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

B\'ezier curves that are close to elastica

We study the problem of identifying those cubic B\'ezier curves that are close in the L2 norm to planar elastic curves. The problem arises in design situations where the manufacturing process produces elastic curves; these are difficult to work with in a digital environment. We seek a sub-class of special B\'ezier curves as a proxy. We identify an easily computable quantity, which we call the lambda-residual, that accurately predicts a small L2 distance. We then identify geometric criteria on the control polygon that guarantee that a B\'ezier curve has lambda-residual below 0.4, which effectively implies that the curve is within 1 percent of its arc-length to an elastic curve in the L2 norm. Finally we give two projection algorithms that take an input B\'ezier curve and adjust its length and shape, whilst keeping the end-points and end-tangent angles fixed, until it is close to an elastic curve.Comment: 13 pages, 15 figure

Interactive design exploration for constrained meshes

In architectural design, surface shapes are commonly subject to geometric constraints imposed by material, fabrication or assembly. Rationalization algorithms can convert a freeform design into a form feasible for production, but often require design modifications that might not comply with the design intent. In addition, they only offer limited support for exploring alternative feasible shapes, due to the high complexity of the optimization algorithm. We address these shortcomings and present a computational framework for interactive shape exploration of discrete geometric structures in the context of freeform architectural design. Our method is formulated as a mesh optimization subject to shape constraints. Our formulation can enforce soft constraints and hard constraints at the same time, and handles equality constraints and inequality constraints in a unified way. We propose a novel numerical solver that splits the optimization into a sequence of simple subproblems that can be solved efficiently and accurately. Based on this algorithm, we develop a system that allows the user to explore designs satisfying geometric constraints. Our system offers full control over the exploration process, by providing direct access to the specification of the design space. At the same time, the complexity of the underlying optimization is hidden from the user, who communicates with the system through intuitive interfaces

Interactive real-time physics: an intuitive approach to form-finding and structural analysis for design and education

Real-time physics simulation has been extensively used in computer games, but its potential has yet to be fully realised in design and education. We present an interactive 3D physics engine with a wide variety of applications. In common with traditional FEM, the use of a local element stiffness matrix is retained. However, unlike typical non-linear FEM routines, elements forces, moments and inertia are appropriately lumped at nodes following the Dynamic Relaxation Method. A semi-implicit time integration scheme updates linear and angular momentum, and subsequently the local coordinate frames of the nodes. The Co-Rotational formulation is used to compute the resultant field of displacements in global coordinates including large deformations. The results obtained compare well against established commercial software. We demonstrate that the method presented allows the making of interactive structural models that can be used in teaching to develop an intuitive understanding of structural behaviour. We also show that the same interactive physics framework allows real-time optimization that can be used for geometric and structural design application

The Sound Manifesto

Computing practice today depends on visual output to drive almost all user interaction. Other senses, such as audition, may be totally neglected, or used tangentially, or used in highly restricted specialized ways. We have excellent audio rendering through D-A conversion, but we lack rich general facilities for modeling and manipulating sound comparable in quality and flexibility to graphics. We need co-ordinated research in several disciplines to improve the use of sound as an interactive information channel. Incremental and separate improvements in synthesis, analysis, speech processing, audiology, acoustics, music, etc. will not alone produce the radical progress that we seek in sonic practice. We also need to create a new central topic of study in digital audio research. The new topic will assimilate the contributions of different disciplines on a common foundation. The key central concept that we lack is sound as a general-purpose information channel. We must investigate the structure of this information channel, which is driven by the co-operative development of auditory perception and physical sound production. Particular audible encodings, such as speech and music, illuminate sonic information by example, but they are no more sufficient for a characterization than typography is sufficient for a characterization of visual information.Comment: To appear in the conference on Critical Technologies for the Future of Computing, part of SPIE's International Symposium on Optical Science and Technology, 30 July to 4 August 2000, San Diego, C

Collaborative Verification-Driven Engineering of Hybrid Systems

Hybrid systems with both discrete and continuous dynamics are an important model for real-world cyber-physical systems. The key challenge is to ensure their correct functioning w.r.t. safety requirements. Promising techniques to ensure safety seem to be model-driven engineering to develop hybrid systems in a well-defined and traceable manner, and formal verification to prove their correctness. Their combination forms the vision of verification-driven engineering. Often, hybrid systems are rather complex in that they require expertise from many domains (e.g., robotics, control systems, computer science, software engineering, and mechanical engineering). Moreover, despite the remarkable progress in automating formal verification of hybrid systems, the construction of proofs of complex systems often requires nontrivial human guidance, since hybrid systems verification tools solve undecidable problems. It is, thus, not uncommon for development and verification teams to consist of many players with diverse expertise. This paper introduces a verification-driven engineering toolset that extends our previous work on hybrid and arithmetic verification with tools for (i) graphical (UML) and textual modeling of hybrid systems, (ii) exchanging and comparing models and proofs, and (iii) managing verification tasks. This toolset makes it easier to tackle large-scale verification tasks

Human Motion Trajectory Prediction: A Survey

With growing numbers of intelligent autonomous systems in human environments, the ability of such systems to perceive, understand and anticipate human behavior becomes increasingly important. Specifically, predicting future positions of dynamic agents and planning considering such predictions are key tasks for self-driving vehicles, service robots and advanced surveillance systems. This paper provides a survey of human motion trajectory prediction. We review, analyze and structure a large selection of work from different communities and propose a taxonomy that categorizes existing methods based on the motion modeling approach and level of contextual information used. We provide an overview of the existing datasets and performance metrics. We discuss limitations of the state of the art and outline directions for further research.Comment: Submitted to the International Journal of Robotics Research (IJRR), 37 page