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The following is a multipoint aerodynamic shape optimization case for the NACA0012 airfoil at low speed. We optimize the weighted drag coefficient considering three different flight conditions, i.e., CL=0.3, 0.7, and 0.5 (nominal). The weight of these three conditions are 0.25, 0.25, and 0.5, respectively.

Case: Airfoil aerodynamic optimization 
Geometry: NACA0012
Objective function: Weighted drag coefficient (CD)
Lift coefficient (CL): 0.3, 0.7, and 0.5 (nominal)
Weights: 0.25, 0.25, 0.5 (nominal)
Design variables: 20 free-form deformation (FFD) points moving in the y direction, three angle of attacks
Constraints: Symmetry, volume, thickness, and lift constraints (total number: 34)
Mach number: 0.02941 (10 m/s)
Reynolds number: 0.6667 million
Mesh cells: ~4,000
Solver: DASimpleFoam

Fig. 1. Mesh and FFD points for the NACA0012 airfoil

The runScript.py is based on the one used in the NACA0012 low speed case with the following modifications:

  • In the global parameters, we define the number of multipoint configurations nMultiPoints = 3, and set far field boundary conditions and flight conditios for all the three configurations. MPWeights, U0, alpha0, and CL_target are the weights, far field velocity, angle of attack, and target lift coefficient for the three configurations. Note that we set the nominal conditions CL_target = 0.5 as the last element in the list such that the intermdiate flow fields will be saved for this condition, which facilitates the post-processing using Paraview.

    # global parameters
    nMultiPoints = 3
    MPWeights = [0.25, 0.25, 0.5]
    U0 = [10.0, 10.0, 10.0]
    # we use the first U0 as reference velocity to normalize CD and CL
    URef = U0[0]  
    CL_target = [0.3, 0.7, 0.5]
    alpha0 = [3.008097, 7.622412, 5.139186]
    p0 = 0.0
    nuTilda0 = 4.5e-5
    A0 = 0.1
  • In daOptions we set "multiPoint": True to activatve the multipoint optimization mode, and set "nMultiPoints": nMultiPoints.

  • In the design variable setup section, we add three angle of attack design variables, each for one flight condition. We also append mp0_, mp1_, and mp2_ to the name of the angle of attack variable. Note: we provide a dummy function (dummyFunc) for alpha variable. This is because we will use a function setMultiPointCondition to change the angle of attack later in runScript.py, so there is no need to provide an alpha function for DVGeo.addGeoDVGlobal here.

    for i in range(nMultiPoints):
        DVGeo.addGeoDVGlobal("mp%d_alpha" % i, alpha0[i], dummyFunc, lower=0.0, upper=10.0, scale=1.0)
        # add alpha for designVar
        daOptions["designVar"]["mp%d_alpha" % i] = {
            "designVarType": "AOA",
            "patch": "inout",
            "flowAxis": "x",
            "normalAxis": "y",
  • In the optFuncs setup section. We provide a setMultiPointCondition function to set flow boundary conditions for the three flight configurations. Here setMultiPointCondition uses the design variable dictionary xDV and the multipoint index as the input, and it implements a general method to change the boundary conditions. For this case, we first extract the alpha value from xDV, compute the far field velocity components, replace the primalBC key in DAOption and update its value in DASolver (DASolver.updateDAOption()).

    def setMultiPointCondition(xDV, index):
        aoa = xDV["mp%d_alpha" % index].real * np.pi / 180.0
        inletU = [float(U0[index] * np.cos(aoa)), float(U0[index] * np.sin(aoa)), 0]
        DASolver.setOption("primalBC", {"U0": {"variable": "U", "patch": "inout", "value": inletU}})
  • Next, we define a function (setMultiPointObjFuncs) to combine the objective function values. This function will be used in optFuncs.calcObjFuncValuesMP.

    def setMultiPointObjFuncs(funcs, funcsMP, index):
        for key in funcs:
            if "fail" in key:
            elif "DVCon" in key:
                funcsMP[key] = funcs[key]
            elif "CD" in key:
                    funcsMP["obj"] += funcs[key] * MPWeights[index]
                except Exception:
                    funcsMP["obj"] = 0.0
                    funcsMP["obj"] += funcs[key] * MPWeights[index]
            elif "CL" in key:
                funcsMP["mp%d_CL" % index] = funcs[key]

