Adding Modules

1. Click Select Modules in the Model Panel. The Physical Model Selection dialog box opens.
2. Select Optimization under Available Modules list, click Add.
3. Select Flow under Available Modules list, click Add.

4. Click Close to close the Physical Model Selection dialog box.
   

Flow module

1. Select Flow in the Model Panel.
2. Enter 1e-05 for the Convergence Criterion.
3. Select 2^nd Order Upwind for Velocity under Numeric Scheme in the Properties Panel.

Global Expression for Optimization

1. Click Edit Expression icon on the View Toolbar to open the Expression Editor dialog box.
2. Copy and paste the expressions written under Description drop-down list for Global Expressions, see Figure 9.27.
   
Description

# parameters

d2r = pi/180 # degree to radian

s_factor = 0.48 # geometry safety factor

# flow speed (m/s) and properties

v_flow =1

density = 1

visc = 0.0005

# geometry of original concentric annulus shape

r_min = 0.05

r_max = 1

dr = r_max - r_min

dz= 0.01

area = dz*r_min

# define bump shape

#=============

# deformed the original half circle to a shape consist of two ellipses, front and rear, connected at x =0.

# the y coordinate is fixed to r_min at x=0. For both ellipses, the major and minor axis are always either align with y axis or parallel to x axis.

# Center of ellipses can move along y axis

# equation (x)^2/a^2 + (y-(1-b))^2/b^2 = 1, where all the length normalized by r_min

# use length (multiplier of r_min) and tip angle iith y-axis (in degree) at x = 0 to define the ellipses shape

fl = optimization.fl

rl = optimization.rl

ftheta_new = optimization.fang

rtheta_new = optimization.rang

# validate input: theta>arctan(2/l)

ftheta_min = atan(2.1/fl)/d2r

rtheta_min = atan(2.1/rl)/d2r

ct1 = ftheta_new-ftheta_min

ct2 = rtheta_new-rtheta_min

ftheta = (ftheta_new>ftheta_min)? ftheta_new : ftheta_min

rtheta = (rtheta_new>rtheta_min)? rtheta_new : rtheta_min

# slope dx/dy = -cot (angle)

fs = -cot(ftheta*d2r)

rs = -cot(rtheta*d2r)

# solve for a, and b for both front and rear ellipses

fb = (fl+fs)/(fl+2*fs)

rb = (rl+rs)/(rl+2*rs)

fa = sqrt(fb^2*fl^2/(2*fb-1))

ra = sqrt(rb^2*rl^2/(2*rb-1))

# define front (upstream) ellipse for y<=0,

flx = fa*r_min

fly = fb*r_min

# define rear (downstream) ellipse for y>=0

rlx = ra*r_min

rly = rb*r_min

# move coordiates of mesh

#=================

# define a radial vector start from origin, and go through current point

# the vector is defined as x=(xp/rp)r, y=(yp/rp)r

xp = x

yp = y

rp = sqrt(xp^2+yp^2)

ff = xp/rp

gg=y/rp

# Radius ratio in original mesh

r_ratio = (r_max-rp)/dr

# find intersection points of the radial vector with both front/back ellipses, still solving equation normalized by r_min

# intersection with front

FA = fb^2*ff^2+fa^2*gg^2

FB = 2*gg*fa^2*(fb-1)

FC = fa^2*(1-2*fb)

# r> 0, so only use positive r results, also convert to real size

fr_min = ((-FB + sqrt(FB^2-4*FA*FC))/(2*FA))*r_min

# intersection with rear

RA = rb^2*ff^2+ra^2*gg^2

RB = 2*gg*ra^2*(rb-1)

RC = ra^2*(1-2*rb)

# r> 0, so only use positive r results, also convert to real size

rr_min = ((-RB + sqrt(RB^2-4*RA*RC))/(2*RA))*r_min

# new r

rn = (xp>0)? r_max - r_ratio*(r_max - fr_min) : r_max - r_ratio*(r_max-rr_min)

xnew = x*rn/rp

ynew = y*rn/rp

znew = z

# output section

# =========

# output total force

drag = -(flow.px@cylinder_min + flow.tx@cylinder_min)

plot.drag = drag

#plot.drag: Drag Force [N]

# drag coefficient

cd = 2*drag/(density*v_flow^2*area)

plot.cd = cd

#plot.cd: CD []

# Reynolds Number

Re = density*v_flow*2*r_min/visc

plot.Re = Re

#plot.Re: Reynolds Number []

3. Click OK to close the Expression Editor dialog box.

Optimization module

1. Select Optimization in the Model Panel.
2. Enter No. of Design of Experiments as 10 in the Model Tab of Properties Panel.
3. Enter No. of Optimization Runs as 50.

4. Enter cd for the Expression under Objective. Make sure the Objective is Minimize.

5. Enter Number of Values as 50 under Objective.

6. Select Create New Constraint under Create New Constraint, if new Constraints must be defined.

7. Enter ct1 for Expression under Constraint 1.

8. Select Greater Equal (≥) in the 2^nd dropdown for Processing method under Constraint 1.

9. Enter ct2 for Expression under Constraint 2.

10. Select Greater Equal (≥) in the 2^nd dropdown for Processing method under Constraint 2.

11. Select Add a Variable under Create New Variable, if new Variables must be defined.
12. Enter fl for the Variable Name under Variable 1.
13. Enter 0.5 for Minimum and 8 for Maximum.
14. Enter fang for the Variable Name under Variable 2.
15. Enter 10 for Minimum and 90 for Maximum.
16. Enter rl for the Variable Name under Variable 3.
17. Enter 0.5 for Minimum and 8 for Maximum.
18. Enter rang for the Variable Name under Variable 4.
19. Enter 10 for Minimum and 90 for Maximum.

Inlet

1. Select cylinder from the Boundaries list under Volumes in the Geometric Entities Panel.
2. Select Specified Pressure Inlet from the Flow drop-down list in the Model Tab of Properties Panel.
3. Enter -v_flow m/s for X under Velocity under the Flow drop-down list.

Symmetry

1. Select dir_max, dir_min, radial_max and radial_min from the Boundaries list under Volumes in the Geometric Entities Panel.

2. Select Symmetry from the Flow drop-down list in the Model Tab of Properties Panel.

Fluid properties

1. Select Volumes in the Geometric Entities Panel.
2. Select Model Tab in the Properties Panel.

3. Enter density kg/m³ for Value under Density for the Common list in Model Tab of the Properties Panel.

4. Enter visc Pa-s for Value under Viscosity in the Flow list .
5. Enter -v_flow m/s for X Velocity under Initial Condition in the Flow list.

Volume Remeshing

1. Select annulus Volumes in the Geometric Entities Panel.
2. Select Volume Remesh under the drop-down list in the Model Tab of Properties Panel.

3. Select Expression under Method drop-down list.

4. Enter the xnew, ynew, znew for X,Y,Z values respectively.