optimal order preconditioners for the morse ingard equations abstract the morse ingard equations of thermoacoustics morse and ingard 1986 are a system of coupled time harmonic equations for the temperature and pressure of an excited gas they form a critical aspect of modeling trace gas sensors in this paper we analyze a reformulation of the system that has a weaker coupling between the equations than the original form we give a garding type inequality for the system that leads to optimal order asymptotic finite element error estimates we also develop preconditioners for the coupled system these are derived by writing the system as a 2 2 block system with pressure and temperature unknowns segregated into separate blocks and then using either the block diagonal or block lower triangular part of this matrix as a preconditioner consequently the preconditioner requires inverting smaller helmholtz like systems individually for the pressure and temperature rigorous eigenvalue bounds are given for the preconditioned system and these are supported by numerical experiments

a systematic development of solid shell element formulations for linear and non linear analyses employing only displacement degrees of freedom in the present contribution we propose a so called solid shell concept which incorporates only displacement degrees of freedom thus some major disadvantages of the usually used degenerated shell concept are overcome these disadvantages are related to boundary conditions the handling of soft and hard support the need for special co ordinate systems at boundaries the connection with continuum elements and in geometrically non linear analyses to a complicated update of the rotation vector first the kinematics of the so called solid shell concept in analogy to the degenerated shell concept are introduced then several modifications of the solid shell concept are proposed to obtain locking free solid shell elements leading also to formulations which allow the use of general three dimensional material laws and which are also able to represent the normal stresses and strains in thickness direction numerical analyses of geometrically linear and non linear problems are finally performed using solely assumed natural shear strain elements with a linear approximation in in plane direction although some considerations are needed to get comparable boundary conditions in the examples analysed the solid shell elements prove to work as good as the degenerated shell elements the numerical examples show that neither thickness nor shear locking are present even for distorted element shapes 1998 john wiley sons ltd

numerical implementation of multiplicative elasto plasticity into assumed strain elements with application to shells at large strains alternative formulations of isotropic large strain elasto plasticity are presented which are especially well suited for the implementation into assumed strain elements based on the multiplicative decomposition of the deformation gradient into elastic and plastic parts three distinct eigenvalue problems related to the reference intermediate and current configuration are investigated these eigenvalue problems are connected by similarity transformations which preserve the eigenvalues they play an important role in the subsequent development of alternative constitutive formulations and the corresponding finite element implementation the developed constitutive procedures rely on the right cauchy green tensor or equivalently on the green lagrangian strain tensor rather than the deformation gradient consequently they can be applied directly to assumed strain elements specifically we are concerned with efficient low order shell elements for which the assumed strain method has proven to be extremely powerful to overcome spurious locking effects

variational and projection methods for the volume constraint in finite deformation elasto plasticity this paper focuses on the treatment of volume constraints which in the context of elasto plasticity typically arise as a result of assuming volume preserving plastic flow projection methods based on the modification of the discrete gradient operator b often proposed on an ad hoc basis are systematically obtained in the variational context furnished by a three field hu washizu principle the fully nonlinear formulation proposed here is based on a local multiplicative split of the deformation gradient into volume preserving and dilatational parts without relying on rate forms of the weak form of momentum balance this approach fits naturally in a formulation of plasticity based on the multiplicative decomposition of the deformation gradient and enables one to exactly enforce the condition of volume preserving plastic flow within the framework proposed in this paper rate forms and incrementally objective algorithms are entirely bypassed

assumed strain stabilization of the eight node hexahedral element an accurate and robust stabilization based on the assumed strain method and an operator orthogonal to constant strain fields is presented for an eight node hexahedral element with uniform reduced integration the stabilization forces depend only on the element geometry and material properties user specified parameters are not needed yet the assumed strain stabilization is nearly as fast as perturbation type hourglass control in an explicit code it has excellent coarse mesh accuracy for linear elastic bending and is easily incorporated into a nonlinear program the assumed strain field is also used with four point integration which does not require stabilization in addition two forms of the b matrix are studied and it is shown that the mean form is more efficient since it passes the patch test in a simplified form

a 3d brick element based on hu washizu variational principle for mesh distortion in order to improve the accuracy of the mixed element for irregular meshes a penalty equilibrating 3d mixed element based on the hu washizu variational principle has been proposed in this paper the key idea in this work is to introduce a penalty term into the hu washizu three field functional which can enforce the stress components to satisfy the equilibrium equations in a weak form compared with the classical hybrid and mixed elements this technique can efficiently reduce the sensitivity of the element to mesh distortion the reason for the better results of this penalty technique has been investigated by considering a simple 2d problem from this investigation it has been found that the penalty parameter here plays the role of a scaling factor to reduce the influence of the parasitic strain or stress which is similar to the devised selective scaling factor proposed by sze furthermore compared with the hybrid stress element the proposed element based on the three field variational principle is more suitable for material non linear analysis numerical examples have demonstrated the improved performance of the present element especially in stress computation when fem meshes are irregular copyright 2002 john wiley sons ltd

