exact energy and momentum conserving algorithms for general models in nonlinear elasticity implicit time integration schemes that inherit the conservation laws of total energy linear and angular momentum are considered for initial boundary value problems in finite deformation elastodynamics conserving schemes are constructed for general hyperelastic material models both compressible and incompressible and are formulated in a way that is independent of spatial discretization three numerical examples for ogden type material models implemented using a finite element discretization in space are given to illustrate the performance of the proposed schemes these examples show that relative to the standard implicit mid point rule the conserving schemes exhibit superior numerical stability properties without a compromise in accuracy

quasi solutions vector lyapunov functions and monotone method the notion of quasi solution which was introduced earlier has been developed in this paper further the notion of coupled quasi solutions and coupled maximal and minimal solutions has been introduced it is shown that the idea of quasi solutions leads to isolated subsystems and has obtained error estimates between solutions and quasi solutions also monotone iterative techniques have been developed to obtain coupled maximal and minimal quasi solutions

existence theorems for a class of functional differential systems in recent papers 2 6 the authors have established existence and comparison theorems for the well known cauchy problem for ordinary diferential equations without using the monotone property on the given system these results are obtained under the conditions of the type which have been considered in the classical paper of miiller 8 1 the interesting feature of the results established in 2 6 is the fact that the solutions of the cauchy problem remain in the given sector in this paper we shall first establish existence and comparison theorems for a class of more general functional differential systems without using monotone property further we develop a monotone iterative technique to establish the existence of minimal and maximal solutions

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

second order functional differential equations with past present and future dependence abstract in this paper we obtain a result on existence of extremal quasisolutions and solutions for second order functional differential equations with functional boundary conditions and past present and future dependence in order to do that we develop a generalized monotone method in presence of lower and upper solutions some examples of application are provided too

a new energy momentum time integration scheme for non linear thermo mechanics abstract the aim of this paper is the design a new one step implicit and thermodynamically consistent energy momentum em preserving time integration scheme for the simulation of thermo elastic processes undergoing large deformations and temperature fields following bonet et al 2020 we consider well posed constitutive models for the entire range of deformations and temperature in that regard the consideration of polyconvexity inspired constitutive models and a new tensor cross product algebra are shown to be crucial in order to derive the so called discrete derivatives fundamental for the construction of the algorithmic derived variables namely the second piola kirchoff stress tensor and the entropy or the absolute temperature the proposed scheme inherits the advantages of the em scheme recently published by franke et al 2018 whilst resulting in a simpler scheme from the implementation standpoint a series of numerical examples will be presented in order to demonstrate the robustness and applicability of the new em scheme although the examples presented will make use of a temperature based version of the em scheme using the helmholtz free energy as the thermodynamical potential and the temperature as the thermodynamical state variable we also include in an appendix an entropy based analogue em scheme using the internal energy as the thermodynamical potential and the entropy as the thermodynamical state variable

w1 p quasiconvexity and variational problems for multiple integrals abstract variational problems for the multiple integral i u g u x dx where r m and u r n are studied a new condition on g called w1 p quasiconvexity is introduced which generalizes in a natural way the quasiconvexity condition of c b morrey it being shown in particular to be necessary for sequential weak lower semicontinuity of i in w 1 p r n and for the existence of minimizers for certain related integrals counterexamples are given concerning the weak continuity properties of jacobians in w 1 p r n p n m an existence theorem for nonlinear elastostatics is proved under optimal growth hypotheses

a monotone iterative scheme for nonlinear reaction diffusion systems having nonmonotone reaction terms abstract a monotone scheme is proposed for the solution of weakly coupled systems of reaction diffusion equations without any monotonicity property of the nonlinear reaction terms the considerations are taken within the framework of weak solutions

cubically hyponormal weighted shifts and their examples we obtain criteria for cubical hyponormality which provide a tool for finding examples being distinct the classes of 2 hyponormal cubically hyponormal and quadratically hyponormal operators

existence of multiple quasifixed points of mixed monotone operators by iterative techniques fixed point theorems of mixed monotone operators by iterative technique in a banach space with the ordering relative to cone k the notion of coupled quasi fixed points is introduced and the main result shows their existence

perturbation of the q numerical radius of a weighted shift operator abstract we formulate the taylor series expansion for the q numerical radius of a weighted shift operator with periodic weights near q 0 coefficients up to the fourth order in the expansion are found via the perturbation theory of hermitian matrices

davis wielandt shell and q numerical range abstract we describe the boundary of the q numerical range of a square matrix using its davis wielandt shell the result is used to generate an algorithm for plotting the q numerical range of the square matrix computations of the q numerical ranges of a special class of matrices are explicitly given

