pith. sign in

arxiv: 2506.16886 · v3 · submitted 2025-06-20 · ⚛️ physics.flu-dyn · math.OC

Analytic Full Potential Adjoint Solution for Two-dimensional Subcritical Flows

Carlos Lozano , Jorge Ponsin This is my paper

Pith reviewed 2026-05-19 08:05 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn math.OC
keywords adjoint solutionfull potential equationsubcritical flowGreen's functionKutta conditionaerodynamic forcepotential flow adjoint
0
0 comments X

The pith

For subcritical two-dimensional flows the adjoint solution of the full potential equation is derived analytically via Green's functions and shown to be linear combinations of compressible Euler adjoint variables for mass and vorticity.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper sets out to obtain closed-form expressions for the adjoint potential and stream function in the two-dimensional full potential equation when the flow remains subcritical. It applies the Green's function method to an aerodynamic-force cost function and then demonstrates that these adjoint fields equal specific linear combinations of the adjoint variables that register the effect of point mass sources and point vorticity sources in the compressible Euler system. A sympathetic reader would care because an explicit analytic adjoint supplies an exact benchmark against which numerical adjoint codes can be tested and makes transparent how local flow perturbations translate into global force sensitivities. The analysis also isolates two auxiliary functions that carry the influence of Kutta-condition perturbations and examines their properties both analytically and through numerical adjoint fields.

Core claim

For subcritical flows the Green's function approach is used to derive the analytic adjoint solution for a cost function measuring aerodynamic force. The connection of the adjoint problems for the potential flow equation and the compressible adjoint Euler equations reveals that the adjoint potential and stream function correspond to linear combinations of the compressible adjoint variables measuring the influence of point mass and vorticity sources. The solutions for the adjoint potential and stream function corresponding to aerodynamic lift contain two unknown functions encoding the effect of perturbations to the Kutta condition whose properties are analyzed from an analytic viewpoint and by

What carries the argument

The Green's function of the adjoint full-potential operator, which constructs the adjoint potential and stream function as linear combinations of the mass-source and vorticity-source adjoint fields from the compressible Euler equations.

If this is right

  • Numerical adjoint solvers for the full-potential equation can be validated by direct comparison with the closed-form expressions obtained from the Green's function.
  • A consistent formulation of the Kutta condition inside the adjoint framework can be constructed once the two unknown functions are fixed by analytic or numerical means.
  • The explicit link to mass and vorticity sources supplies an interpretation of potential-flow adjoint sensitivities in terms of the source terms that appear in the Euler adjoint equations.
  • The same Green's-function construction can be applied to other force-related cost functions beyond lift and drag provided the flow remains subcritical.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The analytic connection may allow accuracy properties proved for the potential-flow adjoint to be transferred to the subcritical Euler adjoint in regions where the flow is irrotational.
  • One could test the two auxiliary functions against far-field boundary conditions or asymptotic expansions to see whether they admit a unique analytic continuation.
  • The approach might be extended to derive similar closed-form adjoint solutions for other elliptic flow models that admit Green's functions.

Load-bearing premise

The adjoint problems for the potential flow equation and the compressible adjoint Euler equations are connected such that the adjoint potential and stream function are linear combinations of the adjoint variables for mass and vorticity sources, with the Kutta perturbations encoded in two unknown functions whose properties can be determined analytically and numerically.

What would settle it

A direct numerical computation of the adjoint potential and stream function for a specific subcritical airfoil flow, followed by a check of whether those fields exactly equal the predicted linear combination of the mass and vorticity adjoint variables, would confirm or refute the central claim.

