Geometric constraint subsets and subgraphs in the analysis of assemblies and mechanisms

Geometric Reasoning ability is central to many applications in CAD/CAM/CAPP environments. An increasing demand exists for Geometric Reasoning systems which evaluate the feasibility of virtual scenes specified by geometric relations. Thus, the Geometric Constraint Satisfaction or Scene Feasibility (GCS/SF) problem consists of a basic scenario containing geometric entities, whose context is used to propose constraining relations among still undefined entities. If the constraint specification is consistent, the answer of the problem is one of finitely or infinitely many solution scenarios satisfying the prescribed constraints. Otherwise, a diagnostic of inconsistency is expected. The three main approaches used for this problem are numerical, procedural or operational and mathematical. Numerical and procedural approaches answer only part of the problem, and are not complete in the sense that a failure to provide an answer does not preclude the existence of one. The mathematical approach previously presented by the authors describes the problem using a set of polynomial equations. The common roots to this set of polynomials characterizes the solution space for such a problem. That work presents the use of Groebner basis techniques for verifying the consistency of the constraints. It also integrates subgroups of the Special Euclidean Group of Displacements SE(3) in the problem formulation to exploit the structure implied by geometric relations. Although theoretically sound, these techniques require large amounts of computing resources. This work proposes Divide-and-Conquer techniques applied to local GCS/SF subproblems to identify strongly constrained clusters of geometric entities. The identification and preprocessing of these clusters generally reduces the effort required in solving the overall problem. Cluster identification can be related to identifying short cycles in the Spatial Constraint graph for the GCS/SF problem. Their preprocessing uses the aforementioned Algebraic Geometry and Group theoretical techniques on the local GCS/SF problems that correspond to these cycles. Besides improving the efficiency of the solution approach, the Divide-and-Conquer techniques capture the physical essence of the problem. This is illustrated by applying the discussed techniques to the analysis of the degrees of freedom of mechanisms.MSC: 68U07La habilidad del Razonamiento Geométrico es central a muchas aplicaciones de CAD/CAM/CAPP (Computer Aided Design, Manufacturing and Process Planning). Existe una demanda creciente de sistemas de Razonamiento Geométrico que evalúen la factibilidad de escenas virtuales, especificados por relaciones geométricas. Por lo tanto, el problema de Satisfacción de Restricciones Geométricas o de Factibilidad de Escena (GCS/SF) consta de un escenario básico conteniendo entidades geométricas, cuyo contexto es usado para proponer relaciones de restricción entre entidades aún indefinidas. Si la especificación de las restricciones es consistente, la respuesta al problema es uno del finito o infinito número de escenarios solución que satisfacen las restricciones propuestas. De otra forma, un diagnóstico de inconsistencia es esperado. Las tres principales estrategias usadas para este problema son: numérica, procedimental y matemática. Las soluciones numérica y procedimental resuelven solo parte del problema, y no son completas en el sentido de que una ausencia de respuesta no significa la ausencia de ella. La aproximación matemática previamente presentada por los autores describe el problema usando una serie de ecuaciones polinómicas. Las raíces comunes a este conjunto de polinomios caracterizan el espacio solución para el problema. Ese trabajo presenta el uso de técnicas con Bases de Groebner para verificar la consistencia de las restricciones. Ella también integra los subgrupos del grupo especial Euclídeo de desplazamientos SE(3) en la formulación del problema para explotar la estructura implicada por las relaciones geométricas. Aunque teóricamente sólidas, estas técnicas requieren grandes cantidades de recursos computacionales. Este trabajo propone técnicas de Dividir y Conquistar aplicadas a subproblemas GCS/SF locales para identificar conjuntos de entidades geométricas fuertemente restringidas entre sí. La identificación y pre-procesamiento de dichos conjuntos locales, generalmente reduce el esfuerzo requerido para resolver el problema completo. La identificación de dichos sub-problemas locales está relacionada con la identificación de ciclos cortos en el grafo de Restricciones Geométricas del problema GCS/SF. Su preprocesamiento usa las ya mencionadas técnicas de Geometría Algebraica y Grupos en los problemas locales que corresponden a dichos ciclos. Además de mejorar la eficiencia de la solución, las técnicas de Dividir y Conquistar capturan la esencia física del problema. Esto es ilustrado por medio de su aplicación al análisis de grados de libertad de mecanismos.MSC: 68U0

