elasticity.py

import numpy as np
import scipy.sparse as sp
from scipy.sparse.linalg import spsolve


def elasticity(x):
    # x (2D array): contains 1's and 0's
    # penal (positive float): affects penalization of cells, but has no effect here since no gray values

    penal = 1
    nely, nelx = np.shape(x)

    ## MATERIAL PROPERTIES
    EO = 1
    Emin = 1e-9
    nu = 0.3

    ## PREPARE FINITE ELEMENT ANALYSIS
    A11 = np.array([[12, 3, -6, -3], [3, 12, 3, 0], [-6, 3, 12, -3], [-3, 0, -3, 12]])
    A12 = np.array([[-6, -3, 0, 3], [-3, -6, -3, -6], [0, -3, -6, 3], [3, -6, 3, -6]])
    B11 = np.array([[-4, 3, -2, 9], [3, -4, -9, 4], [-2, -9, -4, -3], [9, 4, -3, -4]])
    B12 = np.array([[2, -3, 4, -9], [-3, 2, 9, -2], [4, 9, 2, 3], [-9, -2, 3, 2]])

    KE = 1 / (1 - nu**2) / 24 * (np.block([[A11, A12], [np.transpose(A12), A11]]) + nu * np.block([[B11, B12], [np.transpose(B12), B11]]))

    nodenrs = np.arange(1, (1 + nelx) * (1 + nely) + 1).reshape((1 + nely, 1 + nelx), order="F")
    edofVec = np.reshape(2 * nodenrs[:-1, :-1] + 1, (nelx * nely, 1), order="F")
    edofMat = np.tile(edofVec, (1, 8)) + np.tile(np.concatenate(([0, 1], 2*nely+np.array([2,3,0,1]), [-2, -1])), (nelx*nely, 1))

    iK = np.reshape(np.kron(edofMat, np.ones((8, 1))).T, (64 * nelx * nely, 1), order='F') # Need order F to match the reshaping of Matlab
    jK = np.reshape(np.kron(edofMat, np.ones((1, 8))).T, (64 * nelx * nely, 1), order='F')

    ## PERIODIC BOUNDARY CONDITIONS
    e0 = np.eye(3)
    ufixed = np.zeros((8, 3))
    U = np.zeros((2*(nely+1)*(nelx+1), 3))

    alldofs = np.arange(1, 2*(nely+1)*(nelx+1)+1)
    n1 = np.concatenate((nodenrs[-1, [0, -1]], nodenrs[0, [-1,0]]))
    d1 = np.reshape(([[(2*n1-1)], [2*n1]]), (1, 8), order='F')  # four corner points of the object
    n3 = np.concatenate((nodenrs[1:-1, 0].T, nodenrs[-1, 1:-1]))
    d3 = np.reshape(([[(2*n3-1)], [2*n3]]), (1, 2*(nelx+nely-2)), order='F')  # Left and Bottom boundaries
    n4 = np.concatenate((nodenrs[1:-1, -1].flatten(), nodenrs[0, 1:-1].flatten()))
    d4 = np.reshape(([[(2*n4-1)], [2*n4]]), (1, 2*(nelx+nely-2)), order='F')  # Right and Top Boundaries
    d2 = np.setdiff1d(alldofs, np.hstack([d1, d3, d4]))  # All internal nodes in the shape

    for j in range(0, 3):
        ufixed[2:4, j] = np.array([[e0[0, j], e0[2, j] / 2], [e0[2, j] / 2, e0[1, j]]]) @ np.array([nelx, 0])
        ufixed[6:8, j] = np.array([[e0[0, j], e0[2, j]/2], [e0[2, j]/2, e0[1, j]]]) @ np.array([0, nely])
        ufixed[4:6, j] = ufixed[2:4, j]+ufixed[6:8, j]

    wfixed = np.concatenate((np.tile(ufixed[2:4, :], (nely-1, 1)), np.tile(ufixed[6:8, :], (nelx-1, 1))))

    ## INITIALIZE ITERATION
    qe = np.empty((3, 3), dtype=object)
    Q = np.zeros((3, 3))
    xPhys = x

    '''
    # For printing out large arrays
    with np.printoptions(threshold=np.inf):
        K_copy[np.abs(K_copy) < 0.000001] = 0
        print(K_copy)
    '''

    ## FE-ANALYSIS
    sK = (KE.flatten(order='F')[:, np.newaxis] * (Emin+xPhys.flatten(order='F').T**penal*(EO-Emin)))[np.newaxis, :].reshape(-1, 1, order='F')

    K = sp.coo_matrix((sK.flatten(order='F'), (iK.flatten(order='F')-1, jK.flatten(order='F')-1)), shape=(np.shape(alldofs)[0], np.shape(alldofs)[0]))
    K = (K + K.transpose())/2
    K = sp.csr_matrix(np.nan_to_num(K))  # Remove the NAN values and replace them with 0's and large finite values
    K = K.tocsc()  # Converting to csc from csr transposes the matrix to match the orientation in MATLAB


    k1 = K[d2-1][:, d2-1]
    k2 = (K[d2[:,None]-1, d3-1] + K[d2[:,None]-1, d4-1])
    k3 = (K[d3-1, d2[:, None]-1] + K[d4-1, d2[:, None]-1])
    k4 = K[d3-1,d3.T-1] + K[d4-1, d4.T-1] + K[d3-1, d4.T-1] + K[d4-1, d3.T-1]

    Kr_top = sp.hstack((k1, k2))
    Kr_bottom = sp.hstack((k3.T, k4))
    Kr = sp.vstack((Kr_top, Kr_bottom)).tocsc()

    U[d1-1, :] = ufixed
    U[np.concatenate((d2-1, d3.ravel()-1)), :] = spsolve(Kr, ((-sp.vstack((K[d2[:, None]-1, d1-1], (K[d3-1, d1.T-1]+K[d4-1, d1.T-1]).T)))*ufixed-sp.vstack((K[d2-1, d4.T-1].T, (K[d3-1, d4.T-1]+K[d4-1, d4.T-1]).T))*wfixed))
    U[d4-1, :] = U[d3-1, :]+wfixed


    ## OBJECTIVE FUNCTION AND SENSITIVITY ANALYSIS
    for i in range(0, 3):
        for j in range(0, 3):
            U1 = U[:, i] # not quite right
            U2 = U[:, j]
            qe[i, j] = np.reshape(np.sum((U1[edofMat-1] @ KE) * U2[edofMat-1], axis=1), (nely, nelx), order='F') / (nelx * nely)
            Q[i, j] = sum(sum((Emin+xPhys**penal*(EO-Emin))*qe[i, j]))
    Q[np.abs(Q) < 0.000001] = 0

    return Q