---
title: Force Partitioning Method (FPM)
url: https://www.emergentmind.com/topics/force-partitioning-method-fpm
type: topic
---

# Force Partitioning Method (FPM)

Searching arXiv for recent and foundational papers on the Force Partitioning Method and closely related formulations.
arxiv_search(query="Force Partitioning Method aerodynamic load partitioning Q criterion influence potential Menon Mittal", max_results=10, sort_by="relevance")
arxiv_search(query="2205.04400 force partitioning lift low aspect ratio wings", max_results=5, sort_by="relevance")
arxiv_search(query="2304.12544 force and moment partitioning method swept wings", max_results=5, sort_by="relevance")
Force Partitioning Method (FPM) is a fluid-mechanical framework for quantifying how distinct flow features contribute to pressure-induced aerodynamic loads on an immersed body. In the formulation used for incompressible, vortex-dominated flows, an auxiliary harmonic potential encodes the body geometry and the target load direction, and a projection of the Navier–Stokes equations yields additive decompositions of force by physical mechanism, spatial region, and identified flow structure. In the low-aspect-ratio wing study that made the method central to lift attribution, FPM was used to separate spanwise-oriented and cross-span-oriented vorticity, and also rotation-dominated and strain-dominated regions, thereby providing a quantitative account of how different vortex structures generate positive or negative lift [2205.04400]. Closely related formulations extend the same logic to moments, to PIV-based experiments, and to modal decompositions of the $Q$-field in unsteady loading and aeroacoustic problems [2304.12544; 2501.05331].

## 1. Definition and analytical scope

FPM is designed to partition the pressure force into additive components attributable to distinct physical mechanisms, spatial regions, and identified flow structures. In the low-aspect-ratio wing formulation, the method partitions the pressure-induced aerodynamic load into contributions from unsteady acceleration, convection, viscous diffusion, and far-field acceleration; into contributions from regions over the wing, in the near wake, and in the far wake; and into contributions from spanwise-oriented versus cross-span-oriented vortices, as well as from rotation-dominated versus strain-dominated regions [2205.04400].

The method is rigorous in the sense used in the source literature: the partition is obtained from the governing equations rather than from heuristic vortex labeling. Because the relevant force contribution is expressed as a linear volume integral weighted by an influence potential, the contribution from any masked feature or spatial subdomain is additive, and the sum over mutually exclusive partitions recovers the total vortex-induced force. This additivity is the basis for statements such as
$C_L^\omega \approx C_L^\text{span} + C_L^\text{cross} + C_L^\text{outside}$,
where “outside” denotes the residual from outside the analysis subdomain [2205.04400].

In comparative terms, the method differs from classical surface-pressure versus viscous decompositions, which do not isolate distinct volumetric features such as leading-edge vortex cores and their surrounding strain fields, and from impulse-theory or Lamb-vector-based approaches, which provide integral or global measures but can be less straightforward to partition by local features in complex three-dimensional flows. The source literature characterizes the present FPM, rooted in Quartappelle–Napolitano, Wu, and Howe and extended by Chang, Zhang, and Menon–Mittal, as enabling exact volumetric attribution of pressure-induced loads to identified structures via $-\rho\int 2Q\,\phi_i\,dV$, separation of rotation and strain contributions, and consistent additivity across subdomains and features in fully three-dimensional vortex-dominated flows [2205.04400].

## 2. Mathematical foundation

The method is formulated for incompressible flow governed by the Navier–Stokes equations and continuity,
\[
\rho\,\frac{\partial \boldsymbol{u}}{\partial t} + \rho\,(\boldsymbol{u}\cdot\nabla)\boldsymbol{u} = -\nabla p + \mu\,\nabla^2 \boldsymbol{u},
\qquad
\nabla\cdot\boldsymbol{u} = 0.
\]
FPM introduces an auxiliary scalar potential field $\phi_i$ that depends on the body geometry and on the force direction $i$. For lift on a wing, $i=2$. The potential solves a Laplace problem in the fluid domain $V_f$,
\[
\nabla^2 \phi_i = 0 \quad \text{in } V_f,
\]
with mixed Neumann boundary conditions
\[
\boldsymbol{\hat{n}}\cdot\nabla \phi_i =
\begin{cases}
n_i & \text{on } B,\\
0 & \text{on } \Sigma,
\end{cases}
\]
where $B$ is the body surface and $\Sigma$ is the outer boundary [2205.04400].

