GPDOF — A FAST ALGORITHM TO DECOMPOSE UNDER-CONSTRAINED GEOMETRIC CONSTRAINT SYSTEMS: APPLICATION TO 3D MODELING

Our approach exploits a general-purpose decomposition algorithm, called GPDOF, and a dictionary of very efficient solving procedures, called r-methods, based on theorems of geometry. GPDOF decomposes an equation system into a sequence of small subsystems solved by r-methods, and produces a set of input parameters.1. Recursive assembly methods (decomposition-recombination), maximum matching based algorithms, and other famous propagation schema are not well-suited or cannot be easily extended to tackle geometric constraint systems that are under-constrained. In this paper, we show experimentally that, provided that redundant constraints have been removed from the system, GPDOF can quickly decompose large under-constrained systems of geometrical constraints. We have validated our approach by reconstructing, from images, 3D models of buildings using interactively introduced geometrical constraints. Models satisfying the set of linear, bilinear and quadratic geometric constraints are optimized to fit the image information. Our models contain several hundreds of equations. The constraint system is decomposed in a few seconds, and can then be solved in hundredths of seconds.

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GEOMETRICAL CONSTRAINT SYSTEM DECOMPOSITION: A MULTI-GROUP APPROACH

International Journal of Computational Geometry & Applications ◽

10.1142/s0218195906002129 ◽

2006 ◽

Vol 16 (05n06) ◽

pp. 431-442 ◽

Cited By ~ 5

Author(s):

PASCAL SCHRECK ◽

PASCAL MATHIS

Keyword(s):

Geometric Constraint ◽

Transformation Groups ◽

Geometrical Constraint ◽

Group Approach ◽

New Approach ◽

System Decomposition ◽

Constraint System ◽

Constraint Systems ◽

Real Objects

Since they help to specify the shape of real objects, geometric constraint systems encountered in CAD domain are often invariant by isometries. But other transformation groups can be considered to improve the solving process. More precisely, using different transformation groups leads to a new approach of decomposition which generalizes in some sense the classical approaches. This paper presents a method able to perform such a multi-group decomposition.

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Numerical Study of an Impact Oscillator

Volume 3A: 15th Biennial Conference on Mechanical Vibration and Noise — Vibration of Nonlinear, Random, and Time-Varying Systems ◽

10.1115/detc1995-0256 ◽

1995 ◽

Author(s):

Kannan Marudachalam ◽

Faruk H. Bursal

Keyword(s):

Dynamical Systems ◽

Numerical Algorithm ◽

Initial Conditions ◽

Numerical Study ◽

Harmonic Excitation ◽

General Purpose ◽

Constrained Systems ◽

Global Dynamics ◽

Impact Oscillator ◽

Stick Slip

Abstract Systems with discontinuous dynamics can be found in diverse disciplines. Meshing gears with backlash, impact dampers, relative motion of components that exhibit stick-slip phenomena axe but a few examples from mechanical systems. These form a class of dynamical systems where the nonlinearity is so severe that analysis becomes formidable, especially when global behavior needs to be known. Only recently have researchers attempted to investigate such systems in terms of modern dynamical systems theory. In this work, an impact oscillator with two-sided rigid constraints is used as a paradigm for studying the characteristics of discontinuous dynamical systems. The oscillator has zero stiffness and is subjected to harmonic excitation. The system is linear without impacts. However, the impacts introduce nonlinearity and dissipation (assuming inelastic impacts). A numerical algorithm is developed for studying the global dynamics of the system. A peculiar type of solution in which the trajectories in phase space from a certain set of initial conditions merge in finite time, making the dynamics non-invertible, is investigated. Also, the effect of “grazing,” a behavior common to constrained systems, on the dynamics of the system is studied. Based on the experience gained in studying this system, the need for an efficient general-purpose numerical algorithm for solving discontinuous dynamical systems is motivated. Investigation of stress, vibration, wear, noise, etc. that are associated with impact phenomena can benefit greatly from such an algorithm.

