A criterion is given for the convergence of numerical solutions of the Navier-Stokes equations in two dimensions under steady conditions. The criterion applies to all cases, of steady viscous flow in two dimensions and shows that if the local ' mesh Reynolds number ', based on the size of the mesh used in the solution, exceeds a certain fixed value, the numerical solution will not converge.

On the Navier-Stokes equations

Computational geometry

Computer systems commonly cache the values of variables to gain efficiency. In applications where the goal is to track attributes of a continuously moving or deforming physical system over time, caching relations between variables works better than caching individual values. The reason is that, as the system evolves, such relationships are more stable than the values of individual variables.Kinetic data structures (KDSs) are a novel formal framework for designing and analyzing sets of assertions to cache about the environment, so that these assertion sets are at once relatively stable and tailored to facilitate or trivialize the computation of the attribute of interest. Formally, a KDS is a mathematical proof animated through time, proving the validity of a certain computation for the attribute of interest. KDSs have rigorous associated measures of performance and their design shares many qualities with that of classical data structures.The KDS framework has led to many new and promising algorithms in applications where the efficient modeling of motion is essential. Among these are collision detection for moving rigid and deformable bodies, connectivity maintenace in ad-hoc networks, local environment tracking for mobile agents, and visibility/occlusion maintenance. This talk will survey the general ideas behind KDSs and illustrate their application to simple geometric problems that arise in virtual and physical environments.

Data structures for mobile data (invited talk) (abstract only)

Path planning still remains one of the core problems in modern robotic applications, such as the design of autonomous vehicles and perceptive systems. The basic path-planning problem is concerned with finding a good-quality path from a source point to a destination point that does not result in collision with any obstacles. In this article, we chose the roadmap approach and utilized the Voronoi diagram to obtain a path that is a close approximation of the shortest path satisfying the required clearance value set by the user. The advantage of the proposed technique versus alternative path-planning methods is in its simplicity, versatility, and efficiency.

Roadmap-Based Path Planning - Using the Voronoi Diagram for a Clearance-Based Shortest Path

Computer-Aided Geometric Design

Some well-known line-polyhedron intersection methods are summed up and new accelerating modifications presented. Results of comparison of known and newly developed methods are included. New methods use the fact that each line can be described as the intersection of two planes.

https://link.springer.com/content/pdf/10.1007/BF01889980.pdf

3D-line clipping algorithms: a comparative study

The paper introduces a new randomized sampling-based method of motion planning suitable for the problem of narrow passages. The proposed method was inspired by the method of exit points for cavities in protein models and is based on the Rapidly Exploring Random Tree (RRT). Unlike other methods, it can also provide locations of the exact positions of narrow passages. This information is extremely important as it helps to solve this part of space in more detail and even to decide whether a path through this bottleneck exists or not. For data with narrow passages, the proposed method finds more paths in a shorter time, for data without narrow passages, the proposed method is slower but still provides correct paths.

/pdf/narrow-passage-problem-solution-for-motion-planning-2osj8i373l.pdf

Narrow Passage Problem Solution for Motion Planning

Extension of the edge tracing algorithm to disconnected Voronoi skeletons

Encoding the geographic coordinates into the compact string expression is an important problem in geosciences, therefore, several algorithms exist. Such a string is used for several purposes. One of the most frequent is probably its use as a part of a URL or as a hashtag of ad hoc data. An important part of spatially referenced data can be also a timestamp, but current encoding methods do not allow to encode the temporal dimension. In this chapter we propose a new encoding algorithm focused on a point data expressed by latitude, longitude and timestamp coordinates. The algorithm details are described and its use is shown on examples.

STCode: The Text Encoding Algorithm for Latitude/Longitude/Time

Duality is a well-known principle of computational geometry. It is often used for detecting the mutual position of convex geometrical objects, the dimensions of which do not differ more than by one; e.g., a convex polyhedron and a hyperplane or two convex polyhedra of the same dimensions. In this paper, a new possibility of usage is shown for objects with dimensions different by two. The mutual position of a convex polyhedron and a line is tested. The method works as follows. The line is described as an intersection of two planes. With a proper choice of these planes, the original problem of detecting a polyhedron-line intersection is reduced to a one-dimensional range search, which is much simpler to solve. The price to be paid for this reduction is computationally more demanding preprocessing.

Ivana Kolingerová

Papers

3D-line clipping algorithms: a comparative study

Narrow Passage Problem Solution for Motion Planning

Extension of the edge tracing algorithm to disconnected Voronoi skeletons

STCode: The Text Encoding Algorithm for Latitude/Longitude/Time

Convex polyhedron-line intersection detection using dual representation