MITESD_77S10_lec09

MITESD_77S10_lec09 - Multidisciplinary System Design...

Info iconThis preview shows pages 1–9. Sign up to view the full content.

View Full Document Right Arrow Icon
1 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Gradient Calculation and Sensitivity Analysis Lecture 9 Olivier de Weck Karen Willcox Multidisciplinary System Design Optimization (MSDO)
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
2 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Today‟s Topics • Gradient calculation methods – Analytic and Symbolic – Finite difference – Complex step – Adjoint method – Automatic differentiation • Post-Processing Sensitivity Analysis – effect of changing design variables – effect of changing parameters – effect of changing constraints
Background image of page 2
3 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Definition of the Gradient 1 2 n J x J x J x J “How does the function J value change locally as we change elements of the design vector x ?” Compute partial derivatives of J with respect to x i i J x J Gradient vector points normal to the tangent hyperplane of J(x) 1 x 2 x 3 x
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
4 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics 0 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 1 x 2 Contour plot 3.1 3.25 3.5 3.5 4 5 Geometry of Gradient vector (2D) Example function: 1 2 1 2 12 1 , J x x x x xx 2 1 1 2 2 2 1 2 1 1 1 1 J x x x J J x x x Gradient normal to contours
Background image of page 4
5 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Geometry of Gradient vector (3D) 222 1 2 3 J x x x 1 2 3 2 2 2 x Jx x increasing values of J 1 x 2 x 3 x Tangent plane 1 2 3 2 2 2 6 0 x x x 111 T o x o T J x Gradient vector points to larger values of J J=3 Example
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
6 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Other Gradient-Related Quantities Jacobian: Matrix of derivatives of multiple functions w.r.t. vector of variables Hessian: Matrix of second-order derivatives 12 1 1 1 2 2 2 z z z n n n J J J x x x J J J x x x J J J x x x J 1 2 z J J J J z x 1 n x z 2 2 2 2 1 2 2 2 2 2 2 1 2 2 1 2 2 1 2 2 1 2 2 n n n n n x J x x J x x J x x J x J x x J x x J x x J x J J H n x n
Background image of page 6
Why Calculate Gradients • Required by gradient-based optimization algorithms – Normally need gradient of objective function and each constraint w.r.t. design variables at each iteration – Newton methods require Hessians as well • Isoperformance/goal programming • Robust design • Post-processing sensitivity analysis – determine if result is optimal – sensitivity to parameters, constraint values 7 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
8 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Background image of page 8
Image of page 9
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 11/08/2011 for the course AERO 16.851 taught by Professor Ldavidmiller during the Fall '03 term at MIT.

Page1 / 44

MITESD_77S10_lec09 - Multidisciplinary System Design...

This preview shows document pages 1 - 9. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online