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

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

geometrically exact covariant approach for contact between curves for curve to curve contact situation a geometrically exact description in a covariant form is developed the contact kinematics the variational formulation and the constitutive relations for contact tractions are described in a specially defined curvilinear coordinate system based on the closest point projection cpp procedure the fundamental problems about the solvability of the cpp procedure for the curve to curve case are investigated in detail as well the introduced coordinate system is independent of the choice of a contacting curve this allows to describe all possible relative motions of both curves including not only normal and tangential interactions but also rotational interaction between curves representing e g circular cross sections of beams all necessary derivations for the iterative solution method such as linearization of weak forms and the return mapping schemes are fulfilled in a covariant form for the arbitrary distance between the curves this allows to apply the developed theory to both contact cases when bodies are contacting by their edges and to contact between beams another advantage is the complete independence concerning the order of approximation involved in finite element construction the numerical examples are chosen to verify the proposed theory and to compare different cases for both the beam to beam and the edge to edge contact cases as well as to illustrate the possibilities of the theory to describe relative contact kinematics

thermo mechanical contact problems on non matching meshes non matching meshes and domain decomposition techniques based on lagrange multipliers provide a flexible and efficient discretization technique for variational inequalities with interface constraints although mortar methods are well analyzed for variational inequalities its application to dynamic thermo mechanical contact problems with friction is still a field of active research in this work we extend the mortar approach for dynamic contact problems with coulomb friction to the thermo mechanical case we focus on the discretization and on algorithmic aspects of dynamic effects such as frictional heating and thermal softening at the contact interface more precisely we generalize the mortar concept of dual lagrange multipliers to non linear robin type interface conditions and apply local static condensation to eliminate the heat flux numerical examples in the two dimensional and the three dimensional setting illustrate the flexibility of the discretization on non matching meshes

a weighted point based formulation for isogeometric contact abstract in the context of isogeometric contact analysis surfaces of objects can be described smoothly due to the high continuity of the involved shape functions this facilitates construction of a continuous normal field which may be beneficial for modeling of contact problems this property allows for stable collocation of the virtual contact work using considerable less evaluation points than numerical integration while the approximation quality of the computational results is almost the same in this context a weighted point based method pts is introduced as an extension of the point to segment method pts proposed previously by matzen et al 2013 the weights enable transmission of contact stresses instead of contact forces along the contact surfaces in normal direction the non penetration condition is enforced by the lagrange multiplier method combined with the penalty method in tangential direction corresponding to the collocation weights a constant or nurbs lagrange multiplier field is introduced a comparison between the collocation based contact formulations pts pts and a mortar method is drawn showing that weighted collocation is able to achieve almost the same accuracy as the integration based mortar formulation

a modified diffusion coefficient technique for the convection diffusion equation a new modified diffusion coefficient mdc technique for solving convection diffusion equation is proposed the galerkin finite element discretization process is applied on the modified equation rather than the original one for a class of one dimensional convection diffusion equations we derive the modified diffusion coefficient analytically as a function of the equation coefficients and mesh size then prove that the discrete solution of this method coincides with the exact solution of the original equation for every mesh size and or equation coefficients the application of the derived analytic formula of mdc is extended for other classes of convection diffusion equations where the analytic formula is computed locally within each element according to the mesh size and the values of associated coefficients in each direction the numerical results of the proposed approach for two dimensional variable coefficients with boundary layers convection dominated problems show stable and accurate solutions even on coarse grids accordingly multigrid based solvers retain their efficient convergence rates

high order compact exponential finite difference methods for convection diffusion type problems a class of high order compact hoc exponential finite difference fd methods is proposed for solving one and two dimensional steady state convection diffusion problems the newly proposed hoc exponential fd schemes have nonoscillation property and yield high accuracy approximation solution as well as are suitable for convection dominated problems the o h 4 compact exponential fd schemes developed for the one dimensional 1d problems produce diagonally dominant tri diagonal system of equations which can be solved by applying the tridiagonal thomas algorithm for the two dimensional 2d problems o h 4 k 4 compact exponential fd schemes are formulated on the nine point 2d stencil and the line iterative approach with alternating direction implicit adi procedure enables us to deal with diagonally dominant tridiagonal matrix equations which can be solved by application of the one dimensional tridiagonal thomas algorithm with a considerable saving in computing time to validate the present hoc exponential fd methods three linear and nonlinear problems mostly with boundary or internal layers where sharp gradients may appear due to high peclet or reynolds numbers are numerically solved comparisons are made between analytical solutions and numerical results for the currently proposed hoc exponential fd methods and some previously published hoc methods the present hoc exponential fd methods produce excellent results for all test problems it is shown that besides including the excellent performances in computational accuracy efficiency and stability the present method has the advantage of better scale resolution the method developed in this article is easy to implement and has been applied to obtain the numerical solutions of the lid driven cavity flow problem governed by the 2d incompressible navier stokes equations using the stream function vorticity formulation

