Predicting fluid-structure interaction with graph neural networks Rui Gao1and Rajeev K. Jaiman1 Department of Mechanical Engineering University of British Columbia

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Predicting ﬂuid-structure interaction with graph neural networks

Rui Gao1and Rajeev K. Jaiman1

Department of Mechanical Engineering, University of British Columbia

(*Electronic mail: rjaiman@mail.ubc.ca)

(Dated: 17 October 2023)

We present a rotation equivariant, quasi-monolithic graph neural network framework for the

reduced-order modeling of ﬂuid-structure interaction systems. With the aid of an arbitrary

Lagrangian-Eulerian formulation, the system states are evolved temporally with two sub-

networks. The movement of the mesh is reduced to the evolution of several coefﬁcients via

complex-valued proper orthogonal decomposition, and the prediction of these coefﬁcients

over time is handled by a single multi-layer perceptron. A ﬁnite element-inspired hypergraph

neural network is employed to predict the evolution of the ﬂuid state based on the state

of the whole system. The structural state is implicitly modeled by the movement of the

mesh on the solid-ﬂuid interface; hence it makes the proposed framework quasi-monolithic.

The effectiveness of the proposed framework is assessed on two prototypical ﬂuid-structure

systems, namely the ﬂow around an elastically-mounted cylinder, and the ﬂow around a

hyperelastic plate attached to a ﬁxed cylinder. The proposed framework tracks the interface

description and provides stable and accurate system state predictions during roll-out for at

least 2000 time steps, and even demonstrates some capability in self-correcting erroneous

predictions. The proposed framework also enables direct calculation of the lift and drag

forces using the predicted ﬂuid and mesh states, in contrast to existing convolution-based

architectures. The proposed reduced-order model via graph neural network has implications

for the development of physics-based digital twins concerning moving boundaries and

ﬂuid-structure interactions.

arXiv:2210.04193v2 [physics.flu-dyn] 16 Oct 2023

I. INTRODUCTION

Accurate and efﬁcient predictions and control of the spatial-temporal dynamics of ﬂuid-structure

systems are essential in various engineering disciplines. The coupling between ﬂuid and structure

can lead to complex dynamical effects such as vortex-induced vibrations and ﬂutter/galloping

Owing to the highly nonlinear and multiscale characteristics of ﬂuid-structure coupling, state-

of-the-art computational ﬂuid dynamics (CFD) and ﬁnite element analysis (FEA) tools based on

the solution of partial differential equations are considered for engineering analysis and design.

While high-ﬁdelity CFD/FEA simulations can provide accurate prediction, the high computational

cost of these simulations limits their applications for downstream design optimization and control

tasks. High-ﬁdelity CFD/FEA simulations can serve the role of a data generator for data-driven

predictions, with efﬁcient design iterations or active control of coupled ﬂuid-structure systems.

This work is motivated by the need to make coupled ﬂuid-structure simulations efﬁcient for the

digital twin technology, whereby multi-query analysis, optimization and control are required.

This limitation of high-ﬁdelity modeling based on partial differential equations has inspired

the development of the so-called data-driven reduced-order modeling (ROM) techniques. By

constructing low-dimensional models, ROM techniques have the potential to address the limitations

of high-ﬁdelity CFD/FEA models for efﬁcient multi-query analysis, optimization and control tasks.

Such data-driven models can be used in an ofﬂine-online manner, whereby the models can be

trained to learn a low-dimensional representation of the system from the high-dimensional physical

data in the ofﬂine stage, and provide efﬁcient predictions during the online stage. Numerous data-

driven techniques have been developed for low-dimensional modeling and predictive ofﬂine-online

applications. Popular methods like proper orthogonal decomposition (POD)

2,3

and dynamic mode

decomposition (DMD)

, as well as their variants (e.g.,

5–7

), are usually based on the projection into

a linear subspace. However, these methods encounter difﬁculty when applied to scenarios with high

Reynolds numbers and convection-dominated problems, whereas one needs a signiﬁcantly larger

number of linear subspaces to achieve a satisfactory approximation.

