poisson-transform 0.2.2

Poisson Solver in Tranformed 2D Space using Finite Difference

Poisson 2D Transform

This is a finite difference solver for poisson equations in curved 2d space.

Many different poisson solvers exist but few solve for curved-spaces.

This package provides that in a very simple to use interface.

How to use

Install

pip install poisson_transform

Use

First you specify the following inputs:

• f: The RHS of poisson's equation (as a function that takes input coordinates and outputs a value). If empty will set f(x, y)=1.
• g: the Boundary conditions of poisson's equation. If empty will set BCs to Dirichlet.
  • g should be a function that takes in coordinates and returns (a, b, g) for the boundary condition such that a*u + b*∂u/∂n = g on ∂Ω (boundary of domain)
    • For example, returning (1, 0, 0) sets u=0 which is a dirichlet conditions on that point.
    • For example, returning (0, 1, 0) sets ∂u/∂n=0 which is a neumann conditions on that point.
    • For example, returning (1, 2, 3) sets u + 2*∂u/∂n = 3 on that point.
• Specify the transformation to be used T_x, T_y where x = T_x(ksi) and y = T_y(eta). ksi and eta are the identity coordinates: $\xi \in [0, 1]$ and $\eta \in [0, 1]$. If empty then the identity transformation will be used.
  1. To do this, simply obtain the variables ksi, eta with the following line ksi, eta = Transformation.get_ksi_eta()
  2. Then, define an arbitrary transformation on the unit square Tx = ksi**2 + 0.75*ksi ; Ty = (1-eta)*(1.25*ksi) + eta*(2.75-ksi). This step can be arbitrarily complex and uses Sympy under the hood to perform the calculations. See below examples for more complex transformations such as rotated ellipses.
  3. Then, get the transformation object to be given to the solver using transformation = Transformation(ksi, eta, Tx, Ty)
• Finally, call solve_and_plot with the number of point to use for the mesh.
  • Since the above inputs can be left blank, the simplest use case is solve_and_plot(Nx=30, Ny=30) which will solve poisson's equation on the unit square with Dirichlet BCs using a mesh grid of ($30 \times 30$).
  • To provide all the above inputs, simply write solve_and_plot(Nx=29, Ny=29, transformation=transformation, f=f, g=g)

See below for example uses.

Note: This package has not been tested extensively. If you find an issue please report it (or make a pull-request if fixed)

Examples

from poisson_transform import solve_and_plot
solve_and_plot(Nx=30, Ny=30)

22:54:15 Warning: both f and g are None. Are you sure this is what you want?
22:54:15 f is None. Setting f=1
22:54:15 g is None. Setting Dirichlet BCs 
22:54:17 Integral: 0.03500889856987609

from poisson_transform import solve_and_plot
def f(x, y, ksi, eta):
    """Returns f for the equation -∇2 u = f in Ω"""
    if (x - 0.3)**2 + (y - 0.4)**2 < 0.2**2 :
        return 10
    return 0

solve_and_plot(Nx=31, Ny=31, f=f)

22:54:19 g is None. Setting Dirichlet BCs 
22:54:21 Integral: 0.06935111999589494

from poisson_transform import solve_and_plot
def f(x, y, ksi, eta):
    """Returns f for the equation -∇2 u = f in Ω"""
    if 0.45 < x < 0.55:
        return 10
    return 0
def g(x, y, ksi, eta):
    """Returns (a, b, g) for the boundary condition a*u + b*∂u/∂n = g on ∂Ω and (1, 0, g) for dirichlet conditions inside the domain)"""
    if ksi == 0:  # left, neumann = 0
        return (0, 1, 0)
    if eta == 1:  # top, dirichlet = 0
        return (1, 0, 0)
    if ksi == 1:  # right, neumann = 0
        return (0, 1, 0)
    if eta == 0:  # bottom, dirichlet = 0
        return (1, 0, 0)

solve_and_plot(Nx=31, Ny=31, f=f, g=g)

