# Ship-Hull Shape Optimization with a T-Spline based BEM-Isogeometric Solver

Ship-Hull Shape Optimization with a T-Spline based BEM-Isogeometric Solver

The optimization of a hull form with respect to its hydrodynamic performance is considered as a major task in ship design. In our previous works [1][2], IsoGeometric Analysis (IGA) is applied to the solution of the Boundary Integral Equation (BIE) associated with the Neumann–Kelvin problem in the context of shape optimization with respect to the wave resistance of ships. In [1] we presented a ship-hull optimization process combining modern optimization techniques, a NURBS multi-patch parametric ship-hull model and an isogeometric NURBS based Boundary Element Method (BEM) solver for the required calculations. In this work, we present an alternative ship-hull optimization process combining a T-Spline [3] based parametric ship-hull model and a T-Spline based BEM solver for the calculation of ship wave resistance. The ship parametric model is constructed within Rhino® modeling software with the aid of its scripting environment and Autodesk®’s T-Splines® Plug-In for Rhino®. The methodology initiates with a list of suitable parameters and proceeds with the generation of a control cage, which permits embedding basic ship-specific shape features (e.g., parallel mid-ship part, bulbous bow etc). Ship hull generation is based on the aforementioned control cage, which, with the aid of the T-Spline plug-in functionality, yields a single analysis-suitable, cubic T-Spline surface for the representation of the ship hull. Obviously, the resulting surface bases are C2 everywhere except for the vicinity of extraordinary points as opposed to our previous NURBS hull model where the continuity of the associated bases at patch boundaries is limited to C0. Concerning the hydrodynamic solver, we couple collocated BEM with unstructured analysis-suitable T-Spline [4] surfaces for solving a linear BIE arising in the context of the aforementioned Neumann-Kelvin problem, following the formulation by Brard [5] and Baar & Price [6]. The local-refinement capabilities of the adopted T-Spline bases, which are used for representing both the body-geometry and approximating the solution of the associated BIE, in accordance with the Isogeometric concept proposed by Hughes et al. [7], lead to a solver that achieves the same error level for many fewer degrees of freedom as compared with the corresponding NURBS-based isogeometric-BEM solver developed in our previous works. Shape optimization in naval hydrodynamics is a demanding and challenging task, for several reasons: firstly, each cost function evaluation (e.g., wave resistance) is computationally expensive. Moreover, the output of the numerical solver could be noisy, due to geometrical and numerical approximations, which generates spurious local minima and may yield the failure of the optimization process. In this context, the proposed IGA approach is particularly interesting, since the geometry generated by the CAD modeler is directly used by the solver without any geometrical approximations. The shape optimization is demonstrated using the optimization algorithms employed in [1] for the bulbous bow of a simple ship-hull against the criterion of minimum wave resistance.