# Geometry-Aware Score Distillation via 3D Consistent Noising and Gradient
Consistency Modeling

Min-Seop Kwak$\,{}^{\textnormal{1}}$ Donghoon Ahn$\,{}^{\textnormal{1}}$ Inès
Hyeonsu Kim$\,{}^{\textnormal{1}}$
Jin-Hwa Kim $\,{}^{\text{2,3}}$ Seungryong Kim22footnotemark: 2
$\,{}^{\text{1}}$

${}^{\text{1}}$Korea University ${}^{\text{2}}$NAVER AI Lab ${}^{\text{3}}$AI
Institute of Seoul National University
Co-corresponding authors.

###### Abstract

Score distillation sampling (SDS), the methodology in which the score from
pretrained 2D diffusion models is distilled into 3D representation, has
recently brought significant advancements in text-to-3D generation task.
However, this approach is still confronted with critical geometric
inconsistency problems such as the Janus problem. Starting from a hypothesis
that such inconsistency problems may be induced by multiview inconsistencies
between 2D scores predicted from various viewpoints, we introduce GSD, a
simple and general plug-and-play framework for incorporating 3D consistency
and therefore geometry awareness into the SDS process. Our methodology is
composed of three components: 3D consistent noising, designed to produce 3D
consistent noise maps that perfectly follow the standard Gaussian
distribution, geometry-based gradient warping for identifying correspondences
between predicted gradients of different viewpoints, and novel gradient
consistency loss to optimize the scene geometry toward producing more
consistent gradients. We demonstrate that our method significantly improves
performance, successfully addressing the geometric inconsistency problems in
text-to-3D generation task with minimal computation cost and being compatible
with existing score distillation-based models. Our project page is available
at https://ku-cvlab.github.io/GSD/.

Figure 1: Teaser. Our framework incorporates 3D awareness into the score
distillation sampling (SDS) process through a 3D consistent noising, which
induces consistency of the predicted 2D score. As a general plug-and-play
module that can be attached to any SDS-based text-to-3D generation baselines
with little computation cost, it brings about highly enhanced view consistency
and fidelity to 3D generation results across various baselines.

##  1 Introduction

Text-to-3D generation, which is the task of generating a 3D scene from a text
prompt, has seen great advancements in recent years due to the advent of
powerful generative models such as diffusion model [7, 25]. As the main
objective of this task is to generate a high-quality 3D model solely from
user-given text, it enables even non-professional users to create 3D models
easily with little to no handwork. Naturally, advancements in this task have
opened up numerous possibilities in various domains such as VR/AR, computer-
generated graphics, and gaming.

However, due to the comparative lack of size and quantity of existing 3D
ground truth datasets in comparison to 2D images or videos, directly training
a diffusion model to a 3D representation is difficult. To circumvent this
problem, the majority of methods [9, 14, 3, 28] leverage pretrained 2D
diffusion models to optimize a 3D representation [16, 17, 10] through a
methodology named score distillation sampling (SDS) [20], in which the 3D
representation is optimized using the score predicted by the 2D diffusion
model from noised renderings of the scene. Such SDS process is conducted at
various viewpoints of the scene, each process taking place independently.
However, because this methodology inherently relies on 2D diffusion model that
lacks explicit knowledge of 3D domain, it often suffers from geometric
inconsistency problems such as the Janus problem [23, 24], in which
inconsistent, multi-faced geometries appear throughout the 3D scene at
erroneous regions, harming the global geometry and making the generated shape
unsuitable for real-world applications.

To understand and counter this issue, we analyze the SDS process from the
perspective of multiview consistency, hypothesizing that such geometric
inconsistency problem is correlated to the independence of each SDS process,
which in turn causes the lack of multiview consistency between 2D scores
predicted from different viewpoints. More specifically, we focus on the fact
that under the naiv̈e SDS setting [20], a single point in 3D receives vastly
different optimization signals from various viewpoints, resulting artifacts
and geometrically inconsistent geometric features such as Janus problem. Under
this observation, encouraging the multiview consistency of SDS gradients
between nearby viewpoints would lead to reduction in such artifacts – allowing
for robust and geometrically consistent text-to-3D generation.

In this light, we propose a novel methodology, named Geometry-aware Score
Distillation (GSD), which incorporates multiview correspondence awareness to
the SDS process to facilitate multiview consistency of generated gradients, as
described in Fig. 1. Our method is a plug-and-play module that can be attached
to existing SDS-based baselines for enhanced geometric consistency, with
little computation cost and no need for additional networks or modules. Our
method consists of two components. First, to encourage multiview consistency
of predicted 2D scores across viewpoints, we introduce 3D consistent noising,
combining point cloud representation with integral noising [2] to produce 3D
geometry-aware 2D Gaussian noises in SDS process. Our 3D consistent noising
imbues separate SDS denoising processes implicitly with 3D awareness.
Secondly, we propose geometry-based gradient warping to warp the generated
gradient of a viewpoint to other viewpoints, allowing for the comparison of
gradients between corresponding locations across various viewpoints. We
finally leverage the warped gradients for our novel multiview gradient
consistency loss, which helps to regularize and reduce inconsistent scene
features, such as artifacts and regions suffering from Janus problem.

Our experimental results and analysis prove our hypothesis correct, showing
that the application of our methodology strongly benefits the optimization
process across various SDS-based text-to-3D baselines [29, 26, 20]. Our
methodology enhances the geometric consistency and fidelity of the generated
results, resulting 3D scenes competitive to state-of-the-art. Our ablation
study demonstrates that our contributions are strongly interconnected,
justifying the need for all our components to be used in conjunction with one
another.

##  2 Related work

#### Text-to-3D generation.

DreamFusion [20] and SJC [27] introduced an optimization technique called
score distillation sampling (SDS), which leverages pretrained large-scale
text-to-image diffusion models to generate 3D objects. Since its introduction,
SDS has been widely adopted in various text-to-3D generation models. Magic3D
[14] and Fantasia3D [4] employ a coarse-to-fine strategy with SDS
optimization, achieving high-fidelity results. ProlificDreamer [28] has
significantly improved the quality of 3D objects generated from text-to-3D
tasks. This progress is due to treating the model’s 3D parameters as random
variables instead of constants, as in SDS, and developing a gradient-based
update rule using the Wasserstein gradient flow. More recently, models such as
DreamGaussian [26], GSGEN [4], LucidDreamer [13] and GaussianDreamer [29]
incorporates 3D Gaussian Splatting representation into SDS-based text-to-3D
generation frameworks.

#### Geometric inconsistency problem within SDS.

In text-to-3D generation tasks, maintaining 3D geometric consistency is
crucial, yet a geometric inconsistency problem called the Janus problem [28,
24] commonly occurs. Various approaches have been attempted address this.
MVDream [24] and EfficientDreamer [30] fine-tuned a pretrained Stable
Diffusion [21] model using a 3D dataset and enabled the model to generate
orthogonal multi-view images with robust geometric consistency. 3DFuse [23]
proposes a method that injects coarse 3D priors into a pretrained diffusion
model. However, MVDream and EfficientDreamer rely on a large-scale 3D dataset
Objaverse [5] during training, which is limited in terms of asset quality,
causes the model to generate clay-textured images similar to those in the
Objaverse dataset. 3DFuse is also limited in another aspect, still exhibiting
3D geometry inconsistencies in cases when the coarse 3D priors are erroneous
or of low-quality.

##  3 Preliminaries

Figure 2: Overall framework. Our framework consists of three components for
geometry-aware score distillation: 3D consistent noising, geometry-based
gradient warping, and gradient consistency modeling. Through these components,
our framework encourages multiview consistency between predicted 2D scores and
enhances the quality of generated 3D scenes.

Diffusion models have demonstrated impressive capabilities in text-to-image
generation [18, 22, 1]. Building on this achievement, DreamFusion [20]
introduces the score distillation sampling (SDS), which generates plausible 3D
objects by leveraging pretrained text-to-image diffusion models to optimize 3D
representation such as NeRF [16] parameterized by ${\bm{\theta}}$. Similarly,
SJC [27] formulates the SDS based on the assumption that a 3D probability
density of ${\bm{\theta}}$ given prompt $y$, denoted by
$p_{\sigma_{t}}({\bm{\theta}};y)$, is proportional to the expected probability
densities of multiview 2D rendered images $\mathbf{z}_{{\bm{\theta}},\pi}$
over the camera poses $\pi$ sampled from the uniform hemispherical
distribution of the camera viewpoints $\Pi$, denoted by
$p_{\sigma_{t}}(\mathbf{z}_{{\bm{\theta}},\pi};y)$. This can be expressed as
$\mathbb{E}_{t}\big{[}p_{\sigma_{t}}({\bm{\theta}};y)\big{]}\propto\mathbb{E}_{\pi\sim\Pi,t}\big{[}p_{\sigma_{t}}(\mathbf{z}_{{\bm{\theta}},\pi};y)\big{]}$,
where ${\sigma_{t}}$ denotes a noise level at time step $t$. As the score is
the gradient of the log probability density of data, the following equation is
derived using Jensen’s inequality, with
$\log\tilde{p}_{\sigma_{t}}({\bm{\theta}};y)$ as the lower-bound of
$\log{p}_{\sigma_{t}}({\bm{\theta}};y)$:

$\displaystyle\nabla_{{\bm{\theta}}}\mathcal{L}_{\textrm{SDS}}\coloneqq\mathbb{E}_{t}\left[\underbrace{\nabla_{{\bm{\theta}}}\log\tilde{p}_{\sigma_{t}}({\bm{\theta}};y)}_{\text{3D
score}}\right]$
$\displaystyle=\mathbb{E}_{\pi\sim\Pi,t}\left[\underbrace{\nabla_{\mathbf{z}_{{\bm{\theta}},\pi}}\log
p_{\sigma_{t}}(\mathbf{z}_{{\bm{\theta}},\pi};y)}_{\text{2D
score}}\cdot\frac{\partial\mathbf{z}_{{\bm{\theta}},\pi}}{\partial{\bm{\theta}}}\right],$
(1)

where the 2D score, or the gradient of $\log
p_{\sigma_{t}}(\mathbf{z}_{{\bm{\theta}},\pi};y)$, is obtained using
pretrained 2D diffusion models [21].

