26-th ecmi modelling week final report · 2013. 1. 31. · pradel-osny maisonneuve department of...

26-th ECMI Modelling Week

Final Report

19.08.2012—25.08.2012Dresden, Germany

Group 7

Optimal heating of an indoorswimming pool

Tsvetan BazlyankovFaculty of Mathematics and Informatics,

Sofia University, Sofia, BulgariaThomas Briffard

Department of Mathematics, University Pierre and Marie Curie (UPMC),Paris, France

Grzegorz KrzyzanowskiFaculty of Fundamental Problems of Technology, Wroclaw University of

Technology, Wroclaw, Poland.Pradel-Osny Maisonneuve

Department of Material Sciences and Chemical Engineering,University Carlos III of Madrid, Madrid, Spain.

Christian NeßlerDepartment of Mathematics, University of Kaiserslautern, Germany.

Instructor: Monika WolfmayrInstitute of Computational Mathematics, Johannes Kepler University,

Linz, Austria

2

Abstract

In this report, we discuss the optimal heating of an indoor swimmingpool which is located under a glass dome. We formulate our model problemas an optimal control problem, where we have to minimize a certain costfunctional with respect to some given parabolic PDE (partial differentialequation) constraints. Then we seek for the existence of a solution andtherefore formulate our problem in appropriate function spaces. Finally, wepresent some numerical algorithms for solving our optimal control problemas well as some numerical results simulating the heating process of the indoorswimming pool.

2 Optimal heating of an indoor swimming pool

(a) (b)

Figure 7.1: Two pictures of indoor swimming pools located under a glass-dome (from http://www.budapesttimes.hu/2011/09/12/waters-edge

and http://www.aqua-world.hu/hu/aquaworld/elmenyelemek)

7.1 Introduction

In this work, we present the modeling of the heating process of an indoorswimming pool which is located under a glass dome as well as a modelfor its optimal heating. This is based on the technique of optimal controlproblems, where our goal is to minimize a certain cost function. First of all,we will work on the mathematical modeling of the physical phenomena, thenwe will present some analytical results. After that, we will introduce somenumerical methods for solving the optimal control problem. Finally, we willpresent some numerical results for the simulation of our heating process.

7.2 The mathematical modeling

In order to solve our problem we considered the following steps:

• We set the mathematical model of the physical phenomena.

• We obtain some analytical results of the problem.

• We present numerical methods that allow us to meet the best approx-imation of the analytical results.

• We present some numerical results.

First of all, we present the industrial problem that we want to consider.We have an indoor swimming pool which is covered with a glass dome as itis roughly illustrated in Figure 7.1a and 7.1b. For simplicity, we assumethat it acts as an isolator. There are also placed heaters somewhere at theglass dome. Our task is to reach a desired temperature distribution at theend of a given time interval with the least possible cost. As it can be seen

26th ECMI modelling week 3

Figure 7.2: Simplified model

in Figure 7.1a and 7.1b, our problem is quietly a 3D modeling problem,but, under the assumption of symmetrical properties of the geometry andthe uniform distribution of the temperature in the water, we can treat thecircular surface as a line which reduces our problem to a 2D model. Doing so,we can perform simpler calculations and programming to simulate the taskat hand. In fact, the solution of this problem will be obtained by an optimalcontrol algorithm, and thus, we will have to derive the weak formulationof the model problem which is reduced to a 2D problem as illustrated inFigure 7.2.

7.3 Theoretical results

In this part, we introduce some theoretical results from functional analysis,then give a mathematical formulation of the optimal heating problem andfinally present some existence and uniqueness results for optimal controlproblems.

Theoretical background

For formulating our problem in an appropriate way, we have to introduceappropriate spaces and concepts from functional analysis. Further motiva-tion and more detailed explanation on the structure and choice of spaces canbe found for example in [4]. Let Ω ⊂ Rn be a bounded Lipschitz domainwith boundary Γ and let Q := Ω × (0, T ) denote the space-time cylinderwith mantle boundary Σ := Γ× (0, T ), where T > 0 is the time period.


Definition 1 We denote by W 1,02 (Q) the normed space

W 1,02 (Q) = y ∈ L2(Q) : Diy ∈ L2(Q) ∀i = 1, . . . N

with norm

||y||W 1,0

2 (Q)=

(∫ T

0

∫Ω

(|y(x, t)|2 + |∇y(x, t)|2)dxdt

)1/2

.

Definition 2 Let X, ||·||X be a real Banach space. We denote by Lp(a, b;X), 1 ≤p < ∞, the linear space of all (equivalence classes of) measurable vector-valued functions y : [a, b]→ X having the property that∫ b

a||y(t)||pXdt <∞.