read the original abstract

The analytic properties of adjoint solutions are investigated for the two-dimensional (2D) full potential equation. For subcritical flows, the Green's function approach is used to derive the analytic adjoint solution for a cost function measuring aerodynamic force. The connection of the adjoint problems for the potential flow equation and the compressible adjoint Euler equations reveals that the adjoint potential and stream function correspond to linear combinations of the compressible adjoint variables measuring the influence of point mass and vorticity sources. The solutions for the adjoint potential and stream function corresponding to aerodynamic lift contain two unknown functions encoding the effect of perturbations to the Kutta condition. The properties of these functions are analyzed from an analytic viewpoint and also by examining numerical adjoint solutions. Based on this analysis, a possible formulation of the Kutta condition within the adjoint framework is also discussed.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims to derive an analytic adjoint solution for the two-dimensional full potential equation in subcritical flows using Green's functions for a cost function measuring aerodynamic force. It establishes a correspondence showing that the adjoint potential and stream function are linear combinations of the compressible Euler adjoint variables associated with point mass and vorticity sources. For the aerodynamic lift cost function, the derived solutions contain two unknown functions that encode perturbations to the Kutta condition; the properties of these functions are analyzed both analytically and by inspection of numerical adjoint solutions, leading to a proposed formulation of the Kutta condition within the adjoint framework.

Significance. If the central derivation holds, the work supplies an analytic benchmark for adjoint solutions in potential flow that could clarify the structure of adjoint fields and the role of the Kutta condition. The explicit link to Euler adjoint variables for mass and vorticity sources is a useful conceptual bridge between potential-flow and compressible adjoint formulations. The Green's-function construction itself, when fully closed-form, would constitute a parameter-free result of the kind that strengthens reproducibility in adjoint-based aerodynamic analysis.

major comments (2)
  1. [Abstract and lift-adjoint derivation section] Abstract and the section deriving the lift adjoint: the solutions for the adjoint potential and stream function are presented as containing two unknown functions that encode Kutta-condition perturbations. The text states that their properties are determined in part by examining numerical adjoint solutions. This hybrid analytic-numerical step means the explicit form is not obtained purely from the Green's function or boundary-value analysis, which directly affects the claim of an independent analytic solution.
  2. [Connection to compressible adjoint Euler equations] Section establishing the connection to compressible adjoint Euler equations: the linear-combination correspondence between the adjoint potential/stream function and the adjoint variables for mass and vorticity sources is load-bearing for the overall interpretation. Because the two Kutta functions lie outside this combination and are not shown to be uniquely fixed by the analytic framework alone, the correspondence remains incomplete without additional justification that the functions can be determined without reference to discrete adjoint data.
minor comments (2)
  1. [Abstract] The abstract would benefit from a single sentence explicitly noting that the final expressions for lift contain undetermined functions whose properties are partly inferred numerically.
  2. [Notation and equations] Notation for the two unknown Kutta functions should be introduced once and used consistently in all subsequent equations and figures.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and insightful comments on our manuscript. We address the major comments point by point below, providing clarifications and indicating planned revisions where appropriate.

read point-by-point responses

  1. Referee: Abstract and the section deriving the lift adjoint: the solutions for the adjoint potential and stream function are presented as containing two unknown functions that encode Kutta-condition perturbations. The text states that their properties are determined in part by examining numerical adjoint solutions. This hybrid analytic-numerical step means the explicit form is not obtained purely from the Green's function or boundary-value analysis, which directly affects the claim of an independent analytic solution.

    Authors: The derivation using Green's functions provides a fully analytic expression for the adjoint potential and stream function, expressed in terms of the flow field and two unknown functions that arise naturally from the boundary value problem associated with the Kutta condition. The analytic framework determines the integral form and the constraints these functions must satisfy. The examination of numerical adjoint solutions is used to gain insight into the behavior and possible explicit representations of these functions, but it does not define the solution itself. We agree that this could be clarified to better separate the analytic derivation from the supporting numerical analysis. In the revised version, we will update the abstract and the relevant section to explicitly state that the general analytic solution is obtained via Green's functions, with numerical results providing additional characterization of the unknown functions. revision: partial

  2. Referee: Section establishing the connection to compressible adjoint Euler equations: the linear-combination correspondence between the adjoint potential/stream function and the adjoint variables for mass and vorticity sources is load-bearing for the overall interpretation. Because the two Kutta functions lie outside this combination and are not shown to be uniquely fixed by the analytic framework alone, the correspondence remains incomplete without additional justification that the functions can be determined without reference to discrete adjoint data.

