A criterion is given for the convergence of numerical solutions of the Navier-Stokes equations in two dimensions under steady conditions. The criterion applies to all cases, of steady viscous flow in two dimensions and shows that if the local ' mesh Reynolds number ', based on the size of the mesh used in the solution, exceeds a certain fixed value, the numerical solution will not converge.

On the Navier-Stokes equations

Finite element methods for Maxwell's equations.

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Linear And Nonlinear Programming

It is the purpose of this paper to describe some of the recent developments in the mathematical theory of linear and quasilinear elliptic and parabolic systems with nonhomogeneous boundary conditions. For illustration we use the relatively simple set-up of reaction-diffusion systems which are — on the one h and — typical for the whole class of systems to which the general theory applies and — on the other h and — still simple enough to be easily described without too many technicalities. In addition, quasilinear reaction-diffusion equations are of great importance in applications and of actual mathematical and physical interest, as is witnessed by the examples we include.

Nonhomogeneous Linear and Quasilinear Elliptic and Parabolic Boundary Value Problems

Geometry: shape, size, aspect ratio and orientation Flow Type: forced, natural, laminar, turbulent, internal, external Boundary: isothermal (Tw = constant) or isoflux (q̇w = constant) Fluid Type: viscous oil, water, gases or liquid metals Properties: all properties determined at film temperature Tf = (Tw + T∞)/2 Note: ρ and ν ∝ 1/Patm ⇒ see Q12-40 density: ρ ((kg/m) specific heat: Cp (J/kg ·K) dynamic viscosity: μ, (N · s/m) kinematic viscosity: ν ≡ μ/ρ (m/s) thermal conductivity: k, (W/m ·K) thermal diffusivity: α, ≡ k/(ρ · Cp) (m/s) Prandtl number: Pr, ≡ ν/α (−−) volumetric compressibility: β, (1/K)

Convection Heat Transfer

We investigate two control problems related to the aerodynamic optimization of flows around airfoils in high-lift configurations. The first task is the steady state maximization of lift subject to restrictions on the drag. This leads to a boundary control problem for the 2D stationary Navier-Stokes equations with constrained control functions belonging to $\textbf{L}^2(\Gamma)$ under an integral state constraint. We derive optimality conditions and treat the problem numerically by direct solution of the associated nonsmooth optimality system. The second part is based on a k-ω-Wilcox98 turbulence model. To deal with the curse of dimension and the nonlinearity of the system, we discuss a reduced-order model by adapting a small system of ODEs to solutions computed with the full model.

/pdf/optimal-boundary-control-problems-related-to-high-lift-1uzlc8urkd.pdf

Optimal Boundary Control Problems Related to High-Lift Configurations

An optimal boundary control problem for the one-dimensional heat equation is considered. The objective functional includes a standard quadratic terminal observation, a Tikhonov regularization term with regularization parameter $ν$, and the $L^1$-norm of the control that accounts for sparsity. The switching structure of the optimal control is discussed for $ν ≥ 0$. Under natural assumptions, it is shown that the set of switching points of the optimal control is countable with the final time as only possible accumulation point. The convergence of switching points is investigated for $ν \searrow 0$.

On the switching behavior of sparse optimal controls for the one-dimensional heat equation

Traveling localized spots represent an important class of self-organized two-dimensional patterns in reaction–diffusion systems. We study open-loop control intended to guide a stable spot along a desired trajectory with desired velocity. Simultaneously, the spot’s concentration profile does not change under control. For a given protocol of motion, we first express the control signal analytically in terms of the Goldstone modes and the propagation velocity of the uncontrolled spot. Thus, detailed information about the underlying nonlinear reaction kinetics is unnecessary. Then, we confirm the optimality of this solution by demonstrating numerically its equivalence to the solution of a regularized, optimal control problem. To solve the latter, the analytical expressions for the control are excellent initial guesses speeding-up substantially the otherwise time-consuming calculations.

/pdf/control-of-traveling-localized-spots-1ndx1pdq7r.pdf

Control of traveling localized spots

On a Numerical Method for Coercive Parabolic Boundary Control Problems

In this article sufficient optimality conditions are established for optimal control of evolutionary Navier-Stokes equations. The second-order condition requires coercivity of the Lagrange function on a suitable subspace together with first-order necessary conditions. It ensures local optimality of a reference function in a Ls-neighborhood, whereby the underlying analysis allows to use weaker norms than L∞. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Fredi Tröltzsch

Papers

Optimal Boundary Control Problems Related to High-Lift Configurations

On the switching behavior of sparse optimal controls for the one-dimensional heat equation

Control of traveling localized spots

On a Numerical Method for Coercive Parabolic Boundary Control Problems

Second‐order sufficient optimality conditions for the optimal control of instationary Navier‐Stokes equations