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 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

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

a hierarchical approach to adaptive local refinement in isogeometric analysis abstract adaptive local refinement is one of the key issues in isogeometric analysis in this article we present an adaptive local refinement technique for isogeometric analysis based on extensions of hierarchical b splines we investigate the theoretical properties of the spline space to ensure fundamental properties like linear independence and partition of unity furthermore we use concepts well established in finite element analysis to fully integrate hierarchical spline spaces into the isogeometric setting this also allows us to access a posteriori error estimation techniques numerical results for several different examples are given and they turn out to be very promising

numerical methods for systems of nonlinear parabolic equations with time delays abstract the purpose of this paper is to investigate some numerical aspects of a class of coupled nonlinear parabolic systems with time delays the system of parabolic equations is discretized by the finite difference method which yields a coupled system of nonlinear algebraic equations the mathematical analysis of the nonlinear system is by the method of upper and lower solutions and its associated monotone iterations three monotone iterative schemes are presented and it is shown that the sequence of iterations from each one of these iterative schemes converges monotonically to a unique solution of the finite difference system a theoretical comparison result for the various monotone sequences and error estimates for the three monotone iterative schemes are obtained it is also shown that the finite difference solution converges to the classical solution of the parabolic system as the mesh size decreases to zero

proper learning algorithm for functions of k terms under smooth distributions in this paper we introduce a probabilistic distribution called a smooth distribution which is a generalization of variants of the uniform distribution such as q bounded distribution and product distribution then we give an algorithm that under the smooth distribution properly learns the class of functions of k terms given as fk tkn g f1 v fk v g fk f1 fk tn in polynomial time for constant k where fk is the class of all boolean functions of k variables and tn is the class of terms over n variables although class fk tkn was shown by blum and singh to be learned using dnf as the hypothesis class it has remained open whether it is properly learnable under a distribution free setting

accelerated monotone iterative methods for finite difference equations of reaction diffusion this paper is concerned with numerical methods for a finite difference system of reaction diffusion convection equation under nonlinear boundary condition various monotone iterative methods are presented and each of these methods leads to an existence comparison theorem as well as a computational algorithm for numerical solutions the monotone property of the iterations gives improved upper and lower bounds of the solution in each iteration and the rate of convergence of the iterations is either quadratic or nearly quadratic depending on the property of the nonlinear function application is given to a model problem from chemical engineering and some numerical results including a test problem with known analytical solution are presented to illustrate the various rates of convergence of the iterations

mapped b spline basis functions for shape design and isogeometric analysis over an arbitrary parameterization abstract it is well known that b spline surfaces are defined by a regular array of control vertices in case of models with arbitrary topology it is extremely difficult to maintain continuity conditions among neighboring surfaces the scenario is also true for applications in isogeometric analysis iga this paper presents a novel method for shape design and isogeometric analysis from a quadrilateral control mesh of arbitrary topology using mapped b spline basis functions based on an arbitrary input quadrilateral control mesh a global parameterization of the final surface is first defined through a gravity center method gcm a re parameterization method is then applied to map a b spline basis function to others that are explicitly defined and are tailored to each of the control vertices which can be either regular or extraordinary ones the final surface is defined by all the input control vertices with their corresponding mapped basis functions for practical implementation the surface can be evaluated patch by patch depending on the order of the b spline basis function used for mapping to others the global continuity of the resulting surface including at extraordinary points can be arbitrary higher order in the present paper a uniform cubic b spline basis function is used and the resulting surface is globally c 2 continuous several numerical examples are provided to demonstrate the proposed method for both shape design and isogeometric analysis

