The paper is devoted to discrete optimization of the viscous heatconducting uid ows. In the problem under consideration we want to create a ow with desired properties using optimal heating or cooling on some boundary sections. Optimal control approach reduces this problem to the constrained minimization. In this case the cost functional describes the objectives, mathematical model is the constraint and temperature boundary value is the control. If the ow is unsteady and the objectives data is discrete we need to use discrete optimization methods for numerical solution of this problem. We propose new numerical algorithm that does not use the rst order necessary optimality conditions and based on the nite dimensional minimization.
? This work was supported by the Russian Science Foundation (project no. 14-11-00079). Copyright c by the paper's authors. Copying permitted for private and academic purposes. In: A. Kononov et al. (eds.): DOOR 2016, Vladivostok, Russia, published at http://ceur-ws.org In the bounded two-dimensional domain on the time interval (0; tmax) we consider the following dimensionless system of partial di erential equations
1 RePr
T = 0 in
; T jt=0 = T0 in ; T = 0 on in; T = on c;
= 0 on 0 [ out
(
In this paper we shall consider the unsteady ow in the open cavity (see Fig. 1). The boundary consists of four parts: inlet section in, outlet section out, solid walls 0 and control section c. For the velocity vector v we prescribe the in ow parabolic pro le g on in, no-slip boundary condition on 0 [ c and "do nothing" boundary condition on the outlet out. The temperature boundary value on the control section c will play the role of control.
In our optimization problem we want to obtain the velocity eld v close to a given "optimal" vector eld vd using temperature boundary control on c. We assume that values of the vector vd are given only at xed times t1; t2; : : : ; tN = tmax. So we have a discrete set of data and need to solve the discrete optimization problem. 2.1
At the beginning we split the time interval (0; tmax) into N parts (tn 1; tn) of length
n = tn tn 1, n = 1; 2; : : : ; N and write the following semidiscrete approximation for
the problem (
; v0 = v0 in vn = g(tn) on in; vn = 0 on 0 [ c; pnn
T n
T n 1 n + vn 1 rT n
1 RePr
; @n Here vn, pn, T n (n = 1; 2; : : : ; N ) are the new unknown functions that depend only on the space variables. Let us note that we use an implicit di erence scheme for time discretization. It ensures the stability of our numerical solutions even for large time intervals.
Multiplying equations in (
T n 1; s) (vn vn 1; w) + ((vn 1
r)vn; w) + R1e (rvn; rw)
Gr + Re2 (T nj; w) = 0 8w 2 W; ( ; div vn) = 0 8 vn = gn on in; vn = 0 on 0 [ c; + (vn 1
1
rT n; s) + RePr (rT n; rs) = 0 8s 2 S;
T n = 0 on in; T n =
n on c:
(pn; div w)
2 X;
Here the function v0 = v0, 0 = 0 are determined from the initial conditions of the
problem (
Appling the nite element method for space disctretization we can solve the problem
(
We can clearly see a large vortex occupying most of the cavity. Most of the uid will never leave the cavity. In this case we have a stagnant zone in which the suspended particles are accumulated. We want to suppress this vortex. Therefore we choose the potential ow with zero vorticity as the desirable vector eld vd. Usually the moving bottom wall is used for ow control (see [8, 9]). In this study we consider the heatconducting uid an can use the temperature boundary control. This control can be more e ective then tangential velocity because the temperature can a ect the entire uid volume by means the buoyancy e ect.
Let d be the subset of (subdomain, surface or curve in the two-dimensional
case). It will play the role of the observation set in our optimization problem for the
system (
vdnj2d 1 Z 2 d + 2
Z c ( n)2d ; depending on the boundary function n and corresponding velocity vector vn. Here = const 0 is the small regularization parameter.
The following method is based on the main idea of [10] where an optimal boundary control problem for the stationary Navier-Stokes equations was solved using the nite dimensional minimization approach.
We will nd the unknown function n in the following form:
M n = X ki i: i=1
M vn = wn + X kivi: i=1 Here f igiM=1 are the given basic functions on c while ki, i = 1; 2; : : : ; M are the unknown coe cients that must be found at each time step. The corresponding velocity vector then can be written as Here wn is the solution for homogeneous boundary conditions in (15) while vi (i = 1; : : : ; M ) are the solutions corresponding n = i in (15) and homogeneous boundary conditions in (13). Then the functional (16) can be written as
J (vn; n) = jvn vdnj2d ( n)2d =
wn + 2
Z c 1 Z 2 d
M vdn + X kivi i=1 !2 d (16) (17) (18) (20)
M X aijki = bj; j = 1; 2; : : : ; M: i=1 M X ki i i=1 !2 d = 1 XM XM aijkikj 2 i=1 j=1
M 1 X bjkj + c: j=1 2 c Here the coe cients aij = (vi; vj) d + ( i; j) c ; bj = (vdn wn; vj) d ; c = kvd n wnk2 d (19) can be easily calculated by known functions i, vi, vdn and wn.
