isogeometric boundary element analysis using unstructured t splines we couple collocated isogeometric boundary element methods and unstructured analysis suitable t spline surfaces for linear elastostatic problems we extend the definition of analysis suitable t splines to encompass unstructured control grids unstructured meshes and develop basis functions which are smooth rational polynomials defined in terms of the bezier extraction framework and which pass standard patch tests we then develop a collocation procedure which correctly accounts for sharp edges and corners extraordinary points and t junctions this approach is applied to several three dimensional problems including a real world t spline model of a propeller we believe this work clearly illustrates the power of combining new analysis suitable computer aided design technologies with established analysis methodologies in this case the boundary element method

isogeometric analysis using t splines we explore t splines a generalization of nurbs enabling local refinement as a basis for isogeometric analysis we review t splines as a surface design methodology and then develop it for engineering analysis applications we test t splines on some elementary two dimensional and three dimensional fluid and structural analysis problems and attain good results in all cases we summarize the current status of t splines their limitations and future possibilities

analysis of multiple scattering iterations for high frequency scattering problems ii the three dimensional scalar case in this paper we continue our analysis of the treatment of multiple scattering effects within a recently proposed methodology based on integral equations for the numerical solution of scattering problems at high frequencies in more detail here we extend the two dimensional results in part i of this work to fully three dimensional geometries as in the former case our concern here is the determination of the rate of convergence of the multiple scattering iterations for a collection of three dimensional convex obstacles that are inherent in the aforementioned high frequency schemes to this end we follow a similar strategy to that we devised in part i first we recast the iterated neumann multiple scattering series in the form of a sum of periodic orbits of increasing period corresponding to multiple reflections that periodically bounce off a series of scattering sub structures then we proceed to derive a high frequency recurrence that relates the normal derivatives of the fields induced on these structures as the waves reflect periodically and finally we analyze this recurrence to provide an explicit rate of convergence associated with each orbit while the procedure is analogous to its two dimensional counterpart the actual analysis is significantly more involved and perhaps more interestingly it uncovers new phenomena that cannot be distinguished in two dimensional configurations e g the further dependence of the convergence rate on the relative orientation of interacting structures as in the two dimensional case and beyond their intrinsic interest we also explain here how the results of our analysis can be used to accelerate the convergence of the multiple scattering series and thus to provide significant savings in computational times

x fem in isogeometric analysis for linear fracture mechanics the extended finite element method x fem has proven to be an accurate robust method for solving problems in fracture mechanics x fem has typically been used with elements using linear basis functions although some work has been performed using quadratics in the current work the x fem formulation is incorporated into isogeometric analysis to obtain solutions with higher order convergence rates for problems in linear fracture mechanics in comparison with x fem with conventional finite elements of equal degree the nurbs based isogeometric analysis gives equal asymptotic convergence rates and equal accuracy with fewer degrees of freedom dof results for linear through quartic nurbs basis functions are presented for a multiplicity of one or a multiplicity equal the degree

growth functions on fuchsian groups and the euler characteristic let g be a finitely generated group and 2 a finite generating set for g then 2 determines a norm on g called the word norm defined by letting ig be the minimal length ofg as a word in 27 following milnor 15 we define the growth series g z 1 a2z a z where a is the number of elements of g of word norm exactly n much of the early work e g 1 16 17 23 on growth series was concerned with asymptotic properties of the coefficients of g z this work culminated in gromov s theorem 12 that g has polynomial growth if and only if it is virtually nilpotent and in grigorchuk s theorem 11 that there are finitely generated groups whose growth is neither polynomial nor exponential if g is a coxeter group and 2 is the standard generating set for g bourbaki 3 showed that the growth series g z is the series of a rational function f z and serre 20 showed that f 1 i z g where z g is the rational euler characteristic of g note that if g is finite then g 1 order g l z g but g z is not defined at one if g is infinite if g is a compact hyperbolic or irreducible euclidean coxeter group and 27 is the standard generating set for g serre showed that f 1 z f z and so the poles of f are algebraic units in contrast to the work on the asymptotic properties of the coefficients of g z these results raised the possibility of a beautiful theory of the exact structure of growth functions on groups however a decade went by before there was much further work on this subject in two papers 5 and 6 cannon studied growth functions for cocompact hyperbolic groups if g is a cocompact hyperbolic group and x is any finite generating set for g he showed that the coefficients a of the growth series g z satisfy a linear recursion and hence g z is the power series of a rational function f z he also computed some examples when g is a closed surface group or a compact hyperbolic coxeter group if g is a closed surface group and 2 is a geometric generating set see below for a definition for g coming from a one relator presentation then f is reciprocal f z f 1 z f 1 l x g the

