Honeycomb Tessellation Research Articles

Topology optimization for isotropic elastic materials using honeycomb tessell

Topology optimization is gaining popularity as a primary tool for engineers in the initial stages of design. Essentially, the design domain is broken down into individual pixels, with the material density of each element or mesh point serving as a design variable. The optimization problem is then tackled through mathematical programming and optimization methods that rely on analytical gradient calculation. In this study, topology optimization using honeycomb tessellation elements is explored. Hexagonal elements have the ability to flexibly connect two adjacent elements. The use of the hexagonal element limits the occurrence of the checkerboard pattern to the finite elements of the quadrilateral standard Lagrangian type. A mathematical model is developed with the objective function being the minimum compliance value of the design domain. The element stiffness matrix is constructed using the strain-displacement matrix and the constitutive matrix, assuming a unit Young's modulus. Additionally, optimal conditions are established using Lagrangian multipliers. Two sensitivity and density filtering filters are employed to increase optimization efficiency, prevent the algorithm from reaching a local optimal state, and speed up convergence. If the suggested filter is employed, the objective function achieves a value of c=173,0293 and convergence is attained after 200 iterations. In contrast, without using the filter, the objective function has a larger value (c=186,7922) and convergence occurs at the 27th iteration. The results are significant for optimizing topology to meet specific boundary condition requirements. This paper proposes a novel approach using a combination of filters to advance topology optimization using hexagonal elements in future applications.

HoneyTop90: A 90-line MATLAB code for topology optimization using honeycomb tessellation

This paper provides a simple, compact and efficient 90-line pedagogical MATLAB code for topology optimization using hexagonal elements (honeycomb tessellation). Hexagonal elements provide nonsingular connectivity between two juxtaposed elements and, thus, subdue checkerboard patterns and point connections inherently from the optimized designs. A novel approach to generate honeycomb tessellation is proposed. The element connectivity matrix and corresponding nodal coordinates array are determined in 5 (7) and 4 (6) lines, respectively. Two additional lines for the meshgrid generation are required for an even number of elements in the vertical direction. The code takes a fraction of a second to generate meshgrid information for the millions of hexagonal elements. Wachspress shape functions are employed for the finite element analysis, and compliance minimization is performed using the optimality criteria method. The provided Matlab code and its extensions are explained in detail. Options to run the optimization with and without filtering techniques are provided. Steps to include different boundary conditions, multiple load cases, active and passive regions, and a Heaviside projection filter are also discussed. The code is provided in Appendix~A, and it can also be downloaded along with supplementary materials from \url{https://github.com/PrabhatIn/HoneyTop90}.

A topological proof of the modified Euler characteristic based on the orbifold concept.

The notion of the Euler characteristic of a polyhedron or tessellation has been the subject of in-depth investigations by many authors. Two previous papers worked to explain the phenomenon of the vanishing (or zeroing) of the modified Euler characteristic of a polyhedron that underlies a periodic tessellation of a space under a crystallographic space group. The present paper formally expresses this phenomenon as a theorem about the vanishing of the Euler characteristic of certain topological spaces called topological orbifolds. In this new approach, it is explained that the theorem in question follows from the fundamental properties of the orbifold Euler characteristic. As a side effect of these considerations, a theorem due to Coxeter about the vanishing Euler characteristic of a honeycomb tessellation is re-proved in a context which frees the calculations from the assumptions made by Coxeter in his proof. The abstract mathematical concepts are visualized with down-to-earth examples motivated by concrete situations illustrating wallpaper and 3D crystallographic space groups. In a way analogous to the application of the classic Euler equation to completely bounded solids, the formula proven in this paper is applicable to such important crystallographic objects as asymmetric units and Dirichlet domains.

