a critical evaluation of the resolution properties of b spline and compact finite difference methods resolution properties of b spline and compact finite difference schemes are compared using fourier analysis in periodic domains and tests based on solution of the wave and heat equations in finite domains with uniform and nonuniform grids results show that compact finite difference schemes have a higher convergence rate and in some cases better resolution however b spline schemes have a more straightforward and robust formulation particularly near boundaries on nonuniform meshes

an isogeometric method for the reissner mindlin plate bending problem we present a new isogeometric method for the discretization of the reissner mindlin plate bending problem the proposed scheme follows a recent theoretical framework that makes possible the construction of a space of smooth discrete deflections wh and a space of smooth discrete rotations h such that the kirchhoff constraint is exactly satisfied at the limit therefore we obtain a formulation which is natural from the theoretical mechanical viewpoint and locking free by construction we prove that the method is uniformly stable and satisfies optimal convergence estimates finally the theoretical results are fully supported by numerical tests

an automatic quadrature for cauchy principal value integrals an automatic quadrature is presented for computing cauchy principal value integrals q f c faf t t c dt a c b for smooth functions f t after subtracting out the singularity we approximate the function f t by a sum of chebyshev polynomials whose coefficients are computed using the ftt the evaluations of q f c for a set of values of c in a b are efficiently accomplished with the same number of function evaluations numerical examples are also given

locking free isogeometric collocation methods for spatial timoshenko rods in this work we present the application of isogeometric collocation techniques to the solution of spatial timoshenko rods the strong form equations of the problem are presented in both displacement based and mixed formulations and are discretized via nurbs based isogeometric collocation several numerical experiments are reported to test the accuracy and efficiency of the considered methods as well as their applicability to problems of practical interest in particular it is shown that mixed collocation schemes are locking free independently of the choice of the polynomial degrees for the unknown fields such an important property is also analytically proven

mixed finite element methods for elastic rods of arbitrary geometry the boundary value problem for rods having arbitrary geometry and subjected to arbitrary loading is studied within the context of the small strain theory the basic assumptions underlying the rod kinematics are those corresponding to the timoshenko hypotheses in the plane rectilinear case that is plane sections normal to the line of centroids in the undeformed state remain plane but not necessarily normal the problem is formulated in both the standard and mixed variational forms and after establishing the existence and uniqueness of solutions to these equivalent problems the corresponding discrete problems are studied finite element approximations of the mixed problem are shown to be stable and convergent it is shown that the equivalence between the mixed problem and the standard problem with selective reduced integration holds only for the case of rods having constant curvature and torsion though the results of numerical experiments are presented these confirm the convergent behaviour of the mixed problem

an isogeometric collocation approach for bernoulli euler beams and kirchhoff plates abstract in this paper iga collocation methods are for the first time introduced for the solution of thin structural problems described by the bernoulli euler beam and kirchhoff plate models in particular a precise description of the proposed methods of the relevant implementation details and of the strategy to efficiently deal with different combinations of boundary conditions is given finally several numerical experiments confirm that the proposed formulations represent an efficient and geometrically flexible tool for the simulation of thin structures

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

uniform approximations to finite hilbert transform and its derivative interpolatory integration rules of numerical stability are presented for approximating cauchy principal value p v integrals 11 f t t c dt and hadamard finite part f p integrals 11f t t c 2 dt 1 c 1 respectively for a given smooth function f t present quadrature rules consist of interpolating f t at abscissae in the interval of integration 1 1 except for the pole c where neither the function value f c nor its derivative f c is required followed by subtracting out the singularities we demonstrate that the use of both endpoints 1 as abscissae in interpolating f t is essential for uniformly approximating the integrals namely for bounding the approximation errors independently of the values of c in fact for the f p integrals the use of double abscissae at both endpoints 1 as well as simple abscissae in 1 1 enables the uniform approximations while the use of simple abscissae at both endpoints 1 and those in 1 1 is sufficient for the p v integrals these facts suggest that finite hilbert transforms p v integrals and their derivatives f p integrals with varied values of c could be approximated efficiently with the same number of abscissae respectively some numerical examples are given

