# Modal Folding: Discovering Smooth Folding Patterns for Sheet Materials using
Strain-Space Modes

Pengbin Tang  ETH ZürichSwitzerland Université de MontréalCanada
<EMAIL_ADDRESS>,  Ronan Hinchet  ETH ZürichSwitzerland
<EMAIL_ADDRESS>,  Roi Poranne  ETH ZürichSwitzerland University of
HaifaIsrael<EMAIL_ADDRESS>,  Bernhard Thomaszewski  ETH
ZürichSwitzerland<EMAIL_ADDRESS>and  Stelian Coros  ETH
ZürichSwitzerland<EMAIL_ADDRESS>
(2024)

###### Abstract.

Folding can transform mundane objects such as napkins into stunning works of
art. However, finding new folding transformations for sheet materials is a
challenging problem that requires expertise and real-world experimentation. In
this paper, we present Modal Folding—an automated approach for discovering
energetically optimal folding transformations, i.e., large deformations that
require little mechanical work. For small deformations, minimizing internal
energy for fixed displacement magnitudes leads to the well-known elastic
eigenmodes. While linear modes provide promising directions for bending, they
cannot capture the rotational motion required for folding. To overcome this
limitation, we introduce strain-space modes—nonlinear analogues of elastic
eigenmodes that operate on per-element curvatures instead of vertices. Using
strain-space modes to determine target curvatures for bending elements, we can
generate complex nonlinear folding motions by simply minimizing the sheet’s
internal energy. Our modal folding approach offers a systematic and automated
way to create complex designs. We demonstrate the effectiveness of our method
with simulation results for a range of shapes and materials, and validate our
designs with physical prototypes.

Nonlinear Modal Analysis, Computational Design, Folding, Origami

††journalyear: 2024††copyright: acmlicensed††conference: Special Interest
Group on Computer Graphics and Interactive Techniques Conference Conference
Papers ’24; July 27-August 1, 2024; Denver, CO, USA††booktitle: Special
Interest Group on Computer Graphics and Interactive Techniques Conference
Conference Papers ’24 (SIGGRAPH Conference Papers ’24), July 27-August 1,
2024, Denver, CO, USA††doi: 10.1145/3641519.3657401††isbn:
979-8-4007-0525-0/24/07††ccs: Applied computing Computer-aided design††ccs:
Computing methodologies Modeling and simulation Figure 1. Our method
automatically generates a diverse set of folding patterns using strain-space
modes. Here we show a particular folding transformation for a square sheet of
fabric and its physical prototype (bottom right). Figure 2. Periodic folding
patterns obtained using Modal Folding (a-e) with reflection boundary
conditions and their corresponding physical prototypes (right) manufactured
using copper sheets and 3D-printed press dies.

##  1\. Introduction

In the hands of a skilled artist, a plain sheet of fabric can fold into a work
of perplexing beauty. Finding such folding transformations, however, is an art
in itself that has been curiously elusive to science. Nevertheless, systematic
approaches for generating folding patterns would have wide implications beyond
art, including deployable shading systems, architectural facade panels, and
textile metamaterials.

In this work, we present Modal Folding—a novel approach for automatically
discovering folding patterns of sheet materials. Intuitively, folding
transformations correspond to large deformations that require little
mechanical work. Informally, we can ask given a prescribed deformation
magnitude, what is the configuration that minimizes the internal energy?
Formulating this question as a constrained optimization problem reveals direct
ties with the well-known elastic eigenmodes. Indeed, for linear materials and
small deformations, elastic eigenmodes correspond to locally optimal
solutions. However, while linear modes point toward displacements that
prioritize bending over stretching, they are inherently unable to capture the
rotational motion required for folding.

To address this limitation, we introduce _strain-space modes_ as a new
nonlinear analogue of elastic eigenmodes. Instead of following linear modes in
position space, we consider their effect on the sheet’s extrinsic deformation.
In the discrete setting, this gives rise to a set of per-element bending
strains. Collectively, we interpret these per-element strains as directions in
a nonlinear strain space. Any strain-space configuration along such a
direction can be mapped back to position space through closest-point
projection, using the sheet’s elastic energy as the metric. Our key
observation is that this approach—linear modes in strain space combined with
energy-minimizing reconstruction— gives rise to complex nonlinear folding
transformations in position-space. We leverage this insight to explore folding
patterns by tracking strain-space modes in a physics-based and material-aware
way. Modal Folding offers a systematic and automated way to discover complex
designs. We demonstrate folding transformations for various input shapes,
analyze the impact of material properties, and explore different types of
periodic tilings. To illustrate the feasibility of our simulation results, we
build physical prototypes using fabric, paper, and copper sheets.