Projecting the momentum equation onto $\nabla\phi_i$ and integrating over $V_f$ yields a representation of the surface pressure force
\[
F_i = \int_B p\,n_i\,dS.
\]
In the wing formulation, the resulting terms correspond respectively to unsteady acceleration on the body, convective or vortex-induced effects, viscous diffusion, and far-field flow acceleration. For the stationary wings considered there, the unsteady body term is zero and the far-field term is negligible in the large domain; viscous diffusion contributes modestly, about $10\%$ of the total lift, while the convective term dominates the pressure-induced lift [2205.04400].

The connection to vortical structure follows from the pressure Poisson equation,
\[
\nabla^2 p = -\rho\,\frac{\partial u_i}{\partial x_j}\,\frac{\partial u_j}{\partial x_i}
= -\rho\,(S:S - \Omega:\Omega)
= 2\rho\,Q,
\]
where $S$ and $\Omega$ are the strain-rate and rotation-rate parts of $\nabla \boldsymbol{u}$ and
\[
Q = \frac{1}{2}\big(\|\Omega\|^2 - \|S\|^2\big).
\]
Substitution yields the central FPM relation for the vortex-induced component,
\[
F_i^\omega
\equiv
\rho\int_{V_f}\nabla\cdot\big(\boldsymbol{u}\cdot\nabla\boldsymbol{u}\big)\,\phi_i\,dV
=
-\rho\int_{V_f} 2\,Q\,\phi_i\,dV.
\]
Rotation-dominated regions with $Q>0$ and strain-dominated regions with $Q<0$ therefore induce pressure forces of opposite sign. This relation underlies the direct separation of vortex-core and strain-field contributions in later applications [2205.04400].

A closely related pressure-loading partition used in modal force partitioning writes
\[
F^{(i)}_{\text{pressure}} = F^{(i)}_{B} + F^{(i)}_{O} + F^{(i)}_{Q} + F^{(i)}_{\mu},
\]
with
\[
F^{(i)}_{Q} = -\int_{V_f} 2\rho\,Q\,\phi^{(i)}\,dV,
\]
where $F_B$ is the acceleration-reaction term, $F_O$ is the outer-boundary term, and $F_\mu$ is the pressure loading induced by viscous diffusion of momentum. For the moderate-to-high Reynolds number flows considered there, $F_B \approx 0$ for bodies with negligible rigid-body acceleration, $F_O \approx 0$ for steady inflow and distant $\Sigma$, and $F_\mu$ is small compared with $F_Q$ [2501.05331].

## 3. Partitioning workflow and feature classification

The implementation used for low-aspect-ratio wings proceeds by first solving the auxiliary Laplace problem for $\phi_2$ on the CFD mesh. Because $\phi_2$ depends only on geometry and domain, it can be precomputed once per configuration. Velocity gradients are then used to form $S$, $\Omega$, and the scalar field
\[
Q = \tfrac{1}{2}\big(\|\Omega\|^2 - \|S\|^2\big).
\]
To distinguish orientations of vorticity, the study defines the spanwise component of the unit vorticity vector,
\[
\eta_z = \frac{\omega_z}{\sqrt{\omega_x^2 + \omega_y^2 + \omega_z^2}},
\]
and partitions the flow into spanwise-oriented vorticity, defined by $|\eta_z| > \cos(\pi/4)\approx 0.707$, and cross-span-oriented vorticity, defined as the complementary set. Each orientation class is then split into rotation-dominated and strain-dominated regions through the sign of $Q$ [2205.04400].

