Hydroelastic analysis of partially or totally immersed Mindlin plates with a hybrid isogeometric-based approach

Abstract

In this study, the dynamic characteristics of a cantilevered Mindlin plate, partially or totally in contact with fluid on both sides, are examined using the proposed hybrid isogeometric finite element – boundary element framework. The interaction problem is divided into two parts through the implementation of the linear hydroelasticity theory, allowing for the independent treatment of structural and fluid problems. In the structural part of the analysis, the thick plate is modeled as first-order shear deformable by adopting Reissner–Mindlin plate theory, with the material assumed to be homogeneous and isotropic. The resulting eigenvalue problem is then solved by the isogeometric finite element method (IGAFEM). In the second part of the analysis, assuming the fluid is ideal and incorporating the in vacuo dynamic characteristics as boundary conditions, fluid-structure interaction effects are determined by the isogeometric boundary element method (IGABEM) in terms of generalized added mass coefficients. Subsequently, parametric studies for various plate thicknesses and submergence depths are conducted. The analysis reveals that wet natural frequencies differ noticeably from their in vacuo counterparts due to the presence of the free surface of the fluid. In addition, the effect of the radiated free surface waves from the vibrating structure are included into the mathematical model by imposing linearized free surface condition, through the frequency-dependent added mass and hydrodynamic damping effects of the fluid. The numerical accuracy of the proposed approach is verified against the commercial finite element software ANSYS, showing favourable agreement in the predicted natural frequencies and corresponding mode shapes.

Keywords

Fluid–structure interactions Mindlin plate added mass fluid-plate coupling system

1. Introduction

Dynamic characteristics of thin-walled structures in the presence of fluid may vary significantly from those of in the air, depending on the density of the surrounding fluid and the elasticity of a structure. In physical environments where structure and dense fluid interact, such as aerospace, petroleum, and marine engineering applications, the effect of fluid on the dynamic response characteristics of an elastic structure becomes significant. Therefore, it is essential to establish a rigorous model in order to analyze the interaction problem accurately. One of the most significant impacts of surrounding fluid is the added mass effect due to the acceleration of fluid particles, which can sometimes be much larger than the self-weight of the structure. Consequently, with an increase in the total inertia of the system while the stiffness more or less remains unchanged, the natural frequencies of an elastic structure interacting with fluid may be considerably lower compared to those of in ambient air.

A fluid-structure interaction (FSI) problem inherently involves the mutual transfer of loads between structural and fluid domains. To achieve more accurate results, two-way FSI algorithms should be employed. However, when dealing with small structural deformations, one-way FSI assumptions can significantly simplify mathematical models and reduce computation time. Recognizing this advantage, Bishop et al. [1] proposed a novel methodology based on one-way FSI approach. In this methodology, known as linear hydroelasticity theory, problems related to structure and fluid domains are solved by finite element method and boundary element method, respectively. Due to its capability to handle a wide variety of geometries and its favourable computational cost compared to highly time-consuming tools such as computational fluid dynamics (CFD) and smoothed particle hydrodynamics (SPH), linear hydroelasticity theory is still widely used in modern engineering applications. This methodology has been adopted by numerous researchers to perform FSI analysis of various problems (for instance, see Fu and Price [2], Fu et al. [3], Ergin et al. [4], Ergin and Uğurlu [5] and Riggs et al. [6]). In this context, Uğurlu et al. [7] analysed free vibration characteristics of rectangular plates resting on two-parameter elastic foundation and interacting with various levels of fluid using FE-BE methodology. Monterrubio and Krysl [8] carried out a comprehensive study to analyze the natural frequencies and mode shapes of submerged structures with various geometries. In this study, mathematical model based on FE-BE method was adopted; however, for structural and fluid domains, non-matching discretized models were used. Using the same methodology, Ardic et al. [9] obtained the dynamic response characteristics of horizontal cylindrical shells and rectangular plates partially in contact with stationary fluid and discussed the distortions of wet mode shapes compared to those in the ambient air. Recently, Nie et al. [10] performed wet modal analysis of liquid storage tanks with cyclic periodicity using FE-BE approach.

Analytical methods are widely used to analyze vibratory characteristics of elastic structures interacting with fluid (for instance, see Haddara et al. [11], Kwak [12], Meylan [13], Jeong [14], Jeong et al. [15], Askari et al. [16, 17], Chen et al. [18], and Kim and Kwak [19]). Among these, variational principles based on the minimization of the energy functional of fluid-structure system are recognized as well-established tools and have a wide range of applications for numerous FSI configurations. In fact, variational principles [20 –22] are drawing a great deal of interest in the development of efficient models for complex multi-scale, multi-physics problems, such as strain-gradient modeling [23 –25], fracture [26, 27], granular media [28, 29], thermoelasticity [30, 31], and wave propagation [32 –34]. These analytical methods offer significant advantages in terms of their comparably low computational cost and ease of applicability, yet their applicability is primarily limited to simple structural and fluid domains. In this context, Cheung and Zhou [35] studied the vibratory characteristics of rectangular bottom plates interacting with fluid inside the rigid prismatic container by using Rayleigh–Ritz approach. Using the same methodology, Zhou and Cheung [36] examined the wet modal characteristics of vertical rectangular plates partially in contact with water on one side. In both these studies, the effect of various boundary conditions on the natural frequencies of an elastic plate was investigated. By taking into consideration the effect of free surface waves and using combined Rayleigh–Ritz and Galerkin approaches, Zhou and Liu [37] performed three-dimensional hydroelastic analysis of prismatic liquid storage tanks formed by four elastic rectangular plates. Segura et al. [38] examined the vibratory characteristics of simple-supported rectangular plates in the presence of fluid. In this study, the mathematical model of the structural domain is established based on Pagano’s [39] elasticity solution, and the wet mode shapes are obtained directly by solving coupled-problem, without making any a priori assumption. Recently, Verma and Kumar [40] used closed-form mode shape expressions of Euler–Bernoulli beams to obtain natural frequencies and distorted wet mode shapes of rectangular plates partially in contact with liquid. Furthermore, the aforementioned analytical approaches were also used to examine fluid-structure-soil interaction effects. In this direction, Hashemi et al. [41] carried out an analytical study to obtain wet modal characteristics of vertically positioned shear deformable plates in contact with fluid, and comprehensive results were presented for various fluid depth ratios, foundation parameters and aspect ratios.

The utilization of composite materials has experienced a consistent upward trend due to their inherent advantages: superior fatigue and corrosion resistance, enhanced mechanical properties, an optimized strength-to-weight ratio. Notably, over the past decade, there has been a significant increase in research efforts focused on the mechanical performance of composite plates (for instance, see Shahbaztabar and Ranji [42], Canales and Mantari [43], Browning and Askes [44], Thinh et al. [45], Ramian et al. [46], Bendahmane et al. [47], Wang et al. [48], Ramian et al. [49], Li et al. [50], and Shahabad et al. [51]). In line with this reseach trend, Farsani et al. [52] analyzed the effect of thickness variation and porosity on the free vibration characteristics of functionally graded plates partially in contact with fluid. Pham et al. [53] performed a rigorous study to examine the effect of boundary conditions, geometrical configurations, mechanical propertie, and porosity on the vibration characteristics of multi-directional functionally graded plates. In this study, the eigenvalue problem was solved by Galerkin–Vlasov variational method, and the results are presented for different levels of fluid. Ulbricht et al. [54] extended the work of Segura et al. [38] to investigate the effect of fluid presence on the free vibration characteristics of angle-ply laminated composite bottom plates.

Plates reinforced with stiffeners serve as primary components across various engineering disciplines, including civil, marine, naval, aerospace, and mechanical engineering. The incorporation of stiffeners significantly improves the plate’s load-bearing capacity and mitigates buckling risks [55]. However, the dynamic behavior of stiffened plates differs substantially from that of unstiffened plates, particularly when interacting with fluid, a topic that has been extensively studied by researchers. Among these, Takeda and Niwa [56] performed hydroelastic analysis of stiffened vertical plates by using the energy method. The velocity potential of fluid is represented by a series expansion of harmonic waves, and wet natural frequencies were presented by various fluid depth ratios. Using the Rayleigh–Ritz method and Mindlin plate theory, Li et al. [57] performed a hydroelastic modal analysis of stiffened plates in both air and fluid environments, and experiments were conducted for validation. Singh and Pal [58] applied FE method to analyze the forced response of stiffened plates in the dam-reservoir system. In this study, the authors presented results for both undisturbed and linearized fluid-free surface boundary conditions. Moreover, several researchers extended assumed mode method [59] to reveal vibration characteristics of stiffened bottom plates interacting with fluid inside the liquid storage tank [60], and stiffened vertical plates partially in contact with fluid [61]. In all these studies, the stiffened plate is modeled as first-order shear deformable, and zero dynamic pressure on fluid-free surface assumption is made.