However, instead of directly using the rendered image
$\mathbf{z_{{\bm{\theta}},\pi}}$, in order to address the out-of-distribution
problems, the perturb-and-average scoring (PAAS) is required, in which the 2D
noise $\mathbf{n}\sim\mathcal{N}(0,\mathbf{I})$ is added to
$\mathbf{z_{{\bm{\theta}},\pi}}$. Specifically, it defines the denoiser
$\mathcal{D}(\cdot)$ such that
$\mathcal{D}(\mathbf{z}_{{\bm{\theta}},\pi}+\sigma_{t}\mathbf{n};\sigma_{t},y)=(\mathbf{z}_{{\bm{\theta}},\pi}+\sigma_{t}\mathbf{n})-\sigma_{t}\bm{\epsilon_{\phi}}(\mathbf{z}_{{\bm{\theta}},\pi}+\sigma_{t}\mathbf{n},y,t)$
with the rendered image from ${\bm{\theta}}$ at camera pose $\pi$, aggregated
with noise $\mathbf{n}$ scaled by noise level $\sigma_{t}$. The residual noise
$\bm{\epsilon_{\phi}}(\cdot)$ is predicted from a frozen 2D diffusion model
[21] parameterized by $\bm{\phi}$. It then defines a gradient map
$\mathbf{g}_{{\bm{\theta}},\pi}$ representing the 2D score as follows:

$\mathbf{g}_{{\bm{\theta}},\pi}=\frac{\mathcal{D}(\mathbf{z}_{{\bm{\theta}},\pi}+{\sigma_{t}}\mathbf{n};{\sigma_{t}},y)-(\mathbf{z}_{{\bm{\theta}},\pi}+{\sigma_{t}}\mathbf{n})}{{\sigma_{t}}^{2}},$
(2)

and when we compute expectation over these predicted gradients w.r.t random
noise $\mathbf{n}$, it gives us the score, or the update direction, for the
non-noisy rendered image $\mathbf{z}_{{\bm{\theta}},\pi}$ itself:

$\begin{split}\nabla_{\mathbf{z}_{{\bm{\theta}},\pi}}\log
p_{\sqrt{2}\sigma_{t}}(\mathbf{z}_{{\bm{\theta}},\pi})\approx&\mathbb{E}_{\mathbf{n}\sim\mathcal{N}(0,\mathbf{I}),t}\left[\mathbf{g}_{{\bm{\theta}},\pi}\right]\\\
=&\mathbb{E}_{\mathbf{n}\sim\mathcal{N}(0,\mathbf{I}),t}\left[\frac{\mathcal{D}(\mathbf{z}_{{\bm{\theta}},\pi}+\sigma_{t}\mathbf{n};\sigma_{t},y)-\mathbf{z}_{{\bm{\theta}},\pi}}{\sigma_{t}^{2}}\right]-\cancel{\underbrace{\mathbb{E}_{\mathbf{n}\sim\mathcal{N}(0,\mathbf{I}),t}\left[\frac{\mathbf{n}}{\sigma_{t}}\right]}_{=0}},\end{split}$
(3)

where $\log p_{\sqrt{2}\sigma_{t}}(\cdot)$ appears because the diffusion model
predicts the Gaussian noise of already noised
$\mathbf{z}_{{\bm{\theta}},\pi}$, and as
$\mathbb{E}_{\mathbf{n}\sim\mathcal{N}(0,\mathbf{I})}\left[\mathcal{N}(\mathbf{z}_{{\bm{\theta}},\pi}+\sigma_{t}\mathbf{n};\mu,\sigma_{t}^{2}\mathbf{I})\right]=\mathcal{N}(\mathbf{z}_{{\bm{\theta}},\pi};\mu,2\sigma_{t}^{2}\mathbf{I})$,
the variance becomes $2\sigma_{t}^{2}$ in regards to
$\mathbf{z}_{{\bm{\theta}},\pi}$ and thus resulting a logarithm with the base
of $\sqrt{2}\sigma_{t}$ [27].

Relating back to Eq. 1, obtaining a 3D score for optimizing ${\bm{\theta}}$
requires computing the expectation over multiple camera viewpoints $\pi$.
Assuming a rendered image $\mathbf{z}_{{\bm{\theta}},\pi}$ at the viewpoint
$\pi$ that is noised with noise $\mathbf{n}$, the final equation for score
distillation is expressed as follows:

$\nabla_{{\bm{\theta}}}\mathcal{L}_{\textrm{SDS}}\approx\mathbb{E}_{\pi\sim\Pi,\mathbf{n}\sim\mathcal{N}(0,\mathbf{I}),t}\left[\frac{\mathcal{D}(\mathbf{z}_{{\bm{\theta}},\pi}+\sigma_{t}\mathbf{n};\sigma_{t},y)-\mathbf{z}_{{\bm{\theta}},\pi}}{\sigma_{t}^{2}}\cdot\frac{\partial\mathbf{z}_{{\bm{\theta}},\pi}}{\partial{\bm{\theta}}}\right].$
(4)

##  4 Methodology

###  4.1 Motivation and overview

In the standard SDS process [20, 27, 28], the 2D noise $\mathbf{n}$ is sampled
independently per viewpoint. This brings to question cases where the sampled
viewpoints are close to one another, where the rendered images
$\mathbf{z}_{{\bm{\theta}},\pi}$ cover nearby, overlapping regions of the 3D
scene. Under the standard SDS setting, the different renderings of the
overlapping regions would result in largely unrelated 2D scores and therefore
unaligned 3D optimization signals, as the noises $\mathbf{n}$ are sampled
independently from one another. In simpler words, within standard SDS, the
predicted 2D scores of nearby viewpoints lack multiview consistency. In this
light, our work starts from the hypothesis that such a lack of multiview
consistency amongst the predicted 2D scores results in unaligned 3D gradients,
contributing to geometric inconsistency problems such as the aforementioned
Janus problem. We seek to counter this problem by incorporating geometric
awareness into the SDS process [20, 27, 28].

Let us assume a mapping function $\mathcal{W}(\cdot)$ that contains the 3D
correspondence relationship between viewpoints. Because we have explicit 3D
geometry represented by ${\bm{\theta}}$, $\mathcal{W}(\cdot)$ is obtainable by
verifying which locations in 2D renderings are projections of an identical
location in 3D, establishing 3D geometry-based correspondence across different
viewpoints. We can use this $\mathcal{W}(\cdot)$ to map an image of one
viewpoint to another in a geometrically consistent manner - a process known as
warping. Intuitively, applying $\mathcal{W}_{j\rightarrow i}(\cdot)$ to the
noise $\mathbf{n}_{j}\sim\mathcal{N}(0,\mathbf{I})$ at viewpoint $\pi_{j}$ and
mapping it to nearby viewpoint $\pi_{i}$ would result in multiview-consistent
noise $\mathcal{W}_{j\rightarrow i}(\mathbf{n}_{j})$ for
$\mathbf{z}_{{\bm{\theta}},\pi_{i}}$. We hypothesize that this ultimately
results in more similar and aligned 2D scores between the two viewpoints, in
accordance with the following formulation. More specifically, the gradient map
$\mathbf{g}^{\mathrm{w}}_{{\bm{\theta}},\pi_{i}}$ predicted from viewpoint
$\pi_{i}$ is defined as:

$\mathbf{g}^{\mathrm{w}}_{{\bm{\theta}},\pi_{i}}=\sum_{\pi_{j}\in\Pi_{i,j}}\frac{\mathcal{D}(\mathbf{z}_{{\bm{\theta}},\pi_{i}}+\sigma_{t}\mathcal{W}_{j\rightarrow
i}(\mathbf{n}_{j});\sigma_{t},y)-(\mathbf{z}_{{\bm{\theta}},\pi_{i}}+\sigma_{t}\mathcal{W}_{j\rightarrow
i}(\mathbf{n}_{j}))}{\sigma_{t}^{2}},$ (5)

where $\Pi_{i,j}$ denotes the set of camera poses near an anchor pose
$\pi_{i}$. The equation for multiview consistent SDS loss is then defined as
follows:

$\nabla_{{\bm{\theta}}}\mathcal{L}_{\textrm{SDS}}^{\mathrm{w}}\approx\mathbb{E}_{\pi_{i}\sim\Pi,\mathbf{n}_{j}\sim\mathcal{N}(0,\mathbf{I}),t}\left[\frac{\mathcal{D}(\mathbf{z}_{{\bm{\theta}},\pi_{i}}+\sigma_{t}\mathcal{W}_{j\rightarrow
i}(\mathbf{n}_{j});\sigma_{t},y)-\mathbf{z}_{{\bm{\theta}},\pi_{i}}}{\sigma_{t}^{2}}\cdot\frac{\partial\mathbf{z}_{{\bm{\theta}},\pi_{i}}}{\partial{\bm{\theta}}}\right],$
(6)

assuming an ideal case in which the warped noise maps
$\mathcal{W}_{j\rightarrow i}(\mathbf{n}_{j})$ retain the properties of the
standard normal distribution. Note that $\mathbf{z}_{{\bm{\theta}},\pi_{i}}$
can also be approximated as
$\mathbf{z}_{{\bm{\theta}},\pi_{i}}\approx\mathcal{W}_{j\rightarrow
i}(\mathbf{z}_{{\bm{\theta}},\pi_{j}})$. This means that the nearer the
viewpoints are and $\mathcal{W}_{j\rightarrow i}$ approaches identity mapping,
the estimated scores of nearby viewpoints in Eq. 5 and Eq. 6 also increase in
similarity and consistency. Based on Chang et al. [2], which demonstrates that
diffusion models’ video generation quality strongly benefits from
incorporating such correspondence relationship between noise given to each
frame, we hypothesize that such consistency between noises and gradients of
multiple viewpoints would benefit the optimization process, resulting in more
robust and coherent geometry.