Algorithmic Aspects of Cyclic Combinational Circuit Synthesis

Digital circuits are called combinational if they are memoryless: if they have outputs that depend only on the current values of the inputs. Combinational circuits are generally thought of as acyclic (i.e., feed-forward) structures. And yet, cyclic circuits can be combinational. Cycles sometimes occur in designs synthesized from high-level descriptions, as well as in bus-based designs [16]. Feedback in such cases is carefully contrived, typically occurring when functional units are connected in a cyclic topology. Although the premise of cycles in combinational circuits has been accepted, and analysis techniques have been proposed [7], no one has attempted the synthesis of circuits with feedback at the logic level. We have argued the case for a paradigm shift in combinational circuit design [10]. We should no longer think of combinational logic as acyclic in theory or in practice, since most combinational circuits are best designed with cycles. We have proposed a general methodology for the synthesis of multilevel networks with cyclic topologies and incorporated it in a general logic synthesis environment. In trials, benchmark circuits were optimized significantly, with improvements of up to 30%I n the area. In this paper, we discuss algorithmic aspects of cyclic circuit design. We formulate a symbolic framework for analysis based on a divide-and-conquer strategy. Unlike previous approaches, our method does not require ternary-valued simulation. Our analysis for combinationality is tightly coupled with the synthesis phase, in which we assemble a combinational network from smaller combinational components. We discuss the underpinnings of the heuristic search methods and present examples as well as synthesis results for benchmark circuits. In this paper, we discuss algorithmic aspects of cyclic circuit design. We formulate a symbolic framework for analysis based on a divide-and-conquer strategy. Unlike previous approaches, our method does not require ternary-valued simulation. Our analysis for combinationality is tightly coupled with the synthesis phase, in which we assemble a combinational network from smaller combinational components. We discuss the underpinnings of the heuristic search methods and present examples as well as synthesis results for benchmark circuits

Geometric constraint subsets and subgraphs in the analysis of assemblies and mechanisms

Geometric Reasoning ability is central to many applications in CAD/CAM/CAPP environments -- An increasing demand exists for Geometric Reasoning systems which evaluate the feasibility of virtual scenes specified by geometric relations -- Thus, the Geometric Constraint Satisfaction or Scene Feasibility (GCS/SF) problem consists of a basic scenario containing geometric entities, whose context is used to propose constraining relations among still undefined entities -- If the constraint specification is consistent, the answer of the problem is one of finitely or infinitely many solution scenarios satisfying the prescribed constraints -- Otherwise, a diagnostic of inconsistency is expected -- The three main approaches used for this problem are numerical, procedural or operational and mathematical -- Numerical and procedural approaches answer only part of the problem, and are not complete in the sense that a failure to provide an answer does not preclude the existence of one -- The mathematical approach previously presented by the authors describes the problem using a set of polynomial equations -- The common roots to this set of polynomials characterizes the solution space for such a problem -- That work presents the use of Groebner basis techniques for verifying the consistency of the constraints -- It also integrates subgroups of the Special Euclidean Group of Displacements SE(3) in the problem formulation to exploit the structure implied by geometric relations -- Although theoretically sound, these techniques require large amounts of computing resources -- This work proposes Divide-and-Conquer techniques applied to local GCS/SF subproblems to identify strongly constrained clusters of geometric entities -- The identification and preprocessing of these clusters generally reduces the effort required in solving the overall problem -- Cluster identification can be related to identifying short cycles in the Spatial Con straint graph for the GCS/SF problem -- Their preprocessing uses the aforementioned Algebraic Geometry and Group theoretical techniques on the local GCS/SF problems that correspond to these cycles -- Besides improving theefficiency of the solution approach, the Divide-and-Conquer techniques capture the physical essence of the problem -- This is illustrated by applying the discussed techniques to the analysis of the degrees of freedom of mechanism

An Elimination Method for Solving Bivariate Polynomial Systems: Eliminating the Usual Drawbacks

