a new cloud based hp finite element method a hybrid computational method for solving boundary value problems is introduced which combines features of the meshless hp cloud methods with features of conventional finite elements the method admits straightforward nonuniform hp type approximations easy implementation of essential boundary conditions is robust under severe distortions of the mesh and can deliver exponential rates of convergence results of numerical experiments are presented

isogeometric analysis using t splines we explore t splines a generalization of nurbs enabling local refinement as a basis for isogeometric analysis we review t splines as a surface design methodology and then develop it for engineering analysis applications we test t splines on some elementary two dimensional and three dimensional fluid and structural analysis problems and attain good results in all cases we summarize the current status of t splines their limitations and future possibilities

immersed b spline i spline finite element method for geometrically complex domains a novel b spline based immersed finite element method is introduced for the computation of geometrically and topologically complex problems the geometry description and the finite element analysis rely on a block structured logically cartesian mesh which encloses the domain of interest a signed distance function is used for representing the domain on the cartesian mesh whereby the domain boundary is the zeroth level set of the signed distance function away from the domain boundaries the standard b spline basis functions are used for the finite element interpolation close to domain boundaries a new approach has been developed for modifying the b spline basis functions so that they locally interpolate the dirichlet boundary conditions the efficiency and robustness of the proposed approach is demonstrated with a number of one two and three dimensional linear boundary value problems

isogeometric collocation methods with generalized b splines we introduce isogeometric collocation methods based on generalized b splines and we analyze their performance through numerical examples for univariate and multivariate scalar and vector valued problems in particular advection diffusion and linear elasticity model problems are addressed the resulting method combines the favorable properties of isogeometric collocation and the geometrical and analytical problem oriented advantages of generalized b splines

applying the hp d version of the fem to locally enhance dimensionally reduced models abstract a modification of the hp d method is presented which allows to locally enhance a dimensionally reduced finite element approximation with an ansatz based on a more accurate i e dimensionally higher model the procedure is based on a hierarchical domain decomposition which is applied to couple discretizations of different types of mechanical models like beams plates and solids thereby the error of the underlying mathematical model can be reduced in critical regions by introducing locally a physically more sophisticated model two examples are given to demonstrate the basic properties of the proposed method in the first example the solution of a one dimensional beam model is improved locally by a two dimensional finite element computation in the second example the local enhancement of a reissner mindlin plate by a fully three dimensional continuum approach is considered yielding a significant reduction of the model error

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

subdivision surfaces a new paradigm for thin shell finite element analysis we develop a new paradigm for thin shell finite element analysis based on the use of subdivision surfaces for i describing the geometry of the shell in its undeformed configuration and ii generating smooth interpolated displacement fields possessing bounded energy within the strict framework of the kirchhoff love theory of thin shells the particular subdivision strategy adopted here is loop s scheme with extensions such as required to account for creases and displacement boundary conditions the displacement fields obtained by subdivision are h2 and consequently have a finite kirchhoff love energy the resulting finite elements contain three nodes and element integrals are computed by a one point quadrature the displacement field of the shell is interpolated from nodal displacements only in particular no nodal rotations are used in the interpolation the interpolation scheme induced by subdivision is non local i e the displacement field over one element depend on the nodal displacements of the element nodes and all nodes of immediately neighbouring elements however the use of subdivision surfaces ensures that all the local displacement fields thus constructed combine conformingly to define one single limit surface numerical tests including the belytschko et al 10 obstacle course of benchmark problems demonstrate the high accuracy and optimal convergence of the method

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

sequential convergence in fuzzy neighborhood spaces abstract we introduce the notions of sequential convergence and the first axiom of countability in fuzzy neighborhood spaces we show that these notions fulfil good extension criterion attributed to lowen convergent sequences and cauchy sequences are studied in the light of fuzzy probabilistic pseudometric space

multigrid methods for isogeometric analysis with thb splines we propose a geometric multigrid algorithm for the solution of the linear systems which arise when using hierarchical spline spaces for isogeometric discretizations of elliptic partial differential equations in particular we focus on the use of hierarchical b spline hb and truncated hierarchical b spline thb bases to implement adaptively refined isogeometric discretizations we describe an approach for constructing the nested sequences of hierarchical spline spaces which are required for our geometric multigrid solver furthermore we give an algorithm for the computation of the so called prolongation matrices which map coarser space functions to their finer space representations for both hb and thb spline bases in several numerical experiments we study two dimensional poisson problems with singularities and employ a posteriori error estimators for the adaptive refinement of the hierarchical spline spaces using standard multigrid smoothers we observe that using thb spline bases consistently confers a dramatic advantage in terms of iteration numbers over the use of hb spline bases in particular for higher spline degrees when used as a preconditioner for conjugate gradient iteration the proposed multigrid method exhibits practical iteration numbers for two dimensional problems up to the tested case of fourth degree splines