As a result, we have the nite dimensional minimization problem for variables ki, i = 1; 2; : : : ; M . Solution of this problem can be found by solving the following system of linear algebraic equations: Substituting aij and bj in this system and using (18) we obtain the following equalities (vn
vdn; vj ) d + ( n; j ) c = 0; j = 1; 2; : : : ; M: that have a clear meaning. If = 0 then the di erence vn vdn must be orthogonal to all functions vj in subdomain d. Solving the system (20) and substituting ki in (17) we nd the boundary temperature n on the n-th time step. 3.1
Proposed numerical algorithm can be written as follows.
Step 0. Choose M basic functions i on c. Set initial values v0 and T 0. Set n = 1.
Step 1. Assuming that vn 1 and T n 1 are already known, solve M + 1 linearized boundary value problems to nd vi, i = 1; : : : ; M and wn.
Step 2. Calculate aij and bj by (19). Solve linear system (20) to nd ki.
Step 3. Calculate n by (17). Solve linear boundary value problem (
Step 4. If n < N then set n := n + 1 and go to step 1.
Let us note that this algorithm does not use the rst order necessary optimality conditions (see [1, 3, 4, 6, 7]) and it is simpler to implement. It can be e ciently parallelized because M + 1 boundary value problems to nd wn and vi are solved independently. It must be noted that at each time step we nd the control and the state simultaneously. 3.2
Now let us consider some computatiomal results for optimization of viscous heatconducting uid ow in the open cavity (see Fig. 1). In this problem we nd optimal heating or cooling of the control boundary section c to create the velocity v closed to the given vector eld vd in the subdomain d. We want to suppress large vortex (see Fig. 2) in the cavity therefore we choose the potential ow with zero vorticity as the desirable vector eld vd. The observation set d occupies the entire cavity. In these computations Reynolds number Re = 10, Grashof number Gr = 104 and Prandtl number Pr = 7. The dimensionless time interval (0; tmax) was chosen as (0; 20) with N = 100.
The controlled ow at time t = 10 is shown in Fig. 3. It is clearly seen that the vortex is fully suppressed and the main ow covers the whole cavity.
Corresponding temperature values on the bottom boundary of the cavity are shown in Fig. 4. This is a dimensionless temperature deviation from the values given on the inlet boundary section.
Fig. 5 shows obtained temperature controls on the left (solid line) and right (dotted line) boundaries of the cavity. It is easy to notice that temperature on the right part is higher than on the left part.
These gures help us to understand the basic e ect of the temperature boundary control. We see cooling on the left boundary and heating on the right boundary. Therefore the cold uid with a higher density moves downwards along the left wall. The heated uid with lower density goes up along the right wall. Heating and cooling 1.0 0.8 0.6 0.4 0.2 0.2 0.4 0.6 0.8 1.0 change the uid density and thus a ect the uid ow. Let us note that in order to take into account of such e ects we need to consider more sophisticated mathematical model in comparison with the works of other authors devoted to the boundary control problems for Navier-Stokes equations.
The open source software freeFEM++ [11] was used for the discretization and numerical solving boundary value problems by the nite element method. The main goal of the computational experiments was to determine the dependence of the solution accuracy on the choice of the problem parameters. We choose di erent ways to specify the given velocity data vd. For example, the observation domain d can coincide with the whole ow domain or can be some subdomain. The location of the small observation domain with xed size can be also very important. Numerical experiments show that the solution is more accurate if the observation domain d is located closer to the control boundary c. The number of basis functions M also a ects the numerical
bottom 5 -3 -4 -5 -6 0 0.2 0.4
0.6 x 0.8 1 solution. Based on the analysis of computational results we can choose optimal values and develop some recommendations for future applications. 4
In this paper we considered the discrete optimization problem for the unsteady viscous heat-conducting uid ows. This problem consists in creating a ow with desired properties using optimal heating or cooling on some boundary sections. We have proposed new numerical algorithm that does not use the rst order necessary optimality conditions and based on the nite dimensional minimization. Computational experiments have shown the ability to change the ow with small Reynolds numbers by means of the temperature boundary control.
left right 0 0.2 0.8
1 0.4
0.6 y