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# A 165 LINE TOPOLOGY OPTIMIZATION CODE BY NIELS AAGE AND VILLADS EGEDE JOHANSEN, JANUARY 2013 |
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from __future__ import division |
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import numpy as np |
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from scipy.sparse import coo_matrix |
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from scipy.sparse.linalg import spsolve |
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from matplotlib import colors |
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import matplotlib.pyplot as plt |
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# MAIN DRIVER |
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def main(nelx,nely,volfrac,penal,rmin,ft,sysargv): |
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print("Minimum compliance problem with OC") |
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print("ndes: " + str(nelx) + " x " + str(nely)) |
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print("volfrac: " + str(volfrac) + ", rmin: " + str(rmin) + ", penal: " + str(penal)) |
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print("Filter method: " + ["Sensitivity based","Density based"][ft]) |
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# Max and min stiffness |
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Emin=1e-9 |
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Emax=1.0 |
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# dofs: |
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ndof = 2*(nelx+1)*(nely+1) |
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# Allocate design variables (as array), initialize and allocate sens. |
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x=volfrac * np.ones(nely*nelx,dtype=float) |
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xold=x.copy() |
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xPhys=x.copy() |
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g=0 # must be initialized to use the NGuyen/Paulino OC approach |
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dc=np.zeros((nely,nelx), dtype=float) |
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# FE: Build the index vectors for the for coo matrix format. |
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KE=lk() |
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edofMat=np.zeros((nelx*nely,8),dtype=int) |
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for elx in range(nelx): |
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for ely in range(nely): |
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el = ely+elx*nely |
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n1=(nely+1)*elx+ely |
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n2=(nely+1)*(elx+1)+ely |
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edofMat[el,:]=np.array([2*n1+2, 2*n1+3, 2*n2+2, 2*n2+3,2*n2, 2*n2+1, 2*n1, 2*n1+1]) |
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# Construct the index pointers for the coo format |
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iK = np.kron(edofMat,np.ones((8,1))).flatten() |
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jK = np.kron(edofMat,np.ones((1,8))).flatten() |
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# Filter: Build (and assemble) the index+data vectors for the coo matrix format |
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nfilter=int(nelx*nely*((2*(np.ceil(rmin)-1)+1)**2)) |
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iH = np.zeros(nfilter) |
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jH = np.zeros(nfilter) |
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sH = np.zeros(nfilter) |
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cc=0 |
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for i in range(nelx): |
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for j in range(nely): |
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row=i*nely+j |
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kk1=int(np.maximum(i-(np.ceil(rmin)-1),0)) |
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kk2=int(np.minimum(i+np.ceil(rmin),nelx)) |
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ll1=int(np.maximum(j-(np.ceil(rmin)-1),0)) |
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ll2=int(np.minimum(j+np.ceil(rmin),nely)) |
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for k in range(kk1,kk2): |
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for l in range(ll1,ll2): |
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col=k*nely+l |
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fac=rmin-np.sqrt(((i-k)*(i-k)+(j-l)*(j-l))) |
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iH[cc]=row |
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jH[cc]=col |
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sH[cc]=np.maximum(0.0,fac) |
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cc=cc+1 |
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# Finalize assembly and convert to csc format |
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H=coo_matrix((sH,(iH,jH)),shape=(nelx*nely,nelx*nely)).tocsc() |
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Hs=H.sum(1) |
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# BC's and support |
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dofs=np.arange(2*(nelx+1)*(nely+1)) |
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fixed=np.union1d(dofs[0:2*(nely+1):2],np.array([2*(nelx+1)*(nely+1)-1])) |
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free=np.setdiff1d(dofs,fixed) |
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# Solution and RHS vectors |
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f=np.zeros((ndof,1)) |
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u=np.zeros((ndof,1)) |
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# Set load |
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f[1,0]=-1 |
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# Initialize plot and plot the initial design |
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plt.ion() # Ensure that redrawing is possible |
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fig,ax = plt.subplots() |
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im = ax.imshow(-xPhys.reshape((nelx,nely)).T, cmap='gray',\ |
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interpolation='none',norm=colors.Normalize(vmin=-1,vmax=0)) |
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fig.show() |
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# Set loop counter and gradient vectors |
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loop=0 |
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change=1 |
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dv = np.ones(nely*nelx) |
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dc = np.ones(nely*nelx) |
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ce = np.ones(nely*nelx) |
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while change>0.01 and loop<2000: |
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loop=loop+1 |
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# Setup and solve FE problem |
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sK=((KE.flatten()[np.newaxis]).T*(Emin+(xPhys)**penal*(Emax-Emin))).flatten(order='F') |
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K = coo_matrix((sK,(iK,jK)),shape=(ndof,ndof)).tocsc() |
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# Remove constrained dofs from matrix |
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K = K[free,:][:,free] |
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# Solve system |
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u[free,0]=spsolve(K,f[free,0]) |
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|
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# print(f.shape, f) |
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# print(K.shape, K) |
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# print(f[free,0]) |
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# print(u.shape, u) |
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|