On topology optimization with elliptical masks and honeycomb tessellation with explicit length scale constraints

Topology optimization using gradient search with negative and positive elliptical masks and honeycomb tessellation is presented. Through a novel skeletonization algorithm for topologies defined using filled and void hexagonal cells/elements, explicit minimum and maximum length scales are imposed on solid states in the solutions. An analytical example is presented suggesting that for a skeletonized topology, optimal solutions may not always exist for any specified volume fraction, minimum and maximum length scales, and that there may exist implicit interdependence between them. A sequence for length scale (SLS) methodology is proposed wherein solutions are sought by specifying only the minimum and maximum length scales with volume fraction getting determined within a specified range systematically. Through four benchmark problems in small deformation topology optimization, it is demonstrated that solutions by-and-large satisfy the length scale constraints though the latter may get violated at certain local sites. The proposed approach seems promising, noting especially that solutions, if rendered perfectly black and white with minimum length scale explicitly imposed and boundaries smoothened, are quite close in performance compared with the parent topologies. Attaining volume-distributed topologies, wherein members are more or less of the same thickness, may also be possible with the proposed approach.

A cellular sound-absorbing metasurface with subwavelength thickness

The sound absorbing metasurface efficient in the audible frequency range is proposed. The metasurface has a cellular structure, each cell having a hexagonal shape with a hollow tunnel inside. The wall of the hexagonal cell is sub-divided into six hollow chambers connected to the central tunnel via the six thinner channels of different diameters. The hollow chambers act as Helmholtz resonators providing six different resonant frequencies for each cell. The negative effective bulk modulus property of the metasurface allows full adsorption at the resonant frequencies. By carefully designing the size of the connecting channels we can manipulate the desired sound adsorption frequency range. The thickness of the metasurface is in the range of 0.03–0.1 wavelengths for the sound frequency range between 300 and 1000 Hz. Six absorption peaks are achieved for each unit cell providing broader range of absorption. The hexagonal shape of the unit cell allows full utilization of the metasurface volume by a standard honeycomb tessellation of the cells. By designing a metasurface containing individual cells with different inner channel diameters, the absorption peaks can be multiplied or overlapped and further broaden the frequency range with the absorption coefficient higher than the desired value.

Computational Synthesis of Large Deformation Compliant Mechanisms Undergoing Self and Mutual Contact

Topologies of large deformation contact-aided compliant mechanisms (CCMs), with self and mutual contact, exemplified via path generation applications, are designed using the continuum synthesis approach. Design domain is parameterized using honeycomb tessellation. Assignment of material to each cell, and generation of rigid contact surfaces, are accomplished via suitably sizing and positioning negative circular masks using the stochastic hill-climber search. To facilitate contact analysis, boundary smoothing is implemented. Mean value coordinates are employed to compute shape functions, as many regular hexagonal cells get degenerated into irregular, concave polygons as a consequence of boundary smoothing. Both geometric and material nonlinearities are considered. The augmented Lagrange multiplier method with a formulated active set strategy is employed to incorporate both self and mutual contact. Synthesized contact-aided compliant continua trace paths with single, and importantly, multiple kinks and experience multiple contact interactions pertaining to both self and mutual contact modes.

On topology optimization with embedded boundary resolution and smoothing

Honeycomb tessellation, by virtue of the edge connectivity it offers, helps eliminate various geometric singularities and thus shows promise in topology optimization. Numerous V-notches, which are sites of stress concentration and also pose difficulty in manufacturing, get retained on the continuum boundaries. Explicit boundary resolution and smoothing scheme is embedded within each gradient step when performing topology optimization with negative circular masks to assuage these serrations. This procedure, in particular, is deemed necessary when synthesizing contact aided compliant mechanisms. Continuum boundaries of each intermediate design are identified, and multiple smoothing steps are performed on the boundary nodes by shifting them systematically. Consequently, many regular hexagonal cells get morphed into concave ones. Shape functions based on Mean Value Coordinates (MVC) are implemented to cater to hexagonal finite elements of generic shapes. A continuation approach is used on the logistic material assignment model to tend to Heaviside function and thus yield close to black and white solutions in addition to avoiding early convergence and numerical instability. Number of boundary smoothing steps is increased with iterations as well. Deletion/addition of masks is performed to eliminate those not contributing in topology optimization and to introduce others in cases when singularities like local islands or layers prevail. To enhance continuity and to facilitate manufacturing, boundary nodes are interpolated using order m splines. For the examples presented, it is noted that splines of order 4 are adequate in representing the continuum boundaries accurately.