In the last decade, deep neural networks have been explored as alternatives to the aforementioned

techniques. Autoencoders, as a non-linear extension of POD, have been shown to have signiﬁcant

advantages over POD in the compression of data

. Combined with convolution, which supplies

the inductive bias of locality and translational equivariance, the convolutional encoder-propagator-

decoder architecture has been adopted in many applications, including ﬂow over ﬁxed bodies in

both 2D

9–12

and 3D

scenarios, ﬂuid-structure interactions

14,15

, under-water noise propagation

and many more. Leveraging the computational power of modern graphical processing units (GPU),

the computational times needed for these convolutional neural networks are usually reduced to the

level of milliseconds per step, which are orders of magnitude faster than the traditional full-order

CFD simulations.

While convolutional neural networks have achieved numerous successes, they have inherent

limitations and pose challenges in dealing with ﬂuid-structure boundaries. As convolutions have to

be performed on a uniform Cartesian grid, the resolution of the grid on different regions within

the simulation domain has to maintain similar resolutions. Considering the ﬂow past a bluff body

as an example, the region near the body’s surface is resolved by a grid of the same density as the

far ﬁeld. This means that one must either introduce a signiﬁcant number of grids in a far-ﬁeld

region that does not feature much physics, or tolerate a relatively low resolution near the body’s

surface. The former choice of maintaining a dense grid throughout the bulk and interface requires

substantially more computational resources. As a result, the balance usually aligns with the latter,

inevitably leading to the loss of ﬁne physical details and the difﬁculty in extracting important

physical statistics like lift and drag coefﬁcients. Despite many efforts to mitigate this issue, by

forming a hybrid model with traditional machine learning techniques such as proper orthogonal

decomposition

17,18

, interpolation & projection schemes

, or special mesh design

, a completely

satisfactory solution is not yet available.

Graph neural networks, recently introduced as a geometric deep learning framework, have the

ability to address this difﬁculty, although existing applications are mostly restricted to ﬂuid systems

with ﬁxed boundaries. As its name suggests, graph neural networks operate on graphs, which

can be intuitively converted from any mesh, meaning that one can control the resolution of the

different parts of the simulation domain using the strategies adopted in CFD. As a result, graph

neural networks can maintain a signiﬁcantly better resolution for important regions within the

domain with the same mesh size compared with convolutional neural networks. With this clear

advantage over convolutional neural networks, graph neural networks are recently introduced for

modeling ﬂuid ﬂow. Applications include ﬂow around ﬁxed bodies like cylinder or airfoil

20–22

reacting ﬂows

, ﬂow ﬁeld super-resolution

, ﬂow ﬁeld completion

, etc. Additional techniques

and designs like on-the-ﬂy graph adaptation

, multi-graph with different levels of ﬁneness

21,26

rotational equivariance

, quadrature integration-based loss

, and polynomial processors

are also

combined with the graph neural networks to further boost their performance.

While these works have demonstrated the potential of graph neural networks for various ﬂuid

ﬂow applications, no study exists for the application to fully coupled ﬂuid-structure systems.

Notably, Pfaff et al.

simulated the dynamics of ﬂags in their work, but the graph is limited to the

ﬂag itself, and the ﬂuid ﬂow surrounding the ﬂag is not simulated. Li et al.

simulated the ﬂow

ﬂushing around a free-moving rigid box within a container. However, both the ﬂow and the box are

discretized into large particles, and therefore losing the ﬁne physical details. Most other existing

graph neural network-based works on ﬂuid ﬂow modeling, to the best of the authors’ knowledge,

only focus on applications with ﬁxed, rigid body and/or domain boundaries.

