Development of Homotopic Local Piston Theory

Open AccessTechnical NotesDevelopment of Homotopic Local Piston TheoryZou Zhicheng, Liu Chenyu, Xie Changchuan and An ChaoZou Zhicheng https://orcid.org/0000-0002-4470-0100Beihang University, 100191 Beijing, People's Republic of China, Liu ChenyuBeijing Institute of Astronautical Systems Engineering, 100076 Beijing, People's Republic of China, Xie ChangchuanBeihang University, 100191 Beijing, People's Republic of China and An Chao https://orcid.org/0000-0003-3535-5437Beihang University, 100191 Beijing, People's Republic of ChinaPublished Online:21 Sep 2025https://doi.org/10.2514/1.J065694SectionsRead Now ToolsAdd to favoritesDownload citationTrack citations ShareShare onFacebookXLinked InRedditEmail AboutI. Introduction Piston theory has recently been widely used in supersonic aerodynamic calculations for aeroelastic analyses due to its simple form, high computational efficiency, and remarkable accuracy within its applicable range [1]. The local piston theory (LPT) generally denotes a hybrid approach that integrates computational fluid dynamics (CFD) with piston theory. Compared with classical piston theory (CPT), LPT replaces the freestream parameters with local flow field parameters on the surface and substitutes surface perturbations for the downwash velocity. This modification overcomes the limitations of CPT, which is restricted to small angles of attack and thin wings [2], extending its applicable Mach number range to 1.2–10. The essence of LPT is the dynamic linearization of the flow about a mean steady state [1]. The mean steady state of the flow field is derived using high-fidelity CFD techniques, improving computational accuracy and expanding the applicability of the method. Meanwhile, unsteady effects are included as perturbations to the mean steady state flow field, assessed by piston theory, thus enhancing computational efficiency. The concept of LPT was firstly proposed by Morgan et al. [3] in the 1950s. Zhang et al. [4] used Euler-CFD-based LPT to calculate unsteady aerodynamic forces for aeroelastic analysis. In order to perform aeroelastic study of curved panels, Yang et al. [5] updated LPT using flow field characteristics derived from CFD. LPT with viscous correction was introduced by Liu et al. [6] in order to predict hypersonic unsteady aerodynamic loads. Meijer and Dala [7] studied the role of higher-order terms in LPT. By adding a two-shock separation model with blending functions, Sullivan et al. [8] increased the accuracy of LPT. Thayer and McNamara [9] enriched LPT with Unsteady Reynolds-Averaged Navier-Stokes equations (URANS) flow field predictions about rigid and statically deformed geometries. As current discussions on LPT are generally based on single steady reference flowfield, the theoretical foundation of LPT imposes certain limitations, with aerodynamic increments computed as perturbations induced by motion and deformation relative to this baseline. The accuracy of aerodynamic force prediction is inversely correlated with the magnitude of the perturbation. In other words, when using LPT to predict the aerodynamic force of a deviated state relative to a reference state, the larger the deviation, the greater the discrepancy between the predicted aerodynamic force and the actual aerodynamic force of the deviated state. Furthermore, this discrepancy also leads to a fundamental issue: the aerodynamic force computed for the same deviated state varies depending on the choice of reference state. To address the discrepancy and inconsistency of LPT, this Technical Note introduces the concept of homotopic mapping to reconcile the aerodynamic forces computed from different reference states. A novel LPT called homotopic local piston theory (HLPT) incorporating multiple reference states is thus established, and the effectiveness of