the primal dual active set strategy as a semismooth newton method this paper addresses complementarity problems motivated by constrained optimal control problems it is shown that the primal dual active set strategy which is known to be extremely efficient for this class of problems and a specific semismooth newton method lead to identical algorithms the notion of slant differentiability is recalled and it is argued that the max function is slantly differentiable in lp spaces when appropriately combined with a two norm concept this leads to new local convergence results of the primal dual active set strategy global unconditional convergence results are obtained by means of appropriate merit functions

a 3d mortar method for solid mechanics a version of the mortar method is developed for tying arbitrary dissimilar 3d meshes with a focus on issues related to large deformation solid mechanics issues regarding momentum conservation large deformations computational efficiency and bending are considered in particular a mortar method formulation that is invariant to rigid body rotations is introduced a scheme is presented for the numerical integration of the mortar surface projection integrals applicable to arbitrary 3d curved dissimilar interfaces here integration need only be performed at problem initialization such that coefficients can be stored and used throughout a quasi static time stepping process even for large deformation problems a degree of freedom reduction scheme exploiting the dual space interpolation method such that direct linear solution techniques can be applied without lagrange multipliers is proposed this provided a significant reduction in factorization times example problems which touch on the aforementioned solid mechanics related issues are presented published in 2003 by john wiley sons ltd

nonlinear complementarity functions for plasticity problems with frictional contact in this paper a unifying framework for the numerical treatment of frictional contact in combination with elastoplasticity with hardening is presented both kinds of problems lead to the mathematical structure of variational inequalities which are discretized by introducing additional variables for the contact stresses or the deviatoric stresses leading to a mixed formulation by defining a family of nonlinear complementarity functions which can be applied to both frictional and plastic yield conditions these inequality constraints are reformulated as equalities combined with the equilibrium equations a system of nonlinear equations is obtained that is solved in terms of a semismooth newton method to obtain a larger domain of convergence and better stability properties the introduction of additional scaling parameters is motivated and their benefit to the convergence of the scheme is demonstrated due to the locality of the contact and plastic constraints the resulting method can be interpreted as a primal dual active set strategy furthermore for linear hardening laws our method is a generalization of the well known radial return algorithm which is obtained for a special suboptimal choice of the plastic scaling parameters the efficiency and the generality of the proposed quasi newton scheme is demonstrated by several numerical examples including the simulation of a three dimensional contact problem with several elastoplastic bodies exponential isotropic hardening and coulomb friction

elastodynamics by bem a new direct formulation a new direct be formulation is proposed for the solution of elastodynamic problems an analytical regularization procedure is devised via an integration by parts technique without introducing any hypothesis about either the discretization of the boundary geometry or the space time interpolation of the elastic elds involved the only requirement is for continuity in the displacement eld the regularization of the integral equations for static elasticity using the same approach already available in the literature is presented as a particular case of the general procedure introduced herein the numerical implementation of this technique is discussed and two dimensional examples are presented copyright 1999 john wiley sons ltd

restrictions matrices for platonic solids invariance and applications to space time energetic bem abstract geometrical symmetry is found in many linear field problems group representation theory is the only valuable tool for exploiting this property from a computational point of view in this context here we apply a technique for taking into account equivariance in the numerical treatment of space time boundary integral equations which are invariant under a finite group g of congruences of r 3 each related to one of the so called platonic solids this technique is based upon suitable restriction matrices strictly related to a system of unitary irreducible pairwise not equivalent matrix representations of g the main development is expounded in the framework of space time energetic boundary element method applied to neumann exterior wave propagation problems where the discretization matrices have a block lower triangular toeplitz structure and the diagonal block to be inverted at each time step is typically dense several numerical results related to computational saving are presented

power penalty method for solving hjb equations arising from finance nonlinear hamilton jacobi bellman hjb equation commonly occurs in financial modeling implicit numerical scheme is usually applied to the discretization of the continuous hjb so as to find its numerical solution since it is generally difficult to obtain its analytic viscosity solution this type of discretization results in a nonlinear discrete hjb equation we propose a power penalty method to approximate this discrete equation by a nonlinear algebraic equation containing a power penalty term under some mild conditions we give the unique solvability of the penalized equation and show its convergence to the original discrete hjb equation moreover we establish a sharp convergence rate of the power penalty method which is of an exponential order with respect to the power of the penalty term we further develop a damped newton algorithm to iteratively solve the lower order penalized equation finally we present a numerical experiment solving an incomplete market optimal investment problem to demonstrate the rates of convergence and effectiveness of the new method we also numerically verify the efficiency of the power penalty method by comparing it with the widely used policy iteration method

