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On concave constraint functions and duality in predominantly black-and-white topology optimization

机译：关于凹面约束函数和以黑白为主的拓扑优化中的对偶性

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We study the 'classical' discrete, solid-void or black-and-white topology optimization problem, in which minimum compliance is sought, subject to constraints on the available material resource. We assume that this problem is solved using methods that relax the discreteness requirements during intermediate steps, and that the associated programming problems are solved using sequential approximate optimization (SAO) algorithms based on duality. More specifically, we assume that the advantages of the well-known Falk dual are exploited. Such algorithms represent the state-of-the-art in (large-scale) topology optimization when multiple constraints are present; an important example being the method of moving asymptotes (MMA). We depart by noting that the aforementioned SAO algorithms are invariably formulated using strictly convex subproblems. We then numerically illustrate that strictly concave constraint functions, like those present in volumetric penalization, as recently proposed by Bruns and co-workers, may increase the difficulty of the topology optimization problem when strictly convex approximations are used in the SAO algorithm. In turn, volumetric penalization methods are of notable importance, since they seem to hold much promise for generating predominantly solid-void or discrete designs.rnWe then argue that the nonconvex problems we study may in some instances efficiently be solved using dual SAO methods based on nonconvex (strictly concave) approximations which exhibit monotonicity with respect to the design variables.rnIndeed, for the topology problem resulting from SIMP-like volumetric penalization, we show explicitly that convex approximations are not necessary. Even though the volumetric penalization constraint is strictly concave, the maximum of the resulting dual subproblem still corresponds to the optimum of the original primal approximate subproblem.
机译：我们研究“经典”离散，实心空隙或黑白拓扑优化问题，其中寻求最小的依从性，但要受可用材料资源的限制。我们假设使用在中间步骤中放宽离散性要求的方法解决了该问题，并且使用基于对偶性的顺序逼近优化（SAO）算法解决了相关的编程问题。更具体地说，我们假设利用了众所周知的Falk dual的优点。当存在多个约束时，此类算法代表了（大规模）拓扑优化的最新技术。一个重要的例子是移动渐近线（MMA）的方法。我们通过注意到上述SAO算法始终使用严格凸子问题来表示。然后，我们用数值方法说明，像Bruns和同事最近提出的那样，严格的凹约束函数（如体积罚分中存在的约束函数）可能会在SAO算法中使用严格凸近似时增加拓扑优化问题的难度。反过来，体积罚分方法也具有重要意义，因为它们似乎对于产生主要为实心或离散设计具有很大的希望。然后我们认为，在某些情况下，我们可以使用基于SAO的双重SAO方法有效地解决我们研究的非凸问题。对于设计变量而言，具有单调性的非凸（严格凹）逼近。确实，对于类似SIMP的体积罚分法所引起的拓扑问题，我们明确表明凸逼近是不必要的。即使体积罚分约束是严格凹的，所得的双重子问题的最大值仍对应于原始原始近似子问题的最优值。

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