Algorithmic aspects of alternating sum of volumes. Part 2: Nonconvergence and its remedy

The paper is the second part of a 2-part paper. The first part focused on the issues of data structure and fast difference operation. The second studies the non-convergence of the alternating sum of volumes (ASV) process. An ASV is a series of convex components joined by alternating union and difference operations. It is desirable that an ASV series be finite. However, such is not always the case - the ASV algorithm can be nonconvergent. The paper investigates the causes of this nonconvergence, and finds and proves the conditions responsible for it. Linear time algorithms are then developed for detection.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/29257/1/0000314.pd

Faster ASV decomposition for orthogonal polyhedra using the Extreme Vertices Model (EVM)

The alternating sum of volumes (ASV) decomposition is a widely used technique for converting a B-Rep into a CSG model. The obtained CSG tree has convex primitives at its leaf nodes, while the contents of its internal nodes alternate between the set union and difference operators. This work first shows that the obtained CSG tree T can also be expressed as the regularized Exclusive-OR operation among all the convex primitives at the leaf nodes of T, regardless the structure and internal nodes of T. This is an important result in the case in which EVM represented orthogonal polyhedra are used because in this model the Exclusive-OR operation runs much faster than set union and difference operations. Therefore this work applies this result to EVM represented orthogonal polyhedra. It also presents experimental results that corroborate the theoretical results and includes some practical uses for the ASV decomposition of orthogonal polyhedra.Postprint (published version

Algorithmic aspects of alternating sum of volumes. Part 1: Data structure and difference operation

In terms of basic theory, a unique conversion from a boundary representation to a CSG representation is of importance. In terms of application, the extraction of features by convex decomposition is of interest. The alternating sum of volumes (ASV) technique offers both. However, some algorithmic issues are still unresolved. The paper is the first section of a 2-part paper that addresses specialized set operations and the convergence of the ASV process. In the first part, a fast difference operation for the ASV process and a data structure for pseudopolyhedra are introduced.A fast difference operation between an object and its convex hull is made possible by the crucial observation it takes only linear time to distinguish them. However, it takes O(NlogN) time to construct a data structure with the proper tags. The data structure supporting the operation is a pseudopolyhedron, capturing the special relationship between an object and its convex hull. That the data structure is linear in space is also shown.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/29308/1/0000371.pd

A method to convert algebraic boundary representations to CSG representations for three-dimensional solids

Recent advances in reverse engineering have focused on recovering a boundary representation (b-rep) of an object, often for integration with rapid prototyping. This boundary representation may be a 3-D point cloud, a triangulation of points, or piecewise algebraic or parametric surfaces. This paper presents work in progress to develop an algorithm to extend the current state of the art in reverse engineering of mechanical parts. This algorithm will take algebraic surface representations as input and will produce a constructive solid geometry (CSG) description that uses solid primitives such as rectangular block, pyramid, sphere, cylinder, and cone. The proposed algorithm will automatically generate a CSG solid model of a part given its algebraic b-rep, thus allowing direct input into a CAD system and subsequent CSG model generation

Dividing Complex Parts into Multiple Pieces for Advanced Joining and Additive Manufacturing

Poor material utilization is inherent to conventional manufacturing processes, leading to high material waste and machining times. Additive manufacturing processes attempt to solve this issue by allowing production of near-net shapes, but the processes may be too expensive or infeasible. By leveraging both processes in a single part, the waste and cost of manufacturing can be reduced. For example, a complex part could have the main body produced by bar stock and machining, while small protruding features are joined onto the main body by additive manufacturing. This thesis presents a method to divide complex parts into smaller pieces to be built up through advanced joining and additive manufacturing. A beam search algorithm is applied to consider the vast number of manufacturing and joining options and converge on the lowest cost solutions. The algorithm runs by intelligently identifying cutting planes and iteratively applying them to solid geometry to create possible manufacturing alternatives. Each manufacturing alternative is then evaluated based on cost, and the lowest cost alternatives are presented to the designer to aid in determining better manufacturing plans. Application of this method will reduce material waste and machining to offset the added joining costs. This thesis presents the development, implementation, and testing of this approach

Computational Scattering Models for Elastic and Electromagnetic Waves in Particulate Media

