# Poisson 2D Transform
This is a [finite difference](https://en.wikipedia.org/wiki/Finite_difference_method) solver for [poisson equations](https://en.wikipedia.org/wiki/Poisson%27s_equation) 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
```python
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
```python
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
```python
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
```python
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
```python
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
```python
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
```python
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
```python
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
```python
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
```python
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
```python
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
```python
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
```python
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
Raw data
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"keywords": "poisson, solver, finite difference, transformed space, 2D, numerical, mathematics, physics, engineering, computational, science, research",
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"description": "# Poisson 2D Transform\r\n\r\nThis is a [finite difference](https://en.wikipedia.org/wiki/Finite_difference_method) solver for [poisson equations](https://en.wikipedia.org/wiki/Poisson%27s_equation) in **curved 2d space**.\r\n\r\nMany different poisson solvers exist but few solve for curved-spaces.\r\n\r\nThis package provides that in a very simple to use interface.\r\n\r\n# How to use\r\n\r\n### Install\r\n pip install poisson_transform\r\n\r\n### Use\r\n\r\nFirst you specify the following inputs:\r\n\r\n- 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`.\r\n- g: the Boundary conditions of poisson's equation. If empty will set BCs to Dirichlet.\r\n - `g` should be a function that takes in coordinates and returns (a, b, g) for the boundary condition such that `a*u + b*\u2202u/\u2202n = g` on \u2202\u03a9 (boundary of domain) \r\n - For example, returning `(1, 0, 0)` sets `u=0` which is a dirichlet conditions on that point.\r\n - For example, returning `(0, 1, 0)` sets `\u2202u/\u2202n=0` which is a neumann conditions on that point.\r\n - For example, returning `(1, 2, 3)` sets `u + 2*\u2202u/\u2202n = 3` on that point.\r\n- 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.\r\n 1. To do this, simply obtain the variables `ksi, eta` with the following line `ksi, eta = Transformation.get_ksi_eta()`\r\n 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.\r\n 3. Then, get the `transformation` object to be given to the solver using `transformation = Transformation(ksi, eta, Tx, Ty)`\r\n- Finally, call `solve_and_plot` with the number of point to use for the mesh. \r\n - 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 ($30 \\times 30$).\r\n - To provide all the above inputs, simply write `solve_and_plot(Nx=29, Ny=29, transformation=transformation, f=f, g=g)`\r\n\r\nSee below for example uses.