Subject: Advanced Multilevel Techniques to Topology Optimization
Author: Dipl.-Ing. Roman Stainko
Supervisor: O.Univ.Prof. Dipl.-Ing. Dr. Ulrich Langer
External Referee: Prof. Dr. Martin P. Bendsøe

Computer-aided design is nowadays the basis of successful product planning and production control. In almost all sectors of industry faster and faster changing and adapting product specifications demand shorter and shorter production times for competative products. In order to cope with this situation and to realize short development times, enterprises rely on computer simulations. During the last two decades a new field of applied mathematics has reached a state of maturity to enter the world of computer-aided engineering, namely the topology optimization techniques.

the cantilever beam in 3D: coarse and fine grid solution

This work deals with mathematical methods for topology optimization problems. In particular, we focus on two specific design - constraint combinations, namely the maximization of material stiffness at given mass and the minimization of mass while keeping a certain stiffness. Both problems show different properties and will also be treated with different approaches in this work. One characteristics of topology optimization problems is that the set of feasible designs is constrained by a partial differential equation. Moreover, these optimization problems are not well-posed, so regularization techniques have to be applied.

long beam example: 1st component of the displacement

The first combination, also known as the minimal compliance problem, is treated in the framework of an adaptive multi-level approach. Well-posedness of the problem is achieved by applying filter methods to the problem. Such a filter method is used for adaptive mesh refinement along the interface between void and material, i.e. along the boundary of the structure. The resulting optimization problems on each level are solved by the method of moving asymptotes, a well-known optimization technique in the field of topology optimization. In order to ensure an efficient solution of the linear systems, raising from the finite element discretization of the partial differential equations, a multigrid method is applied.

long beam example: 2nd component of the displacement

The treatment of the second combination is by far less understood as the minimal compliance problem. The main source of difficulties is a lack of constraint qualifications for the set of feasible designs, defined by local stress constraints. To overcome these difficulties the set of constraints is reformulated, involving only linear and 0–1 constraints. These are finally relaxed by a Phase–Field approach, which also regularizes the problem. This relaxation scheme results in large-scale optimization problems, which finally solved by an interior-point optimization method.

Most of the computing time of these optimization routines is actually spent in solving linear saddle point problems. In order to speed up computations an efficient solver with optimal complexity for these system is of high importance. Multigrid methods certainly belong to the most efficient methods for solving large-scale systems, e.g., arising from discretized partial differential equations. One of the most important ingredients of an efficient multigrid method is a proper smoother. In this work a multiplicative Schwarz-type smoother is considered, that consists of the solution of several small local saddle point problems, and that leads to an KKT-solver with linear complexity.

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