a 3d isogeometric be fe analysis with dynamic remeshing for the simulation of a deformable particle in shear flows abstract a three dimensional isogeometric coupled boundary element and finite element approach based on analysis suitable t splines is developed for the simulation of deformable capsules suspended in shear flows boundary element analysis is used to solve the fluid stokes equation whereas the hydrodynamic membrane load is computed via isogeometric analysis under the assumption that the membrane is a hyper elastic thin shell with negligible bending resistance the smoothness of the t spline basis functions accommodate large deformations of the capsule without the need for additional smoothing techniques and can be used to accurately compute the membrane load a balanced distribution of membrane elements can be constructed using an unstructured locally refined mesh these properties are coupled with an adaptive temporal implicit integration scheme several benchmark examples are solved to illustrate the accuracy and potential of the method the approach is then applied to simulate the dynamics of a capsule in a real geometry of a brain capillary

interior two dimensional acoustic modelling and modal analysis using isogeometric approach abstract in this paper the interior two dimensional acoustic modelling and modal analysis are presented in the framework of isogeometric analysis iga the acoustic domain governed by the helmholtz equation is parameterized and analysed by non uniform rational lagrange nurl basis which can construct exact geometry model inherently like non uniform rational b spines nurbs and can provide a more straightforward way than nurbs to impose dirichlet boundary conditions the verification and versatility of iga using nurl are assessed in the investigation of a unit square domain and a unit circular domain by comparison with analytical solutions furthermore a quadrilateral domain with concave or convex conics is studied and some innovative results are presented which can be the benchmark data for other algorithm researches the effects of geometric weight and aspect ratio on acoustic eigenvalue and eigenmode of the quadrilateral domain are investigated in detail

plane wave interpolation in direct collocation boundary element method for radiation and wave scattering numerical aspects and applications the classical boundary element formulation for the helmholtz equation is rehearsed and its limitations with respect to the number of variables needed to model a wavelength are explained a new type of interpolation for the potential is then described in which the usual boundary element shape functions are modified by the inclusion of a set of plane waves propagating in a range of directions this is termed the plane wave basis boundary element method the modifications needed to the classical procedures in terms of integration of the element matrices and location of collocation points are described the well known singular value decomposition solution technique which is adopted here for the solution of the system matrix equation in its complex form is briefly outlined the conditioning of the system matrix is analysed for a simple radiation problem the corresponding diffraction problem is also analysed and results are compared with analytical and classical boundary element solutions the chief method is adopted to enhance the quality of the solution particularly in the vicinity of irregular frequencies the plane wave basis boundary element method is then applied to two problems scattering of plane waves by an elliptical cylinder and the multiple circular cylinder plane wave scattering problem in both cases results are compared with analytical solutions the results clearly demonstrate that the new method is considerably more efficient than the classical approach for a given number of degrees of freedom the frequency for which accurate results can be obtained using the new technique can be up to three or four times higher than that of the classical method this makes the method a powerful new addition to our tools for tackling high frequency radiation and scattering problems

extended isogeometric boundary element method xibem for three dimensional medium wave acoustic scattering problems a boundary element method bem based on non uniform rational b splines nurbs is used to find solutions to three dimensional wave scattering problems governed by the helmholtz equation the method is extended in a partition of unity sense multiplying the nurbs functions by families of plane waves this method is called the extended isogeometric boundary element method xibem in this paper the collocation xibem formulation is described and numerical results are given the numerical results are compared against closed form or converged solutions comparisons are made against the conventional boundary element method and the non enriched isogeometric bem igabem when compared to non enriched boundary element simulations using xibem significantly reduces the number of degrees of freedom required to obtain a solution of a given error thus with a fixed computational resource problems of a shorter wavelength can be solved

h and p adaptivity driven by recovery and residual based error estimators for pht splines applied to time harmonic acoustics abstract in this work we demonstrate the application of pht splines for time harmonic acoustic problems modeled by the helmholtz equation solutions of the helmholtz equation have two features global oscillations associated with the wave number and local gradients caused by geometrical irregularities we show that after a sufficient number of degrees of freedom is used to approximate global oscillations adaptive refinement can capture local features of the solution we compare residual based and recovery based error estimators and investigate the performance of p refinement the simulations are done in the context of recently introduced geometry independent field approximation gift where pht splines are only used to approximate the solution while the computational domain is parameterized with nurbs this approach builds on the natural adaptation ability of pht splines and avoids the re parameterization of the nurbs geometry during the solution refinement process

