Broken FEEC projections on polar domains

psydac/feec/polar/examples/analyticalTE.py

@@ -0,0 +1,289 @@
+from sympde.topology import Square
+from sympde.topology.analytical_mapping import PolarMapping
+
+from psydac.api.tests.test_api_feec_2d import add_colorbar
+from psydac.feec.pull_push import push_2d_hcurl, push_2d_l2
+from numpy import pi
+import numpy as np
+import matplotlib.pyplot as plt
+
+class CircularCavitySolution:
+    """
+    Time-harmonic solution of Maxwell's equations in a disk-like domain with
+    perfectly conducting walls. This is a "transverse electric" solution, with
+    E = (Ex, Ey) and B = Bz. Domain is [0, R] x [0, 2pi].
+
+    Parameters
+    ----------
+    R : float
+        domain radius
+
+    c: float
+        Speed of light in arbitrary units
+
+    (m, n): int
+        Mode number. Warning: m > 0, n >= 0
+
+    D: float
+        shift of logical center (in "Target" mapping with c0=D*R2, c1=0, k=0, D=D)
+
+    scale: float
+        Rescaling the values by a real factor. Default = 1
+
+    """
+
+    def __init__(self, R, c, m, n, D=0, scale=1):
+        from numpy import pi
+        from scipy.special import jnp_zeros
+        pnm = jnp_zeros(n, m)[-1]
+        kc = pnm / R
+        omega = c * kc
+
+        phase = pi / 4
+
+        self.c = c
+        self.scale = scale
+        self.n = n
+        self.kc = kc
+        self.omega = omega
+        self.phase = phase
+        self._R = R
+        assert 0 <= D < .5
+        self._D = D
+
+    # Exact solutions for electric and magnetic field with polar parametrization of disk domain
+    def Es_ex(self, t, s, theta):
+        from numpy import sin  # , cos, sqrt, arctan2
+        from scipy.special import jv
+
+        scale = self.scale
+        n = self.n
+        kc = self.kc
+        omega = self.omega
+        phase = self.phase
+        c = self.c
+
+        return - scale * c * n / (s * kc + 1e-10) * sin(n * theta) * jv(n, kc * s) * sin(omega * t + phase)
+
+    def Et_ex(self, t, s, theta, s_factor=True):
+        """
+        if s_factor: multiply by s (as in logical field)
+        """
+        from numpy import sin, cos
+        from scipy.special import jvp
+
+        scale = self.scale
+        n = self.n
+        kc = self.kc
+        omega = self.omega
+        phase = self.phase
+        c = self.c
+
+        val = - scale * c * cos(n * theta) * jvp(n, kc * s) * sin(omega * t + phase)
+        if s_factor:
+            val *= s
+        return val
+
+    def B_ex(self, t, s, theta, s_factor=True):
+        """
+        if s_factor: multiply by s (as in logical field)
+        """
+        from numpy import cos
+        from scipy.special import jv
+
+        scale = self.scale
+        n = self.n
+        kc = self.kc
+        omega = self.omega
+        phase = self.phase
+
+        val = scale * cos(n * theta) * jv(n, kc * s) * cos(omega * t + phase)
+        if s_factor:
+            val *= s
+        return val
+        # The magnitude of B is approximately equal to scale / 3
+
+    def dB_dt_ex(self, t, s, theta):
+        '''
+        = dB/dt
+        '''
+        from numpy import cos, sin
+        from scipy.special import jv
+
+        scale = self.scale
+        n = self.n
+        kc = self.kc
+        omega = self.omega
+        phase = self.phase
+
+        return scale * omega * s * cos(n * theta) * jv(n, kc * s) * sin(omega * t + phase)
+
+    # physical field
+
+    def get_radius_angle(self, x, y):
+        from numpy import sqrt, arctan2  # ,  sin, cos
+        r = sqrt(x * x + y * y)
+        alpha = arctan2(y, x)
+        return r, alpha
+
+    def Ex_ex(self, t, x, y):
+        from numpy import cos, sin
+
+        r, alpha = self.get_radius_angle(x, y)
+        return cos(alpha) * self.Es_ex(t, r, alpha) - sin(alpha) * self.Et_ex(t, r, alpha, s_factor=False)
+
+    def Ey_ex(self, t, x, y):
+        from numpy import cos, sin
+
+        r, alpha = self.get_radius_angle(x, y)
+        return sin(alpha) * self.Es_ex(t, r, alpha) + cos(alpha) * self.Et_ex(t, r, alpha, s_factor=False)
+
+    def Bz_ex(self, t, x, y):
+
+        r, alpha = self.get_radius_angle(x, y)
+        return self.B_ex(t, r, alpha, s_factor=False)
+
+
+def main():
+    """
+    This function is not currently used. It is kept for possible future development.