22:54:24 Integral: 0.0832355038593408

from poisson_transform import Transformation, solve_and_plot, plotGeometry
def f(x, y, ksi, eta):
    """Returns f for the equation -∇2 u = f in Ω"""
    if 0.45 < ksi < 0.55:
        return 100
    return 0
def g(x, y, ksi, eta):
    """Returns (a, b, g) for the boundary condition a*u + b*∂u/∂n = g on ∂Ω and (1, 0, g) for dirichlet conditions inside the domain)"""
    if ksi == 0:  # left, neumann = 0
        return (0, 1, 0)
    if eta == 1:  # top, dirichlet = 0
        return (1, 0, 0)
    if ksi == 1:  # right, neumann = 0
        return (0, 1, 0)
    if eta == 0:  # bottom, dirichlet = 0
        return (1, 0, 0)

ksi, eta = Transformation.get_ksi_eta()
Tx = ksi**2 + 0.75*ksi
Ty = (1-eta)*(1.25*ksi) + eta*(2.75-ksi)
transformation = Transformation(ksi, eta, Tx, Ty)
# plotGeometry(29, 29, transformation)
solve_and_plot(Nx=29, Ny=29, transformation=transformation, f=f, g=g)

22:54:28 Integral: 5.2916463228833726

from poisson_transform import Transformation, plotGeometry, solve_and_plot

ksi, eta = Transformation.get_ksi_eta()
Tx = ksi**2 + 0.1*ksi
Ty = 1/2*ksi*(1-eta) + eta
transformation = Transformation(ksi, eta, Tx, Ty)
plotGeometry(Nx=29, Ny=29, transformation=transformation)
solve_and_plot(Nx=29, Ny=29, transformation=transformation)

22:54:30 Warning: both f and g are None. Are you sure this is what you want?
22:54:30 f is None. Setting f=1
22:54:30 g is None. Setting Dirichlet BCs 
22:54:32 Integral: 0.016063603756651158

import numpy as np
from poisson_transform import Transformation, solve_and_plot
def f(x, y, ksi, eta):
    """Returns f for the equation -∇2 u = f in Ω"""
    if (0.7 < ksi < 0.8 and 0.6 < eta < 0.7) or (0.3 < ksi < 0.4 and 0.3 < eta < 0.4) or (0.1 < ksi < 0.2 and 0.4 < eta < 0.55):
        return 100
    return 0
def g(x, y, ksi, eta):
    """Returns (a, b, g) for the boundary condition a*u + b*∂u/∂n = g on ∂Ω and (1, 0, g) for dirichlet conditions inside the domain)"""
    if ksi == 0:  # left, neumann = 0
        return (0, 1, 0)
    if ksi == 1:  # right, neumann = 0
        return (0, 1, 0)
    if eta == 1:  # top, dirichlet = 0
        return (1, 0, 0)
    if eta == 0:  # bottom, dirichlet = 0
        return (1, 0, 0)

ksi, eta = Transformation.get_ksi_eta()
Tx = ksi
ellipse_bottom = (ksi-0.5)**2 + 0.2
ellipse_top = 1 - ellipse_bottom
Ty = ellipse_bottom*(1-eta) + ellipse_top*eta
rotate_phi = -1.2*np.pi/4
Tx_rot, Ty_rot = np.cos(rotate_phi)*Tx - np.sin(rotate_phi)*Ty, np.sin(rotate_phi)*Tx + np.cos(rotate_phi)*Ty
solve_and_plot(Nx=15, Ny=15, transformation=Transformation(ksi, eta, Tx_rot, Ty_rot), f=f, g=g, contour_levels=20)

22:54:35 Integral: 0.021996048708012562

import importlib
import poisson_transform
importlib.reload(poisson_transform)
import numpy as np
from poisson_transform import Transformation, solve_and_plot
def f(x, y, ksi, eta):
    """Returns f for the equation -∇2 u = f in Ω"""
    # centers = ((0.7, 0.6, 0.1), (0.3, 0.3, 0.1), (0.1, 0.4, 0.1))
    centers = ((0.7, 0.0, 0.1), (0.9, -0.4, 0.1), (0.5, 0.0, 0.1))
    if any((x - xc)**2 + (y - yc)**2 < r**2 for xc, yc, r in centers):
        return 100
    return 0
def g(x, y, ksi, eta):
    """Returns (a, b, g) for the boundary condition a*u + b*∂u/∂n = g on ∂Ω and (1, 0, g) for dirichlet conditions inside the domain)"""
    if ksi == 0:  # left, neumann = 0
        return (0, 1, 0)
    if ksi == 1:  # right, neumann = 0
        return (0, 1, 0)
    if eta == 1:  # top, dirichlet = 0
        return (1, 0, 0)
    if eta == 0:  # bottom, dirichlet = 0
        return (1, 0, 0)

