The dynamical problem of an elastic plate resting on a two-parameter foundation which does not react in tension

Augmented Lagrangian Finite-Elements for Plate Contact Problems

“…The analytical solution of the augmented Lagrangian formulation (17) is determined solving a sequence of differential equations, each one representing a plate in unilateral contact with an elastic Winkler medium. Adopting the iteration procedure obtained from the functional ( 17), the contact pressure after n augmentations is where yf = -k,/D, A ; -' ( X ) is an (n -1)th degree polynomial and C; are constants of integration determined by the boundary conditions, Hence, along the part of plate in contact with the rigid support, the solution is with E:-'(x) a polynomial of (n -1)th degree, different from nl-'(x), and C : are constants of integration determined by the boundary conditions.…”

“…Increasing the value of the penalty coefficient, a better constraint enforcement can be obtained, at the expense of the number of iterations (not reported here) needed to reach the convergence; in particular, for k,h4/D > lop3 the finite-element method does not converge within 20 iterations. Then, the penalty parameter is set as k,h4/D = lop4, and the capacity of the augmented Lagrangian formulations ( 17) and (23) to enforce the constraints is studied. The dimensionless contact pressure pL3/D along the strip plate axis is plotted in Figure 5 Table I11 the contact lengths obtained by the analytical and numerical approaches, via augmented formulation ( 17), are tabulated versus the number of augmentations and are compared with the exact solution.…”

“…Then, the penalty parameter is set as k,h4/D = lop4, and the capacity of the augmented Lagrangian formulations ( 17) and (23) to enforce the constraints is studied. The dimensionless contact pressure pL3/D along the strip plate axis is plotted in Figure 5 Table I11 the contact lengths obtained by the analytical and numerical approaches, via augmented formulation ( 17), are tabulated versus the number of augmentations and are compared with the exact solution. From the results presented, it is possible to conclude that the iterative algorithm relative to the augmented formulation (23) is not appropriate for the solution of contact problems.…”

“…From the results presented, it is possible to conclude that the iterative algorithm relative to the augmented formulation (23) is not appropriate for the solution of contact problems. Hence, only the iterative algorithm relative to the augmented formulation ( 17) is explored in the remaining examples. …”

“…The case of unilateral support is then studied through the augmented Lagrangian formulation (17). setting kljh3/D = IW3.…”