2021

[1]Keith J. Lee

[1]Department of Architecture, Building Technology, Massachusetts Institute of
Technology, 77 Massachusetts Avenue, Cambridge, 02139, Massachusetts, USA

# Spherical harmonic shape descriptors of nodal force demands for quantifying
spatial truss connection complexity

<EMAIL_ADDRESS>Renaud Danhaive<EMAIL_ADDRESS>Caitlin T. Mueller
<EMAIL_ADDRESS>*

###### Abstract

The connections of a spatial truss structure play a critical role in the safe
and efficient transfer of axial forces between members. For discrete
connections, they can also improve construction efficiency by acting as
registration devices that lock members in precise orientations. As more
geometrically complex spatial trusses are enabled by computational workflows
and the demand for material-efficient spanning systems, there is a need to
understand the effects of global form on the demands at the connections. For
large-scale structures with irregular geometry, customizing individual nodes
to meet exact member orientations and force demands may be infeasible;
conversely, standardizing all connections results in oversized nodes and a
compromise in registration potential. We propose a method for quantifying the
complexity of spatial truss designs by the variation in nodal force demands.
By representing nodal forces as a geometric object, we leverage the spherical
harmonic shape descriptor, developed for applications in computational
geometry, to characterize each node by a rotation and translation-invariant
fixed-length vector. We define a complexity score for spatial truss design by
the variance in the positions of the feature vectors in higher-dimensional
space, providing an additional performance metric during early stage design
exploration. We then develop a pathway towards reducing complexity by
clustering nodes with respect to their feature vectors to reduce the number of
unique connectors for design while minimizing the effects of mass
standardization.

###### keywords:

truss, similarity, clustering, connections, spherical harmonics

##  1 Introduction

The complexity of a structural design is often the restricting factor to its
realization. It is also difficult to define and quantify. Variations in
internal forces to be resisted and the optimal member sizes required,
anticipated fabrication time and tooling, and the ease of assembly all
contribute to design complexity. For spatial truss structures, all of these
measures apply. Consider the spatial truss in Figure 1, consisting of 185
nodes and 664 tubular steel elements. Despite a regular planar grid spacing of
nodes and relatively regular member orientations, a significant variation in
axial forces is observed.

Figure 1: A spatial truss design: a) problem formulation, b) displaced shape
with internal force colormap, c) distribution of axial forces

For many similar structures built around the world, two missed opportunities
are often evident: first, members are typically uniformly sized, resulting in
unnecessary material consumption and the increase in structural mass. Often,
this uniform sizing comes from the reduction of material procurement
complexity and resulting cost reduction. Second, the design and detailing of
nodal connections are also uniform, despite an equally large variation in
force transfer requirements. Generally, these nodes are often heavy steel
cylinders or spheres with mitered element ends and welded connections.
Although the connections themselves come from easily fabricated parts,
construction complexity is increased by the lack of inherent registration
capacity of the nodes, as well as the time required to miter weld the
elements. Further, as the building industry pushes towards facilitating the
deconstruction and reuse of building components, providing discrete nodes over
welded connections should be encouraged (Fivet and Brütting, 2020; Brütting et
al, 2019).

Although the optimal sizing and orientation of spatial truss elements has been
the focus of much research, the performance potential of the connections has
been less understood. Spatial truss nodes play an important role, both as
member registration devices that lock member orientations and facilitate
assembly, and as a critical force transfer mechanism between elements. For
shape-optimized trusses, where orientations are often highly irregular and
transferred forces are large, the need for both registration and strength is
critical.

Ideally, all nodes in a spatial truss should be tailored to their connected
members and anticipated forces. However, this adds significant complexity to
all stages of the design, manufacture, and construction process. Conversely,
if a single standardized connection is used, it must be oversized for most of
the structure, and providing registration for elements would be difficult for
non-regular truss geometries. In this case, complexity is inherent in the
variation of the force demands at each node, and represents the lost potential
of customization. If the variations in nodal demands can be quantified,
designers can strategically reduce the anticipated complexity of design and
manufacture by identifying critical nodes that warrant customization, and
nodes where a standardized connector is sufficient.

We develop a method of consistent nodal demand characterization and its
resulting nodal complexity metric to provide this insight to designers. By
representing the forces acting at a node by a single spherical object, we
leverage the spherical harmonic shape descriptor developed in the field of
computational geometry to characterize an arbitrary number of force magnitudes
and orientations with a fixed-length feature vector. The conversion of forces
into a single geometric object resolves challenges in comparing the similarity
of nodes with different numbers of element connectivity, and the use of the
spherical harmonic shape descriptor provides invariance to the orientation of
the node, allowing for recognition of similar nodal demands that are rotated
or translated throughout a spatial truss. The feature vectors of all nodes
provide the basis of a nodal distance matrix, which is then used for a
complexity metric when evaluating multiple designs, and for clustering
analysis to strategically reduce the number of unique connections required for
design and fabrication.

##  2 Related work

Our method builds upon existing research in quantifying design complexity,
design rationalization, computational geometry, and geometric representation
of forces. We present several key works in each field.

###  2.1 Design complexity and rationalization

In mechanical and industrial design, Bashir and Thomson (1999) defines a
quantitative metric of design complexity by the depth and breadth of the
function trees of each component in the product; the authors find a positive
correlation with worker-hours required at surveyed manufacturing plants.
Design for Assembly and Assembly-Oriented Design procedures have been
developed to integrate assembly feasibility in the early design phase of
product design (Boothroyd, 1987; Dalgleish et al, 2000). These procedures
range from qualitative rubrics for engineering feedback to rigorous data
management for communication between engineers, designers, and fabricators
(Demoly et al, 2011).

In the design of buildings, Mitchell (2005) qualitatively defines
architectural complexity as the ratio of the quantity of information required
to geometrically represent the design to the quantity of information required
for fabrication and assembly of the elements. These two measures are presented
as conceptually orthogonal axes, where both components can either be minimized
(highly regular, modular construction) or maximized (highly customized,
complex construction). Scheurer (2010) provides insight on the relationship
between geometric complexity and the technology required for its realization
from the perspective of an architecture practitioner.

Reducing complexity in the manufacturing and construction process has a
significant impact on material consumption. Moynihan and Allwood (2014)
surveyed 23 steel framed buildings in the UK and found an average utilization
rate (design load of member divided by the member capacity) of less than 50%,
indicating significant unnecessary material consumption. Of the 23 buildings,
the designers of 8 noted fabrication and construction efficiency as the
primary driver for oversizing members.

