EL CONTROL DE LAS OFICINAS LIQUIDADORAS

Modelling with finite elements

Similarly to representing structures where the finite element method is accepted as the default approach, FE descriptions of the acoustic space are widely adopted. The main benefits against competing approaches, such as BEM, can be found in FEM’s long history of active use and development. Currently, algorithms dealing with the sparse matrices arising in FEM are highly optimised and arguably more straightforward to manipulate in general [71]. An extensive introduction to the numerical aspects of FEM implementation for sound propagation problems can be found in [72]. Another excellent resource is the recent book by Atalla and Sgard [73], essentially covering all critical aspects of the broad topic of acoustic simulation with deterministic schemes.

n

n

Γ

Γ

Exterior problem Interior problem

Figure 2.2: Interior and exterior acoustic problems. Γ denotes the fluid-structure boundary and n the surface normal

Similarly to solids, finite element representations of fluids are obtained by discretisation with 3-D elements. However, a clear distinction is made on the basis of whether the acoustic space is physically contained within the structure, or encloses it. The two cases are referred to as interior and exterior problems, respectively, and are illustrated on Figure 2.2. Special treatment by means of imposing some form of artificial boundary is needed for the latter one, as it calls for the filling of an infinite space with finite elements. Due to the relevance of exterior problems to vibroacoustic modelling for space applications, this particular case will be discussed in more detail.

Regardless of the problem configuration, in a homogeneous and isotropic medium the propagation of waves is governed by the Helmholtz equation

∇2p(x) + k2p(x) = 0 (2.38) where ∇2 is the Laplace operator, p(x) is the pressure at a point x in space and k is the wavenumber. For an external acoustic medium, a crucial requirement that must be fulfilled is the Sommerfield radiation condition, stating that waves infinitely far away

from the boundary Γ are outgoing. It is written as lim |x|→+∞|x| (d−1)/2 = ∂p(x) ∂ |x| − ikp(x)  (2.39) where d = 1, 2, 3 is the space dimension. A well-posed mathematical description dictates that an appropriate boundary condition (BC) should be imposed on the defined artificial boundary. This proves a challenging task, particularly if a general solution is sought that is expected to be numerically stable, efficient, accurate in its representation of the underlying physics and also valid for a range of media, geometries and wave types. Harari [74] reports the main approaches developed in response to this issue.

Absorbing boundary conditions, PMLs and IEMs

In the pursuit of applying FEM to unbounded domains, several methods have gained widespread approval over the decades. Absorbing boundary conditions (ABCs) were the first to emerge, with significant breakthroughs made in the late 70s. Broadly speaking, the aim of ABCs is to define a specific condition on the outward fluid boundary which forms the edge of the FE space, such that spurious reflections are eliminated. Among the most well-known ones are the ABCs proposed by Bayliss and Turkel [75], with many others to follow or improve on the concept.

In brief, classic ABCs often yield satisfactory results, but this assertion is strongly affected by the specifics of the application [72]. Notwithstanding, they have enabled the definition of the contemporary high-order absorbing boundary conditions, summarised in [76]. The principal advantage of the newer formulations is that solution accuracy up to any desired order can be required without causing an ill-conditioning of the problem.

Absorbing BC Structure Fluid FE Absorbing layer Structure Fluid FE Structure Infinite elements (a) (b) (c) Γ_{𝑅 Γ𝑅} Γ_𝑅

Figure 2.3: Unbounded medium treatments at the artificial boundary ΓR: (a) Absorb-

The concept of absorbing layers was conceived chronologically in parallel to absorbing boundary conditions. Like ABCs, an artificial boundary is defined, but the principal difference is that it is a narrow region of finite extent. The equations of motion undergo certain modifications within that space, devised so that the outgoing waves can be absorbed.

The absorbing layer technique paved the way for the seminal work of Berenger [77], in which the so-called perfectly matched layer (PML) is proposed. A PML is designed to have completely zero reflectivity with respect to any plane wave at the artificial bound- ary ΓR, depicted on Figure 2.3. In addition, the acoustic solution decays exponentially

within the PML region, with waves potentially undergoing multiple reflections between its inner and outer bounds. Waves reaching the fluid-structure interface Γ tend to be of insignificant strength. The extension of PMLs, originally addressing problems in electromagnetism, to acoustics is owed to Turkel and Yefet [78].

An approach rather different to the previously described ones also emerged at the same time. Infinite element methods (IEMs) make use of ’elements’ represented by semi- infinite prisms with associated shape functions attempting to mimic the behaviour of the solution in the far-field. Modern versions trace their foundations to the works of Burnett [79] and Astley et al. [80]. Such IEMs are constructed in separate, usually spheroidal, coordinate systems and employ shape functions that automatically satisfy the Sommerfield radiation condition. Overall, present-day IEMs may be considered a viable alternative to boundary element methods for exterior problems. For an in-depth analysis the reader is referred to the papers of Tsynkov [81] and later Thompson [82], who also cover the related techniques detailed above.