Combined gradient‐stochastic optimization with negative circular masks for large deformation topologies

SUMMARYTopology optimization of large deformation two‐dimensional continua is presented using a combined gradient‐stochastic search with negative circular masks. The possibility of generating perfect black and white topologies is explored while attaining the efficiency of first‐order and second‐order search algorithms. The design region is modeled with honeycomb tessellation to thwart the known connectivity singularities such as the checkerboards and point flexures. Mask shrinkage is incorporated for ease in density transition between gradient and stochastic steps. Notches at continuum boundaries are moderated through multiple use of a simple boundary smoothing method. A neo‐Hookean elasticity model is employed to simulate the material nonlinearities in large displacement continua. With examples on stiff beams and large deformation compliant mechanisms, it is illustrated that perfectly binary, connected and smooth topologies can be obtained within a few hundred design evaluations.Copyright © 2012 John Wiley & Sons, Ltd.

A virtual infrastructure based on honeycomb tessellation for data dissemination in multi-sink mobile wireless sensor networks

A new category of intelligent sensor network applications emerges where motion is a fundamental characteristic of the system under consideration. In such applications, sensors are attached to vehicles, or people that move around large geographic areas. For instance, in mission critical applications of wireless sensor networks (WSNs), sinks can be associated to first responders. In such scenarios, reliable data dissemination of events is very important, as well as the efficiency in handling the mobility of both sinks and event sources. For this kind of applications, reliability means real-time data delivery with a high data delivery ratio. In this article, we propose a virtual infrastructure and a data dissemination protocol exploiting this infrastructure, which considers dynamic conditions of multiple sinks and sources. The architecture consists of 'highways' in a honeycomb tessellation, which are the three main diagonals of the honeycomb where the data flow is directed and event data is cached. The highways act as rendezvous regions of the events and queries. Our protocol, namely hexagonal cell-based data dissemination (HexDD), is fault-tolerant, meaning it can bypass routing holes created by imperfect conditions of wireless communication in the network. We analytically evaluate the communication cost and hot region traffic cost of HexDD and compare it with other approaches. Additionally, with extensive simulations, we evaluate the performance of HexDD in terms of data delivery ratio, latency, and energy consumption. We also analyze the hot spot zones of HexDD and other virtual infrastructure based protocols. To overcome the hot region problem in HexDD, we propose to resize the hot regions and evaluate the performance of this method. Simulation results show that our study significantly reduces overall energy consumption while maintaining comparably high data delivery ratio and low latency.

A Material-Mask Overlay Strategy for Continuum Topology Optimization of Compliant Mechanisms Using Honeycomb Discretization

This paper proposes novel honeycomb tessellation and material-mask overlay methods to obtain optimal single-material compliant topologies free from checkerboard and point-flexure pathologies. The presence of strain-free rotation regions in rectangular cell based discretization is identified to be a cardinal cause for appearance of such singularities. With each hexagonal cell sharing an edge with its neighboring cells, strain-free displacements are not permitted anywhere in the continuum. The new material assignment approach manipulates material within a subregion of cells as opposed to a single cell thereby reducing the number of variables making optimization efficient. Cells are allowed to get filled with only the chosen material or they can remain void. Optimal solutions obtained are free from intermediate material states and can be manufactured requiring no material interpretation and less postprocessing. Though the hexagonal cells do not allow strain-free rotations, some subregions undergoing large strain deformations can still be present within the design. The proposed procedure is illustrated using three classical examples in compliant mechanisms solved using genetic algorithm.