# Conformally Graded Metamaterials for Elastic Wave Guidance

Charles Dorn1, Dennis M. Kochmann1,∗ 1Mechanics & Materials Lab, Department of
Mechanical and Process Engineering, ETH Zürich, 8092 Zürich, Switzerland
∗Corresponding author, email<EMAIL_ADDRESS>
###### Abstract

Although metamaterials have been widely used for controlling elastic waves
through bandgap engineering, the directed guidance of stress waves in non-
periodic structures has remained a challenge. This work demonstrates that
spatially graded metamaterials based on conformal mappings present a rich
design space for controlling and attenuating wave motion –- without the need
for bandgaps. Conformal mappings transform an elementary unit cell by scaling
and rotation into graded lattices with approximately geometrically similar
unit cells. This self-similarity allows for control over the local wave
dispersion throughout the metamaterial. As a key mechanism, it is shown that
elastic waves cannot propagate through graded unit cells with significant size
differences, except at low frequencies. This is exploited to create low-pass
elastic wave guides, extending beyond classical bandgap engineering, since
bandgaps are not required to achieve wave guiding and attenuation. Experiments
confirm the low-pass elastic wave filtering capability of a planar truss
metamaterial with conformal grading. Finally, a systematic design of curved
metamaterial surfaces is presented, providing a flexible framework for
programming low-pass attenuation and wave guiding in three dimensions.

Keywords: Metamaterial, Elasticity, Wave guide, Dispersion relation, Conformal
mapping

##  1 Introduction

Lattices appear naturally in atomic crystals, where wave dispersion has been
studied for decades [1] and plays a fundamental role in lattice vibrations,
electrical and thermal conduction, and the refraction of light. Metamaterials
do what nature cannot do: the careful design of lattices, which enables as-
designed effective properties. Dispersive phenomena observed at the atomic
scales, such as bandgaps (frequency bands where wave propagation is
forbidden), can be achieved and exploited by metamaterials on a wide range of
length scales. The design of periodic mechanical metamaterials, composed of
repeating arrangements of beams, plates, shells, or composite materials, has
been well-studied due to their appealing elastic wave attenuation capability
arising from bandgaps [2]. However, graded metamaterials with smooth spatial
variation of unit cells have remained largely unexplored in the context of
elastic wave propagation, despite the immensely increased design space
compared to periodic architectures [3].

Motivated by the opportunity to leverage bandgaps for vibration suppression
and wave guiding, significant efforts have been made to design and optimize
periodic metamaterials with wide bandgaps that span desired frequencies [4, 5,
6, 7, 8, 9]. By contrast, spatially graded architectures are less explored but
offer a considerably larger design space, since both the unit cell content and
its spatial variation can be controlled. For example, bandgaps of different
unit cells in a graded metamaterial can combine to produce a wider effective
bandgap. This phenomenon, sometimes referred to as rainbow trapping, has been
demonstrated for elastic waves in one-dimensional [10, 11, 12, 13, 14, 15, 16]
and two-dimensional [17, 18, 19] metamaterials (where wave attenuation has
been demonstrated along a specific direction only). While existing research
highlights the promise of graded metamaterials for wave manipulation,
systematic methods that explore the vast design space of spatially graded
lattices have remained an open challenge.

In this work, we present metamaterials graded by conformal mappings, which we
show to be effective for mechanical wave attenuation and wave guiding.
Conformal mappings consist of a locally uniform scaling and rotation, ensuring
that each unit cell in the graded lattice is approximately scaled and rotated.
Hence, all unit cells are approximately geometrically similar. Due to this
self-similarity, we show that waves cannot propagate along trajectories that
see a significant change in unit cell size, except at low frequencies. A
simple yet powerful design principle follows: designing regions with a
significant unit cell size difference ensures that waves cannot propagate
between these regions. As a result, certain regions can be isolated from high-
frequency wave propagation. Additionally, we show that attenuation via spatial
grading is achievable in the absence of bandgaps, as long as neighboring
dispersion surfaces are non-intersecting.

Our use of conformal mappings for mechanical metamaterials is distinct from
their extensive use for optical metamaterials. Transformation optics often
relies on conformal mappings to achieve practical effective material
parameters [20, 21, 22, 23] (similarly for transformation acoustics [24, 25,
26, 27]). However, transformation optics and acoustics typically apply in the
non-dispersive setting [28], requiring sub-wavelength unit cells. Furthermore,
they do not directly generalize to elastodynamics due to a lack of form
invariance [29].

In the elastodynamic setting, conformal maps have primarily been utilized in
the low-frequency and long-wavelength (with respect to a unit cell) regime by
relying on effective continuum properties [30, 31, 32, 33, 34], thus missing
out on the rich dispersive behavior arising in the short-wavelength regime
that is key to mechanical metamaterials. In contrast to previous work, we
utilize conformal mappings to control waves with short wavelengths. To this
end, with inspiration from the discrete geometry literature [35, 36, 37], we
design planar and curved conformally graded metamaterial surfaces. While
conformal mappings represent only a fraction of the spatial grading design
space, we show that their wave dynamics can be engineered using a simple but
general design procedure.

The resulting spatially graded metamaterials appeal to a wide range of
applications across scales. Their broad effective bandgaps are practical for
vibration suppression, e.g., to shield infrastructure from seismic waves [38].
Grading is also appealing for energy harvesting by trapping or focusing waves
[39, 40]. Additionally, frequency-selective sensing and signal processing
capabilities are enabled by spatial grading [41, 42]. The conformal
architectures presented in this work enhance the applicability of graded
metamaterials by offering low-pass attenuation capabilities without relying on
bandgaps. Furthermore, we propose a systematic design method for curved
metamaterial surfaces, which offers a flexible tool for programmable wave
guidance and attenuation.

Figure 1: Planar truss lattice for low-pass attenuation: (a) Reference
periodic truss lattice. (b) Graded truss lattice corresponding to a conformal
mapping of the reference lattice. (c) Dispersion relations of the reference
lattice with unit cell side length $l$ and $\omega_{0}=l^{-1}\sqrt{E/\rho}$.
(d) Second dispersion surface of unit cell 1, and (e) of unit cell 2
highlighted in (b), with the $\omega_{r}$ level set highlighted.

##  2 Conformally Graded Metamaterials

We design spatially graded lattices through a mapping
$f:\mathcal{M}\mapsto\mathbb{R}^{3}$, which transforms a planar, periodic
reference lattice $\mathcal{M}\subset\mathbb{R}^{2}$ into a graded lattice
$\mathcal{M}^{\prime}\subset\mathbb{R}^{3}$. Surface $\mathcal{M}^{\prime}$
can be flat or curved (both cases will be illustrated). We denote points in
the reference lattice by $\bm{x}\in\mathcal{M}$ and their mapped position in
the graded lattice by $\bm{x}^{\prime}=f(\bm{x})\in\mathcal{M}^{\prime}$.

The differential
$\;\\!\mathrm{d}f=\frac{\;\\!\mathrm{d}\bm{x}^{\prime}}{\;\\!\mathrm{d}\bm{x}}$
of the mapping defines how a vector $\bm{v}\in\mathbb{R}^{2}$ maps to the
vector $\bm{v}^{\prime}=\;\\!\mathrm{d}f(\bm{v})\in\mathbb{R}^{3}$ (i.e., it
characterizes how vectors tangent to the reference lattice map onto the
corresponding vectors tangent to the graded lattice). The mapping is
conformal, if
$\;\\!\mathrm{d}f(\bm{v})\cdot\;\\!\mathrm{d}f(\bm{w})=J^{2}\bm{v}\cdot\bm{w}$,
where $J>0$ is the conformal scaling factor.

