# Stochastic Natural Vibration Analyses of Free-Form Shells
## 1 Introduction  
The term free-form shells refers to shells whose geometric shapes cannot be represented by certain mathematical formulas or their combination [1]. When compared with traditional shells, such as cylindrical shells and spherical shells, the geometrical shape of free-from shells is flexible to satisfy certain structural, architectural, and other requirements, which advances the wide application of free-form shells in civil engineering, aerospace engineering, and so on. The mechanical properties of free-form shells are strongly dependent on their geometric shapes. Therefore, shape optimization became an indispensable approach for designing a free-form shell. [PERSON] et al. [2] optimized free-form shells for structural stiffness under given loads while using numerical methods, which were emerged with physical experiments. [PERSON] and [PERSON] [3] combined Non-Uniform Rational B-Spline (NURBS) with genetic algorithm (GA), as well as the gradient method to perform shape optimization of free-from shells, in which the objective functions are set as minimum strain energy and minimum geometric deviation from the prescribed shape. [PERSON] and [PERSON] [4] developed a height adjusting method to obtain optimal shape of free-form shells, with respect to minimum strain energy. [PERSON] and [PERSON] [1] described a conjugate gradient optimization method for the shape design of cable-breaced free-form grid shells, with strain energy as the optimization objective. [PERSON] and [PERSON] [5] conducted a study on local optimal solutions and a modified optimization method, which was applied on the shape optimization of cable-stiffened latticed free-form shells, with theobjective of the minimization of strain energy. It can be seen that, in the past decades, numerical optimization methods have been developed for free-form shells, many of which made efforts to obtain a geometrical shape with minimum strain energy, i.e., largest structural stiffness. There is a relative dearth of research on mechanical properties of free-form shells despite the fast development of shape optimization of free-form shells. Only minimal literature is concerned with that. For example, [PERSON] [6] studied the buckling properties of free-form shells; [PERSON] et al. [7; 8] carried out experimental and numerical studies on the static behaviors of the free-form concrete shells. However, the existing work mainly studies static characteristics and it is limited to the deterministic analysis without any consideration of uncertainty factors, which are unavoidable in practical engineering, such as geometrical imperfection and deviations of material properties. The previous stochastic studies on traditional shells have demonstrated the significant effect of uncertain factors on mechanical properties of structures [9; 10; 11; 12; 13]. The construction process of free-form shells is more complex when compared with traditional shells, which tends to lead more uncertainty factors. This further motivates this study to consider the effects of uncertainty factors on mechanical properties of free-form shells, particularly on the dynamic characteristics. In this study, the stochastic natural vibration analyses of free-form shells is carried out while considering three uncertainty factors, including geometric imperfection, shell thickness deviation, and elastic modulus deviation. Free-form shell models are generated by a self-coded numerical optimization algorithm. Latin hypercube sampling (LHS) is used for the random sampling of three uncertainty factors based on their stochastic probability modeling. The global sensitivity study is conducted to rank the importance of these error sources. The outline of this paper is as follows: in Section 2, the shape generation method of free-form shells is presented and programmed; Section 3 describes the stochastic and sensitivity analysis method; in Section 4, stochastic and sensitivity analyses are carried out on free-form shell, which are designed by the shape generation algorithm presented in Section 2.