Cost Functions

ADflow works in three dimensions (\(x, y, z\)) and application of angles such as alpha and beta change the inflow angle, not the mesh itself. For example, a high lift airfoil at \(\alpha=20^{\circ}\) has the same mesh as that same airfoil at \(\alpha=0^{\circ}\). Similarly, a moth T-foil at a \(\beta=5^{\circ}\) leeway angle has the same mesh at \(\beta=0^{\circ}\).

The relevant subroutines for the evaluation of cost functions are located in surfaceIntegrations.F90 and zipperIntegrations.F90. The primary cost functions that are computed in the Fortran code are defined in the adflowCostFunctions dictionary in pyADflow.py. The following list describes each primary cost function and specifies its units.

lift

Surface stresses (this includes shear/viscous, normal/pressure, and momentum/unsteady stresses) integrated in the direction of the liftIndex but perpendicular to the streamwise direction (i.e., dot product in this direction). To clarify, the angle of attack alpha does affect the direction of the force projection. Units: Newton

drag

Surface stresses (this includes shear/viscous, normal/pressure, and momentum/unsteady stresses) integrated in the streamwise direction (i.e., dot product in this direction). Units: Newton

cl

Lift coefficient computed as \(\frac{L}{qA}\) where \(A\) is areaRef declared in AeroProblem() and \(q\) is dynamic pressure. Units: None

cd

Drag coefficient computed as \(\frac{D}{qA}\) where \(A\) is areaRef declared in AeroProblem() and \(q\) is dynamic pressure. Units: None

clp

Component of the lift coefficient cl from pressure / normal stresses. Units: None

clv

Component of the lift coefficient cl from viscous / shear stresses. Units: None

clm

Momentum component of the lift coefficient cl that stems from time rate of change of velocity (unsteady simulation). Units: None

cdp

Component of the drag coefficient cd from pressure / normal stresses. Units: None

cdv

Component of the drag coefficient cd from viscous / shear stresses. Units: None

cdm

Momentum component of the drag coefficient cd that stems from time rate of change of velocity (unsteady simulation). Units: None

fx

Force from surface stresses (this includes shear/viscous, normal/pressure, and momentum/unsteady stresses) integrated in the global \(x\) direction. This direction does not change based on angle of attack and side slip angle. Units: Newton

fy

Like fx but in the global \(y\) direction Units: Newton

fz

Like fx but in the global \(z\) direction Units: Newton

cfx

Force coefficient in the global \(x\) direction computed as \(\frac{F_x}{qA}\) where \(A\) is areaRef. Units: None

cfxp

Components of the fx coefficient cfx from pressure / normal stresses. Units: None

cfxv

Components of the fx coefficient cfx from viscous / shear stresses. Units: None

cfxm

Momentum component of the fx coefficient cfx that stems from time rate of change of velocity (unsteady simulation). Units: None

cfy

Like cfx but in the global \(y\) direction.

cfyp

Like cfxp but in the global \(y\) direction.

cfyv

Like cfxv but in the global \(y\) direction.

cfym

Like cfxm but in the global \(y\) direction.

cfz

Like cfx but in the global \(z\) direction.

cfzp

Like cfxp but in the global \(z\) direction.

cfzv

Like cfxv but in the global \(z\) direction.

cfzm

Like cfxm but in the global \(z\) direction.

mx

Moment about \(x\) axis, computed at the location (xRef, yRef, zRef) as defined in AeroProblem. Units: Newton * meter

my

Moment about \(y\) axis, computed at the location (xRef, yRef, zRef) as defined in AeroProblem. Units: Newton * meter

mz

Moment about \(z\) axis, computed at the location (xRef, yRef, zRef) as defined in AeroProblem. Units: Newton * meter

cmx

Moment coefficient about \(x\) axis computed as \(\frac{M_x}{qAc_{ref}}\) where \(A\) is areaRef and \(c_{ref}\) is a reference length (AeroProblem.chordRef). Units: None

cmy

Moment coefficient about \(y\) axis computed as \(\frac{M_y}{qAc_{ref}}\) where \(A\) is areaRef and \(c_{ref}\) is a reference length (AeroProblem.chordRef). Units: None

cmz

Moment coefficient about \(z\) axis computed as \(\frac{M_z}{qAc_{ref}}\) where \(A\) is areaRef and \(c_{ref}\) is a reference length (AeroProblem.chordRef). Units: None

cm0

NOTE: Time spectral stability derivatives are broken as of 2023. Moment coefficient about the \(z\) axis at the zero value of the time spectral motion perturbation.

cmzalpha

NOTE: Time spectral stability derivatives are broken as of 2023. Derivative of the moment coefficient about the \(z\) axis with respect to angle of attack.

cmzalphadot

NOTE: Time spectral stability derivatives are broken as of 2023. Derivative of the moment coefficient about the \(z\) axis with respect to the time derivative of angle of attack.

cl0

NOTE: Time spectral stability derivatives are broken as of 2023. Lift coefficient at the zero value of the time spectral motion perturbation.

