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 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

geometry independent field approximation cad analysis integration geometrical exactness and adaptivity in isogeometric analysis iga the same spline representation is employed for both the geometry of the domain and approximation of the unknown fields over this domain this identity of the geometry and field approximation spaces was put forward in the now classic 2005 paper 20 as a key advantage on the way to the integration of computer aided design cad and subsequent analysis in computer aided engineering cae 20 claims indeed that any change to the geometry of the domain is automatically inherited by the approximation of the field variables without requiring the regeneration of the mesh at each change of the domain geometry yet in finite element versions of iga a parameterization of the interior of the domain must still be constructed since cad only provides information about the boundary the identity of the boundary and field representation decreases the flexibility in which this parameterization can be generated and somewhat constrains the modeling and simulation process because an approximation able to represent the domain geometry accurately need not be adequate to also approximate the field variables accurately in particular when the solution is not smooth we propose here a new paradigm called geometry independent field approximation gift where the spline spaces used for the geometry and the field variables can be chosen and adapted independently while preserving geometric exactness and tight cad integration gift has the following features 1 it is possible to flexibly

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

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

high order finite elements for shells in this article the p version finite element method is applied to thin walled structures two different hierarchic element formulations are compared a shell approach as well as a shell like solid formulation both approaches are compared for linear elastic and elastoplastic problems special emphasis is placed on the efficiency as well as on determining the area of application for both formulations

enforcement of constraints and maximum principles in the variational multiscale method abstract we present a new theoretical framework for the enforcement of constraints in variational multiscale vms analysis the theory is first presented in an abstract operator format and subsequently specialized for the steady advection diffusion equation the approach borrows heavily from results in constrained and convex optimization an exact expression for the fine scales is derived in terms of variational derivatives of the constraints lagrange multipliers and a fine scale green s function the methodology described enables the development of numerical methods which satisfy predefined attributes a practical and effective procedure for solving the steady advection diffusion equation is presented based on a vms inspired stabilized method weakly enforced dirichlet boundary conditions and enforcement of a maximum principle and conservation constraint

multigrid methods for isogeometric analysis with thb splines we propose a geometric multigrid algorithm for the solution of the linear systems which arise when using hierarchical spline spaces for isogeometric discretizations of elliptic partial differential equations in particular we focus on the use of hierarchical b spline hb and truncated hierarchical b spline thb bases to implement adaptively refined isogeometric discretizations we describe an approach for constructing the nested sequences of hierarchical spline spaces which are required for our geometric multigrid solver furthermore we give an algorithm for the computation of the so called prolongation matrices which map coarser space functions to their finer space representations for both hb and thb spline bases in several numerical experiments we study two dimensional poisson problems with singularities and employ a posteriori error estimators for the adaptive refinement of the hierarchical spline spaces using standard multigrid smoothers we observe that using thb spline bases consistently confers a dramatic advantage in terms of iteration numbers over the use of hb spline bases in particular for higher spline degrees when used as a preconditioner for conjugate gradient iteration the proposed multigrid method exhibits practical iteration numbers for two dimensional problems up to the tested case of fourth degree splines

banach spaces of universal disposition in this paper we present a method to obtain banach spaces of universal and almost universal disposition with respect to a given class m of normed spaces the method produces among others the only separable banach space of almost universal disposition with respect to the class f of finite dimensional spaces gurari space g or the only under ch banach space with density character the continuum which is of universal disposition with respect to the class s of separable spaces kubis space k we moreover show that k is isomorphic to an ultrapower of the gurari space and that it is not isomorphic to a complemented subspace of any c k space other properties of spaces of universal disposition are also studied separable injectivity partially automorphic character and uniqueness

a separable frechet space of almost universal disposition abstract the gurari space is the unique separable banach space g which is of almost universal disposition for finite dimensional banach spaces which means that for every e 0 for all finite dimensional normed spaces e f for every isometric embedding e e g there exists an e isometric embedding f f g such that f e e we show that g n with a special sequence of semi norms is of almost universal disposition for finite dimensional graded frechet spaces the construction relies heavily on the universal operator on the gurari space recently constructed by garbulinska wegrzyn and the third author in addition we consider a non graded sequence of semi norms on g n with which the space g n is of almost universal disposition for finite dimensional frechet spaces with a fixed sequence of semi norms in both cases this yields in particular that g n is universal in the class of all separable frechet spaces

