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

implicit dynamic analysis using an isogeometric reissner mindlin shell formulation summary in isogeometric analysis identical basis functions are used for geometrical representation and analysis in this work non uniform rational basis splines basis functions are applied in an isoparametric approach an isogeometric reissner mindlin shell formulation for implicit dynamic calculations using the galerkin method is presented a consistent as well as a lumped matrix formulation is implemented the suitability of the developed shell formulation for natural frequency analysis is demonstrated by a numerical example in a second set of examples transient problems of plane and curved geometries undergoing large deformations in combination with nonlinear material behavior are investigated via a zero thickness stress algorithm for arbitrary material models a j2 plasticity constitutive law is implemented in the numerical examples the effectiveness robustness and superior accuracy of a continuous interpolation method of the shell director vector is compared with experimental results and alternative numerical approaches copyright 2016 john wiley sons ltd

geometry independent field approximation cad analysis integration geometrical exactness and adaptivity in isogeometric analysis iga the same spline representation is employed for both the geometry of the domain and approximation of the unknown fields over this domain this identity of the geometry and field approximation spaces was put forward in the now classic 2005 paper 20 as a key advantage on the way to the integration of computer aided design cad and subsequent analysis in computer aided engineering cae 20 claims indeed that any change to the geometry of the domain is automatically inherited by the approximation of the field variables without requiring the regeneration of the mesh at each change of the domain geometry yet in finite element versions of iga a parameterization of the interior of the domain must still be constructed since cad only provides information about the boundary the identity of the boundary and field representation decreases the flexibility in which this parameterization can be generated and somewhat constrains the modeling and simulation process because an approximation able to represent the domain geometry accurately need not be adequate to also approximate the field variables accurately in particular when the solution is not smooth we propose here a new paradigm called geometry independent field approximation gift where the spline spaces used for the geometry and the field variables can be chosen and adapted independently while preserving geometric exactness and tight cad integration gift has the following features 1 it is possible to flexibly

on the use of nurbs based discretizations in the scaled boundary finite element method for wave propagation problems abstract we discuss the application of non uniform rational b splines nurbs in the scaled boundary finite element method sbfem for the solution of wave propagation problems at rather high frequencies we focus on the propagation of guided waves along prismatic structures of constant cross section comparisons are made between nurbs based discretizations and high order spectral elements in terms of the achievable convergence rates we find that for the same order of shape functions nurbs can lead to significantly smaller errors compared with lagrange polynomials the difference becomes particularly important at very high frequencies where spectral elements are prone to instabilities furthermore we analyze the behavior of nurbs for the discretization of curved boundaries where the benefit of exact geometry representation becomes crucial even in the low frequency range

on a splitting type algorithm for the numerical realization of contact problems with coulomb friction this paper presents and analyses an iterative process for the numerical realization of contact problems with coulomb friction which is based on the method of successive approximations combined with a splitting type approach numerical examples illustrate the efficiency of this method

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

a quadrilateral g1 conforming finite element for the kirchhoff plate model abstract a quadrilateral bi cubic g 1 conforming finite element for the analysis of kirchhoff plates is presented the rational version of the gregory patch proposed by loop et al 2009 is the starting point of our formulation this version of the gregory patch consists in rational enhancement of the bi cubic bezier interpolation representing a suitable tool for designing g 1 conforming quadrilateral element on c 0 conforming un structured meshes the element includes as additional degrees of freedom the edge rotations like in the loof formulations but is only displacement based because of the presence of the rational functions the second derivatives of the interpolation present a finite discontinuity at the corners of the element that prevent the element from passing the bending patch test the element so formulated does not present optimal rate of convergence under h refinement operation the formulation is enhanced enforcing these discontinuities to be zero by means of lagrange multipliers it is shown that with these constraints the element passes the patch test and presents optimal rate of convergence for unstructured mesh in this way the rational conforming approximation collapses into a conforming re arrangement of the complete bi cubic bezier interpolation some examples and benchmarks are presented in order to test the performance of the element for the kirchhoff plate model

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

a method for analysis of c 1 continuity of subdivision surfaces a sufficient condition for c1 continuity of subdivision surfaces was proposed by reif comput aided geom design 12 1995 pp 153 174 and extended to a more general setting in d zorin constr approx accepted for publication in both cases the analysis of c1 continuity is reduced to establishing injectivity and regularity of a characteristic map in all known proofs of c1 continuity explicit representation of the limit surface on an annular region was used to establish regularity and a variety of relatively complex techniques were used to establish injectivity we propose a new approach to this problem we show that for a general class of subdivision schemes regularity can be inferred from the properties of a sufficiently close linear approximation and injectivity can be verified by computing the index of a curve an additional advantage of our approach is that it allows us to prove c1 continuity for all valences of vertices rather than for an arbitrarily large but finite number of valences as an application we use our method to analyze c1 continuity of most stationary subdivision schemes known to us including interpolating butterfly and modified butterfly schemes as well as the kobbelt s interpolating scheme for quadrilateral meshes

the inverse integrating factor and the poincare map this work is concerned with planar real analytic differential systems with an analytic inverse integrating factor defined in a neighborhood of a regular orbit we show that the inverse integrating factor defines an ordinary differential equation for the transition map along the orbit when the regular orbit is a limit cycle we can determine its associated poincare return map in terms of the inverse integrating factor in particular we show that the multiplicity of a limit cycle coincides with the vanishing multiplicity of an inverse integrating factor over it we also apply this result to study the homoclinic loop bifurcation we only consider homoclinic loops whose critical point is a hyperbolic saddle and whose poincare return map is not the identity a local analysis of the inverse integrating factor in a neighborhood of the saddle allows us to determine the cyclicity of this polycycle in terms of the vanishing multiplicity of an inverse integrating factor over it our result also applies in the particular case in which the saddle of the homoclinic loop is linearizable that is the case in which a bound for the cyclicity of this graphic cannot be determined through an algebraic method