When project circumstances permit the realization of complex geometries, shape
rationalization is often performed to reduce fabrication and construction
challenges while maintaining architectural intent. In the case of free-formed
structural envelopes and facades, much of the focus is in converting arbitrary
surfaces to be either developable—surfaces with 0 Gaussian curvature that can
be flattened without distortion, or ruled—surfaces that can be made of linear
elements. Pottmann et al (2015) provides an overview of many of the challenges
and solutions towards this discretization, and notes the existing barriers to
make any technique completely generalizable.

Alternatively, designers can limit themselves with tools that only generate
developable and ruled surfaces to ensure their initial intent can be readily
realized. The hyperbolic paraboloid concrete shell structures of Felix Candela
were based on ruled surfaces whose formwork can be assembled with standard
dimensional lumber (Candela, 1955); Glaeser and Gruber (2007) provides a
computational method for the same effect for developable surfaces.

Complexity in design is not always considered detrimental. During the option
evaluation phase of design, complexity can be interpreted as diversity. Brown
and Mueller (2019) investigated multiple metrics of diversity in the context
of parametric design exploration. The values of each design-changing parameter
are represented as components of a point in $\mathbb{R}^{n}$, with different
points in the parameter space representing different designs. Diversity is
then measured by the spread of the design points in parameter space.

Rationalization for building efficiency is not restricted to geometry. Stephen
et al (2018) considered the domain of regional load demands across the United
States as the basis of rationalization. By clustering similar load demand
regions, material-efficient emergency shelters can be designed for a reduced
set of loading requirements, rather than a single oversized shelter.

For topology optimized truss structures, methods of integrating construction
complexity based on unit node and element costs have been developed by
Asadpoure et al (2015) and Torii et al (2016). These methods seek to minimize
the accumulation of lower-impact nodes and elements in the ground-based
topology optimization approach.

For the rationalization of spatial truss nodes, Koronaki et al (2020) pursued
an alternative method of characterizing nodal demands for the purpose of
complexity reduction. A best-fit reference plane is placed among normalized
element vectors acting at a node using Principal Component Analysis, and a
consistent reference element is selected. All other elements are characterized
by the relative angle to this reference element with respect to the reference
plane. Dissimilarity between two nodes is measured by the difference in the
ordered set of element angles. The nodal positions are then optimized to
maximize the number of similar nodes within a set angle tolerance.

###  2.2 Shape similarity

By representing forces as a three dimensional shape, we leverage shape
recognition tools in the field of computational geometry. Shape similarity
plays a key role in fields ranging from molecular biology to 3D animation.
Applications include indexing from expansive catalogs of parts to feature
recognition and shape interpolation for character modeling. Biasotti et al
(2016) provided a state of the art overview of a wide range of proposed shape
similarity descriptors. The output for most shape descriptors is the feature
vector, a fixed-length vector representation of a given shape. Dissimilarity
between two shapes is measured by the distance between feature vectors.

Two general categories of shape descriptors exist: geometry-based and
topology-based. For geometry-based shape similarity, the exact positions of
shape vertices, edges, and faces are analyzed and compared. The challenge for
these methods is controlling invariance, where the same shape in different
orientations is not inherently captured by the analysis. Chen et al (2003)
used a series of evenly spaced virtual cameras around a given 3D object to
find 2D projection images from all angles. An optimal alignment procedure,
akin to rigidly rotating the positions of the virtual cameras, is performed to
compare two shapes with minimum distortion. Advancements in view-based shape
recognition include hypergraph representations of the captured 2D images (Wang
et al, 2016), and various learning-based processes, summarized by Qi et al
(2021).

Topology-based methods consider the underlying connectivity of vertices and
edges of the input shape as the basis of shape characterization; these methods
exploit intrinsic properties of the topology that are independent of reference
frames and orientations. The Heat Kernel Signature, the intrinsic
characterization of simulated heat diffusion in a meshed object, has been used
to recognize shapes in varying non-rigid deformations (Bronstein and Kokkinos,
2010; Raviv et al, 2010), and for recognizing highly detailed features on a
specific shape (Bronstein et al, 2011).

A unique group of shape descriptors are those that exploit orthonormal basis
functions on the circle (2D shapes) and sphere (3D shapes). These methods are
not topology-based, but have intrinsic invariance to object orientation. In
general, the input shapes are transformed into periodic circular or spherical
functions, that are then expanded as sums of basis functions using Fourier
series (2D) or spherical harmonics (3D). the unique coefficients of the
expanded functions allow for a fixed-length vector representation of the shape
they represent. 2D shape recognition using Fourier expansion was first
proposed by Persoon and Fu (1977). The contour of an input shape is
represented as a periodic function, which is then represented as a weighed sum
of cosine functions of increasing frequency; the scalar coefficients of each
term provides the rotation-invariant feature vector. The use of Fourier shape
descriptors have been expanded for finer resolution of more complex shapes
(Zhang and Lu, 2002; Yang and Yu, 2019), as well as for shapes with holes for
spatial geography applications (Xu et al, 2017) .

The 3D equivalent is the decomposition of the input shape into its expanded
spherical harmonics. The spherical harmonic shape descriptor, first proposed
by Kazhdan et al (2003), intersects a given shape with a series of concentric
spherical shells. The points of intersection for a given shell represent the
shell’s spherical function, which can then be decomposed as a weighed sum of
spherical harmonic basis functions; by taking the $L_{2}$-norm of the
frequency components of each expansion series, a rotation-invariant
characterization of the given shape can be determined. This inherent rotation
invariance is the key to consistent characterization and comparisons between
shapes, and is the method chosen for adaption in this paper; the mechanics of
this process is described in detail in Section 3. Advancements on this method
have been proposed by Wang et al (2016), who optimized the expansion
coefficients for more complex shapes with detailed feature, and by Esteves et
al (2018) for applications in convolutional neural networks.

