N体运动的程序模拟

这依然是与《三体》有关的一篇文章。空间中三个星体在万有引力作用下的运动被称之为三体问题，参见我的上一篇文章：三体运动的程序模拟。而这一节，对三体问题进行了扩展，实现了空间中N个星体在万有引力下的运动模拟。

程序中使用了两个物理定律：

（1）万有引力定律

这是牛顿发现的:任意两个质点有通过连心线方向上的力相互吸引。该引力的大小与它们的质量乘积成正比，与它们距离的平方成反比，与两物体的化学本质或物理状态以及中介物质无关。

以数学表示为：

其中：

• : 两个物体之间的引力
• : 万有引力常数
• : 物体1的质量
• : 物体2的质量
• : 两个物体之间的距离

依照国际单位制，F的单位为牛顿(N)，m₁和m₂的单位为千克(kg)，r 的单位为米(m)，常数G近似地等于6.67 × 10⁻¹¹ N m² kg⁻²（牛顿米的平方每千克的平方）。当然程序中G不可能设置为这么小的一个数,用户可以自己设置万有引力常数,以实现合理的运动.

(2)势能与动能守恒定律

引力位能或引力势能是指物体因为大质量物体的万有引力而具有的位能，其大小与其到大质量的距离有关。

其中G为万有引力常数，M、m分别为两物体质量，r为两者距离。

动能，简单的说就是指物体因运动而具有的能量。数值上等于（1/2）Mv^2. 动能是能量的一种，它的国际单位制下单位是焦耳(J)，简称焦。

势能与动能守恒即在整个运动过程中,其总能量是不变的,即N体的动能与势能之和为固定值.

本来我只打算使用万有引力定律,没有考虑势能动能守恒定律.但程序运行后发现,由于没有使用微积分,即使采样间隔时间再小,误差也很大.这让我想起当年学大学物理的时候,大一上学期开高数,然后下学期开了物理.现在对大学物理讲的东西,只记得是用高数中的微积分解物理问题.可那时,我高数学得就是个渣,所以物理也没办法学好.当然就算物理学好了,现在也早忘光了.没有用微积分,这个程序中算法误差不小,模拟一下吧,不要太计较.

核心代码也发一下,希望有兴趣的人可以对其做出优化:

 /****************************************************************

   File name   :  NBodyKernel.h
   Author      :  叶峰
   Version     :  1.0a
   Create Date :  2014/09/19
   Description :  

 *****************************************************************/

 // --------------------------------------------------------------------------------------

 #ifndef _NBodyKernel_H_
 #define _NBodyKernel_H_

 // INCLUDES -----------------------------------------------------------------------------

 #include "..\..\..\Why\YCommon_h\WhyMath.h"
 #include "..\..\..\Why\YInterface\Others\YdD3D9Math.h"

 // --------------------------------------------------------------------------------------

 #define YD_ORBIT_VERTICES_COUNT         1024

 // --------------------------------------------------------------------------------------

 class Star
 {
 public:
     Vector3 vPosition;          // 位置
     Vector3 vVelocity;          // 速度
     Vector3 vForce;             // 受力
     Yreal   fWeight;            // 质量
     Yreal   fRadiusScale;       // 缩放
     Yreal   fKineticEnergy;     // 动能
     Ycolor  color;              // 颜色
     Ybool   bOrbitVisible;      // 轨迹可见
     Yuint   id;
     Vector3 verticesOrbit[YD_ORBIT_VERTICES_COUNT];
     Star*   pNext;

     Star()
     {
         fWeight = 0.0f;
         fRadiusScale = 1.0f;
         fKineticEnergy = 0.0f;
         color = 0xffffffff;
         bOrbitVisible = YD_FALSE;
         id = -;
         memset(verticesOrbit, , sizeof(verticesOrbit));
         pNext = NULL;
     }

     // 根据速度计算动能
     void CalculateKineticEnergyByVelocity()
     {
         Yreal sqv = D3DXVec3LengthSq(&vVelocity);
         fKineticEnergy = 0.5f*fWeight*sqv;
     }

