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题名:
树在风中的摇曳-基于物理的计算机动画
作者: 冯金辉
答辩日期: 1999
专业: 计算机软件
授予单位: 中国科学院软件研究所
授予地点: 中国科学院软件研究所
学位: 博士
关键词: 基于物理的计算机动画 ; 动态模拟 ; 树木 ; 树叶 ; 非线性力学 ; 波动
摘要: 树木在自然景物的构成中占有相当重要的地位,迷人的风景离不开树木的点辍。在相当长的一段时间内,人们投入了很大的精力研究树木的造型和绘制算法。尽管有许多成功的工作,但在影视、广告、游戏、虚拟现实等领域纹理映射仍然是创建树木的主要手段,而树木的动态模拟则更是很少有人涉及。因此,本文对树木的动态模拟进行了研究,提出了一系列行之有效的算法,并基于此制作了动画实例,取得了令人满意的效果。树是一种复杂的自相似结构。从树根到树叶,其物理性质诸如杨氏模量、密度、半径、长度等变化相当大,其相应的力学性质也从线性过程到非线性。这给树的动态模拟带来了极大的困难。针对树不同部分的不同物理性质,我们把树分为三个部分:固定树条、可动枝条以及波动树条。固定枝条 包括主干及它的绝大多数分支,它们在通常风力作用下的运动一般是不为人眼所觉察的,因此在模拟过程中不需要计算它们的运动。我们把这些枝条定义为固定枝条。可动枝条 包括树的末端枝条和它们的一些上级枝条,它们比固定枝条要软,其运动可以很明显地被人眼所觉察。一般来说它们的位移都比较大因而运动是非线性的,经典理论不能很好地解决。波动枝条 通常指树的末端枝条,象柳树的末端枝条在风吹动下的运动实际上是一种波动,在树条中有机械波在传播。可动枝条的运动属于非线性运动,基于非线性理论,我们把可动枝条简化为一条不可伸长的曲杆,并用折线段来逼近。算法首先针对某一折线段进行处理。因为枝条不可伸长,那么它的变形就只有扭转和弯曲,实际上就是相对其初始状态的转动,可采用欧拉角来度量这种转动。在该线段所在局部标架下,如果取沿线段的弧长s作为自变量,该线段的变形即可用欧拉角关于弧长的二阶常微分方程为表示。该常微分方程在初始条件已知的情况下可直接进行数值积分求解。对于整条树枝,若根部边界条件已知,则可首先对其第一个折线段进行数值求解,获得首段尾部边界条件后,把它作为下一段的初始条件即可继续对下一段进行数值求解,如此连续下去,整条树枝即得解决。求解关于枝条的常微分方程需要12个边界条件,而我们已知的边界条件却分布在枝条的两端:在根部和尾部各六个。这样我们实施所谓发射法,即从根部开始首先假定一个边界软件,进行积分至尾部,与尾部边界条件进行比较,然后根据比较的结果对根部边界条件进行修正,再行积分直至尾部边界条件获得满足为止。上述数值方法很容易处理具有复杂物理、几何特性的枝条,因为我们采用折线段来逼近枝条,当折线段取得足够小时对曲线的逼近效果相当理想。 我们认为在每一线段内树枝的物理性质诸如杨氏模量、密度、半径等是常量,而在不同线段之间各物理量是不相同的,这样从根部到尾部枝条物理性质的变化就得到了很好的模拟。对于树的整体运动,我们首先从主干开始对其进行深度优化遍历,并依据深度优先的规则对可动枝条进行顺序数值求解。这样在对树进行深度优先遍历的同时即可对整棵树进行运动计算,这就是我们提出的深度优先的数值积分遍历算法。在分支连接处参数满足平衡条件,在这个条件的约束下,不同分支间的运动是一致的、协调的。对于波动枝条,其边界条件为一端固定一端自由,这样即可从基本波动方程出发直接获得其运动。我们以杨树叶为例分析研究了树叶的运动,把其分为叶柄和叶面两种转动的合成。把叶面抽象为一个具有质量的点,该点与叶柄构成弹簧-质量系统。该系统在叶柄曲线与风力方向所构成的平面内转动。叶面在跟随叶柄运动的同时以叶柄为轴绕叶柄转动。我们构造了两种风力模型:随风和稳定风。在这两个风力模型的作用下,以柳树和杨树为例,本文算法生成了它们在风中的运动:柳树具有全部固定枝条、可动枝条和波动枝条,尤其是柳树的末端枝条在风中的摇曳是相当动人的景象,很有代表性;而杨树则代表了一类具有明显树叶摆动的树种。这些典型运动我们都进行了模拟,并制作了动画实例,效果很好。
英文摘要: Trees, in particular the dynamic waving of trees plays a very important role in the natural scenery of landscape. Unfortunately, due to the high level complexity of the structure of trees and the highly varying properties of trees' components, physically-based simulation of trees' motion becomes extremely a difficult task. As an effort in tackling such a problem, a solution is proposed and implemented in the paper. By this solution in the control and simulation of the motion of trees blown by wind, the behavior of moving branches of trees is simulated based on the contemporary non-linear mechanics, and that of waving branches based on the principle of mechanical wave. Though the structure of a tree is clear, from trunk, big branches to little branches till leaves, organized as a format of what conventionally called the tree structure, the physical properties of trees are however unusually comprehensive. The branches spread out in very random directions with random distribution of density, and the shape of the branches is also highly varying, some in curve and some even in helix. The primary branches are often very thick and hard, while little or end branches are very thin and soft. The radius and the elasticity along the branches are decreasing gradually, and thus the branches at different position perform different physical behavior in mechanics. Furthermore, all those features mentioned are highly varying with the type of tree species. In the approach we proposed, a tree is mainly classified into three parts (take willow as an example) according to their physical properties: non-movable branches, movable branches and waving branches. To those hard branches, including the trunk and its most direct children, their movements under the normal condition of wind are hardly detected by human being. We refer to those branches as non-movable branches. This implies that it is unnecessary to calculate the displacement of those branches, though an accurate solution is always viable in terms of classical mechanics. Movable branches include the tree's last branches and some of their parents. They are softer than non-movable branches, so under the normal condition of wind their displacement would be large enough to be detected. The behavior of this kind of branches in their physical motion is non-linear so the classical mechanics is unable to give the accurate solution. In our solution, a new method is provided to solve the large three-dimensional displacement in this case. For waving branches that are soft and long, their movement under the wind is actually the wave movement and the wave is propagating through the branches. Therefore, the principle of transverse mechanical wave could be applied to solving the problem. A single branch could be assumed as an inextensible rod, so the branch would deform only through bending and twisting. In fact, these kinds of deformations are rotation about their initial state, and could be measured through three Eular angles. In order to perform the numerical computation, we divide a branch into a series of discrete segments. Under the local coordinate of such a segment, its movement becomes three second order differential equations of the Euler angles with their differentiation with respect to arc length. These three equations could be integrated directly using numerical method.
语种: 中文
内容类型: 学位论文
URI标识: http://ir.iscas.ac.cn/handle/311060/5744
Appears in Collections:中科院软件所

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Recommended Citation:
冯金辉. 树在风中的摇曳-基于物理的计算机动画[D]. 中国科学院软件研究所. 中国科学院软件研究所. 1999-01-01.
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