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# AXI-SYMMETRIC NEWTONIAN MODEL ; UPDATED LAGRANGIAN FORMULATION |
# AXI-SYMMETRIC NEWTONIAN MODEL ; UPDATED LAGRANGIAN FORMULATION |
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# step 1 rho*(v_star-v) = dt * (sigma'_ij,j-teta3*p,i+f_i) |
# step 1 rho*(v_star-v) = dt * (sigma'_ij,j-teta3*p,i+f_i) |
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# step 2 dp=-dt*B*(v_j,j+teta1*v_star_j,j-dt*teta1*((1-teta3)*p_,jj+teta2*dp_,jj)) |
# step 2 dp=-dt*B*(v_j,j+teta1*v_star_j,j-dt*teta1*((1-teta3)*p_,jj+teta2*dp_,jj)) |
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# step 3 rho*(v+-v) = -dt*((1-teta3)*p_,jj+teta2*dp_,jj) |
# step 3 rho*(v+-v) = -dt*((1-teta3)*p_,jj+teta2*dp_,jj) |
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# step 4 p+=1/2(p+dp+abs(p+dp)) |
# step 3b p+=1/2(p+dp+abs(p+dp)) |
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# step 4 sigma'i+_ij,j=f(v+,p+,...) |
# step 4 sigma'i+_ij,j=f(v+,p+,...) |
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from esys.escript import * |
from esys.escript import * |
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from esys.escript.linearPDEs import LinearSinglePDE, LinearPDESystem |
from esys.escript.linearPDEs import LinearSinglePDE, LinearPDESystem |
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from esys.finley import Rectangle |
from esys.finley import Rectangle |
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iter = 0 |
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nstep = 3000 |
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w_step=max(int(nstep/50),1)*0+1 |
nel = 20 |
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mstep = 5 |
H = 0.5 |
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H = 0.5 |
L = 1.0 |
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eta = 1.0 # shear viscosity |
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ro = 1.0 |
eta = 1.0 # shear viscosity |
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g = 1.00 |
ro = 1.0 |
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tauy0 = 0.9*ro*g*H/sqrt(3) #0.11/(3*sqrt(3)) |
g = 1.00 |
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Pen = 100 |
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vtop = 0.01 |
alpha_w = 1.00 |
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dt = 10**(-10) |
alpha = 1.00*1000000. |
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small = EPSILON |
Pen=0. |
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alpha = 1.00 |
B=100. |
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nel = 20 |
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# len = sqrt(0.5/nel) |
nstep = 3000 |
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toler = 0.001 |
dt = 1. |
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alphaw = 1.00 |
small = EPSILON |
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L = 1.0 |
w_step=max(int(nstep/50),1)*0+1 |
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teta1 = 0.5 |
toler = 0.001 |
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teta2 = 0.5 |
teta1 = 0.5 |
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teta3 = 0 # =0 split A; =1 split B |
teta2 = 0.5 |
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Etau=1000000000. |
teta3 = 1 # =0 split A; =1 split B |
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# create domain: |
# create domain: |
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dom=Rectangle(int(nel*L/min(L,H)),int(nel*H/min(L,H)),order=1, l0=L, l1=H) |
dom=Rectangle(int(nel*L/min(L,H)),int(nel*H/min(L,H)),order=1, l0=L, l1=H) |
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x=dom.getX() |
x=dom.getX() |
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momentumStep1=LinearPDESystem(dom) # A momentumStep1 |
momentumStep1=LinearPDESystem(dom) |
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momentumStep1.setValue(q=whereZero(x[0])*[1.,0.]) # +whereZero(x[1])*[1.,1.]) |
momentumStep1.setValue(q=whereZero(x[0])*[1.,0.]+whereZero(x[1])*[0.,1.]) # fix x0=0 and x1=0 |
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face_mask=whereZero(FunctionOnBoundary(dom).getX()[1]) |
face_mask=whereZero(FunctionOnBoundary(dom).getX()[1]) |
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# b 1 U1_2={0.0} |
pressureStep2=LinearSinglePDE(dom) |
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# b 1 U1_1={0,0} |
pressureStep2.setReducedOrderOn() |
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# b 4 U1_1={0.0} |
pressureStep2.setValue(q=whereZero(x[0]-L)+whereZero(x[1]-H)) |
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pressureStep2=LinearSinglePDE(dom) # A [reduced] pressureStep2 |
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pressureStep2.setReducedOrderOn() # |
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pressureStep2.setValue(q=whereZero(x[0]-L)+whereZero(x[1]-H)) |
momentumStep3=LinearPDESystem(dom) |
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# b free U1={0.0} ? |
momentumStep3.setValue(q=whereZero(x[0])*[1.,0.]+whereZero(x[1])*[0.,1.]) |
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# |
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# initial values: |
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# |