the proposed theory is validated by comparing the calculation results with those of LPT and CFD through the numerical model of a fin under steady and unsteady conditions.II. Local Piston Theory and Generalized Aerodynamic ForceA. LPT Considering the Three-Dimensional Effect To enhance the applicability of piston theory to complex engineering models, this study builds upon LPT considering the three-dimensional effect developed by Liu et al. [10,11] for further investigation. The discretized LPT pressure formula for a three-dimensional aerodynamic surface is p=pL+nT∘(ALt∂UA∂t+ALx∂UA∂x+ALy∂UA∂y+ALz∂UA∂z)(1) In the equation, p represents the pressure vector, while pL denotes the local pressure vector. UA represents the displacements of the aerodynamic surface, and n is the unit outward normal vector. Additionally, ALt, ALx, ALy, and ALz represent the local parameter matrices. The symbol ∘ denotes the Hadamard product. By integrating the pressure with the area of each element, the discretized aerodynamic force formula based on LPT is derived as FA=n∘Sp(2)where FA is the aerodynamic force vector and S is the grid area matrix.B. Generalized Aerodynamic Force of LPT For generalized aerodynamic force calculation, the relationship between structural displacements US and aerodynamic pressure can be established using the interpolation matrix G: p=pL+nT∘[ALtGU˙S+(ALx∂G∂x+ALy∂G∂y+ALz∂G∂z)US](3) In the equation, U˙S represents the first-order time derivative of US, corresponding to the deformation velocity. Similarly, the aerodynamic force at structural nodes FS can be obtained from FA using the interpolation matrix G: FS=GTFA=GTn∘Sp(4) Substituting Eq. (3) into Eq. (4), the physical aerodynamic force expression of LPT is derived as Q¯=Q¯L+A¯1U˙S+A¯0US(5)where Q¯ represents the aerodynamic force vector. A¯1 denotes the dynamic influence coefficient matrix of physical aerodynamic force, while A¯0 denotes the static influence coefficient matrix of physical aerodynamic force. The structural displacement US can be represented in the form of generalized coordinates: US=Φq(6)where Φ=[φ1φ2⋯φn] represents the generalized coordinates matrix, and q=[ξ1ξ2⋯ξn]T represents the generalized coordinates. Substituting Eq. (6) into Eq. (5) the generalized aerodynamic force Q based on LPT is derived as Q=QL+A1q˙+A0q(7)where q˙ represents the first-order time derivative of q, corresponding to the generalized velocity. QL is the local generalized aerodynamic force generated by local pressure. A1 is the dynamic influence coefficient matrix of generalized aerodynamic force, and A0 is the static influence coefficient matrix of generalized aerodynamic force. As observed from Eq. (7), the expression for the generalized aerodynamic force of LPT is remarkably concise, allowing it to be explicitly represented as a function of the generalized velocity q˙ and the generalized coordinates q. This significantly reduces the complexity of analysis. The subsequent theoretical development will be built upon the foundation of the generalized aerodynamic force formulation in this section.III. Homotopic Local Piston TheoryA. The Concept of Homotopic Mapping Homotopy is a continuous deformation between mappings [12]. Let X and Y be topological spaces, and denote C(X,Y) as the set of all continuous mappings from X to Y. Suppose mappings f,g∈C(X,Y). If there exists a continuous mapping H:X×I→Y such that ∀ x∈X,H(x,0)=f(x),H(x,1)=g(x)(8)then f and g are said to be homotopic, denoted as f≃g. The mapping H is called a homotopy between f and g. Here, X×I={(x,t)|x∈X,t∈I} represents the Cartesian product of X and I=[0,1]. For each t∈I, the homotopy H defines a function ht∈C(X,Y) as ht=H(x,t), thus forming a one-parameter family of continuous mappings {ht|t∈I}, where ht is referred to as the slice of H at t. By definition, h0=f, h1=g. Linear homotopy is the most