a 4 node finite shell element for the implementation of general hyperelastic 3d elasticity at finite strains abstract in this paper a finite shell element for large deformations is presented based on extensible director kinematics the essential feature is an interface to arbitrary three dimensional material laws the non linear lagrangian formulation is based on the three field variational principle parametrized with the displacement vector enhanced green lagrangian strain tensor and second piola kirchhoff stress tensor the developed quadrilateral shell element is characterized by a course mesh accuracy and distortion insensitivity compared with bilinear displacement approaches furthermore plane stress response is approximately recovered in the asymptotic case of vanishing thickness a number of example problems investigating large deformation as well as finite strain applications are presented compressible and incompressible hyperelastic materials of the st venant kirchhoff neo hookean and mooney rivlin type are particularly used

finite element approximation and preconditioners for a coupled thermal acoustic model we consider a time harmonic mathematical model for a coupled thermal acoustic model we demonstrate coercivity of the pde system and hence well posedness and optimal error estimates for standard galerkin methods the large scale and ill conditioning of the discrete system motivates the study of preconditioners three block preconditioners are studied the first two applying block jacobi and block gauss seidel approaches to our problem give provably mesh independent spectral bounds the third preconditioner based on problem specific physical reasoning does not have this property however on moderate sized meshes it considerably outperforms the existing mesh techniques numerical experiments in two and three dimensions confirm the theoretical results

geometrically non linear enhanced strain mixed methods and the method of incompatible modes a class of assumed strain mixed finite element methods for fully non linear problems in solid mechanics is presented which when restricted to geometrically linear problems encompasses the classical method of incompatible modes as a particular case the method relies crucially on a local multiplicative decomposition of the deformation gradient into a conforming and an enhanced part formulated in the context of a three field variational formulation the resulting class of mixed methods provides a possible extension to the non linear regime of well known incompatible mode formulations in addition this class of methods includes non linear generalizations of recently proposed enhanced strain interpolations for axisymmetric problems which cannot be interpreted as incompatible modes elements the good performance of the proposed methodology is illustrated in a number of simulations including 2 d 3 d and axisymmetric finite deformation problems in elasticity and elastoplasticity remarkably these methods appear to be specially well suited for problems involving localization of the deformation as illustrated in several numerical examples

a preconditioned finite element solution of the coupled pressure temperature equations used to model trace gas sensors quartz enhanced photoacoustic spectroscopy qepas is a technique for detecting trace gases which relies on a quartz tuning fork resonator to amplify and measure the weak acoustic pressure waves th

a uniform deformation gradient hexahedron element with artificial hourglass control an 8 noded linear hexahedron element for large strain hyperelastic analysis is presented in this paper the element is based on a constant deformation gradient interpolation and is formulated using a mixed variational principle of the hu washizu type volumetric and isochoric components of the deformation are treated independently to ensure the correct evaluation of the element volume a simple procedure to control the propagation of spurious hourglass deformation modes is also discussed this is based on the addition of artificial hourglass forces which vanish under constant deformation gradient conditions thereby ensuring that the element passes a non linear version of the patch test applications in the field of neo hookean materials and superplastic forming processes are also considered

strichartz inequalities and the nonlinear schrodinger equation on compact manifolds kklwe prove strichartz estimates with fractional loss of derivatives for the schrodinger equation on any riemannian compact manifold as a consequence we infer low regularity local well posedness results in any dimension as well as global existence results for the cauchy problem of nonlinear schrodinger equations on surfaces in the case of defocusing polynomial nonlinearities and on three manifolds in the case of cubic defocusing nonlinearities we also discuss the optimality of these strichartz estimates on spheres

stochastic dissipative pde s and gibbs measures we study a class of dissipative nonlinear pde s forced by a random force omega t x with the space variable x varying in a bounded domain the class contains the 2d navier stokes equations under periodic or dirichlet boundary conditions and the forces we consider are those common in statistical hydrodynamics they are random fields smooth in t and stationary short correlated in time t in this paper we confine ourselves to kick forces of the form where the k s are smooth bounded identically distributed random fields the equation in question defines a markov chain in an appropriately chosen phase space a subset of a function space that contains the zero function and is invariant for the random flow of the equation concerning this markov chain we prove the following main result see theorem 2 2 the markov chain has a unique invariant measure to prove this theorem we present a construction assigning to any invariant measure a gibbs measure for a 1d system with compact phase space and apply a version of ruelle perron frobenius uniqueness theorem to the corresponding gibbs system we also discuss ergodic properties of the invariant measure and corresponding properties of the original randomly forced pde

