Discontinuous Galerkin finite element methods for linear elasticity and quasistatic linear viscoelasticity

Summary. We consider a finite-element-in-space, and quadrature-in-time-discretization of a compressible linear quasistatic viscoelasticity problem. The spatial discretization uses a discontinous Galerkin finite element method based on polynomials of degree r—termed DG(r)—and the time discretization uses a trapezoidal-rectangle rule approximation to the Volterra (history) integral. Both semi- and fully-discrete a priori error estimates are derived without recourse to Gronwall's inequality, and therefore the error bounds do not show exponential growth in time. Moreover, the convergence rates are optimal in both h and r providing that the finite element space contains a globally continuous interpolant to the exact solution (e.g. when using the standard ^k polynomial basis on simplicies, or tensor product polynomials, ^k, on quadrilaterals). When this is not the case (e.g. using ^k on quadri-laterals) the convergence rate is suboptimal in r but remains optimal in h. We also consider a reduction of the problem to standard linear elasticity where similarly optimal a priori error estimates are derived for the DG(r) approximation. Mathematics Subject Classification (2000):65N36Shaw and Whiteman would like to acknowledge the support of the US Army Research Office, Grant #DAAD19-00-1-0421, and the UK EPSRC, Grant #GR/R10844/01. Whiteman would also like to acknowledge support from TICAM in the form of Visiting Research Fellowships.     

Analysis of a quasistatic contact problem for piezoelectric materials

We consider a mathematical model which describes the frictional contact between a piezoelectric body and an electrically conductive foundation. The process is quasistatic, the material behavior is modeled with an electro-viscoelastic constitutive law and the contact is described with subdifferential boundary conditions. We derive the variational formulation of the problem which is in the form of a system involving two history-dependent hemivariational inequalities in which the unknowns are the velocity and electric potential field. Then we prove the existence of a unique weak solution to the model. The proof is based on a recent result on history-dependent hemivariational inequalities obtained in Migórski et al. (submitted for publication) [16].     

A complementarity approach to a quasistatic multi-rigid-body contact problem

In this paper, we study the problem of predicting the quasistatic planar motion of a passive rigid body in frictional contact with a set of active rigid bodies. The active bodies can be thought of as the links of a mechanism or robot manipulator whose positions can be actively controlled by actuators. The passive body can be viewed as a grasped object, which moves only in response to contact forces and other external forces such as those due to gravity. We formulate this problem as a certain uncoupled complementarity problem, and show that it belongs to the class of NP-complete problems. Finally, numerical results of our proposed linear programming-based solution algorithm for this class of problems are presented and compared to the only other currently available solution algorithm.The research of this author was based on work supported by the National Science Foundation under grants DDM-9104078 and CCR-9213739.The research of this author was partially supported by the National Science Foundation under grant IRI-9304734, the Texas Advanced Technology Program under grant 999903-095, and the Texas Advanced Research Program under grant 999903-078.     

Discontinuous Galerkin methods for the chemotaxis and haptotaxis models

In this work, first we formulate and compare three different discontinuous Interior Penalty Galerkin methods for the 2D Keller–Segel chemotaxis model. Keller–Segel chemotaxis model is the important starting step in the modeling of the real biological system. We show in the numerical tests that two of the proposed methods fail to give accurate, oscillation-free solutions.     

A quasistatic contact problem for viscoelastic materials with friction and damage

We consider a mathematical model which describes the frictional contact between a deformable body and a foundation. The process is quasistatic, the material is assumed to be viscoelastic with long memory and the frictional contact is modelled with subdifferential boundary conditions. The mechanical damage of the material is described by the damage function, which is modelled by a nonlinear partial differential equation. We derive the variational formulation of the problem, which is a coupled system of a hemivariational inequality and a parabolic equation. Then we prove the existence of a unique weak solution to the model. The proof is based on arguments of abstract stationary inclusion and a fixed point theorem.     

A quasistatic viscoplastic contact problem with normal compliance and friction

We consider a quasistatic contact problem between a viscoplasticbody and an obstacle, the so-called foundation. The contactis modelled with normal compliance and the associated versionof Coulomb's law of dry friction. We derive a variational formulationof the problem and, under a smallness assumption on the normalcompliance functions, we establish the existence of a weak solutionto the model. The proof is carried out in several steps. Itis based on a time-discretization method, arguments of monotonicityand compactness, Banach fixed point theorem and Schauder fixedpoint theorem.     

Galerkin methods for a semilinear parabolic problem with nonlocal boundary conditions

We formulate and analyze a Crank-Nicolson finite element Galerkin method and an algebraically-linear extrapolated Crank-Nicolson method for the numerical solution of a semilinear parabolic problem with nonlocal boundary conditions. For each method, optimal error estimates are derived in the maximum norm.Dedicated to Professor J. Crank on the occasion of his 80th birthdaySupported in part by the National Science Foundation grant CCR-9403461.Supported in part by project DGICYT PB95-0711.     

A Galerkin method with modified piecewise polynomials for solving a second-order boundary value problem

Summary The classical Ritz-Galerkin method is applied to a linear, second-order, self-adjoint boundary value problem. The coefficient functions of the operator exhibit a piecewise smooth behaviour characteristic of some physical situations. A trial function is constructed using a modified quintic smooth Hermite space , in order to meet some desired regularity conditions for the approximate solution. A collocation technique is used to reduce the amount of computational work. Known convergence properties for the projection method are recalled which, in this particular case, are illustrated by a series of numerical experiments.     

