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accuracyVerification_Steady.py
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accuracyVerification_Steady.py
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'''
Convergence-in-space test
Coupled large strain poroelasticity and chemotaxis
NeoHookean material
P1b-P1-P1 discretisation for displ-porosity-pressure
P1-P1 elements for the chemotaxis
Backward Euler scheme for the time discretisation
Unit square, manufactured solutions
Mixed boundary conditions
'''
from fenics import *
import sympy2fenics as sf
parameters["form_compiler"]["representation"] = "uflacs"
parameters["form_compiler"]["cpp_optimize"] = True
parameters["form_compiler"]["quadrature_degree"] = 3
strain = lambda v: sym(grad(v))
kappa = lambda J,phi: kappa0*(1+ (1-phi0)**2/phi0**3*J*phi**3*(J-phi)**2) # Kozeny-Carman
react_p = lambda phi,cp,cl: phi*cp*(gammap - lmbdrp*cl)
react_l = lambda phi,cp,cl: phi*cp*cl * lmbdpr
ell = lambda p, cp: Sv*Lp0*(1.0+Lbp*cp) * (pc - p - sigma0*(pic-pii)/(1.+Lbp*cp)) \
- l0*(1.+vmax*(p-p0)**nHill/(km**nHill+(p-p0)**nHill))
def str2exp(s):
return sf.sympy2exp(sf.str2sympy(s))
# time constants
dt = 0.01; Tfinal = 3.*dt;
# ********* model constants setting to 1 most of them ******* #
# Poroelasticity
lmbdas = Constant(36.4)
mus = Constant(2.1)
rhos = Constant(1.)
rhof = Constant(1.)
kappa0 = Constant(1.)
alpha = Constant(0.5)
muf = Constant(1.)
# parameters?
Dp = Constant(0.9)
Dl = Constant(0.8)
Lp0 = Constant(1.)
Lbp = Constant(1.)
sigma0 = Constant(1.)
Lbr = Constant(1.)
vmax = Constant(1.)
gammap = Constant(1.)
gammal = Constant(1.)
lmbdrp = Constant(1.)
lmbdpr = Constant(1.)
clmax = Constant(1.)
mul = Constant(1.)
p0 = Constant(1.)
km = Constant(1.)
pic = Constant(1.)
pc = Constant(1.)
l0 = Constant(1.)
pii = Constant(1.)
nHill = Constant(2.)
chi = Constant(1.)
Sv = Constant(1.)
I = Identity(2)
uinf = Constant(0.25)
# ******* Exact solutions for error analysis ****** #
u_str = '(uinf*(sin(pi*x)*cos(pi*y)+x*x*0.5/lmbdas)*t,uinf*(-cos(pi*x)*sin(pi*y)+y*y*0.5/lmbdas)*t)'
dt_u_str = '(uinf*(sin(pi*x)*cos(pi*y)+x*x*0.5/lmbdas),uinf*(-cos(pi*x)*sin(pi*y)+y*y*0.5/lmbdas))'
cp_str = 't*(0.3*exp(x)+0.1*cos(pi*x)*cos(pi*y))'
dt_cp_str = '(0.3*exp(x)+0.1*cos(pi*x)*cos(pi*y))'
cl_str = 't*(0.3*exp(x)+0.1*sin(pi*x)*sin(pi*y))'
dt_cl_str = '0.3*exp(x)+0.1*sin(pi*x)*sin(pi*y)'
p_str = 'sin(pi*x*y)*cos(pi*x*y)*t'
dt_p_str = 'sin(pi*x*y)*cos(pi*x*y)'
phi0_str = '0.6+0.1*sin(x*y)'
