# Fluid-structure interaction simulations of wind gusts impacting a hyperbolic paraboloid tensile structure

Guillaume de Nayer, M. Breuer, A.-K. Goldbach

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Guillaume de Nayer, M. Breuer, A.-K. Goldbach. Fluid-structure interaction simulations of wind gusts impacting a hyperbolic paraboloid tensile structure. Physics of Fluids, 2024, 36, pp.105152. 10.1063/5.0232369. hal-04731899

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# Fluid-structure interaction simulations of wind gusts impacting a hyperbolic paraboloid tensile structure

Special Collection: Fluid-Structure Interaction

G. De Nayer ⚫; M. Breuer ⚫; A.-K. Goldbach

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Physics of Fluids 36, 105152 (2024)

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# Fluid-structure interaction simulations of wind gusts impacting a hyperbolic paraboloid tensile structure

Cite as: Phys. Fluids **36**, 105152 (2024); doi: 10.1063/5.0232369

Submitted: 7 August 2024 · Accepted: 17 September 2024 ·

Published Online: 8 October 2024

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G. De Nayer,<fnref n="1" /> ID M. Breuer,<fnref n="1" />,⁴ and A.-K. Goldbach<fnref n="2" />

## AFFILIATIONS

Note: This paper is part of the special topic, Fluids-Structure Interaction.

a) Author to whom correspondence should be addressed: breuer@hsu-hh.de

## ABSTRACT

The paper focuses on fluid-structure interactions (FSI) between a turbulent, gusty fluid flow, and a membrane structure. Lightweight structures are particularly vulnerable to wind gusts and can be completely destroyed by them, making it essential to develop and evaluate numerical simulation methods suited for these types of problems. In this study, a thin-walled membrane in the shape of a hyperbolic paraboloid (*hypar*) is analyzed as a real-scale example. The membrane structure is subjected to discrete wind gusts of varying strength from two different directions. A partitioned FSI approach is employed, utilizing a finite-volume flow solver based on the large-eddy simulation technique and a finite-element solver developed for shell and membrane structures. A recently proposed source-term formulation enables the injection of discrete wind gusts within the fluid domain in front of the structure. In a step-by-step analysis, first the fluid flow around the structure, initially assumed to be rigid, is investigated, including a grid sensitivity analysis. This is followed by examining the two-way coupled FSI system, taking the flexibility of the membrane into account. Finally, the study aims to assess the impact of wind gusts on the resulting deformations and the induced stresses in the tensile material, with a particular focus on the influence of different wind directions.

© 2024 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0232369

## NOMENCLATURE

A_c Frontal area of the roof (m<fnref n="2" />)
A_g Strength of the wind gust (m/s)

B₀ = (O, e₁, e₂, e₃) Cartesian basis (x,y,z)

c_p = (p - p_∞) / (2 * ρ_{air} * u_∞^2)
Pressure coefficient (-)
c_x = F_s / (2 * ρ_{air} * u_∞^2 * A_c)
Streamwise force coefficient (-)
c_y = F_v / (2 * ρ_{air} * u_∞^2 * A_c)
Spanwise force coefficient (-)
c_z = F_z / (2 * ρ_{air} * u_∞^2 * A_c)
Vertical force coefficient (-)

C_s Smagorinsky constant (-)
d_cable Diameter of the edge cables (m)
E_PES/PVC Young's modulus for the warp and fill of the membrane (kN/m<fnref n="2" />)
E_steel Young's modulus of the edge cables (kN/m<fnref n="2" />)

f₁, f₂ First and second dominant eigenfrequencies of the structure (Hz)
Fₓ, Fᵧ, F_z Fluid force in x-, y-, and z-direction (N)
Fᵢ Gust-related spatial or temporal shape function (-)
F₀cable Initial force in the edge cables (N)
h Height of the lower masts (m)
HShypar Height-to-span ratio (-)
L Projected length and width of the squared-based *hypar* structure (m)
Lφᵢ Characteristic length or time scales associated with the wind gust
Ma Mach number (-)
n₀membrane Pre-stress warp and fill of the membrane (kN/m<fnref n="2" />)
n Surface normal vector of the control volume (-)

<footnote n="1">¹Professur für Strömungsmechanik, Helmut-Schmidt-Universität Hamburg, D-22043 Hamburg, Germany</footnote>

<footnote n="2">²Lehrstuhl für Statik, Technische Universität München, D-80290 München, Germany</footnote>
<table><tr><td>p<sub>&infin;</sub></td><td>Pressure of the undisturbed free-stream airflow (Pa)</td></tr><tr><td>Q</td><td>Q-criterion (s<sup>-2</sup>)</td></tr><tr><td>Q* = Q (I<sup>2</sup>/u<sub>&infin;</sub><sup>2</sup>)</td><td>Dimensionless Q-criterion (-)</td></tr><tr><td>Re<sub>L</sub></td><td>Reynolds number (-)</td></tr><tr><td>S<sub>grid</sub>(t)</td><td>Time-dependent surface of a grid cell (m<sup>2</sup>)</td></tr><tr><td>S<sub>gust</sub><sup>ST</sup></td><td>Momentum source term for discrete gust generation (N)</td></tr><tr><td>SP<sub>hypar</sub></td><td>Shape parameter (-)</td></tr><tr><td>t</td><td>Time (s)</td></tr><tr><td>t<sub>membrane</sub></td><td>Thickness of the membrane (m)</td></tr><tr><td>u = u&#773; + u<sub>g</sub></td><td>Total velocity vector (m/s)</td></tr><tr><td>u&#773;</td><td>Flow velocity vector (variable solved for during the simulation) (m/s)</td></tr><tr><td>u<sub>g</sub></td><td>Gust velocity vector (does not include the background velocity) (m/s)</td></tr><tr><td>u<sup>grid</sup></td><td>Grid velocity vector (m/s)</td></tr><tr><td>u, v, w</td><td>Velocity components in the Cartesian basis B<sub>0</sub> (m/s)</td></tr><tr><td>u<sub>&infin;</sub></td><td>Streamwise velocity of the undisturbed free-stream airflow (m/s)</td></tr><tr><td>u<sub>conv</sub></td><td>Convective velocity for convective outlet boundary condition (m/s)</td></tr><tr><td>V<sup>grid</sup>(t)</td><td>Time-dependent volume of a grid cell (m<sup>3</sup>)</td></tr><tr><td>&alpha;</td><td>Exponent of the power law (-)</td></tr><tr><td>&alpha;<sub>r</sub>, &beta;<sub>r</sub></td><td>Mass- and stiffness-proportional Rayleigh damping parameters (Hz, s)</td></tr><tr><td>&delta;</td><td>Height of the boundary layer (m)</td></tr><tr><td>&Delta;&rho;<sub>net</sub></td><td>Difference in the net pressure coefficient between FSI and CFD data (-)</td></tr><tr><td>&Delta;d</td><td>Instantaneous total displacement of the membrane (m)</td></tr><tr><td>&Delta;t</td><td>Time step (s)</td></tr><tr><td>&Delta;t* = &Delta;t (u<sub>&infin;</sub>/L)</td><td>Dimensionless time step (-)</td></tr><tr><td>&Delta;x, &Delta;z</td><td>Instantaneous displacement of the membrane in x- and z-direction (m)</td></tr><tr><td>&zeta;<sub>1</sub>, &zeta;<sub>2</sub></td><td>Damping ratio of the first and second dominant eigenfrequencies (-)</td></tr><tr><td>&theta;</td><td>Wind angle (&deg;)</td></tr><tr><td>&vartheta;<sub>air</sub></td><td>Temperature of the air (&deg;C)</td></tr><tr><td>&mu;<sub>air</sub></td><td>Dynamic viscosity of the air [kg/(m s)]</td></tr><tr><td>&nu;<sub>PES/PVC</sub></td><td>Poisson's ratio of the membrane (-)</td></tr><tr><td>&rho;<sub>PES/PVC</sub></td><td>Density of the membrane (kg/m<sup>3</sup>)</td></tr><tr><td>&rho;<sub>steel</sub></td><td>Density of the edge cables (kg/m<sup>3</sup>)</td></tr><tr><td>&rho;<sub>air</sub></td><td>Air density (kg/m<sup>3</sup>)</td></tr><tr><td>&sigma;<sub>1</sub></td><td>Maximal principal stress (kN/m<sup>2</sup>)</td></tr><tr><td>&sigma;<sub>2</sub></td><td>Minimal principal stress (kN/m<sup>2</sup>)</td></tr><tr><td>&phi;</td><td>Dummy variable equal to x, y, z or t</td></tr><tr><td>&phi;<sub>g</sub></td><td>Gust-related central value associated with a dummy variable &phi;</td></tr><tr><td>&omega;<sub>1</sub>, &omega;<sub>2</sub></td><td>First and second dominant angular eigenfrequencies of the structure (Hz)</td></tr></table>

# I. INTRODUCTION

Tensile membrane structures have gained widespread use globally in the construction of large-span roofs for public facilities and

eco-friendly buildings. The hyperbolic paraboloid, one of the first membranes built by the lightweight pioneer Frei Otto,¹ remains a popular architectural choice. Due to their extreme slenderness and flexibility, membrane structures are particularly sensitive to wind loads. This sensitivity can lead to undesirable aerodynamic fluid-structure interaction (FSI) phenomena, such as dynamic buffeting, galloping, fluttering, and, in the worst case, structural damage and failure. To mitigate these phenomena and reduce their effects, civil engineers need a deeper understanding of how these flexible roofs respond dynamically to the impact of gusty flows. Consequently, experimental and numerical investigations have been conducted by the research community over the past few years.

The following literature survey is restricted to studies on doubly curved hyperbolic, paraboloid (hypar) structures. The hypar, or four-point sail, is one of the “basic shapes” used to build membrane structures.²,³ Other basic shapes are the ridge-valley tent, the arch-supported surface (or saddle), and the point-supported surface (or cone). While this research focuses on flexible membrane structures, rigid roofs can also be hypar-shaped. The survey begins with studies involving rigid models. These structures can either be enclosed by walls or left open, such as in canopy or tent configurations.

Uematsu et al.⁴ and Takeda et al.⁵ conducted experiments on three different rigid hypar open roofs exposed to a turbulent boundary layer. They provided measurements of aerodynamic forces and moments for various wind directions. In a follow-up investigation, Uematsu et al.⁶ added small holes to the models to introduce porosity, which significantly reduced the wind loads. Xin and Jihong⁷ measured mean values and fluctuations of the pressure on a rigid enclosed hypar model, identifying Gaussian and non-Gaussian regions of the wind pressure distributions. Their origin and characteristics were analyzed, depending on the wind direction. Dong and Ye⁸ experimentally investigated the most critical wind direction for a rigid enclosed roof model, confirming that the wind direction along the connection of the high points is the most unfavorable. Liu et al.⁹ validated this issue in their experimental study by applying six boundary layers of different strength and turbulence intensity on a scaled enclosed model.

Colliers et al.¹⁰ installed various rigid models of hypar glass-fiber reinforced structures in a wind tunnel, including both open and enclosed configurations. These roof models were equipped with pressure taps on both sides to determine the distribution of the mean pressure coefficient $c_p$. Complementary to these experiments, RANS simulations were conducted, showing good agreement with the experimental data. Additional simulations were performed for various wind orientations and surface curvatures, providing $c_p$ distributions and flow patterns. De et al.¹¹ used large eddy simulations (LES) to predict the flow around a rigid enclosed hypar roof and validated their computational framework by comparing their results with those of Colliers et al.¹⁰ In a second step, the framework was applied to rigid, open, or enclosed cone structures to investigate the flow response under varying geometric and inflow conditions. Sun et al.¹² experimentally investigated the wind pressure characteristics of open and enclosed rigid ridge-valley structures. They considered different wind directions and shapes, discussing their influence on the peak wind pressure. Similar to the hypar geometry, the mean wind pressure was found to be significantly affected by the wind direction and the height-to-span ratio. Kandel et al.¹³ conducted wind tunnel experiments on both enclosed and open rigid models of a double saddle arch-type structures. They
provided mean and peak wind pressure coefficients for different wind directions, finding that oblique directions were the most critical.

In Rizzo et al.,¹⁴ the mean pressure coefficient and the wind-pressure correlation coefficient were experimentally evaluated on four rectangular low-rise hyperbolic paraboloid rigid *saddle* models. The study considered different model heights and curvatures, as well as three wind incidence angles. These structures had enclosed walls surrounding the roof. This contribution highlighted the dominance of the wind incidence angle and the dimensions of the building. An oblique wind angle of 45° generated a complex flow field, largely determined by the way the flow detaches from different roof edges.

Due to the rigidity of the aforementioned models, deformations were not taken into account. Since deformations and oscillations of the structures affect the flow and thus the loads, numerous investigations are nowadays based on flexible models.

