def basal_beta():
# surface elevation
s = alpha * H
b = sp.Function("b", real=True)(x)
nx_b, ny_b, nz_b = list(grad(b - z))
db = (s - H).subs(H,
H_exact).factor(fraction=False).diff(x).subs(H_exact, H)
# normalized x component of the downward-pointing normal vector
norm = (db**2 + 1**2)**S("1/2")
nx_n = db / norm
nz_n = -1 / norm
N_base = {nx: nx_n, ny: 0, nz: nz_n}
I_base = 2 * eta(u_exact, v_exact, 3) * M(u_exact, v_exact).row(0) * N
I_base = I_base[0].subs(N_base).doit()
BC_basal = I_base.subs(H, H_exact).doit().subs(H_exact, H)
sp.var("beta", positive=True)
eq_beta = sp.Eq(BC_basal + beta * u, 0)
return solve(eq_beta, beta)[0].subs(u, Q_0 / H)
def basal_bc(u0, v0):
N_bed = {nx: 0, ny: 0, nz: -1}
I = (2 * eta(u0, v0, n) * M(u0, v0).row(0) * N)[0].subs(N_bed)
x_p = sp.symbols("x_p", positive=True)
return I.subs(z, 0).subs(constants()).subs(x, x_p).subs(x_p, x).simplify()
def surface_bc():
u0, v0 = exact()
return (2 * eta(u0, v0, n) * M(u0, v0).row(0) * N)[0].subs({
nx: 0,
ny: 0,
nz: 1
})
def lateral_bc():
u0, v0 = exact()
return (2 * eta(u0, v0, n) * M(u0, v0).row(0) * N)[0].subs({
nx: 1,
ny: 0,
nz: 0,
x: L
})
def surface_bc(u0, v0, surface):
ds = surface.diff(x)
# normalized x component of the downward-pointing normal vector
n_s_norm = (ds**2 + 1**2)**S("1/2")
nx_s = -ds / n_s_norm
ny_s = 0
nz_s = 1 / n_s_norm
N_surface = {nx: nx_s, ny: ny_s, nz: nz_s}
return (2 * eta(u0, v0, n) * M(u0, v0).row(0) * N)[0].subs(N_surface)
def source_lateral():
"Lateral BC"
N_right = {nx: 1, ny: 0, nz: 0}
I = 2 * eta(u_exact, v_exact, 3) * M(u_exact, v_exact).row(0) * N
I = I[0].subs(N_right).subs(H, H_exact).doit()
# tell SymPy that x is positive to make it simplify the expression
I = I.subs(x, x_p)
I = I.subs(x_p, x).subs(H_exact, H)
return I, 0
def surface_bc():
# surface elevation
s = alpha * H
ds = s.diff(x)
# normalized x component of the upward-pointing normal vector
n_s_norm = (ds**2 + 1**2)**S("1/2")
nx_s = -ds / n_s_norm
ny_s = 0
nz_s = 1 / n_s_norm
N_surface = {nx: nx_s, ny: ny_s, nz: nz_s}
f_s = (2 * eta(u_exact, v_exact, 3) * M(u_exact, v_exact).row(0) * N)
f_s = f_s[0].subs(N_surface).subs(H, H_exact).doit()
f_s = f_s.subs(H_exact, H).factor()
return f_s, 0
def lateral_bc(u0, v0):
N_right = {nx: 1, ny: 0, nz: 0}
return (2 * eta(u0, v0, n) * M(u0, v0).row(0) * N)[0].subs(N_right), 0.0