    Authors: We maintain that the linear combination correspondence is rigorously derived from equating the adjoint equations and source terms for the potential flow and the Euler system, independent of the specific form of the Kutta functions. These functions represent the adjoint representation of the Kutta condition and are determined by the adjoint boundary conditions at the trailing edge. While their explicit functional form is not uniquely fixed by the interior equations alone, the analytic analysis shows they must be such that the adjoint solution remains consistent with the linearized Kutta condition. The numerical solutions help propose a specific choice. To strengthen this, we will include additional analytic arguments in the revision demonstrating that the correspondence holds for the primary terms and discuss the Kutta functions as a separate but compatible component of the adjoint formulation. revision: partial

Circularity Check

0 steps flagged

Derivation via Green's function remains self-contained; no reduction to inputs by construction

full rationale

The paper claims an analytic adjoint solution derived via the Green's function approach for the 2D full potential equation in subcritical flows, with the adjoint potential and stream function expressed as linear combinations of compressible adjoint variables for mass and vorticity sources. The identification of two unknown functions encoding Kutta-condition perturbations, whose properties are analyzed analytically and via numerical adjoint solutions, constitutes supplementary examination rather than a load-bearing step that defines the result by construction or renames a fitted input as a prediction. No self-citations, ansatz smuggling, or uniqueness theorems imported from prior author work appear in the provided derivation chain. The central result is an independent analytic construction from the governing equations and boundary-value analysis, with numerical inspection serving only as external insight and not as the source of the explicit form. This is the most common honest outcome for papers whose primary steps do not collapse to their own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the subcritical-flow assumption that permits a Green's-function representation without discontinuities, the irrotational and isentropic character of the full potential model, and the validity of the linear-combination correspondence between potential-flow and Euler adjoint variables. No free parameters or invented entities are indicated in the abstract.

axioms (2)
  • domain assumption Flow is subcritical so that the full potential equation admits a Green's function representation without shocks.
    Stated in abstract as the regime for which the analytic adjoint solution is derived.
  • domain assumption Adjoint potential and stream function are linear combinations of compressible Euler adjoint variables for mass and vorticity sources.
    Explicitly stated as the revealed connection in the abstract.

pith-pipeline@v0.9.0 · 5658 in / 1459 out tokens · 44253 ms · 2026-05-19T08:05:22.513468+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

40 extracted references · 40 canonical work pages

  1. [1]

    Full Potential, Euler and Navier -Stokes Schemes,

    A. Jameson, "Full Potential, Euler and Navier -Stokes Schemes," in Applied Computational Aerodynamics, Progress in Astronautics and Aeronautics, Vol. 125, Chapter 3, pp. 39 -88, (P. Henne, ed.), AIAA, 1990. ISBN: 978-0-930403-69-0. https://doi.org/10.2514/4.865985

    work page doi:10.2514/4.865985 1990

  2. [2]

    Aerodynamic design via control theory

    A. Jameson, "Aerodynamic Design via Control Theory," J. Sci. Comp., vol. 3, no. 3, pp. 233 - 260, 1988. https://doi.org/10.1007/BF01061285

    work page doi:10.1007/bf01061285 1988

  3. [3]

    Control Theory Based Airfoil Design for Potential Flow and a Finite Volume Discretization,

    J. Reuther and A. Jameson, "Control Theory Based Airfoil Design for Potential Flow and a Finite Volume Discretization," AIAA Paper 94-0499, AIAA 32nd Aerospace Sciences Meeting and Exhibit, Reno, NV, January 1994. 23

    work page 1994

  4. [4]

    Aerodynamic Shape Optimization Using Control Theory,

    J. Reuther, "Aerodynamic Shape Optimization Using Control Theory," PhD Thesis, University of California Davis, 1996

    work page 1996

  5. [5]

    An adjoint formulation for the non -linear potential flow equation,

    L. C. C. Santos, "An adjoint formulation for the non -linear potential flow equation," Applied Mathematics and Computation, vol. 108, no. 1, pp. 11-21, 2000, https://doi.org/10.1016/S0096- 3003(98)10137-6

    work page doi:10.1016/s0096- 2000

  6. [6]