As pioneers in this field, Uematsu and Uchiyama¹⁵ conducted wind tunnel tests on various types of hyperbolic paraboloid flexible membranes. The model was enclosed by walls with an open bottom, allowing the internal pressure to adapt to ambient pressure. The membrane response and flow behavior on the membrane were measured using laser distance sensors and pressure transducers. They discovered that self-excited oscillations occurred within a specific flow velocity range for a certain wind direction. Liu et al.¹⁶ conducted an FSI study on the vibrations of enclosed flexible membrane *saddle* models under wind loads. Theoretical models were derived and validated by measurements. Li et al.¹⁷ aimed to investigate the aerodynamic response of hyperbolic paraboloid flexible membrane structures without enclosing walls under both normal and typhoon wind conditions. Particular attention was paid to ensure that the inflow within the wind tunnel effectively replicated the key wind characteristics. The findings revealed that the typhoon-induced suction significantly amplified the maximum deformations and increased the risk of aeroelastic instability. Continuing on the topic of extreme weather conditions, Liu et al.¹⁸ conducted a comprehensive experimental analysis of the impact of hail on a flexible *hypar* membrane structure. In parallel, computational structural dynamics (CSD) simulations were performed. Various configurations, including different shapes and pre-tension levels, were systematically tested.

In several studies, the flexible thin structures are modeled as a tensioned cable net. For instance, Rizzo et al.¹⁹,²⁰ focused on reduced-scale models of hyperbolic paraboloid real roofs to determine their dynamic deformations under wind load within a wind tunnel. These *saddle* models were constructed as cable nets with a membrane cover. In Rizzo et al.,²¹ a construction method for these aeroelastic models of cable net roofs was proposed and validated. Rigid walls enclosed the doubly curved roof to the ground, and diverse shapes and curvatures were presented. Additionally, finite-element (FE) models at the model scale were built to support the calibration of each prototype. The predicted wind-induced displacements closely matched those estimated on the prototype, with a relative error of approximately 10%. Furthermore, they detected that the structural dynamics are strongly modified under wind load, as it acts like an added mass that changes structural frequencies and damping ratios. Relying solely on simulations, a real-scale *saddle*-form flexible roof was investigated by Vassilopoulou and Gantes²² and Vassilopoulou et al.²³ The dynamic responses of the cable net model, involving either rigid end support or a deformable contour ring, were computed under harmonic or wind

load. The study underscored the critical role of the contour ring in the model, as it introduces novel vibration modes capable of fundamentally reshaping the behavior of the cable net. Since *hypar* structures are typically constructed from a continuous sheet of membrane material possessing shear stiffness, computational models with continuous membrane elements provide a more accurate representation than those reduced to cable elements. The former are considered in the following.

To capture the membrane's response to the flow, FSI simulations using a one-way coupling approach provide a cost-effective method. In Luo and Han,²⁴ the flow around a *hypar* canopy model was simulated for different wind directions, and mean $c_p$ distributions were provided. The time-dependent membrane displacements were obtained by solving a non-linear equation of motion, incorporating a constant mean wind load and a variable turbulent wind load.

To investigate the effect of deformation of open *hypar* structures on their associated wind force coefficients, Takeda et al.⁵,²⁵ conducted pure structural analyses and one-way coupled FSI simulations. The simulations relied on RANS predictions with the $k-\epsilon$ turbulence model for the fluid part. After completing the fluid analysis, the fluid loads were transferred to a finite-element solver. Both solvers were not coupled in time, but applied sequentially to achieve convergence. The results were compared with experimental data, and corrected wind force coefficients were suggested.

Based on the flexible four-point *hypar* membrane geometry, Kawulok et al.²⁶ carried out low-cost, steady-state RANS simulations combined with different turbulence models to predict $c_p$ distributions. Subsequently, the static deformations of the roof were deduced. A comparative analysis with the results of Colliers et al.¹⁰ revealed that, for most parts, the resulting errors remained negligible, except for specific wind orientations.

Yang et al.²⁷ performed LES of the flow field around a rigid enclosed *hypar* roof, considering the experimental setup of Liu et al.⁹ They compared predicted pressure distributions with the measurements for three wind directions. In a subsequent analysis, the *hypar* shape was subjected to forced excitation in its first symmetric mode at varying amplitudes. The study investigated motion-induced forces, wind pressures, and flow structures above the roof.

The last part of this literature survey focuses on numerical predictions of the dynamic response of *hypar* membrane structures to wind flows using a two-way fully coupled FSI simulation framework. Michalski et al.²⁸,²⁹ studied the fluid-structure interaction of a large flexible *cone*-shaped umbrella. Transient, fully coupled FSI simulations were conducted based on FE solvers and the Arbitrary Lagrangian-Eulerian (ALE) method. The turbulent flow was predicted using the LES technique. Special attention was laid on a consistent structural modeling of the pre-stressed membrane and on the approaching atmospheric boundary layer. To facilitate comparison with the experimental data, strain gauges and cameras were installed in and around real umbrellas to measure forces, moments, and deformations of the structures. The study achieved good agreement between simulations and measurements.

To assess their fully coupled FSI methodology, Wüchner et al.³⁰ presented flow and deformation results of a *hypar* four-point tent subjected to wind. Initial pre-stresses were applied, and a separate form-finding procedure was employed to determine the equilibrium shape prior to the FSI simulation.³¹ A gust-like wind load was imposed to
replicate time-dependent conditions found in natural environments. Predictions of the unsteady membrane behavior relied on a finite-volume flow solver based on URANS coupled with a FE structural code. Since the primary objective was to validate the FSI framework, a detailed analysis of the underlying physics of the flow around the *hypar* tent was not provided.

Similarly, Zorrilla *et al.*<sup>32</sup> investigated a flexible *hypar* four-point tent to evaluate their FSI framework. Their FE-based software utilized the variational multiscale method proposed by Hughes<sup>33</sup> as a stabilization technique for turbulent flows. The methodology relied on an embedded mesh technique instead of the body-fitted ALE method used in Wüchner *et al.*,<sup>30</sup> facilitating easier solutions for problems involving large structural deformations. However, the main drawback of the embedded method lies in the errors appearing close to the structure at the FSI interface.

In Valdés-Vázquez *et al.*,<sup>34</sup> a realistic double *hypar* roof model was exposed to winds of varying velocities. After the form-finding procedure, coupled unsteady FSI simulations employing the ALE technique and a FE framework were conducted. The predictions were compared with associated static results. Significant deviations between the results of both approaches were observed, indicating that FSI phenomena around this kind of hyperbolic paraboloid membrane structures cannot be accurately captured by a static approximation.

Based on the literature survey, it is evident that investigations accounting for the flexibility of the membrane (I) are essential to accurately capture all relevant FSI phenomena occurring at such *hypar* structures. Moreover, to predict the flow field precisely, modern eddy-resolving schemes (II), such as the LES technique, must be used preferentially. Since the deformed shape of the tent significantly influences the surrounding flow, and the flow affects the structural deflections, a two-way FSI coupling (III) is required. Furthermore, previous studies have not yet addressed the simulation of turbulent flows involving discrete wind gusts (IV) interacting with *hypar* structures. Therefore, the current contribution aims to close a gap in the literature taking all four challenges (I–IV) simultaneously into account. It focuses on advanced coupled simulations of the turbulent gusty flow around the four-point, flexible, membrane, tent model proposed in De Smedt *et al.*<sup>35</sup> during the fourth TensiNet Round Robin Exercise (https://www.tensinet.com: The TensiNet association supports the efforts toward standardizing tensile membrane structures in Europe by defining Round Robin exercises<sup>36,37</sup>). The main objectives of the current study are as follows:

1. To prove the feasibility of wall-resolved fully coupled FSI simulations involving a turbulent flow at a high Reynolds number (Re = 4 × 10<sup>6</sup>).

2. To enhance understanding of the flow field around both rigid and deformable tents under various wind directions.

3. To investigate the impact of wind gusts on both rigid and deformable *hypar* structures, assessing the effect of gusts, which approach from different directions.

4. To determine the effect of the membrane's flexibility on the flow around the *hypar*.

The present contribution is structured as follows: Sec. II introduces the targeted case study involving a *hypar* four-point flexible tent exposed to a gusty boundary layer. This section defines the geometry and the materials of the structure, describes the turbulent boundary layer approaching the tent, and outlines the characteristics of the

discrete wind gusts. Section III provides a detailed overview of the entire numerical simulation framework. It begins with an explanation of the numerical methods employed, followed by a description of the computational setup used for the simulations. Section IV presents the numerical findings, including pure CFD results on the rigid tent model and FSI predictions of the flexible membrane counterpart. For the sake of brevity, the investigations regarding the quality assurance of the fluid and structure models are provided in Appendixes A and B.

## II. FOUR-POINT FLEXIBLE TENT IN CUSTY BOUNDARY LAYER

### A. Geometry and material

The subject of investigation in the latest Round Robin<sup>35</sup> Exercise 4, proposed by the TensiNet association, is a full-scale membrane with four steel edge cables, supported by rigid steel masts at its four corners Ⓐ, Ⓑ, Ⓒ, and Ⓓ, as depicted in Figs. 1(a) and 1(b). The Cartesian basis $B_0 = (O, e_1, e_2, e_3)$ used throughout this study is centered at the midpoint of the *hypar* structure, as sketched in Fig. 1(c). The membrane has a square base with dimensions $L × L = 6 × 6$ m and a thickness of $t_{membrane} = 10^{-3}$ m. The heights of the lower and higher masts are $h = 2$ m and $2h = 4$ m, respectively. Due to its symmetric design, the considered *hypar* tent is characterized by the following shape parameter (SP) (the shape parameter SP is defined as the sag-to-span ratio of the principal curve between the two high and the two low corners, respectively) and height-to-span ratio (HS) (the height-to-span ratio HS is defined as the ratio of the height of the low corners to the span between the two high and the two low corners, respectively):

$$SP_{hypar} = \frac{h}{\sqrt{2} L} \approx 0.236; \quad HS_{hypar} = \frac{h}{\sqrt{2} L} \approx 0.236. \quad (1)$$

The textile material selected for the roof by the TensiNet association is composed of polyester fibers with a polyvinyl chloride (PES/PVC) coating. Both the surface and the bottom wall, where the structure is anchored, are assumed to be ideally smooth. The thin-walled membrane that forms the canopy is uniformly pre-stressed with a membrane stress $n_0^{membrane} = 4 × 10^3$ kN/m<sup>2</sup>, which is considered homogeneous and isotropic. The resulting pre-stress value of $n_0^{membrane} × t_{membrane} = 4$ kN/m was suggested by the Round Robin<sup>35</sup> Exercise 4 and lies well within common design rules of structural membranes, which state that it should be set between 5% and 10% of the respective tensile strength (86 kN/m for the reference material of RR 4), but not above 5 kN/m to ensure constructability (see, e.g., Wagner<sup>38</sup>).

The surrounding cables, with a diameter $d_{cable}$, are made of steel and are initially tensioned to the force $F_0^{cable}$, resulting in a cable prestress of $n_0^{cable}$. All geometric and material parameters are provided in Table I. To conclude, the flexibility of the support structure is taken into account with the exception of the steel masts, which are modeled as rigid for simplification, following the approach in Colliers.<sup>2</sup>

### B. Turbulent boundary layer

Tensile structures like the proposed *hypar* are commonly used as free-standing outdoor roofs providing protection from heat and sun. To model the wind interaction around such a structure, it is placed within a smooth, thick turbulent boundary layer, which can be described by a power law with an exponent $\alpha = 1/7$, as proven by
<figure><img src="image_11.png" /><figcaption>FIG. 1. Four-point hypar membrane structure.</figcaption></figure>

<table><caption>TABLE I. Geometrical and material parameters of the four-point hypar structure.</caption><thead><tr><th>Dimensions</th><th>Membrane material</th><th>Cable material</th></tr></thead><tbody><tr><td>L = 6 m</td><td>ρ<sub>PES/PVC</sub> = 1200 kg/m<sup>3</sup></td><td>ρ<sub>steel</sub> = 8000 kg/m<sup>3</sup></td></tr><tr><td>t<sub>membrane</sub> = 1 mm</td><td>E<sub>PES/PVC</sub> = 6 × 10<sup>5</sup> kN/m<sup>2</sup></td><td>E<sub>steel</sub> = 2.05 × 10<sup>8</sup> kN/m<sup>2</sup></td></tr><tr><td>d<sub>cable</sub> = 12 mm</td><td>ν<sub>PES/PVC</sub> = 0.4</td><td></td></tr><tr><td>h = 2 m</td><td>n<sub>0</sub><sup>membrane</sup> = 4 × 10<sup>3</sup> kN/m<sup>2</sup></td><td>n<sub>0</sub><sup>cable</sup> = 2.652 × 10<sup>5</sup> kN/m<sup>2</sup></td></tr><tr><td>A<sub>c</sub> = 3.81 m<sup>2</sup></td><td></td><td></td></tr></tbody></table>

