Thickness control in structural optimization via a level set method

Abstract

In the context of structural optimization via a level-set method we propose a framework to handle geometric constraints related to a notion of local thickness. The local thickness is calculated using the signed distance function to the shape. We formulate global constraints using integral functionals and compute their shape derivatives. We discuss different strategies and possible approximations to handle the geometric constraints. We implement our approach in two and three space dimensions for a model of linearized elasticity. As can be expected, the resulting optimized shapes are strongly dependent on the initial guesses and on the specific treatment of the constraints since, in particular, some topological changes may be prevented by those constraints.

Minimum length scale in topology optimization by geometric constraints

“…Such treatment seems to be a necessity for the minimum length-scale type control as introduced here and in refs. [17,18,19,20]. The parameter c for the indicator functions is chosen as c = r 4 for all the examples.…”

“…In both papers [17,18], the sensitivity regarding the change of the medial-zone is neglected in the sensitivity analysis. Possible shortcomings of this approach are discussed in detail in [20].…”

“…However, the energy functional requires complex geometric information and there is no explicit way to define the exact length scale. A rigorous mathematic approach for imposing minimum and maximum length scale in level set based topology optimization is proposed in [19,20]. Besides the above approaches, predefined engineering features with length scale can be designed and optimized using a CSG based level set approach as discussed in [21].…”

“…Besides the above approaches, predefined engineering features with length scale can be designed and optimized using a CSG based level set approach as discussed in [21]. Readers who are interested in maximum length scale in topology optimization are referred to the articles [17,18,19,20,22].…”

“…In this work, two geometric constraints are proposed to meet the above condition and minimum length scale is achieved by solving a constrained optimization problem. The approach is inspired by the works in [17,18,19,20] but different from them. Partly only the minimum length-scale problem is considered here and partly the constraints here are formulated based on a filtering-threshold topology optimization scheme and they are differentiable w.r.t.…”