Helmholtz sound-hard scattering problem with absorbing boundary conditions

This example allows to solve the 2d Helmholtz sound-hard (scattering) problem by a R-radius circle. It is useful to understand how to:

Solve PDEs with complex values, i.e. with \(u = u_0 + \imath u_1\)
Handle Robin boundary conditions for PDEs with complex values
Truncate unbounded domains via absorbing boundary conditions

Problem setup

For a wavenumber \(k_0= 2\), we will solve a sound-hard scattering problem for \(u = u^{scat} = u_0 + \imath u_1:`\)

\[- u_{xx}-u_{yy} - k_0^2 u = 0, \qquad \Omega = \overline{B(0,R)}^c\]

with the Neumann boundary conditions

\[\gamma_1 u :=\nabla u \cdot n =- u^{inc}, \qquad (x,y)\in \Gamma^{in} : = \partial (B(0,R))\]

with \(n\), and suitable radiation conditions at infinity. The analytical formula for the scattered field is given by Bessel function (refer to waves-fenicsx).

We decide to truncate the domain and we approximate the radiation conditions by absorbing boundary conditions (ABCs), on a length square \(\Gamma^{out}\). Refer to this recent study for the wavenumber analysis error.

Projection to the real and imaginary axes for \(u = u_0+ \imath * u_1\) leads to:

\[- u_{0,xx}-u_{0,yy} - k_0^2 u_0 = 0 \qquad \text{in}\qquad \Omega \cap D^{out}\]

and

\[- u_{1,xx}-u_{1,yy} - k_0^2 u_1 = 0 \qquad\text{in}\qquad \Omega \cap D^{out}\]

The boundary conditions read:

\[\gamma_1 u = - \gamma_1 u^{inc} \qquad \text{on} \qquad\Gamma^{in}\]

\[\gamma_1 u - \imath k_0 \gamma_0 = 0 \qquad \text{on} \qquad \Gamma^{out}.\]

Absorbing boundary conditions rewrite:

\[\gamma_1 [u_0+ \imath u_1] - \imath k_0 [u_0 + \imath u_1] = 0 \qquad \text{on} \qquad \Gamma^{out}\]

i.e.

\[\gamma_1 u_0 + k_0 u_1 = 0 \qquad\text{on}\qquad \Gamma^{out}\]

\[\gamma_1 u_1 - k_0 u_0 = 0 \qquad \text{on}\qquad\Gamma^{out}.\]

This example is inspired by this Dolfinx tutorial.

Implementation

This description goes through the implementation of a solver for the above scattering problem step-by-step.

First, the DeepXDE and required modules are imported:

import deepxde as dde
import numpy as np
import scipy
from scipy.special import jv, hankel1

Then, we begin by defining the general parameters for the problem. The PINN will be trained over 5000 iterations, we also define the learning rate, the number of dense layers and nodes, and the activation function.

weights = 1
iterations = 10000
learning_rate = 1e-3
num_dense_layers = 3
num_dense_nodes = 350
activation = "sin"

We set the physical parameters for the problem.

k0 = 2
wave_len = 2 * np.pi / k0
length = 2 * np.pi
R = np.pi / 4
n_wave = 20
h_elem = wave_len / n_wave
nx = int(length / h_elem)

We define the geometry (inner and outer domains):

outer = dde.geometry.Rectangle([-length / 2, -length / 2], [length / 2, length / 2])
inner = dde.geometry.Disk([0, 0], R)

geom = outer - inner

We introduce the analytic solution for the sound-hard scattering problem:

def sound_hard_circle_deepxde(k0, a, points):

  fem_xx = points[:, 0:1]
  fem_xy = points[:, 1:2]
  r = np.sqrt(fem_xx * fem_xx + fem_xy * fem_xy)
  theta = np.arctan2(fem_xy, fem_xx)
  npts = np.size(fem_xx, 0)
  n_terms = int(30 + (k0 * a)**1.01)
  u_sc = np.zeros((npts), dtype=np.complex128)
  for n in range(-n_terms, n_terms):
      bessel_deriv = jv(n-1, k0*a) - n/(k0*a) * jv(n, k0*a)
      hankel_deriv = n/(k0*a)*hankel1(n, k0*a) - hankel1(n+1, k0*a)
      u_sc += (-(1j)**(n) * (bessel_deriv/hankel_deriv) * hankel1(n, k0*r) * \
          np.exp(1j*n*theta)).ravel()
  return u_sc