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

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

periodic heterogeneous structures new explicit solutions and effective characteristics of refraction of an imposed field complex variable methods are applied to study three double periodic two phase planar heterogeneous structures explicit expressions of functionals of energy dissipation and of effective resistivities conductivities are found on the basis of strict analytical solutions of a boundary value problem for an arbitrary direction of an imposed external field comparisons with the results of bakhvalov and panasenko homogenization of processes in periodic media nauka moscow 1984 dykhne zh eksp teor fiz 59 1970 pp 110 113 soviet phys jetp 32 1971 pp 63 65 emets zh eksp teor fiz 96 1989 pp 701 711 soviet phys jetp 96 1989 keller j appl phys 34 1963 pp 991 993 and mendelson j appl phys 46 1975 pp 4740 4741 are made in particular the dissipation in the general case is shown essentially to depend on the direction of the external field for a rectangular checkerboard dissipation within the two phases does not coincide it is rigorously pr

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

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

isogeometric analysis using lr b splines abstract the recently proposed locally refined b splines denoted lr b splines by dokken et al 2013 6 may have the potential to be a framework for isogeometric analysis to enable future interoperable computer aided design and finite element analysis in this paper we propose local refinement strategies for adaptive isogeometric analysis using lr b splines and investigate its performance by doing numerical tests on well known benchmark cases the theory behind lr b spline is not presented in full details but the main conceptual ingredients are explained and illustrated by a number of examples

boundary conforming free surface flow computations interface tracking for linear higher order and isogeometric finite elements abstract the simulation of certain flow problems requires a means for modeling a free fluid surface examples being viscoelastic die swell or fluid sloshing in tanks in a finite element context this type of problem can among many other options be dealt with using an interface tracking approach with the deforming spatial domain stabilized space time dsd sst formulation a difficult issue that is connected with this type of approach is the determination of a suitable coupling mechanism between the fluid velocity at the boundary and the displacement of the boundary mesh nodes in order to avoid large mesh distortions one goal is to keep the nodal movements as small as possible but of course still compliant with the no penetration boundary condition standard displacement techniques are full velocity velocity in a specific coordinate direction and velocity in normal direction in this work we investigate how the interface tracking approach can be combined with isogeometric analysis for the spatial discretization if nurbs basis functions of sufficient order are used for both the geometry and the solution both a continuous normal vector as well as the velocity are available on the entire boundary this circumstance allows the weak imposition of the no penetration boundary condition we compare this option with an alternative that relies on strong imposition at discrete points furthermore we examine several coupling methods between the fluid equations boundary conditions and equations for the adjustment of interior control point positions

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

analysis of acoustic networks including cavities by means of a linear finite volume method abstract a procedure allowing for the analysis of complex acoustic networks including three dimensional cavities described in terms of zero dimensional equivalent elements is presented and validated the procedure is based on the linearization of the finite volume method often used in gas dynamics which is translated into an acoustic network comprising multi ports accounting for mass exchanges between the finite volumes and equivalent 2 ports describing momentum exchange across the volume surfaces the application of the concept to a one dimensional case shows that it actually converges to the exact analytical solution when a sufficiently large number of volumes are considered this has allowed the formulation of an objective criterion for the choice of a mesh providing results with a prefixed error up to a certain helmholtz number which has been generalized to three dimensional cases the procedure is then applied to simple but relevant three dimensional geometries in the absence of a mean flow showing good agreement with experimental and other computational results

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

regular hexagonal three phase checkerboard abstract a two dimensional doubly periodic three phase hexagonal structure is considered the flow in the structure is generated by three sets of vortexes sinks sources which are the same in each phase and are located in the centers of the hexagons complex analysis methods are utilized to reduce the doubly periodic r linear conjugation problem to the simpler one riemann hilbert rh problem on a three sheeted riemann surface in turn the latter problem is reduced to a rh problem involving three joined sectors on the plane which was previously investigated in 3 the limiting cases with one non conducting phase and two phases of the same conductivities are investigated all solutions derived are verified both numerically and analytically examples of relevant flow networks streamlines and equipotentials are plotted in the whole structure and separately in each phase

