on certain definite quadrature formulae gauss and lobatto quadrature formulae related to spaces of cubic splines with double and equidistant knots are constructed such quadrature formulae are known as asymptotically optimal definite formulae of order 4 some monotonicity results concerning the associated quadrature processes are established

partial tensor decomposition for decoupling isogeometric galerkin discretizations abstract system matrix assembly for isogeometric i e spline based discretizations of partial differential equations is more challenging than for classical finite elements due to the increased polynomial degrees and the larger and hence more overlapping supports of the basis functions the global tensor product structure of the discrete spaces employed in isogeometric analysis can be exploited to accelerate the computations using sum factorization precomputed look up tables and tensor decomposition we generalize the third approach by considering partial tensor decompositions we show that the resulting new method preserves the global discretization error and that its computational complexity compares favorably to the existing approaches moreover the numerical realization simplifies considerably since it relies on standard techniques from numerical linear algebra

the resultant on compact riemann surfaces we introduce a notion of the resultant of two meromorphic functions on a compact riemann surface and demonstrate its usefulness in several respects for example we exhibit several integral formulas for the resultant relate it to potential theory and give explicit formulas for the algebraic dependence between two meromorphic functions on a compact riemann surface as a particular application the exponential transform of a quadrature domain in the complex plane is expressed in terms of the resultant of two meromorphic functions on the schottky double of the domain

selective and reduced numerical integrations for nurbs based isogeometric analysis we propose a new approach to construct selective and reduced integration rules for isogeometric analysis based on nurbs elements the notion of an approximation space that approximates the target space is introduced we explore the use of various approximation spaces associated with optimal patch wise numerical quadratures that exactly integrate the polynomials in approximation spaces with the minimum number of quadrature points patch rules exploit the higher continuity of spline basis functions the tendency of smooth spline functions to exhibit numerical locking in nearly incompressible problems when using a full gauss legendre quadrature is alleviated with selective or reduced integration stability and accuracy of the schemes are examined analyzing the discrete spectrum in a generalized eigenvalue problem we propose a local algorithm which is robust and computationally efficient to compute element by element the quadrature points and weights in patch rules the performance of the methods is assessed on several numerical examples in two dimensional elasticity and reissner mindlin shell structures

asymptotically optimal definite quadrature formulae of 4th order we construct several sequences of asymptotically optimal definite quadrature formulae of fourth order and evaluate their error constants besides the asymptotical optimality an advantage of our quadrature formulae is the explicit form of their weights and nodes for the remainders of our quadrature formulae monotonicity properties are established when the integrand has a non negative or non positive fourth derivative and a posteriori error estimates are proven

robustness of isogeometric structural discretizations under severe mesh distortion abstract this paper investigates higher order and higher continuity functions in isogeometric structural analysis under distortion of the control and physical meshes first the concepts behind isogeometric analysis are briefly reviewed the effect of perturbing control points upon the geometrical mapping is observed in the one dimensional setting the concept is extended to multiple dimensions with a series of cases where the interior of a cube is distorted in different ways and patch tests are performed to assess accuracy a circular fillet patch is introduced through the appropriate degeneration of a control mesh in order to address the problem of modeling small features in finite element models the fillet patch is tested with two problems of linear elasticity and is shown to be accurate the investigation is further extended to the setting of large deformation nearly incompressible nonlinear elasticity with two examples which make use of the recently proposed f projection method

improved numerical integration for locking treatment in isogeometric structural elements part i beams a general mathematical framework is proposed in this paper to define new quadrature rules in the context of bb spline nurbs based isogeometric analysis high order continuity across the elements within a patch turned out to have higher accuracy than c0c0 finite elements as well as a better time efficiency unfortunately a maximum regularity accentuates the shear and membrane locking in thick structural elements the improved selective reduced integration schemes are given for uni dimensional beam problems with basis functions of order two and three and can be easily extended to higher orders the resulting bb spline nurbs finite elements are free from membrane and transverse shear locking moreover no zero energy modes are generated the performance of the approach is evaluated on the classical test of a cantilever beam subjected to a distributed moment and compared to lagrange under integrated finite elements

highly accurate lagrangian flux calculation via algebraic quadratures on spline approximated donating regions abstract lagrangian flux through a fixed curve segment within a time interval t n t n k can be formulated as an integral at the initial time t n over a compact point set called donating region in which each particle will pass through the curve during the time interval and contribute to the flux based an explicit constructive and analytical solution of the donating region the author proposes algorithms of lagrangian flux calculation lfc as algebraic quadratures over the spline approximated donating regions this generic formulation yields high accuracies up to the 8th order both in time and space as another benefit lfc leads to a conservative semi lagrangian method whose time step size is free of the eulerian stability constraint c r 1 the high accuracy and robustness of the proposed lfc algorithms are demonstrated by various numerical tests with courant numbers ranging from 1 to 10 000 and with accuracies from the 2nd order to the 8th order lfc might also be useful in designing hybrid eulerian methods for complex geometries

on the monotonicity of sequences of quadrature formulae leti f l f k 0 r 0 vk 1 a k f x k be a quadrature formula and let s n f n 1 be successive approximations of the definite integrali f 0 1 f x dx obtained by the composition ofl i e s n f l n where varphi n x frac 1 n sum nolimits k 0 n 1 f left frac k x n right we prove sufficient conditions for monotonicity of the sequence s n f n 1 as particular cases the monotonicity of well known newton cotes and gauss quadratures is shown finally a recovery theorem based on the monotonicity results is presented

gpu accelerated computation of the isogeometric analysis stiffness matrix abstract due to high regularity across mesh elements isogeometric analysis achieves higher accuracy per degree of freedom and improved spectrum properties among others compared with finite element analysis however this inherent feature of isogeometric analysis increases the density of the stiffness matrix and requires more elaborate numerical integration schemes for its computation for these reasons the assembly of the stiffness matrix in isogeometric analysis is a computationally demanding task which needs special attention in order to be affordable for real world applications in this paper we address the computational efficiency of assembling the stiffness matrix using the standard element wise gaussian quadrature a novel approach is proposed for the formulation of the stiffness matrix which exhibits several computational merits among them its amenability to parallelization and the efficient utilization of the graphics processing units to drastically accelerate computations

