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AuthorUgail, Hassan (19)Gonzalez Castro, Gabriela (9)Athanasopoulos, Michael (4)Wilson, M.J. (4)Bloor, M.I.G. (3)Spares, Robert (3)Sweeney, John (3)Whiteside, Benjamin R. (3)Kirmani, N. (1)Kubeisa, S. (1)View MoreSubject

PDE surfaces (19)

Partial differential equations (PDEs) (8)Partial differential equations (3)Boundary representations (2)Cyclic animation (2)Interactive design (2)Parametric design (2)Surface parameterization (2)Surface profiling (2)Aircraft design (1)View MoreDate Issued2000 - 2010 (17)1998 - 1999 (2)
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Shape morphing of complex geometries using partial differential equations.

Gonzalez Castro, Gabriela; Ugail, Hassan (Academy Publisher, 2007)

An alternative technique for shape morphing
using a surface generating method using partial differential
equations is outlined throughout this work. The boundaryvalue
nature that is inherent to this surface generation
technique together with its mathematical properties are
hereby exploited for creating intermediate shapes between
an initial shape and a final one. Four alternative shape
morphing techniques are proposed here. The first one is
based on the use of a linear combination of the boundary
conditions associated with the initial and final surfaces,
the second one consists of varying the Fourier mode for
which the PDE is solved whilst the third results from a
combination of the first two. The fourth of these alternatives
is based on the manipulation of the spine of the surfaces,
which is computed as a by-product of the solution. Results
of morphing sequences between two topologically nonequivalent
surfaces are presented. Thus, it is shown that the
PDE based approach for morphing is capable of obtaining
smooth intermediate surfaces automatically in most of the
methodologies presented in this work and the spine has been
revealed as a powerful tool for morphing surfaces arising
from the method proposed here.

Method of trimming PDE surfaces

Ugail, Hassan (Elsevier, 2006)

A method for trimming surfaces generated as solutions to Partial Differential Equations
(PDEs) is presented. The work we present here utilises the 2D parameter
space on which the trim curves are defined whose projection on the parametrically
represented PDE surface is then trimmed out. To do this we define the trim curves
to be a set of boundary conditions which enable us to solve a low order elliptic
PDE on the parameter space. The chosen elliptic PDE is solved analytically, even
in the case of a very general complex trim, allowing the design process to be carried
out interactively in real time. To demonstrate the capability for this technique we
discuss a series of examples where trimmed PDE surfaces may be applicable.

Time-dependent shape parameterisation of complex geometry using PDE surfaces

Ugail, Hassan (Nashboro Press, 2004)

On the spine of a PDE surface

Ugail, Hassan (Springer, 2003)

The spine of an object is an entity that can characterise the
object¿s topology and describes the object by a lower dimension. It has
an intuitive appeal for supporting geometric modelling operations.
The aim of this paper is to show how a spine for a PDE surface can
be generated. For the purpose of the work presented here an analytic
solution form for the chosen PDE is utilised. It is shown that the spine
of the PDE surface is then computed as a by-product of this analytic
solution.
This paper also discusses how the of a PDE surface can be used to manipulate
the shape. The solution technique adopted here caters for periodic
surfaces with general boundary conditions allowing the possibility of the
spine based shape manipulation for a wide variety of free-form PDE surface
shapes.

Manipulation of PDE surfaces using an interactively defined parameterisation

Ugail, Hassan; Bloor, M.I.G.; Wilson, M.J. (Elsevier, 1999)

Manipulation of PDE surfaces using a set of interactively defined parameters is considered. The PDE method treats surface design as a boundary-value problem and ensures that surfaces can be defined using an appropriately chosen set of boundary conditions and design parameters. Here we show how the data input to the system, from a user interface such as the mouse of a computer terminal, can be efficiently used to define a set of parameters with which to manipulate the surface interactively in real time.

On interactive design using the PDE method.

Ugail, Hassan; Bloor, M.I.G.; Wilson, M.J. (Vanderbilt University Press, 1998)

Parametric design of aircraft geometry using partial differential equations

Athanasopoulos, Michael; Ugail, Hassan; Gonzalez Castro, Gabriela (Elsevier, 2009)

The PDE surface method in higher dimensions.

Woodland, A.; Ugail, Hassan; Labrosse, F. (2007)

This paper presents a method to extend PDE surfaces to high dimensional
spaces. We review a common existing analytic solution, and show
how it can be used straightforwardly to increase the dimension of the
space the surface is embedded within. We then further develop a numerical
scheme suitable for increasing the number of variables that parametrise
the surface, and investigate some of the properties of this solution with a
view to future work.

Cyclic animation using Partial differential Equations

Gonzalez Castro, Gabriela; Athanasopoulos, Michael; Ugail, Hassan; Willis, P.; Sheng, Y (2010)

This work presents an efficient and fast method for achieving cyclic animation using Partial Differential Equations (PDEs). The boundary-value nature associ- ated with elliptic PDEs offers a fast analytic solution technique for setting up a framework for this type of animation. The surface of a given character is thus cre- ated from a set of pre-determined curves, which are used as boundary conditions so that a number of PDEs can be solved. Two different approaches to cyclic ani- mation are presented here. The first consists of using attaching the set of curves to a skeletal system hold- ing the animation for cyclic motions linked to a set mathematical expressions, the second one exploits the spine associated with the analytic solution of the PDE as a driving mechanism to achieve cyclic animation, which is also manipulated mathematically. The first of these approaches is implemented within a framework related to cyclic motions inherent to human-like char- acters, whereas the spine-based approach is focused on modelling the undulatory movement observed in fish when swimming. The proposed method is fast and ac- curate. Additionally, the animation can be either used in the PDE-based surface representation of the model or transferred to the original mesh model by means of
a point to point map. Thus, the user is offered with the choice of using either of these two animation repre- sentations of the same object, the selection depends on the computing resources such as storage and memory capacity associated with each particular application.

Automatic Reconstruction and Web Visualization of Complex PDE Shapes

Pang, M.; Sheng, Y.; Gonzalez Castro, Gabriela; Sourin, A.; Ugail, Hassan (2010)

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