果冻物理学3D

将对象的体积分成许多小块(元素)，通常为 tetrahedra .让我们将这些元素的整个集合称为网格.您实际上将要制作该网格的两个副本.标记一个为静止"网格，另一个标记为世界"网格.我会告诉你为什么.

Break the volume of your object up into many small pieces (elements), usually tetrahedra. Let's call the entire collection of these elements the mesh. You'll actually want to make two copies of this mesh. Label one the "rest" mesh, and the other the "world" mesh. I'll tell you why next.

对于您的世界网格中的每个四面体，测量其相对于其对应的其余四面体的变形程度.将其变形的量度称为应变".通常，这是通过首先测量所谓的变形梯度(通常表示为 F )来实现的.有几个 好 文件，其中介绍了如何执行此操作.一旦有了 F ，定义应变( e )的一种非常典型的方法是: e = 1/2( F ^ T * F )- I .这被称为格林氏菌株.它是旋转不变的，因此非常方便.

For each tetrahedron in your world mesh, measure how deformed it is relative to its corresponding rest tetrahedron. The measure of how deformed it is is called "strain". This is typically accomplished by first measuring what is known as the deformation gradient (often denoted F). There are several good papers that describe how to do this. Once you have F, one very typical way to define the strain (e) is: e = 1/2(F^T * F) - I. This is known as Green's strain. It is invariant to rotations, which makes it very convenient.

使用您要模拟的材料(明胶，橡胶，钢等)的特性，并使用在上述步骤中测量的应变，得出"应力"，每个四面体.

Using the properties of the material you are trying to simulate (gelatin, rubber, steel, etc.), and using the strain you measured in the step above, derive the "stress" of each tetrahdron.

对于每个四面体，访问每个节点(顶点，角，点(这些都表示相同的东西))，并对三个三角形面的面积加权法向矢量(其余形状)取平均值节点.将四面体的应力乘以该平均向量，并且由于该四面体的应力而在该节点上作用有弹力.当然，每个节点都可能属于多个四面体，因此您需要对这些力进行汇总.

For each tetrahedron, visit each node (vertex, corner, point (these all mean the same thing)) and average the area-weighted normal vectors (in the rest shape) of the three triangular faces that share that node. Multiply the tetrahedron's stress by that averaged vector, and there's the elastic force acting on that node due to the stress of that tetrahedron. Of course, each node could potentially belong to multiple tetrahedra, so you'll want to be able to sum up these forces.

集成！有简单的方法可以做到这一点，而有困难的方法.无论哪种方式，您都希望遍历世界网格中的每个节点，并用其质量除以其力来确定其加速度.从这里开始的简单方法是:

Integrate! There are easy ways to do this, and hard ways. Either way, you'll want to loop over every node in your world mesh and divide its forces by its mass to determine its acceleration. The easy way to proceed from here is to:

• 将其加速度乘以较小的时间值 dt .这样可以改变速度 dv .
• 在节点的当前速度上添加 dv 以获得新的总速度.
• 将该速度乘以 dt 即可得到位置变化 dx .
• 将 dx 添加到节点的当前位置以获取新位置.

• Multiply its acceleration by some small time value dt. This gives you a change in velocity, dv.
• Add dv to the node's current velocity to get a new total velocity.
• Multiply that velocity by dt to get a change in position, dx.
• Add dx to the node's current position to get a new position.

这种方法称为明确进行欧拉集成.您必须使用非常小的 dt 值才能使其正常工作而不会炸裂，但是实现起来如此容易，以至于它可以很好地作为起点.

This approach is known as explicit forward Euler integration. You will have to use very small values of dt to get it to work without blowing up, but it is so easy to implement that it works well as a starting point.

根据需要重复执行第2步到第5步.

Repeat steps 2 through 5 for as long as you want.

我遗漏了很多细节和花哨的附加功能，但希望您能推断出我遗漏的很多内容. 这是我第一次使用的一些说明的链接时间我做到了.该网页包含一些有用的伪代码以及一些相关材料的链接.

I've left out a lot of details and fancy extras, but hopefully you can infer a lot of what I've left out. Here is a link to some instructions I used the first time I did this. The webpage contains some useful pseudocode, as well as links to some relevant material.

sealab.cs.utah.edu/课程/CS6967-F08/Project-2/

以下链接也非常有用:

sealab.cs.utah.edu /Courses/CS6967-F08/FE-notes.pdf

这是一个非常有趣的话题，祝您好运！如果您遇到困难，请给我留言.

This is a really fun topic, and I wish you the best of luck! If you get stuck, just drop me a comment.