Isogeometric analysis (IGA), pioneered by Hughes et al. [62] in 2005, is recognized as an efficient method for providing a seamless connection between computer-aided design (CAD) and finite element analysis (FEA). The distinctive feature of IGA arises from its utilization of identical basis functions, typically non-uniform rational B-splines (NURBS), for both geometric representation and solution field approximation. This innovative approach enables precise geometry representation even at the most basic level of discretization, presenting a significant advantage over conventional FEA techniques. IGA incorporates several key benefits compared to traditional methods, including enhanced precision per degree of freedom, greater geometric adaptability, and smooth integration with CAD platforms. Moreover, IGA provides a higher degree of continuity throughout element interfaces, highlighting its merit especially for problems involving higher-order derivatives, such as those arising in shell and plate theories. From a theoretical point of view, IGA is based on the isoparametric principle, employing uniform basis functions for both geometry and analysis. This unified strategy promotes more effective numerical integration and straightforward mesh refinement procedures, like knot insertion and order elevation, without compromising the underlying geometry. Hence, since it was introduced, IGA has been extensively adapted for different problems (for example, see Cazzani et al. [63], Greco et al. [64], Schulte et al. [65], Greco and Cuomo [66], and Obohat et al. [67])

The authors have extensively studied FSI problems involving elastic structures in the context of linear hydroelasticity theory. For example, Ergin and Temarel [68] used the boundary integral equation method to examine FSI issues of a partially filled and submerged horizontal cylindrical shell. In this study, they employed the image method to apply the infinite frequency free surface condition and described FSI effects as generalized added mass coefficients, utilizing a constant source strength distribution across the panels over the wetted surface of the structure. Yildizdag et al. [69] modified this methodology, introducing NURBS-based basis functions in both FE and BE formulations to analyze the vibration characteristics of thin rectangular plates partially/fully submerged into fluid medium. In the following studies by Yildizdag et al. [70, 71], this framework was further extended to examine the hydroelastic vibration characteristics of horizontal cylindrical shells interacting with fluid, by implementing Kirchhoff–Love thin shell formulation, for undisturbed and linearized fluid-free surface boundary conditions, respectively. Recently, Ardic et al. [72] adapted this NURBS-based isogeometric FE-BE methodology to analyze both in vacuo and wet modal characteristics of clamped circular plates partially in contact with fluid. To validate the calculated results, they conducted comprehensive experiments.

This study presents a comprehensive numerical approach for analyzing the natural frequencies and corresponding mode shapes of cantilevered Mindlin plates partially or totally immersed in fluid. The fluid is modeled using potential flow theory, where it is assumed to be inviscid, incompressible, and irrotational. By considering the structure to vibrate at relatively high frequencies and assuming that the resulting fluid pressure is in phase with the structural acceleration, the problem is simplified to a linearized fluid-structure system. The generalized FSI forces are derived from the linear form of Bernoulli’s equation. Utilizing this linearized hydroelasticity theory, the FSI problem is decoupled into two parts: an in vacuo problem and a wet problem, allowing them to be analyzed independently. For the in vacuo analysis, in the absence of external forces and structural damping, the generalized equation of motion is solved using the isogeometric finite element method (IGAFEM). The normal velocity components of the fluid on the wetted surface of the structure are determined by enforcing the continuity condition, which expresses the normal velocities in terms of the in vacuo modal displacements. In the wet analysis, the isogeometric boundary element method (IGABEM) is employed to calculate the hydrodynamic inertia corresponding to each principal mode. This is done by discretizing the wetted part of the structure into hydrodynamic panels. In the absence of axial flow, free surface waves, and viscous effects, the interaction between the structure and the fluid is governed solely by the fluid’s inertial effects, represented as added mass coefficients. The total generalized mass matrix, formed by combining the generalized structural mass and the hydrodynamic added mass matrices, is then used to solve the eigenvalue problem for the fluid-structure system. To evaluate the validity and accuracy of the proposed numerical framework, both the in vacuo and wet dynamic characteristics of the structure are compared with results obtained from the commercial finite element software ANSYS, demonstrating the effectiveness of the method. Subsequently, the study investigates the characteristics and behavior of the added mass and hydrodynamic damping in the low-frequency region, where the effects of the free surface waves due to vibrating structure are significant.

2. Numerical model

2.1. Introduction to NURBS

NURBS serve as a fundamental tool in CAD systems for generating 3D virtual models. To define a NURBS curve, a knot vector, denoted as $Ξ (ξ)$ , is established as a set of non-decreasing real numbers in the form:

Ξ (ξ) = [ξ_{1}, ξ_{2}, . . ., ξ_{n + p + 1}], where ξ_{i} \in R .

(1)

Here, $i$ represents the knot index ranging from $1$ to $n + p + 1$ , where $p$ signifies the polynomial degree and $n$ denotes the number of basis functions composing the B-spline. A $p$ th-degree NURBS curve is defined as:

C (ξ) = \sum_{i = 1}^{n} R_{i}^{p} (ξ) P_{i},

(2)

where $P_{i}$ represents the coordinate of the $i$ th control point, and $R_{i}^{p} (ξ)$ stands for the $p$ th-degree basis function, defined as follows:

R_{i}^{p} (ξ) = \frac{N_{i, p} (ξ) w_{i}}{\sum_{\hat{j} = 1}^{n} N_{\hat{j}, p} (ξ) w_{\hat{j}}} .

(3)

Here, $w_{i}$ denotes the weight of the corresponding control point, and $N_{i, p} (ξ)$ represents the $p$ th-degree B-spline basis function, defined as follows for $p = 0$ :

N_{i, 0} (ξ) = {\begin{matrix} 1 if ξ_{i} \leq ξ \leq ξ_{i + 1}, \\ 0 otherwise, \end{matrix}

and for $p = 1, 2, 3, . . .$ :

N_{i, p} (ξ) = \frac{ξ - ξ_{i}}{ξ_{i + p} - ξ_{i}} N_{i, p - 1} (ξ) + \frac{ξ_{i + p + 1} - ξ}{ξ_{i + p + 1} - ξ_{i + 1}} N_{i + 1, p - 1} (ξ) .

(4)

Similarly, an NURBS surface is defined through the tensor product of 1D basis functions, involving two parametric directions:

S (ξ, η) = \sum_{i = 1}^{n} \sum_{j = 1}^{m} R_{i, j}^{p, q} (ξ, η) P_{i, j},

(5)

where $R_{i, j}^{p, q} (ξ, η)$ is defined as:

R_{i, j}^{p, q} = \frac{N_{i, p} (ξ) M_{j, q} (η) w_{i, j}}{\sum_{î = 1}^{n} \sum_{\hat{j} = 1}^{m} N_{î, p} (ξ) M_{\hat{j}, q} (η) w_{î, \hat{j}}},

(6)

where $P_{i, j}$ represents the net of control points and $w_{i, j}$ represents the weight of the corresponding control point. The terms $N_{i, p} (ξ)$ and $M_{j, q} (η)$ are respectively 1D $p$ th- and $q$ th-degree B-spline basis functions, defined by the following knot vectors:

Ξ = [ξ_{1}, ξ_{2}, \dots, ξ_{n + p + 1}] ℋ = [η_{1}, η_{2}, \dots, η_{m + q + 1}],

(7)

where $n$ and $m$ represent the number of B-spline basis functions in each parametric direction $ξ$ and $η$ , respectively. Figure 1 provides an example of an NURBS surface, illustrating its elements and control net along with its parametric domain defined by knot vectors $Ξ$ and $H$ . In this study, all spline and IGA connectivity algorithms were implemented following Piegl and Tiller [73] and Cottrell et al. [74].

2.2. Isogeometric hybrid FE-BE numerical framework

Let us consider a plate with a thickness $t$ , positioned in the $x - y$ plane, as depicted in Figure 2. In this configuration, the orthonormal vector set ${E_{x}, E_{y}, E_{z}}$ , aligned along the $x$ , $y$ , and $z$ -axes, respectively, establishes a Cartesian coordinate system with its origin located at the mid-surface of the plate. Based on the assumptions of Mindlin plate theory, the displacement vector at any given point is expressed as:

u = z θ_{y} E_{x} - z θ_{x} E_{y} + w (x, y) E_{z}

(8)

where $θ_{x}$ and $θ_{y}$ are the rotations about the $x$ and $y$ -axes, respectively, and $w$ denotes the displacement along the $z$ -axis. The components of the linearized Green–Lagrange strain tensor ( $ε$ ) are defined using displacement vector in equation (8) as follows

ε_{xx} = z \frac{d θ_{y}}{dx},

(9)

ε_{yy} = - z \frac{d θ_{x}}{dy},

(10)

2 ε_{xy} = z \frac{d θ_{y}}{dy} - z \frac{d θ_{x}}{dx},

(11)

2 ε_{yz} = - θ_{x} + \frac{dw}{dy},

(12)

2 ε_{xz} = θ_{y} + \frac{dw}{dx} .

