Boundary integral calculations of scattered fields: Application to a spacecraft launcher

The objective of this article is to illustrate the possibilities of the boundary integral method in the analysis of acoustic scattering by the structure of a space launcher. Such an analysis is important in situations where the sound field reaches critical levels at takeoff. In these circumstances an increase in amplitude associated with reflection and scattering must be estimated with precision. The method is based on integral representations of the wave field and only requires a surface mesh of the solid structure. The number of nodes of such surface meshes is much smaller than the 3-D finite difference or finite element equivalents. Another important advantage of the method is that it automatically incorporates the farfield radiation condition. This condition is only approximated in field methods. The calculations are performed with a code originally developed by Hamdi [Doctoral thesis, Universite de Compi(cid:127)gne (1982) ]. It is here validated for the scattering of a plane wave by a hard sphere. The convergence properties of the method are examined and are found to be acceptable for most practical purposes. Scattering by the Ariane IV fairings (the payload housing) is then specifically considered. It is found that the geometrical shape of the fairings causes local increases of the sound field. This tendency is confirmed by measurements performed on a (cid:127) model of the launcher. Effects of geometrical modifications are also discussed.


INTRODUCTION
Modern space vehicles such as the Space Shuttle and spacecraft launchers such as Ariane rockets generate intense sound fields at liftoff and during the first instants of atmospheric flight. Acoustic loads induced on the structure and on the payload are particularly important when the vehicle is on the launch pad with a propulsion system operating at full power. Extreme acoustic loads are also generated in certain cases at a moderate elevation of the vehicle, corresponding to a strong interaction of the propulsive jets with the structure of the launch pad. The prediction of these loads constitutes an important technological problem. This prediction is commonly performed with semiempirical models based on spectral and spatial decompositions of the sound power radiated by the propulsion system. Simple concepts are used to describe the propagation from the noise sources and take into account ground effects and the lateral deflection of the jets exhausted by the rocket engines. Standard prediction methods and recommended practices are discussed in Ref.

A typical application to the Ariane I launcher is given in Ref. 2.
Theoretical calculations based on these concepts require a fair amount of experimental data and their accuracy is limited. As a consequence subscale testing is extensively used to obtain this experimental input, complement the theoretical evaluation of the acoustic loads, and tune the prediction codes.
One of the shortcomings of the prediction methods is that they do not account for the complicated wave reflections and diffractions on the launch stand and on the vehicle. The geometry is so complex that until recently it has discour-aged any detailed analysis.
Recent progress in computational methods now allows a direct treatment of such problems. An example of this approach is provided by Buell

Now if a linear analysis of the wave motion is acceptable, then most of these difficulties may be avoided with boundary integral methods (BIM). These methods are based on integral representations of the wave field and appear under various names in the technical literature, such as the boundary integral equation method (BIE), the Helmholtz integral equation method (HIE), and boundary element method (BEM). All these formulations rely on the solution of integral equations on the solid boundaries of the propagation domain. Thus, for a 3-D problem, the calculation only requires theJdiscretization of 2-D solid surfaces.
The number of nodes of the surface meshes involved is therefore much smaller than that of 3-D finite difference or finite element meshes required in field methods. It is also much easier to generate surface meshes. Another important advantage of the BIM is that they automatically incorporate radiation conditions. These conditions are only approximated in field methods.
The literature on the BIM is already extensive. Initial i77i applications of these methods to acoustic problems have been made by Chen and Schweikert, 4 Chertock, 5 Copley, 6 Schenck, 7 Burton and Miller, s Kleinmann and Roach, 9 iones, •ø and Fillipi. TM Various integral formulations are proposed in these studies. It is also shown that for certain boundary integral equations the solution of the exterior problem is perturbed by the eigensolutions of the interior problem for the same geometry. This nonuniqueness problem is extensively discussed in these early studies. Improved It is one of our objectives to illustrate this aspect and show that BIM may be used to analyze scattering by complex shapes and multibody configurations. Our study specifically concerns the scattering of incident acoustic waves by the structure of the Ariane IV space launcher. This analysis is important in situations where the sound field reaches critical levels. In these circumstances an increase of the field amplitudes associated with reflections and diffractions must be estimated with precision.
A detailed examination of the field in the vicinity of the launcher may be used to evaluate the acoustic impact of some modifications of the external shape of the fairings.
In this study, we use a boundary integral equation formulation originally developed by Hamdi. •6 The calculations are performed with the REDSTAR code (REflection and Diffraction on the STructure of ARiane launchers). This code contains a boundary integral computational kernel also implemented by Hamdi. •6 We begin by giving a brief theoretical formulation of the method (Sec. I). Limits associated with resources available on current computers are discussed in Sec. II. A surface mesh generation procedure based on CAD for "Computer Aided Design" software is described in Sec. III. Test calculations are examined in Sec. IV and acoustic wave scattering by the Ariane IV launcher is considered in Sec. V.

