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

bicubic b spline surfaces constrained by the biharmonic pde abstract bicubic b spline surface constrained by the biharmonic pde is presented in this paper by representing the biharmonic pde in the form of the bilinear b spline bases we find the regular vector valued coefficients and discover that bicubic b spline surface can satisfy the biharmonic pde when the control points of the boundaries for open or closed surfaces are given the inner control points can be fully determined for each case of the surfaces open in both directions closed in one direction and closed in both directions a linear system for solving inner control points is established some examples show the effectiveness of the given method

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

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

variational formulation and isogeometric analysis for fourth order boundary value problems of gradient elastic bar and plane strain stress problems abstract the fourth order boundary value problems of one parameter gradient elastic bar and plane strain stress models are formulated in a variational form within an h 2 sobolev space setting for both problems the existence and uniqueness of the solution is established by proving the continuity and coercivity of the associated symmetric bilinear form for completeness the full sets of boundary conditions of the problems are derived and in particular the new types of boundary conditions featured by the gradient elastic models are given the additional attributes singly and doubly by utilizing the continuity and coercivity of the continuous problems corresponding error estimates are formulated for conforming galerkin formulations finally numerical results with isogeometric c p 1 continuous discretizations for nurbs basis functions of order p 2 confirm the theoretical results and illustrate the essentials of both static and vibration problems

smooth cubic spline spaces on unstructured quadrilateral meshes with particular emphasis on extraordinary points geometric design and isogeometric analysis considerations abstract we present a framework for geometric design and isogeometric analysis on unstructured quadrilateral meshes acknowledging the differing requirements posed by design e g the convenience of an intuitive control net and analysis e g good approximation behavior we propose the construction of a separate smooth spline space for each while ensuring isogeometric compatibility requiring the geometric models to be members of the analysis suitable spaces the methodology is simple and is presented for bi cubic splines extensions to higher degrees are possible and are briefly discussed the presentation has been structured to show compatibility with t splines a state of the art cad technology but the approach should extend to other locally refinable spline technologies based on local tensor product structures an instantiation of the framework is presented and several numerical tests focused on geometric design and isogeometric analysis demonstrate the versatility of the developed framework and show significantly higher convergence rates than attained previously in the considered setting

injectivity sets for the radon transform over circles and complete systems of radial functions abstract a necessary and sufficient characterization is given that specifies which sets of sums of translations of radial functions are dense in the set of continuous functions in the plane this problem is shown to be equivalent to inversion for the radon transform on circles centered on restricted subsets of the plane the proofs rest on the geometry of zero sets for harmonic polynomials and the microlocal analysis of this circular radon transform a characterization of nodal sets for the heat and wave equation in the plane are consequences of our theorems and questions of pinkus and ehrenpreis are answered

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

isogeometric analysis of the advective cahn hilliard equation spinodal decomposition under shear flow we present a numerical study of the spinodal decomposition of a binary fluid undergoing shear flow using the advective cahn hilliard equation a stiff nonlinear parabolic equation characterized by the presence of fourth order spatial derivatives our numerical solution procedure is based on isogeometric analysis an approximation technique for which basis functions of high order continuity are employed these basis functions allow us to directly discretize the advective cahn hilliard equation without resorting to a mixed formulation we present steady state solutions for rectangular domains in two dimensions and for the first time in three dimensions we also present steady state solutions for the two dimensional taylor couette cell to enforce periodic boundary conditions in this curved domain we derive and utilize a new periodic bezier extraction operator we present an extensive numerical study showing the effects of shear rate surface tension and the geometry of the domain on the phase evolution of the binary fluid theoretical and experimental results are compared with our simulations

