The Geometry of Ships

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30 THE PRINCIPLES OF NAVAL ARCHITECTURE SERIES Curves of intersection arising from the intersections between two surfaces can be recognized as snakes residing on both of the surfaces. The difficulties that can be present in computing such intersections have been discussed above in Section 4.15. 6.2 Applications of Curves on Surfaces. Curves on surfaces can play several roles in definition of ship geometry: • As decorative lines; e.g., cove stripe, boot stripe, hull decorations • As boundaries of subsurfaces and trimmed surfaces; e.g., delineating subdivision of the hull surface into shell plates for fabrication • As a junction between surfaces; e.g., the deck-at-side curve drawn on the hull and used as an edge curve for the weather deck surface • As a trace for a linear feature to be constructed on another surface; e.g., a guard, strake, or bilge keel • As alignment marks to be carried through a plate expansion process. Section 7 Geometry of Solids The history of geometric modeling in engineering design has progressed from “wireframe” models representing curves only, to surface modeling, to solid modeling. Along with the increase in dimensionality, there is a concomitant increase in the level of complexity of representation. Wireframe and surface models have gone a long way toward systematizing and automating design and manufacturing, but ultimately most articles that are manufactured, including ships and their components, are 3-D solids, and there are fundamental benefits in treating them as such. Wireframe representations were the dominant technology of the 1970s; surface modeling became well developed during the 1980s; during the 1990s the focus shifted to solid models as computer speed and storage improved to handle the higher level of complexity, and as the underlying mathematical, algorithmic, and computational tools required to support solids were further developed. We will first briefly review a number of alternative representations of solids, each of which has some advantages and some limited applications. Of these, boundary representation or B-rep solids have emerged as the most successful and versatile solid modeling technology, and they will therefore be the focus of this section. 7.1 Various Solid Representations. 7.1.1 Volume Elements (Voxels). A conceptually simple solid representation is to divide space into a 3-D rectangular array (lattice) of individual cubic volume elements or voxels, and then characterize the contents of each voxel within a domain of interest. This is a 3-D extension of the way 2-D images are represented as arrays of picture elements or “pixels.” For a homogeneous solid, the voxel information can be as little as one bit, i.e., is this voxel occupied by material, or is it empty? Or, if a complex inhomogeneous solid is being described, numerous attributes can be attached to each voxel; e.g., density, temperature, concentration of various chemical species, etc. Voxels are most useful for medium-resolution descriptions of inhomogeneous solids with significant internal structure. The storage requirements and processing effort are high, and increase as the cube of the resolution. For example, a voxel description of the human body at a resolution of 1 mm requires on the order of 10 8 voxels (and of course, 1 mm is still a very coarse resolution for describing most tissues and anatomical structures). 7.1.2 Contours. Contours or level sets on surfaces were described in Section 4.19, and were related to the description of an object as a solid. In naval architecture, transverse sections (contours of the longitudinal coordinate X) are the standard representation of the envelope of a vessel for purposes of hydrostatic analysis. The individual sections are represented as closed polylines. Contours are also used within a hydrostatic model to describe tanks, voids, or compartments inside the vessel. Fig. 27 Offsets representation of a ship as a solid cut by contours (X constant).
THE GEOMETRY OF SHIPS 31 This is often called an “offsets” representation. Offsets are also a suitable representation of form for some types of seakeeping and resistance analyses, primarily those based on quasi-2-D “strip theory” or “slender body theory” approximations (Fig. 27). Contours are a simple and compact solid representation, but obviously they provide very limited detail unless contour intervals are small and points are densely distributed. They do not lend themselves to rendering as a solid. They are best suited for elongated shapes, where the surfaces have small slopes with respect to the longitudinal axis, and where the solid being represented has no relevant internal structure. 7.1.3 Polyhedral Models. A polyhedron is a solid bounded by planar faces. The representation can be, for example, a list of 3-D points (vertices) and a list of faces, each face being a list of vertices which form a closed planar polygon. Triangle meshes, where each face is a triangle, are a commonly used special case. Triangles have the advantage of being automatically planar; this is not generally true of the quadrilateral “panels” formed by the tabulated mesh of a parametric surface. A polyhedral mesh, especially a triangle mesh, is easy to render as a solid. Watertight triangular meshes are widely used for computer-aided manufacturing such as stereolithography, communicated by STL files. Polyhedral models are limited for representing smooth curved surfaces, which require fine subdivision if they are not to appear faceted (Fig. 28). 