# Visual design intuition: Predicting dynamic properties of beams from raw
cross-section images

Philippe M. Wyder12, Hod Lipson1
1Department Of Mechanical Engineering, Columbia University New York, USA
2 Corresponding Author<EMAIL_ADDRESS>
###### Abstract

In this work we aim to mimic the human ability to acquire the intuition to
estimate the performance of a design from visual inspection and experience
alone. We study the ability of convolutional neural networks to predict static
and dynamic properties of cantilever beams directly from their raw cross-
section images. Using pixels as the only input, the resulting models learn to
predict beam properties such as volume maximum deflection and eigenfrequencies
with 4.54% and 1.43% Mean Average Percentage Error (MAPE) respectively,
compared to the Finite Element Analysis (FEA) approach. Training these models
doesn’t require prior knowledge of theory or relevant geometric properties,
but rather relies solely on simulated or empirical data, thereby making
predictions based on “experience” as opposed to theoretical knowledge. Since
this approach is over 1000 times faster than FEA, it can be adopted to create
surrogate models that could speed up the preliminary optimization studies
where numerous consecutive evaluations of similar geometries are required. We
suggest that this modeling approach would aid in addressing challenging
optimization problems involving complex structures and physical phenomena for
which theoretical models are unavailable.

###### Index Terms:

artificial design intuition, Shape optimization, biomimetics, deep learning,
modal analysis, frequency analysis, surrogate model, geometric optimization

##  I Goals and Motivations

Arriving at an optimal design of a part, component, or structure often
requires evaluating countless candidate solutions with similar geometries.
This is commonly accomplished by performing finite element analysis (FEA) for
each candidate design. This process is computationally expensive and time
consuming, especially when the design problem involves complex geometries and
complex nonlinear physics, and when sufficiently accurate theoretical models
are unavailable. As finite element models cannot learn, but rather solve each
problem anew, they cannot benefit from prior solutions of geometrically
similar problems.

To overcome the computational burden of full FEA during optimization,
engineers sometimes use surrogate models trained on sample cases to quickly
predict approximate mechanical properties of candidate solutions without
requiring a complete analysis. Once established, surrogate models can be used
in the early exploratory phase of an optimization process.

Surrogate models are commonly created using machine learning (ML) techniques,
allowing them to predict approximate structural properties when trained on
examples from a relevant design domain[1, 2, 3]. Typically, an ML algorithm
receives the relevant physical and geometric design variables as input, and
predicts the desired properties as outputs. For example, to predict beam
deflection, the model would be provided with design variables such as beam
length, material properties, and second moments. Using a dataset of input
variables and corresponding outputs, an ML model can be trained to represent a
function that approximates the resulting beam deflection.

Unfortunately, the relevant physical and geometric input variables for
predicting a particular property are often not known in advance, especially
when modeling complex phenomena, such as the energy absorption of a vehicle
crash box, or the lift of a flexing insect wing. On the other hand, it is
common knowledge that the second moment is a relevant input variable for
predicting a beam’s natural frequencies. In design spaces where both
analytical models and knowledge of relevant design variables are lacking,
surrogate models that learn the relevant variables would allow us to explore
and optimize structures.

This was the motivation behind the present study, as a part of which an ML
algorithm was adopted in an attempt to create a surrogate model from raw data
inputs (such as image pixels) alone, i.e., without prior identification of the
relevant input variables. If successful, the proposed strategy could be
extended to tackle other problems for which the relevant input variables are
unknown, such as those involving complex, nonlinear interactions: buckling
under impact, aero-elastic vibrations, or the dynamic behavior of nano-
reinforced micro structures with complex geometries. For example, the analysis
of mechanical properties of micro-scale cantilever beams with carbon nanotube
(CNT) reinforcements or structures with embedded CNTs or silica carbide tubes
require FEA, since they are impossible to analyze with analytical methods [5,
4]. Thus, having a modeling approach that is able to identify the relevant
parameters from a visual representation could facilitate the exploration of
these design spaces where humans lack visual intuition.

Based on the results presented in this work, we believe that distilling
”visual design intuition” through deep learning could lead to faster and more
automated design optimization, especially for recurring problems that lack
precise mathematical formulation.

###  I-A Deep neural network models for visual intuition

Deep learning methods, and in particular Convolutional Neural Networks (CNNs),
have been successfully used in a wide range of design applications, from
generating car silhouettes to predicting molecule structures [6, 7, 8, 9].
Furthermore, CNNs have the capacity to model complex phenomena, such as
classifying objects in images, recognizing actions in videos, and predicting
the dynamic behavior of a lava lamp or a candle flame, as well as to predict
the behavior of another agent in a game of hide and seek [11, 12, 10, 13].
However, in most cases, CNN models are trained on raw image data, without
access to user-selected relevant variables or analytical models.

As humans, we have a naturally developed non-parametric visual intuition for
our environment and the materials and structures that we typically interact
with. This allows us to appreciate, for example, that a thinner beam has a
lower natural frequency than a thicker beam of the same length, and that the
tip of a longer beam will deflect more than that of a shorter beam with the
same cross-section. Our intuition for shapes and materials similarly allows us
to rule out flawed designs without relying on theoretical knowledge or
analytical models.