I. THEORETICAL BACKGROUND
This section gives a brief description of the boundary integral formulation used in the present study. Further details and an application to radiation problems may be found If the surface S is rigid then, the acoustic velocity normal to S vanishes and the boundary condition on $ is =0, onS.

The fieldps is the solution of the Helmholtz equation in f•, its normal derivative on S is fixed by (4), and it satisfies
where VM and V e are the gradient operators at points r M and re, respectively. Once/• is determined, the scattered field Ps (rst) in the exterior domain • may be obtained from Eq. (6).
At points belonging to S this expression must be replaced by 1 fs/• •G (r•4,re) dSe ß (12)

Ps (re) = -•-/• (r•4) + (re) 8n e
In practice, the search of/• is conducted in discrete form. A surface mesh of M triangular panels comprising N nodes is first defined and the potential/• is evaluated at these nodes. Second-order interpolating functions are used to approximate the integrands appearing in the variational principle (10) and ( 11 ).
The nodal values of the potential are obtained by solving this linear system of complex symmetric equations.

II. COMPUTATIONAL LIMITATIONS
The computational resources available on current computers set upper limits to the size of the problems that may be solved with the BIM described above.
Numerical experiments indicate that the panel charac-tedstic dimension h must be chosen less than or equal to a quarter of the wavelength This sets a lower bound to the wavelength:

,4/(MSIZE) 1/2</• 2. (18)
For instance, consider the Ariane IV fairings of radius a = 2 m and length 1 = 8.6 m, including a portion of the third stage of length 1' = 14.4 m, and let MSIZE_• 1 440 000 (the maximum memory size available for the user on the CRAY-1 computer operating at ONERA). In this situation A > 2 m, which corresponds to a frequency limitation of 170 Hz.
It is difficult to give a definite statement about the degree of precision of the results obtained. However, the test calculations performed for known configurations indicate that an acceptable accuracy is reached if condition (17) is satisfied. Finally, it is worth noting that large-scale calculation (more than 1000 nodes) requires the use of 64-bit words.

Another limitation to the problem size is set by the CPU time required. Test calculations carried out on the CRAY-1 computer indicate that this time is essentially determined by the node number N and by the number of points NP where the pressure field is evaluated:
Tee v ,•' 1.2 X 10-7N 3 _[_ 5.4 X 10-4N 2 + 1.5 X 10 -3 NPXN.
If NP is moderately large and Nis large it is obvious that the first term becomes rapidly dominant and that as a consequence the computation time increases as the sixth power of the frequency ( Tcpv o: f6). The calculations pertaining to the Ariane IV launcher typically required the maximum memory size available and about 45 min of CPU time on the CRAY-1 computer.