##  2\. Related Work

#### Folding Thin Sheets

The problem of folding thin sheet materials has been studied intensively. A
central challenge lies in capturing sharp creases that can emerge during
folding (Schreck et al., 2016), which may require adapting mesh resolution and
edge orientations (Narain et al., 2012).

Origami, the ancient art of paper folding, continues to inspire applications
in engineering (Meloni et al., 2021; Callens and Zadpoor, 2018). While
traditional Origami designs are made from straight creases, curved folding
opens up an even larger and fascinating design space (Kilian et al., 2008,
2017; Rabinovich et al., 2019; Jiang et al., 2019; Mundilova, 2019; Tahouni et
al., 2020). Despite the prevalence of folding with sharp creases, smooth
folding techniques also enjoy a multitude of applications, including robotic
manipulation of clothes (Li et al., 2015; De Gusseme and Wyffels, 2022). Zhu
et al. (2013) proposed a computational approach to soft folding that draws on
principles from Origami. Folding can also be used to modulate the mechanical
properties of fabrics. In this context, Ren et al. (2024) introduced a
parametric design tool for creating so-called smocking patterns. Our approach
likewise falls into the category of soft folding. However, instead of relying
on heuristics or user-provided guidance, our method automatically discovers
smooth folding transformation from a single input shape.

#### Eigenmodes in Animation and Design

Eigenmodes have been used extensively in animation, especially in the context
of subspace simulation. While linear modes are ideal for vibration analysis
with small amplitude oscillations, they are notorious for generating artifacts
for larger deformations. Numerous approaches have been proposed to improve
upon linear modes, including modal derivatives (Barbič and James, 2005),
higher-order descent directions (Hildebrandt et al., 2011), and correction of
linearization artifacts (Barbič et al., 2012; Choi and Ko, 2005; Pan et al.,
2015; Huang et al., 2011). Besides subspace simulation, linear modes have also
been used to generate fracture animations (Sellán et al., 2023).

Linear eigenmodes have also been widely used to design physical artifacts that
involve small-amplitude vibrations, including metallophones (Umetani et al.,
2010; Bharaj et al., 2015), wind instruments (Umetani et al., 2016), and
acoustic filters (Li et al., 2016). Zhou et al. (2013) and Liu et al.(2022)
leveraged linear eigenmodes to analyze and improve the stability of 3D printed
designs. Similar in spirit, Zehnder et al. (2016) employed sparse linear
eigenanalysis to detect unwanted structural flexibility in curve networks.

While the aforementioned methods focus on small displacements, Tang et al.
(2020) developed a method for designing compliant mechanical systems with
periodic large-amplitude oscillations. Also targeting the large-deformation
regime, Dünser et al. (2022) proposed a nonlinear extension to linear
eigenmodes. When applied to thin sheets, these nonlinear compliant modes tend
to generate bending deformations initially, but as we show in our analysis,
they ultimately induce stretching for larger deformations. By contrast, our
approach produces large folding transformations with much lower elastic
energy.

#### Rest-State Modifications & Intrinsic Actuation

The concept of driving elastic deformations through rest state modifications
has been explored in the past (Kondo et al., 2005; Schumacher et al., 2012).
Several works have used volumetric blend shapes to drive rest poses for
physics-based facial animation (Kozlov et al., 2017; Ichim et al., 2017).
Similarly, intrinsic actuation via rest-state modifications has been leveraged
for motion control of deformable characters (Coros et al., 2012; Tan et al.,
2012). Similar to these existing works, our approach builds on rest curvature
modifications to drive folding transformations. To the best of our knowledge,
however, our work is the first to combine the concept of intrinsic actuation
with modal analysis.

