full analytical sensitivities in nurbs based isogeometric shape optimization abstract non uniform rational b spline nurbs has been widely used as an effective shape parameterization technique for structural optimization due to its compact and powerful shape representation capability and its popularity among cad systems the advent of nurbs based isogeometric analysis has made it even more advantageous to use nurbs in shape optimization since it can potentially avoid the inaccuracy and labor tediousness in geometric model conversion from the design model to the analysis model although both positions and weights of nurbs control points affect the shape until very recently usually only control point positions are used as design variables in shape optimization thus restricting the design space and limiting the shape representation flexibility this paper presents an approach for analytically computing the full sensitivities of both the positions and weights of nurbs control points in structural shape optimization such analytical formulation allows accurate calculation of sensitivity and has been successfully used in gradient based shape optimization the analytical sensitivity for both positions and weights of nurbs control points is especially beneficial for recovering optimal shapes that are conical e g ellipses and circles in 2d cylinders ellipsoids and spheres in 3d that are otherwise not possible without the weights as design variables

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

isogeometric boundary element analysis using unstructured t splines we couple collocated isogeometric boundary element methods and unstructured analysis suitable t spline surfaces for linear elastostatic problems we extend the definition of analysis suitable t splines to encompass unstructured control grids unstructured meshes and develop basis functions which are smooth rational polynomials defined in terms of the bezier extraction framework and which pass standard patch tests we then develop a collocation procedure which correctly accounts for sharp edges and corners extraordinary points and t junctions this approach is applied to several three dimensional problems including a real world t spline model of a propeller we believe this work clearly illustrates the power of combining new analysis suitable computer aided design technologies with established analysis methodologies in this case the boundary element method

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

on stable representations of bell elements abstract the paper is concerned with the space of c 1 continuous polynomial splines of degree 5 on general triangulations such that the restriction to each triangle corresponds to a bell element it is investigated how the standard construction of bell elements which is based on prescribing values and derivatives up to order two at the vertices of the triangulation can be used to define different bases the most straightforward one is the nodal basis which lacks stability but it is shown that this can be improved by scaling the interpolation data however the prime focus of the paper is on the construction of b spline like bases that depend on the choice of special domain triangles associated with the vertices of the triangulation a b spline like basis consists of functions that are locally supported and form a partition of unity if the triangulation satisfies certain geometric constraints e g is acute and the domain triangles are chosen suitably the basis is also non negative the stability analysis of the derived bases is addressed theoretically as well as numerically the latter with the examples of least square fitting and pde solving

detection of signals by information theoretic criteria a new approach is presented to the problem of detecting the number of signals in a multichannel time series based on the application of the information theoretic criteria for model selection introduced by akaike aic and by schwartz and rissanen mdl unlike the conventional hypothesis testing based approach the new approach does not requite any subjective threshold settings the number of signals is obtained merely by minimizing the aic or the mdl criteria simulation results that illustrate the performance of the new method for the detection of the number of signals received by a sensor array are presented

a stabilized powell sabin finite element method for the 2d euler equations in supersonic regime in this paper a powell sabin finite element ps fem scheme is presented for the solution of the 2d euler equations in supersonic regime the spatial discretization is based on ps splines that are piecewise quadratic polynomials with a global continuity defined on conforming triangulations some geometrical issues related to the practical construction of the ps elements are discussed in particular the generation of the control triangles and the imposition of the boundary conditions a stabilized formulation is considered and a novel shock capturing technique in the context of continuous finite elements is proposed to reduce oscillations around the discontinuity and compared with the classical technique proposed by tezduyar and senga 2006 the code is verified using manufactured solutions and validated using two challenging numerical examples which allows to evaluate the performance of the ps discretization in capturing the shocks