Next, we express the PDE residual of the Helmholtz equation:

def pde(x, y):
    y0, y1 = y[:, 0:1], y[:, 1:2]

    y0_xx = dde.grad.hessian(y, x, component=0, i=0, j=0)
    y0_yy = dde.grad.hessian(y, x, component=0, i=1, j=1)

    y1_xx = dde.grad.hessian(y, x,component=1, i=0, j=0)
    y1_yy = dde.grad.hessian(y, x,component=1, i=1, j=1)

    return [-y0_xx - y0_yy - k0 ** 2 * y0,
            -y1_xx - y1_yy - k0 ** 2 * y1]

Then, we introduce the exact solution and both Neumann and Robin boundary conditions:

def sol(x):
    result = sound_hard_circle_deepxde(k0, R, x).reshape((x.shape[0],1))
    real = np.real(result)
    imag = np.imag(result)
    return np.hstack((real, imag))

def boundary(x, on_boundary):
    return on_boundary

def boundary_outer(x, on_boundary):
    return on_boundary and outer.on_boundary(x)

def boundary_inner(x, on_boundary):
    return on_boundary and inner.on_boundary(x)

def func0_inner(x):
    normal = -inner.boundary_normal(x)
    g = 1j * k0 * np.exp(1j * k0 * x[:, 0:1]) * normal[:, 0:1]
    return np.real(-g)

def func1_inner(x):
    normal = -inner.boundary_normal(x)
    g = 1j * k0 * np.exp(1j * k0 * x[:, 0:1]) * normal[:, 0:1]
    return np.imag(-g)

def func0_outer(x, y):
    normal = outer.boundary_normal(x)
    result = - k0 * y[:, 1:2]
    return result

def func1_outer(x, y):
    normal = outer.boundary_normal(x)
    result =  k0 * y[:, 0:1]
    return result

bc0_inner = dde.NeumannBC(geom, func0_inner, boundary_inner, component = 0)
bc1_inner = dde.NeumannBC(geom, func1_inner, boundary_inner, component = 1)

bc0_outer = dde.RobinBC(geom, func0_outer, boundary_outer, component = 0)
bc1_outer = dde.RobinBC(geom, func1_outer, boundary_outer, component = 1)

bcs = [bc0_inner, bc1_inner, bc0_outer, bc1_outer]

Next, we define the weights for the loss function and generate the training and testing points.

loss_weights = [1, 1, weights, weights, weights, weights]
data = dde.data.PDE(
    geom,
    pde,
    bcs,
    num_domain=nx**2,
    num_boundary=8 * nx,
    num_test=5 * nx **2,
    solution=sol
)

Next, we choose the network. Here, we use a fully connected neural network of depth 4 (i.e., 3 hidden layers) and width 50. Besides, we choose sin as activation function and Glorot uniform as initializer :

net = dde.maps.FNN(
    [2] + [num_dense_nodes] * num_dense_layers + [2], activation, "Glorot uniform"
)

Now, we have the PDE problem and the network. We build a Model and define the optimizer and learning rate.

model.compile(
    "adam", lr=learning_rate, loss_weights=loss_weights , metrics=["l2 relative error"]
)

We first train the model for 5000 iterations with Adam optimizer:

losshistory, train_state = model.train(iterations=iterations)

Complete code

"""Backend supported: tensorflow.compat.v1, tensorflow, pytorch, paddle"""
import deepxde as dde
import numpy as np
import scipy
from scipy.special import jv, hankel1

# General parameters
weights = 1
iterations = 10000
learning_rate = 1e-3
num_dense_layers = 3
num_dense_nodes = 350
activation = "tanh"

# Problem parameters
k0 = 2
wave_len = 2 * np.pi / k0
length = 2 * np.pi
R = np.pi / 4
n_wave = 20
h_elem = wave_len / n_wave
nx = int(length / h_elem)