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

algebraic multilevel preconditioning in isogeometric analysis construction and numerical studies abstract we present algebraic multilevel iteration amli methods for isogeometric discretization of scalar second order elliptic problems the construction of coarse grid operators and hierarchical complementary operators are given moreover for a uniform mesh on a unit interval the explicit representation of b spline basis functions for a fixed mesh size h is given for p 2 3 4 and for c 0 and c p 1 continuity the presented methods show h and almost p independent convergence rates supporting numerical results for convergence factor and iterations count for amli cycles v linear w nonlinear w are provided numerical tests are performed in two dimensions on square domain and quarter annulus and in three dimensions on quarter thick ring

liquid vapor transformations with surfactants phase field model and isogeometric analysis surfactants are compounds that find energetically favorable to be located at the boundaries between fluids they are able to modify the properties of those interfaces for example reducing surface tension here we propose a new model for liquid vapor flows with surfactants which captures the dynamics of the surfactant and accounts for phase transformations in the fluid the aforementioned model is derived from a free energy functional by using a coleman noll approach the proposed theory emanates from the isothermal navier stokes korteweg equations which describe single component two phase flow and naturally allow for phase transformations we believe that our model has significant potential to study the influence of surfactants in vaporization and condensation processes from a numerical point of view the proposed model poses significant challenges to existing discretization methods including stiffness in space and time internal and boundary layers as well as higher order partial differential operators to overcome these challenges we propose algorithms based on isogeometric analysis which permit an accurate and efficient discretization finally we illustrate the viability of the theoretical framework and the effectiveness of our algorithms by solving several numerical problems in two and three dimensions

four phase checkerboard composites two dimensional periodic rectangular checkerboard media are considered in the situation where mean fluxes are prescribed across the structure the closed form solution is obtained in the general case where the checkerboard is constructed using four rectangular cells each having a different constant resistivity this four cell structure then repeats doubly periodically to cover the whole plane this general solution is then used to calculate the effective properties thus this four phase checkerboard encapsulates many limiting and special cases as a starting point we develop a concise closed form solution to a basic problem involving four joined quarter planes each of a different resistivity subsequent manipulations yield solutions to problems posed in increasingly convoluted domains while retaining the essentially simple structure found for joined quarter planes

an unfitted hp adaptive finite element method based on hierarchical b splines for interface problems of complex geometry generating finite element discretizations with direct interface parameterizations constitutes a considerable computational expense in case of complex interface geometries the paper at hand introduces a b spline finite element method which circumvents parameterization of interfaces and offers fast and easy meshing irrespective of the geometric complexity involved its core idea is the adaptive approximation of discontinuities by hierarchical grid refinement which adds several levels of local basis functions in the close vicinity of interfaces but unfitted to their exact location so that a simple regular grid of knot span elements can be maintained numerical experiments show that an hp refinement strategy which simultaneously increases the polynomial degree of the b spline basis and the levels of refinement around interfaces achieves exponential rates of convergence despite the presence of discontinuities it is also demonstrated that the hierarchical b spline fem can be used to transfer the recently introduced finite cell concept to geometrically nonlinear problems its computational performance imposition of unfitted boundary conditions and fast hierarchical grid generation are illustrated for a set of benchmark problems in one two and three dimensions and the advantages of the regular grid approach for complex geometries are demonstrated by the geometrically nonlinear simulation of a voxel based foam composite

isogeometric schwarz preconditioners for linear elasticity systems isogeometric schwarz preconditioners are constructed and analyzed for both compressible elasticity in primal formulation and almost incompressible elasticity in mixed formulation these preconditioners require the solution of local elasticity problems on overlapping subdomains forming a decomposition of the problem domain and the solution of a coarse elasticity problem associated with the subdomain coarse mesh an h analysis of the preconditioner for the primal formulation of compressible elasticity yields an optimal convergence rate bound that is scalable in the number of subdomains and is linear in the ratio between subdomain and overlap sizes extensive numerical experiments in 2d and 3d confirm this theoretical bound and show that an analogous bound holds for the mixed formulation of almost incompressible elasticity the numerical tests also show the good preconditioner performance with respect to the polynomial degree p and regularity k of the isogeometric basis functions as well as with respect to the presence of discontinuous elastic coefficients in composite materials and to domain deformation