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 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

a generalized finite element formulation for arbitrary basis functions from isogeometric analysis to xfem many of the formulations of cm rent research interest including iosogeometric methods and the extended finite element method use nontraditional basis functions some such as subdivision surfaces may not have convenient analytical representations the concept of an element if appropriate at all no longer coincides with the traditional definition developing a new software for each new class of basis functions is a large research burden especially if the problems involve large deformations non linear materials and contact the objective of this paper is to present a method that separates as much as possible the generation and evaluation of the basis functions from the analysis resulting in a formulation that can be implemented within the traditional structure of a finite clement program but that permits the use of arbitrary sets of basis functions that are defined only through the input file elements ranging from a traditional linear four node tetrahedron through a higher order element combining xfem and isogeometric analysis may be specified entirely through an input file without any additional programming examples of this framework to applications with lagrange elements isogeometric elements and xfem basis functions for fracture are presented

adaptive finite element methods for elliptic equations over hierarchical t meshes isogeometric analysis using nurbs non uniform rational b splines as basis functions gives accurate representation of the geometry and the solution but it is not well suited for local refinement in this paper we use the polynomial splines over hierarchical t meshes pht splines to construct basis functions which not only share the nice smoothness properties as the b splines but also allow us to effectively refine meshes locally we develop a residual based a posteriori error estimate for the finite element discretization of elliptic equations using pht splines basis functions and study their approximation properties in addition we conduct numerical experiments to verify the theory and to demonstrate the effectiveness of the error estimate and the high order approximations provided by the numerical solution

parametric mesh regularization for interpolatory shape design and isogeometric analysis over a mesh of arbitrary topology abstract isogeometric analysis iga over a mesh of arbitrary topology is a challenging task since the concept of iga was proposed in the literature one can find some results in iga over an irregular mesh based on approximating schemes while little work is reported so far using interpolatory schemes this paper presents a parametric mesh regularization pmr method for iga using mapped interpolatory basis functions over a mesh of arbitrary topology given a geometric model defined over a mesh of arbitrary topology a global parametric mesh is first defined from the given control mesh using a gravity center method gcm yuan and ma 2014 the resulting parametric mesh is further regularized such that it ensures absolute interpolatory modeling with mapped interpolatory basis functions necessary geometric rules are developed for the regularization process while maintaining the same mesh topological structure the interpolatory basis function used in this paper is a kind of truncated interpolatory basic function tibf studied in our early work yuan and ma 2013 it can also be applied to other mapped interpolatory basis functions in addition to the capability of interpolatory modeling the proposed pmr method can also be applied to approximating schemes which further enriches the capability of such schemes in both modeling and isogeometric analysis of some models in a more appropriate and convenient manner the proposed pmr method has also the potential in achieving higher order continuity at extraordinary vertices using higher order mapped interpolatory basis functions

a new approach to solid modeling with trivariate t splines based on mesh optimization abstract we present a new method to construct a trivariate t spline representation of complex genus zero solids for the application of isogeometric analysis the proposed technique only demands a surface triangulation of the solid as input data the key of this method lies in obtaining a volumetric parameterization between the solid and the parametric domain the unitary cube to do that an adaptive tetrahedral mesh of the parametric domain is isomorphically transformed onto the solid by applying a mesh untangling and smoothing procedure the control points of the trivariate t spline are calculated by imposing the interpolation conditions on points sited both on the inner and on the surface of the solid the distribution of the interpolating points is adapted to the singularities of the domain in order to preserve the features of the surface triangulation

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

spline based meshfree method summary in the present work the spline based meshfree method sbmfm is presented the sbmfm uses spline basis functions for field variables and boundary representation because of the inherent properties of splines that is nurbs nonuniform rational b splines the analysis domain used in the sbmfm has its own parametric domain unlike the classical concepts in the computer aided design cad community in which the nurbs patches are used to model the body the nurbs patch is treated as a background mesh in the present work the real analysis domain is created by trimming operation so that the domain boundaries are described by spline curves only on a single nurbs patch building the geometry in this way enables a convenient boundary representation of the domains because direct manipulation of spline curves becomes possible from a deformation point of view the mesh distortion problem is effectively avoided even if the analysis domain undergoes a large deformation in the sbmfm boundary spline curves freely move with complete independence from the quadrilateral background nurbs patch the so called element connectivity is unnecessary for geometry description in this paper geometrically nonlinear problems are considered implementation strategy is presented and the effectiveness of the present sbmfm is shown through various numerical examples copyright 2012 john wiley sons ltd