+    """
+
+    # Logical domain is rectangle [0, R] x [0, 2pi]
+    R = 2.0
+
+    # Speed of light equal c and scaling of the fields by a scale factor
+    c = 1
+    scale = 1
+
+    # Mode number
+    # (m, n) = (1, 0)
+    # (m, n) = (2, 1)
+    (m, n) = (2, 3)
+
+    # Exact solution
+    exact_solution = CircularCavitySolution(R=R, c=c, m=m, n=n, scale=scale)
+
+    # Exact fields, as callable functions of (t, s, theta)
+    Es_ex = exact_solution.Es_ex
+    Et_ex = exact_solution.Et_ex
+    B_ex = exact_solution.B_ex
+    dB_dt_ex = exact_solution.dB_dt_ex
+
+    # Logical domain: [0, R] x [0, 2pi]
+    logical_domain = Square('Omega', bounds1=[0, R], bounds2=[0, 2 * pi])
+
+    # Physical domain: disk of radius R obtained as image of the logical_domain
+    # with the analytical mapping of a circle
+    mapping = PolarMapping('F', c1=0, c2=0, rmin=0, rmax=1)
+    domain = mapping(logical_domain)
+    F = mapping.get_callable_mapping()
+
+    # Set time
+    t = 0
+
+    Es = lambda x, y: Es_ex(t, x, y)
+    Et = lambda x, y: Et_ex(t, x, y)
+    B = lambda x, y: B_ex(t, x, y)
+    dB_dt = lambda x, y: dB_dt_ex(t, x, y)
+
+    # Plot of fields
+    N = 100
+
+    rho = np.linspace(1e-20, R, N)
+    theta = np.linspace(0, 2 * pi, N)
+    rho, theta = np.meshgrid(rho, theta, indexing='ij')
+    x, y = F(rho, theta)
+
+    Ex_values = np.empty_like(rho)
+    Ey_values = np.empty_like(rho)
+    B_values = np.empty_like(rho)
+
+    for i, x1i in enumerate(rho[:, 0]):
+        for j, x2j in enumerate(theta[0, :]):
+            Ex_values[i, j], Ey_values[i, j] = \
+                push_2d_hcurl(Es, Et, x1i, x2j, F)
+
+            B_values[i, j] = push_2d_l2(B, x1i, x2j, F)
+
+    fig, axs = plt.subplots(2, 2, figsize=(12, 12))
+    im0 = axs[0, 0].contourf(x, y, Ex_values, 50)
+    im1 = axs[0, 1].contourf(x, y, Ey_values, 50)
+    im2 = axs[1, 0].contourf(x, y, np.sqrt(Ex_values ** 2 + Ey_values ** 2), 50)
+    im3 = axs[1, 1].contourf(x, y, B_values, 50)
+    axs[0, 0].set_title(r'$E_x$')
+    axs[0, 1].set_title(r'$E_y$')
+    axs[1, 0].set_title(r'$||\mathbf{E}||$')
+    axs[1, 1].set_title('$B$')
+    add_colorbar(im0, axs[0, 0])
+    add_colorbar(im1, axs[0, 1])
+    add_colorbar(im2, axs[1, 0])
+    add_colorbar(im3, axs[1, 1])
+
+    # Test: curl E = - d_t B_z with finite Differences
+    # on a tensor grid in the square inscribed the disk
+    N = 160
+    l = R / np.sqrt(2)  # edge length
+
+    x, dx = np.linspace(-l, l, N, retstep=True)
+    y, dy = np.linspace(-l, l, N, retstep=True)
+    x, y = np.meshgrid(x, y, indexing='ij')
+    rho = np.sqrt(x ** 2 + y ** 2)
+    theta = np.arctan2(y, x) % (2 * pi)
+
+    Ex_values = np.empty_like(rho)
+    Ey_values = np.empty_like(rho)
+    Bt_values = np.empty_like(rho)
+
+    ni, nj = rho.shape
+    for i in range(ni):
+        for j in range(nj):
+            x1_ij = rho[i, j]
+            x2_ij = theta[i, j]
+            Ex_values[i, j], Ey_values[i, j] = \
+                push_2d_hcurl(Es, Et, x1_ij, x2_ij, F)
+
+            Bt_values[i, j] = push_2d_l2(dB_dt, x1_ij, x2_ij, F)
+
+    fig, axs = plt.subplots(2, 3, figsize=(15, 15))
+    im1 = axs[0, 0].contourf(x, y, np.sqrt(Ex_values ** 2 + Ey_values ** 2))
+    im2 = axs[0, 1].contourf(x, y, -Bt_values)
+    axs[0, 0].set_title(r'$||\mathbf{E}||$')
+    axs[0, 1].set_title(r'$-\partial_t B$')
+    for axi in axs.flat:
+        axi.set_aspect('equal')
+    add_colorbar(im1, axs[0, 0])
+    add_colorbar(im2, axs[0, 1])
+
+    curlE_values = np.zeros_like(rho)
+    valerr = 0
+    for i in range(1, ni - 1):
+        for j in range(1, nj - 1):
+            curlE_values[i, j] = (Ey_values[i + 1, j] - Ey_values[i - 1, j]) / (2 * dx) \
+                                 - (Ex_values[i, j + 1] - Ex_values[i, j - 1]) / (2 * dy)
+
+            d = np.abs(Bt_values[i, j] + curlE_values[i, j])
+
+            if valerr < d:
+                valerr = d
+
+    skip = (slice(None, None, int(N / 20)), slice(None, None, int(N / 20)))
+    im3 = axs[1, 0].contourf(x, y, curlE_values)
+    axs[1, 0].quiver(x[skip], y[skip], Ex_values[skip], Ey_values[skip])
+    add_colorbar(im3, axs[1, 0])
+    axs[1, 0].set_title(r'curl $\mathbf{E}$')
+    im4 = axs[1, 1].contourf(x, y, -Bt_values)
+    axs[1, 1].quiver(x[skip], y[skip], Ex_values[skip], Ey_values[skip])
+    add_colorbar(im4, axs[1, 1])
+    im5 = axs[0, 2].contourf(x, y, Ex_values)
+    im6 = axs[1, 2].contourf(x, y, Ey_values)
+    add_colorbar(im5, axs[0, 2])
+    add_colorbar(im6, axs[1, 2])
+    print('|curlE + d_t B| <= ', valerr)
+
+
+if __name__ == "__main__":
+    main()
+    plt.show()
+
+