Counihan.³⁹ The thickness $\delta$ of the incoming boundary layer at the inlet of the computational domain (outlined in Sec. III B) corresponds to the height of the taller mast, i.e., $2h = 4$ m [see Fig. 1(c)]. The

undisturbed free-stream flow above the boundary layer has a mean velocity in the x-direction of $u_\infty = 10$ m/s. In this configuration, the approaching velocity at the height of the midpoint of the hypar is
<table><caption>TABLE II. Fluid flow parameters for the four-point *hypar* structure case.</caption><thead><tr><th>Turbulent boundary layer</th><th>Discrete gust</th></tr></thead><tbody><tr><td>α = 1/7</td><td>Strength: near gale to storm</td></tr><tr><td>δ = 2 h = 4 m</td><td>Height: middle of the <i>hypar</i></td></tr><tr><td>u<sub>∞</sub> = 10 m/s</td><td>u<sup>max</sup> ∈ [15, 25 m/s]</td></tr><tr><td>ρ<sub>air</sub> = 1.225 kg/m<sup>3</sup></td><td>...</td></tr><tr><td>μ<sub>air</sub> = 18.27 × 10<sup>-6</sup> kg/(m s)</td><td>...</td></tr><tr><td>θ = {0°, 45°}</td><td>θ = {0°, 45°}</td></tr><tr><td>Re<sub>L</sub> = 4 × 10<sup>6</sup></td><td>...</td></tr></tbody></table>

approximately 9.6 m/s. Under standard atmospheric conditions and at a temperature of $\vartheta_{air} = 20°$ C, the Reynolds number associated with the airflow, based on the roof's width, is $Re_L = \rho_{air} L u_{\infty} / \mu_{air} \approx 4 \times 10^6$. The Reynolds number defined using the height from the ground to the low corner of the structure is about $Re_h \approx 1.34 \times 10^6$. With a Mach number below 0.03, the flow can be considered incompressible. Moreover, the fluid is assumed to be isothermal. In this study, the inflow is modeled without considering turbulence intensity, resulting in a stationary inlet condition. Notably, the pioneering work of Uematsu and Uchiyama¹⁵ showed that a turbulence intensity of 6% in the boundary layer leads to twice as large rms-values of the displacements compared to a smooth flow case. Consequently, turbulence has an effect on the wind-induced structural response, which can be attributed to the corresponding increase in the rms-values of the pressure coefficient on the roof. On the other hand, the study also showed that the mean deflections scale with the square of the incoming flow velocity. For the gust cases analyzed in this study, the incoming velocity rises up to a factor of 2.5 (see Table II). Consequently, the turbulent fluctuations have a much smaller influence on the deformation of the structure than the velocity variations induced by the gust. Similarly, a preliminary study by De Nayer and Breuer¹⁶ demonstrated that the turbulence intensity level has a minor impact on the instantaneous deflections of the membrane compared to the impact of wind gusts. Therefore, turbulent fluctuations in the inflow are omitted here for the purpose of simplification.

The main wind direction is defined by the angle $\theta$ between the free-stream velocity vector and the straight line connecting the two low corners of the membrane, i.e., points Ⓐ and Ⓑ. In others words, a 0° wind direction aligns with points Ⓐ and Ⓑ, while a 90° wind direction aligns with points Ⓐ and Ⓑ. Previous studies by Dong and Ye⁸ and Liu et al.⁹ indicated that the 90° case is the most critical in terms of structural loads. However, their analysis focused on a *hypar* structure enclosed by walls. In the present study, an open canopy is investigated. As a result, the 0° and 90° configurations are antisymmetric if the ground wall and the associated boundary layer are not considered. Thus, the loads obtained for 0° and 90° wind directions are nearly antisymmetric, as demonstrated by De Nayer and Breuer.¹¹ In this study, the 45° case is initially analyzed due to previous findings suggesting that the other cases generate higher loads and potential for significant membrane deformations at least at lower pre-stresses. Subsequently, the 0° scenario is analyzed as the second scenario. The 90° case is excluded from further analysis due to its antisymmetry with the 0° configuration. Table II summarizes the relevant parameters of the airflow.

### C. Discrete wind gusts

Considering the significant impact of wind gusts on the stability and lifespan of membrane structures, their inclusion in the design process and load calculations is crucial. In this study, the modeled gusts, as described in Sec. III A, are considered as deterministic discrete gusts. This approach offers the advantage of full control over their shape, amplitude, and position during the injection phase [see Fig. 1(c)], thereby influencing their effect on the structure. Gusts of varying intensities, ranging from *near gale* to *storm strength* (see Ref. 42), are generated and injected directly in front of the *hypar*, targeting the middle of the membrane to maximize their impact. Furthermore, these gusts are aligned with the main wind direction, specifically in the 0° or 45° wind direction scenarios.

## III. SIMULATION FRAMEWORK

To accurately predict the highly dynamic behavior of a gusty boundary layer flow around the four-point flexible membrane, a fluid-structure interaction simulation framework is applied. This framework combines a high-fidelity finite-volume flow solver, employing the large-eddy simulation technique, with a finite-element structural solver.

### A. Numerical methods

The FSI problem is addressed using a partitioned approach,⁴³ which employs the computational structural dynamics solver Carat++⁴⁴ and the fluid solver FASTEST-3D.⁴⁵ The coupling and surface mapping between the two distinct surface discretizations at the FSI interface are handled using the open-source software EMPIRE.⁴⁶ All data exchange relies on the message-passing interface (MPI) communication.

#### 1. Solver for the structure

Carat++ is a finite-element and isogeometric structural solver designed to predict the geometrically non-linear behavior of thin-walled structures such as shells and membranes.⁴⁷,⁴⁸ The momentum equation, within a Lagrangian frame of reference, is applied to describe the dynamic equilibrium of the structure.

The material modeling assumes a linear-elastic and isotropic behavior, based on the St. Venant–Kirchhoff law, which links the second Piola–Kirchhoff stress tensor with the Green–Lagrange strain tensor. An investigation³ into parameter variations and their impact on the structural response concluded that the level of pre-stress and loading intensity are dominant factors, and that simplifying to isotropic material behavior is acceptable. Additionally, the membrane is analyzed in its fully pre-stressed state, where large deflections but low strains are anticipated, indicating that the linear elastic material model is sufficient. Since the internal structural damping cannot be neglected in this type of lightweight membrane, the well-established Rayleigh damping method is used. According to Chowdhury and Dasgupta,⁴⁹ the constant mass- and stiffness-proportional parameters are computed as follows:

$$ \alpha_r = \frac{2\omega_1 \omega_2}{\omega_2^2 - \omega_1^2} (\omega_2 \zeta_1 - \omega_1 \zeta_2) = 9.0 \text{ Hz}, \quad (2a) $$

$$ \beta_r = \frac{2\omega_1 \omega_2}{\omega_2^2 - \omega_1^2} \left( \frac{\zeta_2}{\omega_1} - \frac{\zeta_1}{\omega_2} \right) = 1 \times 10^{-3} \text{ s}, \quad (2b) $$
based on the first two significant angular eigenfrequencies of the structure $\omega_{1,2} = 2\pi f_{1,2}$ with $f_1 = 14.25$ Hz and $f_2 = 16.00$ Hz (see the Appendix A), and their corresponding damping ratios $\zeta_1$ and $\zeta_2$. Both $\zeta_1$ and $\zeta_2$ are set to 0.095, as this damping ratio is typical for pre-stressed real-scale structures, as investigated by Wagner and Heinlein.⁵⁰

Temporal discretization is carried out using the standard second-order non-linear Newmark scheme.

## 2. Solver for the fluid flow including wind gust modeling

FASTEST-3D, the fluid solver, predicts complex turbulent flows using the large-eddy simulation technique. It solves the filtered Navier-Stokes equations in an Arbitrary Lagrangian-Eulerian formulation. These equations are discretized using the finite-volume technique on a curvilinear, block-structured, body-fitted grid with a collocated variable arrangement. Surface and volume integrals are approximated using the midpoint rule. Most flow variables are linearly interpolated to the cell faces, resulting in a second-order accurate central scheme. To avoid wiggles in the simulation data, the convective fluxes are approximated using flux blending,⁵¹ comprising 5% of a first-order accurate upwind scheme and 95% of a second-order accurate central scheme. To prevent unwanted oscillations on non-staggered grids, the momentum interpolation technique of Rhie and Chow⁵² is applied. To couple the pressure and velocity fields, a semi-implicit predictor-corrector scheme is used to solve the incompressible Navier-Stokes equations. First, the momentum equations are time-marched using a low-storage multi-stage Runge-Kutta method to derive an intermediate velocity. Then, the corrector step ensures mass conservation in the form of a divergence-free velocity field by solving a Poisson equation for the pressure correction. The prediction of the inherent flow turbulence is achieved by the LES technique, where the large scales of turbulence are resolved and the small scales are modeled through a subgrid-scale (SGS) model.⁵³ In this study, the SGS model incorporates the classical Smagorinsky⁵⁴ model with its standard parameter $C_s = 0.1$ (Ref. 27), along with the van-Driest damping function for the near-wall region.

To mimic the extreme wind events within the flow field, a deterministic method is employed to generate discrete gusts. This modeling approach is straightforward and does not rely on a turbulence-based wind description. The shape of the gust is defined in the Cartesian basis $B_0$, as depicted in Fig. 1(c), with the velocity field represented by the mathematical functions ($F_1, F_2, F_3$) in the three spatial directions and $F_t$ in time (see De Nayer et al.⁵⁵ for details about the theory). The gust is injected in the streamwise direction to maximize its impact on the flexible membrane. Consequently, the gust velocity vector $\mathbf{u}_g$ reads

$$ \mathbf{u}_g|_{B_0}(t, x, y, z) = A_g F_t(t) F_1(x) F_2(y) F_3(z) \mathbf{e}_1. \quad (3) $$

According to Eq. (3), the factor $A_g$ allows to control the strength of the wind gust and is referred to as the gust amplitude. The spatial and temporal functions follow the extreme coherent gust (ECG) shape (see Ref. 56). In De Nayer et al.,⁵⁷ the classic mathematical formulation for ECG was adapted to introduce the central value $\phi_g$,

$$ F_i(\phi) = \begin{cases} \frac{1}{2} \left( 1 + \cos \left( \frac{2\pi(\phi - \phi_g)}{L_g^\phi} \right) \right) & \text{for } (\phi - \phi_g) \in \left[ -\frac{L_g^\phi}{2}, \frac{L_g^\phi}{2} \right] \\ 0 & \text{else.} \end{cases} \quad (4) $$

The variable $\phi$ represents either the spatial coordinates $\phi = \{x, y, z\}$ or the time $t$, and the subscript $i$ takes values 1, 2, and 3 corresponding to $x, y, z$, or $t$, respectively. $L_g^\phi$ denotes the characteristic length and time scales associated with the wind gust. The temporal shape function $F_t(t)$ and the spatial shape in the gust direction $F_1(x)$ are interconnected, both following the same analytical function. To introduce the gust velocity vector $\mathbf{u}_g$ into the background flow and thus generate a gust within the computational domain, the source-term formulation originally proposed by De Nayer and Breuer⁵⁸ is employed. According to this approach, only the filtered momentum equation of the Navier-Stokes equations is modified, without adding mass to the flow. For a finite-volume scheme, the integral form of the governing equations is used. By applying the ALE formulation of the Navier-Stokes equations to simulate FSI problems, the governing equations, including the momentum source term (ST) for the discrete gust, read

$$ \begin{aligned} \frac{d}{dt} \int_{V_{\text{grid}}} \rho_{\text{air}} dV + \int_{S_{\text{grid}}} \rho_{\text{air}} (\bar{\mathbf{u}} - \mathbf{u}_{\text{grid}}) \cdot \mathbf{n} \, dS &= 0, \\ \frac{d}{dt} \int_{V_{\text{grid}}} \rho_{\text{air}} \bar{\mathbf{u}} \, dV + \int_{S_{\text{grid}}} \rho_{\text{air}} (\bar{\mathbf{u}} - \mathbf{u}_{\text{grid}}) \bar{\mathbf{u}} \cdot \mathbf{n} \, dS \\ &= S_{\text{gust}}^{\text{ST}} - \int_{S_{\text{grid}}} \bar{\mathbf{\tau}} \cdot \mathbf{n} \, dS - \int_{S_{\text{grid}}} p \mathbf{n} \, dS, \end{aligned} $$

where $\bar{\mathbf{\tau}}$ represents the stress tensor based on the flow velocity $\bar{\mathbf{u}}$, while $\mathbf{n}$ denotes the surface normal vector of the control volume (CV) under consideration. The volume and surface integrals are taken over the current time-dependent volume $V_{\text{grid}}(t)$ and the surface $S_{\text{grid}}(t)$ of a grid cell. $\mathbf{u}_{\text{grid}}$ is the grid velocity of the moving surface of the CV. The momentum source term $S_{\text{gust}}^{\text{ST}}$ describes the gust in terms of the total velocity $\mathbf{u} = \bar{\mathbf{u}} + \mathbf{u}_g$ and is formulated as follows:

$$ S_{\text{gust}}^{\text{ST}} = \int_{V_{\text{grid}}} (\mathbf{u} \cdot \nabla) (\rho \mathbf{u}) \, dV, $$

where $\bar{\mathbf{u}}$ is the primary variable solved for during the simulation. $\mathbf{u}_g$ describes the imposed analytical gust velocity defined by Eq. (3). This method offers the advantage that, following the solution of the Poisson equation for the pressure correction, the flow remains divergence-free, ensuring mass conservation. For further details on the theoretical background, refer to De Nayer and Breuer.⁵⁸ Further applications can be found in De Nayer and Breuer,⁴⁰ De Nayer et al.,⁵⁵ and De Nayer and Breuer.⁵⁹

## 3. Coupling scheme

Simulations of lightweight structures exposed to highly transient turbulent flows require a two-way FSI coupling.⁴³ The simulation methodology offers both loose and strong coupling algorithms between the solvers. Initially, due to the significant disparity in density between air (the fluid medium) and the membrane material, it was assumed that the loose coupling procedure, as experienced in De Nayer et al.,⁵⁵ would provide sufficient stability of the coupling. However, it became evident that, particularly under higher wind gust strengths, this approach was not always adequate. In a loose coupling scheme, the grid is adapted to the deformations of the FSI interface from one time step to next. In scenarios without any gusts, these deformations are small, and the
deformed grid closely resembles the original design intended for the current Reynolds number. Consequently, the CFD solver, using a semi-implicit predictor-corrector time discretization, converges quickly and without issue. However, during the interaction between the gust and the flexible structure, the grid undergoes more significant deformations within a short time interval, while simultaneously local velocities increase sharply due to the gust impact. These combined factors can lead to divergence of the CFD solver in certain gust impact scenario. To address this challenge, a strong two-way coupling scheme is applied. This approach ensures the convergence of the CFD solver within each time step, thereby mitigating the divergence issues encountered with gust interactions.