design of simple low order finite elements for large strain analysis of nearly incompressible solids a simple four node quadrilateral and an eight node hexahedron for large strain analysis of nearly incompressible solids are proposed based on the concept of deviatoric volumetric split and the replacement of the compatible deformation gradient with an assumed modified counterpart the formulation developed is applicable to arbitrary material models the closed form of the corresponding exact tangent stiffnesses which have a particularly simple structure is derived it ensures asymptotically quadratic rates of convergence of the newton raphson scheme employed in the solution of the implicit finite element equilibrium equations from a practical point of view the incorporation of the proposed elements into existing codes is straightforward it requires only small changes in the routines of the standard displacement based 4 node quadrilateral and 8 node brick a comprehensive set of numerical examples involving hyperelasticity as well as multiplicative elasto plasticity is provided it illustrates the performance of the proposed elements over a wide range of applications including strain localisation problems metal forming simulation and adaptive analysis

momentum estimates and ergodicity for the 3d stochastic cubic ginzburg landau equation with degenerate noise abstract in this paper we consider the ergodicity of invariant measures for the stochastic ginzburg landau equation with degenerate random forcing first we show the existence and pathwise uniqueness of strong solutions with h 1 initial data and then the existence of an invariant measure for the feller semigroup by the krylov bogoliubov method then in the case of degenerate additive noise using the notion of asymptotically strong feller property we prove the uniqueness of invariant measures for the transition semigroup

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

assumed deformation gradient finite elements with nodal integration for nearly incompressible large deformation analysis an assumed strain finite element technique for non linear finite deformation is presented the weighted residual method enforces weakly the balance equation with the natural boundary condition and also the kinematic equation that links the elementwise and the assumed deformation gradient assumed gradient operators are derived via nodal integration from the kinematic weighted residual a variety of finite element shapes fits the derived framework four node tetrahedra eight 27 and 64 node hexahedra are presented here since the assumed deformation gradients are expressed entirely in terms of the nodal displacements the degrees of freedom are only the primitive variables displacements at the nodes the formulation allows for general anisotropic materials and no volumetric deviatoric split is required the consistent tangent operator is inexpensive and symmetric furthermore the material update and the tangent moduli computation are carried out exactly as for classical displacement based models the only deviation is the consistent use of the assumed deformation gradient in place of the displacement derived deformation gradient examples illustrate the performance with respect to the ability of the present technique to resist volumetric locking a constraint count can partially explain the insensitivity of the resulting finite element models to locking in the incompressible limit copyright 2008 john wiley sons ltd

mean strain eight node hexahedron with stabilization by energy sampling summary a method for stabilizing the mean strain hexahedron is described that differs from the currently known approaches for simplicity the developments are limited to linear elasticity but with an arbitrarily anisotropic elasticity matrix the technique relies on a sampling of the stabilization energy using two quadrature rules the mean strain quadrature and the full gaussian integration rule the use of two quadrature rules is shown to guarantee consistency and stability the stabilization energy is assumed to be generated by a modified constitutive matrix based on the spectral decomposition the spectral decomposition of the constitutive matrix identifies the stiff and flexible modes of deformation the stiff modes of deformation are only sampled by the mean strain integration which eliminates volumetric locking for isotropic materials as well as locking due to strongly anisotropic material properties the accuracy and convergence characteristics of the present formulations compare favorably with the capabilities of mean strain and other high performance hexahedral elements as implemented in abaqus copyright 2014 john wiley sons ltd

unique ergodicity for a class of stochastic hyperbolic equations with additive space time white noise in this paper we consider a certain class of second order nonlinear pdes with damping and space time white noise forcing posed on the d dimensional torus this class includes the wave equation for d 1 and the beam equation for d le 3 we show that the gibbs measure is the unique invariant measure for this system since the flow does not satisfy the strong feller property we introduce a new technique for showing unique ergodicity this approach may be also useful in situations in which finite time blowup is possible

large elasto plastic finite element analysis of solids and shells with the enhanced assumed strain concept abstract a finite element simulation of arbitrarily large elasto plastic deformations of three dimensional and shell structures is described two numerical aspects originally developed by j c simo have been combined i e the multiplicative large strain plasticity and the enhanced assumed strain finite element concept the constitutive model is worked out in its algorithmic version applying projection methods and consistent linearization the enhanced strain field is added to the conventional hexahedral displacement elements avoiding volumetric locking in the plastic range in addition to solid structures the three dimensional material model without any modifications is also applied to thin walled shell structures here an extended shell formulation a so called seven parameter model is used in which the thickness strain is augmented by a linear term the applicability of the entire formulation is demonstrated by several numerical examples

