Discrete Topology Optimization of Structures Without Uncertainty

Abstract

The objective of topology optimization of a structure is to design its layout optimally.The topology of a structure is defined by its genus or number of handles. When topology of a structure is optimized, its topology might change whenever the material state of the design cell is switched from solid to void or vice versa. This leads to uncertainty fostering ambiguous topology solutions. In order to avoid this uncertainty, discrete topology optimization technique is utilized.In discrete topology optimization procedure all the design cells are considered to be either solid or void, thereby grey cell or intermediate material cells are eliminated and the uncertainty is avoided.Existence of point connections might cause topology uncertainty. This uncertainty can be eradicated through design domain discretization. In optimization process, hexagonal discretization, hybrid discretization, and modified quadrilateral discretization techniques are primarily used to discretize the design domain. In these discretization approaches, any two neighboring cells share an edge and so there will never be point connections. The hybrid discretization is primarily utilized as it has more number of local topology search directions, in comparison to other discretization approaches.iv However, the topology optimization solutions pertaining to structures are uncertain, if the design domain is discretized differently. This uncertainty is called as mesh dependence problem.In order to eliminate it, the degree of genus (DOG) based topology optimization strategy is introduced. In this strategy, the genus of an optimized structure is constrained during topology optimization process. There exists no topology uncertainty problem even if the design domain is discretized with different discretization approaches. This introduced strategy is used for discrete topology optimization of structures that have multiple loading points. The accuracy of this strategy is demonstrated with some examples.v ACKNOWLEDGEMENT

Inequality constraint on the maximum genus for 3D structural compliance topology optimization

“…A precise method is defined as one that satisfies the following two conditions: (a) the genus or the number of enclosed voids in the optimized structure can be calculated directly; (b) the genus or the number of enclosed voids in the optimized structure can be controlled. Precise methods can be subdivided as follows: (a) an inequality constrains the number of enclosed voids; (b) an equality constrains the number of enclosed voids, such as in the application of graph theory and set theory to control the number and sizes of the enclosed voids of topologically optimized structures 2 , the virtual temperature approach 18 , 19 used to eliminate enclosed voids and fulfil the connectivity requirement, and the method of imposing an equality constraint on the number of enclosed voids with discrete sensitivity 20 ; (c) an inequality constrains the genus 3 , 21 ; (d) an equality constrains the genus; and (e) inequality or equality constraints exist for both the genus and the number of enclosed voids in the optimized structure. For the aforementioned five cases of precise methods, the research focused on the last three cases is limited.…”