Ship Hull Representation by Non-Uniform Rational B-Spline Surface Patches

The purpose of this work is to propose a new method for representing the ship hull shape with mathematic surfaces so that geometric data can be generated for any point on the hull where required to assist the production process. An extensive survey of previous work is presented covering both the use of parametric curves and surfaces to model the ship hull and also the most relevant software systems developed for that purpose. The main methods and algorithms available for the generation and edition of curves and surfaces are presented and compared taking into consideration the intended application. From the analysis of the formulations available it was concluded that the most adequate one, which however had not yet been extensively used to model ship hulls was the Non-Uniform Rational B-Splines (NURBS), due to the potential of their capability to represent exactly conic curves and surfaces. Therefore these surfaces were selected as the basis of the method developed in this thesis. A procedure is proposed for the representation of a given hull form in a two step approach, creating first a wireframe model over which the surface patches are generated. Both curves and surfaces are based on the NURBS formulation. To create the wireframe model, first a set of longitudinal boundary lines is selected, dividing the surface into areas of similar shape. Then, these lines are fitted by curves and faired to some extent. Next, transverse sections are defined and split by the boundary lines. Surface patches are then generated over the transverse section curves within the limits of each patch. Finally, to obtain the traditional representation of the ship surface by transverse sections, buttocks and waterlines, contour lines are generated for constant values of x, y and z coordinates. A computer system has been developed incorporating an interface that allows the visualization of the curves and surfaces being modeled. The system incorporates several algorithms for generation and edition of curves and surfaces, in addition to the main contribution of this thesis which is the use of NURBS to represent the ship hull surface. The system also incorporates curve and surface analysis tools and some basic fairing algorithms so that during the several steps of the creation of the model, the fairness of the curves and surfaces can be evaluated and improved to some extent. The procedure is tested and compared with an existing commercial system through some application examples, of a complete hull and in more detail in the bow region, showing that good results can be obtained with the system presented here

Rational quadratic Bézier spirals

A quadratic Bézier representation withholds a curve segment with free from loops, cusps and inflection points. Furthermore, this rational form provides extra freedom to generate visually pleasing curves due to the existence of weights. In this paper, we propose sufficient conditions for rational quadratic Bézier curves to possess monotonic increasing/decreasing curvatures by means of monotone curvature tests which are based on the derivative of curvature functions. We have derived a simple interval of the middle weight that assures the construction of a family of rational quadratic Bézier curves to be planar spirals, which is characterized by the turning angle, end curvatures and the chords of control polygon. The proposed formulation can be used by CAD systems for aesthetic product design, highway/railway design and robot trajectory design avoiding unwanted curvature oscillations

Curvature line parametrized surfaces and orthogonal coordinate systems. Discretization with Dupin cyclides

Cyclidic nets are introduced as discrete analogs of curvature line parametrized surfaces and orthogonal coordinate systems. A 2-dimensional cyclidic net is a piecewise smooth $C^1$-surface built from surface patches of Dupin cyclides, each patch being bounded by curvature lines of the supporting cyclide. An explicit description of cyclidic nets is given and their relation to the established discretizations of curvature line parametrized surfaces as circular, conical and principal contact element nets is explained. We introduce 3-dimensional cyclidic nets as discrete analogs of triply-orthogonal coordinate systems and investigate them in detail. Our considerations are based on the Lie geometric description of Dupin cyclides. Explicit formulas are derived and implemented in a computer program.Comment: 39 pages, 30 figures; Theorem 2.7 has been reformulated, as a normalization factor in formula (2.4) was missing. The corresponding formulations have been adjusted and a few typos have been correcte

Characterizing envelopes of moving rotational cones and applications in CNC machining

Motivated by applications in CNC machining, we provide a characterization of surfaces which are enveloped by a one-parametric family of congruent rotational cones. As limit cases, we also address ruled surfaces and their offsets. The characterizations are higher order nonlinear PDEs generalizing the ones by Gauss and Monge for developable surfaces and ruled surfaces, respectively. The derivation includes results on local approximations of a surface by cones of revolution, which are expressed by contact order in the space of planes. To this purpose, the isotropic model of Laguerre geometry is used as there rotational cones correspond to curves (isotropic circles) and higher order contact is computed with respect to the image of the input surface in the isotropic model. Therefore, one studies curve-surface contact that is conceptually simpler than the surface-surface case. We show that, in a generic case, there exist at most six positions of a fixed rotational cone that have third order contact with the input surface. These results are themselves of interest in geometric computing, for example in cutter selection and positioning for flank CNC machining.RYC-2017-2264