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

skew symmetric nitsche s formulation in isogeometric analysis dirichlet and symmetry conditions patch coupling and frictionless contact abstract a simple skew symmetric nitsche s formulation is introduced into the framework of isogeometric analysis iga to deal with various problems in small strain elasticity essential boundary conditions symmetry conditions for kirchhoff plates patch coupling in statics and in modal analysis as well as signorini contact conditions for linear boundary or interface conditions the skew symmetric formulation is parameter free for contact conditions it remains stable and accurate for a wide range of the stabilization parameter several numerical tests are performed to illustrate its accuracy stability and convergence performance we investigate particularly the effects introduced by nitsche s coupling including the convergence performance and condition numbers in statics as well as the extra outlier frequencies and corresponding eigenmodes in structural dynamics we present the hertz test the block test and a 3d self contact example showing that the skew symmetric nitsche s formulation is a suitable approach to simulate contact problems in iga

high accuracy solutions of incompressible navier stokes equations abstract in recent years we have developed high accuracy finite difference approximations for partial differential equations of elliptic type with particular emphasis on the convection diffusion equation these approximations are of compact type have a local truncation error of fourth order and allow the use of standard iterative schemes to solve the resulting systems of algebraic equations in this paper we extend these high accuracy approximations to the solution of navier stokes equations solutions are obtained for the model problem of driven cavity and are compared with solutions obtained using other approximations and those obtained by other authors it is discovered that the high order approximations do indeed produce high accuracy solutions and have a potential for use in solving important problems of viscous fluid flows

mortar based frictional contact formulation for higher order interpolations using the moving friction cone abstract finite element discretizations including contact usually apply the standard nts node to segment element thus a coupling with higher order solid elements leads to inconsistencies in the transmission of contact stresses one can circumvent this using the mortar method which includes a weak projection of the contact constraints in this paper we present a penalty formulation based on the mortar concept for two dimensional large deformation frictional contact the discretization of the contact surfaces contains quadratic approximations which can be used within a quadratic approximation of the solid elements to obtain a simple and efficient matrix formulation the moving friction cone algorithm is applied

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

a contact smoothing method for arbitrary surface meshes using nagata patches abstract this paper presents a contact surface smoothing method combined with the node to segment discretization technique to solve large deformation frictional contact problems between deformable bodies the nagata patch interpolation is used to smooth the surface mesh providing a master surface with quasi g 1 continuity between patches moreover the local support of the interpolation method allows to deal with surface meshes of arbitrary topology regular and irregular finite element discretizations as well as hybrid meshes the non physical oscillations in the contact force evolution induced by the faceted contact surface representation are reduced using the proposed smoothing method furthermore the smooth representation of the master surface allows a more accurately evaluation of the resulting stresses and forces while providing an important improvement in convergence behaviour four representative numerical examples are used to demonstrate the advantages of the proposed contact smoothing method the results show a significant improvement in the accuracy robustness and performance of the numerical simulations using the smoothing approach when compared with the piecewise faceted contact surface description

matrix dependent prolongations and restrictions in a blackbox multigrid solver multigrid methods are studied for the solution of linear systems resulting from the 9 point discretization of a general linear second order elliptic partial differential equation in two dimensions the rate of convergence of standard multigrid methods often deteriorates when the coefficients in the differential equation are discontinuous or when dominating first order terms are present these difficulties may be overcome by choosing the prolongation and restriction operators in a special way a novel way to do this is proposed as a result a blackbox solver written in standard fortran 77 has been developed numerical experiments for several hard test problems are described and comparison is made with other algorithms the standard mg method and a method introduced by kettler a significant improvement of robustness and efficiency is found