\r\n\r\nNote: This package has not been tested extensively. If you find an issue please report it (or make a pull-request if fixed)\r\n\r\n# Examples\r\n\r\n\r\n```python\r\nfrom poisson_transform import solve_and_plot\r\nsolve_and_plot(Nx=30, Ny=30)\r\n```\r\n\r\n 22:54:15 Warning: both f and g are None. Are you sure this is what you want?\r\n 22:54:15 f is None. Setting f=1\r\n 22:54:15 g is None. Setting Dirichlet BCs \r\n 22:54:17 Integral: 0.03500889856987609\r\n \r\n\r\n\r\n \r\n\r\n \r\n\r\n\r\n\r\n```python\r\nfrom poisson_transform import solve_and_plot\r\ndef f(x, y, ksi, eta):\r\n \"\"\"Returns f for the equation -\u22072 u = f in \u03a9\"\"\"\r\n if (x - 0.3)**2 + (y - 0.4)**2 < 0.2**2 :\r\n return 10\r\n return 0\r\n\r\nsolve_and_plot(Nx=31, Ny=31, f=f)\r\n```\r\n\r\n 22:54:19 g is None. Setting Dirichlet BCs \r\n 22:54:21 Integral: 0.06935111999589494\r\n \r\n\r\n\r\n \r\n\r\n \r\n\r\n\r\n\r\n```python\r\nfrom poisson_transform import solve_and_plot\r\ndef f(x, y, ksi, eta):\r\n \"\"\"Returns f for the equation -\u22072 u = f in \u03a9\"\"\"\r\n if 0.45 < x < 0.55:\r\n return 10\r\n return 0\r\ndef g(x, y, ksi, eta):\r\n \"\"\"Returns (a, b, g) for the boundary condition a*u + b*\u2202u/\u2202n = g on \u2202\u03a9 and (1, 0, g) for dirichlet conditions inside the domain)\"\"\"\r\n if ksi == 0: # left, neumann = 0\r\n return (0, 1, 0)\r\n if eta == 1: # top, dirichlet = 0\r\n return (1, 0, 0)\r\n if ksi == 1: # right, neumann = 0\r\n return (0, 1, 0)\r\n if eta == 0: # bottom, dirichlet = 0\r\n return (1, 0, 0)\r\n\r\nsolve_and_plot(Nx=31, Ny=31, f=f, g=g)\r\n```\r\n\r\n 22:54:24 Integral: 0.0832355038593408\r\n \r\n\r\n\r\n \r\n\r\n \r\n\r\n\r\n\r\n```python\r\nfrom poisson_transform import Transformation, solve_and_plot, plotGeometry\r\ndef f(x, y, ksi, eta):\r\n \"\"\"Returns f for the equation -\u22072 u = f in \u03a9\"\"\"\r\n if 0.45 < ksi < 0.55:\r\n return 100\r\n return 0\r\ndef g(x, y, ksi, eta):\r\n \"\"\"Returns (a, b, g) for the boundary condition a*u + b*\u2202u/\u2202n = g on \u2202\u03a9 and (1, 0, g) for dirichlet conditions inside the domain)\"\"\"\r\n if ksi == 0: # left, neumann = 0\r\n return (0, 1, 0)\r\n if eta == 1: # top, dirichlet = 0\r\n return (1, 0, 0)\r\n if ksi == 1: # right, neumann = 0\r\n return (0, 1, 0)\r\n if eta == 0: # bottom, dirichlet = 0\r\n return (1, 0, 0)\r\n\r\nksi, eta = Transformation.get_ksi_eta()\r\nTx = ksi**2 + 0.75*ksi\r\nTy = (1-eta)*(1.25*ksi) + eta*(2.75-ksi)\r\ntransformation = Transformation(ksi, eta, Tx, Ty)\r\n# plotGeometry(29, 29, transformation)\r\nsolve_and_plot(Nx=29, Ny=29, transformation=transformation, f=f, g=g)\r\n```\r\n\r\n 22:54:28 Integral: 5.2916463228833726\r\n \r\n\r\n\r\n \r\n\r\n \r\n\r\n\r\n\r\n```python\r\nfrom poisson_transform import Transformation, plotGeometry, solve_and_plot\r\n\r\nksi, eta = Transformation.get_ksi_eta()\r\nTx = ksi**2 + 0.1*ksi\r\nTy = 1/2*ksi*(1-eta) + eta\r\ntransformation = Transformation(ksi, eta, Tx, Ty)\r\nplotGeometry(Nx=29, Ny=29, transformation=transformation)\r\nsolve_and_plot(Nx=29, Ny=29, transformation=transformation)\r\n```\r\n\r\n 22:54:30 Warning: both f and g are None. Are you sure this is what you want?