isogeometric analysis of high order partial differential equations on surfaces we consider the numerical approximation of high order partial differential equations pdes defined on surfaces in the three dimensional space with particular emphasis on closed surfaces we consider computational domains that can be represented by b splines or nurbs as for example the sphere and we spatially discretize the pdes by means of nurbs based isogeometric analysis in the framework of the standard galerkin method we numerically solve benchmark laplace beltrami problems of the fourth and sixth order as well as the corresponding eigenvalue problems with the goal of analyzing the role of the continuity of the nurbs basis functions on closed surfaces in this respect we show that the use of globally high order continuous basis functions as allowed by the construction of periodic nurbs leads to the efficient solution of the high order pdes finally we consider the numerical solution of high order phase field problems on closed surfaces namely the cahn hilliard and crystal equations

heisenberg uniqueness pairs and the klein gordon equation a heisenberg uniqueness pair hup is a pair where is a curve in the plane and is a set in the plane with the following property any finite borel measure in the plane supported on which is absolutely continuous with respect to arc length and whose fourier transformb vanishes on must automatically be the zero measure we prove that when is the hyperbola x1x2 1 and is the lattice cross z 0 0 z where are positive reals then is an hup if and only if 1 in this situation the fourier transform b of the measure solves the one dimensional klein gordon equation phrased di erently we show that e i nt e i n t n 2 z span a weak star dense subspace in l 1 r if and only if 1 in order to prove this theorem some elements of linear fractional theory and ergodic theory are needed such as the birkho ergodic theorem an idea parallel to the one exploited by makarov and poltoratski in the context of model subspaces is also needed as a consequence we solve a problem on the density of algebras generated by two inner functions raised by matheson and stessin

heisenberg uniqueness pairs for some algebraic curves in the plane abstract a heisenberg uniqueness pair is a pair where is a curve and is a set in r 2 such that whenever a finite borel measure having support on which is absolutely continuous with respect to the arc length on satisfies 0 then it is identically 0 in this article we investigate the heisenberg uniqueness pairs corresponding to the spiral hyperbola circle and certain exponential curves further we work out a characterization of the heisenberg uniqueness pairs corresponding to four parallel lines in the latter case we observe a phenomenon of interlacing of three trigonometric polynomials

solving the triharmonic equation over multi patch planar domains using isogeometric analysis abstract we present a framework for solving the triharmonic equation over bilinearly parameterized planar multi patch domains by means of isogeometric analysis our approach is based on the construction of a globally c 2 smooth isogeometric spline space which is used as discretization space the generated c 2 smooth space consists of three different types of isogeometric functions called patch edge and vertex functions all functions are entirely local with a small support and numerical examples indicate that they are well conditioned the construction of the functions is simple and works uniformly for all multi patch configurations while the patch and edge functions are given by a closed form representation the vertex functions are obtained by computing the null space of a small system of linear equations several examples demonstrate the potential of our approach for solving the triharmonic equation

isogeometric collocation on planar multi patch domains abstract we present an isogeometric framework based on collocation to construct a c 2 smooth approximation of the solution of the poisson s equation over planar bilinearly parameterized multi patch domains the construction of the used globally c 2 smooth discretization space for the partial differential equation is simple and works uniformly for all possible multi patch configurations the basis of the c 2 smooth space can be described as the span of three different types of locally supported functions corresponding to the single patches edges and vertices of the multi patch domain for the selection of the collocation points which is important for the stability and convergence of the collocation problem two different choices are numerically investigated the first approach employs the tensor product greville abscissae as collocation points and shows for the multi patch case the same convergence behavior as for the one patch case 1 which is suboptimal in particular for odd spline degree the second approach generalizes the concept of superconvergent points from the one patch case cf 2 3 4 to the multi patch case again these points possess better convergence properties than greville abscissae in case of odd spline degree

coxeter system of lines and planes are sets of injectivity for the twisted spherical means abstract it is well known that a line in r 2 is not a set of injectivity for the spherical means for odd functions about that line we prove that any line passing through the origin is a set of injectivity for the twisted spherical means tsm for functions f l 2 c whose each of spectral projection e 1 4 z 2 f k is a polynomial then we prove that any coxeter system of even number of lines is a set of injectivity for the tsm for l p c 1 p 2 further we deduce that certain coxeter system of even number of planes is a set of injectivity for the tsm for l p c n 1 p 2 we observe that a set s r 2 n 1 c is a set of injectivity for the tsm for a certain class of functions on c n 1