Spatial attribution is performed by defining cuboidal integration subdomains. In the wing application, each cuboid spans the analysis subdomain in $y$ and $z$ and has length $1C$ in $x$; the near-body cuboid is centered around mid-chord, and successive cuboids tile the wake downstream in one-chord increments. Partitioned vortex-induced lift is then computed as
\[
F_2^\omega(\mathcal{R}) = -\rho\int_{\mathcal{R}} 2\,Q\,\phi_2\,dV
\]
for each region $\mathcal{R}$ and for each feature mask. The force is converted to a lift coefficient through
\[
C_L = \frac{F_2}{\tfrac{1}{2}\rho U_\infty^2 A},
\qquad
C_L^\omega = \frac{F_2^\omega}{\tfrac{1}{2}\rho U_\infty^2 A}.
\]
The study emphasizes that additivity is guaranteed by linearity of the integral and by the use of mutually exclusive masks [2205.04400].

Moment-oriented extensions use the same architecture with a different influence potential. For spanwise pitching moment about a specified pitch axis, the potential $\phi$ satisfies
\[
\nabla^2 \phi = 0,
\qquad
\frac{\partial \phi}{\partial \boldsymbol{n}}=
\begin{cases}
\big[(\boldsymbol{x}-\boldsymbol{x}_p)\times \boldsymbol{n}\big]\cdot \boldsymbol{e}_z
& \text{on airfoil},\\
0 & \text{on outer boundary},
\end{cases}
\]
and the vorticity-induced moment is
\[
M_v = -2\rho \int_V Q \,\phi \,\mathrm{d}V.
\]
In this formulation the moment density is $-2Q\phi$, which permits direct mapping of positive and negative moment contributions from LEV, TV, and related structures [2304.12544].

Experimental implementations follow the same logic. In a two-dimensional pitching-wing study, the physics-based force and moment partitioning method computes the instantaneous aerodynamic moment directly from measured velocity fields without reconstructing pressure, using the same $\phi$–$Q$ formulation, and supplements it with an added-mass torque obtained from body kinematics and $\phi$ [2301.13373].

## 4. Low-aspect-ratio wings: lift decomposition and physical mechanism

In the principal wing study, the geometry is a rectangular flat-plate wing of thickness $15\%$ of chord $C$, with aspect ratios $AR=2$ and $AR=3$, angles of attack $\alpha=15^\circ$ and $25^\circ$, and incompressible flow at chord-based Reynolds number $Re=1000$. The flow solver is a sharp-interface immersed boundary method with second-order spatial accuracy, Adams–Bashforth second-order time stepping, and multigrid solution of the pressure Poisson equation. The computational domain is $18C \times 20C \times 20C$ with a $192\times128\times256$ grid, and the FPM analysis subdomain is $6C \times 3C \times 3C$, centered about the wing and extending $4.55C$ downstream; that subdomain captures about $95\%$ of the vortex-induced lift [2205.04400].

Within these conditions, $C_L^\omega$ accounts for about $90\%$ of the total lift; viscous diffusion contributes the remainder, about $10\%$, while viscous shear contributes a very small negative amount. The principal comparative result is that in three of four cases cross-span-oriented vorticity contributes more to lift than spanwise-oriented vorticity, contrary to the common expectation that leading-edge, spanwise vortices dominate lift [2205.04400].

| Configuration | Spanwise contribution | Cross-span contribution |
|---|---:|---:|
| $AR=2$, $\alpha=15^\circ$ | $\sim 26\%$ of total, $\sim 29\%$ of vortex-induced lift | $\sim 60\%$ of total, $\sim 66\%$ of vortex-induced lift |
| $AR=2$, $\alpha=25^\circ$ | $\sim 29\%$ of total, $\sim 39\%$ of vortex-induced lift | $\sim 47\%$ of total, $\sim 63\%$ of vortex-induced lift |
| $AR=3$, $\alpha=25^\circ$ | $\sim 34\%$ of total, $\sim 44\%$ of vortex-induced lift | $\sim 43\%$ of total, $\sim 55\%$ of vortex-induced lift |
| $AR=3$, $\alpha=15^\circ$ | $\sim 56\%$ of total, $\sim 69\%$ of vortex-induced lift | $\sim 25\%$ of total, $\sim 31\%$ of vortex-induced lift |