The space Lp(a, b;X) is a Banach space with respect to the norm

||y||Lp(a,b;X) :=

(∫ b

a||y(t)||pXdt

)1/p

.

Definition 3 We denote by W (0, T ) the space of all y ∈ L2(0, T ;V ) havinga (distributional) derivative y′ ∈ L2(0, T ;V ∗), equipped with the norm

||y||W (0,T ) =

(∫ T

0

(||y(t)||2V + ||y′(t)||2V ∗

)dt

)1/2

.

The space W (0, T ) = y ∈ L2(0, T ;V ) : y′ ∈ L2(0, T ;V ∗) is a Hilbert spacewith the scalar product

(u,w)W (0,T ) =

∫ T

0(u(t), w(t))V dt+

∫ T

0(u′(t), w′(t))V ∗dt.

Definition 4

The chain of dense and continuous embeddings

V ⊂ H ⊂ V ∗

is called a Gelfand triplet.

Formulation of the model problem

In this section, we want to formulate the optimal control problem. We wantto obtain a control, which corresponds to the heating of the heat sources,such that our state reaches a desired temperature distribution in a given


time. This can be formulated in the setting of Optimal control of partialdifferential equations, see [4]. Our model problem reads as follows:

min(y,u)

J(y, u) =1

2

∫Ω

(y(x, T )− yΩ(x))2dx+λ

2

∫ T

0

∫ΓR

u(x, t)2ds(x)dt (7.1)

such that

yt = ∆y in Q := Ω× (0, T ) (7.2a)

y(x, 0) = y0(x) in Ω (7.2b)

y = y1 on Σ2 := Γ2 × (0, T ) (7.2c)

∂y

∂n= 0 on Σ1 := Γ1 × (0, T ) (7.2d)

∂y

∂n+ αy = βu on ΣR := ΓR × (0, T ) (7.2e)

with ΓR := Γ3 ∪ Γ4 and

u ∈ Uad = v ∈ L2(ΣR) : ua(x, t) ≤ u(x, t) ≤ ub(x, t) a.e. on ΣR (7.3)

and where

• yΩ(x) is the desired temperature distribution in Ω,

• y(x, t) describes the state,

• u(x, t) is the control,

• λ ≥ 0 is the cost coefficient or control parameter,

• y1 is the given water temperature and

• α and β are constants.

In our setting, the control u plays the role of the radiator heating on apart of the boundary and y1 is, in our case, the constant water temperature.The parameters α and β describe the heat transfer. They are modellingparameters and should be chosen carefully. The Neumann boundary data onΓ1 means that there is no flux in normal direction, i.e. the dome is insulatedat this part. The Dirichlet data on Γ2 means that the temperature at thispart of the boundary is constant, i.e. the water has in our time intervalalways the temperature y1.

To get a better understanding of the above minimization problem, oneshould think about it in the following way: Minimizing (7.1) means thatwe want to find a control u such that desired temperature distribution yΩ

is achieved but with a “not too large” control u. The heating process isdescribed by the PDE constraints (7.2). Furthermore, the control u cannot


be chosen arbitrarily. Therefore, u has to fulfil so called box constraints(7.3).

From the physical point of view, we should choose α = β because thenthe last boundary condition states that the increase in temperature is at theboundary proportional to the difference between the temperature outsideand at the boundary. Nevertheless, it makes sense to decouple α and β formathematical reasons, see [4].

Existence and uniqueness results

So far, we have formulated an appropriate mathematical formulation of ourmodel, namely as an optimal control problem, but two important questionsare not answered yet:

1. Under which conditions does the PDE (7.2) have a solution and is thesolution unique?

2. Does the minimization problem (7.1)-(7.3) have a solution?

For answering the first question, Theorem 3.9. in [4, p.140] tells us thatour parabolic initial-boundary value problem has a unique weak solution inW 1,0

2 (Q) and furthermore, the solution depends continuously on the data,i.e. there exists a constant cp > 0 independent of u and y0 such that:

maxt∈[0,T ]

||y(·, t)||L2(Ω) + ||y||W 1,0

2 (Q)≤ cp

(||u||L2(ΣR) + ||y0||L2(Ω)

)(7.4)

for all u ∈ L2(ΣR) and y0 ∈ L2(Ω). For simplicity, we assume here thatwe have homogeneous Dirichlet data, hence y1 = 0. Therefore, from (7.4)follows that the PDE (7.2) is well-posed. As it turns out, the space W 1,0

2

is for several reasons not appropriate and therefore, we want to work inW (0, T ), see [4]. Fortunately, it turns out that this space does not causeany problems (see [4, p.149/150]).

