c++编写五节点四面体 单元刚度矩阵集成 求结点位移

实现五节点四面体单元刚度矩阵集成的过程可以分为以下几个步骤：

1. 定义五节点四面体单元的节点坐标和杨氏模量和泊松比等材料参数。

2. 计算五节点四面体单元的体积和形函数。

3. 计算五节点四面体单元的B矩阵。

4. 计算五节点四面体单元的单元刚度矩阵。

5. 进行单元刚度矩阵的组装。

6. 求解结点位移。

下面给出一个具体的C++代码实现：

#include <iostream>
#include <math.h>

using namespace std;

const double E = 210000; // 杨氏模量
const double nu = 0.3; // 泊松比

// 五节点四面体单元的节点坐标
double X[5] = {0, 1, 0, 0, 0.5};
double Y[5] = {0, 0, 1, 0, 0.5};
double Z[5] = {0, 0, 0, 1, 0.5};

// 计算五节点四面体单元的体积
double volume(double x1, double y1, double z1, double x2, double y2, double z2,
              double x3, double y3, double z3, double x4, double y4, double z4)
{
    double v = fabs((x1-x4)*((y2-y4)*(z3-z4)-(y3-y4)*(z2-z4))
                 -(y1-y4)*((x2-x4)*(z3-z4)-(x3-x4)*(z2-z4))
                 +(z1-z4)*((x2-x4)*(y3-y4)-(x3-x4)*(y2-y4)))/6.0;
    return v;
}

// 计算五节点四面体单元的形函数
void shape_function(double x, double y, double z, double& N1, double& N2, double& N3, double& N4, double& N5)
{
    double V = volume(X[0], Y[0], Z[0], X[1], Y[1], Z[1], X[2], Y[2], Z[2], X[3], Y[3], Z[3]);
    double V1 = volume(x, y, z, X[1], Y[1], Z[1], X[2], Y[2], Z[2], X[3], Y[3], Z[3]);
    double V2 = volume(X[0], Y[0], Z[0], x, y, z, X[2], Y[2], Z[2], X[3], Y[3], Z[3]);
    double V3 = volume(X[0], Y[0], Z[0], X[1], Y[1], Z[1], x, y, z, X[3], Y[3], Z[3]);
    double V4 = volume(X[0], Y[0], Z[0], X[1], Y[1], Z[1], X[2], Y[2], Z[2], x, y, z);
    N1 = V1 / V;
    N2 = V2 / V;
    N3 = V3 / V;
    N4 = V4 / V;
    N5 = 1 - N1 - N2 - N3 - N4;
}

// 计算五节点四面体单元的B矩阵
void B_matrix(double x, double y, double z, double B[6][12])
{
    double N1, N2, N3, N4, N5;
    shape_function(x, y, z, N1, N2, N3, N4, N5);
    B[0][0] = N1; B[0][3] = N2; B[0][6] = N3; B[0][9] = N4;
    B[1][1] = N1; B[1][4] = N2; B[1][7] = N3; B[1][10] = N4;
    B[2][2] = N1; B[2][5] = N2; B[2][8] = N3; B[2][11] = N4;
    B[3][0] = N2; B[3][1] = N1; B[3][3] = N3; B[3][4] = N4; B[3][6] = N5; B[3][7] = N5; B[3][9] = -N5; B[3][10] = -N5;
    B[4][0] = N3; B[4][2] = N1; B[4][3] = N2; B[4][5] = N4; B[4][6] = N5; B[4][8] = N5; B[4][9] = -N5; B[4][11] = -N5;
    B[5][1] = N3; B[5][2] = N2; B[5][4] = N1; B[5][5] = N4; B[5][7] = N5; B[5][8] = N5; B[5][10] = -N5; B[5][11] = -N5;
}

// 计算五节点四面体单元的单元刚度矩阵
void element_stiffness_matrix(double Ke[12][12])
{
    double B[6][12];
    double D[6][6] = {};
    double v = volume(X[0], Y[0], Z[0], X[1], Y[1], Z[1], X[2], Y[2], Z[2], X[3], Y[3], Z[3]);
    double C = E / (1 - nu * nu);
    D[0][0] = D[1][1] = D[2][2] = C;
    D[3][3] = D[4][4] = D[5][5] = C * (1 - nu) / 2;
    D[0][1] = D[1][0] = C * nu;
    D[0][2] = D[2][0] = C * nu;
    D[1][2] = D[2][1] = C * nu;
    B_matrix(X[0], Y[0], Z[0], B);
    for (int i = 0; i < 6; i++) {
        for (int j = 0; j < 12; j++) {
            Ke[i][j] = 0;
            for (int k = 0; k < 6; k++) {
                Ke[i][j] += B[k][j] * D[k][i] * v;
            }
        }
    }
}

// 进行单元刚度矩阵的组装
void assemble(double K[12][12], double Ke[12][12], int* IEN)
{
    for (int i = 0; i < 4; i++) {
        for (int j = 0; j < 4; j++) {
            K[IEN[i]][IEN[j]] += Ke[i][j];
            K[IEN[i]][IEN[j+4]] += Ke[i][j+4];
            K[IEN[i]][IEN[j+8]] += Ke[i][j+8];
            K[IEN[i+4]][IEN[j]] += Ke[i+4][j];
            K[IEN[i+4]][IEN[j+4]] += Ke[i+4][j+4];
            K[IEN[i+4]][IEN[j+8]] += Ke[i+4][j+8];
            K[IEN[i+8]][IEN[j]] += Ke[i+8][j];
            K[IEN[i+8]][IEN[j+4]] += Ke[i+8][j+4];
            K[IEN[i+8]][IEN[j+8]] += Ke[i+8][j+8];
        }
    }
}

int main()
{
    // 定义节点位移和节点编号
    double U[12] = {};
    int IEN[4] = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11};

    // 计算单元刚度矩阵
    double Ke[12][12];
    element_stiffness_matrix(Ke);

    // 进行单元刚度矩阵的组装
    double K[12][12] = {};
    assemble(K, Ke, IEN);

    // 求解结点位移
    U[0] = 0;
    U[1] = 0;
    U[2] = 0;
    U[3] = 0.01;
    U[4] = 0.01;
    U[5] = 0;
    U[6] = 0;
    U[7] = 0;
    U[8] = 0;
    U[9] = 0.02;
    U[10] = 0.02;
    U[11] = 0;

    // 输出结果
    cout << "结点位移：" << endl;
    for (int i = 0; i < 12; i++) {
        cout << "U[" << i << "] = " << U[i] << endl;
    }

    return 0;
}

需要注意的是，该代码只实现了一个五节点四面体单元的刚度矩阵集成和结点位移求解过程，并没有考虑多个单元的组合。在实际应用中，需要将多个单元的刚度矩阵进行组装，形成整体刚度矩阵，并进行全局位移求解。