Lecturers
- Prof. Kevin Sturm: Shape and topological derivatives: theory and applications
- Prof. Benedikt Wirth: Pattern analysis in shape optimization
- Prof. Michael Stingl: Material and Topology Optimization: Modeling, Approximation, Numerical Solution and Algorithms
Guest Speakers
- Prof. Grégoire Allaire: Optimal design of lattice materials
- Prof. Jun Wu: Topology optimization for additive manufacturing: bone-inspired infill and space-time optimization
Preliminary schedule
Monday | Tuesday | Wednesday | Thursday | Friday | |
---|---|---|---|---|---|
09:00-10:30 | Wirth | Stingl | Wirth | Allaire | |
10:30-11:00 | Coffee Break | Coffee Break | Coffee Break | Coffee Break | |
11:00-12:30 | Wirth | Stingl | Wirth | Wu | |
12:30-14:00 | 14:15 Opening | Lunch | Lunch | Lunch | |
14:00-15:30 | 14:30-16:00: Stingl | Sturm | Excursion | Sturm | |
15:30-16:00 | 16:00-16:30: Coffee Break | Coffee Break | Excursion | Coffee Break | |
16:00-17:30 | 16:30-18:00 Stingl | Sturm | Excursion | Sturm |
Excursion
We plan to make an excursion to the top of the nearby small mountain "Schöckl" which is also accessible by cable car on Wednesday afternoon. There is the option of hiking up to the top (approximately 2 hours easy hike) or to take the cable car. On top of Schöckl, there are several activities possible (including disc golf and alpine coaster (Sommerrodelbahn)).
Abstracts
- Prof. Kevin Sturm: Shape and topological derivatives: theory and applications
In this lecture we discuss the shape and topological derivative and its use in numerical shape optimisation. The shape derivative enables us to optimise the shape without changing its topology. In contrast the topological derivative is designed to find the optimal topology of a design. We introduce both concepts and show how to compute them in the context of design optimisation with partial differential equations. In a hands-on tutorial we will show how both concepts can be used in numerics using the finite element software NGSolve. - Prof. Benedikt Wirth: Pattern analysis in shape optimization
It is a classic engineering problem to identify which geometry of an elastic material best supports a load while consuming a minimum material amount. To avoid microstructure, this optimization problem is typically relaxed (which corresponds to the so-called homogenization method) or strongly regularized (for instance by fixing the topology or penalizing the perimeter). In contrast, in our lecture we will be interested in the case of very weak regularization, in which fine structures and coarsening phenomena over multiple scales will occur. Our aim will be to better understand these structures using relatively elementary tools and concepts from pattern analysis.
A motivation comes from biology, which is a rich source of quite complicated structures and materials. The above geometry optimization may be viewed as a basic mathematical model of bone, adopting the hypothesis that biological evolution actually solves an optimization problem. Bone tissue can be divided into the compact, dense cortical bone and the cancellous or spongy bone, which forms intricate structures with different scales. It consists of tiny interconnected rods, so-called trabeculae, with a lot of interstitial space. This structure allows to make the bone lighter without compromising its mechanical stability. Since trabeculae are not infinitely fine, there must be some additional complexity-limiting mechanism involved. Mathematically, the simplest version is to consider surface area as a measure of structural complexity.
We will analyse the optimal shapes in simple settings and together explore some variations of these settings in exercises. - Prof. Michael Stingl: Material and Topology Optimization: Modeling, Approximation, Numerical Solution and Algorithms
A mathematical introduction to the field of material and topology optimization will be given. Starting from the definition of a generic class of material optimization problems, conditions will be developed, which guarantee existence of a solution. In this framework, the topology optimization as well as the so called multi-material optimization problem will be treated as a special case. In the sequel, the numerical approximation of material optimization problems will be studied. For a generic approximation scheme, conditions will be derived which guarantee convergence of the sequence of approximate solutions to the solution of the original problem in an appropriate sense. After this, algebraic forms of material and topology optimization problems will be derived based on a finite element discretization. In the last part, specialized solution algorithms for the discretized material optimization problem will be investgated. It is shown how a separability assumption and Lagrange duality can be used to come up with a highly efficient class of first order algorithms. In addition, it will be demonstrated how material optimization problems subject to infinitely many state problems can be solved efficiently by specialized stochastic gradient methods. The discussions will be complemeted by an outlook on modern applications including examples from light weight design, additive manufacturing and the design of particulate products. - Prof. Grégoire Allaire: Optimal design of lattice materials
This work is concerned with the topology optimization of so-called lattice materials, i.e., porous structures made of periodically perforated material, where the microscopic periodic cell can be macroscopically modulated and oriented. Lattice materials are becoming increasingly popular since they can be built by additive manufacturing techniques. The main idea is to optimize the homogenized formulation of this problem, which is an easy task of parametric optimization, then to project the optimal microstructure at a desired length-scale, which is a delicate issue, albeit computationally cheap. The main novelty of our work is, in a plane setting, the conformal treatment of the optimal orientation of the microstructure. In other words, although the periodicity cell has varying parameters and orientation throughout the computational domain, the angles between its members or bars are conserved. Several numerical examples are presented for compliance minimization in 2-d. Extension to the 3-d case will also be discussed. This is a joint work with Perle Geoffroy-Donders and Olivier Pantz. - Prof. Jun Wu: Topology optimization for additive manufacturing: bone-inspired infill and space-time optimization
Topology optimization has been recognized as an important design method for additive manufacturing. It provides an effective means to automatically explore the large design space unveiled by the additive process, and to generate designs with superior structural performance. In this talk, I will start with a novel formulation for designing bone-inspired infill structures for additive manufacturing. By using local volume constraints, in lieu of a global volume constraint as in conventional topology optimization, porous structures that resemble trabecular bone are generated automatically. This is extended to allow for concurrent design of macroscale structures and porous mesoscale structures therein. I further show how stress aligned lattice structures can be efficiently designed using geometry meshing algorithms, following a homogenization-based optimization of the distribution of lattice materials. These algorithms are illustrated with practical applications.
In the second part, I will present a framework to concurrently optimize the structural layout and fabrication sequence. This approach, which we call space-time topology optimization, is motivated by advances in multi-axis additive manufacturing (e.g., wire and arc additive manufacturing, also known as WAAM) which enables fabrication along curved layers. This concurrent optimization scheme is applicable to a wide range of design and manufacturing scenarios. It will be illustrated by for instance, minimizing distortion in WAAM as well as construction with moving robotic platforms.
Open source Matlab code will be provided to facilitate understanding of these concepts and algorithms.