dimension and basis construction for c2 smooth isogeometric spline spaces over bilinear like g2 two patch parameterizations abstract a particular class of planar two patch geometries called bilinear like g 2 two patch geometries is introduced this class includes the subclass of all bilinear two patch parameterizations and possesses similar connectivity functions along the patch interface it is demonstrated that the class of bilinear like g 2 two patch parameterizations is much wider than the class of bilinear parameterizations and can approximate with good quality given generic two patch parameterizations we investigate the space of c 2 smooth isogeometric functions over this specific class of two patch geometries the study is based on the equivalence of the c 2 smoothness of an isogeometric function and the g 2 smoothness of its graph surface cf groisser and peters 2015 and kapl et al 2015 the dimension of the space is computed and an explicit basis construction is presented the resulting basis functions possess simple closed form representations have small local supports and are well conditioned in addition we introduce a subspace whose basis functions can be generated uniformly for all possible configurations of bilinear like g 2 two patch parameterizations numerical results obtained by performing l 2 approximation and solving poisson s equation indicate that already the subspace possesses optimal approximation properties

space of c2 smooth geometrically continuous isogeometric functions on planar multi patch geometries dimension and numerical experiments abstract we study the space of c 2 smooth isogeometric functions on bilinearly parameterized multi patch domains r 2 where the graph of each isogeometric function is a multi patch spline surface of bidegree d d d 5 6 the space is fully characterized by the equivalence of the c 2 smoothness of an isogeometric function and the g 2 smoothness of its graph surface cf groisser and peters 2015 kapl et al 2015 this is the reason to call its functions c 2 smooth geometrically continuous isogeometric functions in particular the dimension of this c 2 smooth isogeometric space is investigated the study is based on the decomposition of the space into three subspaces and is an extension of the work kapl and vitrih 2017 to the multi patch case in addition we present an algorithm for the construction of a basis and use the resulting globally c 2 smooth functions for numerical experiments such as performing l 2 approximation and solving triharmonic equation on bilinear multi patch domains the numerical results indicate optimal approximation order

an isogeometric analysis approach to gradient damage models continuum damage formulations are commonly used for the simulation of diffuse fracture processes implicit gradient damage models are employed to avoid the spurious mesh dependencies associated with local continuum damage models the c 0 continuity of traditional finite elements has hindered the study of higher order gradient damage approximations in this contribution we use isogeometric finite elements which allow for the construction of higher order continuous basis functions on complex domains we study the suitability of isogeometric finite elements for the discretization of higher order gradient damage approximations

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

refinable c2 piecewise quintic polynomials on powell sabin 12 triangulations we present a construction of nested spaces of c 2 macro elements of degree 5 on triangulations of a polygonal domain obtained by uniform refinements of an initial triangulation and a powell sabin 12 split

boundaries of a convex set and interpolation sets the structure of the boundary of a convex set c is a field of intensive research in functional analysis the classical boundary is the set ext c of the extreme points of c at least when some compactness is assumed and c is recovered from ext c by means of the integral representation theory in this paper a more general notion of boundary definition 1 i is considered such a boundary needs not contain or even meet the extreme points and thus the classical tools are not available however through r c james s technique and a remarquable result of s simons a convex set can often be recovered in a strong sense from its boundary sect 1 tight connections are established between this notion lacunarity in harmonic analysis sect 2 banach spaces containing or not n sect 3 compactness and duality theory sect 4 finally sect 5 is devoted to miscellaneous results and questions in the spirit of the work and to examples showing the necessity of the assumptions we made as a general remark we did not systematically try to state the results in their most general form as far as it was not necessary to introduce new ideas for doing so this is specially true for the results of sect 2 where we worked in the duality 2 of the little fourier analysis although the techniques we used remain valid within the general frame of locally compact abelian groups

