Multilevel optimization for PDAE-constrained optimal control problems with pointwise constraints on control and state

We have developed a fully adaptive optimization environment suitable to solve complex optimal control problems restricted by partial differential algebraic equations (PDAEs) and pointwise constraints on the control [1, 2]. This contribution is devoted to the inclusion of pointwise constraints on the state within the optimization environment. To this end we first give a brief introduction into the architecture of the environment and the inclusion of pointwise constraints on the state by Moreau-Yosida regularization. Then, we test the new tool by applying it to an optimal boundary control problem for the cooling of hot glass down to room temperature, modeled by radiative heat transfer and semi-transparent boundary conditions. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

?le, Fette, Wachse

Ohne Zusammenfassung     

Laminar thermally developing flow inside right-angularly triangular ducts

An analytical approach based on the generalized integral transform technique is presented, for the solution of laminar forced convection within the thermal entry region of ducts with arbitrarily shaped cross-sections. The analysis is illustrated through consideration of a right triangular duct subjected to constant wall temperature boundary condition. Critical comparisons are made with results available in the literature, from direct numerical approaches. Numerical results for dimensionless average temperature and Nusselt numbers are presented for different apex angles.Nomenclature a,b sides of right triangular duct - A _c cross-sectional area of duct - c _p specific heat of fluid - D _h =4A _c /p hydraulic diameter, with P the wet perimeter - h(z) heat transfer coefficient at duct wall - k thermal conductivity - Pe=c _p D _h /k Peclet number - T(x, y, z) temperature distribution - T _o inlet temperature - T _w prescribed wall temperature - u(x, y); U(X, Y) dimensional and dimensionless velocity profile - average flow velocity - x; X dimensional and dimensionless normal coordinate (Fig. 1) - x ₁(y); X ₁(Y) dimensional and dimensionless position at irregular boundary (Fig. 1) - y; Y dimensional and dimensionless normal coordinate (Fig. 1) - z; Z dimensional and dimensionless axial coordinate Greek letters side of right triangular duct in X direction (dimensionless) - side of right triangular duct in Y direction (dimensionless) - density of fluid - (X, Y, Z) dimensionless temperature distribution - * apex angle of triangular duct (Fig. 1) - ** apex angle of triangular duct (Fig. 1)     

Einfluß der Seitenwände auf das Einsetzen der Konvektion in einer horizontalen Flüssigkeitsschicht

Zusammenfassung Der Einfluß der Seitenwände auf das Einsetzen von Naturkonvektion in einem beheizten Fluid wird untersucht, wobei die Wärme sowohl über die Berandung als auch durch innere Wärmequellen dem Medium zugeführt wird. Betrachtet wird eine Fluidschicht, die inx-Richtung unendlich ausgedehnt und durch vier Seiten begrenzt ist.Eine zweidimensionale Approximation dieses dreidimensionalen Problems wird mit Lösungen verglichen, die sich unter Benutzung einer allgemeinen dreidimensionalen Darstellung eines toroidalen und poloidalen Vektorfeldes für den Fall von freien bzw. festen horizontalen und festen vertikalen Rändern ergeben.Für Höhen- zu BreitenverhältnisseA, die zwischen 0.001 und 100 liegen, unterscheiden sich die kritischen Rayleighzahlen bei der zweidimensionalen Approximation nur wenig von den Ergebnissen des allgemeinen dreidimensionalen Falles. Zusätzlich werden für die zweidimensionale Approximation unter Berücksichtigung innerer Wärmequellen die kritischen externen Rayleighzahlen bestimmt, wobei die Rayleighzahlen für interne Wärmequellen in einem Bereich von 0R₁<10⁶ liegen. Soweit ein Vergleich der hier gewonnenen Ergebnisse mit denen aus Experimenten und früheren theoretischen Untersuchungen möglich ist, wurde gute Übereinstimmung festgestellt.

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Summary The influence of sidewalls on the onset of convection in a differentially and internally heated fluid layer is investigated. A two-dimensional approximation to the three-dimensional problem is compared to solutions using a general three-dimensional representation of the toroidal and poloidal vector fields in the case of free and rigid upper and lower boundaries with rigid sidewalls. For all aspect ratios, 0.001A100, it is found that the critical Rayleigh numbers in the two-dimensional approximation do not differ substantially from the general three-dimensional results. Additionally, in the two-dimensional approximation, the problem is investigated with both internal heating and an imposed temperature difference between the upper and lower boundaries. In this case all boundaries are rigid and the critical Rayleigh numbers, R_E, are obtained for aspect ratios between 0.001 and 100 and internal Rayleigh numbers in the range 0R_I<10⁶. Where comparison of the results with experimental and previous theoretical results is possible, good agreement is found.