一种设计分解的正确性证明

二维变量化设计系统可以用含有n个未知数、m个方程的非线性方程组表示．通过设计分解可以提高几何约束求解的效率和数值稳定性．给出了一种基于图论的设计分解方法及其正确性证明．该方法可以（1）处理结构欠约束系统的分解;（2）检测出冗余约束．分解算法在有限步内终止,其结果是结构相容的     

几何约束求解的简化迭代算法

针对几何约束系统图分解中复合顶点的求解问题,提出复合顶点的图分解算法和等价自由变量的简化迭代求解算法.通过去除复合顶点部分边界约束对复合顶点进行图分解,对求解序列中的欠约束顶点添加等价自由变量、以等价自由变量的部分迭代求解、替代系统的整体数值求解,以提高求解效率和稳定性.该算法具有很强的通用性,并在实际应用中得到验证.     

同伦法在逆向工程约束求解中的应用

逆向工程模型重建过程中,为了再现逼近特征之间固有的几何约束关系,应该在满足特定约束关系下对测量数据进行优化拟合,但在拟合时,由于迭代初值以及约束的不确定性,使得通常的牛顿迭代法有时难以得到收敛解。为解决上述问题,提出了一种应用同伦迭代法求解约束优化拟合方程的策略,该策略首先对部分约束进行合并处理,然后给出欠约束同伦的构造方法,并由局部曲率控制欠约束同伦迭代步长。实例结果显示,该方法能有效完成特定约束条件下数据优化拟合计算。     

一种利用有向图优化约束求解的方法

为克服约束求解的效率问题及可靠性问题，本文提出了一种基于图结构的约束求解方法，它利用图瓣形式来表示几何元素之间的约束关系，使得几何元素的求解从整体下降 至局部，将一个方程组的求解问题论为几个小方程组的求解，大大降低了计算复杂度，进而提高了求解的可靠性。     

三维几何约束系统的等价性分析

针对过约束、完整约束和欠约束三维几何约束系统的求解问题,提出了等价性分析方法.该方法基于三维几何约束系统的内在等价性,充分挖掘几何领域知识,依据拆解约束闭环、缩减约束闭环和析出约束闭环等原则,采用等价约束替换来处理几何约束闭环问题,优化几何约束图的结构,实现几何约束系统的优化分解.最后用多个实例验证了该方法的正确性和有...     

一种基于图分解的几何约束求解方法

为了提高几何约束求解的效率和鲁棒性 ,对基于图的构造方法进行了改进 ,即加入虚约束进行扩展和过约束问题的一致性判定 ,提出了一种基于图分解的方法 ,用此方法可以处理包括完全约束、过约束和欠约束等多种情况的约束求解问题 ,另外 ,在该方法中还通过引入分解树将约束求解的范围由整体下降到局部 ,使大部分求解过程能够采用几何求解实现 ,提高了求解和后续修改的效率 ,通过实验数据测试证明 ,该方法对于大型约束求解问题可以达到实时处理的效果 ,具有较强的实用性     

装配位置约束建模及求解

以基本几何约束组合统一表达装配约束,为提高求解效率,研究了姿态约束和位置约束的可解耦情况下位置约束的解析求解.将基本位置约束映射为移动空间并以参数方程表达,通过移动空间的增量解析求交,满足约束;在姿态约束和位置约束的不可解耦情况,联立基本约束进行整体数值法求解.文中方法保持了基本约束表达的独立性,适合于欠约束系统和完整约束系统.     

基于有向图的二维约束求解算法研究

针对过约束、几何完全定义状态判定和约束求解效率等问题,提出了基于约束图,利用自由度理论和约束冲突机制,通过反向约束方向平衡约束,进而通过排序进行约束求解的算法。算法采用约束图记录约束和几何的关系;通过约束平衡的方法进行过约束和几何完全定义的判定;采用排序求解方法,将庞大计算问题转化为一组相对简单的计算问题。算法已得到初步应用,对过约束和几何完全定义状态的判定有明显的效果,而且提高了约束求解效率。     

约束驱动的二维变量几何造型系统

给出了一种二维变量几何系统的系统结构，它将约束求解分成几何推理与数据求解两个层次进行，并阐述了几何约速推理与数值求解相结合的算法，对约束模式的一致性及约束的局部求也作了相应的研究。     

通用几何约束系统统一建模研究

在几何约束和几何实体的基本约束和欧拉参数表达的基础上,研究了通用几何约束系统的统一建模问题。通过对三维几何实体姿态约束和位置约束解耦性的分析,抽象出球实体、盒体和球盒体三种基本几何实体表达空间几何实体,并以基本约束的组合表达几何约束,形成几何约束模型特有的层次结构;并以有向图管理几何约束系统,可以清晰地反映姿态约束和位置约束的解耦性,实现约束系统的细粒度分解,得到规模更小的求解序列,实现高效求解。方法实现于原型系统WhutVAS中。     

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.     