In this work, we aim to ﬁll this gap in the literature by modeling ﬂuid-structure systems via

graph neural network. Adopting the arbitrary Lagrangian-Eulerian formulation, we propose a

rotation-equivariant quasi-monolithic graph neural network framework for modeling ﬂuid-structure

systems. The recently developed

-GNN

is employed to predict the evolution of the ﬂuid state

based on the state of the whole ﬂuid-structure system. Mesh and solid movements are ﬁrst projected

to a lower dimension via complex-valued POD (CPOD), and then the low-order POD coefﬁcients

in polar coordinates are propagated through time by a multi-layer perceptron. The solid-state is

implicitly modeled by the movement of the mesh on the ﬂuid-solid boundary. The framework is

applied to two prototypical ﬂuid-structure problems: an elastically-mounted cylinder in a uniform

ﬂow, and a hyperelastic plate attached to a ﬁxed cylinder in a channel ﬂow. It is demonstrated that

the framework can generate stable and accurate roll-out predictions over at least thousands of time

steps. More importantly, accurate lift and drag force predictions can be directly extracted from the

predicted system states by simply integrating the Cauchy stress tensor at the surface of the moving

solid body, which is difﬁcult for existing convolution-based frameworks.

This article is organized as follows. In Section II, we describe the full-order ﬂuid-structure

interaction system and discuss the individual components for our proposed quasi-monolithic

graph neural network framework: complex-valued proper orthogonal decomposition, multi-layer

perceptrons, and

-GNN. These individual components are then assembled in Section III to form

a novel quasi-monolithic graph neural network framework. Detailed setup of the experiments

on two prototypical ﬂuid-structure interaction problems, namely the elastically-mounted cylinder

system undergoing vortex-induced vibration, as well as the hyperelastic plate attached to a ﬁxed

cylinder immersed in a channel ﬂow, are covered in Section IV. The results of these experiments

are presented and discussed in Section V. We conclude the work in Section VI.

II. METHODOLOGY

Before presenting our quasi-monolithic methodology, we ﬁrst provide a brief review of the full-

order representation of the ﬂuid-structure system. Subsequently, we will introduce the individual

components that will be assembled into a complete framework in Sec. III.

A. Full-order system, discretization in space and time

We brieﬂy summarize the full-order system comprising the Eulerian ﬂuid and the Lagrangian

solid, together with the traction and velocity continuity conditions at the ﬂuid-solid interface. Under

the arbitrary Lagragian-Eulerian (ALE) framework, for the coupled system between isothermal

incompressible viscous ﬂuid ﬂow and an compressible elastic solid body, the system can be modeled

ρf∂uf

∂t+ρf(uf−w)·∇uf=∇·σf+bfon Ωf,(1a)

∇·uf=0 on Ωf,(1b)

ρs∂2φs

∂t2=∇·(σs) + ρsbson Ωs,(1c)

with boundary conditions on the ﬂuid-solid interface

uf(t) = us(t)on Γf s,(2a)

σs·n=σf·non Γf s,(2b)

along with appropriate boundary conditions on other domain boundaries. The superscripts

(·)f

and

(·)s

denote the state parameters for ﬂuid and solid respectively. The ﬂuid velocity, mesh velocity

and body force are denoted by

and

respectively within the ﬂuid domain

Ωf

, while

φs

σs

and

denote the displacement, the velocity, the stress and the body force for the solid body

Ωs

respectively. The solid and ﬂuid density are denoted by

ρs

and

ρf

respectively. At the solid-ﬂuid

interface

Γf s

denotes the unit outward normal vector. Assuming Newtonian ﬂuid, the Cauchy

stress tensor is written as

σf=−pfI+µf(∇uf+ (∇uf)T),(3)

in which pfdenotes the pressure, and µfdenotes the viscosity of the ﬂuid.

The governing Eqs. (1) and (2) can be re-written in the abstract dynamical form

dt =˜

F(q),(4)

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Predictingfluid-structureinteractionwithgraphneuralnetworksRuiGao1andRajeevK.Jaiman1DepartmentofMechanicalEngineering,UniversityofBritishColumbia(*Electronicmail:rjaiman@mail.ubc.ca)(Dated:17October2023)Wepresentarotationequivariant,quasi-monolithicgraphneuralnetworkframeworkforthereduced-ordermodel...

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Predicting fluid-structure interaction with graph neural networks Rui Gao1and Rajeev K. Jaiman1 Department of Mechanical Engineering University of British Columbia

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