a flexible inner outer preconditioned gmres algorithm a variant of the gmres algorithm is presented that allows changes in the preconditioning at every step there are many possible applications of the new algorithm some of which are briefly discussed in particular a result of the flexibility of the new variant is that any iterative method can be used as a preconditioner for example the standard gmres algorithm itself can be used as a preconditioner as can cgnr or cgne the conjugate gradient method applied to the normal equations however the more appealing utilization of the method is in conjunction with relaxation techniques possibly multilevel techniques the possibility of changing preconditioners may be exploited to develop efficient iterative methods and to enhance robustness a few numerical experiments are reported to illustrate this fact

an adaptive stabilization strategy for enhanced strain methods in non linear elasticity this paper proposes and analyzes an adaptive stabilization strategy for enhanced strain es methods applied to quasistatic non linear elasticity problems the approach is formulated for any type of enhancements or material models and it is distinguished by the fact that the stabilization term is solution dependent the stabilization strategy is first constructed for general linearized elasticity problems and then extended to the non linear elastic regime via an incremental variational principle a heuristic choice of the stabilization parameters is proposed which in the numerical examples proved to provide stable approximations for a large range of deformations different problems and material models we also provide explicit lower bounds for the stabilization parameters that guarantee that the method will be stable these are not advocated since they are generally larger than the ones based on heuristics and hence prone to deteriorate the locking free behavior of es methods numerical examples with two different non linear elastic models in thin geometries and incompressible situations show that the method remains stable and locking free over a large range of deformations finally the method is strongly based on earlier developments for discontinuous galerkin methods and hence throughout the paper we offer a perspective about the similarities between the two

local times for solutions of the complex ginzburg landau equation and the inviscid limit abstract we consider the behaviour of the distribution for stationary solutions of the complex ginzburg landau equation perturbed by a random force it was proved in s kuksin and a shirikyan 2004 4 that if the random force is proportional to the square root of the viscosity 0 then the family of stationary measures possesses an accumulation point as 0 we show that if is such a point then the distributions of the l 2 norm and of the energy possess a density with respect to the lebesgue measure the proofs are based on ito s formula and some properties of local time for semimartingales

exponential mixing for stochastic pdes the non additive case we establish a general criterion which ensures exponential mixing of parabolic stochastic partial differential equations spde driven by a non additive noise which is white in time and smooth in space we apply this criterion on two representative examples 2d navier stokes ns equations and complex ginzburg landau cgl equation with a locally lipschitz noise due to the possible degeneracy of the noise doob theorem cannot be applied hence a coupling method is used in the spirit of kuksin and shirikyan j math pures appl 1 567 602 2002 and mattingly commun math phys 230 421 462 2002

three dimensional extension of non linear shell formulation based on the enhanced assumed strain concept conventional shell formulations such as 3 or 5 parameter theories or even 6 parameter theories including the thickness change as extra parameter require a condensation of the constitutive law in order to avoid a significant error due to the assumption of a linear displacement field across the thickness this means that the normal stress in thickness direction has to either vanish or be constant in general these extra constraints cannot be satisfied explicitly or they lead to elaborate strain expressions the main objective of the present study is to introduce directly a complete 3 d constitutive law without modification therefore a 7 parameter theory is utilized which includes a linear varying thickness stretch as extra variable allowing also large strain effects

an improved one point integration method for large strain elastoplastic analysis abstract in this paper we present a new one point integration method generalizing flanagan and belytschko s method internat j numer methods engrg 17 1981 679 706 and a modification of belytschko and bindeman s method comput methods appl mech engrg 88 1991 311 340 both in the frame of large deformation elastoplastic analysis these stabilization methods are combined with the radial return method used to integrate the constitutive law plane strain problems are first considered and the method is then generalized to axisymmetrical situations the explicit time integration scheme with its critical timestep is also considered a few examples are presented that slow the great time savings that can be obtained with reduced integration without any loss of accuracy and even with a gain in the solution quality since the underintegrated elements prove to be flexurally superconvergent

mean strain eight node hexahedron with optimized energy sampling stabilization for large strain deformation summary a method for stabilizing the mean strain hexahedron for applications to anisotropic elasticity was described by krysl 2015 the technique relied on a sampling of the stabilization energy using the mean strain quadrature and the full gaussian integration rule this combination was shown to guarantee consistency and stability the stabilization energy was expressed in terms of input parameters of the real material and the value of the stabilization parameter was fixed in a quasi optimal manner by linking the stabilization to the bending behavior of the hexahedral element krysl submitted here the formulation is extended to large strain hyperelasticity as an example the formulation allows for inelastic behavior to be modeled the stabilization energy is expressed through a stored energy function and contact with input parameters in the small strain regime is made as for small strain elasticity the stabilization parameter is determined to optimize bending performance the accuracy and convergence characteristics of the present formulations for both solid and thin walled structures shells compare favorably with the capabilities of mean strain and other high performance hexahedral elements described in the open literature and also with a number of successful shell elements copyright 2015 john wiley sons ltd