III. SURFACE MESH GENERATION
A considerable amount of time is usually spent on the generation of the surface meshes. Specific procedures may be devised for simple geometrical configurations, but it is more interesting to develop general methods applicable to moderately complex 3-D configurations. One approach that is particularly well suited to technical applications consists of using a general purpose computer-aided design (CAD) software to define the solid surface geometry and mesh this surface automatically. From the stored graphic model one then retrieves a data file containing the nodal coordinates and nodal connectivities (i.e., the couples of nodes that define the branches of the surface mesh). This procedure is schematically described in Fig. 2.
Further details are given in the remainder of this section. For simplicity we will only consider the case of an axisymmetric solid body.
The for example, possible to create or scan CATIA graphic models by associating this manager with an external FORTRAN code IAMLECT. The output file obtained contains the nodal information and the nodal identifiers. Another external code IAMFICH then generates the connectivity matrix. The output file obtained contains all the necessary information required by an external application such as the boundary integral computer code REDSTAR. The operations described above are interactive and the user may step in the process, for example, to modify certain parts of the mesh.

IV. TEST CALCULATIONS
From among the many test calculations used to validate the REDSTAR code we will only examine the results obtained in the case of scattering of a plane wave by a rigid sphere (Fig. 4). This standard problem has a well-known analytic solution, which may be expressed as a In the immediate vicinity of the sphere, the numerical evaluation of the field produces unreliable results for all values of ka. This is so because the integral appearing in expression (6) becomes singular when the field point rM approaches the scattering surface $. This difficulty only arises in a thin boundary layer adjacent to the body. To obtain accurate values of the field in this layer it is, for example, possible to extrapolate these values from the field estimates obtained outside this region.
It is now worth examining the convergence properties of the BIM used in this study. To this purpose let us again consider the scattering of plane acoustic waves by a rigid sphere.
To estimate the numerical error it is convenient and also more appropriate to compare the calculated and exact scattered fieldsp•n and p•. The exact scattered fieldp• is obtained by summing 90 terms of the infinite spherical harmonic series appearing in expression (21 ). Figure 8

Figure 9 indicates that the relative error decreases rapidly with the ratio h/A. When h/A is less than • the error falls below 9% and the accuracy is sufficient for most engineering purposes.
Another evaluation of the rate of convergence of the method consists of changing the mesh size while keeping ka fixed. This is done in Fig. 10 for ka -2 and for six different mesh sizes corresponding to A/h -3, 6, 9, 12, 16, and 24.

The relative error decreases when h decreases (i.e., when the number of nodes increases). However, it is also found that the error cannot be made infinitely small by augmenting the number of nodes because round-off errors become non-negligible and also because the precision of the linear elements used in the formulation becomes insufficient. This second aspect may be improved if the linear surface elements are replaced by quadratic elements.
As a final point let us consider the evolution of the relative error with the radial distance. The reduced wavenumber is fixed at ka -2 and the error is calculated for five mesh sizes and five values of the radius r/a --1.2, 2, 3, 4, and 5.
For a given mesh the error remains essentially constant (Fig.   11 ). This indicates that the choice of the mesh size essentially determines the accuracy. consists of complementing the BIM with a few null field relations and stating that field vanishes inside the scattering body. This method, 7 which usually eliminates the nonuniqueness problem, is not implemented in the present code.

V. SCATTERING BY THE ARIANE IV STRUCTURE
We now describe some typical results of calculations relating to the Ariane IV launcher. We will first consider the complete launcher and then examine in some detail the field structure in the vicinity of the equipment bay and fairings of the rocket. In all calculations it is assumed that the outer , structure of the launcher is acoustically rigid.