a method of finite element tearing and interconnecting and its parallel solution algorithm a novel domain decomposition approach for the parallel finite element solution of equilibrium equations is presented the spatial domain is partitioned into a set of totally disconnected subdomains each assigned to an individual processor lagrange multipliers are introduced to enforce compatibility at the interface nodes in the static case each floating subdomain induces a local singularity that is resolved in two phases first the rigid body modes are eliminated in parallel from each local problem and a direct scheme is applied concurrently to all subdomains in order to recover each partial local solution next the contributions of these modes are related to the lagrange multipliers through an orthogonality condition a parallel conjugate projected gradient algorithm is developed for the solution of the coupled system of local rigid modes components and lagrange multipliers which completes the solution of the problem when implemented on local memory multiprocessors this proposed method of tearing and interconnecting requires less interprocessor communications than the classical method of substructuring it is also suitable for parallel vector computers with shared memory moreover unlike parallel direct solvers it exhibits a degree of parallelism that is not limited by the bandwidth of the finite element system of equations

discontinuous galerkin isogeometric analysis of elliptic problems on segmentations with non matching interfaces abstract in this paper we develop a discontinuous galerkin isogeometric analysis method for solving elliptic problems on decompositions of the computational domain into volumetric patches with non matching parametrized interfaces we specially focus on high order numerical solutions for complex gap regions and extend ideas from our previous work on simple gap regions for the communication of the numerical solution between the subdomains which are separated by the gap region discontinuous galerkin numerical fluxes are constructed taking into account the diametrically opposite points on the boundary of the gap due to lack of information on the behavior of the solution in the gap region the fluxes coming from the interior of the gap are approximated by taylor expansions with respect to the adjacent subdomain solutions we follow the same ideas of our previous work and show a priori error estimates in the dg norm with respect to the mesh size and the gap distance numerical examples performed for two three and even four dimensional computational domains demonstrate the robustness of the proposed numerical method and validate the estimates predicted by the theory

cut finite element methods for elliptic problems on multipatch parametric surfaces we develop a finite element method for the laplace beltrami operator on a surface described by a set of patchwise parametrizations the patches provide a partition of the surface and each patch is

a nurbs based hybrid collocation galerkin method for the analysis of boundary represented solids abstract the paper is concerned with a new numerical method nurbs based hybrid collocation galerkin method nurbs hcgm to solve the in plane motion problem of elastic solids it combines the merits of the so called scaled boundary finite element method sb fem and the isogeometric collocation method for the analysis the boundary scaling technique of sb fem is adopted it leads to a formulation where only the boundary of a structure is discretized here the nurbs basis functions are employed for the description of the geometry of the boundary as well as for the approximation of the displacements at the boundary this is in accordance with the boundary representation modeling technique which is commonly employed in computer aided design software the inner domain is described by a radial scaling parameter applying the weak form only in circumferential direction the governing partial differential equations of elasticity are transformed to an ordinary differential equation ode of euler type where the unknown displacements are a function of the radial scaling parameter in the present work a nurbs based collocation scheme is introduced to solve this equation nurbs basis functions are suggested for the approximation of the displacements in scaling direction the higher continuity provided by nurbs allows to use collocation to solve the ode directly instead of using the weak form in scaling direction the proposed approach is validated by comparison with the eigenvalue solution of the ode it is remarked that the eigenvalue solution is restricted to linear problems whereas the proposed method could be extended to nonlinear problems in general the presented formulation will allow to model patches bounded by an arbitrary number of contour boundaries the accuracy of the proposed approach is analyzed and estimated with respect to analytical solutions the computational cost is investigated with the help of numerical examples and is compared to isogeometric galerkin approach

isogeometric kirchhoff love shell formulations for general hyperelastic materials abstract we present formulations for compressible and incompressible hyperelastic thin shells which can use general 3d constitutive models the necessary plane stress condition is enforced analytically for incompressible materials and iteratively for compressible materials the thickness stretch is statically condensed and the shell kinematics are completely described by the first and second fundamental forms of the midsurface we use c 1 continuous isogeometric discretizations to build the numerical models numerical tests including structural dynamics simulations of a bioprosthetic heart valve show the good performance and applicability of the presented methods

existence of inverse jacobi multipliers around hopf points in r 3 emphasis on the center problem abstract we consider the center problem at hopf points of analytic systems in r 3 that has a classical solution in the lyapunov center theorem which is given in terms of an analytic first integral here we give a new solution in terms of an analytic inverse jacobi multiplier v the existence of a smooth and non flat inverse jacobi multiplier around a hopf point of saddle focus type is also proved when studying these problems we needed to discuss the relation between inverse jacobi multipliers and center manifolds w c in particular to know under what conditions w c v 1 0 to illustrate our results we solve the center problem for the lu system

an unfitted finite element method based on nitsche s method for elliptic interface problems in this paper we propose a method for the finite element solution of elliptic interface problem using an approach due to nitsche the method allows for discontinuities internal to the elements in the approximation across the interface we show that optimal order of convergence holds without restrictions on the location of the interface relative to the mesh further we derive a posteriori error estimates for the purpose of controlling functionals of the error and present some numerical examples

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

treatment of reissner mindlin shells with kinks without the need for drilling rotation stabilization in an isogeometric framework this work presents a framework for the computation of complex geometries containing intersections of multiple patches with reissner mindlin shell elements the main objective is to provide an isogeometric finite element implementation which neither requires drilling rotation stabilization nor user interaction to quantify the number of rotational degrees of freedom for every node for this purpose the following set of methods is presented control points with corresponding physical location are assigned to one common node for the finite element solution a nodal basis system in every control point is defined which ensures an exact interpolation of the director vector throughout the whole domain a distinction criterion for the automatic quantification of rotational degrees of freedom for every node is presented an isogeometric reissner mindlin shell formulation is enhanced to handle geometries with kinks and allowing for arbitrary intersections of patches the parametrization of adjacent patches along the interface has to be conforming the shell formulation is derived from the continuum theory and uses a rotational update scheme for the current director vector the nonlinear kinematic allows the computation of large deformations and large rotations two concepts for the description of rotations are presented the first one uses an interpolation which is commonly used in standard lagrange based shell element formulations the second scheme uses a more elaborate concept proposed by the authors in prior work which increases the accuracy for arbitrary curved geometries numerical examples show the high accuracy and robustness of both concepts the applicability of the proposed framework is demonstrated