large deformation frictional contact mechanics continuum formulation and augmented lagrangian treatment absract a complete methodology for the formulation and solution of unilateral contact problem with non associated threshold friction between solids undergoing large deformations is presented the approach includes a continuum contact mechanics model combined with an augmented lagrangian method for treating the contact and friction inequality constraints the main developments are a precise statement of the incremental contact minimization problem the exact computations of the first and second variations of the normal gap and the tangential slip velocity and an extension of the augmented lagrangian formulation for frictional contact problems from the discrete to the continuum framework the corresponding saddle point functional first variation mixed virtual work and second variation exact linearization are given the development are illustrated by numerical simulations of frictional contact problems between rigid elastic and elastoplastic bodies a new severe benchmark for large slip frictional contact elements is also proposed

a family of simple two pass dual formulations for the finite element solution of contact problems simple stable unbiased algorithms for two pass two body contact have until now eluded researchers in computational mechanics this work examines the subject starting from an interpolation scheme proposed in an earlier work this method though stable and unbiased results in an exceedingly complex algorithm in three dimensions motivated by the need for simplicity a new formulation of the contact problem is introduced which results in a family of simple stable geometrically unbiased elements the starting point of these elements is any algorithm which is stable for the signorini problem a number of example problems are presented to compare the performance of the elements

an augmented lagrangian method for discrete large slip contact problems reference lmaf article 1993 001view record in web of science record created on 2005 09 14 modified on 2017 05 10

analysis of the sor iteration for the 9 point laplacian the sor iteration for solving linear systems of equations depends upon an overrelaxation factor omega a theory for determining omega was given by young iterative methods for solving partial differential equations of elliptic types trans amer math soc 76 1954 pp 92 111 for consistently ordered matrices here we determine the optimal omega for the 9 point stencil for the model problem of laplace s equation on a square we consider several orderings of the equations including the natural rowwise and multicolor orderings all of which lead to nonconsistently ordered matrices and find two equivalence classes of orderings with different convergence behavior and optimal omega s we compare our results for the natural rowwise ordering to those of garabedian estimation of the relaxation factor for small mesh size math comp 10 1956 pp 183 185 and explain why both results are in a sense correct even though they differ we also analyze a pseudo sor method for the model p

isogeometric finite element data structures based on bezier extraction of t splines we develop finite element data structures for t splines based on bezier extraction generalizing our previous work for nurbs as in traditional finite element analysis the extracted bezier elements are defined in terms of a fixed set of polynomial basis functions the so called bernstein basis the bezier elements may be processed in the same way as in a standard finite element computer program utilizing exactly the same data processing arrays in fact only the shape function subroutine needs to be modified while all other aspects of a finite element program remain the same a byproduct of the extraction process is the element extraction operator this operator localizes the topological and global smoothness information to the element level and represents a canonical treatment of t junctions referred to as hanging nodes in finite element analysis and a fundamental feature of t splines a detailed example is presented to illustrate the ideas copyright 2011 john wiley sons ltd

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

numerical simulation of 2d square driven cavity using fourth order compact finite difference schemes abstract fourth order compact finite difference schemes are employed with multigrid techniques to simulate the two dimensional square driven cavity flow with small to large reynolds numbers the governing navier stokes equation is linearized in streamfunction and vorticity formulation the fourth order compact approximation schemes are coupled with fourth order approximations for velocities and vorticity boundaries numerical solutions are obtained for square driven cavity flow at high reynolds numbers and are compared with solutions obtained by other researchers using other approximation methods

a mortar based frictional contact formulation for large deformations using lagrange multipliers abstract in this work a lagrange multiplier method is proposed to solve 2d coulomb frictional contact problems in the context of large deformations as the proposed formulation is based on the mortar method the constraints are imposed in a weak integral sense along the contact surface in order to compute the contact integrals we use a numerical integration based on the definition of the kinematical variables gap slip and their variations at the quadrature points the linearization of non linear equations virtual work and contact constraints is developed in order to apply a newton s method the examples show that the numerical integration still preserves the optimal rate of convergence of the finite element solution

