local existence of unique strong solution to non isothermal model for incompressible nematic liquid crystals in 3d in this paper we consider the non isothermal model for incompressible flow of nematic liquid crystals in three dimensions and prove the local existence and uniqueness of the strong solution with periodic initial conditions on t 3

decay estimates of solutions for the equations of motion of compressible viscous and heat conductive gases in an exterior domain in ℝ3 we consider the equations of motion of compressible viscous and heat conductive gases in an exterior domain in 3 we give the l q l p estimates for solutions to the linearized equations and show an optimal decay estimate for solutions to the nonlinear problem

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

a nurbs based hybrid collocation galerkin method for the analysis of boundary represented solids abstract the paper is concerned with a new numerical method nurbs based hybrid collocation galerkin method nurbs hcgm to solve the in plane motion problem of elastic solids it combines the merits of the so called scaled boundary finite element method sb fem and the isogeometric collocation method for the analysis the boundary scaling technique of sb fem is adopted it leads to a formulation where only the boundary of a structure is discretized here the nurbs basis functions are employed for the description of the geometry of the boundary as well as for the approximation of the displacements at the boundary this is in accordance with the boundary representation modeling technique which is commonly employed in computer aided design software the inner domain is described by a radial scaling parameter applying the weak form only in circumferential direction the governing partial differential equations of elasticity are transformed to an ordinary differential equation ode of euler type where the unknown displacements are a function of the radial scaling parameter in the present work a nurbs based collocation scheme is introduced to solve this equation nurbs basis functions are suggested for the approximation of the displacements in scaling direction the higher continuity provided by nurbs allows to use collocation to solve the ode directly instead of using the weak form in scaling direction the proposed approach is validated by comparison with the eigenvalue solution of the ode it is remarked that the eigenvalue solution is restricted to linear problems whereas the proposed method could be extended to nonlinear problems in general the presented formulation will allow to model patches bounded by an arbitrary number of contour boundaries the accuracy of the proposed approach is analyzed and estimated with respect to analytical solutions the computational cost is investigated with the help of numerical examples and is compared to isogeometric galerkin approach

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

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

nonparabolic dissipative systems modeling the flow of liquid crystals we study a simplified system which retains most of the interesting mathematical properties of the original ericksen leslie equations for the flow of liquid crystals this is a coupled nonparabolic dissipative dynamic system we derive several energy laws which enable us to prove the global existence of the weak solutions and the classical solutions we also discuss uniqueness and some stability properties of the system 1995 john wiley sons inc

integration by interpolation and look up for galerkin based isogeometric analysis abstract even though isogeometric analysis has a clear advantage regarding the number of degrees of freedom needed to achieve a certain level of accuracy the time needed for matrix assembly by means of numerical integration of stiffness or mass matrix entries constitutes a severe bottleneck in the process in order to overcome this difficulty we propose the new approach of integration by interpolation and lookup iil firstly applying spline interpolation to the common factors in the occurring integrals approximately transforms them into integrals of piecewise polynomial functions whose integrands are expressed in tensor product b spline form the common factors represent the influence of the geometry mapping i e the nurbs domain parameterization and the contributions of possibly non constant material coefficients secondly these integrals are evaluated exactly using pre computed look up tables for integrals of tri products of univariate b splines and their derivatives for the model case of elliptic problems we perform a theoretical analysis to demonstrate that the iil method maintains the overall approximation order of the galerkin discretization provided that the spline interpolation is sufficiently accurate moreover we provide a comparison of the computational complexity of our method with that of a standard gauss quadrature method finally we present experimental results to illustrate the performance of the iil method and to support our theoretical results

the pointwise estimates of diffusion wave for the navier stokes systems in odd multi dimensions we study the dissipation of solutions of the isentropic navier stokes equations in odd multi dimensions pointwise estimates of the time asymptotic shape of the solutions are obtained and shown to exhibit the generalized huygen s principle our approach is based on the detailed analysis of the green function of the linearized system this is used to study the coupling of nonlinear diffusion waves

optimal lp lq convergence rates for the compressible navier stokes equations with potential force in this paper we are concerned with the optimal lp lq convergence rates for the compressible navier stokes equations with a potential external force in the whole space under the smallness assumption on both the initial perturbation and the external force in some sobolev spaces the optimal convergence rates of the solution in lq norm with 2 q 6 and its first order derivative in l2 norm are obtained when the initial perturbation is bounded in lp with 1 p 6 5 the proof is based on the energy estimates on the solution to the nonlinear problem and some lp lq estimates on the semigroup generated by the corresponding linearized operator