    In optFuncs.calcObjFuncValuesMP, we loop over all the flight conditions and call DASolver() to solve the primal and compute the objective dict funcs. Then we call setMultiPointObjFuncs, which takes funcs and multipoint index as input, and outputs the dict funcsMP, where MP means multipoint. funcs contains the objective and constraint function values and a fail flag. Here we create a obj key by combining CD computed from each flight condition. An example is as follows.

    funcs   = {"CD": 0.01, "CL": 0.3, "DVCon_0": 1.0, "fail": False} # flight condition 0
    funcs   = {"CD": 0.03, "CL": 0.7, "DVCon_0": 1.0, "fail": False} # flight condition 1
    funcs   = {"CD": 0.02, "CL": 0.5, "DVCon_0": 1.0, "fail": False} # flight condition 2
    # weights for the three flight conditions are 0.25, 0.25 and 0.5
    funcsMP = {"obj": 0.02, "mp0_CL": 0.3, "mp1_CL": 0.7, "mp2_CL": 0.5, "DVCon_0": 1.0, "fail": False} # funcs for multipoint

    Note that we do not combine CL, instead we keep all the lift coefficients by appending mp0_, mp1_, and mp2_ to the objective function names. This will allow us to set lift constraints for all flight conditions (see optProb.addCon in runScript.py). Also note that the geometry constraints (DVCon) are same for all three flight conditions, therefore there is no need to do weighting or appending any prefix names to DVCon.

  • In addition, we define a function setMultiPointObjFuncsSens to combine the objective function derivatives. This function will be used in optFuncs.calcObjFuncSensMP. In optFuncs.calcObjFuncSensMP, we loop over all the flight conditions and call DASolver.solveAdjoint() to solve the adjoint and compute the objective derivative dict funcsSens. Then setMultiPointObjFuncsSens takes design variable dict xDVs, the mulitpoint objective value dict funcsMP, objective derivative dict funcsSens for flight condition index as input, and outputs the multipoint objective derivative dict funcsSensMP.

    def setMultiPointObjFuncsSens(xDVs, funcsMP, funcsSens, funcsSensMP, index):
        for key in funcsMP:
                keySize = len(funcsMP[key])
            except Exception:
                keySize = 1
            except Exception:
                funcsSensMP[key] = {}
            if "fail" in key:
            elif "DVCon" in key:
                funcsSensMP[key]["mp%d_alpha" % index] = np.zeros((keySize, 1), "d")
                funcsSensMP[key]["shapey"] = funcsSens[key]["shapey"]
            elif "obj" in key:
                funcsSensMP[key]["mp%d_alpha" % index] = funcsSens["CD"]["mp%d_alpha" % index] * MPWeights[index]
                    funcsSensMP[key]["shapey"] += funcsSens["CD"]["shapey"] * MPWeights[index]
                except Exception:
                    funcsSensMP[key]["shapey"] = np.zeros(len(xDVs["shapey"]), "d")
                    funcsSensMP[key]["shapey"] += funcsSens["CD"]["shapey"] * MPWeights[index]
            elif "mp%d_CL" % index in key:
                for alphaI in range(nMultiPoints):
                    if alphaI == index:
                        funcsSensMP[key]["mp%d_alpha" % alphaI] = funcsSens["CL"]["mp%d_alpha" % index]
                        funcsSensMP[key]["mp%d_alpha" % alphaI] = np.zeros((keySize, 1), "d")
                funcsSensMP[key]["shapey"] = funcsSens["CL"]["shapey"]

    The objective derivative dict funcsSens contains the derivatives of all objectives and constraints with respect to all design variables.

    For DVCon, we need to copy the value from funcsSens, while setting all alpha derivatives to zeros. This is because the geometry constraint is independent of the angle of attack.

    For obj, we need to combine the shapey derivatives by the weights, and set the derivative for mp*_alpha individually.