Conformal mappings are chosen specifically, because they consist solely of a
local uniform scaling (since for any vector $\bm{v}$ we have
$|\;\\!\mathrm{d}f(\bm{v})|=J|\bm{v}|$) and a rotation (since the angle
between vectors is unchanged by the mapping) of the unit cells. Fig. 1ab
illustrates a conformal mapping in the case where $\mathcal{M}^{\prime}$ is
planar. Here, the vectors $\bm{v}$ and $\bm{w}$, which are attached to the
reference lattice at point $\bm{x}_{i}$, are uniformly scaled and rotated by
the mapping to vectors $\bm{v}^{\prime}$ and $\bm{w}^{\prime}$, while the
angle between them is preserved. For a more extensive treatment of conformal
mappings, see [43, 44].

We assume that in the transformed lattice the unit cells vary slowly in space,
so that $\;\\!\mathrm{d}f$ can be taken as constant throughout any given unit
cell. Under this assumption, each unit cell undergoes only uniform scaling by
a factor $J$ and a rotation. This assumption allows the graded lattice to be
modeled as locally periodic, so that wave propagation is governed by the local
dispersion relations [3]. That is, the local dispersion relations at a given
unit cell on the graded lattice are calculated assuming periodicity of that
unit cell, which is a valid approximation for slow spatial gradings.

Conformal grading yields a special case of local periodicity since, under the
above assumptions, all unit cells are geometrically similar to the unit cell
of the reference lattice. Consider a reference unit cell with dispersion
relations $\omega(\bm{k})$, relating the wave vector $\bm{k}\in\mathbb{R}^{2}$
to the frequency $\omega$ of an elastic plane wave. A unit cell in the
transformed lattice has local dispersion relations
$\omega^{\prime}(\bm{k}^{\prime})$. Since this unit cell is a scaled and
rotated copy of the reference unit cell, its dispersion relations take the
form

$\omega^{\prime}(\bm{k}^{\prime})=J^{-1}\omega(\bm{k}),$ (1)

where $\bm{k}^{\prime}$ is the wave vector in the transformed coordinates (a
derivation is provided in Section 1 of the Supplementary Material, which holds
generally for linear elastic waves). For the example mapping in Fig. 1, the
dispersion relations of the reference lattice (Fig. 1c) are hence simply re-
scaled to determine the local dispersion relations of any unit cell in the
graded lattice, including unit cells 1 (Fig. 1d) and 2 (Fig. 1e).

To understand how to exploit this inverse frequency scaling relation, we view
wave propagation from the perspective of ray theory. Ray theory has long been
used to model wave propagation in many fields, ranging from optics [45] to
seismology [46], where it provides approximate solutions to a wave equation
along characteristic ray trajectories. We recently extended it to graded
elastic metamaterials [3], under the assumption that wavelengths are
significantly smaller than the length scale associated with the spatial
grading of the unit cells. Ray theory is valid in the context of high
frequencies and short wavelengths, i.e., when unit cells change slowly in
space relative to the wavelength, which is the context considered here.

We rely on two key properties of rays to reveal the special properties of
conformally graded lattices in the linear elastic regime:

  1. (i)

A ray propagates at a fixed frequency.

  2. (ii)

A ray is associated with a particular dispersion surface (i.e., a particular
mode). It cannot switch between modes as long as the unit cells vary slowly in
space and no frequency degeneracies are encountered.

Property (i) holds as long as the local dispersion relations are time-
invariant [47], which is generally the case for elastic media. Property (ii)
is a re-statement of the adiabatic theorem, which is well-known in quantum
mechanics [48] but also applies to elastodynamics [49].

Combining Properties (i) and (ii) with the dispersion scaling law of Eq. (1)
admits a general observation about elastic wave propagation in conformally
graded lattices: high-frequency waves cannot propagate along trajectories that
see significant changes in unit cell size. This becomes evident as follows. By
Eq. (1), local dispersion surfaces everywhere are a scaled copy of the
reference dispersion surfaces. By Properties (i) and (ii), a ray is confined
to a level set of a given mode’s dispersion surface corresponding to the ray’s
fixed frequency $\omega_{r}$ (e.g., in Fig. 1d, a ray’s wave vector must lie
on the intersection of the dispersion surface and the ray’s fixed frequency
$\omega_{r}$). If that dispersion surface has a nonzero minimum frequency, it
is always possible to scale that surface up or down such that it no longer
intersects $\omega_{r}$ (e.g., in Fig. 1e there is no intersection between
$\omega_{r}$ and the scaled dispersion surface). This scaling argument,
however, does not apply at low frequencies of dispersion surfaces emerging
from $\omega=0$ (the so-called acoustic modes).

By this scaling argument, attenuation via spatial grading is achievable when
adjacent dispersion surfaces are non-intersecting. This includes bandgaps,
which ensure that adjacent dispersion surfaces do not intersect. However,
attenuation is also possible in the absence of a bandgap, if adjacent
dispersion surfaces are non-intersecting, as we show in the following
examples.

##  3 A Planar Truss Lattice for Low-Pass Attenuation

An instructive example that demonstrates the use of a conformal grading for
wave attenuation is a planar truss lattice that acts as a low-pass filter of
elastic waves. Consider the conformal mapping

$z^{\prime}=e^{z},$ (2)

where $z^{\prime}=x_{1}^{\prime}+\text{i}x_{2}^{\prime}$,
$z=x_{1}+\text{i}x_{2}$ in 2D, and $\text{i}=\sqrt{-1}$. Fig. 1 illustrates
this mapping, which conformally transforms the reference lattice with square
unit cells (Fig. 1a) to a radial lattice (Fig. 1b). In the transformed
lattice, the outermost unit cells are $J_{\text{o}}=3.51$ times larger than
the innermost ones, which have $J_{\text{i}}=1$, where $J_{\text{o}}$ and
$J_{\text{i}}$ are scaling factors with respect to the reference lattice.

Figure 2: Experimental confirmation of the low-pass attenuation in a
conformally graded truss: truss lattice prototype from the top view (a) and a
close-up view (b). (c) Experimental setup: a piezoelectric transducer excites
the lattice, whose response is measured by a scanning laser Doppler vibrometer
along radial lines. (d) Measured FRF amplitude along a radial line. (e)
Predicted regions of allowed and forbidden propagation based on dispersion
relation scaling.

Each unit cell consists of slender linear elastic beams along its four edges.
Dispersion relations for the reference unit cell are computed numerically
using a beam finite element model of a unit cell (Supplementary Material,
Section 2). The first three dispersion surfaces corresponding to out-of-plane
modes are plotted in Fig. 1c over the first Brillouin zone, with frequencies
normalized by $\omega_{0}=l^{-1}\sqrt{E/\rho}$, where $l$ is the edge length
of the reference unit cell, and $E$ and $\rho$ are Young’s modulus and the
mass density of the beams’ base material. The lowest modes of truss lattices
are decoupled into in- and out-of-plane modes [50]; by considering out-of-
plane excitation, only the out-of-plane modes must be considered.

This conformally graded truss lattice acts as a low-pass filter for waves
applied to the center of the lattice; waves above a cutoff frequency cannot
propagate from the inner to the outer radius. To illustrate this, consider a
mode 2 ray that starts in unit cell 1 of Fig. 1 and propagates radially
outward at the excitation frequency $\omega_{r}$. In unit cell 1, the wave
vector of this ray must lie on the intersection of the mode 2 (red) dispersion
surface and the $\omega_{r}$ level-set (gray), as depicted in Fig. 1d. Moving
to unit cell 2, which is 2.73 times larger than unit cell 1, its dispersion
surface is obtained by scaling the frequency of unit cell 1’s dispersion
surface by a factor of $1/2.73$. At unit cell 2, there is no intersection
between the mode 2’s dispersion surface and the $\omega_{r}$ level set (Fig.
1e). Therefore, a mode 2 ray starting at unit cell 1 with frequency
$\omega_{r}$ is forbidden in unit cell 2. Supplementary Video S1 visualizes
the dispersion surface as the unit cell is re-scaled, highlighting the
critical scaling factor where propagation at frequency $\omega_{r}$ becomes
forbidden.