(13)

Using the mid plane strain field given above, the mid-surface of the plate is represented by a single NURBS surface. Within each element, the deflection and two rotations ( $w$ , $θ_{x}$ , and $θ_{y}$ ) are approximated using NURBS shape functions as follows

w^{e} = \sum_{i = 1}^{p + 1} \sum_{j = 1}^{q + 1} R_{i, j}^{e} (ξ, η) w_{i, j}^{e} \equiv \sum_{I = 1}^{n_{cp}} R_{I}^{e} w_{I}^{e},

(14)

θ_{x}^{e} = \sum_{i = 1}^{p + 1} \sum_{j = 1}^{q + 1} R_{i, j}^{e} (ξ, η) θ_{x i, j}^{e} \equiv \sum_{I = 1}^{n_{cp}} R_{I}^{e} θ_{x I}^{e},

(15)

θ_{y}^{e} = \sum_{i = 1}^{p + 1} \sum_{j = 1}^{q + 1} R_{i, j}^{e} (ξ, η) θ_{y i, j}^{e} \equiv \sum_{I = 1}^{n_{cp}} R_{I}^{e} θ_{y I}^{e} .

(16)

Referring to Section 2.1, in each element, there are $p + 1$ non-zero shape functions in the parametric direction $ξ$ and $q + 1$ non-zero shape functions in the parametric direction $η$ . Therefore, the approximation within each element involves $n_{cp}$ non-zero shape functions ( $R_{I}^{e}$ ), where $n_{cp} = (p + 1) (q + 1)$ . Based on the above approximations, the displacement vector within each element can be expressed as:

u^{e} = [\begin{matrix} z θ_{y}^{e} \\ - z θ_{x}^{e} \\ w^{e} \end{matrix}] = [\begin{matrix} 0 & 0 & z \\ 0 & - z & 0 \\ 1 & 0 & 0 \end{matrix}] [\begin{matrix} w^{e} \\ θ_{x}^{e} \\ θ_{y}^{e} \end{matrix}] \equiv [R] [N_{p}^{e}] [d^{e}] .

(17)

The matrix $[N_{p}^{e}]$ is a $3 \times 3 n_{cp}$ interpolation matrix,

[N_{p}^{e}] = {[R_{1}^{e} I \cdot \cdot \cdot R_{n_{cp}}^{e} I]}_{3 \times 3 n_{cp}}

(18)

where $I$ is a $3 \times 3$ identity matrix, $[d^{e}]$ is a vector of size $3 n_{cp}$ consisting of the independent variables of the displacement field,

[d^{e}] = {[\begin{matrix} d_{1}^{e} \\ . \\ . \\ . \\ d_{n_{cp}}^{e} \end{matrix}]}_{3 n_{cp} \times 1} where d_{I}^{e} = {[\begin{matrix} w_{I}^{e} \\ {θ_{x}^{e}}_{I} \\ {θ_{y}^{e}}_{I} \end{matrix}]}_{3 \times 1}

(19)

Similarly, the normal and tangential components of the strain tensor within each element are written as

ε^{e} = [\begin{matrix} ε_{xx}^{e} \\ ε_{yy}^{e} \\ 2 ε_{xy}^{e} \\ 2 ε_{yz}^{e} \\ 2 ε_{xz}^{e} \end{matrix}] = [\begin{matrix} z & 0 & 0 & 0 & 0 \\ 0 & z & 0 & 0 & 0 \\ 0 & 0 & z & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix}] [\begin{matrix} \frac{d θ_{y}^{e}}{dx} \\ - \frac{d θ_{x}^{e}}{dy} \\ \frac{d θ_{y}^{e}}{dy} - \frac{d θ_{x}^{e}}{dx} \\ - θ_{x}^{e} + \frac{d w^{e}}{dy} \\ θ_{y}^{e} + \frac{d w^{e}}{dx} \end{matrix}] \equiv [S] [B^{e}] [d^{e}] .

(20)

In this equation, $[B^{e}]$ is the strain-displacement matrix with the dimensions $5 \times 3 n_{cp}$ :

[B^{e}] = {[{[B^{e}]}_{1} \cdot \cdot \cdot {[B^{e}]}_{n_{cp}}]}_{5 \times 3 n_{cp}} where {[B^{e}]}_{I} = {[\begin{matrix} 0 & 0 & \frac{{dR}_{I}^{e}}{dx} \\ 0 & - \frac{{dR}_{I}^{e}}{dy} & 0 \\ 0 & - \frac{{dR}_{I}^{e}}{dx} & \frac{{dR}_{I}^{e}}{dy} \\ \frac{{dR}_{I}^{e}}{dy} & - R_{I}^{e} & 0 \\ \frac{{dR}_{I}^{e}}{dx} & R_{I}^{e} & 0 \end{matrix}]}_{5 \times 3}

(21)

By utilizing the strain and kinetic energy functionals of the plate, the virtual work expression for the free vibration problem is written as

\int_{V} ρ_{p} δ u \cdot \overset{\cdot\cdot}{u} dV + \int_{V} δ ε \cdot σ (u) dV = 0,

(22)

where $δ u$ and $δ ε$ are the virtual displacement vector and virtual strain tensor, respectively, $σ$ is the stress tensor, and $ρ_{p}$ is the mass density of the plate. Finally, the element stiffness and mass matrices are derived by substituting strain tensor given in equation (20) into the virtual work epxression (equation (22)) and integrating it along the thickness, as follows

[k^{e}] = \int_{A^{e}} {[B^{e}]}^{T} [\bar{C}] [B^{e}] dA,

(23)

[m^{e}] = \int_{A^{e}} {[N^{e}]}^{T} [\bar{m}] [N^{e}] dA .

(24)

Assuming the material is linear, homogeneous and isotropic; the stresses and strains are linearly related to each other through the constitutive matrix. The matrices $[\bar{C}]$ and $[\bar{m}]$ in equations (23) and (24) are defined as follows:

[\bar{C}] = {[\begin{matrix} D_{b} & D_{b} ν & 0 & 0 & 0 \\ D_{b} ν & D_{b} & 0 & 0 & 0 \\ 0 & 0 & \frac{D_{b} (1 - ν)}{2} & 0 & 0 \\ 0 & 0 & 0 & D_{s} & 0 \\ 0 & 0 & 0 & 0 & D_{s} \end{matrix}]}_{5 \times 5}, [\bar{m}] = ρ_{p} {[\begin{matrix} t & 0 & 0 \\ 0 & \frac{t^{3}}{12} & 0 \\ 0 & 0 & \frac{t^{3}}{12} \end{matrix}]}_{3 \times 3},

(25)

where

D_{b} = \frac{E t^{3}}{12 (1 - ν^{2})} and D_{s} = \frac{λ Et}{2 (1 + ν)} .

(26)

The terms $E$ , $ν$ , and $λ$ denote Young’s modulus, Poisson’s ratio, and shear correction factor, respectively. The global stiffness and mass matrices are formed by assembling element stiffness and mass matrices, which are calculated by numerical integration over each surface element using Gauss quadrature. In the absence of external forces, the differential equations of the motion for an undamped system can be written in the matrix form as

M \overset{\cdot\cdot}{d} (t) + Kd (t) = 0

(27)

where $M$ is the global mass matrix, and $K$ is the global stiffness matrix of the plate. In addition, $d (t)$ and $\overset{\cdot\cdot}{d} (t)$ correspond to the displacement and acceleration vectors of the system, respectively. By assuming that the solution of equation (27) is harmonic with frequency, a trial solution can be written as:

d = \bar{d} e^{i ω t},

(28)

where $ω$ is the frequency of the harmonic oscillation, $t$ is time, and $\bar{d}$ is a vector consisting of displacement and rotation amplitudes. Substituting the trial solution from equation (28) into equation (27), one can obtain the following algebraic eigenvalue problem

(- ω^{2} M + K) \bar{d} = 0,

(29)

equation (29) has a non-trivial solution only if the determinant of the coefficients matrix is zero

\det [- ω^{2} M + K] = 0,

(30)

equation (30) is known as the characteristic equation of the free vibration problem. This characteristic equation has N distinct roots for a system with $N$ degrees of freedom. These roots of the system are denoted as $ω_{1}^{2}$ , $ω_{2}^{2}$ , …, $ω_{N}^{2}$ and the square roots of these quantities are the natural frequencies of the system, namely $ω_{r}$ ( $r$ = 1, 2, …, N). Corresponding to each natural frequency $ω_{r}$ , an eigenvector ${\bar{d}}_{r}$ , representing the displacement amplitudes, can be determined. This eigenvector ${\bar{d}}_{r}$ describes the $r$ th principal mode, which is associated with the $r$ th natural frequency $ω_{r}$ . Using the superposition principle and the orthogonality of the eigenvectors, the nodal distortions of the structure with $N$ degrees of freedom can be written as a linear combination of the principal modes as follows:

d = \sum_{r = 1}^{N} {\bar{d}}_{r} p_{r} (t) = \bar{D} p (t)

(31)

Here, $\bar{D}$ is the $N \times N$ modal matrix consisting of eigenvectors, and $p$ is the $N \times 1$ vector of principal coordinates. Finally, by substituting equation (31) into equation (27) and premultiplying by ${\bar{D}}^{T}$ , the following generalized equation of motion is written in terms of the principal coordinates as follows

(- ω^{2} a + c) p = 0

(32)

where $a$ and $c$ are the generalized structural mass and stiffness matrices, respectively,

a = {\bar{D}}^{T} M \bar{D}, c = {\bar{D}}^{T} K \bar{D} .

