a variationally consistent fractional time step integration method for incompressible and nearly incompressible lagrangian dynamics in this paper we present a fractional time step method for lagrangian formulations of solid dynamics problems the method can be interpreted as belonging to the class of variational integrators which are designed to conserve linear and angular momentum of the entire mechanical system exactly energy fluctuations are found to be minimal and stay bounded for long durations in order to handle incompressibility a mixed formulation in which the pressure appears explicitly is adopted the velocity update over a time step is split into deviatoric and volumetric components the deviatoric component is advanced using explicit time marching whereas the pressure correction for each time step is computed implicitly by solving a poisson like equation once the pressure is known the volumetric component of the velocity update is calculated in contrast with standard explicit schemes where the time step size is determined by the speed of the pressure waves the allowable time step for the proposed scheme is found to depend only on the shear wave speed this leads to a significant advantage in the case of nearly incompressible materials and permits the solution of truly incompressible problems copyright 2005 john wiley sons ltd

an elementary proof of kharitonov s stability theorem with extensions gives an elementary proof of kharitonov s theorem using simple complex plane geometry without invoking the hermite bieler theorem kharitonov s theorem is a stability result for classes of polynomials defined by letting each coefficient vary independently in an arbitrary interval the result states that the whole class is hurwitz if and only if four special well defined polynomials are hurwitz the paper also gives elementary proofs of two previously known extensions for polynomials of degree less than six the requirement is reduced to fewer than four polynomials and the theorem is generalized to polynomials with complex coefficients

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

an energy stable mixed formulation for isogeometric analysis of incompressible hyper elastodynamics we develop a mixed formulation for incompressible hyper elastodynamics based on a continuum modeling framework recently developed in 41 and smooth generalizations of the taylor hood element based on non uniform rational b splines nurbs this continuum formulation draws a link between computational fluid dynamics and computational solid dynamics this link inspires an energy stability estimate for the spatial discretization which favorably distinguishes the formulation from the conventional mixed formulations for finite elasticity the inf sup condition is utilized to provide a bound for the pressure field the generalized method is applied for temporal discretization and a nested block preconditioner is invoked for the solution procedure 42 the inf sup stability for different pairs of nurbs elements is elucidated through numerical assessment the convergence rate of the proposed formulation with various combinations of mixed elements is examined by the manufactured solution method the numerical scheme is also examined under compressive and tensile loads for isotropic and anisotropic hyperelastic materials finally a suite of dynamic problems is numerically studied to corroborate the stability and conservation properties

a computational framework for polyconvex large strain elasticity abstract this paper presents a novel computational formulation for large strain polyconvex elasticity the formulation based on the original ideas introduced by schroder et al 2011 introduces the deformation gradient the fibre map its adjoint the area map and its determinant the volume map as independent kinematic variables of a convex strain energy function compatibility relationships between these variables and the deformed geometry are enforced by means of a multi field variational principle with additional constraints this process allows the use of different approximation spaces for each variable the paper extends the ideas presented in schroder et al 2011 by introducing conjugate stresses to these kinematic variables which can be used to define a generalised convex complementary energy function and a corresponding complementary energy principle of the hellinger reissner type where the new conjugate stresses are primary variables together with the deformed geometry both compressible and incompressible or nearly incompressible elastic models are considered a key element to the developments presented in the paper is the new use of a tensor cross product presented for the first time by de boer 1982 page 76 which facilitates the algebra associated with the adjoint of the deformation gradient for the numerical examples quadratic interpolation of the displacements piecewise linear interpolation of strain and stress fields and piecewise constant interpolation of the jacobian and its stress conjugate are considered for compressible cases in the case of incompressible materials two formulations are presented first continuous quadratic interpolation for the displacement together with piecewise constant interpolation for the pressure and second linear continuous interpolation for both displacement and pressure stabilised via a petrov galerkin technique

to stabilize an interval plant family it suffices to simultaneously stabilize sixty four polynomials it is shown that stability of three specific polynomial families can be deduced from the stability of a finite number of polynomials these polynomial families are the characteristic polynomials of unity feedback loops with the controller in the forward path and where the plant includes a specific form of parameter uncertainty for the first polynomial family the plant has parameter uncertainty in the even or odd terms of the numerator or denominator polynomial for the second polynomial family the plant has a numerator or denominator which is an interval polynomial for the third polynomial family the plant is interval because of the structure of these results it is shown that they lead to robust stabilization results two examples are included the approach employed here was developed for plants with affine uncertainty it is demonstrated that considerable simplification results if the plants under investigation are interval