By applying this scaling argument more generally, rays of all frequencies
(except those approaching $\omega=0$ of mode 1) become forbidden for a
sufficiently large scaling factor, corresponding to when the dispersion
surface is scaled to no longer intersect the frequency of the ray. Thus, rays
starting at unit cell 1 will reach some maximum radius, beyond which
propagation is forbidden. Based on this dispersion scaling argument, the
critical radius is computed across a range of frequencies and plotted in Fig.
2e. Regions where propagation is allowed are highlighted in red, and regions
where a wave starting at the inner radius is forbidden are white. Details of
the scaling analysis are given in Section 2.3 of the Supplementary Material,
where care is taken to account for mode conversion due to frequency
degeneracies where dispersion surfaces intersect.

We experimentally demonstrate the attenuation capability of this geometry,
using the prototype in Fig. 2a, which is made of aluminum and has a tapered
thickness to ensure that all unit cells are geometrically similar (see the
close-up view in Fig. 2b). Broadband excitation was applied to the center by a
piezoelectric (PZT) transducer in the out-of-plane direction. The out-of-plane
displacement was measured with a scanning laser Doppler vibrometer, as
illustrated in Fig. 2c. Details of the fabrication and experiment are
presented in Section 2.2 of the Supplementary Material.

The experimental results in Fig. 2d show the amplitude of the frequency
response function (FRF) [51], denoted $H(r^{\prime},f)$, between the
displacement at radial coordinate $r^{\prime}$ and the inner radius. This
provides a measure of attenuation as a wave travels outward from the inner
radius by interpreting the displacement signal at the inner radius as the
input to the truss. Regions with high FRF amplitude in Fig. 2d agree closely
with allowable propagation regions in Fig. 2e, whereas regions of low values
correspond to the predicted regions of forbidden propagation. Based on the
dispersion relation scaling analysis, the predicted low-pass cutoff frequency
is 14.7 kHz; higher frequencies are predicted to be attenuated before reaching
the outer radius of the lattice. This prediction agrees with the experimental
low-pass cutoff frequency, which is observed to be about 16 kHz.

Figure 3: Conformal bump example. (a) Target scaling factor distribution
plotted on the reference lattice. (b) Optimal conformal truss lattice
realizing the prescribed scaling. Color indicates the maximum amplitude during
a finite element simulation with broadband excitation applied on the lower
left edge (highlighted in red) with amplitude $\hat{u}$.

##  4 Curved Metamaterial Surfaces

In addition to the above plane-to-plane case, the same physical principles
apply to conformal mappings that transform a planar surface into a curved
surface, for which we propose a systematic design framework for curved
conformal lattice surfaces. Based on the guiding principle that waves do not
propagate between unit cells with a significant scaling difference, our
objective is to control the conformal scaling factor distribution in the
lattice in order to either isolate a region from incident high-frequency waves
or to confine waves in a specified region. The advantage of curved surfaces is
that curvature offers a richer design space than planar lattices, allowing for
elaborate scaling factor distributions to be realized as follows. Based on a
desired unit cell scaling distribution, a target conformal scaling factor
distribution $J(\bm{x})$ is first prescribed on a planar reference lattice.
Next, an optimization problem identifies a transformed three-dimensional (3D)
lattice that best realizes the prescribed scaling factor distribution.

We approach this problem from the perspective of discrete differential
geometry, which naturally follows from the discrete nature of the lattice. In
the discrete setting, the target scaling distribution is achieved by re-
scaling the edges from the reference length $l$ to a target length
$\tilde{l}$. The target edge lengths are carefully chosen to satisfy discrete
conformal equivalence [35], as detailed in the Supplemental Material, Section
3.1.

The problem of computing a 3D embedding (i.e., vertex coordinates
$\bm{x}^{\prime}$) of a discrete surface given its discrete metric (i.e., edge
lengths $\tilde{l}$) has been addressed in the discrete differential geometry
literature [52, 53, 54, 55]. Inspired by these approaches, to compute the
transformed vertex coordinates $\bm{x}^{\prime}$ we pose the unconstrained
minimization

$\min_{\bm{x}^{\prime}}E(\bm{x}^{\prime}),$ (3)

where

$E(\bm{x}^{\prime})=\sum_{m}k\big{(}l_{m}(\bm{x}^{\prime})-\tilde{l}_{m}\big{)}^{2}+\sum_{n}\kappa\,\theta_{n}(\bm{x}^{\prime})^{2}.$
(4)

Here, $m$ indexes all edges and $n$ indexes the interior edges, which are not
on the boundary. This optimization can be interpreted as an energy
minimization problem by imagining that an extensional spring is placed on each
edge, with stiffness $k$ and rest length $\tilde{l}$. In practice, a
regularization term is required to achieve smooth solutions, otherwise the
optimal surfaces will be ‘crumpled’ [55]. To this end, the second term in Eq.
(4) effectively places a torsional spring on all interior edges with torsional
stiffness $\kappa$ and zero rest angle (the latter corresponds to a flat
configuration when the faces connected to the edge are co-planar). This term
penalizes crumpled configurations in favor of smoother solutions. Smooth unit
cell variations are further essential to avoid sharp kinks, which may
invalidate Property (ii) by leading to mode conversion or in-plane and out-of-
plane mode coupling.

Figure 4: Conformal spiral example. (a) Target conformal scaling factor
distribution plotted on the reference lattice. (b) Optimal conformal truss
lattice realizing the prescribed scaling. Color indicates the maximum
amplitude during a finite element simulation with a harmonic point excitation
of displacement amplitude $\hat{u}$ perpendicular to the surface (along the
red arrow).

Fig. 3 presents the example of a conformal bump. The reference lattice is
planar with uniform square unit cells comprised of beams along each edge. The
scaling factor is defined according to a radial bump function and is plotted
in Fig. 3a on the reference lattice. It leads to large unit cells in the
center with a maximum scaling factor of $J=3.0$, surrounded by smaller unit
cells of $J=1$ outside of the bump; the optimal 3D geometry is plotted in Fig.
3b. (Constraints were added to fix the four boundaries of the lattice; for
implementation details, see Section 3 of the Supplementary Material.)

The bump in the center of the transformed lattice, which contains large unit
cells, is isolated from high-frequency waves initiated on the boundaries of
the lattice, whose smaller unit cells support those frequencies. To
demonstrate this effect, a transient finite element simulation was performed.
In the simulation, broadband displacement excitation was applied on the lower
left (red) edge in the $x_{3}^{\prime}$-direction with magnitude $\hat{u}$.
Beam elements were used to model the lattice; see the Supplementary Material,
Section 3.4 for details. The suppression of waves in the bump region is
evident from the color map of Fig. 3b, which shows the maximum displacement
magnitude at each point over the duration of the simulation (see also
Supplementary Video S2). Importantly, this mechanism is not restricted to a
narrow range of frequencies (as is often the case when leveraging bandgaps to
manipulate waves). Instead all frequencies above the critical frequency (which
is controlled by the unit cell scaling factor distribution) are attenuated in
the bump region. This effectively extends the benefits of rainbow trapping to,
in principle, arbitrarily defined regions.

This type of architecture is appealing for vibration isolation applications.
For example, sensitive equipment could be mounted to the bump for shielding
from high-frequency vibrations. To potentially achieve full attenuation across
all frequencies, one could consider metamaterials connected to a foundation,
which can exhibit bandgaps starting at zero frequency [56], combined with
conformal grading.

