Numerical analysis of a contact problem including bone remodeling

In this work, a contact problem between an elastic body and a deformable obstacle is numerically studied. The bone remodeling of the material is also taken into account in the model and the contact is modeled using the normal compliance contact condition. The variational problem is written as a nonlinear variational equation for the displacement field, coupled with a first-order ordinary differential equation to describe the physiological process of bone remodeling. An existence and uniqueness result of weak solutions is stated. Then, fully discrete approximations are introduced based on the finite element method to approximate the spatial variable and an Euler scheme to discretize the time derivatives. Error estimates are obtained, from which the linear convergence of the algorithm is derived under suitable regularity conditions. Finally, some 2D numerical results are presented to demonstrate the behavior of the solution.     

Damageable contact between an elastic body and a rigid foundation

In this work, the contact problem between an elastic body and a rigid obstacle is studied, including the development of material damage which results from internal compression or tension. The variational problem is formulated as a first-kind variational inequality for the displacements coupled with a parabolic partial differential equation for the damage field. The existence of a unique local weak solution is stated. Then, a fully discrete scheme is introduced using the finite element method to approximate the spatial variable and an Euler scheme to discretize the time derivatives. Error estimates are derived on the approximate solutions, from which the linear convergence of the algorithm is deduced under suitable regularity conditions. Finally, three two-dimensional numerical simulations are performed to demonstrate the accuracy and the behaviour of the scheme.     

Robust Certified Numerical Homotopy Tracking

We describe, for the first time, a completely rigorous homotopy (path-following) algorithm (in the Turing machine model) to find approximate zeros of systems of polynomial equations. If the coordinates of the input systems and the initial zero are rational our algorithm involves only rational computations, and if the homotopy is well posed an approximate zero with integer coordinates of the target system is obtained. The total bit complexity is linear in the length of the path in the condition metric, and polynomial in the logarithm of the maximum of the condition number along the path, and in the size of the input.     

Convergence of free boundaries in discrete obstacle problems

We show that a piecewise linear finite element approximation of the obstacle problem gives an approximate free boundary converges, in an appropriate distance, to the free boundary of the continuous problem under a stability condition on the obstacle.     

Numerical analysis of a quasi-static contact problem for a thermoviscoelastic beam

In this paper we revisit a quasi-static contact problem of a thermoviscoelastic beam between two rigid obstacles which was recently studied in [1]. The variational problem leads to a coupled system, composed of an elliptic variational inequality for the vertical displacement and a linear variational equation for the temperature field. Then, its numerical resolution is considered, based on the finite element method to approximate the spatial variable and the implicit Euler scheme to discretize the time derivatives. Error estimates are proved from which, under adequate regularity conditions, the linear convergence is derived. Finally, some numerical simulations are presented to show the accuracy of the algorithm and the behavior of the solution.     

A dynamic viscoelastic contact problem with normal compliance,finite penetration and nonmonotone slip rate dependent friction

We consider a mathematical model which describes the dynamic evolution of a viscoelastic body in frictional contact with an obstacle. The contact is modelled with normal compliance and unilateral constraint, associated to a rate slip-dependent version of Coulomb’s law of dry friction. In order to approximate the contact conditions, we consider a regularized problem wherein the contact is modelled by a standard normal compliance condition without finite penetrations. For each problem, we derive a variational formulation and an existence result of the weak solution of the regularized problem is obtained. Next, we prove the convergence of the weak solution of the regularized problem to the weak solution of the initial nonregularized problem. Then, we introduce a fully discrete approximation of the variational problem based on a finite element method and on a second order time integration scheme. The solution of the resulting nonsmooth and nonconvex frictional contact problems is presented, based on approximation by a sequence of nonsmooth convex programming problems. Finally, some numerical simulations are provided in order to illustrate both the behaviour of the solution related to the frictional contact conditions and the convergence result.     

ARTIFICIAL BOUNDARY METHOD FOR THE THREE-DIMENSIONAL EXTERIOR PROBLEM OF ELASTICITY

The exact boundary condition on a spherical artificial boundary is derived for thethree-dimensional exterior problem of linear elasticity in this paper. After this bound-ary condition is imposed on the artificial boundary, a reduced problem only defined in abounded domain is obtained. A series of approximate problems with increasing accuracycan be derived if one truncates the series term in the variational formulation, which isequivalent to the reduced problem. An error estimate is presented to show how the errordepends on the finite element discretization and the accuracy of the approximate problem.In the end, a numerical example is given to demonstrate the performance of the proposedmethod.     