The spatial partition clarifies why the balance changes between over-wing and wake regions. Over the wing, lift is largest and is dominated by spanwise-oriented vortex cores, namely the LEV and leading-edge shear layer. Cross-span vorticity in this region, primarily tip vortices, yields net negative lift due to the strain-dominated regions, consistent with classical downwash-induced lift reduction. In the wake, the total vortex-induced lift shows a non-monotonic trend with downstream distance and becomes negative near $x \approx 2C$ downstream of mid-chord. The magnitude of this negative wake lift is about $27\%$ for $AR=2$, $\alpha=15^\circ$, $33\%$ for $AR=2$, $\alpha=25^\circ$, $6\%$ for $AR=3$, $\alpha=15^\circ$, and $17\%$ for $AR=3$, $\alpha=25^\circ$, each expressed as a fraction of the peak over-wing lift [2205.04400].

FPM identifies the negative wake lift as a spanwise-vorticity effect. The positive contribution from spanwise cores drops rapidly due to tilting, while the negative contribution from spanwise strain regions increases because of growth of vortex-induced strain in the wake. By contrast, cross-span vorticity in the wake produces net positive lift: cross-span cores, identified with streamwise vortices and horseshoe legs, strengthen via tilting and stretching, while their strain regions grow less. The balance between rotation and strain in the two orientation classes, governed by vortex tilting and stretching, explains both why cross-span vorticity contributes more to lift in most cases and why spanwise vorticity in the near wake can produce net negative lift [2205.04400].

## 5. Extensions to moments, oscillations, and modal structure

The force-to-moment extension is usually termed the Force and Moment Partitioning Method (FMPM). In pitching swept wings, FMPM was used to correlate three-dimensional vortex dynamics with the resultant unsteady aerodynamic moment by solving an influence potential for pitching moment about the mid-chord, mid-span axis and integrating $-2\rho Q\phi$ over a multi-layer stereoscopic PIV volume. That study reports that LEV generally contributes positive moment, TV increasingly contributes negative moment with higher sweep, and TEV contributes negligibly because it lies near the trailing edge where $\phi\approx 0$. The same partitioning was used to interpret non-monotonic sweep effects on energy transfer and stability boundaries, including an optimal sweep angle of $\Lambda=10^\circ$ for power extraction and destabilization, and added-mass moment peaks of about $2\%$, $3\%$, and $5\%$ of the total peak moment for $\Lambda=0^\circ$, $10^\circ$, and $20^\circ$, respectively [2304.12544].

A related two-dimensional experimental study of a sinusoidally pitching NACA 0012 wing in quiescent water used the same $\phi$–$Q$ formulation to estimate vortex-induced aerodynamic moment from mid-span PIV. In that case, the method partitions torque into a vorticity-induced contribution,
\[
\tau = -2\rho\int_V Q\,\phi\,dV,
\]
an added-mass term obtained from prescribed kinematics,
\[
\tau_a = -\,\rho\int_S \mathbf{n}\cdot\Bigl(\frac{d\mathbf{U}}{dt}\Bigr)\,\phi\,dS,
\]
and leading- and trailing-edge subdomain contributions. The study reports that the FMPM-estimated moment follows the trend of direct measurements but underestimates the magnitude by about $50\%$, with likely causes identified as near-wall PIV errors and three-dimensional effects missed by 2C2D measurements [2301.13373].

FPM has also been applied to flow-induced vibration of cylinders. In that setting, the method partitions net hydrodynamic force into kinematic, vorticity-induced, viscous, potential, and outer-boundary components, and the vorticity-induced term is further split between a near-body shear-layer region and the wake. The principal physical conclusion is that wake vortex shedding is necessary to initiate oscillations, but it is the vorticity associated with the boundary layer over the cylinder that is responsible for the sustenance of flow-induced vibrations. Quantitatively, the cited cases show positive energy transfer from the shear layer and negative energy transfer from the wake at stationary states [2006.11649].

