Vibrations of Euler-Bernoulli Beams with Pointwise Obstacles

A penalty/Newton/conjugate gradient method for the solution of obstacle problems

“…on Ω}. After the over-relaxation methods with projection and Uzawa type algorithms discussed in, e.g., [8] and [7] proved too slow in the context of problem (1), it was decided to give a chance to an approach combining (exterior) penalty and Newton's method. Actually, with this combination, we have been able to simulate (see [1,5] for details) the vibrations, with obstacles, of strings and beams; however, those were obstacle problems in one space dimension, where the linear systems resulting from penalty/Newton (after appropriate finite element discretizations) could be solved easily by direct methods taking advantage of the sparsity and band structure of the corresponding matrices.…”

“…After the over-relaxation methods with projection and Uzawa type algorithms discussed in, e.g., [8] and [7] proved too slow in the context of problem (1), it was decided to give a chance to an approach combining (exterior) penalty and Newton's method. Actually, with this combination, we have been able to simulate (see [1,5] for details) the vibrations, with obstacles, of strings and beams; however, those were obstacle problems in one space dimension, where the linear systems resulting from penalty/Newton (after appropriate finite element discretizations) could be solved easily by direct methods taking advantage of the sparsity and band structure of the corresponding matrices. Since some of the finite element (or finite difference) meshes associated to the solution of problem (1) involve more than 10 6 grid points (1023 2 to be precise) it is clear that, as of today, direct methods are not a feasible option (for most practitioners, at least) as components of the penalty/Newton solution of problem (2).…”

Numerical solutions of fourth order variational inequalities

“…Variational inequality theory has become an effective and powerful tool for studying obstacle and unilateral problems arising in mathematical and engineering sciences including fluid flow through porous media, elasticity, transportation and economics equilibrium, optimal control, nonlinear optimization, operations research, see, for example [3][4][5][6][7]101.…”