a high order exponential adi scheme for two dimensional time fractional convection diffusion equations in this paper a high order compact exponential alternating direction implicit adi scheme for two dimensional fractional convection diffusion equations is proposed with o t 3 c h 1 4 h 2 4 accuracy where t h 1 h 2 are the temporal and spatial step sizes respectively the convergence of the finite difference scheme is studied using its matrix form by the energy method difficulty arising from the convection term is overcome by an analysis based on the eigenvalue decomposition of nonsymmetric tridiagonal matrices

numerical implementation of two nonconforming finite element methods for unilateral contact we consider the finite element approximation of the unilateral contact problem between elastic bodies we are interested in a practical problem which often occurs in finite element computations concerning two independently discretized bodies in unilateral contact it follows that the nodes of both bodies located on the contact surface do not fit together we present two different approaches in order to define unilateral contact on nonmatching meshes the first is an extension of the mortar finite element method to variational inequalities that defines the contact in a global way on the contrary the second one expresses local node on segment contact conditions in both cases the theoretical approximation properties are given then we implement and compare the two methods

isogeometric contact analysis geometric basis and formulation for frictionless contact this paper introduces an isogeometric framework for the numerical analysis of contact problems in this approach nurbs is utilized as basis for geometric representation and analysis the contact surfaces and the underlying solids are parameterized in an identical paradigm the use of nurbs geometry eliminates at the onset the geometric discontinuities induced by the traditional faceted approximation contact detection is carried out patch wise instead of element wise leading to an efficient and robust numerical algorithm a strategy for smoothing sharp corners is also described the algorithm does not rely on fillets but entails only a minor perturbation to the nurbs parameters a variationally consistent surface to surface formulation is developed in the context of frictionless contact numerical examples are presented to demonstrate the performance and utility of the method

a large deformation frictional contact formulation using nurbs based isogeometric analysis this paper focuses on the application of nurbs based isogeometric analysis to coulomb frictional contact problems between deformable bodies in the context of large deformations a mortar based approach is presented to treat the contact constraints whereby the discretization of the continuum is performed with arbitrary order nurbs as well as c0 continuous lagrange polynomial elements for comparison purposes the numerical examples show that the proposed contact formulation in conjunction with the nurbs discretization delivers accurate and robust predictions results of lower quality are obtained from the lagrange discretization as well as from a different contact formulation based on the enforcement of the contact constraints at every integration point on the contact surface copyright 2011 john wiley sons ltd

cn continuous modelling of smooth contact surfaces using nurbs and application to 2d problems this paper presents a strategy for the finite element implementation of cn continuous contact surfaces for deformable bodies undergoing finite deformations whereby n represents an arbitrary level of continuity the proposed novel approach avoids the non physical oscillations of contact forces which are induced by the traditional enforcement of kinematic contact constraints via faceted surfaces discretizing the interacting boundaries in particular for certain problems the level of continuity may influence the rate of convergence significantly within a non linear solution scheme a hierarchical tree data structure is proposed for an efficient search algorithm to find the neighbour elements on adaptively refined meshes which are involved in the smoothing process of a particular finite element the same data structure is used for the automatic detection of the contact surfaces of a body three representative numerical examples demonstrate the increased rate of convergence the ability to trace the actual surface more accurately and the prevention of pressure jumps of the proposed method copyright 2003 john wiley sons ltd

incorporation of contact for high order finite elements in covariant form the covariant contact description is applied to incorporate the treatment of contact problems into a high order finite element technique a hierarchical enrichment of the shape functions space allows to construct a contact layer finite element combining both exact geometry representation for contact surfaces and a mesh with linear shape functions for the interior of the contacting bodies the developed contact approach can be viewed as a smoothing technique for linear meshes as well as a general application of high order fem the good approximation property for the developed contact layer elements is shown for the classical hertz problem even within a few contact elements discretizing the contact zone

a compact multigrid solver for convection diffusion equations we introduce a multigrid algorithm to solve the convection diffusion equations using a nine point compact difference scheme we test the efficiency of the algorithm with various smoothers and intergrid transfer operators the algorithm displays a grid independent convergence rate and produces solutions with high accuracy numerical results are presented to validate the conclusions

