An accurate and fast regularization approach to thermodynamic topology optimization

authored by: Dustin Roman Jantos, Klaus Hackl, Philipp Junker
Abstract: In a series of previous works, we established a novel approach to topology optimization for compliance minimization based on thermodynamic principles known from the field of material modeling. Hamilton's principle for dissipative processes directly yields a partial differential equation (referred to as the evolution equation) as an update scheme for the spatial distribution of density mass describing the topology. Consequently, no additional mathematical minimization algorithms are needed. In this article, we introduce a regularization scheme by penalization of the gradient of the spatial distribution of mass density. The parabolic evolution equation (owing to a similar structure to the transient heat-conduction equation) is solved most efficiently by an explicit time discretization. The Laplace operator is discretized via a Taylor series expansion yielding an operator matrix that is constant for the entire optimization process. This method shares some similarities to meshless methods and allows for an accurate application also on unstructured finite element meshes. The minimal size of the structure member can directly be controlled, a priori, by a numerical parameter introduced along with the regularization, similar to classical filter radii.
Organisation(s): Institute of Continuum Mechanics
External Organisation(s): Ruhr-Universität Bochum
Type: Article
Journal: International Journal for Numerical Methods in Engineering
Volume: 117
Pages: 991-1017
No. of pages: 27
ISSN: 0029-5981
Publication date: 02.03.2019
Publication status: Published
Peer reviewed: Yes
ASJC Scopus subject areas: Numerical Analysis, Engineering(all), Applied Mathematics
Electronic version(s): https://doi.org/10.1002/nme.5988 (Access: Closed)
: Details in the research portal "Research@Leibniz University"

BibTeX

@article{0d6377c6e7f5490ab5b6926ed022e6d5,
title = "An accurate and fast regularization approach to thermodynamic topology optimization",
abstract = "In a series of previous works, we established a novel approach to topology optimization for compliance minimization based on thermodynamic principles known from the field of material modeling. Hamilton's principle for dissipative processes directly yields a partial differential equation (referred to as the evolution equation) as an update scheme for the spatial distribution of density mass describing the topology. Consequently, no additional mathematical minimization algorithms are needed. In this article, we introduce a regularization scheme by penalization of the gradient of the spatial distribution of mass density. The parabolic evolution equation (owing to a similar structure to the transient heat-conduction equation) is solved most efficiently by an explicit time discretization. The Laplace operator is discretized via a Taylor series expansion yielding an operator matrix that is constant for the entire optimization process. This method shares some similarities to meshless methods and allows for an accurate application also on unstructured finite element meshes. The minimal size of the structure member can directly be controlled, a priori, by a numerical parameter introduced along with the regularization, similar to classical filter radii.",
keywords = "meshless Laplacian, parabolic PDE, regularization, thermodynamic topology optimization, unstructured meshes, variational material modeling",
author = "Jantos, {Dustin Roman} and Klaus Hackl and Philipp Junker",
note = "Publisher Copyright: {\textcopyright} 2018 John Wiley & Sons, Ltd.",
year = "2019",
month = mar,
day = "2",
doi = "10.1002/nme.5988",
language = "English",
volume = "117",
pages = "991--1017",
journal = "International Journal for Numerical Methods in Engineering",
issn = "0029-5981",
publisher = "John Wiley and Sons Ltd",
number = "9",
}