Discrete Differential Geometry

This is the collection of extended abstracts for the 26 lectures and the open problem session at the fourth Oberwolfach workshop on Discrete Differential Geometry

On the Geometries of Conic Section Representation of Noisy Object Boundaries

This paper studies some geometrical properties of conic sections and the utilization of these properties for the generation of conic section representations of object boundaries in digital images. Several geometrical features of the conic sections, such as the chord, the characteristic point, the guiding triangles, and their appearances under the tessellation and noise corruption of the digital images are discussed. The study leads to a noniterative algorithm that takes advantage of these features in the process of formulating the conic section parameters and generating the approximations of object boundaries from the given sequences of edge pixels in the images. The results can be optimized with respect to certain different criteria of the fittings

Modeling and sensitivity analysis of aircraft geometry for multidisciplinary optimization problems

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 2012.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Cataloged from student-submitted PDF version of thesis.Includes bibliographical references (p. 415-421).A new geometry management paradigm for aircraft design utilizes Computer Aided Design (CAD) systems as the source for consistent geometry models across design phases and analysis tools. Yet various challenges inhibit the widespread application of CAD models in aircraft conceptual design because current CAD platforms are not designed for automated shape optimization. In particular, CAD models built with conventional methods can perform poorly in automated design frameworks and their associated CAD systems do not provide shape sensitivities. This thesis aims to remedy these concerns by bridging the computational geometry tools in CAD with aerospace design needs. A methodology for constructing CAD models is presented using concepts of multifidelity/multidisciplinary geometry and design motion. A formalized definition of design intent emerges from this approach that enables CAD models with parameterization flexibility, shape malleability and regeneration robustness for automated design settings. Analytic shape sensitivities are also presented to apply CAD models in gradient-based shape optimization. The parameterization and sensitivities for sketches, extrude, revolve and sweep features are given for mechanical design; shape sensitivities for B-spline curves and surfaces are also presented for airfoil and wing design. Furthermore, analytic methods modeling the sensitivity of intersection edges and nodes in a boundary representation (BRep) are given. Comparisons between analytic and finite-difference gradients show excellent agreement, however an error associated with the finite-difference gradient is found to exist if linearizing the support points of B-spline curves/surfaces and regenerating with a geometry kernel. This important outcome highlights a limitation of the finite-difference method when used on CAD models containing these entities. Finally, various example design problems are shown which highlight the application of the methods presented in the thesis. These include mechanical part design, inverse/forward design of airfoils and wings, and a multidisciplinary design space study. Gradient-based optimization is used in each design problem to compare the impact of analytic and finite-difference geometry gradients on the final designs obtained. With each of these contributions, the application of CAD-by David Sergio Lazzara.Ph.D