on the derivation of highest order compact finite difference schemes for the one and two dimensional poisson equation with dirichlet boundary conditions the primary aim of this paper is to answer the question what are the highest order five or nine point compact finite difference schemes to answer this question we present several simple derivations of finite difference schemes for the one and two dimensional poisson equation on uniform quasi uniform and nonuniform face to face hyperrectangular grids and directly prove the existence or nonexistence of their highest order local accuracies our derivations are unique in that we do not make any initial assumptions on stencil symmetries or weights for the one dimensional problem the derivation using the three point stencil on both uniform and nonuniform grids yields a scheme with arbitrarily high order local accuracy however for the two dimensional problem the derivation using the corresponding five point stencil on uniform and quasi uniform grids yields a scheme with at most second order local accuracy and on nonuniform grids yields at most first order local accuracy when expanding the five poin

exponential higher order compact scheme for 3d steady convection diffusion problem abstract in this paper the exponential higher order compact ehoc finite difference schemes proposed by tian and dai 2007 10 for solving the one and two dimensional steady convection diffusion equations with constant or variable convection coefficients are extended to the three dimensional case the proposed ehoc scheme has the feature that it provides very accurate solution exact in case of constant convection coefficient in 1d of the homogeneous equation that is responsible for the fundamental singularity of the homogeneous solution while approximates the particular part of the solution by fourth order accuracy over the nineteen point compact stencil the key properties of this scheme are its stability accuracy and efficiency so that high gradients near the boundary layers can be effectively resolved even on coarse uniform meshes to validate the present ehoc method three test problems mostly with boundary or internal layers are solved comparisons are made between numerical results for the present ehoc scheme and other available methods in the literature

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

an analysis of dual formulations for the finite element solution of two body contact problems abstract this paper examines the convergence properties of dual finite element formulations of the two dimensional frictionless two body contact problem under the assumption of infinitesimal kinematics the centerpiece of the proposed analysis is the well known babuska brezzi condition suitably adapted to the present problem it is demonstrated for certain canonical geometries that several widely used methods that employ pressure or force interpolations derived from the discretizations of both surfaces violate the babuska brezzi condition thus producing increasingly oscillatory solutions under mesh refinement alternative algorithms are proposed that circumvent this difficulty and are shown to yield convergent solutions

on some aspects for contact with rigid surfaces surface to rigid surface and curves to rigid surface algorithms abstract special algorithms allowing a simplified description of contact between deformable body and rigid surfaces are developed based on the geometrically exact covariant description of contact a special attention is given to various geometric combinations where the contact can be represented as a contact between surfaces and b contact between a curve and a surface for contact between surfaces leading to the segment to analytical surface stas approach two algorithms can be distinguished based on the selection of a coordinate system for the closest point projection cpp procedure a rigid surface is a slave surface and b rigid surface is a master surface a special combination of both contact kinematics for the surface to surface and for the curve to curve approaches is employed for the contact between a curve and a surface leading to the curve to rigid surface ctrs approach the last algorithm is verified with the well known euler formula for the rope cylinder interaction as well as with a new derived generalization into a 3d spiral rope on a cylinder the developed algorithms can be straightforwardly implemented within an iso geometric approach as well as within the conventional finite elements where rigid surfaces are given by cad patches any type of elements can be employed for the contacting deformable surface curve because the algorithms are formulated in a covariant form

a segment to segment mortar contact method for quadratic elements and large deformations abstract this paper describes a new extension of mortar based segment to segment algorithms for large deformation contact analysis permitting their use when quadratic interpolations are used in the underlying finite element method the work builds upon results previously described by puso and laursen m a puso t a laursen a mortar segment to segment contact method for large deformation solid mechanics comput methods appl mech engrg 193 2004 601 629 m a puso t a laursen a mortar segment to segment frictional contact method for large deformations comput methods appl mech engrg 193 2004 601 629 where methods were developed for quasistatic and implicit dynamic analysis in the context of linear element interpolations here we see that many algorithmic ideas from the linear case can be adopted for use with quadratic elements along with different alternative interpolations of the mortar contact multipliers the result is a class of methods appropriate for use for either mesh tying true large sliding simulation frictionless or frictional and readily admits mixing of element types several numerical examples are given to demonstrate the effectiveness of the approach