basic form of homotopy, with an intuitive interpretation that as t varies from 0 to 1, ht transitions linearly from f(x) to g(x). It is expressed as H(x,t)=(1−t)f(x)+tg(x)(9)B. Homotopic Local Piston Theory For LPT, the process of determining aerodynamic force from a given perturbation motion and deformation can be regarded as a mapping. Based on the continuity of physical quantities, this mapping is inherently continuous. Therefore, a homotopy H:f≃g can be constructed, where f and g represent different reference flow fields in LPT. Based on Eq. (7), the two generalized aerodynamic force expressions of LPT corresponding to these two reference flow fields can be formulated as follows. {Qf=QLf+A1fq˙rf+A0fqrfQg=QLg+A1gq˙rg+A0gqrg(10)where the subscript r represents the reference value used to distinguish the characteristic states of the two reference flow fields. For a deviation state with generalized velocity q˙ and coordinates q, the generalized aerodynamic force based on the reference the flow field f is f(q˙,q), while the generalized aerodynamic force based on the reference flow field g is g(q˙,q): {f(q˙,q)=QLf+A1f(q˙−q˙rf)+A0f(q−qrf)g(q˙,q)=QLg+A1g(q˙−q˙rg)+A0g(q−qrg)(11) In the equation, (q˙−q˙r) and (q−qr) represent the generalized perturbation velocity and coordinates of the perturbed state relative to the reference flow fields f and g, respectively. The homotopy H from f to g can be defined as H(q˙,q,t)=[1−σ(t)]f(q˙,q)+σ(t)g(q˙,q)(12) In the equation, t∈[0,1] represents the degree of deviation between the two reference flow fields; σ(t) is a function represents the weight of the two equations. The locality requirement of LPT stipulates that σ(t) is positively correlated with t, and σ(t)={0(t=0)⇒H(q˙,q,0)=f(q˙,q)1(t=1)⇒H(q˙,q,1)=g(q˙,q)(13) Equation (12) is the generalized aerodynamic force expression of HLPT. As shown in Fig. 1, The weight function σ(t) can take various forms, including piecewise, linear, and various nonlinear forms, highlighting the excellent flexibility and versatility of HLPT. The choice of σ(t) is crucial as it determines the form of HLPT, and the choice depends on the distribution of reference states and the expected aerodynamic nonlinearity within the calculation range. For example, the fully piecewise form divides the range between two reference states and applies conventional LPT based on the nearest reference on each side, which is suitable when the reference states are close and the aerodynamic response is approximately linear. In contrast, linear or nonlinear forms of σ(t) are preferred when broader ranges or smoother transitions are required, especially in cases involving aerodynamic nonlinearity.Fig. 1 Different forms of weight functions.C. HLPT with Respect to Angle of Attack HLPT does not impose any specific constraints on the form of generalized motion, enabling the establishment of HLPT models for any type of motion. This section focuses on pitching motion, deriving HLPT in terms of angle of attack (AoA). The entire derivation process can serve as a reference for other forms of motion. The generalized aerodynamic force expression of LPT considering only the AoA α is given by M=ML+A1α˙+A0α(14)where M represents the generalized aerodynamic force corresponding to α. Based on the reference flow fields f and g with reference AoA αrf and αrg, respectively (where the change rate of reference AoA α˙rf=α˙rg=0), the generalized aerodynamic force of a certain deviation state (α˙,α) between f and g can be calculated: {f(α˙,α)=MLf+A1fα˙+A0f(α−αrf)g(α˙,α)=MLg+A1gα˙+A0g(α−αrg)(15) And the generalized aerodynamic force based on HLPT is H(α˙,α,t)=[1−σ(t)]f(α˙,α)+σ(t)g(α˙,α)(16) Figure 2 is the illustration of the forces obtained by HLPT based on different weight functions σ(t). Since LPT retains the reference flow field information, the results of the HLPT at the endpoint