Topology Optimization of transient problems using an adaptive diagonally implicit Runge-Kutta scheme

Abstract

This thesis derives a consistent adjoint sensitivity analysis for an adaptive time-
stepping scheme, the Singly Diagonally Implicit Runge-Kutta scheme and evaluates
its performance by comparing the scheme with a conventional time integration ap-
proach. The goals are to derive the sensitivity analysis and assess whether solution
accuracy can be improved while preserving computational efficiency. Both the Runge-
Kutta scheme and the adjoint sensitivity analysis are implemented within the PETSc
framework to generate optimized designs with respect to transient heat conduction
problems.

Although transient heat conduction has been widely studied in the context of topology
optimization, prior research predominantly relies on basic time integration schemes
and is mostly limited to two-dimensional problems. Gradient-based optimization in
transient settings presents additional challenges, as it requires efficient computation
of gradients of objective and constraint functions. To address this, an adjoint sensi-
tivity analysis for the Runge-Kutta scheme is derived and validated against numerical
gradient approximations.

The performance of the adaptive time-stepping scheme is tested by comparison to the
implicit Euler scheme using two heat source scenarios: one constant and the other
varying with time applied to a simple test case. The results indicate that the Runge-
Kutta scheme is more accurate for the same computational effort, and comparing an
adaptive step size to a non-adaptive one indicates that the smallest error of the solution
comes with the adaptive scheme. Furthermore, to validate the derived sensitivity
analysis and its implementation together with the Runge-Kutta scheme, optimized
designs are generated in three dimensions and compared to results from prior studies
in 2D. The obtained results are somewhat different but some similarities are present
and general characteristics regarding terminal times and appearance are obtained.

Popular Abstract

How can we design materials and structures to effectively manage heat?
This project tackles that question by exploring smarter ways to simulate
and optimize heat flow in engineering systems.

From laptops and smartphones to cars and industrial machines,
managing heat is essential. Overheating can cause device failure,
reduce their efficiency, or make them unsafe to use.
To address this, engineers may use optimization techniques to determine
the best way to place materials in a structure for efficient heat management.

This project used a smarter and adaptive way to simulate heat flow.
By focusing extra computing power on the important moments, like when
temperatures spike and change rapidly, the method delivers more accurate results
while still being efficient. The approach was tested on a 3D heat-dissipating structure
and outperformed conventional simulation techniques, especially in dealing with sudden
temperature changes. When designing an optimized structure to spread heat away from its center,
the designs became more branched and spread out as the working time increased.
Another key finding was how the method could adapt its focus during simulations.
When the temperature increased rapidly, the simulation automatically took smaller steps,
ensuring accurate results during critical moments. This ability to adjust makes the method not only
more reliable, but also more practical for solving real-world engineering problems.

Efficient heat management is essential in many areas, from building faster and safer
electronics to designing greener energy systems. Poor heat handling can lead to wasted energy,
shorter product lifespans, and higher costs. Using advanced computer simulations, engineers can
design smarter solutions that save energy, improve safety, and extend the life of devices
and structures. This research provides a powerful new tool for engineers, enabling them to make
better decisions about material placement and design. Whether it is cooling the next
generation of supercomputers or improving the insulation in homes, these smarter
designs have the potential to make a big impact.

• author: Andersson, Andréas
• supervisor: Gunnar Granlund
• organization: Solid Mechanics; Department of Construction Sciences
• course: FHLM01 20242
• year: 2025
• type: H3 - Professional qualifications (4 Years - )
• subject: Technology and Engineering
• keywords: Topology optimization, Transient heat transfer, Sensitivity analysis, PETSc
• publication/series: TFHF-5000
• report number: TFHF-5263
• language: English
• id: 9181699
• date added to LUP: 2025-01-23 09:13:53
• date last changed: 2025-01-23 09:13:53