Small free vibrations of an infinite cylindrical shell rotating on rollers

Small free vibrations of an infinitely long rotating cylindrical shell being in contact with rigid cylindrical rollers are considered. A system of linear differential equations for the vibrations of such a shell is derived. By using the Fourier transform of the solutions in the circumferential coordinate, a system of algebraic equations for approximately determining the vibration frequencies and mode shapes is obtained. It is shown that, for any number n of uniformly distributed rollers, the approximate values of the first n frequencies and mode shapes can be found explicitly. On the basis of the orthogonal sweep method, an algorithm for numerically solving the boundary value eigenvalue problem describing the vibrations of a rotating shell is developed. Analytical and numerical results are compared. The obtained approximate formulas for frequencies and the numerical algorithm can be used to design centrifugal concentrators for ore enrichment.     

A mixed variational formulation for a class of contact problems in viscoelasticity

We consider a deformable body in frictionless unilateral contact with a moving rigid obstacle. The material is described by a viscoelastic law with short memory, and the contact is modeled by a Signorini condition with a time-dependent gap. The existence and uniqueness results for a weak formulation based on a Lagrange multipliers approach are provided. Furthermore, we discuss an efficient algorithm approximating the weak solution for the more general case of a two-body contact problem including friction. In order to illustrate the theory we present two numerical examples in 3D.     

On the problem of optimizing contact force distributions

The problem of optimizing the distribution of contact forces between a rigid obstacle and a discretized linear elastic body is considered. The design variables are the initial gaps between the potential contact nodal points and the obstacle. Two different cost functionals are investigated: the first reflects the objective of minimizing the maximum contact force; the second is the equilibrium potential energy. Contrary to what has been claimed in the literature, it is shown that these cost functionals do not give, in general, the same optimal design. However, it is also shown that, if a certain frequently realized assumption is met by the system flexibility matrix, then this equality does hold.The min-max cost functional is nonconvex and nondifferentiable, and Clarke's theory of nonsmooth optimization is used to establish a sufficient optimality condition. Investigating its consequences, both necessary and sufficient optimality conditions can be given. The equilibrium potential energy cost functional, on the other hand, turns out to have the remarkable porperties of differentiability and convexity.This work was supported by The Center for Industrial Information Technology (CENHT), Linköping Institute of Technology, Linköping, Sweden.     

Optimal control computation for parabolic systems with boundary conditions involving time delays

In this paper, we consider a class of optimal control problems involving a second-order, linear parabolic partial differential equation with Neumann boundary conditions. The time-delayed arguments are assumed to appear in the boundary conditions. A necessary and sufficient condition for optimality is derived, and an iterative method for solving this optimal control problem is proposed. The convergence property of this iterative method is also investigated.On the basis of a finite-element Galerkin's scheme, we convert the original distributed optimal control problem into a sequence of approximate problems involving only lumped-parameter systems. A computational algorithm is then developed for each of these approximate problems. For illustration, a one-dimensional example is solved.     

A posteriori error analysis for the normal compliance problem

In this work, a contact problem between a linear elastic material and a deformable obstacle is numerically analyzed. The contact is modeled using the well-known normal compliance contact condition. The weak formulation leads to a nonlinear variational equation which is approximated by using the finite element method. A priori error estimates are recalled. Then, we define an a posteriori error estimator of residual type to evaluate the accuracy of the finite element approximation of the problem. Upper and lower bounds of the discretization error are proved for this estimator.     

Motion of a vibrating string in the presence of a convex obstacle: A free boundary problem

Here we study the motion of a vibrating string in the presence of an arbitrary obstacle. We show that if the string always rebounds on the concave parts of the obstacle, it can either rebound or roll on the convex parts. The latter is the case if the velocity of the string is null at the contact point just before contact, or if the contact point propagates at a characteristic speed. Four examples are given. The three first correspond to the same obstacle, a sinusoidal arc, but with different initial conditions. In the first case, the string rebounds on the whole of the obstacle and the motion is explicitly determined when it is periodic. In the second case, the string rolls on the convex part of the obstacle up to the inflexion point and then rebounds on the concave part and unwinds on the convex part. In the third case, the string is initially at rest on the obstacle; then it instantaneously leaves the concave part while it unwinds progressively on the convex part. The fourth case is similar to the third but with a different obstacle; the motion, which is periodic, is determined explicitly.     

A stabilized Lagrange multiplier method for the finite element approximation of contact problems in elastostatics

In this work we consider a stabilized Lagrange (or Kuhn–Tucker) multiplier method in order to approximate the unilateral contact model in linear elastostatics. The particularity of the method is that no discrete inf-sup condition is needed in the convergence analysis. We propose three approximations of the contact conditions well adapted to this method and we study the convergence of the discrete solutions. Several numerical examples in two and three space dimensions illustrate the theoretical results and show the capabilities of the method.     

