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

2003 ◽  
Vol 125 (3) ◽  
pp. 443-451 ◽  
Author(s):  
Bernhard Bettig ◽  
Jami Shah
Keyword(s):  
Linear Equations ◽  
Independent Solution ◽  
Solid Modeling ◽  
Multiple Solution ◽  
Parametric Modeling ◽  
Geometric Constraints ◽  
Basic Solution ◽  
Solution Selection ◽  
Systematic Derivation ◽  
Solution Problem

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 often are frustrated by a problem that unpredictably causes their sketches to become twisted, contorted, or take an unexpected shape. Mathematically, 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. In practice, 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. 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 may prove to be particularly useful.

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

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.

Parametric Solid Modeling

2006 ◽  
Author(s):  
Horea T. Ilies¸
Keyword(s):  
Heuristic Algorithms ◽  
Solid Modeling ◽  
Parametric Modeling ◽  
Geometric Constraints ◽  
Parametric Models ◽  
Open Problems ◽  
Solid Models ◽  
Modeling Systems ◽  
Alternative Approaches ◽  
And Performance

Parametric modeling systems are fundamentally changing the design process practiced in the industry today. Practically all commercial CAD systems combine established solid modeling techniques with constraint solving and heuristic algorithms to create, edit and manipulate solid models, while enforcing the requirement that every such solid model must maintain the validity of the prescribed geometric constraints. However, a number of fundamental (open) problems limit the functionality and performance of these parametric modeling systems. For example, the allowable parametric changes are history dependent; the number of parameters describing even relatively simple parts can quickly become prohibitively large, and commercial constraint solvers are limited today to 2-dimensional geometric constraints. Consequently, current parametric modeling systems do not support many practical design situations due to the associated theoretical and computational difficulties, as well as to the considerable organizational obstacles generated by the need to handle large parametric models. This paper investigates the current practices and limitations of parametric solid modeling systems, and explores some alternative approaches that could complement the identified limitations.

scholarly journals A MATHEMATICAL REVIEW ON THE MULTIPLE-SOLUTION PROBLEM

2011 ◽  
Vol 26 (25) ◽  
pp. 4511-4520 ◽  
Author(s):  
K. ZHU ◽  
X. H. MO ◽  
C. Z. YUAN ◽  
P. WANG
Keyword(s):  
Multiple Solutions ◽  
High Energy Physics ◽  
High Energy ◽  
Multiple Solution ◽  
The Other ◽  
Experimental Measurements ◽  
Mathematical Review ◽  
Analytical Formulae ◽  
Solution Problem ◽  
Energy Physics

The recent multiple-solution problem in extracting physics information from a fit to the experimental data in high energy physics is reviewed from a mathematical viewpoint. All these multiple solutions were previously found via a fit process, while in this paper we prove that if the sum of two coherent Breit–Wigner functions is used to fit the measured distribution, there should be two and only two nontrivial solutions, and they are related to each other by analytical formulae. For real experimental measurements in more complicated situations, we also provide a numerical method to derive the other solution from the already obtained one. The excellent consistency between the exact solution obtained this way and the fit process justifies the method. From our results it is clear that the physics interpretation should be very different depending on which solution is selected. So we suggest that all the experimental measurements with potential multiple solutions be re-analyzed to find the other solution because the result is not complete if only one solution is reported.

Unique and accurate soil parameter identification for air-cushioned robotic vehicles

Robotica ◽  
2015 ◽  
Vol 35 (3) ◽  
pp. 613-635
Author(s):  
Shuo Xu ◽  
Yinan Gu ◽  
Jing Sun ◽  
Dawei Tu
Keyword(s):  
Performance Optimization ◽  
Multiple Solution ◽  
Identification Accuracy ◽  
Soil Conditions ◽  
Soil Parameters ◽  
Vertical Force ◽  
Tractive Force ◽  
Accurate Identification ◽  
Solution Problem ◽  
Accuracy Problem

SUMMARYOn-line identification of soil parameters is a pre-condition of operating performance optimization and control for unmanned ground vehicles (UGV). Inverse calculation from measured vehicular operating parameters is a prevalent methodology. However, it inherently suffers from a multiple-solution problem caused by the coupling of soil parameters in terramechanics equations and an accuracy problem caused by the influences of state noise and measurement noise. These problems in tractive-force-related soil parameters identification were addressed here for air-cushioned vehicles (ACV) by taking advantage of their additional degree of control freedom in vertical force. To be specific, a g-function algorithm was proposed to solve the multiple-solution problem from reproductive tractive force equations; de-noising techniques consisting of mean-effect strategies, sampling points selection and sample rearrangement were employed to solve the accuracy problem. A series of experiments were conducted to evaluate these techniques at different noise levels and in different soil conditions. They got satisfactory results in terms of data utilization ratio, identification accuracy and performance stability. The contribution of the paper lies in inventing a novel algorithm for unique and accurate identification of tractive-force-related soil parameters without making any simplification to the original terramechanics equation and with robustness to variations of noise level and soil condition.