stabilization of bi linear mixed finite elements for tetrahedra with enhanced interpolation using volume and area bubble functions in order to overcome the oscillatory effects of the mixed bi linear galerkin formulation for tetrahedral elements a stabilization approach is presented to this end the mixed method of incompatible modes and the mixed method of enhanced strains are reformulated thus giving both the interpretation of a mixed finite element method with stabilization terms for non linear problems these are non linearly dependent on the current deformation state and therefore are replaced by linearly dependent stabilization terms the approach becomes most attractive for the numerical implementation since the use of quantities related to the previous newton iteration step typically arising for mixed enhanced elements is completely avoided the stabilization matrices for the mixed method of incompatible modes and the mixed method of enhanced strains are obtained with volume and area bubble functions various numerical examples are presented which illustrate successfully the stabilization effect for bi linear tetrahedral elements copyright 2007 john wiley sons ltd

a note on enhanced strain methods for large deformations abstract enhanced strain elements have been applied to simulate geometrically and materially non linear problems their main advantage is due to the fact that they perform very good in the incompressible limit as well as in bending situations however in compressive deformation states these elements depict stability modes which are associated with hour glass forms and thus denote a rank deficiency for such deformations in this paper the appearance of these modes will be investigated analytically by means of a simple representative example

random data cauchy theory for supercritical wave equations ii a global existence result we prove that the subquartic wave equation on the three dimensional ball with dirichlet boundary conditions admits global strong solutions for a large set of random supercritical initial data in bigcap s 1 2 h s theta times h s 1 theta we obtain this result as a consequence of a general random data cauchy theory for supercritical wave equations developed in our previous work 6 and invariant measure considerations inspired by earlier works by bourgain 2 3 on the non linear schrodinger equation which allow us to obtain also precise large time dynamical informations on our solutions

a uniform strain hexahedron and quadrilateral with orthogonal hourglass control the treatment of zero energy modes which arise due to one point integration of first order isoparametric finite elements is addressed a method for precisely isolating these modes for arbitrary geometry is developed two hourglass control schemes viscous and elastic are presented and examined in addition a convenient one point integration scheme which analytically integrates the element volume and uniform strain modes is presented

gibbsian dynamics and ergodicity for the stochastically forced navier stokes equation we study stationary measures for the two dimensional navier stokes equation with periodic boundary condition and random forcing we prove uniqueness of the stationary measure under the condition that all determining modes are forced the main idea behind the proof is to study the gibbsian dynamics of the low modes obtained by representing the high modes as functionals of the time history of the low modes

on a physically stabilized one point finite element formulation for three dimensional finite elasto plasticity abstract in the case of linear elasticity a direct connection between the concept of reduced integration with hourglass stabilization and a mixed method can usually be established in the non linear case this is in general not possible to overcome this difficulty we suggest in this paper a new concept based on a taylor expansion of the constitutively dependent quantities with respect to the centre of the element the push forward of the second linear term of the taylor series for the first piola kirchhoff stress tensor to the current configuration determines the so called hourglass stabilization part of the residual force vector due to the fact that the element uses only one gauss point and the hourglass stabilization part is computed by means of a simple functional evaluation the present element technology is very efficient from the computational point of view in contrast to the 2d case the computation of the jacobi determinant only in the centre of the 3d element does not yield the correct volume if the element shape deviates from being a parallelipiped it is shown in the paper that the error becomes negligibly small for a relatively coarse discretization the formulation is free of volumetric locking and can compete with shell formulations up to an aspect ratio of about hundred for bending dominated problems at least two elements over the thickness are needed in order to compute the onset of plastification correctly the element behaves very robustly in finite elasticity and inelasticity also when large element distortions occur

on distribution of energy and vorticity for solutions of 2d navier stokes equation with small viscosity we study distributions of some functionals of space periodic solutions for the randomly perturbed 2d navier stokes equation and of their limits when the viscosity goes to zero the results obtained give explicit information on distribution of the velocity field of space periodic turbulent 2d flows