Figure 1: Twisted cantilever beam that is clamped at one end: the beam’s first
three eigenfrequencies ($f_{1}$, $f_{2}$, and $f_{3}$) are shown as they were
estimated using FEA, an analytical solution, and our trained model. The FEA
solution serves as the ground truth.

Hence, the aim of the present study is to determine whether CNNs, when used as
surrogate fitness measures for design optimization, can provide a similar
visual design intuition. A trained engineer would struggle to guess the first
eigenfrequency ($f_{1}$) of the beam shown in Fig. 1 solely from seeing its
cross-section, even if provided with the information on the beam’s material
and geometry. On the other hand, our proposed model that capitalizes on
efficient GPU parallelization provided a correct answer to this particular
question in approximately 2.3 ms with 1.61% mean absolute error (based on the
model presented in Appendix F-A). To achieve these results, we trained a CNN
on cross-sectional images of randomly-generated cantilever beams to predict
FEA outcomes, such as volume maximum stress, beam deflection, and
eigenfrequency. The resulting model was subsequently applied as a surrogate
fitness measure to identify beam geometries that fulfill predetermined
configurations, such as a specific set of eigenfrequencies.

In sum, the model presented here was trained on previously computed FEA
solutions and was effectively used as a surrogate fitness measure for design
space exploration. To demonstrate the feasibility of our proposed approach, we
developed the necessary modules to generate data, train a CNN model, predict
mechanical properties, and generate designs for cantilever beams. As a proof
of concept, we used our method to identify cantilever beams with a specific
set of first three eigenfrequencies. The main contribution of this study stems
from a novel method for analyzing and optimizing cantilever beams using a CNN
that infers the beam’s mechanical properties from its cross-section.

##  II Background

Engineers rely on computational methods such as FEM to optimize their designs.
During the design process, analysis results guide engineers’ design decisions.
This process can be enhanced through FEM as it partitions a complex problem
(shape) into a mesh made up of many small sub-problems (shapes) for which
solutions can be easily found, and an overarching solution pertaining to the
whole shape can be derived. As each mesh element needs to be solved
independently, the computational effort required is directly related to the
number of mesh components and their respective degrees of freedom [14]. An
adequately scaled mesh with the right mesh components is therefore crucial to
finding an accurate solution within reasonable time. To facilitate meshing for
common users, modern algorithms generate adaptive meshes that are refined in
areas with more complex geometric features, such as curved surfaces or tight
corners. Adaptive meshes reduce the computation time while improving the
overall result accuracy.

Finding a design that exhibits specific physical behaviors, such as a beam
geometry that resonates at a particular set of frequencies or deflects by a
predetermined amount given a specific load and constraints, is a trial-and-
error process and requires optimization algorithms to complete effectively.
These algorithms require fitness measures to evaluate candidate solutions and
guide their search. Consequently, optimization speed is mainly determined by
the time it takes to evaluate the fitness of a design. Since, FEA requires
extensive computational resources, it makes optimizations time intensive.

In our approach, this issue is overcome by supplementing FEA with a CNN model
that learns from previously evaluated cases to guide future design
exploration. The trained model provides an approximate quantitative measure
for a design’s performance three orders of magnitude faster than an FEA
analysis conducted using the same hardware. Akin to a skilled design engineer,
our model learns from prior experience and gradually gains “design intuition”
which it can apply when presented with a new design.

Given that training deep neural network models tends to be a data-intensive
process, to demonstrate that our model does not suffer from this shortcoming,
we provide evidence confirming that it yields reasonable levels of accuracy
even when trained on a combined training and validation dataset only 80 data
points in size (see section V-C). Further, we show that our cross-section
image resolution can be reduced by a factor of four while still producing
excellent results (see Appendix D). Given the performance measures highlighted
above and the current advances in the use of CNNs for modeling complex
physical phenomena[11], the approach presented in this work has the potential
for use in a wide range of applications, especially those involving more
complex geometries.

Neural networks can produce unexpected results when applied to areas for which
training data is sparse or absent. Consequently, the engineering community has
been justifiably cautious to adopt opaque data-driven models for critical
applications, as false models could result in catastrophic outcomes. This risk
can be mitigated by clearly defining the search space and ensuring that it is
adequately spanned by the dataset. Additionally, it must be noted that models
trained on FEA results are inherently less accurate than FEA. To mitigate such
concerns, in Section V-A, we demonstrate the accuracy of our approach by
comparing its performance with widely accepted analytical solutions. Finally,
we propose trained models as a supplement to FEA. Our approach quickly
provides an acceptable solution that can either be verified or perfected using
numerical models.

##  III Literature Review

Surrogate-based optimization has been extensively studied in engineering
literature and has historically relied on classical machine learning (ML)
methods rather than deep learning models. Yet, despite the adoption of more
powerful ML methods, the implementation steps outlined by Forrester et al.
have remained the same: (1) selection of variables to be optimized; (2)
sampling the design space; (3) surrogate model selection and training; (4)
exploration of the search space through the surrogate model; and (5)
improvements to surrogate model based on the obtained results or verification
of a satisfactory solution against the ground truth[1]. Our proposed approach
is based on this same framework, albeit with changes to the surrogate model
selection and data representation.

