Nicholas M. Patrikalakis

Computation of the solutions of nonlinear polynomial systems

Abstract: A fundamental problem in computer aided design is the efficient computation of all roots of a system of nonlinear polynomial equations inn variables which lie within ann-dimensional @?. We present two techniques designed to solve such problems, which rely on representation of polynomials in the multivariate Bernstein basis and subdivision. In order to isolate all of the roots within the given domain, each method uses a different scheme for constructing a series of bounding @?es; the first method projects control polyhedra onto a set of coordinate planes, and the second employs linear optimization. We also examine in detail the local convergence properties of the two methods, proving that the former is quadratically convergent forn=1 and linearly convergent forn 1, while the latter is quadratically convergent for alln. Worst-case complexity analysis, as well as analysis of actual running times are performed.

Modeling and designing functionally graded material components for fabrication with local composition control

Abstract Solid Freeform Fabrication (SFF) processes have demonstrated the ability to produce parts with locally controlled composition. In the limit, processes, such as three dimensional (3D) Printing can create parts with composition control on a length scale of 100 μm. To exploit this potential, new methods to model, exchange, and process parts with Local Composition Control need to be developed. An approach to modeling a parts geometry, topology, and composition is presented. This approach is based on subdividing the solid model into sub-regions and associating analytic composition blending functions with each region. These blending functions define the composition throughout the model as mixtures of the primary materials avialable to the SFF machine. Design tools based upon distance functions are also introduced, such as the specification of composition as a function of the distance from the surface of a part. Finally, the role of design rules restricting maximum and minimum concentrations is discussed.

An algorithm for the medial axis transform of 3D polyhedral solids

The medial axis transform (MAT) is a representation of an object which has been shown to be useful in design, interrogation, animation, finite element mesh generation, performance analysis, manufacturing simulation, path planning and tolerance specification. In this paper, an algorithm for determining the MAT is developed for general 3D polyhedral solids of arbitrary genus without cavities, with nonconvex vertices and edges. The algorithm is based on a classification scheme which relates different pieces of the medial axis (MA) to one another, even in the presence of degenerate MA points. Vertices of the MA are connected to one another by tracing along adjacent edges, and finally the faces of the axis are found by traversing closed loops of vertices and edges. Representation of the MA and its associated radius function is addressed, and pseudocode for the algorithm is given along with recommended optimizations. A connectivity theorem is proven to show the completeness of the algorithm. Complexity estimates and stability analysis for the algorithms are presented. Finally, examples illustrate the computational properties of the algorithm for convex and nonconvex 3D polyhedral solids with polyhedral holes.

Path Planning of Autonomous Underwater Vehicles for Adaptive Sampling Using Mixed Integer Linear Programming

The goal of adaptive sampling in the ocean is to predict the types and locations of additional ocean measurements that would be most useful to collect. Quantitatively, what is most useful is defined by an objective function and the goal is then to optimize this objective under the constraints of the available observing network. Examples of objectives are better oceanic understanding, to improve forecast quality, or to sample regions of high interest. This work provides a new path-planning scheme for the adaptive sampling problem. We define the path-planning problem in terms of an optimization framework and propose a method based on mixed integer linear programming (MILP). The mathematical goal is to find the vehicle path that maximizes the line integral of the uncertainty of field estimates along this path. Sampling this path can improve the accuracy of the field estimates the most. While achieving this objective, several constraints must be satisfied and are implemented. They relate to vehicle motion, intervehicle coordination, communication, collision avoidance, etc. The MILP formulation is quite powerful to handle different problem constraints and flexible enough to allow easy extensions of the problem. The formulation covers single- and multiple-vehicle cases as well as single- and multiple-day formulations. The need for a multiple-day formulation arises when the ocean sampling mission is optimized for several days ahead. We first introduce the details of the formulation, then elaborate on the objective function and constraints, and finally, present a varied set of examples to illustrate the applicability of the proposed method.