a truly variationally consistent and symmetric mortar based contact formulation for finite deformation solid mechanics abstract in this work two mortar based segment to segment contact formulations will be developed for the frictionless finite deformation case while the first one is derived by consistent variation of all active contributions of a scalar valued potential subject to inequality constraints thus resulting in a truly variationally consistent and symmetric formulation the second approach is designed in such a way that it is less computationally expensive but still conserves important quantities such as linear and angular momentum since both formulations are derived side by side the introduced simplifications and errors can be specifically analyzed and quantified based on a lagrange multiplier approach the corresponding inequality constraint terms are introduced in two popular ways firstly in form of a standard lagrangian formulation and secondly via an augmented lagrangian formulation both variational forms as well as both solution procedures will be consistently linearized and discussed finally the obtained results will be compared to each other as well as to a well established yet slightly inconsistent mortar based contact formulation

nonlinear small data scattering for the wave and klein gordon equation on etudie l operateur diffusion qui appartient a la paire d equations u tt au f u 0 et u tt au 0 a represente l operateur j 1 n 2 x j 2 m 2 m r et f u u 1 u ou r 1

mass concentration phenomena for the l 2 critical nonlinear schrodinger equation in this paper we show that any solution of the nonlinear schr o dinger equation iu t delta u pm u frac 4 n u 0 which blows up in finite time satisfies a mass concentration phenomena near the blow up time our proof is essentially based on the bourgain s one cite mr99f 35184 which has established this result in the bidimensional spatial case and on a generalization of strichartz s inequality where the bidimensional spatial case was proved by moyua vargas and vega cite mr1671214 we also generalize to higher dimensions the results in keraani cite mr2216444 and merle and vega cite mr1628235

a mortar segment to segment contact method for large deformation solid mechanics abstract contact modeling is still one of the most difficult aspects of non linear implicit structural analysis most 3d contact algorithms employed today use node on segment approaches for contacting dissimilar meshes two pass node on segment contact approaches have the well known deficiency of locking due to over constraint in this work we develop and demonstrate a segment to segment contact approach based on the mortar method by eliminating the over constraint this method appears to be much more robust than the node on segment approach furthermore node on segment approaches suffer when individual nodes slide out of contact at contact surface boundaries or when contacting nodes slide from facet to facet this causes jumps in the contact forces due to the discrete nature of the constraint enforcement and difficulties in convergence for implicit solution techniques these jumps in forces are avoided in the segment to segment approach since penetration is measured by the occluded weighted volume which varies smoothly with perturbations until now all mortar schemes have been developed for small deformation sliding and have mainly been applied on flat surfaces here we will present the integration scheme and the linearization approach used for handling the contact surface integrals in the event of large sliding the schemes are designed to conserve linear momentum and can be applied to arbitrarily curved 3d surfaces although expensive our numerical examples demonstrate that the approach is far more robust than even smoothed node on segment contact techniques that is problems previously not solvable with node on segment can now be solved with our segment to segment contact

a primal dual active set algorithm for three dimensional contact problems with coulomb friction in this paper efficient algorithms for contact problems with tresca and coulomb friction in three dimensions are presented and analyzed the numerical approximation is based on mortar methods for nonconforming meshes with dual lagrange multipliers using a nonsmooth complementarity function for the three dimensional friction conditions a primal dual active set algorithm is derived the method determines active contact and friction nodes and at the same time resolves the additional nonlinearity originating from sliding nodes no regularization and no penalization are applied and superlinear convergence can be observed locally in combination with a multigrid method it defines a robust and fast strategy for contact problems with tresca or coulomb friction the efficiency and flexibility of the method is illustrated by several numerical examples

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

improved robustness and consistency of 3d contact algorithms based on a dual mortar approach abstract mortar finite element methods have been successfully applied as discretization scheme to a wide range of contact and impact problems in recent years the 3d finite deformation contact formulation taken up and enhanced in this paper is based on a mortar approach using so called dual lagrange multipliers which substantially facilitate the treatment of interface constraints as compared with standard mortar techniques despite being quite well established in the meantime dual mortar methods may lack robustness or even consistency in certain situations e g when large curvatures occur in the contact zone or when one body slides off another at a sharp edge however since such scenarios are regularly appearing in engineering practice the present contribution provides several new extensions that completely resolve these issues and thus significantly improve the applicability of dual mortar formulations for challenging contact problems the proposed extensions include a consistent biorthogonalization procedure to obtain feasible dual lagrange multiplier shape functions close to boundaries of the contact surfaces an improved conditioning of the global linear system of equations by nodal scaling and a novel approach to unify the advantages of standard and dual mortar methods via a petrov galerkin type of lagrange multiplier interpolation several numerical examples demonstrate the achievable improvements in terms of consistency and robustness for 3d contact analysis including finite deformations