isogeometric collocation for elastostatics and explicit dynamics we extend the development of collocation methods within the framework of isogeometric analysis iga to multi patch nurbs configurations various boundary and patch interface conditions and explicit dynamic analysis the methods developed are higher order accurate stable with no hourglass modes and efficient in that they require a minimum number of quadrature evaluations the combination of these attributes has not been obtained previously within standard finite element analysis

some estimates for h p k refinement in isogeometric analysis in this paper we propose a theoretical study of the approximation properties of nurbs spaces which are used in isogeometric analysis we obtain error estimates that are explicit in terms of the mesh size h the degree p and the global regularity measured by the parameter k our approach covers the approximation with global regularity from c 0 up to c k 1 with 2k 1 p notice that the interesting case of higher regularity up to k p is still open however our results give an indication of the role of the smoothness k in the approximation properties and offer a first mathematical justification of the potential of isogeometric analysis based on globally smooth nurbs

an isogeometric implicit g1 mixed finite element for kirchhoff space rods abstract in the paper we present a multi patch isogeometric mixed element with implicit geometric continuity at the ends for the analysis of non polar kirchhoff space rods rotations are introduced at the ends of the element analogously to the hermitian interpolation performing a change of basis for the configuration space a mixed method in which the internal forces are interpolated by means of an l 2 projection leading to a b formulation is used to cope with the membrane flexural locking that affects the solution the efficiency of the method in avoiding locking and instabilities is investigated the formulation adopted has the advantage with respect to other formulations that a non full global stiffness matrix is obtained this approach appears very appealing for efficient analyses of general assemblies of 3d non polar elements

explicit isogeometric collocation for the dynamics of three dimensional beams undergoing finite motions abstract we initiate the study of three dimensional shear deformable geometrically exact beam dynamics through explicit isogeometric collocation methods the formulation we propose is based on a natural combination of the chosen finite rotations representation with an explicit geometrically consistent lie group time integrator we focus on extending the integration scheme originally proposed for rigid body dynamics to our nonlinear initial boundary value problem where special attention is required by neumann boundary conditions the overall formulation is simple and only relies on a geometrically consistent procedure to compute the internal forces once control angular and linear accelerations of the beam cross sections are obtained from the previous time step the capabilities of the method are shown through numerical applications involving very large displacements and rotations and different boundary conditions

nonlinear isogeometric spatial bernoulli beam abstract a new element formulation of a geometrically nonlinear spatial curved beam assuming bernoulli theory including torsion without warping is proposed the element formulation is derived directly from the 3d continuum by means of nonlinear kinematics thus accounting for large displacements the geometric description of the proposed element is adapted from the spatial rod of greco and cuomo 2013 and extended to a nonlinear element formulation the proposed formulation can handle arbitrary orientations of the cross section along the beam in this publication nurbs are used as basis functions for discretization since they can easily provide the required c 1 continuity between elements the presented element formulation has four degrees of freedom three for displacements and one for the rotation around the center line in order to prove the accuracy of the developed spatial bernoulli beam several numerical examples are presented and compared to analytic solutions and other element formulations

development of a mixed displacement stress formulation for the analysis of elastoplastic structures under small strains application to a locking free nurbs based solid shell element abstract in this paper we develop a general framework for the extension of mixed displacement stress types of formulations to elastoplastic problems under small strains the difficulty with such formulations is that contrary to the more usual mixed methods used for incompressible materials plasticity which involves the deviatoric part of the stress is affected by the static unknown thus the plastic flow is not governed by displacement induced strains alone the approach followed in this paper consists in choosing the total elastic stress as the static unknown of the mixed formulation then one determines a strain field which is used as input for the plastic projection we applied this formalism to the mixed nurbs based solid shell element of bouclier et al 2013 2015 which led to an efficient extension of the element into the elastoplastic domain we were able to verify the ability of this element to overcome shell locking problems through several test cases involving a plastic flow rule with linear isotropic strain hardening

