Geometric Constraint Problems Research Articles

Some five dimensional persistent submanifolds of the Study quadric

This work studies persistent submanifolds of the group of rigid-body displacements. At each point of a submanifold the tangent space, translated to the identity in the group becomes a screw system. For persistent submanifolds these screws systems are mutually conjugate by a rigid-body transformation. We present novel results on persistent submanifolds composed of three subgroups and products of subgroups with persistent submanifolds. We also show that some submanifolds defined as solutions to geometric constraint problems are persistent.Examples of 5-dimensional persistent submanifolds are then studied. We show that many examples can be show to be persistent in several ways. We also show that there are large overlaps between different types of submanifolds. Many constraint varieties can be realised by open-loop kinematic chains and many of these persistent 5-dimensional submanifolds are the intersection of the Study quadric with algebraic hypersurfaces. Many of the examples considered are well known but we also introduce several novel examples of 5-dimensional persistent submanifolds.

Optimality conditions for higher order polyhedral discrete and differential inclusions

The problems considered in this paper are described in polyhedral multi-valued mappings for higher order(s-th) discrete (PDSIs) and differential inclusions (PDFIs). The present paper focuses on the necessary and sufficient conditions of optimality for optimization of these problems. By converting the PDSIs problem into a geometric constraint problem, we formulate the necessary and sufficient conditions of optimality for a convex minimization problem with linear inequality constraints. Then, in terms of the Euler-Lagrange type PDSIs and the specially formulated transversality conditions, we are able to obtain conditions of optimality for the PDSIs. In order to obtain the necessary and sufficient conditions of optimality for the discrete-approximation problem PDSIs, we reduce this problem to the form of a problem with higher order discrete inclusions. Finally, by formally passing to the limit, we establish the sufficient conditions of optimality for the problem with higher order PDFIs. Numerical approach is developed to solve a polyhedral problem with second order polyhedral discrete inclusions.

Influência da Convecção Mista sobre a Otimização Geométrica de um Arranjo Triangular de Cilindros em Escoamentos Laminares

<div><p class="SPabstract">Este estudo numérico consiste em avaliar a forma geométrica de um arranjo triangular de cilindros submetidos a escoamentos transientes, bi-dimensionais, incompressíveis, laminares e com convecção mista por meios do Constructal Design. As simulações foram realizadas para escoamentos com números de Reynolds e Prandtl, de 100 e 0,71, respectivamente, avaliando-se três diferentes números de Richardson: Ri = 0,1; 1,0; e 10,0. As equações de conservação de massa, quantidade de movimento e energia foram resolvidas com o uso do método de volumes finitos (FVM – do inglês: Finite Volume Method). A área ocupada pelos cilindros é uma restrição geométrica do problema. O principal objetivo é avaliar a influência do Ri no efeito da razão S<sub>T</sub>/D (passo transversal sobre o diâmetro) sobre o coeficiente de arrasto (C<sub>D</sub>) e o número de Nusselt (Nu<sub>D</sub>) entre os cilindros e o escoamento circundante, bem como obter as geometrias ótimas do problema multiobjetivo. Em todos os casos estudados a razão <em>S</em><sub>L</sub>/<em>D</em> (passo longitudinal sobre o diâmetro) foi mantida fixa, <em>S</em><sub>L</sub>/<em>D</em> = 1,5. Foi avaliado que a variação do mecanismo motriz do escoamento (causado pela variação do número de Richardson) não somente afetou as magnitudes de C<sub>D</sub> e Nu<sub>D</sub>, mas também alterou o comportamento das camadas limite fluidodinâmica e térmica conduzindo a alterações no efeito da razão <em>S</em><sub>T</sub>/<em>D</em> sobre C<sub>D</sub> e Nu<sub>D</sub>.</p></div>

H-graphs: A new representation for tree decompositions of graphs

In geometric constraint solving, 2D well constrained geometric problems can be abstracted as Laman graphs. If the graph is tree decomposable, the constraint-based geometric problem can be solved by a Decomposition–Recombination planner based solver. In general decomposition and recombination steps can be completed only when steps on which they are dependent have already been completed. This fact naturally defines a hierarchy in the decomposition–recombination steps that traditional tree decomposition representations do not capture explicitly.In this work we introduce h-graphs, a new representation for decompositions of tree decomposable Laman graphs, which captures dependence relations between different tree decomposition steps. We show how h-graphs help in efficiently computing parameter ranges for which solution instances to well constrained, tree decomposable geometric constraint problems with one degree of freedom can actually be constructed.