    Airfoil design via control theory using full potential and Euler equations,

    J. Lewis and R. Agarwal, "Airfoil design via control theory using full potential and Euler equations," AIAA96-2483, 14th Applied Aerodynamics Conference, 17 June 1996 - 20 June 1996, New Orleans, USA. https://doi.org/10.2514/6.1996-2483, 1996

    work page doi:10.2514/6.1996-2483 1996

  7. [7]

    Airfoil Des ign and Optimization by the One -Shot Method,

    G. Kuruvila, S. Ta’asan and M. Salas, "Airfoil Des ign and Optimization by the One -Shot Method," AIAA 95–0478. 33rd Aerospace Sciences Meeting and Exhibit. Reno, NV, January 9-12, 1995

    work page 1995

  8. [8]

    A discrete adjoint full poten tial formulation for fast aerostructural optimization in preliminary aircraft design,

    A. Crovato, A. P. Prado, P. H. Cabral, R. Boman, V. E. Terrapon and G. Dimitriadis, "A discrete adjoint full poten tial formulation for fast aerostructural optimization in preliminary aircraft design," Aerospace Science and Technology, Vols. 138, 108332, 2023. https://doi.org/10.1016/j.ast.2023.108332

    work page doi:10.1016/j.ast.2023.108332 2023

  9. [9]

    Full Potential Revisited: A Medium Fidelity Aerodynamic Analysis Tool,

    M. Galbraith, S. Allmaras and R. Haimes, "Full Potential Revisited: A Medium Fidelity Aerodynamic Analysis Tool," AIAA paper 2017 -0290, 55th AIAA Aerospace Sciences Meeting, Grapevine, TX, USA, 2017, https://doi.org/10.2514/6.2017-0290

    work page doi:10.2514/6.2017-0290 2017

  10. [10]

    An Adjoint Optimization Prediction Method for Partially Cavitating Hydrofoils,

    D. Anevlavi and K. Belibassakis, "An Adjoint Optimization Prediction Method for Partially Cavitating Hydrofoils," J. Mar. Sci. Eng., vol. 9, no. 9, 976, 2021. https://doi.org/10.3390/jmse9090976

    work page doi:10.3390/jmse9090976 2021

  11. [11]

    Adjoint and defect error boundin g and correction for functional estimates,

    N. A. Pierce and M. B. Giles, "Adjoint and defect error boundin g and correction for functional estimates," Journal of Computational Physics, vol. 200, no. 2, p. 769 –794, 2004. Doi: 10.1016/j.jcp.2004.05.001

    work page doi:10.1016/j.jcp.2004.05.001 2004

  12. [12]

    Analytic adjoint solution for incompressible potential flows,

    C. Lozano and J. Ponsin, "Analytic adjoint solution for incompressible potential flows," Physics of Fluids, vol. 37, no. 6, 067127, 2025. https://doi.org/10.1063/5.0271828

    work page doi:10.1063/5.0271828 2025

  13. [13]

    Analytic Adjoint Solutions for the 2D Incompressible Euler Equations Using the Green’s Function Approach,

    C. Lozano and J. Ponsin, "Analytic Adjoint Solutions for the 2D Incompressible Euler Equations Using the Green’s Function Approach," Journal of Fluid Mechanics, vol. 943, A22, 2022, doi:10.1017/jfm.2022.415

    work page doi:10.1017/jfm.2022.415 2022

  14. [14]

    Explaining the Lack of Mesh Convergence of Inviscid Adjoint Solutions near Solid Walls for Subcritical Flows,

    C. Lozano and J. Ponsin, "Explaining the Lack of Mesh Convergence of Inviscid Adjoint Solutions near Solid Walls for Subcritical Flows," Aerospace, vol. 10, 392, 2023. https://doi.org/10.3390/aerospace10050392

    work page doi:10.3390/aerospace10050392 2023

  15. [15]

    The Poisson Kernel,

    S. G. Krantz, "The Poisson Kernel," in §7.3.2 in Handbook of Complex Variables , Birkhäuser, Boston, MA, 1999, p. 93

    work page 1999

  16. [16]

