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"Pironneau, Olivier"
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A BEM Adjoint-Based Differentiable Shape Optimization of a Stealth Aircraft
Modern fighter aircraft have an increasing need for at least a moderate level of stealth, and the shape design must bear a part of this constraint. However, the high frequency of close range radar makes high-fidelity radar cross-section computation methods such as the boundary element method too expensive to use in a gradient-free optimization process. On the other hand, asymptotic methods are not able to accurately predict the RCS of complex shapes such as intake cavities. Hence, the need arises for efficient and accurate methods to compute the gradient of high-fidelity radar cross-section computation methods with respect to shape parameters. In this paper, we propose an adjoint formulation for the boundary element method to efficiently compute these gradients. We present a novel approach to calculate the gradient of high-fidelity radar cross-section computations using the boundary element method. Our method employs an adjoint formulation that allows for the efficient computation of these gradients. This is particularly beneficial in shape optimization problems where accurate and efficient methods are crucial to designing modern fighter aircraft with stealth capabilities. By avoiding the need for solving the actual adjoint problem in certain cases, our formulation provides a more streamlined solution while still maintaining high accuracy. We demonstrate the effectiveness of our method by performing shape optimization on various shapes, including simple geometries like spheres and ellipsoids, as well as complex aircraft shapes with multiple variables.
Journal Article
Gradient-Based Aero-Stealth Optimization of a Simplified Aircraft
Modern fighter aircraft increasingly need to conjugate aerodynamic performance and low observability. In this paper, we showcase a methodology for a gradient-based bidisciplinary aero-stealth optimization. The shape of the aircraft is parameterized with the help of a CAD modeler, and we optimize it with the SLSQP algorithm. The drag, computed with the help of a RANS method, is used as the aerodynamic criterion. For the stealth criterion, a function is derived from the radar cross-section in a given cone of directions and weighed with a function whose goal is to cancel the electromagnetic intensity in a given direction. Stealth is achieved passively by scattering back the electromagnetic energy away from the radar antenna, and no energy is absorbed by the aircraft, which is considered as a perfect conductor. A Pareto front is identified by varying the weights of the aerodynamic and stealth criteria. The Pareto front allows for an easy identification of the CAD model corresponding to a chosen aero-stealth trade-off.
Journal Article
Shape and topology optimization based on the convected level set method
The aim of this research is to construct a shape optimization method based on the convected level set method, in which the level set function is defined as a truncated smooth function obtained by using a sinus filter based on a hyperbolic tangent function. The local property of the hyperbolic tangent function dramatically reduces the generation of red the error between the specified profile of the hyperbolic tangent function and the level set function that is updated using a time evolution equation. In addition, the small size of the error facilitates the use of convective reinitialization, whose basic idea is that the reinitialization is embedded in the time evolution equation, whereas such treatment is typically conducted in a separate calculation in conventional level set methods. The convected level set method can completely avoid the need for additional calculations when performing reinitialization. The validity and effectiveness of our presented method are tested with a mean compliance minimization problem and a problem for the design of a compliant mechanism.
Journal Article
Numerical Analysis of an Electroless Plating Problem in Gas–Liquid Two-Phase Flow
Electroless plating in micro-channels is a rising technology in industry. In many electroless plating systems, hydrogen gas is generated during the process. A numerical simulation method is proposed and analyzed. At a micrometer scale, the motion of the gaseous phase must be addressed so that the plating works smoothly. Since the bubbles are generated randomly and everywhere, a volume-averaged, two-phase, two-velocity, one pressure-flow model is applied. This fluid system is coupled with a set of convection–diffusion equations for the chemicals subject to flux boundary conditions for electron balance. The moving boundary due to plating is considered. The Galerkin-characteristic finite element method is used for temporal and spatial discretizations; the well-posedness of the numerical scheme is proved. Numerical studies in two dimensions are performed to validate the model against earlier one-dimensional models and a dedicated experiment that has been set up to visualize the distribution of bubbles.
Journal Article
Mixing Monte-Carlo and Partial Differential Equations for Pricing Options
There is a need for very fast option pricers when the financial objects are mod-eled by complex systems of stochastic differential equations. Here the authors investigate option pricers based on mixed Monte-Carlo partial differential solvers for stochastic volatility models such as Heston's. It is found that orders of magnitude in speed are gained on full Monte-Carlo algorithms by solving all equations but one by a Monte-Carlo method, and pricing the underlying asset by a partial differential equation with random coefficients, derived by Ito calculus. This strategy is investigated for vanilla options, barrier options and American options with stochastic volatilities and jumps optionally.
Paper
Mini-symposium on automatic differentiation and its applications in the financial industry
Automatic differentiation has been involved for long in applied mathematics as an alternative to finite difference to improve the accuracy of numerical computation of derivatives. Each time a numerical minimization is involved, automatic differentiation can be used. In between formal derivation and standard numerical schemes, this approach is based on software solutions applying mechanically the chain rule formula to obtain an exact value for the desired derivative. It has a cost in memory and cpu consumption. For participants of financial markets (banks, insurances, financial intermediaries, etc), computing derivatives is needed to obtain the sensitivity of their exposure to well-defined potential market moves. It is a way to understand variations of their balance sheets in specific cases. Since the 2008 crisis, regulation demands to compute this kind of exposure to many different cases, to be sure that market participants are aware and ready to face a wide spectrum of market configurations. This paper shows how automatic differentiation provides a partial answer to this recent explosion of computations to be performed. One part of the answer is a straightforward application of Adjoint Algorithmic Differentiation (AAD), but it is not enough. Since financial sensitivities involve specific functions and mix differentiation with Monte-Carlo simulations, dedicated tools and associated theoretical results are needed. We give here short introductions to typical cases arising when one uses AAD on financial markets.
Journal Article
Pricing futures by deterministic methods
In this article we will focus on only a small part of financial mathematics, namely the use of partial differential equations for pricing futures. Even within this narrow range it is hard to be systematic and complete, or even to do better than existing books such as Wilmott, Howison and Dewynne (1995), Achdou and Pironneau (2005), or software manuals such as Lapeyre, Martini and Sulem (2010). So this article may be valuable only to the extent that it reflects ten years of teaching, conferences and interaction with the protagonists of financial mathematics. Also, because the theory of partial differential equations is not always well known, we have chosen a pragmatic approach and left out the details of the theory or the proofs of some results, and refer the reader to other books. The numerical algorithms, on the other hand, are given in detail.
Journal Article
Finite Element Characteristic Methods Requiring no Quadrature
The characteristic methods are known to be very efficient for convection-diffusion problems including the Navier-Stokes equations. Convergence is established when the integrals are evaluated exactly, otherwise there are even cases where divergence has been shown to happen. The family of methods studied here applies Lagrangian convection to the gradients and the function as in Yabe (Comput. Phys. Commun. 66(2–3), 233–242,
1991
); the method does not require an explicit knowledge of the equation of the gradients and can be applied whenever the gradients of the convection velocity are known numerically. We show that converge can be second order in space or more. Applications are given for the rotating bell problem.
Journal Article