(33)

In the next part of the numerical model, fluid actions are introduced and applied to the structure as external loading. The plate is assumed to oscillate in its in vacuo mode shapes when it interact with fluid. By assuming that the surrounding fluid is incompressible, inviscid, and its motion is irrotational; the fluid velocity vector $v (x, t)$ at time $t$ at the point $x = (x, y, z)$ can be described as the gradient of the velocity potential

v (x, t) = \nabla Φ (x, t)

(34)

where $Φ$ is the unsteady velocity potential associated with the perturbation of the flow field due to the nodal distortions of the structure, and governed by Laplace’s equation throughout the fluid domain, that is

\nabla^{2} Φ = 0

(35)

Considering a harmonic oscillatory motion in the surrounding fluid environment, the dynamic response of the vibrating structure can be written as

p (t) = p_{0} e^{i ω t}

(36)

where $p_{0}$ denotes the response amplitude vector and $ω$ is the frequency of the harmonic oscillation of the flexible structure, and $t$ is the time. The velocity potential function due to the vibration of the structure in the $r$ th in vacuo mode can be expressed as follows

Φ_{r} (x, t) = ϕ_{r} (x) p_{0_{r}} e^{i ω t}, r = 1, 2, . . ., M,

(37)

where $M$ represents the number of modes of interest, and $p_{0_{r}}$ is an unknown amplitude for the $r$ th principal coordinate. The terms $ϕ_{r} (x)$ ( $r$ = 1, 2, …) are the radiation potentials due to vibrations of the flexible structure. On the wetted surface of the oscillating plate, the kinematic continuity conditions is satisfied by the velocity potential function. This requires that the structure and the fluid move together at their interface, and therefore, the normal fluid velocity must be equal to the normal velocity of the structure, and this condition for the $r$ th modal vibration can be expressed as

\frac{\partial ϕ_{r}}{\partial n} = i ω u_{r} \cdot n

(38)

where ${\bar{u}}_{r}$ is the $r$ th modal displacement vector determined via in vacuo modal analysis and $n$ is the outward unit normal vector on the wetted surface. The linear wave theory is employed under the assumption that the steepness of free surface waves is significantly small compared to the wavelength. Based on this assumption, the velocity potential must satisfy the linearized free surface boundary condition, which is expressed as

g \frac{\partial ϕ_{r}}{\partial z} - ω^{2} ϕ_{r} = 0 .

(39)

Here, $ω$ represents the frequency of the free surface waves and $g$ is the gravitational acceleration. When the natural frequencies of the elastic structure lie in a high-frequency range, the influence of free surface waves generated by the vibrating structure becomes negligibly small. Consequently, the effect of free surface waves can be effectively ignored. In this scenario, the acceleration of the fluid particles is primarily due to the high-frequency vibrations of the structure, rather than the gravitational effects. Therefore, the first term in the free surface boundary condition, as given in equation (39), vanishes in the high-frequency range, simplifying the free surface boundary condition to

ϕ_{r} = 0

(40)

Using Bernoulli’s equation and neglecting the second-order terms and gravitational effects, the dynamic fluid pressure on the elastic structure due to the $r$ th modal vibration becomes

P_{r} (x, t) = - ρ \frac{\partial Φ_{r}}{\partial t} .

(41)

The $k$ th component of the generalized FSI force amplitude due to the $r$ th in vacuo vibration mode of the structure in terms of the pressure acting on the wetted surface as

\begin{matrix} Z_{kr} = - \int_{S_{w}} P_{r} u_{k} \cdot n dS = \int_{S_{w}} ρ \frac{\partial Φ_{r}}{\partial t} dS = \int_{S_{w}} ρ (i ω) ϕ_{r} u_{k} \cdot n dS \\ = ω^{2} A_{kr} - i ω B_{kr} \end{matrix}

(42)

In the above expression, $A_{kr}$ and $B_{kr}$ , respectively, represent the generalized added mass coefficient in phase with the structural acceleration and the generalized hydrodynamic damping coefficient in phase with the velocity. Namely,

A_{kr} = \frac{ρ}{ω^{2}} Re [\int_{S_{w}} ρ (i ω) ϕ_{r} u_{k} \cdot n],

(43)

B_{kr} = - \frac{ρ}{ω^{2}} Im [\int_{S_{w}} ρ (i ω) ϕ_{r} u_{k} \cdot n] .

(44)

In this study, the velocity potentials are expressed through a distribution of sources over the wetted surfaces of the plate. This is accomplished by applying the following boundary integral equation for potential flow problem

c (Q) ϕ (Q) + \int_{S_{w}} ϕ (P) \frac{\partial G (P, Q)}{\partial n} dS = \int_{S_{w}} G (P, Q) \frac{\partial ϕ}{\partial n} (P) dS, P \in S_{w} .

(45)

Here, the terms $P$ and $Q$ represent the locations of source and field points, respectively. In the equation (45), the Green’s function $G (P, Q)$ , satisfying the free surface boundary condition given in equation (39), is defined as follows Telste and Noblesse [75] and Xie et al. [76]

G (P, Q) = - \frac{1}{4 π} (\frac{1}{R} + \frac{1}{R'}) - \frac{\bar{g} (P, Q)}{4 π} - \frac{i k_{0}}{2} e^{k_{0} Z} J_{0} (k_{0} r) .

(46)

In equation (46), $1 / R$ represents Rankine source whereas $1 / R^{'}$ corresponds to the image source, $P' (x_{P^{'}}, y_{P^{'}}, z_{P^{'}})$ , computed through mirroring the source point with respect to free surface. The term $\bar{g} (P, Q)$ is the free-surface term which takes different representations according to the locations of the source and field positions, and frequency of interest (see Telste and Noblesse [75]). The term $k_{0}$ is the wave number, $k_{0} = ω^{2} / g$ , where $ω$ is the frequency. The term $Z$ denotes the vertical distance between source point ( $P$ ) and its image ( $P'$ ), and $r$ represents the horizontal distance between source ( $P$ ) and field ( $Q$ ) points. Also, $J_{0}$ represents the zeroth-order Bessel function. The distances $R$ , $R'$ , and $r$ are defined as follows

R = \sqrt{{(x_{Q} - x_{P})}^{2} + {(y_{Q} - y_{P})}^{2} + {(z_{Q} - z_{P})}^{2}},

(47)

R' = \sqrt{{(x_{Q} - x_{P})}^{2} + {(y_{Q} - y_{P})}^{2} + {(z_{Q} - z_{P^{'}})}^{2}},

(48)

r = \sqrt{{(x_{Q} - x_{P})}^{2} + {(y_{Q} - y_{P})}^{2}} .

(49)

As mentioned earlier, in the high-frequency region, the boundary condition given in equation (40) applies. The Green’s function that satisfies this condition is a simplified version of equation (46), where the frequency-dependent terms are omitted. This simplified form is written as:

G_{\infty} (P, Q) = - \frac{1}{4 π} (\frac{1}{R} + \frac{1}{R'})

(50)

In the equation (45), the term $c (Q)$ denotes the solid angle, and it is calculated by using the argument based on uniform potential applied to a closed domain does not produce flux [77], i.e.,

c (Q) = \int_{S_{w}} \frac{\partial}{\partial n} (\frac{1}{4 π R}) dS .