Analysis of two frictional viscoplastic contact problems with damage

In this work we study two quasistatic frictional contact problems arising in viscoplasticity including the mechanical damage of the material, caused by excessive stress or strain and modelled by an inclusion of parabolic type. The variational formulation is provided for both problems and the existence of a unique solution is proved for each of them. Then a fully discrete scheme is introduced using the finite element method to approximate the spatial domain and the Euler scheme to discretize the time derivatives. Error estimates are derived and, under suitable regularity assumptions, the linear convergence of the algorithm is deduced. Finally, some numerical examples are presented to show the performance of the method.     

A Simple Semi-Implicit Scheme for Partial Differential Equations with Obstacle Constraints

We develop a simple and efficient numerical scheme to solve a class of obstacle problems encountered in various applications. Mathematically, obstacle problems are usually formulated using nonlinear partial differential equations (PDE). To construct a computationally efficient scheme, we introduce a time derivative term and convert the PDE into a time-dependent problem. But due to its nonlinearity, the time step is in general chosen to satisfy a very restrictive stability condition. To relax such a time step constraint when solving a time dependent evolution equation, we decompose the nonlinear obstacle constraint in the PDE into a linear part and a nonlinear part and apply the semi-implicit technique. We take the linear part implicitly while treating the nonlinear part explicitly. Our method can be easily applied to solve the fractional obstacle problem and min curvature flow problem. The article will analyze the convergence of our proposed algorithm. Numerical experiments are given to demonstrate the efficiency of our algorithm.     

Thermoelastic contact with Barber's heat exchange condition

We consider a nonlinear parabolic problem that models the evolution of a one-dimensional thermoelastic system that may come into contact with a rigid obstacle. The mathematical problem is reduced to solving a nonlocal heat equation with a nonlinear and nonlocal boundary condition. This boundary condition contains a heat-exchange coefficient that depends on the pressure when there is contact with the obstacle and on the size of the gap when there is no contact. We model the heat-exchange coefficient as both a single-valued function and as a measurable selection from a maximal monotone graph. Both of these models represent modified versions of so-called imperfect contact conditions found in the work of Barber. We show that strong solutions exist when the coefficient is taken to be a continuously differentiable function and that weak solutions exist when the coefficient is taken to be a measurable selection from a maximal monotone graph. The proofs of these results reveal an interesting interplay between the regularity of the initial condition and the behavior of the coefficient at infinity.     

Convergence analysis of a conforming adaptive finite element method for an obstacle problem

The adaptive algorithm for the obstacle problem presented in this paper relies on the jump residual contributions of a standard explicit residual-based a posteriori error estimator. Each cycle of the adaptive loop consists of the steps ‘SOLVE’, ‘ESTIMATE’, ‘MARK’, and ‘REFINE’. The techniques from the unrestricted variational problem are modified for the convergence analysis to overcome the lack of Galerkin orthogonality. We establish R-linear convergence of the part of the energy above its minimal value, if there is appropriate control of the data oscillations. Surprisingly, the adaptive mesh-refinement algorithm is the same as in the unconstrained case of a linear PDE—in fact, there is no modification near the discrete free boundary necessary for R-linear convergence. The arguments are presented for a model obstacle problem with an affine obstacle χ and homogeneous Dirichlet boundary conditions. The proof of the discrete local efficiency is more involved than in the unconstrained case. Numerical results are given to illustrate the performance of the error estimator.     

Stress formulation for frictionless contact of an elastic-perfectly-plastic body

A mathematical model for frictionless contact of a deformable body with a rigid moving obstacle is analyzed. The Prandtl–Reuss elastic-perfectly-plastic constitutive law is used to describe the material's behavior, and contact is modeled with a unilateral condition imposed on the surface velocity. The problem is motivated by the process of the plowing of the ground. A variational formulation of the problem is derived in terms of the stresses and the existence of the unique weak solution is proven. The proof is based on arguments for differential inclusions obtained in A. Amassad, M. Shillor and M. Sofonea (2001). A quasistatic contact problem for an elastic perfectly plastic body with Tresca's friction. Nonlin. Anal., 35, 95–109. Finally, a study of the continuous dependence of the solution on the data is presented.     

On possibility of obtaining linear accuracy evaluation of approximate solutions to inverse problems

We consider a concept of linear a priori estimate of the accuracy for approximate solutions to inverse problems with perturbed data. We establish that if the linear estimate is valid for a method of solving the inverse problem, then the inverse problem is well-posed according to Tikhonov. We also find conditions, which ensure the converse for the method of solving the inverse problem independent on the error levels of data. This method is well-known method of quasi-solutions by V. K. Ivanov. It provides for well-posed (according to Tikhonov) inverse problems the existence of linear estimates. If the error levels of data are known, a method of solving well-posed according to Tikhonov inverse problems is proposed. This method called the residual method on the correctness set (RMCS) ensures linear estimates for approximate solutions. We give an algorithm for finding linear estimates in the RMCS.