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

an isogeometric collocation approach for bernoulli euler beams and kirchhoff plates abstract in this paper iga collocation methods are for the first time introduced for the solution of thin structural problems described by the bernoulli euler beam and kirchhoff plate models in particular a precise description of the proposed methods of the relevant implementation details and of the strategy to efficiently deal with different combinations of boundary conditions is given finally several numerical experiments confirm that the proposed formulations represent an efficient and geometrically flexible tool for the simulation of thin structures

isogeometric collocation for elastostatics and explicit dynamics we extend the development of collocation methods within the framework of isogeometric analysis iga to multi patch nurbs configurations various boundary and patch interface conditions and explicit dynamic analysis the methods developed are higher order accurate stable with no hourglass modes and efficient in that they require a minimum number of quadrature evaluations the combination of these attributes has not been obtained previously within standard finite element analysis

isogeometric collocation neumann boundary conditions and contact abstract isogeometric collocation methods have been proposed recently and their accuracy and efficiency demonstrated for elastostatics and explicit dynamics this paper addresses two important aspects in the development of the isogeometric collocation technology namely the imposition of neumann boundary conditions and the enforcement of contact constraints which are both treated within the same framework it is shown that the strong imposition of neumann boundary conditions may lead to a significant loss of accuracy in some situations in particular when non uniform meshes are used two possible remedies are proposed to restore the desired level of accuracy while keeping the computational cost virtually unchanged i e a hybrid collocation galerkin approach and an enhanced collocation ec approach a frictionless contact formulation suitable for the collocation framework is further proposed and shown to pass the contact patch test to machine precision when combined with the ec approach the formulation is shown to deliver accurate results and to perform robustly also for highly non uniform meshes for all the collocation formulations contact pressures are greater than or equal to zero pointwise in contrast with standard lagrange finite elements

explicit isogeometric collocation for the dynamics of three dimensional beams undergoing finite motions abstract we initiate the study of three dimensional shear deformable geometrically exact beam dynamics through explicit isogeometric collocation methods the formulation we propose is based on a natural combination of the chosen finite rotations representation with an explicit geometrically consistent lie group time integrator we focus on extending the integration scheme originally proposed for rigid body dynamics to our nonlinear initial boundary value problem where special attention is required by neumann boundary conditions the overall formulation is simple and only relies on a geometrically consistent procedure to compute the internal forces once control angular and linear accelerations of the beam cross sections are obtained from the previous time step the capabilities of the method are shown through numerical applications involving very large displacements and rotations and different boundary conditions

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

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

isogeometric collocation on planar multi patch domains abstract we present an isogeometric framework based on collocation to construct a c 2 smooth approximation of the solution of the poisson s equation over planar bilinearly parameterized multi patch domains the construction of the used globally c 2 smooth discretization space for the partial differential equation is simple and works uniformly for all possible multi patch configurations the basis of the c 2 smooth space can be described as the span of three different types of locally supported functions corresponding to the single patches edges and vertices of the multi patch domain for the selection of the collocation points which is important for the stability and convergence of the collocation problem two different choices are numerically investigated the first approach employs the tensor product greville abscissae as collocation points and shows for the multi patch case the same convergence behavior as for the one patch case 1 which is suboptimal in particular for odd spline degree the second approach generalizes the concept of superconvergent points from the one patch case cf 2 3 4 to the multi patch case again these points possess better convergence properties than greville abscissae in case of odd spline degree

three dimensional simulation of unstable gravity driven infiltration of water into a porous medium infiltration of water in dry porous media is subject to a powerful gravity driven instability although the phenomenon of unstable infiltration is well known its description using continuum mathematical models has posed a significant challenge for several decades the classical model of water flow in the unsaturated flow the richards equation is unable to reproduce the instability here we present a computational study of a model of unsaturated flow in porous media that extends the richards equation and is capable of predicting the instability and captures the key features of gravity fingering quantitatively the extended model is based on a phase field formulation and is fourth order in space the new model poses a set of challenges for numerical discretizations such as resolution of evolving interfaces stiffness in space and time treatment of singularly perturbed equations and discretization of higher order spatial partial differential operators we develop a numerical algorithm based on isogeometric analysis a generalization of the finite element method that permits the use of globally smooth basis functions leading to a simple and efficient discretization of higher order spatial operators in variational form we illustrate the accuracy efficiency and robustness of our method with several examples in two and three dimensions in both homogeneous and strongly heterogeneous media we simulate for the first time unstable gravity driven infiltration in three dimensions and confirm that the new theory reproduces the fundamental features of water infiltration into a porous medium our results are consistent with classical experimental observations that demonstrate a transition from stable to unstable fronts depending on the infiltration flux

