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Designing Perturbative Metamaterials from Discrete Models: From Veselago lenses to topological insulators
Kathryn H. Matlack, Marc Serra-Garcia, Antonio Palermo, Sebastian D. Huber, Chiara Daraio
TL;DR
Metamaterial discovery lacks systematic approaches, despite discrete models describing complex phenomena. The paper introduces perturbative metamaterials and demonstrates the scheme on Veselago lenses, zero group velocity materials, and topological insulators.
Problem
Metamaterial discovery lacks systematic approaches and is based on intuition, trial and error, or related approaches.
Method
The paper introduces perturbative metamaterials and maps discrete-model building blocks to weakly interacting metamaterial unit cells.
Results
The scheme is demonstrated on a Veselago lens, a zero group velocity material, and a topological insulator.
Takeaways & Limitations
The approach supports systematic exploration of metamaterial designs, including a design space on the order of 10^30 configurations.
Takeaways & Limitations
Aligned plates permit couplings of only one sign, while long-range couplings affect the group velocity in the flat band.
Abstract
from arXiv · showhide
Discrete models provide concise descriptions of complex physical phenomena, such as negative refraction, topological insulators, and Anderson localization. While there are multiple tools to obtain discrete models that demonstrate particular phenomena, it remains a challenge to find metamaterial designs that replicate the behavior of desired nontrivial discrete models. Here we solve this problem by introducing a new class of metamaterial, which we term 'perturbative metamaterial', consisting of weakly interacting unit cells. The weak interaction allows us to associate each element of the discrete model (individual masses and springs) to individual geometric features of the metamaterial, thereby enabling a systematic design process. We demonstrate our approach by designing 2D mechanical metamaterials that realize Veselago lenses, zero-dispersion bands, and topological insulators. While our selected examples are within the mechanical domain, the same design principle can be applied to acoustic, thermal, and photonic metamaterials composed of weakly interacting unit cells.
Introduction
The paper introduces perturbative metamaterials as a systematic way to convert discrete mass-spring models into physical designs. Weakly interacting unit cells make design effects additive, enabling efficient searches that realize Veselago lenses, zero group velocity materials, and topological insulators.
- Motivation: Metamaterial discovery lacks a systematic process for converting discrete models into physical structures because design spaces are large and structure–dynamics relationships are non-trivial.Existing approaches rely on intuition, trial and error, or unguided searches, while dynamic designs must account for interactions among vibrational modes.
- Approach: The proposed tool maps metamaterial building blocks to mass-spring components and searches combinations of building blocks to match a target model.The approach addresses reduced-order-model extraction and seeks design elements whose effects do not interfere with one another.
- Perturbative metamaterials: Perturbative metamaterials use weakly interacting unit-cell modes so design changes have additive effects and the search can be divided into smaller independent sub-spaces.This additive structure yields an exponential speedup of the search process.
- Reduced-order modeling: The method extracts reduced-order models with a Schrieffer-Wolff transformation, isolates modes in the frequency range of interest, and catalogs how geometric changes affect dynamics.The resulting information guides optimization of metamaterial components toward a target mass-spring model.
- Demonstration: A suitable series expansion explores on the order of 10^30 design configurations, beyond current optimization methods, in plates connected by soft beams.The plate-and-beam system achieves the additive property required by the search algorithm.
- Demonstrations: The scheme is demonstrated with increasing unit-cell complexity through a Veselago lens, a zero group velocity material, and a topological insulator.These examples show the approach across several nontrivial mechanical metamaterial targets.
Extracting a reduced order model from a perturbative metamaterial
The paper extracts reduced-order models for weakly interacting metamaterial unit cells by mapping local plate modes and beam-induced couplings into resonators and springs. A first-order perturbative treatment makes geometric contributions additive, enabling accurate model reduction and systematic design exploration.
- Model construction: The reduced-order model represents each relevant plate mode as a local resonator and each beam-induced interaction as a spring coupling.The model distinguishes uncoupled plates from plates connected by beams and separates physical-system coupling V from reduced-model coupling V_R.
- Perturbative approximation: Weak interactions justify neglecting higher powers of ΔK, allowing the coupled system to be treated perturbatively.The approximation is organized around the coupling strength relative to spectral gaps between modes.
- Model reduction: The Schrieffer-Wolff transformation removes coupling between relevant and irrelevant mode spaces by block-diagonalizing the dynamical matrix.For small coupling, its first-order term is a satisfactory approximation; higher orders improve accuracy for stronger coupling.
- Model reduction: The first-order transformation makes beam and hole contributions additive, so complex geometries can be assembled from individually calculated responses.This additivity is the key property used to combine geometric features when constructing metamaterial designs.
