To flatten a diaphragm from 3D space into a 2D plane, a thin film structure surface expansion algorithm based on a spring-particle system is employed. This method begins by defining a parameter domain and spatial domain that maintain the same topological structure. When selecting a plane for projection, if the curvature of the surface is not too significant, it can be directly projected onto the xy-plane to obtain a 2D representation. However, care must be taken to avoid self-intersections or overlapping during the projection process. If such issues arise, an alternative projection plane can be chosen to ensure a clean, non-overlapping planar layout.
A spring-particle system can be constructed using a triangular mesh on the plane. In this model, each triangle node acts as a mass point, while the edges are replaced with springs. This allows physical properties, such as force and deformation energy, to be related to geometric quantities like node spacing and surface shape. The mass at each particle is determined by the area of the cell it resides in. The difference between the original 3D diaphragm and its 2D projection is treated as elastic deformation energy stored in the system.
In the spring-particle system, each particle Pi is connected to neighboring particles Pj via springs. During the deformation process, if the distance between two nodes in the 2D mesh is greater than their corresponding distance in the 3D surface, a tensile force is applied; otherwise, a compressive force is used. This dynamic interaction helps simulate the behavior of the material under tension and compression.
The spring force acting on a particle Pi is defined as: fi = Σ (C * (|PiPj| - dj) * NPiPj), where C is the spring stiffness coefficient, |PiPj| is the distance between Pi and Pj on the 2D mesh, dj represents the actual distance between these points on the 3D surface, and NPiPj is the unit vector pointing from Pi to Pj. m denotes the total number of springs connected to Pi.
To control local accuracy in curved surfaces, different spring stiffness coefficients can be assigned to different regions. This allows precise control over the unfolding process. For instance, to meet boundary compatibility requirements, a higher stiffness can be applied to boundary springs, ensuring that adjacent diaphragms match in length and maintain the required precision.
Example 2: This example applies the proposed algorithm to a first-order tension saddle surface. The surface has equal tension in both the warp and weft directions, with fixed ends on two sides and parabolic curves on the other two. The surface tension is 40N/cm in both directions. The projected lengths on the plane are 18m and 16m respectively. The entire surface is divided into four diaphragms. The pattern development achieves an area accuracy and shape accuracy of 0.3%, with the iteration count limited to 5000 times.
Conclusion: The spatial patch expansion algorithm presented in this paper offers several advantages. First, it is capable of developing complex diaphragms and provides accurate solutions for developable surfaces, while offering approximations for non-developable ones through controlled area and shape errors. Second, by adjusting the spring stiffness, the algorithm enables localized control over the accuracy of the unfolding, which is crucial for ensuring compatibility between cut pieces in membrane structures. Third, since the algorithm avoids solving large systems of equations, it converges quickly, significantly reducing computational time.
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