isogeometric analysis with geometrically continuous functions on two patch geometries we study the linear space of c s smooth isogeometric functions defined on a multi patch domain r 2 we show that the construction of these functions is closely related to the concept of geometric continuity of surfaces which has originated in geometric design more precisely the c s smoothness of isogeometric functions is found to be equivalent to geometric smoothness of the same order g s smoothness of their graph surfaces this motivates us to call them c s smooth geometrically continuous isogeometric functions we present a general framework to construct a basis and explore potential applications in isogeometric analysis the space of c 1 smooth geometrically continuous isogeometric functions on bilinearly parameterized two patch domains is analyzed in more detail numerical experiments with bicubic and biquartic functions for performing l 2 approximation and for solving poisson s equation and the biharmonic equation on two patch geometries are presented and indicate optimal rates of convergence

boundaries of compact convex sets and fragmentability if x is a compact convex set in a real locally convex space b x is said to be its boundary if every affine continuous function on x attains its maximum at some point of b we study relations between fragmentability of b and the whole set x as a byproduct we obtain a characterization of separable asplund spaces we also study the possibility of finding the haar system in a boundary of a metrizable compact convex set

isogeometric analysis for surface pdes with extended loop subdivision abstract we investigate the isogeometric analysis for surface pdes based on the extended loop subdivision approach the basis functions consisting of quartic box splines corresponding to each subdivided control mesh are utilized to represent the geometry exactly and construct the solution space for dependent variables as well which is consistent with the concept of isogeometric analysis the subdivision process is equivalent to the h refinement of nurbs based isogeometric analysis the performance of the proposed method is evaluated by solving various surface pdes such as surface laplace beltrami harmonic biharmonic triharmonic equations which are defined on the limit surfaces of extended loop subdivision for different initial control meshes numerical experiments show that the proposed method has desirable performance in terms of the accuracy convergence and computational cost for solving the above surface pdes defined on both open and closed surfaces the proposed approach is proved to be second order accuracy in the sense of l 2 norm with theoretical and or numerical results which is also outperformed over the standard linear finite element by several numerical comparisons

representation of piecewise biharmonic surfaces using biquadratic b splines piecewise biquadratic b spline surface satisfying biharmonic condition is studied in this paper by applying biharmonic pde biquadratic b spline surface is fully determined by boundary control vertices a linear system for solving inner control vertices is established its coefficient matrix is block tridiagonal and the proof of non singularity of coefficient matrix is presented under the condition that all knots do not coincide with each other a few examples are given to show the effectiveness of piecewise biharmonic biquadratic b spline surface an interpolation method on given boundary points is presented if boundary curves are open

reproducing kernel triangular b spline based fem for solving pdes abstract we propose a reproducing kernel triangular b spline based finite element method fem as an improvement to the conventional triangular b spline element for solving partial differential equations pdes in the latter unexpected errors can occur throughout the analysis domain mainly due to the excessive flexibilities in defining the b spline the performance therefore becomes unstable and cannot be controlled in a desirable way to address this issue the proposed improvement adopts the reproducing kernel approximation in the calculation of b spline kernel function three types of pde problems are tested to validate the present element and compare against the conventional triangular b spline it has been shown that the improved triangular b spline satisfies the partition of unity condition even for extreme conditions including corners and holes

bounding the influence of domain parameterization and knot spacing on numerical stability in isogeometric analysis abstract isogeometric analysis iga was introduced by hughes et al 2005 1 as a new method to bridge the gap between the geometry description and numerical analysis similar to the finite element approach the iga concept to solve a partial differential equation leads to a linear system of equations the condition number of the coefficient matrix is a crucial factor for the stability of the system it depends strongly on the domain parameterization which provides the isogeometric discretization in this paper we derive a bound for the condition number of the stiffness matrix of the poisson equation in particular we investigate the influence of the domain parameterization and the knot spacing on the stability of the numerical system the factors appearing in our bound reflect the stability properties of a given domain parameterization

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

weights of boundaries of compact convex sets let b be a boundary of a compact convex set x we prove that the weight of x equals the weight of b provided b is lindelof or x is a standard compact convex set and b is the set of extreme points of x

