Symbolic Field Notebook

[1]:

%matplotlib widget

import numpy as np
import sympy as sp

from mechpy.core.symbolic.coord import (
    SymbolicCartesianCoordSystem,
    SymbolicCylindricalCoordSystem,
)

from mechpy.core.symbolic.field import (
    SymbolicScalarField,
    SymbolicVectorField,
    SymbolicTensorField,
)

Symbolic Field

Linear Scalar Field

[2]:

data = sp.Array([1, 2, 3])
scalar_field = SymbolicScalarField.create_linear(data=data)
display(scalar_field.data)

$\displaystyle \left[\begin{matrix}x + 2 y + 3 z\end{matrix}\right]$

Non Linear Scalar Field

[3]:

coord_system = SymbolicCartesianCoordSystem()
display(coord_system)

SymbolicCartesianCoordSystem(origin=(0, 0, 0), basis=(x, y, z))

[4]:

x1, x2, x3 = coord_system.basis
f = x1*x1 - x2*x2
display(f)

$\displaystyle x^{2} - y^{2}$

[5]:

data = sp.Array([f])
scalar_field = SymbolicScalarField.create(coord_system=coord_system, data=data)
display(scalar_field)

SymbolicScalarField(
(x, y, z),
[x**2 - y**2],
{})

[6]:

scalar_field.plot()

[7]:

coord_system = SymbolicCylindricalCoordSystem()
display(coord_system)

SymbolicCylindricalCoordSystem(origin=(0, 0, 0), basis=(r, theta, z))

[8]:

x1, x2, x3 = coord_system.basis
f = x1 + x3*x3
display(f)

$\displaystyle r + z^{2}$

[9]:

data = sp.Array([f])
scalar_field = SymbolicScalarField.create(coord_system, data)
display(scalar_field)

SymbolicScalarField(
(r, theta, z),
[r + z**2],
{})

[10]:

scalar_field = scalar_field.to_cartesian()
display(scalar_field)
scalar_field.plot()

SymbolicScalarField(
(x, y, z),
[z**2 + sqrt(x**2 + y**2)],
{})

Linear Vector Field

[11]:

data = sp.Array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
vector_field = SymbolicVectorField.create_linear(data=data)
display(vector_field.data)

$\displaystyle \left[\begin{matrix}x + 2 y + 3 z & 4 x + 5 y + 6 z & 7 x + 8 y + 9 z\end{matrix}\right]$

[12]:

vector_field.plot()

Linear Tensor Field

[13]:

data = sp.Array(
    [
        [[1, 2, 3], [4, 5, 6], [7, 8, 9]],
        [[11, 12, 13], [14, 15, 16], [17, 18, 19]],
        [[21, 22, 23], [24, 25, 26], [27, 28, 29]],
    ]
)
display(data.shape)
tensor_field = SymbolicTensorField.create_linear(data=data)
display(tensor_field.data)

$\displaystyle \left( 3, \ 3, \ 3\right)$

$\displaystyle \left[\begin{matrix}x + 2 y + 3 z & 4 x + 5 y + 6 z & 7 x + 8 y + 9 z\\11 x + 12 y + 13 z & 14 x + 15 y + 16 z & 17 x + 18 y + 19 z\\21 x + 22 y + 23 z & 24 x + 25 y + 26 z & 27 x + 28 y + 29 z\end{matrix}\right]$

Field with params

[14]:

n, m = sp.symbols("n m")
field_params = {
    n: None,
    m: None,
}
data = sp.Array([1 * n, -2 * m, 0])
linear_scalar_field = SymbolicScalarField.create_linear(
    data=data,
    field_params=field_params,
)
display(linear_scalar_field)
linear_scalar_field.subs_field_params({n: 3})
linear_scalar_field.subs_field_params({m: 0})
display(linear_scalar_field)
linear_scalar_field.plot()

SymbolicScalarField(
(x, y, z),
[-2*m*y + n*x],
{n: None, m: None})

SymbolicScalarField(
(x, y, z),
[3*x],
{})

Ploting field with params

[15]:

n, m = sp.symbols("n m")
field_params = {
    n: set(np.arange(-2, 2, 0.3)),
    m: set(np.arange(-2, 2, 0.3)),
}
data = sp.Array([1 * n, -2 * m, 0])
linear_scalar_field = SymbolicScalarField.create_linear(
    data=data,
    field_params=field_params,
)
linear_scalar_field.plot()