finite deformation frictional mortar contact using a semi smooth newton method with consistent linearization a two dimensional finite deformation frictional contact formulation with coulomb s law is presented the approach considers multibody contact and is based on a mortar formulation the enforcement of contact constraints is realized with dual lagrange multipliers these alternative multiplier spaces are constructed in a way that the multipliers can easily be eliminated from the global system of equations by static condensation such that the system size does not increase friction kinematic variables are formulated in an objective way and enter non smooth complementarity functions for expressing the contact constraints an active set strategy is derived by applying a semismooth newton method which treats contact nonlinearities material and geometrical nonlinearities in one single iterative scheme by further carrying out a consistent linearization for both normal and frictional contact forces and constraints a robust and highly efficient algorithm for linear and higher order quadratic interpolation is achieved efficiency of the proposed method and quality of results are demonstrated in serveral exemples

consistent treatment of boundaries with mortar contact formulations using dual lagrange multipliers abstract computational modelling of contact problems raises two basic questions which method should be used to enforce the contact conditions and how should this method be discretised the most popular enforcement methods are the lagrange multiplier method the penalty method and combinations of these two a frequently used discretisation method is the so called node to segment approach however this approach might lead to problems like jumps in contact forces loss of convergence or failure to pass the patch test thus in the last few years several segment to segment contact algorithms based on the mortar method were proposed combination of a mortar discretisation with a penalty based enforcement of the contact conditions leads to unphysical penetrations on the other hand a lagrange multiplier mortar method requires additional unknowns hence condensation of the lagrange multipliers is desirable to preserve the initial size of the system of equations this can be achieved by interpolating the lagrange multipliers with so called dual shape functions discretising two contacting bodies leads to opposed contact surface representations of finite element edges called slave and master elements respectively in current versions of dual lagrange multiplier mortar formulations an inconsistency at the boundary appears when only a part of a slave element instead of the entire element belongs to the contact area we present a modified definition of the dual shape functions in such slave elements the basic idea is to construct dual shape functions that fulfill the so called biorthogonality condition within the contact area this leads to consistent mortar matrices also in the boundary region to avoid ill conditioning of the stiffness matrix the modified mortar matrices are weighted with appropriate weighting factors in doing so the corresponding modified lagrange multiplier nodal values are of the same order as the unmodified ones various examples demonstrate the performance of the modified mortar contact algorithm

an unbiased computational contact formulation for 3d friction to high accuracy as is shown a new 3d friction formulation is also proposed that is a direct extension of the 1d setup expressing the friction variables in the parameter space used for the curvilinear surface description the new formulation resorts to classical expressions in the continuum limit the current approach uses c 1 smooth contact surface representations based on either hermite or nurbs interpolation a penalty regularization is considered for the impenetrability and tangential sticking constraints the new unbiased friction formulation is illustrated by several 2d and 3d examples which include an extensive analysis of the model parameters a convergence study and the comparison with a classical biased master slave contact algorithm

on the blow up phenomenon of the critical nonlinear schrodinger equation abstract in this paper we consider the blow up phenomenon of critical nonlinear schrodinger equations in dimension 1d and 2d we define the minimal mass as the l 2 norm necessary to ignite a wave collapse and we stress its role in the blow up mechanism asymptotic compactness properties and l 2 concentration are proved the proof relies on linear and nonlinear profile decompositions

generalized strichartz inequalities for the wave equation strichartz inequalities for the wave equation u f are estimates of the solution u of the cauchy problem for that equation in the form of space time integral norms in terms of similar norms of the inhomogeneity f and of suitable norms of the initial data

on the solvability of closest point projection procedures in contact analysis analysis and solution strategy for surfaces of arbitrary geometry abstract the uniqueness and existence of the closest point projection procedure widely used in contact mechanics are analyzed in the current article first a projection domain for c 2 continuous surfaces is created based on the geometrical properties of surfaces then any point from the projection domain has a unique projection onto the given surface it is shown that in order to construct a continuous projection domain for arbitrary globally c 1 or c 0 continuous surfaces a projection routine should be generalized and also include a projection onto a curved edge and onto corner points criteria of uniqueness and existence of the corresponding projection routine are given and discussed from the geometrical point of view some examples showing the construction of the projection domain as well as the necessity of a generalized projection routine are given

exploiting partial or complete geometrical symmetry in 3d symmetric galerkin indirect bem formulations procedures based on group representation theory allowing the exploitation of geometrical symmetry in symmetric galerkin bem formulations are investigated in particular this investigation is based on the weaker assumption of partial geometrical symmetry where the boundary has two disconnected components one of which is symmetric e g this can be very useful for defect identification problems the main development is expounded in the context of 3d neumann elastostatic problems considered as model problems and then extended to sgbie formulations for dirichlet and or scalar problems both abelian and non abelian finite symmetry groups are considered the effectiveness of the present approach is demonstrated through numerical examples where both partial and complete symmetry are considered in connection with both abelian and non abelian symmetry groups

