Efficient calculation for evaluating vast amounts of quadrupole sources in BEM using fast multipole method

Transcription

1 PROCEEDINGS of the 22 nd International Congress on Acoustics Boundary Element and Meshless Methods on Acoustics and Vibrations: Paper ICA Efficient calculation for evaluating vast amounts of quadrupole sources in BEM using fast multipole method Takayuki Masumoto (a), Arief Gunawan (b), Masaaki Mori (c), Yosuke Yasuda (d), Takuya Oshima (e), Tetsuya Sakuma (f) (a) Cybernet Systems Co.,LTD., Japan, (b) Cybernet Systems Co.,LTD., Japan, (c) Cybernet Systems Co.,LTD., Japan, (d) Kanagawa University, Japan, (e) Niigata University, Japan, (f) The University of Tokyo, Japan, Abstract There are increasing demands for computational prediction of the propagation of flow-induced noise. As numerical approaches for predicting flow-induced noise, finite-difference method (FDM), finite-element method (FEM) and boundary-element method (BEM) are extensively used to solve the Lighthill's equation or the Curle's equation. Among these approaches, the BEM has a wide field application due to several benefits such as smart modeling of the acoustic radiation field and easy mesh generation. Despite these benefits, both memory requirement and calculation complexity increase by the second power of the number of DOFs in the BEM approach. Therefore the BEM with the application of the fast multipole method (FMBEM) was developed. The FMBEM reduces both memory requirement and calculation complexity to the linear increase. However, when BEM is applied to predict the propagation of flow-induced noise, calculation cost for evaluating quadrupole point sources becomes to be unpractical level. This is due to the fact that the effect of each source should be evaluated at each boundary element in the BEM procedure. Therefore, the calculation complexity and memory increase by the factor of the number of quadrupole sources times the number of boundary elements. To reduce the calculation complexity and memory, the fast multipole method is applied for the quadrupole sources evaluation. Consequently evaluation time was reduced to almost linear manner. In this paper, the analysis method, the validation of various parameter settings and some numerical examples are shown. Keywords: flow-induced noise, boundary element method, fast multipole method

2 1 Introduction An acoustic analogy has been widely used to compute flow-induced noise [1]. In this method, an unsteady flow field must be calculated to obtain the acoustic source information, then the propagation of the far-field acoustic pressure is predicted mainly using the Curle's equation [2], the Ffowcs Williams-Hawkings equation [3] or some numerical methods such as the finite element method (FEM) or the boundary element method (BEM) based on the Lighthill s analogy [4,5]. Among numerical methods, the application of the BEM is increasing in these days because the BEM offers several benefits such as the smart modeling capability of an acoustic radiation field and the feature of easy mesh generation. In addition to these benefits, the application of the fast multipole method (FMM) to the BEM (FMBEM) extends the applicable field of the BEM. Because the FMBEM reduces both memory requirement and calculation complexity to almost linear increase [6,7], the application of the BEM can be extended to higher frequency and larger size of the acoustic field. When the BEM is applied to the flow-induced noise prediction based on the Lighthill s analogy, sound sources generated by the flow (aerodynamic sound sources) should be evaluated at each element position first. Because these sources are normally extracted from the result of computational fluid dynamics (CFD), the number of sources equals to the number of nodes (N s ) used by CFD. Therefore the computational complexity for this process is O(N s N b ) where N b is the number of boundary elements. Considering the fact that the computational complexity for the FMBEM excluding sound source evaluation process is almost O(N b ) and N s >> N b, the computational complexity of evaluating aerodynamic sound source accounts for the large ratio of the whole process and this results in unpractical calculation time. In this paper, to reduce the computational complexity, the FMM is applied to the aerodynamic sound sources evaluation process. 2 Computational procedure 2.1 Aerodynamic sound sources The Lighthill s equation in the frequency domain (1) is derived from the equation of continuity and the compressible Navier-Stokes equation, as follows. (1) (2) 2