a finite deformation mortar contact formulation using a primal dual active set strategy in recent years nonconforming domain decomposition techniques and in particular the mortar method have become popular in developing new contact algorithms here we present an approach for two dimensional frictionless multibody contact based on a mortar formulation and using a primal dual active set strategy for contact constraint enforcement we consider linear and higher order quadratic interpolations throughout this work so called dual lagrange multipliers are introduced for the contact pressure but can be eliminated from the global system of equations by static condensation thus avoiding an increase in system size for this type of contact formulation we provide a full linearization of both contact forces and normal non penetration and tangential frictionless sliding contact constraints in the finite deformation frame the necessity of such a linearization in order to obtain a consistent newton scheme is demonstrated by further interpreting the active set search as a semi smooth newton method contact nonlinearity and geometrical and material nonlinearity can be resolved within one single iterative scheme this yields a robust and highly efficient algorithm for frictionless finite deformation contact problems numerical examples illustrate the efficiency of our method and the high quality of results

a mortar method for energy momentum conserving schemes in frictionless dynamic contact problems summary in the present work the mortar method is applied to planar large deformation contact problems without friction in particular the proposed form of the mortar contact constraints is invariant under translations and rotations these invariance properties lay the foundation for the design of energy momentum timestepping schemes for contact impact problems the iterative solution procedure is embedded into an active set algorithm lagrange multipliers are used to enforce the mortar contact constraints the solution of generalized saddle point systems is circumvented by applying the discrete null space method numerical examples demonstrate the robustness and enhanced numerical stability of the newly developed energymomentum scheme copyright q 2008 john wiley sons ltd

three dimensional mortar based frictional contact treatment in isogeometric analysis with nurbs a three dimensional mortar based frictional contact treatment in isogeometric analysis with nurbs is presented in the finite deformation regime within a setting where the nurbs discretization of the contact surface is inherited directly from the nurbs discretization of the volume the contact integrals are evaluated through a mortar approach where the geometrical and frictional contact constraints are treated through a projection to control point quantities the formulation delivers a non negative pressure distribution and minimally oscillatory local contact interactions with respect to alternative lagrange discretizations independent of the discretization order these enable the achievement of improved smoothness in global contact forces and moments through higher order geometrical descriptions it is concluded that the presented mortar based approach serves as a common basis for treating isogeometric contact problems with varying orders of discretization throughout the contact surface and the volume

existence and nonexistence of bound state solutions for schrodinger systems with linear and nonlinear couplings abstract we study the schrodinger systems with linear and nonlinear coupling terms doubly coupled nonlinear schrodinger system for short which arise naturally in nonlinear optics and in the hartree fock theory for bose einstein condensates among other physical problems first for small linear coupling constant we get existence of a nontrivial bound state solution to the system via perturbation method furthermore we prove each component of the bound state solution is nonnegative by energy estimate second we establish a version of pohozaev nehari identity and prove a nonexistence result for the more general system when the spatial dimension n 4

a non consistent start up procedure for contact problems with large load steps abstract following up to a previous investigation this paper proposes a strategy to deal with frictionless contact problems involving large penetrations in the context of the node to segment formulation and of the penalty method the rationale is based on two main considerations the first one is that within an iteration scheme the use of consistent linearization is only convenient when the field of the unknowns is sufficiently close to the solution point the second one is that if the order of magnitude of the maximum contact pressure can be estimated a priori this information can be exploited to approach the solution in a faster and more reliable way beside being very effective in case of large initial penetrations where convergence using the standard algorithms would either not be achieved at all or achieved with a large number of iterations the strategy can also be used to increase the convergence rate in contact problems with penetrations of usual magnitude the proposed strategy is based on a check of the nodal contact pressure to select the technique that has to be used to perform each iteration if the contact pressure is smaller than a predefined limit the problem is solved in the standard way using consistent linearization and newton s method when the contact pressure exceeds the limit a modified method is used this one is based on the enforcement of a contact pressure limit and on the use of a simplified secant stiffness where the geometric stiffness term is disregarded the strategy has to be integrated with a specific safeguard algorithm to guarantee convergence to the correct solution also in cases where the maximum contact pressure has been underestimated two alternative procedures for this purpose are proposed the solution strategy can be used with any contact formulation here the efficiency and stability of the strategy is illustrated within the framework of the well known node to segment formulation with a 2d implementation several example problems are presented

convergence and performance of iterative methods for solving variable coefficient convection diffusion equation with a fourth order compact difference scheme abstract we conduct convergence analysis on some classical stationary iterative methods for solving the two dimensional variable coefficient convection diffusion equation discretized by a fourth order compact difference scheme several conditions are formulated under which the coefficient matrix is guaranteed to be an m matrix we further investigate the effect of different orderings of the grid points on the performance of some stationary iterative methods multigrid method and preconditioned gmres three sets of numerical experiments are conducted to study the convergence behaviors of these iterative methods under the influence of the flow directions the orderings of the grid points and the magnitude of the convection coefficients