#### Shape Interpolation

Interpolating between a set of input shapes is a basic task in geometry
processing (Alexa et al., 2000). Since the body of literature is too large for
an exhaustive review, we focus on methods that interpolate triangle meshes
using shell-like deformation energies. The common approach in this context is
to first convert shapes into differential coordinates—e.g., dihedral angles
and edge lengths—that are invariant under translation and (often) rotation.
Building on this basis, Fröhlich et al. (2011) linearly interpolated between
differential coordinates of input shapes. The interpolated shape is obtained
by minimizing a discrete shell energy (Grinspun et al., 2003) with rest angles
and edge lengths given by the interpolated differential coordinates.
Similarly, our method also builds on linear interpolation in a space of
nonlinear differential coordinates to which we refer as strain space. However,
instead of interpolating between example shapes, we extrapolate a single shape
using strain-space directions induced by linear eigenmodes.

Another closely related concept is geodesic interpolation in shape space
(Kilian et al., 2007). Heeren et al. (2012) obtained the shortest geodesics
between pairs of shapes by minimizing the total dissipation of the discrete
shell energy along the path. Similarly, Heeren and colleagues (2014) compute
geodesics by minimizing the path energy defined through a sequence of deformed
shapes. In addition, several works leverage Principal Geodesic Analysis (PGA)
for capturing the variability of a set of shapes (Freifeld and Black, 2012;
Heeren et al., 2018; Sassen et al., 2020). All the above methods require
either a second shape or a tangent space direction, which must be specified by
the user. Our approach automatically computes energetically efficient folding
transformations from a single input shape using the eigenstructure of its
Hessian.

##  3\. Method

Our approach leverages the mechanics of thin sheet materials to compute
folding transformations in an automated manner. We begin with a brief review
of linear modes before we proceed to our nonlinear strain-space modes.

###  3.1. Eigenmodes

We use a standard discrete shell model (Gingold et al., 2004; Grinspun et al.,
2003) to describe the mechanics of thin sheet materials (see Supp. 1). Using
this model, we compute energetically optimal folding transformations that
minimize the mechanical work required to achieve deformations of prescribed
magnitudes. Let $\mathbf{X}$ and $\mathbf{x}$ denote vertex positions for
undeformed and deformed states, respectively. Furthermore, let
$\mathbf{f}=-\nabla_{\mathbf{x}}W(\mathbf{X},\mathbf{x})$ denote the vector of
forces and $\mathbf{H}=\nabla^{2}W$ the corresponding Hessian arising from the
discrete total energy $W$. This energy includes internal elastic energy and
external penalty energies for fixing vertices to exclude rigid
transformations. For deformations around the rest state, a second-order
expansion of the elastic energy yields

(1)
$W(\mathbf{X}+\mathbf{u})=W(\mathbf{X})-\mathbf{f}(\mathbf{X})^{\mathsf{T}}\mathbf{u}+\frac{1}{2}\mathbf{u}^{\mathsf{T}}\mathbf{H}\mathbf{u}+O(\mathbf{u}^{3})\approx\frac{1}{2}\mathbf{u}^{\mathsf{T}}\mathbf{H}\mathbf{u}\
,$

where we used the fact that both the energy and its gradient vanish at the
reference configuration. Finding displacements that minimize the energy for
given deformation budgets leads to a constrained optimization problem,

(2)
$\min\frac{1}{2}\mathbf{u}^{\mathsf{T}}\mathbf{H}\mathbf{u}\qquad\text{s.t.}\quad\frac{1}{2}\mathbf{u}^{\mathsf{T}}\mathbf{M}\mathbf{u}=b^{2}\
,$

where $\mathbf{M}$ is the mass matrix and $b$ is a given displacement
magnitude. The Lagrangian for this problem is

(3)
$\mathcal{L}(\mathbf{u},\lambda)=\frac{1}{2}\mathbf{u}^{\mathsf{T}}\mathbf{H}\mathbf{u}+\lambda(\frac{1}{2}\mathbf{u}^{\mathsf{T}}\mathbf{M}\mathbf{u}-b^{2})\
,$

where $\lambda$ is a Lagrange multiplier. The first-order optimality
conditions of this problem require that

(4) $\mathbf{H}\mathbf{u}+\lambda\mathbf{M}\mathbf{u}=0\ .$

It is evident from this expression that solutions to the above optimization
problem are collinear with generalized eigenvectors $\mathbf{e}_{i}$ of the
Hessian $\mathbf{H}$. In particular, the solution corresponding to the
smallest eigenvalue $\lambda_{0}$ minimizes the mechanical work required to
generate a displacement of magnitude $b$. Solutions for other eigenvalues
$\lambda_{i}$ minimize work in the subspace orthogonal to the $i-1$ lower-
order eigenvectors.