a stabilised immersed boundary method on hierarchical b spline grids for fluid rigid body interaction with solid solid contact abstract an accurate efficient and robust numerical scheme is presented for the simulation of the interaction between flexibly supported rigid bodies and incompressible fluid flow with topology changes and solid solid contact the solution of the incompressible navier stokes equations is approximated by employing a stabilised formulation on cartesian grids discretised with hierarchical b splines the solid is modelled as a rigid body and represented by linear segments along its boundary kinematic conditions along the fluid rigid body interface are enforced weakly using nitsche s method while ghost penalty operators are employed to avoid excessive ill conditioning of the system matrix arising from small cut cells a staggered scheme is used for resolving the coupled fluid rigid body interaction the contact between moving or moving and fixed solid bodies is modelled with lagrange multipliers the excellent performance and wide range of applicability of the proposed scheme are demonstrated in a number of benchmark tests as well as industrially relevant model problems the examples cover the galloping phenomena particulate flow hydraulic check valves and a model turbine

3d composite finite elements for elliptic boundary value problems with discontinuous coefficients for scalar and vector valued elliptic boundary value problems with discontinuous coefficients across geometrically complicated interfaces a composite finite element approach is developed composite basis functions are constructed mimicking the expected jump condition for the solution at the interface in an approximate sense the construction is based on a suitable local interpolation on the space of admissible functions we study the order of approximation and the convergence properties of the method numerically as applications heat diffusion in an aluminum foam matrix filled with polymer and linear elasticity of microstructured materials in particular specimens of trabecular bone are investigated furthermore a numerical homogenization approach is developed for periodic structures and real material specimens which are not strictly periodic but are considered as statistical prototypes thereby effective macroscopic material properties can be computed

isogeometric analysis for compound b spline surfaces abstract in this work an isogeometric analysis method for compound b spline surfaces is proposed the main advantage of the proposed method is that it directly utilizes the compound cad surface model generated with surface boolean operations in cad systems to achieve the analysis without additional efforts such as representing the cad model with multiple tensor product patches for this an integration scheme for trimmed elements is presented first which can well handle the geometric complexities of cells within a trimmed element then based on b spline representations of the curves on surfaces the displacement continuity requirement along an intersection curve is realized by the degree of freedom dof coupling of the intersected surfaces in which the dof coupling constraint is enforced by minimizing the augmented potential energy with lagrange multiplier the stationarity of the augmented potential energy yields the equilibrium equations for the displacements of the compound surfaces the validity of the proposed integration scheme is demonstrated with the convergence study on the problem of a hole within an infinite plate and the effectiveness of the proposed isogeometric analysis method is illustrated with an elaborated 3d compound surface shell structure constructed by boolean combination of several surfaces

on fuzzy neighbourhood spaces abstract we study the behaviour of the fuzzy closure operator of a fuzzy neighbourhood space fns for short with regard to the two operations of sum and conjunction on fuzzy subsets we supply sufficient conditions on a cech fuzzy closure operator to be the fuzzy closure operator of a fns finally we study quotient fuzzy neighbourhood systems with the help of the notion of dual fuzzy neighbourhood spaces of 11

optimal and reduced quadrature rules for tensor product and hierarchically refined splines in isogeometric analysis abstract we continue the study initiated in hughes et al 2010 in search of optimal quadrature rules for tensor product and hierarchically refined splines in isogeometric analysis these rules are optimal in the sense that there exists no other quadrature rule that can exactly integrate the elements of the given spline space with fewer quadrature points we extend the algorithm presented in hughes et al 2010 with an improved starting guess which combined with arbitrary precision arithmetic results in the practical computation of quadrature rules for univariate non uniform splines up to any precision explicit constructions are provided in sixteen digits of accuracy for some of the most commonly used uniform spline spaces defined by open knot vectors we study the efficacy of the proposed rules in the context of full and reduced quadrature applied to two and three dimensional diffusion reaction problems using tensor product and hierarchically refined splines and prove a theorem rigorously establishing the stability and accuracy of the reduced rules

