complementarity framework for non linear dynamic analysis of skeletal structures with softening plastic hinges in this paper we describe an algorithm for the incremental state update of elasto plastic systems with softening the algorithm uses a complementary pivoting technique and is based on casting the incremental state update as a complementarity problem in developing the algorithm we take advantage of the special features of solid and structural mechanics problems to achieve good computational performance and hence the ability to compute numerical solutions to practical size problems for example the notion of a tangent stiffness matrix arises numerical examples using models of skeletal structures are presented to demonstrate the practicability of the algorithm the numerical examples also raise some interesting questions about multiplicity of the solutions copyright 2010 john wiley sons ltd

on a time dependent grade two fluid model in two dimensions we prove the existence of a weak solution of a time dependent grade two fluid model in a plane lipschitz domain and uniqueness of the solution in a convex polygon the method of proof is constructive and can be adapted to the numerical analysis of finite element schemes for solving this problem numerically

exact solutions for electro osmotic flow of viscoelastic fluids in rectangular micro channels transient electro osmotic flow of viscoelastic fluids in rectangular micro channels is investigated the general twofold series solution for the velocity distribution of electro osmotic flow of viscoelastic fluids with generalized fractional oldroyd b constitutive model is obtained by using finite fourier and laplace transforms under three limiting cases the generalized oldroyd b model simplifies to newtonian model fractional maxwell model and generalized second grade model where all the explicit exact solutions for velocity distribution are found through the discrete laplace transform of the sequential fractional derivatives these exact solutions may be able to predict the flow behavior of viscoelastic biological fluids in biomems and lab on a chip devices and thus could benefit the design of these devices

damage integration in the strain space the focus is on isotropic elastodamaging softening materials where the damage parameter is expressed as a function of the total strain by integrating the mechanical work in the strain space along a stepwise holonomic loading history an incremental strain energy is obtained a coaxiality condition for the incremental strain energy to be potential is identified and its implications on the associativity of the damage evolution are discussed under some hypotheses the increment of the mechanical work is shown to be minimum along strain radial paths these results are used to construct a multifield variational framework supporting finite element nonlocal formulations

elastoplastic damage modelling including the gradient of damage formulation and computational aspects a framework for continuum elastoplastic damage modelling which employs irreversible thermodynamics and internal state variables is investigated the damage part of the modelling involves the gradient of damage quantity which together with the equations of motion are issued from a new formulation of the principle of virtual power recently proposed it is shown how the plastic part of the modelling takes its standard format next we consider in detail the variational formulation and subsequent numerical implementation of the elastoplastic damage models the development of an algorithm consistent with the present formulation is given where for the plastic part it leaves the standard return mapping algorithms unchanged application is made to different classical rate independent and rate dependent plastic models with the presence of a damage mechanism the numerical implementation in the context of the finite element method is discussed in detail and a set of representative applications and numerical examples is given

multilevel formulations in the limit analysis and design of structures with bilateral contact constraints in this paper we study the rich class of formulations that arise in the limit analysis and design of elastic plastic structures in the presence of contact constraints it is well known that in the absence of contacts both the limit analysis and limit design problems can be written as linear programs however when contact constraints are present the structure effectively exhibits both softening and stiffening behaviour under monotonically increasing loading the resulting limit analysis and limit design problems are non convex and are difficult to solve due to the presence of complementary type of equality constraints we show that by using a mixed form of the minimum principle we can restate the limit analysis and limit design problems as two and three level formulations respectively further under a strong assumption on the problem and solution data we can take advantage of the underlying convexity to reduce both these multilevel formulations to equivalent linear programs while it may not be possible to always verify this assumption in practice we show that a two step iterative procedure is effective in reaching a solution to the limit design problem