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

pole assignment robustness in a specified disk abstract a sufficient condition is proposed to guarantee robust pole location within a specified circular region for a linear uncertain system the explicit bounds on linear time invariant perturbations with highly structured information are obtained under these allowable highly structured perturbations both stability robustness and certain performance robustness will thus be ensured the merit of the proposed new approach is demonstrated by a given example where the results achieved are much better than the ones reported recently

surface fitting using convex powell sabin splines abstract convexity conditions for powell sabin splines are derived and an algorithm is presented for fitting a convex powell sabin spline to a set of scattered data part of this algorithm deals with the elimination of as many redundant conditions as possible the powell sabin splines are defined in terms of basis functions b splines with local support making use of their bernstein bezier representation leads to an easy handling of and an efficient calculating with these b splines numerical examples are supplied to illustrate the usefulness of surface fitting by means of convex powell sabin splines

triangulation based isogeometric analysis of the cahn hilliard phase field model abstract this paper presents triangulation based isogeometric analysis of the cahn hilliard phase field model the cahn hilliard phase field model is governed by a time dependent fourth order partial differential equation the corresponding primal variational form involves second order operators making it difficult to be directly analyzed with traditional c 0 finite element analysis in this paper we construct c 1 bernstein bezier simplicial elements through macro element techniques including various triangle split based macro elements in both 2d and 3d space we extend triangulation based isogeometric analysis to solving the primal variational form of the cahn hilliard equation we validate our method by convergence analysis showing the nodal and degree of freedom advantages over c 0 finite element analysis we then demonstrate detailed system evolution from randomly perturbed initial conditions in periodic two dimensional squares and three dimensional cubes we incorporate an adaptive time stepping scheme in these numerical experiments our numerical study demonstrates that triangulation based isogeometric analysis offers optimal convergence and time step stability is applicable to complex geometry and allows local refinement

bivariate splines of various degrees for numerical solution of partial differential equations bivariate splines with various degrees are considered in this paper a matrix form of the extended smoothness conditions for these splines is presented upon this form the multivariate spline method for numerical solution of partial differential equations pdes proposed by awanou lai and wenston in the multivariate spline method for scattered data fitting and numerical solutions of partial differential equations in wavelets and splines g chen and m j lai eds nashboro press brentwood tn 2006 pp 24 76 is generalized to obtain a new spline method it is observed that combined with prelocal refinement of triangulation and automatic degree raising over triangles of interest the new spline method of bivariate splines of various degrees is able to solve linear pdes very effectively and efficiently

isogeometric shape optimization of photonic crystals via coons patches abstract in this paper we present an approach that extends isogeometric shape optimization from optimization of rectangular like nurbs patches to the optimization of topologically complex geometries we have successfully applied this approach in designing photonic crystals where complex geometries have been optimized to maximize the band gaps salient features of this approach include the following 1 multi patch coons representation of design geometry the design geometry is represented as a collection of coons patches where the four boundaries of each patch are represented as nurbs curves the use of multiple patches is motivated by the need for representing topologically complex geometries the coons patches are used as a design representation so that designers do not need to specify interior control points and they provide a mechanism to compute analytical sensitivities for internal nodes in shape optimization 2 exact boundary conversion to the analysis geometry with guaranteed mesh injectivity the analysis geometry is a collection of nurbs patches that are converted from the multi patch coons representation with geometric exactness in patch boundaries the internal nurbs control points are embedded in the parametric domain of the coons patches with a built in mesh rectifier to ensure the injectivity of the resulting b spline geometry i e every point in the physical domain is mapped to one point in the parametric domain 3 analytical sensitivities sensitivities of objective functions and constraints with respect to design variables are derived through nodal sensitivities the nodal sensitivities for the boundary control points are directly determined by the design parameters and those for internal nodes are obtained via the corresponding coons patches

isogeometric triangular bernstein bezier discretizations automatic mesh generation and geometrically exact finite element analysis isogeometric analysis iga was introduced as a way to bypass the design to analysis bottleneck inherent in the traditional computer aided design cad through finite element analysis fea paradigm however an outstanding problem in the field of iga is that of surface to volume parameterization in cad packages solid objects are represented by a collection of nurbs or t spline bounding surfaces but to perform engineering analysis on real world problems we must find a way to parameterize the volumes of these objects as well this has proven to be difficult using traditional iga as the tensor product nature of trivariate nurbs and t splines limit their ability to create analysis suitable parameterizations of arbitrarily complex volumes to overcome the limitations of trivariate nurbs and t splines we propose the use of rational bernstein bezier tetrahedra to create analysis suitable volumetric parameterizations for isogeometric analysis in this paper which is part one of a two part series we present the methodology for discretizing two dimensional geometries using rational bernstein bezier triangles in addition to presenting finite element analysis methodologies based on rational bernstein bezier triangles we also introduce two new mesh generation strategies for automatically creating high quality geometrically exact curvilinear meshes we assess the quality of our mesh generation schemes using a suite of challenging two dimensional geometries and we verify the accuracy of our proposed numerical discretization approach using the method of manufactured solutions

