Program Overview

Topology Optimization

Course Description

The course teaches the finite element method for partial differential equations and gradient-based constrained numerical optimization. The two topics are combined to learn the topology optimization method. Python and fenics are used to implement solvers for partial differential equations and topology optimization problems. Examples from solid mechanics, heat transfer, and wave propagation will be used.

Course Details

• ECTS: 5
• Forms of instruction: Lecture
• Form of examination: Take-home assignment (Assign)
• Languages of instruction: Danish and English
• Level: Bachelor
• Location: Aarhus

Course Content

The course content includes:

• The finite element method for partial differential equations
• Gradient-based constrained numerical optimization
• Topology optimization method
• Implementation of solvers using Python and fenics
• Examples from solid mechanics, heat transfer, and wave propagation

Description of Qualifications

Topology Optimization is a general method to determine a geometry, which results in an optimal value of some property of a physical problem or process. The classical example is to determine the geometry, which minimizes the deflection of a beam using only a certain amount of material. Although the method has its origin in solid mechanics, it has proven useful in many areas of engineering and science, where the associated physical problem can be stated as a partial differential equation. At the end of the course, the students are expected to be able to:

• Deduce a finite element discretization of a partial differential equation.
• Design an optimization problem, where the solution determines an optimal geometry.
• Recommend a filtering and projection scheme for a specific problem in topology optimization.
• Work out an implementation of a topology optimization problem using Python and fenics.
• Argue for the choice of constrained numerical optimization method.

Academic Prerequisites

• A course in advanced mathematics such as "MTMEMAT1-01 Differential geometry and Partial Differential Equations", “Partial Differential Equations” or similar.
• Basic knowledge about physics of mechanics, e.g., the course “Physics and Mechanics” or similar.
• The course “Theory of Elasticity” or similar is recommended.

Program Details

• Type of course: Ordinary, Exchange
• Primary programme: Bachelor's Degree Programme in Engineering (Mechanical Engineering)
• Related programmes: Bachelor's Degree Programme in Engineering (Mechanical Engineering), Master's Degree Programme in Mechanical Engineering
• Department: Department of Mechanical and Production Engineering
• Faculty: Technical Sciences
• Location: Aarhus
• Maximum number of participants: None
• STADS UVA code: U006

Teaching

• Forms of instruction: Lecture and classroom instruction
• Instructor: Søren Peder Madsen
• Course coordinator: Søren Peder Madsen

Examination

• Form of examination: Take-home assignment (Assign)
• Form of co-examination: Internal co-examination
• Assessment: Passed/failed
• Permitted exam aids: All

Comments on the Form of Instruction

During the teaching period, the students will carry out an individual topology optimization project on a subject of their own choice.

Comments on the Examination

• Take-home assignment: An individual written report (in pdf format) and associated Python program files etc. are submitted at the end of the teaching period.
• Re-exam: A new take-home assignment is defined and submitted before the end of the re-exam period.