An 808 Line Phasor-Based Dehomogenisation Matlab Code For Multi-Scale Topology Optimisation

TL;DR

The paper addresses multi-scale topology optimisation by coupling homogenisation-based macro-model optimization with a phasor-based dehomogenisation that reconstructs a fine-scale, manufacturable design. The main approach combines Rank-2 lamination parameterisation with on-the-fly dehomogenisation, enabling real-time or near-real-time translation of a coarse multi-scale solution into a single-scale structure, while preserving stiffness under a volume constraint. Key contributions include an 808-line educational Matlab tool, robust phase-alignment and branch-closure techniques, and demonstrations across Michell cantilever and MBB/DB models, achieving single-scale performance within about 10% of the multi-scale solution and with orders-of-magnitude speed improvements. The work demonstrates practical impact for engineering design by enabling interactive exploration of manufacturable multi-scale laminates on standard hardware, with extensions to passive variables and multiple loading cases explored for future development.

Abstract

This work presents an 808-line Matlab educational code for combined multi-scale topology optimisation and phasor-based dehomogenisation titled deHomTop808. The multi-scale formulation utilises homogenisation of optimal microstructures to facilitate efficient coarse-scale optimisation. Dehomogenisation allows for a high-resolution single-scale reconstruction of the optimised multi-scale structure, achieving minor losses in structural performance, at a fraction of the computational cost, compared to its large-scale topology optimisation counterpart. The presented code utilises stiffness optimal Rank-2 microstructures to minimise the compliance of a single-load case problem, subject to a volume fraction constraint. By exploiting the inherent efficiency benefits of the phasor-based dehomogenisation procedure, on-the-fly dehomogenisation to a single-scale structure is obtained. The presented code includes procedures for structural verification of the final dehomogenised structure by comparison to the multi-scale solution. The code is introduced in terms of the underlying theory and its major components, including examples and potential extensions, and can be downloaded from https://github.com/peterdorffler/deHomTop808.git.

Figures (23)

• Figure 1: An overview of the example model and the multi-scale topology optimisation and dehomogenisation principles. The model of the baseline problem being optimised, as well as the element and node counting conventions are illustrated in (a). The relation between the FE-grid, FE-element and dehomogenisation grids is illustrated in (b). (c) illustrates the nature of the multi-scale structure being optimised, with element-wise Rank-2 material, and (d) how the multi-scale structure relates to the realised single-scale structure obtained by phasor-based dehomogenisation.
• Figure 2: Final multi-scale structure from running the deHomTop808 code the first time, with lines indicating the Rank-2 orientations and layer widths. The greyscale indicates relative density.
• Figure 3: Final single-scale structure obtained by post dehomogenisation with dmin=0.2 of the multi-scale structure in \ref{['fig:firstRun01']} (top). Plot of the local energy of the post-dehomogenised structure (bottom).
• Figure 4: Result from the second featured call to the deHomTop808 code, with the optimised multi-scale structure from the first run as input. Dehomogenisation and analysis are performed for a halved minimal feature size, dmin=0.1, compared to \ref{['fig:firstRun01']}. The dehomogenised structure (top) as well as the local energy distribution (bottom) of this single-scale structure is plotted.
• Figure 5: Illustrating the relation between the definition of a single phasor kernel and its correspoding spatial signal. (a) presents the anisotropic contour-lines of the Gaussian weights about the kernel centre $\mathbf{\mathring{x}}$ with normal $\mathbf{n}$ and degree of anisotrpy $\mathbf{r}=(1/2,2)^\intercal$. The weighted complex signal emitted from this kernel, for some fixed frequency $\omega$, is separated into its real and imaginary parts in (b) and (c) respectively. (d) presents the real-valued argument of this complex signal, where the weight-induced magnitude variations in the complex field are normalised to form a perfectly contrasting oriented periodic wave. The kernel phase shift $\varphi$ controls the value at the kernel centre, and the wavelength $L/\omega$ is controlled by the input frequency $\omega$ and the span $L$ of the domain sampled.