Sensitivity information is often of interest in engineering applications (e.g., gradient-based optimization). Heat transfer problems frequently involve complicated geometries for which exact solutions cannot be easily derived. As such, it is common to resort to numerical solution methods such as the finite element method. The semi-analytic complex variable method (SACVM) is an accurate and efficient approach to computing sensitivities within a finite element framework. The method is introduced and a derivation is provided along with a detailed description of the algorithm which requires very minor changes to the analysis code. Three benchmark problems in steady-state heat transfer are studied including a nonlinear problem, an inverse shape determination problem, and a reliability analysis problem. It is shown that the SACVM is superior to the other methods considered in terms of computation time and sensitivity to perturbation size.
Issue Section:
Heat and Mass Transfer
Keywords:
Conduction
References
1.
Jin
, W.
, Dennis
, B. H.
, and Wang
, B. P.
, 2010
, “Improved Sensitivity Analysis Using a Complex Variable Semi-Analytical Method
,” Struct. Multidiscip. Optim.
, 41
(3
), pp. 433
–439
.2.
Hu
, R.
, Jameson
, A.
, and Wang
, Q.
, 2012
, “Adjoint-Based Aerodynamic Optimization of Supersonic Biplane Airfoils
,” J. Aircr.
, 49
(3
), pp. 802
–814
.3.
Sherman
, L. L.
, L. L.
, III, T
, A. C.
, Green
, Newman
, P. A.
, Hou
, G. W.
, and Korivi
, V. M.
, 1996
, “First- and Second-Order Aerodynamic Sensitivity Derivatives Via Automatic Differentiation With Incremental Iterative Methods
,” J. Comput. Phys.
, 129
(2
), pp. 307
–331
.4.
Lyness
, J. N.
, and Mohler
, C. B.
, 1967
, “Numerical Differentiation of Analytic Functions
,” SIAM J. Numer. Anal.
, 4
(2
), pp. 202
–210
.5.
Lyness
, J. N.
, 1968
, “Differentiation Formulas for Analytic Functions
,” Math. Comp
, 22
(102
), pp. 352
–362
.6.
Squire
, W.
, and Trapp
, G.
, 1998
, “Using Complex Variables to Estimate Derivatives of Real Functions
,” SIAM Rev.
, 40
(1
), pp. 110
–112
.7.
Newman
, J.
, III, Whitfield
, D.
, and Anderson
, W.
, 1999
, “A Step-Size Independent Approach for Multidisciplinary Sensitivity Analysis and Design Optimization
,” AIAA
Paper No. AIAA-99-3101.https://arc.aiaa.org/doi/abs/10.2514/6.1999-31018.
Anderson
, W. K.
, Newman
, J. C.
, and Nielsen
, E. J.
, 1999
, “Sensitivity Analysis for the Navier–Stokes Equations on Unstructured Meshes Using Complex Variables,” AIAA
Paper No. AIAA-99-3294.9.
Martins
, J. R. R. A.
, Sturdza
, P.
, and Alonso
, J. J.
, 2003
, “The Complex-Step Derivative Approximation
,” ACM Trans. Math. Software
, 29
(3
), pp. 245
–262
.10.
Wang
, B. P.
, and Apte
, A. P.
, 2006
, “Complex Variable Method for Eigensolution Sensitivity Analysis
,” AIAA J.
, 44
(12
), pp. 2958
–2961
.11.
Rodriguez
, D.
, 2000
, “A Multidisciplinary Optimization Method for Designing Inlets Using Complex Variables
,” AIAA
Paper No. AIAA-2000-4875.12.
Errico
, R. M.
, 1997
, “What Is an Adjoint Model?
,” Bull. Am. Meteorol. Soc.
, 78
(11
), pp. 2577
–2591
.13.
Plessix
, R.-E.
, 2006
, “A Review of the Adjoint-State Method for Computing the Gradient of a Functional With Geophysical Applications
,” Geophys. J. Int.
, 167
(2
), pp. 495
–503
.14.
Dowding
, K. J.
, and Blackwell
, B. F.
, 2000
, “Sensitivity Analysis for Nonlinear Heat Conduction
,” ASME J. Heat Transfer
, 123
(1
), pp. 1
–10
.15.
Cao
, Y.
, Li
, S.
, Petzold
, L.
, and Serban
, R.
, 2003
, “Adjoint Sensitivity Analysis for Differential-Algebraic Equations: The Adjoint DAE System and Its Numerical Solution
,” SIAM J. Sci. Comput.
, 24
(3
), pp. 1076
–1089
.16.
Barthelemy
, B.
, and Haftka
, R. T.
, 1988
, “Accuracy of the Semi-Analytical Method for Shape Sensitivity Calculation
,” AIAA
Paper No. 88-2284.https://arc.aiaa.org/doi/10.2514/6.1988-228417.
Olhoff
, N.
, and Rasmussen
, J.
, 1991
, “Study of Inaccuracy in Semi-Analytical Sensitivity Analysis—A Model Problem
,” Struct. Optim.
, 3
(4
), pp. 203
–213
.18.
Cheng
, G.
, Gu
, Y.
, and Wang
, X.
, 1990
, “Improvement of Semi-Analytic Sensitivity Analysis and Mcads
,” International Conference Engineering Optimization in Design Processes
, Karlsruhe, Germany, Sept. 3–4.19.
Chang
, G.
, and Olhoff
, N.
, 1991
, “New Method of Error Analysis and Detection in Semi-Analytical Sensitivity Analysis
,” NATO/DFDASI Berchtesgaden
, pp. 361
–383
.20.
Mlejnek
, H.
, 1992
, “Accuracy of Semi-Analytical Sensitivities and Its Improvement by the Natural Method
,” Struct. Optim.
, 4
(2
), pp. 128
–131
.21.
Oral
, S.
, 1996
, “An Improved Semianalytical Method for Sensitivity Analysis
,” Struct. Optim.
, 11
(1–2
), pp. 67
–69
.22.
Parente
, E.
, and Vaz
, L. E.
, 2001
, “Improvement of Semi-Analytical Design Sensitivities of Non-Linear Structures Using Equilibrium Relations
,” Int. J. Numer. Methods Eng.
, 50
(9
), pp. 2127
–2142
.23.
IEEE, 2008, “IEEE Standard for Floating-Point Arithmetic,” Institute of Electrical and Electronics Engineers, Piscataway, NJ, Standard No.
IEEE
754-2008.24.
Zienkiewicz
, O.
, Taylor
, R.
, and Zhu
, J.
, 2013
, The Finite Element Method: Its Basis and Fundamentals
, 7th ed., Butterworth-Heinemann
, Oxford, UK
.25.
Nocedal
, J.
, and Wright
, S. J.
, 2006
, Numerical Optimization
, 2nd ed., Springer
, New York.26.
Haldar
, A.
, and Mahadevan
, S.
, 2000
, Reliability Assessment Using Stochastic Finite Element Analysis
, 1st ed., Wiley
, New York.Copyright © 2018 by ASME
You do not currently have access to this content.
