deeplearning/modulus/modulus-sym-v110/_modules/modulus/sym/eq/pdes/diffusion.html

Sym v1.1.0

Source code for modulus.sym.eq.pdes.diffusion

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"""Diffusion equation
"""

from sympy import Symbol, Function, Number

from modulus.sym.eq.pde import PDE
from modulus.sym.node import Node


[docs]class Diffusion(PDE): """ Diffusion equation Parameters ========== T : str The dependent variable. D : float, Sympy Symbol/Expr, str Diffusivity. If `D` is a str then it is converted to Sympy Function of form 'D(x,y,z,t)'. If 'D' is a Sympy Symbol or Expression then this is substituted into the equation. Q : float, Sympy Symbol/Expr, str The source term. If `Q` is a str then it is converted to Sympy Function of form 'Q(x,y,z,t)'. If 'Q' is a Sympy Symbol or Expression then this is substituted into the equation. Default is 0. dim : int Dimension of the diffusion equation (1, 2, or 3). Default is 3. time : bool If time-dependent equations or not. Default is True. mixed_form: bool If True, use the mixed formulation of the diffusion equations. Examples ======== >>> diff = Diffusion(D=0.1, Q=1, dim=2) >>> diff.pprint() diffusion_T: T__t - 0.1*T__x__x - 0.1*T__y__y - 1 >>> diff = Diffusion(T='u', D='D', Q='Q', dim=3, time=False) >>> diff.pprint() diffusion_u: -D*u__x__x - D*u__y__y - D*u__z__z - Q - D__x*u__x - D__y*u__y - D__z*u__z """ name = "Diffusion" def __init__(self, T="T", D="D", Q=0, dim=3, time=True, mixed_form=False): # set params self.T = T self.dim = dim self.time = time self.mixed_form = mixed_form # coordinates x, y, z = Symbol("x"), Symbol("y"), Symbol("z") # time t = Symbol("t") # make input variables input_variables = {"x": x, "y": y, "z": z, "t": t} if self.dim == 1: input_variables.pop("y") input_variables.pop("z") elif self.dim == 2: input_variables.pop("z") if not self.time: input_variables.pop("t") # Temperature assert type(T) == str, "T needs to be string" T = Function(T)(*input_variables) # Diffusivity if type(D) is str: D = Function(D)(*input_variables) elif type(D) in [float, int]: D = Number(D) # Source if type(Q) is str: Q = Function(Q)(*input_variables) elif type(Q) in [float, int]: Q = Number(Q) # set equations self.equations = {} if not self.mixed_form: self.equations["diffusion_" + self.T] = ( T.diff(t) - (D * T.diff(x)).diff(x) - (D * T.diff(y)).diff(y) - (D * T.diff(z)).diff(z) - Q ) elif self.mixed_form: T_x = Function("T_x")(*input_variables) T_y = Function("T_y")(*input_variables) if self.dim == 3: T_z = Function("T_z")(*input_variables) else: T_z = Number(0) self.equations["diffusion_" + self.T] = ( T.diff(t) - (D * T_x).diff(x) - (D * T_y).diff(y) - (D * T_z).diff(z) - Q ) self.equations["compatibility_T_x"] = T.diff(x) - T_x self.equations["compatibility_T_y"] = T.diff(y) - T_y self.equations["compatibility_T_z"] = T.diff(z) - T_z self.equations["compatibility_T_xy"] = T_x.diff(y) - T_y.diff(x) self.equations["compatibility_T_xz"] = T_x.diff(z) - T_z.diff(x) self.equations["compatibility_T_yz"] = T_y.diff(z) - T_z.diff(y) if self.dim == 2: self.equations.pop("compatibility_T_z") self.equations.pop("compatibility_T_xz") self.equations.pop("compatibility_T_yz")
[docs]class DiffusionInterface(PDE): """ Matches the boundary conditions at an interface Parameters ========== T_1, T_2 : str Dependent variables to match the boundary conditions at the interface. D_1, D_2 : float Diffusivity at the interface. dim : int Dimension of the equations (1, 2, or 3). Default is 3. time : bool If time-dependent equations or not. Default is True. Example ======== >>> diff = DiffusionInterface('theta_s', 'theta_f', 0.1, 0.05, dim=2) >>> diff.pprint() diffusion_interface_dirichlet_theta_s_theta_f: -theta_f + theta_s diffusion_interface_neumann_theta_s_theta_f: -0.05*normal_x*theta_f__x + 0.1*normal_x*theta_s__x - 0.05*normal_y*theta_f__y + 0.1*normal_y*theta_s__y """ name = "DiffusionInterface" def __init__(self, T_1, T_2, D_1, D_2, dim=3, time=True): # set params self.T_1 = T_1 self.T_2 = T_2 self.D_1 = D_1 self.D_2 = D_2 self.dim = dim self.time = time # coordinates x, y, z = Symbol("x"), Symbol("y"), Symbol("z") normal_x, normal_y, normal_z = ( Symbol("normal_x"), Symbol("normal_y"), Symbol("normal_z"), ) # time t = Symbol("t") # make input variables input_variables = {"x": x, "y": y, "z": z, "t": t} if self.dim == 1: input_variables.pop("y") input_variables.pop("z") elif self.dim == 2: input_variables.pop("z") if not self.time: input_variables.pop("t") # variables to match the boundary conditions (example Temperature) T_1 = Function(T_1)(*input_variables) T_2 = Function(T_2)(*input_variables) # set equations self.equations = {} self.equations["diffusion_interface_dirichlet_" + self.T_1 + "_" + self.T_2] = ( T_1 - T_2 ) flux_1 = self.D_1 * ( normal_x * T_1.diff(x) + normal_y * T_1.diff(y) + normal_z * T_1.diff(z) ) flux_2 = self.D_2 * ( normal_x * T_2.diff(x) + normal_y * T_2.diff(y) + normal_z * T_2.diff(z) ) self.equations["diffusion_interface_neumann_" + self.T_1 + "_" + self.T_2] = ( flux_1 - flux_2 )
© Copyright 2023, NVIDIA Modulus Team. Last updated on Oct 17, 2023.