## 3D Topology Optimization with Fatigue Constraints

Student thesis: Master thesis (including HD thesis)

• Joachim Emil Kokholm Hersbøll
4. term , Design of Mechanical Systems, Master (Master Programme)
This thesis concern the formulation and execution of structural optimization by the use of topology optimization in both 2D and 3D, for large-scale optimization problems dependent on many variables. Topology optimization strives to find the optimal distribution of material for a specific domain, given an objective and subjected to a set of constraints. Here, the objective is to minimize the mass, while being subjected yield and/or fatigue constraints.

First chapter of this thesis deals with the general issues of topology optimization. These issues that contain checkerboarding, mesh dependent solutions, minimum length scale and solid/void structures, are illustrated for the classic topology optimization formulation which seeks to minimize the compliance while being subjected to a volume constraint.

Both the yield and fatigue constraints are dependent upon the stresses which occur in a structure subjected to loading. Thus, the second chapter considers the specific challenges associated with stress based constraints in topology optimization. Initially, the formulation of stress constraint within the context of density based topology optimization is examined, followed by well known issue of singular optima. The problem of singular optima is here solved using the qp approach. Due to the local nature of stress based constraints, and since this thesis is concerning large-scale optimization problems, will the associated computational consequences be addressed by employing an aggregate function. Based on the aforementioned is it possible to formulate a yield constraint, using the von Mises yield criterion.

Besides the yield constraint, is a constraint based on the fatigue of the structure is also formulated. Like yielding, fatigue does depend on the stresses in the structure, thus the previous mentioned challenges are likewise relevant and solved with the same techniques.

The fatigue constraint is formulated based on assumptions of proportional loading with a variable amplitude, as well as linear elastic deformation in the high cycle domain. The load history which the structure is subjected to, will be reduced to reversals using rainflow counting. Each of these reversals produces a multiaxial stress state in the structure, this state is transformed into an equivalent uniaxial amplitude stress with either the Sines criterion or signed von Mises where the mean stress effects are taken into account by using the modified Goodman equation. From this equivalent uniaxial amplitude stress, the number of cycles to failure is estimated using the Basquin curve. By employing Palmgren-Miners linear damage hypothesis a local damage constraint can be formulated, which like the yield constraint is handled by using an aggregate function.

These Topology optimization formulations are lastly solved for a number of standard benchmark examples in both 2D and 3D which illustrates the characteristics of fatigue constraints.
Language English 1 Jun 2018 74
ID: 280227024