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。它们用于分析囊泡和细胞膜等生物结构的行为，模拟织物和复合材料的变形，在外科培训和可视化工具中预测皮肤的疤痕和皱纹，以及分析建筑物和车辆中结构元素的屈曲和皱折。这项研究将建立新的算法来模拟薄弯曲材料的物理行为，如织物、纸张或金属板材，具有前所未有的效率，而目前此类模拟是出了名的困难和计算昂贵，因为像一张纸或布这样的薄物体更容易弯曲，而不是拉伸，而且更喜欢以几何复杂的方式弯曲和折叠，而不是压缩。此外，这种复杂性是不可预测和混乱的；即使对于相同的物体，在相同的载荷下，皱纹的确切图案也可能有很大的不同。最后，预测薄物体在与自身和环境的摩擦接触下的行为尤其具有挑战性；由于薄的几何结构，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。为了使薄材料模拟更实用于工程、设计和机器人应用，其中性能至关重要，该项目将开发一个更高效、更简化的模型来模拟薄材料的变形。借鉴连续介质力学中的应力场方法的关键见解是，薄对象的行为由穿过材料的张力线主导，而由压缩和弯曲引起的精细皱纹解决起来非常昂贵，但对对象的粗略形状或力学行为几乎没有贡献。通过利用这一洞察力并将计算精力集中在跟踪和模拟张力线上，可以在不牺牲精度的情况下显著提高薄壳模拟的性能。换言之，在纯张力区，弹性膜能量是凸的，标准壳有限元方法的性能很好。然而，在纯压缩或拉伸和压缩混合的区域，由于薄材料的压缩和弯曲阻力之间存在尺度分离，因此发生屈曲，并且壳体的后屈曲状态包含许多高度非线性的、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一维曲线的集合，仍然可以近似地表示壳体的粗大形状。详细的工作计划将包括在以张力为主的表面、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智力优势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.