# phi depends on J and on phi0...
nkmax = 4
hh = []; nn = []; ephi = []; rphi = []; eu = []; ru = []; ep = []; rp = [];
ecp = []; rcp = []; ecl = []; rcl = [];
ru.append(0.0); rp.append(0.0); rphi.append(0.0); rcp.append(0.0); rcl.append(0.0)
# ***** Error analysis ***** #
for nk in range(nkmax):
print("....... Refinement level : nk = ", nk)
nps = pow(2,nk+1); mesh = UnitSquareMesh(nps,nps)
n = FacetNormal(mesh)
hh.append(mesh.hmax())
t = 0
phi0 = Expression(str2exp(phi0_str), degree=6, domain=mesh)
cp_ex = Expression(str2exp(cp_str), t = t, degree=6, domain=mesh)
dt_cp_ex = Expression(str2exp(dt_cp_str), t = t, degree=6, domain=mesh)
cl_ex = Expression(str2exp(cl_str), t = t, degree=6, domain=mesh)
dt_cl_ex = Expression(str2exp(dt_cl_str), t = t, degree=6, domain=mesh)
u_ex = Expression(str2exp(u_str), t = t, uinf= uinf, lmbdas=lmbdas, degree=6, domain=mesh)
dt_u_ex = Expression(str2exp(dt_u_str), t = t, uinf= uinf, lmbdas=lmbdas, degree=6, domain=mesh)
p_ex = Expression(str2exp(p_str), t = t, degree=6, domain=mesh)
dt_p_ex = Expression(str2exp(dt_p_str), t = t, degree=6, domain=mesh)
F_ex = I + grad(u_ex)
J_ex = det(F_ex)
invF_ex = inv(F_ex)
#NEO HOOKEAN
PP_ex = mus*(F_ex - invF_ex.T) \
+ lmbdas*ln(J_ex)*invF_ex.T \
- alpha*p_ex*J_ex*invF_ex.T
phi_ex = J_ex - 1 + phi0
dt_phi_ex = J_ex * inner(grad(dt_u_ex).T,invF_ex) # Zheng2020 eq. (43)
bb_ex = - div(PP_ex) / rhos
pflux_ex = phi_ex*J_ex*invF_ex*kappa(J_ex,phi_ex)/muf*invF_ex.T*grad(p_ex)
ll_ex = rhof*J_ex*dt_phi_ex - rhof*J_ex*ell(p_ex,cp_ex) \
- div(pflux_ex)
mp_ex = J_ex*(dt_phi_ex*cp_ex+phi_ex*dt_cp_ex) \
- div(phi_ex*J_ex*invF_ex*Dp*invF_ex.T*grad(cp_ex)) \
- J_ex* react_p(phi_ex,cp_ex,cl_ex)
ml_ex = J_ex*(dt_phi_ex*cl_ex+phi_ex*dt_cl_ex) \
- div(phi_ex*J_ex*invF_ex*Dl*invF_ex.T*grad(cl_ex) \
- phi_ex*chi*cl_ex*J_ex*invF_ex*invF_ex.T*grad(cp_ex)) \
- J_ex * react_l(phi_ex,cp_ex,cl_ex)
# ********* Finite dimensional spaces ********* #
P1 = FiniteElement("CG", mesh.ufl_cell(), 1)
Bub = FiniteElement("Bubble", mesh.ufl_cell(), 3)
P1b = VectorElement(P1 + Bub)
Hh = FunctionSpace(mesh,MixedElement([P1,P1,P1b,P1,P1]))
print("**************** Total Dofs = ", Hh.dim())
nn.append(Hh.dim())
Sol = Function(Hh); dSol = TrialFunction(Hh)
cp,cl,u,phi,p = split(Sol)
wp,wl,v,psi,q = TestFunctions(Hh)
# Essential BCs - product space
bdry = MeshFunction("size_t", mesh, 1)
bdry.set_all(0)
Gamma = CompiledSubDomain(" near(x[0],0.0) && on_boundary")
Sigma = CompiledSubDomain("(near(x[0],1.0) || near(x[1],1.0) || near(x[1],0.0)) && on_boundary")
Gamma.mark(bdry,91); Sigma.mark(bdry,92);
ds = Measure("ds", subdomain_data=bdry)
cpold = interpolate(cp_ex, Hh.sub(0).collapse())
clold = interpolate(cl_ex, Hh.sub(1).collapse())