fast formation of isogeometric galerkin matrices by weighted quadrature abstract in this paper we propose an algorithm for the formation of matrices of isogeometric galerkin methods the algorithm is based on three ideas the first is that we perform the external loop over the rows of the matrix the second is that we calculate the row entries by weighted quadrature the third is that we exploit the local tensor product structure of the basis functions while all ingredients have a fundamental role for computational efficiency the major conceptual change of paradigm with respect to the standard implementation is the idea of using weighted quadrature the test function is incorporated in the integration weight while the trial function the geometry parametrization and the pdes coefficients form the integrand function this approach is very effective in reducing the computational cost while maintaining the optimal order of approximation of the method analysis of the cost is confirmed by numerical testing where we show that for p large enough the time required by the floating point operations is less than the time spent in unavoidable memory operations the sparse matrix allocation and memory write the proposed algorithm allows significant time saving when assembling isogeometric galerkin matrices for all the degrees of the test spline space and paves the way for a use of high degree k refinement in isogeometric analysis

effective elastic behavior of some models for perfect cellular solids abstract the effective elastic behavior of some models for low density cellular solids or solid foams are calculated using analytical and numerical techniques the models are perfect in the sense that imperfections or irregularities as often encountered in real foams have been removed we believe that the present models can serve as references to which more advanced models which include imperfections and irregularities can be compared the work in this paper does not address buckling or yielding in cell walls which play an increasingly important role as foam stresses increase

a binary tree element subdivision method for evaluation of nearly singular domain integrals with continuous or discontinuous kernel abstract an adaptive and efficient volume element subdivision method using binary tree for evaluation of nearly singular domain integrals with continuous or discontinuous kernel in three dimensional 3 d boundary element method bem has been presented in the conventional subdivision method csm for evaluation of nearly singular integrals the patches are obtained by simply connecting the source point with each vertex of the element thus the accuracy of the integral obtained with csm is easily affected by the shape of the element and the location of the source point in contrast the proposed binary tree subdivision method btsm is more convenient to implement and can guarantee successful patch generation under any circumstances for accurate evaluation of nearly singular domain integrals with continuous or discontinuous kernel numerical results for volume elements of arbitrary type with various relative locations of the source point demonstrate robustness accuracy and efficiency of the proposed method

generalized gaussian quadrature rules for systems of arbitrary functions a numerical algorithm is presented for the construction of generalized gaussian quadrature rules originally introduced by s karlin and w studden over three decades ago the quadrature rules to be discussed possess most of the desirable properties of the classical gaussian integration formulae such as positivity of the weights rapid convergence mathematical elegance etc the algorithm is applicable to a wide class of functions including smooth functions not necessarily polynomials as well as functions with end point singularities such as those encountered in the solution of integral equations complex analysis potential theory and several other areas the performance of the algorithm is illustrated with several numerical examples

contact treatment in isogeometric analysis with nurbs abstract we study nurbs based isogeometric analysis of contact problems and compare with standard c 0 continuous lagrange finite elements a knot to surface kts algorithm is developed to treat the contact constraints with nurbs contact surface discretizations qualitative studies deliver satisfactory results for various finite deformation frictionless thermoelastic contact problems quantitative studies based on the hertz problem suggest the need for a relaxation of the mechanical contact constraints that appear in the standard kts approach the improved mortar based kts algorithm delivers robust and accurate results for nurbs discretizations based on numerical examples we conclude that nurbs based isogeometric analysis is a viable technology for contact problems and offers potential accuracy as well as convergence improvements over c 0 continuous finite elements

b and f projection methods for nearly incompressible linear and non linear elasticity and plasticity using higher order nurbs elements abstract this paper presents projection methods to treat the incompressibility constraint in small and large deformation elasticity and plasticity within the framework of isogeometric analysis after reviewing some fundamentals of isogeometric analysis we investigate the use of higher order non uniform rational b splines nurbs within the b projection method the higher continuity property of such functions is explored in nearly incompressible applications and shown to produce accurate and robust results a new non linear f projection method based on a modified minimum potential energy principle and inspired by the b method is proposed for the large deformation case it leads to a symmetric formulation for which the consistent linearized operator for fully non linear elasticity is derived and used in a newton raphson iterative procedure the performance of the methods is assessed on several numerical examples and results obtained are shown to compare favorably with other published techniques

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

efficient mass and stiffness matrix assembly via weighted gaussian quadrature rules for b splines abstract calabro et al 2017 changed the paradigm of the mass and stiffness computation from the traditional element wise assembly to a row wise concept showing that the latter one offers integration that may be orders of magnitude faster considering a b spline basis function as a non negative measure each mass matrix row is integrated by its own quadrature rule with respect to that measure each rule is easy to compute as it leads to a linear system of equations however the quadrature rules are of the newton cotes type that is they require a number of quadrature points that is equal to the dimension of the spline space in this work we propose weighted quadrature rules of gaussian type which require the minimum number of quadrature points while guaranteeing exactness of integration with respect to the weight function the weighted gaussian rules arise as solutions of non linear systems of equations we derive rules for the mass and stiffness matrices for uniform c 1 quadratic and c 2 cubic isogeometric discretizations in each parameter direction our rules require locally only p 1 quadrature points p being the polynomial degree while the nodes cannot be reused for various weight functions as in calabro et al 2017 the computational cost of the mass and stiffness matrix assembly is comparable

hyponormal operators with finite rank self commutators and quadrature domains this paper studies some classes of pure hyponormal operators with finite rank self commutators satisfying the condition that their spectra are unions of a finite collection of the closures of quadrature domains

isogeometric variational multiscale modeling of wall bounded turbulent flows with weakly enforced boundary conditions on unstretched meshes abstract in this work we combine i nurbs based isogeometric analysis ii residual driven turbulence modeling and iii weak imposition of no slip and no penetration dirichlet boundary conditions on unstretched meshes to compute wall bounded turbulent flows while the first two ingredients were shown to be successful for turbulence computations at medium to high reynolds number i akkerman y bazilevs v m calo t j r hughes s hulshoff the role of continuity in residual based variational multiscale modeling of turbulence comput mech 41 2008 371 378 y bazilevs v m calo j a cottrell t j r hughes a reali g scovazzi variational multiscale residual based turbulence modeling for large eddy simulation of incompressible flows comput methods appl mech engrg 197 2007 173 201 it is the weak imposition of no slip boundary conditions on coarse uniform meshes that maintains the good performance of the proposed methodology at higher reynolds number y bazilevs t j r hughes weak imposition of dirichlet boundary conditions in fluid mechanics comput fluids 36 2007 12 26 y bazilevs c michler v m calo t j r hughes weak dirichlet boundary conditions for wall bounded turbulent flows comput methods appl mech engrg 196 2007 4853 4862 these three ingredients form a basis of a possible practical strategy for computing engineering flows somewhere between rans and les in complexity we demonstrate this by solving two challenging incompressible turbulent benchmark problems channel flow at friction velocity reynolds number 2003 and flow in a planar asymmetric diffuser we observe good agreement between our calculations of mean flow quantities and both reference computations and experimental data this lends some credence to the proposed approach which we believe may become a viable engineering tool