In this paper, we propose GSD, a general framework for facilitating the
multiview consistencies of 2D scores predicted through SDS for improvement of
the geometric consistency and fidelity of generated scenes, as shown in Fig.
2. First, we introduce 3D consistent noising in Section 4.2, grounding each
viewpoint’s denoising process on the 3D geometry of the given scene as stated
above. Next, we conduct geometry-based gradient warping between different
viewpoints, in Section 4.3, so that the gradient map generated from a
viewpoint can be mapped to its 3D geometry-wise corresponding location in
another viewpoint, allowing for similarity measurement. Leveraging these
warped gradients, in Section 4.4, we describe our novel correspondence-aware
gradient consistency loss, which effectively regularizes artifacts and
inconsistency-inducing scene features through multiview consistency modeling
of the 2D scores.

###  4.2 3D consistent noising

Figure 3:  PAAS-based illustration of our consistent noising. Introduction of
3D-consistent noising induces more consistent SDS gradient across nearby
viewpoints, whose enhanced consistency allows for coherent geometry.

As explained above, we aim to design a 3D consistent noise that incorporates
3D correspondence prior, which we designate as $\mathbf{n}^{\mathrm{c}}$,
which would induce more consistent 2D scores across different viewpoints, as
described in Fig. 3, facilitating more robust 3D scene generation. A key
factor in designing $\mathbf{n}^{\mathrm{c}}$ is that the 2D noise produced by
consistent noising must possess the characteristics of a standard normal
distribution – namely, its mean and variance being that of
$\mathcal{N}(0,\mathbf{I})$, and the noise values for the pixels must be
independently and identically distributed (i.i.d).

This makes the solution of naïvely warping a 2D noise to another viewpoint
with the warping function $\mathcal{W}(\cdot)$ unsuitable, as the
interpolation (e.g., bilinear, nearest) that takes place during the warping
process harms such properties. To overcome these issues, the warping method
proposed by Chang et al. [2] interprets a noise map as the integral of
conditionally upsampled higher-resolution noise map and achieves ideal noise
warping through integral noising; however, this warping process requires heavy
computation, making it unsuitable for SDS, as it must be conducted at every
iteration.

To this end, we introduce 3D consistent integral noising, which satisfies the
above criteria by leveraging an intermediate 3D point cloud representation
incorporated with conditional noise upsampling and discrete noise integral [2]
into it. We use 3DGS [10] as our 3D representation. The mean locations of the
3D Gaussians can be used to define a point cloud that is always aligned to the
geometry of the 3D scene, as described in Fig. 4(b). We then imbue each point
with a random noise value sampled from a normal distribution, resulting in a
3D noised point cloud $\mathbf{n}^{\mathrm{3D}}$, which will be projected and
aggregated to produce 3D-consistent 2D noise maps, as we describe below.

#### Conditionally upsampled point cloud.

We adopt the conditional upsampling proposed in Chang et al. [2] to 3D point
cloud setting, interpreting each value in 3D point as an integration of
upsampled points within a partitioned volume. Assuming this volume is a
spherical volume surrounding each original point in
$\mathbf{n}^{\mathrm{3D}}$, we generate an upscaled point cloud, whose
locations are sampled from a Gaussian distribution centered around the
original point, as described in (c) of Fig. 4. The upscaling occurs by a
factor of hyperparameter $N$, meaning that $N$ points are newly sampled for
each original point $n^{\mathrm{3D}}\in\mathbf{n}^{\mathrm{3D}}$. Assuming an
original point indexed $k$, whose noise value is $n^{\mathrm{3D}}_{k}$, the
noise values for its $N$ upsampled points, designated
$\mathbf{m}^{\mathrm{3D}}_{k}$, are conditionally sampled from the original
point as follows:

$\mathbf{m}^{\mathrm{3D}}_{k}\sim\mathcal{N}(\bm{\bar{\mu}},\bar{\mathbf{\Sigma}}),\quad\text{with}\quad\bar{\bm{\mu}}=\frac{n^{\mathrm{3D}}_{k}}{N}\mathbf{u},\quad\bar{\mathbf{\Sigma}}=\frac{1}{N}\left(\mathbf{I}_{N}-\frac{1}{N}\mathbf{u}\mathbf{u}^{\top}\right),$
(7)

where $\mathbf{u}=(1,...,1)^{\top}$ whose size is $N$, $\mathbf{I}_{N}$ being
$N\times N$ identity matrix. In implementation, this corresponds to having $N$
noise values sampled from $\mathcal{N}(0,\mathbf{I})$, removing their mean,
and adding to them $n^{\mathrm{3D}}_{k}/{N}$. This conditional sampling is
conducted independently per channel of the noise map.

Figure 4: 3D consistent $\int$-noising. To produce a 3D geometry-aware 2D
noise map that preserves the properties of the standard Gaussian distribution,
we conduct 3D conditional upsampling of point clouds and discrete integral of
projected noise values. Please refer to Sec. 4.2 for more detailed explanation
of the subfigures.

#### Discrete noise integral.

After conditionally upsampling the point cloud, we project its points onto a
pixelized grid for a given viewpoint. As the number of projected points varies
for each pixel, we perform discrete noise integral to aggregate their values,
obtaining a representative value for each pixel, while preserving the Gaussian
properties of the noise map as a whole. Let us denote the set of noise values
$m^{\mathrm{3D}}$ of the projected upsampled noise points
$\mathbf{m}^{\mathrm{3D}}$, projected to a pixel $p$ at viewpoint $\pi$, as
$\Omega({p})$. Our discrete noise is pixelwise aggregated in the following
manner, summed up and normalized to preserve the Gaussian properties of the
noise map:

$n^{\mathrm{c}}({p})=\frac{1}{\sqrt{\left|\Omega({p})\right|}}\sum_{m^{\mathrm{3D}}\in\Omega({p})}m^{\mathrm{3D}},$
(8)

where $n^{\mathrm{c}}({p})$ stands for the final, aggregated noise value for
the pixel $p$ at camera $\pi$, with $|\Omega(p)|$ being the size of the set,
i.e., the total number of points projected to the pixel $p$. The points are
set to have no volumes, forcing each point to be projected to, and thus
contribute to, only a single pixel, allowing the integral process to take
place discretely and thus ensuring complete independence of pixel values. The
final 3D consistent noise map $\mathbf{n}^{\mathrm{c}}$ is built from
$n^{\mathrm{c}}({p})$ for all pixels $p$.

#### Spherical background noising.

We generate 2D noise maps for the foreground and background separately and
combine them to gain the final noise map, as described in Fig 2. For the
foreground process, to make the 2D noises aligned solely with the rendered
surfaces, we take into account only the points that lie within a certain
distance from the rendered depth, preventing self-occluded surfaces from the
other side of the object from influencing the noise integral process. For the
background, we create a spherical point cloud surrounding the scene, which we
noise, upscale, and integrate likewise, and add this noise to the empty
regions of the foreground noise to produce a final, full 2D noise map
retaining standard normal distribution properties.

#### 3D consistent noises and gradients.

Our final gradient map for viewpoint $\pi$ is defined as:

$\mathbf{g}^{\mathrm{c}}_{{\bm{\theta}},\pi}=\frac{\mathcal{D}(\mathbf{z}_{{\bm{\theta}},\pi}+\sigma_{t}\mathbf{n}^{\mathrm{c}};\sigma_{t},y)-(\mathbf{z}_{{\bm{\theta}},\pi}+\sigma_{t}\mathbf{n}^{\mathrm{c}})}{\sigma_{t}^{2}},$
(9)

and our full 3D-consistency aware SDS equation is defined as:

$\nabla_{{\bm{\theta}}}\mathcal{L}_{\textrm{SDS}}^{\mathrm{c}}\approx\mathbb{E}_{\pi\sim\Pi,\mathbf{n}^{\mathrm{c}}\sim\mathcal{N}(0,\mathbf{I}),t}\left[\frac{\mathcal{D}(\mathbf{z}_{{\bm{\theta}},\pi}+\sigma_{t}\mathbf{n}^{\mathrm{c}};\sigma_{t},y)-\mathbf{z}_{{\bm{\theta}},\pi}}{\sigma_{t}^{2}}\cdot\frac{\partial\mathbf{z}_{{\bm{\theta}},\pi}}{\partial{\bm{\theta}}}\right].$
(10)

Our results in Sec. 5.2 show that our 3D consistent noising brings clear
improvements to the overall quality and convergence speed of the optimization
process. As hypothesized, giving 3D-geometry-aware noise to corresponding
pixels in different viewpoints facilitates their SDS gradients to be more
consistent, leading to faster convergence and more high-fidelity generation
results.

Figure 5:  Properties of 3D consistent noise. Our 3D consistent integral
noising preserves the properties of perfect standard Gaussian distribution
that random noise (a) displays, while also demonstrating interpolative
qualities that bilinear interpolation (b) possesses.

