Purpose

Topology optimization of structures under self-weight loading is a challenging problem which has received increasing attention in the past years. The use of standard formulations based on compliance minimization under volume constraint suffers from numerous difficulties for self-weight dominant scenarios, such as non-monotonic behaviour of the compliance, possible unconstrained character of the optimum and parasitic effects for low densities in density-based approaches. This paper aims to propose an alternative approach for dealing with topology design optimization of structures into three spatial dimensions subject to self-weight loading.

Design/methodology/approach

In order to overcome the above first two issues, a regularized formulation of the classical compliance minimization problem under volume constraint is adopted, which enjoys two important features: (a) it allows for imposing any feasible volume constraint and (b) the standard (original) formulation is recovered once the regularizing parameter vanishes. The resulting topology optimization problem is solved with the help of the topological derivative method, which naturally overcomes the above last issue since no intermediate densities (grey-scale) approach is necessary.

Findings

A novel and simple approach for dealing with topology design optimization of structures into three spatial dimensions subject to self-weight loading is proposed. A set of benchmark examples is presented, showing not only the effectiveness of the proposed approach but also highlighting the role of the self-weight loading in the final design, which are: (1) a bridge structure is subject to pure self-weight loading; (2) a truss-like structure is submitted to an external horizontal force (free of self-weight loading) and also to the combination of self-weight and the external horizontal loading; and (3) a tower structure is under dominant self-weight loading.

Originality/value

An alternative regularized formulation of the compliance minimization problem that naturally overcomes the difficulties of dealing with self-weight dominant scenarios; a rigorous derivation of the associated topological derivative; computational aspects of a simple FreeFEM implementation; and three-dimensional numerical benchmarks of bridge, truss-like and tower structures.

中文翻译：

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自重加载三维结构的拓扑优化

目的

自重加载下结构的拓扑优化是一个具有挑战性的问题，近年来受到越来越多的关注。在体积约束下使用基于柔量最小化的标准公式对于自重主导场景会遇到许多困难，例如柔量的非单调行为、最优的可能无约束特征以及基于密度的方法中低密度的寄生效应。本文旨在提出一种替代方法，用于处理自重载荷作用下三个空间维度的结构拓扑设计优化。

设计/方法论/途径

为了克服上述前两个问题，采用了体积约束下经典柔度最小化问题的正则化表述，它具有两个重要特征：（a）它允许施加任何可行的体积约束和（b）标准（原始） ）一旦正则化参数消失，公式就会恢复。由此产生的拓扑优化问题借助拓扑导数方法来解决，该方法自然地克服了上述最后一个问题，因为不需要中间密度（灰度）方法。

发现

提出了一种新颖而简单的方法，用于处理自重载荷作用下三个空间维度的结构拓扑设计优化。提供了一组基准示例，不仅显示了所提出方法的有效性，还强调了自重荷载在最终设计中的作用，这些示例是：（1）桥梁结构承受纯自重荷载; (2) 桁架式结构受到外部水平力（无自重载荷）以及自重和外部水平载荷的联合作用； (3)塔结构主要承受自重荷载。

原创性/价值

合规最小化问题的另一种正则化表述，自然克服了处理自重主导场景的困难；相关拓扑导数的严格推导；简单 FreeFEM 实现的计算方面；以及桥梁、桁架式和塔式结构的三维数值基准。