on existence and scattering with minimal regularity for semilinear wave equations abstract we prove existence and scattering results for semilinear wave equations with low regularity data we also determine the minimal regularity that is needed to ensure local existence and well posedness and we give counterexamples to well posedness more specifically we show that equations of the type u u p with initial data u u t in r n 1 r n have a local solution if p n and we construct counterexamples if p n the existence results rely on mixed norm space time estimates of strichartz type

a semi smooth newton method for orthotropic plasticity and frictional contact at finite strains abstract a new approach for the unified treatment of frictional contact and orthotropic plasticity at finite strains using semi smooth newton methods is presented the contact discretization is based on the well known mortar finite element method using dual lagrange multipliers to facilitate the handling of the additional lagrange multiplier degrees of freedom exploiting the similarity of the typical inequality constraints of plasticity and friction all involved discrete inequalities are reformulated as nonlinear non smooth equations using complementarity functions the resulting set of discrete semi smooth equations can be solved efficiently by a variant of newton s method where all additionally introduced variables are condensed from the global system so that a linear system only consisting of the displacement degrees of freedom has to be solved in each iteration step in contrast to classical radial return mapping methods for computational plasticity the plastic constraints are not required to hold at every iterate in the nonlinear solution procedure but only at convergence this relaxation in the pre asymptotic behavior results in an increased flexibility regarding algorithm design and a potentially higher robustness compared to radial return mapping algorithms the presented elasto plasticity algorithm includes arbitrary isotropic hyperelasticity an anisotropic hill type yield function with isotropic and kinematic hardening plastic spin and appropriate finite element technology for nearly incompressible materials therefore it is well suited for the modeling of sheet metal forming and similar processes several numerical examples underline the robustness of the proposed plasticity algorithm and the efficient treatment of elasto plastic contact problems

nonlinear black scholes equations in finance associated control problems and properties of solutions we study properties of solutions to fully nonlinear versions of the standard black scholes partial differential equation these equations have been introduced in financial mathematics in order to deal with illiquid markets or with stochastic volatility we show that typical nonlinear black scholes equations can be viewed as dynamic programming equation of an associated control problem we establish existence and comparison results and show that the equation induces a convex risk measure on the set of all continuous terminal value claims moreover we study the asymptotic behavior of solutions as market frictions get large finally the pricing of individual contracts relative to a book of derivatives is discussed

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

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

higher order mortar finite element methods in 3d with dual lagrange multiplier bases mortar methods with dual lagrange multiplier bases provide a flexible efficient and optimal way to couple different discretization schemes or nonmatching triangulations here we generalize the concept of dual lagrange multiplier bases by relaxing the condition that the trace space of the approximation space at the slave side with zero boundary condition on the interface and the lagrange multiplier space have the same dimension we provide a new theoretical framework within this relaxed setting which opens a new and simpler way to construct dual lagrange multiplier bases for higher order finite element spaces as examples we consider quadratic and cubic tetrahedral elements and quadratic serendipity hexahedral elements numerical results illustrate the performance of our approach

using symmetrics and antisymmetrics to analyze a parallel multigrid algorithm the elliptic boundary value problem case symmetry and antisymmetry properties of a class of elliptic partial differential equations are exploited to prove when a particular parallel multilevel algorithm is a direct method rather than the usual iterative method no smoothing is required for this result examples are presented including variable coefficient ones a connection between the algorithm in this article and domain decomposition is established even though this algorithm is more general and different the parallel algorithm is also analyzed when it is iterative and it is shown how to increase processor utilization hackbusch s robust multigrid algorithm a new approach to robust multi grid solvers in iciam 87 proceedings of the first international conference on industrial and applied mathematics society for industrial and applied mathematics philadelphia pa 1988 pp 111 126 is analyzed for some model problems and it is shown that the parallel algorithm in this article uses much less computer time with at most the same amount

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

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

a global local approach for hydraulic phase field fracture in poroelastic media abstract in this work phase field modeling of hydraulic fractures in porous media is extended towards a global local approach therein the failure behavior is solely analyzed in a small local domain in the surrounding medium a simplified and linearized system of equations is solved both domains are coupled with robin type interface conditions the fractures inside the local domain are allowed to propagate and consequently both subdomains change within time here a predictor corrector strategy is adopted in which the local domain is dynamically adjusted to the current fracture pattern the resulting framework is algorithmically described in detail and substantiated with some numerical tests

