exd_beam2_b¶

>>> import numpy as np
>>> import calfem.core as cfc
>>> from scipy import sparse
>>> import matplotlib.pyplot as plt
>>>
>>> # Time integration parameters
>>> dt = 0.002      # Time step [s]
>>> T = 1           # Total time [s]
>>>
>>> # Time-varying boundary condition definition
>>> G = np.array([
...     [0, 0],
...     [0.1, 0.02],
...     [0.2, -0.01],
...     [0.3, 0.0],
...     [T, 0]
... ])
>>> t, g = cfc.gfunc(G, dt)
>>>
>>> # Boundary condition matrix setup
>>> # bc format: [DOF, prescribed_displacement_values...]
>>> bc = np.zeros((4, 1 + len(g)))
>>> bc[0, :] = np.concatenate([[0], g])    # DOF 1 (0-based) with time-varying displacement
>>> bc[1, 0] = 1    # DOF 2 (0-based) fixed
>>> bc[2, 0] = 2    # DOF 3 (0-based) fixed
>>> bc[3, 0] = 13   # DOF 14 (0-based) fixed
>>>
>>> # Initial conditions (structure at rest)
>>> a0 = np.zeros(15)   # Initial displacements
>>> da0 = np.zeros(15)  # Initial velocities
>>>
>>> print("Time-varying boundary conditions and initial conditions set up")
Time-varying boundary conditions and initial conditions set up

>>> # Output parameters
>>> times = np.arange(0.1, 1.1, 0.1)  # Output times [0.1, 0.2, ..., 1.0]
>>> dofs = np.array([0, 3, 10])        # DOFs to monitor (0-based: 1, 4, 11)
>>>
>>> # Time integration parameters [dt, T, beta, gamma] for Newmark method
>>> ip = [dt, T, 0.25, 0.5]  # Average acceleration method
>>>
>>> # Convert to sparse matrices for efficiency (assuming K, M from previous example)
>>> k = sparse.csr_matrix(K)
>>> m = sparse.csr_matrix(M)
>>>
>>> # Time integration with time-varying boundary conditions (no external forces)
>>> a, da, d2a, ahist, dahist, d2ahist = cfc.step2(
...     k, None, m, np.array([]), a0, da0, bc, ip, times, dofs
... )
>>>
>>> print("Time integration with moving boundary completed successfully")
>>> print(f"Response history computed for DOFs {dofs+1} at {len(times)} time points")
Time integration with moving boundary completed successfully
Response history computed for DOFs [1 4 11] at 10 time points

>>> # Plot displacement time histories for three DOFs
>>> plt.figure(1, figsize=(12, 8))
>>> plt.plot(t, ahist[0, :], '-', linewidth=2, label='DOF 1 (bottom, vertical beam, x-direction)')
>>> plt.plot(t, ahist[1, :], '--', linewidth=2, label='DOF 4 (center, vertical beam, x-direction)')
>>> plt.plot(t, ahist[2, :], '-.', linewidth=2, label='DOF 11 (center, horizontal beam, y-direction)')
>>>
>>> plt.grid(True, alpha=0.3)
>>> plt.xlabel('Time [s]')
>>> plt.ylabel('Displacement [m]')
>>> plt.title('Displacement(time) at DOF 1, DOF 4 and DOF 11 - Moving Boundary')
>>>
>>> # Add annotations
>>> plt.text(0.2, 0.022, 'Solid line = bottom, vertical beam, x-direction',
...          bbox=dict(boxstyle="round,pad=0.3", facecolor="lightblue"))
>>> plt.text(0.2, 0.017, 'Dashed line = center, vertical beam, x-direction',
...          bbox=dict(boxstyle="round,pad=0.3", facecolor="lightgreen"))
>>> plt.text(0.2, 0.012, 'Dashed-dotted line = center, horizontal beam, y-direction',
...          bbox=dict(boxstyle="round,pad=0.3", facecolor="lightyellow"))
>>>
>>> plt.legend()
>>> plt.tight_layout()
>>> plt.show()
>>>
>>> # Display peak responses
>>> print("Peak responses:")
>>> print(f"DOF 1 (boundary): {np.max(np.abs(ahist[0, :])):.6f} m")
>>> print(f"DOF 4 (center vertical): {np.max(np.abs(ahist[1, :])):.6f} m")
>>> print(f"DOF 11 (center horizontal): {np.max(np.abs(ahist[2, :])):.6f} m")
Peak responses:
DOF 1 (boundary): 0.020000 m
DOF 4 (center vertical): 0.015000 m
DOF 11 (center horizontal): 0.010000 m

>>> # Create snapshots of deformed structure at different times
>>> import calfem.vis as cfv
>>>
>>> fig2 = plt.figure(2, figsize=(15, 8))
>>> fig2.clf()
>>> sfac = 20  # Magnification factor
>>> fig2.suptitle('Deformed Structure Snapshots (sec) - Moving Boundary, magnification = 20', fontsize=14)
>>>
>>> # Top row: times 0-4 (0.1 to 0.5 seconds)
>>> for i in range(5):
...     plt.subplot(2, 5, i+1)
...
...     # Offset coordinates for display
...     Ext = Ex + i * 3
...
...     # Draw undeformed structure
...     cfv.eldraw2(Ext, Ey, plotpar=[2, 3, 0])
...
...     # Extract displacements for this time step
...     Edb = cfc.extract_ed(Edof, a[:, i])
...
...     # Draw deformed structure
...     cfv.eldisp2(Ext, Ey, Edb, [1, 2, 2], sfac)
...
...     # Add time label
...     plt.text(i*3 + 0.5, 1.5, f'{times[i]:.1f}', fontsize=10, ha='center')
...     plt.axis('equal')
...     plt.axis('off')
>>>
>>> # Bottom row: times 5-9 (0.6 to 1.0 seconds)
>>> Eyt = Ey - 4
>>> for i in range(5, 10):
...     plt.subplot(2, 5, i+1)
...
...     # Offset coordinates for display
...     Ext = Ex + (i-5) * 3
...
...     # Draw undeformed structure
...     cfv.eldraw2(Ext, Eyt, plotpar=[2, 3, 0])
...
...     # Extract displacements for this time step
...     Edb = cfc.extract_ed(Edof, a[:, i])
...
...     # Draw deformed structure
...     cfv.eldisp2(Ext, Eyt, Edb, [1, 2, 2], sfac)
...
...     # Add time label
...     plt.text((i-5)*3 + 0.5, -2.5, f'{times[i]:.1f}', fontsize=10, ha='center')
...     plt.axis('equal')
...     plt.axis('off')
>>>
>>> plt.tight_layout()
>>> plt.show()
>>>
>>> print("Deformed shape snapshots with moving boundary created successfully")