from functional analysis to iterative methods we examine condition numbers preconditioners and iterative methods for finite element discretizations of coercive pdes in the context of the fundamental solvability result the lax milgram lemma working in this hilbert space context is justified because finite element operators are restrictions of infinite dimensional hilbert space operators to finite dimensional subspaces moreover useful insight is gained as to the relationship between hilbert space and matrix condition numbers and translating hilbert space fixed point iterations into matrix computations provides new ways of motivating and explaining some classic iteration schemes in this framework the simplest preconditioner for an operator from a hilbert space into its dual is the riesz isomorphism simple analysis gives spectral bounds and iteration counts bounded independent of the finite element subspaces chosen moreover the abstraction allows us to consider not only riesz map preconditioning for convection diffusion equations in h 1 but also operators on other hilbert spaces such as planar elasticity in left h 1 right 2

a large deformation solid shell concept based on reduced integration with hourglass stabilization in this paper a new eight node brick solid shell finite element formulation based on the concept of reduced integration with hourglass stabilization is presented the work focuses on static problems the starting point of the derivation is the three field variational functional upon which meanwhile established 3d enhanced strain concepts are based important additional assumptions are made to transfer the approach into a powerful solid shell first of all a taylor expansion of the first piola kirchhoff stress tensor with respect to the normal through the centre of the element is carried out in this way the stress becomes a linear function of the shell surface co ordinates whereas the dependence on the thickness co ordinate remains non linear secondly the jacobian matrix is replaced by its value in the centre of the element these two assumptions lead to a computationally efficient shell element which requires only two gauss points in the thickness direction and one gauss point in the plane of the shell element additionally three internal element degrees of freedom have to be determined to avoid thickness locking one important advantage of the element is the fact that a fully three dimensional stress state can be modelled without any modification of the constitutive law the formulation has only displacement degrees of freedom and the geometry in the thickness direction is correctly displayed copyright 2006 john wiley sons ltd

exponential mixing properties of stochastic pdes through asymptotic coupling we consider parabolic stochastic partial differential equations driven by white noise in time we prove exponential convergence of the transition probabilities towards a unique invariant measure under suitable conditions these conditions amount essentially to the fact that the equation transmits the noise to all its determining modes several examples are investigated including some where the noise does not act on every determining mode directly

a multiple quadrature eight node hexahedral finite element for large deformation elastoplastic analysis abstract a multiple quadrature underintegrated hexahedral finite element which is free of volumetric and shear locking and has no spurious singular modes is described and implemented for nonlinear analysis finite element formulations are derived in the corotational coordinate system the use of consistent tangent operators for large deformation elastoplasticity with nonlinear isotropic kinematic hardening rules preserves the quadratic rate of convergence of the newton s iteration method in static analysis test problems studied demonstrate the efficiency and effectiveness of this element in solving a wide variety of problems including sheet metal forming processes

a coupling approach to randomly forced nonlinear pde s ii we consider a class of discrete time random dynamical systems and establish the exponential convergence of its trajectories to a unique stationary measure the result obtained applies in particular to the 2d navier stokes system and multidimensional complex ginzburg landau equation with random kick force

randomly forced cgl equation stationary measures and the inviscid limit we study a complex ginzburg landau cgl equation perturbed by a random force which is white in time and smooth in the space variable x assuming that dim x 4 we prove that this equation has a unique solution and discuss its asymptotic in time properties next we consider the case when the random force is proportional to the square root of the viscosity and study the behaviour of stationary solutions as the viscosity goes to zero we show that under this limit a subsequence of solutions in question converges to a nontrivial stationary process formed by global strong solutions of the nonlinear schrodinger equation

exponential convergence for the stochastically forced navier stokes equations and other partially dissipative dynamics we prove that the two dimensional navier stokes equations possess an exponentially attracting invariant measure this result is in fact the consequence of a more general harris like ergodic theorem applicable to many dissipative stochastic pdes and stochastic processes with memory a simple iterated map example is also presented to help build intuition and showcase the central ideas in a less encumbered setting to analyze the iterated map a general doeblin like theorem is proven one of the main features of this paper is the novel coupling construction used to examine the ergodic theory of the non markovian processes

hourglass control in linear and nonlinear problems abstract mesh stabilization techniques for controlling the hourglass modes in under integrated hexahedral and quadrilateral elements are described it is shown that the orthogonal hourglass techniques previously developed can be obtained from simple requirements that insure the consistency of the finite element equations in the sense that the gradients of linear fields are evaluated correctly it is also shown that this leads to an hourglass control that satisfies the patch test the nature of the parameters which relate the generalized stresses and strains for controlling hourglass modes is examined by means of a mixed variational principle and some guidelines for their selection are discussed finally effective means of implementing these hourglass procedure in computer codes are described applications to both the laplace equation and the equations of solid mechanics in 2 and 3 dimensions are considered