- Limitations: Higher-order corrections capture stronger couplings but introduce long-range stiffness terms between plates that are not physically connected by beams.The reduced description therefore becomes less local as coupling strength increases.
- Validation: The method extracts coupling matrices from finite-element modes of coupled plates and compares reduced-order dispersion with finite-element band structures near 145 kHz.The comparison evaluates soft and stronger beam couplings using successive SW transformation orders.
Metamaterial design from a discrete model
The design procedure maps discrete-model degrees of freedom onto plate modes, realizes target inter-plate couplings with beams, tunes local responses with holes, and optimizes the assembled multi-cell structure. Additive perturbative responses make large configuration spaces tractable through staged searches and optimization.
- Design examples: The method is applied to design metamaterials including a Veselago lens and examples based on degenerate plate modes for zero group velocity materials.The selected mapping can use multiple modes of a single plate for designs requiring multiple degrees of freedom.
- Search strategy: Additive first-order responses allow the design space to contain up to 10^30 configurations while separate subspaces contain fewer than 10^10 elements.The approach explores large spaces by optimizing design components separately and combining responses from a limited number of finite-element simulations.
- Discrete-model mapping: The procedure first maps discrete-model degrees of freedom to plate modes, allowing one plate to represent multiple degrees of freedom through degenerate modes.The mapping is manual and can have multiple acceptable realizations.
- Inter-plate couplings: Beam location, thickness, and sometimes angle are selected to introduce desired inter-plate couplings using precomputed coupling-stiffness tables.A combinatorial search selects beam sets whose additive responses match the target coupling, followed by gradient optimization for second-order effects.
- Local response tuning: Holes tune each plate’s local mode frequencies and same-plate modal couplings after accounting for the effects of connected beams.Hole responses are tabulated and combined through a combinatorial search before gradient-based refinement.
- Global optimization: A final gradient-based optimization over multiple unit cells compensates for second-order interactions among beams and holes and for long-range couplings.The optimization includes all geometric features in the assembled system.
Phononic Veselago lens
The authors design a phononic Veselago lens by mapping positive and negative mass-spring couplings to weakly interacting mechanical unit cells. Finite-element results clearly show the lens effect and agree closely with the mass-spring model.
- The target lens is a double-negative medium with negative effective modulus and density embedded in a positive conventional medium.
- Each square-lattice unit cell contains one resonator coupled to its four nearest neighbors with target stiffnesses +K or −K.
- Plate offsets enable both positive and negative couplings, overcoming the single-sign coupling constraint of perfectly aligned plates.
- Hole parameters compensate local beam-induced stiffness so unit cells inside and outside the lens achieve the required resonance frequencies.
- Finite-element simulations at 175.284 kHz clearly reproduce the Veselago lens effect and show excellent agreement with the mass-spring model at 175.204 kHz.
Zero group velocity material
The authors realize a zero group velocity material with a flat band by mapping multiple mass-spring degrees of freedom onto degenerate plate modes. Its dispersion closely follows the target model, although long-range couplings produce slight residual group velocity.
- The zero group velocity material targets a flat band and illustrates how one plate can implement multiple degrees of freedom.
- Two degenerate modes in one plate represent x and y degrees of freedom, while a mode shifted outside the frequency range is removed from the dynamics.
- The design uses combinatorial searches for offsets, beam geometry, and hole locations, followed by gradient-based optimization to reduce second-order stiffness error.
- The metamaterial dispersion shows excellent correspondence to the mass-spring system across the three bands of interest.
- Long-range couplings leave slight non-zero group velocity in the otherwise flat band.
Topological insulator
The authors design a mechanical topological-insulator metamaterial by translating mass-spring degrees of freedom and coupling matrices into plates and beams. The design reproduces bulk and edge modes, including propagation around a defect.
- The target mass-spring model has three two-degree-of-freedom lattice sites per unit cell with nontrivial intercell couplings.
- Each two-degree-of-freedom site is translated into one plate using degenerate plate modes, with three plates coupled by optimized beams.
- Separate coupling optimizations reduce the search from 10^30 configurations to three searches over 10^10 configurations, while beam angles accommodate different plate alignments.
- The dispersion relation shows three bulk bands separated by counter-propagating edge modes crossing at π/3 and 2π/3.
- Finite-element simulations show edge modes and excellent agreement with the target model, while an edge mode persists and propagates around a three-fixed-plate defect.
- The design method is presented as applicable beyond mechanics, including electromagnetic systems and photonic circuits.
Methods
The study combines finite-element simulations and linear-algebra calculations, using COMSOL Multiphysics and MATLAB on cluster resources. Simulations assume linear elasticity with epoxy beams and steel plates.
- Finite-element simulations use COMSOL Multiphysics, while linear-algebra calculations use MATLAB with communication through COMSOL LiveLink.
