Sensitivity Analysis
Sensitivity Analysis: Theoretical Foundations
What is Sensitivity Analysis?
Professor, what is sensitivity analysis?
It quantifies how much a small change in a design variable affects the objective function. It provides information to determine the "direction" for optimization.
Methods for Calculating Sensitivity
| Method | Computational Cost | Accuracy |
|---|---|---|
| Finite Difference Method | Number of design variables × FEM runs | Approximate (depends on step size) |
| Analytical Sensitivity (Direct Method) | 1 FEM run + additional calculation | Exact |
| Adjoint Method (Adjoint) | 1 FEM run + 1 adjoint analysis | Exact. Most efficient when there are many design variables |
Is the adjoint method the most efficient?
For cases with tens of thousands of design variables like topology optimization, the adjoint method can obtain sensitivities for all variables with one additional calculation. The finite difference method would require tens of thousands of FEM runs.
Summary
The Adjoint Method for Sensitivity Analysis is from Fox & Kapoor (1968)
The "Adjoint method" used for sensitivity calculation in structural optimization was first formulated by Fox & Kapoor (1968, AIAA) as eigenvalue sensitivity for vibration problems. The revolutionary aspect of the adjoint method is that the computational cost for sensitivity becomes constant (O(1)) regardless of the number of design variables N_d, allowing efficient sensitivity calculation even for problems with thousands of design variables. This characteristic supports the practical implementation of modern topology optimization (SIMP method), and internally, all sensitivity calculations in OptiStruct and ABAQUS Topology use the Adjoint method.
Computational Methods for Sensitivity Analysis
FEM for Sensitivity Analysis
Summary
Automatic Differentiation (AD) Drastically Reduced Implementation Cost for Sensitivity Analysis
Manually deriving sensitivities (gradients) for complex CAE codes involves enormous cost, so "Automatic Differentiation (AD)" has been in practical use since the 1990s. AD analyzes the computational graph of source code to automate the application of the chain rule numerically. Forward mode is suitable for sensitivity calculation per design variable, while reverse mode (Backpropagation) is suitable for calculating sensitivities of all variables per objective function. The automatic differentiation engines in TensorFlow and PyTorch use the same technology as neural network learning and are also utilized in CAE optimization frameworks like OPENMDAO (NASA).
Sensitivity Analysis in Practice
Sensitivity Analysis in Practice
Check sensitivities as a preliminary step to optimization. Identify which design variables most affect the objective function.
Practical Checklist
Quantify Which Design Variables are Important with Sobol' Indices
In sensitivity analysis practice, identifying "which design variables most affect the objective function" is important, and the variance-based global sensitivity measure "Sobol' indices" has become a standard tool. Proposed by Ilya Sobol' (Russian Academy of Sciences) in 1993, this method decomposes each design variable's contribution rate into 1st order and 2nd order (interaction) components. In Toyota's engine fuel efficiency optimization, Sobol' analysis of 10 variables (compression ratio, ignition timing, injection amount, etc.) was conducted, confirming that 65% of total variation originated from the compression ratio alone, enabling efficient optimization.
Sensitivity Analysis: Software & Solver Comparison
Tools
OpenMDAO was Open-Sourced by NASA in the 2010s
OpenMDAO (Open Multidisciplinary Design, Analysis, and Optimization) is an MDO (Multidisciplinary Design Optimization) framework developed by NASA Ames Research Center in the 2010s and released as open source. It provides adjoint method, automatic differentiation, and complex-step differentiation through a unified API and is widely adopted as a design optimization education tool in major aerospace engineering departments at MIT, Stanford, TUDelft, etc. Boeing disclosed at the 2017 AIAA SciTech Forum that they utilized OpenMDAO for wing shape optimization of the 737 MAX.
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