AE6310 Course Notes

Welcome to AE6310: Optimization for the Design of Engineered Systems.

AE6310 is about the application of numerical optimization methods to the design of engineering systems.

Recommended references

I recommend the following reference textbooks for further information about the course material:

1. Engineering Design Optimization, J.R.R.A. Martins and A. Ning

2. Numerical Optimization, J. Nocedal and S.J. Wright

3. Practical Methods of Optimization, R. Fletcher

4. Practical Optimization, M.H. Wright, P.E. Gill, and W. Murray

5. Trust Region Methods, A.R. Conn, N.I.M. Gould, and P.L. Toint

6. Convex Optimization, S. Boyd and L. Vandenberghe

7. Numerical Optimization Techniques for Engineering Design, G.N. Vanderplaats

8. Multidisciplinary Design Optimization supported by Knowledge Based Engineering, Sobieski, Moris, and van Tooren

Engineering Design using Optimization

In this course, we will look at how to apply optimization methods to engineering problems. Below are two examples of complex design problems that are solved using numerical optimization methods.

Aeroelastic example

Aircraft wings are flexible and deform under aerodynamic load, twisting and bending relative to their undeformed shape. The aerodynamic loads, in turn, depend on the deformed shape of the wing. As a result, there is a tight coupling between aerodynamics and structures. To find the equilibrium shape of the wing, we have to couple aerodynamic and structural disciplines together into a single analysis.

Aerostructural optimization consists of finding the wing planform, shape and structural sizing variables that optimize the performance of the aircraft. The following is a simple aerostructural optimization from the open source tool OpenAeroStruct.

A key consideration in multidisciplinary design problems is connecting the different disciplines together for analysis and design. The extended design structure matrix (XDSM) diagram is a convenient way to visualize how the different analysis components interact together. The following XDSM diagram is from the above aerostructural optimization problem.

Structural topology optimization example

In structural design you often do not know in advance the best structural shape or topology to meet design requirements. Topology optimization methods are designed to generate an optimized structure, free from restrictions on the shape or topology.

This image illustrates the results of a topology optimization case where the the objective is to minimize the mass of the structure subject to a constraint on the approximate maximum stress. Topology optimization problems may involve millions of design variables and require the solution of a challenging finite-element simulation. This course does not cover topology optimziation specifically, however this type of application can be tackled with the knowledge and theory developed in this course.

Table of Contents

• Introduction to Python
  • Python basics
  • NumPy basics
  • matplotlib basics
• Introduction to Optimization
  • Design formulation and parametrization
  • Simulation-based optimization
  • Types of optimization algorithms and their performance
  • One-dimensional minimization from calculus
  • Linear Algebra Background
  • Definitions
  • Calculus background
• Unconstrained Optimization Theory
  • What is an optimal solution?
  • First and Second-order necessary conditions for unconstrained optimization
  • Second-order sufficient conditions for unconstrained optimization
  • Using Unconstrained Optimization Theory
  • Live Demo of Unconstrained Optimization
• Minimization of Quadratic Functions
  • Examples
• Optimization algorithms: Fundamentals
  • Line Search Methods
  • Trust Region Methods
• Line Search Methods
  • Interactive Line Search for the Rosenbrock Function
  • Example: 1D line search
  • Example: Rosenbrock function
  • Exact minimizers for line searches
  • Graphical line search
  • Inexact line search criteria
  • Wolfe conditions
  • Second Wolfe condition
  • Strong Wolfe conditions
  • Backtracking line search
  • Interpolation-based backtracking
  • Strong Wolfe condition line search
• Line Search Algorithms
  • Steepest descent method
  • Conjugate Gradient Method
  • Newton’s Method
  • Quasi-Newton Methods
• Trust region methods
  • Interactive Trust Region for the Rosenbrock Function
  • Basic trust region algorithm
  • The Cauchy point
  • Steihaug–Toint algorithm
  • Exact trust region subproblem solution
  • Example: Exact step computation for 2D problems
• Constrained Optimization Theory
  • Active constraints
  • Descent and Feasible directions
  • The Karush-Kuhn-Tucker conditions: First-order necessary conditions
  • Second-order necessary conditions
  • Second-order sufficient conditions
• Constrained Optimization Algorithms
  • Penalty Methods and Augmented Lagrangian Methods
  • Sequential Quadratic Optimization (Sequential Quadratic Programming)
  • Primal-Dual Interior-Point Methods
  • Key Features of Constrained Optimization
  • Pointers for using Constrained Optimization Algorithms
• Simulation-based Optimization
  • Background example: Lifting line analysis of a wing
  • Optimization problem formulations for simulation-based design
  • Derivative evaluation methods for simulation-based optimization
  • Finite-difference methods
  • The complex-step method
  • The Adjoint Method
  • The Adjoint and Direct Methods
  • Implementation of the Adjoint Method
  • Automatic differentiation
  • AD in practice: Autograd
• Case Study: Trajectory Optimization
  • Optimization problem formulation
  • Discretization in time
  • Adjoint method
  • Full-space method
  • Automatic differentation
  • Minimum Time to Climb Problem
  • The Brachistochrone
• Case Study: PDE Constrained Optimization
  • The continuous adjoint
  • The discrete adjoint
  • Dual consistency
  • Nonlinear PDE-constrained optimization
  • Finite element equations
  • Optimization problem
  • Nozzle optimization
• Multidisciplinary Optimization
  • Multidisciplinary analysis case study: Aeroelastic analysis
  • Components of Aeroelastic Analysis and Design Optimization
  • Multidisciplinary Feasible Architecture
  • XDSM Diagrams: Visualizing and representing MDO problems
  • OpenMDAO: Complex multidisciplinary problems
  • Aerodynamic analysis
  • Structural analysis
  • Load and displacement transfer
  • Aeroelastic Model
• Surrogate modeling
  • Surrogate modeling applications and terminology
  • Steps in surrogate model construction
  • Regularization
  • Polynomials in multiple dimensions
  • Radial basis functions
  • Design of experiments
  • Neural Networks
  • Principal Component Analysis
  • Optimization Under Uncertainty
• Derivative Free Optimization
  • Terminology
  • Genetic Optimization Algorithms
  • Particle Swarm Optimization
  • Coordinate and pattern search methods
  • Model-based DFO
• 2026 Class Project: Race Car Minimum Lap Time
  • Introduction
  • Track Model
  • Vehicle Model
  • Optimal Control Problem
  • Vehicle Parameters
  • Pacejka Tire Parameters
  • Numerical Method
  • Assignment
  • Report Requirements
  • References
• 2025 Class Project: Minimum time to climb problem
  • Assignment
  • Report requirements

Indices and tables

• Index

• Module Index

• Search Page