Beyond merely attenuating waves, conformal grading also provides a means of
designing wave guides. A second example of a conformal spiral is presented to
demonstrate wave guiding. In this example, a region of small unit cell scaling
in the shape of an Archimedean spiral is prescribed, which is plotted on the
reference lattice in Fig. 4a. In the spiral region, the smallest scaling
factor is $J=0.47$, while a factor of $J=1$ is prescribed outside of the
spiral region. The optimal lattice surface is plotted in Fig. 4b, which takes
the form of a wavy surface to accommodate the smaller unit cells along the
spiral.

In this example, the path of small unit cells along the spiral is capable of
confining waves. We demonstrate this with a transient finite element
simulation, where a harmonic displacement point excitation of magnitude
$\hat{u}$ is applied at the center of the spiral in the direction locally
normal to the surface at that point. The frequency of the excitation is chosen
to be near the top of the lowest dispersion surface, such that only a small
increase in unit cell size forbids propagation. The resulting wave emanating
from the point source follows the path of small unit cells and is thus guided
along the spiral. The coloring of Fig. 4b shows the maximum displacement
magnitude throughout the lattice during the simulation (see also Supplementary
Video S3). This example demonstrates that the spatially graded lattice can be
leveraged to effectively steer waves by designing regions of forbidden
propagation.

While both examples confirm that waves are attenuated in regions with a large
unit cell scaling factor compared to the excitation location, perfect
attenuation is not achieved, as may be expected for a number of reasons. The
assumption of local periodicity is an approximation, as is the assumption that
the ‘locally flat’ dispersion relations are valid for curved lattice surfaces.
Finally, the unit cells are not exactly geometrically similar in these two
examples. Nevertheless, our assumptions are sufficiently valid to produce the
desired wave confinement and attenuation behavior.

##  5 Conclusion

We have shown that conformally graded metamaterials effectively attenuate
mechanical waves above a controllable cut-off frequency and hence can be
designed to behave as effective waveguides—without requiring bandgaps. While
metamaterials with broadband attenuation capability have been pursued by a
growing body of research, conformally graded architectures go beyond broadband
to achieve low-pass attenuation. Furthermore, our dispersion scaling analysis
is more general than tracking how bandgaps shift with spatial unit cell
variations, which is typically done to study attenuation in graded
architectures [17, 18, 19]. Specifically, a bandgap between two dispersion
surfaces is not required to achieve attenuation, but only the less strict
requirement that there is no intersection between neighboring dispersion
surfaces. For example, there is no bandgap between the lowest two dispersion
surfaces of Fig. 1c; yet, since there is no intersection between them,
attenuation via spatial grading is achieved.

While an experimental demonstration confirms the low-pass attenuation
capability of a planar lattice, curved lattice surfaces offer a wide design
space for conformal grading, whose systematic inverse design shows promise for
vibration isolation applications across length scales. This work represents
only a small step into the vast and largely unexplored design space of graded
and curved metamaterial surfaces. The effectiveness of conformal grading
suggests that more general types of grading are promising for linear elastic
wave manipulation.

## Acknowledgments

This work was supported by an ETH Zürich Postdoctoral Fellowship. The authors
thank Vignesh Kannan for assistance with the experimental setup and Bastian
Telgen for assistance with dispersion relation computations.

Supplementary material for: “Conformally Graded Metamaterials for Elastic Wave
Guidance”
by Charles Dorn and Dennis M. Kochmann

##  S6-5 Dispersion relation scaling derivation

A key observation explored in this work is that geometrically similar unit
cells have dispersion relations that scale inversely with the unit cell’s
size, which is stated in Eq. (1) of the main text. Here, we provide a
derivation of this result. To this end, we briefly review elastic wave
propagation in periodic solids, before focusing on changes to the dispersion
relations when geometrically transforming the unit cell.

###  S6-5.1 Waves in periodic solids

Let $\mathcal{B}\subset\mathbb{R}^{3}$ be a linear elastic body with spatial
coordinates $\bm{x}\in\mathcal{B}$. The mass density $\rho(\bm{x})$ and
fourth-order elasticity tensor $\bm{C}(\bm{x})$ are periodic with a unit cell
$\Omega$. The governing elastodynamic equation of linear momentum balance is

$\displaystyle\rho\ddot{u}_{i}=\left(C_{ijkl}u_{k,l}\right)_{,j},$ (S1)

where $\bm{u}(\bm{x},t):\mathcal{B}\times\mathbb{R}\rightarrow\mathbb{R}^{3}$
is the displacement field and $t$ is time. Here and in the following, we use
Einstein’s summation convention, and a comma index denotes differentiation;
i.e., $(\cdot)_{,i}=\frac{\partial(\cdot)}{\partial x_{i}}$.

When using Bloch wave analysis, only a single unit cell needs to be
considered. A displacement ansatz is made for one unit cell in the form of a
plane wave

$\bm{u}(\bm{x},t)=\bm{A}e^{\text{i}\varphi},$ (S2)

where $\varphi=\bm{k}\cdot\bm{x}-\omega t$ is the phase,
$\bm{A}\in\mathbb{C}^{3}$ is the complex-valued amplitude vector, and
$\text{i}=\sqrt{-1}$. Inserting this ansatz into Eq. (S1) and simplifying
leads to the Bloch eigenvalue problem

$\displaystyle\bm{\mathcal{L}}^{B}[\bm{A}]+\omega^{2}\rho\bm{A}=\bm{0},$ (S3)

where the Bloch operator is defined as

$\displaystyle\mathcal{L}^{B}_{i}[\bm{A}]=\frac{\partial C_{ijkl}}{\partial
x_{j}}\left(\frac{\partial}{\partial
x_{l}}+\text{i}k_{l}\right)A_{k}+C_{ijkl}\left(\frac{\partial^{2}}{\partial
x_{l}\partial x_{j}}+\text{i}k_{j}\frac{\partial}{\partial
x_{l}}+\text{i}k_{l}\frac{\partial}{\partial x_{j}}-k_{l}k_{j}\right)A_{k}.$
(S4)

A standard way to compute the dispersion relations is to convert Eq. (S3) to
the weak form, apply Bloch boundary conditions, and solve the eigenvalue
problem numerically using, for example, a finite element model of a unit cell.
This process is described, e.g., in [2].

###  S6-5.2 Dispersion relations of a transformed unit cell

We proceed to derive how the dispersion relations are related between two unit
cells that are geometrically similar. Specifically, we will show that the
Bloch eigenvalue problem preserves its form under uniform scaling and
rotation, but with the eigenvalues scaled inversely by the uniform scaling
factor. Lattices that are periodic in two dimensions are considered, though we
model the unit cells as three-dimensional as we consider out-of-plane
displacements.

Figure S1: Reference unit cell $\Omega$ alongside the scaled and rotated unit
cell $\Omega^{\prime}$.

The unit cell $\Omega$ is assumed periodic in coordinates $x_{1}$ and $x_{2}$,
with $x_{3}$ representing the out-of-plane coordinate. A second unit cell
$\Omega^{\prime}$ is periodic in coordinates $x^{\prime}_{1}$ and
$x^{\prime}_{2}$ with out-of-plane coordinate $x^{\prime}_{3}$. Unit cell
$\Omega^{\prime}$ is geometrically similar to $\Omega$; its geometry is
obtained by uniformly scaling and rotating $\Omega$. The transformation from
$\Omega$ to $\Omega^{\prime}$ can be expressed through a coordinate
transformation $\bm{x}^{\prime}=\bm{x}^{\prime}(\bm{x})$, whose gradient
$\bm{F}\in\mathbb{R}^{3\times 3}$ is constant over the unit cell and consists
of a scaling factor $J\in\mathbb{R}$ and a rotation matrix
$\bm{R}\in\mathbb{R}^{3\times 3}$, such that

$\displaystyle\bm{F}=\frac{d\bm{x}^{\prime}}{d\bm{x}}=J\bm{R}.$ (S5)

Fig. S1 shows an example of two such unit cells, for the case of a square unit
cell with a beam along each edge.

