Smooth boundary topology optimization using B-spline and hole generation

Abstract

A new topology optimization methodology, named smooth boundary topology optimization, is proposed in this paper to overcome the shortcomings of cell-based approaches, where the number of design variables tends to be too large, and a filtering process or an image process must be followed for an applicable result. The domain boundary is initially represented by a geometric function, B-spline curves composed of a number of control points, and the density of each finite element is not considered; only the location of the control point is considered. After calculating two kinds of design sensitivities, the topological sensitivity and the shape sensitivity of the B-spline’s control points, the design is improved by either creating a hole in the domain or moving control points to change the boundary, or both if necessary. These two kinds of design sensitivities are verified by comparing with the numerical sensitivity. Selection criterion, SC, is introduced to determine the moment of creating a hole. SC is defined as the sensitivity of an objective function divided by the sensitivity of a constraint function, and it means the ratio of improving the objective function to sacrificing the constraint function. Two kinds of SCs on creating a hole and moving boundary are calculated and compared to determine the direction of design modification. Once it is determined to create a hole, its boundary is represented by a new B-spline with a sufficient number of control points and recognized as a new boundary of the design domain. Additionally, some manipulative topics on programming are covered related to boundary rendering: the side constraints of each design variable (the location of control point) to prevent curve entangling phenomenon, and the hole merging functionality in case an intersection between two separate holes occurs. The feasibility of this methodology is shown with several applications where compliance or natural frequency is chosen as a performance measure. Using...