improved versions of assumed enhanced strain tri linear elements for 3d finite deformation problems abstract improved three dimensional tri linear elements for finite deformation problems are developed based on an assumed enhanced strain methodology which in the linear regime incorporates the classical method of incompatible modes as a particular case three crucial modifications of a recently proposed element which reduces to wilson s brick in the linear regime are introduced to prevent locking response in distorted configurations and to maintain proper rank while preserving excellent performance in bending dominated and localization problems i a modified quadrature rule ii an additional enhancement of the divergence term and iii a modification of the isoparametric shape function derivatives for the three dimensional problem moreover these modified shape function derivatives are shown to improve the performance of the standard tri linear brick in distorted configurations in addition a strategy is described to circumvent the memory storage requirements in the static condensation procedure of the enhanced strain parameters the excellent performance of the improved methodology is illustrated in representative numerical simulations

a multigrid method enhanced by krylov subspace iteration for discrete helmholtz equations standard multigrid algorithms have proven ineffective for the solution of discretizations of helmholtz equations in this work we modify the standard algorithm by adding gmres iterations at coarse levels and as an outer iteration we demonstrate the algorithm s effectiveness through theoretical analysis of a model problem and experimental results in particular we show that the combined use of gmres as a smoother and outer iteration produces an algorithm whose performance depends relatively mildly on wave number and is robust for normalized wave numbers as large as 200 for fixed wave numbers it displays grid independent convergence rates and has costs proportional to the number of unknowns

finite element concepts for finite elastoplastic strains and isotropic stress response in shells theoretical and computational analysis this paper presents appropriate finite element concepts accounting for finite elastoplastic strains and isotropic stress response in arbitrary shells three parameterization strategies for the calculation of thin and thick smooth shells as well as shell intersections are discussed in detail in this context the importance of the deformation gradient for an efficient implementation is especially emphasized according to the parameterizations the computational plasticity algorithms are derived in general three dimensional 3 d form for thick shell structures and a restricted 2 d form satisfying the plane stress state underlying thin shells in order to describe arbitrary shell geometries the isoparametric concept is applied to all proposed finite element formulations these are based on quadrilateral 4 node mixed finite shell elements the variational approaches of the enhanced assumed strain method the assumed natural strain concept and the reduced integration technique are accurately considered by means of representative numerical examples the performance of the three different finite element concepts is demonstrated furthermore their individual properties are addressed with respect to efficiency and numerical accuracy thus crucial consequences for an optimal modelling of finite plastic strains in shells can be outlined

optimal solid shells for non linear analyses of multilayer composites ii dynamics abstract we are presenting a simple low order solid shell element formulation having only displacement degrees of freedom dofs i e without rotational dofs that has an optimal number of parameters to pass the patch tests and thus allows for efficient and accurate analyses of large deformable multilayer shell structures using elements at extremely high aspect ratio with the dynamics referred to a fixed inertial frame the elements can be used to analyze multilayer shell structures undergoing large overall motion the formulation of this element is based on the mixed hu washizu variational principle leading to a novel enhancing strain tensor enhanced assumed strain eas method that renders the computation particularly efficient with improved in plane and out of plane bending behavior poisson thickness locking especially in refined analyses of composite structures involving a large number of high aspect ratio layers the energy momentum conserving algorithm in the context of current solid shell element is presented we discuss the eas formulation based on the displacement gradient and its complexity compared to formulation on the green lagrange strain shear locking and curvature thickness locking are treated using the assumed natural strain ans method the element has an optimal combination of the ans method and the minimal number of eas parameters required to pass the plate bending patch test numerical examples involving dynamic analyses with conservation of energy and momentum of multilayer shell structures having a large range of element aspect ratios are presented several implicit direct integration methods with without numerical dissipation are used and compared in terms of the accuracy stability and cost in multilayer shell structure finally we note that the topic in this paper is a fitting dedication to professor ekkehard ramm who has made important pioneering contributions in this research direction

mixed finite element methods reduced and selective integration techniques a unification of concepts the equivalence of certain classes of mixed finite element methods with displacement methods which employ reduced and selective integration techniques is established this enables one to obtain the accuracy of the mixed formulation without incurring the additional computational expense engendered by the auxiliary field of the mixed method applications and numerical examples are presented for problems with constraints which can be difficult to enforce in finite element approximations and have often dictated the use of mixed principles these include thin beams and plates and linear and nonlinear incompressible and nearly incompressible continuum problems in solid and fluid mechanics

