As an alternative to solving the wave equation, we may assume that the pressure field is time-harmonic, i.e., P(x, t) = Ψ(x, ω)eιωt, where ω = 2πν is the angular frequency. As per the Fourier theorem, any periodic functionP can be represented as a superposition of sinusoidal functions. Therefore, by using a sufficiently long period, we may approximate any pressure field as a superposition of time-harmonic pressure fields Ψ. This reduces the problem to that of solving the Helmholtz equation [59]:
∇2Ψ +k2Ψ = ˜F , (2.5)
where k = 2λπ is the wavenumber. There are multiple numerical methods for solving the Helmholtz equation. Since they operate on a Fourier decomposition of the pressue field, they are also called frequency-domain methods.
2.2.1
Finite Element Method
In thefinite element method (FEM) [81], the domain is discretized using an irregular mesh, whose elements may be of any shape. Typically, though, simple shapes, such as tetrahedra in 3D, are used. With each vertex, or node, of the mesh, we associate a basis function φi. These basis functions may be of any form as long their partial
derivatives can be defined. In practice, though, simple linear functions are commonly used.
The pressure field at any point x ∈ D is defined as a linear combination of the basis functions:
Ψ(x) =X
i
ciφi(x). (2.6)
Combining the above equation with the Helmholtz equation and the boundary conditions yields a system of linear equations. Moreover, since the basis functions are defined to have compact support, i.e., φi non-zero only for the mesh elements
adjacent to node i, the linear system can be represented using a sparse matrix. This sparse matrix solve is performed once for each frequency.
The computational and storage complexity of FEM are dominated by the sparse matrix linear solve. The number of elements in the spatial discretization must be
n = O(ν3). Since the linear solve must be repeated for each frequency, the storage
complexity of FEM is O(ν4), as is its computational complexity.
One of the main advantages of FEM over, say, FDTD, is that the mesh may be irregular, and hence may better approximate complex boundaries.
2.2.2
Boundary Element Method
The boundary element method (BEM) [20] is based on the Helmholtz-Kirchhoff in- tegral theorem, according to which, the pressure at any point in the interior of the domain can be uniquely determined from the values of pressure (or its normal deriva- tive) at each point on the boundary of the domain. Hence, BEM proceeds in two steps.
First, the boundary is discretized using a surface mesh, typically a triangle mesh. With each node of the boundary, we associate a basis function φi. The pressure at
any point x∈∂D is again defined as a linear combination of the basis functions:
Ψ(x) =X
i
ciφi(x). (2.7)
Combining the above equation with the integral form of the Helmholtz equation and the boundary conditions yields a system of linear equations, which must be represented using a dense matrix, since the equations describe propagation between each pair of surface mesh elements. Solving this system yields the pressure on the boundary.
Finally, the value of pressure at any interior point is computed by evaluating an integral as per the Helmholtz-Kirchhoff integral theorem. Since the domain itself is not discretized in BEM, the numerical errors are significantly reduced.
by the complexity of solving the linear system. The number of elements in the surface mesh must be O(ν2) to satisfy the Nyquist condition. This results in an
n ×n dense matrix, where n = O(ν2). Since the linear solve must be repeated
for each frequency, the storage complexity (i.e., memory required while calculating the solution) of BEM is O(ν5), and its computational complexity is O(ν7) if direct
matrix inversion is used. This complexity can be reduced to O(ν4) per frequency using iterative Krylov subspace solvers.
This complexity can be further reduced usingfast multipole methods (FMM) [30]. FMM approximates the interactions between groups of mesh nodes. The main result of this approximation is that the size of the dense matrix is reduced tok×n, wherek
is a large constant and n=O(ν2). Therefore, the storage complexity of FMM-BEM
is O(ν3), and its computational complexity is O(ν2logν) per frequency.
2.2.3
Equivalent Source Method
Theequivalent source method (ESM) [55] is closely related to the BEM, in that it also solves the integral form of the Helmholtz equation. However, instead of solving for pressure sampled on the domain boundary, pressure is sampled on an offset surface of the boundary. The pressure on the offset surface is then expressed as a weighted sum of elementary point sources φi, which are Green’s functions for the Helmholtz
equation. The main advantage of sampling pressure on the offset surface is that fewer basis functions (point sources) are required to express the pressure with the same degree of accuracy.
The basis expansion of the pressure field is then combined with the integral form of the Helmholtz equation and the boundary conditions, resulting in a system of linear equations, which must be represented using a dense matrix. As a result, the asymptotic complexity of ESM is the same as that of BEM, i.e., the storage
complexity of ESM is O(ν5), and its computational complexity is O(ν7).
Recently, a precomputation-based algorithm has been developed based on ESM, which uses transfer operators similar to those defined in Chapter 3. This approach al- lows scattering and diffraction from discrete objects to be simulated in real-time [52].