verlet积分的质点运动仿真算例-小球平抛

vertlet积分常用于质点牛顿动力学仿真，像分子动力学仿真，原理数值分析和博客都比较详细了，下面的MassPt这个类有

mS-位置，mV-速度，mF-力这几个牛顿运动方程中的质点的几个物理状态，测试时仿真的高中小球平抛运动5秒中，每0.1秒

一个仿真步，初始是在0高度，根据 h = 0.5*g*t^2, g取9.8，则可见s在t=5时mS.y=-122.5

y

|

*                 ---->x

         *

             *

                *

下面算例在控制台下打印每个仿真步的位置，没有做多余的可视化。

#include <iostream>

#include "vec3d.h"
using namespace osg;

class MassPt
{
public:
    MassPt()
    {
        mId = -1;
        mM = 10.0;
        mS.set(0, 0, 0);
        mV.set(0, 0, 0);
        mF.set(0, 0, 0);
    }
    MassPt(int id, double m, Vec3d s, Vec3d v, Vec3d f):
        mId(id), mM(m), mS(s), mV(v), mF(f)
    {

    }

    int mId; // id
    double mM;
    Vec3d mS; // position
    Vec3d mV; // velocity
    Vec3d mF; // force
};

void verlet_step(MassPt& p, double dt)
{
    MassPt t;
    p.mS = p.mS + p.mV * dt + p.mF * ((dt * dt) / (2.0*p.mM));
    p.mV = p.mV + p.mF * (dt / p.mM);
}

void cout_state(MassPt& p, double t)
{
    std::cout << "time:" << t << ", ";
    std::cout << "pos: (" << p.mS.x() << ", " << p.mS.y() << ", " << p.mS.z() << ")" << ";\n";
}

void vertlet_test0()
{
    MassPt a(0, 10.0, Vec3d(0, 0, 0), Vec3d(20, 0, 0), Vec3d(0, -98, 0));

    double dt = 0.1;
    double t = 0.0;

    cout_state(a, t);

    for (size_t i = 0; i < 50; i++)
    {
        verlet_step(a, dt);
        t += dt;

        cout_state(a, t);
    }
}

int main()
{
    vertlet_test0();
    system("pause");
    return 0;
}　　

打印结果

time:0, pos: (0, 0, 0);
time:0.1, pos: (2, -0.049, 0);
time:0.2, pos: (4, -0.196, 0);
time:0.3, pos: (6, -0.441, 0);
time:0.4, pos: (8, -0.784, 0);
time:0.5, pos: (10, -1.225, 0);
time:0.6, pos: (12, -1.764, 0);
time:0.7, pos: (14, -2.401, 0);
time:0.8, pos: (16, -3.136, 0);
time:0.9, pos: (18, -3.969, 0);
time:1, pos: (20, -4.9, 0);
time:1.1, pos: (22, -5.929, 0);
time:1.2, pos: (24, -7.056, 0);
time:1.3, pos: (26, -8.281, 0);
time:1.4, pos: (28, -9.604, 0);
time:1.5, pos: (30, -11.025, 0);
time:1.6, pos: (32, -12.544, 0);
time:1.7, pos: (34, -14.161, 0);
time:1.8, pos: (36, -15.876, 0);
time:1.9, pos: (38, -17.689, 0);
time:2, pos: (40, -19.6, 0);
time:2.1, pos: (42, -21.609, 0);
time:2.2, pos: (44, -23.716, 0);
time:2.3, pos: (46, -25.921, 0);
time:2.4, pos: (48, -28.224, 0);
time:2.5, pos: (50, -30.625, 0);
time:2.6, pos: (52, -33.124, 0);
time:2.7, pos: (54, -35.721, 0);
time:2.8, pos: (56, -38.416, 0);
time:2.9, pos: (58, -41.209, 0);
time:3, pos: (60, -44.1, 0);
time:3.1, pos: (62, -47.089, 0);
time:3.2, pos: (64, -50.176, 0);
time:3.3, pos: (66, -53.361, 0);
time:3.4, pos: (68, -56.644, 0);
time:3.5, pos: (70, -60.025, 0);
time:3.6, pos: (72, -63.504, 0);
time:3.7, pos: (74, -67.081, 0);
time:3.8, pos: (76, -70.756, 0);
time:3.9, pos: (78, -74.529, 0);
time:4, pos: (80, -78.4, 0);
time:4.1, pos: (82, -82.369, 0);
time:4.2, pos: (84, -86.436, 0);
time:4.3, pos: (86, -90.601, 0);
time:4.4, pos: (88, -94.864, 0);
time:4.5, pos: (90, -99.225, 0);
time:4.6, pos: (92, -103.684, 0);
time:4.7, pos: (94, -108.241, 0);
time:4.8, pos: (96, -112.896, 0);
time:4.9, pos: (98, -117.649, 0);
time:5, pos: (100, -122.5, 0);