###  2.3 Geometric representation of force

Geometric representation of force for analysis and design is most recognized
in the field of Graphic Statics. Developed in its current form in the 19th
century, graphic statics represents the axial forces of linear elements of a
form diagram with a reciprocal force diagram (Maxwell, 1864; Rankine, 1864).
In 2D, the reciprocal force diagram represents a loaded linear element with an
equivalent line whose length is proportional to the internal force; in 3D, it
is represented by a 2D polygon whose area is force-proportional. Nodal
equilibrium is represented as a closed polygon (2D) or polyhedron (3D).
Although initially used as a procedural method of analysis and design, recent
algebraic formalizations of graphic statics principals have allowed
computational graphic statics to resurge as a modern tool in design and
analysis (Van Mele and Block, 2014; Akbarzadeh, 2016; Hablicsek et al, 2019;
Nejur and Akbarzadeh, 2021; Mozaffari et al, 2021).

###  2.4 Research gap and contributions

In Brown and Mueller (2019), a quantifiable measure of design variation was
presented. However, the method quantifies inter-design variation, rather than
variation within a single design, and does not consider internal force
magnitudes. We develop a similar metric of design complexity, but focus on the
internal force demands of a single design.

In Koronaki et al (2020), a method of spatial truss node rationalization was
developed. One drawback is the omission of force magnitude when characterizing
a structural node. We present an alternative method that includes force
magnitudes, and further, does not require a reference frame alignment process
for each node.

We adapt the spherical harmonic shape descriptor method by Kazhdan et al
(2003) by converting nodal force demands into geometric spherical objects, and
extending the resulting shape descriptors to useful applications in the
building design, both as a performance metric in the early design phase, and
as the basis of clustering for design rationalization. We bypass the
concentric shells method and directly convert nodal forces to a single
spherical function. Along with reduced computation, it also omits the need to
calibrate the number and spacing of these shells to fully characterize a given
shape.

Although the equilibrium shapes of graphic statics are the most natural
geometric representations of force demand, current computational tools for the
analysis of spatial truss structures are limited to self-stressed networks or
specific typologies. Further, using polygons and polyhedra requires either
secondary processing to ensure rotation invariance during comparison, or the
concentric shell method if using spherical harmonic descriptors. We present an
alternative geometric representation of force that streamlines inter-nodal
comparisons.

Our methodology is summarized in Figure 2.

Figure 2: Methodology overview: 1. the nodal force demands are converted to
spherical functions that encode force magnitudes and orientations, 2. the
force functions are expanded into spherical harmonic basis functions, 3. the
feature vector is extracted from the harmonic decomposition, 4. the distance
matrix between all feature vectors is determined, 5. clustering analysis and
complexity metrics are enabled by the distance matrix.

##  3 Methodology

This section provides a detailed overview of the conversion of an arbitrary
spatial truss node and its force demands to a rotation-invariant feature
vector using the spherical harmonic shape descriptor developed by Kazhdan et
al (2003). We modify the process to be readily applicable for spatial truss
analysis by providing a deterministic method of converting nodal force demands
to a single spherical force function, and calibrate the resolution of the
feature vector to ensure the force function is well represented. Multiple
visualization methods are also developed to provide designers with rapid
insight on nodal complexity for a given spatial truss form. The process is
summarized in Figure 3.

Figure 3: Overview of the conversion of nodal force demands to a rotation-
invariant fixed-length feature vector.

The examples provided in this section are based on the structure in Figure 1;
multiple variations of this spatial truss are also analyzed for comparison
throughout this paper. All variations contain the same topology, number of
nodes and elements, and overall enclosed planar area. Linear elastic analysis
is performed to extract all internal forces. In total, 100 variations were
generated using Latin hypercube sampling of a parameterized model (Danhaive
and Mueller, 2021). The model has six design variables which correspond to the
z-coordinates of control points of a bilaterally symmetric NURBS surface used
to define the upper curved geometry of the two-layer trussed roof. In this
model, the bottom layer and positions of the columns and supports remain
constant. The parametric model is intended to generate a range of designs of
varying efficiency achieved through structural morphology, as a proxy for
design alternatives that would be considered in early-stage structural
exploration.

###  3.1 Spherical function representation

The spherical harmonic shape descriptor was initial developed to be
generalizable to any 3D object, where the intersection of spheres of
increasing radius with the given object creates binary spherical functions.
This step takes added computation to sample the intersection points, and
requires calibration of the number of spheres required to capture small
details. We bypass this step by directly creating a single representative
spherical function that captures all information of nodal force demand.

We first convert the set of forces acting at a node to its equivalent function
by considering the spherical coordinates of entry (for compression forces) and
exit (for tension forces). At each coordinate, the scalar function value is
equal to the respective axial force magnitude. To provide the smooth, square-
integrable function required for spherical harmonic expansion, we convert the
singular axial force values as Gaussian distributions on the surface of the
sphere:

$f_{i}(\theta,\phi)=f(\theta_{i},\phi_{i})e^{-\delta\left((\theta-\theta_{i})^{2}+(\phi-\phi_{i})^{2}\right)}$
(1)

Where $f(\theta_{i},\phi_{i})$ is the magnitude of force $i$ acting at
$\theta_{i},\phi_{i}$. We take the physics convention of $\theta$ as the angle
from the polar axis $z$, and $\phi$ as the angle from the $x$ axis in the $xy$
plane. Because the end result of the spherical harmonic shape descriptor is
invariant to rotation, a consistent reference axis is not required when
determining the spherical force function, but is taken as the global XYZ axes
for convenience. For a node with $n$ forces, the spherical force function is
then the sum of all Gaussian force distributions:

$f(\theta,\phi)=\sum_{i}^{n}f_{i}(\theta,\phi)$ (2)

The selection of the variation factor $\delta$ requires calibration. Large
values approaches a dirac delta function of a singular spike at the location
of the force, resulting in poor approximations when expanding into spherical
harmonics; small values smooth the forces over large regions of the sphere,
and result in the loss of individual force delineation. We show a range of
values for $\delta$ in Figure 4. We take $\delta=20$ for all subsequent
analyses for good delineation between close force vectors and good
approximation by spherical harmonic expansion (Section 3.2). Structures with
closely packed elements at nodes may require a larger value for better
individual force delineation.

Figure 4: Spherical functions of varying values of $\delta$. We take
$\delta=20$ for subsequent analyses for good representation of all force sets
while remaining computationally efficient.

The conversion of force demands into spherical functions is performed for all
nodes of the structure; a sample is provided in Figure 5. The depth of the
indentations and protrusions are proportional to the forces within each node,
but are not to scale when comparing two nodes. This deformation of the
spherical surface is intended for better visualization, but the force function
itself remain scalar-valued one-to-one functions on the surface of the unit
sphere, $S^{2}$.