a new fuzzy compactness defined by fuzzy nets compactness is one of the most important notions in topology and is thoroughly investigated in general topology but it seems that the research about this notion in the area of fuzzy topology is not so effective as in general topology in fact it would seem that there was not even a good definition of compactness in this area until recently indeed soon after the famous article written by zadeh 8 was published chang 1 introduced the notion of fuzzy topological spaces and at the same time a certain fuzzy compactness was given from then on a series of different notions of fuzzy compactness have been discussed 2 3 71 and in 5 they were compared in detail but it seems unsatisfactory that all these fuzzy compactness ideas are defined for the whole fuzzy topological space but not for an arbitrary fuzzy subset therefore they are far from the crisp compactness for example it is impossible to discuss the hereditary property with respect to closed subsets of fuzzy compactness in 1977 pu and liu introduced the notion of fuzzy nets and a new concept of so called q neighborhood was given which could reflect the features of neighborhood structure in fuzzy topological spaces and by this new neighborhood structure the moore smith convergence theory was established spendidly 6 it furnishes us a basic tool to investigate the fuzzy compactness in a more general way to investigate the compactness of arbitrary fuzzy subsets in this paper a new fuzzy compactness defined by fuzzy nets we shall call it n compactness is given but we introduce the notion of r neighborhood see section 2 instead of qneighborhood to establish the moore smith convergence theory for it is perhaps more convennient to use n compactness possesses the following properties 1 n compactness is defined for arbitrary fuzzy subsets 2 for n compactness the hereditary property with respect to closed subsets holds 3 n compactness is a good extension 5 4 the continuous images of ncompact sets are n compact 5 for n compactness the tychonoff product theorem holds 6 if an n compact fuzzy topological space is t2 6 then it will satisfy a stronger separation axiom which will be called t4 in this paper 1 0022 247x 83 3 00

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

arc transitive abelian regular covers of cubic graphs abstract quite a lot of attention has been paid recently to the construction of edge or arc transitive covers of symmetric graphs in most cases the approach has involved voltage graph techniques which are excellent for finding regular covers in which the group of covering transformations is either cyclic or elementary abelian or more generally homocyclic but are not so easy to use when the covering group has other forms even when it is abelian but not homocyclic in this paper a different approach is introduced that can be used more widely this new approach takes a universal group for the action of the automorphism group of the base graph and uses reidemeister schreier theory to obtain a presentation for a universal covering group and some representation theory and other methods for determining suitable quotients this approach is then used to find all arc transitive abelian regular covers of k 4 k 3 3 the cube q 3 and the petersen graph a sequel will do the same for the heawood graph

reduced bezier element quadrature rules for quadratic and cubic splines in isogeometric analysis abstract we explore the use of various element based reduced quadrature strategies for bivariate and trivariate quadratic and cubic spline elements used in isogeometric analysis the rules studied encompass tensor product gauss and gauss lobatto rules and certain so called monomial rules that do no possess a tensor product structure the objective of the study is to determine quadrature strategies which enjoy the same accuracy and stability behavior as full gauss quadrature but with significantly fewer quadrature points several cases emerge that satisfy this objective and also demonstrate superior efficiency compared with standard c 0 continuous finite elements of the same order

sequential compactness in fuzzy neighborhood spaces abstract the present work deals with introducing and studying the notions of n compactness and n sequential compactness in fuzzy neighbourhood spaces we show that the notion of n sequential compactness fulfils a good extension criterion attributed to lowen j math anal appl 64 1978 446 454 and agrees it well some properties of these concepts are discussed

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

analysing finite locally s arc transitive graphs we present a new approach to analysing finite graphs which admit a vertex intransitive group of automorphisms g and are either locally g s arc transitive for s 2 or g locally primitive such graphs are bipartite with the two parts of the bipartition being the orbits of g given a normal subgroup n which is intransitive on both parts of the bipartition we show that taking quotients with respect to the orbits of n preserves both local primitivity and local s arc transitivity and leads us to study graphs where g acts faithfully on both orbits and quasiprimitively on at least one we determine the possible quasiprimitive types for g in these two cases and give new constructions of examples for each possible type the analysis raises several open problems which are discussed in the final section

a posteriori error estimation and adaptivity for elastoplasticity using the reciprocal theorem we present a posteriori error estimators and adaptive methods for the finite element approximation of nonlinear problems and especially elastoplasticity the main characteristic of the proposed method is the introduction of duality techniques or in other notions the reciprocal theorem for error estimation at an equilibrium point the nonlinear boundary value problem and an additional linearized dual problem are considered the loading of the dual problem is specifically designed for capturing the influence of the errors of the entire domain to the considered variable our approach leads to easy computable refinement indicators for locally or integrally defined variables for instationary problems as elastoplasticity in a first step we neglect the errors due to time discretization and evaluate the error indicators within each time step for a stationary problem the versatility of the presented framework is demonstrated with numerical examples

