Local constraints in topology optimization for minimum and overhang constraints

Abstract

As the aerospace industry continues to be in the vanguard of technology development, one of the current hot topics is the search for new lightweight structures and materials that enable the reduction of fuel consumption and environmental impact. In recent years, a powerful tool aimed to tackle this problem has been consistently making its way into every design process for every new part that is being designed. That tool is known as Topology Optimization (TO) and through the use of numerical methods, it is capable of the reduction of weight as well as optimizing material layout and minimizing design costs and time. Despite that, topology optimization has one disadvantage, which is related to its manufacturability limitations. For a topology optimized design, it is not unusual to develop complex shapes with long and thin bars and/or tiny holes. This has led to the creation of new techniques that take into consideration Additive Manufacturing (AM) constraints. This project is aimed to address the issues stated above with respect to manufacturing limitations in topology optimization, by means of developing local formulation for minimum length scale. An introduction to the theoretical background is exposed, covering topics such as Finite Element Methods (FEM) in the classical elastic problem, a concise yet comprehensive overview of current topology optimization techniques and the most common solvers used in this field to solve the minimization problems. It is then followed by a basic explanation on additive manufacturing techniques and the full formulation of the local constraints for minimum length scale. Two approaches are considered, which consist of the Primal and Dual optimization problem. This will prove useful both for Density and Level Set, which are the common design variables in TO. Lastly, all the simulations are exposed consisting of different benchmark case studies, all focused on demonstrating the feasibility of the results with respect to additive manufacturing. Additionally, the programming part of this work was conducted with the use of Object-Oriented Programming (OOP) and the Git environment. The results have showed promising advancement towards being able to control local features inside a topology optimization problem and have proved feasible for additive manufacturing techniques in a theoretical framework. This was achieved by the creation of new functionals that work using two approaches: the p-norm formulation and the multilevel formulation. The first one is a method that with the use of a parameter p is able to compute the maximum of the functions to which is applied and therefore is able to establish local neighbourhoods inside the domain where local constraints are considered. The local p-norm volume constraint successfully created the desired infill structures, and the local p-norm perimeter constraint allowed for the control of the minimum length scale of the topologies. The other approach ables the user to directly customize specific regions of the domain to which apply the constraints by means of the Lagrange multipliers variations. This method also proved successful in the control over the minimum length scale.