dynamics of the ericksen leslie equations with general leslie stress ii the compressible isotropic case in this article the non isothermal compressible ericksen leslie system for nematic liquid crystals subject to general leslie stress is considered it is shown that this system is locally well posed within the l q setting and that for initial data close to equilibria points which are identical with the ones for the incompressible situation the solution exists globally moreover any global solution which does not develop singularities converges to an equilibrium in the topology of the natural state manifold note that no structural assumptions on the leslie coefficients are imposed and in particular parodi s relation is not being assumed the results can be viewed as an extension of the studies in hieber and pruss math ann 369 977 996 2017 for the incompressible case to the compressible situation

on large time behavior of solutions to the compressible navier stokes equations in the half space in r3 the navier stokes equation for compressible viscous fluid is considered on the half space in r3 under the zero dirichlet boundary condition for the momentum with initial data near an arbitrarily given equilibrium of positive constant density and zero momentum time decay properties in l2 norms for solutions of the linearized problem are investigated to obtain the rate of convergence in l2 norms of solutions to the equilibrium when initial data are sufficiently close to the equilibrium in some lower bounds are derived for solutions to the linearized problem one of which indicates a nonlinear phenomenon not appearing in the case of the cauchy problem on the whole space

isogeometric collocation methods for the reissner mindlin plate problem abstract within the general framework of isogeometric methods collocation schemes have been recently proposed as a viable and promising low cost alternative to standard isogeometric galerkin approaches in this paper isogeometric collocation methods for the numerical approximation of reissner mindlin plate problems are proposed for the first time locking free primal and mixed formulations are herein considered and the potential of isogeometric collocation as a geometrically flexible and computationally efficient simulation tool for shear deformable plates is shown through the solution of several numerical tests

an ultra weak finite element method as an alternative to a monte carlo method for an elasto plastic problem with noise an efficient method for obtaining numerical solutions of an elasto plastic oscillator with noise is considered since monte carlo simulations for the underlying stochastic process are too slow as an alternative approximate solutions of the partial differential equation defining the invariant measure of the process are studied the regularity of the solution of that partial differential equation is not sufficient to employ a standard finite element method to overcome the difficulty an ultra weak finite element method has been developed and successfully implemented

asymptotic behavior of solutions of the compressible navier stokes equations on the half space asymptotic behavior of solutions of the compressible navier stokes equations on the half space r n n 2 is considered around a given constant equilibrium a solution formula for the linearized problem is derived and l p estimates for solutions of the linearized problem are obtained for 2 p it is shown that as in the case of the cauchy problem the leading part of the solution of the linearized problem is decomposed into two parts one that behaves like diffusion waves and the other one purely diffusively there however are some aspects different from the cauchy problem especially in considering spatial derivatives it is also shown that the solution of the linearized problem approaches for large times the solution of the nonstationary stokes problem in some l p spaces and as a result a solution formula for the nonstationary stokes problem is obtained large time behavior of solutions of the nonlinear problem is then investigated in l p spaces for 2 p by applying the results on the linearized analysis and the weighted energy method the results indicate that there may be some nonlinear interaction phenomena not appearing in the cauchy problem

on the asymptotic convergence of spline collocation methods for partial differential equations we examine the asymptotic accuracy of the method of collocation for the approximate solution of linear elliptic partial differential equations specifically we consider the nodal collocation of a second order equation in the plane with biperiodicity conditions using tensor product smooth splines of odd degree as trial functions we prove optimal rates of convergence in l 2 for partial derivatives of the approximate solution which are of order at least two in one variable while the solution itself and its gradient converge in l 2 at rates less than the optimal approximation theoretic results

global well posedness and decay of smooth solutions to the non isothermal model for compressible nematic liquid crystals abstract the cauchy problem for the three dimensional non isothermal model for compressible nematic liquid crystals is considered existence of global in time smooth solutions is established provided that the initial datum is close to a steady state 0 d by using the l q l p estimates and the fourier splitting method if the initial perturbation is small in h 3 norm and bounded in l q q 1 6 5 norm we obtain the optimal decay rates for the first and second order spatial derivatives of solutions in addition the third and fourth order spatial derivatives of director field d in l 2 norm are achieved

analysis of the dpg method for the poisson equation we give an error analysis of the recently developed dpg method applied to solve the poisson equation and a convection diffusion problem we prove that the method is quasioptimal error estimates in terms of both the mesh size h and the polynomial degree p for various element shapes can be derived from our results results of extensive numerical experiments are also presented

bezier projection a unified approach for local projection and quadrature free refinement and coarsening of nurbs and t splines with particular application to isogeometric design and analysis abstract we introduce bezier projection as an element based local projection methodology for b splines nurbs and t splines this new approach relies on the concept of bezier extraction and an associated operation introduced here spline reconstruction enabling the use of bezier projection in standard finite element codes bezier projection exhibits provably optimal convergence and yields projections that are virtually indistinguishable from global l 2 projection bezier projection is used to develop a unified framework for spline operations including cell subdivision and merging degree elevation and reduction basis roughening and smoothing and spline reparameterization in fact bezier projection provides a quadrature free approach to refinement and coarsening of splines in this sense bezier projection provides the fundamental building block for h p k r adaptivity in isogeometric analysis

