CHS: Small: Efficient Simulation of Thin Materials With Discrete Tension Field Theory

薄壳模拟是科学计算的基础工具。它们被用于分析生物结构（如囊泡和细胞膜）的行为，模拟织物和复合材料的变形，预测外科训练和可视化工具中皮肤的疤痕和皱纹，以及分析建筑物和车辆中结构元件的屈曲和皱缩。这项研究将建立新的算法，以前所未有的效率模拟薄弯曲材料（如织物、纸或金属板）的物理行为，而这种模拟目前是出了名的困难和计算昂贵，因为薄物体（如纸或布）弯曲比拉伸更容易，并且更倾向于以几何复杂的方式弯曲和皱褶，而不是压缩。此外，这种复杂性是不可预测和混乱的；即使是相同的物体，在相同的载荷下，褶皱的确切形状也会有很大的不同。最后，预测薄物体在与自身和环境摩擦接触下的行为尤其具有挑战性；由于薄的几何形状，必须使用昂贵的碰撞检测和响应算法来确保薄部件不会相互穿过隧道，无论它们被推到一起的速度或力量有多快。为了使薄材料模拟在工程、设计和机器人应用中更加实用，这些应用的性能至关重要，该项目将开发一种更有效、更简化的模型来模拟薄材料的变形。从连续介质力学的张力场方法中借鉴的关键见解是，薄物体的行为是由材料中的张力线主导的，而由压缩和弯曲引起的细尺度褶皱的解决成本非常高，而且对物体的粗尺度形状或机械行为贡献很小。通过利用这一见解，并将计算精力集中在跟踪和模拟张力线上，薄壳模拟的性能可以在不牺牲精度的情况下得到大幅提高。换句话说，在纯拉伸区域，弹性膜能量是凸的，标准壳有限元方法表现良好。然而，在纯压缩或混合拉伸和压缩区域，屈曲发生是因为薄材料对压缩和弯曲的阻力之间存在尺度分离，并且壳体的后屈曲状态包含许多高度非线性，复杂的皱纹和折痕，然而，通过忽略皱纹并将壳体视为与壳体上的拉应力方向一致的1D曲线的集合，可以近似地描述壳体的粗糙形状。详细的工作计划将包括对张力为主的表面、混合应力表面以及接触和摩擦的工作。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Thin shell simulations are a foundational tool in scientific computing. They are used to analyze the behavior of biological structures such as vesicles and cell membranes, to simulate deformation of fabrics and composites, to predict scarring and wrinkling of skin in surgery training and visualization tools, and to analyze buckling and crumpling of structural elements in buildings and vehicles. This research will establish new algorithms for simulating the physical behavior of thin curved materials such as fabric, paper, or sheet metal, with unprecedented efficiency, whereas such simulations currently are notoriously difficult and computationally expensive because thin objects like a sheet of paper or cloth bend far more readily than they stretch and will prefer to buckle and crumple in geometrically complex ways rather than compress. Moreover, this complexity is unpredictable and chaotic; the exact pattern of wrinkles can vary wildly even for identical objects under identical loads. Finally, predicting how a thin object behaves under frictional contact with itself and the environment is especially challenging; due to the thin geometry, expensive collision detection and response algorithms must be used to ensure that thin parts do not tunnel through each other, no matter how quickly or forcefully they are pushed together.To make thin material simulations more practical for use in engineering, design, and robotics applications, where performance is critical, this project will develop a more efficient, simplified model for how to simulate deformation of thin materials. The key insight, borrowed from the tension field approach in continuum mechanics, is that the behavior of thin objects is dominated by lines of tension through the material, while fine-scale wrinkles induced by compression and bending are extraordinarily expensive to resolve yet contribute little to the object's coarse-scale shape or mechanical behavior. By exploiting this insight and focusing computational effort on tracking and simulating the lines of tension, the performance of thin shell simulations can be substantially improved without sacrificing accuracy. Put another way, in regions of pure tension the elastic membrane energy is convex and standard shell finite element methods perform well. Whereas in regions of pure compression, or mixed tension and compression, buckling occurs since there is a scale separation between the resistance of thin materials to compression and to bending, and the post-buckled state of the shell contains many highly nonlinear, complex wrinkles and creases, yet the coarse shape of the shell can nevertheless be approximated by ignoring the wrinkles and treating the shell as a collection of 1D curves aligned to the tensile stress directions on the shell. The detailed work plan will comprise work on tension-dominated surfaces, mixed-stress surfaces, and contact and friction.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.