7.1.4 Parametric Solids. Parametric solids are an extension of the concept of the parametric curve x(t) and the parametric surface x(u, v) to one more dimension. A parametric solid is described by a 3-D vector function of three dimensionless parameters: x(u, v, w). Each parameter has a nominal range, e.g., [0, 1]. Under moderate conditions on the function x, as a moving point assumes all values in the 3-D parameter space [0, 1] [0, 1] [0, 1], it sweeps out a 3-D solid in physical space. Each of the six faces of the solid is a parametric surface in two of the parameters. A simple but often useful example is a ruled solid between two surfaces x 1 (u, v) and x 2 (u, v): x(u, v, w) (1 w) x 1 (u, v) wx 2 (u, v) (45) For example, if x 1 is the hull outer surface and x 2 is an offset surface a uniform distance inside x 1 , the ruled solid between them represents the shell as a finitethickness solid. A 3-D mesh in this solid made by uniformly subdividing the u, v, and possibly w parameters creates a set of curvilinear finite elements suitable for analyzing the hull as a thick shell. Figure 29 shows an integral tank modeled as a ruled solid between a subsurface on the hull and its vertical projection onto a horizontal plane at the level of the tank top. Another readily understood and useful parametric solid is the B-spline solid x(u, v, w) = i j X ijk B i (u) B j (v) B k (w) (46) k where the X ijk are a 3-D net of control points, and the B i , B j , B k are sets of B-spline basis functions for the three parametric directions. Each of the six faces of a B-spline solid is a B-spline surface in two of the parameters, based on the set of control points that make up the corresponding “face” of the control point net. Parametric solids are very easily implemented within the framework of relational geometry, so the solid updates when any ancestor entity is changed. Compared with B-rep solids, their data structures and required programming support are very simple and reliable. They also have a relatively large amount of internal structure, i.e., an embedded body-fitted 3-D coordinate system u, v, w which allows the unique identification of any interior point, and a way to describe any variation of properties within the solid. Parametric solids are very useful for generating “block-structured” grids and finite elements for various forms of discretized analysis in volumes. Fig. 28 A triangle mesh model of a ship hull surface. This could be suitable for hydrostatic analysis or for subdivision for capacity calculations. Fig. 29 An integral tank defined as a parametric solid.
Page 2 and 3: The Principles of Naval Architectur
Page 4 and 5: Nomenclature m A A
Page 6 and 7: x PREFACE computations are presente
Page 8 and 9: Section 1 Geometric Modeling for Ma
Page 10 and 11: THE GEOMETRY OF SHIPS 3 Although fi
Page 12 and 13: THE GEOMETRY OF SHIPS 5 and their f
Page 14 and 15: THE GEOMETRY OF SHIPS 7 The concept
Page 16 and 17: THE GEOMETRY OF SHIPS 9 while an un
Page 18 and 19: 10 THE PRINCIPLES OF NAVAL ARCHITEC
Page 26 and 27: THE GEOMETRY OF SHIPS 17 plicit sur
Page 28 and 29: THE GEOMETRY OF SHIPS 19 Fig. 13 Pa
Page 30 and 31: THE GEOMETRY OF SHIPS 21 4.9 B-spli
Page 32 and 33: THE GEOMETRY OF SHIPS 23 Fig. 19 A
Page 34 and 35: THE GEOMETRY OF SHIPS 25 A typical
Page 36 and 37: THE GEOMETRY OF SHIPS 27 Section 5
Page 38 and 39: THE GEOMETRY OF SHIPS 29 A curve ly
Page 46: Fig. 32 The lines plan of a cargo s
Page 49 and 50: THE GEOMETRY OF SHIPS 37 curves, ty
Page 51 and 52: THE GEOMETRY OF SHIPS 39 tration of
Page 53 and 54: THE GEOMETRY OF SHIPS 41 Alternativ
Page 55 and 56: THE GEOMETRY OF SHIPS 43 The genera
Page 57 and 58: THE GEOMETRY OF SHIPS 45 where: L a
Page 61 and 62: THE GEOMETRY OF SHIPS 47 Define the
Page 63 and 64: THE GEOMETRY OF SHIPS 49 Molded dis
Page 65 and 66: THE GEOMETRY OF SHIPS 51 11.2 Bonje
Page 68 and 69: THE GEOMETRY OF SHIPS 53 Section 12
Page 70 and 71: THE GEOMETRY OF SHIPS 55 Fig. 41 A
Page 74 and 75: THE GEOMETRY OF SHIPS 57 surface ab
Page 76 and 77: INDEX Note: Page numbers followed b
Page 78 and 79: Index Terms Links center of buoyanc
Page 80 and 81: Index Terms Links coordinates, homo
Page 82 and 83: Index Terms Links displacement 3 4
Page 84 and 85: Index Terms Links geometry definiti
Page 86 and 87: Index Terms Links isoparms 17 K kee
Page 88 and 89: Index Terms Links model (Cont.) sca
Page 90 and 91:
Index Terms Links pole 18 polygon 1
Page 92 and 93:
Index Terms Links seakeeping 2 31 3
Page 94 and 95:
Index Terms Links surfaces (Cont.)
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Index Terms Links triangle mesh 27f
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The Geometry of Ships

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