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# Objective and sensitivity |
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ce[:] = (np.dot(u[edofMat].reshape(nelx*nely,8),KE) * u[edofMat].reshape(nelx*nely,8) ).sum(1) |
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obj=( (Emin+xPhys**penal*(Emax-Emin))*ce ).sum() |
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dc[:]=(-penal*xPhys**(penal-1)*(Emax-Emin))*ce |
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dv[:] = np.ones(nely*nelx) |
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# Sensitivity filtering: |
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if ft==0: |
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dc[:] = np.asarray((H*(x*dc))[np.newaxis].T/Hs)[:,0] / np.maximum(0.001,x) |
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elif ft==1: |
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dc[:] = np.asarray(H*(dc[np.newaxis].T/Hs))[:,0] |
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dv[:] = np.asarray(H*(dv[np.newaxis].T/Hs))[:,0] |
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# Optimality criteria |
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xold[:]=x |
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(x[:],g)=oc(nelx,nely,x,volfrac,dc,dv,g) |
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# Filter design variables |
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if ft==0: xPhys[:]=x |
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elif ft==1: xPhys[:]=np.asarray(H*x[np.newaxis].T/Hs)[:,0] |
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# Compute the change by the inf. norm |
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change=np.linalg.norm(x.reshape(nelx*nely,1)-xold.reshape(nelx*nely,1),np.inf) |
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# Plot to screen |
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im.set_array(-xPhys.reshape((nelx,nely)).T) |
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fig.canvas.draw() |
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# Write iteration history to screen (req. Python 2.6 or newer) |
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print("it.: {0} , obj.: {1:.3f} Vol.: {2:.3f}, ch.: {3:.3f}".format(\ |
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loop,obj,(g+volfrac*nelx*nely)/(nelx*nely),change)) |
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plt.show() |
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|
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# np.savetxt("./results/top88_output" + str(loop) + '.csv', xPhys.reshape((nelx,nely)).T, delimiter=',') |
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# np.save('results/top88_output' + str(loop) + '.npy',xPhys.reshape((nelx,nely)).T) |
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# plt.savefig('images/top88_output' + str(loop) + '.png') |
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|
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# print(f.shape, f) |
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# print(K.shape, K) |
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# print(f[free,0]) |
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# print(u.shape, u) |
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print(u.reshape(nelx+1,nely+1,2)) |
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|
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# Make sure the plot stays and that the shell remains |
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np.save('results/top88_xPhys' + '_' +str(sysargv[1]) + '_' + str(sysargv[2]) + '.npy',xPhys.reshape((nelx,nely)).T) |
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np.save('results/top88_u' + '_' +str(sysargv[1]) + '_' + str(sysargv[2]) + '.npy',u) |
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np.savetxt('results/top88_xPhys' + '_' +str(sysargv[1]) + '_' + str(sysargv[2]) + '.csv', xPhys.reshape((nelx,nely)).T, delimiter=',') |
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plt.savefig('results/top88_xPhys' + '_' +str(sysargv[1]) + '_' + str(sysargv[2]) + '.png') |
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plt.show() |
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print("Press any key...") |
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#element stiffness matrix |
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def lk(): |
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E=1 |
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nu=0.3 |
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k=np.array([1/2-nu/6,1/8+nu/8,-1/4-nu/12,-1/8+3*nu/8,-1/4+nu/12,-1/8-nu/8,nu/6,1/8-3*nu/8]) |
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KE = E/(1-nu**2)*np.array([ [k[0], k[1], k[2], k[3], k[4], k[5], k[6], k[7]], |
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[k[1], k[0], k[7], k[6], k[5], k[4], k[3], k[2]], |
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[k[2], k[7], k[0], k[5], k[6], k[3], k[4], k[1]], |
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[k[3], k[6], k[5], k[0], k[7], k[2], k[1], k[4]], |
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[k[4], k[5], k[6], k[7], k[0], k[1], k[2], k[3]], |
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[k[5], k[4], k[3], k[2], k[1], k[0], k[7], k[6]], |
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[k[6], k[3], k[4], k[1], k[2], k[7], k[0], k[5]], |
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[k[7], k[2], k[1], k[4], k[3], k[6], k[5], k[0]] ]); |
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return (KE) |
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# Optimality criterion |
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def oc(nelx,nely,x,volfrac,dc,dv,g): |
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l1=0 |
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l2=1e9 |
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move=0.2 |
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# reshape to perform vector operations |
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xnew=np.zeros(nelx*nely) |
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while (l2-l1)/(l1+l2)>1e-3: |
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lmid=0.5*(l2+l1) |
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xnew[:]= np.maximum(0.0,np.maximum(x-move,np.minimum(1.0,np.minimum(x+move,x*np.sqrt(-dc/dv/lmid))))) |
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gt=g+np.sum((dv*(xnew-x))) |
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if gt>0 : |
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l1=lmid |
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else: |
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l2=lmid |
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return (xnew,gt) |
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# The real main driver |
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if __name__ == "__main__": |
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# Default input parameters |
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nelx=180 |
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nely=60 |
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volfrac=0.4 |
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rmin=5.4 |
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penal=3.0 |
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ft=1 # ft==0 -> sens, ft==1 -> dens |
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import sys |
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if len(sys.argv)>1: nelx =int(sys.argv[1]) |
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if len(sys.argv)>2: nely =int(sys.argv[2]) |
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if len(sys.argv)>3: volfrac=float(sys.argv[3]) |
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if len(sys.argv)>4: rmin =float(sys.argv[4]) |
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if len(sys.argv)>5: penal =float(sys.argv[5]) |
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if len(sys.argv)>6: ft =int(sys.argv[6]) |
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main(nelx,nely,volfrac,penal,rmin,ft,sys.argv) |
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