a fully locking free isogeometric approach for plane linear elasticity problems a stream function formulation in this paper we study plane incompressible elastic problems by means of a stream function formulation such that a divergence free displacement field can be computed from a scalar potential the numerical scheme is constructed within the framework of nurbs based isogeometric analysis and we take advantage of the high continuity guaranteed by nurbs basis functions in order to obtain the displacement field from the potential differentiation as a consequence the obtained numerical scheme is automatically locking free in the presence of the incompressibility constraint a discontinuous galerkin approach is proposed to deal with multiple mapped possibly multiply connected domains extensive numerical results are provided to show the method capabilities 2007 elsevier b v all rights reserved

uniform approximation to fractional derivatives of functions of algebraic singularity fractional derivative d qf x 0 1 with g x being a well behaved function we propose a quadrature method for uniformly approximating d q x ag x the present method consists of interpolating g x at abscissae in 0 1 by a finite sum of chebyshev polynomials it is shown that the use of the lower endpoint x 0 as an abscissa is essential for the uniform approximation namely to bound the approximation errors independently of x 0 1 numerical examples demonstrate the performance of the present method

an algorithm based on the fft for a generalized chebyshev interpolation an algorithm for a generalized chebyshev interpolation procedure increasing the number of sample points more moderately than doubling is pre sented the fft for a real sequence is incorporated into the algorithm to enhance its efficiency numerical comparison with other existing algorithms is given

duality and unified analysis of discrete approximations in structural dynamics and wave propagation comparison of p method finite elements with k method nurbs we study the discretization behavior of classical finite element and nurbs approximations on problems of structural vibrations and wave propagation we find that on the basis of equal numbers of degrees of freedom and bandwidth nurbs have superior approximation properties in fact we observe that the high mode behavior of classical finite elements is divergent with the order of approximation a surprisingly negative result on the other hand nurbs offer almost spectral approximation properties and all modes converge with increasing order of approximation

reduced bezier element quadrature rules for quadratic and cubic splines in isogeometric analysis abstract we explore the use of various element based reduced quadrature strategies for bivariate and trivariate quadratic and cubic spline elements used in isogeometric analysis the rules studied encompass tensor product gauss and gauss lobatto rules and certain so called monomial rules that do no possess a tensor product structure the objective of the study is to determine quadrature strategies which enjoy the same accuracy and stability behavior as full gauss quadrature but with significantly fewer quadrature points several cases emerge that satisfy this objective and also demonstrate superior efficiency compared with standard c 0 continuous finite elements of the same order

chebyshev series expansion of inverse polynomials the chebyshev series expansion n 0 antn x of the inverse of a polynomial j 0kbjtj x is well defined if the polynomial has no roots in 1 1 if the inverse polynomial is decomposed into partial fractions the an are linear combinations of simple functions of the polynomial roots also if the first k of the coefficients an are known the others become linear combinations of these derived recursively from the bj s on a closely related theme finding a polynomial with minimum relative error towards a given f x is approximately equivalent to finding the bj in f x 0k bj tj x 1 k 1 an tn x a newton algorithm produces these if the chebyshev expansion of f x is known

a quadrature rule of interpolatory type for cauchy integrals abstract we construct a quadrature rule of interpolatory type based on a trigonometric interpolation for cauchy integrals we obtain the same error bound o ann as hasegawa and torii math comp 56 1991 741 754 numerical examples are included to substantiate the performance of the present method

on the asymptotic convergence of spline collocation methods for partial differential equations we examine the asymptotic accuracy of the method of collocation for the approximate solution of linear elliptic partial differential equations specifically we consider the nodal collocation of a second order equation in the plane with biperiodicity conditions using tensor product smooth splines of odd degree as trial functions we prove optimal rates of convergence in l 2 for partial derivatives of the approximate solution which are of order at least two in one variable while the solution itself and its gradient converge in l 2 at rates less than the optimal approximation theoretic results

an embedded formula of the chebyshev collocation method for stiff problems abstract in this study we have developed an embedded formula of the chebyshev collocation method for stiff problems based on the zeros of the generalized chebyshev polynomials a new strategy for the embedded formula using a pair of methods to estimate the local truncation error as performed in traditional embedded runge kutta schemes is proposed the method is performed in such a way that not only the stability region of the embedded formula can be widened but by allowing the usage of larger time step sizes the total computational costs can also be reduced in terms of concrete convergence and stability analysis the constructed algorithm turns out to have an 8th order convergence and it exhibits a stability through several numerical experimental results we have demonstrated that the proposed method is numerically more efficient compared to several existing implicit methods