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

reduced bezier element quadrature rules for quadratic and cubic splines in isogeometric analysis abstract we explore the use of various element based reduced quadrature strategies for bivariate and trivariate quadratic and cubic spline elements used in isogeometric analysis the rules studied encompass tensor product gauss and gauss lobatto rules and certain so called monomial rules that do no possess a tensor product structure the objective of the study is to determine quadrature strategies which enjoy the same accuracy and stability behavior as full gauss quadrature but with significantly fewer quadrature points several cases emerge that satisfy this objective and also demonstrate superior efficiency compared with standard c 0 continuous finite elements of the same order

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

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

spherical nilpotent orbits and the kostant sekiguchi correspondence let g be a connected linear semisimple lie group with lie algebra g and let k c aut p c be the complexified isotropy representation at the identity coset of the corresponding symmetric space the kostant sekiguchi correspondence is a bijection between the nilpotent k c orbits in p c and the nilpotent g orbits in g we show that this correspondence associates each spherical nilpotent k c orbit to a nilpotent g orbit that is multiplicity free as a hamiltonian k space the converse also holds

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

liquid vapor phase transition thermomechanical theory entropy stable numerical formulation and boiling simulations abstract we develop a new continuum mechanics modeling framework for liquid vapor flows with particular focus on the van der waals fluid by invoking microforce theory the coleman noll procedure is generalized to derive consistent constitutive relations in the presence of non local effects a new thermodynamically consistent algorithm for the van der waals model is designed using functional entropy variables and a new temporal scheme employing a family of new quadrature rules we show that the resulting fully discrete scheme is unconditionally stable in entropy and second order time accurate isogeometric analysis is utilized for spatial discretization the analytical properties of the formulation are corroborated by benchmark problems three sets of application problems are simulated to demonstrate the capability of the model and the algorithm our methodology provides a particularly useful predictive tool for boiling flows

superconvergent gradient smoothing meshfree collocation method abstract a superconvergent meshfree collocation method with smoothed nodal gradients is presented in this method the first order smoothed gradients of meshfree shape function are constructed through a meshfree interpolation of the standard derivatives of meshfree shape function while the second order smoothed gradients are computed through directly differentiating the first order smoothed gradients it is noted that the second order smoothed gradients evaluated at meshfree nodes can be conveniently expressed as two successive first order gradient smoothing operations on the meshfree shape function which facilitates a trivial numerical implementation subsequently an employment of the second order smoothed gradients in the strong form of a given problem leads to the gradient smoothing meshfree collocation method purely using nodes as the collocation points based upon a local truncation error analysis it is systematically shown that the proposed meshfree collocation method yields superconvergent solutions for odd degree basis functions a key ingredient attributed to this superconvergence property is that the second order smoothed gradients meet the consistency conditions which go one order beyond the original basis degree of meshfree approximation another interesting fact is that the present formulation enables a convergent collocation scheme when the linear basis function is used in meshfree approximation which is non feasible in the conventional collocation formulation the effectiveness of the proposed methodology is validated by numerical examples for both potential and elasticity problems numerical results well demonstrate the superconvergence and higher efficiency of the present gradient smoothing meshfree collocation method