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

global existence of weak solutions of the nematic liquid crystal flow in dimension three for any bounded smooth domain 3 or 3 we establish the global existence of a weak solution u d 0 3 s2 of the initial boundary value or the cauchy problem of the simplified ericksen leslie system llf modeling the hydrodynamic flow of nematic liquid crystals for any initial and boundary or cauchy data u0 d0 h h1 s2 with d0 s 2 the upper hemisphere furthermore u d satisfies the global energy inequality 2016 wiley periodicals inc

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

analytical and computational assessment of locking in the hp finite element method locking is the phenomenon by which the numerical approximation of parameter dependent problems deteriorates for values of the parameter close to a limiting value in this paper we give a definition of locking and develop precise computable and analytic ways to quantify it using the example of nearly incompressible elasticity we show by means of computational and theoretical results the difference between the h version and php version in combatting locking our results establish the superiority of high order elements both h p and hp when the standard variational form is used we also discuss other issues such as curved elements mixed methods and locking phenomena for problems over anisotropic materials and over thin domains

singularity formation to the two dimensional compressible non isothermal nematic liquid crystal flows in a bounded domain abstract we study the singularity formation of strong solutions to the two dimensional 2d compressible non isothermal nematic liquid crystal flows in a bounded domain under a geometric condition of the initial orientation field we show that the strong solution exists globally if the temporal integral of the maximum norm of the divergence of the velocity is bounded our method relies on critical sobolev inequalities of logarithmic type and delicate energy estimates

liquid crystal flows in two dimensions this paper is concerned with a simplified hydrodynamic equation proposed by ericksen and leslie modeling the flow of nematic liquid crystals in dimension two we establish both interior and boundary regularity theorems for such a flow under smallness conditions as a consequence we establish the existence of global in time weak solutions on a bounded smooth domain in mathbb r 2 which are smooth everywhere with possible exceptions of finitely many singular times

classical solutions for the compressible liquid crystal flows with nonnegative initial densities abstract in this paper we study the cauchy problem of the simplified ericksen leslie system modeling compressible nematic liquid crystal flows in r 3 we first prove the local existence of the strong solutions provided that the data satisfies a natural compatibility condition then by deriving the smoothing effect of the solution in t 0 we conclude that it is indeed classical on some time interval 0 t 0 here we do not need the condition that initial density is bounded below away from zero

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

landau de gennes theory of nematic liquid crystals the oseen frank limit and beyond we study global minimizers of a continuum landau de gennes energy functional for nematic liquid crystals in three dimensional domains subject to uniaxial boundary conditions we analyze the physically relevant limit of small elastic constant and show that global minimizers converge strongly in w1 2 to a global minimizer predicted by the oseen frank theory for uniaxial nematic liquid crystals with constant order parameter moreover the convergence is uniform in the interior of the domain away from the singularities of the limiting oseen frank global minimizer we obtain results on the rate of convergence of the eigenvalues and the regularity of the eigenvectors of the landau de gennes global minimizer we also study the interplay between biaxiality and uniaxiality in landau de gennes global energy minimizers and obtain estimates for various related quantities such as the biaxiality parameter and the size of admissible strongly biaxial regions

overlapping schwarz preconditioners for isogeometric collocation methods isogeometric collocation methods are very recent and promising numerical schemes that preserve the advantages of isogeometric analysis but often exhibit better performances than their galerkin counterparts in the present paper an additive overlapping schwarz method for isogeometric collocation discretizations is introduced and studied the resulting preconditioner accelerated by gmres is shown to be scalable with respect to the number of subdomains and very robust with respect to the isogeometric discretization parameters such as the mesh size and polynomial degree as well as with respect to the presence of discontinuous elliptic coefficients and domain deformations

isogeometric collocation for phase field fracture models abstract phase field models based on the variational formulation for brittle fracture have recently been shown capable of accurately and robustly predicting complex crack behavior their numerical implementation requires costly operations at the quadrature point level which may include finding eigenvalues and forming tensor projection operators we explore the application of isogeometric collocation methods for the discretization of second order and fourth order phase field fracture models we show that a switch from isogeometric galerkin to isogeometric collocation methods has the potential to significantly speed up phase field fracture computations due to a reduction of point evaluations we advocate a hybrid collocation galerkin formulation that provides a consistent way of weakly enforcing neumann boundary conditions and multi patch interface constraints is able to handle the multiple boundary integral terms that arise from the weighted residual formulation and offers the flexibility to adaptively improve the crack resolution in the fracture zone we present numerical examples in one and two dimensions that illustrate the advantages of our approach