Numerical models were developed to simulate the propagation of elastic and electromagnetic waves in an arbitrary, dense dispersion of spherical particles. The scattering interactions were modeled with vector multipole fields using pure-orbital vector spherical harmonics, and solved using the full vector form of the boundary conditions. Multiple scattering was simulated by translating the scattered wave fields from one particle to another with the use of translational addition theorems, summing the multiple-scattering contributions, and recalculating the scattering in an iterative fashion to a convergent solution. The addition theorems were rederived in this work using an integral method, and were shown to be numerically equivalent to previously published theorems. Both ordered and disordered collections of up to 5,000 spherical particles were used to demonstrate the ability of the scattering models to predict the spatial and frequency distributions of the transmitted waves. The results of the models show they are qualitatively correct for many particle configurations and material properties, displaying predictable phenomena such as refractive focusing, mode conversion, and photonic band gaps. However, the elastic wave models failed to converge for specific frequency regions, possibly due to resonance effects. Additionally, comparison of the multiple-scattering simulations with those using only single-particle scattering showed the multiple-scattering computations are quantitatively inaccurate. The inaccuracies arise from nonconvergence of the translational addition theorems, introducing errors into the translated fields, which minimize the multiple-scattering contributions and bias the field amplitudes towards single-scattering contributions. The addition theorems are shown to converge very slowly, and to exhibit plateaus in convergence behavior that can lead to false indications of convergence. The theory and algorithms developed for the models are broad-based, and can accommodate a variety of structures, compositions, and wave modes. The generality of the approach also lends itself to the modeling of static fields and currents. Suggestions are presented for improving and implementing the models, including extension to nonspherical particles, efficiency improvements for the algorithms, and specific applications in a variety of fields

Automated decomposition of complex parts for manufacturing with advanced joining processes

Recent advancements in joining operations and additive manufacturing now allow complex metal parts to be built up from raw materials, as opposed to being machined down from solid blocks. This not only opens up the design space, but also allows for much more efficient manufacturing. By decomposing a complex part into simpler subparts, material waste and machining time can be reduced. This can significantly lower the production cost. However, if decomposed into too many pieces, the cost of the additional required processing steps will outweigh these savings. The primary challenge in designing with advanced joining is in deciding how to divide complex parts into subparts to be cost effective and feasible. It can be difficult to intuitively decide how to split up a part, and currently it requires hours of an engineer’s time to make comparisons between different combinations of stock materials and joining operations. The methods presented in this dissertation employ artificial intelligence (AI) search methods to optimize the decomposition of these complex parts and automatically generate manufacturing plans. In addition, since closed-die forging is a common solution for metal aerospace parts, an automated closed-die forging design method is presented. This approach allows engineers to compare traditional and new manufacturing methods to make better manufacturing decisions early in the design phase

A knowledge-based approach for the extraction of machining features from solid models

Computer understanding of machining features such as holes and pockets is essential for bridging the communication gap between Computer Aided Design and Computer Aided Manufacture. This thesis describes a prototype machining feature extraction system that is implemented by integrating the VAX-OPS5 rule-based artificial intelligence environment with the PADL-2 solid modeller. Specification of original stock and finished part geometry within the solid modeller is followed by determination of the nominal surface boundary of the corresponding cavity volume model by means of Boolean subtraction and boundary evaluation. The boundary model of the cavity volume is managed by using winged-edge and frame-based data structures. Machining features are extracted using two methods : (1) automatic feature recognition, and (2) machine learning of features for subsequent recognition. [Continues.

Numerical algebraic geometry approach to polynomial optimization, The

2017 Summer.Includes bibliographical references.Numerical algebraic geometry (NAG) consists of a collection of numerical algorithms, based on homotopy continuation, to approximate the solution sets of systems of polynomial equations arising from applications in science and engineering. This research focused on finding global solutions to constrained polynomial optimization problems of moderate size using NAG methods. The benefit of employing a NAG approach to nonlinear optimization problems is that every critical point of the objective function is obtained with probability-one. The NAG approach to global optimization aims to reduce computational complexity during path tracking by exploiting structure that arises from the corresponding polynomial systems. This thesis will consider applications to systems biology and life sciences where polynomials solve problems in model compatibility, model selection, and parameter estimation. Furthermore, these techniques produce mathematical models of large data sets on non-euclidean manifolds such as a disjoint union of Grassmannians. These methods will also play a role in analyzing the performance of existing local methods for solving polynomial optimization problems

System- and Data-Driven Methods and Algorithms

An increasing complexity of models used to predict real-world systems leads to the need for algorithms to replace complex models with far simpler ones, while preserving the accuracy of the predictions. This two-volume handbook covers methods as well as applications. This first volume focuses on real-time control theory, data assimilation, real-time visualization, high-dimensional state spaces and interaction of different reduction techniques