Note that the conformal mappings discussed in the main text do not dictate how
the thickness of the unit cell transforms. We choose to enforce that unit cell
thickness also scales with $J$ to ensure geometric similarity of all unit
cells. Consequently, all examples in this work involve graded lattices with
variable thickness (i.e., both the in-plane dimensions and the out-of-plane
thickness scale with $J$).

The rotation matrix $\bm{R}$ represents a rotation in the plane of the unit
cell and has the form

$\displaystyle\bm{R}=\begin{bmatrix}Q_{11}&Q_{12}&0\\\ Q_{21}&Q_{22}&0\\\
0&0&1\end{bmatrix},$ (S6)

where $\bm{Q}$ is a two-dimensional rotation matrix.

Using the transformation between $\Omega$ and $\Omega^{\prime}$, the Bloch
eigenvalue problem for the reference unit cell can be rewritten in the
transformed coordinates. Our aim is to show that this is identical to the
Bloch eigenvalue problem written directly for the transformed unit cell but
with frequency scaled by $J^{-1}$. First, we define how the spatial
derivatives and wave vector transform. In terms of $\bm{F}$, we can relate
spatial derivatives and wave vector in the reference domain to those in the
transformed domain as

$\displaystyle\frac{\partial}{\partial x_{i}}$
$\displaystyle=\frac{\partial}{\partial x_{i^{\prime}}^{\prime}}\frac{\partial
x_{i^{\prime}}^{\prime}}{\partial x_{i}}=\frac{\partial}{\partial
x_{i^{\prime}}^{\prime}}F_{i^{\prime}i}$ (S7)

and

$\displaystyle k_{i}$ $\displaystyle=\frac{\partial\varphi}{\partial
x_{i}}=\frac{\partial\varphi}{\partial x_{i^{\prime}}^{\prime}}\frac{\partial
x_{i^{\prime}}^{\prime}}{\partial
x_{i}}=F_{i^{\prime}i}k^{\prime}_{i^{\prime}}.$ (S8)

Primed indices are associated with the transformed coordinates. Using Eqs.
(S7) and (S8) in Eq. (S3), the Bloch eigenvalue problem for the reference unit
cell rewritten in transformed coordinates is

$\displaystyle(C_{ijkl}F_{l^{\prime}l}F_{j^{\prime}j})_{,j^{\prime}}(A_{k,l^{\prime}}+\text{i}k^{\prime}_{l^{\prime}}A_{k})+C_{ijkl}F_{l^{\prime}l}F_{j^{\prime}j}(A_{k,l^{\prime}j^{\prime}}+\text{i}k^{\prime}_{j^{\prime}}A_{k,l^{\prime}}+\text{i}k^{\prime}_{l^{\prime}}A_{k,j^{\prime}}$
$\displaystyle-A_{k}k^{\prime}_{j^{\prime}}k^{\prime}_{l^{\prime}})$ (S9)
$\displaystyle+\rho\omega^{2}\delta_{ik}A_{k}=0,$

where $\delta_{ik}$ denotes the Kronecker delta. We now define the amplitude
$\bm{A}^{\prime}$ in the transformed coordinates such that
$A_{k}=F_{k^{\prime}k}A^{\prime}_{k^{\prime}}$. Multiplying Eq. (S9) by
$F_{i^{\prime}i}$ yields

$\displaystyle(C_{ijkl}F_{i^{\prime}i}F_{j^{\prime}j}F_{k^{\prime}k}F_{l^{\prime}l})_{,j^{\prime}}(A^{\prime}_{k^{\prime},l^{\prime}}+\text{i}k^{\prime}_{l^{\prime}}A^{\prime}_{k^{\prime}})$
(S10)
$\displaystyle+C_{ijkl}F_{i^{\prime}i}F_{j^{\prime}j}F_{k^{\prime}k}F_{l^{\prime}l}(A^{\prime}_{k^{\prime},l^{\prime}j^{\prime}}+\text{i}k^{\prime}_{j^{\prime}}A^{\prime}_{k^{\prime},l^{\prime}}+\text{i}k^{\prime}_{l^{\prime}}A^{\prime}_{k^{\prime},j^{\prime}}-A^{\prime}_{k^{\prime}}k^{\prime}_{j^{\prime}}k^{\prime}_{l^{\prime}})+\rho\omega^{2}\delta_{ik}F_{i^{\prime}i}F_{k^{\prime}k}A^{\prime}_{k^{\prime}}=0.$

This can be simplified by assuming that each point within the unit cell has
isotropic material properties. The corresponding elastic tensor takes the form
$C_{ijkl}=\lambda\delta_{ij}\delta_{kl}+\mu(\delta_{ik}\delta_{jk}+\delta_{il}\delta_{jk})$,
where $\lambda$ and $\mu$ are the first and second Lamé parameters,
respectively. Using Eq. (S5) and the fact that rotation matrices are
orthogonal ($R_{ij}R_{kj}=\delta_{ik}$), the terms in Eq. (S10) simplify to

$C_{ijkl}F_{i^{\prime}i}F_{j^{\prime}j}F_{k^{\prime}k}F_{l^{\prime}l}=J^{4}R_{i^{\prime}i}R_{j^{\prime}j}R_{k^{\prime}k}R_{l^{\prime}l}\left[\lambda\delta_{ij}\delta_{kl}+\mu(\delta_{ik}\delta_{jk}+\delta_{il}\delta_{jk})\right]=J^{4}C_{i^{\prime}j^{\prime}k^{\prime}l^{\prime}}$
(S11)

and

$\displaystyle
F_{i^{\prime}i}F_{k^{\prime}k}\delta_{ik}=J^{2}R_{i^{\prime}i}R_{k^{\prime}k}\delta_{ik}=J^{2}\delta_{i^{\prime}k^{\prime}}.$
(S12)

Inserting Eqs. (S11) and (S12) into Eq. (S10), we obtain

$\displaystyle
J^{4}C_{i^{\prime}j^{\prime}k^{\prime}l^{\prime},j^{\prime}}(A^{\prime}_{k^{\prime},l^{\prime}}+\text{i}k^{\prime}_{l^{\prime}}A^{\prime}_{k^{\prime}})+J^{4}C_{i^{\prime}j^{\prime}k^{\prime}l^{\prime}}(A^{\prime}_{k^{\prime},l^{\prime}j^{\prime}}+\text{i}k^{\prime}_{j^{\prime}}A^{\prime}_{k^{\prime},l^{\prime}}+\text{i}k^{\prime}_{l^{\prime}}A^{\prime}_{k^{\prime},j^{\prime}}-$
$\displaystyle
A^{\prime}_{k^{\prime}}k^{\prime}_{j^{\prime}}k^{\prime}_{l^{\prime}})$ (S13)
$\displaystyle+J^{2}\rho\omega^{2}A^{\prime}_{i^{\prime}}=0,$

which can be rewritten as

$\displaystyle\bm{\mathcal{L}^{\prime}}^{B}[\bm{A}^{\prime}]+J^{-2}\omega^{2}\rho\bm{A}^{\prime}=\bm{0}.$
(S14)

Here, $\bm{\mathcal{L}^{\prime}}^{B}$ represents the Bloch operator of Eq.
(S4) written in the transformed coordinates. Finally, upon defining
$\omega^{\prime}=J^{-1}\omega$, Eq. (S15) recovers the Bloch eigenvalue
problem in the transformed coordinates:

$\displaystyle\bm{\mathcal{L}^{\prime}}^{B}[\bm{A}^{\prime}]+\omega^{\prime}\rho\bm{A}^{\prime}=\bm{0},$
(S15)

which is identical to what we would obtain if we directly wrote the eigenvalue
problem for the scaled and rotated unit cell. Therefore, the dispersion
relations $\omega^{\prime}(\bm{k}^{\prime})$ of a scaled and rotated unit cell
are directly obtained from the dispersion relations $\omega(\bm{k})$ of the
reference unit cell by

$\displaystyle\omega^{\prime}(\bm{k}^{\prime})=J^{-1}\omega(\bm{k}).$ (S16)

While this derivation holds strictly for periodic lattices, the results also
apply to the local dispersion relations of locally periodic lattices under the
assumption that $\bm{F}$ is (approximately) constant throughout each unit
cell. This means that in a conformally graded metamaterial, assumed to exhibit
geometrically similar unit cells, the dispersion relations of a single
reference unit cell describes the wave propagation throughout the entire
metamaterial. All local dispersion relations can be obtained directly from
those of the reference unit cell.

