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

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

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

spline collocation for strongly elliptic equations on the torus we prove convergence and error estimates in sobolev spaces for the collocation method with tensor product splines for strongly elliptic pseudodifferential equations on the torus examples of applications include elliptic partial differential equations with periodic boundary conditions but also the classical boundary integral operators of potential theory on torus shaped domains in three or more dimensions for odd degree splines we prove convergence of nodal collocation for any strongly elliptic operator for even degree splines and midpoint collocation we find an additional condition for the convergence which is satisfied for the classical boundary integral operators our analysis is a generalization to higher dimensions of the corresponding analysis of arnold and wendland 4

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

modular centralizer algebras corresponding to p groups abstract we study the loewy structure of the centralizer algebra k p q for p a p group with subgroup q and k a field of characteristic p here k p q is a special type of hecke algebra the main tool we employ is the decomposition k p q k c p q i of k p q as a split extension of a nilpotent ideal i by the group algebra k c p q we compute the loewy structure for several classes of groups investigate the symmetry of the loewy series and give upper and lower bounds on the loewy length of k p q several of these results were discovered through the use of magma especially the general pattern for most of our computations as a final application of the decomposition we determine the representation type of k p q

searching for more general weight conjectures using the symmetric group as an example abstract let k be a field of characteristic p and let s n be the symmetric group of degree n assume that n p let a be the centralizer in ks n of ks n 1 we find all simple a modules and relate the number of non projective simple a modules to p local information

on the variational formulation of stress constraints in isogeometric design the isogeometric paradigm is aimed at unifying the geometric and analysis descriptions of engineering problems this unification is brought about by employing the same basis functions to describe exact geometry and to approximate the physical response the isogeometric framework has recently been extended to include design optimisation a novel variational stress constraint formulation is proposed in this paper that further extends recent efforts in isogeometric design the weak form of the constraint is obtained by utilising the same basis functions not only to represent geometry response and design variables but also lagrange multipliers in this form the proposed variational formulation fits seamlessly with the underlying analysis and design environment the variational constraint formulation is compared to the widely utilised stress averaging and collocation approaches to this end volume minimisation of elastic arches subjected to stress constraints is considered results indicate that the proposed variational approach has the best convergence rates and utilises a much smaller number of discrete constraints than competing methods

an isogeometric analysis approach to gradient damage models continuum damage formulations are commonly used for the simulation of diffuse fracture processes implicit gradient damage models are employed to avoid the spurious mesh dependencies associated with local continuum damage models the c 0 continuity of traditional finite elements has hindered the study of higher order gradient damage approximations in this contribution we use isogeometric finite elements which allow for the construction of higher order continuous basis functions on complex domains we study the suitability of isogeometric finite elements for the discretization of higher order gradient damage approximations

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

long term memory in stock market prices a test for long term memory that is robust to short range dependence is developed it is a modification of the r s statistic and the relevant asymptotic sampling theory is derived via functional central limit theory contrary to previous findings when applied to daily and monthly stock returns indexes over several time periods this test yields no evidence of long range dependence once short range dependence is accounted for monte carlo experiments show that the modified r s test has power against at least two specific models of long term memory suggesting that models with short range dependence may adequately capture the behavior of historical stock returns copyright 1991 by the econometric society

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

block theory branching rules and centralizer algebras this paper is one in a series 8 11 exploring the algebra kgh the centralizer in the group algebra kg of the subalgebra kh where k is a field of characteristic not zero g is a finite group and h is a subgroup of g all these papers search for theorems similar to alperin s weight conjecture 2 the immediate goal is to find results that relate information about blocks of kgh or simple modules over kgh to p local information the ultimate goal is to gain insight into alperin s conjecture see the introductions to 10 and 11 for a detailed description of the program

