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

薄壳模拟是科学计算中的基础工具。它们用于分析生物结构的行为，如囊泡和细胞膜，模拟织物和复合材料的变形，预测手术培训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起皱。这项研究将建立新的算法来模拟薄的弯曲材料的物理行为，如织物，纸张或金属片，具有前所未有的效率，而目前这种模拟是众所周知的困难和计算昂贵，因为薄的物体，如一张纸或布弯曲远比它们拉伸更容易，并且更喜欢以几何复杂的方式弯曲和起皱，而不是压缩。 此外，这种复杂性是不可预测和混乱的;即使是在相同的负载下，相同的物体，皱纹的确切模式也会有很大的不同。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，以确保薄部件不会彼此穿透，无论它们被推到一起有多快或多有力。为了使薄材料模拟在工程、设计和机器人应用中更实用，在性能至关重要的情况下，该项目将开发一种更有效的简化模型，用于模拟薄材料的变形。 从连续介质力学的张力场方法中借用的关键见解是，薄物体的行为由通过材料的张力线主导，而由压缩和弯曲引起的细尺度褶皱解决起来非常昂贵，但对物体的粗尺度形状或机械行为贡献甚微。 通过利用这一洞察力，并将计算工作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，并且标准壳有限元方法表现良好。 而在纯压缩或拉伸和压缩混合的区域中，屈曲发生，因为薄材料对压缩和弯曲的抵抗力之间存在尺度分离，并且壳体的屈曲后状态包含许多高度非线性的复杂褶皱和折痕，然而，通过忽略褶皱并将壳体视为对齐的1D曲线的集合，壳体上的拉应力方向。 详细的工作计划将包括张力主导表面、混合应力表面以及接触和摩擦方面的工作。该奖项反映了NSF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.