查找对象浮动状态的优化方法

Question

要解决的问题是找到浮体的浮动状态,考虑其重量和重心.

我使用的功能计算给定下沉,鞋跟和修剪的身体的移位体积和中心.下沉是长度单位,后跟/修剪的角度限制在-90到90之间.

当移位的体积等于重量并且重心处于具有中心的垂直线时发现浮动状态.

我将此实现为具有3个变量(下沉,修剪,后跟)和3个方程的非线性Newton-Raphson根发现问题.这种方法有效,但需要良好的初步猜测.所以我希望找到一个更好的方法,或者找到初始值的好方法.

下面是用于Newton-Raphson迭代的牛顿和雅可比算法的代码.功能体积采用参数下沉,鞋跟和修剪.并返回体积,以及bouyancy中心的坐标.

我还包括了maxabs和GSolve2算法,我相信这些算法来自Numerical Recipies.

void jacobian(float x[], float weight, float vcg, float tcg, float lcg, float jac[][3], float f0[]) {
    float h = 0.0001f;
    float temp;
    float j_volume, j_vcb, j_lcb, j_tcb;
    float f1[3];

    volume(x[0], x[1], x[2], j_volume, j_lcb, j_vcb, j_tcb);
    f0[0] = j_volume-weight;
    f0[1] = j_tcb-tcg;
    f0[2] = j_lcb-lcg;

    for (int i=0;i<3;i++) {
        temp = x[i];
        x[i] = temp + h;
        volume(x[0], x[1], x[2], j_volume, j_lcb, j_vcb, j_tcb);

        f1[0] = j_volume-weight;
        f1[1] = j_tcb-tcg;
        f1[2] = j_lcb-lcg;
        x[i] = temp;

        jac[0][i] = (f1[0]-f0[0])/h;
        jac[1][i] = (f1[1]-f0[1])/h;
        jac[2][i] = (f1[2]-f0[2])/h;
    }
}


void newton(float weight, float vcg, float tcg, float lcg, float &sinkage, float &heel, float &trim) {
    float x[3] = {10,1,1};

    float accuracy = 0.000001f;
    int ntryes = 30;
    int i = 0;
    float jac[3][3];
    float max;
    float f0[3];
    float gauss_f0[3];

    while (i < ntryes) {
        jacobian(x, weight, vcg, tcg, lcg, jac, f0);

        if (sqrt((f0[0]*f0[0]+f0[1]*f0[1]+f0[2]*f0[2])/2) < accuracy) {
            break;
        }

        gauss_f0[0] = -f0[0];
        gauss_f0[1] = -f0[1];
        gauss_f0[2] = -f0[2];

        GSolve2(jac, 3, gauss_f0);

        x[0] = x[0]+gauss_f0[0];
        x[1] = x[1]+gauss_f0[1];
        x[2] = x[2]+gauss_f0[2];

        // absmax(x) - Return absolute max value from an array
        max = absmax(x);
        if (max < 1) max = 1;

        if (sqrt((gauss_f0[0]*gauss_f0[0]+gauss_f0[1]*gauss_f0[1]+gauss_f0[2]*gauss_f0[2])) < accuracy*max) {
            x[0]=x2[0];
            x[1]=x2[1];
            x[2]=x2[2];
            break;
        }

        i++;
    }

    sinkage = x[0];
    heel = x[1];
    trim = x[2];
}

int GSolve2(float a[][3],int n,float b[]) {
    float x,sum,max,temp;
    int i,j,k,p,m,pos;

    int nn = n-1; 

    for (k=0;k<=n-1;k++)
    {
        /* pivot*/
        max=fabs(a[k][k]);
        pos=k;


        for (p=k;p<n;p++){
            if (max < fabs(a[p][k])){
                max=fabs(a[p][k]);
                pos=p;
            }
        }

        if (ABS(a[k][pos]) < EPS) {
            writeLog("Matrix is singular");
            break;
        }

        if (pos != k) {
            for(m=k;m<n;m++){
                temp=a[pos][m];
                a[pos][m]=a[k][m];
                a[k][m]=temp;
            }
        }

        /*   convert to upper triangular form */
        if ( fabs(a[k][k])>=1.e-6)
        {
            for (i=k+1;i<n;i++)
            {
            x = a[i][k]/a[k][k];
            for (j=k+1;j<n;j++) a[i][j] = a[i][j] -a[k][j]*x;
            b[i] = b[i] - b[k]*x;
            }
        }
        else
        {
            writeLog("zero pivot found in line:%d",k);
            return 0;
       }
     }

/*   back substitution */
     b[nn] = b[nn] / a[nn][nn];
     for (i=n-2;i>=0;i--)
     {
        sum = b[i];
         for (j=i+1;j<n;j++) 
           sum = sum - a[i][j]*b[j];
        b[i] = sum/a[i][i];
     }
     return 0;
}

float absmax(float x[]) {
    int i = 1;
    int n = sizeof(x);
    float max = x[0];
    while (i < n) {
        if (max < x[i]) {
            max = x[i];
        }
        i++;
    }
    return max;
}

Answer 1

您是否考虑过一些随机搜索方法来找到初始值，然后用 Newton Raphson 进行微调？一种可能性是进化计算，您可以使用 Inspyred 包。对于在很多方面与您描述的问题类似的物理问题，请查看以下示例：http ://inspyred.github.com/tutorial.html#lunar-explorer