The fluid loads, comprising pressure and shear stresses, are predicted by FASTEST-3D and exchanged either once per sub-iteration (strong coupling) or once per time step (loose coupling) with Carat++, using the coupling tool EMPIRE.⁴⁶ In response, the displacements of the nodes at the FSI interface, computed by Carat++, are transmitted back to FASTEST-3D within the same sub-iteration or time step. EMPIRE facilitates the mapping between the non-matching CFD and CSD grids and manages the data exchange between its clients. Details about the standard mortar mapping technique currently applied can be found in Wang et al.⁶⁰ and Apostolatos et al.⁶¹ This mapping technique ensures consistency of the displacements and the fluid loads along the FSI interface. The relative error in mapped displacements at the FSI interface is minimal (under 0.01%) except at the curved sides of the *hypar*, where it increases slightly but remains below 0.2%. This slight increase is due to the fact that the fluid's FSI interface does not perfectly match the structural counterpart. The relative error in the total fluid force computed during the transfer remains below 0.15%.

The structural deflections, mapped at the FSI interface, are propagated within the fluid domain using a fast procedure that relies on both linear and transfinite interpolations (TFI). For cases involving larger displacements, a hybrid grid adaptation method, originally developed for LES by Sen et al.,⁶² becomes preferable. This hybrid technique employs the inverse distance weighting (IDW) method at block boundaries of the block-structured grid, while the deformation within each block is based on TFI. Although this approach maintains the high grid quality during its adaptation, it is still computationally intensive. To optimize the balance between grid quality and CPU consumption, a mixed approach is implemented, applying the hybrid method only every 100 time steps. However, in the current study, the fast TFI procedure has proven adequate for achieving satisfactory results.

This two-way fully coupled FSI simulation framework has been validated, among others, based on various experimental FSI benchmarks.⁶³⁻⁶⁷ These validations includes complementary experimental and numerical studies, enabling detailed comparisons between measured and predicted data. Overall, a good agreement between measurements and predictions was observed, validating the two-way coupled FSI simulation framework. Additionally, its performance for gusty flow conditions has been extensively tested, particularly for a wall-mounted, synclastic, air-inflated, flexible hemisphere exposed to various wind events.⁴⁰,⁵⁵,⁵⁸,⁵⁹

## B. Computational setup

### 1. Grids

For the structure, which includes the membrane and the four edge cables, finite-element grids of varying resolution (coarse, medium, fine, and finest) have been generated. A grid sensitivity study in Appendix A, based on both static and transient test cases, indicates that a grid consisting of 40 × 40 × 2 = 3200 membrane *Constant Strain Triangle* (CST) elements and 160 cable elements (40 per edge) ensures reliable results for wind speeds up to 25 m/s. This specific grid configuration, depicted in Fig. 2, is referred to as the *fine* grid in subsequent discussions. In this fine grid configuration, all nodes are free to move, except for the four nodes attached to the masts, which are restricted from translational movement. The membrane is exposed to the airflow on both its lower and upper surfaces, as accounted for by the fluid solver, and both sides constitute a part of the FSI interface.

The computational fluid domain is a rectangular region with dimensions of 5.5 L in length, 5 L in width, and 2.33 L in height [see Fig. 3(a)]. The leading edge of the structure is positioned 1.5 L from the inlet plane, while the trailing edge is 3 L from the outlet. The origin of the coordinate system is located at the center of the tensile roof [see Fig. 3(b)]. A block-structured fine grid, referred to the *reference grid*, has been generated, containing approximately 138.24 × 10⁶ control volumes (CVs). The grid serves as a reference for the simulations. The FSI interface on the fluid side, depicted in Fig. 3(b), includes 8 × 10⁴ CVs, summing up both the upper and lower membrane surfaces. To comply with the recommendations of Piomelli and Chasnov⁶⁸ for wall-resolved LES around the structure, the first cell center is located at a wall-normal distance of 2.1 × 10⁻⁵ L from the membrane, ensuring dimensionless values (e.g., z⁺) remain below 2. Due to the very fine grid close to the corners [see Figs. 3(c) and 3(d)], the aspect ratios of the CVs in all directions are between 1 and 10 and their geometric stretching ratios are

<figure><img src="image_12.png" /><figcaption>FIG. 2. FEM grid used for the structure simulation (membrane elements in black and cable elements in orange).</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>FIG. 3. Finite-volume reference grid used for the LES predictions with a 45° wind direction.</figcaption></figure>

kept below 1.07. The CVs located near the middle of the membrane edges exhibit aspect ratios below 150 in the wall-parallel direction and below 500 in the wall-normal direction. Recognizing that the boundary layer on the ground wall has less impact on the FSI behavior of the membrane, the grid resolution in that region is coarsened accordingly. The first cell center is located at a wall-normal distance of $2.1 \times 10^{-4} L$ from the ground, resulting in $z^+$ values below 15.

A second CFD grid, termed the *default grid*, is derived by coarsening the previous fine grid. The number of CVs is halved in all three directions for each block, resulting in $17.28 \times 10^6$ control volumes. This grid configuration achieves non-dimensional wall-normal distance values below 5 on the membrane, except in small areas near the corners, and below 30 on the ground wall. The FSI interface in this grid setup is reduced to $2 \times 10^4$ CVs from both sides of the membrane.

The boundary conditions are specified as follows: both the ground wall and the membrane are considered as no-slip walls. Symmetry boundary conditions are applied on the sides and the top of the rectangular computational domain. Preliminary investigations using periodic conditions on the sides showed no discernible differences in the results. Therefore, symmetry boundary conditions were chosen for the final setup. At the outlet, where large turbulent vortices need to pass the outlet without perceptible disturbances, a convective boundary condition is employed with a constant convective velocity $u_{\text{conv}} = 0.8 u_{\infty} = 8$ m/s.

## 2. Time steps

For the CFD predictions without any wind gusts, a dimensionless time step size of $\Delta t^* = \Delta t u_{\infty} / L = 13 \times 10^{-6}$ is used for both the *default* and *reference* grids. This corresponds to a Courant–Friedrich–Lewy number of approximately 0.08 and 0.2, calculated using the local velocity and the length of the cell for the *default* and *reference* grids,

respectively. First CFD simulations involving wind gusts, a smaller dimensionless time step of $\Delta t^* = 3.34 \times 10^{-6}$ is necessary to capture the rapid changes in flow and ensure stability in the simulations. In coupled FSI simulations involving wind gusts, both solvers use the same dimensionless time step, $\Delta t^* = 3.34 \times 10^{-6}$, to guarantee a stable coupling scheme using the *default* CFD grid.

## 3. Gust parameters

In the course of this study, gusts with varying amplitudes ($0.5 \le A_g/u_{\infty} \le 1.5$), corresponding to wind conditions ranging from *near gale* to *storm*, are taken into consideration for the CFD and FSI simulations. Since these gusts are injected in the same direction as the wind, the streamwise velocity in the gusty region raises significantly reaching maximal values of $u^{\text{max}} = u_{\infty} + A_g$. The size of each gust is kept constant to reduce the number of simulations required. Each gust has a length equal to the length of the structure $L_g^x/L = 1$, while its width and height are smaller, set at $L_g^y/L = 0.5$ and $L_g^z/L = 0.5$, respectively, ensuring that the gust stays sufficiently distant from the ground wall. The precise position, where the gust impacts the structure, significantly influences its structural response. Thus, the position vector of the gust during its injection ($x_g^0, y_g^0, z_g^0$) is another crucial input parameter. In this study, $x_g^0/L$ is set to -1.5, allowing the gust to be fully developed before reaching the flexible structure. Both the spanwise position $y_g^0$ and the altitude $z_g^0$ are set to 0, ensuring that the gust impacts the middle of the structure, maximizing its effect on the structural response.

## 4. High-performance computing (HPC) aspects

The fluid simulation, parallelized using domain decomposition and MPI communications, is running on 1008 cores (14 nodes) for the
fine reference grid and on 180 cores (6 nodes) for the default grid. The structural simulation is less computationally demanding and runs on a single core. In scenarios involving FSI predictions, the coupling tool EMPIRE utilizes an additional core for managing the interaction between the fluid and structural solvers.

The CFD simulations for both the rigid model and the coupled FSI simulations for the flexible membrane were conducted on HSUper, the central computer cluster of the Helmut-Schmidt-University Hamburg. For the CFD grid sensitivity study, presented in Appendix B, the CFD predictions at a wind angle of 45° carried out on both the reference and the default grid for a long dimensionless time interval of 50 took about 1500 and 550 wall-clock hours, respectively. A typical coupled simulation on the default grid without gust combined with the loose coupling scheme ran about 150 h for a dimensionless time interval of 4. In the case of wind gusts impacting the flexible structure, the strong coupling scheme is applied and 1200 h are required for the same dimensionless time interval. Typically, one FSI simulation generates about 500 GB of data for post-processing, highlighting the significant demands in terms of CPU-time and storage. Overall, these simulations demonstrate the substantial computational resources required to accurately model and analyze complex interactions between fluid dynamics and structural responses, especially under dynamic wind conditions and FSI scenarios.

## IV. RESULTS

The outcome of the simulations is presented systematically in three primary steps. In the first step (Sec. IV A), the flow around the structure is analyzed in detail for the case with the smooth, thick boundary layer without any gust. The goal is to understand the characteristic flow phenomena occurring at the membrane and in its wake. For this purpose, the flow around the rigid model is considered. These simulations are conducted using the reference grid, ensuring a detailed and high-resolution investigation of the flow. In addition to instantaneous flow features and force coefficients, time-averaged quantities are also taken into account. The time-averaging is performed over a time interval of 50 dimensionless time units (time normalized by $u_\infty$ and L), ensuring converged statistics.

In the second step (Sec. IV B), the impact of gusts is taken into account, analyzing the effect on the flow field and the resulting forces. These simulations provide insights into how gusts alter the airflow behavior around the structure, including transient flow characteristics and dynamic loads.

Finally, in Sec. IV C, the flexibility of the membrane and the induced deformations and stresses are examined. For these studies, the default grid is applied for the fluid flow, while the fine grid is used for the structure. This approach is justified by the grid sensitivity study (Appendix B) and the significant resources required by FSI simulations.

This multi-step approach—starting with the flow analysis around the rigid structure, progressing to gust effects, and finally examining flexible membrane deformations—offers a comprehensive understanding of the FSI problem.

### A. Flow around the rigid model without gust

#### 1. Wind direction of 45°

Figure 4(a) depicts a three-dimensional snapshot of the instantaneous flow field around the rigid *hypar* at a 45° wind direction. To

visualize the characteristic vortical structures of the flow, iso-surfaces of the dimensionless Q-criterion are shown, normalized by $u_\infty^2/L^2$. The main flow features visible are the tip vortices that develop at all four edges. To understand this feature, the situation at the corner Ⓐ [top left, see Fig. 1(b)] can serve as an example. The thin structure in the vicinity of the corner acts like an inclined plate or airfoil on the flow. On the lower side, the pressure is high in this region, while on the upper side, the flow separates at the sharp edge of the membrane, generating a recirculation with a low pressure zone. This pressure difference induces the first tip vortex at corner Ⓐ, which rotates clockwise in the downstream viewing direction. At corner Ⓐ, the opposite occurs: the high-pressure region is on the upper side, while the flow separation with the low-pressure zone appears on the underside. Looking again in the downstream direction, the resulting tip vortex also rotates clockwise. In this respect, the current flow differs significantly from the flow around a wing, where the two tip vortices rotate in opposite directions, leading to a downwash velocity and the appearance of the induced drag. Nevertheless, the tip vortices are also very important in the current case, as they induce a velocity field as well that influences the pressure distribution and resulting fluid forces acting on the structure itself.