#### Analysis.

The validity of our method is demonstrated in Fig. 5, where we compare the
3D-consistent noise $\mathbf{n}^{\mathrm{c}}_{i}$ at pose $\pi_{i}$ produced
by our method with other methods, such as warping and random noising. To this
end, we compute the covariance of the produced noise, its cross-covariance
with the noise of nearby viewpoint $\mathbf{n}^{\mathrm{c}}_{j}$ at pose
$\pi_{j}$, and the distribution of the generated noise values. As expected,
random noising (a) shows no correlation with nearby viewpoints, while the
distributions of bilinear warping (b) and nearest warping (c) show a
discrepancy with standard normal distribution, with (b) especially lacking the
i.i.d characteristic, as shown in the covariance matrix. 2D integral noising
[2] (d) is flawless in every quality, but its heavy computation limits its
usage within SDS, as the warping process must occur multiple times within a
single iteration. In comparison, our method preserves the Gaussian properties
such as mean, variance, and its i.i.d nature, as well as accurately
representing the interpolative correlation between viewpoints, resulting an
ideal 3D-consistent noise map, while being computationally efficient.

###  4.3 Geometry-based gradient warping

As the results for 3D consistent noising imply that encouraging multiview
consistency between the SDS gradients through 3D consistent noising benefits
the optimization process, we aim to strengthen this gradient similarity with
an additional loss. To achieve this, we first find a mapping between
3D-corresponding locations across multiple viewpoints and use this mapping to
compare the gradients generated at different viewpoints. To this end, we
conduct geometry-based warping of the SDS gradient maps,
$\mathbf{g}^{\mathrm{c}}_{i}$, and we exploit the rendered depth of the 3D
scene to warp the 2D gradient map of one viewpoint to another in a
geometrically consistent manner [12], as described in detail below.

We leverage the depth map rendered from the 3D representation, in our case
3DGS baseline [10], which we note as $\mathbf{d}$. The rendered depth map can
be used to establish a geometry-based correspondence relation between the
pixel locations of two different viewpoints, which in turn is utilized to warp
an image from one viewpoint to another. In our case, the targets of geometry-
based warping are the gradient maps at $\pi_{1}$ and $\pi_{2}$, which are
$\mathbf{g}^{\mathrm{c}}_{1}$ and $\mathbf{g}^{\mathrm{c}}_{2}$. Assuming two
viewpoints $\pi_{1}$ and $\pi_{2}$, with the viewpoint difference from former
to latter being $R_{1\rightarrow 2}$, a corresponding location for a pixel
$p_{1}$ in the image $\mathbf{g}^{\mathrm{c}}_{2}$ at viewpoint $\pi_{2}$,
written as $p_{1\rightarrow 2}$, is determined as follows:

$p_{1\rightarrow 2}=KR_{1\rightarrow 2}{\mathbf{d}}_{1}(p_{1})K^{-1}p_{1},$
(11)

in which $\mathbf{d}_{1}(p_{1})$ is the depth rendered from viewpoint
$\pi_{1}$ at pixel $p_{1}$, and $K$ is the intrinsic matrix. Using this, we
define a 3D geometry-based mapping function $\mathcal{W}_{1\rightarrow
2}(\cdot)$ that contains correspondence information between every pixel at
viewpoint $i$ and the locations at $\mathbf{g}^{\mathrm{c}}_{2}$. Applying
$\mathcal{W}_{1\rightarrow 2}(\cdot)$ at $\mathbf{g}^{\mathrm{c}}_{2}$ allows
us to obtain the warped gradient map $\mathbf{g}^{\mathrm{c}}_{2\rightarrow
1}$, and vice versa :

${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\mathbf{g}^{\mathrm{c}}_{2\rightarrow
1}}(p_{1})=\mathrm{sampler}({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\mathbf{g}^{\mathrm{c}}_{2}};\mathcal{W}_{1\rightarrow
2}(p_{1})),$ (12)

where $\mathrm{sampler}(\cdot)$ is a nearest sampling operator for an inverse
warping.

###  4.4 Correspondence-aware gradient consistency loss

We introduce an additional loss, dubbed correspondence-aware gradient
consistency loss, where we penalize the dissimilarity between the gradients
that have a 3D-correspondence mapping to guide the scene toward a more robust
and consistent appearance and geometry. The motivation for such a loss is
intuitive. Equation 6 shows that using 3D consistent noise removes much of the
randomness that the noising process brought upon the SDS process, which in
turn indicates that the differences between generated gradients are
predominantly caused by variations in appearance and geometry.

As we are comparing the gradients generated from nearby viewpoints with nearby
camera pose differences, heavy differences between corresponding gradients are
highly likely to be caused by a sharp change in appearance or geometry. These
sharp changes can generally be attributed to artifacts [12, 11] and
geometrically inconsistent features, such as Janus problems, produced on the
3D scene. In this light, a similarity loss that forces the corresponding
gradients to be more similar to one another has a regularizing effect,
reducing the artifacts and geometrical inconsistencies.

Let us assume that we have a gradient map $\mathbf{g}^{\mathrm{c}}_{i}$ at the
viewpoint $\pi_{i}$ and a warped gradient map
$\mathbf{g}^{\mathrm{c}}_{j\rightarrow i}$ from the viewpoint $\pi_{j}$.
Because $\mathbf{g}^{\mathrm{c}}_{j\rightarrow i}$ has been warped according
to 3D geometric correspondence, the consistency loss between two adjacent
viewpoints $\pi_{i}$ and $\pi_{j}$, in which $\mathbf{g}^{\mathrm{c}}_{j}$ has
been warped to $\pi_{i}$, is defined as follows:

$\mathcal{L}_{\mathrm{sim}}\coloneqq\sum_{\pi_{i}\in\Pi}\sum_{\pi_{j}\in\Pi_{i,j}}\sum_{p}\mathbf{o}_{j\rightarrow
i}(p)\cdot\left(1-\frac{\mathbf{g}^{\mathrm{c}}_{i}(p)\cdot\mathbf{g}^{\mathrm{c}}_{j\rightarrow
i}(p)}{\|\mathbf{g}^{\mathrm{c}}_{i}(p)\|\|\mathbf{g}^{\mathrm{c}}_{j\rightarrow
i}(p)\|}\right),$ (13)

where $\mathbf{o}_{j\rightarrow i}$ stands for self-occlusion mask adopted
from [12], which masks out erroneously warped locations at
$\mathbf{g}^{\mathrm{c}}_{j\rightarrow i}$. Note that we back-propagate this
loss only to the rendered depth $\mathbf{d}$ which was used in warping image
$\mathbf{g}^{\mathrm{c}}_{j}$ to $\pi_{i}$, as this loss is essentially a
geometry regularizing loss. Our experimental result at 5.3 demonstrates the
effectiveness of our loss in reducing geometric inconsistencies such as Janus
problems as well as aiding the generation of more fine-detailed geometry, and
also shows that our loss must be used in conjunction with 3D consistent
noising for proper effectiveness.

##  5 Experiments

GaussianDreamer |  |  |  |  |  |
---|---|---|---|---|---|---
+GSD |  |  |  |  |  |
|  “a cat wearing
a bee costume” |  “a turtle” |  “a goose made out of gold”
GaussianDreamer |  |  |  |  |  |
+GSD |  |  |  |  |  |
|  “a product photo
of a cat-shaped toy” |  “a mysterious
LEGO wizard” |  “a snail”
GaussianDreamer |  |  |  |  |  |
+GSD |  |  |  |  |  |
|  “a zoomed out DSLR photo
of a ceramic lion” |  “an old vintage car” |  “a DSLR photo
of origami motorcycle”
Figure 6: Qualitative improvement over GaussianDreamer [29] baseline. The
incorporation of GSD framework drastically enhances the 3D consistency and
fidelity of generated scenes.

###  5.1 Implementation details

We have implemented our method using the PyTorch framework, and all our
experiments were conducted with the Stable Diffusion model based on LDM [21].
The majority of our implementations were conducted on the Threestudio [6]
baseline of GaussianDreamer [29], and we utilized the off-the-shelf Point-E
[19] module to obtain the initial point cloud for 3D Gaussian Splatting [10].
For hyperparameters, our noised point cloud upsampling ratio $N=9$, and for
each iteration of the optimization process, we render images separated by 5∘
from each other and proceed with our consistent noising and gradient
consistency modeling process.

Dreamfusion \+ 3DFuse |  |  |  |  |  |
---|---|---|---|---|---|---
+GSD |  |  |  |  |  |
|  “a zoomed out DSLR photo
of a ceramic lion” |  “a peacock with a crown” |  “a DSLR photo of
an ironman figure”
ProlificDreamer \+ 3DFuse |  |  |  |  |  |
+GSD |  |  |  |  |  |
|  “a beautiful mermaid” |  “a majestic eagle” |  “a DSLR photo of a blue bird
with a beak and feathered wings”
Figure 7: Qualitative improvement over Dreamfusion [20] and ProlificDreamer
[28] baselines combined with 3DFuse [23]. To demonstrate the effectiveness of
our approach on other SDS methodologies, we apply GSD to 3DFuse [23]-combined
Dreamfusion [20] and ProlificDreamer [28]. Our method successfully improves
upon overall generation, removing artifacts and view inconsistencies that
remain despite using 3DFuse [23].

###  5.2 Qualitative analysis

#### Improvement over 3DGS baseline.

Fig 6 shows the improvement that GSD brings to its baseline model, which is
the Threestudio [6]-based GaussianDreamer [26] model. Note that we do not
apply the VSD loss [28], but the standard SDS loss with CFG = 100. Our
experiments reveal that 3DGS-based baselines are fallible to highly saturated
textures and geometrically inconsistent artifacts caused by a high CFG. We
demonstrate that our method counters such errors and geometric inconsistencies
successfully, reducing multi-faced Janus problems drastically as well as
fixing incoherent geometries such as multiple beaks on “a goose made out of
gold” or two heads appearing on “a turtle”.