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

optimal and reduced quadrature rules for tensor product and hierarchically refined splines in isogeometric analysis abstract we continue the study initiated in hughes et al 2010 in search of optimal quadrature rules for tensor product and hierarchically refined splines in isogeometric analysis these rules are optimal in the sense that there exists no other quadrature rule that can exactly integrate the elements of the given spline space with fewer quadrature points we extend the algorithm presented in hughes et al 2010 with an improved starting guess which combined with arbitrary precision arithmetic results in the practical computation of quadrature rules for univariate non uniform splines up to any precision explicit constructions are provided in sixteen digits of accuracy for some of the most commonly used uniform spline spaces defined by open knot vectors we study the efficacy of the proposed rules in the context of full and reduced quadrature applied to two and three dimensional diffusion reaction problems using tensor product and hierarchically refined splines and prove a theorem rigorously establishing the stability and accuracy of the reduced rules

a subdivision based implementation of the hierarchical b spline finite element method abstract a novel technique is presented to facilitate the implementation of hierarchical b splines and their interfacing with conventional finite element implementations the discrete interpretation of the two scale relation as common in subdivision schemes is used to establish algebraic relations between the basis functions and their coefficients on different levels of the hierarchical b spline basis the subdivision projection technique introduced allows us first to compute all element matrices and vectors using a fixed number of same level basis functions their subsequent multiplication with subdivision matrices projects them during the assembly stage to the correct levels of the hierarchical b spline basis the proposed technique is applied to convergence studies of linear and geometrically nonlinear problems in one two and three space dimensions

an isogeometric design through analysis methodology based on adaptive hierarchical refinement of nurbs immersed boundary methods and t spline cad surfaces we explore hierarchical refinement of nurbs as a basis for adaptive isogeometric and immersed boundary analysis we use the principle of b spline subdivision to derive a local refinement procedure which combines full analysis suitability of the basis with straightforward implementation in tree data structures and simple generalization to higher dimensions we test hierarchical refinement of nurbs for some elementary fluid and structural analysis problems in two and three dimensions and attain good results in all cases using the b spline version of the finite cell method we illustrate the potential of immersed boundary methods as a seamless isogeometric design through analysis procedure for complex engineering parts defined by t spline cad surfaces specifically a ship propeller and an automobile wheel we show that hierarchical refinement considerably increases the flexibility of this approach by adaptively resolving local features

the role of bezier extraction in adaptive isogeometric analysis local refinement and hierarchical refinement summary we present two adaptive refinement techniques namely adaptive local refinement and adaptive hierarchical refinement for isogeometric analysis an element wise point of view is adopted exploiting bezier extraction which facilitates the implementation of adaptive refinement in isogeometric analysis locally refined t splines lr t splines and hierarchical t splines are employed for the description of the geometry as well as for the approximation of the solution space in the analysis the refinement is carried out with the aid of a subdivision operator which is computed by again exploiting the bezier extraction operator the concept and algorithm of an element based adaptive isogeometric analysis are illustrated numerical examples are given to examine the accuracy the convergence and the condition number this article is protected by copyright all rights reserved

on the completeness of hierarchical tensor product b splines abstract given a grid in r d consisting of d bi infinite sequences of hyperplanes possibly with multiplicities orthogonal to the d axes of the coordinate system we consider the spaces of tensor product spline functions of a given degree on a multi cell domain such a domain consists of finite set of cells which are defined by the grid a piecewise polynomial function belongs to the spline space if its polynomial pieces on adjacent cells have a contact according to the multiplicity of the hyperplanes in the grid we prove that the connected components of the associated set of tensor product b splines whose support intersects the multi cell domain form a basis of this spline space more precisely if the intersection of the support of a tensor product b spline with the multi cell domain consists of several connected components then each of these components contributes one basis function in order to establish the connection to earlier results we also present further details relating to the three dimensional case with single knots only a hierarchical b spline basis is defined by specifying nested hierarchies of spline spaces and multi cell domains we adapt the techniques from giannelli and juttler 2013 to the more general setting and prove the completeness of this basis in the sense that its span contains all piecewise polynomial functions on the hierarchical grid with the smoothness specified by the grid and the degrees under certain assumptions on the domain hierarchy finally we introduce a decoupled version of the hierarchical spline basis that allows to relax the assumptions on the domain hierarchy in certain situations such as quadratic tensor product splines the decoupled basis provides the completeness property for any choice of the domain hierarchy

mixed stress displacement isogeometric collocation for nearly incompressible elasticity and elastoplasticity abstract we propose a mixed stress displacement isogeometric collocation method for nearly incompressible elastic materials and for materials exhibiting von mises plasticity the discretization is based on isogeometric analysis iga with non uniform rational b splines nurbs as basis functions as compared to conventional iga galerkin formulations isogeometric collocation methods offer a high potential of computational cost reduction for higher order discretizations as they eliminate the need for quadrature in the proposed mixed formulation both stress and displacement fields are approximated as primary variables with the aim of treating volumetric locking and instability issues which occur in displacement based isogeometric collocation for nearly incompressible elasticity and von mises plasticity the performance of the proposed approach is demonstrated by several numerical examples