These considerations suggest that generalized eigenvectors of the energy
Hessian are promising directions for folding transformations, i.e., large-
magnitude displacements that require little mechanical work. Indeed, when
analyzing eigenvectors for the discrete shell energies described in Supp. 1,
we see that low-order eigenvectors are normal to the surface for flat sheets.
Since normal displacements do not generate stretching to first order, these
eigenvectors correspond to bending deformations, which are energetically
favorable. However, simply following these eigenvectors along linear
trajectories,

(5) $\Psi_{i}^{\mathrm{LM}}(t)=\mathbf{X}+t\mathbf{e}_{i}\ ,$

does not lead to physically meaningful results: although linear modes produce
bending initially, they tend to induce stretching for larger displacements,
resulting in a steep increase in energy (see Fig. 3). This behavior is
explained by the fact that the discrete shell energy is highly nonlinear and
the quadratic approximation (1), from which eigenvectors are computed, is only
valid for small displacements.

Many extensions of linear eigenmodes to the finite-deformation range have been
explored in the past. We experimented with the two variants that we believe to
be the most promising options. However, as our analysis in Sec. 4 shows,
neither of them are able to produce the low-energy transformations that we
seek to achieve. For this reason, we develop a new formulation that we
describe next.

###  3.2. Strain-Space Modes

Linear modes ultimately suffer from a lack of rotational invariance: a linear
trajectory in position-space cannot generate pure bending deformations. Our
formulation, instead, leverages the rotational invariance of discrete shell
energies. By modulating per-element rest curvatures, we can draw the
equilibrium configuration of the thin sheet towards desired states. Let
$\bar{\boldsymbol{\kappa}}=(\kappa_{0},\ldots,\kappa_{m})^{\mathsf{T}}$ denote
the vector of per element rest curvatures, which can either be per-edge
dihedral angles $\kappa_{i}=\theta_{i}$ (Grinspun et al., 2003) or per-
triangle shape operators $\kappa_{i}=\mathbf{S}_{i}$ (Gingold et al., 2004),
depending on the type of bending element used (see Supp. 1). Let
$W^{\mathcal{M}}(\mathbf{X},\mathbf{x})$ and
$W^{\mathcal{B}}(\mathbf{X},\bar{\boldsymbol{\kappa}},\mathbf{x})$ denote the
discrete membrane and bending energies, respectively. Minimizing elastic
energy
$W(\mathbf{X},\bar{\boldsymbol{\kappa}},\mathbf{x})=W^{\mathcal{M}}(\mathbf{X},\mathbf{x})+W^{\mathcal{B}}(\mathbf{X},\bar{\boldsymbol{\kappa}},\mathbf{x})$
for given rest curvatures $\bar{\boldsymbol{\kappa}}$ will produce the
deformed configuration $\mathbf{x}^{*}$ that best approximates the target
curvatures,

(6)
$\mathbf{x}^{*}=\operatorname*{arg\,min}_{\mathbf{x}}W(\mathbf{X},\bar{\boldsymbol{\kappa}},\mathbf{x})\
.$

We note that the minimizer will have zero energy if and only if the target
curvatures are compatible in the sense that they can be obtained without
stretching the sheet. Otherwise, the minimizer will balance in-plane
deformation and deviation from target angles in a material-aware manner.