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momentumStep3=LinearPDESystem(dom) # A momentumStep3 |
U=Vector(0.,Solution(dom)) |
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momentumStep3.setValue(q=whereZero(x[0])*[1.,0.]) # +whereZero(x[1])*[1.,1.]) |
p=ro*g*(L-ReducedSolution(dom).getX()[0])*(H-ReducedSolution(dom).getX()[1])/3 |
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# b 1 U1_2={0.0} |
# p=ro*g*(H-ReducedSolution(dom).getX()[1]) |
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# b 1 U1_1={0,0} |
stress=Tensor(0.,Function(dom)) |
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# b 4 U1_1={0.0} |
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t=dt |
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istep=0 |
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# A deformation |
while istep < nstep: |
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# e V700= [grad] V100 |
istep=istep+1 |
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# e V901= V701+V704 +V101/(X1+small) |
print "time step :",istep," t = ",t |
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# e V601=sqrt(2*((V701)^2+(V704)^2+(V101/(X1+small))^2+\ |
r=Function(dom).getX()[0] |
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# (V702+V703)^2/2)) |
r_b=FunctionOnBoundary(dom).getX()[0] |
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# e U1={abs(V901)/(V601+small)} |
print " volume : ",integrate(r) |
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# |
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# step 1: |
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# calculate normal |
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# A stress |
n_d=dom.getNormal() |
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# e V700= [grad] V100 |
t_d=matrixmult(numarray.array([[0.,-1.],[1.,0]]),n_d) |
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# e V901= V701+V704 +V101/(X1+small) |
sigma_d=(sign(inner(t_d,U))*alpha_w*t_d-n_d)*Pen*clip(inner(n_d,U),0.) |
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# e V601=sqrt(2*((V701-V901)^2+(V704-V901)^2+(V101/(X1+small)-V901)^2+\ |
print " sigma_d =",inf(sigma_d),sup(sigma_d) |
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# (V702+V703)^2/2)) |
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# e V602=((eta*V601<alpha*V401)*eta+(eta*V601>(alpha*V401-small))*(alpha*V401/(V601+small))) |
momentumStep1.setValue(D=ro*kronecker(dom), |
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# e V704=2*V602*V704-V401 |
Y=ro*U+dt*((stress[:,0]-p*teta3*kronecker(dom)[:,0])/r+[0.,-ro*g]), |
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# b 3 [integrated] {V704} |
# Y=r*ro*U+dt*r*[0.,-ro*g], |
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# |
X=-dt*(stress-teta3*p*kronecker(dom)), |
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# A surface |
y=sigma_d*face_mask) |
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# b 3 [integrated] {1} |
U_star=momentumStep1.getSolution() |
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# |
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# step 2: |
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U=Vector(0.,Solution(dom)) # V100=0 |
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p=ro*g*(L-ReducedSolution(dom).getX()[0])*(H-ReducedSolution(dom).getX()[1])/3 # V401=ro*g*(L-X1)*(H-X2)/3 |
# U2=U+teta1*(U_star-U) |
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t=dt |
U2=U+teta1*U_star |
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istep=0 |
gg2=grad(U2) |
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En=0 |
div_U2=gg2[0,0]+gg2[1,1]+U2[0]/r |
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grad_p=grad(p) |
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while istep < nstep: |
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istep=istep+1 |
pressureStep2.setValue(A=r*dt*B*teta1*teta2/ro*dt*kronecker(dom), |
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print "time step :",istep," t = ",t |
D=r, |
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Y=-dt*B*r*div_U2, |
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n_d=dom.getNormal() |
X=-r*B*dt**2/ro*teta1*(1-teta3)*grad_p) |
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t_d=matrixmult(numarray.array([[0.,1.],[-1.,0]]),n_d) |
dp=pressureStep2.getSolution() |
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# |
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s=-En*inner(U,n_d)*face_mask |
# step 3: |
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nStressn=s*wherePositive(s) |
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u_boundary=Etau*inner(U,t_d)*face_mask |
p2=(1-teta3)*p+teta2*dp |
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m=whereNegative(u_boundary-nStressn) |
grad_p2=grad(p2) |
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nStressTau=nStressn*(1.-m)+u_boundary*m |
momentumStep3.setValue(D=r*ro*kronecker(dom), |
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print nStressn |
Y=r*(ro*U_star-dt*teta2*grad_p2)) |
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# print nStressn*n_d+nStressTau*t_d |
U_new=momentumStep3.getSolution() |
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if istep == 20: |
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# print nStressTau |