clalpha

NOTE: Time spectral stability derivatives are broken as of 2023. Derivative of the lift coefficient with respect to angle of attack.

clalphadot

NOTE: Time spectral stability derivatives are broken as of 2023. Derivative of the lift coefficient with respect to the time derivative of angle of attack.

cfy0

NOTE: Time spectral stability derivatives are broken as of 2023. Force coefficient in the \(y\) axis direction at the zero value of the time spectral motion perturbation.

cfyalpha

NOTE: Time spectral stability derivatives are broken as of 2023. Derivative of the force coefficient in the \(y\) axis direction with respect to angle of attack.

cfyalphadot

NOTE: Time spectral stability derivatives are broken as of 2023. Derivative of the force coefficient in the \(y\) axis direction with respect to the time derivative of angle of attack.

cd0

NOTE: Time spectral stability derivatives are broken as of 2023. Drag coefficient at the zero value of the time spectral motion perturbation.

cdalpha

NOTE: Time spectral stability derivatives are broken as of 2023. Derivative of the drag coefficient with respect to angle of attack.

cdalphadot

NOTE: Time spectral stability derivatives are broken as of 2023. Derivative of the drag coefficient with respect to the time derivative of angle of attack.

cmzq

NOTE: Time spectral stability derivatives are broken as of 2023. Derivative of the moment coefficient about the \(z\) axis with respect to pitch rate.

cmzqdot

NOTE: Time spectral stability derivatives are broken as of 2023. Derivative of the moment coefficient about the \(z\) axis with respect to the time derivative of pitch rate.

clq

NOTE: Time spectral stability derivatives are broken as of 2023. Derivative of the lift coefficient with respect to pitch rate.

clqdot

NOTE: Time spectral stability derivatives are broken as of 2023. Derivative of the lift coefficient with respect to the time derivative of pitch rate.

cbend

NOTE: Broken as of 2023. Root bending moment coefficient.

sepsensor

The separation values for the given surface is provided by this cost function. See Kenway and Martins [1] for more details.

sepsensorks

The separation sensor value based on the KS aggregation for the given surface is provided by this cost function. We first compute the deviation of the local velocity from the projected freestream on the desired surface. Then, we use a trigonometric function to compute the sensor by providing an allowable flow deviation angle from this projected vector. As a result, the sensor provides values ranging from -1 to a positive number, which depends on sepSensorKsPhi angle selection. Any values that are greater than 0 are out of the allowable flow deviation. Thus, to constraint the separation, we use KS aggregation to find the maximum value in the sensor and can constraint it to be less than or equal to 0. See Abdul-Kaiyoom et al. [2] and Abdul-Kaiyoom [3] for more details.

sepsensorksarea

The area separated based on the KS aggregation approach. This sensor provides the total area of the cells, where sepsensorks is greater than 0 in those cells. This area is computed by considering two heaviside smoothing function with KS aggregation approach. It is recommended to start the initial angle of attack in the separated region to compute this value because of the sensor’s highly nonlinear behaviour. See Abdul-Kaiyoom et al. [2] and Abdul-Kaiyoom [3] for more details.

sepsensoravgx

The separation sensor average in x direction. The sensor times the distance in x direction value is computed here.

sepsensoravgy

The separation sensor average in y direction. The sensor times the distance in y direction value is computed here.

sepsensoravgz

The separation sensor average in z direction. The sensor times the distance in z direction value is computed here.

cavitation

Cavitation sensor, not to be confused with the cavitation number. It is a modified Heaviside function accounting for how much \(-C_p\) exceeds the cavitation number over the given surface family. The computeCavitation flag must be set to True because this introduces additional computations. The most common use case of this cost function involves constraining it over a specific surface family, not all walls. See Liao et al. [4] for more details. Units: None

cpmin

Minimum coefficient of pressure (\(C_p\)) over the given surface family. This function is computed with Kreisselmeier-Steinhauser (KS) function aggregation resulting in a conservative constraint because the cpmin outputted will always be more negative than the true \(C_{p,min}\). The computeCavitation flag must be set to True because this introduces additional computations involving global communications across processors. If False, the returned value is zero. Units: None

mdot

Mass flow rate through the integration surface. Units: kg / s

mavgptot

Mass flow rate averaged total pressure. Units: Pascal

aavgptot

Area averaged total pressure. Units: Pascal

aavgps

Area averaged static pressure. Units: Pascal

mavgttot

Mass flow rate averaged total temperature. Units: Kelvin

mavgps

Mass flow rate averaged static pressure. Units: Pascal

mavgmn

Mass flow rate averaged Mach number. Units: None

area

The area of the integrated surface. Units: meter^2

axismoment

Moments about the axis given by momentAxis defined in the AeroProblem(). Units: Newton * meter

flowpower

Added power by actuator region to the flow volume computed from volume integration. Units: Watt