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

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

block tridiagonal reduction of perturbed normal and rank structured matrices abstract it is well known that if a matrix a c n n solves the matrix equation f a a h 0 where f x y is a linear bivariate polynomial then a is normal a and a h can be simultaneously reduced in a finite number of operations to tridiagonal form by a unitary congruence and moreover the spectrum of a is located on a straight line in the complex plane in this paper we present some generalizations of these properties for almost normal matrices which satisfy certain quadratic matrix equations arising in the study of structured eigenvalue problems for perturbed hermitian and unitary matrices

explicit gaussian quadrature rules for c 1 cubic splines with symmetrically stretched knot sequences we provide explicit expressions for quadrature rules on the space of c 1 cubic splines with non uniform symmetrically stretched knot sequences the quadrature nodes and weights are derived via an explicit recursion that avoids an intervention of any numerical solver and the rule is optimal that is it requires minimal number of nodes numerical experiments validating the theoretical results and the error estimates of the quadrature rules are also presented the paper provides expressions for quadrature rules on the space of c 1 cubic splines with non uniform symmetrically stretched knot sequences any function from the space is exactly integrated by the rule the quadrature nodes and weights are derived via explicit recursion formulae with no intervention of any numerical solver the rule is optimal i e it requires minimal number of nodes

gaussian quadrature rules for c1 quintic splines with uniform knot vectors abstract we provide explicit quadrature rules for spaces of c 1 quintic splines with uniform knot sequences over finite domains the quadrature nodes and weights are derived via an explicit recursion that avoids numerical solvers each rule is optimal that is requires the minimal number of nodes for a given function space for each of n subintervals generically only two nodes are required which reduces the evaluation cost by 2 3 when compared to the classical gaussian quadrature for polynomials over each knot span numerical experiments show fast convergence as n grows to the two third quadrature rule of hughes et al 2010 for infinite domains

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

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

a quadrature based approach to improving the collocation method the collocation method is a popular method for the approximate solution of boundary integral equations but typically does not achieve the high order of convergence reached by the galerkin method in appropriate negative norms in this paper a quadrature based method for improving upon the collocation method is proposed and developed in detail for a particular case that case involves operators with even symbol such as the logarithmic potential operating on 1 periodic functions a smooth spline trial space of odd degree with constant mesh spacingh 1 n and a quadrature rule with 2n points where ann point quadrature rule would be equivalent to the collocation method in this setting it is shown that a special quadrature rule which depends on the degree of the splines and the order of the operator can yield a maximum order of convergence two powers ofh higher than the collocation method

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

reduced integration at superconvergent points in isogeometric analysis abstract we propose a new reduced integration rule for isogeometric analysis iga based on the concept of variational collocation it has been recently shown that if a discrete space is constructed by smooth and pointwise non negative basis functions there exists a set of points named cauchy galerkin cg points such that collocation performed at these points can reproduce the galerkin solution of various boundary value problems exactly since cg points are not known a priori estimates are necessary in practice and can be found based on superconvergence theory in this contribution we explore the use of estimated cg points i e superconvergent points as numerical quadrature points to obtain an efficient and stable reduced quadrature rule in iga we use the weighted residual formulation as basis for our new quadrature rule so that the proposed approach can be considered intermediate between the standard accurately integrated galerkin variational formulation and the direct evaluation of the strong form in collocation approaches the performance of the method is demonstrated by several examples for odd degrees of discretization we obtain spatial convergence rates and accuracy very close to those of accurately integrated standard galerkin with a quadrature rule of two points per parametric direction independently of the degree

invariant properties of the stress in plane elasticity and equivalence classes of composites attention is drawn to the invariance of the stress field in a two dimensional body loaded at the boundary by fixed forces when the compliance tensor scr g chi is shifted uniformly by ell text i lambda lambda where lambda is an arbitrary constant and scr g text i kappa mu is the compliance tensor of a isotropic material with two dimensional bulk and shear moduli kappa and mu this invariance is explained from two simple observations first that in two dimensions the tensor scr g text i frac 1 2 frac 1 2 acts to locally rotate the stress by 90 degrees and the second that this rotated field is the symmetrized gradient of a vector field and therefore can be treated as a strain for composite materials the invariance of the stress field implies that the effective compliance tensor ell ast also gets shifted by scr g text i lambda lambda when the constituent moduli are each shifted by ell text i lambda lambda this imposes constraints on the functional dependence of ell ast on the material moduli of the components applied to an isotropic composite of two isotropic components it implies that when the inverse bulk modulus is shifted by the constant 1 lambda and the inverse shear modulus is shifted by 1 lambda then the inverse effective bulk and shear moduli undergo precisely the same shifts in particular it explains why the effective young s modulus of a two dimensional media with holes does not depend on the poisson s ratio of the matrix material

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

a high order solver for the heat equation in 1d domains with moving boundaries we describe a fast high order accurate method for the solution of the heat equation in domains with moving dirichlet or neumann boundaries and distributed forces we assume that the motion of the boundary is prescribed our method extends the work of greengard and strain comm pure appl math xliii 1990 pp 949 963 our scheme is based on a time space chebyshev pseudo spectral collocation discretization which is combined with a recursive product quadrature rule to accurately and efficiently approximate convolutions with green s function for the heat equation we present numerical results that exhibit up to eighth order convergence rates assuming n time steps and m spatial discretization points the evaluation of the solution of the heat equation at the same number of points in space time requires mathcal o n m log m work thus our scheme can be characterized as fast that is it is work optimal up to a logarithmic factor

a quadrature based superconvergent isogeometric frequency analysis with macro integration cells and quadratic splines abstract a quadrature based superconvergent isogeometric method is presented for efficient frequency analysis with macro integration cells and quadratic splines the present development particularly refers to the isogeometric analysis of scalar wave equations the single element two element and three element macro integration cells are proposed to develop a set of explicit superconvergent quadrature rules with a 6th order frequency accuracy in contrast to the 4th order frequency accuracy associated with the standard isogeometric formulation with consistent mass matrix the 1d quadrature rules with various macro integration cells are derived from the condition of exact integration of the higher order mass matrix as well as the stiffness matrix consequently 3 point 5 point and 7 point quadrature rules with identical precision are established for the single element two element and three element macro integration cells respectively by construction these quadrature rules exactly recover the isogeometric higher order mass matrix formulation with frequency superconvergence in 1d case the 2d and 3d superconvergent quadrature rules with versatile integration cells are directly constructed through the tensor product formulation of the 1d integration algorithms it is shown that the multidimensional isogeometric analysis employing the proposed quadrature rules for both mass and stiffness matrices does produce the frequency superconvergence simultaneously without the wave propagation direction dependence problem which needs special treatments for the multidimensional higher order mass matrix formulation the proposed approach is featured by its simplicity for numerical implementation and efficiency using macro integration cells numerical examples confirm the efficacy of the present methodology

