The sensitivity equation method in fluid mechanics

Authors

  • Dominique Pelletier Ecole Polytechnique de Montréal C.P. 6079, Succursalle Centre-Ville Montréal, QC, Canada, H3C 3A7
  • Alexander Hay Ecole Polytechnique de Montréal C.P. 6079, Succursalle Centre-Ville Montréal, QC, Canada, H3C 3A7
  • Stéphane Etienne Ecole Polytechnique de Montréal C.P. 6079, Succursalle Centre-Ville Montréal, QC, Canada, H3C 3A7
  • Jeff Borggaard ICAM Virginia Tech, Blacksburg, VA 24061

DOI:

https://doi.org/10.13052/REMN.17.31-61

Keywords:

flow sensitivities, Navier-Stokes, optimal design, nearby flows, uncertainty analysis

Abstract

We present the sensitivity Equation Method (SEM) as a complementary tool to adjoint based optimisation methods. Flow sensitivities exist independently of a design problem and can be used in several non-optimization ways: characterization of complex flows, fast evaluation of flows on nearby geometries, and input data uncertainties cascade through the CFD code to yield uncertainty estimates of the flow field. The Navier-Stokes and sensitivity equationssensitivity are solved by an adaptive finite element method.

Downloads

Download data is not yet available.

References

A.Griewank, Evaluating Derivatives, SIAM, Philadelphia, PA, 2000.

Anderson W. K., Newman J. C.,Whitfield D. L., Nielsen E. J., “ Sensitivity Analysis of Navier-

Stokes Equations on Unstructured Meshes unsing Complex Variables”, AIAA Journal, vol.

, no 1, p. 56-53, 2001.

Blackwell B. F., Dowding K. J., Cochran R. J., Dobranich D., “ Utilization of sensitivity coefficients

to guide the design of a Thermal battery”, Proceedings of the 1998 ASME/IMECE,

ASME, Anaheim, CA, p. 73-82, 1998. HTD-Vol. 561-5.

Borggaard J., Burns J., “ A PDE sensitivity equation method for optimal aerodynamic design”,

J. Comput. phys., vol. 136, n◦ 2, p. 367-384, 1997.

Borggaard J., Pelletier D., “ Optimal Shape Design in Forced Convection Using Adaptive Finite

Elements”, 36th AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, January, 1998.

AIAA Paper 98-0908.

Borggaard J., Pelletier D., Vugrin K., “ On Sensitivity Analysis for Problems with Numerical

Noise”, 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization,

Atlanta, GA, September, 2002. AIAA Paper 2002-5553.

Dennis Jr. J., Schnabel R., Numerical Methods for Unconstrained Optimization and Nonlinear

Equations, Prentice-Hall, 1983.

Duvigneau R., Pelletier D., “ Evaluation of Nearby Flows by a Shape Sensitivity Equation

Method”, 43th AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, Jan., 2005a.

AIAA Paper 2005–0127.

Duvigneau R., Pelletier D., “ On Accurate Boundary Conditions for a Shape Sensitivity Equation

Method”, International Journal for Numerical Methods in Fluids, vol. 50, p. 1417-164,

b.

Godfrey A. G., Cliff E. M., “ Direct Calculation of Aerodynamic Force Derivatives: A

Sensitivity-Equation Approach”, 36th AIAA Aerospace Sciences Meeting and Exhibit,

Reno, NV, January, 1998. AIAA Paper 98-0393.

Godfrey A. G., Cliff E. M., “ Sensitivity Equations for Turbulent Flows”, 39th AIAA Aerospace

Sciences Meeting and Exhibit, Reno, NV, Jan., 2001. AIAA Paper 2001-1060.

Gunzburger M. D., Perspectives in Flow Control and Optimization, SIAM, 2002.

Haug E. J., Choi K., Komkov V., Design sensitivity analysis of structural systems, vol. 177 of

Mathematics in science and engineering, Academic Press, Orlando, 1986.

Hay A., Etienne S., Duvigneau R., Pelletier D., “ Evaluation of Flows on Nearby Geometries by

a Shape Sensitivity EquationMethod”, 44th AIAA Aerospace Sciences Meeting and Exhibit,

Reno, NV, 9-12 January, 2006. AIAA Paper 2006-1296.

Ilinca F., Pelletier D., “ A Continuous shape sensitivity equation method for unsteady laminar

flows”, European Conference on Computational Fluid Dynamics, ECCOMAS CFD 2006,

TU Delft, The Netherlands, september, 2006.

Ilinca F., Pelletier D., Borggaard J., “ A Continuous Second Order Sensitivity Equation Method

for Time-Dependent Incompressible Laminar Flows”, 35th AIAA Fluid Dynamics Conference

and Exhibit, Toronto, ON, June, 2005. AIAA Paper 2005-5252.

KleiberM., Antúnez H., Hien T. D., Kowalczyk P.,Parameter Sensitivity in Nonlinear Mechanics,

John Wiley and Sons, 1997.

Lépine J., Guibault J., Trépanier J.-Y., Pépin F., “ Optimized NonUniform Rational B-Spline

Geometrical Representation for Aerodynamic Design of Wings”, AIAA Journal, vol. 39,

n◦ 11, p. 2033-2041, 2001.

Limache A., Aerodynamic Modeling Using Computational Fluid Dynamics and Sensitivity

Equations, PhD thesis, Virginia Polytechnic Institute and State University, Blacksburg, VA,

Lu S.-Y., Sagaut P., “ Direct sensitivity analysis for smooth unsteady compressible flows using

complex differentiation Design”, International Journal for Numerical Methods in Fluids,

Lyness J., Moler C., “ Numerical differentiation of analytic functions”, SIAM J. Numer. Anal,

vol. 4, n◦ 2, p. 202-210, June, 1967.