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

finite element analysis on implicitly defined domains an accurate representation based on arbitrary parametric surfaces in this paper we present some novel results and ideas for robust and accurate implicit representation of geometric surfaces in finite element analysis the novel contributions of this paper are threefold 1 describe and validate a method to represent arbitrary parametric surfaces implicitly 2 represent arbitrary solids implicitly including sharp features using level sets and boolean operations 3 impose arbitrary dirichlet and neumann boundary conditions on the resulting implicitly defined boundaries the methods proposed do not require local refinement of the finite element mesh in regions of high curvature ensure the independence of the domain s volume on the mesh do not rely on boundary regularization and are well suited to methods based on fixed grids such as the extended finite element method xfem numerical examples are presented to demonstrate the robustness and effectiveness of the proposed approach and show that it is possible to achieve optimal convergence rates using a fully implicit representation of object boundaries this approach is one step in the desired direction of tying numerical simulations to computer aided design cad similarly to the isogeometric analysis paradigm

isogeometric analysis for sixth order boundary value problems of gradient elastic kirchhoff plates abstract sixth order boundary value problems of a one parameter gradient elastic kirchhoff plate model are formulated in a weak form within an h 3 sobolev space setting with the corresponding equilibrium equations and general boundary conditions the corresponding conforming galerkin method is proposed with error estimates for discretizations satisfying c 2 continuity requirements continuity coercivity and consistency of the corresponding bilinear form are utilized for proving the theoretical results numerical computations with conforming isogeometric discretizations of c p 1 continuous nurbs basis functions of order p 3 confirm the theoretical results and illustrate the features of the problem for both statics and free vibrations in particular the effects of the additional boundary conditions and parameter dependent boundary layers corresponding to the gradient elasticity theory are addressed by the numerical examples

explicit bezier control net of a pde surface abstract the pde under study here is a general fourth order linear elliptic partial differential equation having prescribed the boundary control points we provide the explicit expression of the whole control net of the associated pde bezier surface in other words we obtain the explicit expressions of the interior control points as linear combinations of free boundary control points the set of scalar coefficients of these combinations works like a mould for pde surfaces thus once this mould has been computed for a given degree real time manipulation of the resulting surfaces becomes possible by modifying the prescribed information

injectivity sets for spherical means on the heisenberg group in this paper we prove that cylinders of the form r s r r where s r is the sphere z c n z r are injectivity sets for the spherical mean value operator on the heisenberg group h n in l p spaces we prove this result as a consequence of a uniqueness theorem for the heat equation associated to the sub laplacian a hecke bochner type identity for the weyl transform proved by d geller and spherical harmonic expansions are the main tools used

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

from heisenberg uniqueness pairs to properties of the helmholtz and laplace equations abstract the aim of this paper is to establish uniqueness properties of solutions of the helmholtz and laplace equations in particular we show that if two solutions of such equations on a domain of r d agree on two intersecting d 1 dimensional submanifolds in generic position then they agree everywhere

topologies weaker than the weak topology of a banach space abstract in this paper we investigate some properties of the compact subsets of banach spaces x endowed with topologies of the kind x b where b is a norming subset of the dual unit ball b x assuming that b x is sequentially compact we prove that the krein smulian theorem holds for norm bounded x b compact subsets of x and we state that the convex x b compact subsets of x have the weak radon nikod m property when b x is sequentially compact and x has either the separable complementation property or x is weakly lindelof for instance when b x is corson compact we prove that the x b compact subsets resp x b compact convex subsets of x are fragmented by the norm of x resp have the radon nikod m property so if b x is a corson compact then the compact subsets of the space x x b are radon nikod m compact and thus sequentially compact we apply the previous results to prove that if b x is sequentially compact and b is assumed to be a boundary of b x then the norm bounded x b compact subsets of x are weakly compact which partially answers a problem posed by g godefroy we give among others applications to spaces of vector valued bochner integrable functions as well as to spaces of countably additive measures

