large strain viscoelastic constitutive models this paper discusses a new continuum formulation for viscoelastic materials at finite strains the model proposed is based on the multiplicative decomposition of the isochoric component of deformation gradient into elastic and viscous contribution and the generalized maxwell rheological model the inelastic or viscous components of the deformation gradient provide the internal variables required for the irreversible thermo mechanical model nonlinear rate type of evolution equations are then proposed for the internal variables these are based on a particular linear relaxation form of the generalized maxwell model which leads to a viscoelastic formulation that can be seen as a particular case of a large strain viscoplastic model based on maximum plastic dissipation in addition to the rate evolution equations simple incremental stress update equations are proposed these closely resemble the radial return algorithms used in von mises plasticity finally a spatial form of the viscoelastic formulation is presented for isotropic materials this formulation is based on principal directions and logarithmic stretches again incremental equations will be considered in order to permit subsequent computational implementations

material state remapping in computational solid mechanics a computational procedure for remapping material state information from one finite element mesh to another is described the procedure is useful in connection with evolving meshes for inelastic problems as for example occur in the context of fracture simulation and adaptive mesh refinement the proposed method is based on weak enforcement of equality between corresponding fields on the two meshes where piecewise constant fields on both meshes are generalized from the quadrature point values the essential algorithmic problem is that of calculating the volume partition of an arbitrary convex region with respect to a covering set of disjoint convex regions instead of geometrically resolving the associated intersections the problem is herein approximated by a constrained optimization problem which may be readily and efficiently solved computationally this formulation is a main contribution of the paper computational examples are given that illustrate the effectiveness of the proposed procedure copyright 2002 john wiley sons ltd

multiscale phenomena green s functions the dirichlet to neumann formulation subgrid scale models bubbles and the origins of stabilized methods abstract an approach is developed for deriving variational methods capable of representing multiscale phenomena the ideas are first illustrated on the exterior problem for the helmholtz equation this leads to the well known dirichlet to neumann formulation next a class of subgrid scale models is developed and the relationships to bubble function methods and stabilized methods are established it is shown that both the latter methods are approximate subgrid scale models the identification for stabilized methods leads to an analytical formula for the intrinsic time scale whose origins have been a mystery heretofore

finite element formulation for modeling nonlinear viscoelastic elastomers nonlinear viscoelastic response of reinforced elastomers is modeled using a three dimensional mixed finite element method with a nonlocal pressure field a general second order unconditionally stable exponential integrator based on a diagonal pade approximation is developed and the bergstrom boyce nonlinear viscoelastic law is employed as a prototype model an implicit finite element scheme with consistent linearization is used and the novel integrator is successfully implemented finally several viscoelastic examples including a study of the unit cell for a solid propellant are solved to demonstrate the computational algorithm and relevant underlying physics

variationally consistent modeling of finite strain plasticity theory with non linear kinematic hardening abstract variational constitutive updates provide a physically and mathematically sound framework for the numerical implementation of material models in contrast to conventional schemes such as the return mapping algorithm they are directly and naturally based on the underlying variational principle hence the resulting numerical scheme inherits all properties of that principle in the present paper focus is on a certain class of those variational methods which relies on energy minimization consequently the algorithmic formulation is governed by energy minimization as well accordingly standard optimization algorithms can be applied to solve the numerical problem a further advantage compared to conventional approaches is the existence of a natural distance semi metric induced by the minimization principle such a distance is the foundation for error estimation and as a result for adaptive finite elements methods though variational constitutive updates are relatively well developed for so called standard dissipative solids i e solids characterized by the normality rule the more general case i e generalized standard materials is far from being understood more precisely int j sol struct 2009 46 1676 1684 represents the first step towards this goal in the present paper a variational constitutive update suitable for a class of nonlinear kinematic hardening models at finite strains is presented two different prototypes of armstrong frederick type are re formulated into the aforementioned variationally consistent framework numerical tests demonstrate the consistency of the resulting implementation

a variational formulation of constitutive models and updates in non linear finite viscoelasticity the purpose of this article is to present a general framework for constitutive viscoelastic models in finite strain regime the approach is qualified as variational since the constitutive updates obey a minimum principle within each load increment the set of internal variables is strain based and employs according to the specific model chosen a multiplicative decomposition of strain into elastic and viscous components the present approach shares the same technical procedures used for analogous models of plasticity or viscoplasticity such as the solution of a minimization problem to identify inelastic updates and the use of exponential mapping for time integration however instead of using the classical decomposition of inelastic strains into amplitude and direction we take advantage of a spectral decomposition that provides additional facilities to accommodate into simple analytical expressions a wide set of specific models moreover appropriate choices of the constitutive potentials allow the reproduction of other formulations in the literature the final part of the paper presents a set of numerical examples in order to explore the characteristics of the formulation as well as its applicability to usual large scale fem analyses copyright 2005 john wiley sons ltd

the hele shaw asymptotics for mechanical models of tumor growth models of tumor growth now commonly used present several levels of complexity both in terms of the biomedical ingredients and the mathematical description our first goal here is to formulate a free boundary model of hele shaw type a variant including growth terms starting from the description at the cell level and passing to the stiff limit in the pressure law of state in contrast with the classical hele shaw problem here the geometric motion governed by the pressure is not sufficient to completely describe the dynamics a complete description requires the equation on the cell number density we then go on to consider a more complex model including the supply of nutrients through vasculature and we study the stiff limit for the involved coupled system

