VALORACIONES

Coupled elasto-acoustic FEM-FEM

In the comparatively simple case of a sufficiently rigid structure, the presence of the fluid does not significantly modify its modes of vibration and dynamic response. A stationary body can be specified purely as an incident field scatterer and the resulting problem is entirely a fluid one, with a Neumann BC ∀x ∈ Γ : v_fn(x) = 0. Alternatively, a vibrating solid on which the acoustic medium has only a marginal effect represents a radiation problem. Both settings are subject to efficient treatment by deterministic methods. For radiation, one need first compute the normal surface velocity at the interface via standard FE analysis, and then use it to impose a Neumann boundary condition for the acoustic problem. In brief, such uncoupled problems lend themselves to an uncomplicated sequential solution for the two domains.

Unfortunately, such a procedure is commonly not acceptable. Aerospace applications are characterised by lightweight, thin structures and high-energy broadband excitations. This combination dictates the need to simultaneously consider both physical phenom- ena. In this coupled context, a traditional procedure of describing structure-acoustic interactions involves modelling both problems with finite elements ([116], [117]). The general form of the governing equation can be expressed as:

K_s+ iωCs− ω2Ms Tsf ρω2T_sfT Kf+ iωCf − ω2Mf  | {z } Z u p  = fs ff  (2.49)

Here, u and p indicate nodal displacement and pressure. K, C and M have the usual meaning of stiffness, damping and mass, with subscripts s and f denoting structural and fluid quantities, respectively. The corresponding nodal loads are given by fs and ff,

while Tsf is a coupling matrix mapping the structural to the acoustic DOFs. Finally,

Z is the global dynamic stiffness matrix.

Note that (2.49) is dependent on ω, and a separate solution is needed at each frequency point. Of course, the same applies to BEM equations, such as (2.48). Various proce- dures exist for solving (2.49). Without delving into excessive detail on their execution, the most common strategies will be summarised next, particularly for the relevant aeroacoustic loads introduced in Section 2.4.1.

General solution strategy

Denoting the total nodal load vector by x, the corresponding output y is given by

y = Hx (2.50)

where H = Z−1is the input-output transfer function. As the matrix H tends to be very large, solutions using direct inversion of Z, or decomposition-based variants thereof,

are not normally viable. Modal procedures, on the contrary, successfully mitigate that problem. Ideally, H would be symmetric or Hermitian. Then powerful algorithms, such as the Lanczos method, can be employed to rapidly solve (2.49). Otherwise, more general routines exist which can still make use of the sparsity of Z. The reader is referred to the book by Golub and Van Loan [118] for an expansive resource that covers the relevant topics in linear algebra.

Several works in which symmetrisation of Z is explicitly targeted also exist in litera- ture. In [119], the authors devised such an approach and an apparent good agreement was indicated for a simple one dimensional problem against an analytical solution. More recently, Ding and Chen [120] exercised a similar approach for modelling thin- walled acoustic cavities. Again, reasonably good correlation to closed-form solutions was shown for simple problems.

Weakly stationary random process excitation

In a general vibroacoustic setting, it could be expected that the fluid domain requires some form of explicit modelling, similarly to the structural one. However, this is not necessarily the case with acoustic fields amenable to treatment as weakly stationary random processes, such as those outlined in Section 2.4.1. In fact, application of the DSF or TBL directly as a distributed random excitation to a standard structural FE model presents an opportunity for a simplified, streamlined analysis.

Indeed, for a weak coupling scenario, S´eon and Roy [121] suggested such a scheme in which only the elastic FE matrix is built and no acoustic space discretisation is done. The authors claimed acceptable agreement of their model’s predictions compared to experimental results, the case study being the design of a re-entry vehicle. Nevertheless, they did recognise the importance of the coupling strength on the applicability of such techniques. At this stage it must be pointed out that the procedures described next have been derived with a full elasto-acoustic FEM model in mind, as per (2.49). However, they remain completely unaltered and valid for an elastic-only dynamic stiffness Z, as was the case with their deployment in [121].

To apply a distributed random load, a nodal pressure cross-spectral density matrix SP is initially constructed. In essence, it is a discrete version of Spq(ω), introduced in

equations (2.34) and (2.35) for the cases of DSF and TBL, respectively. More general theory on the numerical synthesis of correlated random pressure fields can be found in [122]. Entries of SP assume non-zero values only when the corresponding pair of nodes

is on the fluid-structure interface. Through the coupling matrix Tsf,

Sx = TsfSPTsfT (2.51)

conversion of SP to a nodal loads matrix Sx is facilitated. Then the matrix Sy of PSD

outputs is given by

Sy = HSxHT = HTsfSPTsfTHT (2.52)

Efficient evaluation of Sy

For realistic problems, direct evaluation of Sy may be undesirable or even impossible

on acceptable timescales. Two conventional paths of dealing with this obstacle can be identified. The first one involves a preliminary dynamic reduction, representing (2.49) in modal space on a truncated basis of eigenvectors. The method follows the standard logic of performing modal condensations to reduce computing costs, as described in Section 2.2.2. Following some analytical steps shown in [117], equation (2.52) can be rewritten in terms of modal quantities, indicated by a subscript Φ:

SΦy = HΦTΦsfSPT

T ΦsfH

T

Φ (2.53)

The modal receptance and PSD output matrices HΦ and Sy are of size r × r, where n

is the number of physical degrees of freedom, r the number of eigenvectors in the basis Φ and r  n. Furthermore, HΦ is usually diagonal.

Another option is an evaluation of Sy in physical coordinates via a ’pseudo load-cases’

approach, proposed by Coyette et al. [117], and later refined in [123]. Taking a decom- position of the form

SP ≈ QSPDSPQ

T

SP (2.54)

where the diagonal matrix D_SP contains only the r dominant eigenvalues of SP. Sub-

stituting (2.54) into (2.52), the approximation Sy ≈ (HTsfQSP)DSP(HTsfQSP)

T

= XD_SPXT (2.55) is obtained. This decomposition can be interpreted as presenting the random process algebraically as a set of r uncorrelated load cases associated with the columns of Q_SP. A slightly different method involves starting by factorising Sx instead of SP. The com-

putation cost of X is determined by r, and tends to rise for turbulent boundary layer excitations and at higher frequencies. To avoid this setback, Coyette and Meerbergen [123] devised a procedure that involves a direct partial decomposition of Sy and demon-

strated that its efficiency surpasses the pseudo load-case method, since lower ranks r are required irrespectively of the solution frequency.

Even more recent takes on the problem are outlined in [124] and [125]. An algebraic sampling procedure was proposed for the generation of realisations of the random pro- cess determined by SP. A Cholesky decomposition SP = LSPLTSP is taken instead of

(2.54). Samples S_Ps are extracted from

S_Ps = L_SPζ, Ck= eiθk, θk∈ [0, 2π] (2.56)

where ζ is a vector of random angles with k-th entry θk. The overall response y is built

by statistical manipulation of a sufficiently large pool of realisations.

In addition, both papers covered the idea of direct sampling, in which the diffuse sound field is approximated by a set of discrete plane waves with random phase angles. Each of them produces a blocked pressure excitation along the fluid-structure interface, and the final response is also obtained via statistical manipulation of the associated load cases. An updated spatial correlation function for the DSF has been derived, in order to aid the selection of appropriate phases for the plane waves.