Geometry (e.g., curves, surfaces, solids) is pervasive throughout the airplane industry. At The Boeing Company, the prevalent way to model geometry is the parametric representation. For example, a parametric surface, S, is the image of a function
S:D → ℝ³
whereD ≔ [0..1]×[0..1]is the parameter domain.
Here S denotes the parametrization, as well as the (red) surface itself.
A geometry’s parametric representation is not unique and the accuracy of analysis tools is often sensitive to its quality. In many cases, the best parametrization is one that preserves lengths, areas, and angles well, i.e., a parametrization that is nearly isometric. Nearly isometric parametrizations are used, for example, when designing non-flat parts that will be constructed or machined flat.
Figure 1. Parts that are nearly developable on one side are often machined on a flat table; then re-formed.
Another area where geometry parametrization is especially important is shape optimization activities that involve isogeometric analysis. In these cases, getting a “good enough” parametrization very efficiently is crucial, since the geometry varies from one iteration to another.
In this project, the students will research, discuss, and propose potential measures of “isometricness” and algorithms for obtaining them. Example problems will be available on which to test their ideas.
1. Michael S. Floater, Kai Hormann, Surface parametrization: a tutorial and survey, Advances in Multiresolution for Geometric Modeling, (2005) pp 157—186.
2. J. Gravesen, A. Evgrafov, Dang-Manh Nguyen, P.N. Nielsen, Planar parametrization in isogeometric analysis, Lecture Notes in Computer Science, Volume 8177 (2014) pp 189—212.
3. T-C Lim, S. Ramakrishna, Modeling of composite sheet forming: a review, Composites: Part A, Volume 33 (2002) pp 515—537.
4. Yaron Lipman, Ingrid Daubechies, Conformal Wasserstein distances: comparing surfaces in polynomial time, Advances in Mathematics, vol. 227 (2010) pp. 1047—1077.
In real-life applications critical areas are often non- accessible for measurement and thus for inspection and control. For proper and safe operations one has to estimate their condition and predict their future alteration via inverse problem methods based on accessible data. Typically such situations are even complicated by unreliable or flawed data such as sensor data rising questions of reliability of model results. We will analyze and mathematically tackle such problems starting with physical vs. data driven modeling, numerical treatment of inverse problems, extension to stochastic models and statistical approaches to gain stochastic distributions and confidence intervals for safety critical parameters.
As project example we consider a blast furnace producing iron at temperatures around 2,000 °C. It is running several years without stop or any opportunity to inspect its inner geometry coated with firebrick. Its inner wall is aggressively penetrated by physical and chemical processes. Thickness of the wall, in particular evolvement of weak spots through wall thinning is extremely safety critical. The only available data stem from temperature sensors at the outer furnace surface. They have to be used to calculate wall thickness and its future alteration. We will address some of the numerous design and engineering questions such as placement of sensors, impact of sensor imprecision and failure.
1. F. Bornemann, P. Deuflhard, A. Hohmann, "Numerical Analysis”, de Gruyter, 1995
2. A. C. Davison,” Statistical Models”, Cambridge University Press, 2003
3. William H. Press, “Numerical Recipes in C”, Cambridge University Press, 1992
Computer programming experience in a language like C or C++; Knowledge about Numerical Linear Algebra,
Stochastic and Statistics (see references)