a cartesian parametrization for the numerical analysis of material instability summary we examine four parametrizations of the unit sphere in the context of material stability analysis by means of the singularity of the acoustic tensor we then propose a cartesian parametrization for vectors that lie a cube of side length two and use these vectors in lieu of unit normals to test for the loss of the ellipticity condition this parametrization is then used to construct a tensor akin to the acoustic tensor it is shown that both of these tensors become singular at the same time and in the same planes in the presence of a material instability the performance of the cartesian parametrization is compared against the other parametrizations with the results of these comparisons showing that in general the cartesian parametrization is more robust and more numerically efficient than the others copyright 2016 john wiley sons ltd

error estimation and adaptive meshing in strongly nonlinear dynamic problems we present work aimed at developing a general framework for mesh adaption in strongly nonlinear possibly dynamic problems we begin by showing that the solutions of the incremental boundary value problem for a wide class of materials including nonlinear elastic materials compressible newtonian fluids and viscoplastic solids obey a minimum principle provided that the constitutive updates are formulated appropriately this minimum principle can be taken as a basis for asymptotic error estimation in particular we chose to monitor the error of a lower order projection of the finite element solution the optimal mesh size distribution then follows from a posteriori error indicators which are purely local i e can be computed element by element we demonstrate the robustness and versatility of the computational framework with the aid of convergence studies and selected examples of application

a method of linearization for painleve equations painleve iv v abstract a method to linearize the initial value problem of the painleve equations iv v is given the procedure involves formulating a riemann hilbert boundary value problem on intersecting lines for the inverse monodromy problem this boundary value problem is reduced to a sequence of standard problems on single lines in a certain range of parameter space schlesinger transformations allow one to completely cover the parameter space special solutions are constructed from special cases of the riemann problem as well

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 simple error estimator and adaptive procedure for practical engineerng analysis a new error estimator is presented which is not only reasonably accurate but whose evaluation is computationally so simple that it can be readily implemented in existing finite element codes the estimator allows the global energy norm error to be well estmated and alos gives a good evaluation of local errors it can thus be combined with a full adaptive process of refinement or more simply provide guidance for mesh redesign which allows the user to obtain a desired accuracy with one or two trials when combined with an automatic mesh generator a very efficient guidance process to analysis is avaiable estimates other than the energy norm have successfully been applied giving for instance a predetermined accuracy of stresses

an adaptive procedure based on background cells for meshless methods in practical implementations meshless methods using moving least squares mls approximation and global galerkin formulation require a background mesh for domain integration an adaptive procedure based on background mesh is developed for meshless methods using mls it comprises a cell energy error estimate and a local domain refinement technique the error estimate differs from conventional pointwise approaches in that it evaluates error based on individual cells instead of points for each cell a computed cell energy and a reference cell energy are generated based on a same stress field by using two different integration schemes the difference between the two energy values is used as the basic measure of error estimation the advantage of this strategy is that it requires only one stress field and no reference field needs to be furnished this has significance as the stress field generated by meshless methods is already very smooth and traditional error estimates based on stress smoothing techniques for finite element methods are not applicable to achieve high efficiency in domain refinement a local technique based on the delaunay algorithm is implemented in this technique each node is assigned a scaling factor to control local nodal density refinement of the neighborhood of a node is accomplished simply by adjusting its scaling factor the numerical experiments in this paper show that the proposed adaptive procedure is simple effective and efficient the limitations of the approach are also discussed

h adaptive finite element methods for dynamic problems with emphasis on localization various types of error criteria are examined and it is shown that for problems involving plastic response or localization an error criterion based on an l 2 projection of strains is the most effective for the constant strain elements considered here examples of one dimensional and two dimensional localization shear band formation problems are given

a finite strain incremental secant homogenization model for elasto plastic composites abstract this paper presents a finite strain extension of the incremental secant mean field homogenization mfh formulation for two phase elasto plastic composites the formulation of the local finite strain elasto plastic constitutive equations of each phase is based on a multiplicative decomposition of the deformation gradient as suggested by simo in computer methods in applied mechanics and engineering 99 1 61 112 1992 the latter has proposed algorithms which preserve the classical return mapping schemes of the infinitesimal theory by using principal kirchhoff stresses and logarithmic eigenvalues of the left elastic cauchy green strain relying on this property we show that by considering a quadratic logarithmic free energy and j2 flow theory at the local level infinitesimal strain incremental secant mfh is readily extended to finite strains the proposed formulation and corresponding numerical algorithms are then presented finally the predictions are illustrated with several numerical simulations which are verified against full field finite element simulations of composite cells demonstrating that the micro mechanically based approach is able to predict the influence of the micro structure and of its evolution on the macroscopic properties in a very cost effective manner

schlesinger transformations of painleve ii v the explicit form of the schlesinger transformations for the second third fourth and fifth painleve equations is given

singular limit of the porous medium equation with a drift abstract we study the stiff pressure limit of a nonlinear drift diffusion equation where the density is constrained to stay below the maximal value one the challenge lies in the presence of a drift and the consequent lack of monotonicity in time in the limit a hele shaw type free boundary problem emerges which describes the evolution of the congested zone where density equals one we discuss pointwise convergence of the densities as well as the bv regularity of the limiting free boundary