3 where is the sound pressure, is the wave number, is the density, is the speed of sound, and are the i-th and j-th component of the flow velocity respectively, is the Kronecker s delta and are coefficients of the Lighthill stress tensor. To obtain in frequency domain, a discrete Fourier transform (DFT) has been applied. The acoustic source term is evaluated by the following formula. (3) (4) where and are position vectors of the aerodynamic sound source and observation point respectively, i is the imaginary unit and is the angular velocity. 2.2 Efficient evaluation of aerodynamic sound sources Concept of multilevel fast multipole method (FMM) Multipole expansion expressions Figure 1 illustrates two types of sound fields: Fields 1 and 2. In Field 1, all source points are closer to the origin than observation points, and the field holds the Sommerfeld radiation condition. In this case, multipole expansion can be used to express the potential at an observation point by: origin sources observation point r sources observation point r origin Field 1 Field 2 Figure 1: Illustration of two types of sound field. (5) where r is the position vector of the observation points, and represent the angles in the spherical coordinate, r=, is the multipole expansion coefficient, is the spherical Hankel function of the first kind, is the spherical harmonics function, and is the spherical wave function of singular type. In Field 2, observation points 3

4 are closer to the origin than all source points. Local expansion can be used to express potential at an observation point by: (6) where is the local expansion coefficient, is the spherical Bessel function, and is the spherical wave function of regular type. Equations (5) and (6) are expressions with the origin as the expansion point. As long as the relationship illustrated in Figure 1 is satisfied, the expansion points can be moved arbitrarily by: (7) where and are the position vectors of expansion points before and after translation, respectively, and are the expansion coefficient vectors at the expansion points and is the dense translation matrix. FMM Whereas the matrix-vector product in conventional BEM procedure is performed with respect to all elements, in the FMBEM, elements are grouped into cells and contributions of element groups are accumulated as multipole expansions of the cells representative points, and the influence between cells is analyzed. This considerably reduces the amount of calculation and required memory size. Further efficiency can be achieved by introducing a hierarchical cell structure. Figure 2 shows a hierarchical cell structure for a two-dimensional problem and how influences of the elements are translated. First, a level 0 cell that contains the whole boundary is defined, and then lower-level cells are stratified. Contributions of elements are accumulated as the multipole expansion at the boundary Level 0 Step 1 Step 2 Step 3 level 2 level 3 level 1 Step 4 Step 5 Figure 2: A hierarchical cell structure (for a 2D problem). center point of the lowest-level cell (Step 1), in the same way, the contributions are translated and accumulated at the center point of the higher-level cell (Step 2). The accumulated contribution is translated to the center point of a distant cell and accumulated (Step 3). Then, the contribution is translated to the center points of the lower-level cells (Step 4), all the way until it is translated to the elements of the lowest level cell (Step 5). Finally, influence from neighbour cells that cannot be handled with multipole expansion is directly evaluated between the elements (Step 6). 4

5 Efficient translation method and analysis frequency range As shown by equation (7), however, in order to translate multipole expansion coefficients a translation matrix needs to be created which size is, where is the truncation order of the multipole expansions. For the efficient translation of expansion coefficients, Rokhlin proposed a method [8], in which the far-field transform with spherical harmonics is used to translate expansion coefficients into a directivity function on the unit sphere (far-field signature function). Applying this method for the translation, diagonalized equation (8) is obtained. (8) where and are far-field signature functions corresponding to the multipole and local coefficients at a point r respectively, M and L are multipole and local expansion points, respectively,,,. Though diagonalization of the translation improves calculation efficiency, it is known that this translation is unstable when kd is considerably low (kd << 10). Thus, this method is unavailable for calculation in the low-frequency range. In this paper we use this method under the condition of kd > FMM for evaluating aerodynamic sound sources Because the aerodynamic sound sources are quadrupole, the difference in computational procedure between evaluation of element contributions and aerodynamic sound sources is in step 1. As the step 1 of FMM for the aerodynamic sound source evaluation, the following equation is evaluated instead of equation (34) in [6]. (9) where is the translated and accumulated contribution of the aerodynamic sound source of the cell ml at the lowest level, is the wave number vector at the lowest level, is the group of aerodynamic sources defined at the CFD node locations included in the cell and is the position vector from the location of the aerodynamic sound source l to the center of the cell Handling of aerodynamic sound sources located outside of the bounding box of the boundary element mesh When aerodynamic sound sources are located at the outside of the cell structure defined for the boundary element mesh (the acoustic cell structure), the contributions from these sources cannot be obtained through the cell structure. Therefore the new cell structure that 5