Authors of recent studies recognize the power of deep learning in the context
of structural simulation. For example, Rizzo et al. predicted the critical
flutter velocity of suspension bridges using an artificial neural network
(ANN)[15]. Similarly, Le et al. used an ANN as a surrogate model of fragility
to improve the performance assessment of vertical structures under severe wind
conditions[16]. In both cases, adoption of neural networks yielded significant
performance improvements over numerical analysis. Although CNNs are mainly
used in image classification and segmentation, as well as natural language
processing, they have been found to perform better than other types of ANNs in
engineering applications. For example, Oh et al. and Youqui Zhang et al.
demonstrated that CNNs can be effectively employed for structural health
monitoring using data in both time and frequency domain [17, 18]. CNNs can
accept vectors of time series data as input in order to learn the relevant
features, and then effectively make use of the inherent intertemporal patterns
to output the predicted labels.

In their study, Finol et al. used both 1-dimensional and 2-dimensional CNNs to
effectively model eigenvalue problems in mechanics[19]. While an eigenvalue
problem for the prediction of the first three eigenfrequencies of cantilever
beams is also in focus of the present investigation, unlike Finol et al., we
provide our CNN with only a gray-scale image of the beam cross-section,
without predefining the relevant input variables. As noted earlier, neural
networks can be adopted to effectively model complex systems. However,
training them requires considerable data resources, without making any
inferences from known laws of physics. Karpatne et al. attempted to overcome
these drawbacks by introducing physics-guided neural networks (PGNNs), which
not only produce more physically accurate results, but also require less data
to train[20]. In a similar vein, Ruiyang Zhang et al. proposed a physics-
guided convolutional neural network (PhyCNN) for seismic response
modeling[21]. In both cases, the proposed models were better at generalizing
while helping to counteract overfitting. Moreover, Ruiyang Zhang et al. showed
that, by presenting frequency data as images in the frequency domain, these
can serve as CNN input. Thus, they applied their PhyCNN as a surrogate model
for structural response prediction. In this work, we show that even a less
complex CNN, without incorporating laws of physics or sophisticated encodings,
can be used as a surrogate fitness model for geometric optimization. As
humans, we rely on our inherent ability to recognize shapes in our estimation
of the physical properties of our environment. We mimic this capacity by using
a bitmap image of the beam cross-section as our representation for our model
input. A bitmap image can be kept constant in size while representing any
shape independent of the number of vertices used to describe the cross-
section.

##  IV Materials and Methods

The remainder of this article is organized into four sections: section IV-A
Dataset Generation, section IV-B Datasets, section IV-C Neural Network, and
section IV-D Optimization Algorithm. Thus, after presenting our data
generation approach, we introduce the datasets used in our experiments. Next,
we explain the different neural network configurations explored, as well as
the architecture of the model carried forward. Finally, we address the
adoption of a random search algorithm for geometric optimization of beam
cross-sections in combination with our model as a surrogate fitness measure.

###  IV-A Dataset Generation

Our data generation process comprises random beam generation, static analysis,
and frequency analysis, for which we adopted the FreeCAD API to generate a
Computer Aided Design (CAD) model of each random beam[22], followed by static
and eigenfrequency analysis of each beam using COMSOL Multiphysics 5.4[23].
More detailed information on data generation, as well as our source code, can
be found in Appendix H.

All beams considered in our simulations were assumed to be manufactured using
aluminium with Young’s modulus $E=70e^{9}N/mm^{2}$ and density
$\rho=2.7e^{-6}kg/mm^{3}$. This was an arbitrary choice, given that the study
aim was to demonstrate a CNN-based modeling approach, rather than examine
behaviors of cantilever beams. As the goal of the analyses was to
automatically process a range of beams, geometric non-linearity was enabled in
the solver to increase the accuracy for outliers that may violate the
geometric linearity assumption.

####  IV-A1 Random Beam Generation

Our investigation commenced with the generation of a random polygonal cross-
section for each beam, which was then copied, scaled, rotated, and offset to
create linearly extruded, tapered, and twisted beams using the loft command.
The resulting beams were stored in the dataset as 3D CAD models, while the
original cross-section was stored as a bitmap image. When generating the
cross-section shape, number of vertices, average radius, variance in angular
spacing between vertices (irregularity), and variance in radii between
vertices of the same shape (spikiness) were provided. The vertices of each
cross-section in our datasets have a uniformly randomly sampled radius and
number of vertices in the 24-63 mm and 3-30 range, respectively. The polar
coordinate angle offsets and the radii of the polygon coordinates were sampled
from a uniform distribution and a trimmed normal distribution, respectively.
We adjusted the data generation process to minimize meshing errors during the
automated FEA. The complete cross-section generation algorithm can be found in
Appendix B.

####  IV-A2 Static Analysis

For each beam in the dataset, we performed static analysis on a clamped beam
with a load at its tip. To automate the data analysis, we made use of the
native COMSOL meshing functionality, which generates an adaptive tetrahedral
mesh. We automated the static analysis for our datasets in our custom COMSOL
app (see Appendix I), using the load cases presented in Table I.

####  IV-A3 Frequency Analysis

Similar to the approach adopted for static analysis, when performing frequency
analysis, we sequentially iterated over each beam in a custom COMSOL
application (see Appendix I), with the beam clamped at one end (see fig. 1).
For each beam, we computed the first three eigenfrequencies and normalized
participation factors (npf).