plane wave enriched partition of unity isogeometric analysis puiga for 2d helmholtz problems abstract in general numerical concerns in simulation of time harmonic acoustic waves are mainly due to the dispersion errors and the demands in modeling the problems with higher number of degrees of freedom per wavelength to obtain acceptable results as a remedy plane wave enriched partition of unity finite element method pufem was earlier proposed by melenk and babuska 1996 even though pufem gives accurate results with lesser degrees of freedom and shows exponential convergence it suffers from the approximate representation of geometry and leads to the boundary representation induced errors in this regard plane wave enriched pufem under isogeometric framework named partition of unity isogeometric analysis puiga is proposed for accurate geometry representation and smooth solution approximation which can lead to a better solution several benchmark problems are considered in the present study to illustrate the computational effectiveness of puiga

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

optimal convergence for adaptive iga boundary element methods for weakly singular integral equations in a recent work feischl et al in eng anal bound elem 62 141 153 2016 we analyzed a weighted residual error estimator for isogeometric boundary element methods in 2d and proposed an adaptive algorithm which steers the local mesh refinement of the underlying partition as well as the multiplicity of the knots in the present work we give a mathematical proof that this algorithm leads to convergence even with optimal algebraic rates technical contributions include a novel mesh size function which also monitors the knot multiplicity as well as inverse estimates for nurbs in fractional order sobolev norms

an isogeometric approach of two dimensional acoustic design sensitivity analysis and topology optimization analysis for absorbing material distribution abstract a study of structural shape optimization and absorbing material distribution optimization of noise barrier structures based on the recently proposed isogeometric analysis method with exact geometric definitions is presented the acoustic scattering is approximated using the basis functions that represent the geometry a fast multipole method is applied to accelerate the solution of the boundary element method bem the burton miller formulation is used to overcome the fictitious frequency problem when a single helmholtz boundary integral equation is used for the exterior boundary value problem the strongly singular integrals in the burton miller formulation using the isogeometric bem are evaluated explicitly and directly particularly for the sensitivity formulation with a hyper singular integral the optimality criteria method is used for two types of optimization analyses shape optimization and material distribution topology optimization for the shape optimization the design variables can be set to the locations of the control points because the control points determine the shape of structure for the second optimization a new material interpolation scheme for acoustic problems based on the solid isotropic material with penalization simp method is given where the interpolation variable is not the real structural density used in a conventional simp but a fictitious material density that determines the normalized surface admittance several examples are presented to demonstrate the validity and efficiency of the proposed algorithm

a self adaptive co ordinate transformation for efficient numerical evaluation of general boundary element integrals almost all general purpose boundary element computer packages include a curved geometry modelling capability thus numerical quadrature schemes play an important role in the efficiency of programming the technique the present work discusses this problem in detail and introduces efficient means of computing singular or nearly singular integrals currently found in two dimensional axisymmetric and three dimensional applications emphasis is given to a new third degree polynomial transformation which was found greatly to improve the accuracy of gaussian quadrature scheme s within the near singularity range the procedure can easily be implemented into existing be codes and presents the important feature of being self adaptive i e it produces a variable lumping of the gauss stations toward the singularity depending on the minimum distance from the source point to the element the self adaptiveness of the scheme also makes it inactive when not useful large source distances which makes it very safe for general usage

optimal additive schwarz preconditioning for adaptive 2d iga boundary element methods abstract we define and analyze local multilevel diagonal preconditioners for isogeometric boundary elements on locally refined meshes in two dimensions hypersingular and weakly singular integral equations are considered we prove that the condition number of the preconditioned systems of linear equations is independent of the mesh size and the refinement level therefore the computational complexity when using appropriate iterative solvers is optimal our analysis is carried out for closed and open boundaries and numerical examples confirm our theoretical results