Clearly, different target curvatures will lead to different equilibrium
shapes. Since linear modes are energetically optimal for small displacements,
we choose the per-element curvatures induced by linear modes at the rest
configuration as target curvatures. To this end, we ask that the initial
tangent of the system’s trajectory $\mathbf{x}(t)$ should coincide with the
eigenvector $\mathbf{e}_{i}$, i.e., $\dot{\mathbf{x}}(0)=\mathbf{e}_{i}$. The
position-space velocity induces a rate of change in curvatures according to

(7)
$\dot{\boldsymbol{\kappa}}=\frac{\partial\boldsymbol{\kappa}}{\partial\mathbf{x}}\frac{\partial\mathbf{x}}{\partial
t}=\mathbf{J}\mathbf{e}_{i}\ ,$

where the Jacobian $\mathbf{J}$ maps position-space directions into
corresponding strain-space directions. Following the modal directions
$\mathbf{J}\mathbf{e}_{i}$ defines linear strain-space trajectories
$\bar{\boldsymbol{\kappa}}(t)=\bar{\boldsymbol{\kappa}}(0)+t\mathbf{J}\mathbf{e}_{i}$.
We can reconstruct points on these strain-space trajectories into position-
space configurations by using them as rest curvatures in (6). The
corresponding nonlinear position-space trajectories are obtained by solving
minimization problems of the form

(8)
$\Psi^{\mathrm{SSM}}_{i}(t)=\operatorname*{arg\,min}_{\mathbf{x}}W(\mathbf{X},\bar{\boldsymbol{\kappa}}(t),\mathbf{x})\
.$

The strain-space modes defined in this way build on changes in rest curvature
to drive shape transformation. This intrinsic actuation mechanism is
rotationally invariant by construction. As we show in Sec. 4, this property
makes strain-space modes ideal for exploring folding transformations of thin
sheet materials.

#### Multi-Dimensional Extension

Just as their linear counterparts, one-dimensional strain-space modes can be
extended to multi-dimensional nonlinear subspaces. To this end, we generalize
the definition of rest curvatures,

(9)
$\bar{\boldsymbol{\kappa}}(\mathbf{c})=\bar{\boldsymbol{\kappa}}(0)+\sum_{j}c_{j}\mathbf{J}\mathbf{e}_{j}\
,$

where $\mathbf{c}=(c_{1},\ldots,c_{m})$ is a vector of modal coefficients. The
above definition in (8) gives rise to a nonlinear subspace,

(10)
$\Phi^{\mathrm{SSM}}(\mathbf{c})=\operatorname*{arg\,min}_{\mathbf{x}}W(\mathbf{X},\bar{\boldsymbol{\kappa}}(\mathbf{c}),\mathbf{x})\
.$

In Sec. 4 we illustrate the potential of this nonlinear subspace on an inverse
folding example. Finally, we note that the above definition includes blended
modes, i.e., generalized one-dimensional strain-space modes,

(11)
$\Psi^{\mathrm{SSM}}(\hat{\mathbf{c}},t)=\Phi^{\mathrm{SSM}}(t\hat{\mathbf{c}})\
,$

where $\hat{\mathbf{c}}$ is a unit-length vector of modal coefficients.

#### Implementation

We compute discrete strain-space modes $\Psi^{\mathrm{SSM}}_{i,j}$ by
minimizing (8) with rest curvatures $\bar{\boldsymbol{\kappa}}=\Delta
t\mathbf{J}\mathbf{e}_{i}$ ramping up according to the step size $\Delta t$.
We solve these unconstrained minimization problems using a standard Newton’s
method with adaptive regularization and line search. To incorporate self-
collision handling, we add the barrier potential proposed by Li et al. (2020).
To remove translational and rotational null spaces, we furthermore add
quadratic penalties to fix selected vertices. Since the resulting minimization
problems are nonlinear and non-convex, they generally have multiple local
minima. To avoid jumping between minima when computing the discrete
trajectory, we initialize the Newton solve for a given next state
$\Psi^{\mathrm{SSM}}_{i,j+1}$ with the previous state
$\Psi^{\mathrm{SSM}}_{i,j}$. We found that this simple approach leads to
smooth trajectories for all examples that we considered in this work.

Figure 3. Comparing internal energy and maximum in-plane strain along the
first non-rigid mode computed using linear eigenmodes, nonlinear compliant
modes (NCM), rotation-strain modes (RSM), and our strain-space modes.

##  4\. Results

We evaluate our method on a diverse set of examples and compare folding
patterns generated through simulation with corresponding physical prototypes.
We start by comparing strain-space modes to other alternatives. We then
explore periodic patterns and demonstrate multi-dimensional strain-space modes
on an inverse folding example. Unless otherwise specified, we use hinge-based
bending elements (Grinspun et al., 2003) as the default shell model for all
experiments.