To avoid inflated eigenfrequency results, the mesh used for FEA needs to be as
fine as possible. COMSOL Multiphysics offers nine preset mesh refinement
levels, ranging from Level 1 (extremely fine) to Level 9 (coarse). Through
repeated experiments, we found that Level 3 (finer) yielded the best trade-off
between accuracy and computation time. We found that the computation time
quadruples when the mesh level is reduced from 3 to 2, while the increase in
accuracy is comparatively small.

###  IV-B Datasets

Two datasets—SlenderBeamDS and TwistedBeamDS—were initially generated,
respectively consisting of beams with linearly extruded cross-sections and
beams that are both extruded and twisted. This distinction was made as the
deflection and eigenfrequency of the beams comprising the SlenderBeamDS
dataset can be determined analytically, whereas a common closed-form solution
for twisted beams is not available. To study the influence of beam geometry on
model performance, we generated six additional datasets using the first 5,001
cross-sections stored in the TwistedBeamDS dataset, each of which contains
beams with a different amount of twist and/or taper, as shown in Table I. For
example, TW30TA50_DS contains beams with a $30^{\circ}$ twist and a 0.5 taper
factor, i.e., the cross-section is scaled by 50% over the length of the beam.
Thus, beams within the same dataset differ from each other only in their
cross-section geometry, as described in Appendix G.

TABLE I: Dataset specification table Dataset |  Linearly extruded (SlenderBeamDS) |  30 deg. twisted (TwistedBeamDS) |  Linearly extruded (Linear_DS) |  50% tapered (TA50_DS) |  15 deg. twisted (TW15_DS) |  15 deg. twisted and 50% tapered (TW15TA50_DS) |  30 deg. twisted and 50% tapered (TW30TA50_DS)
---|---|---|---|---|---|---|---
Size | 17501 | 17501 | 5001 | 5001 | 5001 | 5001 | 5001
Twist Angle [deg] | 0 | 30 | 0 | 0 | 15 | 15 | 30
Taper Factor* | 1 | 1 | 1 | 0.5 | 1 | 0.5 | 0.5
Irregularity | 0.4 | 0.4 | 0.4 | 0.4 | 0.4 | 0.4 | 0.4
Spikiness | 0.1 | 0.15 | 0.15 | 0.15 | 0.15 | 0.15 | 0.15
Load Case [N] | [2000,0,0] | [1750,0,0] | [1750,0,0] | [1750,0,0] | [1750,0,0] | [1750,0,0] | [1750,0,0]
* scale factor used to multiply the radii of the bottom cross-section vertices to obtain the top cross-section of the beam.

###  IV-C Neural Network

Figure 2: Neural network diagram of ConvNet Extended for predicting the first
three eigenfrequencies of a cantilever beam.

As the objective of the present study is to propose a deep learning approach
for predicting static and dynamic behaviors of cantilever beams from a single
2D cross-section image, the model is required to infer all other relevant
variables, such as beam length or material properties. Thus, to limit the
search space, all variables except the cross-section geometry were assumed to
be constant for the entire dataset. In the analyses, three neural
networks—ConvNet (see Appendix A), ConvNet Extended, and Fully Connected (see
Appendix A)—were explored. ConvNet and ConvNet Extended differ only in the
number of convolutional layers. The Fully Connected network used in our tests
consists of four fully connected layers. We found that the ConvNet Extended
(Fig. 2), the deepest CNN we tested, outperformed the other two architectures
at predicting the first three eigenfrequencies (see Table I in Appendix A).
The neural network was programmed in such a way that the output layer
dimension changes if the network is trained to predict a different number of
properties. For example, if the network were solely predicting the maximum
beam deflection, the output layer would have one rather than three dimensions.

Unlike classical computer vision methods, CNNs learn the relevant features for
predicting the output during training. Each layer is computed from the
previous one by sliding a ”window” called the kernel over the input tensor and
applying the convolution operation. Moreover, as CNNs are able to extract
learned features from data, they do not require any feature engineering, as
described in detail by LeCun et al. and O’Shea et al. [24, 25].

All our models were implemented in PyTorch and were trained and run on a
desktop computer with Nvidia RTX2080Ti GPUs. Moreover, network size was
adjusted based on the input image dimensions ($img\\_size$) and the number of
labels ($number\\_of\\_labels$) to be predicted.

Each dataset was segregated into training, evaluation, and testing subsets in
a 64:16:20 ratio. For a dataset comprising 17,500 datapoints, this resulted in
a 11,200:2,800:3,500 split. We trained each network using the Adam optimizer
in combination with Mean Squared Error (MSE) (eqn. 1) as the loss
function[26]. Unless specified otherwise, all models were trained with a
learning rate (LR) of 0.0001.

$\displaystyle\frac{1}{n}\sum_{i=1}^{n}(Y_{i}-\hat{Y_{i}})^{2}$ (1)

####  IV-C1 ConvNet Extended Architecture Details

The ConvNet Extended network comprises five convolutional layers and two fully
connected layers and takes as an anti-aliased gray-scale image of the beam
cross-section as input. The first three convolutional layers have 32 5×5
convolutions of stride 1 with padding 2, followed by batch normalization. The
fourth convolutional layer has 64 5×5 convolutions of stride 1 with padding 2,
followed by batch normalization and a max pool layer with a 2×2 convolution
applied at stride 1. Finally, the fifth convolutional layer has 32 5×5
convolutions of stride 1 with padding 2, followed by batch normalization and a
max pool layer with a 2×2 convolution applied at stride 1. All convolutional
layers use ReLU activations[27]. The sixth layer is a
$32*img\\_size/4*img\\_size/4$ to $1024$ size fully connected layer and the
final layer is a $1024$ to $number\\_of\\_labels$ fully connected layer.