finite element and boundary element modelling for the acoustic wave transmission in mean flow medium the frequency characteristics of the acoustic wave transmission in a medium with mean flow are considered one approach is to solve the helmholtz equation with mean flow medium in original co ordinates which is directly discretized for the one dimensional and the axisymmetric fem another approach is to transform the equation into the standard helmholtz equation which is discretized for the axisymmetric fem and the three dimensional bem the numerical models are examined first for a straight circular duct the solutions by the numerical approaches are compared with the analytical solution the examination is then extended to the case when the mean flow is locally present in the muffler with expansion chamber to model the spatial mean flow in the bem model the partitioned domain approach is also developed no shear effect between the two regions are included

exact solution of a boundary value problem for a rectangular checkerboard field a class of two phase composite materials with a biperiodic structure is investigated by the methods of complex analysis two interface conditions continuity of normal component of a desired vector w and tangential component of w at the contact boundary as well as the double periodicity condition are involved in rigorous form the exact analytic solution of the corresponding generalized riemann boundary value problem is obtained the explicit values of the effective parameters namely effective resistivity and dissipation of energy of an elementary cell and resistivities along the symmetry axes are calculated in closed analytic form the coincidence of our formulae with the well known effective resistivity conductivity formula of keller 1964 dykhne 1970 and mendelson 1975 and the dissipation formula of dykhne 1970 is shown in the case of square checkerboard field the keller 1963 identity is generalized for the heterogeneous structure studied

a three phase tessellation solution and effective properties two dimensional doubly periodic three phase structures are considered in the situation where mean fluxes are applied across the structure the approach is to utilize complex variables and to use a mapping that reduces the doubly periodic problem to a m uch simpler one involving joined sectors this is a model composite structure in electrostatics and mathematically analogous areas such as porous media anti plane elasticit y heat conduction and we find various eectiv e parameters and investigate limiting cases the structure is also amenable to asymptotic methods in the case of highly varying composition and we provide these solutions partly as a check upon our analysis and partly as they are useful in their own right

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

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

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

development of a non reflecting boundary condition for multidimensional nonlinear duct acoustic computation the quality of any time domain nonlinear cfd simulation of silencers is determined by the modeling procedure used for the anechoic termination at the boundary end of the computational domain in this study a novel anechoic boundary condition based on the characteristic theory combined with the linear relaxation method has been developed in an open source cfd code wave propagation in ducts and mufflers has been analyzed and the effect of the damping properties of the boundary condition has been studied the study shows that the explicit damping of the solution vector produces good performance at all the frequencies with acoustic waves having varying incidence to the boundary also results indicate that numerical reflection of acoustic waves with high incident angles is reduced if the damping is applied at the boundary within the semi implicit solution of the governing equations finally the boundary condition has been applied together with a nonlinear solver to evaluate the acoustic attenuation performance of circular flow reversing chambers and single plug perforated mufflers with and without mean flow velocity of the gas code validation has been carried out against experimental data

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

spline based meshfree method summary in the present work the spline based meshfree method sbmfm is presented the sbmfm uses spline basis functions for field variables and boundary representation because of the inherent properties of splines that is nurbs nonuniform rational b splines the analysis domain used in the sbmfm has its own parametric domain unlike the classical concepts in the computer aided design cad community in which the nurbs patches are used to model the body the nurbs patch is treated as a background mesh in the present work the real analysis domain is created by trimming operation so that the domain boundaries are described by spline curves only on a single nurbs patch building the geometry in this way enables a convenient boundary representation of the domains because direct manipulation of spline curves becomes possible from a deformation point of view the mesh distortion problem is effectively avoided even if the analysis domain undergoes a large deformation in the sbmfm boundary spline curves freely move with complete independence from the quadrilateral background nurbs patch the so called element connectivity is unnecessary for geometry description in this paper geometrically nonlinear problems are considered implementation strategy is presented and the effectiveness of the present sbmfm is shown through various numerical examples copyright 2012 john wiley sons ltd