a theoretical and computational model for isotropic elastoplastic stress analysis in shells at large strains the article presents new aspects of large strain isotropic elastoplastic analysis of shells on the shell theory side we consider several possible continuum based parametrizations of the shell deformation and investigate their applicability with respect to the implementation of general three dimensional local constitutive models as an aspect of computational shell analysis we then discuss in detail the numerical implementation of a shell based on the parametrization of the top and bottom surface in terms of a new 8 node brick type mixed finite shell element on the basis of assumed strain and enhanced strain variational approaches on the side of plasticity theory we adopt a recently proposed intermediate configuration free formulation of large strain plasticity based on the notion of a plastic metric and consider its representation with respect to the parameter manifold of the shell this frame is formulated in terms of principal strains and principal stresses with respect to dual co and contra variant eigenvector triads associated with a mixed variant elastic strain tensor the new aspect of computational plasticity is the development of a stress update algorithm for the plasticity model mentioned above which includes a general return mapping in the eigenvalue space the algorithm can be applied to general coordinate charts and is particularly convenient when formulated with respect to the space of the curvilinear coordinates of the shell parametrization we demonstrate the performance of the finite element formulation and constitutive algorithms by means of several numerical examples

a class of mixed assumed strain methods and the method of incompatible modes a three field mixed formulation in terms of displacements stresses and an enhanced strain field is presented which encompasses as a particular case the classical method of incompatible modes within this frame work incompatible elements arise as particular compatible mixed approximations of the enhanced strain field the conditions that the stress interpolation contain piece wise constant functions and be l2 ortho gonal to the enhanced strain interpolation ensure satisfaction of the patch test and allow the elimination of the stress field from the formulation the preceding conditions are formulated in a form particularly convenient for element design as an illustration of the methodology three new elements are developed and shown to exhibit good performance a plane 3d elastic plastic quad an axisymmetric element and a thick plate bending quad the formulation described herein is suitable for non linear analysis

exponential mixing of the 2d stochastic navier stokes dynamics we consider the navier stokes equation on a two dimensional torus with a random force which is white noise in time and excites only a finite number of modes the number of excited modes depends on the viscosity v and grows like v 3 when v goes to zero we prove that this markov process has a unique invariant measure and is exponentially mixing in time

a model adaptive hanging node concept based on a new non linear solid shell formulation in this contribution we propose an innovative approach to automatically refine shell finite element meshes in thickness direction the procedure employs a new eight node solid shell element formulation based on the concept of reduced integration with hourglass stabilization apart from the important advantages of computational efficiency and robustness this element formulation offers the possibility to work with a variable mesh density in thickness direction in the present paper we suggest an adaptive approach using the hanging node concept to avoid additional mesh distortion among different stress and strain based error criteria the warping of the cross section and the residual error in the solution of the strong form of the balance of linear momentum have been proven to be suitable error measures the former represents the deviation of the realistically non linear dependence of the characteristic strain coefficient from its corresponding linear distribution applications show that the mesh refines as expected in loading domains in the neighbourhood of bearings or regions with strongly accumulating inelastic deformation on the other hand the discretization remains rather coarse in uncritical parts of the structure which decreases the computational effort noticeably

fast iterative solution of stabilised stokes systems part i using simple diagonal preconditioners mixed finite element approximation of the classical stokes problem describing slow viscous incompressible flow gives rise to symmetric indefinite systems for the discrete velocity and pressure variables iterative solution of such indefinite systems is feasible and is an attractive approach for large problems the use of stabilisation methods for convenient but unstable mixed elements introduces stabilisation parameters we show how these can be chosen to obtain rapid iterative convergence we propose a conjugate gradient like method the method of preconditioned conjugate residuals which is applicable to symmetric indefinite problems describe the effects of stabilisation on the algebraic structure of the discrete stokes operator and derive estimates of the eigenvalue spectrum of this operator on which the convergence rate of the iteration depends here we discuss the simple case of diagonal preconditioning our results apply to both locally and globally stabilised mixed elements as well as to elements which are inherently stable we demonstrate that convergence rates comparable to that achieved using the diagonally scaled conjugate gradient method applied to the discrete laplacian are approachable for the stokes problem

development of a one point quadrature shell element for nonlinear applications with contact and anisotropy a general purpose shell element for nonlinear applications including sheet metal forming simulation is developed based on reduced integration with one point quadrature the developed shell element has five degrees of freedom and four nodes it covers flexible warping behavior without artificial warping correction a physical stabilization scheme with the assumed natural strain method is employed to derive a strain field that can be decomposed into the sum of a constant and a linear term with respect to the natural coordinates the rigid body projection is introduced to treat rigid body rotations effectively the shell element incorporates elasto plastic planar anisotropic material models based on the incremental deformation theory linear and nonlinear patch tests are performed and the results are compared with analytical or previously reported results simulations that include impact and deformable body contact are performed to show the robustness of the contact algorithm finally to demonstrate the capability of handling anisotropic materials the developed shell element is used for the circular cup drawing process simulation in order to predict the earing profile of al 2008 t4 alloy sheet