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

isogeometric collocation using analysis suitable t splines of arbitrary degree abstract this paper deals with the use of analysis suitable t splines of arbitrary degree in combination with isogeometric collocation methods for the solution of second and fourth order boundary value problems in fact analysis suitable t splines appear to be a particularly efficient locally refinable basis for isogeometric collocation able to conserve the cost of only one point evaluation per degree of freedom typical of standard nurbs based isogeometric collocation furthermore t splines allow to easily create highly non uniform meshes without introducing elements with high aspect ratios this makes it possible to avoid the numerical instabilities that may arise in the case of problems characterized by reduced regularity when neumann boundary conditions are imposed in strong form and elements with high aspect ratio are used the local refinement properties of t splines can be also successfully exploited to approximate problems where point loads are applied finally several numerical tests are herein presented in order to confirm all the above mentioned features as well as the good overall convergence properties of the combination of isogeometric collocation and analysis suitable t splines

an isogeometric collocation method using superconvergent points abstract we develop an iga collocation method modified by collocating at points other than the standard greville abscissae the method is related to orthogonal collocation used for solving differential equations and to the superconvergence theory therefore we refer to this method as super collocation iga sc by carefully choosing the collocation points it can be seen that the iga sc converges in the first derivative energy norms at rates similar to that of the galerkin solution this is different from the collocation at greville abscissae iga c where the convergence in energy norm for odd polynomial degrees is typically suboptimal the method is tested on 1d 2d and 3d numerical examples in which it is compared to iga c and galerkin s method iga g the comparison includes a detailed cost vs accuracy analysis which shows an improved efficiency of the proposed method in particular for odd polynomial degrees

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

a collocated c0 finite element method reduced quadrature perspective cost comparison with standard finite elements and explicit structural dynamics summary we demonstrate the potential of collocation methods for efficient higher order analysis on standard nodal finite element meshes we focus on a collocation method that is variationally consistent and geometrically flexible converges optimally embraces concepts of reduced quadrature and leads to symmetric stiffness and diagonal consistent mass matrices at the same time it minimizes the evaluation cost per quadrature point thus reducing formation and assembly effort significantly with respect to standard galerkin finite element methods we provide a detailed review of all components of the technology in the context of elastodynamics that is weighted residual formulation nodal basis functions on gauss lobatto quadrature points and symmetrization by averaging with the ultra weak formulation we quantify potential gains by comparing the computational efficiency of collocated and standard finite elements in terms of basic operation counts and timings our results show that collocation is significantly less expensive for problems dominated by the formation and assembly effort such as higher order elastostatic analysis furthermore we illustrate the potential of collocation for efficient higher order explicit dynamics throughout this work we advocate a straightforward implementation based on simple modifications of standard finite element codes we also point out the close connection to spectral element methods where many of the key ideas are already established copyright 2014 john wiley sons ltd

superconvergence in the generalized finite element method in this paper we address the problem of the existence of superconvergence points of approximate solutions obtained from the generalized finite element method gfem of a neumann elliptic boundary value problem gfem is a galerkin method that uses non polynomial shape functions and was developed in babuska et al in siam j numer anal 31 945 981 1994 babuska et al in int j numer meth eng 40 727 758 1997 melenk and babuska in comput methods appl mech eng 139 289 314 1996 in particular we show that the superconvergence points for the gradient of the approximate solution are the zeros of a system of non linear equations this system does not depend on the solution of the boundary value problem for approximate solutions with second derivatives we have also characterized the superconvergence points of the second derivatives of the approximate solution as the roots of a system of non linear equations we note that smooth generalized finite element approximation is easy to construct

local maximum entropy approximation schemes a seamless bridge between finite elements and meshfree methods we present a one parameter family of approximation schemes which we refer to as local maximum entropy approximation schemes that bridges continuously two important limits delaunay triangulation and maximum entropy max ent statistical inference local max ent approximation schemes represent a compromise in the sense of pareto optimality between the competing objectives of unbiased statistical inference from the nodal data and the definition of local shape functions of least width local max ent approximation schemes are entirely defined by the node set and the domain of analysis and the shape functions are positive interpolate affine functions exactly and have a weak kronecker delta property at the boundary local max ent approximation may be regarded as a regularization or thermalization of delaunay triangulation which effectively resolves the degenerate cases resulting from the lack or uniqueness of the triangulation local max ent approximation schemes can be taken as a convenient basis for the numerical solution of pdes in the style of meshfree galerkin methods in test cases characterized by smooth solutions we find that the accuracy of local max ent approximation schemes is vastly superior to that of finite elements