Figure 5: Spherical force functions for varying nodal force demands.

###  3.2 Spherical harmonic expansion

With suitable spherical representation of nodal force demands, we then
decompose each function into a weighed sum of the spherical harmonic basis
functions. The spherical harmonics are the eigenfunction solutions to
Laplace’s equation on the sphere: $\nabla^{2}f(\theta,\phi)=0$, where
$\nabla^{2}$ is the Laplace Operator, the divergence of the gradient of the
scalar function $f$. The solutions, denoted $Y_{l}^{m}(\theta,\phi)$, are
grouped into frequencies (degree), $l$, and harmonics (mode), $m$, and form a
complete orthonormal set of basis functions on the sphere. Frequencies span
from zero to infinity in integer increments, with each frequency containing
$2l+1$ modes that span from $[-l,l]$; the spherical harmonics of the first 8
frequencies and their corresponding harmonics are shown in Figure 6, with red
regions indicating negative function values and blue regions indicating
positive values. The shapes are determined by scaling the radial component of
the spherical function, $r$, by the magnitude of the function value.

Figure 6: Spherical harmonics for $l\in[0,7]$; $m\in[-l,l]$

As the spherical harmonics form a complete orthonormal set of basis functions,
any square-integrable function $f(\theta,\phi)$ can be represented as a
summation of these harmonics by:

$f(\theta,\phi)=\sum_{l=0}^{l_{max}}\sum_{m=-l}^{l}a_{lm}Y_{l}^{m}(\theta,\phi)$
(3)

Where $a_{lm}$ is the associated scalar coefficient specific to the input
function $f$. As $l_{max}\rightarrow\infty$, an identical representation of
the input function is determined. The precision of the harmonic expansion of a
spherical function depends on the resolution of $l_{max}$ taken by the
summation in Equation 3. The relative error of this resolution for a given
spherical force function $f$ is taken as:

$\text{error}_{l_{max}}=\frac{\lVert
f-\sum_{l=0}^{l_{max}}\sum_{m=-l}^{l}a_{lm}Y_{l}^{m}\rVert_{2}}{\lVert
f\rVert_{2}}$ (4)

We expand a representative nodal force function to varying degrees of
$l_{max}$ and observe the resolution error in Figure 7a. Steady reduction in
error is observed with increasing resolution, especially in the early stages,
with a slower convergence to zero past $l_{max}\approx 14$. We perform this
error analysis on all 185 nodes of the example structure in Figure 7b, and
observe similar frequency-error relationships. We take $l_{max}=16$, or the
first 17 frequencies, for all subsequent analyses, with a mean relative error
of 1%.

Figure 7: The effect of $l_{max}$ on the effective representation of a
spherical force function using spherical harmonic expansion: a) for a single
node, b) for 185 nodes.

There is a circular relationship between the variance factor $\delta$ used in
the force function representation of nodal demand, and the error of
approximation when choosing $l_{max}$. Larger values of $\delta$ will sharpen
the positions of the force values and require more terms during the spherical
harmonic expansion to approximate its shape. As noted in Section 3.1, the
value of $\delta$ will require verification depending on the design of the
specific spatial truss structure, and accordingly, $l_{max}$ may have to be
adjusted. Our selection of $l_{max}=16$ equals the value taken in the original
method by Kazhdan et al (2003).

###  3.3 Frequency accumulation and feature vector representation

We continue to follow the spherical harmonic shape descriptor process by
summing the weighed basis functions in each frequency, $l$, to create a single
spherical function for each frequency:

$f_{l}=\sum_{m=-l}^{l}a_{lm}Y_{l}^{m}$ (5)

This summation provides the key characteristic of rotation-invariance. The set
of all harmonic functions within a fixed frequency,
$Y_{l}^{-l},...,Y_{l}^{l}$, is a representation of the rotation group $SO(3)$.
Any partial feature on the input function that is represented by the specific
set of harmonic functions of frequency $l$ is represented by the same set of
functions irrespective of its orientation in space. Although the corresponding
coefficients of the harmonic functions $a_{lm}$ may be rearranged depending on
the orientation of the reference axes, the overall summation within the
frequency does not change. As such, the overall contribution of a given
frequency in representing a shape can be defined as the $L_{2}$-norm of the
summed function, or its energy. This scalar positive energy value then creates
the components of the rotation-invariant feature vector:

$FV(f)=\left[\lVert f_{0}\rVert_{2},...,\lVert f_{n}\rVert_{2}\right]$ (6)

The feature vector is the final outcome of the conversion process, and the
arbitrary sets of forces acting on different structural nodes can now be
compared.

###  3.4 Measuring dissimilarity

By expanding all nodal functions to the same value of harmonic frequencies, a
fixed-length feature vector representation of each nodal force function is
determined. The dissimilarity between two nodes is captured by the
$L_{2}$-norm of the difference between their respective feature vectors:

$d(\text{node}_{1},\text{node}_{2})=\lVert FV_{1}-FV_{2}\rVert_{2}$ (7)

This distance can be measured between any two nodes, regardless of the number
of connected elements and its position on the structure, and represents both
force orientations and magnitudes. By taking all inter-nodal distances within
a structure, a symmetric distance (or dissimilarity) matrix is created, as
seen in Figure 8.

Figure 8: Distance matrix of all nodes. Darker regions indicate nodes that are
dissimilar from each other.

The distance matrix provides a visual overview of the variation of nodal force
demands across the structure. Rows (or columns) with consistently large
distances indicate connections with highly unique force demands, and are ideal
targets for more intensive connection design and detailing. The dark bands of
consistent dissimilarity in Figure 8 represent the four support connections at
the base of the example structure. An alternative visualization to represent
the dissimilarity of structural nodes is through classical multidimensional
scaling (MDS) of the $n$-dimensional feature vectors into parsable lower
dimensions. Classical MDS is a dimension reduction technique that embeds
higher dimensional points into an arbitrarily lower dimension while best
preserving the initial distance matrix (Torgerson, 1952).