Structural-Electromagnetic Simulation Coupling and Conformal Antenna Design Tool

Airborne and spaceborne radar has long been an effective tool for remote sensing, surveillance, and reconnaissance. Most airborne systems utilize antenna arrays that are installed inside the moldline of the aircraft or in radomes that protect the array from in-flight loads. While externally-mounted arrays can offer the advantage of larger apertures, sensor-vehicle interactions often result in performance degradation of both systems. The presence of an externally-mounted array will increase the vehicle’s drag and potentially affect the dynamics and control of the vehicle. In addition, in-flight structural loads will deform the array, thus resulting in relative phase errors. While there exist a multitude of physics-based simulation tools to determine the effects of the array on the aircraft performance, existing tools are not sufficient for generating deformed arrays necessary for determining in-flight array performance. In response to this need, a computer tool for analyzing antennas undergoing structural loads is developed. The Antenna Deformation Tool (ADT) has two primary uses: generating deformed geometry from the output of a structural Finite Element Model (FEM) for use in an Electromagnetic (EM) simulation, and designing conformal antenna arrays. The two commercial software packages ADT is optimized for are MSC NASTRAN and ANSYS HFSS. Specifically, ADT is designed to generate a deformed 3D Computer Aided Design (CAD) model from a NASTRAN structural mesh. The resulting CAD model is compatible with HFSS electromagnetic simulation software for the assessment of the effects of loads on performance. The main purpose for the development of ADT is to facilitate studies of how structural deformations affect airborne antenna arrays performance and to provide the capability to perform studies easily and quickly using different antennas on the same structural model. ADT capabilities are demonstrated using several representative airborne antenna array structures. ADT is also demonstrated in the design of conformal antenna arrays. ADT can import CAD geometry and deform it according to a prescribed deformation field. The deformation field can either be determined from structural simulations or be provided by the user. This functionality allows the user to take an existing planar antenna design and conform it to a desired shape. Within the scope of airborne antenna arrays, this would allow an engineer to conform the antenna to the moldline of the aircraft or other support structure. Currently, ADT can interpret only quad and triangular 2D elements from NASTRAN. In addition, its ability to interpret a surface from a point cloud is limited to surface meshes in which there are exactly four explicit vertices, or surfaces which have a fairly even boundary with no major discontinuities and can be divided into four even segments. ADT is tested on NASTRAN structural models of small to medium complexity, and the geometry generated from simple models is used in HFSS simulations with success (with occasional post processing required). The antenna deformation submodule shows favorable performance with sheet and solid CAD geometry, though post-processing is required in the case of the latter. Results of some deformed antennas simulated with HFSS in the 200 MHz range are presented. The surface error of the geometry produced by ADT varies with the type of input mesh, with curved meshes and surfaces having higher errors. In terms of average element edge length, the maximum surface error is up to 1% for surfaces with no to small curvatures, and up to 3.6% for highly curved surfaces. This translates to about 0.17% of the mesh diagonal. ADT contains a set of classes and functions which provide ample capabilities for surface generation from meshes, and the process implemented is mostly automatic, requiring minimal user intervention. Due to ADT defining deformed geometry purely on separate meshes, adjacent surfaces are not associative and continuity between them is not guaranteed, which inherently can result in small intersections. These intersections can cause meshing problems with HFSS; however, these issues can be mitigated by adding a small offset. While demonstrated applications are still limited, ADT promises to substantially contribute to the design of aircraft-integrated antennas and multifunctional structures. With very limited capabilities for generating and assessing deformed antenna geometry currently existing, ADT represents a unique tool. ADT could be used not only in developing the next-generation of airborne remote sensing technologies, but to characterize in-flight performance of existing systems as well

CAD interface and framework for curve optimisation applications

Computer Aided Design is currently expanding its boundaries to include more design features in its processes. Design is identified as an iterative process converging to solutions satisfying a set of constraints. Its close relation with optimisation indicate that there is strong potential for the integration of optimisation and CAD. The problem addressed in this thesis lies in interfacing the geometric representation of design with other non-geometric aspects. The example of free-form curve modelling is taken to investigate such relationships. Assumptions are made that Optimisation is powered by Evolutionary Computing algorithms like Genetic Algorithms (GA). The geometric definition of curves is commonly supported by NURBS, whose construction constraints are defined locally at the data points. Here the NURBS formulation is used with GA in an attempt to provide complementary handles on the curves shape other than the usual data point coordinates and control points weights. Differential properties are used for optimising NURBS, Hermite interpolation allows for the definition of higher order constraints (tangent, normal, bi-normal) at data points. The assignment of parameter values at the data points, known as parameterisation also provides control of the curve’s shape. Curve optimisation is also performed at the geometric modelling level. Old mathematical theorems established by Frénet and further developed by other mathematicians provide means of defining a curve’s shape with it’s intrinsic equations. Such representation is possible by using Function Representation (F-rep) algebra available in the ACIS software. Frep allows more generic and exact means of interfacing with the curve’s geometry and new functionality for curve inspection and optimisation are proposed in this thesis. The integration of optimisation findings and CAD are documented in the definition of a framework. The framework architecture proposed reconstructs a new CAD environment from separate elements bolted together in a generic Application Programming Interface (API) named “Oli interface”. Functionality created to interface optimisation and CAD makes a requirement list of the work that both sides should undertake to achieve design optimisation in the CAD environment.EThOS - Electronic Theses Online ServiceGBUnited Kingdo