amalgamation and ramsey properties of lp spaces abstract we study the dynamics of the group of isometries of l p spaces in particular we study the canonical actions of these groups on the space of isometric embeddings of finite dimensional subspaces of l p 0 1 into itself and we show that for every real number 1 p with p 4 6 8 they are e transitive provided that is small enough we achieve this by extending the classical equimeasurability principle of plotkin and rudin we define the central notion of a fraisse banach space which underlies these results and of which the known separable examples are the spaces l p 0 1 p 4 6 8 and the gurarij space we also give a proof of the ramsey property of the classes l p n n p 2 viewing it as a multidimensional borsuk ulam statement we relate this to an arithmetic version of the dual ramsey theorem of graham and rothschild as well as to the notion of a spreading vector of matousek and rodl finally we give a version of the kechris pestov todorcevic correspondence that links the dynamics of the group of isometries of an approximately ultrahomogeneous space x with a ramsey property of the collection of finite dimensional subspaces of x

a stabilized mixed finite element method for finite elasticity formulation for linear displacement and pressure interpolation abstract a stabilized mixed finite element method for finite elasticity is presented the method circumvents the fulfillment of the ladyzenskaya babuska brezzi condition by adding mesh dependent terms which are functions of the residuals of the euler lagrange equations to the usual galerkin method the weak form and the linearized weak form are presented in terms of the reference and current configuration numerical experiments using a tetrahedral element with linear shape functions for the displacements and for the pressure show that the method successfully yields a stabilized element

liquid vapor phase transition thermomechanical theory entropy stable numerical formulation and boiling simulations abstract we develop a new continuum mechanics modeling framework for liquid vapor flows with particular focus on the van der waals fluid by invoking microforce theory the coleman noll procedure is generalized to derive consistent constitutive relations in the presence of non local effects a new thermodynamically consistent algorithm for the van der waals model is designed using functional entropy variables and a new temporal scheme employing a family of new quadrature rules we show that the resulting fully discrete scheme is unconditionally stable in entropy and second order time accurate isogeometric analysis is utilized for spatial discretization the analytical properties of the formulation are corroborated by benchmark problems three sets of application problems are simulated to demonstrate the capability of the model and the algorithm our methodology provides a particularly useful predictive tool for boiling flows

a generalization of kharitonov s theorem robust stability of interval plants the robust stability problem is considered for interval plants in the case of single input multioutput or single output multi input systems a necessary and sufficient condition for the robust stabilization of such plants is developed using a generalization of v l kharitonov s theorem 1978 the generalization given provides necessary and sufficient conditions for the stability of a family of polynomials delta s q sub 1 s p sub 1 s q sub m s p sub m s where the q sub i are fixed and the p sub i are interval polynomials the coefficients of which are regarded as a point in parameter space which varies within a prescribed box this generalization called the box theorem reduces the question of the stability of the box in parameter space to the equivalent problem of the stability of a prescribed set of line segments it is shown that for special classes of polynomials q sub i s the set of line segments collapses to a set of points and this version of the box theorem in turn reduces to kharitonov s original theorem

a new finite element formulation for computational fluid dynamics x the compressible euler and navier stokes equations a space time element method is presented for solving the compressible euler and navier stokes equations the proposed formulation includes the variational equation predictor multi corrector algorithms and boundary conditions the variational equation is based on the time discontinuous galerkin method in which the physical entropy variables are employed a least squares operator and a discontinuity capturing operator are added resulting in a high order accurate and unconditionally stable method implicit explicit predictor multi corrector algorithms applicable to steady as well as unsteady problems are presented techniques are developed to enhance their efficiency implementation of boundary conditions is addressed in particular a technique is introduced to satisfy nonlinear essential boundary conditions and a consistent method is presented to calculate boundary fluxes numerical results are presented to demonstrate the performance of the method

mixed spectral hp element formulation for nonlinear elasticity we present a mixed formulation for high order spectral hp element methods and investigate its stability in large strain analysis of nearly incompressible solids specifically we employ the displacement pressure formulation and use a pressure projection method along with general jacobi polynomial bases to derive the discrete equations we first study the convergence of the method for smooth and singular solutions of elastic and hyperelastic problems for various combinations of mixed elements subsequently we investigate numerically the validity of ellipticity and inf sup conditions for various test problems including cases with large deformation we find that the inf sup condition is satisfied even when the pressure interpolation order op is equal to the displacement order ou but to resolve the locking free behavior it is required that op ou 1 and this choice also gives the best convergence rate in large deformation analysis surprisingly the results are more stable for ou op ou k because the ellipticity in kernel is better satisfied with higher k as ou increases compared to the displacement only formulation the mixed formulation seems to have superior accuracy and it is overall more efficient especially for problems with high bulk modulus

