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

shape optimization using the cut finite element method we present a cut finite element method for shape optimization in the case of linear elasticity the elastic domain is defined by a level set function and the evolution of the domain is obtained by

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

transient three dimensional domain decomposition problems frame indifferent mortar constraints and conserving integration the present work deals with transient large deformation domain decomposition problems the tying of dissimilar meshed grids is performed by applying the mortar method in this connection a reformulation of the original linear mortar constraints is proposed which retains frame indifference for arbitrary discretizations of the interface furthermore a specific coordinate augmentation technique is proposed to make possible the design of an energy momentum scheme numerical examples demonstrate the robustness and enhanced numerical stability of the newly developed energy momentum scheme for three dimensional problems copyright 2009 john wiley sons ltd

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

singularities of the hypergeometric system associated with a monomial curve we compute using d module restrictions the slopes of the irregular hypergeometric system associated with a monomial curve we also study rational solutions and reducibility of such systems

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

irregular modified a hypergeometric systems a modified a hypergeometric system is a system of differential equations for the function f tw x where f y is a solution of an a hypergeometric system in n variables and w is an n dimensional integer vector which is called the weight vector we study the irregularity of modified systems by adapting to this case the notion of umbrella introduced by m schulze and u walther especially we study slopes and gevrey series solutions we develop some applications of this study under some conditions we give laplace integral representations of divergent series solutions of the modified system and we show that certain gevrey series solutions of the original a hypergeometric system along coordinate varieties are gevrey asymptotic expansions of holomorphic solutions of the a hypergeometric system

the mortar finite element method with lagrange multipliers the present paper deals with a variant of a non conforming domain decomposition technique the mortar finite element method in the opposition to the original method this variant is never conforming because of the relaxation of the matching constraints at the vertices and the edges in 3d of subdomains it is shown that written under primal hybrid formulation the approximation problem issued from a discretization of a second order elliptic equation in 2d is nonetheless well posed and provides a discrete solution that satisfies optimal error estimates with respect to natural norms finally the parallelization advantages consequence of this variant are also addressed

studies of refinement and continuity in isogeometric structural analysis we investigate the effects of smoothness of basis functions on solution accuracy within the isogeometric analysis framework we consider two simple one dimensional structural eigenvalue problems and two static shell boundary value problems modeled with trivariate nurbs solids we also develop a local refinement strategy that we utilize in one of the shell analyses we find that increased smoothness that is the k method leads to a significant increase in accuracy for the problems of structural vibrations over the classical c0c0 continuous p method whereas a judicious insertion of c0c0 continuous surfaces about singularities in a mesh otherwise generated by the k method usually outperforms a mesh in which all basis functions attain their maximum level of smoothness we conclude that the potential for the k method is high but smoothness is an issue that is not well understood due to the historical dominance of c0c0 continuous finite elements and therefore further studies are warranted

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

irregular hypergeometric systems associated with a singular monomial curve in this paper we study irregular hypergeometric systems defined by one row specifically we calculate slopes of such systems in the case of reduced semigroups we generalize the case studied by castro and takayama in all the cases we find that there always exists a slope with respect to a hyperplane of this system only in the case of an irregular system defined by a 1 x 2 integer matrix we might need a change of coordinates to study slopes at infinity in the other cases slopes are always at the origin defined with respect to a hyperplane we also compute all the l characteristic varieties of the system so we have a section of the grobner fan of the module defined by the hypergeometric system

gevrey solutions of irregular hypergeometric systems in two variables we describe the gevrey series solutions at singular points of the irregular hypergeometric system gkz system associated with an affine plane monomial curve we also describe the irregularity complex of such a system with respect to its singular support and in particular we prove using elementary methods that this irregularity complex is a perverse sheaf as assured by a theorem of z mebkhout

irregularity of hypergeometric systems via slopes along coordinate subspaces we study the irregularity sheaves attached to the a hypergeometric d module ma introduced by gel fand et al where a z d n is pointed of full rank and c more precisely we investigate the slopes of this module along coordinate subspaces in the process we describe the associated graded ring to a positive semigroup ring for a filtration defined by an arbitrary weight vector l on torus equivariant generators to this end we introduce the a l umbrella a simplicial complex determined by a and l and identify its facets with the components of the associated graded ring we then establish a correspondence between the full a l umbrella and the components of the l characteristic variety of ma we compute in combinatorial terms the multiplicities of these components in the l characteristic cycle of the associated euler koszul complex identifying them with certain intersection multiplicities we deduce from this that slopes of ma are combinatorial independent of and in one to one correspondence with jumps of the a l umbrella this confirms a conjecture of sturmfels and gives a converse of a theorem of hotta ma is regular if and only if a defines a projective variety 1991 mathematics subject classification 13n10 14m25 16w70

on irregular binomial d modules we prove that a holonomic binomial d module m a i is regular if and only if certain associated primes of i determined by the parameter vector beta in mathbb c d are homogeneous we further describe the slopes of m a i along a coordinate subspace in terms of the known slopes of some related hypergeometric d modules that also depend on when the parameter is generic we also compute the dimension of the generic stalk of the irregularity of m a i along a coordinate hyperplane and provide some remarks about the construction of its gevrey solutions