dispersion optimized quadratures for isogeometric analysis abstract we develop and analyze quadrature blending schemes that minimize the dispersion error of isogeometric analysis up to polynomial order seven with maximum continuity in the span the schemes yield two extra orders of convergence superconvergence on the eigenvalue errors while the eigenfunction errors are of optimal convergence order both dispersion and spectrum analysis are unified in the form of a taylor expansion for eigenvalue errors the resulting schemes increase the accuracy and robustness of isogeometric analysis for wave propagation as well as the differential eigenvalue problems we also derive an a posteriori error estimator for the eigenvalue error based on the superconvergence result we verify with numerical examples the analysis of the performance of the proposed schemes

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

isogeometric analysis of the isothermal navier stokes korteweg equations this paper is devoted to the numerical simulation of the navier stokes korteweg equations a phase eld model for water water vapor two phase ows we develop a numerical formulation based on isogeometric analysis that permits straightforward treatment of the higher order partial dierenti al operator that represents capillarity we introduce a new renement methodology that desensitizes the numerical solution to the computational mesh and achieves mesh invariant solutions finally we present several numerical examples in two and three dimensions that illustrate the eectiveness and robustness of our approach

fast isogeometric solvers for explicit dynamics in finite element analysis solving time dependent partial differential equations with explicit time marching schemes requires repeatedly applying the inverse of the mass matrix for mass matrices that can be expressed as tensor products of lower dimensional matrices we present a direct method that has linear computational complexity i e o n o n where nn is the total number of degrees of freedom in the system we refer to these matrices as separable matrices for non separable mass matrices we present a preconditioned conjugate gradient method with carefully designed preconditioners as an alternative we demonstrate that these preconditioners which are easy to construct and cheap to apply o n o n can deliver significant convergence acceleration the performances of these preconditioners are independent of the polynomial order p independence and mesh resolution h independence for maximum continuity b splines as verified by various numerical tests

real separated algebraic curves quadrature domains ahlfors type functions and operator theory abstract the aim of this paper is to inter relate several algebraic and analytic objects such as real type algebraic curves quadrature domains functions on them and rational matrix functions with special properties and some objects from operator theory such as vector toeplitz operators and subnormal operators our tools come from operator theory but some of our results have purely algebraic formulation we make use of xia s theory of subnormal operators and of the previous results by the author in this direction we also correct in section 5 some inaccuracies in the works of d v yakubovich subnormal operators of finite type i xia s model and real algebraic curves in c 2 rev mat iberoamericana 14 1998 95 115 d v yakubovich subnormal operators of finite type ii structure theorems rev mat iberoamericana 14 1998 623 681 by the author

gaussian quadrature for c1 cubic clough tocher macro triangles abstract a numerical integration rule for multivariate cubic polynomials over n dimensional simplices was designed by hammer and stroud 1956 the quadrature rule requires n 2 quadrature points the barycentre of the simplex and n 1 points that lie on the connecting lines between the barycentre and the vertices of the simplex in the planar case this particular rule belongs to a two parameter family of quadrature rules that admit exact integration of bivariate polynomials of total degree three over triangles we prove that this rule is exact for a larger space namely the c 1 cubic clough tocher spline space over macro triangles if and only if the split point is the barycentre this results in a factor of three reduction in the number of quadrature points needed to integrate the clough tocher spline space exactly

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

spherical contractions and interpolation problems on the unit ball in this note fractional representations of multipliers on vector valued functional hilbert spaces are used to give a proof of arveson s version of von neumann s inequality for n contractions on the unit ball we prove a commutant lifting theorem for operators on the classical hardy space over the unit ball in mathbb c n as applications we obtain interpolation results for functions in the schur class we deduce a toeplitz corona theorem on the unit ball and we give a simplified definition of arveson s curvature invariant for n contractions with finite dimensional defect space in the final part we describe a solution of the operator valued nevanlinna pick problem with uniform bounds on uniqueness sets in the unit ball

superconvergent isogeometric analysis of natural frequencies for elastic continua with quadratic splines abstract a superconvergent isogeometric formulation is presented to accurately analyze the natural frequencies for elastic continua this formulation is realized by a set of superconvergent quadrature rules which are designed for the numerical integration of isogeometric mass and stiffness matrices in order to obtain these quadrature rules a natural frequency error measure for elastic continua is systematically deduced using the quadratic basis functions where the mass and stiffness matrices are formulated by the assumed quadrature rules in contrast to the quadrature based superconvergent isogeometric formulation for the scalar valued wave equations it is shown that herein different quadrature rules are required for the mass and stiffness matrices to achieve the superconvergence of natural frequency computation for the vector valued elastic continuum problems consequently the superconvergent quadrature rules are established through optimizing the natural frequency accuracy it turns out that with these quadrature rules the accuracy of natural frequencies for elastic continua is improved by two orders compared with the standard isogeometric formulation employing the consistent mass matrix meanwhile it is found that the superconvergent quadrature rules involve the wave propagation angle and consequently simplified quadrature rules without the angle dependence are further proposed for straightforward practical applications numerical results reveal the superconvergence of the proposed method regarding the natural frequencies for elastic continua

hyponormal operators with rank two self commutators abstract in this paper it is shown that if t l h satisfies i t is a pure hyponormal operator ii t t is of rank two and iii ker t t is invariant for t then t is either a subnormal operator or the putinar s matricial model of rank two more precisely if t ker t t has a rank one self commutator then t is subnormal and if instead t ker t t has a rank two self commutator then t is either a subnormal operator or the kth minimal partially normal extension t k k of a k 1 hyponormal operator t k which has a rank two self commutator for any k z hence in particular every weakly subnormal or 2 hyponormal operator with a rank two self commutator is either a subnormal operator or a finite rank perturbation of a k hyponormal operator for any k z