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

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

linear criteria for lifting automorphisms of elementary abelian regular coverings abstract for a given finite connected graph a group h of automorphisms of and a finite group a a natural question can be raised as follows find all the connected regular coverings of having a as its covering transformation group on which each automorphism in h can be lifted in this paper we investigate the regular coverings with a z p n an elementary abelian group and get some new matrix theoretical characterizations for an automorphism of the base graph to be lifted as one of its applications we classify all the connected regular covering graphs of the petersen graph satisfying the following two properties 1 the covering transformation group is isomorphic to the elementary abelian p group z p n and 2 the group of fibre preserving automorphisms of a covering graph acts arc transitively as a byproduct some new 2 and 3 arc transitive graphs are constructed

arc transitive elementary abelian covers of the complete graph k5 abstract all pairwise non isomorphic connected arc transitive p elementary abelian covers of the complete graph k 5 are constructed using the techniques developed by malnic marusic and potocnik as a result six infinite families of graphs for different values of p two sporadic graphs for p 2 and 12 sporadic graphs for p 5 are described in terms of voltage assignments valued in the elementary abelian group z p k

an unfitted hp adaptive finite element method based on hierarchical b splines for interface problems of complex geometry generating finite element discretizations with direct interface parameterizations constitutes a considerable computational expense in case of complex interface geometries the paper at hand introduces a b spline finite element method which circumvents parameterization of interfaces and offers fast and easy meshing irrespective of the geometric complexity involved its core idea is the adaptive approximation of discontinuities by hierarchical grid refinement which adds several levels of local basis functions in the close vicinity of interfaces but unfitted to their exact location so that a simple regular grid of knot span elements can be maintained numerical experiments show that an hp refinement strategy which simultaneously increases the polynomial degree of the b spline basis and the levels of refinement around interfaces achieves exponential rates of convergence despite the presence of discontinuities it is also demonstrated that the hierarchical b spline fem can be used to transfer the recently introduced finite cell concept to geometrically nonlinear problems its computational performance imposition of unfitted boundary conditions and fast hierarchical grid generation are illustrated for a set of benchmark problems in one two and three dimensions and the advantages of the regular grid approach for complex geometries are demonstrated by the geometrically nonlinear simulation of a voxel based foam composite

imposing essential boundary conditions in mesh free methods imposing essential boundary conditions is a key issue in mesh free methods the mesh free interpolation does not verify the kronecker delta property and therefore the imposition of prescribed values is not as straightforward as for the finite element method the aim of this paper is to present a general overview on the existing techniques to enforce essential boundary conditions in galerkin based mesh free methods special attention is paid to the mesh free coupling with finite elements for the imposition of prescribed values and to methods based on a modification of the galerkin weak form particular examples are used to analyze and compare their performance in different situations

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

the generalized finite element method this paper describes a pilot design and implementation of the generalized finite element method gfem as a direct extension of the standard finite element method sfem or fem which makes possible the accurate solution of engineering problems in complex domains which may be practically impossible to solve using the fem the development of the gfem is illustrated for the laplacian in two space dimensions in domains which may include several hundreds of voids and or cracks for which the construction of meshes used by the fem is practically impossible the two main capabilities are 1 it can construct the approximation using meshes which may overlap part or all of the domain boundary 2 it can incorporate into the approximation handbook functions which are known analytically or are generated numerically and approximate well the solution of the boundary value problem in the neighborhood of corner points voids cracks etc the main tool is a special integration algorithm which we call the fast remeshing approach which is robust and works for any domain with arbitrary complexity the incorporation of the handbook functions into the gfem is done by employing the partition of unity method pum the presented formulations and implementations can be easily extended to the multi material medium where the voids are replaced by inclusions of various shapes and sizes and to the case of the elasticity problem this work can also be understood as a pilot study for the feasibility and demonstration of the capabilities of the gfem which is needed before analogous implementations are attempted in the three dmensional and nonlinear cases which are the cases of main interest for future work

the hp d adaptive finite cell method for geometrically nonlinear problems of solid mechanics summary the finite cell method fcm combines the fictitious domain approach with the p version of the finite element method and adaptive integration for problems of linear elasticity it offers high convergence rates and simple mesh generation irrespective of the geometric complexity involved this article presents the integration of the fcm into the framework of nonlinear finite element technology however the penalty parameter of the fictitious domain is restricted to a few orders of magnitude in order to maintain local uniqueness of the deformation map as a consequence of the weak penalization nonlinear strain measures provoke excessive stress oscillations in the cells cut by geometric boundaries leading to a low algebraic rate of convergence therefore the fcm approach is complemented by a local overlay of linear hierarchical basis functions in the sense of the hp d method which synergetically uses the h adaptivity of the integration scheme numerical experiments show that the hp d overlay effectively reduces oscillations and permits stronger penalization of the fictitious domain by stabilizing the deformation map the hp d adaptive fcm is thus able to restore high convergence rates for the geometrically nonlinear case while preserving the easy meshing property of the original fcm accuracy and performance of the present scheme are demonstrated by several benchmark problems in one two and three dimensions and the nonlinear simulation of a complex foam sample copyright 2011 john wiley sons ltd

