Why Checkerboarding Localizes in Shear Regions under SIMP
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The checkerboard instability in topology optimization using a SIMP-type (solid isotropic material with penalization) density parametrization of material stiffness is well known when bilinear displacement interpolation is employed. The phenomenon has been interpreted from two different perspectives: as a mixed-FE / LBB instability and as a locking effect in local representative element layouts. We argue that neither viewpoint alone fully explains when and where checkerboarding occurs in SIMP formulations. Through systematic computational experiments on classical benchmark structures under various loading conditions, we show that checkerboarding localizes in shear-dominated regions and disappears in primarily axially loaded zones. This obsevation indicates that analyses based on local representative element layouts are insufficient. Instead, understanding the phenomenon requires the consideration of the entire structural desnity distribution and all occurring local element layouts. Moreover, we argue that the LBB perspective cannot be directly applied to classical density-based topology optimization formulations decoupling the displacement and density updates and, therefore, preventing a true mixed-FE problem. Our findings clarify the mechanism responsible for checkerboarding in SIMP-type density interpolation and motivate the development of improved displacement interpolations capable of suppressing these locking phenomena.
