Alan Ball

Conditions for tangent plane continuity over recursively generated B-spline surfaces

The continuity properties of recursively generated B-spline surfaces over an arbitrary topology have been related to the eigenproperties of the local subdivision transformation. In this paper a discrete Fourier transform technique is employed to derive these eigenproperties for a general choice of subdivision weightings. Conditions on these weightings are identified for tangent plane continuity at the extraordinary points and a geometric interpretation is given.

CONSURF. Part 3: how the program is used

Abstract This paper is the last of a series describing the surface lofting program CONSURF, and outlines how the program is used. The overall approach is geometrical and is modelled closely on manual lofting. The program user must have a practical understanding of shape and be able to visualize the surfaces he defines. He must also be numerate, but he does not need to understand the surface mathematics which is confined to the software.

The generalised Cornu spiral and its application to span generation

Abstract A Cornu spiral is a plane curve having a linear curvature profile. This paper considers plane curves having rational linear curvature profiles. These curves are defined to be generalised Cornu spirals (GCS) and are quality curves in the sense that they are continuous and smooth, can contain one inflection at most, and have a bounded and monotonic curvature profile. In addition, the GCS has an extra degree of freedom over the Cornu spiral that is available for shape control. Starting from the intrinsic equation of the GCS, the technique of curve synthesis is used to design a quality curve that can be applied to a wide range of span generation problems.

A matrix approach to the analysis of recursively generated B-spline surfaces

Abstract The analysis of recursively generated B-spline surfaces has been related to the eigenproperties of the local subdivision matrix. In this paper a direct approach is employed to establish general expressions for the eigenvalues by manipulating the characteristic determinant. The results are compared to those of Doo and Sabin and the implications of varying the subdivision weightings are considered.

Control point surfaces over non-four-sided areas

Abstract This paper constructs control point surfaces of arbitrary degree over 3-, 5- and 6-sided areas. These surface patches behave like rectangular Bezier surface patches along their boundaries and can be connected smoothly with surrounding rectangular Bezier patches.

An investigation of curvature variations over recursively generated B-spline surfaces

The continuity properties of recursively generated B-spline surfaces over an arbitrary topology have been related to the eigenproperties of the local subdivision tranformation, and conditions have been established on the subdivision weightings for tangent plane continuity at extraordinary points. In this paper, curves through an extradordinary point, which align in both the tagent and binormal direction, are identified, and their curvatures are compared either side of the point. Further restrictions on the subdivision weightings are derived to optimize the curvature properties of the surface. In general continuity of curvature is not attained.

Design of an n -sided surface patch from Hermite boundary data

Abstract The design of surface patches of other than four sides has traditionally been a time-consuming problem. In this paper it is demonstrated that a B-spline subdivision patch can be fitted into a general n-sided area of a bicubic surface with at least tangent plane continuity on the boundary. One degree of freedom is identified and related to shape control.

Recursively generated B-spline surfaces

The paper reports the progress of an SERC sponsored project on recursive subdivision over arbitrary topologies. The limit surface is generally a standard B-spline surface except at the so called ‘extraordinary points’ where, it is reported, tangent plane continuity can be guaranteed and discrepancies in the curvature can be confined to 10% or less. The optimal subdivision procedure is currently being implemented in a designer-oriented form in collaboration with CADCentre Ltd Cambridge.

Surface modelling and mesh generation for simulating superplastic forming

Abstract This paper presents an approach which enables surface modelling, mesh generation and the Finite Element (FE) analysis to be integrated together to simulate superplastic forming processes for complex shaped components. Techniques have been developed to generate an FE mesh over non-four-sided surface areas, the boundaries of which are Bezier curves of arbitrary degree m, using a consistent expression. Theoretical evidence is given to determine the number of Bezier triangular patches required for accurately re-constructing die surfaces within a commercial FE solver. The techniques developed have been successfully used in determining the process parameters for forming a 3D rectangular box.

Modelling of hardening due to grain growth for a superplastic alloy

Abstract Combined analytical and numerical optimisation procedures have been developed to determine the material constants in superplastic constitutive equations. The conventional grain growth rate equation has been modified to enable accurate modelling of the isothermal and plastic-strain-induced grain growths for different microstructures of a titanium alloy. A set of unified viscoplastic constitutive equations for Ti–6Al–4V at 927°C, which incorporates isotropic hardening and grain growth, has been fully determined from experimental data for different initial grain sizes and strain rates. Close agreement between the predicted and experimental stress–strain relationships has been achieved. In addition, the contributions of hardening constituents, such as strain-rate hardening, isotropic hardening and the hardening due to grain growth have been modelled.