(51)

In the case where the wetted surface $S_{w}$ is not closed, e.g., partially immersed structure, an imaginary surface patch is introduced to compute the solid angle terms. In this study, for the numerical evaluation of the Green’s function and its gradient, the numerical algorithm presented by Telste and Noblesse [75] is adopted. Equation (45) is solved numerically by IGABEM, using NURBS representation of the wetted surface. The velocity potential is approximated considering multi-patch representation of wetted surface as follows

ϕ^{s} = \sum_{i = 1}^{n} \sum_{j = 1}^{m} R_{i, j}^{s} ϕ_{i, j}^{s} \equiv \sum_{I = 1}^{n_{cp}} R_{I}^{e} w_{I}^{e},

(52)

where indice $s$ is the patch number. Then, a system of equations when equation (45) is written for each field point are as follows:

c_{i, j}^{s} ϕ_{i, j}^{s} + \sum_{s = 1}^{n_{p}} (\sum_{e = 1}^{n_{e}} \sum_{k = 1}^{n} \sum_{l = 1}^{m} \int_{S_{w}^{e}} R_{k, l}^{s} ϕ_{k, l}^{s} \frac{\partial G}{\partial n} d S^{e}) = \sum_{s = 1}^{n_{p}} (\sum_{e = 1}^{n_{e}} \sum_{k = 1}^{n} \sum_{l = 1}^{m} \int_{S_{w}^{e}} R_{k, l}^{s} G \frac{\partial ϕ_{k, l}^{s}}{\partial n} d S^{e}) .

(53)

Here, $n_{p}$ is the number of NURBS patches considered in the IGABEM analysis. Therefore, $c_{i, j}^{s}$ and $ϕ_{i, j}^{s}$ are the solid angle and velocity potentials, respectively, for the control points belong to $s$ th NURBS patch. $R_{k, l}^{s}$ and $ϕ_{k, l}^{s}$ represent the rational basis functions and velocity potentials at control points of $s$ th NURBS patch, respectively. The term $S_{w}^{e}$ is the area of the corresponding element, where the numerical integration is performed. Equation (53) yields the following system of equations

[H] {ϕ} = [G] {q} .

(54)

The flux terms in the right hand side of equation (53), $\partial ϕ_{i, j}^{s} / \partial n$ , are defined by using the kinematic boundary condition provided in equation (38). The calculation of wet natural frequencies and corresponding mode shapes is based on high-frequency assumptions. As previously mentioned, in the high-frequency region, the free-surface boundary condition in equation (40) applies, resulting in zero hydrodynamic damping while the added mass reaches its high-frequency limit. Under these conditions, the velocity potentials are calculated by solving the equation given in equation (54) using the Green’s function from equation (50). Subsequently, the infinite-frequency generalized added mass coefficients are computed using equation (43). Therefore, in the generalized equation of motion of the plate interacting with fluid, the fluid effects is represented solely by the infinite-frequency generalized added mass coefficients, and the equation of motion can be expressed as

[- ω^{2} (a + A_{\infty}) + c] p_{0} = 0

(55)

Here, $A_{\infty}$ is the infinite-frequency generalized added mass matrix. The wet natural frequencies and associated mode shapes of the plate in contact with fluid are determined by solving the eigenvalue problem given in equation (55). For each wet natural frequency $ω_{r}$ , there is a corresponding wet eigenvector $p_{0 r}$ = ${p_{r 1}, p_{r 1}, p_{r 1}, \dots, p_{rM}}$ that satisfies the equation (55). The wet mode shapes of the cantilevered Mindlin plate, either partially or totally immersed in fluid, can be expressed in the following manner

{\bar{u}}_{r} (x, y, z) = \sum_{j = 1}^{N} u_{r} (x, y, z) p_{r_{j}}

(56)

where $u_{r} (x, y, z)$ represents the mode shapes of the cantilevered Mindlin plate under in vacuo conditions, and M is the number of in vacuo mode shapes used to generate the wet mode shapes. In order to elucidate the behavior of added mass and hydrodynamic damping coefficients in the low-frequency region, additional computations are performed to obtain frequency-dependent velocity potential. For this purpose, the system of equations given in equation (54) are solved by using free-surface Green’s function provided in equation (46). Subsequently, the generalized added mass and hydrodynamic damping coefficients are obtained based on the expressions given in equations (43) and (44), respectively.

3. Results and discussion

In this section, the presented hybrid isogeometric FE-BE framework is applied to a cantilevered rectangular plate, whose dynamic response characteristics are available in the open literature. Initially, the natural frequencies and corresponding mode shapes of the cantilevered rectangular plate are investigated in the absence of a surrounding fluid environment (i.e., in vacuo or dry analysis). The accuracy of these results is verified by comparing them with those obtained from analytical and experimental studies. Following this verification, a parametric study is conducted for various length-to-thickness ratios. Subsequently, the surrounding fluid environment is introduced to investigate its impact on the dynamic characteristics of the test plate. The effect of immersion depth is also examined for several submergence levels (i.e., wet analysis). The accuracy of the wet natural frequencies and corresponding mode shapes is assessed through comparison with results from the literature and simulations using the commercial software ANSYS. Finally, comprehensive results for various length-to-thickness ratios are presented, and the characteristics of hydrodynamic coefficients are analyzed across a certain frequency spectrum.

To evaluate the accuracy, reliability, and effectiveness of the presented IGAFEM framework, we examine an isotropic clamped plate, whose dry dynamic characteristics are obtained via experimental measurements by Lindholm et al. [78]. The material properties of the test specimen are as follows: Young’s modulus $E =$ 206.8 GPa, Poisson’s ratio $ν =$ 0.3, and mass density $ρ_{s} =$ 7830 kg/m³. The test specimen is a square plate with edge lengths of $a = b =$ 10 x 10 m and a thickness $t =$ 0.238 m. This plate is clamped along one edge between two smooth surfaces, while the other edges are free. The convergence of the IGAFEM results is assessed under four different levels of idealization. For the coarsest level of idealization, the number of elements along all edges of the square plate is set to four, and this number is doubled for each subsequent level of idealization.

Table 1 illustrates the convergence characteristics of the first six in vacuo natural frequencies obtained via the IGAFEM approach. As shown, the fastest convergence rates are observed for refined discretizations, and monotonic convergence characteristics are evident, with negligible differences between the results for the third and fourth levels of idealization. In particular, excellent accuracy is achieved with IGAFEM even at the coarsest level of idealization, where the maximum deviation from the analytical results is only about 1.0% . Furthermore, the dry natural frequencies predicted by the proposed numerical framework show favorable agreement with those obtained analytically and experimentally. The discrepancies in the natural frequencies between those obtained by the proposed IGAFEM approach and the analytical method are all below 1.1%, while the discrepancies with the experimental results are limited to 4.3%. Furthermore, the results of the present numerical methodology are much closer to the analytical results than those measured by the experimental setup. It is important to note that the IGAFEM results are lower than the experimental measurements, since any deviation in the rigidity of the clamped edge from the theoretical infinite value leads to a decrease in the measured natural frequencies.

Table 1.

Convergence of dry natural frequencies (rad/s) for Lindholm’s plate [78].

	$t =$ 0.238 m. ( $a / t =$ 42)
Mode no.	16 elements	64 elements	256 elements	1024 elements	Analytical [79]	Experiment [78]
	(4×4)	(8×8)	(16×16)	(32×32)
1(S)	12.83	12.83	12.83	12.83	12.92	12.30
2(A)	31.36	31.27	31.22	31.22	31.21	30.78
3(S)	78.50	78.45	78.43	78.43	78.77	75.46
4(S)	100.19	100.07	99.99	99.99	101.07	99.84
5(A)	113.82	113.51	113.35	113.35	114.10	110.57
6(S)	198.47	197.90	197.53	197.52	198.67	-

After ensuring the validity of the IGAFEM results, a parametric study is carried out for in vacuo conditions. Throughout this analysis, the edge lengths of the plate remain unchanged while the thickness is systematically increased. The parametric study was carried out for four different ratios of plate length to thickness ( $a / t$ ), with the thickness set as multiples of the thickness of the Lindholm plate ( $t =$ 0.238 m). Commercial software ANSYS was utilized to compare the results of the parametric study. To obtain more accurate results for the thick plate in ANSYS, the structure was meshed using a three-dimensional SOLID186 element, a 20-node hexahedral element with quadratic displacement behavior, with two elements assigned through the thickness. The results of the parametric study for the first six in vacuo natural frequencies are presented in Table 2. At the top of each subtable, the thickness of the plate and the length-to-thickness ratio are indicated. It is evident that the IGAFEM results compare very well with those obtained by ANSYS, with differences below 1%, even when the $a / t$ ratio is below 10.

Table 2.

In vacuo natural frequencies (rad/s) of the cantilevered square plate for various length-to-thickness ratios.