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

multi degree smooth polar splines a framework for geometric modeling and isogeometric analysis we develop a multi degree polar spline framework with applications to both geometric modeling and isogeometric analysis first multi degree splines are introduced as piecewise non uniform rational b splines nurbs of non uniform or variable polynomial degree and a simple algorithm for their construction is presented then an extension to two dimensional polar configurations is provided by means of a tensor product construction with a collapsed edge suitable combinations of these basis functions encoded in a so called isogeometric analysis suitable extraction operator yield ckck smooth polar splines for any k 0k 0 we show that it is always possible to construct a set of smooth polar spline basis functions that form a convex partition of unity and possess locality explicit constructions for k 0 1 2 k 0 1 2 are presented optimal approximation behavior is observed numerically and examples of applications to free form design smooth hole filling and high order partial differential equations demonstrate the applicability of the developed framework

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

a third order partial differential equation for isotropic boundary based triangular bezier surface generation abstract we approach surface design by solving a linear third order partial differential equation pde we present an explicit polynomial solution method for triangular bezier pde surface generation characterized by a boundary configuration the third order pde comes from a symmetric operator defined here to overcome the anisotropy drawback of any operator over triangular bezier surfaces

boundaries for banach spaces determine weak compactness a boundary for a real banach space is a subset of the dual unit sphere with the property that each element of the banach space attains its norm on an element of that subset trivially the pointwise convergence with respect to such a boundary is coarser than the weak topology on the banach space the boundary problem asks whether nevertheless both topologies have the same norm bounded compact sets

isogeometric analysis based on extended loop s subdivision in this paper we present a new approach of isogeometric analysis iga based on the extended loop s subdivision scheme this strategy allows us to integrate geometric modeling and physical simulation the geometries can be open and with holes our proposed method performs geometric modeling via the extended loop s subdivision which allows arbitrary topological structure treats concave convex vertices and has at least c 1 continuity everywhere it is capable of handling domains with arbitrary shaped boundary represented by piecewise cubic b spline curves we apply an efficient integration technique to the domain elements with a fast evaluation technique for closed loop s subdivision surfaces as an example the poisson equation is solved on three planar domains we develop the approximate estimation of finite element in the limit function space of the extended loop s subdivision a detailed study on the convergence character is given with the comparison to the classical finite element analysis fea with linear elements numerical experiments are consistent with our theoretical results it shows that compared with the fea with linear elements the iga scheme based on extended loop s subdivision converges faster and behaves more robustly with respect to the mesh quality

the application of cubic trigonometric b spline to the numerical solution of the hyperbolic problems in this paper a collocation finite difference scheme based on new cubic trigonometric b spline is developed and analyzed for the numerical solution of a one dimensional hyperbolic equation wave equation with non local conservation condition the usual finite difference scheme is used to discretize the time derivative while a cubic trigonometric b spline is utilized as an interpolation function in the space dimension the scheme is shown to be unconditionally stable using the von neumann fourier method the accuracy of the proposed scheme is tested by using it for several test problems the numerical results are found to be in good agreement with known exact solutions and with existing schemes in literature

afem for the laplace beltrami operator on graphs design and conditional contraction property we present an adaptive finite element method afem of any polynomial degree for the laplace beltrami operator on c 1 graphs in r d d 2 we first derive residual type a posteriori error estimates that account for the interaction of both the energy error in h 1 and the surface error in w 1 1 due to approximation of we devise a marking strategy to reduce the total error estimator namely a suitably scaled sum of the energy geometric and inconsistency error estimators we prove a conditional contraction property for the sum of the energy error and the total estimator the conditional statement encodes resolution of in w 1 1 we conclude with one numerical experiment that illustrates the theory

functions and their fourier transforms with supports of finite measure for certain locally compact groups abstract it is well known that if the supports of a function f l1 rd and its fourier transform tf are contained in bounded rectangles then f 0 almost everywhere in 1974 benedicks relaxed the requirements for this conclusion by showing that the supports of f and tf need only have finite measure in this paper we extend the validity of this property to a wide variety of locally compact groups these include rd k where k is a compact connected lie group the motion group the affine group the heisenberg group sl 2 r and all noncompact semisimple groups with some additional restrictions on the functions f