##  S7-5 Planar truss lattice

This section provides details about the geometry, experimental setup, and
numerical modeling for the planar truss lattice of Figs. 1 and 2 of the main
text.

###  S7-5.1 Conformal scaling factor

From the complex representation of the conformal mapping $z^{\prime}=\exp(z)$,
Eq. (2) of the main text, the scaling factor $J$ can be computed. The scaling
factor of a conformal transformation in terms of complex function
$z^{\prime}(z)$ is given by [43]

$\displaystyle J=\left|\frac{dz^{\prime}}{dz}\right|.$ (S17)

Upon inserting Eq. (2) of the main text, this becomes

$\displaystyle J=\left|e^{z}\right|=\left|e^{\ln
z^{\prime}}\right|=r^{\prime},$ (S18)

where $r^{\prime}=\sqrt{(x^{\prime}_{1})^{2}+(x^{\prime}_{2})^{2}}$ is the
radial coordinate in the transformed geometry. Thus, the scaling factor
increases linearly with the radius.

###  S7-5.2 Experiment

Here, we describe the details of the fabrication of the truss lattice
prototype of Fig. 2a and of the experiment that produced the results of Fig.
2d in the main text.

####  S7-5.2.1 Fabrication

A prototype of the planar truss lattice was fabricated from an aluminum alloy
5083 plate with dimensions 63$\times$63$\times$0.4 cm. The prototype consists
of a truss region that ranges from an inner radius of
$r^{\prime}_{\text{i}}=7.96$ cm to an outer radius of
$r^{\prime}_{\text{o}}=29.77$ cm. Within the truss region, there are 100 unit
cells around the circumference and 21 unit cells along the radial direction.
The regions inside $r^{\prime}_{\text{i}}$ and outside $r^{\prime}_{\text{o}}$
are solid material. Dimensions of the prototype were scaled such that the
innermost unit cell (at $r^{\prime}=r^{\prime}_{\text{i}}$) has a width of 0.5
cm and a beam width of 0.5 mm.

Since the unit cell scaling factor varies linearly with radius, as evident
from Eq. (S18), the thickness of the prototype was linearly tapered to ensure
that all unit cells are geometrically similar. The outer region of the
prototype ($r^{\prime}>r^{\prime}_{\text{o}}$) has a thickness of 4 mm,
resulting in a thickness of 1.07 mm in the inner region
($r^{\prime}<r_{\text{i}}$). Thus, the lattice region between
$r^{\prime}_{\text{o}}$ and $r^{\prime}_{\text{i}}$ exhibits a linear
variation in thickness along the radial coordinate from 1.07 to 4 mm, which
can be seen in the close-up photo of the prototype in Fig. 2b of the main
text.

Aluminum was chosen for the material because it exhibits low damping. This is
beneficial for our experimental demonstration to ensure that the observed wave
attenuation is primarily due to the geometric structure of the lattice as
opposed to material loss.

Fabrication of the prototype lattice was performed as a two-step process.
First, the radial thickness taper between $r^{\prime}_{\text{i}}$ and
$r^{\prime}_{\text{o}}$ was cut into the plate by CNC milling, resulting in a
solid plate with a thickness taper on one side and a flat surface on the other
side. Then, the truss section between $r^{\prime}_{\text{i}}$ and
$r^{\prime}_{\text{o}}$ was cut by water jet to complete the fabrication
process.

####  S7-5.2.2 Measurement and data processing

The experimental setup for measuring wave propagation in the prototype is
shown in Fig. S2 and illustrated in Fig. 2c of the main text. To excite out-
of-plane elastic waves, a piezoelectric transducer (Thorlabs PA3BCW) was glued
to the center of the prototype. A function generator (Teledyne LeCroy T3AFG40)
was used to generate an input signal, which was sent through an amplifier
(Thorlabs HVA200) to the transducer. The input signal was a swept sine ranging
from 50 to 300 kHz with a linear sweep rate and a duration of 100 $\mu$s. The
swept sine was shifted to have a minimum value of zero to avoid applying
negative voltage to the transducer. This input signal was observed to provide
sufficient broadband excitation in the 0$-$300 kHz range in the truss region.

A scanning laser Doppler vibrometer (Optomet SWIR) was used to measure the
response of the prototype. Measurements were collected at 100 evenly spaced
points along radial lines from $r^{\prime}_{\text{i}}$ to
$r^{\prime}_{\text{o}}$. At each measurement point, the input signal was
repeatedly applied 1 second apart to allow the vibrations to decay before the
next measurement. For each measurement, the displacement signal was recorded
with a sampling rate of 4 MHz over a duration of 16.384 ms. To reduce noise,
100 independent measurements were averaged at each measurement point. The
maximum observed displacement throughout all measurements was 18.0 nm. No
significant differences in the results were observed along different radial
lines.

Figure S2: Experimental setup, consisting of a piezoelectric transducer that
excites out-of-plane waves in the prototype, whose response is measured by a
laser Doppler vibrometer.

To quantify the wave attenuation along a radial line, we estimate the
frequency response function (FRF) between location $r^{\prime}$ and the inner
radius $r^{\prime}_{\text{i}}$. We interpret the displacement signal at
$r^{\prime}_{\text{i}}$ as an input to the truss region, and we observe the
attenuation of this input using the FRF. To estimate the FRF, following [51],
the power spectral density of the displacement signal at
$r^{\prime}_{\text{i}}$ is denoted
$S_{r^{\prime}_{\text{i}}r^{\prime}_{\text{i}}}$ and the cross power spectral
density between the response at $r^{\prime}_{\text{i}}$ and $r^{\prime}$ is
denoted $S_{r^{\prime}_{\text{i}}r^{\prime}}$. The spectral densities are
computed using overlap averaging (Welch’s) method with temporal segments of 2
ms, an overlap of 1 ms, and a Tukey window applied to each segment (with the
flat region making up 75% of the window). The FRF, denoted $H(r^{\prime},f)$
is estimated from the spectral densities by

$\displaystyle
H(r^{\prime},f)=\frac{S_{r^{\prime}_{\text{i}}r^{\prime}}}{S_{r^{\prime}_{\text{i}}r^{\prime}_{\text{i}}}}.$
(S19)

In Fig. 2d of the main text, the amplitude of the FRF is plotted in decibels,
which is $20\log_{10}|H(r^{\prime},f)|$.

###  S7-5.3 Numerical modeling

To model the truss lattice numerically, we perform a dispersion relation
scaling analysis (Section S7-5.3.1) based on numerically computed dispersion
relations to predict at which radius each frequency becomes forbidden. This
scaling analysis is validated by comparing it to a transient finite element
simulation of the entire truss lattice (Section S7-5.3.2). Finally, we discuss
the use of higher-fidelity modeling using 3D finite elements instead of beam
elements (Section S7-5.3.3). All finite element computations are performed
using the state-of-the-art finite element code ae108 [57].