skew symmetric nitsche s formulation in isogeometric analysis dirichlet and symmetry conditions patch coupling and frictionless contact abstract a simple skew symmetric nitsche s formulation is introduced into the framework of isogeometric analysis iga to deal with various problems in small strain elasticity essential boundary conditions symmetry conditions for kirchhoff plates patch coupling in statics and in modal analysis as well as signorini contact conditions for linear boundary or interface conditions the skew symmetric formulation is parameter free for contact conditions it remains stable and accurate for a wide range of the stabilization parameter several numerical tests are performed to illustrate its accuracy stability and convergence performance we investigate particularly the effects introduced by nitsche s coupling including the convergence performance and condition numbers in statics as well as the extra outlier frequencies and corresponding eigenmodes in structural dynamics we present the hertz test the block test and a 3d self contact example showing that the skew symmetric nitsche s formulation is a suitable approach to simulate contact problems in iga

an efficient and robust rotational formulation for isogeometric reissner mindlin shell elements abstract this work is concerned with the development of an efficient and robust isogeometric reissner mindlin shell formulation for the mechanical simulation of thin walled structures such structures are usually defined by non uniform rational b splines nurbs surfaces in industrial design software the usage of isogeometric shell elements can avoid costly conversions from nurbs surfaces to other surface or volume geometry descriptions the shell formulation presented in this contribution uses a continuous orthogonal rotation described by rodrigues tensor in every integration point to compute the current director vector the rotational state is updated in a multiplicative manner large deformations and finite rotations can be described accurately the proposed formulation is robust in terms of stable convergence behavior in the nonlinear equilibrium iteration for large load steps and geometries with large and arbitrary curvature and in terms of insensitivity to shell intersections with kinks under small angles three different integration schemes and their influence on accuracy and computational costs are assessed the efficiency and robustness of the proposed isogeometric shell formulation is shown with the help of several examples accuracy and efficiency is compared to an isogeometric shell formulation with the more common discrete rotational concept and to lagrange based finite element shell formulations the competitiveness of the proposed isogeometric shell formulation in terms of computational costs to attain a pre defined error level is shown

tricritical point of the j1 j2 ising model on a hyperbolic lattice the ferromagnetic paramagnetic phase transition of the two dimensional frustrated ising model on a hyperbolic lattice is investigated by use of the corner transfer matrix renormalization group method the model contains a ferromagnetic nearest neighbor interaction j 1 and a competing antiferromagnetic interaction j 2 a mean field like second order phase transition is observed when the ratio ensuremath kappa j 2 j 1 is less than 0 203 in the region 0 203l ensuremath kappa l1 4 the spontaneous magnetization is discontinuous at the transition temperature such tricritical behavior suggests that the phase transitions on hyperbolic lattices need not always be mean field like

large elasto plastic finite element analysis of solids and shells with the enhanced assumed strain concept abstract a finite element simulation of arbitrarily large elasto plastic deformations of three dimensional and shell structures is described two numerical aspects originally developed by j c simo have been combined i e the multiplicative large strain plasticity and the enhanced assumed strain finite element concept the constitutive model is worked out in its algorithmic version applying projection methods and consistent linearization the enhanced strain field is added to the conventional hexahedral displacement elements avoiding volumetric locking in the plastic range in addition to solid structures the three dimensional material model without any modifications is also applied to thin walled shell structures here an extended shell formulation a so called seven parameter model is used in which the thickness strain is augmented by a linear term the applicability of the entire formulation is demonstrated by several numerical examples

analytic normalization of analytic integrable systems and the embedding flows abstract this paper provides the normal forms of analytic integrable differential systems and diffeomorphisms via analytic normalizations furthermore we consider the existence of embedding flows of an analytic integrable diffeomorphism

hierarchical nurbs in frictionless contact abstract this work investigates mortar based frictionless contact in the context of nurbs discretizations that are subjected to local hierarchical refinement the investigations emphasize three sets of choices which lead to different contact algorithms that have distinct advantages and disadvantages first on the optimization side both exterior and interior point methods are applied thus spanning inexact constraint enforcement algorithms of the penalty or barrier type as well as exact ones of the primal dual type second on the discretization side the hierarchical basis set of the mortar variables is inherited either directly from the discretization of the slave surface or after an intermediate normalization step to satisfy the partition of unity third in interaction with both optimization and discretization the kinematic mortar variable is recovered from the actual normal gap through the full or lumped solution of a linear system of equations the implications of different choices are highlighted through benchmark problems which monitor the solution quality at the global level through the structural force evolution and at the local level through the contact pressure distribution

isogeometric dual mortar methods for computational contact mechanics in recent years isogeometric analysis iga has received great attention in many fields of computational mechanics research especially for computational contact mechanics an exact and smooth surface representation is highly desirable as a consequence many well known finite element methods and algorithms for contact mechanics have been transferred to iga in the present contribution the so called dual mortar method is investigated for both contact mechanics and classical domain decomposition using nurbs basis functions in contrast to standard mortar methods the use of dual basis functions for the lagrange multiplier based on the mathematical concept of biorthogonality enables an easy elimination of the additional lagrange multiplier degrees of freedom from the global system this condensed system is smaller in size and no longer of saddle point type but positive definite a very simple and commonly used element wise construction of the dual basis functions is directly transferred to the iga case the resulting lagrange multiplier interpolation satisfies discrete inf sup stability and biorthogonality however the reproduction order is limited to one in the domain decomposition case this results in a limitation of the spatial convergence order to o h3 2 in the energy norm whereas for unilateral contact due to the lower regularity of the solution optimal convergence rates are still met numerical examples are presented that illustrate these theoretical considerations on convergence rates and compare the newly developed isogeometric dual mortar contact formulation with its standard mortar counterpart as well as classical finite elements based on first and second order lagrange polynomials