- The simulations ran on ETH Euler cluster nodes with access to up to 400 GB of RAM.
- All simulations use a linear elastic model with epoxy resin beams and steel plates.
Coupling Matrix Extraction
The coupling matrix is extracted by projecting coupled-plate eigenmodes onto a free-plate modal basis, then combining modal eigenvalues and frequencies into a reduced model. The calculation uses finite-element data and a least-squares approximation because the finite basis cannot exactly reproduce coupled vibration profiles.
- Mode sampling: 80 coupled-system eigenmodes are sampled over the plate test area and stored as displacement vectors for coupling-matrix extraction.The sampled displacements include x, y, and z components at 2268 points on each plate.
- Numerical accuracy: Small beam-induced frequency shifts require a highly refined finite-element mesh because small eigenfrequency errors can produce large coupling-matrix errors.The beam-induced shifts are much smaller than the plate resonance frequencies.
- Modal basis: The coupled-plate displacement is represented using the first 40 normal modes of an unperturbed free plate.The basis is selected so the coupled system has twice as many modes as the individual plate, while excluding beam resonances and inconsistently selected degenerate families.
- Modal projection: The Moore-Penrose pseudoinverse provides a least-squares approximation because the finite basis cannot exactly reproduce coupled vibration profiles.The method projects coupled-system modes into the free-plate modal basis using the sampled displacement matrix.
- Matrix construction: The coupling matrix is calculated as V = P D P^-1 − H from the projected modes and the coupled-system eigenvalue matrix.D is an 80x80 matrix whose diagonal elements contain coupled-system eigenvalues, while H contains unperturbed-plate eigenfrequencies.
- Reduced-order interpretation: The extraction method is equivalent to the first-order Schrieffer-Wolff transformation in the low-energy space of the first 40 unperturbed-plate modes.Higher-order terms are then used to obtain a reduced-order model containing only the required modes.
Optimization Process
The optimization combines exhaustive search over beam geometries with gradient refinement to match calculated coupling matrices to objective matrices. A kernel-vector adjustment preserves the coupling matrix while reducing unwanted modes.
- Offset sweep: The optimization is repeated for plate offsets from 2 mm to 4 mm in 0.2 mm increments, using coupling matrices tabulated at 0.2 mm beam width.The resulting table relates coupling matrices to beam location.
- Exhaustive search: Beam locations are searched exhaustively in 0.1 mm steps, while widths range from 0.1 m to 0.5 mm in 0.01 mm steps.The stated width range includes the reported lower bound of 0.1 m.
- Objective matching: Calculated coupling matrices are compared with objective matrices after discarding combinations whose norm differs by more than 50% from the target.The norm can later be finely adjusted by scaling beam widths because coupling varies approximately linearly with width.
- Gradient refinement: Gradient optimization follows the exhaustive search to refine beam parameters and account for interactions between beams.Each iteration evaluates a reference configuration and small perturbations around it.
- Jacobian construction: The Jacobian is estimated from perturbed coupling matrices, using 0.04 mm location, 0.01 mm thickness, and 2-degree angle perturbations.Its columns represent responses to perturbations in configuration-vector components.
- Mode suppression: The kernel vector is chosen to leave the coupling matrix unchanged while minimizing unwanted-mode participation.Its contribution reduces unwanted modes by 30% to 50%.
Finite Element Simulations
Finite-element simulations evaluate the designed Veselago lens, zero-group-velocity material, and topological insulator using dynamic condensation, Floquet boundaries, and full-system models. The simulations extract displacements, energies, dispersion, and edge-mode localization from these designs.
- Veselago lens: The Veselago lens contains 100x100 unit cells and is analyzed by solving force balance with unit-strength driving at the interface between two specified cells.After solving displacements, the RMS amplitude of every unit cell is calculated.
- Dynamic condensation: Dynamic condensation relates boundary displacements and forces through a transfer matrix using 117 degrees of freedom for each connecting-beam cross section.The unit-cell force-balance equation uses mass, stiffness, and damping matrices obtained from COMSOL.
- Zero-group-velocity material: The zero-group-velocity metamaterial is simulated in COMSOL with a unit cell subject to Floquet boundary conditions at half the beam length.The unit-cell model contains 1.02M elements.
- Topological-insulator dispersion: The topological-insulator dispersion relation is modeled using four unit cells, comprising 12 plates coupled by beams, with Floquet boundaries in one dimension.Fixed boundaries are applied at both ends of the beams in the finite dimension.
- Edge modes: Edge-mode polarization is determined by identifying the model side where stored energy density is localized around crossing points.The crossing-point locations depend on which plate in the three-plate unit cell is connected to the fixed boundary.
- Finite-size model: The finite-size topological insulator is simulated directly in COMSOL using 2.6M elements.