Mode	IGAFEM	ANSYS	Difference %
(a) $t =$ 0.476 m. ( $a / t =$ 21)
1(S)	25.60	25.64	0.17
2(A)	61.74	61.81	0.11
3(S)	155.21	155.45	0.16
4(S)	197.47	197.61	0.07
5(A)	222.57	222.90	0.15
6(S)	384.55	385.13	0.15
(b) $t =$ 0.714 m. ( $a / t =$ 14)
1(S)	38.28	38.36	0.21
2(A)	91.32	91.45	0.14
3(S)	229.05	229.56	0.22
4(S)	290.97	291.28	0.11
5(A)	325.60	326.28	0.21
6(S)	556.30	557.50	0.22
(c) $t =$ 0.952 m. ( $a / t =$ 10.5)
1(S)	50.84	50.96	0.23
2(A)	119.76	119.98	0.18
3(S)	298.89	299.80	0.30
4(S)	379.40	379.99	0.15
5(A)	421.00	422.18	0.28
6(S)	710.30	712.41	0.30
(d) $t =$ 1.190 m. ( $a / t =$ 8.4)
1(S)	63.23	63.41	0.29
2(A)	146.92	147.23	0.21
3(S)	364.00	365.43	0.39
4(S)	462.10	463.05	0.20
5(A)	508.04	509.85	0.36
6(S)	846.27	849.49	0.38

Figure 3 presents the first six in vacuo mode shapes of the cantilevered square plate with a thickness of $t =$ 0.238 m, obtained using the proposed IGAFEM approach and the ANSYS software. An RGB-based color scale is utilized to represent the modal displacements: red for positive, green for zero, and blue for negative. These mode shapes can be categorized by their symmetric or antisymmetric nature with respect to the vertical axis passing through the plate’s centroid. Specifically, the first, third, fourth, and sixth modes are symmetric, while the second and fifth modes are antisymmetric. Furthermore, those belonging to the symmetric group can also be classified as bending modes, whereas the others are classified as torsional modes. In this context, the abbreviations S and A in Figure 3 denote symmetry and antisymmetry of the mode shapes. It can be observed from Figure 3 that the deflection patterns of the in vacuo modes obtained by IGAFEM and ANSYS show excellent agreement in terms of the locations of the nodal lines and antinodes.

In the second part of the numerical study, referred to as wet analysis, the surrounding fluid environment is introduced into the problem, where the fluid loading is represented by the added mass coefficients in the generalized equation of motion. The efficiency of this approach was demonstrated in the previous studies published by the authors for the free vibration problem of elastic structures whose in vacuo natural frequencies lie within a relatively high range. In such cases, the perturbations of the fluid free surface are expected to be minimal, and therefore it is reasonable to assume that hydrodynamic damping and restoring effects are negligible. To validate the isogeometric-based hybrid methodology developed for the wet analysis, the cantilevered squared plate, whose in vacuo dynamic characteristics were obtained in the previous part, is investigated. The wet problem involves a surface-piercing vertical square plate that is partially or completely submerged and cantilevered from its upper edge (Figure 4), where $d$ denotes the depth of submergence and the ratio $d / a$ represents the ratio of the height of the surrounding fluid to the height of the plate. The density of the surrounding fluid is taken as 1000 kg/m³.

The validity and convergence of the wet natural frequencies are analyzed using the results obtained by Fu and Price [2] using the conventional FE-BE methodology for the half-submerged case ( $d / a =$ 0.50). To elucidate the convergence characteristics of the IGABEM results, four different groups of idealizations of hydrodynamic panels are considered. In the first group of idealizations, the two opposite wetted faces of the plate are divided into four and eight equally spaced, square-shaped hydrodynamic panels in the vertical and horizontal directions, respectively, while the number of hydrodynamic panels through the thickness of the plate is set to one. In the second to fourth group of idealizations, the panel distribution is refined by doubling the number of panels in all directions for each case. Consequently, the total number of hydrodynamic panels increases progressively, resulting in 80, 288, 1088, and 4352 panels for the first, second, third, and fourth groups of idealizations, respectively. For each scenario, the in vacuo dynamic characteristics of the structure were obtained by discretizing the plate into a total number of 400 isogeometric finite elements. In Table 3, the first six wet natural frequencies identified by IGABEM for four different groups of idealizations are presented comparatively with those obtained by Fu and Price [2]. Favourable convergence characteristics are evident in Table 3, where the discrepancies between the results for the third and fourth groups of idealizations are negligible. It is clear that the wet natural frequencies of the plate in contact with fluid are significantly lower than those of ambient air, primarily due to the increase in mass loading contributed by fluid inertia. The discrepancy between the dry and wet natural frequencies is more pronounced for lower modes; for instance, the drop rate in the fundamental natural frequency is about 56%, and it gradually decreases to 14% for the sixth mode. Moreover, the IGABEM results presented in Table 3 show excellent agreement with those reported by Fu and Price [2], with a maximum discrepancy of less than 2.8%, even for the second group of idealizations where 288 hydrodynamic panels are used.

Table 3.

Convergence of the wet natural frequencies (rad/s) obtained by the IGABEM ( $d / a =$ 0.50).

	$t =$ 0.238 m. ( $a / t =$ 42)
Mode no.	80 elements	288 elements	1088 elements	4352 elements	Fu and Price [2]
	(8×4)	(16×8)	(32×16)	(64×32)
1(S)	9.51	8.34	8.26	8.27	8.22
2(A)	24.81	22.39	22.23	22.25	21.85
3(S)	70.55	66.26	65.47	65.47	65.40
4(S)	85.21	78.02	76.67	76.64	76.68
5(A)	104.32	99.13	98.02	97.99	98.15
6(S)	185.79	177.27	174.83	174.51	-

The same problem for the depth ratio $d / a =$ 0.50 was also studied using commercial software, ANSYS. Throughout the analyses, a mathematical model based on acoustic fluid-structural interaction was used, where the equations of motion of the structure were considered along with the acoustic wave equation. The sonic velocity of fluid is taken as 1507 m/s. A two-way FSI algorithm was adopted, coupling the displacement and pressure degrees of freedom of the structure and fluid across the wetted surface. The other nodes belonging to the fluid domain, referred to as uncoupled nodes, had only one degree of freedom for pressure. The entire structural domain was meshed into eight-node hexahedral elements using the element type of SOLID185, while 3D acoustic brick elements of FLUID30 were used to simulate the fluid domain. The surrounding fluid environment was generated by extending the solid domain in all three spatial coordinates. The computational grid is illustrated in Figure 5. To ensure numerical precision, the fluid environment was modeled with sufficiently large dimensions, with edge lengths of 3 $b$ , 2 $d$ , and 4 $a$ in the $x$ , $y$ , and $z$ directions, respectively, to minimize boundary effects and obtain accurate results (see Figure 4 for the nomenclature of $a$ , $b$ , and $d$ ). As the structure vibrates in a relatively high frequency region, the problem of slosh dynamics was not taken into account, and therefore a zero-pressure boundary condition was imposed on the nodes along the free surface of the fluid. The resulting unsymmetric eigenvalue problem for the acoustic fluid-structural interaction is solved using the subspace approach based on a method designated as the frequency derivative method [80].

To test the convergence of the ANSYS results, various numbers of finite elements were distributed along the structural and fluid domains, and the accuracy of the results was then assessed by comparing them with those presented by Fu and Price [2]. A methodology similar to that used for the convergence analyses of IGABEM results was employed for ANSYS simulations. Specifically, the convergence of the first six wet natural frequencies of the plate, with edge lengths of $a = b =$ 10 m and thickness $t =$ 0.238 m for the submergence ratio of $d / a =$ 0.50, was investigated using four distinct sets of idealizations. In the first set of idealizations, the element length for the entire structural domain was set equal to the thickness of the plate, and it was systematically reduced to $t / 2$ , $t / 3$ and $t / 4$ for the second, third and fourth set of idealizations, respectively. For each group of idealizations, the element lengths in the fluid medium in close proximity to the plate were matched to those of the structural domain and were gradually increased toward the exterior boundaries of the fluid environment to enhance the computational efficiency. The results of the convergence analysis for these four different sets of idealizations are tabulated in Table 4, side by side with those of Fu and Price [2]. For each set of idealizations, the total number of elements is provided at the top of each column, along with the number specifically used to model the structural domain (indicated in parentheses). As can be observed, the natural frequencies for the third set of idealizations have satisfactorily converged, showing only minor discrepancies with those from Fu and Price [2]. Consequently, the element length of $t / 3 =$ 80 mm, corresponding to the third set of idealizations, will be used for further parametric studies in ANSYS. Furthermore, comparison of the results of ANSYS and IGABEM for the same problem (Tables 3 and 4) reveals excellent agreement for the third and fourth sets of idealizations, with differences limited to 1.3% and 0.7%, respectively. This strong agreement also validates the numerical accuracy of the proposed hybrid isogeometric approach.