an improved nurbs based isogeometric analysis with enhanced treatment of essential boundary conditions the nurbs based isogeometric analysis offers a seamless integration between the cad and subsequent finite element analysis and has been shown to be very effective for wide classes of problems due to its accurate geometric description however similar to the moving least square or reproducing kernel meshfree shape functions the nurbs basis functions generally are not interpolatory functions in this work it is shown that the direct imposition of the inhomogeneous essential boundary conditions to the nurbs control points may lead to significant errors with deteriorated rates of convergence consequently an improved formulation for nurbs based isogeometric analysis is proposed this is fulfilled by employing a transformation method that relates the control variables to the collocated nodal values at the essential boundary by using open knot vectors the resulting nurbs basis functions associated with the interior control points vanish at the boundary thus unlike the meshfree approximation the nurbs control points can be clearly partitioned into boundary and interior groups therefore the transformation method can be only applied to the boundary control points via invoking collocation of physical values at a set of boundary points the nurbs approximation enhanced with transformed physical boundary variables is kinematically admissible and the essential boundary conditions can be enforced straightforwardly like the finite element method several potential and elasticity problems evince that much higher solution accuracy with optimal convergence rates can be achieved by the present improved formulation compared with the method with direct imposition of essential boundary conditions to control variables

bezier projection a unified approach for local projection and quadrature free refinement and coarsening of nurbs and t splines with particular application to isogeometric design and analysis abstract we introduce bezier projection as an element based local projection methodology for b splines nurbs and t splines this new approach relies on the concept of bezier extraction and an associated operation introduced here spline reconstruction enabling the use of bezier projection in standard finite element codes bezier projection exhibits provably optimal convergence and yields projections that are virtually indistinguishable from global l 2 projection bezier projection is used to develop a unified framework for spline operations including cell subdivision and merging degree elevation and reduction basis roughening and smoothing and spline reparameterization in fact bezier projection provides a quadrature free approach to refinement and coarsening of splines in this sense bezier projection provides the fundamental building block for h p k r adaptivity in isogeometric analysis

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

optimizing domain parameterization in isogeometric analysis based on powell sabin splines we address the problem of constructing a high quality parameterization of a given planar physical domain defined by means of a finite set of boundary curves we look for a geometry map represented in terms of powell sabin b splines powell sabin splines are c 1 quadratic splines defined on a triangulation and thus the parameter domain can be any polygon the geometry map is generated by the following three step procedure first the shape of the parameter domain and a corresponding triangulation are determined in such a way that its number of corners matches the number of corners of the physical domain second the boundary control points related to the powell sabin b spline representation are chosen so that they parameterize the boundary curve of the physical domain third the remaining inner control points are obtained by solving a nimble optimization problem based on the winslow functional the proposed domain parameterization procedure is illustrated numerically in the context of isogeometric galerkin discretizations based on powell sabin splines it turns out that the flexibility rising from the generality of the parameter domain has a beneficial effect on the quality of the parameterization and also on the accuracy of the computed approximate solution

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

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

robust controller design for uncertain linear systems with circular pole constraints in this paper the problem of robust controller design for uncertain linear continuous and discrete time systems with circular pole constraints is considered the problem we address is the design of robust state feedback controllers such that all poles of the closed loop systems are placed within a prescribed circular region sufficient conditions for the existence of the desired controllers are given based on the generalized inverse theory the analytical expression of the set of desired controllers is also presented it is emphasized that when the uncertainties are absent the results include the characterization of all state feedback controllers achieving the specified circular pole constraints an illustrative example is given to show the applicability of the proposed design procedure accepted in final revised form 7 february 1996 communicated by professor h kimura

from nurbs to nurps geometries abstract quadratic powell sabin splines and their rational extension the so called nurps surfaces are an interesting alternative for classical tensor product nurbs in the context of isogeometric analysis because they allow the use of local refinements while retaining a b spline like representation and exact description of conic sections in this paper we present a simple and effective strategy to convert a given planar geometry defined by a quadratic nurbs representation into a nurps representation suitable for the analysis

