def DDG_uv(U, V, dV):
u = U[0]
v = U[1]
u_ = V[0]
v_ = V[1]
du = dV[0]
dv = dV[1]
em = emin(u, v)
eM = emax(u, v)
abs_u = pow((pow(u[0], 2) + pow(u[1], 2)), 1 / 2)
martEnergy_uu = -3. / 2. * (1. / 3. * v / abs_u - 1. / 2.) * inner(u_, du)
stiffEnergy_uu = -3./2.*(1./3.*v/abs_u - 1./2.)*inner(u_, du) \
+ 1./2.*((v/abs_u + 1./2.))*inner(du, u_)
martEnergy_vv = 1. / 3. * dv * v_
stiffEnergy_vv = 1. / 3. * dv * v_ + dv * v_
isLiquid = And(And(gt(em, -1 / 3), lt(eM, -2 * em)), gt(eM, -em / 2))
isMartensite = And(And(lt(em, -1 / 3), lt(eM, -2 * em)),
gt(eM, -em / 2 + 1 / 2))
isStiff = conditional(
And(And(lt(em, -1 / 3), gt(eM, -em / 2.)), lt(eM, -em / 2 + 1 / 2)),
stiffEnergy_uu + stiffEnergy_vv, 0)
isSolid = conditional(isMartensite, martEnergy_uu + martEnergy_vv, isStiff)
eff_density_uv = conditional(isLiquid, 0., isSolid)
return eff_density_uv * dx
def DG_uv(U, V):
u = U[0]
v = U[1]
u_ = V[0]
v_ = V[1]
em = emin(u, v)
eM = emax(u, v)
abs_u = pow((pow(u[0], 2) + pow(u[1], 2)), 1 / 2)
martEnergy_u = 3. / 2. * (em - 1. / 3.) * 2 * (-1. / 2.) * inner(
u, u_) / abs_u
stiffEnergy_u = 3./2.*(em - 1./3.)*2*(-1./2.)*inner(u, u_)/abs_u \
+ (em + 2.*eM-1)*inner(u, u_)/abs_u
martEnergy_v = 3. / 2. * (em - 1. / 3.) * 2 * 1. / 3. * v_
stiffEnergy_v = 3./2.*(em - 1./3.)*2*1./3.*v_ \
+ (em + 2.*eM-1)*v_
isLiquid = And(And(gt(em, -1 / 3), lt(eM, -2 * em)), gt(eM, -em / 2))
isMartensite = And(And(lt(em, -1 / 3), lt(eM, -2 * em)),
gt(eM, -em / 2 + 1 / 2))
isStiff = conditional(
And(And(lt(em, -1 / 3), gt(eM, -em / 2.)), lt(eM, -em / 2 + 1 / 2)),
stiffEnergy_u + stiffEnergy_v, 0)
isSolid = conditional(isMartensite, martEnergy_u + martEnergy_v, isStiff)
eff_density_uv = conditional(isLiquid, 0., isSolid)
return eff_density_uv * dx
def G(U):
v = U[1]
stiffEnergy = 1 / 2 * 2. / 3. * pow((v - 1.), 2.)
martEnergy = 1 / 2 * 2. / 3. * pow((v + 1. / 2.), 2.)
# isLiquid = And(And(gt(em, -1/3), lt(eM,-2.*em)), gt(eM, - em/2.))
isLiquid = And(gt(v, -1 / 2), lt(v, 1.))
isElastic = gt(v, 1.)
isMartensite = lt(v, -1. / 2.)
isSolid = conditional(isMartensite, martEnergy, stiffEnergy)
eff_density = conditional(isLiquid, 0., isSolid)
return eff_density * dx
def DDG_uv(U, V, dV):
v = U[1]
v_ = V[1]
dv = dV[1]
stiffEnergy = 2. / 3. * dv * v_
martEnergy = 2. / 3. * dv * v_
# isLiquid = And(And(gt(em, -1/3), lt(eM,-2.*em)), gt(eM, - em/2.))
isLiquid = And(gt(v, -1 / 2), lt(v, 1.))
isElastic = gt(v, 1.)
isMartensite = lt(v, -1. / 2.)
isSolid = conditional(isMartensite, martEnergy, stiffEnergy)
eff_density = conditional(isLiquid, 0., isSolid)
return eff_density * dx
def DG_uv(U, V):
v = U[1]
v_ = V[1]
# em = emin(u, v)
# eM = emax(u, v)
stiffEnergy = 2. / 3. * (v - 1.) * v_
martEnergy = 2. / 3. * (v + 1. / 2.) * v_
# isLiquid = And(And(gt(em, -1/3), lt(eM,-2.*em)), gt(eM, - em/2.))
isLiquid = And(gt(v, -1 / 2), lt(v, 1.))