isogeometric mortar methods the application of mortar methods in the framework of isogeometric analysis is investigated theoretically as well as numerically for the lagrange multiplier two choices of uniformly stable spaces are presented both of them are spline spaces but of a different degree in one case we consider an equal order pairing for which a cross point modification based on a local degree reduction is required in the other case the degree of the dual space is reduced by two compared to the primal this pairing is proven to be inf sup stable without any necessary cross point modification several numerical examples confirm the theoretical results and illustrate additional aspects

isogeometric analysis and domain decomposition methods in the present work we use the mortar finite element method for the coupling of nonconforming discretized sub domains in the framework of nonlinear elasticity the mortar method has been shown to preserve optimal convergence rates see laursen 2002 25 for details and is variationally consistent we show that the method can be applied to isogeometric analysis with little effort once the framework of nurbs based shape functions has been implemented furthermore a specific coordinate augmentation technique allows the design of an energy momentum scheme for the constrained mechanical system under consideration the excellent performance of the redesigned mortar method as well as the energy momentum scheme is illustrated in representative numerical examples

a natural framework for isogeometric fluid structure interaction based on bem shell coupling abstract the interaction between thin structures and incompressible newtonian fluids is ubiquitous both in nature and in industrial applications in this paper we present an isogeometric formulation of such problems which exploits a boundary integral formulation of stokes equations to model the surrounding flow and a non linear kirchhoff love shell theory to model the elastic behavior of the structure we propose three different coupling strategies a monolithic fully implicit coupling a staggered elasticity driven coupling and a novel semi implicit coupling where the effect of the surrounding flow is incorporated in the non linear terms of the solid solver through its damping characteristics the novel semi implicit approach is then used to demonstrate the power and robustness of our method which fits ideally in the isogeometric paradigm by exploiting only the boundary representation b rep of the thin structure middle surface

mixed stress displacement isogeometric collocation for nearly incompressible elasticity and elastoplasticity abstract we propose a mixed stress displacement isogeometric collocation method for nearly incompressible elastic materials and for materials exhibiting von mises plasticity the discretization is based on isogeometric analysis iga with non uniform rational b splines nurbs as basis functions as compared to conventional iga galerkin formulations isogeometric collocation methods offer a high potential of computational cost reduction for higher order discretizations as they eliminate the need for quadrature in the proposed mixed formulation both stress and displacement fields are approximated as primary variables with the aim of treating volumetric locking and instability issues which occur in displacement based isogeometric collocation for nearly incompressible elasticity and von mises plasticity the performance of the proposed approach is demonstrated by several numerical examples

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

inverse jacobian multipliers and hopf bifurcation on center manifolds abstract in this paper we consider a class of higher dimensional differential systems in r n which have a two dimensional center manifold at the origin with a pair of pure imaginary eigenvalues first we characterize the existence of either analytic or c inverse jacobian multipliers of the systems around the origin which is either a center or a focus on the center manifold later we study the cyclicity of the system at the origin through hopf bifurcation by using the vanishing multiplicity of the inverse jacobian multiplier

two new triangular g1 conforming finite elements with cubic edge rotation for the analysis of kirchhoff plates abstract two triangular g 1 conforming elements based on the triangular gregory patch suitable for the analysis of the kirchhoff plate model are presented both have cubic normal derivative along the sides so that they can be effectively used in combination with generalized hermitian elements the gregory patch consists in a rational enhancement of the base polynomial spaces useful to design g 1 conforming elements on general c 0 conforming unstructured meshes because of the presence of the rational functions the second derivatives at the corners of the element present finite discontinuities that prevent the elements from passing the bending patch test the discontinuities are removed using a constrained version of the gregory patch with lagrange multipliers in this way the rational conforming space collapses into a conforming rearrangement of the original polynomial interpolant spaces the proposed formulation design elements that pass the bending patch test and present optimal rate of convergence on general unstructured meshes

on methods for stabilizing constraints over enriched interfaces in elasticity summary enriched finite element approaches such as the extended finite element method provide a framework for constructing approximations to solutions of non smooth problems internal features such as boundaries are represented in such methods by using discontinuous enrichment of the standard finite element basis within such frameworks however imposition of interface constraints and or constitutive relations can cause unexpected difficulties depending upon how relevant fields are interpolated on un gridded interfaces this work address the stabilized treatment of constraints in an enriched finite element context both the lagrange multiplier and penalty enforcement of tied constraints for an arbitrary boundary represented in an enriched finite element context can lead to instabilities and artificial oscillations in the traction fields we demonstrate two alternative variational methods that can be used to enforce the constraints in a stable manner in a bubble stabilized approach fine scale degrees of freedom are added over elements supporting the interface the variational form can be shown to have a similar form to a second approach we consider nitsche s method with the exception that the stabilization terms follow directly from the bubble functions in this work we examine alternative variational methods for enforcing a tied constraint on an enriched interface in the context of two dimensional elasticity we examine several benchmark problems in elasticity and show that only nitsche s method and the bubble stabilization approach produce stable traction fields over internal boundaries we also demonstrate a novel difference between the penalty method and nitsche s method in that the latter passes the patch test exactly regardless of the stabilization parameter s magnitude results for more complicated geometries and triple interface junctions are also presented copyright 2008 john wiley sons ltd