towards variational constitutive updates for non associative plasticity models at finite strain models based on a volumetric deviatoric split in this paper an enhanced variational constitutive update suitable for a class of non associative plasticity theories at finite strain is proposed in line with classical numerical formulations for plasticity models such as the by now established return mapping algorithm variational constitutive updates represent a numerical method for computing the unknown state variables however in contrast to conventional algorithms variational constitutive updates are fully variational i e all unknown variables follow jointly from minimizing a certain potential in addition to the physical and mathematical elegance of these variational schemes they show several practical advantages as well for instance numerically efficient and robust optimization schemes can be directly employed for solving the resulting minimization problem since mathematically plasticity is a non smooth problem and often it leads to highly singular systems of equations as known from single crystal plasticity a robust implementation is of utmost importance so far variational constitutive updates have been developed for different classes of standard dissipative solids i e solids characterized by associative evolution equations and flow rules in the present paper this framework is extended to a certain class of non associative plasticity models at finite strain all models falling into this class show a volumetric deviatoric split of the helmholtz energy and the yield function typical prototypes are drucker prager or mohr coulomb models playing an important role in soil mechanics the efficiency and robustness of the resulting algorithmic formulation is demonstrated by means of selected numerical examples

homogenization of the hele shaw problem in periodic spatiotemporal media we consider the homogenization of the hele shaw problem in periodic media that are inhomogeneous both in space and time after extending the theory of viscosity solutions into this context we show that the solutions of the inhomogeneous problem converge in the homogenization limit to the solution of a homogeneous hele shaw type problem with a general possibly nonlinear dependence of the free boundary velocity on the gradient moreover the free boundaries converge locally uniformly in hausdorff distance

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

time discretized variational formulation of non smooth frictional contact the present work extends the non smooth contact class of algorithms introduced by kane et al to the case of friction the formulation specifically addresses contact geometries e g involving multiple collisions between tightly packed non smooth bodies for which neither normals nor gap functions can be properly defined a key aspect of the approach is that the incremental displacements follow from a minimum principle the objective function comprises terms which account for inertia strain energy contact friction and external forcing the euler lagrange equations corresponding to this extended variational principle are shown to be consistent with the equations of motion of solids in frictional contact in addition to its value as a basis for formulating numerical algorithms the variational framework offers theoretical advantages as regards the selection of trajectories in cases of non uniqueness we present numerical and analytical examples which demonstrate the good momentum and energy conservation characteristics of the numerical algorithms as well as the ability of the approach to account for stick and slip conditions

a variationally consistent mesh adaptation method for triangular elements in explicit lagrangian dynamics in this paper a variational formulation for mesh adaptation procedures involving local mesh changes for triangular meshes is presented such local adaptive changes are very well suited for explicit methods as they do not involve significant computational expense they also greatly simplify the projection of field variables from the old to the new meshes crucially the variational nature of the formulation used to derive the equilibrium equations at steps where adaptation takes place ensures that conservation of linear and angular momentum is obtained int j numer meth engng 2000 49 1295 1325 several examples in 2 d showing the application of the proposed adaptive algorithms are used to demonstrate the validity of the methodology proposed copyright 2009 john wiley sons ltd

on the implementation of rate independent standard dissipative solids at finite strain variational constitutive updates this paper is concerned with an efficient variationally consistent implementation for rate independent dissipative solids at finite strain more precisely focus is on finite strain plasticity theory based on a multiplicative decomposition of the deformation gradient adopting the formalism of standard dissipative solids which allows to describe constitutive models by means of only two potentials being the helmholtz energy and the yield function or equivalently a dissipation functional finite strain plasticity is recast into an equivalent minimization problem in contrast to previous models the presented framework covers isotropic and kinematic hardening as well as isotropic and anisotropic elasticity and yield functions based on this approach a novel numerical implementation representing the main contribution of the paper is given in sharp contrast to by now classical approaches such as the return mapping algorithm and analogously to the theoretical part the numerical formulation is variationally consistent i e all unknown variables follow naturally from minimizing the energy of the considered system consequently several different numerically efficient and robust optimization schemes can be directly employed for solving the resulting minimization problem extending previously published works on variational constitutive updates the advocated model does not rely on any material symmetry and therefore it can be applied to a broad range of different plasticity theories as two examples an anisotropic hill type and a barlat type model are implemented numerical examples demonstrate the applicability and the performance of the proposed implementation

adaptive mesh refinement in strain localization problems an adaptive meshing method tailored to problems of strain localization is given the adaption strategy consists of equi distributing the variation of the velocity field over the elements of the mesh a heuristic justification for the use of variations as indicators is advanced and possible connections with interpolation error bounds are discussed meshes are constructed by delaunay triangulation it is shown how the hu washizu principle determines a consistent transfer operator for the state variables examples of application are given which demonstrate the versatility of the method

quasiconvexity at the boundary positivity of the second variation and elastic stability for one dimensional problems of the calculus of variations there is a well developed theory relating the positivity of the second variation at a critical point i e linearized stability to strong relative minima these results largely due to weierstrass are usually proved by the method of fields of extremals as is described in bolza 1904 and morrey 1966 or by an argument involving a careful use of taylor series as in hestenes 1966 chapter 3 14

