## Viscous Inverse Design¶

This notebook demonstrates the use of gradients from viiflow for fully viscous inverse design. It defines a target pressure distribution from one airfoil and, coming from another airfoil, tries to find the shape necessary to arrive at this target pressure. It uses virtual displacements, which do not necessitate the recalculation of the panel operator. Instead, it uses the same model used for the effect of boundary layer thickness onto the flow for modification of the airfoil shape.

The heart of this notebook is a Gauss-Newton iteration which solves for these virtual displacements. Instead of trying to solve the pressure distribution exactly, the iteration sovles a least-squares problem that joins the pressure difference with regularizing terms. Fully viscous inverse design is not a straightforward problem. There are several ways an optimizer may cheat, for example

• The velocity is defined by the inviscid solution of the airfoil shape plus boundary layer thickness. An optimizer can therefore choose to reduce the thickness of the airfoil if for some reason a thick boundary layer leads to the target velocity distribution.
• Kinks in the desired velocity are, in the case below, due to laminar-turbulent transition. However, an optimizer can choose to model this kink by an actual kink in the airfoil.

To alleviate this, the pressure error is appended by a regularizing term that penalizes non-smooth displacements - simply by adding $\frac{\mathrm{d}^2}{\mathrm{d}^2 s} \delta_{virtual}(s)$ at every point along the airfoil surface coordinate $s$ to the Least-Squares problem.

The parameters chosen to increrase/decrease the penalties were chosen ad-hoc by trial and error. In addition, the nodes very close to the stagnation point are not modified.

In addition, the residual $r$ of the viiflow solver itself is added to the Least-Squares problem and scaled such that at convergence its error is sufficiently low. Every iteration then performs for dispalcements $y$ and the viiflow variables $x$ $$y^{k+1} = y^k - \lambda {\Delta y}^k\\ x^{k+1} = x^k - \lambda {\Delta x}^k\\ {\Delta y}^k, {\Delta x}^k = \min_{\Delta y,\Delta x} \| F(y^k,x^k) - \frac{\partial F}{\partial y}(y^k,x^k) \Delta y - \frac{\partial F}{\partial x}(y^k,x^k) \Delta x\|^2\\ \|F(y,x)\|^2 = \gamma_{cp}^2\|ue(y)-ue_{target}\|^2 + \gamma_y^2\| \frac{\mathrm{d}^2}{\mathrm{d}^2 s} y \|^2 + \gamma_r^2 \|r(y,x)\|^2$$ This may seem like a large problem, but the effort for solving the overdetermined least-squares problem grows largely with the degrees of freedom, not the amount of equations.

Below, this procedure is used to morph the S805 airfoil into the S825 airfoil. Even with the regularizing terms, little dips that enforce the laminar-turbulent transition can still be seen when zooming in.

While this solves for an airfoil shape of a specified pressure distribution, it is probably not a very smart idea to use this for actual design. A better idea is to use first an inviscid inverse design method, e.g. conformal mapping [1, 2], and remove the discrepancies using a fully viscid iteration. The benefit of this Gauss-Newton approach is how straightforward additional constraints can be included, e.g. only fit the suction side from .1c onwards or fit multiple target distributions at multiple angles of attack.

In [1]:
import viiflow as vf
import viiflowtools.vf_tools as vft
import viiflowtools.vf_plots as vfp
import numpy as np
import matplotlib
import matplotlib.pyplot as plt

In [21]:
# Analysis Settings
RE = 1e6
ncrit =9
Mach = 0.0
alpha = 2.0

N = 200

# Solve target for our target cp (or more precisely edge velocity)
s = vf.setup(Re=RE,Ma=Mach,Ncrit=ncrit,Alpha=alpha)

# Internal iterations
s.Itermax = 100

# Set-up and initialize based on inviscid panel solution
[p,bl,x] = vf.init([TARGET],s)
res = None

# Solve aerodynamic problem of target airfoil
vf.iter(x,bl,p,s,None,None)
XT0 = p.foils[0].X[0,:].copy()
UT = p.gamma_viscid[0:p.foils[0].N].copy()