Corresponding tip vortices also form at corners Ⓐ and Ⓑ. Astonishingly, all tip vortices rotate in the same direction. Notably, the tip vortices from corners Ⓐ and Ⓑ are straighter and less disturbed compared to those at Ⓐ and Ⓑ. This can be attributed to the fact that the tip vortices at the rear of the membrane are more affected by the flow recirculation and the wake. As visible in the y-z-plane in Fig. 4(a), these tip vortices also influence the development of the boundary layer along the ground wall. Furthermore, the anti-symmetry of the *hypar* shape at this wind angle, combined with the bottom wall, leads to an asymmetric wake flow.

In addition to the four tip vortices, other dominating flow features are the two flow separation regions at the leading edge. These occur on the upper side for $y/L > 0$ and the counterpart on the lower side for $y/L < 0$. Figures 4(b) and 4(c) illustrate this phenomenon using an iso-contour of the streamwise velocity, where a value of $u/u_\infty = -0.01$ is chosen to mark the region of back flow at the arbitrary instant in time. Additionally, Fig. 4(a) shows the flow structures that develop in the shear layer behind the trailing edge of the membrane. Here, the fluid streams along the lower and the upper side of the membrane meet again, forming a shear layer in the wake. This feature can also be clearly recognized in the time-averaged statistical data discussed in Appendix B, indicated by high values of the normal Reynolds stresses ($u'u', \nabla^2\bar{v}$, $\bar{w}'\bar{w}'$) and the shear component ($\bar{u}'\bar{w}'$).

As mentioned earlier, the discussion of the flow features is based on the predicted results using the reference grid. However, as demonstrated in Appendix B, which includes a grid sensitivity study of the CFD predictions using both the default and the reference grids, the evaluation of the time-averaged flow fields shows only minor deviations between the two solutions. Furthermore, it should be noted that the flow phenomena discussed in this section, based on the instantaneous flow, are also clearly identifiable in the time-averaged flow fields presented in Appendix B.

#### 2. Wind direction of 0°

As indicated above, it is expected that flow cases where the wind direction is aligned with one of the diagonals of the *hypar* structure
<figure><img src="image_14.png" /><figcaption>(a) Key flow characteristics visualized by an iso-surface of the dimensionless Q-criterion ($Q^* = 72$) and the dimensionless streamwise velocity in the crosswise $y-z$-plane at $x/L = 2$.</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>(b) Downward viewing direction: Flow separation on the upper side of the membrane visualized by an iso-surface of the negative streamwise velocity $u/u_\infty = -0.01$. The dimensionless streamwise velocity is depicted in the $x-z$-plane at $y/L = 0.5$.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>(c) Upward viewing direction: Flow separation on the lower side of the membrane visualized by an iso-surface of the negative streamwise velocity $u/u_\infty = -0.01$. The dimensionless streamwise velocity is depicted in the $x-z$-plane at $y/L = 0.5$.</figcaption></figure>

FIG. 4. Instantaneous flow field around the rigid model at a wind direction of 45° (reference grid).

will result in larger forces, and consequently, larger deformations of the structure. As an example, the 0° wind direction along the two lower corners is discussed here. Due to the aforementioned antisymmetry of the structure, the 90° case is not included in this discussion.

A three-dimensional snapshot of the instantaneous flow field around the *hypar* structure for this case is shown in Fig. 5(a). The key flow features are again highlighted by iso-surfaces of the dimensionless Q-criterion. Two large conical vortices, previously observed in the experiments carried out by Xin and Jihong, begin to develop from the

leading corner on the lower side. They are mirrored with respect to the x-axis. Each conical vortex consists of a primary vortex and a secondary vortex. These flow structures can also be seen in the wall streamlines depicted in Fig. 6(b). Starting from corner , the flow separates at the sharp edge of the *hypar* and rolls up over the lower side, forming the large primary vortex. A smaller counter-rotating secondary vortex develops between this main vortex and the rigid *hypar* structure, close to the sharp edge, as highlighted by the streamwise vorticity computed based on the time-averaged flow in Fig. 6(a). This
<figure><img src="image_17.png" /></figure>

(a) Upward viewing direction: Key flow characteristics visualized by an iso-surface of the dimensionless Q-criterion ($Q^* = 72$) and the dimensionless streamwise velocity in the crosswise $y-z$-plane at $x/L = 2$.

<figure><img src="image_18.png" /></figure>

(b) Downward viewing direction: Flow separation on the upper side of the membrane visualized by an iso-surface of the negative streamwise velocity $u/u_\infty = -0.01$. The dimensionless streamwise velocity is depicted in the $y-z$-plane at $x/L = 0.8$.

<figure><img src="image_19.png" /><figcaption>FIG. 5. Instantaneous flow field around the rigid model at a wind direction of 0° (reference grid).</figcaption></figure>

(a) Upward viewing direction: Streamwise vorticity in the cross-flow planes at $x/L = \{-0.25, 0, 0.25\}$.

<figure><img src="image_20.png" /></figure>

(b) Upward viewing direction: Wall streamlines on the lower side of the membrane.

<caption>FIG. 6. Visualization of the conical vortices based on the time-averaged flow field around the rigid model at a wind direction of 0° (reference grid).</caption>

kind of primary and secondary coherent vortex structures is well known from the flow around delta wings at high angles of attack.⁶⁹ The conical vortices continue to grow in the streamwise direction, as visible in the $y-z$-plane in Fig. 5(a). They influence the development of

the boundary layer at the ground wall, as shown in Fig. 5. Another interesting phenomenon inherent to these conical vortices is a shear layer instability described by Gursul et al.⁶⁹ A series of vortical substructures were observed in the outer shear layer that rolls up to form
the primary vortex. Subsequently, they were shed and convected downstream and around the primary vortex. This shear layer instability is also evident in animations of the flow around the *hypar* structure, inducing pressure fluctuations near the structure at a dominant frequency of approximately 7.3 Hz (Strouhal number $St = 4.38$ based on $u_\infty$ and $L$). These fluctuations are also reflected in the integral forces on the structure.

In addition to the conical vortices, a flow separation is observed on the upper side of the membrane close to the trailing corner ⑤. As in the 45° case, an iso-contour of the streamwise velocity with a value of $u/u_\infty = -0.01$ is selected to visualize the separation bubble in Fig. 5(b). Contrary to the 45° case, where the flow separates directly at the front edge on the lower and upper sides and reattaches after a distance of about 0.22 $L$, the separation of the flow at 0° wind direction occurs further downstream and is confined to the upper side, centered around the streamwise axis. Similar to the 45° scenario, a shear layer forms behind the trailing edges and corner ⑤, containing small-scale flow structures, as seen in Fig. 5(a).

In analogy to the previous wind direction, a detailed comparison of the results obtained on the *default* and the *reference* grid was also carried out for the 0° wind direction. Again, minor deviations were observed between the two solutions for the time-averaged flow fields. Therefore, for brevity, these results are not presented here.

### 3. Fluid forces acting on the rigid model

The distribution of the pressure and shear stress on the tent, and the resulting fluid forces are particularly relevant in civil engineering. Figure 7 illustrates the pressure coefficient distribution $c_p$ of the time-averaged flow on the upper (top) and lower (bottom) part of the *hypar* roof predicted for both wind directions. In addition, the resulting pressure coefficient determined by subtracting the lower pressure coefficient from the upper one, denoted *net pressure coefficient*, is depicted. At a wind direction of 45°, the two characteristic low and high pressure regions described in Sec. IV A 1 are clearly visible. In the 0° case, the footprints of the two conical vortices are evident in the pressure distribution. Furthermore, the flow accelerates over the curved upper surface, leading to low pressure values, whereas the opposite takes place on the lower side. Direct comparisons with results from the literature is not possible since the current *hypar* setup (geometry and boundary conditions) has not been previously investigated. However, Colliers² presented time-averaged pressure coefficient distributions obtained by RANS simulations using either the $k-\epsilon$ or the $k-\omega$ SST model for squared *hypar* canopies of varying shape parameters and height-to-span ratios. Comparing the current results ($SP = HS \approx 0.236$) with Colliers' data for a *hypar* defined by $HS = 0.23$ and $SP = 0.27$, subjected to uniform flow of 15 m/s, shows good agreement in $c_p$ extrema for both wind directions. Despite differences in the setups, the $c_p$ distributions on both sides align well for the 0° configuration. Notably, due to the wall-resolved LES applied in the present work, the low-pressure zone at the front associated with the conical vortices is predicted with higher resolution. However, at the 45° angle, some discrepancies in the $c_p$ distributions are visible in the flow separation areas near the leading edge. These differences may be attributed to the curved edges in the current geometry compared to the squared edges in Colliers' simulation. Additionally, the RANS model is known to predict flow separation regions not very accurately.

Table III summarizes the time-averaged values and standard deviations of the streamwise ($c_x$), spanwise ($c_y$), and vertical ($c_z$) force coefficients for the cases without gusts. The streamwise force coefficient is approximately 13% higher for the 0° configuration compared to the 45° case. The *hypar* geometry results in low spanwise force coefficients $\bar{c}_y$ close to zero for both wind directions. In the 0° configuration, the structure is symmetric with respect to the streamwise axis, leading to a value of $\bar{c}_y$ close to zero. Conversely, in the 45° setup, the antisymmetry of the *hypar* is disrupted by the bottom wall and the tip vortices rotating in the same direction, resulting in a slightly negative value for $\bar{c}_y$.

The vertical force coefficient $\bar{c}_z$ is also close to zero for the 45° case, which is again due to the antisymmetry of the structure, as observed in the nearly mirrored pressure distribution in Fig. 7(a). In contrast, for the 0° configuration, the *hypar* exhibits a high value of $\bar{c}_z$, attributed to the shape of the rigid model possessing a maximal curvature at this wind angle. The pressure distributions on the upper and lower sides do not cancel out but add instead [see Fig. 7(b)]. It is noteworthy that for both wind directions, the majority of fluid forces originate from pressure effects, as shown in Table VI. However, the shear stress contributions, particularly in the streamwise direction, are not negligible. Furthermore, the force coefficients exhibit only mild fluctuations, leading to relatively small standard deviations.

### B. Wind gusts impacting the rigid model

In Sec. III B 3, the sizes and the positions of the deterministic discrete gust were previously specified and justified ($L_g^x/L = 1$, $L_g^y/L = L_g^z/L = 0.5$, and $z_g^0/L = 0$). The strength of the gust was varied in the current study. However, for brevity, only the case of $A_g/u_\infty = 1.5$ is presented, which results in the strongest forces and deflections. As the gust represents a short-term yet substantial velocity variation, a red iso-surface of the streamwise velocity $u/u_\infty = 1.4$ is chosen for its visualization and path tracking in Figs. 8 and 9.

#### 1. Wind direction of 45°

At a wind direction of 45°, the head of the gust hits the thin rigid structure at the middle of the leading edge, as depicted in Fig. 8(a). This leading edge acts like a knife, splitting the wind gust in two equal parts [see Fig. 8(b)]. Each part follows the shape of the *hypar* on one side, resulting in a slight clockwise rotation during the first half of the roof. In the second half, the rotation direction reverses. As the head of each part reaches the trailing edge, it begins to dissipate due to the strong mixing taking place in the subsequent shear layer, as illustrated in Fig. 8(c). Notably, due to the curvature of the *hypar*'s shape, the two parts are not directly aligned vertically. The upper part shifts slightly toward the positive *y*-direction, while the lower part shifts toward the negative *y*-direction. To clearly illustrate this temporal evolution of the gust shape, Fig. 8(c) includes a complementary schematic sketch of the scenario from both a side and a top view. The upper and lower parts of the gust are shown in different colors to enhance the visibility of these shifts in the positive and negative *y*-direction. Subsequently, as seen in Fig. 8(d), the shape of the gust expands and becomes more chaotic as it travels through the wake, affecting the boundary layer behind the roof. At this stage, the original shape of the gust is lost and it completely mixes with the surrounding fluid field. Again, a schematic
<figure><img src="image_21.png" /><figcaption>FIG. 7. Pressure coefficient of the time-averaged flow on the rigid model for the upper side (top), the lower side (middle), and the resulting net pressure coefficient (bottom) predicted by LES for wind directions of 45° and 0° (reference grid).</figcaption></figure>

<table><caption>TABLE III. Fluid force coefficients of the rigid *hypar* model (reference grid).</caption><thead><tr><th rowspan="2">Case</th><th colspan="3">Mean</th><th colspan="3">Standard deviation</th></tr><tr><th>c&#773;<sub>x</sub></th><th>c&#773;<sub>y</sub></th><th>c&#773;<sub>z</sub></th><th>STD(c<sub>x</sub>)</th><th>STD(c<sub>y</sub>)</th><th>STD(c<sub>z</sub>)</th></tr></thead><tbody><tr><td>0°</td><td>0.326</td><td>4.75 &times; 10<sup>-5</sup></td><td>1.551</td><td>2.53 &times; 10<sup>-3</sup></td><td>2.06 &times; 10<sup>-3</sup></td><td>2.02 &times; 10<sup>-2</sup></td></tr><tr><td>45°</td><td>0.295</td><td>-0.011</td><td>0.042</td><td>1.04 &times; 10<sup>-3</sup></td><td>1.24 &times; 10<sup>-3</sup></td><td>7.03 &times; 10<sup>-3</sup></td></tr></tbody></table>
<figure><img src="image_22.png" /><figcaption>FIG. 8. Wind gust around the rigid model highlighted by a red iso-surface of the streamwise velocity $u/u_{\infty} = 1.4$ at a wind direction of 45° (reference grid). Key characteristics of the surrounding flow are visualized by an iso-surface of the dimensionless Q-criterion ($Q^* = 72$) and the dimensionless streamwise velocity in the crosswise y-z-plane at $x/L = 2$.</figcaption></figure>

sketch of this scenario further aids in understanding the complex development of the gust.