ullemar s formula for the moment map ii we prove the complex analogue of ullemar s formula for the jacobian of the complex moment mapping this formula was previously established in the real case

finite element and nurbs approximations of eigenvalue boundary value and initial value problems abstract we study the spectral approximation properties of finite element and nurbs spaces from a global perspective we focus on eigenfunction approximations and discover that the l 2 norm errors for finite element eigenfunctions exhibit pronounced spikes about the transition points between branches of the eigenvalue spectrum this pathology is absent in nurbs approximations by way of the pythagorean eigenvalue error theorem we determine that the squares of the energy norm errors of the eigenfunctions are the sums of the eigenvalue errors and the squares of the l 2 norm eigenfunction errors the spurious behavior of the higher eigenvalues for standard finite elements is well known and therefore inherited by the energy norm errors along with the spikes in the l 2 norm of the eigenfunction errors the eigenvalue pathology is absent for nurbs the implications of these results to the corresponding elliptic boundary value problem and parabolic and hyperbolic initial value problems are discussed

on numerical quadrature for c1 quadratic powell sabin 6 split macro triangles abstract the quadrature rule of hammer and stroud 1956 for cubic polynomials has been shown to be exact for a larger space of functions namely the c 1 cubic clough tocher spline space over a macro triangle if and only if the split point is the barycentre of the macro triangle kosinka and barto 2018 we continue the study of quadrature rules for spline spaces over macro triangles now focusing on the case of c 1 quadratic powell sabin 6 split macro triangles we show that the 3 node gaussian quadrature s for quadratics can be generalised to the c 1 quadratic powell sabin 6 split spline space over a macro triangle for a two parameter family of inner split points not just the barycentre as in kosinka and barto 2018 the choice of the inner split point uniquely determines the positions of the edge split points such that the whole spline space is integrated exactly by a corresponding polynomial quadrature consequently the number of quadrature points needed to exactly integrate this special spline space reduces from twelve to three for the inner split point at the barycentre we prove that the two 3 node quadratic polynomial quadratures of hammer and stroud exactly integrate also the c 1 quadratic powell sabin spline space if and only if the edge split points are at their respective edge midpoints for other positions of the inner and edge split points we provide numerical examples showing that three nodes suffice to integrate the space exactly but a full classification and a closed form solution in the generic case remain elusive

a nonlinear optimization procedure for generalized gaussian quadratures we present a new nonlinear optimization procedure for the computation of generalized gaussian quadratures for a broad class of square integrable functions on intervals while some of the components of this algorithm have been previously published we present a simple and robust scheme for the determination of a sparse solution to an underdetermined nonlinear optimization problem which replaces the continuation scheme of the previously published works the new algorithm successfully computes generalized gaussian quadratures in a number of instances in which the previous algorithms fail four applications of our scheme to computational physics are presented the construction of discrete plane wave expansions for the helmholtz green s function the design of linear array antennae the computation of a quadrature for the discretization of laplace boundary integral equations on certain domains with corners and the construction of quadratures for the discretization of laplace and helmholtz boundary integral equations on smooth surfaces

exit criteria and monotonicity in compound quadrature of gaussian type in this short note for compound quadrature rules of gaussian type we prove stopping rules and monotonicity results based on peano kernel methods

free vibration of cellular structures using continuum modes this paper is concerned with free vibration of elastic structures made up of cellular material a network of cells in a plane is considered cell walls are modelled as beams the problem of calculating the first few natural frequencies and the corresponding mode shapes is addressed an approximate method based on the assumption that cellular solids behave as a continuum for low frequency dynamics is presented a direct application of the assumed modes method incorporating continuum modes as basis leads to difficulties the source of this difficulty is identified a method that uses inverse power iterations is proposed to pre condition the assumed modes appropriately the proposed method of using continuum modes as basis for model order reduction leads to substantial computational saving while maintaining good accuracy two examples are given to illustrate the proposed procedure of using the continuum modes in conjunction with pre conditioning

isogeometric collocation for kirchhoff love plates and shells abstract with the emergence of isogeometric analysis iga the galerkin rotation free discretization of kirchhoff love shells is facilitated enabling more efficient thin shell structural analysis high order shape functions used in iga also allow the collocation of partial differential equations avoiding the time consuming numerical integration of the galerkin technique the goal of the present work is to apply this method to nurbs based isogeometric kirchhoff love plates and shells under the assumption of small deformations since kirchhoff love plate theory yields a fourth order formulation two boundary conditions are required at each location on the contour generating some conflicts at the corners where there are more equations than needed to remedy this overdetermination we provide priority and averaging rules that cover all the possible combinations of adjacent edge boundary conditions i e the clamped simply supported symmetric and free supports greville and alternative superconvergent points are used for nurbs basis of even and odd degrees respectively for square circular and annular flat plates convergence orders are found to be in agreement with a priori error estimates the proposed isogeometric collocation method is then validated and benchmarked against a galerkin implementation by studying a set of problems involving kirchhoff love shells

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

extremal solutions of the two dimensionall problem of moments by using the theory of the principal function of a hyponormal operator with rank one self commutator one classifies some extremal points of the solutions of the truncatedl problem of moments in two real variables these extremal elements called degenerated solutions of thel problem are proved to be in a natural bijective correspondence with the bounded quadrature domains in the complex plane the main result of the paper characterizes these degenerated solutions from three distinct perspectives complex analysis moment problems and operator theory the interplay between the three points of view each with specific techniques constitutes the main body of the paper

integration by interpolation and look up for galerkin based isogeometric analysis abstract even though isogeometric analysis has a clear advantage regarding the number of degrees of freedom needed to achieve a certain level of accuracy the time needed for matrix assembly by means of numerical integration of stiffness or mass matrix entries constitutes a severe bottleneck in the process in order to overcome this difficulty we propose the new approach of integration by interpolation and lookup iil firstly applying spline interpolation to the common factors in the occurring integrals approximately transforms them into integrals of piecewise polynomial functions whose integrands are expressed in tensor product b spline form the common factors represent the influence of the geometry mapping i e the nurbs domain parameterization and the contributions of possibly non constant material coefficients secondly these integrals are evaluated exactly using pre computed look up tables for integrals of tri products of univariate b splines and their derivatives for the model case of elliptic problems we perform a theoretical analysis to demonstrate that the iil method maintains the overall approximation order of the galerkin discretization provided that the spline interpolation is sufficiently accurate moreover we provide a comparison of the computational complexity of our method with that of a standard gauss quadrature method finally we present experimental results to illustrate the performance of the iil method and to support our theoretical results

