Reg. non-square system of matrix

[Himanshu Mishra]

Hello everyone,

I am solving an IVP problem in a Cartesian box with dimension (x,y,z), periodic in x and y. I am solving 3 equations along with the boundary conditions. I am getting an error, 

"raise ValueError("Non-square system: group={}, I={}, J={}".format(self.group, valid_eqn.nnz, valid_var.nnz))

ValueError: Non-square system: group=(0, 0, None), I=368, J=1017

" 

My code snippet looks like,

#fields################################################

p = dist.Field(name='p', bases=(xbasis,ybasis,zbasis))
T = dist.Field(name='T', bases=(xbasis,ybasis,zbasis))
u = dist.VectorField(coords, name='u', bases=(xbasis,ybasis,zbasis))
us= dist.VectorField(coords, name='us', bases=(xbasis,ybasis,zbasis))
ff=dist.VectorField(coords, name='ff', bases=(xbasis,ybasis,zbasis))
gg=dist.VectorField(coords, name='gg', bases=(xbasis,ybasis,zbasis))


# tau variables

tau_u1=dist.VectorField(coords, name='tau_u1', bases=(xbasis,ybasis))
tau_u2=dist.VectorField(coords, name='tau_u2', bases=(xbasis,ybasis))
tau_p=dist.Field( name='tau_p') #used in continuity eq
tau_T4 = dist.Field(name='tau_T4', bases=(xbasis,ybasis))
tau_T5 = dist.Field(name='tau_T5', bases=(xbasis,ybasis))

#substitutions##########################################

# defining the derivative
dx=lambda A: d3.Differentiate(A, coords['x'])
dy=lambda A: d3.Differentiate(A, coords['y'])
dz=lambda A: d3.Differentiate(A, coords['z'])

# defining the grad_var
ex,ey,ez=coords.unit_vector_fields(dist)
lift_basis=zbasis.derivative_basis(1)
lift= lambda A: d3.Lift(A, lift_basis,-1)

#distribution of local grid for defining the real space variables
x, y, z = dist.local_grids(xbasis, ybasis, zbasis)
us['g'][0]=a_s* np.exp(2 * k_s *z)
us['g'][1]=0
us['g'][2]=0

ff['g'][1]=0
ff['g'][0]=0
ff['g'][2]=f
gg['g'][2]=g
gg['g'][0]=0
gg['g'][1]=0


Gt=dz(T)-lift(tau_T4)
Gu=d3.grad(u)-ez*lift(tau_u1)

problem= d3.IVP([p,T,u,us,ff,gg,tau_u1,tau_u2,tau_p,tau_T4,tau_T5], namespace=locals())

#continuity eq
 
problem.add_equation("trace(Gu)+tau_p = 0")

#governing equations
problem.add_equation("dt(T)-kappa*(lap(T))+lift(tau_T4)+lift(tau_T5)=-(u+us)@grad(T)")

#momentum equations
problem.add_equation("dt(u)+(1/rho_0)*grad(p)-nu*div(Gu)+lift(tau_u2)=-u@grad(u)+cross(us,curl(u))-cross(ff,(u+us))+gg*alpha*(T-T_0)")

#boundary conditions
problem.add_equation("dz(T)(z=0)=0") 
problem.add_equation("dz(u@ex)(z=-Lz)=0")   
problem.add_equation("dz(u@ey)(z=0)=0")
problem.add_equation("dz(T)(z=-Lz)=-q/(kappa*c_p*rho_0)")
problem.add_equation("dz(u@ex)(z=0)=tau/(nu*rho_0)") 
problem.add_equation("dz(u@ey)(z=-Lz)=0")
problem.add_equation("u(z=0)@ez=0") 

#pressure gauge choice
problem.add_equation("integ(p) = 0")

My number of field variables equals to number of "add equations", however, I am not sure why I am not getting square matrices? I will greatly appreciate any suggestions.

Thank You

Himanshu Mishra

[Daniel Lecoanet]

Hi Himanshu,

The issue is that you have 6 (x,y,z) dependent variables (p, T, u, us, ff, gg), 4 (x, y) dependent tau variables (tau_u1, tau_u2, tau_T4, tau_T5), and one spatially-independent variable (tau_p). For a square system, that requires 6 equations, 2 scalar boundary conditions, 2 vector boundary conditions, and 1 gauge condition. You have entered 3 equations, 2 scalar BCs, not quite 2 vector BCs (you only specify BCs on 5 components of the velocity, not 6) and 1 gauge condition. So your system has many more variables than equations.

Daniel

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