isogeometric dual mortar methods for computational contact mechanics in recent years isogeometric analysis iga has received great attention in many fields of computational mechanics research especially for computational contact mechanics an exact and smooth surface representation is highly desirable as a consequence many well known finite element methods and algorithms for contact mechanics have been transferred to iga in the present contribution the so called dual mortar method is investigated for both contact mechanics and classical domain decomposition using nurbs basis functions in contrast to standard mortar methods the use of dual basis functions for the lagrange multiplier based on the mathematical concept of biorthogonality enables an easy elimination of the additional lagrange multiplier degrees of freedom from the global system this condensed system is smaller in size and no longer of saddle point type but positive definite a very simple and commonly used element wise construction of the dual basis functions is directly transferred to the iga case the resulting lagrange multiplier interpolation satisfies discrete inf sup stability and biorthogonality however the reproduction order is limited to one in the domain decomposition case this results in a limitation of the spatial convergence order to o h3 2 in the energy norm whereas for unilateral contact due to the lower regularity of the solution optimal convergence rates are still met numerical examples are presented that illustrate these theoretical considerations on convergence rates and compare the newly developed isogeometric dual mortar contact formulation with its standard mortar counterpart as well as classical finite elements based on first and second order lagrange polynomials

biorthogonal bases with local support and approximation properties we construct locally supported basis functions which are biorthogonal to conforming nodal finite element basis functions of degree p in one dimension in contrast to earlier approaches these basis functions have the same support as the nodal finite element basis functions and reproduce the conforming finite element space of degree p 1 working with gaus lobatto nodes we find an interesting connection between biorthogonality and quadrature formulas one important application of these newly constructed biorthogonal basis functions are two dimensional mortar finite elements the weak continuity condition of the constrained mortar space is realized in terms of our new dual bases as a result local static condensation can be applied which is very attractive from the numerical point of view numerical results are presented for cubic mortar finite elements

a point to segment contact formulation for isogeometric nurbs based finite elements formulation of isogeometric finite elements has received a great deal of attention in the recent past the present study deals with the treatment of problems in structural mechanics including large elastic deformations and contact one decisive difference between isogeometric finite elements based on nurbs functions and standard finite elements fe based on lagrange polynomials is higher inter element continuity this is a promising characteristics to model contact problems as all issues associated with kinks between elements are naturally avoided particularly in cases with large sliding this has the potential to be an attractive feature we present a bilateral isogeometric collocation contact formulation for geometrically non linear two dimensional problems using greville and botella points to collocate the contact integrals different methods to obtain accurate and physically meaningful stress distributions are investigated and compared the results show that the higher inter element continuity in context of non smooth problems may imply undesired effects in the numerical solution such as unphysical stress oscillations similar effects have been observed in standard p fem numerical experiments indicate that these oscillations may be avoided if the basis functions of contact and non contact zones are separated by knot relocation and knot repetition

the global cauchy problem for the non linear klein gordon equation on demontre l unicite de solutions faibles et l existence et l unicite de solutions globales fortement continues dans l espace energie

isogeometric mortar methods the application of mortar methods in the framework of isogeometric analysis is investigated theoretically as well as numerically for the lagrange multiplier two choices of uniformly stable spaces are presented both of them are spline spaces but of a different degree in one case we consider an equal order pairing for which a cross point modification based on a local degree reduction is required in the other case the degree of the dual space is reduced by two compared to the primal this pairing is proven to be inf sup stable without any necessary cross point modification several numerical examples confirm the theoretical results and illustrate additional aspects

isogeometric analysis and domain decomposition methods in the present work we use the mortar finite element method for the coupling of nonconforming discretized sub domains in the framework of nonlinear elasticity the mortar method has been shown to preserve optimal convergence rates see laursen 2002 25 for details and is variationally consistent we show that the method can be applied to isogeometric analysis with little effort once the framework of nurbs based shape functions has been implemented furthermore a specific coordinate augmentation technique allows the design of an energy momentum scheme for the constrained mechanical system under consideration the excellent performance of the redesigned mortar method as well as the energy momentum scheme is illustrated in representative numerical examples

dual quadratic mortar finite element methods for 3d finite deformation contact mortar finite element methods allow for a flexible and efficient coupling of arbitrary nonconforming interface meshes and are by now quite well established in nonlinear contact analysis in this pa

isogeometric analysis for explicit elastodynamics using a dual basis diagonal mass formulation abstract we propose a method to obtain diagonal mass matrices for nurbs based approximation spaces by a dual lumping method the use of lumped mass matrices is of great importance in elastodynamics problems as they can be employed in explicit time integration schemes which do not require the solution of a linear system in finite elements several well established methods such as row sum diagonal scaling or nodal quadrature methods have been used to obtain lumped mass matrices for different applications however for higher order and higher continuity approximation spaces such as those derived from nurbs these approaches have only limited second order accuracy in this work we derive a dual basis which has optimal approximation and dispersion properties while maintaining local support the dual space has discontinuities at the element boundaries knots and it is used to provide the test functions in the context of a petrov galerkin method this results in a general framework for the study of lumped mass matrices which can be employed in explicit time integration schemes with high order accuracy numerical experiments are presented to demonstrate the applicability of the method to problems with smooth solutions as well as to wave propagation problems with reduced regularity