a method of grid optimization for finite element methods a computationally useful criterion for grid optimization is derived based on a measure of the interpolation error associated with the finite element model the result is intended to be used to improve the quality of finite element solutions by changing the location of the nodes within a fixed number of degrees of freedom examples of the application of the criterion are provided the finite element method has become an important tool of engineering analysis and design much of the popularity of the method is due to the freedom that it allows in the construction of the discretized model it has long been realized however that this freedom must be carefully exercised since the quality of the finite element solution greatly depends on how the discretization is performed several methods have been proposed to improve the discretized model in an iterative manner one such method is presented in this paper a natural way of improving the quality of finite element solutions is to increase the number of degrees of freedom the process is normally performed after an initial solution is already available several schemes have been devised to introduce the new degrees of freedom in a selective manner in order to produce the greatest possible improvement of the previous solution this calls for the definition of criteria to identify the regions of the domain where the finite element approximation is poorer the new degrees of freedom are added in these regions by either increasing the order of polynomial approximation inside elements the so called p method or by subdivision of elements or h method the process is continued until a specified accuracy is achieved the quality of the finite element solution may be improved also by optimizing the disposition of the nodes analysts often rely on their experience to construct grids that make an efficient use of the available degrees of freedom it is also possible to improve the quality of existing meshes iteratively using predefined guidelines for the redistribution of the nodes this paper considers the development of such guidelines

a variational constitutive update algorithm for a set of isotropic hyperelastic viscoplastic material models abstract in three dimensional formulations of plasticity the rate of plastic deformation is usually decomposed in direction and amplitude in the most simple cases of von mises type flows this allows to assume a known radial flow direction and computations are reduced to the determination of the amplitude of the plastification in the context of a variational formulation of finite plasticity it is verified that this decomposition between constitutive and kinematic aspects is accomplished by the choice of a quadratic elastic potential hencky model the aim of this paper is to show that a wide set of simple hyperelastic plastic isotropic models not restricted to quadratic elastic behavior can be constructed by relaxing the classical decomposition amplitude direction by the sum of spectral quantities in addition this approach allows for a unified and efficient numerical implementation in which a particular hyperelastic viscoplastic behavior is obtained by a specific choice of the elastic plastic and dissipative pseudo potentials

modeling of impact problems using an h adaptive explicit lagrangian finite element method in three dimensions abstract an h adaptive algorithm was developed to enhance modeling of 3d impact problems the algorithm utilizes a recovery based error indicator involving the root mean square error of the green lagrange strain this indicator proved effective for identifying refinements in the contact region which lead to improved solution accuracy the effectiveness of the algorithm is demonstrated for both a reverse taylor impact test and a ballistic penetration problem both the algorithm and implementation issues are discussed herein

an explicit discontinuous galerkin method for non linear solid dynamics formulation parallel implementation and scalability properties an explicit dynamics spatially discontinuous galerkin dg formulation for non linear solid dynamics is proposed and implemented for parallel computation dg methods have particular appeal in problems involving complex material response e g non local behavior and failure as even in the presence of discontinuities they provide a rigorous means of ensuring both consistency and stability in the proposed method these are guaranteed the former by the use of average numerical fluxes and the latter by the introduction of appropriate quadratic terms in the weak formulation the semi discrete system of ordinary differential equations is integrated in time using a conventional second order central difference explicit scheme a stability criterion for the time integration algorithm accounting for the influence of the dg discretization stability is derived for the equivalent linearized system this approach naturally lends itself to efficient parallel implementation the resulting dg computational framework is implemented in three dimensions via specialized interface elements the versatility robustness and scalability of the overall computational approach are all demonstrated in problems involving stress wave propagation and large plastic deformations copyright 2007 john wiley sons ltd

numerical solution of the scalar double well problem allowing microstructure the direct numerical solution of a non convex variational problem p typically faces the difficulty of the finite element approximation of rapid oscillations although the oscillatory discrete minimisers are properly related to corresponding young measures and describe real physical phenomena they are costly and difficult to compute in this work we treat the scalar double well problem by numerical solution of the relaxed problem rp leading to a degenerate convex minimisation problem the problem rp has a minimiser u and a related stress field dw u which is known to coincide with the stress field obtained by solving p in a generalised sense involving young measures if u h is a finite element solution h dw u h is the related discrete stress field we prove a priori and a posteriori estimates for h in l 4 3 and weaker weighted estimates for u u h the a posteriori estimate indicates an adaptive scheme for automatic mesh refinements as illustrated in numerical experiments

incompressible limit of the navier stokes model with a growth term starting from isentropic compressible navier stokes equations with growth term in the continuity equation we rigorously justify that performing an incompressible limit one arrives to the two phase free boundary fluid system

second order second degree painleve equations related with painleve i vi equations and fuchsian type transformations one to one correspondence between the painleve i vi equations and certain second order second degree equations of painleve type is investigated the transformation between the painleve equations and second order second degree equations is the one involving the fuchsian type equation