# Set-up and initialize based on inviscid panel solution
[p,bl,x0] = vf.init([BASE],s)
res = None

# Solve aerodynamic problem of current airfoil and save for later plotting
XC0 = p.foils[0].X[0,:].copy()
UC = p.gamma_viscid[0:p.foils[0].N].copy()

# To interpolate from one grid to the next, suction and pressure side must have unique grid points
# That is why below a grid is created where the pressure side is appended with *-1 at the nose
XT = XT0.copy()
XC = XC0.copy()
XT[np.argmin(XT0)+1::] = 2*XT0[np.argmin(XT0)]-XT0[np.argmin(XT0)+1::]
XC[np.argmin(XC0)+1::] = 2*XC0[np.argmin(XC0)]-XC0[np.argmin(XC0)+1::]

# Interpolate target pressure onto current airfoil grid
UT = np.interp(-XC.flatten(),-XT.flatten(),np.asarray(UT[:,0]).flatten())

Iteration 11, |res| 0.000090, lam 1.000000
Iteration 12, |res| 0.000012, lam 1.000000

In [22]:
# Weighting factors for Gauss-Newton
facx = 500 # Penalty for smooth dioscplacement
fac_err = 5 #Weighting of cp error w.r.t. above penalties
fac_res = 1e4

NAERO = x.shape[0]
NVD = len(XC)

# Set-up and initialize based on inviscid panel solution
[p,bl,x0] = vf.init([BASE],s)
res = None

# Solve aerodynamic problem to convergence
[x,_,_,_,_] = vf.iter(x0,bl,p,s,None,None)
fprev = np.inf
# Find ST and do not change near there
II = np.logical_and(np.fabs(XT-XT[bl[0].sti])>0.001,p.foils[0].X[0,:].ravel()>np.amin(p.foils[0].X[0,:].ravel()))
II[0]=False
II[NVD-1]=False
iter = 0
lam = 1.0
y = np.zeros(NVD)
while True:
iter+=1

# Solve Aerodynamic problem
s.Itermax = 0
s.Silent = True

# Residual
RESy = fac_err*(p.gamma_viscid[0:p.foils[0].N].A1-UT)

# Penalty for thick boundary layer
#REGdelta = bl[0].bl_fl.nodes.delta*facx

# Penalty for smooth displacement
difforder = 2
REGdelta = np.diff(y,difforder)*facx

# Gauss-Newton step from all terms
F = np.r_[RESy,REGdelta,res*fac_res]
fcurr = np.sqrt(F.T@F)

y0 = y
fprev = fcurr
# Find ST and do not change near there
II = np.logical_and(np.fabs(XT-XT[bl[0].sti])>0.001,p.foils[0].X[0,:].ravel()>np.amin(p.foils[0].X[0,:].ravel()))
II[0]=False
II[NVD-1]=False

dF = np.c_[dFdy[:,II],dFdx]
dX = -np.linalg.lstsq(dF,F,rcond=None)[0]
dy = dX[0:np.sum(II)]
dx = dX[np.sum(II)::]
lam = 1

# Print
resaero = np.sqrt(np.matmul(res,res.T))

for k in range(len(dy)):
lam = np.fmin(lam,0.003/abs(dy[k])) # Do not move virtual displacement more than 1mm
for k in range(len(x)):
lam = np.fmin(lam,.2/(abs(dx[k]/x[k])))

print("iter %u res p:%f resaero: %f dvd:%f lam:%f"%(iter, np.sqrt(np.matmul(F,F.T)), \
resaero,np.sqrt(np.matmul(dy,dy.T)),lam))

