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Register a new userControl of bifurcation structures using shape optimization
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Added by
Nicolas Boull\\'e
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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Many problems in engineering can be understood as controlling the bifurcation structure of a given device. For example, one may wish to delay the onset of instability, or bring forward a bifurcation to enable rapid switching between states. We propose a numerical technique for controlling the bifurcation diagram of a nonlinear partial differential equation by varying the shape of the domain. Specifically, we are able to delay or advance a given bifurcation point to a given parameter value, often to within machine precision. The algorithm consists of solving a shape optimization problem constrained by an augmented system of equations, the Moore--Spence system, that characterize the location of the bifurcation points. Numerical experiments on the Allen--Cahn, Navier--Stokes, and hyperelasticity equations demonstrate the effectiveness of this technique in a wide range of settings.
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We consider a coupled physical-biological model describing growth of microalgae in a raceway pond cultivation process, accounting for hydrodynamics. Our approach combines a biological model (based on the Han model) and shallow water dynamics equations that model the fluid into the raceway pond. We present an optimization procedure dealing with the topography to maximize the biomass production over one lap or multiple laps with a paddle wheel. On the contrary to a widespread belief in the microalgae field, the results show that a flat topography is optimal in a periodic regime. In other frameworks, non-trivial topographies can be obtained. We present some of them, e.g., when a mixing device is included in the model.
We study an optimization problem that aims to determine the shape of an obstacle that is submerged in a fluid governed by the Stokes equations. The mentioned flow takes place in a channel, which motivated the imposition of a Poiseuille-like input function on one end and a do-nothing boundary condition on the other. The maximization of the vorticity is addressed by the $L^2$-norm of the curl and the {it det-grad} measure of the fluid. We impose a Tikhonov regularization in the form of a perimeter functional and a volume constraint to address the possibility of topological change. Having been able to establish the existence of an optimal shape, the first order necessary condition was formulated by utilizing the so-called rearrangement method. Finally, numerical examples are presented by utilizing a finite element method on the governing states, and a gradient descent method for the deformation of the domain. On the said gradient descent method, we use two approaches to address the volume constraint: one is by utilizing the augmented Lagrangian method; and the other one is by utilizing a class of divergence-free deformation fields.
This paper focuses on mixing strategies to enhance the growth rate in an algal raceway system. A mixing device, such as a paddle wheel, is considered to control the rearrangement of the depth of the algae cultures hence the light perceived at each lap. The dynamics of the photosystems after a rearrangement is accounted for by the Han model. Our approach consists in considering permanent regimes where the strategy is parametrized by a permutation matrix which modifies the order of the layers at the beginning of each lap. It is proven that the dynamics of the photosystems is then periodic, with a period corresponding to one lap of the raceway whatever the order of the considered permutation matrix is. An objective function related to the average growth rate over one lap is then introduced. Since N ! permutations (N being the number of considered layers) need to be tested in the general case, it can be numerically solved only for a limited number of layers. Consequently, we propose a second optimization problem associated with a suboptimal solution of the initial problem, which can be determined explicitly. A sufficient condition to characterize cases where the two problems have the same solution is given. Some numerical experiments are performed to assess the benefit of optimal strategies in various settings.
We study the problem of finding the nearest $Omega$-stable matrix to a certain matrix $A$, i.e., the nearest matrix with all its eigenvalues in a prescribed closed set $Omega$. Distances are measured in the Frobenius norm. An important special case is finding the nearest Hurwitz or Schur stable matrix, which has applications in systems theory. We describe a reformulation of the task as an optimization problem on the Riemannian manifold of orthogonal (or unitary) matrices. The problem can then be solved using standard methods from the theory of Riemannian optimization. The resulting algorithm is remarkably fast on small-scale and medium-scale matrices, and returns directly a Schur factorization of the minimizer, sidestepping the numerical difficulties associated with eigenvalues with high multiplicity.
This paper gives a unified convergence analysis of additive Schwarz methods for general convex optimization problems. Resembling to the fact that additive Schwarz methods for linear problems are preconditioned Richardson methods, we prove that additive Schwarz methods for general convex optimization are in fact gradient methods. Then an abstract framework for convergence analysis of additive Schwarz methods is proposed. The proposed framework applied to linear elliptic problems agrees with the classical theory. We present applications of the proposed framework to various interesting convex optimization problems such as nonlinear elliptic problems, nonsmooth problems, and nonsharp problems.
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