###  IV-D Optimization Algorithm

To assess our model’s effectiveness as a surrogate model, we used a trained
model in combination with random search to identify beams that exhibit a
specific set of first three eigenfrequencies ($[f_{1},f_{2},f_{3}]$) and
confirmed our findings using FEA.

For the optimization, the best performing model (manually selected from
multiple training sessions) was trained on the TwistedBeamDS dataset with an
average MAPE of 2.03%. The training graph of the model used in this
experiment, as well as a table showing a detailed breakdown of the model’s
performance can be found in Appendix F-A. During the optimization with the
random search algorithm, we randomly generated new beam cross-section vertices
using Algorithm 2 (see Appendix B). These vertices were subsequently used to
create a cross-section image, as was done during data generation. After
evaluating the resulting cross-section image using our model, we computed the
MSE between the desired values for $[f_{1},f_{2},f_{3}]$ and the predicted
values, saving the best-performing cross-section geometry after each
iteration.

##  V Results

The results yielded by our analyses are presented under the following five
headings: Model Performance, Geometric Optimization of Beam Cross Sections,
Data Efficiency, and Generalizability. In the section related to model
performance, we show that our approach is not only superior to a closed-form
solution, but can be applied to scenarios for which no readily available
closed-form solution exists. The Optimization Algorithm section is designated
for our experimental setup and the results yielded by applying our approach to
identify specific beam configurations with desired properties. Next, in the
Data Efficiency section, we discuss the data requirements of our model, and
present the results obtained by training the model on various problems and
datasets without tuning the relevant network variables in the Generalizability
section.

###  V-A Model Performance

To demonstrate the practical utility of the proposed approach, we compared the
accuracy of our model to that of the analytical solution for both volume
maximum beam deflection and eigenfrequency. As a closed-form solution for
total volume maximum displacement and eigenfrequencies of a twisted cantilever
beam is highly complex, the formula for linearly extruded beams was used in
comparisons related to both linear and twisted beams[28]. Unlike the
analytical solution, our finite element analysis was not constrained by the
assumption of geometric linearity. As a result, larger discrepancies between
the analytical and the finite element solution were expected for beams that do
not satisfy the geometric linearity criterion.

$\displaystyle\theta_{p}=\frac{1}{2}tan^{-1}[\frac{2I_{xy}}{I_{y}-I_{x}}]$
(2a) $\displaystyle
I_{x_{p}}=I_{x}cos^{2}\theta_{p}+I_{y}sin^{2}\theta_{p}-I_{xy}sin2\theta_{p}$
(3a) $\displaystyle
I_{y_{p}}=I_{x}sin^{2}\theta_{p}+I_{y}cos^{2}\theta_{p}+I_{xy}sin2\theta_{p}$
(4a) $\displaystyle I_{p}=\begin{bmatrix}I_{x_{p}}&0&0\\\ 0&I_{y_{p}}&0\\\
0&0&1\end{bmatrix}$ (5a)

To analytically compute the maximum deflection and the first three
eigenfrequencies of each beam, the second moment along the principal axes must
be considered, which was computed with respect to the base coordinate frame
during data generation. This allowed the equations 2a through 5a to be applied
to compute the second moment along the principal axes, as well as the z-axis
rotation needed to transform the applied force vector into the principal axis
coordinate frame.

####  V-A1 Total Deflection

(a)

(b)

Figure 3: Comparison of errors generated by the analytical solution (blue bar)
and the model prediction (orange bar) of total displacement with respect to
the ground truth FEA solution based on test data only, Fig. 3a and 3b show the
results related to the linearly extruded beams comprising the SlenderBeamDS
dataset and the twisted beams included in the TwistedBeamDS dataset,
respectively (see Table I for dataset details). The error bars represent the
95% confidence interval. The graphs confirm that our approach outperforms the
analytical solution for free vibration in both scenarios.

We solved for the maximum deflection $\delta_{max}$ of a clamped cantilever
beam subjected to a point load at its tip using equation 6, where $F_{p}$ is
the point force applied at the beam tip, $I_{p}$ is the second moment along
the principal axis, and $E$ is the Young’s modulus, as described in section
IV-B. The load case considered in this scenario can be found in Table I. A
derivation of equation 6 is provided by Gere et al. in ”Mechanics Of
Materials”[29]. The point force $F_{p}$ was attained by representing the load
case used in the FEA analysis in terms of the eigenbasis of the second moment.