a mixture of thermoelastic solids with two temperatures abstract in this work we study from the numerical point of view a problem involving one dimensional thermoelastic mixtures with two different temperatures that is when each component of the mixture has its own temperature the mechanical problem consists of two hyperbolic equations coupled with two parabolic equations the variational problem is derived in terms of product variables an existence and uniqueness result and an energy decay property are stated then fully discrete approximations are introduced using the finite element method and the backward euler scheme a discrete stability property is proved and a priori error estimates are obtained from which the linear convergence is deduced finally some numerical simulations are described to show the accuracy of the approximation and the behavior of the solution

a quasistatic frictional problem with normal compliance a friction and contact problem will be considered for an elastic solid body which under the influence of volume and surface forces is in contact with a rigid foundation

a finite element method to model progressive fracturing abstract in this paper a theoretical basis is established for a finite element iterative algorithm for load displacement analysis of elasticity problems in which the elastic body fractures under load location of fractures is determined by testing a coulomb friction criterion along inter element edges convergence of the algorithm to a physically meaningful solution is proved and is illustrated by results for a cantilever type problem

gradient enhanced damage for quasi brittle materials summary conventional continuum damage descriptions of material degeneration suffer from loss of well posedness beyond a certain level of accumulated damage as a consequence numerical solutions are obtained which are unacceptable from a physical point of view the introduction of higher order deformation gradients in the constitutive model is demonstrated to be an adequate remedy to this deficiency of standard damage models a consistent numerical solution procedure of the governing partial differential equations is presented which is shown to be capable of properly simulating localization phenomena

iterative lcp solvers for non local loading unloading conditions this paper deals with the finite element analysis of a certain class of non local dissipative constitutive models where the canonical pointwise backward euler scheme cannot be employed for satisfying the loading unloading conditions in the presence of a non local dissipation the admissibility conditions in a point depend on the inelastic strain increment of the surrounding points and can be cast as a linear complementarity problem lcp involving all gauss points of the process zone in order to actually solve the lcp the use of iterative algorithms that can be easily embodied into existing fe codes is discussed the performance of the proposed algorithms is tested in 1d and 2d examples for both elastoplastic and damaging materials copyright 2003 john wiley sons ltd

compact finite difference scheme and rbf meshless approach for solving 2d rayleigh stokes problem for a heated generalized second grade fluid with fractional derivatives abstract this article is devoted to the comparison between a high order difference scheme and radial basis functions rbfs meshless approach for the numerical solution of two dimensional fractional rayleigh stokes problem the fractional derivative of problem is described in the riemann liouville sense in the high order difference scheme we discretize the spaces derivatives with a fourth order compact scheme and use the grunwald letnikov discretization of the riemann liouville derivatives to obtain a fully discrete implicit scheme also in the rbf meshless method we discretize the time fractional derivatives of the mentioned equation by integrating both sides of it then we will use the kansa s approach to approximate the spatial derivatives we prove the stability and convergence for high order difference scheme using fourier analysis and for time discrete scheme in rbf method via energy approach we compare the results of compact finite difference method and rbf meshless approach in terms of accuracy and cpu time also we compare the numerical results of compact difference scheme with other methods in the literature to show the high accuracy and efficiency of the proposed method

two fully discrete schemes for fractional diffusion and diffusion wave equations with nonsmooth data we consider initial boundary value problems for the subdiffusion and diffusion wave equations involving a caputo fractional derivative in time we develop two fully discrete schemes based on the piecewise linear galerkin finite element method in space and convolution quadrature in time with the generating function given by the backward euler method second order backward difference method and establish error estimates optimal with respect to the regularity of problem data these two schemes are first and second order accurate in time for both smooth and nonsmooth data extensive numerical experiments for two dimensional problems confirm the convergence analysis and robustness of the schemes with respect to data regularity

variational and numerical analysis of a quasistatic viscoelastic problem with normal compliance friction and damage we consider a model for quasistatic frictional contact between a viscoelastic body and a foundation the material constitutive relation is assumed to be nonlinear the mechanical damage of the material caused by excessive stress or strain is described by the damage function the evolution of which is determined by a parabolic inclusion the contact is modeled with the normal compliance condition and the associated version of coulomb s law of dry friction we derive a variational formulation for the problem and prove the existence of its unique weak solution we then study a fully discrete scheme for the numerical solutions of the problem and obtain error estimates on the approximate solutions