Constraint-Based Solid Modeling in a Virtual Reality Environment

1998 ◽  
Author(s):  
Shuming Gao ◽  
Huagen Wan ◽  
Qunsheng Peng
Keyword(s):  
Virtual Reality ◽  
Geometric Model ◽  
Rapid Development ◽  
Solid Modeling ◽  
Geometric Constraints ◽  
Solid Model ◽  
Prototype System ◽  
Virtual Reality Environment ◽  
Location Pattern ◽  
Cad System

Abstract With the rapid development of the virtual reality technology, the research on the integration of virtual reality with CAD system becomes more and more popular. The VR-based CAD system allows the designer to make design in a more intuitive and more convenient manner. But, on the negative side, it has difficulty in creating the precise geometric model of a product which is one of the most important part of the product model. This paper presents an approach for constraint based solid modeling in a virtual reality environment. The approach not only allows the designer to create, edit and visualize the design by direct 3D manipulations in a virtual environment, but also makes the created solid model precise by recognizing and solving the geometric constraints involved in the initial design. To effectively support direct 3D manipulations and fast change propagation, we adopt a new constraint based solid model which, besides the typical constituents, includes the Shape Control Point, Location Pattern, Explicit Shape Constraint, etc. Guided by the Location Pattern of each primitive, recognizing and solving of location constraints in our approach reaches real-time. The proposed approach has been implemented and a prototype system has been developed.

Unpacking The Relation Between Spatial Abilities and Creativity in Geometry

2021 ◽  
Vol 4 (3) ◽  
pp. 307-328
Author(s):  
Panagiotis Gridos ◽  
◽  
Evgenios Avgerinos ◽  
Eleni Deliyianni ◽  
Iliada Elia ◽  
...  
Keyword(s):  
Teaching And Learning ◽  
Essential Role ◽  
Solution Process ◽  
Spatial Relations ◽  
Spatial Visualization ◽  
Spatial Abilities ◽  
Multiple Solution ◽  
Ninth Graders ◽  
Problem Solution ◽  
Solution Problem

This study aims to examine the relation between spatial ability and creativity in Geometry. Data was collected from 94 ninth graders. Three spatial abilities were investigated: spatial visualization, spatial relations and closure flexibility. As for students' creativity, it was examined through a multiple solution problem in Geometry focusing on three components of creativity: fluency, flexibility, and originality. The results revealed that spatial visualization predicted flexibility and originality while closure flexibility predicted all creativity components. Additionally, it was deduced that auxiliary constructions played an essential role in the problem-solution process. Finally, further study opportunities for the teaching and learning of Geometry are discussed.

Function Generation With Finitely-Separated Precision Points Using Geometric Constraint Programming

2006 ◽  
Author(s):  
Edward C. Kinzel ◽  
James P. Schmiedeler ◽  
Gordon R. Pennock
Keyword(s):  
Constraint Programming ◽  
Computer Aided Design ◽  
Linear Equations ◽  
Geometric Constraints ◽  
Geometric Constraint ◽  
Function Generation ◽  
Design Software ◽  
Aided Design ◽  
Four Bar Linkage ◽  
Non Linear Equations

This paper explains how Geometric Constraint Programming can be applied to solve function generation problems with finitely-separated positions using a number of different mechanisms. Geometric Constraint Programming uses the sketching mode of commercial parametric computer-aided design software to create kinematic diagrams whose elements are parametrically related so that when a parameter is changed, the design is modified automatically. Geometric constraints are imposed graphically through the user interface, and the numerical solvers integrated into the software solve the relevant systems of non-linear equations without the user explicitly formulating those equations. A key advantage of using Geometric Constraint Programming for function generation is that the same approach can be applied to any mechanism, so no unique algorithms are required. Furthermore, because the implementation is relatively straightforward regardless of the chosen mechanism, the designer can quickly and easily generate solutions for a large number of precision points and/or with complex mechanisms to provide a very accurate match to the desired function. Examples of function generation with a four-bar linkage, a six-bar linkage, and a seven-bar linkage illustrate the benefits of the proposed methodology.

INTERACTION WITH CONSTRAINTS IN 3D MODELING

1991 ◽  
Vol 01 (04) ◽  
pp. 405-425 ◽  
Author(s):  
WOLFGANG SOHRT ◽  
BEAT D. BRÜDERLIN
Keyword(s):  
3D Modeling ◽  
Degrees Of Freedom ◽  
Solid Modeling ◽  
Geometric Constraints ◽  
Geometric Constraint ◽  
Interactive Modeling ◽  
Constraint Solver ◽  
Graphical Interaction ◽  
Definition Of

This paper presents an implementation of an interactive solid modeling system that integrates 1) the definition of objects by graphical interaction and 2) the specification of objects by geometric constraints. In this system, interactive modeling operations for constructing assemblies automatically generate constraints to maintain the properties intended by their invocation, and constraints, in turn, determine the degrees of freedom for further interactive mouse-controlled modeling operations. A symbolic geometric constraint solver is employed for solving systems of simultaneous constraints. Group hierarchies are utilized for representing dependencies and for localizing systems of constraints.

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