Numerical Algebraic Geometry Boot Camp
Daniel Bates, Colorado State University
The goal of this talk is to introduce the fundamental concepts of numerical algebraic geometry, specifically as they are implemented in Bertini, so that workshop participants have a common language and are all on the same page. Topics for this talk will include the basics of homotopy continuation (homotopy construction, some basic theory, numerical considerations, etc.), fundamental data types such as witness sets, a brief overview of some of the more recent techniques in the field (deflation, regeneration, etc.), and a first introduction to Bertini. No prior knowledge of the area will be assumed.
Applications of Bertini in Kinematics, Robotics, and Machine Design
Andrew Murray, University of Dayton
Problems in kinematics, from robotics to machine design, are frequently modeled by a system of algebraic equations. The standard for solving these systems was either clever geometric insights or traditional numerical methods including optimization. The clever insights aren't always there and the traditional numerical methods can stumble finding just a few solutions. Bertini changes this by finding every solution to a polynomial system. This talk will present some history, show a variety of examples from kinematic analysis and synthesis that readily yield to solution by Bertini, and discuss the rapidly rising degrees that this new tool forces us to consider.
The Development of Bertini 2
Daniel Brake, University of Notre Dame,
Jeb Collins, West Texas A&M University,
Tim Hodges, Colorado State University, and
Alan Liddell, University of Notre Dame
Bertini 2 is the redevelopment of Bertini in C++. It replaces and upgrades the original C implementation of the multiprecision polynomial homotopy continuation engine. The project goes beyond simply re-implementing with a clean interface and thorough example-driven documentation of internals. Since it is intended for use as a library, Bertini 2 provides Python bindings for the entire body of code and aims to play nicely with as many other software packages as possible. Interfaces to other languages and packages are planned as well, such as Singular, Polymake, and Julia. We will also offer methods for extending Bertini, and providing custom packages, in any supported language.
In this talk, several of the main package contributors will give a status update of the project, with the goal of giving the audience an opportunity to influence its direction and capabilities.
Printing Algebraic Geometry
Daniel Brake, University of Notre Dame
3D printing continues to advance as a hobby and professional tool. As an artform, printing enables individuals to make whatever they can dream of. As a mathematician, this new form of fabrication can realize mathematical principles for the classroom and public consumption. In turn, it brings the subject closer than ever to mainstream popularity.
In this talk, I will talk about the tools of my arsenal for printing mathematical objects, including Bertini_real, and the other programs in the tool chain. Particularly, Bertini_real decomposes algebraic surfaces in any number of dimensions, in a format which permits relatively easy 3D printing. I will also discuss other methods for generating objects to print, and some of the challenges to printing arbitrary mathematical objects.
Some applications and approaches for certification via α-theory
Nickolas Hein, University of Nebraska, Kearney and Alan Liddell, University of Notre Dame
Smale's α-theory, also known as point-estimation theory, is a collection of results regarding the convergence of Newton's method. Using this theory, one may certify that a point quadratically converges to a solution of a nonlinear system knowing only the point and the system. Consequently, one may study solutions to polynomial systems without exactly determining the solutions.
This allows one to use numerical methods of computation in scenarios requiring provable output.
We may also ensure continuity of solution paths of homotopies.
In this talk, we will give a general overview of α-theory and provide several examples of its utility, including examples from real Schubert Calculus and certified path tracking.
Exploiting multi variable-group structure
Jose Rodriguez, University of Chicago
Systems in applications typically have additional structure arising from groups of variables. Bezout homotopies do not exploit this structure which means that many paths can go to infinity. By utilizing the "multi variable-group structure" of the target system, the system can be solved more efficiently. This talk will discuss both computing isolated solutions and solution components for such systems and conclude with an illustrative example from kinematics.
Revisiting Nonlinear Equations in Classical Thermodynamic Modeling: A Numerical Algebraic Geometry Perspective
Hythem Sidky, University of Notre Dame
Modeling and calculating phase equilibrium is a cornerstone of modern engineering thermodynamics. Although the theoretical foundation is well established and has remained largely unchanged for decades, the development of algorithms for solving phase equilibrium problems is still an active area of research today. The naturally arising equations are often highly nonlinear multidimensional surfaces where a global minimum or a specific solution is sought. As a result, traditional second convergence methods are combined with heuristics and engineering inputs in an attempt to produce both expedient and reliable algorithms. Especially difficult problems often incorporate stochastic optimization methods as well. In this talk we look at some of the recent work in applying numerical algebraic geometry (NAG) techniques to phase equilibrium. The application of numerical polynomial homotopy continuation was found to be reliable in solving for multicomponent mixture critical points. Exploiting the geometric structure of state-of-the-art fundamental equations of state has also lead to the development of a heuristic free method for solving the saturation equations. We further discuss the fine balance between speed and reliability, requirements which are starkly different for large-scale natural gas reservoir simulations and scientific application. Finally, we demonstrate some of the useful tools used in solving these problems, and interfacing with the popular NAG package, Bertini, using familiar engineering software such as MATLAB and Mathematica.
