Analytical Mapping

Analytical Mappings are provided as symbolic expressions, which allow us to compute automatically their jacobian matrices and all related geometrical information.

Available mappings

SymPDE objects	Physical Dimension	Logical Dimension	Description
IdentityMapping	1D, 2D, 3D	1D, 2D, 3D	Identity Mapping object \(\begin{cases} x&=x_1, \\ y&=x_2, \\ z &= x_3 \end{cases} \)
AffineMapping	1D, 2D, 3D	1D, 2D, 3D	Affine Mapping object \(\begin{cases} x &= c_1 + a_{11} x_1 + a_{12} x_2 + a_{13} x_3, \\ y &= c_2 + a_{21} x_1 + a_{22} x_2 + a_{23} x_3, \\ z &= c_3 + a_{31} x_1 + a_{32} x_2 + a_{33} x_3 \end{cases}\)
PolarMapping	2D	2D	Polar Mapping object (Annulus) \( \begin{cases} x &= c_1 + (r_{min} (1-x_1)+r_{max} x_1) \cos(x_2), \\ y &= c_2 + (r_{min} (1-x_1)+r_{max} x_1) \sin(x_2) \end{cases} \)
TargetMapping	2D	2D	Target Mapping object \(\begin{cases} x &= c_1 + (1-k) x_1 \cos(x_2) - D x_1^2, \\ y &= c_2 + (1+k) x_1 \sin(x_2) \end{cases}\)
CzarnyMapping	2D	2D	Czarny Mapping object \(\begin{cases} x &= \frac{1}{\epsilon}(1 - \sqrt{ 1 + \epsilon (\epsilon + 2 x_1 \cos(x_2)) }), \\ y &= c_2 + \frac{b}{\sqrt{1-\epsilon^2/4}} \frac{ x_1 \sin(x_2)}{2 - \sqrt{ 1 + \epsilon (\epsilon + 2 x_1 \cos(x_2)) }} \end{cases}\)
CollelaMapping2D	2D	2D	Collela Mapping object \(\begin{cases} x &= 2 (x_1 + \epsilon \sin(2 \pi k_1 x_1) \sin(2 \pi k_2 x_2)) - 1, \\ y &= 2 (x_2 + \epsilon \sin(2 \pi k_1 x_1) \sin(2 \pi k_2 x_2)) - 1 \end{cases}\)
TorusMapping	3D	3D	Parametrization of a torus (or a portion of it) \(\begin{cases} x &= (R_0 + x_1 \cos(x_2)) \cos(x_3), \\ y &= (R_0 + x_1 \cos(x_2)) \sin(x_3), \\ z &= x_1 \sin(x_2) \end{cases}\)
TorusSurfaceMapping	3D	2D	surface obtained by “slicing” the torus above at r = a \(\begin{cases} x &= (R_0 + a \cos(x_1)) \cos(x_2), \\ y &= (R_0 + a \cos(x_1)) \sin(x_2), \\ z &= a \sin(x_1) \end{cases}\)
TwistedTargetSurfaceMapping	3D	2D	surface obtained by “twisting” the TargetMapping out of the (x, y) plane \(\begin{cases} x &= c_1 + (1-k) x_1 \cos(x_2) - D x_1^2 \\ y &= c_2 + (1+k) x_1 \sin(x_2) \\ z &= c_3 + x_1^2 \sin(2 x_2) \end{cases}\)
TwistedTargetMapping	3D	3D	volume obtained by “extruding” the TwistedTargetSurfaceMapping along z \(\begin{cases} x &= c_1 + (1-k) x_1 \cos(x_2) - D x_1^2, \\ y &= c_2 + (1+k) x_1 \sin(x_2), \\ z &= c_3 + x_3 x_1^2 \sin(2 x_2) \end{cases}\)
SphericalMapping	3D	3D	Parametrization of a sphere (or a portion of it) \(\begin{cases} x &= x_1 \sin(x_2) \cos(x_3), \\ y &= x_1 \sin(x_2) \sin(x_3), \\ z &= x_1 \cos(x_2) \end{cases}\)

Examples

from sympde.topology      import Square
from sympde.topology      import PolarMapping

ldomain = Square('A',bounds1=(0., 1.), bounds2=(0, np.pi))
mapping = PolarMapping('M1',2, c1= 0., c2= 0., rmin = 0., rmax=1.)

domain = mapping(ldomain)

x,y = domain.coordinates

mapping_1 = IdentityMapping('M1', 2)
mapping_2 = PolarMapping   ('M2', 2, c1 = 0., c2 = 0.5, rmin = 0., rmax=1.)
mapping_3 = AffineMapping  ('M3', 2, c1 = 0., c2 = np.pi, a11 = -1, a22 = -1, a21 = 0, a12 = 0)

A = Square('A',bounds1=(0.5, 1.), bounds2=(-1., 0.5))
B = Square('B',bounds1=(0.5, 1.), bounds2=(0, np.pi))
C = Square('C',bounds1=(0.5, 1.), bounds2=(np.pi-0.5, np.pi + 1))

D1     = mapping_1(A)
D2     = mapping_2(B)
D3     = mapping_3(C)

connectivity = [((0,1,1),(1,1,-1)), ((1,1,1),(2,1,-1))]
patches = [D1, D2, D3]
domain = Domain.join(patches, connectivity, 'domain')

A = Square('A',bounds1=(0.2, 0.6), bounds2=(0, np.pi))
B = Square('B',bounds1=(0.2, 0.6), bounds2=(np.pi, 2*np.pi))
C = Square('C',bounds1=(0.6, 1.), bounds2=(0, np.pi))
D = Square('D',bounds1=(0.6, 1.), bounds2=(np.pi, 2*np.pi))

mapping_1 = PolarMapping('M1',2, c1= 0., c2= 0., rmin = 0., rmax=1.)
mapping_2 = PolarMapping('M2',2, c1= 0., c2= 0., rmin = 0., rmax=1.)
mapping_3 = PolarMapping('M3',2, c1= 0., c2= 0., rmin = 0., rmax=1.)
mapping_4 = PolarMapping('M4',2, c1= 0., c2= 0., rmin = 0., rmax=1.)

D1     = mapping_1(A)
D2     = mapping_2(B)
D3     = mapping_3(C)
D4     = mapping_4(D)

connectivity = [((0,1,1),(1,1,-1)), ((2,1,1),(3,1,-1)), ((0,0,1),(2,0,-1)),((1,0,1),(3,0,-1))]
patches = [D1, D2, D3, D4]
domain = Domain.join(patches, connectivity, 'domain')