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Much work has been done in topology optimization of multiscale structures for maximum stiffness or minimum compliance design. Such approaches date back to the original homogenization-based work by Bends{\o}e and Kikuchi from 1988, which lately has been revived due to advances in manufacturing methods like additive manufacturing. Orthotropic microstructures locally oriented in principal stress directions provide for highly efficient stiffness optimal designs, whereas for the pure stiffness objective, porous isotropic microstructures are sub-optimal and hence not useful. It has, however, been postulated and exemplified that isotropic microstructures (infill) may enhance structural buckling stability but this has yet to be directly proven and optimized. In this work, we optimize buckling stability of multiscale structures with isotropic porous infill. To do this, we establish local density dependent Willam-Warnke yield surfaces based on local buckling estimates from Bloch-Floquet-based cell analysis to predict local instability of the homogenized materials. These local buckling-based stress constraints are combined with a global buckling criterion to obtain topology optimized designs that take both local and global buckling stability into account. De-homogenized structures with small and large cell sizes confirm validity of the approach and demonstrate huge structural gains as well as time savings compared to standard singlescale approaches., Comment: Published in Computer Methods in Applied Mechanics and Engineering

This paper proposes a methodology for architecting microstructures with extremal stiffness, yield, and buckling strength using topology optimization. The optimized microstructures reveal an interesting transition from simple lattice like structures for yield-dominated situations to hierarchical lattice structures for buckling-dominated situations. The transition from simple to hierarchical is governed by the relative yield strength of the constituent base material as well as the volume fraction. The overall performances of the optimized microstructures indicate that maximum strength is determined by the buckling strength at low volume fractions and yield strength at higher volume fractions, regardless of the base material's relative yield strength. The non-normalized properties of the optimized microstructures show that higher base material Young's modulus leads to both higher Young's modulus and strength of the architected microstructures. Furthermore, the polynomial order of the maximum strength lines with respect to mass density obtained from the optimized microstructures reduces as base material relative yield strength decreases, reducing from 2.3 for buckling dominated Thermoplastic Polyurethane to 1 for yield dominated steel microstructures.

Buckling strength estimation of architected materials has mainly been restricted to load cases oriented along symmetry axes. However, realistic load scenarios normally exhibit more general stress distributions. In this paper we propose a simple yet accurate method to estimate the buckling strength of stretch-dominated lattice structures based on individual member analysis. As an integral part of the method, the yield strength is also determined. This simplified model is verified by rigorous numerical analysis. In particular, we efficiently compute the complete buckling strength surfaces of an orthotropic bulk modulus optimal plate lattice structure and isotropic stiffness optimal plate and truss lattice structures subjected to rotated uni-axial loads, where the ratio between the highest and lowest buckling strength is found to be 1.77, 2.11 and 2.41, respectively. For comparison, we also provide their yield strength surfaces, where the corresponding ratios are 1.84, 1.16 and 1.79. Furthermore, we use the knowledge gained from the simplified model to create a new configuration of the isotropic plate lattice structure with a more isotropic buckling strength surface and buckling strength ratio of 1.24, without deterioration of the stiffness or yield strength. The proposed method provides a valuable tool to quickly estimate the microstructural buckling strength of stretch-dominated lattice structures, especially for applications where the stress state is non-uniform such as infill in additive manufacturing., Comment: 19 pages with 5 figures

Engineered micro- and nanomechanical resonators with ultra-low dissipation constitute the ideal systems for applications ranging from high-precision sensing such as magnetic resonance force microscopy, to quantum transduction between disparate quantum systems. Traditionally, the improvement of the resonator's performance - often quantified by its Qf product (where Q is quality factor and f is frequency) - through nanomechanical engineering such as dissipation dilution and strain engineering, has been driven by human intuition and insight. Such an approach is inefficient and leaves aside a plethora of unexplored mechanical designs that potentially achieve better performance. Here, we use a computer-aided inverse design approach known as topology optimization to structurally design mechanical resonators with optimal performance of the fundamental mechanical mode. Using the outcomes of this approach, we fabricate and characterize ultra-coherent nanomechanical resonators with record-high Qf products, entering a quantum coherent regime where coherent oscillations are observed at room temperature. Further refinements to the model describing the mechanical system are likely to improve the Qf product even more. The proposed approach - which can be also used to improve phononic crystal and coupled-mode resonators - opens up a new paradigm for designing ultra-coherent micro- and nanomechanical resonators for cutting-edge technology, enabling e.g. novel experiments in fundamental physics (e.g. search for dark matter and quantum nature of gravity) and extreme sensing of magnetic fields, electric fields and mass with unprecedented sensitivities at room temperature.

This paper presents a class of 3D single-scale isotropic materials with tunable stiffness and buckling strength obtained via topology optimization and subsequent shape optimization. Compared to stiffness-optimal closed-cell plate material, the material class reduces the Young's modulus to a range from 79% to 58%, but improves the uniaxial buckling strength to a range from 180% to 767%. Based on small deformation theory, material stiffness is evaluated using the homogenization method. Buckling strength under a given macroscopic stress state is estimated using linear buckling analysis with Block-Floquet boundary conditions to capture both short and long wavelength buckling modes. The 3D isotropic single-scale materials with tunable properties are designed using topology optimization, and are then further simplified using shape optimization. Both topology and shape optimized results demonstrate that material buckling strength can be significantly enhanced by hybrids between truss and variable thickness plate structures.

