Biobjective Shape Optimization Using Weighted Sums: Probability of Failure versus Costs.
Gradient-Based Biobjective Shape Optimization: Probability of Failure versus Cost.
Both talks were held on the same shape optimization problem and presented different biobjective solution approaches that were explored in the last year.
The MCDM Conference is a biannual conference run by the International Society of MCDM (Multi Criteria Decision Making) . It featured 222 talks and had 261 participants from nearly 40 different countries.
This conference has a broad spectrum of attendees which include people with a background in mathematics, computer science, engineering and psychology coming from academia, research laboratories and industry.
Participants of the GIVEN team meeting at the Siemens gas turbines factories in Berlin
The GIVEN consortium visited Siemens in Berlin for its third scientific workshop. Participants from the Universities of Wuppertal and Trier and the Siemens AG came together and discussed the latest progress of the project.
This year the community convened in the city of Spokane, WA. The audience was naturally interdiscplinary, ranging from mathematicians to engineers with attendees from academia, research laboratories and industry.
Johanna Schultes of the GIVEN consortium participated at the workshop Women in Optimization and presented a poster about results of current research with the topic: “Biobjective Shape Optimization to Improve Reliability and Costs of Ceramic Components using Weighted Sums”.
The workshop series “Women in Optimization” is every two years and addresses young female researchers, doctoral students and mathematical students with focus on optimization and covers the wide scientific spectrum of mathematical optimization and its numeric. The aim of the workshop is the further education of young researchers as well as the opportunity of networking in the optimization community.
This year, the workshop took place at the University of Bonn. It combined lectures by experienced scientists with poster presentation by junior researches and offered a lot of space for academic discussions.
Visualisation of the Stress Field in one Component
The GIVEN-Consortium participated in the workshop „Recent Advances in
Multi-Objective Optimization 2018“ (RAMOO 2018) in Nantes, France with
the contribution „Gradient-Based Biobjective Shape Optimization of
Ceramic Components: Probability of Failure versus Cost“. This workshop
is a recurring format in the multi-objective optimization community and
focuses on the latest advances in exact methods in multi-objective
(mixed) integer optimization.
The workshop aims to foster technical discussions and interactions
between the workshop attendees who range from young researchers (master
and doctoral students) to senior researchers (academic researchers and
practitioners). This specialization in multi-objective optimization
topics and the low barriers for discussions help to boost the whole
field of multi-objective optimization.
For further information about this workshop visit the official website
of RAMOO 2018.
The GIVEN consortium met on the last days of November 2018 in Trier for its second scientific workshop. 17 participants from the Universities of Wuppertal and Trier, DLR and Siemens AG came together for two days to discuss progress in the GIVEN project and plan future activities.
The workshop program included 9 scientific talks and several discussion rounds. As it is customary in Trier, also a dinner and a wine tasting have been part of the workshop program.
There were given more than 330 presentations on various topics from Numerical Analysis, Stochastics, Optimization or Modeling.
The GIVEN-Consortium participated in the 20th European Conference on Mathematics in Industry (ECMI) in Budapest, Hungary with the contribution “Shape Optimization for a Stochastic Objective Functional“.
The series of European Consortium for Mathematics in Industry (ECMI) conferences are devoted to enforce the interaction between academy and industry, leading to innovations in both fields. These events have attracted leading experts from business, science and academia, and have promoted the application of novel mathematical technologies to industry.
ECMI conferences are organised under the auspices of the European Consortium for Mathematics in Industry, an association of academic institutions and industrial companies that acts co-operatively with the following aims:
To promote and support the use of mathematical modelling, simulation, and optimization in any activity of social or economic importance.
To educate Industrial Mathematicians to meet the growing demand for such experts.
To operate on a European scale.
For further information about this conference visit the official website of the ECMI.
Prototype of a rotor shaft from one of Ellings designs in 1925
The GIVEN consortium participated in the world’s largest annual turbomachinery conference with the contribution „Adjoint Method to Calculate Shape Gradients of Failure Probabilaties for Turbomachinery Components“ (see publications).
This year’s TurboExpo organized by the American Society of Mechanical Engineers (ASME) is held in Lillestrøm, where the Norwegian inventor Aegidius Elling constructed the first gas turbine in 1903. The photograph shows a prototype of a rotor shaft from one of Ellings designs in 1925 with a one stage axial compressor and a one stage turbine presented at the TurboExpo. In the background Ellings lab log in blue color.
3 000 participants of the Turbo Expo attend to more than 1 000 technical presentations, panel sessions and discussions between industry decision makers.
Movie: Shape flow for a 2D rod towards optimal reliability.
As three working groups (Optimization, High Performance Computing and Stochastics) at the IMACM in Wuppertal, we develop probabilistic shape optimization models for different failure mechanisms, notably for low cycle fatigue (LCF) problems, with gradient based methods. Low cycle fatigue is a surface driven failure mechanism that occurs under cyclic loading of technical units made of polycrystalline metal. To calculate the shape gradients, we discretize the component via finite elements and then use an adjoint approach.
Objective functions based on the peak stress that stem from mechanical integrity are in general not differentiable, since the component always deterministically fails at the point with the lowest longevity and the highest loading. So we replace the deterministic approach by a probabilistic approach to enable the calculation of shape gradients (sensitivities). With these sensitivities we are able to apply gradient based optimization using an adjoin approach without compromising the structural integrity. In our work we focus on 3D turbo-machinery components, i.e. LCF problems, and 2D ceramic geometries under tensile load.
In these settings, we can assume that the material follows a linear isotropic material law. Thus we can constrain our problem with the usual state equation of linear elasticity in the weak formulation with cyclic or mixed Dirichlet-Neumann boundary conditions. The analysis of the probability of failure (PoF) leads to Weibull type objective functionals allowing us to calculate the shape derivatives. We then first discretize the problem via finite elements and then use the Lagrangian approach to avoid the “curse of dimension”.
To solve the state equation and for calculating the adjoint state, the finite element solver ABAQUS is used for the LCF case. To validate the computation we first calculated the shape gradients for a predictable geometry, in this case a bended rod ( Figure 1) which is fixed at the left-hand boundary and surface forces act on the right-hand boundary. With this preliminary case study, we were able to apply our approach to a 3D aero gas turbine 1st stage compressor blisk model (Figure 2).
Figure 1: Shape derivative towards lower fatigue failure probability for a bended rod
Figure 2: Shape derivative towards higher fatigue failure probability for a turbine blisk geometry under centrifugal load
We also applied this approach to simple 2D ceramic geometries. We were able to construct shape flows to improved shapes in these cases. Throughout this work it became clear that multi-objective optimization would be a desirable extension as the minimization of PoF may lead to shapes that have a poor performance in other relevant criteria, like for example efficiency, volume and costs. The first results of this multi-objective optimization for a 2D ceramic case study with objective functions PoF and material consumption (volume), i.e. a part of the Pareto-front of shapes with a tradeoff between these objectives, is shown in Figure 3.
Figure 3: Approximate Pareto front for a 2D bended rod with (negative, logarithmic) probability of survival (x-axis) and material consumption (y-axis) as objective functions
In the upcoming GIVEN collaboration with V. Schulz at U Trier, Siemens Energy and DLR Cologne and also the numerics group (M. Ehrhardt and M. Günther) at BUW, we will extend this multi-objective results to 3D LCF models where the objectives PoF and aerodynamic efficiency, which is computed using adjoint CFD turbo machinery solver TRACE, are considered and integrated to a multicriteria gradient based 3D optimization process.