    For each mp*_CL, we need to assign shapey derivative from each flight condition, and assign mp*_alpha that corresponds to this flight condition and zero out the alpha derivatives from other flight conditions. This is because the angle of attack from other flight conditions does not impact CL for the current flight condition.

    As an example, the funcsSens for each condition and the combined funcsSensMP may look like this:

    funcsSens = { 
        "CD": {"shapey": {0.01, 0.01}, "mp0_alpha": 0.001, "mp1_alpha": 0, "mp2_alpha": 0}, 
        "CL": {"shapey": {0.1, 0.1}, "mp0_alpha": 0.01, "mp1_alpha": 0, "mp2_alpha": 0},
        "DVCon_0": {"shapey": {0.1, 0.1}, "mp0_alpha": 0, "mp1_alpha": 0, "mp2_alpha": 0} 
    } # flight condition 0
    funcsSens = { 
        "CD": {"shapey": {0.03, 0.03}, "mp0_alpha": 0, "mp1_alpha": 0.003, "mp2_alpha": 0}, 
        "CL": {"shapey": {0.3, 0.3}, "mp0_alpha": 0, "mp1_alpha": 0.03, "mp2_alpha": 0},
        "DVCon_0": {"shapey": {0.1, 0.1}, "mp0_alpha": 0, "mp1_alpha": 0, "mp2_alpha": 0} 
    } # flight condition 1
    funcsSens = { 
        "CD": {"shapey": {0.02, 0.02}, "mp0_alpha": 0, "mp1_alpha": 0, "mp2_alpha": 0.002}, 
        "CL": {"shapey": {0.2, 0.2}, "mp0_alpha": 0, "mp1_alpha": 0, "mp2_alpha": 0.02},
        "DVCon_0": {"shapey": {0.1, 0.1}, "mp0_alpha": 0, "mp1_alpha": 0, "mp2_alpha": 0} 
    } # flight condition 2
    # weights for the three flight conditions are 0.25, 0.25 and 0.5
    funcsSensMP = { 
        "obj": {"shapey": {0.02, 0.02}, "mp0_alpha": 0.00025, "mp1_alpha": 0.0075, "mp2_alpha": 0.001}, 
        "mp0_CL": {"shapey": {0.1, 0.1}, "mp0_alpha": 0.01, "mp1_alpha": 0, "mp2_alpha": 0},
        "mp1_CL": {"shapey": {0.3, 0.3}, "mp0_alpha": 0, "mp1_alpha": 0.03, "mp2_alpha": 0},
        "mp2_CL": {"shapey": {0.2, 0.2}, "mp0_alpha": 0, "mp1_alpha": 0, "mp2_alpha": 0.02},
        "DVCon_0": {"shapey": {0.1, 0.1}, "mp0_alpha": 0, "mp1_alpha": 0, "mp2_alpha": 0} 
    } # multipoint function derivatives
  • After the above two functions are set, we need to initialize them in optFuncs

    optFuncs.setMultiPointCondition = setMultiPointCondition
    optFuncs.setMultiPointObjFuncs = setMultiPointObjFuncs
    optFuncs.setMultiPointObjFuncsSens = setMultiPointObjFuncsSens
  • Finally, in the optimization task, we need to provide multipoint value and derivative computation functions by setting objFun=optFuncs.calcObjFuncValuesMP and sens=optFuncs.calcObjFuncSensMP. Also, we need to set three lift constraints

    for i in range(nMultiPoints):
        optProb.addCon("mp%d_CL" % i, lower=CL_target[i], upper=CL_target[i], scale=1)

To run this case, first download tutorials and untar it. Then go to tutorials-master/NACA0012_Airfoil/multipoint and run the preProcessing.sh script to generate the mesh:


Then, use the following command to run the optimization with 4 CPU cores:

mpirun -np 4 python runScript.py 2>&1 | tee logOpt.txt

The case ran for 50 steps and took about 50 minutes using Intel 3.0 GHz CPU with 4 cores. According to logOpt.txt and opt_SLSQP.txt, the initial combined drag is 0.021973710 and the optimized drag is 0.017867543 with a drag reduction of 18.7%.

The evolution of pressure and shape during the optimization is as follows.

Fig. 2. Pressure and shape evolution during the optimization process