# update: |
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saveVTK("stress.vtu",s=(nStressn*n_d+nStressTau*t_d)) |
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1/0 |
p+=dp |
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# print nStressTau |
U=U_new |
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print " U:",inf(U),sup(U) |
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r=dom.getX()[0] |
print " P:",inf(p),sup(p) |
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# A viscosity |
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gg=grad(U) # V700= [grad] V100 |
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vol=gg[0,0]+gg[1,1]+U[0]/(r+small) # V901= V701+V704 +V101/(X1+small) |
p_pos=clip(p,small) |
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tau=sqrt(2*((gg[0,0]-vol/3)**2+(gg[1,1]-vol/3)**2+(U[0]/(r+small)-vol/3)**2+(gg[1,0]+gg[0,1])**2/2)) |
gg=grad(U) |
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# V601=sqrt(2*((V701-V901/3)^2+(V704-V901/3)^2+(V101/(X1+small)-V901/3)^2+(V702+V703)^2/2)) |
vol=gg[0,0]+gg[1,1]+U[0]/r |
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m=whereNegative(eta*tau-alpha*p) # eta*V601<alpha*V401 |
gamma=sqrt(2*((gg[0,0]-vol/3)**2+(gg[1,1]-vol/3)**2+(U[0]/r-vol/3)**2+(gg[1,0]+gg[0,1])**2/2)) |
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eta_d=m*eta+(1.-m)*alpha*p/(tau+small) # |
m=whereNegative(eta*gamma-alpha*p_pos) |
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# U1={(eta*V601<alpha*V401)*eta+(eta*V601>(alpha*V401-small))*(alpha*V401/(V601+small))} |
eta_d=m*eta+(1.-m)*alpha*p_pos/(gamma+small) |
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# viscosity 1 100 V100 200 V200 300 V300 400 V400 |
print " viscosity =",inf(eta_d),sup(eta_d) |
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# solve |
stress=eta_d*(symmetric(gg)-2./3.*vol*kronecker(dom)) |
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print " viscosity =",inf(eta_d),sup(eta_d) # show V801 |
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stress=eta_d*(symmetric(gg)-2./3.*vol*kronecker(dom)) |
# step size control: |
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# solve momentum equationStep1 |
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# momentumStep1 1 100 V100 200 V200 300 V300 400 V400 |
len=inf(dom.getSize()) |
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momentumStep1.setValue(D=ro/dt*kronecker(dom), # {ro/dt}U1_i |
dt1=inf(dom.getSize()/(length(U)+small)) |
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Y=stress[:,0]/(r+small)+[0.,-ro*g], # -{0.0,ro*g}_i |
dt2=inf(0.5*ro*(len**2)/eta_d) |
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X=-(stress+p*kronecker(dom)), # -D_j{V602}D_j{V100}_i-D_i{V602}D_j{V100}_j+D_i{(2/3)*V602}D_j{V100}_j-D_i{V401} |
dt=dt1*dt2/(dt1+dt2) |
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y=-(nStressn*n_d+nStressTau*t_d)) |
print " new step size = ",dt |
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dU2=momentumStep1.getSolution() |
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# solve |
# update geometry |
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# dU2-> V100 |
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# U -> V200 |
dom.setX(dom.getX()+U*dt) |
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# p -> V500 |
t=t+dt |
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if (istep-1)%w_step==0:saveVTK("u.%d.xml"%((istep-1)/w_step),p=p,eta=eta_d,U=U_star,U_star=U_star,gamma=gamma) |
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# pressureStep2 1 100 V100 200 V200 300 V300 400 V500 |
if istep == 3: 1/0 |
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gg2=grad(dU2) |
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div_dU2=gg2[0,0]+gg2[1,1]+dU2[0]/(r+small) |
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pressureStep2.setValue(A=r*teta1*teta2*dt**2*kronecker(dom), # -D_j{teta1*teta2*dt^2}D_jU1 |
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D=r*ro, # {ro/Pen}U1 |
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Y=-r*(ro*dt*vol+teta1*ro*dt*div_dU2)) # -{ro*dt}D_j{V200}_j-{teta1*ro*dt}D_j{V100}_j |
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# solve |
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dp=pressureStep2.getSolution() |
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# dp -> V100 |
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# dU2 -> V200 |
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# U -> V300 |
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# p -> V600 |
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p+=dp # V601=V601+V101 |
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# V701=V101 : dp -> V700 |
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p=(p+abs(p))/2+small # V601=cut V601 |
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# V601=V601+small |
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print " pressure increment :",inf(dp),sup(dp) # show V601 |
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print " pressure range :",inf(p),sup(p) # show V601 |
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# popp |
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# dU2 -> V100 |
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# U -> V200 |
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# p -> V500 |