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

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

stable lagrange multipliers for quadrilateral meshes of curved interfaces in 3d for the discretization of coupled problems by means of mortar finite elements the choice of the discrete lagrange multiplier space is of crucial importance for the performance of the resulting numerical scheme dual lagrange multipliers offer several advantages since they can be locally eliminated in this paper the lowest order dual lagrange multipliers are extended to arbitrary quadrilateral or triangular surface grids in a first step two alternatives are provided for the case of planar interfaces for the approximation of vector fields two more modifications are given for use on curvilinear interfaces a priori results are obtained and several numerical examples are shown

biorthogonal splines for optimal weak patch coupling in isogeometric analysis with applications to finite deformation elasticity abstract a new construction of biorthogonal splines for isogeometric mortar methods is proposed the biorthogonal basis has a local support and at the same time optimal approximation properties which yield optimal results with mortar methods we first present the univariate construction which has an inherent crosspoint modification the multivariate construction is then based on a tensor product for weighted integrals whereby the important properties are inherited from the univariate case numerical results including large deformations confirm the optimality of the newly constructed biorthogonal basis

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

the global cauchy problem for the critical non linear wave equation abstract we study the global cauchy problem for the non linear wave equation p 1 0 for the critical value p n 2 n 2 in space dimension n 3 we identify a weak space time integrability property stip of the solutions and prove that it is sufficient to ensure the uniqueness of weak solutions the global existence of finite energy solutions with the naturally associated stip and the global existence of regular solutions with some n dependent restrictions on the regularity for spherically symmetric solutions we prove that the previous crucial stip follows from the morawetz inequality actually in a much stronger form than necessary thereby proving that all the previous results hold in the spherically symmetric case

oscillations and concentration effects in semilinear dispersive wave equations let un be a sequence of smooth solutions to a dispersive nonlinear wave equation formula inr1 3 with uniformly compactly supported cauchy data converging weakly to 0 inh1 r3 l2 r3 let vn be the sequence of solutions to the linear wave equation with the same cauchy data we show thatun vngoes strongly to 0 in the energy spacec 0 t h1 c1 0 t l2 if f is a subcritical nonlinearity in the critical casef u u5 we show that this property is equivalent tovn 0 inl 0 t l6 then we give sharp sufficient conditions on microlocal measures associated to the data the proof relies on a microlocal version of p l lions concentration compacity soit un une suite de solutions regulieres d une equation d ondes non lineaire dispersive formula dansr1 3 avec donnees de cauchy supportees dans un compact fixe et convergeant faiblement vers 0 dansh1 r3 l2 r3 soit vn la suite des solutions de l equation des ondes lineare avec les memes donnees de cauchy on montre queun vnconverge fortement vers 0 dans l espace d energiec 0 t h1 c1 0 t l2 sifest une non linearite sous critique dans le cas critiquef u u5 on montre que cette propriete equivaut a la convergence forte devnvers 0 dansl 0 t l6 on en donne alors des conditions suffisantes optimales en termes de mesures microlocales associees aux donnees la demonstration est basee sur une version microlocale de la concentration compacite de p l lions

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

stabilization and control for the nonlinear schrodinger equation on a compact surface in this paper we study the stabilization property and the exact controllability for the nonlinear schrodinger equation on a two dimensional compact riemannian manifold without boundary we use a pseudo differential dissipation the proofs are based on a result of propagation of singularities and on recent dispersion estimates strichartz type inequalities due to n burq p gerard and n tzvetkov

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

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

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

high frequency approximation of solutions to critical nonlinear wave equations this work is devoted to the description of bounded energy sequences of solutions to the equation 1 u u 4 0 in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i up to remainder terms small in energy norm and in every strichartz norm the proof relies on scattering theory for 1 and on a structure theorem for bounded energy sequences of solutions to the linear wave equation in particular we infer the existence of an a priori estimate of strichartz norms of solutions to 1 in terms of their energy