blended b spline construction on unstructured quadrilateral and hexahedral meshes with optimal convergence rates in isogeometric analysis abstract we present a novel blended b spline method to construct bicubic tricubic splines over unstructured quadrilateral and hexahedral meshes for isogeometric analysis c 1 and truncated c 2 b spline functions are used in regular elements whereas c 0 and truncated c 1 b spline functions are adopted in boundary elements and interior irregular elements around extraordinary edges vertices the truncation mechanism is employed for a seamless transition from irregular to regular elements the resulting smoothness of the blended construction is c 2 continuous everywhere except c 0 continuous around extraordinary edges and c 1 continuous across the interface between irregular and regular elements the blended b spline construction yields consistent parameterization during refinement and exhibits optimal convergence rates spline functions in the blended construction form a non negative partition of unity are linearly independent and support bezier extraction such that the construction can be used in existing finite element frameworks several examples provide numerical evidence of optimal convergence rates

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

shape optimisation with multiresolution subdivision surfaces and immersed finite elements abstract we develop a new optimisation technique that combines multiresolution subdivision surfaces for boundary description with immersed finite elements for the discretisation of the primal and adjoint problems of optimisation similar to wavelets multiresolution surfaces represent the domain boundary using a coarse control mesh and a sequence of detail vectors based on the multiresolution decomposition efficient and fast algorithms are available for reconstructing control meshes of varying fineness during shape optimisation the vertex coordinates of control meshes are updated using the computed shape gradient information by virtue of the multiresolution editing semantics updating the coarse control mesh vertex coordinates leads to large scale geometry changes and conversely updating the fine control mesh coordinates leads to small scale geometry changes in our computations we start by optimising the coarsest control mesh and refine it each time the cost function reaches a minimum this approach effectively prevents the appearance of non physical boundary geometry oscillations and control mesh pathologies like inverted elements independent of the fineness of the control mesh used for optimisation on the immersed finite element grid the domain boundary is always represented with a relatively fine control mesh of fixed resolution with the immersed finite element method there is no need to maintain an analysis suitable domain mesh in some of the presented two and three dimensional elasticity examples the topology derivative is used for introducing new holes inside the domain the merging or removing of holes is not considered

arc transitive elementary abelian covers of the octahedron graph abstract in this paper we construct the pairwise non congruent elementary abelian covers of the octahedron graph o 6 which admit a lift of an arc transitive group of automorphisms of o 6 it shows that if the covering transformation group is a 2 group then its rank is less than or equal to 7 and there exist exactly 14 non congruent covering projections in total which admit lifts of arc transitive subgroups of the full automorphism group of o 6 if the covering transformation group is an odd prime p group then its rank is 1 3 4 6 or 7 and there exist p 4 such non congruent covering projections in total

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

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

stable lagrange multipliers for quadrilateral meshes of curved interfaces in 3d for the discretization of coupled problems by means of mortar finite elements the choice of the discrete lagrange multiplier space is of crucial importance for the performance of the resulting numerical scheme dual lagrange multipliers offer several advantages since they can be locally eliminated in this paper the lowest order dual lagrange multipliers are extended to arbitrary quadrilateral or triangular surface grids in a first step two alternatives are provided for the case of planar interfaces for the approximation of vector fields two more modifications are given for use on curvilinear interfaces a priori results are obtained and several numerical examples are shown

adaptive fem based nonrigid image registration using truncated hierarchical b splines we present an efficient approach of finite element method fem based nonrigid image registration in which the spatial transformation is constructed using truncated hierarchical b splines thb splines the image registration framework minimizes an energy functional using an fem based method and thus involves solving a large system of linear equations this framework is carried out on a set of successively refined grids however due to the increased number of control points during subdivision large linear systems are generated which are generally demanding to solve instead of using uniform subdivision an adaptive local refinement scheme is carried out only refining the areas of large change in deformation of the image by incorporating the key advantages of thb spline basis functions such as linear independence partition of unity and reduced overlap into the fem based framework we improve the matrix sparsity and computational efficiency the performance of the proposed method is demonstrated on 2d synthetic and medical images

