# Preconditioning the Quad Dominant Mesh Generator for Ship Structural Analysis
## 1 Introduction  
This paper is concerned with automatic mesh generation in the process of the structural analysis of ships and off-shore structures. The ship structural analysis by the finite element method is governed by the acceptance criteria of the classification societies [1]. One of the possibilities to describe an input geometry, for a large class of such structures, is by the use of a dictionary of elements. The dictionaries which we will be considering in this paper consist of points, rods, surfaces, and openings. The points in the dictionary describe positions where the loads (for the finite element analysis) are going to be applied or the measurements are going to be taken. Rods describe the panel stiffeners or pillars. The set of surfaces will consist of three types of entities: web surfaces (parallelograms with one dimension much smaller than the other), regular surfaces (convex quadrilaterals defined by four co-planar nodes), and warped surfaces (closed loop surface defined by four corners which need not be co-planar) [2, 3]. The preconditioning method which we are going to describe extends and improves the meshing algorithm from [4]. The algorithm from [4], called pyREMAKEmsh, is designed for the dictionaries that do not contain the warped surfaces. The present algorithm is extended to include the warped surfaces and also to improve the quality of the mesh by further preconditioning of the geometry. Our algorithm is intended for full automatic meshing of sections of ship structures (such as superstructures, decks,...) according to the rules of classification societies. In a typical situation, a description of a geometry by a dictionary of elements contains many modeling inconsistencies such as overlapping plates or plates not in contact. Note that a stiffened panel as a constructive element is modeled by a plate (shell) in contact with a rod or a web surface. The constructed mesh should not contain any hanging nodes. However, it can happen, due to modeling inconsistency that e.g., a stiffener or a web surface does not overlap with a panel (model by a regular surface) which it is supposed tostiffen. Removing such modeling irregularities is fundamental both for the use of meshing algorithms which require that the surfaces are described as closed loop surfaces as well as by the subsequent finite element discretization procedures which require that there are no hanging nodes in the mesh [2; 5]. The algorithm from [4] utilized boolean operations (with a tolerance) from Open CASCADE geometric kernel to solve this problem. The points in the dictionary represent locations where loads are applied or measurements are taken and so they need to be a part of the final mesh. This further complicates the meshing task. The requirements on a mesher are taken from ([6], Section 1.7) and can be summed up as:  
* The elements with angles less than \(45^{\circ}\) and more than \(135^{\circ}\) should be avoided;
* In the high stress areas such as e.g., web surfaces, the use of triangular elements should be avoided. * The use of triangular shell elements should be kept to the minimum. * Quadrilateral shell elements with a high aspect ratio as well as distorted elements should be avoided--aspect ratio for quadrilaterals is to be kept close to 1 but should not exceed 3 for 4 node elements and should not exceed 5 for three node elements (triangles). * Web surfaces should be modeled with at least four elements along the shorter dimension, and ideally with precisely four elements. * The element size should be at least \(t\times t\), where \(t\) is the thickness of the element. Let us emphasize that classification societies use results of finite element analysis to accept or reject a ship's design. The rules of the classification societies such as ([6], Section 1.7) specify the protocol for performing the sip structural analysis.