GAMES103 cloth 隐式积分法

简介

隐式积分法

显示积分简单而言是通过, 过去的求解未来. 而隐式积分, 简单而言是我要求解现在, 但是我的未知量中也有现在的未知量. 简单而言就是需要通过方程组的思想来进行求解.

参考文献

代码参考师弟 ~~

对于cloth问题, 简而言之, 有两个变量需要我们求解. 即速度v和位置x.

\[\left\{\begin{array}{l}\mathbf{x}^{[1]}=\mathbf{x}^{[0]}+\Delta t \mathbf{v}^{[0]}+\Delta t^{2} \mathbf{M}^{-1} \mathbf{f}^{[1]} \\ \mathbf{v}^{[1]}=\left(\mathbf{x}^{[1]}-\mathbf{x}^{[0]}\right) / \Delta t\end{array}\right.
\]

上述为了求解\(x^{[1]}\), 我们用到了\(\mathbf{f}^{[1]}\), 两个都是未来的变量, 需要通过方程组来进行求解.

数学家将隐式积分的问题转换成求解

\[\mathbf{x}^{[1]}=\operatorname{argmin} F(\mathbf{x}) \quad$ for $\quad F(\mathbf{x})=\frac{1}{2 \Delta t^{2}}\left\|\mathbf{x}-\mathbf{x}^{[0]}-\Delta t \mathbf{v}^{[0]}\right\|_{\mathbf{M}}^{2}+E(\mathbf{x})
\]

碰撞检测

使用基于脉冲法. 其实在lab2.pdf中也有讲解

\[\mathbf{v}_{i} \leftarrow \mathbf{v}_{i}+\frac{1}{\Delta t}\left(\mathbf{c}+r \frac{\mathbf{x}_{i}-\mathbf{c}}{\left\|\mathbf{x}_{i}-\mathbf{c}\right\|}-\mathbf{x}_{i}\right), \quad \mathbf{x}_{i} \leftarrow \mathbf{c}+r \frac{\mathbf{x}_{i}-\mathbf{c}}{\left\|\mathbf{x}_{i}-\mathbf{c}\right\|}
\]

求解函数

\[\left(\frac{1}{\Delta t^{2}} \mathbf{M}+\mathbf{H}\left(\mathbf{x}^{(k)}\right)\right) \Delta \mathbf{x}=-\frac{1}{\Delta t^{2}} \mathbf{M}\left(\mathbf{x}^{(k)}-\mathbf{x}^{[0]}-\Delta t \mathbf{v}^{[0]}\right)+\mathbf{f}\left(\mathbf{x}^{(k)}\right)
\]

中的\(\Delta \mathbf{x}\)来计算新的坐标和位移

A

A 就是 \(\left(\frac{1}{\Delta t^{2}} \mathbf{M}+\mathbf{H}\left(\mathbf{x}^{(k)}\right)\right)\)

其中的 hessian 矩阵比较难求. 作者通过简化A

A =

\[\frac{1}{\Delta t^{2}} m_{i}+4 k
\]

G 梯度

G 其实是 -b

为什么梯度就是 -b 呢??

因为作者使用的是牛顿迭代法

牛顿迭代法有一个特性

\[0=F^{\prime}(x) \approx F^{\prime}\left(x^{(k)}\right)+F^{\prime \prime}\left(x^{(k)}\right)\left(x-x^{(k)}\right)
\]

一个函数的一阶导数等于其一阶导数+二阶导数×偏差.

也就是 \(-F^{\prime}(x) = F^{\prime \prime}\left(x^{(k)}\right)\left(x-x^{(k)}\right)\)

其中\(-F^{\prime}(x)\)

就是

\[\nabla F\left(\mathbf{x}^{(k)}\right)=\frac{1}{\Delta t^{2}} \mathbf{M}\left(\mathbf{x}^{(k)}-\mathbf{x}^{[0]}-\Delta t \mathbf{v}^{[0]}\right)-\mathbf{f}\left(\mathbf{x}^{(k)}\right)
\]

而 b 就是 \(-F^{\prime}(x)\)

G 中包含了 \(\mathbf{f}\left(\mathbf{x}^{(k)}\right)\) 包含两个力, 一个是重力, 另一个是弹簧的弹力.