Leading a continuation method by geometry for solving geometric constraints

Geometric constraint problems arise in domains such as CAD, Robotics, Molecular Chemistry, whenever one expects 2D or 3D configurations of some geometric primitives fulfilling some geometric constraints. Most well-constrained 3D problems are resistant to geometric knowledge based systems. They are often solved by purely numerical methods that are efficient but provide only one solution. Finding all the solutions can be achieved by using, among others, generic homotopy methods, that become costly when the number of constraints grows. This paper focuses on using geometric knowledges to specialize a so-called coefficient parameter continuation to 3D geometric constraint systems. Even if the proposed method does not ensure obtaining all the solutions, it provides several real ones. Geometric knowledges are used to justify it and lead the search of new solutions.

Hierarchy and Adaptive Size Particle Swarm Optimization Algorithm for Solving Geometric Constraint Problems

Geometric constraint problems are equivalent to a series of nonlinear equations which are constraint-meeting. Thus, it is a significant issue to improve solving efficiency of the nonlinear equations. This paper proposes Hierarchy and Adaptive Size Particle Swarm Optimization (HASPSO) algorithm for solving geometric constraint problems, and its aim is to greatly improving solving efficiency. This is the basic idea: according to individual extremum, making a comparison between each particle and its members in the direct next hierarchy, then based on transmission principle, taking the best particle’s personal optimal position as its own to do subsequent iterations. Meanwhile, it depends on the natural principle of Fibonacci sequence, by simulating biological reproduction behavior, to make the algorithm adaptively expand its population size from a single individual to appropriate numbers of ones for subsequent hierarchies. If a particle still has not found precision-meeting optimal solution after a T times of iterations, then our approach judge whether the particle is new reproduced individual before the iterations, if so, it continues next T times of iterations, otherwise it produces one new individual in its direct next hierarchy, and reinitializes its position and velocity. HASPSO is able to rank population based on the form of adaptively increasing size by degrees. Theoretical Analysis and experiment show that compared with traditional particle swarm optimization (PSO) algorithm, it can make solution efficiency greatly improved and is an effective method for solving geometric constraint problems.

Global solutions of well-constrained transcendental systems using expression trees and a single solution test

We present an algorithm which is capable of globally solving a well-constrained transcendental system over some sub-domain D⊂Rn, isolating all roots. Such a system consists of n unknowns and n regular functions, where each may contain non-algebraic (transcendental) functions like sin, exp or log. Every equation is considered as a hyper-surface in Rn and thus a bounding cone of its normal (gradient) field can be defined over a small enough sub-domain of D. A simple test that checks the mutual configuration of these bounding cones is used that, if satisfied, guarantees at most one zero exists within the given domain. Numerical methods are then used to trace the zero. If the test fails, the domain is subdivided. Every equation is handled as an expression tree, with polynomial functions at the leaves, prescribing the domain. The tree is processed from its leaves, for which simple bounding cones are constructed, to its root, which allows to efficiently build a final bounding cone of the normal field of the whole expression. The algorithm is demonstrated on curve–curve intersection, curve–surface intersection, ray-trap and geometric constraint problems and is compared to interval arithmetic.