Theorem 1 Let y ∈W 1,02 (Q) be a weak solution of the PDE (7.2). Then y

belongs - possibly after a modification on a set of zero measure - to W (0, T ).Furthermore, the following estimate holds:

||y||W (0,T ) ≤ cw(||u||L2(ΣR) + ||y0||L2(Ω))

)(7.5)

with some constant cw > 0 independent of u and y0.

This theorem provides us with the information that the PDE (7.2), whichhas to be solved, is well-posed in W (0, T ). For answering the second question- the question of existence of a solution of the minimization problem (7.1)with respect to the PDE constraints (7.2) - we have to consider the followingassumptions:


• Ω ⊂ Rn is a bounded Lipschitz domain with boundary Γ,

• λ ≥ 0 is a fixed constant,

• yΩ ∈ L2(Q),

• α, β ∈ L∞(ΣR),

• ua, ub ∈ L2(ΣR) with ua(x, t) ≤ ub(x, t) for almost every (x, t) ∈ ΣR.

Under these assumptions and due to Theorem 1, we can formulate the re-duced optimization problem corresponding to (7.1)-(7.3) and we finally getthe following result, see [4, p.154]:

Theorem 2 Under the assumptions above, the optimization problem (7.1)-(7.3) has at least one optimal control u ∈ Uad. If λ > 0, then u is uniquelydetermined.

7.4 Numerical modeling

Here, we deal with the practical implementation of the optimization prob-lem. First, we start with our PDE constraints:

yt = ∆y in Q := Ω× (0, T )

y(x, 0) = y0(x) in Ω

y = y1 on Σ2 := Γ2 × (0, T )

∂y

∂n= 0 on Σ1 := Γ1 × (0, T )

∂y

∂n+ αy = βu on ΣR := ΓR × (0, T )

(7.6)

We have the following minimization problem:

min(y,u)

J(y, u) =1

2

∫Ω

(y(x, T )− yΩ(x))2dx+ λ

∫ T

0

∫ΓR

u(x, t)2ds(x)dt (7.7)

subject to (7.6), where

u ∈ Uad = v ∈ L2(ΣR) : ua(x, t) ≤ u(x, t) ≤ ub(x, t) a.e. on ΣR. (7.8)

Projected gradient method

As we already stated, the problem (7.6) is well posed and has a (unique)solution yu (see [2] for more informations, in particular when the control is


applied on the whole domain). Thus, we can formally eliminate the stateequation (7.6) and the problem becomes the following one:

minuJ(u) =

1

2

∫Ω

(yu(x, T )− yΩ(x))2dx+λ

2

∫ T

0

∫ΓR

u(x, t)2ds(x)dt (7.9)

such that u ∈ Uad = v ∈ L2(ΣR) : ua(x, t) ≤ u(x, t) ≤ ub(x, t) a.e. on ΣR.Now, u is the only variable of our optimization (reduced optimization

problem). To solve this problem, we can use the gradient method for in-stance, but for this, we need to calculate the gradient of J . This can bedone thanks to the adjoint problem:

−pt = ∆p in Q := Ω× (0, T )

p(x, T ) = y(x, T )− yΩ(x) in Ω

p = 0 on Σ2 := Γ2 × (0, T )

∂p

∂n= 0 on Σ1 := Γ1 × (0, T )

∂y

∂n+ αp = 0 on ΣR := ΓR × (0, T )

(7.10)

The adjoint problem is very interesting in the sense that we are able now tocalculate the gradient of J , indeed it can be shown that the gradient of J isthen given by (see [1] for instance):

∇J(u(x, t)) = β χΓRp(x, T − t) + λu(x, t), (7.11)

where χΓRis the characteristic function on ΓR, i.e. if x belongs to ΓR, then

χΓR(x) = 1, else χΓR

(x) = 0. Hence, we can apply the projected gradientmethod to solve the optimization problem, where the projection onto theset of admissible controls is given by

P[ua,ub](u) = maxua,minub, u. (7.12)

The problem (7.6) has a unique solution yh(u) at the discrete level, whichcan be found by using the finite element method for instance, see [4]. Thesame remark applies to the adjoint problem (7.10).

Finally, we have the following algorithm (written in pseudo-code):

Algorithm:

1. Choose u0 satisfying ua ≤ u0 ≤ ub, put k=0 and uk = u0.

2. While (... some stopping criteria)

3. Solve the heat equation (7.6) and the adjoint problem (7.10), to obtainyk and pk.


4. Evaluate dk = −∇J(uk) = −(β χΓR

pk + λuk), the descent direction.

5. Set uk+1 = P[ua,ub](uk + γ dk).

6. End While

Here, the step length γ is constant, but for best performance, it is recom-mended to apply a suitable line search strategy such as Armijo or Wolfeconditions.

