公差分析中自由度管理的多面体尺寸调整
Adapting polytopes dimension for managing degrees of freedom in tolerancing analysis
摘要：
在公差分析中，几何或接触规范可以用多面体表示。由于曲面的不变性和关节的自由度，这些操作数多面体在大多数情况下最初是无界的（即多面体）。Homri等人提出在每个多面体的无界位移上引入虚边界（称为帽半空间），将它们变成6-多面体。这个决定是由R6中多面体操作的复杂性所激发的。然而，该策略在计算Minkowski和时必须面对帽半空间数目的乘法。一般来说，用于计算帽刻面的时间大于用于计算表示有界位移的真实极限的刻面的时间。为了解决这个问题，本文提出了使用螺旋理论来确定一组位移，定义一个表面相对于另一个表面的定位。这组位移定义了R6的子空间，其中各个表面的多面体必须被投影和操作，以避免沿着无界位移的方向计算小平面和顶点。利用这种新的策略，可以通过减少操作数的维数来降低Minkowski和的复杂性，从而减少计算时间。一个例子说明了该方法，并显示在计算过程中的时间减少。
英文摘要：
In tolerancing analysis, geometrical or contact specifications can be represented by polytopes. Due to the degrees of invariance of surfaces and that of freedom of joints, these operand polytopes are originally unbounded in most of the cases (i.e. polyhedra). Homri et al. proposed the introduction of virtual boundaries (called cap half-spaces) over the unbounded displacements of each polyhedron to turn them into 6-polytopes. This decision was motivated by the complexity that operating on polyhedra in R6 supposes. However, that strategy has to face the multiplication of the number of cap half-spaces during the computation of Minkowski sums. In general, the time for computing cap facets is greater than for computing facets representing real limits of bounded displacements. In order to deal with that, this paper proposes the use of the theory of screws to determine the set of displacements that defines the positioning of one surface in relation to another. This set of displacements defines the subspace of R6 in which the polytopes of the respective surfaces have to be projected and operated to avoid calculating facets and vertices along the directions of unbounded displacements. With this new strategy it is possible to decrease the complexity of the Minkowski sums by reducing the dimension of the operands and consequently reducing the computation time. An example illustrates the method and shows the time reduction during the computations.
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