bivariate spline spaces and minimal determining sets abstract the paper describes bivariate spline spaces and minimal determining sets shows applications of these concepts presents two new triangular finite elements describes public domain software for the analysis of spline spaces and concludes with a concise history of these subjects the software described in this paper is available on http www math utah edu alfeld mds that page also contains links to colored versions of the figures in this paper

numerical solution of partial differential equations with powell sabin splines powell sabin splines are piecewise quadratic polynomials with global c 1 continuity they are defined on conforming triangulations of two dimensional domains and admit a compact representation in a normalized b spline basis recently these splines have been used successfully in the area of computer aided geometric design for the modelling and fitting of surfaces in this paper we discuss the applicability of powell sabin splines for the numerical solution of partial differential equations defined on domains with polygonal boundary a galerkin type pde discretization is derived for the variable coefficient diffusion equation special emphasis goes to the treatment of dirichlet and neumann boundary conditions finally an error estimator is developed and an adaptive mesh refinement strategy is proposed we illustrate the effectiveness of the approach by means of some numerical experiments

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

rotation free bernstein bezier elements for thin plates and shells development and validation abstract a new finite element procedure for thin plates and shells is presented it combines a geometrically non linear rotation free kirchhoff love formulation with triangular and quadrilateral bernstein bezier elements and c 1 and g 1 inter element continuity conditions as well as boundary conditions for clamping and for symmetry the formulation is free from transverse shear locking and relies on a high polynomial degree to mitigate membrane locking bernstein bezier elements are as opposed to nurbs suitable for arbitrary triangulations unlike with hermite elements no stress concentrations occur if the boundary is partially clamped and the formulation can be potentially extended to stiffened plates and shells and to step wise changes of thickness and material properties the convergence behaviour is demonstrated and the computational efficiency is compared with that of c reissner mindlin elements on several numerical examples

continuity and convergence in rational triangular bezier spline based isogeometric analysis abstract this paper presents a method for isogeometric analysis using rational triangular bezier splines rtbs where optimal convergence rates are achieved in this method both the geometry and the physical field are represented by bivariate splines in bernstein bezier form over the triangulation of a domain from a given physical domain bounded by nurbs curves a parametric domain and its triangulation are constructed by imposing continuity constraints on bezier ordinates we obtain a set of global c r smooth basis functions convergence analysis shows that isogeometric analysis with such c r rtbs basis can deliver the optimal rate of convergence provided that the c r geometric map remains unchanged during the refinement process this condition can be satisfied by constructing a pre refinement geometric map that is sufficiently smooth numerical experiments verify that optimal rates of convergence are achieved for poisson and linear elasticity problems

isogeometric analysis with bezier tetrahedra abstract this paper presents an approach for isogeometric analysis of 3d objects using rational bezier tetrahedral elements in this approach both the geometry and the physical field are represented by trivariate splines in bernstein bezier form over the tetrahedrangulation of a 3d geometry given a nurbs represented geometry either untrimmed or trimmed we first convert it to a watertight geometry represented by rational triangular bezier splines rtbs for trimmed geometries a compatible subdivision scheme is developed to guarantee the watertightness the rtbs geometry preserves exactly the original nurbs surfaces except for an interface layer between trimmed surfaces where controlled approximation occurs from the watertight rtbs geometry a bezier tetrahedral partition is generated automatically by imposing continuity constraints on bezier ordinates of the elements we obtain a set of global c r smooth basis functions and use it as the basis for analysis numerical examples demonstrate that our method achieves optimal convergence in c r spaces and can handle complicated geometries

isogeometric analysis using t splines on two patch geometries we develop optimal approximation estimates for t splines in the case of geometries obtained by gluing two standard tensor product patches we derive results both for the t spline space in the parametric domain and the mapped t nurbs in the physical one a set of numerical tests in complete accordance with the theoretical developments is also presented

isogeometric analysis with powell sabin splines for advection diffusion reaction problems abstract this paper presents the use of powell sabin splines in the context of isogeometric analysis for the numerical solution of advection diffusion reaction equations powell sabin splines are piecewise quadratic c 1 functions defined on a given triangulation with a particular macro structure we discuss the galerkin discretization based on a normalized powell sabin b spline basis we focus on the accurate detection of internal and boundary layers and on local refinements we apply the approach to several test problems and we illustrate its effectiveness by a comparison with classical finite element and recent isogeometric analysis procedures