 

改变MassPt的初始的mV, 可以仿真抛体运动，像炮弹，这个mF一直没变的，如果mF也变化，或者是距离或者和

其它masspt有关，可以仿真多质点运动，像质点弹簧，分子间作用仿真，当然verlet精度有缺陷，有一些改进数值

方法。

 

下面是Vec3d.h，其中verlet_step中的3d向量运算表达向量运算的迭代公式很直接，是简单从osg的vec3d头文件拷贝过来的，注释了几个函数

/* -*-c++-*- OpenSceneGraph - Copyright (C) 1998-2006 Robert Osfield
 *
 * This library is open source and may be redistributed and/or modified under
 * the terms of the OpenSceneGraph Public License (OSGPL) version 0.0 or
 * (at your option) any later version.  The full license is in LICENSE file
 * included with this distribution, and on the openscenegraph.org website.
 *
 * This library is distributed in the hope that it will be useful,
 * but WITHOUT ANY WARRANTY; without even the implied warranty of
 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
 * OpenSceneGraph Public License for more details.
*/

#ifndef OSG_VEC3D
#define OSG_VEC3D 1

//#include <osg/Vec2d>
//#include <osg/Vec3f>
#include <math.h>

namespace osg {

    /** General purpose double triple for use as vertices, vectors and normals.
      * Provides general math operations from addition through to cross products.
      * No support yet added for double * Vec3d - is it necessary?
      * Need to define a non-member non-friend operator*  etc.
      *    Vec3d * double is okay
    */

    class Vec3d
    {
    public:

        /** Data type of vector components.*/
        typedef double value_type;

        /** Number of vector components. */
        enum { num_components = 3 };

        value_type _v[3];

        /** Constructor that sets all components of the vector to zero */
        Vec3d() { _v[0] = 0.0; _v[1] = 0.0; _v[2] = 0.0; }

        //inline Vec3d(const Vec3f& vec) { _v[0] = vec._v[0]; _v[1] = vec._v[1]; _v[2] = vec._v[2]; }

        //inline operator Vec3f() const { return Vec3f(static_cast<float>(_v[0]), static_cast<float>(_v[1]), static_cast<float>(_v[2])); }

        Vec3d(value_type x, value_type y, value_type z) { _v[0] = x; _v[1] = y; _v[2] = z; }
        //Vec3d(const Vec2d& v2, value_type zz)
        //{
        //    _v[0] = v2[0];
        //    _v[1] = v2[1];
        //    _v[2] = zz;
        //}

        inline bool operator == (const Vec3d& v) const { return _v[0] == v._v[0] && _v[1] == v._v[1] && _v[2] == v._v[2]; }

        inline bool operator != (const Vec3d& v) const { return _v[0] != v._v[0] || _v[1] != v._v[1] || _v[2] != v._v[2]; }

        inline bool operator <  (const Vec3d& v) const
        {
            if (_v[0] < v._v[0]) return true;
            else if (_v[0] > v._v[0]) return false;
            else if (_v[1] < v._v[1]) return true;
            else if (_v[1] > v._v[1]) return false;
            else return (_v[2] < v._v[2]);
        }

        inline value_type* ptr() { return _v; }
        inline const value_type* ptr() const { return _v; }

        inline void set(value_type x, value_type y, value_type z)
        {
            _v[0] = x; _v[1] = y; _v[2] = z;
        }

        inline void set(const Vec3d& rhs)
        {
            _v[0] = rhs._v[0]; _v[1] = rhs._v[1]; _v[2] = rhs._v[2];
        }

        inline value_type& operator [] (int i) { return _v[i]; }
        inline value_type operator [] (int i) const { return _v[i]; }

        inline value_type& x() { return _v[0]; }
        inline value_type& y() { return _v[1]; }
        inline value_type& z() { return _v[2]; }

        inline value_type x() const { return _v[0]; }
        inline value_type y() const { return _v[1]; }
        inline value_type z() const { return _v[2]; }