###  4.1. Comparing Modes

As a first experiment that allows for both quantitative and qualitative
evaluation, we compare the lowest-energy mode produced by different methods
for a square sheet with side lengths of $20cm$. We set Young’s modulus to
$2.9GPa$, Poisson’s ratio to $0.3$, and use sheets of $1mm$ thickness. As can
be seen from the energy plot shown on the left of Fig. 3, linear modes induce
stretching even for small deformations, leading to a steep energy profile. The
nonlinear compliant modes by Dünser et al. (2022) perform much better in
comparison, showing a slower increase in energy for small to moderate
deformations. However, as the diagonal tips of the sheet approach each other,
the external actuation force aligns with the sheet’s tangent space. This
alignment induces stretching, which leads to a sharp increase in energy. The
reason for this behavior is that nonlinear compliant modes rely on extrinsic
actuation using an external force collinear with the linear mode. While a
force with constant direction can create pure bending initially, it will
induce stretching eventually. In contrast, our strain-space modes build on an
intrinsic actuation method that modulates rest curvatures of discrete shell
elements. The resulting actuation forces remain (approximately) normal to the
surface even for large deformations. As a result, our strain space modes
produce large bending deformations with virtually zero stretching and,
consequently, a much slower increase in energy. We furthermore experimented
with another nonlinear extension of eigenmodes, the rotation-strain modes by
Huang et al. (2011). As can be seen from Fig. 3, the lowest-order rotation-
strain mode initially shows a slower increase in energy than its linear
counterpart. However, it clearly exhibits double curvature, which translates
into nonzero in-plane deformation and, consequently, a steep increase in
energy beyond the small displacement regime. We conjecture that the least-
squares geometry reconstruction step, which is at the heart of rotation-strain
modes, does not distinguish between bending and stretching, leading to
configurations that exhibit physically unrealistic levels of in-plane
deformation.

As a second basic example, we compare the four different mode types on a
simple origami design111https://origamisimulator.org/ consisting of a square
sheet divided into 3$\times$3 sub-squares, each with a side length of $7.2cm$.
The creased sheet exhibits a single degree of freedom, corresponding to a
nonlinear twist motion of the center square. To emulate the effect of pre-
scoring, we reduce the bending stiffness for crease edges by a factor of 10.
As shown on the right of Fig. 3, only our method successfully reproduces the
twisting behavior; see also the accompanying video. Additionally, the energy
and in-plane strain plots show that our method achieves the final twisted
state with low energy and small maximum in-plane strain.

These two examples underscore the observation that, unlike existing nonlinear
modes, our strain-space modes give rise to energetically efficient folding
transformations with very low levels of stretching. In the following examples,
we further explore strain-space modes and their applications.

###  4.2. Exploring Strain-Space Modes

Strain-space modes can generate diverse and interesting folding
transformations for a large variety of input shapes and materials. Fig. 4
shows a selection of strain-space modes for the square sheet used in the
initial example of Fig. 3. Please see the accompanying video for an exhaustive
tour through the modal shapes up to mode index 99. It can generally be noted
that the folding transformations computed with our method lead to large
bending deformations with negligible stretching. Strain-space modes
furthermore reflect the symmetries of their linear counterparts, but the
nonlinearity of our approach gives rise to a rich space of intricately folded
shapes.

To validate that the folding transformations generated by our method are
physically feasible, we build physical prototypes using thin cotton sheets. To
this end, we select and attach several pairs of points in close proximity such
that the fabric maintains the desired folded state at equilibrium. We
developed a simple interactive tool for this purpose in which users select
pairs of vertices that are attached using zero-length springs. The static
solver then provides feedback on the deviation between equilibrium and target
states. We repeat the procedure until a desired shape accuracy is obtained.
See the accompanying video for a demonstration of this process. We create
physical prototypes for folded configurations using adhesive to connect the
pairs of points selected by the user. As can be seen from the example shown in
Fig. 1, our prototypes successfully reproduce the folded shapes predicted in
simulation.

Mode 25 | Mode 30 | Mode 35
---|---|---
|  |
Mode 43 | Mode 50 | Mode 57
|  |
Figure 4. Modal folding patterns for a square sheet.

