless integration of the cad boundary representation of the domain and the finite element method fem the importance of the geometrical model in finite element simulations is addressed and the benefits and potential of nefem are discussed and compared with respect to other curved finite element techniques

image based goal oriented adaptive isogeometric analysis with application to the micro mechanical modeling of trabecular bone abstract isogeometric analysis iga of geometrically complex three dimensional objects is possible when used in combination with the finite cell method fcm in this contribution we propose a computational methodology to automatically analyze the effective elastic properties of scan based volumetric objects of arbitrary geometric and topological complexity the first step is the reconstruction of a smooth geometry from scan based voxel data using a b spline level set function the second step is a goal oriented adaptive isogeometric linear elastic analysis elements are selected for refinement using dual weighted residual shape function indicators and hierarchical splines are employed to construct locally refined spline spaces the proposed methodology is studied in detail for various numerical test cases including the computation of the effective young s modulus of a trabecular bone micro structure reproduced from ct scan data

goal oriented error estimation and adaptivity for free boundary problems the domain map linearization approach in free boundary problems the accuracy of a goal quantity of interest depends on both the accuracy of the approximate solution and the accuracy of the domain approximation we develop duality based a posteriori error estimates for functional outputs of solutions of free boundary problems that include both sources of error the derivation of an appropriate dual problem linearized adjoint is however nonobvious for free boundary problems to derive an appropriate dual problem we present the domain map linearization approach in this approach the free boundary problem is first transformed into an equivalent problem on a fixed reference domain after which the dual problem is obtained by linearization with respect to the domain map we show for a bernoulli type free boundary problem that this dual problem corresponds to a poisson problem with a nonlocal robin type boundary condition furthermore we present numerical experiments that demonstrate the effectivity of the dual based error estimate and its usefulness in goal oriented adaptive mesh refinement

topological equivalence of linear representations for cyclic groups i in the two parts of this paper we prove that the reidemeister torsion invariants determine topological equivalence of g representations for g a finite cyclic group

the completion of locally refined simplicial partitions created by bisection recently in found comput math 7 2 2007 245 269 we proved that an adaptive finite element method based on newest vertex bisection in two space dimensions for solving elliptic equations which is essentially the method from sinum 38 2000 466 488 by morin nochetto and siebert converges with the optimal rate the number of triangles n in the output partition of such a method is generally larger than the number m of triangles that in all intermediate partitions have been marked for bisection because additional bisections are needed to retain conforming meshes a key ingredient to our proof was a result from numer math 97 2004 219 268 by binev dahmen and devore saying that n n 0 cm for some absolute constant c where n 0 is the number of triangles from the initial partition that have never been bisected in this paper we extend this result to bisection algorithms of n simplices with that generalizing the result concerning optimality of the adaptive finite element method to general space dimensions

simple a posteriori error estimators in adaptive isogeometric analysis in this article we propose two simple a posteriori error estimators for solving second order elliptic problems using adaptive isogeometric analysis the idea is based on a serendipity11according to wikipedia serendipity means a fortunate happenstance or pleasant surprise it was coined by horace walpole in 1754 one aspect of walpole s original definition of serendipity is the need for an individual to be sagacious enough to link together apparently innocuous facts in order to come to a valuable conclusion we feel that this applies for the present discovery but it is of course up to the readers to judge pairing of discrete approximation spaces s h p k m s h p 1 k 1 m where the space s h p 1 k 1 m is considered as an enrichment of the original basis of s h p k m by means of the k refinement a typical unique feature available in isogeometric analysis the space s h p 1 k 1 m is used to obtain a higher order accurate isogeometric finite element approximation and using this approximation we propose two simple a posteriori error estimators the proposed a posteriori error based adaptive h refinement methodology using lr b splines is tested on classical elliptic benchmark problems the numerical tests illustrate the optimal convergence rates obtained for the unknown as well as the effectiveness of the proposed error estimators

mechanically based models adaptive refinement for b spline finite element this article presents two new methods for adaptive refinement of a b spline finite element solution within an integrated mechanically based computer aided engineering system the proposed techniques for adaptively refining a b spline finite element solution are a local variant of np refinement and a local variant of h refinement the key component in the np refinement is the linear co ordinate transformation introduced into the refined element the transformation is constructed in such a way that the transformed nodal configuration of the refined element is identical to the nodal configuration of the neighbour elements therefore the assembly proceeds as with classic finite elements while the solution approximation conforms exactly along the inter element boundaries for the h refinement this transformation is introduced into a construction that merges the super element from the finite element world with the hierarchical b spline representation from the computational geometry in the scope of developing sculptured surfaces the proposed approach supports c0 as well as the hermite b spline c1 continuous shapes for sculptured solids c0 continuity only is considered in this article the feasibility of the proposed methods in the scope of the geometric design is demonstrated by several examples of creating sculptured surfaces and volumetric solids numerical performance of the methods is demonstrated for a test case of the two dimensional poisson equation copyright 2003 john wiley sons ltd