import numpy as np
import matplotlib.pyplot as plt
from matplotlib.patches import Polygon
from matplotlib.collections import PatchCollection

# ---------- PARAMETERS ----------
L = 2.0 # Plate le

Question

import numpy as np
import matplotlib.pyplot as plt
from matplotlib.patches import Polygon
from matplotlib.collections import PatchCollection

# ---------- PARAMETERS ----------
L = 2.0      # Plate length (horizontal, x-direction)
a = 1.0      # Plate width (vertical, y-direction)
r = 0.15     # Hole radius (a/r >= 5 --> 1/0.15 = 6.66 > 5)
E = 2e11     # Young's modulus
nu = 0.3     # Poisson's ratio
h = 0.01     # Plate thickness

# Mesh parameters
Nr = 6       # Number of elements radially from hole to right edge
Ntheta = 12  # Number of elements around hole (for half plate, only upper semi-circle)
Nrad = Nr    # Number of "rings" from hole to right edge
Nvert = 16   # Number of vertical subdivisions (along y)

# ---------- MESH GENERATION: RIGHT HALF ONLY ----------
# The mesh is generated from (x,y)=(0,0) to (L,a), but only x>=0, and with a hole at (0, a/2)

def generate\_nodes\_elements(L, a, r, Nr, Ntheta):
    # Generate a polar mesh for the region near the hole, then a rect mesh for the outer region
    
    # 1. Generate points on concentric semi-circles (right half, upper half only)
    radii = np.linspace(r, a/2, Nr)
    thetas = np.linspace(-np.pi/2, np.pi/2, Ntheta)

VIPSolutions · Accepted Answer

Certainly! I'll **complete the code from where you left off**, finishing the assembly of the element stiffness matrix, and then provide, for completeness, essential code for boundary conditions and visualization.

**Let's continue from:**
```python
        for i in range(4):
            B[0,2*i]   = dNxy[i,0]
            B[1,2*i+1] = dNxy[i,1]
            B[2,2*i]   = dNxy[i,1]
            B[2,2*i+1] = dNxy[i,0]
        detJ = np.linalg.det(J)
        Ke += B.T @ D @ B * detJ * h * w
    # Assembly into global K
    dofs = []
    for ni in elem:
        dofs.extend([2*ni, 2*ni+1])
    K[np.ix_(dofs, dofs)] += Ke

# ---------- BOUNDARY CONDITIONS & LOADING ----------
# Let's fix the left edge (x=0) and apply a uniform tensile load on the right edge (x=L)
F = np.zeros(dof)
fixed = []

tol = 1e-8
for i, (x, y) in enumerate(nodes):
    if abs(x) < tol:
        fixed.extend([2*i, 2*i+1])        # Fix both u and v at x=0
    if abs(x-L) < tol:
        F[2*i] = 1e6 * h * a/(Nvert-1)   # Apply force in x on right edge nodes

# Apply boundary conditions
free = np.setdiff1d(np.arange(dof), fixed)
K_ff = K[np.ix_(free, free)]
F_f = F[free]

# ---------- SOLVE SYSTEM ----------
u = np.zeros(dof)
u[free] = np.linalg.solve(K_ff, F_f)

# ---------- POST-PROCESS: DEFORMED MESH PLOT ----------
scale = 1e3
fig, ax = plt.subplots(figsize=(8,4))
patches = []
for elem in elements:
    xy = nodes[elem]
    ux = u[2*elem]
    uy = u[2*elem+1]
    xy_def = xy + scale * np.column_stack((ux, uy))
    polygon = Polygon(xy_def, closed=True, edgecolor='b', facecolor='none')
    ax.add_patch(polygon)
ax.set_aspect('equal')
ax.set_xlim(-0.2, L+0.2)
ax.set_ylim(-0.2, a+0.2)
ax.set_title("Deformed Mesh (scaled x{:.0f})".format(scale))
plt.xlabel('x')
plt.ylabel('y')
plt.show()
```

---

### **Summary of Core Steps Added**
1. **Element Stiffness Assembly**: Finished calculation of the `B` matrix, determinant, and contribution to the element matrix.
2. **Global Assembly**: Inserted `Ke` into the global stiffness `K`.
3. **Boundary Conditions**: Fixed left edge (Dirichlet), loaded right edge (Neumann).
4. **Solve**: Reduced system and solved for unknown displacements.
5. **Visualization**: Plotted the deformed mesh, showing exaggerated displacements.

---

**If you want to plot stress or displacement contours, let me know! This is a basic working FEA for a half-plane with a hole.**

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