monotone finite difference schemes for nonlinear systems with mixed quasi monotonicity monotone finite difference schemes are proposed for nonlinear systems with mixed quasi monotonicity two monotone iteration processes for the corresponding discrete problems are presented which converge monotonically to the quasi solutions of the discrete problems the limits are the exact solutions under some conditions a monotone finite difference scheme on uniform mesh with the accuracy of fourth order is constructed the numerical results coincide with theoretical analysis

dynamics of nonlinear parabolic systems with time delays abstract the dynamics of a coupled system of semilinear parabolic equations with discrete time delays is investigated using the method of upper and lower solutions it is shown that if the reaction function in the system possesses a mixed quasimonotone property and the corresponding elliptic system has a pair of coupled upper and lower solutions then there is a monotone iteration process which yields a pair of quasisolutions of the elliptic system and the sector between the quasisolutions is an attractor of the delayed parabolic system under some additional conditions this sector is a global attractor and the solution of the parabolic system converges to a true solution of the elliptic system the same conclusions are obtained for a coupled system of parabolic ordinary equations with time delays applications are given to three model problems arising from ecology and nuclear engineering these model problems possess multiple steady state solutions and sufficient conditions are given to ensure the stability and instability of these solutions

finite difference reaction diffusion systems with coupled boundary conditions and time delays abstract this paper is concerned with finite difference solutions of a coupled system of reaction diffusion equations with nonlinear boundary conditions and time delays the system is coupled through the reaction functions as well as the boundary conditions and the time delays may appear in both the reaction functions and the boundary functions the reaction diffusion system is discretized by the finite difference method and the investigation is devoted to the finite difference equations for both the time dependent problem and its corresponding steady state problem this investigation includes the existence and uniqueness of a finite difference solution for nonquasimonotone functions monotone convergence of the time dependent solution to a maximal or a minimal steady state solution for quasimonotone functions and local and global attractors of the time dependent system including the convergence of the time dependent solution to a unique steady state solution also discussed are some computational algorithms for numerical solutions of the steady state problem when the reaction function and the boundary function are quasimonotone all the results for the coupled reaction diffusion equations are directly applicable to systems of parabolic ordinary equations and to reaction diffusion systems without time delays

monotone explicit iterations of the finite element approximations for the nonlinear boundary value problem in this paper we consider monotone explicit iterations of the finite element schemes for the nonlinear equations associated with the boundary value problem u bu 2 based on piecewise linear polynomials and the lumping operator these iterations construct the monotonically decreasing and increasing sequences and convergence proofs are given finally we present some numerical examples verifying the effectiveness of the theory

efficient numerical methods for spatially extended population and epidemic models with time delay abstract reaction diffusion models with time delay have been widely applied in population biology as well as epidemiology this type of models can possibly exhibit complex dynamical behaviors such as traveling wave self organized spatial pattern or chaos numerical methods play an essential role in the study of these dynamical behaviors this paper concerns the finite element approximation for reaction diffusion models with time delay two fully discrete schemes and corresponding a priori error estimates are derived generally the research on evolution of population and epidemic needs to survey long time dynamical behaviors of these models so that it is important to improve the speed of numerical simulation to this end interpolation technique is used in our schemes to avoid numerical integration of reaction term an outstanding advantage of using interpolation of reaction term is that it improves the operation speed greatly meanwhile does not reduce convergence order applications are given to some model problems arising from population biology and epidemiology from these simulations some interesting phenomena can be found and we try to explain them in biological significance

generalized b splines as a tool in isogeometric analysis the concept of isogeometric analysis has been proposed in 13 where nurbs are considered as basis of the analysis thanks to their ability to construct an exact geometric model in several practical applications and to their popularity in commercial cad systems in this paper we propose an alternative to the rational model presenting an isogeometric analysis approach based on generalized b splines geometric models exactly represented by generalized b splines include those generated by nurbs moreover generalized b splines possess all fundamental properties of algebraic b splines and nurbs including classical refinement processes as h p k refinements finally since generalized b splines are not confined to rational functions they behave completely similar to algebraic b splines with respect to differentiation and integration this seems to be of interest in the treatment of some relevant problems