# Computational domain
outer = dde.geometry.Rectangle([-length / 2, -length / 2], [length / 2, length / 2])
inner = dde.geometry.Disk([0, 0], R)

geom = outer - inner

# Exact solution
def sound_hard_circle_deepxde(k0, a, points):

    fem_xx = points[:, 0:1]
    fem_xy = points[:, 1:2]
    r = np.sqrt(fem_xx * fem_xx + fem_xy * fem_xy)
    theta = np.arctan2(fem_xy, fem_xx)
    npts = np.size(fem_xx, 0)
    n_terms = int(30 + (k0 * a) ** 1.01)

    u_sc = np.zeros((npts), dtype=np.complex128)
    for n in range(-n_terms, n_terms):
        bessel_deriv = jv(n - 1, k0 * a) - n / (k0 * a) * jv(n, k0 * a)
        hankel_deriv = n / (k0 * a) * hankel1(n, k0 * a) - hankel1(n + 1, k0 * a)
        u_sc += (
            -((1j) ** (n))
            * (bessel_deriv / hankel_deriv)
            * hankel1(n, k0 * r)
            * np.exp(1j * n * theta)
        ).ravel()
    return u_sc


# Definition of the pde
def pde(x, y):
    y0, y1 = y[:, 0:1], y[:, 1:2]

    y0_xx = dde.grad.hessian(y, x, component=0, i=0, j=0)
    y0_yy = dde.grad.hessian(y, x, component=0, i=1, j=1)

    y1_xx = dde.grad.hessian(y, x, component=1, i=0, j=0)
    y1_yy = dde.grad.hessian(y, x, component=1, i=1, j=1)

    return [-y0_xx - y0_yy - k0**2 * y0, -y1_xx - y1_yy - k0**2 * y1]


def sol(x):
    result = sound_hard_circle_deepxde(k0, R, x).reshape((x.shape[0], 1))
    real = np.real(result)
    imag = np.imag(result)
    return np.hstack((real, imag))


# Boundary conditions
def boundary(x, on_boundary):
    return on_boundary


def boundary_outer(x, on_boundary):
    return on_boundary and outer.on_boundary(x)


def boundary_inner(x, on_boundary):
    return on_boundary and inner.on_boundary(x)


def func0_inner(x):
    normal = -inner.boundary_normal(x)
    g = 1j * k0 * np.exp(1j * k0 * x[:, 0:1]) * normal[:, 0:1]
    return np.real(-g)


def func1_inner(x):
    normal = -inner.boundary_normal(x)
    g = 1j * k0 * np.exp(1j * k0 * x[:, 0:1]) * normal[:, 0:1]
    return np.imag(-g)


def func0_outer(x, y):
    result = -k0 * y[:, 1:2]
    return result


def func1_outer(x, y):
    result = k0 * y[:, 0:1]
    return result


# ABCs
bc0_inner = dde.NeumannBC(geom, func0_inner, boundary_inner, component=0)
bc1_inner = dde.NeumannBC(geom, func1_inner, boundary_inner, component=1)

bc0_outer = dde.RobinBC(geom, func0_outer, boundary_outer, component=0)
bc1_outer = dde.RobinBC(geom, func1_outer, boundary_outer, component=1)

bcs = [bc0_inner, bc1_inner, bc0_outer, bc1_outer]

loss_weights = [1, 1, weights, weights, weights, weights]

data = dde.data.PDE(
    geom,
    pde,
    bcs,
    num_domain=nx**2,
    num_boundary=8 * nx,
    num_test=5 * nx**2,
    solution=sol,
)
net = dde.maps.FNN(
    [2] + [num_dense_nodes] * num_dense_layers + [2], activation, "Glorot uniform"
)
model = dde.Model(data, net)

model.compile(
    "adam", lr=learning_rate, loss_weights=loss_weights, metrics=["l2 relative error"]
)
losshistory, train_state = model.train(iterations=iterations)

dde.saveplot(losshistory, train_state, issave=True, isplot=True)