We present an exact and complete algorithm to isolate the real solutions of a zero-dimensional bivariate polynomial system. The proposed algorithm constitutes an elimination method which improves upon existing approaches in a number of points. First, the amount of purely symbolic operations is significantly reduced, that is, only resultant computation and square-free factorization is still needed. Second, our algorithm neither assumes generic position of the input system nor demands for any change of the coordinate system. The latter is due to a novel inclusion predicate to certify that a certain region is isolating for a solution. Our implementation exploits graphics hardware to expedite the resultant computation. Furthermore, we integrate a number of filtering techniques to improve the overall performance. Efficiency of the proposed method is proven by a comparison of our implementation with two state-of-the-art implementations, that is, LPG and Maple's isolate. For a series of challenging benchmark instances, experiments show that our implementation outperforms both contestants.Comment: 16 pages with appendix, 1 figure, submitted to ALENEX 201

Pattern matching : a sheaf-theoretic approach

A general theory of pattern matching is presented by adopting an extensional, geometric view of patterns. The extension of the matching relation consists of the occurrences of all possible patterns in a particular target. The geometry of the pattern describes the structure of the pattern and the spatial relationships among parts of the pattern. The extension and the geometry, when combined, produce a structure called a sheaf. Sheaf theory is a well developed branch of mathematics which studies the global consequences of locally defined properties. For pattern matching, an occurrence of a pattern, a global property of the pattern, is obtained by gluing together occurrences of parts of the pattern, which are locally defined properties.A sheaf-theoretic view of pattern rnatching provides a uniforrn treatrnent of pattern matching on any kind of data structure-strings, trees, graphs, hypergraphs, and so on. Such a parametric description is achieved by using the language of category theory, a highly abstract description of commonly occurring structures and relationships in mathematics.A generalized version of the Knuth-Morris-Pratt pattern matching algorithm is derived by gradually converting the extensional description of pattern rnatching as a sheaf into an intensional description. The algorithm results from a synergy of four very general program synthesis/transformation techniques: (1) Divide and conquer: exploit the sheaf condition; assemble a full match by gluing together partial matches; (2) Finite differencing: collect and update partial matches incrementally while traversing the target; (3) Backtracking: instead of saving all partial matches, save just one; when this partial match cannot be extended, fail back to another; (4) Partial evaluation: precompute pattern-based (and therefore constant) computations.The derivation is carried out in a general frarnework using Grothendieck topologies. By appropriately instantiating the underlying data structures and topologies, the sarne scheme results in matching algorithms for patterns with variables and with multiple patterns. Slight variations of the derivation result in Earley's algorithm for context-free parsing, and Waltz filtering, a relaxation algorithm for providing 3-D interpretations to 2-D irnages.Other applications of a geometric view of patterns are briefly considered: rewrites, parallel algorithms, induction and computability

Index-3 divide-and-conquer algorithm for efficient multibody system dynamics simulations: theory and parallel implementation

[Abstract] There has been a growing attention to efficient simulations of multibody systems, which is apparently seen in many areas of computer-aided engineering and design both in academia and in industry. The need for efficient or real-time simulations requires high fidelity techniques and formulations that should significantly minimize computational time. Parallel computing is one of the approaches to achieve this objective. This paper presents a novel index-3 divide-and-conquer algorithm for efficient multibody dynamics simulations that elegantly handles multibody systems in generalized topologies through the application of the augmented Lagrangian method. The proposed algorithm exploits a redundant set of absolute coordinates. The trapezoidal integration rule is embedded into the formulation and a set of nonlinear equations need to be solved every time instant. Consequently, the Newton–Raphson iterative scheme is applied to find the system coordinates and joint constraint loads in an efficient and highly parallelizable manner. Two divide-and-conquer based mass-orthogonal projections are performed then to circumvent the effect of constraint violation errors at the velocity and acceleration level. Sample open- and closed-loop multibody system test cases are investigated in the paper to confirm the validity of the approach. Challenging simulations of multibody systems featuring long kinematic chains are also performed in the work to demonstrate the robustness of the algorithm. The details of OpenMP-based parallel implementation on an eight-core shared memory computer are presented in the text and the parallel performance results are extensively discussed. Significant speedups are obtained for the simulations of small- to large-scale multibody open-loop systems. The mentioned features make the proposed algorithm a good general purpose approach for high-fidelity, efficient or real-time multibody dynamics simulations.Ministerio de Economía y Competitividad; JCI-2012-12376Poland. National Science Center; DEC-2012/07/B/ST8/0399