an isogeometric symmetric galerkin boundary element method for two dimensional crack problems abstract the isogeometric analysis is applied for the weakly singular symmetric galerkin boundary element method sgbem to analyze quasi static elastic problems including crack problems in two dimensional domains this method takes the advantages from the common boundary representation of the isogeometric analysis and the boundary element method the background of the developed method is to use non uniform rational b splines nurbs for the galerkin approximation of both geometry and field variables i e the displacement and traction on the boundary the basic ingredient of the method is a pair of weakly singular weak form integral equations for the displacement and traction on the boundary these integral equations contain at most weakly singular kernels of ln r where r is the distance from a source point to a field point various numerical examples are examined to validate the accuracy and efficiency of the proposed method a model of crack propagation is also discussed to illustrate the use of the method for crack growth simulation through the numerical examples it is observed that the isogeometric sgbem produces highly accurate results yet it is simple to implement

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

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

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

on hp error estimation in the bem for a three dimensional helmholtz exterior problem the purpose of the paper is to give an a priori convergence estimate for the hp adaptive boundary element method the method is used to solve an exterior helmholtz boundary value problem with the robin s boundary condition in the three dimensional space the mathematical model describes an acoustic wave scattering problem on a bounded elastic solid where time harmonic process is assumed the estimate consists of three parts which correspond to influence of three errors caused by interpolation of functions geometry and an incident wave

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

perturbation technique for non self adjoint systems with repeated eigenvalues the reanalysis of non self adjoint systems is computationally very expensive in the design of structural dynamics an effective even if approximate evaluation of the eigensolutions of a modified system is valuable to designers in a paper by meirovitch and ryland meirovitch l and ryland g a perturbation technique for gyroscopic systems with small internal and external damping journal of sound and vibration vol 100 no 3 1985 pp 393408 a perturbation technique was used to derive the perturhed solutions of a damped gyroscopic system with distinct eigenvalues that approach is extended to the case of systems with repeated eigenvalues a numerical example is presented to verify the present method and satisfactory results are observed

a nurbs based immersed methodology for fluid structure interaction abstract we introduce an isogeometric immersed and fully implicit formulation for fluid structure interaction fsi the method focuses on viscous incompressible flows and nonlinear hyperelastic incompressible solids which are a common case in various fields such as for example biomechanics in our fsi method we utilize an eulerian mesh on the whole domain and a lagrangian mesh on the solid domain the lagrangian mesh is arbitrarily located on top of the eulerian mesh in a non conforming fashion due to the formulation of our problem based on the immersed finite element method ifem we do not need mesh update or remeshing algorithms the fluid structure interface is the boundary of the lagrangian mesh but cuts arbitrarily the eulerian mesh the generalized method is used for time discretization and nurbs based isogeometric analysis is employed for the spatial discretization on both fluid and solid domains the information transfer between the two meshes is carried out using the nurbs functions which avoids the use of the so called discretized delta functions the higher order and especially the higher continuity of nurbs functions allow us to deal with severe mesh distortion in the lagrangian mesh in comparison with classical c 0 linear piecewise functions as we prove numerically our numerical solutions attain good agreement with theoretical data for free falling objects in two and three dimensions which confirms the feasibility of our methodology

a performance study of nurbs based isogeometric analysis for interior two dimensional time harmonic acoustics abstract this work evaluates the performance of a nurbs based isogeometric finite element formulation for solving stationary acoustic problems in two dimensions an initial assessment is made by studying eigenvalue problems for a square and a circular domain the spectral approximation properties of nurbs functions of varying order are compared to those of conventional polynomials and are found to be superior yielding more accurate representations of eigenvalues as well as eigenmodes the higher smoothness of nurbs shape functions yields better approximations over an extended frequency range when compared to conventional polynomials two numerical case studies including a geometrically complex domain are used to benchmark the method versus the traditional finite element method a convergence analysis confirms the higher efficiency of the isogeometric method on a per degree of freedom basis simulations over a wider frequency range also illustrate that the method suffers less from the dispersion effects that deteriorate the acoustic response towards higher frequencies the tensor product structure of nurbs however also imposes practical considerations when modelling a complex geometry consisting of multiple patches

convergence estimates for a plane elasticity problem solved by the galerkin boundary integral formulation with nurbs abstract a classical mixed boundary value problem of linear elasticity in two dimensions in the galerkin boundary integral formulation is considered we prove a priori error estimates of the solution to this problem by the isogeometric adaptive method using nurbs the estimates include approximation of the boundary of the domain

a natural framework for isogeometric fluid structure interaction based on bem shell coupling abstract the interaction between thin structures and incompressible newtonian fluids is ubiquitous both in nature and in industrial applications in this paper we present an isogeometric formulation of such problems which exploits a boundary integral formulation of stokes equations to model the surrounding flow and a non linear kirchhoff love shell theory to model the elastic behavior of the structure we propose three different coupling strategies a monolithic fully implicit coupling a staggered elasticity driven coupling and a novel semi implicit coupling where the effect of the surrounding flow is incorporated in the non linear terms of the solid solver through its damping characteristics the novel semi implicit approach is then used to demonstrate the power and robustness of our method which fits ideally in the isogeometric paradigm by exploiting only the boundary representation b rep of the thin structure middle surface

