This research is a continuation of the Algebraic 3D Graphic Statics Methods that addressed the reciprocal constructions in an earlier publication (Hablicsek et al. 2019). It provides algorithms and (numerical) methods to geometrically control the magnitude of the internal and external forces in the reciprocal diagrams of 3D/Polyhedral Graphic statics. 3D graphic statics (3DGS) is a recently rediscovered method of structural form-finding based on a 150-year old proposition by Rankine and Maxwell in Philosophical Magazine. In 3DGS, the form of the structure and its equilibrium of forces are represented by two polyhedral diagrams that are geometrically and topologically related. The areas of the faces of the force diagram represent the magnitude of the internal and external forces in the members of the form diagram. The proposed method allows the user to control and constrain the areas and edge lengths of the faces of general polyhedrons that can be convex, self-intersecting, or concave in a group of aggregated polyhedral cells. In this method, a quadratic formulation is introduced to compute the area of a face based on its edge lengths only. This quadratic function is then turned into a linear formulation to facilitate the non-trivial computation of reciprocal polyhedral diagrams. The approach is applied to force diagrams, including a group of polyhedral cells, to manipulating the face geometry with a predefined area and the edge lengths. The method is implemented as a multi-step algorithm where each step includes computing the geometry of a single face with a target area and updating the polyhedral geometry. One of the remarkable results of this framework is to control the construction of the zero-area faces as proposed by McRobie (2017b). The zero-area faces represent a member with zero force in the form diagram. This research shows how self-intersecting faces, including the zero-area faces, can be constructed with additional edge constraints in a group of polyhedral cells without breaking the reciprocity of the form and force diagrams. Thus, it provides more hints on the generalization of the principle of the equilibrium of polyhedral frames. It also suggests a design approach where the boundary conditions and internal forces of compression-only systems can be manipulated to the design systems with both compression and tensile forces with no change in the geometry or the faces’ planarity of the form diagram.

这项研究是代数 3D 图形静力学方法的延续，该方法解决了早期出版物（Hablicsek 等人，2019 年）中的互易构造问题。它提供算法和（数值）方法来几何控制 3D/多面体图形静力学倒数图中的内力和外力的大小。3D 图形静力学 (3DGS) 是最近重新发现的一种结构找形方法，该方法基于 Rankine 和 Maxwell 在哲学杂志上提出的 150 年历史。在 3DGS 中，结构的形式及其力的平衡由两个几何和拓扑相关的多面体图表示。力图的面面积表示形式图成员中内力和外力的大小。所提出的方法允许用户控制和约束一组聚合多面体单元中的一般多面体的面的面积和边长，这些面可以是凸面、自相交或凹面。在这种方法中，引入了二次公式来仅根据边长计算人脸的面积。然后将该二次函数转化为线性公式，以促进倒数多面体图的非平凡计算。该方法应用于力图，包括一组多面体单元，以操纵具有预定义区域和边长的面几何形状。该方法作为多步算法实现，其中每一步都包括计算具有目标区域的单个面的几何形状并更新多面体几何形状。该框架的显着结果之一是控制了 McRobie (2017b) 提出的零面积面的构建。零面积面代表形式图中具有零力的构件。这项研究展示了如何在不破坏形状和力图的互易性的情况下，在一组多面体单元中使用附加边约束构造自相交面，包括零面积面。因此，它为多面体框架平衡原理的推广提供了更多提示。它还提出了一种设计方法，其中可以将仅压缩系统的边界条件和内力操纵到具有压缩力和拉伸力的设计系统，而不会改变形状图的几何形状或面的平面度。