$\displaystyle\delta_{max}=\frac{(F_{p}L)^{3}}{3EI_{p}}$ (6)

As shown in figure 3a, our model is over four percentage points more accurate
than the analytical solution for linearly extruded beams. Moreover, when
applied to both linearly extruded and twisted beams, our model is
approximately 0.2 mm more accurate in predicting volume maximum total
displacement (see Fig. 3). One advantage of our approach stems from the
assumptions imposed on the analytical solution related to the beam shape and
material composition, leading to constraints such as linear bending. Thus, for
beams that violate these assumptions, the analytical solution is naturally
less accurate. Most importantly, our approach in combination with numerical
approximations is a viable strategy for exploring the design spaces of complex
problems that lack a known closed-form solution.

####  V-A2 First Three Eigenfrequencies ($[f_{1},f_{2},f_{3}]$)

The eigenfrequency models used in this experiment were trained with 0.00001
learning rate and 100 batch size, which were determined based on the
validation set performance.

(a)

(b)

Figure 4: Comparison of errors generated by the analytical solution (blue bar)
and the model prediction (orange bar) of total displacement with respect to
the ground truth FEA solution based on test data only, Fig. 4a and 4b show the
results yielded by applying our model to linearly extruded beams included in
the SlenderBeamDS dataset and the twisted beams comprising the TwistedBeamDS
dataset, respectively (see Table I for dataset details). The error bars
represent the 95% confidence interval. The graphs show that our model
outperforms the analytical solution for free vibration in both scenarios.

We next solved for the eigenfrequencies using the Euler-Bernoulli equations
for free vibration of a cantilever beam using eqns. 7a to 9a in combination
with the root solutions provided in Table II in Appendix C. $\rho$ denotes
material density, $A$ stands for cross section area, and $L$ is the beam
length. For a detailed derivation please see Han et al. ”Dynamics of
Transversely Vibrating Beams”[30].

$\displaystyle\cosh{\beta_{n}}\cos{\beta_{n}}+1=0$ (7a)
$\displaystyle\lambda_{m}=A\rho$ (8a)
$\displaystyle\omega_{n}=\frac{\beta_{n}^{2}}{L^{2}}\sqrt{\frac{EI_{p}}{\lambda_{m}}}$
(9a) $\displaystyle f_{n}=\frac{\omega_{n}}{2\pi}$ (10a)

Our analyses revealed that the first three eigenmodes of the beams in our
datasets occur along the two principal axes tangential to the beam length.
Thus, we solved eqn. 9a analytically in both directions and selected the
lowest three eigenfrequencies among those found. It is worth noting that
computing more than the first three eigenfrequencies may require solutions for
all principal directions.

Our findings indicate that the proposed trained model, $ConvNetExtended$
outperforms the analytical solution for eigenfrequencies not only when applied
to twisted beams, but also linearly extruded beams, as shown in figure 4. The
trained model predicted the total displacement and the first three
eigenfrequencies with half the percentage error of the analytical model when
applied to linearly extruded beams (see Fig 3a and 4a). When applied to
twisted beams, the trained model outperformed the analytical solution in terms
of evaluating total displacement by over three percentage points (see Fig.
3b). When predicting the first three eigenfrequencies of twisted beams, the
trained model outperformed the analytical solution by a small percentage
margin, while being significantly more accurate with respect to the MAE (see
Fig. 4b). In sum, our approach is not only more accurate than the analytical
solution, but also more versatile, as it can be trained on diverse datasets
and produces accurate results for various types of beam geometries that may
not have an obvious closed-form solution, such as beams that are both tapered
and twisted, as demonstrated in Section V-D.

###  V-B Geometric Optimization Of Beam Cross Sections

In this section, we study the performance of our models when employed as a
surrogate fitness measure in the search for a beam cross-section geometry that
exhibits a specific set of first three eigenfrequencies. For this purpose, we
used a model that was trained to predict the first three eigenfrequencies
based on the TwistedBeamDS dataset as our surrogate fitness model. Next, a
random search algorithm was applied to identify cross-section designs that
correspond to twisted beams with a desired set of first three
eigenfrequencies. All geometries that met these criteria after a fixed number
of evaluations were assessed via FEA and were used to compute the accuracy of
our approach.

First, we randomly selected 100 sets of first three eigenfrequencies
$[f_{1},f_{2},f_{3}]$ from beams in the TwistedBeamDS test set. As our model
was trained on the TwistedBeamDS dataset, this strategy guaranteed that a beam
that satisfies the desired eigenfrequency configuration exists, while
excluding the possibility that the model had previously been presented with
this particular set of eigenfrequencies. Second, we ran 10 optimization
searches for each set of eigenfrequencies using a random search algorithm.
Each search was terminated after 100,000 evaluations. Finally, the identified
beam cross-sections were extruded into 3D CAD beams that were analyzed in
COMSOL Multiphysics to determine their respective true first three
eigenfrequencies.

Figure 5: Three examples of the beam configurations yielded by the
optimization experiment.

Our findings confirmed that the model developed as a part of this study was
effective at guiding the random search algorithm toward beam cross-section
geometries that provided a good fit to the target eigenfrequencies. The
proposed solutions cumulatively deviated by 8.18 Hz on average over the three
target eigenfrequencies resulting in 2.39% MAPE. The identified cross-section
geometries, once extruded with a 30 degree twist and analyzed using FEA,
exhibited eigenfrequencies very close to the specified target. The three
examples in Fig. 5 show that the search produced cross-sections that differ in
geometry, size, and vertex count to attain a beam geometry that best matches
the target eigenfrequencies. The samples were manually selected from different
frequency ranges to provide insight into the pertinent cross-section
geometries.