initial value boundary value problems for fractional diffusion wave equations and applications to some inverse problems abstract we consider initial value boundary value problems for fractional diffusion wave equation t u x t l u x t where 0 2 where l is a symmetric uniformly elliptic operator with t independent smooth coefficients first we establish the unique existence of the weak solution and the asymptotic behavior as the time t goes to and the proofs are based on the eigenfunction expansions second for 0 1 we apply the eigenfunction expansions and prove i stability in the backward problem in time ii the uniqueness in determining an initial value and iii the uniqueness of solution by the decay rate as t iv stability in an inverse source problem of determining t dependent factor in the source by observation at one point over 0 t

an analysis of the rayleigh stokes problem for a generalized second grade fluid we study the rayleigh stokes problem for a generalized second grade fluid which involves a riemann liouville fractional derivative in time and present an analysis of the problem in the continuous space semidiscrete and fully discrete formulations we establish the sobolev regularity of the homogeneous problem for both smooth and nonsmooth initial data v v including v in l 2 omega v l2 a space semidiscrete galerkin scheme using continuous piecewise linear finite elements is developed and optimal with respect to initial data regularity error estimates for the finite element approximations are derived further two fully discrete schemes based on the backward euler method and second order backward difference method and the related convolution quadrature are developed and optimal error estimates are derived for the fully discrete approximations for both smooth and nonsmooth initial data numerical results for one and two dimensional examples with smooth and nonsmooth initial data are presented to illustrate the efficiency of the method and to verify the convergence theory

an enumerative method for the solution of linear complementarity problems abstract in this paper an enumerative method for the solution of the linear complementarity problem lcp is presented this algorithm either finds all the solutions of any general lcp that is no assumption is made concerning the class of the matrix or it establishes that no such solution exists the method is extended to the second linear complementarity problem slcp a new problem which has been introduced for the solution of a general quadratic program

an fe mpec approach for limit load evaluation in the presence of contact and displacement constraints abstract this paper describes a mathematical programing based approach for the direct limit load evaluation of a structural system under simultaneous contact and limited displacement conditions the contact model we adopt can simulate either a classical unilateral nonassociative coulomb friction situation or a cohesive fracture idealization at the potential discontinuity interface between contacting bodies the discrete fe model is constructed using locking free mixed finite elements the main feature and novelty of our proposed approach is to compute in a single step the maximum load capacity of the structure such that both the imposed displacement limitations and the contact conditions are satisfied in essence the formulation is a nontrivial extension of classical limit analysis the analysis is cast in its most natural form namely in mixed static kinematic variables and leads to what is known in the mathematical programing literature as a mathematical program with equilibrium constraints or mpec a nonlinear programing nlp based algorithm is proposed to solve the mpec two examples are provided to illustrate application of the proposed scheme and some comments regarding the various advantages of the adopted mathematical programing framework are made

error estimates for a semidiscrete finite element method for fractional order parabolic equations we consider the initial boundary value problem for a homogeneous time fractional diffusion equation with an initial condition v x and a homogeneous dirichlet boundary condition in a bounded convex polygonal domain omega we study two semidiscrete approximation schemes i e the galerkin finite element method fem and lumped mass galerkin fem using piecewise linear functions we establish almost optimal with respect to the data regularity error estimates including the cases of smooth and nonsmooth initial data i e v in h 2 omega cap h 1 0 omega and v in l 2 omega for the lumped mass method the optimal l 2 norm error estimate is valid only under an additional assumption on the mesh which in two dimensions is known to be satisfied for symmetric meshes finally we present some numerical results that give insight into the reliability of the theoretical study

a numerical method for a partial integro differential equation a method is considered for the integration in time of a partial integro differential equation the discretization technique employed is patterned after an idea of ch lubich error bounds are derived for both smooth and nonsmooth initial data

