Course Syllabus

AOE 5064 Structural Optimization (Spring 2019)

Instructor Information

Prof. Canfield, 214 Randolph Hall, 231-5981,

  • Class hours: 9:00-9:50 AM MF, RAND 2087, W HOLD 303, Zoom: Structural Optimization
  • Office hours: 2:30PM - 4:00PM MWF, RAND 214 (or by appointment)

Emergency Preparedness

Catalog Description

Structural optimization via [multivariable] calculus (of variations). Application of techniques of mathematical programming to optimize trusses, beams, frames, columns, and other structures. Sensitivity calculation of structural response. Approximation techniques and dual and optimality criteria methods.

• Solid Mechanics (undergraduate)
• MATLAB programming

Required Textbook

Elements of Structural Optimization, Raphael T. Haftka and Zafer Gürdal,
Kluwer Academic Publishers, 3rd Revised and Expanded Edition, 1992


Numerical Optimization Techniques for Engineering Design: With Applications, Garret N. Vanderplaats, McGraw Hill, 1984 (
Introduction to Optimum Design, Jasbir S. Arora, 3rd Edition, Elsevier Academic Press, 2012

The textbook will be supplemented with reading material from seminal journal papers (found on Scholar Resources).

Course Objectives

The objectives of this course are for each student to …
  • Understand how to formulate a structural optimization problem, including defining appropriate design variables, constraints, and objective functions.
  • Understand how structural analysis methods are integrated with optimization methods to synthesize a structural design.
  • Distinguish among sizing, shape, and topology optimization as classes of structural optimization.
  • Comprehend how to calculate sensitivity derivatives.
  • Apply various approximation methods to construct a sequence of approximate structural design problems appropriate for static strength, natural frequencies, buckling, and dynamic response.
  • Solve structural design problems according to either the primal or dual method.
  • Evaluate the results of a structural optimization using optimality criteria to determine the nature of the solution.
  • Apply appropriate algorithms for discrete design variables and multi-objective optimization problems.
  • Complete a structural optimization design project.

Course Schedule (Syllabus)

Hour Topic Paper Haftka
1. Administration & Introduction  
2. Structural Synthesis Schmit 1
3. Elasticity Review & Axial Bar   1.2­–1.3
4. Finite Element Stiffness of Axial Bar   3.3
5. Finite Element Assembly of Trusses   6.7
6. Finite Element Analysis of Trusses   6.7
7. Calculus Review   2.1
8. Method of Lagrange Multipliers   2.3
9. Karush-Kuhn-Tucker Optimality Conditions   5.1
10. Sensitivity Analysis of Discrete Systems Martins 7.1
11. Direct Method for Sensitivity Derivatives Vanderplaats 7.2.1
12. Adjoint Method for Sensitivity Derivatives Arora & Haug 7.2.1
13. Sensitivity of Natural Frequencies   7.3.1
14. Complex Step Derivatives Martins  
15. Linear Programming (LP)   3.3­–3.5
16. Simplex Algorithm for LP   3.6
18. Sequential Linear Programming (SLP)   6.3
19. Numerical Optimization   4.2–4.3
20. Fully Stressed Design (FSD)   9.1
21. Optimality Criterion (OC) Methods Venkayya 9.3
22. Midterm Exam on Lessons 2–19    
23. Topology Optimization   6.5
24. Solid Isotropic Material with Penalization Rozvany, Sigmund  
25. Sequential Quadratic Programming (SQP) Schittkowski 5.2, 5.9
26. Approximation Concepts Schmit, Farshi 6
28. Reciprocal Approximation   6.1.1
29. Force Approximation Vanderplaats  
30. Rayleigh Quotient Approximation (RQA) Canfield 6.2.2
31. Dual Formulation Fleury 9.2
32. Conservative Approximation Starnes 6.1.1
33. Method of Moving Asymptotes (MMA) Svanberg  
34. Two-Point Exponential Approximation (TPEA) Fadel 6.1.2
35. Move Limits & Constraint Deletion Schmit 6.4
36. Trust Regions Alexandrov 4.3.3
37. Shape Optimization Hansen 6.5
38. Multi-Objective (Pareto) Optimization   1.2.2
39. Genetic Algorithms Hajela 4.4
40. Discrete Variables: Branch & Bound   3.9
41. End-of-Term Exam  
42. Final Project Presentations  

Administrative dates and information for this class are provided by the Office of the Registrar: 

Grading Policy

 Homework & Class Participation
 In-Class Test 1
 In-Class Test 2
 Final Project

Tests will be graded on a standard numeric scale. If the mean is below a B, then scaled T-scores* may be used to determine letter grade for the test. 