analytic integrable systems analytic normalization and embedding flows abstract in this paper we mainly study the existence of analytic normalization and the normal form of finite dimensional complete analytic integrable dynamical systems more details we will prove that any complete analytic integrable diffeomorphism f x b x f x in c n 0 with b having eigenvalues not modulus 1 and f x o x 2 is locally analytically conjugate to its normal form meanwhile we also prove that any complete analytic integrable differential system x a x f x in c n 0 with a having nonzero eigenvalues and f x o x 2 is locally analytically conjugate to its normal form furthermore we will prove that any complete analytic integrable diffeomorphism defined on an analytic manifold can be embedded in a complete analytic integrable flow we note that parts of our results are the improvement of moser s one in j moser the analytic invariants of an area preserving mapping near a hyperbolic fixed point comm pure appl math 9 1956 673 692 and of poincare s one in h poincare sur l integration des equations differentielles du premier order et du premier degre ii rend circ mat palermo 11 1897 193 239 these results also improve the ones in xiang zhang analytic normalization of analytic integrable systems and the embedding flows j differential equations 244 2008 1080 1092 in the sense that the linear part of the systems can be nonhyperbolic and the one in n t zung convergence versus integrability in poincare dulac normal form math res lett 9 2002 217 228 in the way that our paper presents the concrete expression of the normal form in a restricted case

coupling adaptively refined multi patch spline discretizations via boundary compatibility abstract the present paper studies adaptive refinement on multi patch domains in isogeometric analysis in particular we investigate the gluing construction for adaptively refined spline spaces to obtain discretizations that are c 0 smooth across interfaces we will see that this is closely related to the concept of boundary compatibility of an adaptive spline construction given a spline basis or more generally a generating system if linear independence is not guaranteed on a d dimensional box domain there are two possibilities for constructing the spline basis on the domain boundary firstly one can simply restrict the basis functions to the boundary secondly one may restrict the underlying mesh to the boundary and construct the spline basis on the resulting mesh the two constructions do not necessarily produce the same set of functions if they do then the spline bases are said to be compatible we study this property for hierarchical hb and truncated hierarchical b splines thb splines and identify sufficient conditions these conditions are weaker for thb than for hb splines finally we demonstrate the importance of boundary compatibility for geometric modeling and for adaptive refinement in isogeometric analysis in particular when considering multi patch domains

local enrichment of nurbs patches using a non intrusive coupling strategy geometric details local refinement inclusion fracture in this work we apply a non intrusive global local coupling strategy for the modelling of local phenomena in a nurbs patch the idea is to consider the nurbs patch to be enriched as the global model this results in a simple flexible strategy first the global nurbs patch remains unchanged which completely eliminates the need for costly re parametrization procedures even if the local domain is expected to evolve then easy merging of a linear nurbs code with any other existing robust codes suitable for the modelling of complex local behaviour is possible the price to pay is the number of iterations of the non intrusive solver but we show that this can be strongly reduced by means of acceleration techniques the main development for nurbs is to be able to handle non conforming geometries only slight changes in the implementation process including the setting up of suitable quadrature rules for the evaluation of the interface reaction forces are made in response to this issue a range of numerical examples in two dimensional linear elasticity are given to demonstrate the performance of the proposed methodology and its significant potential to treat any case of local enrichment in a nurbs patch simply

isogeometric analysis of large deformation thin shells using rht splines for multiple patch coupling we present an isogeometric thin shell formulation for multi patches based on rational splines over hierarchical t meshes rht splines nitsche s method is employed to efficiently couple the patches the rht splines have the advantages of allowing a computationally feasible local refinement are free from linear dependence possess high order continuity and satisfy the partition of unity and non negativity in addition the c1c1 continuity of the rht splines avoids the rotational degrees of freedom the good performance of the present method is demonstrated by a number of numerical examples

isogeometric shape optimization of shells using semi analytical sensitivity analysis and sensitivity weighting abstract we present isogeometric shape optimization for shell structures applying sensitivity weighting and semi analytical analysis we use a rotation free shell formulation and all involved geometry models i e initial design analysis model optimization model and final design use the same geometric basis in particular nurbs a sensitivity weighting scheme is presented which eliminates certain effects of the chosen discretization on the design update a multilevel design approach is applied such that the design space can be chosen independently from the analysis space the use of semi analytical sensitivities allows having different polynomial degrees for design and analysis model different numerical examples are performed which confirm the applicability of the proposed method furthermore a shape optimization example with an exact solution is presented which can serve as general benchmark for shape optimization methods

biorthogonal splines for optimal weak patch coupling in isogeometric analysis with applications to finite deformation elasticity abstract a new construction of biorthogonal splines for isogeometric mortar methods is proposed the biorthogonal basis has a local support and at the same time optimal approximation properties which yield optimal results with mortar methods we first present the univariate construction which has an inherent crosspoint modification the multivariate construction is then based on a tensor product for weighted integrals whereby the important properties are inherited from the univariate case numerical results including large deformations confirm the optimality of the newly constructed biorthogonal basis

a nitsche type formulation and comparison of the most common domain decomposition methods in isogeometric analysis this paper provides a detailed elaboration and assessment of the most common domain decomposition methods for their application in isogeometric analysis the methods comprise a penalty approach lagrange multiplier methods and a nitsche type method for the nitsche method a new stabilized formulation is developed in the context of isogeometric analysis to guarantee coercivity all these methods are investigated on problems of linear elasticity and eigenfrequency analysis in 2d in particular focus is put on non uniform rational b spline patches which join nonconformingly along their common interface thus the application of isogeometric analysis is extended to multi patches which can have an arbitrary parametrization on the adjacent edges moreover it has been shown that the unique properties provided by isogeometric analysis that is high order functions and smoothness across the element boundaries carry over for the analysis of multiple domains copyright 2013 john wiley sons ltd

the embedding flows of c hyperbolic diffeomorphisms abstract in weigu li j llibre xiang zhang extension of floquet s theory to nonlinear periodic differential systems and embedding diffeomorphisms in differential flows amer j math 124 2002 107 127 we proved that for a germ of c hyperbolic diffeomorphisms f x a x f x in r n 0 if a has a real logarithm with its eigenvalues weakly nonresonant then f x can be embedded in a c autonomous differential system its proof was very complicated which involved the existence of embedding periodic vector field of f x and the extension of the floquet s theory to nonlinear c periodic differential systems in this paper we shall provide a simple and direct proof to this last result next we shall show that the weakly nonresonant condition in the last result on the real logarithm of a is necessary for some c diffeomorphisms f x a x f x to have c embedding flows finally we shall prove that a germ of c hyperbolic diffeomorphisms f x a x f x with f x o x 2 in r 2 0 has a c embedding flow if and only if either a has no negative eigenvalues or a has two equal negative eigenvalues and it can be diagonalizable