exponential map algorithm for stress updates in anisotropic multiplicative elastoplasticity for single crystals this paper presents a new stress update algorithm for large strain rate independent single crystal plasticity the theoretical frame is the well established continuum slip theory based on the multiplicative decomposition of the deformation gradient into elastic and plastic parts a distinct feature of the present formulation is the introduction and computational exploitation of a particularly simple hyperelastic stress response function based on a further multiplicative decomposition of the elastic deformation gradient into spherical and unimodular parts resulting in a very convenient representation of the schmid resolved shear stresses on the crystallographic slip systems in terms of a simple inner product of eulerian vectors the key contribution of this paper is an algorithmic formulation of the exponential map exp sl 3 sl 3 for updating the special linear group sl 3 of unimodular plastic deformation maps this update preserves exactly the plastic incompressibility condition of the anisotropic plasticity model under consideration the resulting fully implicit stress update algorithm treats the possibly redundant constraints of single crystal plasticity by means of an active set search it exploits intrinsically the simple representation of the schmid stresses by formulating the return algorithm and the associated consistent elastoplastic moduli in terms of eulerian vectors updates the performance of the proposed algorithm is demonstrated by means of a representative numerical example

a level set approach to eulerian lagrangian coupling we present a numerical method for coupling an eulerian compressible flow solver with a lagrangian solver for fast transient problems involving fluid solid interactions such coupling needs arise when either specific solution methods or accuracy considerations necessitate that different and disjoint subdomains be treated with different eulerian or lagrangian schemes the algorithm we propose employs standard integration of the eulerian solution over a cartesian mesh to treat the irregular boundary cells that are generated by an arbitrary boundary on a structured grid the eulerian computational domain is augmented by a thin layer of cartesian ghost cells boundary conditions at these cells are established by enforcing conservation of mass and continuity of the stress tensor in the direction normal to the boundary the description and the kinematic constraints of the eulerian boundary rely on the unstructured lagrangian mesh the lagrangian mesh evolves concurrently driven by the traction boundary conditions imposed by the eulerian counterpart several numerical tests designed to measure the rate of convergence and accuracy of the coupling algorithm are presented as well general problems in one and two dimensions are considered including a test consisting of an isotropic elastic solid and a compressible fluid in a fully coupled setting where the exact solution is available

an energy momentum conserving algorithm using the variational formulation of visco plastic updates in this paper we use the variational formulation of elasto plastic updates proposed by ortiz and stainier comput methods appl mech eng 1999 171 419 444 in the context of consistent time integration schemes we show that such a formulation is well suited to obtain a general expression of energy momentum conserving algorithms moreover we present numerical examples that illustrate the efficiency of our developments copyright 2005 john wiley sons ltd

monodromy of certain painleve vi transcendents and reflection groups we study the global analytic properties of the solutions of a particular family of painleve vi equations with the parameters 0 1 2 and 2 2 1 2 with arbitrary 2 we introduce a class of solutions having critical behaviour of algebraic type and completely compute the structure of the analytic continuation of these solutions in terms of an auxiliary reflection group in the three dimensional space the analytic continuation is given in terms of an action of the braid group on the triples of generators of the reflection group we show that the finite orbits of this action correspond to the algebraic solutions of our painleve vi equation and use this result to classify all of them we prove that the algebraic solutions of our painleve vi equation are in one to one correspondence with the regular polyhedra or star polyhedra in the three dimensional space

first degree birational transformations of the painleve equations and their contiguity relations we present a consistent truncation allowing us to obtain the first degree birational transformation found by okamoto for the sixth painleve equation the discrete equation arising from its contiguity relation is then just the sum of six simple poles an algebraic solution is presented which is equivalent to but simpler than the umemura solution finally the well known confluence provides a unified picture of all first degree birational transformations for the lower painleve equations ranging them in two distinct sequences

on the solvability of the painleve vi equation a rigorous method was introduced by fokas and zhou 1992 for studying the riemann hilbert problem associated with the painleve ii and iv equations the same methodology has been applied to the painleve i iii and v equations in this paper we will apply the same methodology to the painleve vi equation we will show that the cauchy problem for the painleve vi equation admits in general a global meromorphic solution in t furthermore the special solution which can be written in terms of a hypergeometric function is obtained via solving the special case of the riemann hilbert problem

implementation of incremental variational formulations based on the numerical calculation of derivatives using hyper dual numbers abstract in this paper novel implementation schemes for the automatic calculation of internal variables stresses and consistent tangent moduli for incremental variational formulations ivfs describing inelastic material behavior are proposed ivfs recast inelasticity theory as an equivalent optimization problem where the incremental stress potential within a discrete time interval is minimized in order to obtain the values of internal variables in the so called multilevel newton raphson method for the inelasticity theory this minimization problem is typically solved by using second derivatives with respect to the internal variables in addition to that to calculate the stresses and moduli further second derivatives with respect to deformation tensors are required compared with classical formulations such as the return mapping method the ivfs are relatively new and their implementation is much less documented furthermore higher order derivatives are required in the algorithms demanding increased implementation efforts therefore even though ivfs are mathematically and physically elegant their application is not standard here novel approaches for the implementation of ivfs using hdns of second and higher order are presented to arrive at a fully automatic and robust scheme with computer accuracy the proposed formulations are quite general and can be applied to a broad range of different constitutive models which means that once the proposed schemes are implemented as a framework any other dissipative material model can be implemented in a straightforward way by solely modifying the constitutive functions these include the helmholtz free energy function the dissipation potential function and additional side constraints such as e g the yield function in the case of plasticity its uncomplicated implementation for associative finite strain elasto plasticity and performance is illustrated by some representative numerical examples