An example of how the distance matrix can be used for visual understanding of
nodal force demand variation, as well as a Classical MDS embedment in
$\mathbb{R}^{2}$, is shown in Figure 9. Four designs with the same topology,
planar dimensions, and materials as in Figure 1 are shown along with their
distance matrices and the 2 dimensional embedment of their nodal feature
vectors. The absolute coordinates of the embedded points hold no significance,
but the distances between them are representative of the Euclidean distances
in $\mathbb{R}^{n}$. By visual inspection, it is evident that designs with
greater inter-nodal dissimilarity have more dark regions in the distance
matrices, as well as a greater dispersion of embedded feature vector points.
During the project ideation phase, designers can look to both the distance
matrix and embedded point visualizations of the nodal feature vectors to
indicate designs that are better or worse suited for either mass customization
or standardization of the joints. Larger dispersions of the embedded points
(or darker distance matrices) indicate a greater penalty in standardization.

Figure 9: Distance matrix and 2D embedment of feature vectors using MDS for
four design alternatives.

Geometric analysis of forces allow for a wealth of visualizations during each
stage of the analysis process. We show 5 equivalent representations of nodal
force demands in Figure 10, each useful for varying stages of analyses or
visual understanding of force demand variance. We introduce the superimposed
parallel coordinate plot of feature vectors in 10e as an alternative to the
MDS embedded points as a method of visualizing the degree of similarity
between nodes. The visually distinct line group in the parallel coordinate
plot again represent the unique force demands acting at the compression
supports in Figure 1.

Figure 10: Equivalent representations: a) force vectors acting on node, b)
spherical force functions, c) feature vectors, d) embedded points in
$\mathbb{R}^{2}$, e) superimposed parallel coordinate plots of feature
vectors.

###  3.5 Characteristics of the feature vector

The feature vector representation of nodal force demands has two key
characteristics. First, the feature vectors scale linearly to the input
function:

$FV(cf)=cFV(f)$

When only member orientations are of concern when calculating inter-nodal
dissimilarity, force demands can first be normalized before harmonic expansion
and analysis. Second, we observe that smooth variations of the input spherical
force functions result in smooth changes in the output feature vector. From a
representative node and its associated forces, we move one force along an arc
(shown in Figure 11), and plot both its feature vector in parallel coordinate
form and its relative $\mathbb{R}^{2}$ embedded position to all other sampled
force positions. All other forces acting on the node remain at the same
position, but their magnitudes are changed at each iteration to maintain
equilibrium. We observe small changes in the feature vector with small changes
of the force position, best observed by the incremental distances between the
embedded feature vector points in Figure 11b. This gradual increase is smooth,
but not predictable; the start and end positions of the moving force vector
have greater similarity than regions in between.

Figure 11: Smooth changes of input: a) the transition path of a single force
in a nodal force set, b) the resulting positions of the feature vectors
embedded in $\mathbb{R}^{2}$ using MDS, c) the spherical force functions of
the altered force directions, d) parallel plot of changing feature vectors

Third, as normed frequency energies are taken as the components of the feature
vector, they are invariant to direction of the force acting at the node:

$FV(f)=FV(-f)$

This represents a loss of information once a feature vector is calculated for
a given node, resulting in a non-bijective map from nodal force demands to the
space of feature vectors. It is not possible to invert the feature vector to
the input distribution of compression and tension forces. This limitation
means that tasks such as interpolating new force vector sets is not possible
directly, but could be achieved through iterative optimization methods in
future work.

Finally, it is emphasized that the components of the feature vector do not
directly correspond to a specific force acting on a node. Rather, they
represent overall regional features of the force function captured by a given
frequency. A large component in the feature vector should not be interpreted
as a particularly large, singular force. However, this lack of direct
correspondence between the feature vector and its associated node is key to
its function: any truss node, with an arbitrary number of elements and
orientations, can be represented by a vector of the same dimension.

###  3.6 Methodology summary

The procedure to characterize and compare a set of structural nodes and their
force demands require the following steps: 1. conversion into a spherical
force function, $f(\theta,\phi)$, created as a sum of Gaussian distributions
of individual force components on the surface of the sphere; 2. expansion into
spherical harmonic basis functions; 3. the summation and $L_{2}$-norm of the
frequency energies,
$FV_{l}(f)=\lVert\sum_{m=-l}^{l}a_{lm}Y_{l}^{m}\rVert_{2}$. This feature
vector is of dimension $n=l_{max}$, chosen by the user, and taken by the
authors as $16$. Each feature vector can be considered as points in $n$
dimensional space, and the Euclidean distance between any two feature vectors
are representative of the dissimilarity between the force demands of their
respective structural nodes, both in magnitude and orientation.

##  4 Applications in spatial truss design

We present two immediate applications of the feature vector representation of
nodal force demands and its corresponding distance matrix. First, as a
performance metric of design complexity, measured in proxy as the magnitude of
distribution of the feature vectors. Second, as the basis of clustering
analysis for strategic reduction of unique connections that are designed and
manufactured.

###  4.1 Complexity and structural performance

When evaluating multiple design options, a metric of complexity may help
reduce design, fabrication, and construction costs. In the case of spatial
truss nodes, designers may seek to limit the variations in nodal force demands
if limited to a single standardized connection. Conversely, designers should
be informed when a large standardization penalty would incur, and suggest the
selection of alternative designs.

By considering the nodal feature vectors as embedded points in
$\mathbb{R}^{17}$, we take the radius of the minimal bounding hypersphere as
the metric of nodal dissimilarity within a given spatial truss. Comparing the
size of the bounding hyperspheres from different designs provides a numeric
measure of node complexity, and provides an additional performance metric to
work in tandem with structural performance. We note this measure of complexity
should only be used comparatively, as the feature vectors do not have direct
physical meaning.

A comparison of three alternative designs, as well as their 2D embedded points
and minimal bounding circle, is shown in Figure 12. In the follow examples,
the radius of the 2D minimal bounding sphere is exactly the radius of the
bounding sphere in the full feature vector space ($\mathbb{R}^{17}$). We
observe significant variation in the complexity scores of each design, despite
equal numbers of nodes and elements and overall enclosed area. Designers
seeking to minimize standardization penalties for connections or who seek to
create a reduced number of unique joints should seek designs with smaller
bounding circles.

Figure 12: Comparing three alternative designs with their nodal complexity
scores, measured by the radius of the minimal bounding hyperspheres of the
nodal feature vectors in $\mathbb{R}^{17}$

Node complexity is not the only metric of design performance. More commonly,
the global structural performance, measured by the structural material
quantity required to withstand the expected loads, drives the design process.
We analyzed 100 variations of the initial example structure to investigate the
relationship between nodal complexity and structural performance, with the bi-
objective plot shown in Figure 13.