bloch theorem for isogeometric analysis of periodic problems governed by high order partial differential equations abstract with the emergence of isogeometric analysis iga numerical rotation free solutions to high order partial differential equations pdes are facilitated indeed iga allows the use of high order continuous basis functions through the domain for periodic systems the bloch theorem is useful for analyzing wave propagation restricting the analysis to a single unit cell and applying periodic boundary conditions however in the literature boundary conditions are only accounting for c 0 continuity and we propose in the present paper to enforce the g 1 continuity as well or to maintain the angle in case of unit cells meeting at kinks while g 1 continuity is enforced by maintaining the collinearity between adjacent control points in the presence of kinks variations of the field derived with respect to opposite in plane edge normals are linked by the rodrigues rotation matrix this method applied at collocation points belonging to the unit cell edges can be viewed as a novel approach of coupling multi patch geometries with kinks avoiding the use of nonlinear multipliers or additional bending strip elements both transfer matrix and indirect methods of the bloch theorem are investigated for respectively quasi one and quasi two dimensional wave propagation in periodic structures discretized by nurbs based iga dispersion curves of a set of numerical problems treating isogeometric rotation free extensible elastica beams and kirchhoff love shells are then analyzed

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

hybrid collocation galerkin approach for the analysis of surface represented 3d solids employing sb fem abstract this paper presents a numerical method to solve the three dimensional elasticity problem of surface represented solids a surface oriented formulation is derived in which the three dimensional solid is described by its boundary surfaces and a radial scaling center scaling the boundary surfaces with respect to the scaling center yields a parameterization of the complete solid the definition of the boundary surfaces is sufficient for the description of the solid thus the formulation conforms ideally to the boundary representation modeling technique used in cad in the present approach no tensor product structure of three dimensional objects is exploited to parameterize the physical domain the weak form of the equation of motion and the weak form of the neumann boundary conditions are enforced only at the boundary surfaces it leads to a transformation of the governing partial differential equations of elasticity to an ordinary differential equation ode of euler type solving the ode leads to the displacement in the radial scaling direction in the present approach the isogeometric galerkin finite element method is employed to describe the geometry of the boundary surfaces and also to approximate the displacement response of the boundary surfaces it exploits two dimensional nurbs objects to parameterize the boundary surfaces the final euler type ode is solved by the nurbs based collocation method the displacement response in the radial scaling direction is approximated by the nurbs basis functions hence the method can be extended to nonlinear problems the accuracy of the method is validated against analytical solutions in general the proposed formulation is able to model solids bounded by an arbitrary number of surfaces

hybrid non uniform recursive subdivision with improved convergence rates abstract this paper introduces a new non uniform subdivision surface representation called hybrid non uniform subdivision surface for short hnuss the subdivision scheme is constructed through two steps the first step inserts a set of edges and converts a valence n extraordinary point into a valence n face the second step combines both primal and dual subdivision schemes to define the subdivision rules the developed subdivision scheme generalizes bi cubic nurbs to arbitrary topology and is proved to be g 1 continuous for any valence extraordinary points and any non negative knot intervals the hnuss limit surface has comparable shape quality as non uniform subdivision via eigen polyhedron li et al 2016 and has better shape quality than all the other subdivision schemes in addition numerical experiments show that hnuss based isogeometric analysis yields improved convergence rates compared to any existing non uniform subdivision schemes

isogeometric analysis with geometrically continuous functions on planar multi patch geometries abstract we generate a basis of the space of bicubic and biquartic c 1 smooth geometrically continuous isogeometric functions on bilinear multi patch domains r 2 the basis functions are obtained by suitably combining c 1 smooth geometrically continuous isogeometric functions on bilinearly parameterized two patch domains cf 18 they are described by simple explicit formulas for their spline coefficients these c 1 smooth isogeometric functions possess potential for applications in isogeometric analysis which is demonstrated by several examples such as the biharmonic equation in particular the numerical results indicate optimal approximation power

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

majority vote model on hyperbolic lattices we study the critical properties of a nonequilibrium statistical model the majority vote model on heptagonal and dual heptagonal lattices such lattices have the special feature that they only ca

a scaled boundary isogeometric formulation for the elasto plastic analysis of solids in boundary representation abstract this contribution deals with the nonlinear analysis of boundary represented solids with elasto plastic material behavior based on the so called scaled boundary isogeometric formulation sb iga the proposed approach combines the features of the scaled boundary finite element method and isogeometric analysis based on the original boundary representation of the cad model a formulation is provided where the geometrical description of the boundary is sufficient to define the entire surface the domain is parameterized by a radial scaling parameter emanating from a scaling center and a parameter in circumferential direction along the boundary non star shaped domains are tackled by standard sub structuring here conforming discretizations are considered for the two dimensional case according to the isogeometric paradigm nurbs basis functions are employed for the approximation of the solution the displacement response is derived based on a multiplicative decomposition of the approximation in circumferential and radial scaling direction the boundary value problem is solved with the galerkin method the newton raphson iterative scheme is employed to obtain the nonlinear response several benchmark tests demonstrate the accuracy and computational efficiency of the formulation

space of c 2 smooth geometrically continuous isogeometric functions on two patch geometries abstract the space of c 2 smooth geometrically continuous isogeometric functions on bilinearly parameterized two patch domains is considered the investigation of the dimension of the spaces of biquintic and bisixtic c 2 smooth geometrically continuous isogeometric functions on such domains is presented in addition c 2 smooth isogeometric functions are constructed to be used for performing l 2 approximation and for solving triharmonic equation on different two patch geometries the numerical results indicate optimal approximation order