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

a fictitious domain distributed lagrange multiplier based fluid structure interaction scheme with hierarchical b spline grids abstract we present a numerical scheme for fluid structure interaction based on hierarchical b spline grids and fictitious domain distributed lagrange multipliers the incompressible navier stokes equations are solved over a cartesian grid discretised with b splines the fluid grid near the immersed solids is refined locally using hierarchical b splines the immersed solid is modelled as geometrically exact beam discretised with standard linear lagrange shape functions the kinematic constraint at the fluid solid interface is enforced with distributed lagrange multipliers the unconditionally stable and second order accurate generalised method is used for integration in time for both the fluid and solid domains a fully implicit and fully coupled solution scheme is developed by using the newton raphson method to solve the non linear system of equations obtained with galerkin weak formulation first the spatial and temporal convergence of the proposed scheme is assessed by studying steady and unsteady flow past a fixed cylinder then the scheme is applied to several benchmark problems to demonstrate the efficiency and robustness of the proposed scheme the results obtained with the present scheme are compared with the reference values

truncated hierarchical tricubic c0 spline construction on unstructured hexahedral meshes for isogeometric analysis applications abstract we present a new method for truncated hierarchical tricubic spline th spline3d construction to enable adaptive isogeometric analysis on unstructured hexahedral meshes taking the input unstructured hexahedral mesh as the control mesh we first develop blending functions with the aid of tricubic bernstein polynomials this development is an extension of defining bicubic bernstein polynomial blending functions over quadrilateral meshes we further build the hierarchical structure and apply the truncation mechanism to the developed blending functions for highly localized refinement during the refinement of th spline3d high level blending functions are added in the solution space whereas certain low level ones are discarded or truncated depending on the high level subdomain the blending functions are piecewise polynomials that form a partition of unity their support overlapping is also reduced due to the truncation mechanism resulting in sparser stiffness matrices compared to the classical hierarchical refinement th spline3d supports bezier extraction such that it can be easily incorporated into existing finite element frameworks several examples are used to demonstrate the analysis suitability and efficiency of the proposed method

the locally 2 arc transitive graphs admitting a ree simple group abstract in this paper three infinite families of locally 2 arc transitive graphs are constructed which are vertex intransitive regular and all vertex stabilizers are conjugate to the best of our knowledge these are the first infinite families of graphs with these properties in particular they are semi symmetric it is then shown that the only locally 2 arc transitive graphs admitting a ree simple group are i the graphs in these three families ii vertex transitive 2 arc transitive graphs admitting a ree simple group previously classified by the first and third authors and iii standard double covers of the graphs in ii this is the first complete classification of locally 2 arc transitive graphs for an infinite family of simple groups

multigrid methods with web splines we describe and analyze a multigrid algorithm for finite element approximations of second order elliptic boundary value problems with weightedextended b splines web splines this new technique provides high accuracy with relatively low dimensional subspaces does not require any grid generation and is ideally suited for hierarchical solution techniques in particular we show that the standard w cycle yields uniform convergence i e the required number of iterations is bounded independent of the grid width

on error estimates and adaptivity in elastoplastic solids applications to the numerical simulation of strain localization in classical and cosserat continua the a posteriori error estimates based on the post processing approach are introduced for elastoplastic solids the standard energy norm error estimate established for linear elliptic problems is generalized here to account for the presence of internal variables through the norm associated with the complementary free energy this is known to represent a natural metric for the class of elastoplastic problems of evolution in addition the intrinsic dissipation functional is utilized as a basis for a complementary a posteriori error estimates a posteriori error estimates and adaptive refinement techniques are applied to the finite element analysis of a strain localization problem as a model problem the constitutive equations describing a generalization of standard j2 elastoplasticity within the cosserat continuum are used to overcome serious limitations exhibited by classical continuum models in the post instability region the proposed a posteriori error estimates are appropriately modified to account for the cosserat continuum model and linked with adaptive techniques in order to simulate strain localization problems superior behaviour of the cosserat continuum model in comparison to the classical continuum model is demonstrated through the finite element simulation of the localization in a plane strain tensile test for an elastopiastic softening material resulting in convergent solutions with an h refinement and almost uniform error distribution in all considered error norms

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

a stabilised immersed boundary method on hierarchical b spline grids abstract in this work an immersed boundary finite element method is proposed which is based on a hierarchically refined cartesian b spline grid and employs the non symmetric and penalty free version of nitsche s method to enforce the boundary conditions the strategy allows for h and p refinement and employs a so called ghost penalty term to stabilise the cut cells an effective procedure based on hierarchical subdivision and sub cell merging which avoids excessive numbers of quadrature points is used for the integration of the cut cells a basic laplace problem is used to demonstrate the effectiveness of the cut cell stabilisation and of the penalty free nitsche method as well as their impact on accuracy the methodology is also applied to the incompressible navier stokes equations where the supg pspg stabilisation is employed simulations of the lid driven cavity flow and the flow around a cylinder at low reynolds number show the good performance of the methodology excessive ill conditioning of the system matrix is robustly avoided without jeopardising the accuracy at the immersed boundaries or in the field