Figure 13: Bi-objective plot of structural material quantity vs. hypersphere
radius for 100 spatial truss designs; deflected shapes and internal forces of
three key designs highlighted.

Each structure is given the same initial loading condition and loads as in
Figure 1. An iterative member sizing process is performed for all structural
members by sizing each hollow tube section to withstand its internal load. As
cross section properties change the stiffness and force distribution in the
structure, this process was repeated until convergence. A minimum cross
section area of $400mm^{2}$ was chosen to represent the smallest standard
structural hollow tube section (HSS42.2X3.6). The total mass of this structure
was then divided by the enclosure area of 100$m^{2}$ to provide a normalized
score of structural performance. The minimal bounding hypersphere found from
the nodal feature vectors of each plot make up the other axis. We highlight
the highest performing structure for each objective, as well as a
simultaneously poorly performing structure. The bounds of the figure have been
adjusted to exclude severe outliers that perform poorly for both objectives,
with the largest offender having a hypersphere radii of approximately 15000
and a minimum mass of 150$kg/m^{2}$. We observe that the design with the best
complexity performance does not perform well structurally, but achieves its
high degree of nodal similarity due to relatively parallel top and bottom
chord planes. The most structurally performant design is more conventional,
with a deeper cross section at the support region that tapers towards the
cantilever.

###  4.2 Clustering

A method of reducing the standardization penalty without extensive
customization is by strategically designing fewer connections to meet the
demand of multiple configurations. To achieve this, similar nodal demands must
be identified to minimize the variation within each design target. We use
K-means clustering on the nodal feature vectors analyzed in Section 3 to
reduce the nodes into 10 similar groups; the results are shown in Figure 14.
The 2D embedded points are coloured by their cluster assignments along with
the individual bounding spheres for each cluster. An immediate measure of the
reduction in nodal complexity is the change in size of the overall bounding
hypersphere to those of the individual clusters. The superimposed feature
vectors, as well as highlighted feature vectors for each cluster are shown as
parallel coordinate plots.

Figure 14: Clustering structural nodes into 10 similar groups. Clusters are
ordered in increasing cluster complexity (hypersphere radius of clustered
feature vectors). Clusters 1 and 2 contain identical nodal demands, and have a
complexity score of 0.

The symmetry of the design and its loading condition is recognized, with each
feature vector having at least one identical feature vector across the
symmetry plane between the supports. Due to the odd nodal grid spacing
transverse to this plane, nodes along the centerline between the supports do
not have a symmetric twin, but are clustered with their adjacent nodes as
their geometries and internal forces remain similar. Symmetric node groups
with highly unique force demands, such as the large compression forces
experienced by the dark blue supports, are isolated in their own cluster, with
an associated zero radius hypersphere due to their identical loading
conditions. Groups with higher numbers indicate larger variation within the
cluster, but all feature vectors show similar patterns of peaks and troughs in
their highlighted parallel coordinate plots, with small variations of
magnitude in each component.

An additional four structures and their clustered node groups are shown in
Figure 15. Clusters are coloured in the same increasing hypersphere radius
order as shown in Figure 14. Symmetry is again recognized, with the two
compression supports being uniquely separated in all cases. During design
iterations of the same topology, observing this consistent grouping provides
an indicator that certain regions of the structure are deserving of
specialized connection design. This is further evident by observing the
embedded 2D feature vector points in Figure 14, where simply providing two
connection designs—one for the compression supports and one for all other
nodes–would drastically reduce the overall nodal complexity score for the
entire structure. It is also from the colour ordering of the clusters that
despite a consistent topology and loading condition, the most similar nodes
are not consistently located across the structure. For example, the green
clusters shift from the central region to the structure to the cantilevered
corners between two design iterations.

Figure 15: Four designs with 10 nodal clusters. The colours of the clusters
are ordered in increasing complexity as in Figure 14. Despite the same
topology, best performing clusters are not consistently located in the
structure.

Despite significant variation of performance observed in Figure 13, we note a
dense region of designs with both similar complexity performance and
structural performance, 16 representative samples are shown in Figure 16. It
is clear that even with an added performance constraint when evaluating
designs, a large variation in architectural form can be achieved, as observed
by the relatively small changes in both performance scores. The nodal
complexity analysis can help designers narrow down and refine a wide range of
designs while still maintaining design freedom.

Figure 16: 16 designs with a narrow range of both structural perfomance and
nodal complexity. Designers are not restricted in achievable forms when
considering performance metrics; rather, they can make more informed decisions
when selecting a final design.

We perform another clustering analysis on a higher density structure with 1548
nodes and 6427 elements in Figure 17 (design from Tam et al (2018)). The
structure was designed for 3D printed linear extrusion using PLA, with overall
dimensions of $40\times 5\times 8.5$cm. A 1N load was placed at each node for
structural analysis, and supports are placed along the two lowest lines of
nodes (captured by Cluster 4, red).

Figure 17: Clustering analysis of 3D printed lattice structure with 1548 nodes
and 6427 elements. At least three distinct and suitable candidates for
clustering are observed by the distinct bands in the superimposed parallel
coordinate plot. They are captured by Clusters 1,4/5, and 10.

We observe again one cluster of identical nodes, as well as the detection of
structural symmetry. Further, we observe three natural candidates for
clustering by the bunched parallel coordinate plot lines that are distinct
from other feature vectors. By visual inspection of the parallel coordinate
plots, it is possible to deduce the ideal starting number of clusters to
minimize joint complexity while designing as few unique joints as possible. In
the case of 3D printed lattice structures that lack discrete joints, the nodal
demand analysis and subsequent clustering may be used in identifying critical
joint regions and optimizing the extrusion path to reinforce these joints with
varying amounts of material.

###  4.3 Other applications

The presented method can be easily extended to multiple load cases when a more
in-depth analysis is performed. Nodal feature vectors for each load case can
be concatenated into feature matrices, with the same $L_{2}$-norm distance
matrix. The feature matrix is more akin to the initial methodology by Kazhdan
et al (2003), where each shape contains multiple spherical functions that
capture different features of the input geometry.