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

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

mixed isogeometric finite cell methods for the stokes problem we study the application of the isogeometric finite cell method iga fcm to mixed formulations in the context of the stokes problem we investigate the performance of the iga fcm when utilizing some isogeometric mixed finite elements namely taylor hood sub grid raviart thomas and nedelec elements these element families have been demonstrated to perform well in the case of conforming meshes but their applicability in the cut cell context is still unclear dirichlet boundary conditions are imposed by nitsche s method numerical test problems are performed with a detailed study of the discrete inf sup stability constants and of the convergence behavior under uniform mesh refinement

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

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

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

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

a robust non linear mixed hybrid quadrilateral shell element in the paper a non linear quadrilateral shell element for the analysis of thin structures is presented the variational formulation is based on a hu washizu functional with independent displacement stress and strain fields the interpolation matrices for the mid surface displacements and rotations as well as for the stress resultants and strains are specified restrictions on the interpolation functions concerning fulfillment of the patch test and stability are derived the developed mixed hybrid shell element possesses the correct rank and fulfills the in plane and bending patch test using newton s method the finite element approximation of the stationary condition is iteratively solved our formulation can accommodate arbitrary non linear material models for finite deformations in the examples we present results for isotropic plasticity at finite rotations and small strains as well as bifurcation problems and post buckling response the essential feature of the new element is the robustness in the equilibrium iterations it allows very large load steps in comparison to other element formulations copyright 2005 john wiley sons ltd

a mortar finite element method using dual spaces for the lagrange multiplier the mortar finite element method allows the coupling of different discretization schemes and triangulations across subregion boundaries in the original mortar approach the matching at the interface is realized by enforcing an orthogonality relation between the jump and a modified trace space which serves as a space of lagrange multipliers in this paper this lagrange multiplier space is replaced by a dual space without losing the optimality of the method the advantage of this new approach is that the matching condition is much easier to realize in particular all the basis functions of the new method are supported in a few elements the mortar map can be represented by a diagonal matrix in the standard mortar method a linear system of equations must be solved the problem is considered in a positive definite nonconforming variational as well as an equivalent saddle point formulation

isogeometric reissner mindlin shell analysis with exactly calculated director vectors abstract an isogeometric reissner mindlin shell derived from the continuum theory is presented the geometry is described by nurbs surfaces the kinematic description of the employed shell theory requires the interpolation of the director vector and of a local basis system hence the definition of nodal basis systems at the control points is necessary for the proposed formulation the control points are in general not located on the shell reference surface and thus several choices for the nodal values are possible the proposed new method uses the higher continuity of the geometrical description to calculate nodal basis system and director vectors which lead to geometrical exact interpolated values thereof thus the initial director vector coincides with the normal vector even for the coarsest mesh in addition to that a more accurate interpolation of the current director and its variation is proposed instead of the interpolation of nodal director vectors the new approach interpolates nodal rotations account is taken for the discrepancy between interpolated basis systems and the individual nodal basis systems with an additional transformation the exact evaluation of the initial director vector along with the interpolation of the nodal rotations lead to a shell formulation which yields precise results even for coarse meshes the convergence behavior is shown to be correct for k refinement allowing the use of coarse meshes with high orders of nurbs basis functions this is potentially advantageous for applications with high numerical effort per integration point the geometrically nonlinear formulation accounts for large rotations the consistent tangent matrix is derived various standard benchmark examples show the superior accuracy of the presented shell formulation a new benchmark designed to test the convergence behavior for free form surfaces is presented despite the higher numerical effort per integration point the improved accuracy yields considerable savings in computation cost for a predefined error bound

irregular hypergeometric d modules abstract we study the irregularity of hypergeometric d modules m a via the explicit construction of gevrey series solutions along coordinate subspaces in x c n as a consequence we prove that along coordinate hyperplanes the combinatorial characterization of the slopes of m a given by m schulze and u walther 2008 in 23 still holds for any full rank integer matrix a we also provide a lower bound for the dimensions of the spaces of gevrey solutions along coordinate subspaces in terms of volumes of polytopes and prove the equality for very generic parameters holomorphic solutions of m a at nonsingular points can be understood as gevrey solutions of order one along x at generic points and so they are included as a particular case

an isogeometric indirect boundary element method for solving acoustic problems in open boundary domains abstract this work introduces an indirect boundary element method bem within a nurbs based isogeometric framework for solving three dimensional acoustic problems in the frequency domain developments of isogeometric boundary elements so far have focused on the direct bem yet a multitude of common problems in acoustics involves open boundary surfaces which require a more involved indirect boundary element formulation the current work presents an indirect variational bem which makes use of nurbs shape functions additionally a novel technique for coupling strongly non conforming patches is introduced to allow the analysis of more complex geometries the proposed isogeometric indirect boundary element method is verified against analytical solutions and benchmarked against the conventional polynomial based indirect bem also two open boundary problems are studied including analyses over wider frequency ranges and one industrial type complex geometry containing multiple non conforming patches the proposed method is found to be not only significantly more efficient than its polynomial based counterpart but also very robust against strong non conformities in the nurbs descriptions

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

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

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