6 circumscribes the mesh used by CFD is firstly defined. In order to perform the FMM process efficiently, the higher level cells (level -1, level ) are defined recursively based on the level 0 acoustic cell until the new cell structure includes completely the fluid mesh (Figure 3). Secondly aerodynamic sound source contributions are translated and accumulated for all the lowest cells through the step 1 and those are translated to the higher level cells by the step 2. Thirdly the contributions are translated to the cells indicated by pink boxes in Figure 4 at the same hierarchical cell level by the step 3. Lastly, the contributions are translated to each boundary elements through the step 4 and the step 5 (Figure 4). Acoustic boundary element mesh Level 0 Level -1 Fluid mesh Level -2 Figure 3 : Conceptual diagram of the new cell structure layout in two dimension. (A case that a hierarchical cell level of a root cell for a fluid mesh is -2) Level -1 This cell includes whole fluid mesh Level 0 This cell includes whole acoustic mesh Level 1 Level 2 Level 3 (Lowest level) Boundary elements Step 1 Step 2 Step 3 Aerodynamic sound sources Step 4 Step 5 Step 6 Cells including boundary elements and aerodynamic sound sources Cells including aerodynamic sound sources Figure 4: Conceptual diagram of a translation of aerodynamic sound source contributions. (A case of hierarchical cell level being set to 3 and fluid mesh being circumscribed by level -1 cell. Note that some of the step 3 process at level 2 and 3 and the step 6 process at the lowest level are omitted to be illustrated for simplicity.) 6

7 2.3 Numerical results To confirm the accuracy of transformation by the FMM, the contribution computed by the FMM is compared to that of calculated by the direct evaluation of the equation (3) with Model setup is set to 1. The quadrupole source is located at the origin and the evaluation points are located on the circumference of a circle whose center is the origin. The radius of the circle ( ) and the wave number are set to be k =2. The level 0 cell is defined to circumscribe the evaluation circle and the k of the level 0 cell is consequently, where is the diagonal length of the level n cell. The hierarchical cell level is set to 4 resulting in the k of the lowest cell being Figure 5 shows the pressure amplitude distribution at the evaluation points. From T 11 in xy-plane y From T 12 in xy-plane y From T 13 in zx-plane x x x z From T 22 in xy-plane From T 23 in yz-plane From T 33 in zx-plane y z x x y z Direct evaluation FMM evaluation Figure 5: Comparison of the translation of sound source contribution. 7

8 3 Application to the practical model 3.1 Low Mach number flow past the 3D circular cylinder Transient CFD simulation The turbulence vortex shedding from a 3D circular cylinder of diameter D = 10 mm and span L = 20D is simulated at Re = 26,000 and M = (U = 40 m/s). The unsteady flow field is calculated using the CFD code ANSYS CFX [9] version 15.0 and its incompressible LES (Dynamic Smagorinsky model) calculation features. There are 1,550,000 cells and 1,588,730 nodes in the computational domain (14D 10D 20D). A steady velocity is imposed on the inflow boundary. No-slip conditions are applied on the cylinder and side walls. Slip-conditions are applied on the top and bottom boundaries. On the other boundary, a zero pressure outflow condition is applied. The transient simulation was performed with the time step size = 5e-5 s. Figure 6 shows the geometry and the result. In this figure an instantaneous snapshot of an isocontour of the Q criterion factor is shown Acoustic simulation side wall M= side wall Figure 6: Instantaneous snapshot of an isocontour of the Q criterion (Q=500000) The number of boundary elements and aerodynamic sound sources is 1,200 and 1,643,169 respectively. These aerodynamic sound source informations (volume integral part in equation (3)) are extracted from the CFD result. The hierarchical cell level is set to 4, it also circumscribe the fluid mesh. Responses are calculated at 1710 Hz by conventional approach (direct evaluation of the sources and conventional BEM) and FMM approach (FMM evaluation of the sources and FMBEM). Amplitude [Pa] 250 Figure 7 shows acoustic pressure distribution on the 200 boundary element mesh. The difference of maximum acoustic pressure in amplitude between 100 both results is 0.07 pa. This is 0.03% of the maximum pressure from the conventional approach. 0 Figure 7: Acoustic pressure distribution at 1710 Hz Computational cost Right: conventional approach, Left: FMBEM Table 1 lists the computational time only for the aerodynamic sound source evaluation. In this case the aerodynamic sound source evaluation time by FMM approach is 2.9 times faster than that of direct evaluation approach. top bottom 8