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

pollution studies for high order isogeometric analysis and finite element for acoustic problems abstract it is well known that galerkin finite element methods suffer from pollution error when solving wave problems to reduce the pollution impact on the solution different approaches were proposed to enrich the finite element method with wave like functions so that the exact wavenumber is incorporated into the finite element approximation space solving wave problems with isogeometric analysis was also investigated in the literature where the superior behaviour of isogeometric analysis due to higher continuity in the underlying basis has been studied recently a plane wave enriched isogeometric analysis was introduced for acoustic problems however it remains unquantified the impact of these different approaches on the pollution or how they perform compared to each other in this work we show that isogeometric analysis outperforms finite element method in dealing with pollution we observe similar behaviour when both the methods are enriched with plane waves using higher order polynomials with fewer enrichment functions seems to improve the pollution compared to lower order polynomials with more functions however the latter still leads to smaller errors using similar number of degrees of freedom in conclusion we propose that partition of unity isogeometric analysis can be an efficient tool for wave problems as enrichment eliminates the need for domain re meshing at higher frequencies and also due to its ability to capture the exact geometry even on coarse meshes as well as its improved pollution behaviour

an isogeometric boundary element method for three dimensional potential problems abstract isogeometric analysis iga coupled with boundary element method i e igabem received a lot of attention in recent years in this paper we extend the igabem to solve 3d potential problems this method offers a number of key improvements compared with conventional piecewise polynomial formulations firstly the models for analysis in the igabem are exact geometrical representation no matter how coarse the discretization of the studied bodies is thus the igabem ensures that no geometrical errors are produced in the analysis process secondly a meshing process is no longer required which means redundant computations are eliminated to allow analysis to be carried out with greatly reduced pre processing to accurately evaluate the singular integrals appearing in our method the power series expansion method is employed the integration surface is on the real surface of the model rather than the interpolation surface i e no geometrical errors thus the value of integral is more accurate than the traditional boundary element method which can improve the computation accuracy of the igabem some numerical examples for 3d potential problems are used to validate the solutions of the present method with analytical and numerical solutions available

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

multi patch nonsingular isogeometric boundary element analysis in 3d abstract a multi patch nonsingular isogeometric boundary element method igabem for 3d problems is presented that provides accurate solutions for multi patch igabem in order to conveniently implement this method based on the greville abscissae a new collocation method moves the first and the last collocation points of each parametric direction inside of their patches and a simple method for merging equations handles the extra equations the numerical results verify the accuracy and efficiency of the present method by comparing it to the conventional igabem

nurbs enhanced line integration boundary element method for 2d elasticity problems with body forces abstract a nurbs enhanced boundary element method for 2d elasticity problems with body forces is proposed in this paper the non uniform rational b spline nurbs basis functions are applied to construct the geometry and the model can be reproduced exactly at all stages since the refinement will not change the shape of the boundary both open curves and closed curves are considered the fields are approximated by the traditional lagrangian basis functions in parameter space rather than by the same nurbs basis functions for geometry approximation the parametric boundary elements and collocation nodes are defined from the knot vector of the curve and the refinement of the nurbs curve is easy boundary conditions can be imposed directly since the lagrangian basis functions have the property of delta function in addition most methods for the treatment of singular integrals in traditional boundary element method can be applied in the proposed method to overcome the difficulty for evaluation of the domain integrals in problems with body forces a line integration method is further applied in this paper to compute the domain integrals without additional volume discretizations numerical examples have shown the accuracy of the proposed method

numerical evaluation of the two dimensional partition of unity boundary integrals for helmholtz problems there has been considerable attention given in recent years to the problem of extending finite and boundary element based analysis of helmholtz problems to higher frequencies one approach is the partition of unity method which has been applied successfully to boundary integral solutions of helmholtz problems providing significant accuracy benefits while simultaneously reducing the required number of degrees of freedom for a given accuracy these benefits accrue at the cost of the requirement to perform some numerically intensive calculations in order to evaluate boundary integrals of highly oscillatory functions in this paper we adapt the numerical steepest descent method to evaluate these integrals for two dimensional problems the approach is successful in reducing the computational effort for most integrals encountered the paper includes some numerical features that are important for successful practical implementation of the algorithm

a hybrid igafem igabem formulation for two dimensional stationary magnetic and magneto mechanical field problems abstract isogeometric analysis iga can bridge the gap between geometrical and numerical modelling to this end the same functions used in computer aided design are applied to represent geometry and approximate field variables in the numerical model the concept has already been implemented to solve field problems using finite element fem and boundary element methods bem but coupling of both methods has not been applied in the current work an isogeometric fem bem coupling is proposed and applied to two dimensional stationary magnetic field problems while fem is used to model magnetisable bodies allowing for heterogeneous structures and non linear constitutive behaviour the bem domain accounts for the surrounding free space both methods are coupled on the surface of the magnetisable body due to this hybrid igafem bem approach no meshing of free space is necessary and truncation errors are avoided for problems to be solved on open infinite or semi infinite domains once the solution for the magnetic problem is obtained using the hybrid method igafem is used to solve a magneto mechanical field problem with one sided coupling in a subdomain of the magnetic problem this one sided coupling is realised by a magnetic stress tensor computed from the solution of the stationary magnetic field problem from the comparison of error norms and convergence rates for nurbs and discretisations based on lagrangian polynomials smaller errors and similar convergence rates are found for the proposed method for the same polynomial order of the basis functions and a comparable mesh size