semilinear subdiffusion with memory in the one dimensional case abstract for 0 1 we consider the semilinear integro differential equation on the one dimensional domain a b in the unknown u u x t d t u l 1 u 0 t k t s l 2 u s d s f x t u g x t where d t is the caputo fractional derivative and l 1 and l 2 are uniform elliptic operators with time dependent smooth coefficients under certain structural conditions on the nonlinearity f the global existence and uniqueness of classical solutions to the related initial boundary value problems are established via the so called continuation argument approach the key point is looking for suitable a priori estimates of the solution in the fractional holder spaces

a fourier method and an extrapolation technique for stokes first problem for a heated generalized second grade fluid with fractional derivative in this article we consider stokes first problem for a heated generalized second grade fluid with fractional derivative sfp hgsgf implicit and explicit numerical approximation schemes for the sfp hgsgf are presented the stability and convergence of the numerical schemes are discussed using a fourier method in addition the solvability of the implicit numerical approximation scheme is also analyzed a richardson extrapolation technique for improving the order of convergence of the implicit scheme is proposed finally a numerical test is given the numerical results demonstrate the good performance of our theoretical analysis

numerical analysis of the rayleigh stokes problem for a heated generalized second grade fluid with fractional derivatives australian research council dp0559807 national natural science foundation of china 10271098 nsf of china tianyuan fund for mathematics 10726061

on the solution of a minimum weight elastoplastic problem involving displacement and complementarity constraints this paper deals with a special class of minimum weight design problem involving discretized structures holonomic reversible plasticity and constraints on displacements a key feature of the optimization problem is the presence of complementarity conditions involving the orthogonality of two sign constrained vectors this problem falls within the important class of so called mathematical programs with equilibrium constraints mpecs two simple algorithms are proposed to solve our synthesis problem and application is illustrated by some examples concerning truss like structures

a frictionless contact problem for elastic viscoplastic materials with normal compliance numerical analysis and computational experiments in this paper we consider a frictionless contact problem between an elastic viscoplastic body and an obstacle the process is assumed to be quasistatic and the contact is modeled with normal compliance we present a variational formulation of the problem and prove the existence and uniqueness of the weak solution using strongly monotone operators arguments and banach s fixed point theorem we also study the numerical approach to the problem using spatially semi discrete and fully discrete finite elements schemes with implicit and explicit discretization in time we show the existence of the unique solution for each of the schemes and derive error estimates on the approximate solutions finally we present some numerical results involving examples in one two and three dimensions

a simple and efficient bem implementation of quasistatic linear visco elasticity abstract a simple yet efficient procedure to solve quasistatic problems of special linear visco elastic solids at small strains with equal rheological response in all tensorial components utilizing boundary element method bem is introduced this procedure is based on the implicit discretisation in time the so called rothe method combined with a simple algebraic transformation of variables leading to a numerically stable procedure proved explicitly by discrete energy estimates which can be easily implemented in a bem code to solve initial boundary value visco elastic problems by using the kelvin elastostatic fundamental solution only it is worth mentioning that no inverse laplace transform is required here the formulation is straightforward for both 2d and 3d problems involving unilateral frictionless contact although the focus is to the simplest kelvin voigt rheology a generalization to maxwell boltzmann jeffreys and burgers rheologies is proposed discussed and implemented in the bem code too a few 2d and 3d initial boundary value problems one of them with unilateral frictionless contact are solved numerically

duality methods for solving variational inequalities abstract methods of maximal monotone operators are used in order to study from a general point of view duality numerical algorithms for solving variational inequalities with classical algorithms such as uzawa s method for the standard and augmented lagrangian this paper presents some new algorithms which appear to have very good numerical performances