accurate efficient and iso geometrically flexible collocation methods for phase field models we propose new collocation methods for phase field models our algorithms are based on isogeometric analysis a new technology that makes use of functions from computational geometry such as for example non uniform rational b splines nurbs nurbs exhibit excellent approximability and controllable global smoothness and can represent exactly most geometries encapsulated in computer aided design cad models these attributes permitted us to derive accurate efficient and geometrically flexible collocation methods for phase field models the performance of our method is demonstrated by several numerical examples of phase separation modeled by the cahn hilliard equation we feel that our method successfully combines the geometrical flexibility of finite elements with the accuracy and simplicity of pseudo spectral collocation methods and is a viable alternative to classical collocation methods

functional entropy variables a new methodology for deriving thermodynamically consistent algorithms for complex fluids with particular reference to the isothermal navier stokes korteweg equations we propose a new methodology for the numerical solution of the isothermal navier stokes korteweg equations our methodology is based on a semi discrete galerkin method invoking functional entropy variables a generalization of classical entropy variables and a new time integration scheme we show that the resulting fully discrete scheme is unconditionally stable in energy second order time accurate and mass conservative we utilize isogeometric analysis for spatial discretization and verify the aforementioned properties by adopting the method of manufactured solutions and comparing coarse mesh solutions with overkill solutions various problems are simulated to show the capability of the method our methodology provides a means of constructing unconditionally stable numerical schemes for nonlinear non convex hyperbolic systems of conservation laws

effects of parameterization and knot placement techniques on primal and mixed isogeometric collocation formulations of spatial shear deformable beams with varying curvature and torsion abstract we present a displacement based and a mixed isogeometric collocation iga c formulation for free form three dimensional shear deformable beams with high and rapidly varying curvature and torsion when such complex shapes are concerned the approach used to build the iga geometric model becomes relevant although iga c has been so far successfully applied to a wide range of problems the effects that different parameterization and knot placement techniques may have on the accuracy of collocation based formulations is still an unexplored field to fill this gap primal and mixed formulations are used combining two parameterization methods chord length and equally spaced with two knot placement techniques uniformly spaced and de boor with respect to the space varying frenet local frame we derive the strong form of the governing equations in a compact form through the definition of two matrix operators conveniently used to perform first and second order derivatives of the vector fields involved in the formulations this approach is very efficient and easy to implement within a collocation based scheme several challenging numerical experiments allow to test the different considered parameterizations and knot placement techniques revealing in particular that with the primal formulation an equally spaced parameterization is definitively the most recommended choice and it should always be used with an approximation degree of at least p 6 although some caution must be adopted when very high jacobians and small curvatures occur the same holds for the mixed formulation with the difference that p 4 is enough to yield accurate results

isogeometric collocation for large deformation elasticity and frictional contact problems abstract isogeometric collocation methods have been recently proposed as an alternative to standard galerkin approaches as they provide a significant reduction in computational cost for higher order discretizations in this work we explore the application of isogeometric collocation to large deformation elasticity and frictional contact problems we first derive the non linear governing equations for the elasticity problem with finite deformation kinematics and provide details on their consistent linearization some numerical examples demonstrate the performance of collocation in its basic and enhanced versions differing by the enforcement of neumann boundary conditions for problems with strong singularities enhanced collocation is shown to outperform basic collocation and to lead to a spatial convergence behavior very similar to galerkin whereas for weaker or no singularities enhanced and basic collocation may give very similar results a large deformation contact formulation is subsequently developed and tested in the frictional setting where collocation confirms the excellent performance already obtained for the frictionless case finally it is shown that the contact formulation in the collocation framework passes the contact patch test to machine precision in a three dimensional setting with arbitrarily inclined non matching discretizations thus outperforming most of the available contact formulations and all those with pointwise enforcement of the contact constraints