impressions of convexity an illustration for commutator bounds abstract we determine the sharpest constant c p q r such that for all complex matrices x and y and for schatten p q and r norms the inequality xy yx p c p q r x q y r is valid the main theoretical tool in our investigations is complex interpolation theory

a locking free approximation of curved rods by straight beam elements we consider an elastic model for a curved rod with arbitrary three dimensional geometry incorporating shear and membrane as well as bending and torsion effects we define an approximation procedure based on a discretization by linear timoshenko beam elements introducing an equivalent mixed problem we establish optimal error estimates independent of the thickness thereby proving that shear and membrane locking is avoided the approximation scheme is tested on specific examples and the numerical results confirm the estimates obtained by theory

on the asymptotic convergence of collocation methods we prove quasioptimal and optimal order estimates in various sobolev norms for the approximation of linear strongly elliptic pseudodifferential equations in one independent variable by the method of nodal collocation by odd degree polynomial splines the analysis pertains in particular to many of the boundary element methods used for numerical computation in engineering applications equations to which the analysis is applied include fredholm integral equations of the second kind certain first kind fredholm equations singular integral equations involving cauchy kernels a variety of integro differential equations and two point boundary value problems for ordinary differential equations the error analysis is based on an equivalence which we establish between the collocation methods and certain nonstandard galerkin methods we compare the collocation method with a standard galerkin method using splines of the same degree showing that the galerkin method is quasioptimal in a sobolev space of lower index and furnishes optimal order approximation for a range of sobolev indices containing and extending below that for the collocation method and so the standard galerkin method achieves higher rates of convergence

algorithms for approximating finite hilbert transform with end point singularities and its derivatives abstract algorithms are proposed for the numerical evaluation of cauchy principal value integrals 1 1 w t f t t x d t 1 x 1 with weight functions of jacobi type singularities w t 1 t 1 t where 1 2 and 1 2 for a given function f t and hadamard finite part integrals 1 1 w t f t t x 2 d t the function f is interpolated by using a finite sum of chebyshev polynomials the present algorithms require o n log n arithmetic operations where n is the order of the interpolation polynomial it is shown that the present scheme gives uniform approximations namely the errors are bounded independently of x and is very efficient for smooth f further we discuss approximations of hyper singular integrals 1 1 w t f t t x n d t n 3 and show their uniform convergences numerical examples are given to demonstrate the performance of the present schemes

an error embedded method based on generalized chebyshev polynomials in this paper we develop an error embedded method based on generalized chebyshev polynomials for solving stiff initial value problems the solution and the error at each integration step are calculated by generalized chebyshev polynomials of two consecutive degrees having overlapping zeros which enables us to minimize overall computational costs further the errors at each integration step are embedded in the algorithm itself in terms of concrete convergence and stability analysis the constructed algorithm turns out to have the 6th order convergence and an almost l stability we assess the proposed method with several numerical results showing that it uses larger time step sizes and is numerically more efficient

a collocated c0 finite element method reduced quadrature perspective cost comparison with standard finite elements and explicit structural dynamics summary we demonstrate the potential of collocation methods for efficient higher order analysis on standard nodal finite element meshes we focus on a collocation method that is variationally consistent and geometrically flexible converges optimally embraces concepts of reduced quadrature and leads to symmetric stiffness and diagonal consistent mass matrices at the same time it minimizes the evaluation cost per quadrature point thus reducing formation and assembly effort significantly with respect to standard galerkin finite element methods we provide a detailed review of all components of the technology in the context of elastodynamics that is weighted residual formulation nodal basis functions on gauss lobatto quadrature points and symmetrization by averaging with the ultra weak formulation we quantify potential gains by comparing the computational efficiency of collocated and standard finite elements in terms of basic operation counts and timings our results show that collocation is significantly less expensive for problems dominated by the formation and assembly effort such as higher order elastostatic analysis furthermore we illustrate the potential of collocation for efficient higher order explicit dynamics throughout this work we advocate a straightforward implementation based on simple modifications of standard finite element codes we also point out the close connection to spectral element methods where many of the key ideas are already established copyright 2014 john wiley sons ltd

