Variational approach to relaxed topological optimization: Closed form solutions for structural problems in a sequential pseudo-time framework

Abstract

The work explores a specific scenario for structural computational optimization based on the following elements: (a) a relaxed optimization setting considering the ersatz (bi-material) approximation, (b) a treatment based on a nonsmoothed characteristic function field as a topological design variable, (c) the consistent derivation of a relaxed topological derivative whose determination is simple, general and efficient, (d) formulation of the overall increasing cost function topological sensitivity as a suitable optimality criterion, and (e) consideration of a pseudo-time framework for the problem solution, ruled by the problem constraint evolution. In this setting, it is shown that the optimization problem can be analytically solved in a variational framework, leading to, nonlinear, closed-form algebraic solutions for the characteristic function, which are then solved, in every time-step, via fixed point methods based on a pseudo-energy cutting algorithm combined with the exact fulfillment of the constraint, at every iteration of the non-linear algorithm, via a bisection method. The issue of the ill-posedness (mesh dependency) of the topological solution, is then easily solved via a Laplacian smoothing of that pseudo-energy. In the aforementioned context, a number of (3D) topological structural optimization benchmarks are solved, and the solutions obtained with the explored closed-form solution method, are analyzed, and compared, with their solution through an alternative level set method. Although the obtained results, in terms of the cost function and topology designs, are very similar in both methods, the associated computational cost is about five times smaller in the closedform solution method this possibly being one of its advantages. Some comments, about the possible application of the method to other topological optimization problems, as well as envisaged modifications of the explored method to improve its performance close the work.

Topology optimization of thermal problems in a nonsmooth variational setting: closed-form optimality criteria

“…Alternatively, the topology can be implicitly defined through a smooth function (termed discrimination function in Oliver et al [19]) ψ(x) : Ω → R, ψ ∈ H 1 (Ω), defined as…”

“…where λ stands for a Lagrange multiplier enforcing restriction C(χ) = 0, and L stands for the Lagrangian function of the optimization problem (see Oliver et al [19] for additional information). Then, a closed-form solution for the topology in equation ( 4) can be computed as…”

“…where ξ(χ, x) is termed the pseudo-energy 9 and it shall be specifically derived for each considered problem. Equations (11) constitute a closed-form-algebraic (non-linear fixed-point equation) solution of the problem, which are solved, for χ(x) and λ, via the Cutting&Bisection algorithm proposed in [19]. The resulting global algorithm is sketched in Box I, where the constraint equation is expressed in terms of the pseudo-time t ∈ [0, T ], in the context of a time advancing strategy.…”

“…The algorithm to obtain the optimal characteristic function distribution, χ(x), 13 is based on the Cutting&Bisection technique, shown in Algorithm 2, in the context of the pseudo-time-advancing strategy. The strategy, described in Oliver et al [19], is sketched in Algorithm 1. The number of time-steps of this methodology is related to the robustness and computational cost of the problem: the more time-steps, the more robust the solution is, although the computational cost of the optimization is higher.…”

“…In addition, the procedure to compute the Lagrange multiplier, imposing the constraint equation of ( 12)-(a), is illustrated in Figure 4. A modified Marching Cubes method, detailed in Oliver et al [19], is used to numerically compute the 0-level iso-surface of the discrimination function, ψ. Through this technique, the element hard-phase volume can be obtained, along with the constraint value, C.…”