two aggregate function based algorithms for analysis of 3d frictional contact by linear complementarity problem formulation three dimensional frictional contact is formulated as linear complementarity problem lcp by using the parametric variational principle and quadratic programming method two aggregate function based algorithms called respectively as self adjusting interior point algorithm and aggregate function smoothing algorithm are proposed for the solution of the lcp derived from the contact problems a nonlinear finite element code is developed for numerical analysis of 3d multi body contact problems four numerical examples are computed to demonstrate the applicability and computational efficiency of the methods proposed

a frictionless contact problem for elastic viscoplastic materials with normal compliance and damage abstract we study a quasistatic frictionless contact problem with normal compliance and damage for elastic viscoplastic bodies the mechanical damage of the material caused by excessive stress or strain is described by a damage function whose evolution is modelled by an inclusion of parabolic type we provide a variational formulation for the mechanical problem and sketch a proof of the existence of a unique weak solution to the model we then introduce and study a fully discrete scheme for the numerical solutions of the problem an optimal order error estimate is derived for the approximate solutions under suitable solution regularity numerical examples are presented to show the performance of the method

an improved tau method for the multi dimensional fractional rayleigh stokes problem for a heated generalized second grade fluid abstract we develop efficient algorithms based on the legendre tau approximation for one and two dimensional fractional rayleigh stokes problems for a generalized second grade fluid the time fractional derivative is described in the riemann liouville sense discussions on the l 2 convergence of the proposed method are presented numerical results for one and two dimensional examples with smooth and nonsmooth solutions are provided to verify the validity of the theoretical analysis and to illustrate the efficiency of the proposed algorithms

analysis of dynamic nonlinear thermoviscoelastic beam problems abstract in this paper we study a dynamic contact problem between a nonlinear viscoelastic beam and two rigid obstacles thermal effects are also taken into account and the contact is modeled using the classical signorini condition the existence of solutions is proved by considering approximate problems from a penalization procedure proving some a priori estimates and passing to the limit an exponential decay property is also obtained then fully discrete approximations of the approximate problem are provided using the finite element method for the spatial approximation and the implicit euler scheme for the discretization of the time derivatives a stability property is obtained and a priori error estimates are proved from which under some additional regularity conditions the linear convergence of the algorithm is deduced fully discrete approximations of the signorini problem are then introduced proceeding in a similar way for which a stability property is obtained finally some numerical simulations are performed to demonstrate the accuracy of the approximation and the behavior of the solution

numerical algorithm for the space time fractional fokker planck system with two internal states the fractional fokker planck system with multiple internal states is derived in xu and deng math model nat phenom mathbf 13 10 2018 where the space derivative is laplace operator if the jump length distribution of the particles is power law instead of gaussian the space derivative should be replaced with fractional laplacian this paper focuses on solving the two state fokker planck system with fractional laplacian we first provide a priori estimate for this system under different regularity assumptions on the initial data then we use l 1 scheme to discretize the time fractional derivatives and finite element method to approximate the fractional laplacian operators furthermore we give the error estimates for the space semidiscrete and fully discrete schemes without any assumption on regularity of solutions finally the effectiveness of the designed scheme is verified by one and two dimensional numerical experiments

error analysis of a finite element method for the space fractional parabolic equation we consider an initial boundary value problem for a one dimensional fractional order parabolic equation with a space fractional derivative of riemann liouville type and order alpha in 1 2 we study a spatial semidiscrete scheme using the standard galerkin finite element method with piecewise linear finite elements as well as fully discrete schemes based on the backward euler method and the crank nicolson method error estimates in the l 2 d and h alpha 2 d norm are derived for the semidiscrete scheme and in the l 2 d norm for the fully discrete schemes these estimates cover both smooth and nonsmooth initial data and are expressed directly in terms of the smoothness of the initial data extensive numerical results are presented to illustrate the theoretical results

feasible descent algorithms for mixed complementarity problems in this paper we consider a general algorithmic framework for solving nonlinear mixed complementarity problems the main features of this framework are a it is well defined for an arbitrary mixed complementarity problem b it generates only feasible iterates c it has a strong global convergence theory and d it is locally fast convergent under standard regularity assumptions this framework is applied to the path solver in order to show viability of the approach numerical results for an appropriate modification of the path solver indicate that this framework leads to substantial computational improvements