fast formation and assembly of finite element matrices with application to isogeometric linear elasticity abstract recently a new formation and assembly strategy was proposed in 1 which resulted in significant speedups in the formation and assembly time of the galerkin mass matrix in isogeometric analysis the strategy relies on three key ingredients 1 assembly row by row instead of element by element and an efficient formation strategy based on 2 sum factorization and 3 weighted quadrature that is applied to each specific row of the matrix compared to traditional element procedures applied to three dimensional problems the computational complexity is lowered from o p 9 per degree of freedom to o p 4 where p is the order of polynomials this is close to the theoretical minimum of o p 3 attained by for example collocation consequently this type of formation and assembly scales favorably with polynomial degree which opens the way for high order isogeometric analysis employing k refinement that is use of maximally smooth higher order splines in this work we discuss various important details for the practical implementation of the weighted quadrature formation strategy proposed in 1 specifically we extend the weighted quadrature scheme to accurately integrate the elements of the stiffness matrix in linear elasticity and propose a means of distributing quadrature points for non uniform mixed continuity spline spaces furthermore we discuss efficient access and assignment into the prevalent sparse matrix data structures namely compressed sparse row csr and compressed sparse column csc in particular row by row or column by column assembly allows matrix rows or columns respectively to be formed contiguously in the storage order of the sparse matrix thereby minimizing the memory overhead and eliminating the addition assignment operation on sparse matrices several three dimensional benchmark problems illustrate the efficiency and efficacy of the proposed formation and assembly technique applied to isogeometric linear elasticity we show that the accuracy of full gauss quadrature is maintained while the computational burden of forming the matrix equations is significantly reduced

on the computational efficiency of isogeometric methods for smooth elliptic problems using direct solvers summary we compare the computational efficiency of isogeometric galerkin and collocation methods for partial differential equations in the asymptotic regime we define a metric to identify when numerical experiments have reached this regime we then apply these ideas to analyze the performance of different isogeometric discretizations which encompass c0 finite element spaces and higher continuous spaces we derive convergence and cost estimates in terms of the total number of degrees of freedom and then perform an asymptotic numerical comparison of the efficiency of these methods applied to an elliptic problem these estimates are derived assuming that the underlying solution is smooth the full gauss quadrature is used in each non zero knot span and the numerical solution of the discrete system is found using a direct multi frontal solver we conclude that under the assumptions detailed in this paper higher continuous basis functions provide marginal benefits copyright 2014 john wiley sons ltd

dispersion minimizing quadrature rules for c1 quadratic isogeometric analysis abstract we develop quadrature rules for the isogeometric analysis of wave propagation and structural vibrations that minimize the discrete dispersion error of the approximation the rules are optimal in the sense that they only require two quadrature points per element to minimize the dispersion error barto et al 2017 1 and they are equivalent to the optimized blending rules we recently described our approach further simplifies the numerical integration instead of blending two three point standard quadrature rules we construct directly a single two point quadrature rule that reduces the dispersion error to the same order for uniform meshes with periodic boundary conditions also we present a 2 5 point rule for both uniform and non uniform meshes with arbitrary boundary conditions consequently we reduce the computational cost by using the proposed quadrature rules various numerical examples demonstrate the performance of these quadrature rules

isogeometric finite element data structures based on bezier extraction of t splines we develop finite element data structures for t splines based on bezier extraction generalizing our previous work for nurbs as in traditional finite element analysis the extracted bezier elements are defined in terms of a fixed set of polynomial basis functions the so called bernstein basis the bezier elements may be processed in the same way as in a standard finite element computer program utilizing exactly the same data processing arrays in fact only the shape function subroutine needs to be modified while all other aspects of a finite element program remain the same a byproduct of the extraction process is the element extraction operator this operator localizes the topological and global smoothness information to the element level and represents a canonical treatment of t junctions referred to as hanging nodes in finite element analysis and a fundamental feature of t splines a detailed example is presented to illustrate the ideas copyright 2011 john wiley sons ltd

a finite strain eulerian formulation for compressible and nearly incompressible hyperelasticity using high order b spline finite elements summary we present a numerical formulation aimed at modeling the nonlinear response of elastic materials using large deformation continuum mechanics in three dimensions this finite element formulation is based on the eulerian description of motion and the transport of the deformation gradient when modeling a nearly incompressible solid the transport of the deformation gradient is decomposed into its isochoric part and the jacobian determinant as independent fields a homogeneous isotropic hyperelastic solid is assumed and b splines based finite elements are used for the spatial discretization a variational multiscale residual based approach is employed to stabilize the transport equations the performance of the scheme is explored for both compressible and nearly incompressible applications the numerical results are in good agreement with theory illustrating the viability of the computational scheme copyright 2011 john wiley sons ltd

nonlinear optimization quadrature and interpolation we present a nonlinear optimization procedure for the design of generalized gaussian quadratures for a fairly broad class of functions while some of the components of the algorithm have been published previously we introduce an improved procedure for the determination of an acceptable initial point for the continuation scheme that stabilizes the newton type process used to find the quadratures the resulting procedure never failed when applied to chebyshev systems for which the existence and uniqueness of generalized gaussian quadratures are well known it also worked for many non chebyshev systems for which the generalized gaussian quadratures are not guaranteed to exist the performance of the algorithm is illustrated with several numerical examples some of the presented quadratures integrate efficiently large classes of singular functions

dispersion optimized quadrature rules for isogeometric analysis modified inner products their dispersion properties and optimally blended schemes abstract this paper introduces optimally blended quadrature rules for isogeometric analysis and analyzes the numerical dispersion of the resulting discretizations to quantify the approximation errors when we modify the inner products we generalize the pythagorean eigenvalue theorem of strang and fix the proposed blended quadrature rules have advantages over alternative integration rules for isogeometric analysis on uniform and non uniform meshes as well as for different polynomial orders and continuity of the basis the optimally blended schemes improve the convergence rate of the method by two orders with respect to the fully integrated galerkin method the proposed technique increases the accuracy and robustness of isogeometric analysis for wave propagation problems