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

isogeometric collocation cost comparison with galerkin methods and extension to adaptive hierarchical nurbs discretizations abstract we compare isogeometric collocation with isogeometric galerkin and standard c 0 finite element methods with respect to the cost of forming the matrix and residual vector the cost of direct and iterative solvers the accuracy versus degrees of freedom and the accuracy versus computing time on this basis we show that isogeometric collocation has the potential to increase the computational efficiency of isogeometric analysis and to outperform both isogeometric galerkin and standard c 0 finite element methods when a specified level of accuracy is to be achieved with minimum computational cost we then explore an adaptive isogeometric collocation method that is based on local hierarchical refinement of nurbs basis functions and collocation points derived from the corresponding multi level greville abscissae we introduce the concept of weighted collocation that can be consistently developed from the weighted residual form and the two scale relation of b splines using weighted collocation in the transition regions between hierarchical levels we are able to reliably handle coincident collocation points that naturally occur for multi level greville abscissae the resulting method combines the favorable properties of isogeometric collocation and hierarchical refinement in terms of computational efficiency local adaptivity robustness and straightforward implementation which we illustrate by numerical examples in one two and three dimensions

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

a robust all at once multigrid method for the stokes control problem in this paper we present an all at once multigrid method for a distributed stokes control problem velocity tracking problem for solving such a problem we use the fact that the solution is characterized by the optimality system karush kuhn tucker system the discretized optimality system is a large scale linear system whose condition number depends on the grid size and on the choice of the regularization parameter forming a part of the problem recently block diagonal preconditioners have been proposed which allow to solve the problem using a krylov space method with convergence rates that are robust in both the grid size and the regularization parameter or cost parameter in the present paper we develop an all at once multigrid method for a stokes control problem and show robust convergence more precisely we show that the method converges with rates which are bounded away from one by a constant which is independent of the grid size and the choice of the regularization or cost parameter

isogeometric large deformation frictionless contact using t splines t spline based isogeometric analysis is applied to frictionless contact problems between deformable bodies in the context of large deformations the continuum is discretized with cubic t splines and cubic nurbs a gauss point to surface formulation is combined with the penalty method to treat the contact constraints in the discretized setting it is demonstrated that analysis suitable t splines coupled with local refinement accurately approximate contact pressures with far fewer degrees of freedom than nurbs both two and three dimensional examples are presented additionally all t spline analysis models are generated using commercially available t spline modeling software without intermediate mesh generation or geometry clean up steps

the variational collocation method we propose the variational collocation method for the numerical solution of partial differential equations the conceptual basis is the establishment of a direct connection between the galerkin method and the classical collocation methods with the perspective of achieving the accuracy of the former with a computational cost of one point evaluation per degree of freedom as in the latter variational collocation requires a discrete space constructed by smooth and pointwise non negative basis functions which makes the approach immediately applicable to isogeometric analysis and some meshfree methods in this paper we concentrate on isogeometric analysis and demonstrate that there exists a set of points such that collocation of the strong form at these points produces the galerkin solution exactly we provide an estimate of these points and show that applying isogeometric collocation at the estimated points completely solves the well known odd even discrepancy in the order of spatial convergence we demonstrate the potential of variational collocation with examples of linear and non linear elasticity as well as kirchhoff plates

remarks on rate constitutive equations for finite deformation problems computational implications it is explicitly shown that if the spatial elasticity tensor of an elastic material is taken as isotropic for all possible configurations then its coefficients cannot be constants they must depend nontrivially on the jacobian determinant of the deformation gradient moreover the assumption typically made for computational purposes that its coefficients remain constant for all possible configurations is incompatible with elasticity it is further shown that an assumption widely used in the computational literature in the context of finite deformation plasticity namely relating an objective stress rate to the rate of deformation tensor through a fourth rank constant isotropic tensor is also incompatible with elasticity thus furnishing an example of an hypoelastic material which is not elastic

isogeometric divergence conforming b splines for the unsteady navier stokes equations divergence conforming b splines are developed for application to the incompressible navier stokes equations on geometrically mapped domains these enable smooth pointwise divergence free solutions and thus satisfy mass conservation in the strongest possible sense semi discrete methods based on divergence conforming b splines are shown to conserve linear and angular momentum and satisfy balance laws for energy vorticity enstrophy and helicity these are geometric structure preserving quantities and numerical simulations that are sensitive to them are shown to be qualitatively correct and quantitatively accurate the methods developed are anticipated to open new doors to the practical calculation of complex flows and to studies of their physical behavior