In Fig. 7, to show our method’s universal effectiveness across various SDS-
based methodologies, we combine GSD with other Instant-NGP [17] based baseline
methods [20, 28] and observe the effects. As our methodology requires a point
cloud aligned with scene geometry, we leverage 3DFuse [23], which conditions
scene optimization on a point cloud. As the generated scene geometry closely
follows the point cloud, we leverage this point cloud to conduct 3D consistent
integral noising. Our results reveal that despite using 3DFuse, which is
designed to enhance view consistency of generated 3D scenes, artifacts and
view inconsistency problems such as the Janus problem persist in numerous
generated results. Application of our approach brings about clear enhancements
in these aspects, resulting in more geometrically robust and well-textured 3D
scenes.

Figure 8: Comparison to multiview generation model. We compare the generation
results of our framework with that of MVDream [24] to demonstrate the
superiority in geometric and textural detail of our GSD-combined SDS baseline,
ProlificDreamer [28], while also displaying strong geometric consistency.

#### Comparison to multiview generation framework.

We compare the generation results of our framework with MVDream [24], a
multiview generation diffusion model fine-tuned on a 3D dataset, Objaverse
[5]. This family of text-to-3D generation models [15, 24] is capable of
directly predicting novel viewpoints of a given image or text, allowing for
faster generation speed that SDS-based frameworks, with MVDream nearly
completely free from view inconsistency problems. However, such fine-tuning on
Objaverse, which is limited in diversity and quality of its 3D assets, causes
its generation results to be constrained by having claylike, low-fidelity
textures, as demonstrated in Fig. 8. In comparison, we show that GSD combined
with SDS methodologies (GaussianDreamer and ProlificDreamer in given results)
is capable of creating scenes of highly detailed geometry and fidelity, fully
leveraging the generative capability of a pretrained 2D diffusion that has not
been fine-tuned to Objaverse, while also demonstrating strong geometric
robustness and consistency as our GSD encourages view-consistent generation
through score distillation process itself.

###  5.3 Ablation study and analysis

#### Enhancement in convergence speed.

Figure 9: Convergence speed comparison. Comparison between naïve noising and
3D-aware noising shows that our method of 3D consistent noising and similarity
loss achieves quicker convergence over baseline, GaussianDreamer [29]. The
prompt “a full body of a cat with a hat” is used.

We find that our method particularly improves the convergence speed of SDS. As
shown in Fig. 9, changing the noise sampling strategy and applying gradient
consistency loss with the same prompt and seed results in faster optimization,
forming hats and faces at early steps when generating with the prompt “a full
body of a cat with a hat”. This can be interpreted as a verification of our
hypothesis regarding multiview consistency of noise maps and generated 2D
scores, as providing the noise maps that are in alignment with the 3D geometry
induces more consistent gradient maps, resulting in more aligned gradients
regarding the 3D scores and facilitating the 3D representation to converge
much more easily.

#### Ablation on 3D consistent noising and gradient consistency loss.

Figure 10:  Ablation. Our experiments show that without 3D consistent noising,
our consistency loss shows little to no effect on the generation process. The
prompt is a “a cute meercat”.

We conduct an ablation study regarding our 3D consistent noising and the
gradient consistency loss in Fig. 10. Our experimental results show that when
the two components are used in conjunction, it brings about enhancement in
geometric robustness and increased fidelity from the naiv̈e result (a), as
clearly shown in (d). However, when the consistency loss is used without
consistent noising, its effects are diminished, as shown in (b). Sole usage of
3D consistent noise $\mathbf{n}^{\mathrm{c}}$ brings about only limited
improvement as well, observable in (c). This indicates that gradient
similarity incurred by 3D consistent noising is crucial for gradient
consistency modeling in allowing meaningful geometry regularization to take
place with consistency loss.

Figure 11: 360∘ visualization. 360∘ comparison between the baseline model,
which is ProlificDreamer [28] combined with 3DFuse [23], and results with GSD
added, along with 3D consistent noise visualization.

#### 360∘ visualization of 3D scene and 3D consistent noise.

Fig. 11 displays a 360∘ comparison of our methodology with that of baseline,
which shows drastic improvement induced by the application of GSD. The
experiment shows an interesting case demonstrating how our method functions:
even though the conditioning geometry is completely identical due to
constraint by 3DFuse, the incorporation of our methodology encourages a more
view-consistent and realistic interpretation of this given geometry,
outputting a drastically enhanced 3D scene optimization result, as well
displayed.

Figure 12: CLIP similarities between each rendered image and view-augmented prompt and images. We compute CLIP similarities for each image and view-augmented prompt (e.g., “front view of” and “back view of”). The x-axis value (image index) corresponds to the azimuth, where 0 stands for the front view and 60 for the back view. The baseline used for this experiment is GaussianDreamer [29].  Table 1: User study. The user study is conducted by surveying 39 participants to evaluate 3D coherence, prompt adherence, and rendering quality. Method | 3D | Prompt | Overall
---|---|---|---
coherence | adherence | quality
Baseline + GSD (Ours) | 65.4% | 68.4% | 61.5%
Baseline | 34.6 % | 31.6 % | 38.4 %

#### User study.

We conducted a user study with 39 participants, whose results are displayed on
Tab. 1. The participants are given six randomly sampled multiview renderings
of the baseline results, produced from baselines GaussianDreamer and
ProlificDreamer, compared with GSD-combined results rendered from the same
angles. We question the participants in three aspects: i) which 3D model
exhibits more realistic 3D geometry, ii) which 3D model adheres more closely
to the given prompt, and iii) which 3D model is superior in overall quality.
The results show that the users prefer the results combined with GSD to those
of the raw baseline model by a significant margin, demonstrating that our
model notably improves performance in all three aspects.

#### Consistency analysis with CLIP similarity.

To measure the consistency of generated 3D objects, we follow previous work
[8] of measuring the CLIP similarity between the generated images and the
front view and back view prompts across various prompts and provide the result
at Fig. 12. However, we do not find a significant correlation between the view
prompts and the images corresponding to each view. This appears to be
partially because the CLIP model, being discriminative, does not accurately
evaluate the similarity between detailed prompts and images.

##  6 Conclusion

Our proposed methodology, GSD, which integrates geometry-based correspondence
prior to the SDS process, significantly enhances the multiview consistency of
generated gradients, thereby improving the geometric fidelity and consistency
of text-to-3D generation. By introducing 3D consistent integral noising,
geometry-based gradient warping, and a novel multiview gradient consistency
loss, we address the critical issue of geometric inconsistencies without
requiring additional training or external modules. Our method demonstrates
notable improvements in the optimization process, achieving competitive
results with state-of-the-art models in both qualitative and quantitative
assessments. The ablation study further confirms the interdependence of our
components, validating the comprehensive efficacy of our approach in enhancing
SDS-based text-to-3D generation.

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# Sustained oscillation of flexible cantilevers without vortex shedding

Shayan Heydari1 and R. K. Jaiman1<EMAIL_ADDRESS>1Department of
Mechanical Engineering, The University of British Columbia, Vancouver, BC
Canada V6T 1Z4

###### Abstract

The present work investigates the fluid-structure interaction (FSI) of a
flexible cylindrical cantilever beam at subcritical Reynolds numbers ($Re$). A
fully-coupled fluid-structure solver based on the three-dimensional (3D)
incompressible Navier-Stokes equations and Euler-Bernoulli beam theory is
employed to numerically examine the coupled dynamics of the beam. We assess
the extent to which such a flexible cylindrical beam could sustain
oscillations in this $Re$ regime when it is either exposed to a steady
upstream wake (i.e., tandem cylinder configuration) or subjected to an
externally applied base excitation. Our results indicate that within a
particular range of reduced velocity parameter ($U^{*}$), the beam experiences
sustained oscillations in both scenarios, leading to periodic vortex shedding
downstream. The mechanism governing the sustained oscillations is
characterized as synchronization, during which the frequency of the cross-flow
fluid loading matches the beam’s first-mode natural frequency. When the beam
is subjected to base excitation, the critical Reynolds number for vortex
shedding ($Re_{c}$) is found to reduce to $Re_{c}\approx 5$. Above this
threshold, vortex shedding is found to occur by stimulating the pair of
counter-rotating vortices in the near-wake region. For the tandem cylinder
configuration, the beam is shown to exhibit figure-eight-shaped tip motion
trajectories during its oscillatory response. However, various patterns of tip
motion trajectories, including figure-eight, and chaotic-type responses, are
observed when the beam is under external base excitation. The findings of this
work aim to generalize our understanding of sustained oscillation in flexible
cylindrical cantilevers and have relevance to the development of bio-inspired
cantilever flow sensors.

##  1 Introduction

Fluid-structure interactions are ubiquitous in nature and various engineering
applications. One intriguing manifestation of FSIs occurs when a flexible body
with a bluff cross-section is exposed to a steady incident flow perpendicular
to its length. When the Reynolds number, which is based on the body’s
characteristic length ($D$) and free-stream velocity ($U_{0}$), is above a
critical value, often approximated as $Re_{c}\approx 45$ (Jackson (1987)), the
shedding of von Kármán vortices can induce sustained oscillations in the body,
a phenomenon widely known as vortex-induced vibrations (VIVs). The study of
VIVs has garnered substantial attention over the past few decades primarily
owing to the intricate vortex dynamics and nonlinear physics that manifest
during VIVs. For a bluff body experiencing VIVs, studies have shown that there
is a particular range of system parameters, known as the synchronization (also
sometimes called lock-in) regime, within which the frequency of vortex
shedding deviates from Strouhal’s relationship, which determines the vortex-
shedding frequency of the body’s stationary counterpart, and becomes equal or
close to the body’s first-mode natural frequency. During VIVs, the vibration
amplitudes are known to exhibit bell-shaped trends as functions of the reduced
velocity parameter $U^{*}$ and have been shown to be of the order $O(D)$ in
the cross-flow direction within the synchronization regime, as reviewed by
Williamson & Govardhan (2004) and Sarpkaya (2004).

