Topology optimization of modulated and oriented periodic microstructures by the homogenization method in 2-d and in 3-d

Duration: 59 mins 26 secs

Share this media item:

Embed this media item:

<iframe width="_width_" height="_height_" src="https://sms.cam.ac.uk/media/3001613/embed" frameborder="0" scrolling="no" allowfullscreen></iframe>

Choose size:

About this item

Available Formats

About this item

Description:	Geoffroy, P Tuesday 11th June 2019 - 14:30 to 15:30


Created:	2019-06-12 08:41
Collection:	New trends and challenges in the mathematics of optimal design
Publisher:	Isaac Newton Institute
Copyright:	Geoffroy, P
Language:	eng (English)


Abstract:	The work presented here is motivated by the optimization of so-called lattice materials which are becoming increasingly popular in the context of additive manufacturing. We propose a method for topology optimization of structures made of periodically perforated material, where the microscopic periodic cell can be macroscopically modulated and oriented in the working domain. This method is made of three steps. The first step amounts to compute the homogenized properties of an adequately chosen parametrized microstructure (here, a cubic lattice with varying bar thicknesses). The second step optimizes the homogenized formulation of the problem, which is a classical problem of parametric optimization. The third, and most delicate, step projects the optimal oriented microstructure at a desired length scale. In 2-d case, rotations are parametrized by a single angle, to which a conformality constraint can be applied. A conformal diffeomorphism is then computed from the orientation field, thanks which each periodic cell is well oriented in the final structure. The 3-d case is more involved and requires new ingredients. In particular, the full rotation matrix is regularized (instead of just one angle in 2-d) and the projection map which deforms the periodic lattice is computed component by component.

Abstract:

The work presented here is motivated by the optimization of so-called lattice materials which are becoming increasingly popular in the context of additive manufacturing. We propose a method for topology optimization of structures made of periodically perforated material, where the microscopic periodic cell can be macroscopically modulated and oriented in the working domain. This method is made of three steps. The first step amounts to compute the homogenized properties of an adequately chosen parametrized microstructure (here, a cubic lattice with varying bar thicknesses). The second step optimizes the homogenized formulation of the problem, which is a classical problem of parametric optimization. The third, and most delicate, step projects the optimal oriented microstructure at a desired length scale. In 2-d case, rotations are parametrized by a single angle, to which a conformality constraint can be applied. A conformal diffeomorphism is then computed from the orientation field, thanks which each periodic cell is well oriented in the final structure. The 3-d case is more involved and requires new ingredients. In particular, the full rotation matrix is regularized (instead of just one angle in 2-d) and the projection map which deforms the periodic lattice is computed component by component.

Available Formats

Format	Quality	Bitrate	Size
MPEG-4 Video	640x360	1.94 Mbits/sec	864.94 MB	View	Download
WebM	640x360	548.94 kbits/sec	239.02 MB	View	Download
iPod Video	480x270	522.11 kbits/sec	227.28 MB	View	Download
MP3	44100 Hz	249.74 kbits/sec	108.84 MB	Listen	Download
Auto *	(Allows browser to choose a format it supports)