image

核心公式

code

using System.Collections;
using System.Collections.Generic;
using UnityEngine;

public class implicit_model : MonoBehaviour
{
	float 		t 		= 0.0333f;
	float 		mass	= 1;
	float		damping	= 0.99f;
	float 		rho		= 0.995f;
	float 		spring_k = 8000;
	int[] 		E; // 边对
	float[] 	L; // 初始边长度
	Vector3[] 	V;
	Vector3 g = new Vector3(0, -9.8f, 0); // 重力
	float r = 2.7f; // 球半径

    // Start is called before the first frame update
    void Start()
    {
		Mesh mesh = GetComponent<MeshFilter> ().mesh;

		//Resize the mesh.
		int n=21;
		Vector3[] X  	= new Vector3[n*n];
		Vector2[] UV 	= new Vector2[n*n];
		int[] triangles	= new int[(n-1)*(n-1)*6];
		for(int j=0; j<n; j++)
		for(int i=0; i<n; i++)
		{
			X[j*n+i] =new Vector3(5-10.0f*i/(n-1), 0, 5-10.0f*j/(n-1));
			UV[j*n+i]=new Vector3(i/(n-1.0f), j/(n-1.0f));
		}
		int t=0;
		for(int j=0; j<n-1; j++)
		for(int i=0; i<n-1; i++)
		{
			triangles[t*6+0]=j*n+i;
			triangles[t*6+1]=j*n+i+1;
			triangles[t*6+2]=(j+1)*n+i+1;
			triangles[t*6+3]=j*n+i;
			triangles[t*6+4]=(j+1)*n+i+1;
			triangles[t*6+5]=(j+1)*n+i;
			t++;
		}
		mesh.vertices=X;
		mesh.triangles=triangles;
		mesh.uv = UV;
		mesh.RecalculateNormals ();

		//Construct the original E
		int[] _E = new int[triangles.Length*2];
		for (int i=0; i<triangles.Length; i+=3)
		{
			_E[i*2+0]=triangles[i+0];
			_E[i*2+1]=triangles[i+1];
			_E[i*2+2]=triangles[i+1];
			_E[i*2+3]=triangles[i+2];
			_E[i*2+4]=triangles[i+2];
			_E[i*2+5]=triangles[i+0];
		}
		//Reorder the original edge list
		for (int i=0; i<_E.Length; i+=2)
			if(_E[i] > _E[i + 1])
				Swap(ref _E[i], ref _E[i+1]);
		//Sort the original edge list using quicksort
		Quick_Sort (ref _E, 0, _E.Length/2-1);

		int e_number = 0;
		for (int i=0; i<_E.Length; i+=2)
			if (i == 0 || _E [i + 0] != _E [i - 2] || _E [i + 1] != _E [i - 1])
					e_number++;

		E = new int[e_number * 2];
		for (int i=0, e=0; i<_E.Length; i+=2)
			if (i == 0 || _E [i + 0] != _E [i - 2] || _E [i + 1] != _E [i - 1])
			{
				E[e*2+0]=_E [i + 0];
				E[e*2+1]=_E [i + 1];
				e++;
			}

		L = new float[E.Length/2];
		for (int e=0; e<E.Length/2; e++)
		{
			int v0 = E[e*2+0];
			int v1 = E[e*2+1];
			L[e]=(X[v0]-X[v1]).magnitude;
		}

		V = new Vector3[X.Length];
		for (int i=0; i<V.Length; i++)
			V[i] = new Vector3 (0, 0, 0);
    }

    void Quick_Sort(ref int[] a, int l, int r)
	{
		int j;
		if(l<r)
		{
			j=Quick_Sort_Partition(ref a, l, r);
			Quick_Sort (ref a, l, j-1);
			Quick_Sort (ref a, j+1, r);
		}
	}

	int  Quick_Sort_Partition(ref int[] a, int l, int r)
	{
		int pivot_0, pivot_1, i, j;
		pivot_0 = a [l * 2 + 0];
		pivot_1 = a [l * 2 + 1];
		i = l;
		j = r + 1;
		while (true)
		{
			do ++i; while( i<=r && (a[i*2]<pivot_0 || a[i*2]==pivot_0 && a[i*2+1]<=pivot_1));
			do --j; while(  a[j*2]>pivot_0 || a[j*2]==pivot_0 && a[j*2+1]> pivot_1);
			if(i>=j)	break;
			Swap(ref a[i*2], ref a[j*2]);
			Swap(ref a[i*2+1], ref a[j*2+1]);
		}
		Swap (ref a [l * 2 + 0], ref a [j * 2 + 0]);
		Swap (ref a [l * 2 + 1], ref a [j * 2 + 1]);
		return j;
	}