isElastic = gt(v, 1.)
isMartensite = lt(v, -1. / 2.)
isSolid = conditional(isMartensite, martEnergy, stiffEnergy)
eff_density = conditional(isLiquid, 0., isSolid)
return eff_density * dx
def phase(u, v):
# phase of dofs
em = emin(u, v)
eM = emax(u, v)
isLiquid = And(And(gt(em, -1 / 3), lt(eM, -2 * em)), gt(eM, -em / 2))
isMartensite = And(And(lt(em, -1 / 3), lt(eM, -2 * em)),
gt(eM, -em / 2 + 1 / 2))
isStiff = conditional(
And(And(lt(em, -1 / 3), gt(eM, -em / 2.)), lt(eM, -em / 2 + 1 / 2)), 3,
-1)
isSolid = conditional(isMartensite, 2, isStiff)
# phase = conditional(isLiquid, 1, isSolid)
phase = conditional(isLiquid, 1, 0)
return phase
def H(v):
em = v / 6.
eM = conditional(gt(-v, 0), -v / 3., +v / 6.)
martEnergy = 3. / 2. * pow((em - 1. / 3.), 2.)
stiffEnergy = 3. / 2. * pow((em - 1. / 3.), 2.) + 1. / 2. * pow(
(em + 2. * eM - 1.), 2.)
isLiquid = And(And(gt(em, -1 / 3), lt(eM, -2 * em)), gt(eM, -em / 2))
isMartensite = And(And(lt(em, -1 / 3), lt(eM, -2 * em)),
gt(eM, -em / 2 + 1 / 2))
isStiff = conditional(
And(And(lt(em, -1 / 3), gt(eM, -em / 2.)), lt(eM, -em / 2 + 1 / 2)),
stiffEnergy, 0)
isSolid = conditional(isMartensite, martEnergy, isStiff)
eff_density = conditional(isLiquid, 0., isSolid)
return eff_density * dx + lmbda / (2. * mu) * v * v * dx
def G(U):
u = U[0]
v = U[1]
em = emin(u, v)
eM = emax(u, v)
martEnergy = 3. / 2. * pow((em + 1. / 3.), 2.)
stiffEnergy = 3. / 2. * pow((em + 1. / 3.), 2.) + 1. / 2. * pow(
(em + 2. * eM - 1.), 2.)
isLiquid = And(And(gt(em, -1 / 3), lt(eM, -2. * em)), gt(eM, -em / 2.))
isMartensite = And(And(lt(em, -1 / 3), lt(eM, -2 * em)),
gt(eM, -em / 2 + 1 / 2))
isStiff = conditional(
And(And(lt(em, -1 / 3), gt(eM, -em / 2.)), lt(eM, -em / 2 + 1 / 2)),
stiffEnergy, 0)
isSolid = conditional(isMartensite, martEnergy, isStiff)
eff_density = conditional(isLiquid, 0., isSolid)
return eff_density * dx
def G(self, U):
u = U[0]
v = U[1]
em = self.emin(u, v)
eM = self.emax(u, v)
martEnergy = 3. / 2. * pow((em + 1. / 3.), 2.)
stiffEnergy = 3. / 2. * pow((em + 1. / 3.), 2.) + 1. / 2. * pow(
(em + 2. * eM - 1.), 2.)
isLiquid = And(And(gt(em, -1 / 3), lt(eM, -2. * em)), gt(eM, -em / 2.))
isMartensite = And(And(lt(em, -1 / 3), lt(eM, -2 * em)),
gt(eM, -em / 2 + 1 / 2))
isStiff = conditional(
And(And(lt(em, -1 / 3), gt(eM, -em / 2.)),
lt(eM, -em / 2 + 1 / 2)), stiffEnergy, 0)
isSolid = conditional(isMartensite, martEnergy, isStiff)
eff_density = conditional(isLiquid, 0., isSolid)
_lmbda = project(Constant(self.lmbda),
FunctionSpace(self.mesh, 'DG', 0))
return eff_density * self.dx + _lmbda / (2. *
self.mu) * v * v * self.dx
def isElastic(em, eM):
return And(And(lt(em, -1 / 3), gt(eM, -em / 2.)), lt(eM, -em / 2 + 1 / 2))
def isMartensite(em, eM):
return And(And(lt(em, -1 / 3), lt(eM, -2 * em)), gt(eM, -em / 2 + 1 / 2))
def isLiquid(em, eM):
return And(And(gt(em, -1 / 3), lt(eM, -2. * em + DOLFIN_EPS)),
gt(eM, -em / 2. - DOLFIN_EPS))
