A survey of partial differential equations in geometric design

YesComputer aided geometric design is an area where the improvement of surface generation techniques is an everlasting demand since faster and more accurate geometric models are required. Traditional methods for generating surfaces were initially mainly based upon interpolation algorithms. Recently, partial differential equations (PDE) were introduced as a valuable tool for geometric modelling since they offer a number of features from which these areas can benefit. This work summarises the uses given to PDE surfaces as a surface generation technique togethe

Fast generation of 3D deformable moving surfaces

Dynamic surface modeling is an important subject of geometric modeling due to their extensive applications in engineering design, entertainment and medical visualization. Many deformable objects in the real world are dynamic objects as their shapes change over time. Traditional geometric modeling methods are mainly concerned with static problems, therefore unsuitable for the representation of dynamic objects. Apart from the definition of a dynamic modeling problem, another key issue is how to solve the problem. Because of the complexity of the representations, currently the finite element method or finite difference method is usually used. Their major shortcoming is the excessive computational cost, hence not ideal for applications requiring real-time performance. We propose a representation of dynamic surface modeling with a set of fourth order dynamic partial differential equations (PDEs). To solve these dynamic PDEs accurately and efficiently, we also develop an effective resolution method. This method is further extended to achieve local deformation and produce n-sided patches. It is demonstrated that this new method is almost as fast and accurate as the analytical closed form resolution method and much more efficient and accurate than the numerical methods

Method of boundary based smooth shape design

The discussion in this paper focuses on how boundary based smooth shape design can be carried out. For this we treat surface generation as a mathematical boundary-value problem. In particular, we utilize elliptic Partial Differential Equations (PDEs) of arbitrary order. Using the methodology outlined here a designer can therefore generate the geometry of shapes satisfying an arbitrary set of boundary conditions. The boundary conditions for the chosen PDE can be specified as curves in 3-space defining the profile geometry of the shape. We show how a compact analytic solution for the chosen arbitrary order PDE can be formulated enabling complex shapes to be designed and manipulated in real time. This solution scheme, although analytic, satisfies exactly, even in the case of general boundary conditions, where the resulting surface has a closed form representation allowing real time shape manipulation. In order to enable users to appreciate the powerful shape design and manipulation capability of the method, we present a set of practical example

Uncovering the specificities of CAD tools for industrial design with design theory – style models for generic singularity

International audienceAccording to some casual observers, computer-aided design (CAD) tools are very similar. These tools are used to design new artifacts in a digital environment; hence, they share typical software components, such as a computing engine and human-machine interface. However, CAD software is dedicated to specific professionals—such as engineers, three-dimensional (3D) artists, and industrial designers (IDs)—who claim that, despite their apparent similarities, CAD tools are so different that they are not substitutable. Moreover, CAD tools do not fully meet the needs of IDs. This paper aims at better characterizing CAD tools by taking into account their underlying design logic, which involves relying on recent advances in design theory. We show that engineering CAD tools are actually modeling tools that design a generic variety of products; 3D artist CAD tools not only design but immediately produce single digital artefacts; and ID CAD tools are neither a mix nor an hybridization of engineering CAD and 3D artist CAD tools but have their own logic, namely to create new conceptual models for a large variety of products, that is, the creation of a unique original style that leads to a generic singularity. Such tools are useful for many creative designers beyond IDs

An efficient active B-spline/nurbs model for virtual sculpting

This thesis presents an Efficient Active B-Spline/Nurbs Model for Virtual Sculpting. In spite of the on-going rapid development of computer graphics and computer-aided design tools, 3D graphics designers still rely on non-intuitive modelling procedures for the creation and manipulation of freeform virtual content. The ’Virtual Sculpting' paradigm is a well-established mechanism for shielding designers from the complex mathematics that underpin freeform shape design. The premise is to emulate familiar elements of traditional clay sculpting within the virtual design environment. Purely geometric techniques can mimic some physical properties. More exact energy-based approaches struggle to do so at interactive rates. This thesis establishes a unified approach for the representation of physically aware, energy-based, deformable models, across the domains of Computer Graphics, Computer-Aided Design and Computer Vision, and formalises the theoretical relationships between them. A novel reformulation of the computer vision approach of Active Contour Models (ACMs) is proposed for the domain of Virtual Sculpting. The proposed ACM-based model offers novel interaction behaviours and captures a compromise between purely geometric and more exact energy-based approaches, facilitating physically plausible results at interactive rates. Predefined shape primitives provide features of interest, acting like sculpting tools such that the overall deformation of an Active Surface Model is analogous to traditional clay modelling. The thesis develops a custom-approach to provide full support for B-Splines, the de facto standard industry representation of freeform surfaces, which have not previously benefited from the seamless embodiment of a true Virtual Sculpting metaphor. A novel generalised computationally efficient mathematical framework for the energy minimisation of an Active B-Spline Surface is established. The resulting algorithm is shown to significantly reduce computation times and has broader applications across the domains of Computer-Aided Design, Computer Graphics, and Computer Vision. A prototype ’Virtual Sculpting’ environment encapsulating each of the outlined approaches is presented that demonstrates their effectiveness towards addressing the long-standing need for a computationally efficient and intuitive solution to the problem of interactive computer-based freeform shape design

Generalized partial differential equations for interactive design

This paper presents a method for interactive design by means of extending the PDE based approach for surface generation. The governing partial differential equation is generalized to arbitrary order allowing complex shapes to be designed as single patch PDE surfaces. Using this technique a designer has the flexibility of creating and manipulating the geometry of shape that satisfying an arbitrary set of boundary conditions. Both the boundary conditions which are defined as curves in 3-space and the spine of the corresponding PDE are utilized as interactive design tools for creating and manipulating geometry intuitively. In order to facilitate interactive design in real time, a compact analytic solution for the chosen arbitrary order PDE is formulated. This solution scheme even in the case of general boundary conditions satisfies exactly the boundary conditions where the resulting surface has an closed form representation allowing real time shape manipulation. In order to enable users to appreciate the powerful shape design and manipulation capability of the method, we present a set of practical examples

Virtual sculpting : an investigation of directly manipulated free-form deformation in a virtual environment

This thesis presents a Virtual Sculpting system, which addresses the problem of Free-Form Solid Modelling. The disparate elements of a Polygon-Mesh representation, a Directly Manipulated Free-Form Deformation sculpting tool, and a Virtual Environment are drawn into a cohesive whole under the mantle of a clay-sculpting metaphor. This enables a user to mould and manipulate a synthetic solid interactively as if it were composed of malleable clay. The focus of this study is on the interactivity, intuitivity and versatility of such a system. To this end, a range of improvements is investigated which significantly enhances the efficiency and correctness of Directly Manipulated Free-Form Deformation, both separately and as a seamless component of the Virtual Sculpting system