αrf and αrg are identical to those of LPT. The calculation accuracy of HLPT in the intermediate region is related to the selection of σ(t). The following examples illustrate some forms of σ(t). The fully piecewise weight function σ(t) is as follows: σ(t)={0(α≤αp)1(α>αp)(17)where αp is a certain AoA between two reference flow fields. The generalized aerodynamic force of the fully piecewise HLPT is given by H(α˙,α)={MLf+A1fα˙+A0f(α−αrf)(α≤αp)MLg+A1gα˙+A0g(α−αrg)(α>αp)(18) Piecewise HLPT can be regarded as applying LPT for calculation within the corresponding range of AoA. The construction process of piecewise HLPT is simple and intuitive, and its aerodynamic expression retains the piecewise linear characteristics about α, which is easy to couple with the structural dynamic models. However, the generalized aerodynamic forces calculated based on two independent LPT models are generally not equal, leading to a discontinuous change of piecewise HLPT with respect to AoA.The linear weight function σ(t) is as follows: {t=α−αrfαrg−αrfσ(t)=t(19) Then the generalized aerodynamic force of the linear HLPT is obtained: H(α˙,α)=αrgMLf−αrfMLg+αrgαrf(A0g−A0f)αrg−αrf+αrgA1f−αrfA1gαrg−αrfα˙+MLg−MLf+(αrg+αrf)(A0f−A0g)αrg−αrfα+A1g−A1fαrg−αrfα˙α+A0g−A0fαrg−αrfα2(20) As observed from Eq. (20), establishing a linear HLPT model may result in a nonlinear aerodynamic form, so the green line in Fig. 2 is not a straight line.If the exact generalized aerodynamic force M1 for a certain deviation state α1 is known, σ(t) can be constructed in the form of a polynomial function by introducing the unknown coefficients c1 and c3. By adjusting c1 and c3, the generalized aerodynamic force M1 at α1 calculated by the HLPT can be accurate. {t=α−αrfαrg−αrfσ(t)=c1t+c3t3(21)Fig. 2 Generalized aerodynamic forces based on different weight functions. In the equation, c1 and c3 must satisfy the following conditions: {σ(1)=c1+c3=1t1=α1−αrfαrg−αrf⇒σ(t1)=c1t1+c3t13⇒H(α˙1,α1,t1)=M1(22)where the first equation is the definition of the homotopy mapping, while the second equation represents the constraint condition ensuring that AoA and generalized aerodynamic force pass through the given state point (α1,M1).Then c1 and c3 can be solved as {c1=t13(f1−g1)−f1+M1(t13−t1)(f1−g1)c3=t1(g1−f1)+f1−M1(t13−t1)(f1−g1)(23) Combining the above equations, the generalized aerodynamic force of nonlinear HLPT can be obtained, whose highest order is related to α4: H(α˙,α)=AN+BNα+CNα2+DNα3+ENα4+α˙(FN+GNα+HNα2+INα3)(24)where the coefficients AN to IN in the equation are related to the selected reference states. The advantage of the nonlinear HLPT is that it can not only pass through the endpoint states, but it can also pass through a given deviation state point. Since nonlinear high-order terms are introduced into the aerodynamic expression, this method can consider certain aerodynamic nonlinear effects. Accordingly, this method increases the complexity of the aerodynamic model and the amount of calculation. In summary, HLPT offers diverse and flexible formulations. In practical applications, the appropriate construction form should be selected according to the specific analysis requirements. Typically, when the known states are relatively close and the required calculation range is small, the piecewise HLPT is preferred, which is easy to construct. However, for larger calculation ranges, the linear HLPT is a more suitable choice. In cases where the aerodynamic forces exhibit strong nonlinear characteristics within the selected range, nonlinear HLPT should be employed to achieve more accurate results. Moreover, for unsteady aerodynamic simulations that require continuous time-domain calculations, it is necessary to employ either the linear or nonlinear HLPT models to avoid the errors caused by the discontinuities inherent in the piecewise HLPT.D. Integration with the CFD