implementation of regularized isogeometric boundary element methods for gradient based shape optimization in two dimensional linear elasticity the present work addresses shape sensitivity analysis and optimization in two dimensional elasticity with a regularized isogeometric boundary element method igabem non uniform rational b splines are used both for the geometry and the basis functions to discretize the regularized boundary integral equations with the advantage of tight integration of design and analysis the application of igabem in shape optimization reduces the mesh generation regeneration burden greatly the work is distinct from the previous literatures in igabem shape optimization mainly in two aspects 1 the structural and sensitivity analysis takes advantage of the regularized form of the boundary integral equations eliminating completely the need of evaluating strongly singular integrals and jump terms and their shape derivatives which were the main implementation difficulty in igabem and 2 although based on the same computer aided design cad model the mesh for structural and shape sensitivity analysis is separated from the geometrical design mesh thus achieving a balance between less design variables for efficiency and refined mesh for accuracy this technique was initially used in isogeometric finite element method and was incorporated into the present igabem implementation

pde triangular bezier surfaces harmonic biharmonic and isotropic surfaces we approach surface design by solving second order and fourth order partial differential equations pdes we present many methods for designing triangular bezier pde surfaces given different sets of prescribed control points and including the special cases of harmonic and biharmonic surfaces moreover we introduce and study a second order and a fourth order symmetric operator to overcome the anisotropy drawback of the harmonic and biharmonic operators over triangular bezier surfaces

an extended isogeometric thin shell analysis based on kirchhoff love theory abstract an extended isogeometric element formulation xiga for analysis of through the thickness cracks in thin shell structures is developed the discretization is based on non uniform rational b splines nurbs the proposed xiga formulation can reproduce the singular field near the crack tip and the discontinuities across the crack it is based on the kirchhoff love theory where c 1 continuity of the displacement field is required this condition is satisfied by the nurbs basis functions hence the formulation eliminates the need of rotational degrees of freedom or the discretization of the director field facilitating the enrichment strategy the performance and validity of the formulation is tested by several benchmark examples

fast isogeometric boundary element method based on independent field approximation abstract an isogeometric boundary element method for problems in elasticity is presented which is based on an independent approximation for the geometry traction and displacement field this enables a flexible choice of refinement strategies permits an efficient evaluation of geometry related information a mixed collocation scheme which deals with discontinuous tractions along non smooth boundaries and a significant reduction of the right hand side of the system of equations for common boundary conditions all these benefits are achieved without any loss of accuracy compared to conventional isogeometric formulations the system matrices are approximated by means of hierarchical matrices to reduce the computational complexity for large scale analysis for the required geometrical bisection of the domain a strategy for the evaluation of bounding boxes containing the supports of nurbs basis functions is presented the versatility and accuracy of the proposed methodology are demonstrated by convergence studies showing optimal rates and real world examples in two and three dimensions

construction of triharmonic bezier surfaces from boundary conditions abstract the surface of partial differential equation pde surface is a surface that satisfies the pde with boundary conditions which can be applied in surface modeling and construction in this paper the construction of tensor product bezier surfaces of triharmonic equation from different boundary conditions is presented the internal control points of the resulting triharmonic bezier surface can be obtained uniquely by the given boundary condition some representative examples show the effectiveness of the presented method

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

heisenberg uniqueness pairs corresponding to a finite number of parallel lines abstract in this paper we study the heisenberg uniqueness pairs corresponding to a finite number of parallel lines we give a necessary condition and a sufficient condition for a subset of r 2 so that becomes a hup

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

on fourier transforms of functions supported on sets of finite lebesgue measure soit g un groupe abelien localement compact et son groupe dual soient m et m les mesures de haar sur g et on demontre soit f l 1 r n et f sa transformee de fourier et soit a x r n f x 0 et b r n f 0 alors m a et m b f 0 a e m ici m et m denotent la mesure de lebesgue sur r n resp r n