# Dynamic Pressure Analysis of Hemispherical Shell Vibrating in Unbounded Compressible Fluid
## 1 Introduction  
A hemispherical shell is widely used in large span structures, which have been constructed in large public buildings such as gymnasiums, exhibition centers, and opera houses [1, 2]. Such curved shapes are often used for steel shells and rod structure as they are naturally strong structures, allowing wide areas to be spanned without the use of internal supports, giving an open, unobstructed interior. Figure 1 shows a geodesic dome for digital artwork that was established in 2014 [3]. The use of steel as a building material reduces both material costs and construction costs, as steel is relatively inexpensive and easily made into compound curves. The shell structure tends to be immensely strong and safe. For example, modern monolithic dome houses have resisted hurricanes and fires. In recent literature, there has been renewed interest in the problem of a hemispherical shell vibrating in contact with air or other fluids [4]. The vibrations of a structure induce local air movement, while the moving airand global atmospheric pressure affect the structure at the same time. Such phenomenon is called fluid-structure interaction (FSI) effect. There are some publications in the literature devoted to the interaction of fluid and shell by theoretical approaches [5; 6]. In the way of potential flow theory, [PERSON] [7] derived the added mass ratio for a rectangular plate under the assumption that the vibration mode is simple harmonic. Actually, the researcher utilized the incompressible assumption so the obtained result was an added mass ratio. [PERSON] [8] reanalyzed the same problem with the assumption that the structure vibrates as a quadratic function and raised another added mass formula. In both papers, the problem was actually treated as a one-dimensional model so that their method cannot be applied to two-dimensional (2D) and three-dimensional (3D) problems. [PERSON] [9] studied the effect of the added mass of a thin shell vibrating in a stationary fluid through using the energy method, and the pressure of the shell was calculated by the thin-airfoil theory. [PERSON] and [PERSON] [10] studied the problem of a beam vibrating in the fluid based on the classical Fourier series, and reported the effects of the added mass ratio on the one-dimensional beam vibrating in non-viscous fluid. Most of the research about the vibrations of shell in fluids commonly studied them by a numerical method, especially by the computation fluid dynamics (CFD) method [11; 12]. [PERSON] and [PERSON] [13] studied the effect of the generation and transmission of the vibro-acoustic energy in an elastic cylindrical shell filled with water. They focused more on vibro-acoustic energy than the pressure. [PERSON] et al. [14] studied the Guangzhou International Convention and Exhibition Center with dimensions that were 210 m wide and 457 m long. They studied the pressure distribution of the exhibition according to the wind tunnel test. [PERSON] [15] conducted some research studies on the pressure of compressible squeeze film. In the study, the CFD method was used. [PERSON] and [PERSON] [16; 17] developed a Rayleigh-Ritz method to study a proportional hydroelastic vibration of two annular plates coupled with a bounded fluid. They considered the effects of the compressibility of the fluid and obtained a theoretical result that could predict the fluid-coupled frequencies well. [PERSON] et al. [18] conducted a number of studies on a cylindrical shell submerged in an unbounded medium. They also assumed the fluid to be an incompressible fluid. [PERSON] [19] developed a differential quadrature method (DQM) to investigate the problem of the free vibration of circular plates in contact with fluid. [PERSON] and [PERSON] [20; 21] investigated a simple-based monolithic implicit method for the strong-couple FSI problem. Their method may be used to analyze the vibration of the shell. These methods consist of several numerical approaches, but they offered little findings on a basis of theoretical analysis.