Database In the preliminary design stage of aircraft, due to the inability to calculate and store continuous aerodynamic data, for analyses like flight dynamics, the existing method relies on a large number of discrete state points from steady and unsteady CFD databases, and the preparation of the CFD database consumes considerable computing resources. By adopting HLPT, the calculation demand of the CFD database can be significantly reduced, as shown in Fig. 3.Fig. 3 Comparison of database required by HLPT and traditional methods. Compared with traditional CFD databases, HLPT only needs steady CFD results to calculate the unsteady aerodynamic forces near the steady state, omitting the time-consuming unsteady CFD calculations, thus significantly reducing the workload of CFD database preparation. Additionally, HLPT can predict the aerodynamic force near the reference state, reducing the required distribution density of database state points and thereby lowering the overall demand for the number of CFD database state points. Compared with LPT based on the same database state points, HLPT can eliminate discontinuities and significantly enhance the prediction accuracy of aerodynamic force. In addition, in traditional analyses, data between discrete state points are obtained through interpolation. In cases where aerodynamic nonlinearity becomes increasingly obvious, even seventh-order polynomial fitting may be required [13]. Regardless of the interpolation method employed, this approach lacks physical interpretability, making it unsuitable for mechanism analysis. Moreover, the selection of fitting parameters is largely experience-dependent, and the sources of error remain unidentified, further limiting its reliability and applicability. Unlike direct interpolation, HLPT, which is a continuous transformation between aerodynamic force models, establishes a theoretical framework that links multiple steady state points, enhancing the capacity to compute continuous aerodynamic forces between discrete states. This method guarantees physical interpretability based on generalized aerodynamic force, a feature that traditional interpolation techniques intrinsically lack.IV. Results and Discussion To verify the effectiveness of HLPT, numerical calculations are conducted under both steady and unsteady conditions, and the calculation results of three aerodynamic calculation methods, LPT, HLPT, and CFD, are compared. Among them, the high-precision CFD results serve as reference values to analyze the relative errors of other methods. Additionally, LPT based on different reference states can also be regarded as fully piecewise HLPT. The calculations in this Note are based on the fin model studied by Liu et al. and Wang et al. [10,11]. The LPT and CFD models are shown in Fig. 4. The CFD model is a structured mesh with refinement around the fin to capture flow features accurately. A no-slip, adiabatic wall boundary condition is applied to the fin surface. For the LPT model, the mesh is surface-fitted, including only the upper and lower surfaces.Fig. 4 Aerodynamic models of the fin. The working condition is shown in Table 1. Commercial software ANSYS Fluent is employed to perform CFD calculations, using density-based solver with an implicit formulation to solve the Reynolds-averaged Navier–Stokes equations. Spatial discretization is performed using the second-order upwind scheme for flow and the Least-Squares Cell-based scheme for gradients. The air in the flow field is modeled as ideal gas.A. Steady Aerodynamic Force Validation Both linear and nonlinear HLPT models with different distance are constructed and examined in this section, where distance denotes the magnitude of the difference between two physical quantities: Short-distance model: The linear HLPT model is constructed using 4° and 8° AoA as reference states (endpoints), while the nonlinear HLPT model is constructed using 4°, 6°, and 8° AoA as reference states.Long-distance model: The linear HLPT model is based on the flow fields of 0° and 12° AoA, while the nonlinear HLPT model is based on the flow fields of 0°, 6°, and 12° AoA. The formulations of the HLPT models are shown in Table 2, where the subscripts S and L denote the short-distance and long-distance models respectively. Since this case is steady, α˙ in the formulations are zero. The results obtained by linear interpolation between the corresponding CFD data points are also included in this part. The steady normal force coefficient Cn calculated by different methods are compared, and the results at 6° AoA (midpoint) are listed to assess accuracy. The comparison of results is presented in Fig. 5, and the errors relative to CFD at 6° AoA are shown in Table 3. It can be observed that the steady aerodynamic coefficients computed by LPT based on different flow fields differ at 6°, confirming the potential discontinuity in aerodynamic forces predicted by fully piecewise HLPT. The discrepancy of the coefficients calculated in the long-distance case is more significant. This indicates that as the distance between the reference states increases, the discrepancy between aerodynamic forces predicted by LPT based on different flow fields becomes more obvious, thereby limiting the applicability of LPT.Fig. 5 Comparison of steady aerodynamic coefficients. As shown in Table 3, HLPT models exhibit significantly higher accuracy than LPT and CFD interpolation in both cases, eliminating the discontinuity of LPT at midpoint. The result of nonlinear HLPT at 6° is the same with the CFD result, since the midpoint is one of its reference states. The aerodynamic coefficients calculated by both HLPT models at other points also closely match the CFD results, further validating the superior global accuracy of HLPT. Since the aerodynamic nonlinearity in the test case is relatively weak, the differences between the results obtained using the nonlinear and linear HLPT models are minor. Moreover, as shown in Fig. 5a, the HLPT models produce results closer to the CFD predictions at 0° and 12° AoA compared with LPT. This indicates that HLPT can not only enhance the accuracy of aerodynamic forces within the endpoints, but also improve the accuracy to some extent beyond the range of the endpoints. It is also worth noting that the linear HLPT curves in above figures exhibit nonlinear characteristics that closely approximate the CFD data. Linear HLPT is capable of capturing nonlinear aerodynamic forces consistent with CFD results based on only two reference states. This represents the fundamental distinction and improvement over traditional methods that directly interpolate aerodynamic derivatives.B. Unsteady Aerodynamic Force Validation In this section, the fin undergoes simple harmonic oscillation, with reference AoA α0=6° and amplitude of oscillation Aα=6°. Short-distance linear and nonlinear HLPT models based on the flow fields of 4° and 8° AoA, and long-distance linear and nonlinear HLPT models based on the flow fields of 0° and 12° AoA are established, respectively. The α1 of both nonlinear models is set to be 6°. Accordingly, LPT, or piecewise HLPT, calculates the unsteady aerodynamic forces based on the flow fields of corresponding AoA. The αp of piecewise HLPT is set to 6°. The unsteady normal force coefficient Cn calculated by four methods are compared. The formulations of the HLPT models are summarized in Table 4. The curves of the unsteady aerodynamic coefficients computed by the four aerodynamic methods are shown in Fig. 6, while the amplitudes of the coefficients and their errors relative to CFD results are presented in Table 5. As can be seen from the figure, the discontinuities of the LPT results become more obvious