interface material failure modeled by the extended finite element method and level sets multi phase materials are characterized by the fact that they possess a specific heterogeneous microstructure this feature often necessitates complex formulations in order to obtain realistic mechanical behavior with macroscopic material models furthermore such complex models cause substantial difficulties when the corresponding material parameters need to be identified and their physical meaning interpreted in multiscale analyses usually the microstructure of a material point is resolved directly thus the material behavior at the material point can be modeled in an elegant natural and simple way the present contribution aims at a detailed geometric modeling of multi phase materials as well as at a local mechanical modeling of material interfaces and interfacial failure the geometry of the material distribution within the microstructure is described through level set functions the mechanical modeling of material interfaces and interfacial cracks is accomplished by the extended finite element method x fem the combination of both the level set method and the x fem allows one to model such internal features of the microstructure without the adaptation of the mesh thus expensive meshing procedures can be avoided and considerable flexibility is obtained with respect to the variation of the material design in the x fem the finite element approximation is enriched by appropriate functions through the concept of partition of unity remarkably the level set method cannot only be applied to describe the geometry of the material interfaces and cracks respectively but also to support and even to develop the corresponding enrichment function

truncated hierarchical catmull clark subdivision with local refinement in this paper we present a new method termed truncated hierarchical catmull clark subdivision thccs which generalizes truncated hierarchical b splines to control grids of arbitrary topology thccs basis functions satisfy partition of unity are linearly independent and are locally refinable thccs also preserves geometry during adaptive hh refinement and thus inherits the surface continuity of catmull clark subdivision namely c2c2 continuous everywhere except at the local region surrounding extraordinary nodes where the surface continuity is c1c1 adaptive isogeometric analysis is performed with thccs basis functions on a benchmark problem with extraordinary nodes local refinement on complex surfaces is also studied to show potential wide application of the proposed method

adaptive finite element methods in computational mechanics we present a general approach to adaptivity for finite element methods and give applications to linear elasticity non linear elasto plasticity and nonlinear conservation laws including numerical results

a posteriori analysis of a b spline based finite element method for the stationary quasi geostrophic equations of the ocean abstract we provide a posteriori error analysis of a b spline based finite element method for the streamfunction formulation of the stationary quasi geostrophic equations to model the large scale wind driven ocean circulation the analysis gives rise to a posteriori error estimator for more efficient and accurate modeling of a strong western boundary layer via adaptive local refinement of the computational geometries to verify the capability of the error estimator we use hierarchical b splines in conjunction with dorfler marking strategy numerical studies are performed for several benchmark examples on rectangular and l shape ocean geometries

divergence conforming discretization for stokes problem on locally refined meshes using lr b splines abstract to solve the incompressible flow problems using isogeometric analysis the div compatible spline spaces were originally introduced by buffa et al 2011 and later developed by evans 2011 in this paper we extend the div compatible spline spaces with local refinement capability using locally refined lr b splines over rectangular domains we argue that the spline spaces generated on locally refined meshes will satisfy compatibility provided they span the entire function spaces as governed by mourrain 2014 dimension formula we will in this work use the structured refined lr b splines as introduced by johannessen et al 2014 further we consider these div compatible lr b spline spaces to approximate the velocity and pressure fields in mixed discretization for stokes problem and a set of standard benchmark tests are performed to show the stability efficiency and the conservation properties of the discrete velocity fields in adaptive isogeometric analysis

a fictitious domain approach with spread interface for elliptic problems with general boundary conditions the aim of this article is to solve second order elliptic problems in an original physical domain using a fictitious domain method with a spread interface approach the main idea of the fictitious domain approach consists in immersing the original domain of study into a geometrically bigger and simpler one called fictitious domain as the spatial discretization is being performed in the fictitious domain this method allows the use of structured meshes the discretization is not boundary fitted to the original physical domain this paper describes several ways to impose dirichlet robin or neumann boundary conditions on a spread immersed interface without locally modifying the numerical scheme and without using lagrange multipliers the numerical applications focus on difiusion and convection problems in the unit disk with dirichlet or robin boundary conditions for such problems analytical solutions can be determined for a correctly chosen source term the numerical resolution is performed using a q1 finite element scheme the spread interface approach is then combined with a local adaptive mesh refinement algorithm in order to increase the precision in the vinicity of the immersed boundary the l2 norm of the errors is computed in order to evaluate the capability of the method immersed boundaries are found in many industrial applications like two phase flow simulations fluid structure interaction etc this article represents a first step towards the simulation of these kinds of applications

n sequential compactness abstract in this paper we first recall the pathological property of sequentially compact fts defined in 1 and define n sequential compactness in this way the so called general sequential compactness is a special kind of n sequential compactness then we discuss the properties of n sequential compactness finally it is pointed out that n sequantial compactness is a good extension 5 of sequential compactness

truncated t splines fundamentals and methods in this paper we present truncated t splines as a new type of t splines suitable for both geometric design and analysis supporting highly localized refinement truncated t spline basis functions are piece wise polynomials that are linearly independent and form a partition of unity refinement of truncated t splines produces nested spline spaces furthermore we study truncated t splines and local refinement on the general domain 2 manifold with extraordinary points in the t mesh g1g1 continuity is attained around extraordinary points by properly capping quartic bezier patches where a constrained optimization problem is solved in the end we study benchmark problems using truncated t splines in the context of isogeometric analysis we also apply truncated t splines to complex geometries to show the smooth surfaces and simulation results under local refinement