Geometric Constraint Solving in Parametric Computer-Aided Design

With parametric computer-aided design (CAD) software, designers can create geometric models that are easily updated (within limits) by modifying the values of controlling parameters. These numeric and non-numeric parameters control the geometry in two ways: parametric operations and geometric constraint solving. This paper examines the advances over the last decade in the representation of parametric operations and of solving geometric constraint problems. An extensive literature has grown up surrounding geometric constraint solving and there has been substantial progress in the types of objects and constraints that can be handled robustly. Yet parametric operations have remained largely within the same conceptualization and begin to limit the flexibility of CAD systems, since they still do not align well with a systematic design process.

Quadratic Constraints on Rigid-Body Displacements

In this work, the solution to certain geometric constraint problems are studied. The possible rigid displacements allowed by the constraints are shown to be intersections of the Study quadric of rigid-body displacements with quadratic hypersurfaces. The geometry of these constraint varieties is also studied and is found to be isomorphic to products of subgroups in many cases. This information is used to find extremely simple derivations for general solutions to some problems in kinematics. In particular, the number of assembly configurations for RRPS and RRRS mechanisms are found in this way. In order to treat planes and spheres on an equal footing, the Clifford algebra for the Möbius group is introduced.

DECOMPOSITION OF GEOMETRIC CONSTRAINT SYSTEMS: A SURVEY

Significant progress has been accomplished during the past decades about geometric constraint solving, in particular thanks to its applications in industrial fields like CAD and robotics. In order to tackle problems of industrial size, many solving methods use, as a preprocessing, decomposition techniques that transform a large geometric constraint system into a set of smaller ones. In this paper, we propose a survey of the decomposition techniques for geometric constraint problems . We classify them into four categories according to their modus operandi, establishing some similarities between methods that are traditionally separated. We summarize the advantages and limitations of the different approaches, and point out key issues for meeting industrial requirements such as generality and reliability.

WELL-CONSTRAINED COMPLETION AND DECOMPOSITION FOR UNDER-CONSTRAINED GEOMETRIC CONSTRAINT PROBLEMS

In this paper, we consider the optimal well-constrained completion problem, that is, for an under-constrained geometric constraint problem, add automatically new constraints in such a way that the new constraint problem G is well-constrained and the set of equations to be solved simultaneously in order to solve G has the smallest size. We propose a polynomial time algorithm which gives a partial solution to the above problem.

Summary of basic 2D constraint solving

This summary reviews basic techniques that are widely available for solving 2D geometric constraint problems. Special attention is devoted to a graph-based approach; different phases of this approach are described and some analysis of them is done. Specifically, we review the triangle decomposition of the graph analysis phase discussing the Church-Rosser property it satisfies and pointing out how to extend the analysis phase when adding more complex primitives, especially variable-radius circles.

AN EFFICIENT METHOD TO DETERMINE THE INTENDED SOLUTION FOR A SYSTEM OF GEOMETRIC CONSTRAINTS

The number of solutions of a geometric constraint problem is generally exponential to the number of geometric elements in the problem. Finding a single intended solution, satisfying additional criteria, typically results in an NP-complete problem. A prototype-based selection scheme is presented here that avoids this problem. First, a resemblance relation between configurations is formally defined. This relation should be satisfied between the intended solution and a prototype configuration. The resemblance relation is in our approach satisfied by applying selection rules to subproblems in a bottom-up solving approach. The resulting solving algorithm is polynomial, because the selection rules are not used as search heuristic, but to unambiguously select a single solution such that no backtracking search is needed. For many applications, in particular CAD, this solution is both meaningful and intuitive.

A C-tree decomposition algorithm for 2D and 3D geometric constraint solving

In this paper, we propose a method which can be used to decompose a 2D or 3D constraint problem into a C-tree. With this decomposition, a geometric constraint problem can be reduced into basic merge patterns, which are the smallest problems we need to solve in order to solve the original problem in certain sense. Based on the C-tree decomposition algorithm, we implemented a software package MMP/Geometer. Experimental results show that MMP/Geometer finds the smallest decomposition for all the testing examples efficiently.

Solving geometric constraints with genetic simulated annealing algorithm.