reissner mindlin shell implementation and energy conserving isogeometric multi patch coupling summary a shear flexible isogeometric reissner mindlin shell formulation using non uniform rational b splines basis functions is introduced which is used for the demonstration of a coupling approach for multiple non conforming patches the six degrees of freedom formulation uses the exact surface normal vectors and curvature the shell formulation is implemented in an isogeometric analysis framework for computation of structures composed of multiple geometric entities to enable local model refinement as well as non matching domains coupling a conservative multi patch approach using lagrange multipliers for structured non uniform rational b splines patches is presented here an additional border frame mesh is used to couple geometries during structural analyses this frame interface approach avoids the problem of excessive constraints when multiple patches are coupled at one point first the shell formulation is verified with several reference cases then the influence of the frame interface discretization and frame penalty stiffness on the smoothness of the results is investigated the effects of the perturbed lagrangian method in combination with the frame interface approach is shown in addition results of models with t joint interface connections and perpendicular stiffener patches are presented copyright 2016 john wiley sons ltd

multiple hopf bifurcation in r3 and inverse jacobi multipliers abstract in this paper we study the maximum number of limit cycles that can bifurcate from a singular point of saddle focus type of an analytic autonomous differential system in r 3 under any analytic perturbation that keeps the location and nature of the singularity we only consider those foci on center manifolds having associated two nonzero purely imaginary and one nonzero real eigenvalues our approach is different from the classical one in the sense that we do not use any center manifold reduction to compute poincare lyapunov constants instead we study the multiple hopf bifurcation first doing a lyapunov schmidt reduction to the associated poincare map obtaining in this way an analytic reduced displacement map next we prove that the order of this displacement map coincides with the vanishing multiplicity denoted m of any locally smooth and non flat inverse jacobi multiplier finally the cyclicity of the focus is given in terms of m

geometrically nonlinear formulation for thin shells without rotation degrees of freedom abstract a finite element formulation is presented for modelling geometrically nonlinear thin shells which exploits standard lagrange finite element basis functions without introducing rotation degrees of freedom the classical regularity requirements associated with thin bending problems are circumvented by introducing special integrals over inter element edges the use of lagrange finite element basis functions and the absence of rotation degrees of freedom make the formulation relatively simple and discontinuities in material properties and non smooth shell geometry can be incorporated trivially the variational problem can be exactly linearised leading to an efficient newton raphson solution process the performance of the approach is demonstrated via a range of numerical benchmarks for both geometrically linear and nonlinear problems it is shown that cubic elements perform particularly well

viscous fluid fingering on a negatively curved surface viscous fingering formation in flat hele shaw cells is a classical and widely studied fluid mechanical problem we examine the development of viscous fluid fingering on a two dimensional surface of constant negative gaussian curvature the hyperbolic plane h 2 a perturbative mode coupling formalism is applied to study the influence of the negative surface curvature on the two most important pattern formation mechanisms of the system fingertip splitting and finger competition we also report on a time dependent control strategy placed on the injection rate which is able to minimize viscous fingering growth on h 2

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

a perturbed lagrangian formulation for the finite element solution of contact problems making use of a perturbed lagrangian formulation a finite element procedure for contact problems is developed for the general case in which node to node contact no longer holds the proposed procedure leads naturally to a discretization of the contact interface into contact segments within the context of a bilinear interpolation for the displacement field a mixed finite element approximation is introduced by assuming discont naous contact pressure constant on the contact segment because of this piece wise constant approximation the gap function enters into the formulation in an average sense instead of through a point wise definition numerical examples are presented that illustrate the performance of the proposed procedure current finite element formulations for contact problems based on either the classical lagrange parameter procedure l 3 12 203 or the penalty function method 4 6 11 are characterized by a point wise enforcement of the contact constraint condition in the sense that penetration of the bodies is established on a nodal basis moreover in this methods the recovery of the contact pressure over the element from the contact nodal forces generally requires an additional procedure within the framework of classical lagrange multiplier methods the contact condition is exactly satisfied by transforming the constrained problem into an unconstrained one with the introduction of additional variables lagrange multipliers these extra variables add computational effort to the solution process which often requires special procedures to handle the presence of zero diagonal terms penalty methods on the other hand enable one to transform the constrained problem into an unconstrained one without introducing additional variables the constraint condition is now satisfied only approximately for finite values of the penalty parameter the main difhculty associated with these methods however lies in the poor conditioning of the problem as the penalty is increased to more accurately enforce the constraint condition this is a well understood phenomenon particularly in the context of the incompressible and nearly incompressible problem in solid and fluid mechanics e g see l 22 25 for a review recently augmented lagrangian procedures have been proposed as a promising way to partially overcome these difficulties and regularize the penalty formulation e g see the survey in 7 8 within the framework of zinearited e u c it is possible to restrict the finite element formulation of contact problems by assuming that node to node contact occurs this is in fact the case often considered in the literature l 2 4 6 12 20 211 in the general context of fully

weak impositions of dirichlet boundary conditions in solid mechanics a critique of current approaches and extension to partially prescribed boundaries abstract in this article we first review various approaches developed to date for the weak imposition of dirichlet boundary conditions in fictitious domain analysis for elasticity problems the hellinger reissner h r principle the linked lagrange multiplier llm method the implicit boundary method and the fat boundary method are discussed along with the well known lagrange multiplier penalty and nitsche s methods we state these approaches in a common form starting with energy functionals and weak forms and discretise using the fictitious domain finite element method previous formulations of these methods were in general developed for full prescription along the dirichlet boundary which generally implies no local effect of boundary inclination however partially prescribed conditions such as the structural roller boundary condition with inclination have wide practical applications in engineering here we provide techniques of imposing such boundary conditions in these methods in detail for those methods that contain algorithmic parameters such as the penalty and nitsche s methods extra computation or empirical estimation is necessary to decide values of the parameters and hence we discuss parametric and convergence behaviours through numerical examples to provide guidance on the choice of parameters