hele shaw limit for a system of two reaction cross diffusion equations for living tissues multiphase mechanical models are now commonly used to describe living tissues including tumour growth the specific model we study here consists of two equations of mixed parabolic and hyperbolic type which extend the standard compressible porous medium equation including cross reaction terms we study the incompressible limit when the pressure becomes stiff which generates a free boundary problem we establish the complementarity relation and also a segregation result several major mathematical difficulties arise in the two species case firstly the system structure makes comparison principles fail secondly segregation and internal layers limit the regularity available on some quantities to bv thirdly the aronson benilan estimates cannot be established in our context we are lead as it is classical to add correction terms this procedure requires technical manipulations based on bv estimates only valid in one space dimension another novelty is to establish an l1 version in place of the standard upper bound

chazy s second degree painleve equations we examine two sets of second degree painlev equations derived by chazy in 1909 denoted by systems ii and iii the last member of each set is a second degree version of the painlev vi equation and there are no other second order second degree painlev equations in the polynomial class with this property we map the last member of system ii into the fokas yortsos equation and demonstrate how both schlesinger and okamoto transformations for painlev vi can be read off the chazy equation the 24 fundamental schlesinger transformations were known to garnier in 1943 while the 64 okamoto transformations date from 1987 in an appendix we gather together the solutions of the five members of system ii system iii is better known being equivalent to jimbo and miwa s equations for the logarithmic derivatives of the tau functions of the six painlev transcendents the last member known to painlev in 1906 was written in a manifestly symmetric form by jimbo and miwa suggesting many induced symmetries for painlev vi in particular schlesinger and okamoto transformations for painlev vi can be read off immediately

a new viscoelastic constitutive model for continuous media at finite thermomechanical changes abstract this paper is concerned with the continuum formulation of a fully coupled thermomechanical constitutive model for highly deformable bodies with viscous dissipation the threedimensional material law is applicable to the thermomechanical characterization of elastomeric high polymeric solids in the finite strain domain under varying temperature the description is based on the concept of internal state variables and rational thermodynamics the main goal of the presentation is to develop consistent constitutive equations for the stress entropy and independent internal variables such that the second law of thermodynamics in the form of the clausius duhem inequality issatisfied motivated by the significant difference in bulk and shear response of elastomers the model employs a local decomposition of the deformation into a dilatational and an isochoric part the framework of nonlinear thermoviscoelasticity presented herein is formulated entirely in the reference configuration and provides a sound continuum basis for approximation techniques such as the finite element method copyright 1996 elsevier science ltd

new contiguity relation of the sixth painleve equation from a truncation for the master painleve equation p6 u we define a consistent method adapted from the weiss truncation for partial differential equations which allows us to obtain the first degree birational transformation of okamoto two new features are implemented to achieve this result the first one is the homography between the derivative of the solution u and a riccati pseudopotential the second one is an improvement of a conjecture by fokas and ablowitz on the structure of this birational transformation we then build the contiguity relation of p6 which yields one new second order non autonomous discrete equation

a novel variational algorithmic formulation for wrinkling at finite strains based on energy minimization application to mesh adaption this paper is concerned with an efficient novel algorithmic formulation for wrinkling at finite strains in contrast to previously published numerical implementations the advocated method is fully variational more precisely the parameters describing wrinkles or slacks together with the unknown deformation mapping are computed jointly by minimizing the potential energy of the considered mechanical system furthermore the wrinkling criteria are naturally included within the presented variational framework the proposed method allows to employ three dimensional constitutive models without any additional modification i e a projection in plane stress space is not required analogously to the wrinkling parameters the non vanishing out of plane component of the strain tensor results conveniently from relaxing the respective helmholtz energy of the membrane the proposed framework is very general and does not rely on any assumption regarding the symmetry of the material i e arbitrary anisotropic hyperelastic models can be consistently taken into account the advantages associated with such a variational method are manifold for instance it opens up the possibility of applying standard optimization algorithms to the numerical implementation this is especially important for highly non linear or singular problems such as wrinkling on the other hand minimization principles provide a suitable basis for a posteriori error estimation and thus for adaptive finite element formulations as a prototype a variational error indicator leading to an efficient h adaption for wrinkling is briefly discussed the performance of the wrinkling approach is demonstrated by selected finite element analyses

three dimensional adaptive meshing by subdivision and edge collapse in finite deformation dynamic plasticity problems with application to adiabatic shear banding this paper is concerned with the development of a general framework for adaptive mesh refinement and coarsening in three dimensional finite deformation dynamic plasticity problems mesh adaption is driven by a posteriori global error bounds derived on the basis of a variational formulation of the incremental problem the particular mesh refinement strategy adopted is based on rivara s longest edge propagation path lepp bisection algorithm our strategy for mesh coarsening or unrefinement is based on the elimination of elements by edge collapse the convergence characteristics of the method in the presence of strong elastic singularities are tested numerically an application to the three dimensional simulation of adiabatic shear bands in dynamically loaded tantalum is also presented which demonstrates the robustness and versatility of the method copyright c 2001 john wiley sons ltd

