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

Triangular meshes are important in many fields in both basic and applied research that rely on their correctness
and accuracy. Many operations with meshes can lead to undesirable situations and the resulting models can be
damaged and further unusable. Self-intersection and mesh-to-mesh intersection are types of operations that are
often present and can cause such problems. We propose an accurate geometry-based method for local repair
of intersecting meshes. The state-of-the-art methods either solve the problem inaccurately, or use methods
such as arbitrary precision arithmetic or virtual perturbation to deal with the troublesome boundary cases. Our
method represents a robust way to repair intersecting meshes accurately without the need to manipulate with
the input data or to employ arbitrary precision arithmetic. The correct solution is obtained through a careful
classification of the cases that could result from a numerical imprecision of the floating point arithmetic.

A Simple and Robust Approach to Computation of Meshes Intersection

This paper presents a method for the metamorphosis of genus-0 triangle meshes based on their intersection. It is an extension of our previous 2D algorithm [Malkova, 2007]. Our algorithm is designed to simulate growing processes, therefore it is useful for morphing objects, where the user expects some parts of the latter object to grow out from the former one (e.g. a head with and without horns). The user can influence the algorithm’s behavior by changing the mutual position of the objects, while the results are easily predictable. CR Categories: I.3.5 [Computational Geometry and Object Modeling]: Boundary representations— [I.3.7]: Three-Dimensional Graphics and Realism—Animation

https://www.ep.liu.se/ecp/034/011/ecp083411.pdf

Core-based morphing algorithm for triangle meshes

Finding which triangle in a planar or 2.5D triangle mesh contains a query point (so-called point location problem) is a frequent task in geosciences, especially when working with triangulated irregular network models. Usually, a large number of point locations has to be performed, and so there is a need for fast algorithms having minimal additional memory requirements and resistant to changes in the triangulation. So-called walking algorithms offer low complexity, easy implementation, and negligible additional memory requirements, which makes them suitable for such applications. In this article, we focus on these algorithms, summarize, and compare them with regard to their use in geosciences. Since such a summary has not been done yet, our article should serve those who are dealing with this problem in their application to decide which algorithm would be the best for their solution. Moreover, the influence of the triangulation type on the number of the visited triangles is discussed.

Walking algorithms for point location in TIN models

Triangle surface models belong to the most popular type of geometric objects description in computer graphics. Therefore, the problem of fast visualization of this type of data is often solved. One of popular approaches is stripification, i.e., a conversion of triangle surface into strips of triangles. This enables to reduce the rendering time by reduction data size and by avoiding of redundant lighting and transformations computations. In this paper we present a new stripification algorithm for static fully triangulated meshes. Our new algorithm is based on the dual graph of triangulation. The experimental results show that our stripification produces much lower number of triangle strips than other stripification algorithms (except one)

Multi-path algorithm for triangle strips

Curvature is an important geometric property in computer graphics that provides information about the behavior

of object surfaces. The exact curvature can only be calculated for a limited set of surfaces description.

Most of the time, we deal with triangles, point sets or some other discrete representation of the surface. For

those, curvature computation is problematic. Moreover, most of existing algorithms were developed for static

geometry and can be slow for interactive modeling. This paper proposes a screen space method which estimates

the mean and Gaussian curvature at interactive rates. The algorithm uses positions and normals to

estimate the curvature from the second fundamental form matrix. Using the screen space has advantages over

the classical approach: low-poly geometry can be used and additional detail can be added with normal and

bump maps. The screen space curvature can be easily added to existing rendering pipelines. The proposed

algorithm was tested on several models and it outperforms current state-of-the-art GPU approaches.

Ivana Kolingerová

Papers

A Simple and Robust Approach to Computation of Meshes Intersection

Core-based morphing algorithm for triangle meshes

Walking algorithms for point location in TIN models

Multi-path algorithm for triangle strips

Fast Screen Space Curvature Estimation on GPU