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Re-parameterization reduces irreducible geometric constraint systems

Computer-Aided Design ◽

10.1016/j.cad.2015.07.011 ◽

2016 ◽

Vol 70 ◽

pp. 182-192 ◽

Cited By ~ 3

Author(s):

Hichem Barki ◽

Lincong Fang ◽

Dominique Michelucci ◽

Sebti Foufou

Keyword(s):

Geometric Constraint ◽

Constraint Systems

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A formalization of geometric constraint systems and their decomposition

Formal Aspects of Computing ◽

10.1007/s00165-009-0117-8 ◽

2009 ◽

Vol 22 (2) ◽

pp. 129-151 ◽

Cited By ~ 8

Author(s):

Pascal Mathis ◽

Simon E. B. Thierry

Keyword(s):

Geometric Constraint ◽

Constraint Systems

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Solution Selectors: A User-Oriented Answer to the Geometric Constraint Multiple Solution Problem

Volume 2A: 27th Design Automation Conference ◽

10.1115/detc2001/dac-21029 ◽

2001 ◽

Author(s):

Bernhard Bettig ◽

Jami Shah

Keyword(s):

Linear Equations ◽

Independent Solution ◽

Solid Modeling ◽

Multiple Solution ◽

Parametric Modeling ◽

Geometric Constraints ◽

Geometric Constraint ◽

Basic Solution ◽

Solution Selection ◽

Solution Problem

Abstract The development of solid modeling to represent the geometry of designed parts and the development of parametric modeling to control the size and shape have had significant impacts on the efficiency and speed of the design process. Designers now rely on parametric solid modeling, but surprisingly often are frustrated by a problem that unpredictably causes their sketches to become twisted and contorted. This problem, known as the “multiple solution problem” occurs because the dimensions and geometric constraints yield a set of non-linear equations with many roots. This situation occurs because the dimensioning and geometric constraint information given in a CAD model is not sufficient to unambiguously and flexibly specify which configuration the user desires. This paper first establishes that only explicit, independent solution selection declarations can provide a flexible mechanism that is sufficient for all situations of solution selection. The paper then describes the systematic derivation of a set of “solution selector” types by considering the occurrences of multiple solutions in combinations of mutually constrained geometric entities. The result is a set of eleven basic solution selector types and two derived types that incorporate topological information. In particular, one derived type “concave/convex” is user-oriented and thought to be very useful.

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4MDS: A Geometric Constraint Based Motion Design Software for Synthesis and Simulation of Planar Four-Bar Linkages

Volume 5B: 38th Mechanisms and Robotics Conference ◽

10.1115/detc2014-35235 ◽

2014 ◽

Cited By ~ 2

Author(s):

Anurag Purwar ◽

Abhijit Toravi ◽

Q. J. Ge

Keyword(s):

Real Time ◽

Geometric Constraints ◽

Geometric Constraint ◽

Design System ◽

Dimensional Synthesis ◽

Complete Control ◽

Dimensional Changes ◽

Design Software ◽

Four Bar Linkage ◽

Motion Design

This paper presents our recent work on designing and developing a geometric constraint based motion design software system for planar four-bar linkages. Given a motion task, the software computes possible four-bar linkage topologies as well as its dimensions. This capability to analyze the given task and find the best type of the linkage and the dimensions simultaneously sets it apart from any other linkage design software. The Four-Bar Motion Design System (4MDS) makes the synthesis and simulation capabilities available to mechanism designers in an intuitive graphical user interface (GUI) environment. Instead of taking a black box approach to mechanism design, wherein the user simply enters the motion requirements and the software outputs parameters of mechanisms, this software facilitates a dialog with the designer by providing various paths to simulation and synthesis in a design session. The designer has complete control over the specification of motion task, interactive tweaking of the motion, choice of linkage topology computed, dimensional changes, and their apparent effect on motion, all done in real time. This interactivity enhances designers kinematic experience. The underlying theoretical foundation of this paper is based on our earlier work on a task-driven approach to unified type and dimensional synthesis of planar four-bar linkage mechanisms. Instead of treating a planar four-bar mechanism as a set of connected rigid links and joints, we treat them as line or circle constraint generators. With that view, the synthesis problem is reduced to extracting geometric constraints hidden in a given motion task and the simulation is reduced to assembling constraints realizable by mechanical dyads. The algorithm employed is simple and efficient and permits real-time computation, and thus precludes using a limiting database-oriented approach. This tool should make innovation of mechanical motion generating devices accessible to novice and experienced designers alike.

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