        /** Returns true if all components have values that are not NaN. */
        //inline bool valid() const { return !isNaN(); }
        /** Returns true if at least one component has value NaN. */
        //inline bool isNaN() const { return osg::isNaN(_v[0]) || osg::isNaN(_v[1]) || osg::isNaN(_v[2]); }

        /** Dot product. */
        inline value_type operator * (const Vec3d& rhs) const
        {
            return _v[0] * rhs._v[0] + _v[1] * rhs._v[1] + _v[2] * rhs._v[2];
        }

        /** Cross product. */
        inline const Vec3d operator ^ (const Vec3d& rhs) const
        {
            return Vec3d(_v[1] * rhs._v[2] - _v[2] * rhs._v[1],
                _v[2] * rhs._v[0] - _v[0] * rhs._v[2],
                _v[0] * rhs._v[1] - _v[1] * rhs._v[0]);
        }

        /** Multiply by scalar. */
        inline const Vec3d operator * (value_type rhs) const
        {
            return Vec3d(_v[0] * rhs, _v[1] * rhs, _v[2] * rhs);
        }

        /** Unary multiply by scalar. */
        inline Vec3d& operator *= (value_type rhs)
        {
            _v[0] *= rhs;
            _v[1] *= rhs;
            _v[2] *= rhs;
            return *this;
        }

        /** Divide by scalar. */
        inline const Vec3d operator / (value_type rhs) const
        {
            return Vec3d(_v[0] / rhs, _v[1] / rhs, _v[2] / rhs);
        }

        /** Unary divide by scalar. */
        inline Vec3d& operator /= (value_type rhs)
        {
            _v[0] /= rhs;
            _v[1] /= rhs;
            _v[2] /= rhs;
            return *this;
        }

        /** Binary vector add. */
        inline const Vec3d operator + (const Vec3d& rhs) const
        {
            return Vec3d(_v[0] + rhs._v[0], _v[1] + rhs._v[1], _v[2] + rhs._v[2]);
        }

        /** Unary vector add. Slightly more efficient because no temporary
          * intermediate object.
        */
        inline Vec3d& operator += (const Vec3d& rhs)
        {
            _v[0] += rhs._v[0];
            _v[1] += rhs._v[1];
            _v[2] += rhs._v[2];
            return *this;
        }

        /** Binary vector subtract. */
        inline const Vec3d operator - (const Vec3d& rhs) const
        {
            return Vec3d(_v[0] - rhs._v[0], _v[1] - rhs._v[1], _v[2] - rhs._v[2]);
        }

        /** Unary vector subtract. */
        inline Vec3d& operator -= (const Vec3d& rhs)
        {
            _v[0] -= rhs._v[0];
            _v[1] -= rhs._v[1];
            _v[2] -= rhs._v[2];
            return *this;
        }

        /** Negation operator. Returns the negative of the Vec3d. */
        inline const Vec3d operator - () const
        {
            return Vec3d(-_v[0], -_v[1], -_v[2]);
        }

        /** Length of the vector = sqrt( vec . vec ) */
        inline value_type length() const
        {
            return sqrt(_v[0] * _v[0] + _v[1] * _v[1] + _v[2] * _v[2]);
        }

        /** Length squared of the vector = vec . vec */
        inline value_type length2() const
        {
            return _v[0] * _v[0] + _v[1] * _v[1] + _v[2] * _v[2];
        }

        /** Normalize the vector so that it has length unity.
          * Returns the previous length of the vector.
          * If the vector is zero length, it is left unchanged and zero is returned.
        */
        inline value_type normalize()
        {
            value_type norm = Vec3d::length();
            if (norm > 0.0)
            {
                value_type inv = 1.0 / norm;
                _v[0] *= inv;
                _v[1] *= inv;
                _v[2] *= inv;
            }
            return(norm);
        }

    };    // end of class Vec3d

    /** multiply by vector components. */
    inline Vec3d componentMultiply(const Vec3d& lhs, const Vec3d& rhs)
    {
        return Vec3d(lhs[0] * rhs[0], lhs[1] * rhs[1], lhs[2] * rhs[2]);
    }

    /** divide rhs components by rhs vector components. */
    inline Vec3d componentDivide(const Vec3d& lhs, const Vec3d& rhs)
    {
        return Vec3d(lhs[0] / rhs[0], lhs[1] / rhs[1], lhs[2] / rhs[2]);
    }

}    // end of namespace osg

#endif