Surface-to-surface intersections

Techniques for computing intersections of algebraic surfaces with piecewise rational polynomial parametric surface patches and intersections of two piecewise rational polynomial parametric surface patches are discussed. The techniques are classified using four categories-lattice evolution methods, marching methods, subdivision methods, and analytic methods-and their principal features are discussed. It is shown that some of these methods also apply to the general parametric surface-intersection problem.<<ETX>>

Cooperative AUV Navigation using a Single Maneuvering Surface Craft

In this paper we describe the experimental implementation of an online algorithm for cooperative localization of submerged autonomous underwater vehicles (AUVs) supported by an autonomous surface craft. Maintaining accurate localization of an AUV is difficult because electronic signals, such as GPS, are highly attenuated by water. The usual solution to the problem is to utilize expensive navigation sensors to slow the rate of dead-reckoning divergence. We investigate an alternative approach that utilizes the position information of a surface vehicle to bound the error and uncertainty of the on-board position estimates of a low-cost AUV. This approach uses the Woods Hole Oceanographic Institution (WHOI) acoustic modem to exchange vehicle location estimates while simultaneously estimating inter-vehicle range. A study of the system observability is presented so as to motivate both the choice of filtering approach and surface vehicle path planning. The first contribution of this paper is to the presentation of an experiment in which an extended Kalman filter (EKF) implementation of the concept ran online on-board an OceanServer Iver2 AUV while supported by an autonomous surface vehicle moving adaptively. The second contribution of this paper is to provide a quantitative performance comparison of three estimators: particle filtering (PF), non-linear least-squares optimization (NLS), and the EKF for a mission using three autonomous surface craft (two operating in the AUV role). Our results indicate that the PF and NLS estimators outperform the EKF, with NLS providing the best performance.

An Automatic Coarse and Fine Surface Mesh Generation Scheme Based on Medial Axis Transform: Part I Algorithms

We present an algorithm for the generation of coarse and fine finite element (FE) meshes on multiply connected surfaces, based on the medial axis transform (MAT). The MAT is employed to automatically decompose a complex shape into topologically simple subdomains, and to extract important shape characteristics and their length scales. Using this technique, we can create a coarse subdivision of a complex surface and select local element size to generate fine triangular meshes within those subregions in an automated manner. Therefore, this approach can lead to integration of fully automatic FE mesh generation functionality into FE preprocessing systems.

Analysis and applications of pipe surfaces

Abstract A pipe (or tubular) surface is the envelope of a one-parameter family of spheres with constant radii r and centers C (t) . In this paper we investigate necessary and sufficient conditions for the nonsingularity of pipe surfaces. In addition, when C (t) is a rational function, we develop an algorithmic method for the rational parametrization of such a surface. The latter is based on finding two rational functions α(t) and β(t) such that ¦ C ′ (t)¦ 2 = α 2 (t) + β 2 (t) (Lu and Pottmann, 1996).

Differential and topological properties of medial axis transforms

Abstract The medial axis transform is a representation of an object which has been shown to be useful in design, interrogation, animation, finite element mesh generation, performance analysis, manufacturing simulation, path planning, and tolerance specification. In this paper, the theory of the medial axis transform for 3-D objects is developed. For objects with piecewise C 2 boundaries, relationships between the curvature of the boundary and the position of the medial axis are developed. For n -dimensional submanifolds of R n with boundaries which are piecewise C 2 and completely G 1 , a deformation retract is set up between each object and its medial axis, which demonstrates that if the object is path connected, then so is its medial axis. Finally, it is proven that path connected polyhedral solids without cavities have path connected medial axes.

Topological and differential-equation methods for surface intersections

Abstract An implicit representation of the intersection of two parametric surface patches is formulated in terms of the zero level set of the oriented distance function of one surface from the other. The topological properties of the vector field defined as the gradient of the oriented distance function (and specifically the concepts of the rotation number of a vector field and the Poincare index theory ) are next used to formulate a condition for the detection of critical points of the field, which may be used to identify internal loops and singularities of the solution. An adaptive search guided by this condition and direct numerical techniques are used to detect and compute critical points. Tensorial differential equations that rely on the implicit representation of the intersection are then developed, and are used to trace intersection segments. The tracing scheme relies on a characterization procedure for singular points that allows the computation of the tangent directions to the intersection curve at such points. The method also allows the tracing of degenerate intersections between surfaces that have a common tangent plane along the intersection curve. Examples drawn from rational spline surface intersections illustrate the above techniques.