Optimal Control of a Drying Process with Constraints to Avoid Cracks

Transcription

1 Optimal Control of a Drying Process with Constraints to Avoid Cracks A.Galant, C. Grossmann, M.Scheffler and J.Wensch Technical University Dresden Department of Mathematics Department of Solid Mechanics Contact: Christian.Grossmann@tu-dresden.de 5-th German Polish Conference on Optimization - Methods and Applications

2 Scope of the Talk Models for the Direct Problem Optimal Drying with Crack Prevention Penalty Treatment of Stress Bounds Discretization of the Control Problem Gradient Evaluation via Adjoints Numerical Examples Outlook

3 1. Models for the Direct Problem The model bases upon a heat-and-mass transport model developed in the thesis of Vögel[1989], simplified.

4 Assumptions: The drying process below the boiling-point. Thermodiffusion is neglected. Isotropic material assumed. Heat-and-mass transition on the surface by heat conduction, radiation. Additive overlay of boundary and evaporation enthalpy. Enthalpy by convective transport is ignored inside. Enthalpy for phase change of water at boundary only. No inner heat and humidity sources. Moisture change considered as uniform diffusion process. Heat conduction coefficient independent of moisture content.

5 Ω R n, (n {2, 3}), bounded convex domain. Γ = Ω its piecewise smooth boundary. T > 0 terminal time, fixed. X = X(x, t) moisture content, ϑ = ϑ(x, t) temperature (x, t) Ω (0, T ] Internal stress occurs by local volume changes which depend on changes of the moisture content field X. Local changes of the moisture content are influenced by the temperature field ϑ. Under the made assumptions this leads to a coupled system of nonlinear parabolic differential equations.

6 Parabolic system X t = (D(X, ϑ) X), ϑ t = 1 ρ dt c p (λ(x, ϑ) ϑ) with initial conditions in Ω (0, T ], (1) X(, 0) = X 0, ϑ(, 0) = ϑ 0 on Ω (2) and boundary conditions. In (1), (2) denotes: D coefficient of moisture diffusion λ coefficient of heat conduction ρ dt, c p constant parameters of the considered material, namely the density of bone-dry material, the specific heat coefficient, respectively.

7 Replacing the heat conduction λ by temperature conduction λ ρ dt c p, we obtain the state equations: X t (D(X, ϑ) X) = 0, in Ω (0, T ] ϑ t (λ(x, ϑ) ϑ) = 0, in Ω (0, T ] D(X, ϑ) X ν = g 1 (X, ϑ; v L, Y L, ϑ L ), on Γ (0, T ] λ(x, ϑ) ϑ ν = g 2 (X, ϑ; v L, Y L, ϑ L ), on Γ (0, T ] X(, 0) = X 0, on Ω ϑ(, 0) = ϑ 0, on Ω (3)

8 where g 1, g 2 are derived from the boundary condition. This yields g 1 (X, ϑ; v L, Y L, ϑ L ) = α(v L) c pl ρ dt (Y (X, ϑ) Y L ) g 2 (X, ϑ; v L, Y L, ϑ L ) = α(v L ) ρ dt c p (X, ϑ) (ϑ L ϑ) β(v L) ρ dt c p (X, ϑ) (Y (X, ϑ) Y L)h v (X, ϑ) g 2 (,,, ) = α(v { L) (ϑ L ϑ) h } v(x, ϑ) (Y (X, ϑ) Y L ). ρ dt c p (X, ϑ) c pl

9 The wood drying can be actively influenced by the surrounding air via: Y L - absolute humidity, ϑ L - temperature, v L - convection velocity. Abstract form of the state equation y t + A(y) y = 0, in Ω (0, T ] B(y) y = g(y, u), on Γ (0, T ] y(, 0) = y 0 on Ω (4)

10 2. An Optimal Control Problem Goals to be achieved target state y d, regularization target u d for controls (minimal energy consumption) v L,d 0 m/s, ϕ L,d 60% (rel.humidity), ϑ L,d 20 o C. Restrictions to be considered controls and states are connected via the state equation (4), box constraints for the controls, internal stresses have to be limited to avoid cracks.

11 Notations: U := L (0, T ) 3 - control space U ad := {u U : u a u( ) u b a.e. in (0, T )} - admissible controls, u a, u b R 3, u a < u b given bounds for the control, abbreviation for the state equation F (y, u) = 0, σ = σ(y(x, t)) - internal stresses, σ a, σ b R, σ a < σ b given bounds for stress.

12 Objective J(y, u) := 1 2 y(, T ) y d 2 L 2 (Ω) n + α 2 u u d 2 L 2 (0,T ) 3 Control problem J(y, u) > min! s.t. F (y, u) = 0, u U ad, σ a σ(y) σ b (5)

13 Evaluation of the stress: ε obs := l obs /l 0 - observed strain (dilatation) ε free := l free /l 0 - unrestricted strain (dilatation) ε := ( l free l obs )/(l 0 l free ) ( l free l obs )/l 0 = ε free ε obs - occurring strain Hooke s law σ(x) = E(x) ε(x)= E(x)(ε free (x) ε obs ) Since drying takes place without external forces we have σ(x) dx = 0. Hence, Ω Ω ε obs = E(x)ε free(x) dx E(x) dx Ω

14 Structural assumption due to Welling E(X = 0.08), X 0.08 E(X) = E B (ρ dt )( (X 0.12)), 0.08 < X < 0.18 E(X = 0.18), X 0.18 where E B (ρ dt ) = ρ 2 dt. Assumption proposed by Kollmann α(x = 0.3) α(x), X < 0.3 ε free (X) = 1 α(x = 0.3) 0, otherwise with source mass α(x) = 3 ρ dt X ρ dt ρ 2. dt

15 Finally, the moisture field influences the elasticity tensor E as well as ε free, i.e. E(x, t) = E(X(x, t)), ε free (x, t) = ε free (X(x, t)). Treatment of the state constraints σ a σ(y) σ b by the smoothed exact penalty P (ξ, a, b; s) := η (ξ a) 2 + s 2 + (ξ b) 2 + s 2 η ( ξ a + ξ b ), s > 0 - penalty parameter (limit s 0+), η > 0, fixed.

16 Augmented (smoothed exact penalty) objective T J(y, u; s) := J(y, u) + 0 Ω P (σ(x(x, t)), σ a, σ b ; s) dx dt. Augmented control problem J(y, u; s) > min! s.t. F (y, u) = 0, u U ad. (6)

17 3. Numerical Experiments

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