Although VIVs have been extensively studied at supercritical Reynolds numbers
(i.e., $Re>Re_{c}$), recent research has unveiled intriguing phenomena in the
subcritical regime of Reynolds number (i.e., $Re<Re_{c}$). In particular,
studies on the coupled dynamics of flexible bluff bodies, including flexible
circular cylinders (Heydari et al. (2022); Bourguet (2020); Buffoni (2003)),
and elastically-mounted rigid cylinders (Miyanawala & Jaiman (2019); Boersma
et al. (2021); Cossu & Morino (2000); Mittal & Singh (2005); Dolci & Carmo
(2019); Meliga & Chomaz (2011)), have revealed that, in some scenarios,
sustained oscillations and periodic vortex shedding could occur in the
subcritical regime of $Re$. For instance, in previous research (Heydari et al.
(2022)), we investigated the FSI of a long flexible cylindrical cantilever
beam for $Re\in[20,40]$ using high-fidelity numerical simulations. The results
of our numerical experiments revealed that the flexible cantilever could
experience sustained oscillations, with a periodic vortex shedding pattern
evident downstream, for Reynolds numbers as low as $Re=22$ within the
synchronization regime.

The findings regarding the sustained oscillation of flexible cylindrical
bodies at subcritical Reynolds numbers challenge the conventional
understanding that vortex shedding exclusively occurs in the supercritical
regime of $Re$ and emphasize the need to expand our knowledge of flow-induced
vibrations, especially at subcritical Reynolds numbers. Such knowledge is not
only relevant to engineering applications but also holds significant promise
within the context of biological flow sensing. Many biological species employ
mechanosensory hairs and other flow-sensing mechanisms to detect flow features
and minute changes in flow patterns (Beem & Triantafyllou (2015)). By
investigating the complexities of sustained oscillations in these structures,
it becomes possible to inform the design of artificial flow sensors tasked
with measuring flow information and differentiating between various flow
patterns. In this context, it is essential to understand the role of wake
interference and base excitation on the sustained oscillation of flexible
cylinders in the subcritical $Re$ regime.

This study examines the coupled dynamics of a flexible cylindrical cantilever
beam, as a canonical model of a biological flow sensor, targeting the
subcritical regime of $Re$. Two distinct configurations will be explored: (i)
a tandem cylinder arrangement, where two cylinders are placed in line, and
(ii) an isolated beam subjected to external base excitation. These
configurations have the potential to significantly influence the oscillatory
response of the beam and alter the wake dynamics in the subcritical $Re$
regime, as demonstrated by Bourguet (2023) in the case of an elastically-
mounted rigid cylinder under forced rotation. The current research is expected
to expand our knowledge of sustained oscillation in flexible cantilevers and
is relevant to biological flow sensing and engineered flow sensors.

##  2 Problem description and numerical methodology

To investigate the coupled dynamics of the flexible cantilever, we utilize a
3D computational framework. A schematic of the computational domain with
details of the domain size and boundary conditions is given in Fig. (1a). The
cantilever is taken as a circular cylinder of diameter $D$ and length
$L=100D$. For the tandem cylinder arrangement, a rigid stationary cylinder is
positioned in the upstream region at a streamwise distance $x_{0}$ from the
cantilever. The no-slip boundary condition is enforced on the surface of the
cantilever, denoted as $\Gamma^{\mathrm{fs}}$, and the surface of the rigid
stationary cylinder. A uniform flow of velocity
$\boldsymbol{u}^{\mathrm{f}}=(U_{0},0,0)$ is applied at the inflow surface and
the slip boundary condition is imposed at the side, top, and bottom surfaces
of the computational domain, as illustrated in Fig. (1a). For the outflow
surface, the traction-free boundary condition, given as
$\boldsymbol{\sigma}^{\mathrm{f}}.\boldsymbol{n}^{\mathrm{f}}=\boldsymbol{0}$,
is specified, where $\boldsymbol{\sigma}^{\mathrm{f}}$ is the Cauchy stress
tensor for a Newtonian fluid and $\boldsymbol{n}^{\mathrm{f}}$ is the unit
normal vector to the outflow surface. The unsteady 3D Navier–Stokes equations
are employed to predict the flow dynamics. Using an arbitrary Lagrangian-
Eulerian (ALE) reference frame on the fluid domain $\Omega^{\mathrm{f}}(t)$,
the Navier–Stokes equations are written as:

$\displaystyle\rho^{\mathrm{f}}\frac{\partial\boldsymbol{u}^{\mathrm{f}}}{\partial
t}\bigg{|}_{\hat{x}^{\mathrm{f}}}+\rho^{\mathrm{f}}(\boldsymbol{u}^{\mathrm{f}}-{{\boldsymbol{u}^{\mathrm{m}}}})\cdot\nabla\boldsymbol{u}^{\mathrm{f}}$
$\displaystyle=\nabla\cdot\boldsymbol{\sigma}^{\mathrm{f}}+\boldsymbol{b}^{\mathrm{f}}\
\ \ \mathrm{on\ \ \Omega^{\mathrm{f}}(t)},$ (1)
$\displaystyle\nabla\cdot\boldsymbol{u}^{\mathrm{f}}$ $\displaystyle=0\ \ \
\mathrm{on\ \ \Omega^{\mathrm{f}}(t)},$ (2)

where $\rho^{\mathrm{f}}$ is the density of the fluid,
$\boldsymbol{u}^{\mathrm{f}}=\boldsymbol{u}^{\mathrm{f}}(\boldsymbol{x}^{\mathrm{f}},t)$
and
$\boldsymbol{u}^{\mathrm{m}}=\boldsymbol{u}^{\mathrm{m}}(\boldsymbol{x}^{\mathrm{f}},t)$
are the fluid and mesh velocities defined for each spatial point
$\boldsymbol{x}^{\mathrm{f}}\in\Omega^{\mathrm{f}}(t)$, respectively,
$\boldsymbol{b}^{\mathrm{f}}$ is the body force applied to the fluid and
$\boldsymbol{\sigma}^{\mathrm{f}}$ is the Cauchy stress tensor, given as
$\boldsymbol{\sigma}^{\mathrm{f}}=-p\boldsymbol{I}+\mu^{\mathrm{f}}(\nabla\boldsymbol{u}^{\mathrm{f}}+(\nabla\boldsymbol{u}^{\mathrm{f}})^{T})$,
where $p$ denotes the fluid pressure, and $\mu^{\mathrm{f}}$ is the dynamic
viscosity of the fluid. The first term in Eq. (1) represents the partial
derivative of $\boldsymbol{u}^{\mathrm{f}}$ with respect to time while the ALE
referential coordinate $\hat{x}^{\mathrm{f}}$ is kept fixed. The fluid forcing
acting on the cantilever’s surface is calculated by integrating the surface
traction on the fluid-structure interface $\Gamma^{\mathrm{fs}}$. The
instantaneous coefficients of the lift and drag forces are then computed as:

$\displaystyle
C_{\mathrm{L}}=\frac{1}{\frac{1}{2}\rho^{\mathrm{f}}U_{\mathrm{0}}^{\mathrm{2}}DL}\int_{\Gamma^{\mathrm{fs}}}(\boldsymbol{\sigma}^{\mathrm{f}}\cdot\boldsymbol{n})\cdot\boldsymbol{n}_{\mathrm{y}}\mathrm{d\Gamma},\quad
C_{\mathrm{D}}=\frac{1}{\frac{1}{2}\rho^{\mathrm{f}}U_{\mathrm{0}}^{\mathrm{2}}DL}\int_{\Gamma^{\mathrm{fs}}}(\boldsymbol{\sigma}^{\mathrm{f}}\cdot\boldsymbol{n})\cdot\boldsymbol{n}_{\mathrm{x}}\mathrm{d\Gamma},$
(3)

where $\boldsymbol{n}_{\mathrm{x}}$ and $\boldsymbol{n}_{\mathrm{y}}$ are the
Cartesian components of the unit outward normal vector $\boldsymbol{n}$. A
stabilized Petrov-Galerkin finite element method is employed to discretize the
fluid domain $\Omega^{f}$ into $n_{el}$ non-overlapping finite elements
$\Omega^{e}$ in space such that $\Omega^{f}=\bigcup_{e=1}^{n_{el}}\Omega^{e}$.
The employed computational grid, as shown in Fig. (1b), consists of
unstructured prism elements, with a boundary layer mesh around the cylinders.
The accuracy of the computational grid and the validity of the numerical
solver have been substantiated in our prior studies (Joshi & Jaiman (2017);
Heydari et al. (2022)).

The flexible cantilever is taken as a slender structure with small lateral
motions, hence, its dynamic response is modeled using the Euler-Bernoulli beam
theory (Blevins (2016)).

Figure 1: Problem specification: (a) schematic of the computational domain
with details of the domain size and boundary conditions (For the isolated beam
configuration, the upstream cylinder is omitted from the computational domain,
and the beam is positioned at distances $15D$ and $45D$ from the inflow and
outflow surfaces, respectively.), (b) the unstructured finite element grid
with a close-up view of the boundary layer mesh.