Analysis and Design of Compliant Mechanisms using Pseudo-Rigid-Body Models
Venkatasubramanian Kalpathy Venkiteswaran , Ohio State University
Compliant Mechanisms are widely used for many applications in robotics and precision engineering due to their inherent properties of stiffness and damping, self-assembly and accuracy of movement, while eliminating friction and wear, especially in smaller length scales. Conventionally, beam theory methods, stiffness matrices or Finite Element methods are used to analyze and design compliant mechanisms. Pseudo-rigid-body models are numerical approximations for compliant elements that capture their stiffness and geometric properties. They can represent various types of compliant elements in a single framework, which gives them an advantage over conventional methods. They can also offer computational advantages over beam theory or FEA.
This presentation will focus on the development and implementation of methods for analysis and synthesis of compliant mechanisms using pseudo-rigid-body (PRB) models. A framework has been developed for uniform representation of PRB models and the optimization of parameters to achieve minimal error when compared to conventional methods. Several types of PRB models have been listed and studied to create a library as a guide for other researchers. Using the results of this framework, a topology optimization scheme is being established for design of compliant mechanisms. It utilizes a simple graph theory approach to set up the adjacency matrix, and the PRB models convert the system into nonlinear algebraic equations. A genetic algorithm-based optimization routine is used with dynamic penalty functions to find the optimal topology based on the design space and other constraints. A few preliminary results will be shown and the limitations and future work will be discussed. It is hoped that this approach opens the door to easy design of compliant mechanisms using various types of compliant elements and also looks into extending this to develop multi-material compliant mechanisms.
Jieyu Wang, Heriot-Watt University
Parallel mechanisms with multiple operation modes need fewer actuators and less time for changeover than the existing reconfigurable parallel mechanisms. There is no need to disassemble a parallel mechanism when switching it from one (operation) mode to another in the process of reconfiguration although it needs to go through a constraint singular configuration in the reconfiguration. Firstly, a 3-DOF (degrees-of-freedom) parallel mechanism with 15 3-DOF operation modes, including four translationaSl modes, six planar modes, four zero-torsion-rate motion modes and one spherical mode proposed in the literature will be presented. Secondly, a 2-DOF rolling single-loop 8-bar linkage and a 16-bar spatial rolling mechanism, which can be deformed into spherical mechanisms or planar mechanisms will be introduced. Further, the mechanisms can both be folded onto a plane. Finally, a mobile robot with eight modes based on 3-URU PMwill be presented. The robot incorporates the kinematic properties of sphere robots, squirming robots, tracked robots, wheeled robots and biped robots. The somersaulting and turning modes are also explored. The numerical algebraic geometry may help meet the challenge encountered in the reconfiguration analysis of multi-mode mechanisms.
Generalized coordinate partitioning for complex mechanisms based on kinematic substructuring
Kristopher Wehage, University of California, Davis
While the kinematic and dynamic behavior of constrained mechanical systems is well–understood, the numerical solution of kinematic posture equations and the dynamic equations of motion remains challenging. The sheer number and complexity of approaches that have been devised over the years highlight the difficulty in achieving efficient and robust numerical solutions for general problems.
In this talk, a systematic method based on graph theoeretic concepts is presented that allows setting up a general mechanism’s governing equations for a wide range of parametric and topological variations. The algorithms and methods described are designed to be both fully automatic – requiring minimal supervision from an analyst for successful execution, robust – capable of handling instantaneous bifurcations and end-of-stroke conditions, and numerically efficient – through the application of numerical reduction strategies, custom sparse matrix methods and vectorization.
One of the primary challenges associated with solving a system of dynamics equations for multiloop systems is that the system's constraint equations, i.e. the Jacobian matrix, is almost never full-rank. Therefore, Generalized Coordinate Partitioning (GCP), a numerical method based on LU decomposition is applied to the Jacobian matrix to find the optimal set of independent, generalized coordinates to describe the system. To increase the efficiency of the GCP algorithm, a new general purpose graph-partitioning algorithm, referred to as Kinematic Substructuring is introduced and numerical results are provided.
On the synthesis problem of spatial serial 3R triads
Martin Pfurner, University Innsbruck
The geometry of small matrix multiplication
Fulvio Gesmundo, Texas A&M University
Decomposition of polynomials with symmetry
Yonghui Guan, Texas A&M University
Synthesis of overconstrained closed spatial single-loop chains with 6 revolute joints
Martin Pfurner, University Innsbruck