Solvers for 3D acoustic problems



Principle ideas of our methodology

Formulation of the problem.
Let U(x,y,z) be an acoustic field generated by a plane wave G and an obstacle D. The so called scattered wave u = U - G satisfies the Helmholtz wave equation
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with the Dirichlet boundary condition u = -G on the obstacle boundary and the Sommerfeld radiation condition at infinity. Here k is the wave number.


Spherical locally fitted grids.
The finite element technique with the spherical locally fitted grids is used for the problem approximation. The traces of such grids on two model obstacles are shown below.

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Solver behavior for different wave numbers.
Let the characteristic size of the model obstacle be D, the wavelenth be L and the plane wave G be propagated in (0,1,1)-direction. The problem is solved by the fictitious domain method in a subspace of small dimension. Let N denote the problem size and n be the dimension of the subspace. The number of GMRES iterations (itr) needed to decrease the initial residual error in 106 times are given in the following table: 

     Union of 3 ellipsoids       Tube-like obstacle

D/L      N      n/N    itr         N      n/N    itr
1.0   4.0x104   0.44    8       6.5x104   0.38    30 
1.2   4.7x104   0.39    7       7.4x104   0.35    19
1.5   1.0x105   0.31   12       1.6x105   0.29    40
2.0   2.1x105   0.24   25       3.5x105   0.24    16
3.0   1.0x105   0.31   10       1.6x105   0.29    39
6.0   7.1x105   0.17   10       1.0x106   0.18    15
The isolines of the real part of the total acoustic field in the XZ plane cross section are shown below.

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Far field patterns.
One of the important characteristics of the scattered wave is its far field pattern. The far field pattern demonstrates the energy distribution in the scattered wave (u = U - G). For the highest frequency waves scattered by the above obstacles we get the following diagrams in the XZ-plane.

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References.