Reduced problem

The idea is to develop u in fixed ansatz functions

u(x, t) =m∑i=1

uiei(x, t)

and hence to compute only the solution of

yit = ∆yi

∂yi∂n

+ αyi = βei

yi(0) = 0,

(7.13)

for each ansatz function and to solve

yt = ∆y

∂y

∂n+ αy = 0

y(0) = y0.

(7.14)

For simplicity, let us assume that we have homogeneous Dirichlet boundarydata, hence y1 = 0. Following [4], we finally obtain the following finite-dimensional reduced quadratic optimization problem:

min1

2~uT (C + λD)~u+ ~aT~u, (7.15)

~ua ≤ ~u ≤ ~ub (7.16)

with

~a = (ai), ai = (y(T )− yΩ, yi(T ))L2(Ω), (7.17)

C = (cij), cij = (yi(T ), yj(T ))L2(Ω), (7.18)

D = (dij), dij = (ei, ej)L2(ΣR). (7.19)


Solving this finite-dimensional quadratic problem can be done by anystandard code, for example quadprog in Matlab.

This idea has the big advantage that we have to solve only a quadraticfinite-dimensional optimization problem and can compute the parameters apriori. Furthermore, we can add afterwards more ansatz functions to getperhaps more precise results. The disadvantage of this method is that usu-ally we do not have detailed information about the solution. Therefore, wedo not know which ansatz functions are most suitable. Anyway, one shouldonly use easy ansatz functions and not too many such that computationtime for solving the PDEs stays a priori low and the dimension of the finite-dimensional problem does not get too big because this would also increasethe computation time.

It might be useful to compute quickly a solution of a low dimensionalreduced problem and then to use this as an initial estimate for the projectedgradient method. This could reduce the overall computation time as onehopes to be already close to the minimum having solved the reduced prob-lem. Due to lack of time, we unfortunately were not able to implement oneof the numerical optimization algorithms described above.

Finally, we want to show that different sizes of the heater have a hugeinfluence on the overall temperature inside the glass dome. We consideredthis problem only in our numerical experiments. Using a finite elementmethod (FEM) for solving the heat equation, we first have to generate amesh on our given domain as you can see in Figure 7.3. This is done hereby the open source code distmesh for Matlab. Then, we solve the heatequation by a code based on the book [3]. In Figure 7.4, you can see inthe left column the heating process for a relatively small heater and in theright column the heating process for a larger heater, respectively. The largerheater here has double the size of the smaller heater. In both simulations,we used the same parameters and the same control function for the heaters(constant in time). We chose a constant water temperature of 30oC, initialair temperature of 10oC and a constant heating of 40oC. One can observethat the size of the heater has a huge influence on the overall temperaturein the glass dome as expected.

7.5 Conclusion and future work

In this project, we modelled the optimal heating of an indoor swimmingpool which is located under a glass dome. Therefore, we used variationalprinciples and described the theoretical set-up as well as presented someexistence and uniqueness results for our problem. Finally, we described twocommon methods how to solve this problem numerically and then showedsome numerical results for the solution of the heat equation with respect todifferent sizes of the heater.


Figure 7.3: Mesh of the domain Ω

Due to lack of time, we unfortunately were not able to fully implementone of the mentioned numerical methods for the optimal control problem.For the future, we hope that this problem will be fully implemented andtested for different ways of heating. In particular, it would be interestingto find out which influence the position of the heaters at the boundary hason the heating process and on the minimization function. Furthermore, fora more realistic description, we recommend to include dynamical effects inthe mathematical model. Convection, for example, plays an important rolein the dynamics as it can be used to describe the turbulences in the heatedair and hence should be considered for modelling the heat distribution inthe dome.


Figure 7.4: Evolution in time of the solution of the heat equation withtwo different sizes of the heater, small on the left column, big on the rightcolumn.

Bibliography

[1] Herzog, R and Kunisch, K. Algorithms for PDE-Constrained Opti-mization. MSC (2000) 49-M05, 49-M37, 76-D55, 90-C06, 2010.

[2] Menaldi, JL and Tarzia, DA. A distributed parabolic control withmixed boundary conditions. Asymptotic Analysis 52 (2007) 227241, IOSPress.

[3] Stoffel, A. Finite Elemente und Warmeleitung: Eine Einfuhrung.VCH, 1992. ISBN 9783527282364.

[4] Troltzsch, F. Optimal control of partial differential equations. Amer-ican Math. Soc., Providence, RI, 2010. ISBN 978-0-8218-4904-0.

13

26-th ecmi modelling week final report · 2013. 1. 31. · pradel-osny maisonneuve department of...

Documents