A general approach for high order absorbing boundary conditions for the Helmholtz equation
Introduction
Describing the behavior of scattered waves about a body is essential in many fields, whether it be the reflections of sonar wave from a submarine, the reflection of radar waves from an airplane or the reflection of microwaves from a cellular phone. The behavior of the wave is described by the wave equation:Separating variables:we get the equations:
Eq. (1) is known as the Helmholtz equation. Thus, the Helmholtz equation represents the spatial behavior of a wave with frequency . We note that given a condition for the wave equation, one can always formally get one for the Helmholtz equation by . The reverse is true only if the operator is a rational function of . Otherwise, it leads to pseudo-differential operators.
We consider the following exterior problem: a wave propagates through a medium, impacts a given body and scatters. To model this, we consider an unbounded domain, , with an inner boundary . We denote the propagating wave as , and the scattered wave as . Hence, we wish to solve the Helmholtz equation for on , using a boundary condition on (we present a Neumann condition, but any boundary condition can be applied):where n is the normal to .
We consider a planar incident wave of the form:where is the wave vector, with wave number and angle . This incident wave is a solution to the Helmholtz equation with wavenumber and thus we can write the scattered formulation of our problem:
This formulation has only one boundary condition - the inner boundary condition (2). In order for the problem to be well-posed in the unbounded domain, we impose another boundary condition ”at infinity” - the Sommerfeld condition:
This condition implies that the solution involves only outgoing waves.
Section snippets
Exact solutions
We now present some exact solutions for the homogeneous Helmholtz equation. We recall that the scattering problem reduces to the homogeneous case if the incident wave has wavenumber . These solutions will allow us to compare the numerical results we will obtain later to exact solutions.
When the inner boundary is circular, it is natural to work with polar coordinates. The incident wave is given by:
Separating variables, a series of Bessel functions results:
Absorbing boundary conditions
For the wave equation an absorbing boundary condition was developed by Engquist and Majda [6] based on a splitting of the wave equation. This led to a pseudo-differential equation which was approximated by either a Taylor or rational approximation. This was later extended to higher order approximations by Hagstrom, Givoli et al. in a series of papers [7], [8], [11], [12]. These can all be formally extended to the Helmholtz equation by the formal substitution . A different approach was
A general high order boundary condition
In order to construct an ABC for the Helmholtz equation, we first consider a more general case. When solving a PD by separation of variables, one sets and gets:
Usually, has a countable set of values (). Then, the solution can be represented as a series:We consider such a series whereIt should be noted that is not the operator forming the PDE, but only an operator for which the are its
Deriving ABCs for 2D Helmholtz equation
We now derive different ABCs based on the method proposed in the previous section. We examine different representations of the solution to the Helmholtz equation and derive different ABCs.
Polar spherical coordinates, using Hankel functions and spherical harmonics
We consider the solution of the 3D Helmholtz equation:where are spherical harmonics of degree j, and solves the three dimensional Bessel equation and so we solve:To fit our notations we consider the following:We derive the recursive functions:We then construct the operators by
Numerical results
In this section we present numerical results of the various old methods and compare them to the new ABCs.
Conclusion
The Helmholtz equation describes wave propagation in the frequency domain. This is important both in acoustics and electromagnetism. Since the domain is unbounded, a Sommerfeld radiation condition is required to make it well-posed. For a numerical approximation we need to replace the unbounded domain by a finite domain bounded by an artificial boundary. One then needs to impose a boundary condition on that surface to make it both well-posed and accurate. One of the original approaches was that
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2017, Computer Methods in Applied Mechanics and EngineeringSolving Helmholtz equation at high wave numbers in exterior domains
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