Table 4.

Convergence of the wet natural frequencies (rad/s) obtained by the ANSYS ( $d / a =$ 0.50).

	$t =$ 0.238 m. ( $a / t =$ 42)
Mode no.	143950 elem.	1078600 elem.	3372300 elem.	7804700 elem.	Fu and Price [2]
	(3784 for plate)	(14620 for plate)	(65024 for plate)	(114920 for plate)
1(S)	9.24	8.19	8.31	8.27	8.22
2(A)	23.22	22.28	22.30	22.29	21.85
3(S)	74.71	66.16	65.81	65.69	65.40
4(S)	81.97	74.58	76.47	76.10	76.68
5(A)	104.61	97.57	98.04	97.92	98.15
6(S)	182.39	172.58	174.67	174.03	-

Figure 6 presents the first six wet mode shapes of the cantilevered plate, with edge lengths of $a = b =$ 10 m and thickness $t =$ 0.238 m for the submergence ratio of $d / a =$ 0.50, obtained via the proposed IGABEM approach and ANSYS software. A comparison between Figures 3 and 6 reveals that the wet mode shapes exhibit only slight variations from the in vacuo modes. This minimal difference suggests weak hydrodynamic coupling between the in vacuo modes, indicating that, for this particular problem, the presence of fluid has a negligible effect on the mode shape patterns. This phenomenon was elucidated by the experimental measurements and analytical results presented by Faria and Inman [81] for an Euler-Bernoulli beam immersed in fluid. Additionally, the wet mode shapes presented in Figure 6, obtained using the proposed IGABEM approach and ANSYS, show complete agreement in the locations of the peak deflections and nodal lines. Similar distortional mode shapes were also observed for the depth ratios of $d / a =$ 0.25, 0.75, 1.00; hence, they are not included here.

Following the validation of the proposed methodology and ANSYS simulations for specific submergence ratio, parametric studies were performed for various plate length-to-thickness ratios ( $a / t$ ) and submergence ratios ( $d / a$ ). These studies involved a series of calculations using both the proposed hybrid methodology and ANSYS software. Five different plate length-to-thickness ratios ( $a / t =$ 42, 21, 14, 10.5, and 8.4) and four submergence ratios ( $d / a =$ 0.25, 0.50, 0.75, and 1.00) were examined, while maintaining constant plate edge lengths of $a = b =$ 10 m. Throughout the analysis, the number of panels along the horizontal and vertical directions on the wetted surface of the plate was specified as (32 x 8), (32 x 16), (32 x 24), and (32 x 32) for the submergence ratios, $d / a =$ 0.25, 0.50, 0.75, and 1.00, respectively. In the ANSYS simulations, the edge length of the element for the structural domain was uniformly set to 80 mm.

Table 5 illustrates the variation in the first six wet natural frequencies for different plate length-to-thickness ratios and depths of submergence, as obtained using the IGABEM approach and ANSYS software. The results indicate that the wet natural frequencies deviate significantly from their in vacuo counterparts, with the reduction becoming more pronounced as the depth of the submergence increases. For instance, comparing the results in Tables 1 and 5, for a plate with a thickness ratio of $a / t =$ 42, the difference between the fundamental mode frequencies for the in vacuo and wet conditions is 21% at $d / a =$ 0.25 and increases to 43% at $d / a =$ 1.00. In addition, the sensitivity of the natural frequencies to the surrounding fluid gradually decreases with increasing mode number, with the largest discrepancies between the dry and wet natural frequencies occurring in the fundamental modes. A detailed comparison of Tables 1, 2 and 5 reveals that these discrepancies reduce as the thickness of the plate increases. For example, the percentage difference in the fundamental mode frequencies at $d / a =$ 0.50 is 36% for the thinnest plate ( $a / t =$ 42), but this reduces to 12% for the thickest plate ( $a / t =$ 8.4). Furthermore, Table 5 shows that the results obtained from the presented hybrid isogeometric approach and ANSYS simulations are in excellent agreement, with a maximum percentage difference of only 1.4%, observed for the fundamental mode frequency of the plate with $a / t =$ 42 at $d / a =$ 1.00. These findings validated that the presented hybrid isogeometric approach can accurately predict the dynamic response characteristics of thin and moderately thick plates interacting with fluid.

Table 5.

First six wet natural frequencies (rad/s) of a cantilevered square plate for various length-to-thickness ratios and submergence depths.

		length-to-thickness ratio
$d / a$	Mode	$a / t$ = 42			$a / t$ = 21		$a / t$ = 14	$a / t$ = 10.5		$a / t$ = 8.4
		IGABEM	ANSYS	IGABEM	ANSYS	GABEM	ANSYS	IGABEM	ANSYS	IGABEM	ANSYS
0.25	1(S)	10.18	10.17	22.42	22.46	34.84	34.94	47.29	47.42	59.58	59.79
	2(A)	26.12	26.17	55.87	56.06	85.12	85.48	113.49	113.96	140.58	141.23
	3(S)	72.23	72.69	147.72	148.52	220.91	222.19	290.60	292.35	355.67	357.96
	4(S)	86.91	86.69	182.53	182.98	275.24	276.51	363.51	365.48	446.12	448.93
	5(A)	104.27	104.69	211.36	212.43	313.36	315.25	408.47	411.06	495.38	498.73
	6(S)	183.59	183.67	367.34	368.71	537.67	540.47	691.43	695.42	827.57	832.25
0.50	1(S)	8.26	8.35	19.48	19.54	31.38	31.44	43.38	43.50	55.38	55.60
	2(A)	22.23	22.36	50.36	50.58	78.97	79.19	106.77	107.21	133.60	134.27
	3(S)	65.47	66.33	140.39	141.04	213.74	214.74	283.40	285.02	348.66	350.99
	4(S)	76.67	76.63	168.83	168.96	260.39	261.14	347.85	349.26	430.35	432.49
	5(A)	98.02	98.45	204.99	205.82	307.24	308.96	402.25	404.95	489.34	493.09
	6(S)	174.83	174.77	358.95	359.89	529.80	532.47	683.83	687.31	820.51	825.51
0.75	1(S)	7.57	7.66	18.26	18.32	29.80	29.86	41.53	41.65	53.45	53.54
	2(A)	20.87	21.03	48.20	48.45	76.36	76.59	103.84	104.30	130.68	131.14
	3(S)	54.47	54.87	124.49	124.79	195.57	196.09	264.04	264.87	329.03	329.92
	4(S)	70.91	71.19	160.25	160.41	250.32	250.69	336.98	337.46	419.47	419.66
	5(A)	83.50	83.83	185.68	186.32	286.24	287.37	380.61	382.50	468.12	470.28
	6(S)	151.74	151.58	329.60	330.10	498.90	500.09	652.91	654.58	790.93	792.14
1.00	1(S)	7.36	7.46	17.89	18.00	29.21	29.36	40.84	41.05	52.63	52.86
	2(A)	20.61	20.79	47.78	48.10	75.60	76.09	103.03	103.72	129.75	130.49
	3(S)	50.14	50.49	117.26	117.73	186.21	186.64	253.47	253.32	317.57	315.67
	4(S)	69.83	70.24	158.25	158.73	247.26	247.22	333.47	333.48	415.54	414.77
	5(A)	78.95	79.37	178.62	179.59	277.28	278.64	370.76	372.18	457.67	458.37
	6(S)	144.53	144.38	318.92	319.68	485.87	486.99	638.87	636.88	776.36	771.19

Table 6 presents the generalized added mass coefficients for the first ten distortional mode shapes of the cantilevered square plate with edge lengths of $a = b =$ 10 m and thickness $t =$ 0.238 m, evaluated at various depths of submergence. These coefficients were derived from in vacuo modes normalized to a generalized mass of 1 kgm². The table denotes symmetry (S) and antisymmetry (A) of mode shapes relative to the vertical axis at the plate’s centroid. Analysis of Table 6 reveals minimal hydrodynamic coupling between symmetric and antisymmetric modes, although certain pairs of symmetric-antisymmetric modes exhibit strong coupling. As the submergence depth increases, diagonal terms progressively dominate over off-diagonal elements. In particular, the magnitude of the generalized added mass coefficients is smaller at $d / a =$ 0.25 compared to the higher submergence ratios, indicating a reduced influence of the fluid on the dynamic response of the structure at lower depths. In contrast, deeper depths of submergence lead to increased fluid inertia, resulting in a marked decrease in the wet natural frequencies, consistent with the findings reported in Table 5. Importantly, the magnitude of these coefficients does not directly correlate with the overall complexity of the in vacuo mode shapes, but is primarily related to the displacement patterns of the modes below the free surface.