a nurbs based galerkin approach for the analysis of solids in boundary representation abstract the paper is concerned with a new numerical method to solve the elasticity problem of solids in boundary representation a formulation is derived where the geometrical description of the boundary is sufficient for defining the equations of elasticity of the complete solid while the interior of the domain is described by a radial scaling parameter the scaling of the boundary with respect to the specified scaling center leads to the complete solid this idea fits perfectly to the boundary representation modeling technique commonly employed in cad in the present approach the tensor product structure of the solid will be reduced by one dimension to parametrize the physical domain i e the three dimensional solid exploits only two dimensional nurbs objects which parametrize the boundary surfaces for the analysis the weak form of the equilibrium equations is enforced for the entire solid in particular the weak form is employed in the scaling direction and the circumferential direction applying the isogeometric paradigm the nurbs functions that describe the boundary of the geometry form also the basis for the approximation of the displacement at the boundary the displacement response in the radial scaling direction on the other hand is approximated by a one dimensional nurbs overall the galerkin projection of the weak form yields a linear system of equilibrium equations whose solution gives rise to the displacement response the accuracy of the method is validated by means of analytical reference data in conclusion the proposed method allows the analysis of solids which are bounded by an arbitrary number of contour boundaries

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

dual and approximate dual basis functions for b splines and nurbs comparison and application for an efficient coupling of patches with the isogeometric mortar method abstract this contribution defines and compares different methods for the computation of dual basis functions for b splines and non uniform rational b splines nurbs they are intended to be used as test functions for the isogeometric mortar method but other fields of application are possible too three different concepts are presented and compared the first concept is the explicit formula for the computation of dual basis functions for nurbs proposed in the work of carl de boor these dual basis functions entail minimal support i e the support of the dual basis functions is equal to the support of the corresponding b spline basis functions in the second concept dual basis functions are derived from the inversion of the gram matrix these dual basis functions have global support along the interface the third concept is the use of approximate dual basis functions which were initially proposed for the use in harmonic analysis the support of these functions is local but larger than the support of the associated b spline basis functions we propose an extension of the approximate dual basis functions for nurbs basis functions after providing the general formulas we elaborate explicit expressions for several degrees of spline basis functions all three approaches are applied in the frame of the mortar method for the coupling of non conforming nurbs patches a method which allows complex discretizations with multiple intersecting interfaces is presented numerical examples show that the explicitly defined dual basis functions with minimal support severely deteriorate the global stress convergence behavior of the mechanical analysis this fact is in accordance with mathematical findings in literature which state that the optimal reproduction degree of arbitrary functions is not possible without extending the support of the dual basis functions the dual basis functions computed from the inverse of the gram matrix yield accurate numerical results but the global support yields significantly higher computational costs in comparison to computations of conforming meshes only the approximate dual basis functions yield accurate and efficient computations where neither accuracy nor efficiency is significantly deteriorated in comparison to computations of conforming meshes all basic cases of t intersections and star intersections are studied furthermore an example which combines all basic cases in a complex discretization is given the applicability of the presented method for the nonlinear case and for shell formulations is shown with the help of one numerical example

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

the weak substitution method an application of the mortar method for patch coupling in nurbs based isogeometric analysis in this contribution a mortar type method for the coupling of non conforming nurbs surface patches is proposed the connection of non conforming patches with shared degrees of freedom requires mutual refinement which propagates throughout the whole patch due to the tensor product structure of nurbs surfaces thus methods to handle non conforming meshes are essential in nurbs based isogeometric analysis the main objective of this work is to provide a simple and efficient way to couple the individual patches of complex geometrical models without altering the variational formulation the deformations of the interface control points of adjacent patches are interrelated with a master slave relation this relation is established numerically using the weak form of the equality of mutual deformations along the interface with the help of this relation the interface degrees of freedom of the slave patch can be condensated out of the system a natural connection of the patches is attained without additional terms in the weak form the proposed method is also applicable for nonlinear computations without further measures linear and geometrical nonlinear examples show the high accuracy and robustness of the new method a comparison to reference results and to computations with the lagrange multiplier method is given

properties of center manifolds the center manifold has a number of puzzling properties associated with the basic questions of existence uniqueness differentiability and analyticity which may cloud its profitable application in e g bifurcation theory this paper aims to deal with some of these subtle properties regarding existence and uniqueness it is shown that the cut off function appear ing in the usual existence proof is responsible for the selection of a single center manifold thereby hiding the inherent nonuniqueness conditions are given for different center manifolds at an equilibrium point to have a nonempty intersection this intersection will include at least the families of stationary and periodic solutions crossing through the equilibrium in the case of nonuniqueness the difference between any two center manifolds will be less than el exp e2x l with el and e2 constants which explains why the formal taylor expansions of different center manifolds are the same while the expansions do not converge the differentiability of a center manifold will in certain cases decrease when moving out of the origin and a simple example shows how the differentiability may be lost center manifolds are usually not analytic however an analytic manifold may exist which contains all periodic solutions of a certain type but which may otherwise not be invariant using this manifold a new and very simple proof of the lyapunov subcenter theorem is given

strong multipatch c1 coupling for isogeometric analysis on 2d and 3d domains abstract the solution spaces of isogoemetric analysis iga constructed from p degree basis functions allow up to c p 1 continuity within one patch however for a multi patch domain the continuity is only c 0 at the boundaries between the patches in this study we present the construction of basis functions of degree p 2 which are c 1 continuous across the common boundaries shared by the patches the new basis functions are computed as a linear combination of the c 0 basis functions on the multi patch domains an advantage of the proposed method is that for the new basis functions the continuity within a patch is preserved without additional treatment of the functions in the interior of the patch we apply continuity constraints to the new basis functions to enforce c 1 continuity where the constraints are developed according to the concept of matched g k constructions always yield c k continuous isogeometric elements discussed in groisser and peters 2015 however for certain geometries the over constrained solution space will lead to c 1 locking collin and sangalli 2016 we discuss and show the usage of partial degree elevation to overcome this problem we demonstrate the potential of the c 1 basis functions for iga applications through several examples involving biharmonic equations