We found that on average meshing and analyzing one beam in COMSOL Multiphysics
takes 4.3 seconds on an AMD Ryzen Threadripper 2950X CPU using meshing
refinement level 3, while evaluating a new beam cross-section using our model
on an Nvidia RTX2080Ti GPU was completed within 2.3 milliseconds. As a result,
using FEA instead of our trained model for this parameter search would extend
the 4-minute duration of each 100,000-step optimization to a week.
Additionally, we were able to shorten the time required for the 100
optimization runs further by running seven surrogate model instances in
parallel on one Nvidia RTX2080Ti. The time comparison between GPU parallelized
code and code that runs on the CPU is not an even one, but is used in this
instance to show the significant performance difference of the two methods on
desktop computer hardware.

Analyzing the frequency response and tuning the components to resonate or not
resonate at particular frequencies is relevant in many practical applications,
from civil engineering to space travel. For example, our approach can be
utilized when working on a mechanical resonator design, as it can identify
geometries that vibrate at specific frequencies. Similarly, this approach can
speedup the search for geometries that avoid specific eigenfrequencies, for
example when designing machine parts that must avoid resonance with motor
vibrations.

###  V-C Data Efficiency: Prediction accuracy as a function of dataset size

Thus far, we assessed the performance of our approach using a large dataset.
However, for design intuition to be useful, it should be adequately trained
with a reasonably small amount of data. In a surrogate-based optimization
approach as outlined in section III, the data quantity required to train the
model determines its practical utility. Thus, in this section, we show that
our model can be trained effectively on relatively small datasets without
compromising its high prediction accuracy.

Figure 6: Barplot of the model’s mean absolute percentage error versus the
dataset size used to train the model.

For this purpose, we partitioned the TwistedBeamDS dataset into differently
sized subsets, and trained four models on each subset to predict the first
three eigenfrequencies. We reserved 20% (i.e., 3,500 data points) of the
TwistedBeamDS for testing, and computed the mean average percentage error for
each model based on the models’ performance on that test set. Each subset was
split in 64:16:20 ratio: 64% of the data was used for training, 16% for
validation, and the remaining 20% was omitted, since we used the previously
selected 3,500 data point test set for testing. The Dataset Size in Figure 6
refers to the total number of data points in the subset; assuming a 64:16:20
split, for a subset consisting of 100 samples, 80 would be designated for
training and validation.

The obtained results are presented in Figure 6. As indicated by the green
trend line, the models trained on a larger dataset generally outperformed
those trained on less data. However, it is also apparent that our approach
performed less than 2.3 percentage points worse with 8.89% MAPE when trained
on 80 data points relative to the full dataset comprising 17,500 data points.
These findings confirm that even a small initial dataset is sufficient to
provide intuition about the design space, and can be a useful starting point
for further design exploration.

###  V-D Generalizability: Model performance on different labels and beams
types

To determine whether our approach can be applied to model various beam
geometries, we trained three models on every combination of 12 labels and 7
datasets and reported the results in Table III in Appendix E. Based on our
previous tests, we found that our approach performs well even if the input
image is of low resolution (see Appendix D). For this evaluation, we trained
the models using images with 64×64 pixel resolution to reduce the memory
requirements and the overall computation time. The results show the average
MAPE and the standard deviation for all three models based on all
dataset/label combinations.

Overall, our approach achieved an average of 4.07% MAPE, considering the best
performing model for each dataset and label combination. This high level of
accuracy was attained without tuning the training-relevant variables of the
CNN for the respective labels. This is a particularly impressive achievement
considering that it is not possible to deduce from a cross-section image
whether a beam is twisted or tapered, or both. Twisted beams tend to bend out
of plane, and thus produce different TotDisp results. Similarly, tapered beams
exhibit a different frequency profile from linearly extruded beams due to the
difference in mass and shifted center of mass. Hence, these results confirm
that our approach can accurately predict both static and dynamic behaviors of
linearly extruded, tapered, twisted, and tapered and twisted cantilever beams.

We also found that our model’s performance suffers when trained on non-
normalized labels that span a wide range of values. This is not unusual for
machine learning models and can be addressed by scaling or normalizing the
model inputs and outputs. To mitigate this issue, we scaled the values for
curl displacement in the y-direction (CurlDisp_Y) and principal strain in the
x-direction (PrincStrain_X) and achieved performance comparable to that
obtained for other labels (see Appendix E). Moreover, even though our model’s
performance on volume maximum total displacement (TotDisp) and total Von Mises
stress (TotVonMises) was sufficiently accurate, careful input and output
scaling or normalization would improve the accuracy even further.

Frequency analysis helps engineers identify eigenfrequencies with significant
mass participation to make design decisions or to inform further analysis.
Hence, identifying a quantitative measure of an eigenfrequency’s impact on the
analyzed structure can inform further analyses. For this purpose, we included
the root mean squared (RMS) of the normalized participation factor (npf1_RMS
to npf3_RMS) as a label in our analysis as a proxy measure for the mass
participation of each predicted eigenfrequency. Our model successfully
predicted the root mean squared (RMS) of the normalized participation factor
with an average MAPE of 2.11%.