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

a two dimensional isogeometric boundary element method for elastostatic analysis the concept of isogeometric analysis where functions that are used to describe geometry in cad software are used to approximate the unknown fields in numerical simulations has received great attention in recent years the method has the potential to have profound impact on engineering design since the task of meshing which in some cases can add significant overhead has been circumvented much of the research effort has been focused on finite element implementations of the isogeometric concept but at present little has been seen on the application to the boundary element method the current paper proposes an isogeometric boundary element method bem which we term igabem applied to two dimensional elastostatic problems using non uniform rational b splines nurbs we find it is a natural fit with the isogeometric concept since both the nurbs approximation and bem deal with quantities entirely on the boundary the method is verified against analytical solutions where it is seen that superior accuracies are achieved over a conventional quadratic isoparametric bem implementation

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

nurbs based parametric mesh free methods a mesh free error reproducing kernel method erkm has recently been proposed by a shaw d roy a nurbs based error reproducing kernel method with applications in solid mechanics comput mech 40 1 2007 127 148 the erkm is based on an initial approximation of the target function by non uniform rational b splines nurbs followed by reproduction of the error via a family of non nurbs basis functions however specifications of the window supports of non nurbs basis functions remain a tricky issue moreover nurbs in higher dimensions 1 is generally defined over rectangular cuboidal in 3d grid structures and thus in many problems of practical interest the geometric complexity of the domain would prevent making use of nurbs in the erkm presently we develop a parametric reformulation of the erkm purely using nurbs this reformulation allows the method to be applicable to non rectangular non cuboidal in 3d physical domains in two or still higher dimensions without a need to explicitly specify the support size of the window a key feature of this development is a geometric map that provides a local bijection between the physical domain and rectangular cuboidal parametric domain the shape functions and their derivatives are then constructed over the parametric domain so that polynomial reproduction and interpolation properties are satisfied over the physical domain with the geometric map being simultaneously preserved a couple of new schemes are also proposed to empower the shape functions with the interpolation property that in turn enables a precise imposition of dirichlet boundary conditions we illustrate the parametric mesh free method in the context of strong weak solutions of a few linear and non linear boundary and initial value problems of engineering interest

extended isogeometric boundary element method xibem for two dimensional helmholtz problems isogeometric analysis is a topic of considerable interest in the field of numerical analysis the boundary element method bem requires only the bounding surface of geometries to be described this makes non uniform rational b splines nurbs which commonly describe the bounding curve or surface of geometries in cad software appear to be a natural tool for the approach this isogeometric analysis bem igabem provides accuracy benefits over conventional bem schemes due to the analytical geometry provided by nurbs when applied to wave problems it has been shown that enriching bem approximations with a partition of unity basis in what has become known as the pu bem allows highly accurate solutions to be obtained with a much reduced number of degrees of freedom this paper combines these approaches and presents an extended isogeometric bem xibem which uses partition of unity enriched nurbs functions this new approach provides benefits which surpass those of both the igabem and the pu bem two numerical examples are given a single scattering cylinder and a multiple scatterer made up of two capsules and a cylinder

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

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

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 nurbs based galerkin approach for the analysis of solids in boundary representation abstract the paper is concerned with a new numerical method to solve the elasticity problem of solids in boundary representation a formulation is derived where the geometrical description of the boundary is sufficient for defining the equations of elasticity of the complete solid while the interior of the domain is described by a radial scaling parameter the scaling of the boundary with respect to the specified scaling center leads to the complete solid this idea fits perfectly to the boundary representation modeling technique commonly employed in cad in the present approach the tensor product structure of the solid will be reduced by one dimension to parametrize the physical domain i e the three dimensional solid exploits only two dimensional nurbs objects which parametrize the boundary surfaces for the analysis the weak form of the equilibrium equations is enforced for the entire solid in particular the weak form is employed in the scaling direction and the circumferential direction applying the isogeometric paradigm the nurbs functions that describe the boundary of the geometry form also the basis for the approximation of the displacement at the boundary the displacement response in the radial scaling direction on the other hand is approximated by a one dimensional nurbs overall the galerkin projection of the weak form yields a linear system of equilibrium equations whose solution gives rise to the displacement response the accuracy of the method is validated by means of analytical reference data in conclusion the proposed method allows the analysis of solids which are bounded by an arbitrary number of contour boundaries

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

localisation property of battle lemarie wavelets sums abstract explicit formulae are given for a type of battle lemarie scaling functions and related wavelets compactly supported sums of their translations are established and applied to alternative norm characterization of sequence spaces isometrically isomorphic to nikolskii besov spaces on r

acoustic isogeometric boundary element analysis an isogeometric boundary element method based on t splines is used to simulate acoustic phenomena we restrict our developments to low frequency problems to establish the fundamental properties of the proposed approach using t splines the computer aided design cad and boundary element analysis are integrated without recourse to geometry clean up or mesh generation a regularized burton miller formulation is used resulting in integrals which are at most weakly singular we employ a collocation based approach to generate the linear system of equations the method is verified against closed form solutions and direct comparisons are made with conventional lagrangian discretizations it is demonstrated that the superior accuracy of the isogeometric approach emanates from the exact geometric description encapsulated in the t spline the method is then applied to a real world application to illustrate the potential for integrated engineering design and analysis

universal perturbation technique for reanalysis of non self adjoint systems a novel matrix perturbation technique for modal dynamic reanalysis of non self adjoint systems is presented this technique is established first by performing complex eigensubspace condensation and orthogonal decomposition procedure the lower order perturbations of eigensolutions i e complex eigenvalues complex left and right eigenvectors are obtained by solving two reduced eigenvalue problems the higher order perturbations of eigensolutions are then derived by implementing the successive approximation process the proposed technique is a kind of universal dynamic reanalysis method because it is universally applicable to systems with all of the three cases distinct repeated and closely spaced eigenvalues illustrative examples are given to verify the present method and satisfactory results are observed