ksi, eta = Transformation.get_ksi_eta()
Tx = ksi
ellipse_bottom = (ksi-0.5)**2 + 0.2
ellipse_top = 1 - ellipse_bottom
Ty = ellipse_bottom*(1-eta) + ellipse_top*eta
rotate_phi = -1.2*np.pi/4
Tx_rot, Ty_rot = np.cos(rotate_phi)*Tx - np.sin(rotate_phi)*Ty, np.sin(rotate_phi)*Tx + np.cos(rotate_phi)*Ty
solve_and_plot(Nx=15, Ny=15, transformation=Transformation(ksi, eta, Tx_rot, Ty_rot), f=f, g=g, contour_levels=20)

22:54:37 Integral: 0.20123610428427863

import sympy
from poisson_transform import Transformation, solve_and_plot, plotGeometry
def f(x, y, ksi, eta):
    """Returns f for the equation -∇2 u = f in Ω"""
    return 1
def g(x, y, ksi, eta):
    """Returns (a, b, g) for the boundary condition a*u + b*∂u/∂n = g on ∂Ω and (1, 0, g) for dirichlet conditions inside the domain)"""
    if eta == 1:  # top, neumann = 0
        return (0, 1, 0)
    if ksi == 0:  # left, dirichlet = 0
        return (1, 0, 0)
    if ksi == 1:  # right, dirichlet = 0
        return (1, 0, 0)
    if eta == 0:  # bottom, dirichlet = 0
        return (1, 0, 0)

ksi, eta = Transformation.get_ksi_eta()
Tx = (2*ksi+0.25-eta)
bottom_curve = 0.5*sympy.Abs(Tx+1e-6)  # add 1e-6 to avoid derivative at kink
Ty = bottom_curve + (1.8 - bottom_curve)*eta
plotGeometry(38, 38, Transformation(ksi, eta, Tx, Ty))
solve_and_plot(Nx=38, Ny=38, transformation=Transformation(ksi, eta, Tx, Ty), f=f, g=g)

22:54:52 Integral: 0.46734117404100667

import importlib
import poisson_transform
importlib.reload(poisson_transform)
import sympy
from poisson_transform import Transformation, solve_and_plot, plotGeometry
def f(x, y, ksi, eta):
    """Returns f for the equation -∇2 u = f in Ω"""
    return 1
def g(x, y, ksi, eta):
    """Returns (a, b, g) for the boundary condition a*u + b*∂u/∂n = g on ∂Ω and (1, 0, g) for dirichlet conditions inside the domain)"""
    if eta == 1:  # top, neumann = 0
        return (0, 1, 0)
    if ksi == 0:  # left, neumann = 0
        return (0, 1, 0)
    if ksi == 1:  # right, dirichlet = 0
        return (1, 0, 0)
    if eta == 0:  # bottom, dirichlet = 0
        return (1, 0, 0)

ll, cc, hh = 3, 0.5, 1
bb = np.sqrt((1/2*ll - hh + cc)**2 - cc**2)
ksi, eta = Transformation.get_ksi_eta()
Tx = ksi*bb
Ty = cc*ksi*(1-eta)+hh*eta
# plotGeometry(21, 21, Transformation(ksi, eta, Tx, Ty))
solve_and_plot(Nx=38, Ny=38, transformation=Transformation(ksi, eta, Tx, Ty), f=f, g=g)

22:54:57 Integral: 0.06464775169432217

from poisson_transform import Transformation, solve_and_plot
def f(x, y, ksi, eta):
    """Returns f for the equation -∇2 u = f in Ω"""
    return 0
def g(x, y, ksi, eta):
    """Returns (a, b, g) for the boundary condition a*u + b*∂u/∂n = g on ∂Ω and (1, 0, g) for dirichlet conditions inside the domain)"""
    if ksi == 0:  # left, dirichlet = 0.5
        return (1, 0, 0.5)
    if ksi == 1:  # right, dirichlet = 1.2
        return (1, 0, 1.2)
    if eta == 0:  # bottom, dirichlet = -0.75
        return (1, 0, -0.75)
    if eta == 1:  # top, dirichlet = -1
        return (1, 0, -1)

ksi, eta = Transformation.get_ksi_eta()
Tx = 12*ksi - 6
Ty = 6*eta - 3
solve_and_plot(Nx=30, Ny=30, transformation=Transformation(ksi, eta, Tx, Ty), f=f, g=g)