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

overlapping schwarz methods for isogeometric analysis we construct and analyze an overlapping schwarz preconditioner for elliptic problems discretized with isogeometric analysis the preconditioner is based on partitioning the domain of the problem into overlapping subdomains solving local isogeometric problems on these subdomains and solving an additional coarse isogeometric problem associated with the subdomain mesh we develop an h analysis of the preconditioner showing in particular that the resulting algorithm is scalable and its convergence rate depends linearly on the ratio between subdomain and overlap sizes for fixed polynomial degree p and regularity k of the basis functions numerical results in two and three dimensional tests show the good convergence properties of the preconditioner with respect to the isogeometric discretization parameters h p k number of subdomains n overlap size and also jumps in the coefficients of the elliptic operator

weak dirichlet boundary conditions for wall bounded turbulent flows abstract in turbulence applications strongly imposed no slip conditions often lead to inaccurate mean flow quantities for coarse boundary layer meshes to circumvent this shortcoming weakly imposed dirichlet boundary conditions for fluid dynamics were recently introduced in y bazilevs t j r hughes weak imposition of dirichlet boundary conditions in fluid mechanics comput fluids 36 2007 12 26 in the present work we propose a modification of the original weak boundary condition formulation that consistently incorporates the well known law of the wall to compare the different methods we conduct numerical experiments for turbulent channel flow at reynolds number 395 and 950 in the limit of vanishing mesh size in the wall normal direction the weak boundary condition acts like a strong boundary condition accordingly strong and weak boundary conditions give essentially identical results on meshes that are stretched to better capture boundary layers however on uniform meshes that are incapable of resolving boundary layers weakly imposed boundary conditions deliver significantly more accurate mean flow quantities than their strong counterparts hence weakly imposed boundary conditions present a robust technique for flows of industrial interest where optimal mesh design is usually not feasible and resolving boundary layers is prohibitively expensive our numerical results show that the formulation that incorporates the law of the wall yields an improvement over the original method

indefinite integration of oscillatory functions by the chebyshev series expansion abstract an automatic quadrature scheme is presented for evaluating the indefinite integral of oscillatory function x 0 t e i t dt 0 x 1 of a given function t which is usually assumed to be smooth the function t is expanded in the chebyshev series to make an efficient evaluation of the indefinite integral combining the automatic quadrature method obtained and sidi s extrapolation method makes an effective quadrature scheme for oscillatory infinite integral a x cos x d x for which numerical examples are also presented

isogeometric configuration design sensitivity analysis of finite deformation curved beam structures using jaumann strain formulation abstract using an isogeometric approach a continuum based configuration design sensitivity analysis dsa method is developed for curved kirchhoff beams with multi patch junctions under the total lagrangian formulation large deformations considering the initial curvature of curved beams are described by geometrically exact beam theory gebt and jaumann strain formulation in the isogeometric approach the higher order continuity and the exact description of initial geometry are naturally embedded using nurbs basis functions in multi patch models c 0 continuity of physical displacement or c 1 continuity of displacement component at junction is weakly imposed using the lagrange multiplier method the superior accuracy of isogeometric analysis iga is verified through the comparison with the results of finite element analysis fea using cubic hermite interpolation in the dsa a material derivative is utilized and the kinematical description of gebt is consistently employed to express orientation design variations contrary to the iga based dsa the hermite basis function explicitly depends on design in the fea based dsa due to its element length parameter moreover since the design velocity field is approximated using the nodal velocity imposed at nodal tangential vector the amount of design perturbations should be very small to obtain precise design sensitivity

positivity of the weights of extended clenshaw curtis quadrature rules we prove that some extended clenshaw curtis quadrature rules have all weights positive we also present extended filippi rules of open type having all weights positive conjectures on the possibility of other positive quadrature rules embedded in clenshaw curtis or filippi rule are suggested

quadrature rule for indefinite integral of algebraic logarithmic singular integrands an automatic quadrature method is presented for approximating the indefinite integral of functions having algebraic logarithmic singularities q x y c f email protected x yf t t c alog t c dt 1 1 within a finite range 1 1 for some smooth function f t that is approximated by a finite sum of chebyshev polynomials we expand the given indefinite integral in terms of chebyshev polynomials by using auxiliary algebraic logarithmic functions present scheme approximates the indefinite integral q x y c f uniformly namely bounds the approximation error independently of the value c as well x and y this fact enables us to evaluate the integral transform q x y c f with varied values of x y and c efficiently some numerical examples illustrate the performance of the present quadrature scheme