A more recent extension is modal force partitioning. When FPM is applied after decomposing the velocity field, the nonlinear dependence of $Q$ on velocity gradients produces inter-modal interaction terms that are difficult to interpret. The proposed remedy is to decompose the $Q$-field directly and evaluate modal forces as
\[
F^{(i)}_{\tilde{Q}_k} = -2\rho \int \tilde{Q}_k\,\phi^{(i)}\,dV.
\]
Applied to a two-dimensional cylinder, a NACA 0015 airfoil, and a three-dimensional revolving wing, this direct $Q$-field decomposition yielded more interpretable force and aeroacoustic partitions. For the revolving wing, the first six $Q$-modes capture about $91.3\%$ of lift variance, and the three leading modes contribute about $69.4\%$ of total acoustic intensity [2501.05331].

## 6. Metrics, validation, limitations, and terminological ambiguity

A data-driven variant has introduced two FPM-derived metrics for unsteady lift attribution in an impulsively started rotor blade at span-based Reynolds number $25{,}500$: a $Q$-strength metric,
\[
\hat{Q} \equiv \int_V \rho\,Q_+\,dV = -\int_V \rho\,Q_-\,dV,
\]
and vortex proximity metrics,
\[
\hat{\phi}_\pm
=
\frac{-2\int_V \rho\,\phi\,Q_\pm\,dV}{\int_V \rho\,Q_\pm\,dV},
\qquad
\Delta\hat{\phi}_\pm = \hat{\phi}_+ - \hat{\phi}_-.
\]
These satisfy
\[
L_Q = \hat{Q}\,\Delta\hat{\phi}_\pm.
\]
In that rotor case, vortex-induced lift contributes about $95.5\%$ of total pressure lift, viscous diffusion about $4.5\%$, the acceleration reaction is non-zero only at the first time step owing to the numerical impulsive start, and Coriolis and centrifugal terms do not contribute to the pressure force when the computational domain is sufficiently large. The reported interpretation is that early lift extrema arise from the interplay between how much vortex content is present and how close positive-$Q$ cores and negative-$Q$ coronae are to the lifting surface [2507.23131].

Across the cited literature, several robustness checks recur. In the low-aspect-ratio wing study, the sum of partitions matches the total vortex-induced lift, and vortex-induced lift plus viscous diffusion agrees with the total pressure-induced lift, with viscous shear very small and negative. In the swept-wing moment study, FMPM moment distributions qualitatively and quantitatively explain measured moment trends across sweep angles, although omitted non-vortex terms can produce magnitude mismatch. In the quiescent pitching-wing study, three phase-averaging strategies yielded similar moments, but near-wall PIV limitations remained a major source of bias [2205.04400; 2304.12544; 2301.13373].

The main limitations stated in the sources are also consistent. FPM depends on accurate velocity-gradient fields, because errors in $Q$ directly affect the load estimate. The auxiliary potential requires a Laplace solve on a sufficiently large domain with appropriate Neumann conditions. For stationary wings the body-acceleration term is zero, but for flapping or pitching wings it must be retained. Orientation thresholds such as $|\eta_z|>\cos(\pi/4)$ worked well for the reported low-aspect-ratio wing flows, but other flows may require tighter cones or modal decompositions. Finite measurement volumes in experiments can truncate far-field contributions, and two-dimensional PIV cannot capture cross-span velocity and gradient components [2205.04400; 2301.13373; 2507.23131].

A recurrent source of confusion is terminological rather than conceptual. In condensed-matter phonon calculations, one paper discusses a “partition of interatomic force constants” and states explicitly that the authors do not coin the term “Force Partitioning Method (FPM).” That work concerns redistribution of supercell force constants in real space and is distinct in scope, equations, and application domain from the fluid-dynamical FPM based on $\phi$ and $Q$ [2003.13344].

Source: https://www.emergentmind.com/topics/force-partitioning-method-fpm