22:55:01 Integral: -29.042871607935275

from poisson_transform import Transformation, solve_and_plot
def f(x, y, ksi, eta):
    """Returns f for the equation -∇2 u = f in Ω"""
    return 0
def g(x, y, ksi, eta):
    """Returns (a, b, g) for the boundary condition a*u + b*∂u/∂n = g on ∂Ω and (1, 0, g) for dirichlet conditions inside the domain)"""
    if ksi == 0:  # left, dirichlet = 0.5
        return (1, 0, 0.5)
    if ksi == 1:  # right, dirichlet = 1.2
        return (1, 0, 1.2)
    if eta == 0:  # bottom, dirichlet = -0.75
        return (1, 0, -0.75)
    if eta == 1:  # top, dirichlet = -1
        return (1, 0, -1)

    # dirichlet conditions inside the domain
    if 1 < x < 1.4 and -0.5 < y < 0.2:
        return (1, 0, 1.5)

ksi, eta = Transformation.get_ksi_eta()
Tx = 12*ksi - 6
Ty = 6*eta - 3
solve_and_plot(Nx=30, Ny=30, transformation=Transformation(ksi, eta, Tx, Ty), f=f, g=g)

22:55:04 Integral: -11.898317261702779

from poisson_transform import Transformation, solve_and_plot
def f(x, y, ksi, eta):
    """Returns f for the equation -∇2 u = f in Ω"""
    return 0
def g(x, y, ksi, eta):
    """Returns (a, b, g) for the boundary condition a*u + b*∂u/∂n = g on ∂Ω and (1, 0, g) for dirichlet conditions inside the domain)"""
    if ksi == 0:  # left, neumann = 0
        return (0, 1, 0)
    if ksi == 1:  # right, neumann = 0
        return (0, 1, 0)
    if eta == 0:  # bottom, dirichlet = 0
        return (1, 0, 0)
    if eta == 1:  # top, neumann = 0
        return (0, 1, 0)

    # dirichlet conditions inside the domain
    if 1 < x < 1.4 and -0.5 < y < 0.2:
        return (1, 0, 1.5)

ksi, eta = Transformation.get_ksi_eta()
Tx = 12*ksi - 6
Ty = 6*eta - 3
solve_and_plot(Nx=50, Ny=50, transformation=Transformation(ksi, eta, Tx, Ty), f=f, g=g)

22:55:10 Integral: 35.2240775563179

from poisson_transform import Transformation, solve_and_plot
def f(x, y, ksi, eta):
    """Returns f for the equation -∇2 u = f in Ω"""
    return 0
def g(x, y, ksi, eta):
    """Returns (a, b, g) for the boundary condition a*u + b*∂u/∂n = g on ∂Ω and (1, 0, g) for dirichlet conditions inside the domain)"""
    if ksi == 0:  # left, neumann = 0
        return (0, 1, 0)
    if ksi == 1:  # right, neumann = 0
        return (0, 1, 0)
    if eta == 0:  # bottom, neumann = 0
        return (0, 1, 0)
    if eta == 1:  # top, neumann = 0
        return (0, 1, 0)

    # dirichlet conditions inside the domain
    if (x)**2 + (y)**2 < 0.4**2:
        return (1, 0, 1)
    elif (x+1.4)**2 + (y)**2 < 0.2**2:
        return (1, 0, -2)
    elif (x-1.4)**2 + (y)**2 < 0.2**2:
        return (1, 0, -2)
    elif -3.5 < x < -2 and -0.25 < y < 0.25:
        return (1, 0, 2)
    elif 2 < x < 3.5 and -0.25 < y < 0.25:
        return (1, 0, 2)

ksi, eta = Transformation.get_ksi_eta()
Tx = 8*ksi - 4
Ty = 8*eta - 4
solve_and_plot(Nx=60, Ny=60, transformation=Transformation(ksi, eta, Tx, Ty), f=f, g=g)

22:55:18 Integral: 64.89227055928428