Towards a Lagrange-Newton approach for PDE constrained shape optimization

The novel Riemannian view on shape optimization developed in [Schulz, FoCM, 2014] is extended to a Lagrange-Newton approach for PDE constrained shape optimization problems. The extension is based on optimization on Riemannian vector space bundles and exemplified for a simple numerical example.Comment: 16 pages, 4 figures, 1 tabl

Shape optimization for the generalized Graetz problem

We apply shape optimization tools to the generalized Graetz problem which is a convection-diffusion equation. The problem boils down to the optimization of generalized eigen values on a two phases domain. Shape sensitivity analysis is performed with respect to the evolution of the interface between the fluid and solid phase. In particular physical settings, counterexamples where there is no optimal domains are exhibited. Numerical examples of optimal domains with different physical parameters and constraints are presented. Two different numerical methods (level-set and mesh-morphing) are show-cased and compared

Optimal damping of a membrane and topological shape optimization

We consider a shape optimization problem of finding the optimal damping set of a two-dimensional membrane such that the energy of the membrane is minimized at some fixed end time. Traditional shape optimization is based on sensitivities of the cost functional with respect to small boundary variations of the shapes. We use an iterative shape optimization scheme based on level set methods and the gradient descent algorithm to solve the problem and present numerical results. The methods presented allow for certain topological changes in the optimized shapes. These changes can be realized in the presence of a force term in the level set equation. It is also observed that the gradient descent algorithm on the manifold of shapes does not require an exact line search to converge and that it is sufficient to perform heuristic line searches that do not evaluate the cost functional being minimized. © 2008 Springer-Verlag

Seasonal changes in plankton respiration and bacterial metabolism in a temperate shelf sea

The seasonal variability of plankton metabolism indicates how much carbon is cycling within a system, as well as its capacity to store carbon or export organic matter and CO2 to the deep ocean. Seasonal variability between November 2014, April 2015 and July 2015 in plankton respiration and bacterial (Bacteria+Archaea) metabolism is reported for the upper and bottom mixing layers at two stations in the Celtic Sea, UK. Upper mixing layer (UML, >75 m in November, 41 - 70 m in April and ~50 m in July) depth-integrated plankton metabolism showed strong seasonal changes with a maximum in April for plankton respiration (1.2- to 2-fold greater compared to November and July, respectively) and in July for bacterial production (2-fold greater compared to November and April). However UML depth-integrated bacterial respiration was similar in November and April and 2-fold lower in July. The greater variability in bacterial production compared to bacterial respiration drove seasonal changes in bacterial growth efficiencies, which had maximum values of 89 % in July and minimum values of 5 % in November. Rates of respiration and gross primary production (14C-PP) also showed different seasonal patterns, resulting in seasonal changes in 14C-PP:CRO2 ratios. In April, the system was net autotrophic (14C-PP:CRO2 > 1), with a surplus of organic matter available for higher trophic levels and export, while in July balanced metabolism occurred (14C-PP:CRO2 = 1) due to an increase in plankton respiration and a decrease in gross primary production. Comparison of the UML and bottom mixing layer indicated that plankton respiration and bacterial production were higher (between 4 and 8-fold and 4 and 7-fold, respectively) in the UML than below. However, the rates of bacterial respiration were not statistically different (p > 0.05) between the two mixing layers in any of the three sampled seasons. These results highlight that, contrary to previous data from shelf seas, the production of CO2 by the plankton community in the UML, which is then available to degas to the atmosphere, is greater than the respiratory production of dissolved inorganic carbon in deeper waters, which may contribute to offshore export