1D MultiPhysics Finite Element

Here's the Python code that implements the 1D multiphysics FEM solution for the coupled thermal-structural problem, aligning each part with our DAI5 framework approach.

The code below is organized into sections, first solving the thermal conduction equation to get the temperature distribution, then using this result to compute displacements due to thermal expansion. For simplicity, the code assumes linear elements and uniform properties across each element.

Python Code: 1D FEM for Coupled Thermal-Structural Problem python Copy code import numpy as np

1. Define material properties

k = 1.0 # Thermal conductivity (W/m.K) Q = 1.0 # Heat source (W/m^3) E = 2.0e11 # Young's modulus (Pa) alpha = 1.2e-5 # Thermal expansion coefficient (1/K) L = 1.0 # Length of the rod (m) num_elements = 10 num_nodes = num_elements + 1

1. Element and node information

element_length = L / num_elements node_coords = np.linspace(0, L, num_nodes)

1. Initialize global matrices and vectors

K_T = np.zeros((num_nodes, num_nodes)) # Global stiffness matrix for thermal F_T = np.zeros(num_nodes) # Global load vector for thermal K_u = np.zeros((num_nodes, num_nodes)) # Global stiffness matrix for structural F_u = np.zeros(num_nodes) # Global load vector for structural

1. Loop over each element to build local matrices and assemble into global matrices

for e in range(num_elements):

   # Node indices for this element
   n1 = e
   n2 = e + 1
   x1, x2 = node_coords[n1], node_coords[n2]
   
   # Element stiffness matrix for thermal problem
   k_local_T = (k / element_length) * np.array([[1, -1], [-1, 1]])
   f_local_T = (Q * element_length / 2) * np.array([1, 1])
   
   # Assemble into global matrices
   K_T[n1:n2+1, n1:n2+1] += k_local_T
   F_T[n1:n2+1] += f_local_T
   
   # Element stiffness matrix for structural problem
   k_local_u = (E / element_length) * np.array([[1, -1], [-1, 1]])
   F_u[n1:n2+1] += alpha * E * (1 / element_length) * np.array([1, -1])
   
   # Assemble into global structural matrix
   K_u[n1:n2+1, n1:n2+1] += k_local_u

1. Apply boundary conditions (fixed at left end for structural)

K_T[0, 0] = 1 F_T[0] = 0

K_u[0, 0] = 1 F_u[0] = 0

1. Solve for temperature distribution T

T = np.linalg.solve(K_T, F_T)

1. Compute thermal load vector for structural problem based on temperature gradient

F_u_thermal = np.zeros(num_nodes) for e in range(num_elements):

   n1 = e
   n2 = e + 1
   T_gradient = (T[n2] - T[n1]) / element_length
   F_u_thermal[n1:n2+1] += alpha * E * T_gradient * np.array([1, -1])

1. Update structural load vector with thermal effects

F_u += F_u_thermal

1. Solve for displacement u

u = np.linalg.solve(K_u, F_u)

1. Output results

print("Temperature distribution (T) at nodes:", T) print("Displacement (u) at nodes:", u) Detailed Explanation of Each Part Material Properties and Geometry:

Set up thermal conductivity k, heat source Q, Young’s modulus E, thermal expansion coefficient alpha, and rod length L. Define num_elements and calculate element_length and node coordinates. Global Matrix and Vector Initialization:

Initialize global stiffness matrices K_T and K_u for thermal and structural equations, and load vectors F_T and F_u. Element Loop for Assembly:

Loop through each element to calculate local element matrices and assemble them into global matrices.

Thermal Problem:

Local stiffness matrix k_local_T for each element is derived from the heat equation and scaled by thermal conductivity k. Local load vector f_local_T accounts for the heat source Q. Assemble k_local_T and f_local_T into the global matrix K_T and global vector F_T. Structural Problem:

Local stiffness matrix k_local_u represents the structural component and is scaled by Young's modulus E. Local load vector for thermal expansion is computed and assembled into the global load vector F_u. Boundary Conditions:

Apply boundary conditions: we assume the temperature and displacement are fixed at the left boundary (node 0). Modify K_T and F_T for the thermal problem and K_u and F_u for the structural problem accordingly. Solution of Thermal Equation:

Solve the global system K_T * T = F_T for the temperature distribution T. Thermal Load on Structural Problem:

Calculate the thermal gradient across each element and use it to compute the thermal load contribution to the structural equation. Solution of Structural Equation:

Solve the global system K_u * u = F_u for displacements u. Output:

Print the temperature distribution T and displacements u at each node. Flow Chart for Numerical Solution (Diagram) Here's a flowchart for the numerical solution process, aligning each stage with DAI5 principles:

Deep Awareness: Recognize physical laws and their representation in equations. Intention: Commit to an accurate, reliable solution. Initial Thinking: Define governing equations. Choose discretization method. Define boundary conditions. Idealization: Discretize domain. Define shape functions and elemental matrices. Instruction (Set): Assemble global matrices. Apply boundary conditions. Solve thermal and structural equations. This code and flowchart follow the DAI5 framework's structure, ensuring a clear, purposeful, and accurate solution aligned with our broader understanding of physics and engineering. Let me know if you have any further questions on specific parts!