maximum margin algorithms with boolean kernels recent work has introduced boolean kernels with which one can learn linear threshold functions over a feature space containing all conjunctions of length up to k for any 1 k n over the original n boolean features in the input space this motivates the question of whether maximum margin algorithms such as support vector machines can learn disjunctive normal form expressions in the probably approximately correct pac learning model by using this kernel we study this question as well as a variant in which structural risk minimization srm is performed where the class hierarchy is taken over the length of conjunctions we show that maximum margin algorithms using the boolean kernels do not pac learn t n term dnf for any t n 1 even when used with such a srm scheme we also consider pac learning under the uniform distribution and show that if the kernel uses conjunctions of length n then the maximum margin hypothesis will fail on the uniform distribution as well our results concretely illustrate that margin based algorithms may overfit when learning simple target functions with natural kernels

numerical methods for semilinear parabolic equations the method of upper lower solutions for continuous parabolic equations is extended to some finite difference system for numerical solutions the idea of this method is that by using the upper or lower solution as the initial iteration one can obtain a monotone sequence that converges to a unique solution of the problem the aim of this paper is to present two iterative schemes for the construction of the monotone sequence and to show that both schemes are numerically stable these two schemes are modified jacobi method and gauss seidel method for nonlinear algebraic equations an advantage of this approach is that each of the two methods yields an error estimate between the true solution and the computed mth iteration on the other hand the standard picard type of iterative scheme is used to show that the finite difference system converges to the continuous parabolic equations

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

isogeometric analysis in electromagnetics b splines approximation abstract we introduce a new discretization scheme for maxwell equations in two space dimension inspired by the new paradigm of isogeometric analysis introduced in hughes et al 2005 16 we propose an algorithm based on the use of bivariate b splines spaces suitably adapted to electromagnetics we construct b splines spaces of variable interelement regularity on the parametric domain these spaces and their push forwards on the physical domain form a de rham diagram and we use them to solve the maxwell source and eigen problem our scheme has the following features i is adapted to treat complex geometries ii is spectral correct iii provides regular e g globally c 0 discrete solutions of maxwell equations

time delayed finite difference reaction diffusion systems with nonquasimonotone functions this paper is concerned with finite difference solutions of a system of reaction diffusion equations with coupled nonlinear boundary conditions and time delays the reaction functions and the boundary functions are not necessarily quasimonotone and the time delays may appear in the reaction functions as well as in the boundary functions the investigation is devoted to the finite difference system for both the time dependent problem and its corresponding steady state problem some monotone iteration processes for the finite difference systems are given and the asymptotic behavior of the time dependent solution in relation to the steady state solution is discussed the asymptotic behavior result leads to some local and global attractors of the time dependent problem including the convergence of the time dependent solution to a unique steady state solution an application and some numerical results to an enzyme substrate reaction diffusion problem are given all the results are directly applicable to parabolic ordinary systems and to reaction diffusion systems without time delays

adaptive isogeometric analysis by local h refinement with t splines isogeometric analysis based on non uniform rational b splines nurbs as basis functions preserves the exact geometry but suffers from the drawback of a rectangular grid of control points in the parameter space which renders a purely local refinement impossible this paper demonstrates how this difficulty can be overcome by using t splines instead t splines allow the introduction of so called t junctions which are related to hanging nodes in the standard fem obeying a few straightforward rules rectangular patches in the parameter space of the t splines can be subdivided and thus a local refinement becomes feasible while still preserving the exact geometry furthermore it is shown how state of the art a posteriori error estimation techniques can be combined with refinement by t splines numerical examples underline the potential of isogeometric analysis with t splines and give hints for further developments

on learning monotone dnf formulae under uniform distributions abstract we show how to learn in polynomial time monotone d term dnf formulae formulae in disjunctive normal form with at most d terms using positive examples drawn from a distribution that is a generalization of the uniform distribution

