J. Zeman

T. W. J. de Geus, J. Vondrejc, J. Zeman et al.

Computational micromechanics and homogenization require the solution of the mechanical equilibrium of a periodic cell that comprises a (generally complex) microstructure. Techniques that apply the Fast Fourier Transform have attracted much attention as they outperform other methods in terms of speed and memory footprint. Moreover, the Fast Fourier Transform is a natural companion of pixel-based digital images which often serve as input. In its original form, one of the biggest challenges for the method is the treatment of (geometrically) non-linear problems, partially due to the need for a uniform linear reference problem. In a geometrically linear setting, the problem has recently been treated in a variational form resulting in an unconditionally stable scheme that combines Newton iterations with an iterative linear solver, and therefore exhibits robust and quadratic convergence behavior. Through this approach, well-known key ingredients were recovered in terms of discretization, numerical quadrature, consistent linearization of the material model, and the iterative solution of the resulting linear system. As a result, the extension to finite strains, using arbitrary constitutive models, is at hand. Because of the application of the Fast Fourier Transform, the implementation is substantially easier than that of other (Finite Element) methods. Both claims are demonstrated in this paper and substantiated with a simple code in Python of just 59 lines (without comments). The aim is to render the method transparent and accessible, whereby researchers that are new to this method should be able to implement it efficiently. The potential of this method is demonstrated using two examples, each with a different material model.

J. Vondřejc, J. Zeman, I. Marek

The focus of this paper is on the analysis of the Conjugate Gradient method applied to a non-symmetric system of linear equations, arising from a Fast Fourier Transform-based homogenization method due to (Moulinec and Suquet, 1994). Convergence of the method is proven by exploiting a certain projection operator reflecting physics of the underlying problem. These results are supported by a numerical example, demonstrating significant improvement of the Conjugate Gradient-based scheme over the original Moulinec-Suquet algorithm.

M. Jílek, K. Stránská, M. Somr et al.

The emerging field of passive macro-scale tile-based self-assembly (TBSA) shows promise in enabling effective manufacturing processes by harnessing TBSA's intrinsic parallelism. However, current TBSA methodologies still do not fulfill their potentials, largely because such assemblies are often prone to errors, and the size of an individual assembly is limited due to insufficient mechanical stability. Moreover, the instability issue worsens as assemblies grow in size. Using a novel type of magnetically-bonded tiles carried by bristle-bot drives, we propose here a framework that reverses this tendency; i.e., as an assembly grows, it becomes more stable. Stability is achieved by introducing two sets of tiles that move in opposite directions, thus zeroing the assembly net force. Using physics-based computational experiments, we compare the performance of the proposed approach with the common orbital shaking method, proving that the proposed system of tiles indeed possesses self-stabilizing characteristics. Our approach enables assemblies containing hundreds of tiles to be built, while the shaking approach is inherently limited to a few tens of tiles. Our results indicate that one of the primary limitations of mechanical, agitation-based TBSA approaches, instability, might be overcome by employing a swarm of free-running, sensorless mobile robots, herein represented by passive tiles at the macroscopic scale.