9 Table 1: Computational Time for aerodynamic sound source evaluation. (Num. of aerodynamic sound source : 1,643,169, Num. of element : 1,200) 3.2 Side view mirror model Transient CFD simulation The turbulence vortex shedding from a side view mirror of cylinder body diameter D = 0.2 m and height H = 0.3 m is simulated at Re = 532,608 and M = (U = 38.9 m/s). The unsteady flow field is calculated using the CFD code ANSYS Fluent [10] version 15.0 and its incompressible LES (Dynamic Smagorinsky model) calculation features. There are Direct FMM s s 3,538,170 cells and 2,927,466 nodes in the computational domain. A steady velocity is imposed on the inflow boundary. No-slip conditions are applied on door mirror and the plate. Slipconditions are applied on the top and side boundaries. On the other boundary, a zero pressure outflow condition is applied. The transient simulation was performed with the time step size = 1e-4 s. Figure 8 shows the geometry and the result. In this figure an instantaneous snapshot of the pressure in the flow field is shown Acoustic simulation 38.9 m/s (140 km/h) Figure 8: Instantaneous snapshot of the pressure The number of boundary elements and aerodynamic sources is 6,330 and 2,927,466 respectively. The hierarchical level is set to 6, and the level -2 cell is used to circumscribe the fluid mesh. The response is calculated at 1670 Hz by the same approaches as the previous case. Figure 9 shows the acoustic pressure distribution on the boundary element mesh. The difference of maximum acoustic pressure in amplitude between both results is 9e-3 pa. This is 0.2% of the maximum pressure from the conventional approach. Sound Pressure Level [db] Figure 9: Acoustic pressure distribution at 1670 Hz. Right: conventional approach, Left: FMBEM 30 9

10 3.2.3 Computational cost Table 2 lists the computational time only for the aerodynamic sound source evaluation. In this case the time for the aerodynamic sound source evaluation by FMM approach is 20.4 times faster than that of direct evaluation approach. Table 2: Computational Time for aerodynamic sound source evaluation. (Num. of aerodynamic sound source: 2,927,466, Num. of element: 10,443) 4 Conclusions Direct FMM s s When the BEM is employed to evaluate the Lighthill equation, contributions from significantly large number of sound sources compared to the number of BEM elements have to be evaluated, and this process takes long time. Therefore to reduce the computational time, FMM is applied to this process. As a result, 3 to 20 times faster calculation is achieved. As a future work, we will apply FMM for the evaluation of aerodynamic sound source at observation points, and for the other translation technique which can be applied for the low frequency region. References [1] Wagner, C.; Hüttl, T.; Sagaut, P. Large-Eddy Simulation for Acoustics, Cambridge University Press, Ver IL, [2] Curle, N. The influence of solid boundaries on aerodynamic sound. Proc. Roy. Soc. Vol A 231, 1955, pp [3] Ffowcs, Williams, J.E.; Hawkings, D.L. Sound generation by turbulence and surfaces in arbitrary motion. Philosophical Transactions of the Royal Society, Series A 264, 1969, pp [4] Lighthill, M. On sound generated aerodynamically, I. General theory. Proc. Roy. Soc., Vol. A 211, 1952, pp [5] Lighthill, M. On sound generated aerodynamically, II. Turbulence as source of sound. Proc. Roy. Soc., Vol. A 222, 1954, pp [6] T, Sakuma; Y, Yasuda. Fast Multipole Boundary Element Method for Large-Scale Steady-State Sound Field Analysis. Part 1: Setup and Validation. ACTA ACUSTICA UNITED WITH ACUSTICA, Vol. 88, 2002, pp [7] Y, Yasuda; T, Sakuma. Fast Multipole Boundary Element Method for Large-Scale Steady-State Sound Field Analysis. Part 2:Examination of Numerical Items. ACTA ACUSTICA UNITED WITH ACUSTICA, Vol. 89, 2003, pp [8] V, Rokhlin. Diagonal forms of translation operators for the Helmholtz equation in three dimensions. Appl. and Comput. Harm. Anal., Vol. 1, 1993, pp [9] ANSYS CFX version 15.0 User's Guide, ANSYS Inc. [10] ANSYS Fluent version 15.0 User's Guide, ANSYS Inc. 10