quasi banach spaces of almost universal disposition we show that for each p 0 1 p 0 1 there exists a separable p banach space gpgp of almost universal disposition that is having the following extension property for each e 0e 0 and each isometric embedding g x yg x y where y is a finite dimensional p banach space and x is a subspace of gpgp there is an e isometry f y gpf y gp such that x f g x x f g x for all x xx x such a space is unique up to isometries does contain an isometric copy of each separable p banach space and has the remarkable property of being locally injective amongst p banach spaces we also present a nonseparable generalization which is of universal disposition for separable spaces and separably injective no separably injective p banach space was previously known for p 1p 1

a nitsche type formulation and comparison of the most common domain decomposition methods in isogeometric analysis this paper provides a detailed elaboration and assessment of the most common domain decomposition methods for their application in isogeometric analysis the methods comprise a penalty approach lagrange multiplier methods and a nitsche type method for the nitsche method a new stabilized formulation is developed in the context of isogeometric analysis to guarantee coercivity all these methods are investigated on problems of linear elasticity and eigenfrequency analysis in 2d in particular focus is put on non uniform rational b spline patches which join nonconformingly along their common interface thus the application of isogeometric analysis is extended to multi patches which can have an arbitrary parametrization on the adjacent edges moreover it has been shown that the unique properties provided by isogeometric analysis that is high order functions and smoothness across the element boundaries carry over for the analysis of multiple domains copyright 2013 john wiley sons ltd

isogeometric analysis for geometric modelling and acoustic attenuation performances of reactive mufflers abstract in the present study the acoustic attenuation performances of mufflers are predicted by employing three dimensional isogeometric analysis iga and multi patch technique for the first time circular elliptical and conical reactive expansion chamber mufflers are exactly parameterized by non uniform rational b splines nurbs and the helmholtz equation governing the interior acoustic field is also solved by nurbs compared with traditional finite element method fem which is rooted in lagrange and hermite functions iga can effectively avoid the time consuming meshing and totally preserve exact geometrical information during the modelling analysis process the computational effectiveness of multi patch iga for calculating transmission loss is verified by comparing with experimental approach boundary element method and other numerical methods from available research articles the effects of geometry such as shapes of the expansion cavity and lengths of the extended inlet outlet are discussed in detail

point collocation scheme in silencers with temperature gradient and mean flow this work presents a mathematical approach based on the point collocation technique to compute the transmission loss of perforated dissipative silencers with transversal temperature gradients and mean flow three dimensional wave propagation is considered in silencer geometries with arbitrary but axially uniform cross section to reduce the computational requirements of a full multidimensional finite element calculation a method is developed combining axial and transversal solutions of the wave equation first the finite element method is employed in a two dimensional problem to extract the eigenvalues and associated eigenvectors for the silencer cross section mean flow as well as transversal temperature gradients and the corresponding thermal induced material heterogeneities are included in the model in addition an axially uniform temperature field is taken into account its value being the inlet outlet average a point collocation technique is then used to match the acoustic fields pressure and axial acoustic velocity at the geometric discontinuities between the silencer chamber and the inlet and outlet pipes transmission loss predictions are compared favorably with a general three dimensional finite element approach offering a reduction in the computational effort sound attenuation in perforated dissipative silencers including temperature gradients and mean flow is computed a two dimensional finite element eigenvalue problem is solved for a silencer cross section with transversal thermal variations a point collocation scheme is presented to match the acoustic pressure and axial velocity at the silencer geometrical discontinuities a significant reduction in the computational requirements is obtained compared to a full three dimensional finite element approach

a local solution approach for adaptive hierarchical refinement in isogeometric analysis abstract adaptively refining the state variables and fast solving the equations are two key issues in isogeometric analysis currently existing research more focuses on the first issue while the efficiency issue is not elaborated when the overall equations are solved at each refinement step this article develops an approach to promote the efficiency by locally solving the equations that govern the refined variables at each refinement step here hierarchical refinement of the state variables in nurbs is adopted and transformational matrix that transforms the basis functions from the current level to the refined level is utilized to handle the relations between the equations at different levels by dividing the coefficient matrix into blocks and estimating the error of state variables for different blocks we represent the global solution method in a hierarchical structure and separately construct and solve equations to obtain the solution for each level in the structure we validate the proposed method by some steady heat transfer and structural analysis problems in domains with different dimensionality we have compared the numerical result accuracy as well as the time cost of the proposed method with those of current available refinement approaches and it turns out to be very promising