The shape of the input mesh has a significant impact on the resulting linear
modes, and this effect translates directly to strain-space modes. For example,
compared to the square sheet, the higher symmetry of the disc-shaped sheet
shown in Fig. 5 leads to a different and more symmetric set of modal shapes.

Mode 16 | Mode 20 | Mode 21
---|---|---
|  |
Mode 24 | Mode 25 | Mode 30
|  |
Figure 5. Symmetric modal folding patterns for a disc-shaped sheet.

In addition to input geometry, modal shapes are also affected by internal
structure. For example, when removing material from a solid sheet, our method
can leverage the void regions to induce in-plane deformations that lead to
more efficient transformations. To illustrate this effect, we use a thin sheet
cut into the shape of a so called life flower pattern as shown in Fig. 6.
Prototypes are fabricated using thick nylon fabric that is laser cut, folded,
and fixed into position using instant glue. The physical shapes obtained in
this way closely match the results obtained from the simulations.

Rest State |  |
---|---|---
Mode 9 |  |
Mode 13 |  |
Mode 24 |  |
Figure 6. Folding patterns for a thin sheet cut into the pattern of a life
flower.

###  4.3. Periodic Folding Patterns

Periodicity plays a central role in many folding patterns, offering both
aesthetic appeal and functional advantages. Examples include smocking patterns
for cloth design, self-folding periodic origami, and seamlessly tilable panels
for architectural facades. In the following, we explore two types of periodic
folding patterns based on translation and reflection tilings of the plane.

#### Translation Tilings.

A translation tiling of the plane with a single shape means that pairs of
opposite tile boundaries—left and right, top and bottom—have to match without
gaps and overlaps. To enforce this periodicity condition, we ask that vertices
$\mathbf{x}_{ik}$ on boundary $i$ should have the same displacement
$\mathbf{T}_{ij}$ as their counterparts $\mathbf{x}_{jk}$ on the opposite
boundary $j$, i.e, $\mathbf{x}_{jk}=\mathbf{x}_{ik}+\mathbf{T}_{ij}$ for all
boundary vertices $k$. Following previous work (Schumacher et al., 2018; Tang
et al., 2023; Li et al., 2023), we implement these constraints by eliminating
degrees of freedom from one side of each boundary pair and adding the constant
displacement between the two sides as additional variables. This
transformation yields an unconstrained minimization problem whose Hessian
gives rise to a set of eigenvectors that automatically satisfy the periodicity
conditions. We convert these eigenvectors into strain-space modes as
before—see Fig. 7 for results.

Figure 7. Strain-space modes for a square sheet subject to periodic boundary
conditions.

#### Reflection Tilings.

Whereas translation tilings ask that a given boundary matches with its
opposite side, reflection tilings require each boundary to tile with itself.
To this end, all vertices of a given boundary must lie in a common reflection
plane. We focus on flat rectangular tiles with axis-aligned $xz$\- and $yz$\-
reflection planes. This setup leaves two degrees of freedom per boundary pair,
i.e., $x=\pm a$ and $y=\pm b$, where $a$ and $b$ denote the distance of the
reflection plane from the origin of the patch; see Fig. 8. Removing all in-
plane degrees of freedom for boundary vertices and adding the two variables
$a$ and $b$, we obtain an unconstrained minimization problem, from which we
compute periodic modal tilings in analogy to the translation-only case.

Figure 8. Folding results for a square sheet tiled with reflections at mode
23. We reflect the unit cell (the top-right tiling) to other tilings according
to reflection planes $x=a$ and $y=b$.

Examples of periodic folding patterns with reflection boundaries are shown in
Figs. 2 and 8. We use a copper-like material with $0.01mm$ thickness,
resulting in somewhat sharper creases compared to cotton. To validate our
simulation results, we fabricate several prototypes using thin copper sheets.
This process involves matching 3D-printed molds that are tightly pressed
against the copper sheets to shape them into the desired pattern. It can be
seen from the results shown in Fig. 2 that our physical prototypes reproduce
the folding patterns obtained in simulation without noticeable artifacts.

###  4.4. Inverse Design

Figure 9. Inverse design for fitting a flat disc of radius $10cm$ into a
sphere with a radius of $6cm$. Using the first 50 strain-space modes we obtain
a folding pattern on the left. In comparison, we gradually decrease the radius
of a sphere and obtain a crumpled folding on the right.

