# Preconditioning the Quad Dominant Mesh Generator for Ship Structural Analysis
## 1 Introduction
This protocol includes restriction of the size and the type of finite element's used and the bullet list provided above is a summary of the relevant requirements from the perspective of a finite element mesh generator. Let us point out that an analysis which does not adhere to the classification rules such as [6] cannot lead to acceptance of the ships' design and as such can incur severe opportunity costs on a user of a meshing tool. These requirements pose a challenge for implementing fully automatic meshing routines. Namely, these rules most often pose local linear constraints on the mesh and as such clash with the geometry optimization algorithms which search for globally optimal tessellations, see [5]. We relax those restrictions in order to accommodate more complex geometries typically encountered in ship structural analysis. Let us briefly review the available results concerning fully automatic mesh generation for ship structural analysis. Meshes based on quadrilateral elements can either be generated directly by an advancing front algorithm, see [7] or by a hierarchical grid-based subdivision methods [8]. In this paper, we explore a third option, which is the construction of a quadrilateral dominated mesh by a recombination of triangular mesh [9; 10]. An algorithm based on the preconditioned Blossom recombination approach [10] has been implemented in the pyREMAKEmsh algorithm presented in [4]. The method was designed for dictionaries which only contained co-planar plates. It was based on the control of the cross field, [11], using the insertion of virtual stiffeners. These stiffeners are used to split a plate with which they intersect but are not included in the dictionary as rod elements. In this approach, it is necessary that the original triangular mesh be generated so that a recombination yields a regular quadrilateral mesh. Quadrilateral elements in gmsh are generated indirectly by the perfect matching algorithm based on the graph theoretic Blossom algorithm [10; 12]. A default implementation starts from a standard Delaunay advancing front algorithm implemented in the \(L^{2}\) metric. An important extension and improvement to the Blossom algorithm based quadrilateral mesh generation is realized by adapting the triangular advancing front mesher so that the generated triangular mesh can be efficiently matched into a regular quadrilateral mesh. It is achieved by modifying the advancing front Delaunay mesh generator with the change of norm from \(L^{2}\) to the \(L^{\infty}\) norm. It is a geometric fact that an equilateral triangle in the \(L^{\infty}\) norm is in fact the right angle isosceles triangle, whereas an equilateral triangle in the \(L^{2}\) (Euclidean) norm has internal angles equal to \(60^{\circ}\), see [13]. With this change of norm, the triangles which are generated by the Delaunay algorithm will tend to be closer to right angle triangles and so will be amenable to good recombination into a regular quadrilateral mesh, see [13]. In this paper, we will present a geometry improvement algorithm which can tackle geometries containing warped plates. The algorithm is based on the processing pipeline which utilizes boolean operations in the 3D geometry kernel Open CASCADE to correct the geometry errors and to enforce local linear constraints. Furthermore, we will introduce a preprocessing step to make the results of the meshing algorithm repeatable and to control the mesh quality. The makeup of the paper is as follows: We will first present materials and methods where we will describe the challenges and the necessary modifications to the algorithm necessary to be able to tackle warped plates. We will then present results of the applications of these algorithm to two characteristic examples from engineering practice. In the Discussion section, we will present statistical evaluation of the preprocessing algorithm.