论文标题
代数3D图形静态：约束区域
Algebraic 3D Graphic Statics: Constrained Areas
论文作者
论文摘要
这项研究提供了算法和数值方法，以几何控制3D/多面体图形静态（3DGS）的相互图中的内部和外部力量的大小。在3DG中，结构及其力平衡的形式由两个在几何和拓扑相关的多面体图表示。力图的面部面积代表系统中内部和外部力的大小。这项研究的方法首次允许用户控制和约束一般多面体的面部面积和边缘长度，这些面孔和边缘长度可以是凸，自我隔离或凹面。结果，设计师可以明确控制力图中的力幅度，并探索各种压缩和张力混合的拟议拟合结构形式的平衡。在这种方法中，使用二次公式根据其边缘长度计算单个面的面积。该方法适用于用预定义的区域和边缘长度操纵面部几何形状。随后，多面体的几何形状通过新更改的面更新。这种方法是一种多步算法，其中每个步骤都包括计算单个面的几何形状并更新多面体几何形状。该框架的独特结果之一是构建零区域，自我切割面的面孔，在该面孔中，自我隔离面的签名区域的总和为零，代表形式图中的零力的成员。这项研究的方法可以阐明某些系统以前无法使用互惠多面体图是合理的系统的均衡。因此，它概括了多面体框架平衡的原理，并在超出仅压缩系统的高度熟悉的拟南芥多面体结构的设计中打开了全新的地平线。
This research 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 (3DGS). In 3DGS, the form of the structure and its equilibrium of forces is 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 system. For the first time, the methods of this research allow 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. As a result, a designer can explicitly control the force magnitudes in the force diagram and explore the equilibrium of a variety of compression and tension-combined funicular structural forms. In this method, a quadratic formulation is used to compute the area of a single face based on its edge lengths. The approach is applied to manipulating the face geometry with a predefined area and the edge lengths. Subsequently, the geometry of the polyhedron is updated with newly changed faces. This approach is a multi-step algorithm where each step includes computing the geometry of a single face and updating the polyhedral geometry. One of the unique results of this framework is the construction of the zero-area, self-intersecting faces, where the sum of the signed areas of a self-intersecting face is zero, representing a member with zero force in the form diagram. The methodology of this research can clarify the equilibrium of some systems that could not be previously justified using reciprocal polyhedral diagrams. Therefore, it generalizes the principle of the equilibrium of polyhedral frames and opens a completely new horizon in the design of highly-sophisticated funicular polyhedral structures beyond compression-only systems.