well posedness for the heat flow of harmonic maps and the liquid crystal flow with rough initial data we investigate the well posedness of 1 the heat flow of harmonic maps from mathbb r n to a compact riemannian manifold n without boundary for initial data in bmo and 2 the hydrodynamic flow u d of nematic liquid crystals on mathbb r n for initial data in bmo 1 bmo

global existence of small solutions to a class of nonlinear evolution equations on etudie le comportement global des solutions d une classe d equations d evolution non lineaires on considere le probleme aux valeurs initiales pour des equations quasilineaires qui sont des perturbations d equations lineaires dissipatives

a collocated isogeometric finite element method based on gauss lobatto lagrange extraction of splines generalizing the concept of bezier extraction we introduce an extraction operator that links c0c0 gauss lobatto lagrange functions with smooth splines this opens the door for collocated isogeometric analysis that combines the accuracy of the galerkin method with collocation type formation and assembly procedures we present the key ingredients of the technology i e integration by parts and the weighted residual form the interaction of gauss lobatto lagrange extraction with gauss lobatto quadrature and symmetrization with the ultra weak formulation we compare the new method with standard isogeometric galerkin and isogeometric point collocation methods for spline discretizations in three dimensions

long time behavior of solution for the compressible nematic liquid crystal flows in r 3 abstract in this paper we investigate the global existence and long time behavior of classical solution for the compressible nematic liquid crystal flows in three dimensional whole space first of all the global existence of classical solution is established under the condition that the initial data are close to the constant equilibrium state in h n r 3 n 3 framework then one establishes algebraic time decay for the classical solution by weighted energy method finally the algebraic decay rate of classical solution in l p r 3 norm with 2 p and optimal decay rate of their spatial derivative in l 2 r 3 norm are obtained if the initial perturbation belong to l 1 r 3 additionally

geometric modeling isogeometric analysis and the finite cell method the advent of isogeometric analysis iga using the same basis functions for design and analysis constitutes a milestone in the unification of geometric modeling and numerical simulation however an important class of geometric models based on the csg constructive solid geometry concept such as trimmed nurbs surfaces do not fully support the isogeometric paradigm since basis functions do not explicitly represent the boundary the finite cell method fcm is a high order fictitious domain method which offers simple meshing of potentially complex domains into a structured grid of cuboid cells while still achieving exponential rates of convergence for smooth problems in the present paper we first discuss the possibility to directly couple the finite cell method to csg without any necessity for meshing the three dimensional domain and then explore a combination of the best of the two approaches iga and fcm closely following ideas of the recently introduced shell fcm the resulting finite cell extension to isogeometric analysis achieves a truly straightforward transfer of a trimmed nurbs surface into an analysis suitable nurbs basis while benefiting from the favorable properties of the high order and high continuity basis functions accuracy and efficiency of the new approach are demonstrated by a numerical benchmark and its versatility is outlined by the analysis of different trimmed design variants of a brake disk

a note on the preconditioned gauss seidal gs method for linear systems in this note recent comparison results for preconditioned gauss seidel gs methods are discussed a new strict comparison result between two different preconditioned gs methods is given some errors in a recent article by niki et al j comput appl math 164 165 2004 587 are pointed out and a new proof for the corresponding results in niki et al is presented

strong solutions of the compressible nematic liquid crystal flow abstract we study strong solutions of the simplified ericksen leslie system modeling compressible nematic liquid crystal flows in a domain r 3 we first prove the local existence of a unique strong solution provided that the initial data 0 u 0 d 0 are sufficiently regular and satisfy a natural compatibility condition the initial density function 0 may vanish on an open subset i e an initial vacuum may exist we then prove a criterion for possible breakdown of such a local strong solution at finite time in terms of blow up of the quantities l t l x and d l t 3 l x

consistency and convergence properties of the isogeometric collocation method abstract isogeometric collocation iga c method has shown its superior behavior over galerkin method in terms of accuracy to computational time ratio and other aspects however relatively little has been published about numerical analysis of the iga c method this paper develops theoretical results on consistency and convergence of the iga c method to a generic boundary initial problem it shows that the iga c method is convergent when differential operator of the boundary initial problem is stable or strongly monotone finally we show some concrete examples whose differential operators are strongly monotone and the iga c method is convergent moreover 2d and 3d numerical examples are presented