Advances in manufacturing techniques may now realize virtually any imaginable microstructures, paving the way for architected materials with properties beyond those found in nature. This has lead to a quest for closing gaps in property-space by carefully designed metamaterials. Development of mechanical metamaterials has gone from open truss lattice structures to closed plate lattice structures with stiffness close to theoretical bounds. However, the quest for optimally stiff and strong materials is complex. Plate lattice structures have higher stiffness and (yield) strength but are prone to buckling at low volume fractions. Hence here, truss lattice structures may still be optimal. To make things more complicated, hollow trusses or structural hierarchy bring closed-walled microstructures back in the competition. Based on analytical and numerical studies of common microstructures from the literature, we provide higher order interpolation schemes for their effective stiffness and (buckling) strength. Furthermore, we provide a case study based on multi-property Ashby charts for weight-optimal porous beams under bending, that demonstrates the intricate interplay between structure and microarchitecture that plays the key role in the design of ultimate load carrying structures. The provided interpolation schemes may also be used to account for microstructural yield and buckling in multiscale design optimization schemes., Comment: 18 pages main manuscript with 6 figures and 10 pages appendix with 2 figures

Designing photonic topological insulators is highly non-trivial because it requires inversion of band symmetries around the band gap, which was so far done using intuition combined with meticulous trial and error. Here we take a completely different approach: we consider the design of photonic topological insulators as an inverse design problem and use topology optimization to maximize the transmission through an edge mode with a sharp bend. Two design domains composed of two different, but initially identical, C$_\text{6v}$-symmetric unit cells define the geometrical design problem. Remarkably, the optimization results in a photonic topological insulator reminiscent of the shrink-and-grow approach to quantum-spin-Hall photonic topological insulators but with notable differences in the topology of the crystal as well as qualitatively different band structures and with significantly improved performance as gauged by the band-gap sizes, which are at least 50 \% larger than previous designs. Furthermore, we find a directional beta factor exceeding 99 \%, and very low losses for sharp bends. Our approach allows for the introduction of fabrication limitations by design and opens an avenue towards designing PTIs with hitherto unexplored symmetry constraints., Comment: 7 pages, 5 figures

An acoustic topological insulator (TI) is synthesized using topology optimization, a free material inverse design method. The TI appears spontaneously from the optimization process without imposing requirements on the existence of pseudo spin-1/2 states at the TI interface edge, or the Chern number of the topological phases. The resulting TI is passive; consisting of acoustically hard members placed in an air background and has an operational bandwidth of $\approx$12.5\% showing high transmission. Further analysis demonstrates confinement of more than 99\% of the total field intensity in the TI within at most six lattice constants from the TI interface. The proposed design hereby outperforms a reference from recent literature regarding energy transmission, field confinement and operational bandwidth., Comment: 6 pages, 5 figures

Small manufacturing-tolerant photonic crystal cavities are systematically designed using topology optimization to enhance the ratio between quality factor and mode volume, Q/V. For relaxed manufacturing tolerance, a cavity with bow-tie shape is obtained which confines light beyond the diffraction limit into a deep-subwavelength volume. Imposition of a small manufacturing tolerance still results in efficient designs, however, with diffraction-limited confinement. Inspired by numerical results, an elliptic ring grating cavity concept is extracted via geometric fitting. Numerical evaluations demonstrate that for small sizes, topology-optimized cavities enhance the Q/V-ratio by up to two orders of magnitude relative to standard L1 cavities and more than one order of magnitude relative to shape-optimized L1 cavities. An increase in cavity size can enhance the Q/V-ratio by an increase of the Q-factor without significant increase of V. Comparison between optimized and reference cavities illustrates that significant reduction of V requires big topological changes in the cavity.

We present numerical studies of two photonic crystal membrane microcavities, a short line-defect cavity with relatively low quality ($Q$) factor and a longer cavity with high $Q$. We use five state-of-the-art numerical simulation techniques to compute the cavity $Q$ factor and the resonance wavelength $\lambda$ for the fundamental cavity mode in both structures. For each method, the relevant computational parameters are systematically varied to estimate the computational uncertainty. We show that some methods are more suitable than others for treating these challenging geometries., Comment: Revised and final version for publication. 28 pages, 10 figures, 7 tables

A common strategy to compensate for losses in optical nanostructures is to add gain material in the system. By exploiting slow-light effects it is expected that the gain may be enhanced beyond its bulk value. Here we show that this route cannot be followed uncritically: inclusion of gain inevitably modifies the underlying dispersion law, and thereby may degrade the slow-light properties underlying the device operation and the anticipated gain enhancement itself. This degradation is generic; we demonstrate it for three different systems of current interest (coupled resonator optical waveguides, Bragg stacks, and photonic crystal waveguides). Nevertheless, a small amount of added gain may be beneficial.