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Higher Derivative Approximations • General form of Cauchy’s Integral, f (n) (z) = n! f(ξ) 2πi (ξ − z) n+1dξ. • Discretize using a trapezoidal-rule, integrate around circle z + rei , f (n) (z) ≈ n! m−1 f z + re mr i2πj m . Γ j=0 ei2πjn m • Fixed r, use more points for better approximation. Bound on error. • With finite-difference, have to decrease h for better accuracy. • First derivative formula can be derived from this... ...but it is the only one that does not involve subtraction. – Joaquim Martins – January 12, 2000 – 7
Complex Functions and Operators in Fortran • Relational operators – Used with if statements to direct the execution thread. – Complex algorithm must follow same thread. – Therefore, compare only the real parts. – Also, max, min, etc. • Arithmetic functions and operators: – Most of these have a mathematical standard definition that is analytic. – Some of them are implemented in Fortran. – Exception: abs ∂u ∂x = ∂v ∂y = abs(x + iy) = −1 ⇐ x < 0 +1 ⇐ x > 0 −x − iy ⇐ x < 0 +x + iy ⇐ x ≥ 0 . – Joaquim Martins – January 12, 2000 – 8
Page 1 and 2: AN AUTOMATED METHOD FOR SENSITIVITY
Page 3 and 4: Sensitivity Analysis Methods • Fi
Page 5 and 6: Cauchy-Riemann Equations • Extens
Page 7: Simple Numerical Example • Estima
Page 11 and 12: Automatic Implementation • Cookbo
Page 13 and 14: Sensitivity Estimates vs. Step Size
Page 15 and 16: Aerodynamic Sensitivities: FLO82 RA
Page 17 and 18: Residual Convergence of Sensitivity
Page 19 and 20: Drag Coefficient Sensitivity 11 10
Page 21: Further Work • Continue to work o

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