####  S7-5.3.1 Dispersion relation scaling analysis

This section describes the dispersion relation scaling analysis that produces
Fig. 2e of the main text. The aim is to compute how far a wave of each
frequency starting at the inner radius can propagate radially, before the unit
cell scaling forbids propagation at that frequency.

We begin by numerically computing the dispersion relations of the innermost
unit cell using a finite element model, approximating the unit cell as a
square. To match the experimental dimensions (see Section S7-5.2.1), we take a
unit cell with an edge length of $l=0.5$ cm, a thickness of 1.07 mm, and 0.5
mm wide beams along its edges. The material is modeled as homogeneous,
isotropic linear elastic with the following properties to model aluminum:
Young’s modulus $E=69.9$ GPa, Poisson’s ratio $\nu=0.33$, and mass density
$\rho=2700$ kg/m3. Each beam is discretized by 40 Timoshenko beam finite
elements.

To compute the dispersion relations from the finite element model of the unit
cell, we follow the formulation presented in [50]. For simplicity, we consider
only out-of-plane displacements since in- and out-of-plane modes are decoupled
for planar truss lattices [50]. That is, in response to out-of-plane
excitation we can expect predominately out-of-plane displacements. The first
eight out-of-plane dispersion surfaces are plotted in Fig. S3a.

To proceed with the dispersion relation scaling analysis, we first note that
the unit cells in the truss lattice do not vary circumferentially. Thus, the
local dispersion relations are axisymmetric. We also consider a point
excitation at the origin, so that the loading is also axisymmetric. As a
result, waves can only propagate radially. Since the unit cell is aligned with
the radial and circumerential coordinates (we take $k_{1}$ to align with the
radial coordinate), $k_{2}=0$ due to axisymmetry. The $k_{2}=0$ plane is
shaded in gray in Fig. S3a. The corresponding slice of the dispersion
relations in the $k_{2}=0$ plane is plotted in Fig. S3b.

Figure S3: (a) Dispersion relations of the innermost unit cell of the radial
lattice, with the $k_{2}=0$ plane shaded gray. (b) Slice of the dispersion
relations intersecting $k_{2}=0$.

With the dispersion curves in the $k_{2}=0$ plane in hand, we perform a
scaling analysis. Consider a specific frequency $f_{r}$ that intersects mode 2
in Fig. S3b. The maximum frequency of mode 2’s dispersion curve is
$f_{2,max}=174.37$ kHz. Therefore, if the unit cell is scaled up by a factor
of $J>f_{2,max}/f_{r}$, mode 2’s dispersion curve will scale down and no
longer intersect $f_{r}$ and this frequency is forbidden. Using Eq. (S18), the
critical frequency is converted to a critical radius, beyond which propagation
is forbidden.

A similar analysis is performed for mode 1, except that frequencies below some
low-pass cutoff $f_{c}$ will reach the outer radius. The outermost unit cell
is $J_{\text{o}}=3.51$ times larger than the innermost one (this value is
taken to match the prototype). Since the maximum frequency of mode 1 in Fig.
S3b is $f_{1,max}=51.66$ kHz, it follows that frequencies below the cutoff
$f_{c}=f_{1,max}/J_{\text{o}}=14.72$ kHz have a critical scaling factor larger
than $J_{\text{o}}$ and will reach the outer radius.

Extra care must be taken in the scaling analysis of modes 3 to 7, because
there are degeneracies (repeated eigenfrequencies for a given wave vector)
where dispersion surfaces intersect. Property (ii) of the main text is
violated for degenerate frequencies, and we assume that mode conversion can
happen between modes 3-7. Instead of scaling each individual mode, we scale
the block of modes 3 to 7, where sufficient scaling is required such that
$f_{r}$ does not intersect the range of frequencies spanned by modes 3 to 7.
Following this procedure, we obtain the uppermost region of allowed
propagation in Fig. 2e of the main text.

####  S7-5.3.2 Transient finite element simulation

For validation of the dispersion scaling analysis, we performed a transient
finite element simulation of the truss lattice. This simulation provides
validation of the assumptions of the dispersion relation scaling analysis
(local periodicity and idealization of unit cells as squares) in a controlled
and noise-free setting to complement the experimental analysis.

The truss lattice geometry (see Fig. 1b of the main text) was discretized with
Timoshenko beam elements. The out-of-plane displacement of all nodes on the
inner radius was prescribed to provide broadband excitation. Specifically, a
swept sine ranging from 50 to 300 kHz with a linear sweep rate and a duration
of 100 $\mu$s was applied. Using a Newmark-beta solver for implicit time
integration, the solution was computed for a duration of 800 $\mu$s with
timesteps of 0.4 $\mu s$.

Displacement signals along one radial line were processed using the same
procedure as for the experiment (Section S7-5.2.2) to compute the FRF with
respect to the displacement applied to the inner radius. Fig. S4 shows the FRF
amplitude as a function of frequency and radius. The result is comparable to
the analogous experimental result shown in Fig. 2d of the main text. However,
cleaner results are available here, since there is no noise, perfect symmetry,
and an ideal broadband excitation was applied directly to the inner radius of
the truss. Results show close agreement between the dispersion scaling
analysis (Fig. 2e of the main text) and Fig. S4. Note that in the transient
finite element simulation, some frequencies propagate slightly further than
predicted by the dispersion scaling analysis. This is due to the fact that the
wave is not immediately attenuated when it reaches a forbidden region;
instead, this is where the wave begins to exponentially decay.

Figure S4: Transient finite element simulation results for the planar truss
lattice. The FRF magnitude along one radial line is plotted.

####  S7-5.3.3 Comparison to higher fidelity modeling

The above numerical analysis relied on modeling truss lattices as networks of
beams. The chosen beam model captured the behavior observed in the experiment
while also providing a computationally efficient model. However, higher-
fidelity modeling is possible using 3D finite elements at the cost of
computational efficiency. We show here that there are differences in the
dispersion relations computed by solid 3D elements versus beam finite elements
[58], but these differences have a minimal effect on the wave attenuation
behavior of the planar truss lattice.

To compute the dispersion relations of a square unit cell (with the same
geometry and material properties as used in Section S7-5.3.1) using 3D finite
elements, the unit cell is discretized by tetrahedral elements, using 7
elements through the width and 14 through the thickness of each beam—following
a mesh convergence study, resulting in a mesh with 61,923 elements in total.
Bloch boundary conditions are applied, and the Bloch eigenvalue problem is
solved for points from $k_{1}=0$ to $k_{1}=\pi l$ (where $l$ is the unit cell
edge length) with $k_{2}=0$. For each value of $\bm{k}$, the computation time
was approximately 10 minutes (in contrast to a 0.1 second computation time for
each $\bm{k}$-point when using the beam model). The resulting dispersion
relations are plotted in Fig. S5a. Here, only modes with mode shapes
exhibiting a nonzero out-of-plane component are shown; all purely in-plane
modes are omitted (the validity of this assumption and the primary relevance
of the out-of-plane modes is confirmed by the experimental data).

Figure S5: (a) Dispersion relations for the same unit cell as in Fig. S3
computed using a 3D finite element model along the $k_{2}=0$ plane. (b)
Regions of forbidden (white) and allowed (red) linear elastic wave
propagation, computed based on the dispersion relations of (a).