A. Geometrical configuration
The geometry of the Ariane IV launcher is shown in Fig.   12; the definitions of the different elements of this vehicle and a plan view of the launch pad and gas trenches are also given. Among the different types of fairings that may be mounted on the launcher we will only consider the short version corresponding to a single payload. In this case, the fairings length is 8.6 m.
For a frequencyf less than or equal to 63 Hz, it is possible to discretize the complete launcher, including the four lateral boosters, and obtain a mesh that satisfies condition (17). Beyond this frequency the analysis must be restricted to a portion of the launcher. The most sensitive ensemble is  Fig. 13 (a) Because the geometry of the equipment bay base plate has a notable influence on the sound field, we will also consider a modified version of this element. In this version the base plate is replaced by a toric panel, which assures a smooth transition between the fairings and the third stage.

the fairings and equipment bay. To perform calculations of the field in the vicinity of these elements it is necessary to represent a portion of the third stage. The mesh used in the calculations is displayed in
The mesh corresponding to this modified configuration is shown in Fig. 13 (b). The mesh used in the complete launcher calculation is displayed in Fig. 13 (c).

B. Results of calculations for the complete launcher
The results presented here correspond to a frequency f= 63 Hz. For this frequency, the Strouhal number based on the nozzle diameter and jet exhaust velocity is St =fd/ uj =0.025. This number is below that corresponding to the maximum of the spectral density of the acoustic power radiated by the launcher (Stmax •0.1 ). From model scale experiments it is known that noise at this frequency is mainly radiated from a point located at about 40 nozzle diameters.
When the rocket is on the launch pad, the noise sources corresponding to f= 63 Hz are located at Xs_•26.5 m, ys=11 m, andzs= --7 m andxs= --21.7 m, ys=17.2 m, and Zs --• --7 m. We only consider the first source and we use a point source to represent the noise radiation. This is justified since we are not trying to predict the acoustic environment of the launcher, but only wish to examine scattering effects associated with the presence of the vehicle in its own sound field. For this study, it is also natural to normalize the total pressure field Pt = Pi -•-Ps by the incident field Pi. The amplitude of the reduced pressure field is then evaluated in decibels, thus yielding a reflection-diffraction index:  Another view of the pressure field is obtained by plotting the RDI in a horizonatal plane (Fig. 15). This plane, located at z = 8 m, cuts the lateral boosters near their middle. The diffraction index reaches local maximums on the booster P 1, which directly faces the noise source, and in the regions situated between the central body and boosters P2 and P4. The shadow region formed behind the vehicle is also well defined.

C. Results of calculations for the launcher fairings
We now consider the acoustic environment of the rocket fairings. We will only examine results obtained at a frequency f= 125 Hz, which corresponds to a Strouhal number St=0.05. This number is close to that corresponding to the maximum radiated power. When the launcher is on the ground this frequency is mainly radiated by points located in the gas trenches at a radial distance rs --• 29 m and in a plane To allow a qualitative comparison with the calculations we have plotted in Fig. 18 some of the experimental data obtained for different elevations of the launcher and a single propulsion configuration.
The amplitude of the sound field below the equipment bay is found to exceed the general level observed on the fairings and on the third stage by 5-6 dB.
Another interesting view of the sound field structure near the base is shown in Fig. 19. The plot corresponds to a plane perpendicular to the rocket axis located at a close distance from the base of the equipment bay. Figure 19 indicates that the field is enhanced on the incidence side and in the two quadrants adjacent to this direction. The extension of the region of increased diffraction index is strongly influenced by the presence of the base plate.

From a technical point of view it is interesting to see if a
modification of the geometry of the base may induce a decrease of the field amplitude near the equipment bay. The modification considered consists of placing an inclined torus below the base and in this way assuring a smooth transition between the fairings and the third stage. Figure 20 shows the RDI distribution corresponding to this new geometry. The general structure of the sound field is not profoundly modified in this configuration. The most important changes are observed near the transition element. The diffracted wave that was generated in the base region has also nearly vanished.
A view of the field structure plotted in the same horizontal plane as Fig. 15 shows some significant changes in the RDI distribution (Fig. 21 ). The maximum of this index is about 4 dB. On the shadow side, the RDI index passes below --3dB.
These results indicate that the modified geometry leads to a 3-dB decrease of the sound amplitude in the vicinity of the equipment bay.