\r\n 22:54:30 f is None. Setting f=1\r\n 22:54:30 g is None. Setting Dirichlet BCs \r\n 22:54:32 Integral: 0.016063603756651158\r\n \r\n\r\n\r\n \r\n\r\n \r\n\r\n\r\n\r\n \r\n\r\n \r\n\r\n\r\n\r\n```python\r\nimport numpy as np\r\nfrom poisson_transform import Transformation, solve_and_plot\r\ndef f(x, y, ksi, eta):\r\n \"\"\"Returns f for the equation -\u22072 u = f in \u03a9\"\"\"\r\n 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):\r\n return 100\r\n return 0\r\ndef g(x, y, ksi, eta):\r\n \"\"\"Returns (a, b, g) for the boundary condition a*u + b*\u2202u/\u2202n = g on \u2202\u03a9 and (1, 0, g) for dirichlet conditions inside the domain)\"\"\"\r\n if ksi == 0: # left, neumann = 0\r\n return (0, 1, 0)\r\n if ksi == 1: # right, neumann = 0\r\n return (0, 1, 0)\r\n if eta == 1: # top, dirichlet = 0\r\n return (1, 0, 0)\r\n if eta == 0: # bottom, dirichlet = 0\r\n return (1, 0, 0)\r\n\r\nksi, eta = Transformation.get_ksi_eta()\r\nTx = ksi\r\nellipse_bottom = (ksi-0.5)**2 + 0.2\r\nellipse_top = 1 - ellipse_bottom\r\nTy = ellipse_bottom*(1-eta) + ellipse_top*eta\r\nrotate_phi = -1.2*np.pi/4\r\nTx_rot, Ty_rot = np.cos(rotate_phi)*Tx - np.sin(rotate_phi)*Ty, np.sin(rotate_phi)*Tx + np.cos(rotate_phi)*Ty\r\nsolve_and_plot(Nx=15, Ny=15, transformation=Transformation(ksi, eta, Tx_rot, Ty_rot), f=f, g=g, contour_levels=20)\r\n```\r\n\r\n 22:54:35 Integral: 0.021996048708012562\r\n \r\n\r\n\r\n \r\n\r\n \r\n\r\n\r\n\r\n```python\r\nimport importlib\r\nimport poisson_transform\r\nimportlib.reload(poisson_transform)\r\nimport numpy as np\r\nfrom poisson_transform import Transformation, solve_and_plot\r\ndef f(x, y, ksi, eta):\r\n \"\"\"Returns f for the equation -\u22072 u = f in \u03a9\"\"\"\r\n # centers = ((0.7, 0.6, 0.1), (0.3, 0.3, 0.1), (0.1, 0.4, 0.1))\r\n centers = ((0.7, 0.0, 0.1), (0.9, -0.4, 0.1), (0.5, 0.0, 0.1))\r\n if any((x - xc)**2 + (y - yc)**2 < r**2 for xc, yc, r in centers):\r\n return 100\r\n return 0\r\ndef g(x, y, ksi, eta):\r\n \"\"\"Returns (a, b, g) for the boundary condition a*u + b*\u2202u/\u2202n = g on \u2202\u03a9 and (1, 0, g) for dirichlet conditions inside the domain)\"\"\"\r\n if ksi == 0: # left, neumann = 0\r\n return (0, 1, 0)\r\n if ksi == 1: # right, neumann = 0\r\n return (0, 1, 0)\r\n if eta == 1: # top, dirichlet = 0\r\n return (1, 0, 0)\r\n if eta == 0: # bottom, dirichlet = 0\r\n return (1, 0, 0)\r\n\r\nksi, eta = Transformation.get_ksi_eta()\r\nTx = ksi\r\nellipse_bottom = (ksi-0.5)**2 + 0.2\r\nellipse_top = 1 - ellipse_bottom\r\nTy = ellipse_bottom*(1-eta) + ellipse_top*eta\r\nrotate_phi = -1.2*np.pi/4\r\nTx_rot, Ty_rot = np.cos(rotate_phi)*Tx - np.sin(rotate_phi)*Ty, np.sin(rotate_phi)*Tx + np.cos(rotate_phi)*Ty\r\nsolve_and_plot(Nx=15, Ny=15, transformation=Transformation(ksi, eta, Tx_rot, Ty_rot), f=f, g=g, contour_levels=20)\r\n```\r\n\r\n 22:54:37 Integral: 0.20123610428427863\r\n \r\n\r\n\r\n \r\n\r\n \r\n\r\n\r\n\r\n```python\r\nimport sympy\r\nfrom poisson_transform import Transformation, solve_and_plot, plotGeometry\r\ndef f(x, y, ksi, eta):\r\n \"\"\"Returns f for the equation -\u22072 u = f in \u03a9\"\"\"\r\n return 1\r\ndef g(x, y, ksi, eta):\r\n \"\"\"Returns (a, b, g) for the boundary condition a*u + b*\u2202u/\u2202n = g on \u2202\u03a9 and (1, 0, g) for dirichlet conditions inside the domain)\"\"\"\r\n if eta == 1: # top, neumann = 0\r\n return (0, 1, 0)\r\n if ksi == 0: # left, dirichlet = 0\r\n return (1, 0, 0)\r\n if ksi == 1: # right, dirichlet = 0\r\n return (1, 0, 0)\r\n if eta == 0: # bottom, dirichlet = 0\r\n return (1, 0, 0)\r\n\r\nksi, eta = Transformation.get_ksi_eta()\r\nTx = (2*ksi+0.25-eta)\r\nbottom_curve = 0.5*sympy.Abs(Tx+1e-6) # add 1e-6 to avoid derivative at kink\r\nTy = bottom_curve + (1.8 - bottom_curve)*eta\r\nplotGeometry(38, 38, Transformation(ksi, eta, Tx, Ty))\r\nsolve_and_plot(Nx=38, Ny=38, transformation=Transformation(ksi, eta, Tx, Ty), f=f, g=g)\r\n```\r\n\r\n 22:54:52 Integral: 0.46734117404100667\r\n \r\n\r\n\r\n \r\n\r\n \r\n\r\n\r\n\r\n \r\n\r\n \r\n\r\n\r\n\r\n```python\r\nimport importlib\r\nimport poisson_transform\r\nimportlib.reload(poisson_transform)\r\nimport sympy\r\nfrom poisson_transform import Transformation, solve_and_plot, plotGeometry\r\ndef f(x, y, ksi, eta):\r\n \"\"\"Returns f for the equation -\u22072 u = f in \u03a9\"\"\"\r\n return 1\r\ndef g(x, y, ksi, eta):\r\n \"\"\"Returns (a, b, g) for the boundary condition a*u + b*\u2202u/\u2202n = g on \u2202\u03a9 and (1, 0, g) for dirichlet conditions inside the domain)\"\"\"\r\n if eta == 1: # top, neumann = 0\r\n return (0, 1, 0)\r\n if ksi == 0: # left, neumann = 0\r\n return (0, 1, 0)\r\n if ksi == 1: # right, dirichlet = 0\r\n return (1, 0, 0)\r\n if eta == 0: # bottom, dirichlet = 0\r\n return (1, 0, 0)\r\n\r\nll, cc, hh = 3, 0.5, 1\r\nbb = np.sqrt((1/2*ll - hh + cc)**2 - cc**2)\r\nksi, eta = Transformation.get_ksi_eta()\r\nTx = ksi*bb\r\nTy = cc*ksi*(1-eta)+hh*eta\r\n# plotGeometry(21, 21, Transformation(ksi, eta, Tx, Ty))\r\nsolve_and_plot(Nx=38, Ny=38, transformation=Transformation(ksi, eta, Tx, Ty), f=f, g=g)\r\n```\r\n\r\n 22:54:57 Integral: 0.06464775169432217\r\n \r\n\r\n\r\n \r\n\r\n \r\n\r\n\r\n\r\n```python\r\nfrom poisson_transform import Transformation, solve_and_plot\r\ndef f(x, y, ksi, eta):\r\n \"\"\"Returns f for the equation -\u22072 u = f in \u03a9\"\"\"\r\n return 0\r\ndef g(x, y, ksi, eta):\r\n \"\"\"Returns (a, b, g) for the boundary condition a*u + b*\u2202u/\u2202n = g on \u2202\u03a9 and (1, 0, g) for dirichlet conditions inside the domain)\"\"\"\r\n if ksi == 0: # left, dirichlet = 0.5\r\n return (1, 0, 0.5)\r\n if ksi == 1: # right, dirichlet = 1.2\r\n return (1, 0, 1.2)\r\n if eta == 0: # bottom, dirichlet = -0.75\r\n return (1, 0, -0.75)\r\n if eta == 1: # top, dirichlet = -1\r\n return (1, 0, -1)\r\n\r\nksi, eta = Transformation.get_ksi_eta()\r\nTx = 12*ksi - 6\r\nTy = 6*eta - 3\r\nsolve_and_plot(Nx=30, Ny=30, transformation=Transformation(ksi, eta, Tx, Ty), f=f, g=g)\r\n```\r\n\r\n 22:55:01 Integral: -29.042871607935275\r\n \r\n\r\n\r\n \r\n\r\n \r\n\r\n\r\n\r\n```python\r\nfrom poisson_transform import Transformation, solve_and_plot\r\ndef f(x, y, ksi, eta):\r\n \"\"\"Returns f for the