Table 6.

The infinite-frequency generalized added mass coefficients (kgm²) of the cantilevered square plate for the submergence ratios: $d / a =$ 0.25, 0.50, 0.75 and 1.00.

Mode no.	1(S)	2(A)	3(S)	4(S)	5(A)	6(S)	7(S)	8(A)	9(A)	10(A)
(a) $d / a =$ 0.25
1(S)	0.59	0.00	−0.34	−0.04	0.00	−0.07	0.13	0.00	0.00	0.00
2(A)	0.00	0.42	0.00	0.00	−0.31	0.00	0.00	−0.09	−0.11	−0.02
3(S)	−0.33	0.00	0.24	−0.06	0.00	0.11	−0.10	0.00	0.00	0.00
4(S)	−0.04	0.00	−0.06	0.32	0.00	−0.25	−0.07	0.00	0.00	0.00
5(A)	0.00	−0.31	0.00	0.00	0.25	0.00	0.00	0.10	0.11	0.04
6(S)	−0.07	0.00	0.11	−0.24	0.00	0.22	0.02	0.00	0.00	0.00
7(S)	0.12	0.00	−0.10	−0.07	0.00	0.02	0.11	0.00	0.00	0.00
8(A)	0.00	−0.09	0.00	0.00	0.11	0.00	0.00	0.31	−0.03	0.23
9(A)	0.00	−0.10	0.00	0.00	0.11	0.00	0.00	−0.03	0.11	−0.06
10(A)	0.00	−0.02	0.00	0.00	0.04	0.00	0.00	0.23	−0.06	0.19
(b) d/a = 0.50
1(S)	1.41	0.00	0.19	0.14	0.00	−0.03	−0.35	0.00	0.00	0.00
2(A)	0.00	0.98	0.00	0.00	0.22	0.00	0.00	−0.09	−0.28	0.07
3(S)	0.19	0.00	0.43	0.11	0.00	−0.13	0.25	0.00	0.00	0.00
4(S)	0.15	0.00	0.12	0.73	0.00	−0.19	−0.18	0.00	0.00	0.00
5(A)	0.00	0.22	0.00	0.00	0.36	0.00	0.00	−0.13	0.20	0.03
6(S)	−0.03	0.00	−0.13	−0.19	0.00	0.30	−0.01	0.00	0.00	0.00
7(S)	−0.36	0.00	0.24	−0.18	0.00	−0.01	0.44	0.00	0.00	0.00
8(A)	0.00	−0.09	0.00	0.00	−0.14	0.00	0.00	0.57	0.13	−0.20
9(A)	0.00	−0.29	0.00	0.00	0.20	0.00	0.00	0.13	0.46	−0.07
10(A)	0.00	0.07	0.00	0.00	0.03	0.00	0.00	−0.20	−0.07	0.25
(c) d/a = 0.75
1(S)	1.87	0.00	−0.30	0.42	0.00	0.07	−0.33	0.00	0.00	0.00
2(A)	0.00	1.25	0.00	0.00	−0.12	0.00	0.00	−0.20	−0.29	0.00
3(S)	−0.30	0.00	1.10	−0.15	0.00	−0.14	0.09	0.00	0.00	0.00
4(S)	0.43	0.00	−0.16	1.07	0.00	0.10	−0.11	0.00	0.00	0.00
5(A)	0.00	−0.12	0.00	0.00	0.85	0.00	0.00	0.03	0.13	0.11
6(S)	0.08	0.00	−0.14	0.11	0.00	0.71	0.01	0.00	0.00	0.00
7(S)	−0.35	0.00	0.09	−0.11	0.00	0.01	0.65	0.00	0.00	0.00
8(A)	0.00	−0.21	0.00	0.00	0.03	0.00	0.00	0.75	0.14	0.01
9(A)	0.00	−0.31	0.00	0.00	0.12	0.00	0.00	0.14	0.64	0.01
10(A)	0.00	0.00	0.00	0.00	0.12	0.00	0.00	0.01	0.01	0.59
(d) d/a = 1.00
1(S)	2.03	0.00	−0.53	0.55	0.00	0.12	−0.11	0.00	0.00	0.00
2(A)	0.00	1.30	0.00	0.00	−0.23	0.00	0.00	−0.26	−0.20	−0.05
3(S)	−0.54	0.00	1.50	−0.36	0.00	−0.18	−0.33	0.00	0.00	0.00
4(S)	0.57	0.00	−0.36	1.21	0.00	0.21	0.11	0.00	0.00	0.00
5(A)	0.00	−0.23	0.00	0.00	1.09	0.00	0.00	0.17	−0.10	0.22
6(S)	0.13	0.00	−0.19	0.22	0.00	0.90	0.08	0.00	0.00	0.00
7(S)	−0.13	0.00	−0.33	0.11	0.00	0.07	1.15	0.00	0.00	0.00
8(A)	0.00	−0.28	0.00	0.00	0.18	0.00	0.00	0.85	0.01	0.12
9(A)	0.00	−0.21	0.00	0.00	−0.10	0.00	0.00	0.01	0.89	−0.06
10(A)	0.00	−0.05	0.00	0.00	0.23	0.00	0.00	0.13	−0.06	0.78

The computed frequency-dependent generalized added mass and hydrodynamic damping coefficients for the first and second elastic modes at various submergence ratios are shown in Figure 7. These coefficients were calculated over the frequency range of 0 < ω < 12.0 (rad/s). The generalized added mass coefficients begin with non-zero positive values at very low frequencies, peak in the low frequency region, and then decrease slightly below the infinite frequency limit value. At higher frequencies, that is to say, ω > 8 rad/s, all generalized added mass coefficients stabilize and attain constant values. In contrast, when the driving frequency is very small, the values of damping coefficients are almost zero as no free surface waves are generated. Similar to the behaviour of generalized added mass, the hydrodynamic damping coefficients reach their maximum values in the low frequency region and vanish at very high frequencies because no waves are set up. Furthermore, Figure 7 shows that for both the first and second distortional modes, the generalized added mass and hydrodynamic damping coefficients have their highest values at the submergence ratio of d/a = 0.50.

4. Conclusion

This work presents a hybrid numerical approach for free vibration analysis of cantilevered Mindlin plates in ambient air and when surrounded by a fluid environment of comparable density. The structural model of the plate with isotropic and homogeneous material properties is based on Reissner–Mindlin plate theory, while the surrounding fluid medium is modeled based on potential flow theory, i.e., assuming the fluid as inviscid, incompressible and irrotational. Both the in vacuo and wet modal analyses are performed by the modal superposition approach, and the wet mode shapes of the partially immersed plate are defined as a linear combination of the dry modes. The validity and robustness of the proposed hybrid methodology were demonstrated by comparing the results with those obtained by the commercial finite element software ANSYS, both with and without a surrounding fluid environment. Both the presented hybrid isogeometric methodology and the finite element technique were capable of predicting natural frequencies and the corresponding modes of the cantilevered Mindlin plate. Additionally, parametric studies were conducted to assess the influence of factors such as plate thickness and submergence depth on the dynamic characteristics of the aforesaid cantilevered Mindlin plates. Subsequently, the frequency dependence of the hydrodynamic characteristics of a fluid-coupled plate, namely added mass and hydrodynamic damping, was studied by imposing a linearized free surface boundary condition in the fluid model. Based on the findings of this study, the conclusions are as follows:

The presented isogeometric-based hybrid methodology successfully predicts the natural frequencies and corresponding mode shapes of thin and moderately thick plates, with a plate length-to-thickness ratio is up to 8.

Minimal discrepancies were observed between the in vacuo and wet mode shapes; however, significant reductions in the wet natural frequencies were noticeable compared to their in vacuo counterparts. Such a reduction differently affected the modes; these reductions become more pronounced for thinner plates.

Submergence depth significantly impacts the natural frequencies of the plate, with wet natural frequencies decreasing steadily as the level of surrounding fluid increases. This reduction rate is most pronounced at lower immersion depths.

Added mass and hydrodynamic damping exhibit strong frequency dependence in the low frequency region, reaching their maximum values. The frequency-dependent added mass plots converge to infinite frequency added mass coefficients in the high frequency region, where the free surface perturbations of the fluid become negligible. This finding aligns with the assumptions made in the wet modal analysis, where the free surface effects are neglected, and the added mass coefficients are assumed to be frequency independent.

These findings suggest that the proposed numerical framework can be further applied to analyze the harmonic and transient responses of elastic structures vibrating near or at the free surface.

Footnotes

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

I Tugrul Ardic

M Erden Yildizdag

Ahmet Ergin

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