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# dp -> V600 |
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# momentumStep3 1 100 V100 200 V200 300 V300 400 V600 |
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A2=Tensor4(0.0,Function(dom)) |
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for i in range(dom.getDim()): |
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for j in range(dom.getDim()): |
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A2[i,i,j,j]+=1 |
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momentumStep3.setValue(A=Pen*A2, # -D_i{Pen}D_jU1_j |
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D=ro/dt*kronecker(dom), # {ro/dt}U1_i |
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Y=(ro/dt*(dU2+U)-teta2*grad(dp))) |
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# Y=r*(ro/dt*(dU2+U)-teta2*dp*[1.,0]), # {ro/dt}{V200}_i+{ro/dt}{V100}_i |
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# X=r*teta2*dp**kronecker(dom)) # -D_i{teta2*V401} |
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U, U_old=momentumStep3.getSolution(), U |
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# U -> V100 |
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# dU2 -> V200 |
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# U_old -> V300 |
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# p -> V600 |
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# dp -> V700 |
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# V400=V300 |
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# V300=V100 |
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# popp |
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# popp |
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# U -> V100 |
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# U_old -> V200 |
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# p -> V400 |
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# dp -> V500 |
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# deformation |
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# solve |
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# show ratio |
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# show V101 |
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# popp |
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print " new U:",inf(U),sup(U) # show v100 |
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print " old U",inf(U_old),sup(U_old) # show v200 |
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print " relative change:",Lsup(U_old)/Lsup(U) # test=L1 V200 |
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# test0=L1 V100 |
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# show test/test0 |
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# black |
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# arrow |
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vmax=Lsup(U) # vmax1=abs(max V100) |
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# vmax2=abs(min V100) |
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# show vmax1 |
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# show vmax2 |
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# vmax=vmax1 |
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# if vmax < vmax2 |
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# vmax=vmax2 |
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# endif |
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print " max velocity = ",vmax # show vmax |
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len=inf(dom.getSize()) |
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dt1=len/vmax # Courant condition |
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dt2=0.5*ro*(len**2)/eta |
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dt=dt1*dt2/(dt1+dt2) |
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En=1/dt |
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t=t+dt |
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print " time , time step ",t,dt # show t |
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# show dt |
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dom.setX(dom.getX()+U*dt) # mapm 500 |
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# V500=V500+V100*dt |
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# V600=V500 |
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# mapm 500 |
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print " volume : ",integrate(r)# vol=Out |
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# show vol |
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# dU -> V100 |
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# U -> V200 |
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# p -> V400 |
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# dp -> V600 |
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# clear |
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# prim |
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# saveVTK("u.%d.xml"%(istep),p=p,eta=eta_d,U=U,dU2=dU2,tau=tau,dp=dp) |
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if (istep-1)%w_step==0:saveVTK("u.%d.xml"%((istep-1)/w_step),p=p,eta=eta_d,U=U,dU2=dU2,tau=tau,dp=dp) |
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