As the spherical harmonic node descriptor captures both force magnitude and
geometry, it can also be used to capture the force capacity and ideal member
orientations of existing structural components. Amtsberg et al (2021)
processed the forks of felled trees to act as structural nodes in a timber
grid shell. The matching of tree fork geometries to the nodes of the design
required all tree forks to be of the same valence as the nodes of the
structure (three branches), and did not consider the comparison of fork force
capacity and the expected internal forces of the design. By representing tree
fork capacities and geometries using the presented method, both the feature
vectors of the tree forks can be matched to the feature vectors of the
designed structural nodes irrespective of valence and with consideration of
force magntiudes.

For conventional spatial trusses, the nodes of a decommissioned structure can
be indexed by their representative feature vectors before storage, and can
later be recalled to optimally fit the expected forces and member orientations
of a new spatial truss design. An optimal assignment of an existing node to
the proposed structure is one that minimizes the distance between the nodal
capacity feature vector of the existing node and the nodal demand feature
vector of the new design. Because only the feature vector is required to
capture both force magnitudes and orientations, databases of existing
structural components can be readily maintained with low memory storage
requirements.

##  5 Conclusion and future work

We present an application of the spherical harmonic shape descriptor in the
domain of nodal force demands in a spatial truss structure. By representing an
arbitrary number of forces acting at a node as a single spherical surface, and
by characterizing this surface with the spherical harmonic feature vector, we
develop a consistent characterization of nodal force demands and a measure of
inter-nodal dissimilarity. By representing the range of this dissimilarity by
the size of the minimal bounding hypersphere in $n$ dimensional feature vector
space, we provide a metric of relative nodal complexity when evaluating
multiple designs. Further, we use the distance matrix of the nodal feature
vectors in a given structure as the basis of node clustering, which can
minimize the penalty of connection standardization and allow for the strategic
design of a reduced set of customized components.

Two questions are left to be resolved, and will be the focus of future work.
First, the practical materialization of a single custom connection for
similar, but not identical, nodal force demands is not trivial. Additive
manufacturing in conjunction with multiple load case topology optimization
should be investigated as method of designing and manufacturing truss nodes
with slight variations in force magnitudes and orientations. Second, as
discussed in Section 3.3, the map of nodal force demands to feature vectors is
not bijective, and thus computational optimization in the feature vector space
is not straightforward. We intend to investigate the efficiency of numeric
autodifferentiation tools to provide a similar pathway of altering a design to
reduce nodal complexity as in Koronaki et al (2020).

The ability to characterize structural nodes with arbitrary loading conditions
while capturing both force magnitude and relative member orientations allows
for greater insight during the design of spatial truss structures. As digital
tools facilitate rapid design iterations, they should also provide more
metrics of performance that recognize practical constraints of time, tooling,
and cost. To minimize material consumption and enable a circular economy of
building materials, we expect the proliferation of form and element optimized
spatial truss structures with discrete nodal connections. These nodes will
play a critical role in improving construction efficiency, by acting as
registration devices, and in deconstruction and reuse, by reducing permanent
miter-and-weld connections. We provide a method of characterizing the
complexity in the demands of these discrete nodal connectors, and methods
towards its strategic reduction.

## Statements and Declarations

The authors have no competing interests, financial or otherwise, pertaining to
this work.