as the distance of reference states increases, which will lead to substantial error accumulation in long-term unsteady simulations. The HLPT curves exhibit better consistency with the CFD curves, with long-distance model demonstrating particularly strong agreement. For the short-distance case, both linear and nonlinear HLPT models improve the amplitude accuracy compared with LPT, demonstrating superior accuracy. For long-distance case, since both HLPT and LPT are based on flow fields at 0° and 12° AoA, they yield identical aerodynamic coefficient amplitude. As both methods are based on the steady CFD flow fields at the endpoints, the errors of amplitudes are minimal, with discrepancies primarily arising from unsteady effects.Fig. 6 Comparison of unsteady aerodynamic coefficients.V. Conclusions Based on the concepts of generalized aerodynamic force and homotopic mapping, HLPT incorporating multiple reference states is established in this Technical Note. Compared with LPT, HLPT can effectively address discontinuities of aerodynamic force between different reference flow fields and improve the computational accuracy at deviation states. In addition, HLPT can effectively reduce the reliance on CFD databases, thereby lowering the computational cost required for analyses. Numerical case studies validate that the HLPT enhances the accuracy of both steady and unsteady aerodynamic calculations. HLPT demonstrates high accuracy in predicting aerodynamic forces for both in-range and out-of-range state points. Moreover, HLPT enables the representation of nonlinear aerodynamic forces while mitigating discontinuities between state points. This approach facilitates efficient and high-precision aerodynamic calculations in the supersonic regime, offering broad application potential.S. Shin, Associate EditorAcknowledgments This work was financially supported by Young Elite Scientists Sponsorship Program of China Association for Science and Technology (Grant No. YESS20230417) and Young Elite Scientists Sponsorship Program of Beijing Association for Science and Technology (Grant No. BYESS2023345). References [1] Meijer M. C. and Dala L., "Generalized Formulation and Review of Piston Theory for Airfoils," AIAA Journal, Vol. 54, No. 1, 2016, pp. 17–27. https://doi.org/10.2514/1.J054090 LinkGoogle Scholar[2] Liu C., Xie C., Yan Y., Man Y., Zhang M. and Yang H., "An Engineering Analysis for Panel Flutter of a Supersonic Inlet," International Forum on Aeroelasticity and Structural Dynamics, IFASD 2019, Savannah, GA, 2019, pp. 2229–2241. Google Scholar[3] Morgan H. G., Runyan H. L. and Huckel V., "Theoretical Considerations of Flutter at High Mach Numbers," Journal of the Aerospace Sciences, Vol. 25, No. 6, 1958, pp. 371–381. https://doi.org/10.2514/8.7688 LinkGoogle Scholar[4] Zhang W., Ye Z., Zhang C. and Liu F., "Supersonic Flutter Analysis Based on a Local Piston Theory," AIAA Journal, Vol. 47, No. 10, 2009, pp. 2321–2328. https://doi.org/10.2514/1.37750 LinkGoogle Scholar[5] Yang Z., Zhou J. and Gu Y., "Integrated Analysis on Static/Dynamic Aeroelasticity of Curved Panels Based on a Modified Local Piston Theory," Journal of Sound and Vibration, Vol. 333, No. 22, 2014, pp. 5885–5897. https://doi.org/10.1016/j.jsv.2014.06.035 CrossrefGoogle Scholar[6] Liu W., Zhang C., Han H. and Wang F., "Local Piston Theory with Viscous Correction and Its Application," AIAA Journal, Vol. 55, No. 3, 2017, pp. 942–954. https://doi.org/10.2514/1.J055207 LinkGoogle Scholar[7] Meijer M. C. and Dala L., "Role of Higher-Order Terms in Local Piston Theory," Journal of Aircraft, Vol. 56, No. 1, 2019, pp. 388–391. https://doi.org/10.2514/1.C034920 LinkGoogle Scholar[8] Sullivan B. T., Bodony D. J., Whalen T. and Laurence S., "Direct Simulation of Fluid–Structure Interaction in a Hypersonic Compression-Ramp Flow," AIAA Journal, Vol. 58, No. 11, 