convolution quadrature and discretized operational calculus ii operational quadrature rules are applied to problems in numerical integration and the numerical solution of integral equations singular integrals power and logarithmic singularities finite part integrals multiple timescale convolution volterra integral equations wiener hopf integral equations frequency domain conditions which determine the stability of such equations can be carried over to the discretization

numerical solving of frictionless contact problems in perfectly plastic bodies abstract we consider the frictionless unilateral contact problems of a perfectly plastic solid with a rigid foundation and two perfectly plastic solids under an analogous variational formulation for each one we introduce a discrete approximation via the finite element method and in due course prove convergence also we introduce a converging iterative algorithm to the solution of the discrete problems and present some numerical experiences

theoretical and computational aspects of a thermodynamically consistent framework for geometrically linear gradient damage this paper presents the theory and the numerics of an isotropic gradient damage formulation within a thermodynamical background the main motivation is provided by localization computations whereby classical local continuum formulations fail to produce physically meaningful and numerically converging results we propose a formulation in terms of the helmholtz free energy incorporating the gradient of the damage field a dissipation potential and the postulate of maximum dissipation as a result the driving force conjugated to damage evolution is identified as the quasi nonlocal energy release rate which essentially incorporates the divergence of a vectorial damage flux besides the strictly local energy release rate on the numerical side besides balance of linear momentum the algorithmic consistency condition must be solved in weak form thereby the crucial issue is the selection of active constraints which is solved by an active set search algorithm borrowed from convex nonlinear programming in the examples we compare the behavior in local damage with the performance of the gradient formulation

damage gradient of damage and principle of virtual power a theory of continuum damage mechanics is developed within the framework of the principle of virtual power because the damage in a solid results from microscopic movements we decided to include the power of these microscopic movements in the power of the internal forces the microscopic velocities are related to the damage rate the power of the internal forces we choose depends on the damage velocity and its gradient to take into account the interactions models issued from this theory are presented they are coherent from the mechanical and mathematical points of view the numerical computations show no mesh sensitivity they describe with good agreement the main experimental properties concrete is chosen as an example to illustrate the theory a model using two damage quantities is also presented it yields the correct description of the unilateral phenomenon observed in concrete finally an extension of the existing models is proposed to describe the fatigue damage behaviour

analysis of a multidimensional thermoviscoelastic contact problem under the green lindsay theory abstract in this paper we investigate the existence the stability and the numerical approximation of a multidimensional dynamic contact problem modeling the evolution of displacement and temperature in a viscoelastic body that may come into contact with a deformable foundation the viscoelastic body is assumed to behave according to kelvin voigt constitutive law with added thermal effects under the green lindsay theory we prove that the presence of viscoelastic terms in the equations provides additional regularity and then an existence and uniqueness result is obtained using the faedo galerkin method an energy decay property is also shown under the assumption of radial symmetry then a numerical approximation based on the finite element method is proposed a stability result is proved from which the decay of the discrete energy is deduced a priori error estimates are shown from which the linear convergence is derived under suitable additional regularity conditions finally some numerical experiments are described to support our results

analysis of a contact problem involving thermoelastic mixtures abstract in this work we study a dynamic contact problem between a thermoelastic mixture and a deformable obstacle the classical normal compliance condition is used for modeling the contact the variational formulation of this problem is written as a nonlinear coupled system of three parabolic variational equations an existence and uniqueness result is proved using the faedo galerkin method then fully discrete approximations are introduced by using the finite element method and the implicit euler scheme a discrete stability property and a priori error estimates are proved from which the decay of the discrete energy and the linear convergence of the algorithm are deduced finally some numerical simulations are presented to show the convergence and the behavior of the solution

an analysis of the modified l1 scheme for time fractional partial differential equations with nonsmooth data we introduce a modified l1 scheme for solving time fractional partial differential equations and obtain error estimates for smooth and nonsmooth initial data in both homogeneous and inhomogeneous c