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

the value of continuity refined isogeometric analysis and fast direct solvers abstract we propose the use of highly continuous finite element spaces interconnected with low continuity hyperplanes to maximize the performance of direct solvers starting from a highly continuous isogeometric analysis iga discretization we introduce c 0 separators to reduce the interconnection between degrees of freedom in the mesh by doing so both the solution time and best approximation errors are simultaneously improved we call the resulting method refined isogeometric analysis riga to illustrate the impact of the continuity reduction we analyze the number of floating point operations flops computational times and memory required to solve the linear system obtained by discretizing the laplace problem with structured meshes and uniform polynomial orders theoretical estimates demonstrate that an optimal continuity reduction may decrease the total computational time by a factor between p 2 and p 3 with p being the polynomial order of the discretization numerical results indicate that our proposed refined isogeometric analysis delivers a speed up factor proportional to p 2 in a 2 d mesh with four million elements and p 5 the linear system resulting from riga is solved 22 times faster than the one from highly continuous iga in a 3 d mesh with one million elements and p 3 the linear system is solved 15 times faster for the refined than the maximum continuity isogeometric analysis

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

divergence conforming discretization for stokes problem on locally refined meshes using lr b splines abstract to solve the incompressible flow problems using isogeometric analysis the div compatible spline spaces were originally introduced by buffa et al 2011 and later developed by evans 2011 in this paper we extend the div compatible spline spaces with local refinement capability using locally refined lr b splines over rectangular domains we argue that the spline spaces generated on locally refined meshes will satisfy compatibility provided they span the entire function spaces as governed by mourrain 2014 dimension formula we will in this work use the structured refined lr b splines as introduced by johannessen et al 2014 further we consider these div compatible lr b spline spaces to approximate the velocity and pressure fields in mixed discretization for stokes problem and a set of standard benchmark tests are performed to show the stability efficiency and the conservation properties of the discrete velocity fields in adaptive isogeometric analysis

a hierarchical approach to the a posteriori error estimation of isogeometric kirchhoff plates and kirchhoff love shells abstract this work focuses on the development of a posteriori error estimates for fourth order elliptic partial differential equations in particular we propose a novel algorithm to steer an adaptive simulation in the context of kirchhoff plates and kirchhoff love shells by exploiting the local refinement capabilities of hierarchical b splines the method is based on the solution of an auxiliary residual like variational problem formulated by means of a space of localized spline functions this space is characterized by c 1 continuous b splines with compact support on each active element of the hierarchical mesh we demonstrate the applicability of the proposed estimator to kirchhoff plates and kirchhoff love shells by studying several benchmark problems which exhibit both smooth and singular solutions in all cases we obtain optimal asymptotic rates of convergence for the error measured in the energy norm and an excellent approximation of the true error

adaptive isogeometric analysis by local h refinement with t splines isogeometric analysis based on non uniform rational b splines nurbs as basis functions preserves the exact geometry but suffers from the drawback of a rectangular grid of control points in the parameter space which renders a purely local refinement impossible this paper demonstrates how this difficulty can be overcome by using t splines instead t splines allow the introduction of so called t junctions which are related to hanging nodes in the standard fem obeying a few straightforward rules rectangular patches in the parameter space of the t splines can be subdivided and thus a local refinement becomes feasible while still preserving the exact geometry furthermore it is shown how state of the art a posteriori error estimation techniques can be combined with refinement by t splines numerical examples underline the potential of isogeometric analysis with t splines and give hints for further developments