schlesinger transformations for the second members of pii and piv hierarchies in this paper we give a method to obtain the schlesinger transformations for the second members of second and fourth painlev e hierarchies the procedure involves formulating a riemann hilbert problem for a transformation matrix which transforms the solution of the linear problem but leaves the associated monodromy data the same

a first order energy dissipative momentum conserving scheme for elasto plasticity using the variational updates formulation abstract in a previous paper l noels l stainier j p ponthot an energy momentum conserving algorithm using the variational formulation of visco plastic updates int j numer methods engrg 65 2006 904 942 the authors demonstrated the efficiency of the variational formulation of elasto plastic updates to develop energy momentum conserving time integration algorithms indeed within such a framework the stress tensor always derives from an incremental potential even when plastic behavior is considered therefore the verification of the conservation of energy in the non linear range can easily be demonstrated the sum of the reversible stored energy and irreversible dissipated energy exactly corresponds to the work of the external forces applied to the structure although this formulation was shown to be accurate and robust the introduction of numerical dissipation for high frequency numerical modes can be necessary to simulate complex phenomena in this work we propose a modification of the variational updates framework to introduce this numerical property leading to a new energy dissipative momentum conserving time integration algorithm for elasto plasticity

a variational formulation for finite deformation wrinkling analysis of inelastic membranes this paper is concerned with a novel fully variational formulation for finite deformation analysis of inelastic membranes with wrinkling in contrast to conventional approaches every aspect of the physical problem derives from minimization of suitable energy functionals a variational formulation of finite strain plasticity theory which leads to a minimization problem for the constitutive updates serves as the starting point for the derivations in order to take into account the kinematics induced by wrinkles and slacks a relaxed version of the finite strain functional is postulated in effect the local incremental stress strain relations are established via differentiation of the relaxed energy functional with respect to the strains hence the presented formulation is fully analogous to that of hyperelasticity with the sole exception that the aforementioned functional depends on history variables and accordingly it is path dependent the advantages associated with the developed variational method are manifold from a practical point of view the possibility of applying standard optimization algorithms to solve the minimization problem describing inelastic membranes is remunerative from a mathematical point of view on the other hand the energy of the system induces some sort of natural metric representing an essential requirement for error estimation and thus for adaptive finite element methods the presented derivation of the model allows to consider possible material symmetries in the elastic as well as plastic response of the material as a prototype a von mises type model is implemented the efficiency and performance of the resulting algorithm are demonstrated by means of numerical examples

uniqueness and existence results on the hele shaw and the stefan problems in this paper we introduce a notion of viscosity solutions for the one phase hele shaw and stefan problems when there is no surface tension we prove the uniqueness and existence of the solutions for both problems and the uniform convergence of solutions of porous medium equation to those of the hele shaw problem

second order second degree painleve equations related with painleve i ii iii equations the algorithmic method introduced by fokas and ablowitz to investigate the transformation properties of painleve equations is used to obtain a one to one correspondence between the painleve i ii and iii equations and certain second order second degree equations of painleve type

mesh refinement strategies without mapping of nonlinear solutions for the generalized and standard fem analysis of 3 d cohesive fractures summary a robust and efficient strategy is proposed to simulate mechanical problems involving cohesive fractures this class of problems is characterized by a global structural behavior that is strongly affected by localized nonlinearities at relatively small sized critical regions the proposed approach is based on the division of a simulation into a suitable number of sub simulations where adaptive mesh refinement is performed only once based on refinement window s around crack front process zone s the initialization of newton raphson nonlinear iterations at the start of each sub simulation is accomplished by solving a linear problem based on a secant stiffness rather than a volume mapping of nonlinear solutions between meshes the secant stiffness is evaluated using material state information stored read on crack surface facets which are employed to explicitly represent the geometry of the discontinuity surface independently of the volume mesh within the generalized finite element method framework moreover a simplified version of the algorithm is proposed for its straightforward implementation into existing commercial software data transfer between sub simulations is not required in the simplified strategy the computational efficiency accuracy and robustness of the proposed strategies are demonstrated by an application to cohesive fracture simulations in 3 d copyright 2016 john wiley sons ltd

optimal transportation meshfree approximation schemes for fluid and plastic flows we develop an optimal transportation meshfree otm method for simulating general solid and fluid flows including fluid structure interaction the method combines concepts from optimal transportation theory with material point sampling and max ent meshfree interpolation the proposed otm method generalizes the benamou brenier differential formulation of optimal mass transportation problems to problems including arbitrary geometries and constitutive behavior the otm method enforces mass transport and essential boundary conditions exactly and is free from tension instabilities the otm method exactly conserves linear and angular momentum and its convergence characteristics are verified in standard benchmark problems we illustrate the range and scope of the method by means of two examples of application the bouncing of a gas filled balloon off a rigid wall and the classical taylor anvil benchmark test extended to the hypervelocity range

on the solvability of painleve ii and iv we introduce a rigorous methodology for studying the riemann hilbert problems associated with certain integrable nonlinear ordinary differential equations for concreteness we investigate the painleve ii and painleve iv equations we show that the cauchy problems for these equations admit in general global meromorphic in t solutions furthermore for special relations among the monodromy data and for t on stokes lines these solutions are bounded for finite t