     // 根据动能计算速度
     void CalculateVelocityByKineticEnergy()
     {
         float v = sqrt(*fKineticEnergy/fWeight);
         float w = D3DXVec3Length(&vVelocity);
         vVelocity *= v/w;
     };
 };

 // --------------------------------------------------------------------------------------

 class CNBodyKernel
 {
 public:
     CNBodyKernel();

     ~CNBodyKernel();

     // 更新N体
     void        UpdateNBody(Yreal deltaTime);

     // 清空N体
     void        Clear();

     // 设置引力系数
     void        SetGravitationCoefficient(Yreal c);

     // 获得引力系数
     Yreal       GetGravitationCoefficient() const
     {
         return m_fGravitationCoefficient;
     }

     // 返回星体数目
     Yuint       GetStarsCount() const
     {
         return m_starsCount;
     }

     // 返回星体链表
     const Star* GetStarsList() const
     {
         return m_starsList;
     }

     // 返回星体对象
     const Star* GetStar(Yuint index) const;

     // 移除星体
     bool        RemoveStar(Yuint index);

     // 添加星体
     void        AddStar(
         const Vector3& vPosition,
         const Vector3& vVelocity,
         const Yreal fWeight,
         Ycolor  color);

     // 设置星体的重量
     void        SetStarWeight(Yuint index, Yreal weight);

     // 设置星体的轨迹是否可见
     void        SetStarOrbitVisible(Yuint index, bool visible);

     // 设置星体的颜色
     void        SetStarColor(Yuint index, Ycolor color);

     Yuint       GetCurrentOrbit() const
     {
         return m_currentOrbit;
     }

     const Vector3* GetStarPosition(Yuint index) const;

     Yreal       GetStarWeight(Yuint index) const;

     bool        GetStarOrbitVisible(Yuint index) const;

     Ycolor      GetStarColor(Yuint index) const;

     Yuint       GetStarID(Yuint index) const;

     void        GetSpaceCenter(Vector3& vCenter) const;

     void        GetWeightCenter(Vector3& vCenter) const; 

 private:
     // 更新星体的位置,速度,动能
     void        UpdateStar(Star& star, Yreal t);

     // 根据动能重新计算速度
     void        RecalculateStarVelocity(Star& star);

     // 计算每个星体的当前受力
     void        CalculateStarsForce();

     // 计算总势能
     void        CalculateAmountOfPotentialEnergy();

     // 计算总动能
     void        CalculateAmountOfKineticEnergy();

     // 计算总能量
     void        CalculateAmountEnergy()
     {
         CalculateAmountOfPotentialEnergy();
         CalculateAmountOfKineticEnergy();
         m_fAmountEnergy = m_fAmountOfPotentialEnergy + m_fAmountOfKineticEnergy;
     }

 private:
     Yreal m_fGravitationCoefficient;    // 引力系数
     Yreal m_fAmountOfPotentialEnergy;   // 当前总势能
     Yreal m_fAmountOfKineticEnergy;     // 当前总动能
     Yreal m_fAmountEnergy;              // 当前能量m_fAmountOfPotentialEnergy + m_fAmountOfKineticEnergy;

     Star* m_starsList;                  // N体链表
     Yuint m_starsCount;                 // N体数目

     Yuint m_currentOrbit;
 };

 // --------------------------------------------------------------------------------------

 #endif

 /****************************************************************

   File name   :  NBodyKernel.cpp
   Author      :  叶峰
   Version     :  1.0a
   Create Date :  2014/09/19
   Description :  

 *****************************************************************/

 // --------------------------------------------------------------------------------------

 #include "..\..\..\Why\YCommon_h\WhyMemoryDefines.h"
 #include "NBodyKernel.h"
 #include <assert.h>

 // --------------------------------------------------------------------------------------

 #define YD_ESP                          0.00001f

 // --------------------------------------------------------------------------------------