if np.sqrt(np.matmul(dy,dy.T))<1e-4 and resaero<1e-4:
print("Converged")
break

if iter>100:
print("Not Converged (iteration)")
break

j =0
for k in np.argwhere(II):
y[k] += lam*dy[j]
j+=1
x += lam*dx

Iteration 11, |res| 0.000001, lam 1.000000
iter 1 res p:11.577981 resaero: 0.000001 dvd:0.203364 lam:0.082208
iter 2 res p:12.242767 resaero: 0.000604 dvd:0.186259 lam:0.087117
iter 3 res p:13.764105 resaero: 0.000972 dvd:0.172358 lam:0.089896
iter 4 res p:15.959540 resaero: 0.001324 dvd:0.156115 lam:0.062213
iter 5 res p:71.926046 resaero: 0.007145 dvd:0.130364 lam:0.077548
iter 6 res p:68.975690 resaero: 0.006854 dvd:0.107978 lam:0.079449
iter 7 res p:65.172692 resaero: 0.006477 dvd:0.096770 lam:0.123907
iter 8 res p:62.255992 resaero: 0.006190 dvd:0.082984 lam:0.202047
iter 9 res p:61.330019 resaero: 0.006106 dvd:0.086396 lam:0.113290
iter 10 res p:76.589723 resaero: 0.007641 dvd:0.076817 lam:0.172184
iter 11 res p:70.039507 resaero: 0.006988 dvd:0.068556 lam:0.129219
iter 12 res p:141.566505 resaero: 0.014151 dvd:0.069234 lam:0.197901
iter 13 res p:121.521239 resaero: 0.012147 dvd:0.062641 lam:0.172071
iter 14 res p:109.742756 resaero: 0.010969 dvd:0.052044 lam:0.207561
iter 15 res p:98.896272 resaero: 0.009885 dvd:0.050741 lam:0.182481
iter 16 res p:90.686274 resaero: 0.009065 dvd:0.039147 lam:0.109423
iter 17 res p:84.303067 resaero: 0.008426 dvd:0.033613 lam:0.425446
iter 18 res p:53.892269 resaero: 0.005384 dvd:0.021228 lam:0.296543
iter 19 res p:50.403255 resaero: 0.005035 dvd:0.021737 lam:0.229477
iter 20 res p:54.547165 resaero: 0.005450 dvd:0.015607 lam:0.271277
iter 21 res p:98.782778 resaero: 0.009876 dvd:0.017351 lam:0.195581
iter 22 res p:89.550980 resaero: 0.008953 dvd:0.012224 lam:0.350165
iter 23 res p:62.245737 resaero: 0.006221 dvd:0.007629 lam:0.252056
iter 24 res p:56.161477 resaero: 0.005612 dvd:0.010964 lam:0.199732
iter 25 res p:50.147343 resaero: 0.005010 dvd:0.004864 lam:0.285473
iter 26 res p:44.736075 resaero: 0.004469 dvd:0.010437 lam:0.181381
iter 27 res p:42.782146 resaero: 0.004273 dvd:0.004076 lam:0.354992
iter 28 res p:37.753943 resaero: 0.003770 dvd:0.010514 lam:0.171915
iter 29 res p:38.570121 resaero: 0.003852 dvd:0.004110 lam:0.423972
iter 30 res p:33.292955 resaero: 0.003323 dvd:0.010861 lam:0.169187
iter 31 res p:36.440627 resaero: 0.003638 dvd:0.004262 lam:0.429909
iter 32 res p:31.877562 resaero: 0.003181 dvd:0.010389 lam:0.175141
iter 33 res p:36.104248 resaero: 0.003605 dvd:0.004207 lam:0.380945
iter 34 res p:33.241461 resaero: 0.003318 dvd:0.009930 lam:0.174998
iter 35 res p:36.186311 resaero: 0.003613 dvd:0.004385 lam:0.386185
iter 36 res p:32.431413 resaero: 0.003237 dvd:0.010219 lam:0.172703
iter 37 res p:35.533034 resaero: 0.003547 dvd:0.004379 lam:0.402085
iter 38 res p:31.368106 resaero: 0.003130 dvd:0.010337 lam:0.173141
iter 39 res p:35.090364 resaero: 0.003503 dvd:0.004342 lam:0.397248
iter 40 res p:31.401306 resaero: 0.003133 dvd:0.010295 lam:0.173363
iter 41 res p:35.125890 resaero: 0.003507 dvd:0.004981 lam:0.569446
iter 42 res p:28.086721 resaero: 0.002801 dvd:0.010232 lam:0.210095
iter 43 res p:39.346780 resaero: 0.003930 dvd:0.004195 lam:0.446515
iter 44 res p:38.096186 resaero: 0.003804 dvd:0.005808 lam:0.249143
iter 45 res p:37.087915 resaero: 0.003703 dvd:0.005365 lam:0.137129
iter 46 res p:33.165589 resaero: 0.003311 dvd:0.008467 lam:0.219475
iter 47 res p:27.696130 resaero: 0.002763 dvd:0.004633 lam:0.404528
iter 48 res p:18.239240 resaero: 0.001814 dvd:0.003554 lam:0.195756
iter 49 res p:15.510942 resaero: 0.001540 dvd:0.004151 lam:0.766077
iter 50 res p:9.532581 resaero: 0.000935 dvd:0.003084 lam:0.569825
iter 51 res p:10.508204 resaero: 0.001034 dvd:0.007085 lam:0.244038
iter 52 res p:13.966531 resaero: 0.001385 dvd:0.001612 lam:0.583469
iter 53 res p:15.817808 resaero: 0.001571 dvd:0.005633 lam:0.249901
iter 54 res p:22.617723 resaero: 0.002255 dvd:0.003520 lam:0.553135
iter 55 res p:23.389172 resaero: 0.002332 dvd:0.002448 lam:0.282489
iter 56 res p:20.213085 resaero: 0.002014 dvd:0.001932 lam:0.681607
iter 57 res p:13.974007 resaero: 0.001387 dvd:0.005423 lam:0.421108
iter 58 res p:20.861572 resaero: 0.002080 dvd:0.003567 lam:0.626103
iter 59 res p:19.241706 resaero: 0.001917 dvd:0.003650 lam:0.479908
iter 60 res p:16.196001 resaero: 0.001612 dvd:0.001652 lam:0.601457
iter 61 res p:10.780398 resaero: 0.001067 dvd:0.004025 lam:0.708433
iter 62 res p:7.547511 resaero: 0.000740 dvd:0.000650 lam:0.727598
iter 63 res p:10.283271 resaero: 0.001017 dvd:0.000367 lam:1.000000
iter 64 res p:53.346335 resaero: 0.005333 dvd:0.003852 lam:0.700823
iter 65 res p:10.819402 resaero: 0.001073 dvd:0.000736 lam:1.000000
iter 66 res p:7.308343 resaero: 0.000717 dvd:0.000306 lam:1.000000
iter 67 res p:1.560742 resaero: 0.000070 dvd:0.000102 lam:1.000000
iter 68 res p:1.396445 resaero: 0.000004 dvd:0.000012 lam:1.000000
Converged