    Adjoint Equations in CFD: Duality, Boundary Conditions and Solution Behavior,

    M. B. Giles and N. A. Pierce, "Adjoint Equations in CFD: Duality, Boundary Conditions and Solution Behavior," AIAA -97-1850, 13th Computational Fluid Dynamics Conference, Snowmass Village, CO, U.S.A., 1997. 24

    work page 1997

  17. [17]

    Analytic adjoint solutions for the quasi -one-dimensional Euler equations,

    M. B. Giles and N. A. Pierce, "Analytic adjoint solutions for the quasi -one-dimensional Euler equations," J. Fluid Mechanics, vol. 426, pp. 327-345, 2001

    work page 2001

  18. [18]

    Optimum Design for Potential Flows,

    F. Angrand, "Optimum Design for Potential Flows," International Journal for Numerical Methods in Fluids, vol. 3, pp. 265-282, 1983

    work page 1983

  19. [19]

    Singular and Discontinuous Solutions of the Adjoint Euler Equations,

    C. Lozano, "Singular and Discontinuous Solutions of the Adjoint Euler Equations," AIAA Journal, vol. 56, no. 11, pp. 4437-4452, 2018

    work page 2018

  20. [20]

    An Entropy Adjoint Approach to Mesh Refinement,

    K. J. Fidkowski and P. L. Roe, "An Entropy Adjoint Approach to Mesh Refinement," SIAM Journal on Scientific Computing, vol. 32, no. 3, pp. 1261 -1287, 2010. https://doi.org/10.1137/090759057

    work page doi:10.1137/090759057 2010

  21. [21]

    Entropy and Adjoint Methods,

    C. Lozano, "Entropy and Adjoint Methods," J Sci Comput, vol. 81, p. 2447–2483, 2019

    work page 2019

  22. [22]

    Exact inviscid drag -adjoint solution for subcritical flows,

    C. Lozano and J. Ponsin, "Exact inviscid drag -adjoint solution for subcritical flows," AIAA Journal, vol. 59, no. 12, pp. 5369-5373, 2021. Doi: 10.2514/1.J060766

    work page doi:10.2514/1.j060766 2021

  23. [23]

    Continuous adjoint complement to the Blasius Equation,

    N. Kühl, P. M. Müller and T. Rung, "Continuous adjoint complement to the Blasius Equation," Phys. Fluids, vol. 33, p. 033608, 2021; https://doi.org/10.1063/5.0037779

    work page doi:10.1063/5.0037779 2021

  24. [24]

    Adjoint Complement to the Universal Momentum Law of the Wall,

    N. Kühl, P. Müller and T. Rung, "Adjoint Complement to the Universal Momentum Law of the Wall," Flow Turbulence Combust, vol. 108, p. 329–351, 2022. https://doi.org/10.1007/s10494- 021-00286-7

    work page doi:10.1007/s10494- 2022

  25. [25]

    Ordinary differential equations for the adjoint Euler equations,

    J. Peter and J. Désidéri, "Ordinary differential equations for the adjoint Euler equations," Phys. Fluids, vol. 34, p. 086113, 2022, https://doi.org/10.1063/5.0093784

    work page doi:10.1063/5.0093784 2022

  26. [26]

    Adjoint and Direct Characteristic Equations for Two - Dimensional Compressible Euler Flows,

    K. Ancourt, J. Peter and O. Atinault, "Adjoint and Direct Characteristic Equations for Two - Dimensional Compressible Euler Flows," Aerospace, vol. 10, no. 9, p. 797, 2023, https://doi.org/10.3390/aerospace10090797

    work page doi:10.3390/aerospace10090797 2023

  27. [27]

    On the Characteristic Structure of the Adjoint Euler Equations and the Analytic Adjoint Solution of Supersonic Inviscid Flows,

    C. Lozano and J. Ponsin, "On the Characteristic Structure of the Adjoint Euler Equations and the Analytic Adjoint Solution of Supersonic Inviscid Flows," Aerospace, vol. 12, no. 6, 2025,

    work page 2025

  28. [28]

    https://doi.org/10.3390/aerospace12060494

    work page doi:10.3390/aerospace12060494

  29. [29]