 CNBodyKernel::CNBodyKernel()
 {
     m_fGravitationCoefficient = 100.0f;
     m_fAmountOfPotentialEnergy = 0.0f;
     m_fAmountOfKineticEnergy = 0.0f;
     m_fAmountEnergy = 0.0f;

     m_starsList = NULL;
     m_starsCount = ;

     m_currentOrbit = ;
 } 

 CNBodyKernel::~CNBodyKernel()
 {
     Clear();
 }

 // 清空N体
 void        CNBodyKernel::Clear()
 {
     Star* pDel = m_starsList;
     Star* pNext;
     while (pDel)
     {
         pNext = pDel->pNext;
         YD_DELETE(pDel);
         pDel = pNext;
     }

     m_starsList = NULL;
     m_starsCount = ;
 }

 // 获得引力系数
 void        CNBodyKernel::SetGravitationCoefficient(Yreal c)
 {
     m_fGravitationCoefficient = c;

     // 计算总能量
     CalculateAmountEnergy();
 }

 // 计算每个星体的当前受力
 void        CNBodyKernel::CalculateStarsForce()
 {
     // 清空所有引力
     Star* pStar = m_starsList;
     while (pStar)
     {
         pStar->vForce = Vector3(0.0f, 0.0f, 0.0f);
         pStar = pStar->pNext;
     }

     // 计算引力
     Star* pCurrent = m_starsList;
     Star* pNext;
     while (pCurrent)
     {
         pNext = pCurrent->pNext;
         while (pNext)
         {
             Vector3 vAB = pNext->vPosition - pCurrent->vPosition;
             float disSqAB = D3DXVec3LengthSq(&vAB);
             if (disSqAB < YD_ESP)
             {
                 disSqAB = YD_ESP;
             }
             float disAB = sqrt(disSqAB);
             Vector3 vForceDir = vAB/disAB;
             Yreal fForce = m_fGravitationCoefficient*pCurrent->fWeight*pNext->fWeight/disSqAB;
             Vector3 vForce = vForceDir*fForce;

             pCurrent->vForce += vForce;
             pNext->vForce -= vForce;

             pNext = pNext->pNext;
         }
         pCurrent = pCurrent->pNext;
     }
 }

 void        CNBodyKernel::UpdateStar(Star& star, Yreal t)
 {
     // 加速度
     const Vector3 acc = star.vForce/star.fWeight;

     star.vPosition.x = star.vPosition.x + star.vVelocity.x*t + 0.5f*acc.x*t*t;
     star.vPosition.y = star.vPosition.y + star.vVelocity.y*t + 0.5f*acc.y*t*t;
     star.vPosition.z = star.vPosition.z + star.vVelocity.z*t + 0.5f*acc.z*t*t;

     star.vVelocity.x += acc.x*t;
     star.vVelocity.y += acc.y*t;
     star.vVelocity.z += acc.z*t;

     // 重新计算动能
     star.CalculateKineticEnergyByVelocity();
 }

 void        CNBodyKernel::UpdateNBody(Yreal deltaTime)
 {
     // 计算每个星体的当前受力
     CalculateStarsForce();

     // 更新星体的位置与速度
     Star* pStar = m_starsList;
     while (pStar)
     {
         UpdateStar(*pStar, deltaTime);
         pStar = pStar->pNext;
     }

     // 能量守恒
     CalculateAmountOfKineticEnergy();
     CalculateAmountOfPotentialEnergy();

     // 理论上的动能
     Yreal fKineticEnergy = m_fAmountEnergy - m_fAmountOfPotentialEnergy;
     if (fKineticEnergy < 0.0f)
     {
         fKineticEnergy = 0.0f;
     }

     Yreal r = fKineticEnergy/m_fAmountOfKineticEnergy;
     pStar = m_starsList;
     while (pStar)
     {
         if (r > YD_ESP)
         {
             pStar->fKineticEnergy *= r;
         }
         else
         {
             pStar->fKineticEnergy = 1.0f;
         }
         // 根据动能计算速度
         pStar->CalculateVelocityByKineticEnergy();
         pStar = pStar->pNext;
     }

     // 更新轨迹点
     pStar = m_starsList;
     while (pStar)
     {
         pStar->verticesOrbit[m_currentOrbit] = pStar->vPosition;
         pStar = pStar->pNext;
     }
     m_currentOrbit++;
     if (m_currentOrbit >= YD_ORBIT_VERTICES_COUNT)
     {
         m_currentOrbit = ;
     }
 }