## 2. Wind direction of 0°

In the 0° setup, similar to the previous case, the gust is split up by the thin structure starting at the lowest point Ⓐ, as depicted in Fig. 9(a). However, in this configuration, the midpoint of the gust is positioned slightly above the corner Ⓐ. As a result, the major part of the gust is directed toward the upper side of the *hypar*. Unlike the 45° setup, the

*hypar's* shape in the 0° wind direction is symmetric with respect to the streamwise x-axis. Therefore, there is no induced rotation of the gust around the streamwise axis. Both parts of the gust closely follow the curved roof, as seen in Fig. 9(b). When the gust leaves the structure at corner Ⓐ, which has a lower height, the lower tiny part of the gust is directed toward the bottom wall due to the curvature of the structure. The upper part of the gust then splits: one portion also moves toward the bottom wall, while the remainder travels directly into the wake, interacting with the recirculation areas behind the structure and the top of the boundary layer [see Fig. 9(c) including the complementary
<figure><img src="image_23.png" /><figcaption>**FIG. 9.** Wind gust around the rigid model highlighted by a red iso-surface of the streamwise velocity $u/u_\infty = 1.4$ at a wind direction of 0° (reference grid). Key characteristics of the surrounding flow are visualized by an iso-surface of the dimensionless Q-criterion ($Q^* = 72$) and the dimensionless streamwise velocity in the y-z-plane at $x/L = 2$.</figcaption></figure>

schematic sketch of the scenario]. Subsequently, both parts of the gust continue their path through the wake. One part stays near the bottom wall, while the other follows the height of the high points Ⓐ and Ⓑ, illustrated as part 1 and part 2 in Fig. 9(d) and its complementary sketch, respectively. Similar to the 45° case, the gust eventually dissipates, mixing completely with the surrounding fluid flow.

**3. Fluid forces acting on the rigid model impacted by gusts**

Introducing the strong wind gust in front of the rigid model, as specified in Sec. IV B, results in short but intense variations of the fluid forces, as depicted in Fig. 10. For the 45° case, the streamwise force coefficient $c_x$ increases by approximately 26%, while the vertical force coefficient $c_z$ experiences a significant rise of more than 1000% during the gust impact. However, since the baseline value of $c_z$ without the

gust is nearly zero, the absolute variation in amplitude due to the gust is limited. At this wind angle, the head of the gust splitting on the first half of the structure [see Fig. 8(a)] leads to a negative spanwise force coefficient $c_y$. As the head of the gust reaches the second half of the *hypar*, $c_y$ becomes positive [see Fig. 8(b)]. It is important to note that the resulting forces in y-direction are about an order of magnitude smaller than those in the other two directions. At the moment when the gust is exactly centered above the structure, the two previously described effects cancel out, returning $c_y$ to values near zero. As the gust begins to leave the structure [see Fig. 8(c)], another minor peak in $c_y$ appears, likely induced by the anti-symmetric shape of the *hypar* and the resulting asymmetric wake flow immediately behind the trailing edge.

For the 0° wind direction, an increase in up to 130% is observed for $c_\infty$, reaching a maximum of 0.73. This absolute increase in $c_x$ is significantly higher in the case of 0° compared to 45°, because the gust
<figure><img src="image_24.png" /><figcaption>FIG. 10. Fluid force coefficients resulting from the impact of a gust on the rigid *hypar* model (reference grid).</figcaption></figure>

impacts an inclined region of the tent structure. Due to the symmetry of the shape at 0° and the position of the gust injection, the spanwise force coefficient is hardly affected by the gust. Notably, a remarkable variation of the vertical force coefficient $c_z$ is observed, ranging from downward to the upward directions with extreme values between -0.347 at $t^* = 0.45$ [see Fig. 9(a)] and 2.804 at $t^* = 0.83$ [see Fig. 9(b)]. It can be concluded that a discrete gust, with a scale smaller than the *hypar* roof but causing a 150% increase in flow velocity for a brief period, results in doubling the wind suction load for the 0° setup.

## C. Fluid-structure interaction of the flexible membrane

After thoroughly analyzing the flow field around the rigid *hypar* structure, in a step-by-step procedure the flexibility of the membrane is taken into account next. The coupled FSI simulations are initiated by the fully developed flow fields discussed in Sec. IV A. During the release phase, the membrane responds to the suddenly occurring forces on its surface, especially the pressure forces, and starts to oscillate. To pass this transition phase without numerical instabilities, the fluid forces applied are linearly increased from zero to the computed value for a short dimensionless time interval of 0.033. Nevertheless, the membrane oscillates for a dimensionless time interval of about 1.67 before reaching a quasi-steady deformation state. Only minor deflections on the order of $2 \times 10^{-3}$ m and $6 \times 10^{-3}$ m are observed locally during the starting phase for the 45° and 0° wind directions, respectively. Subsequently, the resulting deformations become even smaller. This behavior is attributed to the high pre-stresses in both the membrane and the steel cables defined in Sec. II A. These pre-stresses, chosen for the real-scale design, allow only minor deflections. Indeed, based on the first dominant eigenfrequency of the structure $f_1$, the value of the non-dimensional flow velocity $u_\infty/(f_1 L)$ is about 0.12 and thus rather low. With their flexible membrane mounted in a slightly different setup, Uematsu and Uchiyama¹⁵ found a quasi-linear relation between the membrane deformation and the dynamic pressure: A deflection of about $3 \times 10^{-3}$ m is expected for $u_\infty/f_1 L \approx 0.12$. Consequently, in the absence of gusts, the flow fields around the flexible membrane and their associated force coefficients exhibit only marginal deviations from those of the rigid model. Thus, they are neither depicted nor discussed further. The discussion now proceeds directly to the cases where a discrete wind gust impacts the flexible structure.

### 1. Wind direction of 45°

At a wind angle of 45°, the gust impacts the middle of the structure horizontally and is split into half. Due to the antisymmetric shape of the structure with respect to the streamwise axis, the predicted global vertical forces remain small. Moreover, the structure does not offer a large projected area relative to the mean flow direction. As a result, the local streamwise forces resulting from the gust are also limited (see Sec. IV B 1). Given the minor deviations compared to the rigid case depicted in Fig. 10, the force coefficients of the flexible case are not displayed here. However, the wind gust injected into the flow has a strength of $A_g/u_\infty = 1.5$, causing the local streamwise velocities to increase from 10 to 25 m/s. Despite this circumstance, no significant local deformations are observed during the gust impact due to the high pre-stresses in the membrane and cables, combined with the gust's extension being only half the width of the *hypar* structure. The internal stresses remain close to the pre-stress level ($4 \times 10^3$ kN/m²), varying within a range of -0.5% to +0.6% of the minimum and maximum internal stress values of the flexible membrane when exposed to the constant inflow without gust. This minor variation reflects the robustness of the pre-stressed membrane under gust conditions, preventing large deformations despite the significant velocity increases associated with the gust.

### 2. Wind direction of 0°

At a wind angle of 0°, the gust impacts the membrane slightly above the low corner ® at the front. A minor portion of the gust is directed toward the underside of the structure, while the majority remains on the upper side [see Fig. 12(a)]. The resulting fluid forces, integrated over the *hypar* area, are shown in Fig. 11 and compared to those obtained from the rigid counterpart. Figure 12 illustrates the position of the gust and its effects on the flexible membrane at three specific time instants corresponding to the first peak of $c_x$ ($t^* \approx 0.45$), the second peak of $c_z$ ($t^* \approx 0.83$), and the moment when the gust leaves the *hypar* ($t^* \approx 1.5$). Additionally, Fig. 13 presents the total displacements and the principal inner stresses of the membrane at the same time instants.

At $t^* \approx 0.45$, when the gust impacts the front of the membrane, regions of a negative resulting pressure coefficient [Fig. 12(a)] are found near the corner ®. This causes local deformations close to this corner in the positive streamwise direction [Fig. 12(b)] and in the negative vertical direction [Fig. 12(c)]. These streamwise and vertical
<figure><img src="image_25.png" /><figcaption>FIG. 11. Fluid force coefficients resulting from the impact of a gust on the flexible membrane and the rigid *hypar* model (default grid).</figcaption></figure>

<figure><img src="image_26.png" /><figcaption>FIG. 12. Instantaneous net pressure coefficient (left), streamwise deflections of the FSI interface (middle), and vertical deflections (right) at three time instants for the 0° configuration. The position of the wind gust is highlighted by a semi-transparent red iso-surface of the streamwise velocity $u/u_\infty = 1.4$. For the illustration of the shape, the deformations are magnified by a factor of 100 and outlined by two white lines along the diagonals and the additional sketch above right.</figcaption></figure>

deflections are 360% and 420% larger than the quasi-steady deformed state obtained from the FSI simulation of the flexible membrane exposed to constant inflow without gust, respectively. However, the absolute magnitudes remain limited, with values of +3.2 × 10⁻³ m in the streamwise direction and -9.2 × 10⁻³ m in the vertical direction,

leading to a total displacement of about 1 cm [Fig. 13(a)]. These distortions are again associated with inner stress values close to the pre-stress level (4 × 10³ kN/m²). The impact of the gust generates a crosswise-oriented tension chord (about 5% higher than the pre-stress level) located at the front [red area in Fig. 13(b)] just behind the region
<figure><img src="image_27.png" /><figcaption>FIG. 13. Instantaneous total displacement and principal stress distributions at three time instants for the 0° configuration.</figcaption></figure>
<figure><img src="image_28.png" /><figcaption>**FIG. 14.** Membrane deflection along its diagonal (left) and differences in the net pressure coefficient between the FSI results and the CFD prediction of the rigid model (right) at three time instants during the gust impact for the 0° configuration.</figcaption></figure>

of maximum total displacements. This implies that a significant portion of the membrane becomes slack behind the tension chord [blue area in Fig. 13(b)].

Subsequently, the wind gust continues to follow the curvature of the *hypar*. At $t^* \approx 0.83$, the gust is positioned entirely above the flexible structure. The highest flow velocities are found at the front of the gust, resulting in an increased pressure coefficient on the FSI interface over a large region beneath the gust [Fig. 12(d)]. This pressure distribution induces an increase in the vertical deflection of the membrane [Fig. 12(f)] that is 230% larger than the quasi-steady deformed state ($9 \times 10^{-3}$ m instead of $3.9 \times 10^{-3}$ m). Streamwise deflections [Fig. 12(e)], which are about 170% larger than the quasi-steady deformed state, are observed at the front ($0.9 \times 10^{-3}$ m instead of $0.5 \times 10^{-3}$ m) and the tail of the gust ($1.5 \times 10^{-3}$ m instead of $0.9 \times 10^{-3}$ m). The peak total displacement (about $10 \times 10^{-3}$ m) shifts to the region beneath the head of the gust [Fig. 13(d)]. The corresponding inner stress distribution evolves. A tension chord, i.e., an

area of increased tensile stresses [red area in Fig. 13(e)] is aligned in the streamwise direction, with a magnitude about 5% higher than the pre-stress level. At the same time, a crosswise-oriented compression arc, i.e., an area of decreased tensile stress, about 6% lower than the pre-stress level, is observed in Fig. 13(f) (blue area).

As the gust leaves the *hypar* at $t^* \approx 1.5$ [Fig. 12(g)], the streamwise and vertical deflections begin to attenuate, as shown in Figs. 12(h) and 12(i). Consequently, the stress levels of the tension chord and compression arc also decrease [Figs. 13(h) and 13(i)].

After leaving the canopy, the gust propagates and dissipates in the wake. The membrane gradually returns to its quasi-steady deformed state. Throughout the process, the gust induces deflections of the FSI interface, resulting in local deformations ranging from -6 to +3 mm compared to the quasi-steady state.