an energy momentum consistent method for transient simulations with mixed finite elements developed in the framework of geometrically exact shells summary in this paper a mixed variational formulation for the development of energy momentum consistent emc time stepping schemes is proposed the approach accommodates mixed finite elements based on a hu washizu type variational formulation in terms of displacements green lagrangian strains and conjugated stresses the proposed discretization in time of the mixed variational formulation under consideration yields an emc scheme in a natural way the newly developed methodology is applied to a high performance mixed shell finite element the previously observed robustness of the mixed finite element formulation in equilibrium iterations extends to the transient regime because of the emc discretization in time copyright 2016 john wiley sons ltd

algorithms for coupled problems that preserve symmetries and the laws of thermodynamics part i monolithic integrators and their application to finite strain thermoelasticity abstract we propose discretizations for a general class of nonlinear coupled thermomechanical problems of evolution the most salient feature of the new methods is that they rigorously preserve the two laws of thermodynamics as well as the symmetries of the systems they model to formulate such methods we exploit the geometric structure afforded by the generic formalism of non equilibrium thermodynamics and we follow a systematic methodology that results in discrete evolution equations which mimic the generic structure as an illustration a complete discretization of finite strain thermoelasticity is presented using finite elements in space and a monolithic integrator in time simulations are provided which demonstrate the conservation features of the algorithm as well as its remarkable robustness

the constrained bilinear form and the c numerical range abstract let v be an n dimentional unitary space with inner product and s the set x v x x 1 for any a hom v v and q c with q 1 we define w a q ax y x y s x y q if q 1 then w a q is just the classical numerical range ax x x s the convexity of which is well known another generalization of the numerical range is the c numerical range which is defined to be the set w c a tr cu au u unitary where c hom v v in this note we prove that w a q is always convex and that wc a is convex for all a if rank c 1 or n 2

positive solutions for nonlinear operator equations and several classes of applications in this paper we study a class of nonlinear operator equations x ax x 0 on ordered banach spaces where a is a monotone generalized concave operator using the properties of cones and monotone iterative technique we establish the existence and uniqueness of solutions for such equations in particular we do not demand the existence of upper lower solutions and compactness and continuity conditions as applications we study first order initial value problems and two point boundary value problems with the nonlinear term is required to be monotone in its second argument in the end applications to nonlinear systems of equations and to nonlinear matrix equations are also considered

a mixed variational framework for the design of energy momentum schemes inspired by the structure of polyconvex stored energy functions abstract a new approach to the design of energy momentum em consistent algorithms for nonlinear elastodynamics is proposed the underlying mixed variational formulation is motivated by the structure of polyconvex stored energy functions and benefits from the notion of a tensor cross product for second order tensors the structure preserving discretization in time of the mixed variational formulation yields an em consistent semi discrete formulation the semi discrete formulation offers several options for the discretization in space in the special case of a purely displacement based method a new form of the algorithmic stress formula is obtained several numerical examples are presented to evaluate the performance of the newly developed schemes

coupled fixed points of multivalued operators and first order odes with state dependent deviating arguments abstract we establish a coupled fixed point theorem for a meaningful class of mixed monotone multivalued operators and then we use it to derive some results on the existence of quasisolutions and unique solutions to first order functional differential equations with state dependent deviating arguments our results are very general and can be applied to functional equations featuring discontinuities with respect to all of their arguments but we emphasize that they are new even for differential equations with continuously state dependent delays

monotone iterative techniques of v lakshmikantham for a boundary value problem for systems of impulsive differential difference equations by means of the monotone iterative techniques of v lakshmikantham a couple of minimal and maximal quasisolutions of a two point boundary value problem for a system of impulsive differential difference equations are constructed

on a tensor cross product based formulation of large strain solid mechanics abstract this paper describes in detail the formulation of large strain solid mechanics based on the tensor cross product originally presented by r de boer 1982 and recently re introduced by bonet et al 2015a and bonet et al 2015b the paper shows how the tensor cross product facilitates the algebra associated with the area and volume maps between reference and final configurations these maps together with the fibre map make up the fundamental kinematic variables in polyconvex elasticity the algebra proposed leads to novel expressions for the tangent elastic operator which neatly separates material from geometrical dependencies the paper derives new formulas for the spatial and material stress and their corresponding elasticity tensors these are applied to the simple case of a mooney rivlin material model the extension to transversely isotropic material models is also considered