 // 返回星体对象
 const Star* CNBodyKernel::GetStar(Yuint index) const
 {
     if (index >= m_starsCount)
     {
         return NULL;
     }

     const Star* pStar = m_starsList;
     while (index && pStar)
     {
         index--;
         pStar = pStar->pNext;
     }

     return pStar;
 }

 // 移除星体
 bool        CNBodyKernel::RemoveStar(Yuint index)
 {
     if (index >= m_starsCount)
     {
         return false;
     }

     if (index == )
     {
         Star* pStar = m_starsList->pNext;
         YD_DELETE(m_starsList);
         m_starsList = pStar;
     }
     else
     {
         Star* pStar = m_starsList;
         Yuint t = index - ;
         while (t && pStar)
         {
             t--;
             pStar = pStar->pNext;
         }
         assert(pStar);

         Star* pDel = pStar->pNext;
         pStar->pNext = pDel->pNext;
         YD_DELETE(pDel);
     }

     m_starsCount--;

     Yuint id = ;
     Star* pStar = m_starsList;
     while (pStar)
     {
         pStar->id = id;
         id++;
         pStar = pStar->pNext;
     }

     // 重算能量
     CalculateAmountEnergy();

     return true;
 }

 // 添加星体
 void        CNBodyKernel::AddStar(
                     const Vector3& vPosition,
                     const Vector3& vVelocity,
                     const Yreal fWeight,
                     Ycolor  color)
 {
     Star* pNewStar = YD_NEW(Star);
     pNewStar->vPosition = vPosition;
     pNewStar->vVelocity = vVelocity;
     pNewStar->fWeight = fWeight;
     pNewStar->fRadiusScale = yf_pow(fWeight, 1.0f/3.0f);
     pNewStar->CalculateKineticEnergyByVelocity();
     pNewStar->color = color;
     pNewStar->id = m_starsCount;
     for (Yuint i = ; i < YD_ORBIT_VERTICES_COUNT; i++)
     {
         pNewStar->verticesOrbit[i] = vPosition;
     }

     if (!m_starsList)
     {
         m_starsList = pNewStar;
     }
     else
     {
         Star* pStar = m_starsList;
         while (pStar->pNext)
         {
             pStar = pStar->pNext;
         }
         pStar->pNext = pNewStar;
     }

     m_starsCount++;

     // 重算能量
     CalculateAmountEnergy();
 }

 // 设置星体的重量
 void        CNBodyKernel::SetStarWeight(Yuint index, Yreal weight)
 {
     if (index >= m_starsCount)
     {
         return;
     }

     Star* pStar = m_starsList;
     while (index && pStar)
     {
         index--;
         pStar = pStar->pNext;
     }

     pStar->fWeight = weight;
     pStar->fRadiusScale = yf_pow(weight, 1.0f/3.0f);
     pStar->CalculateKineticEnergyByVelocity();

     // 重算能量
     CalculateAmountEnergy();
 }

 // 设置星体的轨迹是否可见
 void        CNBodyKernel::SetStarOrbitVisible(Yuint index, bool visible)
 {
     if (index >= m_starsCount)
     {
         return;
     }

     Star* pStar = m_starsList;
     while (index && pStar)
     {
         index--;
         pStar = pStar->pNext;
     }

     pStar->bOrbitVisible = visible;
 }

 // 设置星体的颜色
 void        CNBodyKernel::SetStarColor(Yuint index, Ycolor color)
 {
     if (index >= m_starsCount)
     {
         return;
     }

     Star* pStar = m_starsList;
     while (index && pStar)
     {
         index--;
         pStar = pStar->pNext;
     }

     pStar->color = color;
 }

 // 计算总动能
 void        CNBodyKernel::CalculateAmountOfKineticEnergy()
 {
     // 动能
     m_fAmountOfKineticEnergy = 0.0f;
     Star* pStar = m_starsList;
     while (pStar)
     {
         //pStar->CalculateKineticEnergyByVelocity();
         m_fAmountOfKineticEnergy += pStar->fKineticEnergy;
         pStar = pStar->pNext;
     }
 }

 // 计算总势能
 void       CNBodyKernel::CalculateAmountOfPotentialEnergy()
 {
     m_fAmountOfPotentialEnergy = 0.0f;

