Cyrille Allery

A priori reduction method for solving the two-dimensional Burgers’ equations

Abstract The two-dimensional Burgers’ equations are solved here using the A Priori Reduction method. This method is based on an iterative procedure which consists in building a basis for the solution where at each iteration the basis is improved. The method is called a priori because it does not need any prior knowledge of the solution, which is not the case if the standard Karhunen–Loeve decomposition is used. The accuracy of the APR method is compared with the standard Newton–Raphson scheme and with results from the literature. The APR basis is also compared with the Karhunen–Loeve basis.

An adaptive ROM approach for solving transfer equations

In this article, we present an adaptive method for solving transfer equations. The method consists in projecting the discretized problem on a basis we have defined in order to obtain a reduced model that can be quickly and accurately solved with classic numerical schemes. The originality of the methods stays in the way of the basis is constructed. At each iteration of computation, the basis is adapted: first the old basis is improved using a Karhunen-Loève decomposition whereas in a second phase the improved basis is expanded with Krylov vectors. The example we study is the one-dimension Burgers’ equation. The results we obtained were compared to the Newton-Raphson method: whereas the accuracy is not better than the Newton-Raphson method, we show that the computationnal time is drastically reduced. In addition, the basis we obtain shows a great ability to represent the long-time dynamics of the system, as shown in the last part of the paper.

Optimal flow control using a POD-based reduced-order model

ABSTRACT The flow control constitutes a challenging research topic with numerous applications in aeronautics, aerodynamics, fluid mechanics, civil engineering, and so on. Due to the use of iterative algorithms, it is costly in both CPU time and memory requirements. In this paper, an optimal control strategy with reduced-order model built with proper orthogonal decomposition approach is proposed. The methodology is developed to control an anisothermal flow in a lid-driven cavity with one control parameter and then extended to at two control parameters. The control is realized in quasi-real time, which opens the way to promising practical applications.

Anisothermal Flow Control by Using Reduced-Order Models

Despite constantly improving computer capabilities, classical numerical methods (DNS, LES,…) are still out of reach in fluid flow control strategies. To make this problem tractable almost in real-time, reduced-order models are used here. The spatial basis is obtained by POD (Proper Orthogonal Decomposition), which is the most commonly used technique in fluid mechanics. The advantage of the POD basis is its energetic optimality: few modes contain almost the totality of energy. The ROM is achieved with the recent developed optimal projection [1], unlike classical methods which use Galerkin projection. This projection method is based on the minimization of the residual equations in order to have a stabilizing effect. It enables moreover to access pressure field. Here, the projection method is slightly different from [1]: a formulation without the Poisson equation is proposed and developed. Then, the ROM obtained by optimal projection is introduced within an optimal control loop. The flow control strategy is illustrated on an isothermal square lid-driven cavity and an anisothermal square ventilated cavity. The aim is to reach a target temperature (or target pollutant concentration) in the cavity, with an interior initial temperature (or initial pollutant concentration), by adjusting the inlet fluid flow rate.Copyright

A fast and robust sub-optimal control approach using reduced order model adaptation techniques

Abstract Classical adjoint-based optimization approach for the optimal control of partial differential equations is known to require a large amount of CPU time and memory storage. In this article, in order to reduce these requirements, a posteriori and a priori model order reduction techniques such as POD (Proper Orthogonal Decomposition) and PGD (Proper Generalized Decomposition) are used. As a matter of fact, these techniques allows a fast access to the temporal dynamics of a solution approximated in a suitable subspace of low dimension, spanned by a set of basis functions that form a reduced basis. The costly high fidelity model is then projected onto this basis and results in a system of ordinary differential equations which can be solved in quasi-real time. A disadvantage of considering a fixed POD basis in a suboptimal control loop, is basically the dependence of such bases on a posteriori information coming from high fidelity simulations. Therefore, a non robustness of the POD basis can be expected for certain perturbations in the original parameter for which it was built. As a result, update the reduced bases with respect to each variation in the control parameter using the POD method is still costly. To get over this difficulty, we equip the usual reduced optimal control algorithm with an intermediate basis adaptation step. The first proposed approach consists in adapting the reduced basis for a new control parameter by interpolating over a set of POD bases previously computed for a range of control parameters. To achieve that, an interpolation technique based on properties of the tangent subspace of the Grassmann manifold (ITSGM) is considered. The second approach is the PGD method, which by nature, enrich a space time decomposition trying to enhance the approximation by learning from its own errors. Relaying on this property, this method is employed in the control loop as a basis corrector, in such a way the given spatial basis is adapted for the new control parameter by performing just few enrichments. These two approaches are applied in the sub-control of the two dimensional non-linear reaction-diffusion equations and Burgers equations.

Numerical divergent series resummation in fluid flow simulation

The perturbation theory has proved to be an efficient tool for the numerical resolution of non-linear problems in mechanics. However, it is not suitable for singular problems, for which the series solution is divergent. We propose to use the Borel-Laplace series resummation method for the resolution of such a problem. The resulting algorithm is applied to some model problems in fluid mechanics.