Grading Scale
 A  >= 93
 A–  >= 90
 B+  >= 87
 B  >= 83
 B–  >= 80
 C+  >= 77
 C  >= 73
 C–  >= 70
 D+  >= 67
 D  >= 63
 D–  >= 60
 F  >= 0

Each homework problem and the final project will be graded on the following letter grade scale. The homework letter grades will be converted to a numeric score, according the the test grading scale, incorporating class participation.

Homework and Project Grades
A Correct Answer and Method
B Correct Method
C Nice Try
D No Clue
F No Attempt

Students are encouraged to discuss homework problems with one another and may compare approach and results, but they are expected to turn in their own, individual work. Do not share computer code. Submit only assignments that are your own work.

The final project grade will be based on a class presentation (40%) and a written report (60%).

**The article, Testing Memo 6: What kind of Grades Should be Averaged?, explains T-scores.

Honor System

Students shall work independently on tests, exams, and projects and submit only their own work. Students are permitted to discuss homework problems in groups, but they are expected to turn in their own, individual work. Computer programming code may not be shared, copied, or distributed among students before an assignment is due.

Student and instructor behavior in this class is governed by the Virginia Tech Honor Code and its core values:

  • Mutual Trust
  • Intellectual Honesty
  • Honesty and Integrity promote quest for Truth
All assignments submitted shall be considered graded work and shall be completed on an individual basis, unless otherwise stated. While discussing assignments and getting help outside of class is both authorized and encouraged, copying solutions from any source is considered a violation, as is sharing or re-use of an unauthorized (i.e., not provided or authorized by the instructor) computer file in full or in part. Honesty in your academic work will develop into professional integrity. The faculty and students of Virginia Tech will not tolerate academic dishonesty. It is your responsibility to seek clarification if there is a question about how the Honor Code applies to a given assignment. Suspected violations of the Honor Code will be processed and dealt with as recommended by the Honor Court.

Attendance and Classroom Behavior

Virginia Tech has a class attendance policy. Class meetings are an integral part of most courses and are the central component of many. Students and faculty are expected to attend class at all regularly scheduled times, except for cancellations announced on a university wide basis by the appropriate authority. When students cannot attend a class, it is their responsibility, as soon as possible, to consult with the course instructor about missed work or tests.

Students are expected to respect one another and the instructors in and outside the classroom. Computers may be used in the classroom only for viewing material for this course or for taking notes. Accessing audio, images, or videos during class may be distracting to other students and is strictly prohibited. Cell phone use is prohibited, except as a student response system.

Policy for Making Up Assignments

Requests to make up for a missed test or assignment must be made by notifying the instructor in advance of the scheduled due date. Requests due to absences shall be documented through the the Dean of Students, who can verify your absence and notify all of your instructors at once. In case of absences for health reasons, Schiffert Health Center medical staff can verify that you have been given medical treatment. For family or personal emergencies, students should consult with Dean of Students Office, who will notify the Associate Dean of Academic Affairs for Engineering. Circumstances out of the student's control, such as illness, a death in the family, or making a presentation at a professional conference, shall normally be considered an excused absence that justifies the request. Circumstances within the student's control are normally considered an unexcused absence for which a make-up request may be denied.

Unscheduled or undocumented requests will be granted or denied at the discretion of the instructor. The make-up test may be an oral exam.

Students may not discuss with other students a test being made up in between the time the student and the class take the test. 

Students with Special Needs

Reasonable and appropriate academic accommodations will be made for students who provide documentation of disability and request for such accommodation.

Please inform the instructor and teaching assistant within the first two weeks of class of potential conflicts that may arise due to participation in religious or ethnic holiday events.

Final Project (Exam)

The final project report will be due on Monday, May 13, 2019 during final exam week.

Course Summary:

Date Details