This paper applies genetic simulated annealing algorithm (SAGA) to solving geometric constraint problems. This method makes full use of the advantages of SAGA and can handle under-/over- constraint problems naturally. It has advantages (due to its not being sensitive to the initial values) over the Newton-Raphson method, and its yielding of multiple solutions, is an advantage over other optimal methods for multi-solution constraint system. Our experiments have proved the robustness and efficiency of this method.

Making constraint solvers more usable: overconstraint problem

The richness and expressive power of geometric constraints causes unintended ambiguities and inconsistencies during their solution or realization. For example, geometric constraint problems may turn out to be overconstrained requiring the user to delete one or more of the input constraints, and the solutions must then be dynamically updated. Without proper guidance by the constraint solver, the user must have profound insight into the mathematical nature of constraint systems and understand the internals of the solver algorithm. But a general user is most likely unfamiliar with those problems, so that the required interaction with the constraint solver may well be beyond the user's ability. In this paper, we present strategies and techniques to empower the user to deal effectively with the overconstraint problem while not requiring him or her to become an expert in the mathematics of constraint solving. We formulate this problem as a series of formal requirements that gel with other essentials of constraint solvers. We then give algorithmic solutions that are both general and efficient (running time typically linear in the number of relevant constraints).

Solving spatial basic geometric constraint configurations with locus intersection

A basic idea of geometric constraint solving (GCS) is to decompose the constraint problem into smaller ones according to some basic configurations. In this paper, we find all spatial basic configurations involving points, lines, and planes containing up to six geometric primitives in an automated way. Many of these basic configurations still resist effective analytical solutions. We propose the locus intersection method (LIM) for GCS, a hybrid method based on geometric computation and numerical search that can be used to find all the solutions for a geometric constraint problem. We show that the LIM can be used to solve all the above basic configurations.

A hybrid approach to geometric constraint solving with graph analysis and reduction

In this paper, a graph constructive approach to solving geometric constraint problems is being described. Usually, the graph constructive approach is efficient; however, it has its limitations in scope: it cannot handle ruler-and-compass non-constructible configurations, and under-constrained problems. To overcome these limitations, a proposed algorithm that isolates ruler-and-compass non-constructible configurations from ruler-and-compass constructible configurations is made. Numerical calculation methods are applied to solve them separately. This separation can maximize the efficiency and robustness of a geometric constraint solver. Moreover, the solver can handle under-constrained problems by classifying under-constrained subgraphs to simplified cases by applying classification rules. Then, it decides the calculating sequence of the geometric entities in each classified case, and calculates the geometric entities by adding appropriate assumptions or constraints. By extending the clustering types, and defining several rules, the proposed approach can overcome the limitations of previous graph constructive approaches. Therefore, an efficient and robust geometric constraint solver using this approach can be made.

A Constraint Solving-Based Approach to Analyze 2D Geometric Problems With Interval Parameters

Many applications of geometric nature can be modeled by geometric problems defined by constraints in which the constraint parameters have interval uncertainty. In a previous work, we developed a method for solving geometric constraint problems where parameters are narrow intervals in the domain of the geometric problem. Based on this work, we present a new approach to solve more general problems with non-trivial-width interval parameters that may not necessarily be in the domain of the problem. We show how our approach is successfully applied to a number of problems like solving geometric problems with tolerances, checking constraint feasibility and analyzing link motion of planar mechanisms.

A complete algorithm for automated discovering of a class of inequality-type theorems

Making use of the discriminant sequence for polynomials, WR algorithm, Wu’s elimination and a partial cylindrical algebraic decomposition, we present here a practical algorithm for automated inequality discovering which can discover new inequalities automatically without requiring to put forward any conjectures beforehand. That is complete for an extensive class of inequality-type theorems. Also this algorithm is applied to the classification of the real physical solutions of geometric constraint problems. Many inequalities with various backgrounds have been discovered or rediscovered by our program, DISCOVERER, which implements the algorithm in Maple.