2020, pp. 4848–4865. https://doi.org/10.2514/1.J059072 LinkGoogle Scholar[9] Thayer J. D. and McNamara J. J., "Effect of Model Fidelity on High-Speed Aeroelastic Behavior of a Cantilever Plate," AIAA Journal, Vol. 62, No. 10, 2024, pp. 3881–3892. https://doi.org/10.2514/1.J063758 LinkGoogle Scholar[10] Liu C., Xie C., Meng Y. and Bai L., "Experimental and Numerical Flutter Analysis Using Local Piston Theory with Viscous Correction," Aerospace, Vol. 10, No. 10, 2023, p. 870. https://doi.org/10.3390/aerospace10100870 CrossrefGoogle Scholar[11] Wang G., Xie C., Liu C. and An C., "Hypersonic Flutter Analysis Based on Three-Dimensional Local Piston Theory," International Forum on Aeroelasticity and Structural Dynamics, Vol. 92, 2024, pp. 17–21. Google Scholar[12] You C., Lectures on Basic Topology, Peking Univ. Press, Beijing, 1997, pp. 103–109. Google Scholar[13] Chang S. and Li D., "Homotopy-Analysis-Method-Based Solutions of Quintic Nonlinear Angular Motion for Projectiles," Journal of Spacecraft and Rockets, Vol. 55, No. 3, 2018, pp. 669–680. https://doi.org/10.2514/1.A33864 LinkGoogle ScholarTablesTable 1 The working condition of the finVelocity, MaHeight, kmUnit Reynolds number2.5111.9E+07Table 2 The formulations of HLPT of the steady case Short-distance modelLong-distance modelLinear HLPTH(α˙,α)=8ML4−4ML8+32(A08−A04)4+ML8−ML4+32(A04−A08)4α+A08−A044α2H(α˙,α)=8ML012+ML12−ML012α+A012−A0012α2Nonlinear HLPTH(α˙,α)=ANS+BNSα+CNSα2+DNSα3+ENSα4H(α˙,α)=ANL+BNLα+CNLα2+DNLα3+ENLα4Table 3 Comparison of steady aerodynamic coefficients at 6°MethodShort-distance caseMethodLong-distance caseCnRelative error of CnCnRelative error of CnLinear HLPT1.640E-010.01%Linear HLPT1.638E-01−0.11%Nonlinear HLPT1.640E-010.00%Nonlinear HLPT1.640E-010.00%LPT (4°)1.591E-01−2.99%LPT (0°)1.521E-01−7.21%LPT (8°)1.689E-013.01%LPT (12°)1.754E-017.00%CFD interpolation1.645E-010.30%CFD interpolation1.679E-012.38%CFD1.640E-01 CFD1.640E-01 Table 4 The formulations of HLPT of the unsteady case Short-distance modelLong-distance modelLinear HLPTH(α˙,α)=8ML4−4ML8+32(A08−A04)4+8A14−4A184α˙+ML8−ML4+32(A04−A08)4α+A18−A144α˙α+A08−A044α2H(α˙,α)=8ML012+ML12−ML012α+12A1012α˙+A112−A1012α˙α+A012−A0012α2Nonlinear HLPTH(α˙,α)=ANS+BNSα+CNSα2+DNSα3+ENSα4+α˙(FNS+GNSα+HNSα2+INSα3)H(α˙,α)=ANL+BNLα+CNLα2+DNLα3+ENLα4+α˙(FNL+GNLα+HNLα2+INLα3)Table 5 Comparison of unsteady aerodynamic coefficient amplitudesMethodShort-distance caseMethodLong-distance caseCn amplitudeRelative error of Cn amplitudeCn amplitudeRelative error of Cn amplitudeLinear HLPT1.691E-012.42%Linear HLPT1.677E-011.60%Nonlinear HLPT1.681E-011.84%Nonlinear HLPT1.677E-011.60%LPT, 4°/8°1.583E-01−4.12%LPT (0°/12°)1.677E-011.60%CFD1.651E-01 CFD1.651E-01 Previous article Next article FiguresReferencesRelatedDetails What's Popular Articles in Advance Altmetrics Metrics Downloaded 10 times 0 citation in Crossref CrossmarkInformationCopyright © 2025 by the American Institute of Aeronautics and Astronautics, Inc. 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TopicsAerodynamic PerformanceAerodynamicsAeroelasticityAerospace SciencesAircraft DesignAircraft Operations and TechnologyAircraft Wing DesignAirfoilComputational Fluid DynamicsFinite Element MethodFluid DynamicsNumerical Analysis KeywordsPiston TheoryAerodynamic CoefficientsComputational Fluid DynamicsReynolds Averaged Navier StokesInfluence CoefficientsAerodynamic PerformanceStructural DisplacementNumerical AnalysisAeroelastic AnalysisMechanical and Structural VibrationsAcknowledgmentsThis work was financially supported by Young Elite Scientists Sponsorship Program of China Association for Science and Technology (Grant No. YESS20230417) and Young Elite Scientists Sponsorship Program of Beijing Association for Science and Technology (Grant No. BYESS2023345).Digital Received31 March 2025Accepted28 August 2025Published online21 September 2025