Abstract The purpose of this research is to demonstrate that the numerical issue associated with the checkerboard patterns can be entirely controlled by applying the finite-volume theory. Usually, in the gradient-based topology optimization algorithms, it is common to occur some problems associated with numerical instabilities, such as checkerboard pattern, mesh dependence, and local minima. The occurrence of checkerboard subdomains is directly related to the assumptions of the finite-element method, as the satisfaction of equilibrium equations and continuity conditions through the element nodes. Differently, the finite-volume theory satisfies the equilibrium equations at the subvolume level, and the continuity conditions are established through the subvolumes adjacent interfaces, as expected from the Continuum Mechanics point of view. Thus, a topology optimization approach based on the standard (or zeroth order) finite-volume theory for linear elasticity is proposed, resulting in a numerically efficient computational modeling, able to obtain checkerboard free topologies in the absence of filtering techniques. A sensitivity filtering technique is employed to solve issues associated with mesh dependence and length scale in the finite-volume approach, providing optimized topologies with desired manufacturing features.

中文翻译：

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应用有限体积理论的符合性最小化的棋盘自由拓扑优化

摘要 本研究的目的是证明可以通过应用有限体积理论完全控制与棋盘图案相关的数值问题。通常，在基于梯度的拓扑优化算法中，经常会出现一些与数值不稳定性相关的问题，例如棋盘格图案、网格依赖和局部极小值。棋盘子域的出现与有限元方法的假设直接相关，即通过单元节点满足平衡方程和连续性条件。不同的是，有限体积理论在子体积水平上满足平衡方程，并且连续性条件通过子体积相邻界面建立，正如从连续介质力学的角度所预期的那样。因此，提出了一种基于线性弹性的标准（或零阶）有限体积理论的拓扑优化方法，从而产生了数值高效的计算建模，能够在没有过滤技术的情况下获得无棋盘格拓扑。采用灵敏度过滤技术来解决有限体积方法中与网格相关性和长度尺度相关的问题，提供具有所需制造特征的优化拓扑。