energy conserving and decaying algorithms in non linear structural dynamics a generalized formulation of the energy momentum methodwill be developed within the framework of the generalizedmethodwhich allows at the same time guaranteed conservation or decay of total energy and controllable numerical dissipation of unwanted high frequency response furthermore the latter algorithm will be extended by the consistently integrated constraints of energy and momentum conservation originally derived for the constraint energy momentum algorithm the goal of this general approach of implicit energyconserving and decaying time integration schemes is to compare these algorithms on the basis of an equivalent notation by the means of an overall algorithmic design and hence to investigate their numerical properties numerical stability and controllable numerical dissipation of high frequencies will be studied in application to non linear structural dynamics among the methods considered will be the newmark method the classical methods the energy momentum methodwith and without numerical dissipation the constraint energymomentum algorithm and the constraint energy method copyright 1999 john wiley sons ltd

constraint energy momentum algorithm and its application to non linear dynamics of shells abstract a modified time integration method is developed for non linear structural dynamics combining controllable algorithmic dissipation of higher modes and at the same time conservation of energy as well as linear and angular momentum although these features seem to be actually inconsistent numerical stability with large time steps for dynamic buckling and snap through problems as well as for dynamical systems with smooth solutions is guaranteed arbitrary implicit one step time integration schemes with newmark approximations are suitable for use as basic algorithms for the proposed method the desired conservation attributes are obtained if the basic algorithm is augmented by energy and momentum constraints the potential of the suggested algorithm is demonstrated by selected applications to non linear dynamics of shell structures

energy momentum consistent finite element discretization of dynamic finite viscoelasticity this paper is concerned with energy momentum consistent time discretizations of dynamic finite viscoelasticity energy consistency means that the total energy is conserved or dissipated by the fully discretized system in agreement with the laws of thermodynamics the discretization is energy momentum consistent if also momentum maps are conserved when group motions are superimposed to deformations the performed approximation is based on a three field formulation in which the deformation field the velocity field and a strain like viscous internal variable field are treated as independent quantities the new non linear viscous evolution equation satisfies a non negative viscous dissipation not only in the continuous case but also in the fully discretized system the initial boundary value problem is discretized by using finite elements in space and time thereby the temporal approximation is performed prior to the spatial approximation in order to preserve the stress objectivity for finite rotation increments incremental objectivity although the present approach makes possible to design schemes of arbitrary order the focus is on finite elements relying on linear lagrange polynomials for the sake of clearness the discrete energy momentum consistency is based on the collocation property and an enhanced second piola kirchhoff stress tensor the obtained coupled non linear algebraic equations are consistently linearized the corresponding iterative solution procedure is associated with newly proposed convergence criteria which take the discrete energy consistency into account the iterative solution procedure is therefore not complicated by different scalings in the independent variables since the motion of the element is taken into account for solving the viscous evolution equation representative numerical simulations with various boundary conditions show the superior stability of the new time integration algorithm in comparison with the ordinary midpoint rule both the quasi rigid deformations during a free flight and large deformations arising in a dynamic tensile test are considered copyright 2009 john wiley sons ltd

definite triples of hermitian matrices and matrix polynomials let a b and c be three n n nonzero hermitian matrices the triple a b c is called defnite if the convex hull of the joint numerical range f a b c x ax x bx x cx r3 x cn x x 1 does not contain 0 0 0 if the triple a b c is nondefinite then the numerical ranges of the matrix polynomials q a v3 b v2 c v1 v3 v2 v1 0 and l a 2 b ic 1 2 1 0 coincide with the whole complex plane providing no information as a consequence it is of particular interest to characterize a definite triple a b c and find the distance between 0 0 0 and the boundary f a b c the distance between a nondefinite triple a b c and the nearest definite triples with specified properties is also investigated moreover applications of definite triples on matrix polynomials of special interest are presented

existence theorems of coupled fixed points abstract by associating with a mixed monotone operator a monotone operator on product space several existence theorems of coupled fixed points are proved

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

convexity of the joint numerical range topological and differential geometric viewpoints abstract we investigate the convexity of the joint numerical range of m tuples of n n hermitian matrices the methods come from differential geometry and the differential and algebraic topology our main result is a sufficient condition for convexity of the joint numerical range for arbitrary m and n modulo a mild technical assumption this condition is formulated in terms of the largest eigenvalue of an associated family of hermitian matrices parameterized by the m 1 dimensional unit sphere the condition is that the eigenvalue has constant multiplicity we show that the constant multiplicity condition is in fact a criterion for the stable convexity of numerical ranges as a byproduct of our main result we obtain a new proof of the celebrated toeplitz hausdorff theorem and of its extension the numerical range of any triple of hermitian n n matrices is convex if n 2

the q numerical radius of weighted shift operators with periodic weights we deal with the q numerical radius of weighted unilateral and bilateral shift operators in particular the q numerical radius of weighted shift operators with periodic weights is discussed and computed