a vertex centred finite volume jameson schmidt turkel jst algorithm for a mixed conservation formulation in solid dynamics a vertex centred finite volume algorithm is presented for the numerical simulation of fast transient dynamics problems involving large deformations a mixed formulation based upon the use of the linear momentum the deformation gradient tensor and the total energy as conservation variables is discretised in space using linear triangles and tetrahedra in two dimensional and three dimensional computations respectively the scheme is implemented using central differences for the evaluation of the interface fluxes in conjunction with the jameson schmidt turkel jst artificial dissipation term the discretisation in time is performed by using a total variational diminishing tvd two stage runge kutta time integrator the jst algorithm is adapted in order to ensure the preservation of linear and angular momenta the framework results in a low order computationally efficient solver for solid dynamics which proves to be very competitive in nearly incompressible scenarios and bending dominated applications

functional entropy variables a new methodology for deriving thermodynamically consistent algorithms for complex fluids with particular reference to the isothermal navier stokes korteweg equations we propose a new methodology for the numerical solution of the isothermal navier stokes korteweg equations our methodology is based on a semi discrete galerkin method invoking functional entropy variables a generalization of classical entropy variables and a new time integration scheme we show that the resulting fully discrete scheme is unconditionally stable in energy second order time accurate and mass conservative we utilize isogeometric analysis for spatial discretization and verify the aforementioned properties by adopting the method of manufactured solutions and comparing coarse mesh solutions with overkill solutions various problems are simulated to show the capability of the method our methodology provides a means of constructing unconditionally stable numerical schemes for nonlinear non convex hyperbolic systems of conservation laws

on the thirty two virtual polynomials to stabilize an interval plant this paper studies the conservatism of the 32 virtual polynomials to stabilize an interval plant it is shown that working with the 32 virtual vertices is generally less conservative than with the kharitonov polynomials of the smallest interval polynomial containing the characteristic polynomial polytope by means of the former it is possible to find all the controllers such that the value set of the polytope of characteristic polynomials is applied in two quadrants as a maximum for each spl omega while using the latter only some of them can be found the cases in which both methods coincide are also analyzed and the conditions on the numerator and denominator of the controller are developed thus this coincidence can be known a priori from the characteristics of the coefficients of the numerator and denominator of the controller it is shown that these conditions are satisfied by the first order controllers

a first order hyperbolic framework for large strain computational solid dynamics part i total lagrangian isothermal elasticity abstract this paper introduces a new computational framework for the analysis of large strain fast solid dynamics the paper builds upon previous work published by the authors gil et al 2014 48 where a first order system of hyperbolic equations is introduced for the simulation of isothermal elastic materials in terms of the linear momentum the deformation gradient and its jacobian as unknown variables in this work the formulation is further enhanced with four key novelties first the use of a new geometric conservation law for the co factor of the deformation leads to an enhanced mixed formulation advantageous in those scenarios where the co factor plays a dominant role second the use of polyconvex strain energy functionals enables the definition of generalised convex entropy functions and associated entropy fluxes for solid dynamics problems moreover the introduction of suitable conjugate entropy variables enables the derivation of a symmetric system of hyperbolic equations dual of that expressed in terms of conservation variables third the new use of a tensor cross product 61 greatly facilitates the algebraic manipulations of expressions involving the co factor of the deformation fourth the development of a stabilised petrov galerkin framework is presented for both systems of hyperbolic equations that is when expressed in terms of either conservation or entropy variables as an example a polyconvex mooney rivlin material is used and for completeness the eigen structure of the resulting system of equations is studied to guarantee the existence of real wave speeds finally a series of numerical examples is presented in order to assess the robustness and accuracy of the new mixed methodology benchmarking it against an ample spectrum of alternative numerical strategies including implicit multi field fraeijs de veubeke hu washizu variational type approaches and explicit cell and vertex centred finite volume schemes

on the sixty four polynomials of djaferis to stabilize an interval plant this paper shows that 32 of the 64 polynomials developed by djaferis to stabilize an interval plant with a controller are superfluous thus it is sufficient to stabilize 32 polynomials simultaneously moreover the construction developed in this note allows the implementation of these polynomials in a simple way

remarks on rate constitutive equations for finite deformation problems computational implications it is explicitly shown that if the spatial elasticity tensor of an elastic material is taken as isotropic for all possible configurations then its coefficients cannot be constants they must depend nontrivially on the jacobian determinant of the deformation gradient moreover the assumption typically made for computational purposes that its coefficients remain constant for all possible configurations is incompatible with elasticity it is further shown that an assumption widely used in the computational literature in the context of finite deformation plasticity namely relating an objective stress rate to the rate of deformation tensor through a fourth rank constant isotropic tensor is also incompatible with elasticity thus furnishing an example of an hypoelastic material which is not elastic

to stabilize a k real parameter affine family of plants it suffices to simultaneously stabilize 4 4 polynomials a controller stabilizes an entire family of plants with affine uncertainty if it simultaneously stabilizes a finite number of polynomials an upper bound for this number is 4k