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

overlapping schwarz preconditioners for isogeometric collocation methods isogeometric collocation methods are very recent and promising numerical schemes that preserve the advantages of isogeometric analysis but often exhibit better performances than their galerkin counterparts in the present paper an additive overlapping schwarz method for isogeometric collocation discretizations is introduced and studied the resulting preconditioner accelerated by gmres is shown to be scalable with respect to the number of subdomains and very robust with respect to the isogeometric discretization parameters such as the mesh size and polynomial degree as well as with respect to the presence of discontinuous elliptic coefficients and domain deformations

the importance of the exact satisfaction of the incompressibility constraint in nonlinear elasticity mixed fems versus nurbs based approximations in the present paper we investigate the capability of finite element methods to correctly reproduce the stability range of finite strain problems in the incompressible regime to this end we develop a numerical scheme obtained combining a stream function formulation with an isogeometric nurbs approach which is able to sharply estimate the stability limits of the continuous problem using such a method we show a pair of benchmark problems on which various well known finite element methods largely fail in approximating the correct stability range

hilbert and hadamard transforms by generalized chebyshev expansion abstract an automatic quadrature is presented for approximating hadamard finite part fp integrals of a smooth function with a double pole singularity within the range of integration the quadrature rule is derived from the differentiation of an approximation to a cauchy principal value integral or the hilbert transform the approximation to the fp integral is represented as a function of the value of pole by using chebyshev polynomials of the second kind since the error can be estimated independently of the value of pole a set of integrals for a set of values of pole can be efficiently approximated to a required tolerance with the same number of function evaluations numerical examples are also included to illustrate the performance of the methods

application of a modified fft to product type integration abstract an automatic integration scheme is proposed for evaluating the so called product type indefinite integral q k y x k t t dt 1 x y where t is assumed to be a smooth function and k t are some singular or badly behaved functions typical examples for k t are 1n t c t c 1 cauchy principal value 1 t c and ei t 1 the function t is approximated by a truncated chebyshev series pn t of degree n whose coefficients are efficiently computed using the fft the approximation qn k to the integral q k is given by q k pn the sequence pn t is recursively generated until the required tolerance for the integral is satisfied to enhance the efficiency of the automatic quadrature the degree n is increased more slowly than doubling which is usually the case the evaluations of qn k q k pn for a set of x y c can be efficiently made by using recurrence relations for the singular kernels k t above numerical examples for the algebraic singular kernel k t t c 1 are included

a collocated isogeometric finite element method based on gauss lobatto lagrange extraction of splines generalizing the concept of bezier extraction we introduce an extraction operator that links c0c0 gauss lobatto lagrange functions with smooth splines this opens the door for collocated isogeometric analysis that combines the accuracy of the galerkin method with collocation type formation and assembly procedures we present the key ingredients of the technology i e integration by parts and the weighted residual form the interaction of gauss lobatto lagrange extraction with gauss lobatto quadrature and symmetrization with the ultra weak formulation we compare the new method with standard isogeometric galerkin and isogeometric point collocation methods for spline discretizations in three dimensions

generalized chebyshev interpolation and its application to automatic quadrature a generalized chebyshev interpolation procedure increasing a fixed number of sample points at a time is developed and analvzed it is incorporated into an efficient automatic quadrature scheme of clenshaw curtis tvpe numerical examples indicate that the present method is efficient not onlv for well behaved functions but for those w ith discontinuous low order derivatives by virtue of adequate error estimation as wvell as saving of sample points