nonsmooth data error estimates for approximations of an evolution equation with a positive type memory term we study the numerical approximation of an integro differential equation which is intermediate between the heat and wave equations the proposed discretization uses convolution quadrature based on the first and second order backward difference methods in time and piecewise linear finite elements in space optimal order error bounds in terms of the initial data and the inhomogeneity are shown for positive times without assumptions of spatial regularity of the data

nonholonomic elastoplastic analysis involving unilateral frictionless contact as a mixed complementarity problem this paper deals with the elastoplastic analysis of suitably discretized elastoplastic structures under frictionless unilateral contact conditions whilst the mathematical programming formulation we adopt can cater equally for hardening as well as softening path dependent nonholonomic plasticity the state problem involving softening is by far the more difficult to solve in view of the inherent nonconvex nature of the underlying mathematical program we propose a novel description of a fairly general class of piecewise linear softening laws and investigate use of an approximate stepwise holonomic path independent approach for capturing the evolution of the structure with softening and or hardening constitutive laws this finite step problem is formulated in a specific way to take advantage of the use of a robust and industry standard complementarity solver named path from within the gams modeling environment we illustrate its application on truss like structures

superconvergence of a discontinuous galerkin method for fractional diffusion and wave equations we consider an initial boundary value problem for partial tu partial t alpha nabla 2u f t that is for a fractional diffusion 1 alpha 0 or wave 0 alpha 1 equation a numerical solution is found by applying a piecewise linear discontinuous galerkin dg method in time combined with a piecewise linear conforming finite element method in space the time mesh is graded appropriately near t 0 but the spatial mesh is quasi uniform previously we proved that the error measured in the spatial l 2 norm is of order k 2 alpha h 2 ell k uniformly in t where k is the maximum time step h is the maximum diameter of the spatial finite elements alpha min alpha 0 le0 and ell k max 1 log k here we prove convergence of order k 3 2 alpha ell k h 2 at each time level t n for 1 alpha 1 thus if 1 2 alpha 1 then the dg solution is superconvergent which generalizes a known result for the classical heat equation i e the case alpha 0 a

numerical analysis of a frictionless viscoelastic contact problem with normal damped response abstract we consider a mathematical model which describes the frictionless contact between a viscoelastic body and a deformable foundation we model the material s behavior with a nonlinear kelvin voigt constitutive law the process is assumed to be quasistatic and the contact is modeled with a general normal damped response condition we present the variational formulation of the problem including the existence of a unique weak solution to the model we then study the numerical approach to the problem using a fully discrete finite elements scheme with an explicit discretization in time we state the existence of the unique solution for the scheme and derive an error estimate on the approximate solutions finally we present some numerical results involving examples in one and two dimensions

evolutionary variational inequalities arising in viscoelastic contact problems we consider a class of evolutionary variational inequalities arising in various frictional contact problems for viscoelastic materials under the smallness assumption of a certain coefficient we prove an existence and uniqueness result using banach s fixed point theorem we then study two numerical approximation schemes of the problem a semidiscrete scheme and a fully discrete scheme for both schemes we show the existence of a unique solution and derive error estimates finally all these results are applied to the analysis and numerical approximations of a viscoelastic frictional contact problem with the finite element method used to discretize the spatial domain

numerical analysis and simulations of a dynamic frictionless contact problem with damage the process of dynamic frictionless contact between a viscoelastic body and a reactive foundation which includes material damage is modelled numerically analyzed and simulated contact is modelled with the normal compliance condition the damage of the material resulting from tension or compression is described by the damage field which measures the pointwise fractional decrease in the load carrying capacity of the material the evolution of the damage is governed by a parabolic differential inclusion a fully discrete scheme for the problem is proposed and error estimates on the numerical solutions derived the scheme has been implemented and numerical simulations of the evolution of the mechanical state of the system and its material damage are presented