	void Swap(ref int a, ref int b)
	{
		int temp = a;
		a = b;
		b = temp;
	}

	void Collision_Handling()
	{
		Mesh mesh = GetComponent<MeshFilter> ().mesh;
		Vector3[] X = mesh.vertices;
		GameObject sphere = GameObject.Find("Sphere");
		Vector3 c = sphere.transform.position;
		//Handle colllision.
		for(int i=0; i<X.Length; i++)
        {
			if (i == 0 || i == 20) continue;
			float distance = (X[i] - c).magnitude;
			if(distance < r)
            {
				V[i] += (c + r * (X[i] - c) / distance - X[i]) / t; // 需要一个很小的速度偏差
				X[i] = c + r * (X[i] - c) / distance;
			}
        }
		mesh.vertices = X;
	}

	void Get_Gradient(Vector3[] X, Vector3[] X_hat, float t, Vector3[] G)
	{
		//Momentum and Gravity.
		for(int i=0; i<G.Length; i++)
        {
			G[i] = (X[i] - X_hat[i]) * mass / (t * t) - g * mass;
        }
		//Spring Force.

		for(int ei = 0; ei < L.Length; ei++)
        {
			int i = E[ei * 2];
			int j = E[ei * 2 + 1];
			Vector3 xi_xj = X[i] - X[j];
			float len_ij = L[ei] / xi_xj.magnitude;
			Vector3 t_ij = spring_k * (1 - len_ij) * xi_xj;
			G[i] += t_ij;
			G[j] -= t_ij;
        }

	}

    // Update is called once per frame
	void Update ()
	{
		Mesh mesh = GetComponent<MeshFilter> ().mesh;
		Vector3[] X 		= mesh.vertices;
		Vector3[] last_X 	= new Vector3[X.Length];
		Vector3[] X_hat 	= new Vector3[X.Length]; // x = x + vi * t
		Vector3[] G 		= new Vector3[X.Length]; // Gradient
		Vector3[] delta_x = new Vector3[X.Length];
		Vector3[] last_delta_x = new Vector3[X.Length];
		Vector3[] old_delta_x = new Vector3[X.Length];
		//Initial Setup.
		// 更新速度和位置
		for (int i=0; i<X.Length; i++)
        {
			V[i] *= damping;
			last_X[i] = X[i];
			X_hat[i] = X[i] + t * V[i];
			X[i] = X_hat[i];
        }

		for(int k=0; k<32; k++)
		{
			Get_Gradient(X, X_hat, t, G);

			// chebyshev
			float A = mass / (t * t) + 4 * spring_k;
			float w = 0;
			float rest = 0;
			for(int i = 0; i<X.Length; i++)
            {
				delta_x[i] = new Vector3(0, 0, 0);
				last_delta_x[i] = new Vector3(0, 0, 0);
            }

			// iter
			for(int j=0; j<50; j++)
            {
				rest = 0;
				for(int i=0; i<X.Length; i++)
                {
					rest += (-delta_x[i] * A - G[i]).sqrMagnitude;
                }
				if(Mathf.Sqrt(rest) < 0.00001)
                {
					Debug.Log("Iter :" + k);
					break;
                }

				if(j == 0)
                {
					w = 1;
                }else if(j == 1)
                {
					w = 2 / (2 - rho * rho);
                }else
                {
					w = 4 / (4 - rho * rho * w);
                }

				for(int i=0; i<X.Length; i++)
                {
					old_delta_x[i] = delta_x[i];
					delta_x[i] += -delta_x[i] - G[i] / A;
					delta_x[i] = w * delta_x[i] + (1 - w) * last_delta_x[i];
					last_delta_x[i] = old_delta_x[i];
                }
            }
			Debug.Log("rest:" + rest);
			//Update X by gradient.
			for(int i=0; i<X.Length; i++)
            {
				if (i == 0 || i == 20) continue;
				X[i] += delta_x[i];
            }
		}

		//Finishing.
		for(int i=0; i<V.Length; i++)
        {
			V[i] += (X[i] - X_hat[i]) / t;
        }
		mesh.vertices = X;

		Collision_Handling ();

		mesh.RecalculateNormals ();
	}
}