Concepts and Applications. Engineering Optimization. Fred van Keulen Matthijs Langelaar CLA H21.1 A.vanKeulen@tudelft.nl. Contents. Sensitivity analysis. Sensitivity of system response (state variables). u constant. s i constant. Sensitivity analysis.
Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.
Sensitivity of system response (state variables)
u constant
si constant
Sensitivity analysisTotal derivative
Partial derivatives
Design variable
Aspects of sensitivity analysisExact
Numericalmodel
Implementation Efficiency
Very easy Terrible*
Moderate As good
as it gets
Lots of work As good
as it gets
Governingequations
Discrete
Discretization
Differentiation
x
Governingequations
Continuum
Differentiation
Discretization
Sensitivity analysis approaches (2)Schematically:
f
x
Model
x
GFD

+
f+Df
x+Dx
Model
Derivativecode
Automaticdifferentiation
Analysiscode
(central)
Condition error
Truncation error
Finite difference derivatives (3)Normalized stress constraint
Hole radius
Load
Displacement
Solution obtained by Newton iterations:
(e.g. )
For small design perturbation, this approaches:
Originalresidual
Nonlinear pathindependent casei.e. subtract original residual from new residual.
State variable vector sensitivity
Discrete derivativesDecomposed Kalready available (direct solver)!
State variable sensitivityPseudoload vector
Already decomposed tangent matrix
(direct solver)
State variable sensitivity (2)Options for calculation of pseudoload vector:
a) Analytical differentiation (lots of work)
b) Automated differentiation (code generator programs)
c) Finite difference approach
Discrete derivativesCHEAP!!
Sensitivity analysis much cheaper than analysis itself!
SEE APPENDIX
Difference consists of order of computations:
Adjoint vs. directGoverningequations
Discretization
Differentiation
Discrete derivative summaryGoverningequations
Differentiation
Discretization
q(x)
I(x,s)
x
Governing equation:
+ boundary conditions
Continuum derivativesGoverning equation
Sensitivity equation
Continuum derivatives (2)Points of attention
Implementation
Adjoint mode
Efficiency
Accuracy
Finite difference
Discrete derivativesSemianalytical
Continuum derivatives