forcexpressure

Pressure component of force in the global \(x\) direction. The pressure is calcualted as the difference of the pressure on the wall and the free stream pressure. For closed surfaces, the free stream pressure delta will cancel and the resulting force calculation is correct. However, integrating the force on open surfaces will not result in the free stream static pressure contribution cancelling out. Dimensional cfxp Units: Newton

forceypressure

Like forcexpressure but in the global \(y\) direction. Units: Newton

forcezpressure

Like forcexpressure but in the global \(z\) direction. Units: Newton

forcexviscous

Viscous component of force in x direction. Dimensional version of cfxv. Units: Newton

forceyviscous

Viscous component of force in y direction. Units: Newton

forcezviscous

Viscous component of force in z direction. Units: Newton

forcexmomentum

Momentum component of force in x direction. Units: Newton

forceymomentum

Momentum component of force in y direction. Units: Newton

forcezmomentum

Momentum component of force in z direction. Units: Newton

dragpressure

Pressure drag. Units: Newton

dragviscous

Viscous drag. Units: Newton

dragmomentum

Momentum drag from time rate of change of velocity (unsteady simulations). Dimensional cdm Units: Newton

liftpressure

Pressure component of lift. Units: Newton

liftviscous

Viscous component of lift. Units: Newton

liftmomentum

Momentum lift (due to changing momentum of flow in unsteady simulation). Units: Newton

mavgvx

Mass-averaged \(x\) velocity (i.e., \(\Sigma \dot{m}_x / \Sigma \dot{m}\)). Units: m / s

mavgvy

Mass-averaged \(y\) velocity. Units: m / s

mavgvz

Mass-averaged \(z\) velocity. Units: m / s

mavgvi

A derived velocity average.

cperror2

The square of the difference between computed cp and target cp* for inverse design. See setTargetCp() call.

cofxx

The following center of force cost functions first list the force component (e.g., Fx) and then the coordinate (e.g., x coordinate). These cost functions look at the sum of all forces. Center of x force, x coordinate. Units: Meter

cofxy

Center of x force, y coordinate. See cofxx description for more details. Units: Meter

cofxz

Center of x force, z coordinate. See cofxx description for more details. Units: Meter

cofyx

Center of y force, x coordinate. See cofxx description for more details. Units: Meter

cofyy

Center of y force, y coordinate. See cofxx description for more details. Units: Meter

cofyz

Center of y force, z coordinate. See cofxx description for more details. Units: Meter

cofzx

Center of z force, x coordinate. See cofxx description for more details. Units: Meter

cofzy

Center of z force, y coordinate. See cofxx description for more details. Units: Meter

cofzz

Center of z force, z coordinate. See cofxx description for more details. Units: Meter

colx

Center of lift force, x coordinate. Units: Meter

coly

Center of lift force, y coordinate. Units: Meter

colz

Center of lift force, z coordinate. Units: Meter

References

[1]

Gaetan K. W. Kenway and Joaquim R. R. A. Martins. Buffet-onset constraint formulation for aerodynamic shape optimization. AIAA Journal, 55(6):1930–1947, June 2017. doi:10.2514/1.J055172.

[2] (1,2)

Mohamed Arshath Saja Abdul-Kaiyoom, Anil Yildirim, Alasdair C. Gray, and Joaquim R. R. A. Martins. Airfoil separation constraint formulation for aerodynamic shape optimization. Journal of Aircraft, 2024. (In press). doi:10.2514/1.C037365.

[3] (1,2)

Mohamed Arshath Saja Abdul-Kaiyoom. Numerical Methods for Coupled Aeropropulsive Design Optimization. PhD thesis, University of Michigan, Ann Arbor, MI, 2025. doi:.

[4]

Yingqian Liao, Joaquim R. R. A. Martins, and Yin Lu Young. 3-D high-fidelity hydrostructural optimization of cavitation-free composite lifting surfaces. Composite Structures, 268:113937, July 2021. doi:10.1016/j.compstruct.2021.113937.

[5]

James G. Coder, Thomas H. Pulliam, David Hue, Gaetan K. W. Kenway, and Anthony J. Sclafani. Contributions to the 6th AIAA CFD Drag Prediction Workshop using structured grid methods. In AIAA SciTech Forum. American Institute of Aeronautics and Astronautics, January 2017. doi:10.2514/6.2017-0960.

[6]

Nitin Garg, Gaetan K. W. Kenway, Joaquim R. R. A. Martins, and Yin Lu Young. High-fidelity multipoint hydrostructural optimization of a 3-D hydrofoil. Journal of Fluids and Structures, 71:15–39, May 2017. doi:10.1016/j.jfluidstructs.2017.02.001.

[7]

Mads H. Aa. Madsen, Frederik Zahle, Niels N. Sørensen, and Joaquim R. R. A. Martins. Multipoint high-fidelity CFD-based aerodynamic shape optimization of a 10 MW wind turbine. Wind Energy Science, 4:163–192, April 2019. doi:10.5194/wes-4-163-2019.