目的 布料仿真是计算机动画领域的一个研究热点和难点，基于粒子系统的模型快速高效却难以准确描述布料真实的物理特性，这些物理属性可以由连续介质的弹性模型利用有限元方法来实现，但所需网格单元较多，求解复杂且耗时。现有方法通常将布料建模为曲面和壳模型，对较厚布料的仿真能力较弱。针对这些常用模型在几何建模、物理真实性和仿真速度上难以兼顾的问题，提出了一种带厚度的布料仿真模型，能够更真实地表达不同厚度布料的形变行为，并利用等几何分析方法进行基于物理的快速动态仿真模拟。 方法 将有厚度的布料建模为三变量B样条体表示的薄板模型，利用等几何-伽辽金方法，推导出在给定边界条件下三维连续介质线弹性力学方程的弱形式，将B样条体的控制网格作为计算网格进行仿真分析，最后在隐式动力系统框架下对线性方程组直接求解。 结果 对本文布料模型进行了多方面的讨论和分析。首先，与几种主流的离散模型下的模拟效果进行了光滑度的对比，本文方法的NURBS建模有着明显的光滑性优势；并在不同自由度下比较了与经典有限元连续介质模型的计算时间，结果表明在模拟结果的RMSE小于0.04时，本文方法至多能减少90.23%的自由度和99.43%的计算时间；与同厚度面料的连续介质壳模型相比，计算时间能提高约30%。其次，对于经典场景如悬布、旗帜和接触问题，实现了逼真快速的动态模拟效果。此外，还展示和讨论了控制网格的密度、基函数的阶数和物理参数等的选择对模拟效果的影响，验证了通过适当的几何参数和物理参数，使用更高分辨率的控制网格或更高阶的基函数将会促进更多的模拟细节效果。 结论 本文提出的等几何方法模拟的厚布料模型是同时满足仿真效果和速度基本要求的有效方法，并且样条基能保持布料的光滑度，实现了更高的动态模拟效率。
Simulation of cloth with thickness based on isogeometric continuum elastic model
Ren Jingwen, Lin Hongwei, Dai Junfei(Zhejiang University)
Objective Cloth simulation is a research hotspot and difficulty in the field of computer animation. It can be seen in a variety of topics such as visual effects, game development, industrial design and interactive virtual environments, etc. With the demand of high quality of experience from the users today, various models have been proposed to improve the simulation performance. Although the models based on the particle system are fast and efficient, they are hard to accurately capture the behaviors in accordance with the real physical properties of the cloth. These physical properties can be described by the elastic model of continuum employing finite element method (FEM). However, solving with FEM in cloth simulation requires a number of degrees of freedom (or elements) and it is much more complex and time-consuming. Therefore, existing methods usually model cloth as a surface or a shell, which leads to weak simulation ability of thick cloth. In order to ease the awkwardness of compromising the geometric modeling, physical authenticity and computation speed in these models, a new cloth simulation model with thickness is proposed, which describes the deformation behavior of the cloth with different thickness more appropriately, and a fast dynamic physically based cloth simulation algorithm is carried out by isogeometric analysis. Isogeometric analysis (IGA) treats the physical domain (the geometry) as the computational field, avoiding the process of mesh generation that has approximating error and is time-consuming in classical FEM. It uses the nonuniform rational B-splines (NURBS) basis functions for both the physical domain and the solution field, which has the merit of higher order continuous solution than the traditional linear basis. The direct computation on the control mesh of the physical domain makes it more accurate and faster to solve the physical problems. Method The thick cloth is initially modeled as a very thin plate expressed by a trivariate B-spline solid. The weft direction and the warp direction of the fabric are free to design, while the basis for the thickness direction is usually linear to decrease the degrees of freedom, or higher order for thicker cloth. The deformations of the B-spline solid with elasticity will represent the behaviors of displacement of the cloth. Focused on IGA-Galerkin method, the weak form of the linear elastic equations of three-dimensional continuum is derived under the given boundary conditions. Then, the integrals in the weak form are computed by Gauss quadrature. By assembling the global stiffness matrix from the local element matrices, a linear system is yielded. The Dirichlet boundary conditions are dealt with Gauss elimination, and the pre-processing of the matching between the index of the local basis and the index of the global basis is also needed. The control mesh is simulated and analyzed as the computing mesh, so the unknowns in the linear system are the control coefficients in the B-splines-expressed solution of the displacement. The damping behavior caused by the dissipation of the energy of the system is modeled as the damping coefficient to the velocity of the control points to simplify the simulation. Considering the dynamic process, the time integration is realized by Newmark implicit method to allow a larger time step and enhance the stability of the system. Finally, the linear equations are solved directly due to the less degrees of freedom compared with other models, and the displacements, velocities and accelerations of the control points of the cloth are updated for each time step. The current state of the cloth can be visualized through the current positions of the control points. Result Our IGA continuum elastic cloth model is discussed in various aspects. Firstly, the smoothness of the simulation results is compared with commonly used discrete models, which displays significant smoothness advantage due to NURBS construction, and the computational time compared with the classical finite element continuum model is compared at different degrees of freedom, which shows that when the RMSE of the simulation results of the two models is less than 0.04, the method can reduce at most 90.23% of the degrees of freedom and 99.43% of the computational time. Compared with the continuum shell-based model of the same thickness, the computational time can be improved by about 30%. Secondly, for classic scenarios such as the hanging cloth and the falling flag, and contact problems, realistic and fast dynamic simulation effects are achieved. In addition, the influences of the density of the control mesh, the order of the basis function, and the selection of physical parameters on the simulation effect are also demonstrated and discussed. It was verified that using higher resolution control mesh or higher order basis with appropriate geometric and physical parameters will promote more detailed simulation effects. Conclusion In this study, we propose an IGA-Galerkin based cloth model with thickness to improve the physically based simulation, which is very intuitive and easy to implement. The trivariate B-splines-expressed model of a very thin plate can keep the smoothness of the cloth and uses less degrees of freedom and elements. The focus on solving the elastic equilibrium equations of the continuum makes it possible to match the simulated cloth with the fabrics in the real world. The proposed IGA-Galerkin cloth model is an effective approach to meet the basic requirements of simulation accuracy and speed, which achieves a higher dynamic physical simulation efficiency.