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

high order compact scheme for the steady stream function vorticity equations a higher order compact scheme that is o h4 on the nine point 2 d stencil is formulated for the steady stream function vorticity form of the navier stokes equations the resulting stencil expressions are presented and hence this new scheme can be easily incorporated into existing industrial software we also show that special treatment of the wall boundary conditions is required the method is tested on representative model problems and compares very favourably with other schemes in the literature

a perturbed lagrangian formulation for the finite element solution of contact problems making use of a perturbed lagrangian formulation a finite element procedure for contact problems is developed for the general case in which node to node contact no longer holds the proposed procedure leads naturally to a discretization of the contact interface into contact segments within the context of a bilinear interpolation for the displacement field a mixed finite element approximation is introduced by assuming discont naous contact pressure constant on the contact segment because of this piece wise constant approximation the gap function enters into the formulation in an average sense instead of through a point wise definition numerical examples are presented that illustrate the performance of the proposed procedure current finite element formulations for contact problems based on either the classical lagrange parameter procedure l 3 12 203 or the penalty function method 4 6 11 are characterized by a point wise enforcement of the contact constraint condition in the sense that penetration of the bodies is established on a nodal basis moreover in this methods the recovery of the contact pressure over the element from the contact nodal forces generally requires an additional procedure within the framework of classical lagrange multiplier methods the contact condition is exactly satisfied by transforming the constrained problem into an unconstrained one with the introduction of additional variables lagrange multipliers these extra variables add computational effort to the solution process which often requires special procedures to handle the presence of zero diagonal terms penalty methods on the other hand enable one to transform the constrained problem into an unconstrained one without introducing additional variables the constraint condition is now satisfied only approximately for finite values of the penalty parameter the main difhculty associated with these methods however lies in the poor conditioning of the problem as the penalty is increased to more accurately enforce the constraint condition this is a well understood phenomenon particularly in the context of the incompressible and nearly incompressible problem in solid and fluid mechanics e g see l 22 25 for a review recently augmented lagrangian procedures have been proposed as a promising way to partially overcome these difficulties and regularize the penalty formulation e g see the survey in 7 8 within the framework of zinearited e u c it is possible to restrict the finite element formulation of contact problems by assuming that node to node contact occurs this is in fact the case often considered in the literature l 2 4 6 12 20 211 in the general context of fully

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

smooth c1 interpolations for two dimensional frictional contact problems finite deformation contact problems are associated with large sliding in the contact area thus in the discrete problem a slave node can slide over several master segments standard contact formulations of surfaces discretized by low order finite elements leads to sudden changes in the surface normal field this can cause loss of convergence properties in the solution procedure and furthermore may initiate jumps in the velocity field in dynamic solutions furthermore non smooth contact discretizations can lead to incorrect results in special cases where a good approximation of the contacting surfaces is needed in this paper a smooth contact discretization is developed which circumvents most of the aformentioned problems a smooth deformed surface with no slope discontinuities between segments is obtained by a c1 continuous interpolation of the master surface different forms of discretizations are possible among these are bezier hermitian or other types of spline interpolations in this paper we compare two formulations which can be used to obtain smooth normal and tangent fields for frictional contact of deformable bodies the formulation is developed for two dimensional applications and includes finite deformation behaviour examples show the performance of the new discretization technique for contact copyright 2001 john wiley sons ltd

geometrically exact theory for contact interactions of 1d manifolds algorithmic implementation with various finite element models abstract the intuitive understanding of contact between bodies is based on the geometry of adjoining bodies a more sophisticated approach of an advanced analysis including the application of various numerical methods is to take advantage of the geometry of an analyzed object and describe the problem in the best coordinate system the best coordinate system to describe contact interaction in all its geometrical details is a coordinate system attached to the geometrical features of contacting bodies this leads to a systematical analysis of geometrical situations leading to contact pairs surface to surface line to surface point to surface line to line point to line each contact pair is inherited with a special coordinate system based on its geometrical properties the current contribution is concentrating on contact between 1d manifolds in 3d space this is the majority of edge to edge beam to beam cable to edge etc contact cases the geometrically exact curve to curve contact approach is then systematically combined together with various finite element approaches classical finite elements isogeometric beam finite elements and also with a new developed solid beam approach examples illustrating the diversity of various finite element combinations e g within knot mechanics are shown