Successive search methods for solving a canonical DC programming problem

In this article, we propose two successive search methods for solving a canonical DC programming problem constrained by the difference set between two compact convex sets in the case where the dimension number is greater than or equal to three. In order to find feasible solutions, the algorithms generate the directions based on a branch and bound procedure, successively. By exploring the provisional solutions throughout the intersection of the boundaries of two compact convex sets, both algorithms calculate an approximate solution.     

A New Discontinuous Galerkin Method for Parabolic Equations with Discontinuous Coefficient

In this paper, a new discontinuous Galerkin method is developed for the parabolic equation with jump coefficients satisfying the continuous flow condition. Theoretical analysis shows that this method is $L^2$ stable. When the finite element space consists of interpolative polynomials of degrees $k$, the convergent rate of the semi-discrete discontinuous Galerkin scheme has an order of$\mathcal{O}(h^k)$. Numerical examples for both 1-dimensional and 2-dimensional problems demonstrate the validity of the new method.     

Numerical analysis and simulations of a quasistatic frictional contact problem with damage in viscoelasticity

We consider a quasistatic problem which models the bilateral contact between a viscoelastic body and a foundation, taking into account the damage and the friction. The damage which results from tension or compression is then involved in the constitutive law and it is modelled using a nonlinear parabolic inclusion. The variational problem is formulated as a coupled system of evolutionary equations for which we state the existence of a unique solution. Then, we introduce a fully discrete scheme using the finite element method to approximate the spatial variable and the Euler scheme to discretize the time derivatives. Error estimates are derived and, under suitable regularity hypotheses, the convergence of the numerical scheme obtained. Finally, a numerical algorithm and results are presented for some two-dimensional examples.     

Numerical methods for solving the coefficient inverse problem

Translated from Metody Matematicheskogo Modelirovaniya i Vychislitel'noi Diagnostika, pp. 35–45, Izd. Moskovskogo Universiteta, Moscow, 1990.     

Best estimates of RBF-based meshless Galerkin methods for Dirichlet problem

In this paper, Galerkin methods based on the radial basis functions to deal with the partial differential equations are discussed. The best error estimates for this method are obtained.     

A quasistatic contact problem with slip‐dependent coefficient of friction

We consider a mathematical model which describes the bilateral quasistatic contact of a viscoelastic body with a rigid obstacle. The contact is modelled with a modified version of Coulomb's law of dry friction and, moreover, the coefficient of friction is assumed to depend either on the total slip or on the current slip. In the first case, the problem depends upon contact history. We present the classical formulations of the problems, the variational formulations and establish the existence and uniqueness of a weak solution to each of them, when the coefficient of friction is sufficiently small. The proofs are based on classical results for elliptic variational inequalities and fixed point arguments. We also study the dependence of the solutions on the perturbations of the friction coefficient and obtain a uniform convergence result. Copyright © 1999 John Wiley & Sons, Ltd.     

Variational and numerical analysis of a quasistatic viscoelastic contact problem with normal compliance and friction

In this paper, a class of generalized evolution variational inequalities arising in quasistatic friction contact problem for viscoelastic materials is introduced and studied. Under some suitable assumptions, we obtain an existence and uniqueness theorem of the solution for the generalized evolution variational inequalities by using Banach’s fixed point theorem. Moreover, we study two numerical approximation schemes of the problem: semidiscrete scheme and fully discrete scheme. For both schemes, we prove the existence of the solution and derive the error estimations.     

Galerkin and weighted Galerkin methods for a forward-backward heat equation

Summary. Galerkin and weighted Galerkin methods are proposed for the numerical solution of parabolic partial differential equations where the diffusion coefficient takes different signs. The approach is based on a simultaneous discretization of space and time variables by using continuous finite element methods. Under some simple assumptions, error estimates and some numerical results for both Galerkin and weighted Galerkin methods are presented. Comparisons with the previous methods show that new methods not only can be used to solve a wider class of equations but also require less regularity for the solution and need fewer computations. Received March 3, 1995     

Local discontinuous Galerkin methods with explicit-implicit-null time discretizations for solving nonlinear diffusion problems

In this paper,we discuss the local discontinuous Galerkin methods coupled with two specific explicitimplicit-null time discretizations for solving one-dimensional nonlinear diffusion problems Ut=(a(U)Ux)x.The basic idea is to add and subtract two equal terms a0 Uxx the right-hand side of the partial differential equation,then to treat the term a0 Uxx implicitly and the other terms(a(U)Ux)x-a0 Uxx explicitly.We give stability analysis for the method on a simplified model by the aid of energy analysis,which gives a guidance for the choice of a0,i.e.,a0≥max{a(u)}/2 to ensure the unconditional stability of the first order and second order schemes.The optimal error estimate is also derived for the simplified model,and numerical experiments are given to demonstrate the stability,accuracy and performance of the schemes for nonlinear diffusion equations.     

Structural Stability of Discontinuous Galerkin Schemes

The goal of this work is to determine classes of traveling solitary wave solutions for a differential approximation of a discontinuous Galerkin finite difference scheme by means of an hyperbolic ansatz. It is shown that spurious solitary waves can occur in finite-difference solutions of nonlinear wave equation. The occurence of such a spurious solitary wave, which exhibits a very long life time, results in a non-vanishing numerical error for arbitrary time in unbounded numerical domain. Such a behavior is referred here to have a structural instability of the scheme, since the space of solutions spanned by the numerical scheme encompasses types of solutions (solitary waves in the present case) that are not solutions of the original continuous equations. This paper extends our previous work about classical schemes to discontinuous Galerkin schemes (David and Sagaut in Chaos Solitons Fractals 41(4):2193?C2199, 2009; Chaos Solitons Fractals 41(2):655?C660, 2009).