effects of parameterization and knot placement techniques on primal and mixed isogeometric collocation formulations of spatial shear deformable beams with varying curvature and torsion abstract we present a displacement based and a mixed isogeometric collocation iga c formulation for free form three dimensional shear deformable beams with high and rapidly varying curvature and torsion when such complex shapes are concerned the approach used to build the iga geometric model becomes relevant although iga c has been so far successfully applied to a wide range of problems the effects that different parameterization and knot placement techniques may have on the accuracy of collocation based formulations is still an unexplored field to fill this gap primal and mixed formulations are used combining two parameterization methods chord length and equally spaced with two knot placement techniques uniformly spaced and de boor with respect to the space varying frenet local frame we derive the strong form of the governing equations in a compact form through the definition of two matrix operators conveniently used to perform first and second order derivatives of the vector fields involved in the formulations this approach is very efficient and easy to implement within a collocation based scheme several challenging numerical experiments allow to test the different considered parameterizations and knot placement techniques revealing in particular that with the primal formulation an equally spaced parameterization is definitively the most recommended choice and it should always be used with an approximation degree of at least p 6 although some caution must be adopted when very high jacobians and small curvatures occur the same holds for the mixed formulation with the difference that p 4 is enough to yield accurate results

avoiding shear locking for the timoshenko beam problem via isogeometric collocation methods in this work we study isogeometric collocation methods for the timoshenko beam problem considering both mixed and displacement based formulations in particular we show that locking free solutions are obtained for mixed methods independently on the approximation degrees selected for the unknown fields moreover several numerical tests are provided in order to support our theoretical results and to show the good behavior and the flexibility of isogeometric collocation methods in this context

non prismatic timoshenko like beam model the present paper combines an effective beam theory with a simple and accurate numerical technique opening the door to the prediction of the structural behavior of planar beams characterized by a continuous variation of the cross section geometry that in general deeply influences the stress distribution and therefore leads to non trivial constitutive relations accounting for these peculiar aspects the beam theory is described by a mixed formulation of the problem represented by six linear ordinary differential equations odes with non constant coefficients depending on both the cross section displacements and the internal forces due to the odes complexity the solution can be typically computed only numerically also for relatively simple geometries loads and boundary conditions however the use of classical numerical tools for this problem like a six field mixed finite element approach might entail several issues e g shear locking ill conditioned matrices etc conversely the recently proposed isogeometric collocation method consisting of the direct discretization of the odes in strong form and using the higher continuity properties typical of spline shape functions allows an equal order approximation of all unknown fields without affecting the stability of the solution this makes such an approach simple robust efficient and particularly suitable for solving the system of odes governing the non prismatic beam problem several numerical experiments confirm that the proposed mixed isogeometric collocation method is actually cost effective and able to attain high accuracy

low rank tensor methods in galerkin based isogeometric analysis abstract the global patch wise geometry map which describes the computational domain is a new feature in isogeometric analysis this map has a global tensor structure inherited from the parametric spline geometry representation the use of this global structure in the discretization of partial differential equations may be regarded as a drawback at first glance as opposed to the purely local nature of high order classical finite elements in this work we demonstrate that it is possible to exploit the regularity of this structure and to identify the great potential for the efficient implementation of isogeometric discretizations first we formulate tensor product b spline bases as well as the corresponding mass and stiffness matrices as tensors in order to reveal their intrinsic structure second we derive an algorithm for the separation of variables in the integrands arising in the discretization this is possible by means of low rank approximation of the integral kernels we arrive at a compact separated representation of the integrals the separated form implies an expression of galerkin matrices as kronecker products of matrix factors with small dimensions this representation is very appealing due to the reduction in both memory consumption and computation times our benchmarks performed using the c library g smo demonstrate that the use of tensor methods in isogeometric analysis possesses significant advantages

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

efficient quadrature for nurbs based isogeometric analysis abstract we initiate the study of efficient quadrature rules for nurbs based isogeometric analysis a rule of thumb emerges the half point rule indicating that optimal rules involve a number of points roughly equal to half the number of degrees of freedom or equivalently half the number of basis functions of the space under consideration the half point rule is independent of the polynomial order of the basis efficient rules require taking into account the precise smoothness of basis functions across element boundaries several rules of practical interest are obtained and a numerical procedure for determining efficient rules is presented we compare the cost of quadrature for typical situations arising in structural mechanics and fluid dynamics the new rules represent improvements over those used previously in isogeometric analysis