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

preconditioned iterative methods and finite difference schemes for convection diffusion we conduct experimental study on numerical solution of the two dimensional convection diffusion equation discretized by three finite difference schemes the traditional central difference scheme the standard upwind scheme and the fourth order compact scheme we study the computed accuracy achievable by each scheme the algebraic properties of the coefficient matrices arising from different schemes and the performance of the gauss seidel iterative method the preconditioned gmres iterative method and the multigrid method for solving linear systems arising from these schemes

iterative methods for cyclically reduced non self adjoint linear systems we study iterative methods for solving linear systems of the type arising from two cyclic discretizations of non self adjoint two dimensional ellip tic partial differential equations a prototype is the convection diffusion equa tion the methods consist of applying one step of cyclic reduction resulting in a reduced system of half the order of the original discrete problem com bined with a reordering and a block iterative technique for solving the reduced system for constant coefficient problems we present analytic bounds on the spectral radii of the iteration matrices in terms of cell reynolds numbers that show the methods to be rapidly convergent in addition we describe numerical experiments that supplement the analysis and that indicate that the methods compare favorably with methods for solving the unreduced system

contact treatment in isogeometric analysis with nurbs abstract we study nurbs based isogeometric analysis of contact problems and compare with standard c 0 continuous lagrange finite elements a knot to surface kts algorithm is developed to treat the contact constraints with nurbs contact surface discretizations qualitative studies deliver satisfactory results for various finite deformation frictionless thermoelastic contact problems quantitative studies based on the hertz problem suggest the need for a relaxation of the mechanical contact constraints that appear in the standard kts approach the improved mortar based kts algorithm delivers robust and accurate results for nurbs discretizations based on numerical examples we conclude that nurbs based isogeometric analysis is a viable technology for contact problems and offers potential accuracy as well as convergence improvements over c 0 continuous finite elements

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

a novel three dimensional contact finite element based on smooth pressure interpolations this article proposes a new three dimensional contact finite element which employs continuous and weakly coupled pressure interpolations on each of the interacting boundaries the resulting formulation circumvents the geometric bias of one pass methods as well as the surface locking of traditional two pass node on surface methods a lagrange multiplier implementation of the proposed element is validated for frictionless quasi static contact by a series of numerical simulations published in 2001 by john wiley sons ltd

a mixed formulation of mortar based frictionless contact abstract a class of mortar based frictionless contact formulations is derived based on a classical three field mixed variational framework within a penalty regularization complemented by uzawa augmentations discrete mortar constraints are naturally induced by the variational setting major aspects of earlier mortar approaches are obtained through constrained lumped or unconstrained recovery procedures for the mixed kinematic and kinetic mortar quantities from their projected counterparts two and three dimensional examples at the infinitesimal and finite deformation regimes highlight the local and global quality of the contact interactions

a primal dual active set strategy for non linear multibody contact problems abstract non conforming domain decomposition methods provide a powerful tool for the numerical approximation of partial differential equations for the discretization of a non linear multibody contact problem we use the mortar approach with a dual lagrange multiplier space to handle the non linearity of the contact conditions we apply a primal dual active set strategy to find the actual contact zone the algorithm can be easily generalized to multibody contact problems a suitable basis transformation guarantees the same algebraic structure in the multibody situation as in the one body case using an inexact primal dual active set strategy in combination with a multigrid method yields an efficient iterative solver different numerical examples for one and multibody contact problems illustrate the performance of the method

compact h4 finite difference approximations to operators of navier stokes type a method of obtaining compact finite difference approximations of h4 accuracy to operators of navier stokes type is considered the basic procedure is developed for operators in one space dimension and subsequently applied to problems in more space dimensions and in time four illustrative numerical examples are given which indicate clearly in various cases that excellent accuracy may be obtained using the methods comparisons with previous results and with the results of h2 accurate computations are made