equation -\u22072 u = f in \u03a9\"\"\"\r\n return 0\r\ndef g(x, y, ksi, eta):\r\n \"\"\"Returns (a, b, g) for the boundary condition a*u + b*\u2202u/\u2202n = g on \u2202\u03a9 and (1, 0, g) for dirichlet conditions inside the domain)\"\"\"\r\n if ksi == 0: # left, dirichlet = 0.5\r\n return (1, 0, 0.5)\r\n if ksi == 1: # right, dirichlet = 1.2\r\n return (1, 0, 1.2)\r\n if eta == 0: # bottom, dirichlet = -0.75\r\n return (1, 0, -0.75)\r\n if eta == 1: # top, dirichlet = -1\r\n return (1, 0, -1)\r\n\r\n # dirichlet conditions inside the domain\r\n if 1 < x < 1.4 and -0.5 < y < 0.2:\r\n return (1, 0, 1.5)\r\n\r\nksi, eta = Transformation.get_ksi_eta()\r\nTx = 12*ksi - 6\r\nTy = 6*eta - 3\r\nsolve_and_plot(Nx=30, Ny=30, transformation=Transformation(ksi, eta, Tx, Ty), f=f, g=g)\r\n```\r\n\r\n 22:55:04 Integral: -11.898317261702779\r\n \r\n\r\n\r\n \r\n\r\n \r\n\r\n\r\n\r\n```python\r\nfrom poisson_transform import Transformation, solve_and_plot\r\ndef f(x, y, ksi, eta):\r\n \"\"\"Returns f for the equation -\u22072 u = f in \u03a9\"\"\"\r\n return 0\r\ndef g(x, y, ksi, eta):\r\n \"\"\"Returns (a, b, g) for the boundary condition a*u + b*\u2202u/\u2202n = g on \u2202\u03a9 and (1, 0, g) for dirichlet conditions inside the domain)\"\"\"\r\n if ksi == 0: # left, neumann = 0\r\n return (0, 1, 0)\r\n if ksi == 1: # right, neumann = 0\r\n return (0, 1, 0)\r\n if eta == 0: # bottom, dirichlet = 0\r\n return (1, 0, 0)\r\n if eta == 1: # top, neumann = 0\r\n return (0, 1, 0)\r\n\r\n # dirichlet conditions inside the domain\r\n if 1 < x < 1.4 and -0.5 < y < 0.2:\r\n return (1, 0, 1.5)\r\n\r\nksi, eta = Transformation.get_ksi_eta()\r\nTx = 12*ksi - 6\r\nTy = 6*eta - 3\r\nsolve_and_plot(Nx=50, Ny=50, transformation=Transformation(ksi, eta, Tx, Ty), f=f, g=g)\r\n```\r\n\r\n 22:55:10 Integral: 35.2240775563179\r\n \r\n\r\n\r\n \r\n\r\n \r\n\r\n\r\n\r\n```python\r\nfrom poisson_transform import Transformation, solve_and_plot\r\ndef f(x, y, ksi, eta):\r\n \"\"\"Returns f for the equation -\u22072 u = f in \u03a9\"\"\"\r\n return 0\r\ndef g(x, y, ksi, eta):\r\n \"\"\"Returns (a, b, g) for the boundary condition a*u + b*\u2202u/\u2202n = g on \u2202\u03a9 and (1, 0, g) for dirichlet conditions inside the domain)\"\"\"\r\n if ksi == 0: # left, neumann = 0\r\n return (0, 1, 0)\r\n if ksi == 1: # right, neumann = 0\r\n return (0, 1, 0)\r\n if eta == 0: # bottom, neumann = 0\r\n return (0, 1, 0)\r\n if eta == 1: # top, neumann = 0\r\n return (0, 1, 0)\r\n\r\n # dirichlet conditions inside the domain\r\n if (x)**2 + (y)**2 < 0.4**2:\r\n return (1, 0, 1)\r\n elif (x+1.4)**2 + (y)**2 < 0.2**2:\r\n return (1, 0, -2)\r\n elif (x-1.4)**2 + (y)**2 < 0.2**2:\r\n return (1, 0, -2)\r\n elif -3.5 < x < -2 and -0.25 < y < 0.25:\r\n return (1, 0, 2)\r\n elif 2 < x < 3.5 and -0.25 < y < 0.25:\r\n return (1, 0, 2)\r\n\r\nksi, eta = Transformation.get_ksi_eta()\r\nTx = 8*ksi - 4\r\nTy = 8*eta - 4\r\nsolve_and_plot(Nx=60, Ny=60, transformation=Transformation(ksi, eta, Tx, Ty), f=f, g=g)\r\n```\r\n\r\n 22:55:18 Integral: 64.89227055928428\r\n \r\n\r\n\r\n \r\n\r\n \r\n\r\n",
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