optimal quadrature for univariate and tensor product splines abstract numerical integration is a core subroutine in many engineering applications including the finite element method fem isogeometric analysis is a fem technology that uses smooth b spline and nurbs basis functions traditionally gaussian quadrature was used for numerical integration but this yields suboptimal performance this is attributed to the fact that gaussian quadrature rules do not take inter element smoothness of the spline basis into account resulting in over integration the equations for exact quadrature are well known but prove notoriously hard to solve due to their nonlinear nature these generalized gauss rules were first introduced in hughes et al 2010 in the context of isogeometric analysis where newton iteration was utilized to study and tabulate several cases later homotopy continuation barto and calo 2015 was used as an alternative strategy for finding these rules in this paper we describe an algorithm to generalize on both of these techniques it is optimal in the sense that it will integrate a space of dimension n using no more than n 1 2 quadrature points the algorithm works on arbitrary nonuniform knot vectors of any polynomial degree and continuity and is demonstrated on polynomial orders up to 15 it extends to 2d and 3d integrals by tensor product

isogeometric thickness stretchable shell efficient formulation for nonlinear dynamic problems an isogeometric shell element with through thickness stretch is introduced and applied to quasi static and dynamic problems the shell element supports full three dimensional constitutive laws ie the plane stress assumption is not required an updated lagrangian rate formulation is adopted and biquadratic spline based interpolation functions are used for in plane interpolation the concept of fiber mass scaling is proposed to lower the highest eigenfrequencies to improve the performance of the formulation clamped and unclamped knot vectors are compared and the advantages of using unclamped knot vectors are demonstrated the shell element is validated using several benchmark tests which indicate good performance of the proposed formulation

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

sum factorization techniques in isogeometric analysis abstract the fast assembling of stiffness and mass matrices is a key issue in isogeometric analysis particularly if the spline degree is increased we present two algorithms based on the idea of sum factorization one for matrix assembling and one for matrix free methods and study the behavior of their computational complexity in terms of the spline order p opposed to the standard approach these algorithms do not apply the idea element wise but globally or on macro elements if this approach is applied to gauss quadrature the computational complexity grows as p d 2 instead of p 2 d 1 as previously achieved

optimal quadrature rules for odd degree spline spaces and their application to tensor product based isogeometric analysis abstract we introduce optimal quadrature rules for spline spaces that are frequently used in galerkin discretizations to build mass and stiffness matrices using the homotopy continuation concept barto and calo 2016 that transforms optimal quadrature rules from source spaces to target spaces we derive optimal rules for splines defined on finite domains starting with the classical gaussian quadrature for polynomials which is an optimal rule for a discontinuous odd degree space we derive rules for target spaces of higher continuity we further show how the homotopy methodology handles cases where the source and target rules require different numbers of optimal quadrature points we demonstrate it by deriving optimal rules for various odd degree spline spaces particularly with non uniform knot sequences and non uniform multiplicities we also discuss convergence of our rules to their asymptotic counterparts that is the analogues of the midpoint rule of hughes et al 2010 that are exact and optimal for infinite domains for spaces of low continuities we numerically show that the derived rules quickly converge to their asymptotic counterparts as the weights and nodes of a few boundary elements differ from the asymptotic values

linear analysis of quadrature domains iii the linear data of a quadrature domain introduced in the first paper in this series is further investigated the matricial structure of all hyponormal operators with finite rank self commutators which is inductively constructed from such linear data is discussed in particular a conservation law of a determinantal function in the iterative process of defining these matrices is exploited the spectrum of the determining left upper corner of some of these operators is analyzed in connection with the existence of generalized quadrature formulas on planar domains

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

variationally consistent domain integration for isogeometric analysis abstract spline type approximations for solving partial differential equations are the basis of isogeometric analysis while the common approach of using integration cells defined by single knot spans using standard e g gaussian quadrature rules is sufficient for accuracy more efficient domain integration is still in high demand the recently introduced concept of variational consistency provides a guideline for constructing accurate and convergent methods requiring fewer quadrature points than standard integration techniques in this work variationally consistent domain integration is proposed for isogeometric analysis test function gradients are constructed to meet the consistency conditions which only requires solving small linear systems of equations the proposed approach allows for significant reduction in the number of quadrature points employed while maintaining the stability accuracy and optimal convergence properties of higher order quadrature rules several numerical examples are provided to illustrate the performance of the proposed domain integration technique

multi patch discontinuous galerkin isogeometric analysis for wave propagation explicit time stepping and efficient mass matrix inversion abstract we present a class of spline finite element methods for time domain wave propagation which are particularly amenable to explicit time stepping the proposed methods utilize a discontinuous galerkin discretization to enforce continuity of the solution field across geometric patches in a multi patch setting which yields a mass matrix with convenient block diagonal structure over each patch we show how to accurately and efficiently invert mass matrices in the presence of curved geometries by using a weight adjusted approximation of the mass matrix inverse this approximation restores a tensor product structure while retaining provable high order accuracy and semi discrete energy stability we also estimate the maximum stable timestep for spline based finite elements and show that the use of spline spaces results in less stringent cfl restrictions than equivalent c 0 or discontinuous finite element spaces finally we explore the use of optimal knot vectors based on l 2 n widths we show how the use of optimal knot vectors can improve both approximation properties and the maximum stable timestep and present a simple heuristic method for approximating optimal knot positions numerical experiments confirm the accuracy and stability of the proposed methods

gaussian quadrature for splines via homotopy continuation we introduce a new concept for generating optimal quadrature rules for splines to generate an optimal quadrature rule in a given target spline space we build an associated source space with known optimal quadrature and transfer the rule from the source space to the target one while preserving the number of quadrature points and therefore optimality the quadrature nodes and weights are considered as a higher dimensional point a zero of a particular system of polynomial equations as the space is continuously deformed by changing the source knot vector the quadrature rule gets updated using polynomial homotopy continuation for example starting with c 1 cubic splines with uniform knot sequences we demonstrate the methodology by deriving the optimal rules for uniform c 2 cubic spline spaces where the rule was only conjectured to date we validate our algorithm by showing that the resulting quadrature rule is independent of the path chosen between the target and the source knot vectors as well as the source rule chosen