Parallel Sparse LU Decomposition on a Mesh Network of Transputers

A parallel algorithm is presented for the LU decomposition of a general sparse matrix on a distributed-memory MIMD multiprocessor with a square mesh communication network. In the algorithm, matrix elements are assigned to processors according to the grid distribution. Each processor represents the nonzero elements of its part of the matrix by a local, ordered, two-dimensional linked-list data structure. The complexity of important operations on this data structure and on several others is analysed. At each step of the algorithm, a parallel search for a set of m compatible pivot elements is performed. The Markowitz counts of the pivot elements are close to minimum, to preserve the sparsity of the matrix. The pivot elements also satisfy a threshold criterion, to ensure numerical stability. The compatibility of the m pivots enables the simultaneous elimination of m pivot rows and m pivot columns in a rank-m update of the reduced matrix. Experimental results on a network of 400 transputers are presented for a set of test matrices from the Harwell–Boeing sparse matrix collection

Exact Motion Planning Amidst Fat Obstacles: An Overview Of Results

The efficiency of algorithms for the exact solution of the motion planning problem depends, to a large extent, on the complexity of the free space. Unfortunately, the theoretical free space complexity can be very high. In many practical cases, however, the actual complexity tends to remain far below the theoretical worst-case bounds. It turns out that the complexity is only linear (in the number of obstacles) if the density of the obstacles in the workspace is small. Such a low obstacle density follows, for example, if the obstacles are `fat' and the robot is not too large compared to the obstacles. These realistic circumstances lead to remarkable efficiency gains for a number of existing planar motion planning problems. More importantly, the assumptions induce a specific structure of the free space that allows to reduce the problem of decomposing the free space to the problem of finding some constrained workspace partition. The approach leads to efficient algorithms for various instan..

Bounding the Locus of the Center of Mass for a Part with Shape Variation

The shape and center of mass of a part are crucial parameters to algorithms for planning automated manufacturing tasks. As industrial parts are generally manufactured to tolerances, the shape is subject to variations, which, in turn, also cause variations in the location of the center of mass. Planning algorithms should take into account both types of variation to prevent failure when the resulting plans are applied to manufactured incarnations of a model part. We study the relation between variation in part shape and variation in the location of the center of mass for a part with uniform mass distribution. We consider a general model for shape variation that only assumes that every valid instance contains a shape PI while it is contained in another shape PE. We characterize the worst-case displacement of the center of mass in a given direction in terms of PI and PE. The characterization allows us to determine an adequate polytopic approximation of the locus of the center of mass. We also show that the worst-case displacement is small if PI is convex and fat and the distance between the boundary of PE and PI is bounded

Range Searching and Point Location among Fat Objects

We present a data structure that can store a set of disjoint fat objects in d-space such that point location and bounded-size range searching with arbitrarily-shaped ranges can be performed efficiently. The structure can deal with either arbitrary (fat) convex objects or non-convex polytopes. The multi-purpose data structure supports point location and range searching queries in time O(log d\Gamma1 n) and requires O(n log d\Gamma1 n) storage, after O(n log d\Gamma1 n log log n) preprocessing. The data structure and query algorithm are rather simple. 1 Introduction Fatness turns out to be an interesting phenomenon in computational geometry. Several papers present surprising combinatorial complexity reductions [3, 15, 22, 26, 32] and efficiency gains for algorithms [1, 4, 19, 28, 33] if the objects under consideration have a certain fatness. Fat objects are compact to some extent, rather than long and thin. Fatness is a realistic assumption, since in many practical instances of ..

Supporting Cuts and Finite Element Deformation in Interactive Surgery Simulation

Interactive surgery simulations have conflicting requirements of speed and accuracy. In this paper we show how to combine a relatively accurate deformation model---the Finite Element (FE) method---and interactive cutting without requiring expensive matrix updates or precomputation. Our approach uses an iterative algorithm for an interactive linear FE deformation simulation. The iterative process requires no global precomputation, so run-time changes of the mesh---that is, cuts--- can be simulated efficiently. Cuts are performed along faces of the mesh; this prevents growth of the mesh without violating mass preservation laws. We present a robust method for changing the mesh topology, and a satisfactory heuristic for determining along which faces to perform cuts. Nodes within the mesh are relocated to align the mesh with a virtual scalpel. This prevents a jagged surface appearance. On the other hand it generates degeneracies, which are removed afterward

Interactive needle insertions in 3D nonlinear material

In this paper, we show an interactive simulation of needle insertions in both 2D and 3D and soft tissue. The approach is based on the Finite Element Method (FEM) and uses quasistatic stick-slip friction for needle/tissue interactions. The FEM equations are solved using an iterative method, and the mesh is refined adaptively near the needle trajectory. The boundary formed by the needle surface is not represented explicitly in the mesh, but its geometry is accounted for in the friction forces. Since the surface is not represented explicitly, the quality of the initial mesh can be maintained by using a simple refinement scheme. This approach can also be applied to both the 3D situation and nonlinear material models. We present results of computational experiments of the 2D simulation, and show samples of the 3D implementation

On the Design of Traps for Feeding 3D Parts on Vibratory Tracks

Abstract-In the context of automated feeding (orienting) of industrial parts, we study the algorithmic design of traps in the bowl feeder track that filter out all but one orientation of a given polyhedral part. We propose a new class of traps that removes a V-shaped portion of the track. The proposed work advances the state-of-the-art in algorithmic trap design by extending earlier work [1], [3], [11]-which focuses solely on 2D parts-to 3D parts, and by incorporating a more realistic part motion model in the design algorithm. The presented complete design algorithm takes as input any polyhedral part, along with its center of mass, and reports all valid trap designs that feed the given part

Maintaining Mesh Connectivity Using a Simplex-Based Data Structure

Many applications in visualization and computer graphics use meshes composed of triangles or tetrahedra, and need to store the connectivity of these meshes. In some applications meshes must be changed at run-time. In this case, it is necessary to update mesh connectivity too. This paper presents a data structure with two operations that can express any mesh change. The prime advantage of this approach is that all low-level code to maintain mesh connectivity is contained in these two routines, which promotes a modular program design. The dat