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

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

a hybrid variational collocation immersed method for fluid structure interaction using unstructured t splines summary we present a hybrid variational collocation immersed and fully implicit formulation for fluid structure interaction fsi using unstructured t splines in our immersed methodology we define an eulerian mesh on the whole computational domain and a lagrangian mesh on the solid domain which moves arbitrarily on top of the eulerian mesh mathematically the problem reduces to solving three equations namely the linear momentum balance mass conservation and a condition of kinematic compatibility between the lagrangian displacement and the eulerian velocity we use a weighted residual approach for the linear momentum and mass conservation equations but we discretize directly the strong form of the kinematic relation deriving a hybrid variational collocation method we use t splines for both the spatial discretization and the information transfer between the eulerian mesh and the lagrangian mesh t splines offer us two main advantages against non uniform rational b splines they can be locally refined and they are unstructured the generalized method is used for the time discretization we validate our formulation with a common fsi benchmark problem achieving excellent agreement with the theoretical solution an example involving a partially immersed solid is also solved the numerical examples show how the use of t junctions and extraordinary nodes results in an accurate efficient and flexible method copyright 2015 john wiley sons ltd

a splitting theorem for the space of smooth functions we will show using purely linear functional analytic methods that each exact complex0 f c s0 p0 c s1 p1 c s2 p2 wherepiare matrices of convolution operators in particular linear differential operators with constant coefficients splits fromp1on in the category of topological vector spaces moreover we characterize when these complexes split completely the obtained result covers all known particular cases as well as some more general cases independently of the analytic nature ofpi the result is a consequence of studying splitting of short exact sequences0 f jx qg 0 wheref x gare graded frechet spaces i e frechet spaces equipped with fixed reduced spectra representing them andj qare consistent with the graded structures we characterize splitting of in case i forgare spacesc where is an open subset in rn or ii gis the space of all sequences

three dimensional manifolds kleinian groups and hyperbolic geometry 1 a conjectural picture of 3 manifolds a major thrust of mathematics in the late 19th century in which poincare had a large role was the uniformization theory for riemann surfaces that every conformai structure on a closed oriented surface is represented by a riemannian metric of constant curvature for the typical case of negative euler characteristic genus greater than 1 such a metric gives a hyperbolic structure any small neighborhood in the surface is isometric to a neighborhood in the hyperbolic plane and the surface itself is the quotient of the hyperbolic plane by a discrete group of motions the exceptional cases the sphere and the torus have spherical and euclidean structures three manifolds are greatly more complicated than surfaces and i think it is fair to say that until recently there was little reason to expect any analogous theory for manifolds of dimension 3 or more except perhaps for the fact that so many 3 manifolds are beautiful the situation has changed so that i feel fairly confident in proposing the

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

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

automatically stable discontinuous petrov galerkin methods for stationary transport problems quasi optimal test space norm we investigate the application of the discontinuous petrov galerkin dpg finite element framework to stationary convection diffusion problems in particular we demonstrate how the quasi optimal test space norm improves the robustness of the dpg method with respect to vanishing diffusion we numerically compare coarse mesh accuracy of the approximation when using the quasi optimal norm the standard norm and the weighted norm our results show that the quasi optimal norm leads to more accurate results on three benchmark problems in two spatial dimensions we address the problems associated to the resolution of the optimal test functions with respect to the quasi optimal norm by studying their convergence numerically in order to facilitate understanding of the method we also include a detailed explanation of the methodology from the algorithmic point of view

streamline upwind petrov galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier stokes equations abstract a new finite element formulation for convection dominated flows is developed the basis of the formulation is the streamline upwind concept which provides an accurate multidimensional generalization of optimal one dimensional upwind schemes when implemented as a consistent petrov galerkin weighted residual method it is shown that the new formulation is not subject to the artificial diffusion criticisms associated with many classical upwind methods the accuracy of the streamline upwind petrov galerkin formulation for the linear advection diffusion equation is demonstrated on several numerical examples the formulation is extended to the incompressible navier stokes equations an efficient implicit pressure explicit velocity transient algorithm is developed which accomodates several treatments of the incompressibility constraint and allows for multiple iterations within a time step the effectiveness of the algorithm is demonstrated on the problem of vortex shedding from a circular cylinder at a reynolds number of 100