## 1 Introduction  
With a view to reducing the greenhouse gas (GHG) emission from newly built ships, the International Maritime Organization (IMO) established the Energy Efficiency Design Index (EEDI) regulation, which mandates stepwise reinforcement of GHG emission allowance every five years. As of 2025, the attained EEDI level should be reduced more than 30% from the baseline level set in 2013. Through this and other regulatory measures, IMO envisages halving the global maritime GHG emission in 2050 compared to that in 2008. Consequently, the development of energy efficient ships using every possible means has become a matter of grave concern for all shipbuilders. Hull form optimization toward minimum resistance is one of relevant strategies for the development of energy efficient ships. However, the hull form optimization is a very complicated task because it involves the variation of sectional area curve (SAC), waterline shape, longitudinal section shape, profile shape, etc. under the design constraints such as principal particulars and the general arrangement of the ship. The skill and experience of designers significantly affect the hydrodynamic performance of hull form. In addition, the resistance performance evaluation requires time-consuming computational fluid dynamics (CFD) simulation. Thus, the simulation-based design (SBD) has long been chosen as the methodology for hull form optimization [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31]. The process of hull form SBD largely consists of three elements: firstly, hull form variation, secondly, performance evaluation of modified hull using numerical simulation, and finally, optimization algorithm. The hull form variation methods can be classified into three categories. The first is the vertex control, in which the vertices of the hull surface defined in terms of either NURBS (Non-Uniform Rational B-Spline) or Bezier surface are directly translated [1; 3; 8; 9; 13]. When applied to complicated hull form, this method requires a prohibitively large number of control operations. Thus, it is basically suitable to simple hull forms like Wigley and Series 60. Recently, an improved variant of vertex control named free form deformation (FFD) has been proposed [23; 24; 25; 26]. Here, the vertex control region is assigned as a 3D volume and the displacement of each vertex is determined parametrically. The second category is called as hull form parameter method [10; 11; 22; 27; 32; 33; 34; 35; 36], which modifies hull form by means of varying hull form parameters. [PERSON] [32] was the first to attempt to define and modify hull form mathematically. The generation of B-spline curves and surfaces using hull form parameters was first proposed by [PERSON] [33] and subsequently adopted by other researchers like [PERSON] [34], [PERSON] [35], [PERSON] and [PERSON] [36], and [PERSON] and [PERSON] [37]. The hull form can be defined solely by such parameters as sectional area, centroid of sectional area, curvatures and tangent angles at the beginning and end point of hull form defining curves, etc. [11; 27]. This is convenient especially in the absence of the original hull form, but it is difficult to define parameters which can reconstruct the complicated hull form precisely. The third category is using parametric modification function, which quantifies the incremental change from the initial hull form. Being an indirect modification, this method is capable of guaranteeing the fairness of the resulting modified hull form as long as the fairness of the parametric modification function is maintained [38]. Furthermore, the modification function can become parametrized as design variable so that transform into diverse hull forms is enabled. The most famous example in this category is the Lackenby method [39]. [PERSON] et al. [4] applied the parametric modification function to the optimization process for Wigley hull. [PERSON] [15] and [PERSON] et al.