existence of stiffness matrix integrals for singularly parameterized domains in isogeometric analysis abstract isogeometric analysis is a numerical simulation method which uses the nurbs based representation of cad models nurbs stands for non uniform rational b splines and is a generalization of the concept of b splines the isogeometric method uses the tensor product structure of 2 or 3 dimensional nurbs functions to parameterize domains which are structurally equivalent to a rectangle or a hexahedron the special case of singularly parameterized nurbs surfaces and nurbs volumes is used to represent non quadrangular or non hexahedral domains without splitting which leads to a very compact and convenient representation if the parameterization of the physical domain is available the test functions for the isogeometric analysis are obtained by composing the inverse of the domain parameterization with the nurbs basis functions in the case of singular parameterizations however some of the resulting test functions are not well defined at the singular points and they do not necessarily satisfy the required integrability assumptions consequently the stiffness matrix integrals which occur in the numerical discretizations may not exist after summarizing the basics of the isogeometric method we discuss the existence of the stiffness matrix integrals for 1 2 and 3 dimensional second order elliptic partial differential equations we consider several types of singularities of nurbs parameterizations and derive conditions which guarantee the existence of the required integrals in addition we present cases with diverging integrals and show how to modify the test functions in these situations

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

on the asymptotic convergence of collocation methods with spline functions of even degree we investigate the collocation of linear one dimensional strongly elliptic integro dif ferential or more generally pseudo differential equations on closed curves by even degree polynomial splines the equations are collocated at the respective midpoints subject to uniform nodal grids of the even degree s splines we prove quasioptimal and optimal order asymptotic error estimates in a scale of sobolev spaces the results apply in particular to boundary element methods used for numerical computations in engineering applications the equations considered include fredholm integral equations of the second and the first kind singular integral equations involving cauchy kernels and integro differential equations having convolutional or constant coefficient principal parts respectively the error analysis is based on an equivalence between the collocation and certain varia tional methods with different degree splines as trial and as test functions we further need to restrict our operators essentially to pseudo differential operators having convolutional prin cipal part this allows an explicit fourier analysis of our operators as well as of the spline spaces in terms of trigonometric polynomials providing babuska s stability condition based on strong ellipticity our asymptotic error estimates extend partly those obtained by d n arnold and w l wendland from the case of odd degree splines to the case of even degree splines 1 introduction in this paper we investigate the asymptotic convergence of the collocation method using even degree polynomial splines applied to strongly elliptic systems of pseudo differential equations on closed curves with convolutional prin cipal part the collocation here employs the gauss points of one point integration as collocation points which corresponds to the usual boundary element collocation in applications 5 12 17 19 26 31 34 this is in contrast to the method investigated by g schmidt in 29 where one collocates at the break points of even order splines the asymptotic convergence properties for the standard galerkin method with splines of arbitrary orders are well known 22 for the collocation method however asymptotic convergence up to now has been shown for strongly elliptic systems only in the case of odd order splines by d n arnold and w l wendland 7

a phase field description of dynamic brittle fracture abstract in contrast to discrete descriptions of fracture phase field descriptions do not require numerical tracking of discontinuities in the displacement field this greatly reduces implementation complexity in this work we extend a phase field model for quasi static brittle fracture to the dynamic case we introduce a phase field approximation to the lagrangian for discrete fracture problems and derive the coupled system of equations that govern the motion of the body and evolution of the phase field we study the behavior of the model in one dimension and show how it influences material properties for the temporal discretization of the equations of motion we present both a monolithic and staggered time integration scheme we study the behavior of the dynamic model by performing a number of two and three dimensional numerical experiments we also introduce a local adaptive refinement strategy and study its performance in the context of locally refined t splines we show that the combination of the phase field model and local adaptive refinement provides an effective method for simulating fracture in three dimensions

numerical efficiency locking and unlocking of nurbs finite elements abstract the strategy of using finite elements with nurbs shape functions for approximation of both geometry and displacements isogeometric approach is investigated from the point of view of finite element technology convergence rates are compared to those of classical finite element approaches utilizing standard lagrange shape functions moreover typical locking phenomena are examined it is found that higher order inter element continuity within the nurbs approach results in identical convergence rates but smaller absolute errors compared to c 0 continuous approaches however nurbs finite elements suffer from the same locking problems as finite elements using lagrange shape functions the discrete shear gap dsg method a general framework for formulation of locking free elements is applied to develop a new class of nurbs finite elements the resulting nurbs dsg elements are absolutely free from locking and preserve the property of improved accuracy compared with standard locking free finite elements the method is exemplified for the timoshenko beam model but may be applied to more general cases