To quantitatively illustrate the membrane deflection during the gust impact, Fig. 14 shows the deformation of the membrane along its diagonal (from ⊕ to ⊘) at the three specific time instants discussed
earlier. Additionally, the differences in the net pressure coefficient between the FSI simulation of the flexible membrane and the pure CFD prediction of the rigid model are depicted. At $t^* \approx 0.45$, when the first peak of $c_x$ appears, the net pressure difference is positive in the region around the corner Ⓓ [Fig. 14(b)]. This indicates that the deformation of the membrane due to the impacting gust results in a higher downward-directed pressure force compared to the rigid counterpart. The higher peak values in the fluid force coefficients were already evident in Fig. 11. This increase in the net pressure coefficient and the resulting force can be clearly attributed to the change in the shape. The membrane initially has a concave shape along its diagonal. When the gust impacts the structure, the front part deforms and develops a convex shape, as shown in Fig. 14(a).

At $t^* \approx 0.83$, the gust is on the structure, causing the second peak of the vertical force coefficient $c_z$, as described earlier. Due to the deformation of the membrane, the curvature in the middle of the structure is attenuated, as visible in Fig. 14(c). Consequently, the difference in the net pressure coefficient [Fig. 14(d)] is negative in this region, indicating a stronger suction pressure in the center of the arched membrane. Nevertheless, a wave-like distribution of the net pressure difference is evident in this predominantly blue area. Additionally, differences in the conical vortices lead to deviations in the net pressure coefficient along the development direction of these vortical flow structures.

At $t^* \approx 1.5$, when the gust leaves the *hypar*, the deformation has already attenuated again, as shown in Fig. 14(e). Consequently, the largest deviations of the net pressure coefficient, depicted in Fig. 14(f), are now observed near the corner Ⓓ, where the gust exits the trailing edge. Otherwise, the regions influenced by the spreading of the conical vortices become noticeable again.

In summary, comparing the flexible and rigid case reveals that the deformation of the membrane during the gust impact augments the extrema of the net pressure difference in several exposed local areas. These areas include the corner region Ⓓ during the gust's arrival, the middle of the *hypar* when the gust is on the structure, and the corner region Ⓓ when the gust leaves the influence area of the structure. These changes account for the higher peak values of the force coefficient observed in Fig. 11.

## V. CONCLUSIONS

The effect of wind gusts impacting a real-scale hyperbolic parabola tensile structure was numerically studied using a partitioned FSI approach. This method couples an eddy-resolving finite-volume fluid solver with a finite-element solver developed for shell and membrane structures. To ensure appropriate grid resolution, the fluid flow around the rigid model was analyzed using a highly refined reference grid with $138 \times 10^6$ CVs, designed for accurate wall-resolved LES with an inflow boundary layer and a free-stream velocity of 10 m/s, corresponding to $Re \approx 4 \times 10^6$. In addition, a default grid with $17 \times 10^6$ CVs was constructed to meet the high computational demands of the coupled FSI simulations. The analysis was based on both time-averaged and instantaneous data. Time-averaged pressure distributions on the rigid model were presented for wind angles of 45° and 0°. Furthermore, the time-averaged first and second moments of the flow field at 45° wind angle were provided. The findings indicated that the default grid could capture all important flow features, including tip and conical vortices, flow separation regions and the wake, with deviations from the reference grid results being within acceptable limits. Since these flow

phenomena affect the resulting forces on the structure and the internal stresses in the material, their accurate reproduction by the simulation is of central importance for the design process. The corresponding grid sensitivity study for the structure revealed that the fine grid ensures accurate results. Since only wind gusts of considerable strength result in perceptible membrane deflections, one specific scenario was investigated for the two wind directions, leading to the following conclusions:

* The *hypar* design demonstrates its robustness when subjected to wind conditions ranging from near-gale to storm levels. Even at the critical wind direction of 0°, the extremes of the local internal stresses remain well below the material's tensile strength.

* For the present configuration, characterized by the high pre-stress in the structure and the moderate inflow velocity, the predicted membrane deflections remain below 1 cm. An FSI framework relying on static load and deformation, as used in Kawulok et al.²⁶ or in Colliers,² would be sufficient to accurately predict the deformations and inner stresses in the material. However, to capture the *instantaneous effects of a discrete wind gust* on the membrane's deformation and stresses, a fully coupled computational framework is required, as applied in the present study. This two-way coupled methodology is also necessary to assess how the deflections influence the surrounding flow and the resulting local and global forces.

* The case investigated in the current study, as described in the Round Robin³⁵ Exercise 4, involves realistic structural and material parameters. Therefore, under mild wind conditions, significant deflections of the flexible membrane are not expected. A direct comparison with the results of Zorrilla et al.,³² who observed much larger deflections, is not feasible because they considered a larger tent and a free-stream velocity five times higher. Furthermore, key material properties of the membrane, such as the Young's modulus, differ significantly. The same arguments also preclude a meaningful comparison with the result of Colliers,² who applied a pre-stress three times lower and an atmospheric boundary layer with a free-stream velocity of about 35 m/s.

* The wind gust applied in this study augments the local velocities by a factor of 2.5 (from 10 to 25 m/s), leading to significant changes in the integrated forces in both the streamwise and vertical directions. However, because the three-dimensional wind gust is smaller than the characteristic length $L$ of the *hypar* structure, its overall impact on the flexible structure is limited. The effect of wider and longer gusts remains a subject of ongoing investigations.

* Gusts of greater strength ($A_g/u_\infty > 1.5$) cannot be accurately predicted using the current grids designed for the present wall-resolved LES. Such stronger gusts would result in much higher Reynolds numbers, necessitating significantly finer grid resolution, particularly in the near-wall regions, to capture the increased velocity gradients.

* Obviously, the wind direction has a significant effect on the peak loads induced during the gust impact. In the 0° case, noticeably higher forces are generated compared to the 45° case due to the more direct interaction of the gust with the membrane. Consequently, this results in much larger deflections of the membrane and increased stresses in the material.
* When comparing the rigid and the flexible structure, it is evident that the deflection of the membrane during the gust impact leads to larger extremes in the local net pressure coefficient and higher peak values in the global fluid forces. Consequently, for design considerations, this means that one cannot rely solely on the measured or calculated data from rigid models; the flexibility of the structure must also be accounted for to ensure accurate predictions.

* Despite using the default CFD grid instead of the reference grid, the FSI configuration of this *hypar* setup remains a challenging case, demanding substantial CPU time and storage resources.

* The simulation methodology presented can, of course, be applied to other architectural designs, provided that high-quality grids can be generated.

In summary, a high-fidelity simulation environment was developed to study the effect of highly dynamic wind gusts on membrane structures, applied to a realistic scenario at high Reynolds number. In the future, such developments of the simulation methodology will enable precise predictions of maximum loads and material stresses during strong wind gusts, well before the construction of new design begins, thereby helping to ensure structural stability.

As a direction for future work on the simulation side, incorporating inflow turbulence should be considered. While its influence on the deformation of the structure under gust loads may be minor, testing this through appropriate simulations could provide valuable insights. Another important topic is the need for more experimental validation data, which are not yet available to a sufficient extent, particularly for the simultaneous analysis of the flow field and the structural deformation.

## ACKNOWLEDGMENTS

The authors want to thank the Chair of Structural Analysis of the Technical University of Munich for providing the computational structure dynamics solver Carat++$^{44}$ and the open-source coupling software EMPIRE.$^{46}$ Additionally, the authors are grateful to J.N. Wood for his support with the CAD model used in this study. The majority of the costly FSI simulations were conducted on HSUper, the central computer cluster of the Helmut-Schmidt-Universität Hamburg. The HPC cluster HSUper has been provided by the project hpc.bw, funded by dtecbw (https://dtecbw.de/)—Digitalization and Technology Research Center of the Bundeswehr. dtecbw is funded by the European Union–NextGenerationEU.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

G. De Nayer: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). M. Breuer: Conceptualization (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Software (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). A.-K. Goldbach: Formal analysis (supporting); Investigation (supporting); Methodology (supporting); Validation (supporting); Visualization (supporting); Writing – original draft (supporting).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## APPENDIX A: CSD GRID SENSITIVITY STUDY BASED ON GEOMETRICALLY NON-LINEAR ANALYSIS

A plane-stress membrane element is used for the finite-element discretization. Three levels of refinement are investigated under a uniformly applied wind suction load of 245, 551, 980, and 1531 N/m$^2$, resulting from wind gust speeds of 10, 15, 20, and 25 m/s. These exemplary loads are calculated based on the assumption that the static pressure at the leading edge is approximately 0.5 $\rho_{\text{air}} u_{\infty}^2$. The *hypar* geometry involves curved shapes resembling airfoils. Typically, for an airfoil, a variation of the pressure coefficient in the range +1 and -3 is observed between the stagnation region and the suction side, respectively. As a result, the wind suction loads applied for this static case are approximated as 4 · 0.5 $\rho_{\text{air}} u_{\infty}^2$. The resulting vertical displacements $\Delta z$ at the *hypar*'s center are given in Table IV. The load-controlled geometrically non-linear analysis yields very similar results for the coarse, medium, and fine grids, consisting of 200, 800, and 3200 triangular membrane elements, respectively (additional cable elements are

<table><caption>TABLE IV. Vertical displacements $\Delta z$ of the *hypar*'s center for different grids under increasing wind suction load.</caption><thead><tr><th rowspan="2">Load</th><th colspan="2">Grid</th><th colspan="2">Coarse</th><th colspan="2">Medium</th><th colspan="2">Fine</th><th>Finest (reference)</th></tr><tr><th>Wind speed (m/s)</th><th>Wind suction load (N/m²)</th><th>Δz (m)</th><th>Error (%)</th><th>Δz (m)</th><th>Error (%)</th><th>Δz (m)</th><th>Error (%)</th><th>Δz (m)</th></tr></thead><tbody><tr><td>10</td><td></td><td>245</td><td>0.019 26</td><td>-1.48</td><td>0.019 47</td><td>-0.41</td><td>0.019 53</td><td>-0.10</td><td>0.019 55</td></tr><tr><td>15</td><td></td><td>551</td><td>0.043 80</td><td>-1.42</td><td>0.044 26</td><td>-0.38</td><td>0.044 39</td><td>-0.09</td><td>0.044 43</td></tr><tr><td>20</td><td></td><td>980</td><td>0.079 74</td><td>-1.76</td><td>0.080 77</td><td>-0.49</td><td>0.081 06</td><td>-0.14</td><td>0.081 17</td></tr><tr><td>21.25</td><td></td><td>1301</td><td>0.108 17</td><td>-2.54</td><td>0.110 07</td><td>-0.83</td><td>0.110 77</td><td>-0.20</td><td>0.110 99</td></tr><tr><td>25</td><td></td><td>1531</td><td>0.129 44</td><td>-7.42</td><td>0.129 48</td><td>-7.39</td><td>0.139 91</td><td>0.07</td><td>0.139 81</td></tr></tbody></table>
<figure><img src="image_29.png" /><figcaption>FIG. 15. Vertical deformations and principal stresses of the *hypar* tent on grids of 200, 800, and 3200 membrane elements (coarse, medium, and fine) for an exemplary wind suction load of 980 N/m².</figcaption></figure>
<figure><img src="image_30.png" /><figcaption>**FIG. 16.** Comparison of the time-averaged velocity components, crosswise vorticity, and Reynolds stresses in the x–z-plane at y/L = 0.3 predicted by LES for a wind direction of 45° on the default and reference grid for the rigid model.</figcaption></figure>
<figure><img src="image_31.png" /><figcaption>**FIG. 17.** Comparison of the time-averaged velocity components, crosswise vorticity, and Reynolds stresses in the x–z-plane at y/L = -0.3 predicted by LES for a wind direction of 45° on the default and reference grid for the rigid model.</figcaption></figure>
<figure><img src="image_32.png" /><figcaption>FIG. 18. Comparison of the time-averaged streamlines, velocity components, and Reynolds stresses in the y-z-plane at x/L = 1.0 predicted by LES for a wind direction of 45° on the default and reference grid for the rigid model.</figcaption></figure>
<table><caption>TABLE V. First dominant frequencies associated with the vertical displacement of the *hypar*'s center reacting to a constant load.</caption><thead><tr><th rowspan="2">Frequencies</th><th colspan="2">Coarse</th><th colspan="2">Medium</th><th colspan="2">Fine</th><th>Finest (reference)</th></tr><tr><th>f (Hz)</th><th>error (%)</th><th>f (Hz)</th><th>error (%)</th><th>f (Hz)</th><th>error (%)</th><th>f (Hz)</th></tr></thead><tbody><tr><td>f<sub>1</sub></td><td>14.50</td><td>1.75</td><td>14.25</td><td>0</td><td>14.25</td><td>0</td><td>14.25</td></tr><tr><td>f<sub>2</sub></td><td>16.25</td><td>1.56</td><td>16.00</td><td>0</td><td>16.00</td><td>0</td><td>16.00</td></tr></tbody></table>

positioned at the edges but not listed here). Additional computations conducted on a grid consisting of 10 000 quadrilateral elements show displacements that are nearly identical to those from the fine grid, as indicated in the last column of Table IV. This table also includes the relative errors with respect to the results obtained on the finest grid. It is evident that spatial convergence has been achieved with the fine grid and for all applied loads.