In [23]:
%matplotlib inline
%config InlineBackend.figure_format = 'svg'
matplotlib.rcParams['figure.figsize'] = [11, 5.5]
fig,ax = plt.subplots(1,1)
ax.plot(p.foils[0].X[0,:],np.power(UC,2)-1,'-k')
ax.plot(p.foils[0].X[0,:],np.power(p.gamma_viscid[0:p.foils[0].N].A1,2)-1,'-',color=(0.6,0.6,0.6))
ax.plot(p.foils[0].X[0,:],np.power(UT,2)-1,'2k')
ax.legend(['Initial Pressure','Found Pressure','Target Pressure'])
xlim = ax.get_xlim()

fig,ax = plt.subplots(1,1)
lines = None
ax.plot(TARGET[0,:],TARGET[1,:],'2k')
lines = vfp.plot_geometry(ax,p,bl,lines)
ax.legend(['Target Airfoil','Initial Geometry','Found Geometry'])
ax.set_xlim(xlim)

Out[23]:
(-0.050014908717028954, 1.0500007099389062)

[1] Selig, Michael S., and Mark D. Maughmer. Generalized multipoint inverse airfoil design. AIAA journal 30.11 (1992): 2618-2625.

[2] Drela, Mark. XFOIL: An analysis and design system for low Reynolds number airfoils. Low Reynolds number aerodynamics. Springer, Berlin, Heidelberg, 1989. 1-12.