strichartz estimates and wave equation in a conic singular space consider the metric cone x c y 0 infty r times y with metric g dr 2 r 2h where the cross section y is a compact n 1 dimensional riemannian manifold y h let delta g be the positive friedrichs extension laplacian on x and let delta h be the positive laplacian on y and consider the operator mathcal l v delta g v 0 r 2 where v 0 in mathcal c infty y such that delta h v 0 n 2 2 4 is a strictly positive operator on l 2 y in this paper we prove global in time strichartz estimates without loss regularity for the wave equation associated with the operator mathcal l v it verifies a conjecture in wang remark 2 4 in ann inst fourier 56 1903 1945 2006 for wave equation the range of the admissible pair is sharp and the range is influenced by the smallest eigenvalue of delta h v 0 n 2 2 4 to prove the result we show a sobolev inequality and a boundedness of a generalized riesz transform in this setting in addition as an application we study the well posed theory and scattering theory for energy critical wave equation with small data on this setting of dimension n ge 3

a high order finite difference method for option valuation in this paper we propose the use of an efficient high order finite difference algorithm to price options under several pricing models including the blackscholes model the mertons jumpdiffusion model the hestons stochastic volatility model and the nonlinear transaction costs or illiquidity models we apply a local mesh refinement strategy at the points of singularity usually found in the payoff of most financial derivatives to improve the accuracy and restore the rate of convergence of a non uniform high order five point stencil finite difference scheme for linear models the time stepping is dealt with by using an exponential time integration scheme with carathodoryfejr approximations to efficiently evaluate the product of a matrix exponential with a vector of option prices nonlinear blackscholes equations are solved using an efficient iterative scheme coupled with a richardson extrapolation our numerical experiments clearly demonstrate the high order accuracy of the proposed finite difference method making the latter a method of choice for solving both linear and nonlinear partial differential equations in financial engineering problems

control of the black scholes equation the purpose of this work is to apply the results developped by j y chemin and cl david to the black scholes equation this latter equation being directly linked to the heat equation it enables us to propose a new approach allowing to control properties of the solution by means of a shape parameter

variationally consistent inertia templates for b spline and nurbs based fem inertia scaling and customization abstract in this contribution variationally consistent inertia templates for b spline and nurbs based finite elements are proposed as a unified concept for two different purposes customization of the template allows construction of masses and reciprocal masses with desired properties like higher order accuracy or improved dispersion behavior inertia scaling allows substantial speed up for explicit dynamics by increased critical time steps the derivation of the template is based on a three field parametrized functional as in previous works of the authors group but with modified primary variables namely displacement velocity and mass specific linear momentum the latter allows for mass preservation for non constant density throughout the domain and is therefore an enhancement to the formulation proposed in tkachuk and bischoff 2015 with the focus on b spline and nurbs based finite elements the proposed template provides alternatives to the row sum lumped mass matrix which is only 2nd order accurate independent of the polynomial order earlier proposed algebraically constructed higher order masses from the literature can be reconstructed in the variational setting described here as special instances furthermore higher order reciprocal masses can be constructed from the template they are especially attractive for explicit dynamics as there is no extra expense per time step compared with lumped mass for linear problems for non linear problems only small overhead is expected but this paper focuses on linear problems only and mainly undistorted meshes tuning the method towards inertia scaling a reduction of the maximum eigenfrequency by 25 40 is obtained in the examples herein whereas the accuracy is higher than for lumped or consistent mass

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

a mortar finite element method using dual spaces for the lagrange multiplier the mortar finite element method allows the coupling of different discretization schemes and triangulations across subregion boundaries in the original mortar approach the matching at the interface is realized by enforcing an orthogonality relation between the jump and a modified trace space which serves as a space of lagrange multipliers in this paper this lagrange multiplier space is replaced by a dual space without losing the optimality of the method the advantage of this new approach is that the matching condition is much easier to realize in particular all the basis functions of the new method are supported in a few elements the mortar map can be represented by a diagonal matrix in the standard mortar method a linear system of equations must be solved the problem is considered in a positive definite nonconforming variational as well as an equivalent saddle point formulation

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

an augmented lagrangian technique combined with a mortar algorithm for modelling mechanical contact problems summary a finite element formulation for three dimensional 3d contact mechanics using a mortar algorithm combined with a mixed penalty duality formulation from an augmented lagrangian approach is presented in this method no penalty parameter is introduced for the regularisation of the contact problem the contact approach based on the mortar method gives a smooth representation of the contact forces across the bodies interface and can be used in arbitrarily curved 3d configurations the projection surface used for integrating the equations is built using a local cartesian basis defined in each contact element a unit normal to the contact surface is defined locally at each element simplifying the implementation and linearisation of the equations the displayed examples show that the algorithm verifies the contact patch tests exactly and is applicable to large displacements problems with large sliding motions copyright 2012 john wiley sons ltd