on a unified approach to transformations and elementary solutions of painleve equations an algorithmic method is developed for investigating the transformation properties of second order equations of painleve type this method which utilizes the singularity structure of these equations yields explicit transformations which relate solutions of the painleve equations ii vi with different parameters these transformations easily generate rational and other elementary solutions of the equations the relationship between painleve equations and certain new equations quadratic in the second derivative of painleve type is also discussed

on the initial value problem of the second painleve transcendent the initial value problem associated with the second painleve transcendent is linearized via a matrix discontinuous homogeneous riemann hilbert rh problem defined on a complicated contour six rays intersecting at the origin this problem is mapped through a series of transformations to three different simple riemann hilbert problems each of which can be solved via a system of two fredholm integral equations the connection of these results with the inverse scattering transform in one and two dimensions is also pointed out

schlesinger transformations for painleve vi equation a method to obtain the schlesinger transformations for painleve vi equation is given the procedure involves formulating a riemann hilbert problem for a transformation matrix which transforms the solution of the linear problem but leaves the associated monodromy data the same

monodromy preserving deformation of linear ordinary differential equations with rational coefficients iii abstract a unified treatment of monodromy and spectrum preserving deformations is presented in particular a general procedure is described to reduce the latter into the former consistently the concept of the function previously introduced for the former 2 is extended to the isospectral context it is shown that the general monodromy and spectrum preserving deformation equations can be written as hirota s bilinear differential equations by using the functions as dependent variables

nonlinear elliptic parabolic problems we introduce a notion of viscosity solutions for a general class of elliptic parabolic phase transition problems these include the richards equation which is a classical model in filtration theory existence and uniqueness results are proved via the comparison principle in particular we show existence and stability properties of maximal and minimal viscosity solutions for a general class of initial data these results are new even in the linear case where we also show that viscosity solutions coincide with the regular weak solutions introduced in alt and luckhaus math z 183 311 341 1983

the variational formulation of viscoplastic constitutive updates we present a class of constitutive updates for general viscoplastic solids including such aspects of material behavior as finite elastic and plastic deformations non newtonian viscosity rate sensitivity and arbitrary flow and hardening rules the distinguishing characteristic of the proposed constitutive updates is that by construction the corresponding incremental stress strain relations derive from a pseudo elastic strain energy density this in turn confers the incremental boundary value problem a variational structure in particular the incremental deformation mapping follows from a minimum principle this minimum principle may conveniently be taken as a basis for error estimation and mesh adaption the accuracy and robustness of the variational constitutive updates is demonstrated with the aid of convergence tests involving the finitely deforming mises solid and ductile single crystals the ability of the updates to resolve the complex patterns of slip activity which arise in the latter application is particularly noteworthy

lagrangian finite element analysis of newtonian fluid flows a fully lagrangian nite element method for the analysis of newtonian ows is developed the approach furnishes in eect a lagrangian implementation of the compressible navier stokes equations as the ow proceeds the mesh is maintained undistorted through continuous and adaptive remeshing of the uid mass the principal advantage of the present approach lies in the treatment of boundary conditions at material surfaces such as free boundaries uid uid or uid solid interfaces in contrast to eulerian approaches boundary conditions are enforced at material surfaces ab initio and therefore require no special attention consistent tangents are obtained for lagrangian implicit analysis of a newtonian uid ow which may exhibit compressibility eects the accuracy of the approach is assessed by comparison of the solution for a sloshing problem with existing numerical results and its versatility demonstrated through a simulation of wave breaking the nite element mesh is maintained undistorted throughout the computation by recourse to frequent and adaptive remeshing 1998 john wiley sons ltd

modelling strong discontinuities in solid mechanics via strain softening constitutive equations part 2 numerical simulation on the basis of the strong discontinuity analysis of standard local stress strain constitutive equations a finite element framework for the simulation of strong discontinuities which belongs to the family of assumed enhanced strain methods is presented taking the standard linear triangle as the underlying element an additional incompatible mode leads to the formulation of an enriched strain field which is shown to be able to appropriately capture strong discontinuities the presented numerical simulations show that mesh size and mesh alignment dependencies can be completely removed

square lattice ising model susceptibility series expansion method and differential equation for χ 3 in a previous paper 2004 j phys a math gen 37 9651 68 we have given the fuchsian linear differential equation satisfied by 3 the three particle contribution to the susceptibility of the isotropic square lattice ising model this paper gives the details of the calculations with some useful tricks and tools allowing one to obtain long series in polynomial time the method is based on series expansion in the variables that appear in the n 1 dimensional integrals representing the n particle contribution to the isotropic square lattice ising model susceptibility the integration rules are straightforward due to remarkable formulae we derived for these variables we obtain without any numerical approximation 3 as a fully integrated series in the variable w s 2 1 s2 where s sh 2k with k j kt the conventional ising model coupling constant we also give some perspectives and comments on these results

a hele shaw problem for tumor growth abstract we consider weak solutions to a problem modeling tumor growth under certain conditions on the initial data solutions can be obtained by passing to the stiff incompressible limit in a porous medium type problem with a lotka volterra source term describing the evolution of the number density of cancerous cells we prove that such limit solutions solve a free boundary problem of hele shaw type we also obtain regularity properties both for the solution and for its free boundary the main new difficulty arises from the competition between the growth due to the source which keeps the initial singularities and the free boundary which invades the domain with a regularizing effect new islands can be generated at singular times