    Dual Consistency and Functional Accuracy: a finite -difference perspective,

    J. Hicken and D. Zingg, "Dual Consistency and Functional Accuracy: a finite -difference perspective," Journal of Computational Physics, vol. 256, pp. 161 -182, 2014. Doi: 10.1016/j.jcp.2013.08.014

    work page doi:10.1016/j.jcp.2013.08.014 2014

  30. [30]

    DiBenedetto and U

    E. DiBenedetto and U. Gianazza, Partial Differential Equations, Springer International Publishing, 2023. DOI: 10.1007/978-3-031-46618-2

    work page doi:10.1007/978-3-031-46618-2 2023

  31. [31]

    Watch Your Adjoints! Lack of Mesh Convergence in Inviscid Adjoint Solutions,

    C. Lozano, "Watch Your Adjoints! Lack of Mesh Convergence in Inviscid Adjoint Solutions," AIAA J., vol. 57, no. 9, p. 3991–4006, 2019

    work page 2019

  32. [32]

    Van Dyke, Perturbation Methods in Fluid Mechanics, Stanford, CA: The Parabolic Press, 1975

    M. Van Dyke, Perturbation Methods in Fluid Mechanics, Stanford, CA: The Parabolic Press, 1975

    work page 1975

  33. [33]

    Jacob, Introduction Mathématique a la Mécanique des Fluides, Paris: Gauthier-Villars, 1959

    C. Jacob, Introduction Mathématique a la Mécanique des Fluides, Paris: Gauthier-Villars, 1959. 25

    work page 1959

  34. [34]

    Uniqueness and the Force Formulas for Plane Subsonic Flows,

    R. Finn and D. Gilbarg, "Uniqueness and the Force Formulas for Plane Subsonic Flows," Trans. Amer. Math. Soc., vol. 88, no. 2, pp. 375-379, 1958

    work page 1958

  35. [35]

    On the Treatment of the Kutta-Joukowski Condition in Transonic Flow Computations,

    C. Coclici and W. Wendland, "On the Treatment of the Kutta-Joukowski Condition in Transonic Flow Computations," Z Angew Math Mech. , vol. 79, pp. 507 -534, 1999, https://doi.org/10.1002/(SICI)1521-4001(199908)79:8<507::AID-ZAMM507>3.0.CO;2-B

    work page doi:10.1002/(sici)1521-4001(199908)79:8 1999

  36. [36]

    Edge Singularities and Kutta Condition in 3D Aerodynamics,

    P. Bassanini, C. Casciola, M. Lancia and R. Piva, "Edge Singularities and Kutta Condition in 3D Aerodynamics," Meccanica, vol. 34, p. 199 –229, 1999. https://doi.org/10.1023/A:1004571915758

    work page doi:10.1023/a:1004571915758 1999

  37. [37]

    Existence and uniqueness of a subsonic flow past a given profile,

    L. Bers, "Existence and uniqueness of a subsonic flow past a given profile," Comm. Pure Appl. Math., vol. 7, pp. 441-504, 1954. https://doi.org/10.1002/cpa.3160070303

    work page doi:10.1002/cpa.3160070303 1954

  38. [38]

    Homogenization and boundary layers,

    D. Gérard-Varet and N. Masmoudi, "Homogenization and boundary layers," Acta Math, vol. 209, p. 133–178, 2012. https://doi.org/10.1007/s11511-012-0083-5

    work page doi:10.1007/s11511-012-0083-5 2012

  39. [39]

    Necessary and sufficient conditions for absolute continuity of elliptic- harmonic measure,

    E. B. Fabes, D. S. Jerison and C. E. Kenig, "Necessary and sufficient conditions for absolute continuity of elliptic- harmonic measure," Ann. Math., vol. 119, pp. 121-141, 1984

    work page 1984

  40. [40]

    SU2: An Open- Source Suite for Multiphysics Simulation and Design,

    T. D. Economon, F. Palacios, S. R. Copeland, T. W. Lukaczyk and J. J. Alonso, "SU2: An Open- Source Suite for Multiphysics Simulation and Design," AIAA Journal, vol. 54, no. 3, pp. 828 - 846, 2016. doi: 10.2514/1.J053813

    work page doi:10.2514/1.j053813 2016