The structural domain $\Omega^{\mathrm{s}}$ consists of structure coordinates
$\boldsymbol{x}^{\mathrm{s}}=(x,y,z)$. The transverse displacements
$\boldsymbol{w}^{\mathrm{s}}(z,t)$ are solved using the Euler-Bernoulli beam
equation excited by a distributed unsteady fluid loading per unit length
$\boldsymbol{f}^{\mathrm{s}}$. Neglecting the damping and shear effects, the
equation of motion of the beam is written as:

$\displaystyle{m}\frac{\partial^{2}\boldsymbol{w}^{\mathrm{s}}(z,t)}{\partial
t^{2}}+EI\frac{\partial^{4}\boldsymbol{w}^{\mathrm{s}}(z,t)}{\partial
z^{4}}=\boldsymbol{f}^{\mathrm{s}}(z,t),$ (4)

where $m=\rho^{\mathrm{s}}A$ is the mass per unit length of the beam, with
$\rho^{\mathrm{s}}$ and $A$ being the density of the structure and the cross-
sectional area, respectively. The Young’s modulus and second moment of area of
the beam are denoted by $E$ and $I$, respectively. For the cases with base
excitation, the transverse displacements $\boldsymbol{w}^{\mathrm{s}}(z,t)$
are computed as:

$\displaystyle\boldsymbol{w}^{\mathrm{s}}(z,t)=\boldsymbol{w}_{rel}^{\mathrm{s}}(z,t)+\boldsymbol{w}_{b}^{\mathrm{s}}(t),$
(5)

where $\boldsymbol{w}_{b}^{\mathrm{s}}(t)$ is the displacement of the beam at
its base and $\boldsymbol{w}_{rel}^{\mathrm{s}}(z,t)$ is the relative
displacement to the base. Hence, Eq. (4) is generalized as:

$\displaystyle{m}\left(\frac{\partial^{2}\boldsymbol{w}_{rel}^{\mathrm{s}}(z,t)}{\partial
t^{2}}+\frac{\partial^{2}\boldsymbol{w}_{b}^{\mathrm{s}}(t)}{\partial
t^{2}}\right)+EI\frac{\partial^{4}\boldsymbol{w}_{rel}^{\mathrm{s}}(z,t)}{\partial
z^{4}}=\boldsymbol{f}^{\mathrm{s}}(z,t).$ (6)

The boundary conditions at the fixed end of the cantilever beam are given as
$\boldsymbol{w}_{rel}^{\mathrm{s}}(0,t)=0$ and
$\frac{\partial\boldsymbol{w}_{rel}^{\mathrm{s}}(z,t)}{\partial
z}\bigg{|}_{z=0}=0$. To find a solution for Eq. (6), we consider a mode
superposition approach. We define the solution to the beam’s relative
displacement as:

$\displaystyle{\boldsymbol{w}}_{rel}^{\mathrm{s}}(z,t)=\sum_{\mathrm{i}=1}^{\infty}{\boldsymbol{\gamma}}_{\mathrm{i}}(t){S}_{\mathrm{i}}(z),$
(7)

in which $\mathrm{i}$ is the mode number,
${\boldsymbol{\gamma}}_{\mathrm{i}}(t)$ is a vector containing unknown time
functions, and $S_{\mathrm{i}}$ is the beam’s i-th natural vibration mode. For
the considered cantilever beam, the natural vibration modes are taken as the
sums of sine, cosine, sinh, and cosh functions written as:

$\displaystyle
S_{\mathrm{i}}\left(z\right)=\cosh\left(\frac{\lambda_{\mathrm{i}}z}{L}\right)-\cos\left(\frac{\lambda_{\mathrm{i}}z}{L}\right)-\sigma_{\mathrm{i}}\sinh\left(\frac{\lambda_{\mathrm{i}}z}{L}\right)+\sigma_{\mathrm{i}}\sin\left(\frac{\lambda_{\mathrm{i}}z}{L}\right),$
(8)

where $\lambda_{\mathrm{i}}$ and $\sigma_{\mathrm{i}}$ are dimensionless
parameters dependent on the mode number (see Blevins (2016) for values of
$\lambda_{\mathrm{i}}$ and $\sigma_{\mathrm{i}}$). In our analysis, we
consider a sinusoidal base excitation given as
$\boldsymbol{w}_{b}^{\mathrm{s}}(t)=\boldsymbol{F}_{b}\sin(2\pi f_{b}t)$,
where $\boldsymbol{F}_{b}$ is a vector containing the components of the base
excitation amplitude in the transverse directions and $f_{b}$ is the frequency
of the base excitation. Substituting Eq. (7) into Eq. (6) and considering the
orthogonality conditions, Eq. (6) is rewritten as:

$\displaystyle\frac{\partial^{2}{\boldsymbol{\gamma}}_{\mathrm{i}}(t)}{\partial
t^{2}}+\omega_{\mathrm{i}}^{2}{\boldsymbol{\gamma}}_{\mathrm{i}}(t)=\frac{1}{mL}\int_{0}^{L}{\boldsymbol{f}}^{\mathrm{s}}(z,t)S_{\mathrm{i}}(z)~{}\mathrm{d}z-\frac{1}{L}\frac{\partial^{2}{\boldsymbol{w}}_{b}^{\mathrm{s}}(t)}{\partial
t^{2}}\int_{0}^{L}S_{\mathrm{i}}(z)~{}\mathrm{d}z,$ (9)

where $\omega_{\mathrm{i}}$ is defined as
$\omega_{\mathrm{i}}=\frac{{\lambda_{\mathrm{i}}}^{2}}{L^{2}}\sqrt{\frac{EI}{m}}$.
The solution to Eq. (9) is obtained using the trapezoidal integration rule.
The relative displacements are then calculated using Eq. (7). Finally, the
beam’s displacement is computed using Eq. (5). The continuity of the velocity
and traction is also satisfied at the fluid-structure interface. For a
comprehensive review of the numerical algorithm and implementation details,
the reader is referred to Jaiman & Joshi (2022).

The dimensionless parameters relevant to this study are mass ratio
($m^{\mathrm{*}}$), Reynolds number $Re$, and reduced velocity
$U^{\mathrm{*}}$ defined as:

$\displaystyle m^{\mathrm{*}}=\frac{4m}{{\pi D^{2}\rho^{\mathrm{f}}}},\qquad
Re=\frac{\rho^{\mathrm{f}}U_{\mathrm{0}}D}{\mu^{\mathrm{f}}},\qquad
U^{\mathrm{*}}$ $\displaystyle=\frac{U_{\mathrm{0}}}{f_{\mathrm{n}}D},$ (10)

where $f_{\mathrm{n}}$ is the first-mode natural frequency of the beam taking
into account the effect of added fluid mass. In our present study, we set the
mass ratio at $m^{*}=1$, Reynolds number $Re\leq 40$, and reduced velocity
$U^{*}\in[5,19]$. The reduced velocity is modified by altering the velocity of
the free-stream flow while maintaining a constant Reynolds number value.

##  3 Sustained oscillations at subcritical Reynolds number

The response characteristic of the cantilever beam in the tandem cylinder
configuration is provided in § 3.1. Vibration amplitudes and frequencies, as a
function of the amplitude and frequency of the base excitation, are presented
in § 3.2. The flow visualization and the vorticity dynamics are discussed in §
3.3.

###  3.1 Response dynamics of tandem cylinders

First, we investigate the coupled dynamics of the cantilever beam in the
tandem cylinder configuration. The streamwise distance between the two
cylinders, denoted by $x_{0}$, is set at either $x_{0}=5D$ or $x_{0}=10D$.
Figure (2a) shows the variation of the root-mean-square (rms) value of the
cross-flow vibration amplitude ($A_{y}^{rms}/D$) at the free end of the
cantilever for $U^{*}\in[5,19]$ at $Re=40$.

Figure 2: Response characteristics of the cantilever beam at $Re=40$: (a)
cross-flow vibration amplitude at the free end of the cantilever (i.e.,
$z/L=1$) as a function of $U^{*}$, (b) variations of the cross-flow vibration
frequency at $z/L=1$ and the lift coefficient frequency with respect to
$U^{*}$ (The area in gray indicates $y$-axis values between $[0.9,1.1]$ and
the dashed blue line depicts the calculated vortex-shedding frequency,
determined using a Strouhal number of 0.12.), (c-d) power spectra of the
vibration amplitudes and fluid loading for $x_{0}=5D$ at $U^{*}=9$, (e) tip
motion trajectories at $U^{*}=9$, (f) $xy$-plane view of the $z$-vorticity
contours at $z/L=1$ and $U^{*}=11$. The flow is from left to right.

Compared to the isolated cantilever beam, where the oscillations occur for
$U^{*}>5$ (Heydari et al. (2022)), the cantilever beam in the tandem
arrangement starts to exhibit sustained oscillations for $U^{*}>7$. The
oscillations are present within $U^{*}\in(7,17)$ for the studied tandem
configurations. In the case of an elastically-mounted rigid cylinder in a
tandem arrangement, previous research (Mysa et al. (2016)) has demonstrated
that at $Re=100$ the cylinder maintains a constant amplitude of oscillation,
larger than its isolated counterpart, for reduced velocities $U^{*}\geq 15$.
However, our present results indicate that within the subcritical regime of
the Reynolds number, sustained oscillations are not present at such high
values of $U^{*}$. As shown in Fig. (2a), when $U^{*}\geq 17$, the cantilever
beam in the tandem arrangement remains in its steady position (i.e., no
sustained oscillations) at both $x_{0}=5D$ and $x_{0}=10D$. It is important to
highlight that for the considered tandem arrangements, oscillations are absent
across all studied $U^{*}$ values when the Reynolds number is set at $Re=30$
or below.

In all the instances where the cantilever beam exhibits sustained
oscillations, there is a frequency match between the frequency of the lift
coefficient and the frequency of the cross-flow oscillations, as shown in Fig.
(2b). The power spectra of the cross-sectional drag and lift coefficients,
along with the power spectra of the vibration amplitudes are shown in Figs.
(2c) and (2d) for the tandem cylinder configuration at $x_{0}=5D$ and
$U^{*}=9$. As seen in these figures, there is a frequency match between the
frequency of the cross-flow (streamwise) oscillation and the frequency of the
lift (drag) coefficient. According to Figs. (2c) and (2d), the streamwise
oscillation frequency is twice the frequency of the cross-flow vibrations.
This 2:1 ratio between the frequency of the streamwise-to-cross-flow
oscillations is observed across all $U^{*}$ values where the beam exhibits
sustained oscillations. A typical figure-eight type of tip motion trajectory,
often associated with this 2:1 ratio, is hence evident during the beam’s
oscillatory response, as shown in Fig. (2e) for the tandem cylinder
configurations at $U^{*}=9$. Within the range of $U^{*}$ values where the beam
exhibits sustained oscillations, a periodic vortex shedding pattern is evident
downstream. For the studied tandem cylinder arrangements, the wake of the
upstream cylinder remains nearly steady and symmetric across all examined
$U^{*}$ values, as shown in Fig. (2f) for the cases at $U^{*}=11$. This steady
upstream wake is found to delay the formation of vortices behind the
cantilever beam and contribute to stabilizing the beam’s oscillatory motion in
the subcritical $Re$ regime. Consequently, the range of $U^{*}$ values
associated with sustained oscillations is narrower for the tandem cylinder
arrangement, compared to the isolated cantilever beam, as evident in Figs.
(2a) and (2b).