遗传模拟退火算法在约束求解中的应用

将遗传模拟退火算法应用于约束求解中 ,提高了约束系统求解的鲁棒性和效率 .与 Newton- Raphson数值方法相比 ,由于遗传模拟退火算法是一种单纯的数值迭代方法 ,不涉及到矩阵求逆 ,因此克服了 Newton- Raphson法对初始值敏感的缺点 ,具有很强的鲁棒性 ;与其他利用 BFGS的优化算法相比 ,由于遗传模拟退火算法是在一个初始的解空间中搜索所有可能的解 ,因此克服了 BFGS优化算法对良约束多解情况只能求出一个解的缺点 ;由于遗传模拟退火算法是将约束问题转化为优化问题后才进一步求解 ,因此其可以处理过约束一致和欠约束的问题     

Geometric constraint satisfaction using optimization methods

The numerical approach to solving geometric constraint problems is indispensable for building a practical CAD system. The most commonly-used numerical method is the Newton–Raphson method. It is fast, but has the instability problem: the method requires good initial values. To overcome this problem, recently the homotopy method has been proposed and experimented with. According to the report, the homotopy method generally works much better in terms of stability. In this paper we use the numerical optimization method to deal with the geometric constraint solving problem. The experimental results based on our implementation of the method show that this method is also much less sensitive to the initial value. Further, a distinctive advantage of the method is that under- and over-constrained problems can be handled naturally and efficiently. We also give many instructive examples to illustrate the above advantages.     

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.     

Singularity Analysis of Geometric Constraint Systems

Singularity analysis in an important subject of the geometric constraint satisfaction problem.In this paper,three kinds of singularities are described and corresponding identifcation methods are presented for both under0constrained systems and over-constrained systems,Another special but common singularity for under-constrained geometric systems,pseudo-singularity,is analyzed.Pseudo-singularity is caused by a variety of constraint mathching of under-constrained systems and can be removed by improving constraint distribution.To avoid pseudo-singularity and decide redundant constraints adaptively,a differentiaiton algorithm is proposed in the paper.Its corrctness and effciency have been validated through its practical applications in a 2D/3D geometric constraint solver CBA.     

A constructive approach to solving 3-D geometric constraint systems using dependence analysis

Solving geometric constraint systems in 3-D is much more complicated than that in 2-D because the number of variables is larger and some of the results valid in 2-D cannot be extended for 3-D. In this paper, we propose a new DOF-based graph constructive method to geometric constraint systems solving that can efficiently handle well-, over- and under-constrained systems based on the dependence analysis. The basic idea is that the solutions of some geometric elements depend on some others because of the constraints between them. If some geometric elements depend on each other, they must be solved together. In our approach, we first identify all structurally redundant constraints, then we add some constraints to well constrain the system. And we prove that the order of a constraint system after processing under-constrained cases is not more than that of the original system multiplied by 5. After that, we apply a recursive searching process to identify all the clusters, which is shown to be capable of getting the minimum order-reduction result of a well-constrained system. We also briefly describe the constraint evaluation phase and show the implementation results of our method.     

几何约束有向图的规划分解研究

在基于有向图表达的几何约束系统中,几何约束的匹配方向、分布状态以及有向图中强连通分量的规模直接影响到整个约束系统的求解;如何对几何约束系统进行合理规划,得到正确有效的求解序列,是目前约束分解研究的重要内容。该文提出了一个规划分解算法,它针对欠约束几何系统的特点,能够优化约束的初始匹配方向,对于约束匹配过程中生成的强连通子图,通过调整约束匹配方向,自适应地改善约束分布,从而减小强连通子图的规模,以求得到几何约束系统正确而高效的求解序列。同时,基于规划分解算法,完成了约束的奇异性分析,提供了面向分解的奇异性分析算法。