uold = project(u_ex, Hh.sub(2).collapse())
phiold = project(phi_ex, Hh.sub(3).collapse())
pold = interpolate(p_ex, Hh.sub(4).collapse())
# ******** Weak form ********** #
F = I + grad(u); J = det(F); invF = inv(F)
P_eff = mus*(F - invF.T) + lmbdas*ln(J)*invF.T
F1 = inner(P_eff - alpha*p*J*invF.T, grad(v)) * dx \
+ psi*(J-1-phi+phi0)*dx \
+ rhof*J/dt*(phi-phiold)*q * dx \
+ dot(phi*J*invF*kappa(J,phi)/muf*invF.T*grad(p),grad(q)) * dx \
- rhos*dot(bb_ex,v)* dx \
- dot(PP_ex*n,v) * ds(92) \
- rhof*J*ell(p,cp)*q*dx \
- ll_ex * q * dx \
- dot(pflux_ex,n) * q * ds(91)
F2 = J/dt*((phi-phiold)*cp+phi*(cp-cpold))*wp*dx \
+ dot(phi*J*invF*Dp*invF.T*grad(cp),grad(wp)) * dx \
- J*react_p(phi,cp,cl)*wp*dx \
- mp_ex*wp*dx\
+ J/dt*((phi-phiold)*cl+phi*(cl-clold))*wl*dx \
+ dot(phi*J*invF*Dl*invF.T*grad(cl),grad(wl)) * dx \
- dot(phi*chi*cl*J*invF*invF.T*grad(cp),grad(wl)) * dx \
- J*react_l(phi,cp,cl)*wl*dx \
- ml_ex*wl*dx
FF = F1 + F2
Tang = derivative(FF,Sol,dSol)
# ********* Time loop ************* #
while (t < Tfinal):
t += dt
print("t=%.2f" % t)
cp_ex.t = t; dt_cp_ex.t = t;
cl_ex.t = t; dt_cl_ex.t = t;
u_ex.t = t; p_ex.t = t; dt_p_ex.t = t;
dt_u_ex.t = t;
bcCp = DirichletBC(Hh.sub(0), cp_ex, 'on_boundary')
bcCl = DirichletBC(Hh.sub(1), cl_ex, 'on_boundary')
bcP = DirichletBC(Hh.sub(4), p_ex, bdry, 92)
u_D = project(u_ex, Hh.sub(2).collapse())
bcU = DirichletBC(Hh.sub(2), u_D, bdry, 91)
bcH = [bcCp,bcCl,bcU,bcP]
solve(FF == 0, Sol, J=Tang, bcs=bcH, \
solver_parameters={'newton_solver':{'linear_solver':'mumps',\
'absolute_tolerance':1.0e-6,\
'relative_tolerance':1.0e-6}})
cph,clh,uh,phih,ph = Sol.split()
assign(cpold,cph); assign(clold,clh); assign(uold,uh)
assign(phiold,phih); assign(pold,ph)
ecp.append(errornorm(cp_ex,cph,'H1'))
ecl.append(errornorm(cl_ex,clh,'H1'))
eu.append(errornorm(u_ex,uh,'H1'))
ephi.append(pow(assemble((phi_ex-phih)**2*dx),0.5))
ep.append(errornorm(p_ex,ph,'H1'))
if(nk>0):
rcp.append(ln(ecp[nk]/ecp[nk-1])/ln(hh[nk]/hh[nk-1]))
rcl.append(ln(ecl[nk]/ecl[nk-1])/ln(hh[nk]/hh[nk-1]))
ru.append(ln(eu[nk]/eu[nk-1])/ln(hh[nk]/hh[nk-1]))
rphi.append(ln(ephi[nk]/ephi[nk-1])/ln(hh[nk]/hh[nk-1]))
rp.append(ln(ep[nk]/ep[nk-1])/ln(hh[nk]/hh[nk-1]))
# ******** Generating error history **** #
print('nn & hh & e(cp) & r(cp) & e(cl) & r(cl) & e(u) & r(u) & e(phi) & r(phi) & e(p) & r(p) ')
print('==========================================================================')
for nk in range(nkmax):
print('%d & %4.4g & %4.4g & %4.4g & %4.4g & %4.4g & %4.4g & %4.4g & %4.4g & %4.4g & %4.4g & %4.4g \\\ ' % (nn[nk], hh[nk], ecp[nk], rcp[nk], ecl[nk], rcl[nk], eu[nk], ru[nk], ephi[nk], rphi[nk], ep[nk], rp[nk]))
# ************* End **************** #