The examples presented so far focused on forward exploration of one-
dimensional modes. However, strain-space modes and their multi-dimensional
extensions can also be used in inverse design problems. For the purpose of
illustration, we consider an inverse folding problem where the goal is to find
a folding transformation for a fabric disc of $10cm$ radius such that it fits
into a sphere with radius of $6cm$. To this end, we define an objective
function that penalizes mesh vertices outside the target sphere. We use the
first 50 strain-space modes to define a nonlinear subspace as in (10) and
optimize for modal coefficients such as to minimize the objective function;
see Supp. 2 for details. For comparison, we use IPC (Li et al., 2020) to
simulate the contact-induced compaction of the disc by gradually decreasing
the radius of the sphere from $10.1cm$ to $6.0cm$ with a step size of $0.1cm$.
As can be seen from the result shown in Fig. 9, inverse design with strain-
space modes leads to a smooth and three-fold symmetric folding pattern. The
contact-driven variant, in contrast, appears crumpled rather than folded.

###  4.5. Additional Results and Experiments

#### Performance.

We show the performance of our method by measuring the average time spent on
computing each state using Newton’s solver, considering 10 modes (from mode 6
to 15) and 10 states for each mode with an extrapolation step of 0.1. We test
examples on a machine with an Intel Core i7-13700F 2.1GHz processor and 32 GB
of RAM. Statistics are in Tab. 1.

Table 1. Performance of our method for the life flower, a disc sheet, and square sheets. Example | life flower | disc | square 40x40 | square 60x60
---|---|---|---|---
#nodes | 1975 | 817 | 1681 | 3721
#elements | 2262 | 1536 | 3200 | 7200
Time[s] | 3.48 | 1.53 | 3.87 | 7.96

In addition to the examples presented in this section, we evaluate the impact
of mesh resolution and discretization on the resulting modal shapes in Supp.
4. We furthermore demonstrate in Supp. 3 that strain-space modes reflect the
mechanical properties of the underlying material, including thickness-to-width
ratio and isotropic vs. orthotropic bending stiffness.

##  5\. Conclusions

We presented a new method to automatically discover folding patterns for thin
sheet materials. The technical core of our approach is formed by strain-space
modes that track the nonlinear bending strains induced by linear eigenmodes.
We showed that strain-space modes lead to energetically favorable deformations
compared to existing alternatives. Our examples reveal a rich space of novel
folding patterns that can be realized as physical surfaces made from fabric,
paper, or sheet metal.

###  5.1. Limitations and Future Work

#### Stretching.

Although strain-space modes are driven through curvatures, they are not
automatically developable: per-element bending strains can combine into non-
zero Gaussian curvature at vertices, which in turn induces stretching.
Empirically, we observed that stretching is generally small (Supp. 3, Fig. 1).
Nevertheless, developing a formulation for modes that are isometric by design
would be an interesting subject for future work.

#### Mesh Dependence.

Our method behaves consistently under refinement, i.e., subdividing a mesh of
given topology will generally not lead to drastic changes in modal shapes
(Supp. 4, Fig. 3). By contrast, changing mesh topology for a given resolution
can change individual strain-space modes substantially (Supp. 4, Fig. 4).
Energy-minimizing remeshing could be a promising strategy for mitigating this
effect.

#### Blending Modes.

We selected modes that we found aesthetically pleasing, interesting, and
diverse. We did not perform in-depth experimentation with mode blending (11)
as the resulting modes tend to have lower symmetry. Nevertheless, our inverse
design example suggests that mode blending can be an effective means of
discovering efficient folding transformations for high-level objectives.

#### Sharp Creases.

Our method creates folds whose smoothness depend on the underlying material.
Thinner sheets will generally produce sharper folds (Supp. 3, Fig. 1).
Nevertheless, it would be interesting to directly generate creases through
modal folding. One promising approach to this goal might be to replace the
elastic energy with a version that eliminates resistance to bending for larger
curvatures, thus promoting creases.

###### Acknowledgements.

The authors thank the anonymous reviewers for their valuable feedback. This
work was supported by the European Research Council (ERC) under the European
Union’s Horizon 2020 research and innovation program (grant agreement
No.866480), and the Swiss National Science Foundation through SNF project
grant 200021_200644.

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