exploiting symmetry in boundary element methods linear operator equations mathcal l f g are considered in the context of boundary element methods where the operator mathcal l is equivariant i e commutes with the actions of a given finite symmetry group by introducing a generalization of reynolds projectors a decomposition of the identity operator is constructed which in turn allows the decomposition of the problem mathcal l f g into a finite number of symmetric subproblems the data function g does not need to possess any symmetry properties it is shown that analogous reductions are possible for discretizations an explicit construction of the corresponding reduced system matrices is given this effects a considerable reduction in the computational complexity for example in the case of the isometry group of the 3 cube the computational complexity of a direct linear equation solver for full matrices is reduced by 99 65 percent specific decompositions of the identity are given for most of the significant finite isometry group

a dual mortar approach for 3d finite deformation contact with consistent linearization in this paper an approach for three dimensional frictionless contact based on a dual mortar formulation and using a primal dual active set strategy for direct constraint enforcement is presented we focus on linear shape functions but briefly address higher order interpolation as well the study builds on previous work by the authors for two dimensional problems first and foremost the ideas of a consistently linearized dual mortar scheme and of an interpretation of the active set search as a semi smooth newton method are extended to the 3d case this allows for solving all types of nonlinearities i e geometrical material and contact within one single newton scheme owing to the dual lagrange multiplier approach employed this advantage is not accompanied by an undesirable increase in system size as the lagrange multipliers can be condensed from the global system of equations moreover it is pointed out that the presented method does not make use of any regularization of contact constraints numerical examples illustrate the efficiency of our method and the high quality of results in 3d finite deformation contact analysis

minimal initial data for potential navier stokes singularities abstract assuming some initial data u 0 h 1 2 r 3 lead to a singularity for the 3d navier stokes equations we show that there are also initial data with the minimal h 1 2 norm which will produce a singularity

a nonsmooth version of newton s method newton s method for solving a nonlinear equation of several variables is extended to a nonsmooth case by using the generalized jacobian instead of the derivative this extension includes the b derivative version of newton s method as a special case convergence theorems are proved under the condition of semismoothness it is shown that the gradient function of the augmented lagrangian forc2 nonlinear programming is semismooth thus the extended newton s method can be used in the augmented lagrangian method for solving nonlinear programs

dual and approximate dual basis functions for b splines and nurbs comparison and application for an efficient coupling of patches with the isogeometric mortar method abstract this contribution defines and compares different methods for the computation of dual basis functions for b splines and non uniform rational b splines nurbs they are intended to be used as test functions for the isogeometric mortar method but other fields of application are possible too three different concepts are presented and compared the first concept is the explicit formula for the computation of dual basis functions for nurbs proposed in the work of carl de boor these dual basis functions entail minimal support i e the support of the dual basis functions is equal to the support of the corresponding b spline basis functions in the second concept dual basis functions are derived from the inversion of the gram matrix these dual basis functions have global support along the interface the third concept is the use of approximate dual basis functions which were initially proposed for the use in harmonic analysis the support of these functions is local but larger than the support of the associated b spline basis functions we propose an extension of the approximate dual basis functions for nurbs basis functions after providing the general formulas we elaborate explicit expressions for several degrees of spline basis functions all three approaches are applied in the frame of the mortar method for the coupling of non conforming nurbs patches a method which allows complex discretizations with multiple intersecting interfaces is presented numerical examples show that the explicitly defined dual basis functions with minimal support severely deteriorate the global stress convergence behavior of the mechanical analysis this fact is in accordance with mathematical findings in literature which state that the optimal reproduction degree of arbitrary functions is not possible without extending the support of the dual basis functions the dual basis functions computed from the inverse of the gram matrix yield accurate numerical results but the global support yields significantly higher computational costs in comparison to computations of conforming meshes only the approximate dual basis functions yield accurate and efficient computations where neither accuracy nor efficiency is significantly deteriorated in comparison to computations of conforming meshes all basic cases of t intersections and star intersections are studied furthermore an example which combines all basic cases in a complex discretization is given the applicability of the presented method for the nonlinear case and for shell formulations is shown with the help of one numerical example

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

the weak substitution method an application of the mortar method for patch coupling in nurbs based isogeometric analysis in this contribution a mortar type method for the coupling of non conforming nurbs surface patches is proposed the connection of non conforming patches with shared degrees of freedom requires mutual refinement which propagates throughout the whole patch due to the tensor product structure of nurbs surfaces thus methods to handle non conforming meshes are essential in nurbs based isogeometric analysis the main objective of this work is to provide a simple and efficient way to couple the individual patches of complex geometrical models without altering the variational formulation the deformations of the interface control points of adjacent patches are interrelated with a master slave relation this relation is established numerically using the weak form of the equality of mutual deformations along the interface with the help of this relation the interface degrees of freedom of the slave patch can be condensated out of the system a natural connection of the patches is attained without additional terms in the weak form the proposed method is also applicable for nonlinear computations without further measures linear and geometrical nonlinear examples show the high accuracy and robustness of the new method a comparison to reference results and to computations with the lagrange multiplier method is given