Checkerboard patterns are a well-known numerical artifact in density-based topology optimization using the Solid Isotropic Material with Penalization (SIMP) method and linear finite elements. Existing explanations based on mixed-field incompatibility or locking-induced stiffness overestimation explain the artificial stiffness of checkerboard layouts but do not clarify their characteristic spatial localization. In this work, we show that checkerboard patterns systematically emerge in multiaxial load-transfer regions, whereas predominantly uniaxial stress regions remain checkerboard-free. Through systematic numerical investigations, we demonstrate that checkerboarding originates where continuous intermediate densities are mechanically favorable for multiaxial load transfer but are suppressed by SIMP penalization. Due to the characteristic behavior of linear elements, checkerboard layouts provide an artificially stiff discrete substitute for these penalized intermediate-density regions. In contrast, uniaxial load paths naturally favor continuous solid struts, rendering checkerboards mechanically disadvantageous. Our findings provide a unified mechanical interpretation of checkerboarding as the interplay between global stress states, SIMP penalization, and element-level locking, thereby explaining both its origin and the spatial localization.

翻译：棋盘格现象是基于固体各向同性材料惩罚（SIMP）方法与线性有限元的密度型拓扑优化中一种典型的数值伪影。现有基于混合场不相容性或锁闭效应引起的刚度高估解释虽揭示了棋盘格布局的虚假刚度特性，但未能阐明其特有的空间定位规律。本研究证明：棋盘格模式系统性出现在多轴荷载传递区域，而单轴主导应力区则无此现象。通过系统数值实验，我们证实棋盘格源于多轴荷载传递区对连续中间密度的机械需求，这种需求因SIMP惩罚机制受到抑制。受线性单元特征行为影响，棋盘格布局为这些被惩罚的中间密度区域提供了虚假高刚度的离散替代方案。相比之下，单轴荷载路径天然倾向于连续实心杆件，使棋盘格布局在机械性能上处于劣势。本研究将棋盘格现象统一解释为全局应力状态、SIMP惩罚机制与单元级锁闭效应三者相互作用的结果，从而阐明其成因与空间定位规律。