variational multiscale residual based turbulence modeling for large eddy simulation of incompressible flows we present an les type variational multiscale theory of turbulence our approach derives completely from the incompressible navier stokes equations and does not employ any ad hoc devices such as eddy viscosities we tested the formulation on forced homogeneous isotropic turbulence and turbulent channel flows in the calculations we employed linear quadratic and cubic nurbs a dispersion analysis of simple model problems revealed nurbs elements to be superior to classical finite elements in approximating advective and diffusive processes which play a significant role in turbulence computations the numerical results are very good and confirm the viability of the theoretical framework

isogeometric collocation cost comparison with galerkin methods and extension to adaptive hierarchical nurbs discretizations abstract we compare isogeometric collocation with isogeometric galerkin and standard c 0 finite element methods with respect to the cost of forming the matrix and residual vector the cost of direct and iterative solvers the accuracy versus degrees of freedom and the accuracy versus computing time on this basis we show that isogeometric collocation has the potential to increase the computational efficiency of isogeometric analysis and to outperform both isogeometric galerkin and standard c 0 finite element methods when a specified level of accuracy is to be achieved with minimum computational cost we then explore an adaptive isogeometric collocation method that is based on local hierarchical refinement of nurbs basis functions and collocation points derived from the corresponding multi level greville abscissae we introduce the concept of weighted collocation that can be consistently developed from the weighted residual form and the two scale relation of b splines using weighted collocation in the transition regions between hierarchical levels we are able to reliably handle coincident collocation points that naturally occur for multi level greville abscissae the resulting method combines the favorable properties of isogeometric collocation and hierarchical refinement in terms of computational efficiency local adaptivity robustness and straightforward implementation which we illustrate by numerical examples in one two and three dimensions

enriched isogeometric collocation for two dimensional time harmonic acoustics abstract in this work the isogeometric collocation iga c is paired with two types of enrichment plane wave pw and generalized harmonic polynomial ghp functions to solve 2d problems for the helmholtz equation a parametric study is conducted and a detailed assessment of the performance of the method in a number of benchmark problems is provided three different collocation methods are tested for the non enriched formulation namely greville abscissae ga approximated cauchy galerkin acg and superconvergent sc points showing that the acg scheme is the best choice in terms of overall error convergence rate and ease of solving the linear system then the influence of the number of shape functions in the original and enriched basis location and number of collocation points and wave number over both the convergence rate as well as the condition number of the stiffness matrix are studied the numerical results show that 1 there is an improvement over the non enriched formulation 2 the improvement depends on the choice of the number and type of enrichment and 3 the pollution error is not completely alleviated with the enriched formulations

anisotropic nurbs approximation in isogeometric analysis abstract we investigate the approximation error of mapped nurbs spaces such as those used in isogeometric analysis in accordance to the isoparametric paradigm we extend the results of 6 to the case of anisotropic meshes such as those needed in the presence of boundary layers the theoretical results cover anisotropic h refinement and are supported by numerical tests involving different geometries

a simple algorithm for obtaining nearly optimal quadrature rules for nurbs based isogeometric analysis we develop new quadrature rules for isogeometric analysis based on the solution of a local nonlinear problem a simple and robust algorithm is developed to determine the rules which are exact for important b spline spaces of uniform and geometrically stretched knot spacings we consider both periodic and open knot vector configurations and illustrate the efficiency of the rules on selected boundary value problems we find that the rules are almost optimally efficient but much easier to obtain than optimal rules which require the solution of global nonlinear problems that are often ill posed

numerical integration of functions with poles near the interval of integration abstract an automatic quadrature scheme is presented for approximating integrals of functions that are analytic in the interval of integration but contain pole or poles of order 2 i e a double pole on the real axis or a complex conjugate pair of double poles near the interval of integration the present scheme is based on product integration rules of interpolatory type using function values of the abscissae only in the interval of integration the integral is approximated and evaluated by using recurrence relations and some extrapolation method after the smooth part of the integrand is expanded in terms of the chebyshev polynomials the fast fourier transform fft technique is used to generate efficiently the sequence of the finite chebyshev series expansions until an approximation of the integral satisfying the required tolerance is obtained with an adequate estimate of the error numerical examples are included to illustrate the performance of the method