## References

  *   * Akbarzadeh (2016) Akbarzadeh M (2016) 3d graphical statics using reciprocal polyhedral diagrams. PhD thesis, ETH Zurich
  * Amtsberg et al (2021) Amtsberg F, Huang Y, Marshall DJ, et al (2021) Structural up-cycling: Matching digital and natu-ral geometry. Advances in Architectural Geometry 2020
  * Asadpoure et al (2015) Asadpoure A, Guest JK, Valdevit L (2015) Incorporating fabrication cost into topology optimization of discrete structures and lattices. Structural and Multidisciplinary Optimization 51(2):385–396
  * Bashir and Thomson (1999) Bashir HA, Thomson V (1999) Estimating Design Complexity. Journal of Engineering Design 10(3):247–257. 10.1080/095448299261317, URL http://www.tandfonline.com/doi/abs/10.1080/095448299261317
  * Biasotti et al (2016) Biasotti S, Cerri A, Bronstein A, et al (2016) Recent trends, applications, and perspectives in 3d shape similarity assessment. In: Computer Graphics Forum, Wiley Online Library, pp 87–119
  * Boothroyd (1987) Boothroyd G (1987) Design for assembly—The key to design for manufacture. Int J Adv Manuf Technol 2(3):3–11. 10.1007/BF02601481, URL http://link.springer.com/10.1007/BF02601481
  * Bronstein et al (2011) Bronstein AM, Bronstein MM, Guibas LJ, et al (2011) Shape google: Geometric words and expressions for invariant shape retrieval. ACM Trans Graph 30(1):1–20. 10.1145/1899404.1899405, URL https://dl.acm.org/doi/10.1145/1899404.1899405
  * Bronstein and Kokkinos (2010) Bronstein MM, Kokkinos I (2010) Scale-invariant heat kernel signatures for non-rigid shape recognition. In: 2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, IEEE, pp 1704–1711
  * Brown and Mueller (2019) Brown NC, Mueller CT (2019) Quantifying diversity in parametric design: a comparison of possible metrics. AIEDAM 33(1):40–53. 10.1017/S0890060418000033, URL https://www.cambridge.org/core/product/identifier/S0890060418000033/type/journal_article
  * Brütting et al (2019) Brütting J, Desruelle J, Senatore G, et al (2019) Design of truss structures through reuse. In: Structures, Elsevier, pp 128–137
  * Candela (1955) Candela F (1955) Structural applications of hyperbolic paraboloidical shells. In: Journal Proceedings, pp 397–416
  * Chen et al (2003) Chen DY, Tian XP, Shen YT, et al (2003) On Visual Similarity Based 3D Model Retrieval. Computer Graphics Forum 22(3):223–232. https://doi.org/10.1111/1467-8659.00669, URL https://onlinelibrary.wiley.com/doi/abs/10.1111/1467-8659.00669, _eprint: https://onlinelibrary.wiley.com/doi/pdf/10.1111/1467-8659.00669
  * Dalgleish et al (2000) Dalgleish GF, Jared GEM, Swift KG (2000) Design for assembly: Influencing the design process. Journal of Engineering Design 11(1):17–29. 10.1080/095448200261162, URL http://www.tandfonline.com/doi/abs/10.1080/095448200261162
  * Danhaive and Mueller (2021) Danhaive R, Mueller CT (2021) Design subspace learning: Structural design space exploration using performance-conditioned generative modeling. Automation in Construction 127:103,664
  * Demoly et al (2011) Demoly F, Yan XT, Eynard B, et al (2011) An assembly oriented design framework for product structure engineering and assembly sequence planning. Robotics and Computer-Integrated Manufacturing 27(1):33–46
  * Esteves et al (2018) Esteves C, Allen-Blanchette C, Makadia A, et al (2018) Learning so (3) equivariant representations with spherical cnns. In: Proceedings of the European Conference on Computer Vision (ECCV), pp 52–68
  * Fivet and Brütting (2020) Fivet C, Brütting J (2020) Nothing is lost, nothing is created, everything is reused: structural design for a circular economy. The Structural Engineer 98(1):74–81
  * Glaeser and Gruber (2007) Glaeser G, Gruber F (2007) Developable surfaces in contemporary architecture. Journal of Mathematics and the Arts 1(1):59–71
  * Hablicsek et al (2019) Hablicsek M, Akbarzadeh M, Guo Y (2019) Algebraic 3d graphic statics: Reciprocal constructions. Computer-Aided Design 108:30–41
  * Kazhdan et al (2003) Kazhdan M, Funkhouser T, Rusinkiewicz S (2003) Rotation invariant spherical harmonic representation of 3 d shape descriptors. In: Symposium on geometry processing, pp 156–164
  * Koronaki et al (2020) Koronaki A, Shepherd P, Evernden M (2020) Rationalization of freeform space-frame structures: Reducing variability in the joints. International Journal of Architectural Computing 18(1):84–99. 10.1177/1478077119894881, URL https://doi.org/10.1177/1478077119894881, publisher: SAGE Publications
  * Maxwell (1864) Maxwell JC (1864) Xlv. on reciprocal figures and diagrams of forces. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 27(182):250–261
  * Mitchell (2005) Mitchell WJ (2005) Constructing complexity. In: Computer aided architectural design futures 2005. Springer, p 41–50
  * Moynihan and Allwood (2014) Moynihan MC, Allwood JM (2014) Utilization of structural steel in buildings. Proc R Soc A 470(2168):20140,170. 10.1098/rspa.2014.0170, URL https://royalsocietypublishing.org/doi/10.1098/rspa.2014.0170
  * Mozaffari et al (2021) Mozaffari S, Hablicsek M, Akbarzadeh M, et al (2021) Developing a polyhedral graphic statics formulation for tetrahedral truss systems. In: Proceedings of the IASS Annual Symposium 2020/21, Surrey, UK
  * Nejur and Akbarzadeh (2021) Nejur A, Akbarzadeh M (2021) Polyframe, efficient computation for 3d graphic statics. Computer-Aided Design 134:103,003. https://doi.org/10.1016/j.cad.2021.103003, URL https://www.sciencedirect.com/science/article/pii/S0010448521000142
  * Persoon and Fu (1977) Persoon E, Fu KS (1977) Shape discrimination using fourier descriptors. IEEE Transactions on systems, man, and cybernetics 7(3):170–179
  * Pottmann et al (2015) Pottmann H, Eigensatz M, Vaxman A, et al (2015) Architectural geometry. Computers & Graphics 47:145–164. 10.1016/j.cag.2014.11.002, URL https://linkinghub.elsevier.com/retrieve/pii/S009784931400140X
  * Qi et al (2021) Qi S, Ning X, Yang G, et al (2021) Review of multi-view 3D object recognition methods based on deep learning. Displays 69:102,053. 10.1016/j.displa.2021.102053, URL https://linkinghub.elsevier.com/retrieve/pii/S0141938221000639
  * Rankine (1864) Rankine WM (1864) Xvii. principle of the equilibrium of polyhedral frames. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 27(180):92–92
  * Raviv et al (2010) Raviv D, Bronstein MM, Bronstein AM, et al (2010) Volumetric heat kernel signatures. In: Proceedings of the ACM workshop on 3D object retrieval, pp 39–44
  * Scheurer (2010) Scheurer F (2010) Materialising complexity. Architectural Design 80(4):86–93
  * Stephen et al (2018) Stephen C, Brown NC, Mayencourt PL, et al (2018) Clustering analysis of structural loading for post-disaster housing design. In: Proceedings of IASS Annual Symposia, International Association for Shell and Spatial Structures (IASS), pp 1–8
  * Tam et al (2018) Tam KMM, Marshall DJ, Gu M, et al (2018) Fabrication-aware structural optimisation of lattice additive-manufactured with robot-arm. International Journal of Rapid Manufacturing 7(2-3):120–168
  * Torgerson (1952) Torgerson WS (1952) Multidimensional scaling: I. theory and method. Psychometrika 17(4):401–419
  * Torii et al (2016) Torii AJ, Lopez RH, Miguel LF (2016) Design complexity control in truss optimization. Structural and Multidisciplinary Optimization 54(2):289–299
  * Van Mele and Block (2014) Van Mele T, Block P (2014) Algebraic graph statics. Computer-Aided Design 53:104–116
  * Wang et al (2016) Wang D, Sun S, Chen X, et al (2016) A 3D shape descriptor based on spherical harmonics through evolutionary optimization. Neurocomputing 194:183–191. 10.1016/j.neucom.2016.01.081, URL https://linkinghub.elsevier.com/retrieve/pii/S0925231216002435
  * Xu et al (2017) Xu Y, Xie Z, Chen Z, et al (2017) Shape similarity measurement model for holed polygons based on position graphs and Fourier descriptors. International Journal of Geographical Information Science 31(2):253–279. 10.1080/13658816.2016.1192637, URL https://www.tandfonline.com/doi/full/10.1080/13658816.2016.1192637
  * Yang and Yu (2019) Yang C, Yu Q (2019) Multiscale Fourier descriptor based on triangular features for shape retrieval. Signal Processing: Image Communication 71:110–119. 10.1016/j.image.2018.11.004, URL https://linkinghub.elsevier.com/retrieve/pii/S0923596518304521
  * Zhang and Lu (2002) Zhang D, Lu G (2002) Shape-based image retrieval using generic Fourier descriptor. Signal Processing: Image Communication 17(10):825–848. 10.1016/S0923-5965(02)00084-X, URL https://linkinghub.elsevier.com/retrieve/pii/S092359650200084X