有限元接触问题代码
有限元接触问题是一个广泛研究的问题，涉及到不同材料之间的接触力学分析。以下是一个简化的有限元接触问题的代码示例，以说明基本的实现思路。
假设有两个物体A和B，它们之间存在接触。首先需要进行网格划分，将物体A和物体B分别划分为有限元网格。然后，需要定义接触区域的初始条件和材料参数。
```python
import numpy as np
import matplotlib.pyplot as plt
from scipy.sparse import lil_matrix
from scipy.sparse.linalg import spsolve
# 定义网格划分函数，将物体划分为有限元网格
def generate_mesh():
# 生成节点坐标数组
nodes = np.array([[0, 0], [0, 1], [1, 0], [1, 1]])
# 生成单元数组
elements = np.array([[1, 2, 3], [2, 3, 4]])
return nodes, elements
# 定义初始条件和材料参数
def define_parameters():
# 定义材料参数
young_modulus_a = 1
young_modulus_b = 2
poisson_ratio_a = 0.3
poisson_ratio_b = 0.4
# 定义接触区域初始条件
contact_area = np.array([1, 2])
return young_modulus_a, young_modulus_b, poisson_ratio_a, poisson_ratio_b, contact_area
# 构建刚度矩阵和力向量
def assemble_stiffness_matrix(nodes, elements, young_modulus_a, young_modulus_b, poisson_ratio_a, poisson_ratio_b):
num_nodes = len(nodes)
num_elements = len(elements)
stiffness_matrix = lil_matrix((num_nodes, num_nodes))
for i in range(num_elements):
element = elements[i]
node_indices = element - 1
x_coords = nodes[node_indices, 0]
y_coords = nodes[node_indices, 1]
area = (x_coords[1] - x_coords[0]) * (y_coords[2] - y_coords[0])
k = area / 2
# 计算局部刚度矩阵
local_stiffness_matrix_a = (young_modulus_a / (1 - poisson_ratio_a**2)) * np.array([[1, poisson_ratio_a, 0], [poisson_ratio_a, 1, 0], [0, 0, (1 - poisson_ratio_a) / 2]])
local_stiffness_matrix_b = (young_modulus_b / (1 - poisson_ratio_b**2)) * np.array([[1, poisson_ratio_b, 0], [poisson_ratio_b, 1, 0], [0, 0, (1 - poisson_ratio_b) / 2]])
# 计算全局刚度矩阵
stiffness_matrix[node_indices[0], node_indices[0]] += k * local_stiffness_matrix_a[0, 0]
stiffness_matrix[node_indices[0], node_indices[1]] += k * local_stiffness_matrix_a[0, 1]
stiffness_matrix[node_indices[0], node_indices[2]] += k * local_stiffness_matrix_a[0, 2]
stiffness_matrix[node_indices[1], node_indices[0]] += k * local_stiffness_matrix_a[1, 0]
stiffness_matrix[node_indices[1], node_indices[1]] += k * local_stiffness_matrix_a[1, 1]
stiffness_matrix[node_indices[1], node_indices[2]] += k * local_stiffness_matrix_a[1, 2]
stiffness_matrix[node_indices[2], node_indices[0]] += k * local_stiffness_matrix_a[2, 0]
stiffness_matrix[node_indices[2], node_indices[1]] += k * local_stiffness_matrix_a[2, 1]
stiffness_matrix[node_indices[2], node_indices[2]] += k * local_stiffness_matrix_a[2, 2]
stiffness_matrix[node_indices[0], node_indices[0]] += k * local_stiffness_matrix_b[0, 0]
stiffness_matrix[node_indices[0], node_indices[1]] += k * local_stiffness_matrix_b[0, 1]
stiffness_matrix[node_indices[0], node_indices[2]] += k * local_stiffness_matrix_b[0, 2]
stiffness_matrix[node_indices[1], node_indices[0]] += k * local_stiffness_matrix_b[1, 0]
stiffness_matrix[node_indices[1], node_indices[1]] += k * local_stiffness_matrix_b[1, 1]
stiffness_matrix[node_indices[1], node_indices[2]] += k * local_stiffness_matrix_b[1, 2]
stiffness_matrix[node_indices[2], node_indices[0]] += k * local_stiffness_matrix_b[2, 0]
python新手代码你好 stiffness_matrix[node_indices[2], node_indices[1]] += k * local_stiffness_matrix_b[2, 1]
stiffness_matrix[node_indices[2], node_indices[2]] += k * local_stiffness_matrix_b[2, 2]
return stiffness_matrix
# 解决线性方程系统
def solve_linear_system(stiffness_matrix, contact_area, nodes):
num_nodes = len(nodes)
num_unknowns = num_nodes - len(contact_area)
force = np.zeros(num_nodes)
displacement = np.zeros(num_nodes)
# 应用边界条件
for i in range(len(contact_area)):
node_index = contact_area[i] - 1
force[node_index] = 1
# 划分刚度矩阵和力向量
stiffness_matrix_global = stiffness_matrix[0:num_unknowns, 0:num_unknowns]
force_global = force[0:num_unknowns]
# 解线性方程系统
displacement_global = spsolve(stiffness_matrix_global, force_global)
# 汇总解向量
for i in range(num_unknowns):
displacement[i] = displacement_global[i]
for i in range(len(contact_area)):
node_index = contact_area[i] - 1
displacement[node_index] = 0
return displacement
# 绘制位移云图
def plot_displacement(nodes, displacement):
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