The lowest two dispersion curves of Fig. S5a closely match those of the beam
model (Fig. S3b). However, there are differences in the higher dispersion
curves between the 3D and beam models. The prediction for the regions of
allowed and forbidden propagation based on the 3D model’s dispersion relations
are plotted in Fig. S5b. The partial bandgap between 500 and 600 kHz is
present in both models, which results in nearly identical predictions (Figs.
S5b and 2d of the main text) for the regions of allowed and forbidden
propagation. Due to the fact that these predictions are in agreement between
the beam and 3D models, we focus on the beam models for computational
efficiency, though it is important to address the availability of higher-
fidelity modeling, which may become important in other scenarios.

##  S8-5 Curved metamaterial surfaces

This section provides details about the design and simulation of the curved
metamaterial surfaces presented in Figs. 3 and 4 of the main text.

###  S8-5.1 Discrete conformal equivalence

Our objective is to transform the periodic reference lattice into a surface
that realizes a prescribed conformal scaling factor distribution $J(\bm{x})$.
This is achieved by re-scaling the edge lengths $l$ of the reference lattice
to spatially varying target lengths $\tilde{l}$. To ensure that the
transformation is conformal, the notion of conformality must be defined in the
discrete setting. Allowing each edge to scale independently is not
sufficiently restrictive, since it allows for arbitrary scaling.
Alternatively, since conformal mappings are angle preserving in the continuous
setting, one could enforce angle preservation in the discrete setting, but
this is too restrictive since it allows only global scaling and rotation [37].
An appropriate definition of discrete conformality is given by discrete
conformal equivalence [35], which preserves the basic structure of continuous
conformal geometry in the discrete setting.

Consider an edge $ij$ of the reference lattice, which connects vertices $i$
and $j$ and has length $l_{ij}$. The transformed lattice with re-scaled edge
lengths $\tilde{l}_{ij}$ is conformally equivalent to the reference lattice if
all edge lengths satisfy

$\displaystyle\tilde{l}_{ij}=e^{\frac{1}{2}(U_{i}+U_{j})}l_{ij},$ (S20)

where $U$ is the log-scaling factor, which is defined on the vertices ($U_{i}$
and $U_{j}$ being the log-scaling factors of vertices $i$ and $j$).

Thus, we prescribe the target log-scaling factor $U$ on each vertex, from
which the set of target edge lengths $\tilde{l}$ that ensure conformality
follow from Eq. (S20). The resulting conformal scaling factor associated with
each edge is $J_{ij}=e^{\frac{1}{2}(U_{i}+U_{j})}$.

###  S8-5.2 Conformal surface optimization

In the conformal bump example of Fig. 3 in the main text, the log-scaling
factor was prescribed according to a radial bump function of the form

$\displaystyle
U(r)=\begin{cases}3\exp\left(\frac{-1}{(1-\frac{r}{20l})^{2}}\right)&r<20l\\\
0&r\geq 20l.\end{cases}$ (S21)

Here, $r=\sqrt{x_{1}^{2}+x_{2}^{2}}$ is the radial coordinate in the reference
domain and $l$ is the side length of the reference unit cell. The log-scaling
factor is evaluated at each vertex location, i.e., $U_{i}=U(\bm{x}_{i})$.
Then, the target edge lengths $\tilde{l}$ are computed according to Eq. (S20).

In the conformal spiral example of Fig. 4 in the main text, the log-scaling
factor was prescribed along an Archimedean spiral of the form $r=5.75l\theta$.
The reference unit cell has side length $l$, while $r$ and $\theta$ are polar
coordinates in the reference domain. The spiral was given a thickness of 5
unit cells; vertices inside the spiral were prescribed $U=-0.75$ while unit
cells outside were prescribed $U=0$. Finally, a Gaussian filter with a
standard deviation of 1.5$l$ was applied to spatially smooth the distribution
of $U$ on the reference lattice, thus removing sharp jumps in the target
scaling factor.

The definition of conformal equivalence from Eq. (S20) of the main text holds
for triangulated discrete surfaces [35]. The lattices considered, however,
contain quadrilateral unit cells. To ensure that conformal equivalence is
satisfied, we triangulated the lattice during the optimization by adding one
diagonal edge to triangulate each unit cell. This edge is only present during
the optimization; once the optimal geometry is obtained, the triangulated edge
is removed.

For both examples, Matlab’s fminunc utility was used to perform the
optimization using an interior point algorithm with cost function gradients
provided, which were derived analytically. When performing the optimization,
the ratio of the cost function weights in Eq. (4) of the main text was taken
as $\kappa/k=5\times 10^{-5}$ for the conformal bump example, and
$\kappa/k=7.5\times 10^{-5}$ for the conformal spiral example. For the
conformal bump example, the constraints of fixing the boundaries of the
surface were directly applied by eliminating the boundary vertex positions
from the design variables.

###  S8-5.3 Reference dispersion relations

For both examples, the beams that comprise the truss lattice are given a
circular cross-section, which is more straightforward to define on a curved
lattice than the previously considered rectangular cross-sections. To compute
the dispersion relations of a reference unit cell, we take the unit cell size
as 0.5$\times$0.5 cm with the same material properties as in the planar truss
lattice example. The cross-section’s diameter is taken as $0.5$ mm. The
dispersion relations, computed by the same process as outlined in Section
S7-5.3.1 but with a circular cross section, are plotted in Fig. S6.

Figure S6: Dispersion relations at the excitation location of the two curved
surface examples. The plane corresponding to $\omega=30$ kHz is highlighted in
gray.

###  S8-5.4 Transient finite element simulations

Transient finite element simulations were performed to demonstrate the wave
attenuation capabilities of the curved metamaterial surface examples.

The unit cell edge length at the excitation location was set to 0.5 cm for
both the conformal bump and spiral examples, so that the local dispersion
relations at the excitation location are given in Fig. S6. Throughout the
lattice, the cross section of each beam is circular with a diameter of $1/10$
of its length to ensure that all unit cells are approximately geometrically
similar. Again, the material properties of aluminum were used, identical to
those used in the example of Section S7-5.3.1. Timoshenko beam finite elements
were used to discretize the lattice.

For the conformal bump surface example, displacement excitation was applied to
one boundary of the surface (highlighted in red in Fig. 3b of the main text)
in the $\bm{x}^{\prime}_{3}$-direction, which is locally out-of-plane. The
displacement excitation is a swept sine ranging from 50 to 300 kHz with a
linear sweep rate; Fig. S6 highlights the modes present in this frequency
range.

For the conformal spiral surface example, a harmonic point excitation with a
frequency of 30 kHz was applied at the center of the spiral in the locally
out-of-plane direction, as shown in Fig. 4b of the main text. Based on the
dispersion relations of Fig. S6, this frequency only excites mode 1, but it is
near the maximum frequency of the first dispersion surface such that a small
increase in unit cell scaling will forbid propagation.

In both examples, a Newmark-beta time integration scheme was used. For the
conformal bump example, the simulation was performed over a duration of 1.2 ms
with a fixed time step of 2.4 MHz. For the conformal spiral example, the
simulation was performed over a duration of 2.5 ms with a fixed time step of
500 kHz. The maximum value of the norm of the three-dimensional displacement
vector during the simulation is plotted in Figs. 3b and 4b of the main text.
Animations of the simulations are included in Supplementary Videos S2 and S3.

##  S9-5 Supplementary video legends

Video S1: Animation of how a dispersion surface scales with unit cell size.
The second out-of-plane dispersion surface for a square unit cell is shown,
whose frequency scales inversely with the unit cell edge length.

Video S2: Finite element simulation of the conformal bump. Broadband
excitation is applied to the left edge (red) and the resulting displacement
magnitudes are shown. Large unit cells on the bump are isolated from the
incident waves.

Video S3: Finite element simulation of the conformal spiral. Harmonic
excitation is applied locally out-of-plane at the spiral center and the
resulting displacement magnitudes are shown. The wave is follows the path of
small unit cells and is guided along the spiral.

## References

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