Showing papers in "Applications of Mathematics in 1986"

Non-polyconvexity of the stored energy function of a Saint Venant-Kirchhoff material

Abstract: A direct proof of the non-polyconvexity of the stored energy function of a Saint Venant-Kirchhoff material is given by means of a simple counter-example.

Dual method for solving a special problem of quadratic programming as a subproblem at nonlinear minimax approximation

Abstract: The paper describes the dual method for solving a special problem of quadratic programming as a subproblem at nonlinear minimax approximation. Two cases are analyzed in detail, differring in linear dependence of gradients of the active functions. The complete algorithm of the dual method is presented and its finite step convergence is proved.

Variational inequalities in plasticity with strain-hardening — equilibrium finite element approach

Abstract: Summary. The incremental finite element method is applied to find the numerical solution of the plasticity problem with strain-hardening. Following Watwood and Hartz, the stress field is approximated by equilibrium triangular elements with linear functions. The field of the strain- hardening parameter is considered to be piecewise linear. The resulting nonlinear optimization problem with constraints is solved by the Lagrange multipliers method with additional variables. A comparison of the results obtained with an experiment is given. inequality of evolution and to use a penalty method. In the present paper we propose an incremental finite element method, starting from the formulation of the quasi-static problem in terms of stresses and hardening parameters only (3). Whereas in the mixed method of (6), (18), the stresses and hardening parameters are approximated by piecewise constant functions and the displacements by piecewise linear functions, we employ piecewise linear functions for both the stresses and the hardening parameters. The stress approximations consist of Watwood-Hartz equilibriated triangular elements (8). The finite element method will produce approximations to the stresses successively at a finite number of time levels. At each time level one has to solve a constrained nonlinear optimization problem. We also discuss the Lagrange multipliers method with slack variables (9), (7), (21) for solving this problem. With a particular choice of finite element spaces, the optimization problem is solvable. Numerical tests for the method proposed were performed for thin perforated strips of the strain-hardening material subjected to the uniform tension. The stress applied was increased monotonically from the elastic region of loading to values producing an impending plastic flow. The numerical results are in a good agreement with the experiment (10).

Exact solutions to some external mixed problems in potential theory

Abstract: A new and elegant procedure is proposed for the solution of mixed potential problems in a half-space with a circular line of division of boundary conditions. The approach is based on a new type of integral operators with special properties. Two general external problems are solved; i) An arbitrary potential is specified at the boundary outside a circle, and its normal derivative is zero inside; ii) An arbitrary normal derivative is given outside the circle, and be potential is zero inside. Several illustrative examples are considered. Certain methods of application of the proposed technique to the solution of a few complex problems are also discussed.

On numerical evaluation of integrals involving Bessel functions

Abstract: The paper is concerned with the efficient evaluation of the integral $\int^\infty_0 f(x)J_n(rx)dx$, where $J_n$ is the Bessel function of index $n$ and $n$ is a nonnegative integer, for a given sequence of values of a real parameter $r$. Two procedures are proposed and compared. One of them consists in a direct generalization of a procedure for the evaluation of of a similar integral with the weight function exp $(irx), which employs the fast Fourier transform. The other approach is based on the construction of a special Gaussian quadrature formula where $J_n$ appears as a weight. The results of the comparison show that the application of the Gaussian formula is much more efficient.