The overall vertical deformation and the distribution of principal stresses are exemplarily depicted in Fig. 15 for the case of 20 m/s and the different discretizations. At this load intensity, the membrane remains taut with no compressive stresses present up to that state. However, if the load is increased beyond this point, tension is lost in certain areas, leading to wrinkles forming in the membrane. Since the membrane element formulation used in this study does not include any kind of wrinkling model, the results obtained become highly dependent on the discretization when compressive stresses are present, rendering coarse meshes unreliable.⁷⁰ Table IV reflects this issue, as the error of the coarse and medium grids increases disproportionately when the load exceeds 1301 N/m² (corresponding to a wind speed of 21.25 m/s) and reaches an intensity at which wrinkling begins to occur. For loads below this threshold, grid convergence is already achieved with the medium-sized grid. Nonetheless, to ensure a smooth distribution of the stresses across the membrane (as shown in the distributions at the corners in Fig. 15) and to properly detect the appearance of undesired tension loss (wrinkling), the fine grid is employed for the FSI simulations.

In a second step, a constant load is applied to the membrane, and the transient structural response is predicted for the different grid resolutions to evaluate the influence of the grid. The vertical displacement at the center of the *hypar* is monitored. Table V summarizes the two first dominant frequencies associated with Δz, determined through fast Fourier transformations. The results show that the medium grid already provides dominant frequencies comparable to those obtained with the reference grid. Consequently, the

choice to use the fine CSD grid for the FSI simulations, based on the previous static test case, is also validated for the case of the transient structural response.

## APPENDIX B: CFD GRID SENSITIVITY STUDY BASED ON THE TIME-AVERAGED FLOW

To evaluate the quality of the LES results, a sensitivity study is conducted for a wind direction of 45° using the two CFD grids outlined in Sec. III B. The *default* grid, which consists of 17.28 × 10⁶

<figure><img src="image_33.png" /><figcaption>FIG. 20. Comparison of the pressure coefficient distribution of the time-averaged flow on the upper (top) and lower (bottom) part of the *hypar* structure predicted by LES for a wind direction of 45° on the *default* and *reference* grid for the rigid model.</figcaption></figure>

<figure><img src="image_34.png" /><figcaption>FIG. 19. Comparison of the flow separation area visualized by wall-streamlines of the time-averaged flow on the upper part of the rigid *hypar* structure predicted by LES for a wind direction of 45° on the *default* and *reference* grid.</figcaption></figure>
<table><caption>TABLE VI. Time-averaged values of the global fluid force coefficients exerted on the rigid model for a wind direction of 45°. Additionally, the pressure related part $\bar{c}_i^p$ and the shear stress related part $\bar{c}_i^s$ are given. The relative error based on the reference case is computed.</caption><thead><tr><th>Grid</th><th>c&#773;<sub>x</sub></th><th>c&#773;<sub>x</sub><sup>p</sup></th><th>c&#773;<sub>x</sub><sup>s</sup></th><th>c&#773;<sub>y</sub></th><th>c&#773;<sub>y</sub><sup>p</sup></th><th>c&#773;<sub>y</sub><sup>s</sup></th><th>c&#773;<sub>z</sub></th><th>c&#773;<sub>z</sub><sup>p</sup></th><th>c&#773;<sub>z</sub><sup>s</sup></th></tr></thead><tbody><tr><td>Default</td><td>0.316</td><td>0.248</td><td>0.0682</td><td>-0.009 98</td><td>-0.0102</td><td>1.76 &times; 10<sup>-4</sup></td><td>0.0364</td><td>0.0368</td><td>-4.11 &times; 10<sup>-4</sup></td></tr><tr><td>Reference</td><td>0.295</td><td>0.248</td><td>0.0472</td><td>-0.0108</td><td>-0.011</td><td>1.89 &times; 10<sup>-4</sup></td><td>0.0416</td><td>0.0420</td><td>-3.34 &times; 10<sup>-4</sup></td></tr><tr><td>Rel. error (%)</td><td>+ 7.1</td><td>0</td><td>+ 45</td><td>-7.6</td><td>-7.2</td><td>-6.9</td><td>-12.5</td><td>-12.4</td><td>-23</td></tr></tbody></table>

CVs, is used for the fully coupled FSI simulations. This grid is a coarser version of the very reference grid, which includes $138.24 \times 10^6$ CVs, with coarsening applied in all spatial directions.

The grid sensitivity study begins with a detailed comparison of the time-averaged flow fields. For this purpose, the instantaneous flow data are averaged over a long time period of approximately 50 dimensionless time units (time normalized by $u_\infty$ and $L$). Despite the fully three-dimensional nature of the turbulent flow, initial comparisons are made using two-dimensional contour plots at specific slices, as they provide a clear qualitative view of the characteristic flow regions. The velocity field and the associated Reynolds stresses are examined in two x-z-planes (see Figs. 16 and 17) and a spanwise y-z-plane selected in the wake (see Fig. 18). The two x-z-planes, located at $y/L = 0.3$ and $y/L = -0.3$, are chosen to capture the flow separation bubble occurring near the leading edge on the upper surface ($y/L = 0.3$) and on the lower surface of the membrane ($y/L = -0.3$). Additionally, the spanwise y-z-plane highlights the asymmetry of the wake flow and the position of the various tip vortices. Figure 19 provides a three-dimensional view of the time-averaged flow separation on the membrane for both grids. Surface streamlines, based on the time-averaged velocity at the first cell of the upper side of the structure, illustrate the size and shape of the separation bubble. Finally, the distribution of the time-averaged pressure coefficient on both sides is presented in Fig. 20, as this parameter is crucial significantly contributing to the fluid forces acting on the structure.

Upon comparing the LES results from both grids, it is clear that the default grid accurately captures all key features of the flow, which are as follows:

* Due to its asymmetric shape, the structure induces a flow separation bubble on the upper side just after the leading edge for $y/L > 0$. This behavior resembles the flow around a thin airfoil at a high angle of attack. The position and shape of this separation bubble simulated on the default grid show good agreement with the reference case (see Fig. 19). However, the bubble is slightly longer on the default grid. Various flow characteristics in this region, including velocity components, vorticity, and Reynolds stresses (see Fig. 16), closely match the highly resolved results from the reference grid. For $y/L < 0$, the flow field exhibits the opposite behavior, with a separation bubble occurring on the underside near the leading edge. Once again, all key flow features obtained on the default grid align closely with those from the reference case (see Fig. 17).

* Above the recirculation area for $y/L > 0$, a strong shear layer develops, in which all normal Reynolds stresses and the shear stress $u'w'/u_\infty^2$ are large. The same behavior is observed for the region below the recirculation region for $y/L < 0$.

* Another significant phenomenon associated with the flow around the hypar membrane is the formation of tip vortices at the four corners of the structure. The size, position, and strength of these vortices are of great importance for the overall FSI problem. The cross-sectional view of the wake, depicted in Fig. 18, provides a detailed examination of these characteristics. It is evident that all four tip vortices computed on the default grid closely match the reference vortices. Additionally, the interaction between the vortex at the lower corner and the vortex at the upper corner on the same side of the membrane, as visible in the vorticity field, is accurately captured.

* The hyperbolic paraboloid shape of the membrane induces an asymmetric S-shaped wake flow, as shown in Fig. 18. The default grid accurately reproduces the dimensions of this S-shaped flow pattern. Within the core of this S-shaped region, the Reynolds stresses are slightly higher compared to those obtained on the reference grid. However, it is worth to note that these differences are minor.

In fluid-structure interaction studies, the forces exerted by the fluid on the structure are crucial. Therefore, Table VI summarizes the time-averaged global force coefficients on the hypar structure, along with their pressure and shear stress components. The dominant force coefficient is the streamwise force coefficient $\bar{c}_x$. For the default grid, there is an approximate error of about 7% in the LES predictions. As expected, the major part of $\bar{c}_x$ is attributed to the pressure component. It is noteworthy that there is no error observed in the pressure-related part for the default simulation (compare Fig. 20). The error in the streamwise direction arises solely from the shear-stress-related component, which correlates with the observed changes in the size of the recirculation region. The default grid tends to overestimate this value. In contrast, the force coefficient in the spanwise direction, $\bar{c}_y$, exhibits the lowest magnitude. Due to the anti-symmetric shape of the membrane, this force coefficient is about 30 times smaller than the streamwise coefficient. A relative error of approximately 8% is observed in the default grid predictions. Finally, for the simulation relying on the default grid, the vertical force coefficient, $\bar{c}_z$, is slightly underestimated with a relative error of about 13%. This discrepancy is attributed to both force components. However, it is important to note again that the vertical force is approximately an order of magnitude smaller than the streamwise component due to the specific shape of the structure.

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<footnote n="56">⁵⁶International Electrotechnical Commission, “Measurement and assessment of power quality of grid connected wind turbines,” IEC-Standard 61400-21 (2002).</footnote>

<footnote n="57">⁵⁷G. De Nayer, M. Breuer, P. Perali, and K. Grollmann, “Modeling of wind gusts for large-eddy simulations related to fluid-structure interactions,” in *ERCOFTAC Series, Direct and Large-Eddy Simulation XI, 11th International ERCOFTAC Workshop on Direct and Large-Eddy Simulation: DLES-11*, May 29–31, 2017, Vol. 25, edited by M. Salvetti, V. Armenio, J. Fröhlich, B. J. Geurts, and H. Kuerten (Springer Nature Switzerland AG 2019, Pisa, Italy, 2019), pp. 453–459.</footnote>

<footnote n="58">⁵⁸G. De Nayer and M. Breuer, “A source-term formulation for injecting wind gusts in CFD simulations,” *J. Wind Eng. Ind. Aerodyn.* **207**, 104405 (2020).</footnote>

<footnote n="59">⁵⁹G. De Nayer and M. Breuer, “Surrogate-based optimization for the worst-case prediction regarding a flexible structure impacted by wind gusts,” *J. Wind Eng. Ind. Aerodyn.* **243**, 105610 (2023).</footnote>

<footnote n="60">⁶⁰T. Wang, R. Wüchner, S. Sicklinger, and K.-U. Bletzinger, “Assessment and improvement of mapping algorithms for non-matching meshes and geometries in computational FSI,” *Comput. Mech.* **57**, 793–816 (2016).</footnote>

<footnote n="61">⁶¹A. Apostolatos, G. De Nayer, K.-U. Bletzinger, M. Breuer, and R. Wüchner, “Systematic evaluation of the interface description for fluid-structure interaction simulations using the isogeometric mortar-based mapping,” *J. Fluids Struct.* **86**, 368–399 (2019).</footnote>

<footnote n="62">⁶²S. Sen, G. De Nayer, and M. Breuer, “A fast and robust hybrid method for block-structured mesh deformation with emphasis on FSI-LES applications,” *Numer. Methods Eng.* **111**, 273–300 (2017).</footnote>

<footnote n="63">⁶³A. Kalmbach and M. Breuer, “Experimental PIV/V3V measurements of vortex-induced fluid-structure interaction in turbulent flow—A new benchmark FSI-PfS-2a,” *J. Fluids Struct.* **42**, 369–387 (2013).</footnote>

<footnote n="64">⁶⁴G. De Nayer, A. Kalmbach, M. Breuer, S. Sicklinger, and R. Wüchner, “Flow past a cylinder with a flexible splitter plate: A complementary experimental-numerical investigation and a new FSI test case (FSI-PfS-1a),” *Comput. Fluids* **99**, 18–43 (2014).</footnote>

<footnote n="65">⁶⁵G. De Nayer, A. Apostolatos, J. N. Wood, K.-U. Bletzinger, R. Wüchner, and M. Breuer, “Numerical studies on the instantaneous fluid-structure interaction of an air-inflated flexible membrane in turbulent flow,” *J. Fluids Struct.* **82**, 577–609 (2018).</footnote>

<footnote n="66">⁶⁶G. De Nayer and M. Breuer, “Numerical FSI investigation based on LES: Flow past a cylinder with a flexible splitter plate involving large deformations (FSI-PfS-2a),” *Int. J. Heat Fluid Flow* **50**, 300–315 (2014).</footnote>

<footnote n="67">⁶⁷J. N. Wood, M. Breuer, and G. De Nayer, “Experimental studies on the instantaneous fluid-structure interaction of an air-inflated flexible membrane in turbulent flow,” *J. Fluids Struct.* **80**, 405–440 (2018).</footnote>

<footnote n="68">⁶⁸U. Piomelli and J. R. Chasnov, “Large-eddy simulations: Theory and applications,” in *Turbulence and Transition Modeling*, edited by M. Hallbäck, D. S. Henningson, A. V. Johansson, and P. H. Alfredson (Kluwer, 1996), pp. 269–331.</footnote>

<footnote n="69">⁶⁹I. Gursul, R. Gordnier, and M. Visbal, “Unsteady aerodynamics of nonslender delta wings,” *Prog. Aerosp. Sci.* **41**, 515–557 (2005).</footnote>

<footnote n="70">⁷⁰A. Jrusjungkiat, “Nonlinear analysis of pneumatic membranes: “From subgrid to interface”,” Ph.D. thesis (Lehrstuhl für Statik, Technische Universität München, Germany, 2009).</footnote>