Sensitivity computation and shape optimization for a non‐linear arch model with limit‐points instabilities

Abstract

A shape optimization method for geometrically non-linear structural mechanics based on a sensitivity gradient is proposed. This gradient is computed by means of an adjoint state equation and the structure is analysed with a total Lagrangian formulation. This classical method is well understood for regular cases, but standard equations 1 have to be modiÿed for limit points and simple bifurcation points. These modiÿcations introduce numerical problems which occur at limit points. 2 Numerical systems are very sti and the quadratic convergence of Newton-Raphson algorithm vanishes, then higher-order derivatives have to be computed with respect to state variables. 3 A geometrically non-linear curved arch is implemented with a ÿnite element method via a formal calculus approach. Thickness and=or shape for di erentiable costs under linear and non-linear constraints are optimized. Numerical results are given for linear and non-linear examples and are compared with analytic solutions. ? 1998 John Wiley & Sons, Ltd.Example 2.1. For a geometrically non-linear shallow beam in 2-D with a small initial curvature 11 and linear change of curvature: ' = (v; w) ∈ V , ' = ( v; w) ∈ V; depending on the boundary conditions, V may be H 1 0 (0; L) × H 2 0 (0; L) for a beam clamped at both ends; V= {(v; w) ∈ H 1 (0; L) × H 2 (0; L) = v(0) = 0; w(0) = 0; w (0) = 0} for a cantilever beam clamped at the left end. V= {(v; w) ∈ H 1 (0; L) × H 2 (0; L) = v(0) = 0; w(0) = 0; v(L) = 0; w(L) = 0} for a simply supported beam.

A boundary-perturbation finite element approach for shape optimization

“…The last equality in (10) follows from the prescribed condition (5). Now we are in a position to deduce the zeroth-and ÿrst-order perturbation problems associated with the original problem (2)- (5). The zeroth-order problem, denoted P 0 , is simply the original problem with = 0, i.e.…”

“…Now we consider a class of optimization problems which are governed by a linear partial di erential equation. To be speciÿc, we consider the set-up shown in Figure 1, with u satisfying the Helmholtz equation (2) in and the boundary conditions (3) - (5). The shape of the boundary B , deÿned by h(Â) in (1), is unknown, but we assume that it is a small perturbation of the circular boundary B.…”

“…A special optimization scheme is now constructed for the problem of minimizing (26) subject to equations (2)- (5). We rely on the BP form of (2) -(5), given in Section 2.2.…”

“…A shape optimization problem involves a boundary whose shape is not known a priori, which is to be determined based on some optimality criterion under given constraints. Shape optimization is a rich and active ÿeld of research; see, e.g., the recent monograph [1] and papers [2][3][4][5][6].…”