rectified unstructured t splines with dynamic weighted refinement for improvement in geometric consistency and approximation convergence abstract t splines developed on unstructured meshes are called unstructured t splines which have the problem of geometric inconsistency and unfairness of mesh density evolution over regular or irregular elements during the refinement geometric inconsistency means that the geometry defined by an unstructured t spline keeps changing throughout the refinement the unfairness of mesh density evolution pertains to different scalings of regular and irregular elements during the refinement which drags down the convergence of the approximation result in isogeometric analysis this paper proposes rectified unstructured t splines to solve these two problems with the dynamic weighted refinement and rectified t spline basis functions owing to the dynamic weighted refinement the mesh size scaling tends to 1 2 fairly everywhere on unstructured meshes and geometric consistency is improved with the rectified t spline basis functions the approximation convergence is also greatly enhanced for isogeometric analysis from the original unstructured t splines which is quite meaningful for engineering simulations

numerical analysis of coupled systems of nonlinear parabolic equations this paper is concerned with numerical solutions of a general class of coupled nonlinear parabolic equations by the finite difference method three monotone iteration processes for the finite difference system are presented and the sequences of iterations are shown to converge monotonically to a unique solution of the system including an existence uniqueness comparison theorem a theoretical comparison result for the various monotone sequences and an error analysis of the three monotone iterative schemes are given also given is the convergence of the finite difference solution to the continuous solution of the parabolic boundary value problem an application to a reaction diffusion model in chemical engineering and combustion theory is given

boundary element monotone iteration scheme for semilinear elliptic partial differential equations the monotone iteration scheme is a constructive method for solving a wide class of semilinear elliptic boundary value problems with the availability of a supersolution and a subsolution the iterates converge monotonically to one or two solutions of the nonlinear pde however the rates of such monotone convergence cannot be determined in general in addition when the monotone iteration scheme is implemented numerically through the boundary element method error estimates have not been analyzed in earlier studies in this paper we formulate a working assumption to obtain an exponentially fast rate of convergence this allows a margin 6 for the numerical implementation of boundary elements within the range of monotone convergence we then interrelate several approximate solutions and use the aubin nitsche lemma and the triangle inequalities to derive error estimates for the galerkin boundary element iterates with respect to the h r 0 r 2 sobolev space norms such estimates are of optimal order furthermore as a peculiarity we show that for the nonlinearities that are of separable type higher than optimal order error estimates can be obtained with respect to the mesh parameter h several examples of semilinear elliptic partial differential equations featuring different situations of existence nonexistence uniqueness multiplicity and stability are discussed computed and the graphics of their numerical solutions are illustrated

mapping techniques for isogeometric analysis of elliptic boundary value problems containing singularities abstract the method of auxiliary mapping mam introduced by babuska and oh 2 is an effective method for dealing with singularities in elasticity 16 mam was extended to boundary element method bem in the framework of mesh free particle methods 17 reproducing polynomial particle methods 20 and also to infinite domain problems 18 similarly we consider nurbs geometrical mappings that are able to generate crack singularities for isogeometric analysis of elliptic boundary value problems however the mapping techniques proposed in this paper are different from mam in order to generate singular shape functions mam uses conformal mappings that locally change the physical domain whereas the nurbs mappings used for design of engineering system are not allowed to alter the physical domain for isogeometric analysis moreover unlike mam the proposed method makes it possible to independently control the radial and angular direction of the function to be approximated as far as the point singularities are concerned we prove error estimates in sobolev norms and demonstrate that the proposed mapping technique is highly effective for isogeometric analysis of elliptic boundary value problems with singularities mesh refinements to deal with singularities are compared with the mapping technique in the isogeometic analysis framework

circular element isogeometric elements of smooth boundary this article introduces a family of elements of smooth curved geometry unlike traditional approaches based on finite element geometry the proposed method employs rational bezier functions for boundary description and as a result common shapes such as circles and ellipses can be exactly described the interior of the element is parameterized in terms of the boundary control points via basis functions that are smooth extensions of the boundary bezier bases the interior basis functions are constructed over the unit circular domain and applied in an isoparametric setting to other convex domains that are topologically circular this family of elements is particularly suited for analyzing discrete bodies undergoing more or less uniform or regular deformations an example involving the compression of a circular elastic disk in which the disk is modeled as a single 16 point element is presented to demonstrate the application