###  3.2 Effect of base excitation

Next, we investigate how an externally applied base excitation alters the
coupled dynamics of the isolated cantilever beam. Figure (3a) demonstrates the
response characteristics of the beam in terms of the rms value of the cross-
flow vibration amplitude as a function of the reduced velocity parameter
$U^{*}$ in two scenarios: when the beam is not subjected to any base
excitation (i.e., unperturbed configuration) and when it experiences base
excitation in the cross-flow direction. In the cases with base excitation, the
excitation amplitude is set at $A_{b}/D=1.0$, and the excitation frequency is
given as $f_{b}/f_{n}=1.0$.

As shown in Fig. (3a), the beam under the specified base excitation
experiences sustained oscillations of larger amplitude throughout the
evaluated $U^{*}$ range, in comparison to the unperturbed configuration. It is
also observed that the cross-flow vibration amplitude follows a nearly linear
trend with respect to $U^{*}$ in the presence of base excitation. This linear
trend is inversely proportional to $U^{*}$ and differs from the usual bell-
shaped trend seen without the presence of base excitation. Unlike the
unperturbed configuration where the frequency of the cross-flow oscillation
increases with increasing $U^{*}$, the beam under the specified base
excitation oscillates at the frequency of the base excitation for all examined
$U^{*}$ values, as illustrated in Fig. (3b). In the case of the beam under
base excitation, the frequency of the lift coefficient matches the frequency
of the cross-flow oscillation across all examined $U^{*}$ values, as shown in
Fig. (3b). Based on the results provided in Fig. (3b), it is evident that the
imposition of a base excitation influences the wake behavior in the
subcritical $Re$ regime. More specifically, the dominant frequency of the lift
coefficient is found to be dictated by the frequency of the base excitation,
independent of the value of $U^{*}$. A comprehensive examination of the wake
dynamics is presented in § 3.3.

Figure 3: Response characteristics of the isolated beam under base excitation
at $Re=40$: (a) variation of the cross-flow vibration amplitude at $z/L=1$ as
a function of $U^{*}$, (b) variation of the cross-flow vibration frequency at
$z/L=1$ and lift coefficient frequency as a function of $U^{*}$ (The area in
gray indicates $y$-axis values between $[0.9,1.1]$ and the dashed blue line
depicts the calculated vortex-shedding frequency, determined using a Strouhal
number of 0.12.), (c) variation of the cross-flow vibration amplitude as a
function of $A_{b}/D$ and $f_{b}/f_{n}$.

To further assess the effect of the amplitude and frequency of the base
excitation on the response characteristic of the beam, we have examined the
FSI of the beam at $U^{*}=7$ and $U^{*}=19$ under a range of base excitation
conditions. The oscillatory behavior of the beam is studied for
$A_{b}/D\in[0.5,1.5]$ and $f_{b}/f_{n}\in[0.5,1.5]$. The results, expressed as
the rms value of the cross-flow vibration amplitude, are provided in Fig.
(3c). As shown in Fig. (3c), there is a rise in the amplitude of cross-flow
oscillation as the base excitation amplitude is increased. At $U^{*}=7$, the
beam exhibits the largest vibration amplitudes when $f_{b}/f_{n}=1.0$. In
these scenarios, the value of $A_{y}^{rms}/D$ is approximately 1.1-2.2 times
that observed in cases where the excitation frequency is either equal to
$f_{b}/f_{n}=0.5$ or $f_{b}/f_{n}=1.5$. At $U^{*}=19$, the value of
$A_{y}^{rms}/D$ shows minimal sensitivity to the frequency of the base
excitation, resulting in comparable cross-flow vibration amplitudes. It is
noteworthy that, across all examined combinations of ($A_{b}/D,f_{b}/f_{n}$),
the oscillations exhibit lower amplitudes at $U^{*}=19$ compared to those at
$U^{*}=7$. Figure (3c) also provides the values of $A_{y}^{rms}/D$ as a
function of the base excitation parameters, under conditions where the fluid-
structure coupling effects are not present (i.e., uncoupled configuration). In
these scenarios, the slope of $A_{y}^{rms}/D$ with respect to $A_{b}/D$ is
found to be more pronounced, with a value of $1.1$ when $f_{b}/f_{n}=0.5$ and
$3.2$ when $f_{b}/f_{n}=1.5$, compared to values between $(0.6,1.1)$ when the
fluid-structure coupling effects are taken into consideration.

For the cases involving fluid-structure coupling, we find that the base
excitation alters the trajectory of the beam’s tip motion. As depicted in Fig.
(4), depending on the amplitude and frequency of the base excitation, the
trajectory of the beam’s tip motion manifests as various responses. At
$U^{*}=7$, the trajectories resemble figure-eight and chaotic-type responses.
However, at $U^{*}=19$, only a figure-eight type of response is seen. The
distinctive characteristics observed in motion trajectories underscore the
significance of base excitation in cantilever flow sensors. Essentially, each
motion trajectory functions as a flow signature, aiding such sensors in the
discrimination of different flow patterns and the subsequent retrieval of
flow-related data.

Figure 4: Qualitative comparison of the tip motion trajectories at $Re=40$ as
a function of $A_{b}/D$ and $f_{b}/f_{n}$: (a) $U^{*}=7$, (b) $U^{*}=19$.

###  3.3 Vorticity dynamics at subcritical Re

In our earlier study (Heydari et al. (2022)), we demonstrated that for an
isolated cantilever beam, vortex shedding could occur for $Re\geq 22$ within a
particular range of the reduced velocity parameter $U^{*}$. However, in the
case of the tandem cylinder arrangement, vortex shedding is found to be
present only for $Re\geq 40$ within a narrow range of $U^{*}$ where the beam
undergoes sustained oscillations. Additionally, for the cases involving base
excitation, the results of our numerical experiments reveal that the base
motion of the beam could significantly influence the critical Reynolds number
for vortex shedding $Re_{c}$. In such cases, by stimulating the pair of
vortices in the near-wake region, it becomes possible to reduce $Re_{c}$ to
just above $Re_{c}\approx 5$. Figure (5) shows the $z$-vorticity contours for
the isolated beam experiencing base excitation at $A_{b}/D=1.0$ and
$f_{b}/f_{n}=1.0$ across different Reynolds numbers. According to the contour
plots, in the presence of base excitation it becomes possible to induce vortex
shedding even at a low Reynolds number of $Re=10$, where, in the absence of
such excitation, the wake would otherwise remain steady and symmetric.

Figure 5: $xy$-plane view of the $z$-vorticity contours at the mid-section of
the beam at $A_{b}/D=1.0$ and $f_{b}/f_{n}=1.0$: (a) $Re=5$, (b) $Re=10$, (c)
$Re=20$, (d) $Re=40$. (a-d) The top and bottom contours correspond to
$U^{*}=7$ and $U^{*}=19$, respectively. The flow is from left to right.

It is important to note that the existence of a pair of counter-rotating
vortices in the near-wake region, which typically appears for $Re\gtrapprox 5$
(Jackson (1987)), is considered crucial to induce vortex shedding in the
subcritical regime of $Re$, irrespective of the amplitude and frequency of the
base excitation. Further research is needed to pinpoint the critical values
for the base excitation parameters necessary to generate vortex shedding for
each ($Re,U^{*}$) combination.

##  4 Conclusions

In this paper, we utilized a high-fidelity 3D numerical framework to
investigate the FSI of a flexible cylindrical cantilever beam at subcritical
Reynolds numbers. The coupled dynamics of the beam was studied for two
distinct configurations: (i) tandem cylinder arrangement and (ii) isolated
beam under base excitation. The cantilever beam in the tandem arrangement was
shown to experience sustained oscillations at $Re=40$ for a specific range of
$U^{*}$. Figure-eight-shaped tip motion trajectories were observed during the
beam’s oscillatory response, with a periodic vortex shedding pattern evident
downstream. Compared to an isolated cantilever beam, the beam in the tandem
arrangement was shown to experience sustained oscillations for a narrower
range of $U^{*}$. Additionally, the vibrations were found to be absent across
all studied $U^{*}$ values when the Reynolds number was taken as $Re=30$ or
below. The dynamics of the isolated beam was also investigated under an
externally applied base excitation. Our results indicated that the beam under
base excitation could experience sustained oscillations with various patterns
of tip motion trajectories. The frequency of the base excitation was found to
control the dominant frequency of the wake and impact the critical Reynolds
number for vortex shedding, lowering it to values just above $Re_{c}\approx
5$. The presented analysis is aimed to broaden our understanding of sustained
oscillations in flexible cylindrical cantilevers in the subcritical regime of
$Re$ and has relevance to the development of cantilever flow sensors.

[Acknowledgements]The research was enabled in part through computational
resources and services provided by The Digital Research Alliance of Canada
(https://alliancecan.ca/) and the Advanced Research Computing facility at the
University of British Columbia (https://arc.ubc.ca/).

[Funding]The authors would like to acknowledge the Natural Sciences and
Engineering Research Council of Canada (NSERC) for funding the project.

[Declaration of interests]The authors report no conflict of interest.

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