     Star* pCurrent = m_starsList;
     Star* pNext;
     while (pCurrent)
     {
         pNext = pCurrent->pNext;
         while (pNext)
         {
             Vector3 vAB = pCurrent->vPosition - pNext->vPosition;
             float disAB = D3DXVec3Length(&vAB);
             float fPotentialEnergyAB = -m_fGravitationCoefficient*pCurrent->fWeight*pNext->fWeight/disAB;
             m_fAmountOfPotentialEnergy += fPotentialEnergyAB;
             pNext = pNext->pNext;
         }
         pCurrent = pCurrent->pNext;
     }
 }

 const Vector3* CNBodyKernel::GetStarPosition(Yuint index) const
 {
     if (index >= m_starsCount)
     {
         return ;
     }

     const Star* pStar = m_starsList;
     while (index && pStar)
     {
         index--;
         pStar = pStar->pNext;
     }

     assert(pStar);
     return &pStar->vPosition;
 }

 Yreal       CNBodyKernel::GetStarWeight(Yuint index) const
 {
     if (index >= m_starsCount)
     {
         return 0.0f;
     }

     const Star* pStar = m_starsList;
     while (index && pStar)
     {
         index--;
         pStar = pStar->pNext;
     }

     assert(pStar);
     return pStar->fWeight;
 }

 bool        CNBodyKernel::GetStarOrbitVisible(Yuint index) const
 {
     if (index >= m_starsCount)
     {
         return false;
     }

     const Star* pStar = m_starsList;
     while (index && pStar)
     {
         index--;
         pStar = pStar->pNext;
     }

     assert(pStar);
     return pStar->bOrbitVisible != YD_FALSE;
 }

 Ycolor      CNBodyKernel::GetStarColor(Yuint index) const
 {
     if (index >= m_starsCount)
     {
         return ;
     }

     const Star* pStar = m_starsList;
     while (index && pStar)
     {
         index--;
         pStar = pStar->pNext;
     }

     assert(pStar);
     return pStar->color;
 }

 Yuint      CNBodyKernel::GetStarID(Yuint index) const
 {
     if (index >= m_starsCount)
     {
         return ;
     }

     const Star* pStar = m_starsList;
     while (index && pStar)
     {
         index--;
         pStar = pStar->pNext;
     }

     assert(pStar);
     return pStar->id;
 }

 void        CNBodyKernel::GetSpaceCenter(Vector3& vCenter) const
 {
     vCenter = Vector3(0.0f, 0.0f, 0.0f);
     if (!m_starsCount)
     {
         return;
     }

     Star* pStar = m_starsList;
     while (pStar)
     {
         vCenter += pStar->vPosition;
         pStar = pStar->pNext;
     }

     vCenter /= (Yreal)m_starsCount;
 }

 void        CNBodyKernel::GetWeightCenter(Vector3& vCenter) const
 {
     vCenter = Vector3(0.0f, 0.0f, 0.0f);
     if (!m_starsCount)
     {
         return;
     }

     Yreal weights = 0.0f;
     Star* pStar = m_starsList;
     while (pStar)
     {
         vCenter += pStar->vPosition*pStar->fWeight;
         weights += pStar->fWeight;
         pStar = pStar->pNext;
     }

     vCenter /= weights;
 }

程序启动后,会出现4个球体在运动。不知道大家有没有注意到：Win7的开机动画就是四个球体的运动，所以我这程序的默认也是生成四个球体。

通过UI控件，可以添加删除星体。没有上限，当然星体越多程序越慢。

可以显示每一个星体的运动轨迹

用户可以实时修改任意一个星体的质量

鼠标右键用于控制视角
键盘U用于开关UI用户界面.
通过UI用户界面可以设置三个球体的质量,设置万有引力系数,设置天体运行速度,设置球体的显示大小.

键盘0~9用于开关第N个球体运动轨迹的显示

键盘G,用于开关三维网格的显示
键盘C,用于开关坐标轴的显示
键盘P,用于暂停
键盘R,用于重置.

F11 全屏切换

软件下载地址:http://files.cnblogs.com/WhyEngine/NBody.zip