Mahieu J., Etienne S., Pelletier D., Borggaard J., “ A Second-order sensitivity equation method

for laminar flow”, International Journal of Computational Fluid Dynamics, vol. 19, n◦ 2,

p. 143-157, February, 2005.

Martins J. R. R. A., Stradza P., Alonso J. J., “ The complex-step derivative approximation”,

ACM transactions on mathematical software - TOMS, vol. 29, n◦ 3, p. 245-262, 2003.

Pelletier D., “ Adaptive Finite Element Computations of Complex Flows”, International Journal

for Numerical Methods in Fluids, vol. 31, p. 189-202, 1999.

Pelletier D., Ilinca F., “ Adaptive Remeshing for the k−ǫModel of Turbulence”, AIAA Journal,

vol. 35, n◦ 4, p. 640-646, 1997.

Pelletier D., Roache P. J., Verification and Validation in Computational Heat Transfer, Handbook

of Numerical Heat Transfer, 2nd edn,Wiley, Hoboken, New Jersey, chapter 13,March,

Peraire J., Vahdati M., Morgan K., Ziekiewicz O., “ Adaptive Remeshing for Compressible

Flow Computations”, Journal of Computational Physics, vol. 72, n◦ 2, p. 449-466, 1987.

Putko M., Newman P., Taylor A., Green L., “ Approach for uncertainty propagation and robust

design in CFD using sensitivity derivatives”, 15th AIAA Computational Fluid Dynamics

Conference, Anaheim, CA, June, 2001. AIAA Paper 2001-2528.

Roache P. J., Verification and Validation in Computational Science and Engineering, Hermosa

publishers, Albuquerque, NM, 1998.

Sherman L. L., Taylor III A. C., Green L., Newman P. A., Hou G.W., Korivi V. M., “ First- and

second-order aerodynamic sensitivity derivatives via automatic differentiation”, Journal of

Computational Physics, vol. 129, n◦ 2, p. 307-331, 1996.

Stanley L. G., Stewart D. L., Design Sensitivity Analysis: Computational Issues of Sensitivity

Equation Methods, vol. 25 of Frontiers in Applied Mathematics, SIAM, Philadelphia, 2001.

Turgeon E., PelletierD., “ Verification and Validation in CFD Using an Adaptive Finite-Element

Method”, CASI Journal, vol. 48, n◦ 4, p. 219-231, December, 2002.

Turgeon É., Pelletier D., Borggaard J., “ A Continuous Sensitivity Equation Approach to Optimal

Design in Mixed Convection”, 33rd AIAA Thermophysics Conference, Norfolk, VA,

Jun.-Jul., 1999. AIAA Paper 99-3625.

Turgeon É., Pelletier D., Borggaard J., “ A Continuous Sensitivity Equation Method for Flows

with Temperature Dependent Properties”, 8th AIAA/NASA/USAF/ISSMO Symposium on

Multidisciplinary Analysis and Optimization, Long Beach, CA, Sep., 2000a. AIAA Paper

-4821.

Turgeon É., Pelletier D., Borggaard J., “ A General Continuous Sensitivity Equation Formulation

for Complex Flows”, 8th AIAA/NASA/USAF/ISSMO Symposium on Multidisciplinary

Analysis and Optimization, Long Beach, CA, Sep., 2000b. AIAA Paper 2000-4732.

Turgeon É., Pelletier D., Borggaard J., “ A General Purpose Sensitivity Equation Formulation

for Complex Flows”, Proceedings of the 8th Annual Conference of the Computational Fluid

Dynamics Society of Canada, vol. 2, Montréal, Canada, p. 697-704, June 11-13, 2000c.

Turgeon É., PelletierD., Borggaard J., “ Application of a Sensitivity EquationMethod to the k−

ǫ Model of Turbulence”, 15th AIAA Computational Fluid Dynamics Conference, Anaheim,

CA, Jun., 2001a. AIAA Paper 2001-2534.

Turgeon É., Pelletier D., Borggaard J., “ Computation of Airfoil Flow Derivatives Using a

Continuous Sensitivity Equation Method”, 8th CASI Aerodynamics Symposium, Toronto,

Canada, April, 2001b.

Turgeon É., Pelletier D., Borggaard J., “ A General Continuous Sensitivity Equation Formulation

for the k−ǫModel of Turbulence”, 31st AIAA Fluid Dynamics Conferenceand Exhibit,

Anaheim, CA, Jun., 2001c. AIAA Paper 2001-3000.

Turgeon É., Pelletier D., Borggaard J., “ Sensitivity and Uncertainty Analysis for Variable Property

Flows”, 39th AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, Jan., 2001d.

AIAA Paper 2001-0139.

Vanderplaats G. N., Numerical optimization techniques for engineering design, third edn, Vanderplaats

Research and Development, Colorado Springs, CO, 1999.

Zienkiewicz O. C., Zhu J. Z., “ The Superconvergent Patch Recovery and a Posteriori Error

Estimates. Part 1: The Recovery Technique”, International Journal for Numerical Methods

in Engineering, vol. 33, p. 1331-1364, 1992a.

Zienkiewicz O. C., Zhu J. Z., “ The Superconvergent Patch Recovery and a Posteriori Error

Estimates. Part 2: Error Estimates and Adaptivity”, International Journal for Numerical

Methods in Engineering, vol. 33, p. 1365-1382, 1992b.

Downloads

Published

2008-08-15

How to Cite

Pelletier, D. ., Hay, A., Etienne, S. ., & Borggaard, J. . (2008). The sensitivity equation method in fluid mechanics. European Journal of Computational Mechanics, 17(1-2), 31 to 61. https://doi.org/10.13052/REMN.17.31-61

Issue

Section

Original Article