##  VI Challenges and Limitations

We focused our study on eigenfrequencies and maximum deflection to ascertain
if our approach is capable of predicting both static and dynamic behaviors of
different beam geometries. While we recognize that eigenfrequencies are rarely
used in practice in the absence of their corresponding mode shapes, their
exploration is beyond the current study, as it requires a neural network
architecture that differs significantly from the one presented in this paper.

Since our proposed model can’t be more accurate than the data it was trained
on, it is important that the right type of FEA is performed for the problem at
hand. At the micro and nano scale, cantilever beams behave differently than at
the macro scale [31, 32]. Therefore, the correct analysis approach must be
used to get accurate data to train a surrogate model and verify optimization
results.

Further, many situations require engineers to study more than the first three
eigenfrequencies. Although this scenario was not explored in this work, it is
likely that the proposed approach is capable of predicting a larger number of
eigenfrequencies without significant changes to the model.

Finally, based on the results presented in this paper, it cannot be speculated
whether this approach generalizes across other applications, search spaces, or
geometries. Nonetheless, as it is capable of predicting different beam
properties from a variety of beam geometries it has the potential for use in a
wide range applications.

##  VII Conclusion

We have demonstrated that finite element analysis results can be used to
impart visual design intuition to CNNs, and that the resulting models can be
used to optimize the shape of a design and to search for specific physical
properties. Our model effectively learned implicit physical and geometric
characteristics, such as beam length, moment of inertia, and material
properties. Owing to its data efficiency, it can be incorporated into hybrid
surrogate optimization approaches where a search space is randomly sampled to
train a model that subsequently serves as a surrogate fitness measure to find
better design solutions, which are numerically verified before being included
into the dataset to improve the model.

We believe that adoption of our approach can significantly improve
productivity in work environments where FEA is frequently performed on similar
shapes, or a corpus of design-relevant solved analyses already exists. In such
cases, the data pertaining to solved simulations can be used to train CNNs to
expedite the work on future design optimization problems. A variety of cloud
platforms already provide FEA capabilities to their customers, and have the
data to capitalize on this method. By capturing each analysis and the
corresponding results, deeper and more complex models with broader
applications could be trained and used to guide design explorations.

Most importantly, our approach can be applied to poorly understood processes,
as it is capable of implicitly learning to extract and compute the relevant
physical and geometric variables from the raw visual data provided, without
the need for manual feature engineering. Owing to its simplicity, it can serve
not only as a means for furthering design optimization in established fields,
but also as a tool for exploring poorly understood yet important new design
spaces. Our approach could create models that mimic expert intuition in areas
where we lack expertise.

##  VIII Future Outlook

While acknowledging that the work presented in this paper had a very narrow
focus—cantilever beams—the models employed are capable of implicitly learning
relevant variables, such as material, beam length, twist angle, etc. Going
forward, we plan to design deep learning models that accept these design-
relevant parameters as inputs to inform their predictions. We also aim to
improve our approach and increase its ability to generalize by exploring
alternative neural network architectures or using conditional neural networks.
We believe that supplementing FEA with CNNs will accelerate and expand our
ability to explore search spaces and optimize designs.

Ultimately, we would like to apply this approach to optimizing structures with
multiple-members. These are the six steps that we believe are needed to employ
the approach demonstrated in this paper to optimize more complex structures:
(1) Identify the design parameters that will be optimized, and define the
multi-member structure in terms of it. (2) Find an appropriate representation
that can be used as input for the model. While a cross-section image was
sufficient for our application, a more complex structure may require multiple
images, a 3D voxel representation, or an alternative representation all
together. (3) Create a high-fidelity simulation. (4) Generate an initial
dataset that spans the desired design space, by randomly sampling the design
space around the area of interest. (5) Train the CNN surrogate model on the
generated dataset. (6) Validate the found design solution against the high-
fidelity simulation. If the model’s predicted solution does not accurately
match the simulated solution, then sample more data points at and near the
found configuration, add those data points to the generated dataset, and
return to step (5).

##  IX Acknowledgements

We thank Joni Mici and Robert Kwiatkowski for reviewing our work and providing
feedback on this manuscript. We also thank Michael Ounsworth for sharing his
implementation of a polar coordinate-based polygon generation algorithm, which
we integrated into our data generation process: see
https://stackoverflow.com/questions/8997099/algorithm-to-generate-
random-2d-polygon.

##  X Author Contributions

Philippe Wyder - Experiments and Manuscript

Hod Lipson - Conceptualization and Editing

##  XI Data Accessibility

The appendix to this document is in the Supplementary Materials PDF. Link to
data generation repository:
https://github.com/ResearchMedia/CantileverBeamDatasetGenerator

Link to neural network repository:
https://github.com/ResearchMedia/CantileverNN

The data was published published on Mendeley Data. Doi: 10.17632/y3m8xm6kfk.1

##  XII Funding Statement

This work has been funded in part by U.S. Defense Advanced Research Project
Agency (DARPA) TRADES grant number HR0011-17-2-0014.

This work was supported in part by the U.S. National Science Foundation (NSF)
AI Institute for Dynamical Systems (dynamicsai.org), grant 2112085

The authors declare that they have no other known competing financial
interests or personal relationships which have, or could be perceived to have,
influenced the work reported in this article.

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See pages - of VisualDesignIntuition_SupplementaryMaterials.pdf