A "poor man's" approach for high-resolution three-dimensional topology optimization of natural convection problems

This paper treats topology optimization of natural convection problems. A simplified model is suggested to describe the flow of an incompressible fluid in steady state conditions, similar to Darcy's law for fluid flow in porous media. The equations for the fluid flow are coupled to the thermal convection-diffusion equation through the Boussinesq approximation. The coupled non-linear system of equations is discretized with stabilized finite elements and solved in a parallel framework that allows for the optimization of high resolution three-dimensional problems. A density-based topology optimization approach is used, where a two-material interpolation scheme is applied to both the permeability and conductivity of the distributed material. Due to the simplified model, the proposed methodology allows for a significant reduction of the computational effort required in the optimization. At the same time, it is significantly more accurate than even simpler models that rely on convection boundary conditions based on Newton's law of cooling. The methodology discussed herein is applied to the optimization-based design of three-dimensional heat sinks. The final designs are formally compared with results of previous work obtained from solving the full set of Navier-Stokes equations. The results are compared in terms of performance of the optimized designs and computational cost. The computational time is shown to be decreased to around 5-20% in terms of core-hours, allowing for the possibility of generating an optimized design during the workday on a small computational cluster and overnight on a high-end desktop

Solar-Sailing Trajectory Design for Close-Up NEA Observations Mission

Near-Earth Asteroids (NEAs) are an extremely valuable resource to study the origin and evolution of the Solar System more in depth. At the same time, they constitute a serious risk for the Earth in the not-so-remote case of an impact. In order to mitigate the hazard of a potential impact with the Earth, several techniques have been studied so far and, for the majority of them, a good knowledge about the chemical and physical composition of the target object is extremely helpful for the success of the mission. A multiple-rendezvous mission with NEAs, with close-up observations, can help the scientific community to improve the overall knowledge about these objects and to support any mitigation strategy. Because of the cost of this kind of mission in terms of Dv, a solar sail spacecraft is proposed in this study, in order to take advantage of the propellantless characteristic of this system. As part of the DLR/ESA Gossamer roadmap, and thus considering the sailcraft based on this technology, the present work is focused on the search of possible sequences of encounters, with priority on Potentially Hazardous Asteroids (PHAs). Because of the huge amount of NEAs, the selection of the candidates for a multiple rendezvous is firstly a combinatorial problem, with more than a billion of possible sequences for only three consecutive encounters. Moreover, an optimization problem should be solved in order to find a feasible solar-sail trajectory for each leg of the sequence. In order to tackle this mixed combinatorial/optimization problem, the strategy used is divided into two main steps: a sequence search by means of heuristic rules and simplified trajectory models, and a subsequent optimization phase. Preliminary results were presented previously by the authors, demonstrating that this kind of mission is promising. In this paper, we aim to find new sequences by introducing a different approach on the sequence search algorithm and by reducing the area-to-mass ratio of the solar sail. A smaller area-to-mass ratio entails either the possibility to carry on more payload or to reduce the sail area, raising the TRL. A grid search over 10 years of launching dates is carried out, resulting in different sequences of objects depending on the departing date. Two sequences are fully studied and optimized. The mission parameters and trajectories of the sequences found are shown and explained

Combination of direct methods and homotopy in numerical optimal control: application to the optimization of chemotherapy in cancer

We consider a state-constrained optimal control problem of a system of two non-local partial-differential equations, which is an extension of the one introduced in a previous work in mathematical oncology. The aim is to minimize the tumor size through chemotherapy while avoiding the emergence of resistance to the drugs. The numerical approach to solve the problem was the combination of direct methods and continuation on discretization parameters, which happen to be insufficient for the more complicated model, where diffusion is added to account for mutations. In the present paper, we propose an approach relying on changing the problem so that it can theoretically be solved thanks to a Pontryagin Maximum Principle in infinite dimension. This provides an excellent starting point for a much more reliable and efficient algorithm combining direct methods and continuations. The global idea is new and can be thought of as an alternative to other numerical optimal control techniques