a sinh transformation for evaluating nearly singular boundary element integrals an implementation of the boundary element method requires the accurate evaluation of many integrals when the source point is far from the boundary element under consideration a straightforward application of gaussian quadrature suffices to evaluate such integrals when the source point is on the element the integrand becomes singular and accurate evaluation can be obtained using the same gaussian points transformed under a polynomial transformation which has zero jacobian at the singular point a class of integrals which lies between these two extremes is that of nearly singular integrals here the source point is close to but not on the element and the integrand remains finite at all points however instead of remaining flat the integrand develops a sharp peak as the source point moves closer to the element thus rendering accurate evaluation of the integral difficult this paper presents a transformation based on the sinh function which automatically takes into account the position of the projection of the source point onto the element which we call the nearly singular point and the distance from the source point to the element the transformation again clusters the points towards the nearly singular point but does not have a zero jacobian implementation of the transformation is straightforward and could easily be included in existing boundary element method software it is shown that for the two dimensional boundary element method several orders of magnitude improvement in relative error can be obtained using this transformation compared to a conventional implementation of gaussian quadrature asymptotic estimates for the truncation errors are also quoted copyright 2004 john wiley sons ltd

some notes on singular integral techniques in boundary element analysis the paper presented here reviews the numerical techniques used currently to calculate the singular integrals and nearly singular integrals in the boundary element analysis some incorrect algorithms published before are discussed and a new numerical technique to calculate the nearly singular integral is developed the numerical results show a significant improvement in both accuracy and efficiency compared to the traditional adaptive gaussian quadrature and subdivision techniques

large eddy simulation of turbulent taylor couette flow using isogeometric analysis and the residual based variational multiscale method we present an application of the residual based variational multiscale turbulence modeling rbvms methodology to the computation of turbulent taylor couette flow at high reynolds number we show that the rbvms formulation globally conserves angular momentum a feature that is felt to be important for flows dominated by rotation and that is not shared by standard stabilized formulations of fluid flow weak imposition of dirichlet boundary conditions is employed to enhance the accuracy of the rbvms framework in the presence of thin turbulent boundary layers near solid walls calculation of conservative boundary forces and torques is also presented for the case of weakly enforced boundary conditions nurbs based isogeometric analysis is employed for the spatial discretization and mesh refinement is performed to assess the convergence characteristics of the proposed methodology numerical tests show that very accurate results are obtained on relatively coarse grids to the best of the authors knowledge this paper is the first to report large eddy simulation computations of this challenging test case

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

on a class of m estimators for gaussian long memory models summary we consider estimation for parametric stationary gaussian models with long memory the spectral density f x 0 is assumed to be characterised by a vector 0 01 022 03 om such that 02 he 1 1 and f x 0 is proportional to xl 2h as x tends to zero an approximate maximum likelihood estimator based on the autoregressive representation of the process is proposed its asymptotic distribution is derived more generally the approach leads to a class of m estimators for which a central limit theorem holds by choosing an appropriate t function robustness against additive outliers can be achieved while keeping high efficiency under the ideal model this is illustrated by a small simulation study a simple algorithm and an explicit formula for the efficiency and thus for choosing an appropriate tuning parameter are given for hampel s redescending function

on a class of estimation and test for long memory this paper advances a new analysis technology path of estimation and test for long memory time series i propose the definitions of time scale series strong variance scale exponent and weak variance scale exponent and prove the strict mathematical equations that strong and weak variance scale exponent can accurately identify the time series of white noise short memory and long memory especially derive the equation relationships between weak variance scale exponent and long memory parameters i also construct two statistics which slmemory statistic tests for long memory properties the paper further displays monte carlo performance for mse of weak variance scale exponent estimator and the empirical size and power of slmemory statistic giving practical recommendations of finite sample and also provides brief empirical examples of logarithmic return rate series data for sino us stock markets

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