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

fuzzy neighborhood spaces abstract in this paper we introduce a subcategory of the category of fuzzy topological spaces and we prove basic properties of this subcategory the main idea behind these spaces is the notion of fuzzy neighborhoods which enables us to introduce a local theory of fuzzy topologies

subdivision shells with exact boundary control and non manifold geometry we introduce several new extensions to subdivision shells that provide an improved level of shape control over shell boundaries and facilitate the analysis of shells with non smooth and non manifold joints to this end extended subdivision schemes are used that enable to relax the continuity of the limit surface along prescribed crease edges and to create surfaces with prescribed limit positions and normals furthermore shells with boundaries in the form of conic sections such as circles or parabolas are represented with rational subdivision schemes which are defined in analogy to rational b splines in terms of implementation the difference between the introduced and conventional subdivision schemes is restricted to the use of modified subdivision stencils close to the mentioned geometric features hence the resulting subdivision surface is in most parts of the domain identical to standard smooth subdivision surfaces the particular subdivision scheme used in this paper constitutes an improved version of the original loop s scheme and is as such based on triangular meshes as in the original subdivision shells surfaces created with the modified scheme are used for interpolating the reference and deformed shell configurations at the integration points the subdivision surface is evaluated using a newly developed discrete parameterization approach in the resulting finite elements the only degrees of freedom are the mid surface displacements of the nodes and additional lagrange parameters for enforcing normal constraints the versatility of the newly developed elements is demonstrated with a number of geometrically nonlinear shell examples copyright 2011 john wiley sons ltd

on sequentially closed graph mappings in fuzzy neighborhood spaces abstract two notions of closed graph and strongly closed graph in fuzzy neighbourhood spaces have been introduced one in fuzzy sets and system 48 1992 377 382 and the other by the author we prove that those notions are equivalent we introduce the notion of sequential convergence in fuzzy neighbourhood space

fundamental aspects of shape optimization in the context of isogeometric analysis we develop a mathematical foundation for shape optimization problems under state equation constraints where both state and control are discretized by b splines or nurbs in other words we use isogeometric analysis iga for solving the partial differential equation and a nodal approach to change domains where control points take the place of nodes and where thus a quite general class of functions for representing optimal shapes and their boundaries becomes available the minimization problem is solved by a gradient descent method where the shape gradient will be defined in isogeometric terms this gradient is obtained following two schemes optimize first discretize then and reversely discretize first optimize then we show that for isogeometric analysis the two schemes yield the same discrete system moreover we also formulate shape optimization with respect to nurbs in the optimize first ansatz which amounts to finding optimal control points and weights simultaneously numerical tests illustrate the theory

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

imposing dirichlet boundary conditions with nitsche s method and spline based finite elements a key challenge while employing non interpolatory basis functions in finite element methods is the robust imposition of dirichlet boundary conditions the current work studies the weak enforcement of such conditions for b spline basis functions with application to both second and fourth order problems this is achieved using concepts borrowed from nitsche s method which is a stabilized method for imposing constraints on surfaces conditions for the stability of the system of equations are derived for each class of problem stability parameters in the nitsche weak form are then evaluated by solving a local generalized eigenvalue problem at the dirichlet boundary the approach is designed to work equally well when the grid used to build the splines conforms to the physical boundary of interest as well as to the more general case when it does not through several numerical examples the approach is shown to yield optimal rates of convergence copyright 2010 john wiley sons ltd

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

extended truncated hierarchical catmull clark subdivision abstract in this paper we present an extended truncated hierarchical catmull clark subdivision ethccs method which improves the efficiency of local refinement in truncated hierarchical catmull clark subdivision thccs based on the stam s method we first build a set of basis functions over arbitrary quadrilateral meshes and apply them to thccs then a new basis function insertion scheme is developed with the aid of the truncation mechanism which refines one ring neighboring elements rather than two ring neighborhoods therefore ethccs significantly improves the efficiency of local refinement compared with thccs as demonstrated by one benchmark problem and several complex models moreover ethccs is also proved to preserve the input geometry and produce nested spaces