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

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

a wavenumber independent boundary element method for an acoustic scattering problem in this paper we consider the impedance boundary value problem for the helmholtz equation in a half plane with piecewise constant boundary data a problem which models for example outdoor sound propagation over inhomogeneous flat terrain to achieve good approximation at high frequencies with a relatively low number of degrees of freedom we propose a novel galerkin boundary element method using a graded mesh with smaller elements adjacent to discontinuities in impedance and a special set of basis functions so that on each element the approximation space contains polynomials of degree nu multiplied by traces of plane waves on the boundary we prove stability and convergence and show that the error in computing the total acoustic field is cal o n nu 1 log 1 2 n where the number of degrees of freedom is proportional to n log n this error estimate is independent of the wavenumber and thus the number of degrees of freedom required to achieve a prescribed level of accuracy does not increase as the wavenumber tends to infinity

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

towards a full bandwidth numerical acoustic model prediction models are at the heart of modern acoustic engineering current commercial room acoustic simulation software almost exclusively approximates the propagation of sound geometrically as rays or beams these assumptions yield efficient algorithms but the maximum accuracy they can achieve is limited by how well the geometric assumption represents sound propagation in a given space this comprises their accuracy at low frequencies in particular methods that directly model wave effects are more accurate but they have a computational cost that scales with problem size and frequency effectively limiting them to small or low frequency scenarios this paper will report the results of initial research into a new full bandwidth model which aims to be accurate and efficient for all frequencies the name proposed for this is the wave matching method this builds on the boundary element method with the premise that if an appropriate interpolation scheme is designed then the model will become geometrically dominated at high frequencies other propagation modes may then be removed without significant error yielding an algorithm which is accurate and efficient this paper will present the general concepts of wave matching and the results from some numerical test cases

the envelope of tridiagonal toeplitz matrices and block shift matrices abstract the envelope of a square complex matrix is a spectrum encompassing region in the complex plane it is contained in and is akin to the numerical range in the sense that the envelope is obtained as an infinite intersection of unbounded regions contiguous to cubic curves rather than half planes in this article the geometry and properties of the envelopes of special matrices are examined in particular symmetries of the envelope of a tridiagonal toeplitz matrix are obtained and the envelopes of block shift matrices jordan blocks and 2 2 matrices are explicitly characterized

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

novel basis functions for the partition of unity boundary element method for helmholtz problems the bem is a popular technique for wave scattering problems given its inherent ability to deal with infinite domains in the last decade the partition of unity bem in which the approximation space is enriched with a linear combination of plane waves has been developed this significantly reduces the number of dofs required per wavelength it has been shown that the element ends are more susceptible to errors in the approximation than the mid element regions in this paper the authors propose that this is due to the use of a collocation approach in combination with a reduced order of continuity in the lagrangian shape function component of the basis functions it is demonstrated using numerical examples that choosing trigonometric shape functions rather than classical polynomial shape functions quadratic in this case provides accuracy benefits collocation schemes are investigated it is found that the somewhat arbitrary choice of collocating at equally spaced points about the surface of a scatterer is better than schemes based on the roots of polynomials or consideration of the fock domain

isogeometric boundary element methods for three dimensional static fracture and fatigue crack growth abstract we present a novel numerical method to simulate crack growth in 3d directly from the computer aided design cad geometry of the component without any mesh generation the method is an isogeometric boundary element method igabem based on non uniform rational b splines nurbs nurbs basis functions are used for the domain and crack representation as well as to approximate the physical quantities involved in the simulations a stable quadrature scheme for singular integration is proposed to enhance the robustness of the method in dealing with highly distorted elements convergence studies in the crack opening displacement is performed for a penny shaped crack and an elliptical crack two approaches to extract stress intensity factors sifs the contour m integral and the virtual crack closure integral are compared using dual integral equations the results show remarkable accuracy in the computed sifs leading to smooth crack paths and reliable fatigue lives without requiring the generation of any mesh from the cad model of the component under consideration