Page 24 - Spring 2007

vibrating structure can be derived in the form of an integral equation, which is an equation with an unknown function under an integral sign. Even though we cannot actually solve the integral equation for the unknown function explicitly for any but the simplest problems, it is still useful to know the form of the solution in terms of physical variables. Also, the solution provides great insight into how the problem might be solved numerically, and, as such, the Kirchhoff-Helmholtz integral equation (KHIE) commonly provides the starting point for many numerical formulations:
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In the KHIE, r represents a point in the acoustic fluid and q is a point on the boundary of the fluid S, or boundary surface. In the context of sound radiation from structural vibrations, the boundary surface is the outer surface of the structure in contact with the acoustic medium. The function G stands for Green’s function, and G and dG/dn in the KHIE represent the acoustic fields of simple (monopole) and dipole sources, respectively, distributed along the boundary surface. The dipole sources are aligned in the direction perpendicular to the boundary surface. The monopole and dipole sources are functions of the point q where they are located and the field point r. The KHIE shows that the acoustic field depends only on what happens at the boundaries since the surface vibrations cause the radiated acoustic field.
By studying the solution for the acoustic field given by the KHIE, much can be learned about how to derive a numerical solution. Unfortunately, its simplicity is deceiving. The acoustic fields of simple and dipole sources are singular functions of the source and field point locations, such that they become infinite when the two points coincide (when r 􏰀 q ). However, in the KHIE, these singular functions add together to yield nonsingular pressure and velocity fields. Taking R to be the distance between r and q, the singularities for the different sources can be categorized as:
hyper-singular function integrated over a surface to yield a finite value. Indeed, the only reason the velocity field of a dipole source yields a finite value is because the surface pres- sure and normal velocity, which are weighting functions for simple and dipole sources, are related through a gradient operation. To illustrate, consider a right-angled corner, where the surface normal is discontinuous. As we travel along the surface towards the corner, the pressure field must change such that the gradient operation produces the correct veloci-
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(1) pressure of a simple source G ~1/R,
(2) velocity of a simple source dG/dn0 ~ 1/R ,
(3) pressure of a dipole source dG/dn ~ 1/R2,
(4) velocity of a dipole source d(dG/dn0)/dn ~1/R3.
The process of taking a derivative makes a singular func-
numerical solutions for many problems.
After constructing and solving the matrix system, the
pressures and normal velocities are known for each element of the boundary surface. The KHIE can then be used to directly compute the pressure at each desired field point loca- tion. The overall sound power output can be computed by setting up a grid of field point locations on a surface enclos- ing the structure and numerically integrating the acoustic intensity over the surface. In theory, the power output can also be computed directly using the pressure and normal velocity on the boundary surface, but this is problematic because this is where the largest errors tend to occur in a boundary element solution.
Aside from the mathematical details in evaluating the integrals, most recent innovations in boundary element methods have concerned ways to increase the speed of the computations and reduce storage requirements. As with all numerical computations, boundary element methods have
tion more singular. For example, taking the derivative of r-1 -2
with respect to r gives -r . When r is less than unity, the func- tion r-1 is smaller than r-2, so that r-2 goes to infinity faster as r 􏰀 0. A function that depends on 1/R is called weakly sin- gular, one that depends on 1/R2 is called strongly singular, and one that depends on 1/R3 is called hyper-singular.
How is it possible, then, for singular functions to add up to give finite pressure and velocity fields? First, integration is a “smoothing” operation. Thus, it tends to reduce the level of singularity, the same way differentiation increases it. Because the functions are integrated over a surface, this process essen- tially reduces the level of singularity by two orders. Thus, it makes sense that weakly and strongly singular functions should yield finite values. However, we would not expect a
22 Acoustics Today, April 2007
The level of continuity in the pressure field is also important. In the exact solution, the surface pressure is an absolutely continuous function, such that the function itself, as well as all of its derivatives, are con- tinuous. This level of continuity is impossible to duplicate with simple polynomial approximations. However, the pres- sure and surface velocity, taken together, must enforce - iωρ v = - p as we travel along the boundary. Thus, it is impossi- ble to duplicate the level of continuity in the actual surface pressure distribution with simple interpolation functions and it is also impossible to exactly enforce the specified boundary
ty on the other side of the corner.
conditions!
From a practical point of view, how is all this relevant?
First, many of the research papers written about boundary element methods in the last ten years are concerned with sorting out the mathematical details of the integrals. These papers are written by, and for, people writing their own boundary element codes. A novice would find it very difficult to understand all the mathematical complexities. (Admittedly, even after years of dedicated effort, we find many of the papers incomprehensible.) A reader interested in a simple explanation of boundary element methods will find the earliest papers on the subject, written in the 60’s, much
20, 21, 22
Also, for the casual users who are not trying to write their own boundary element code, it is only important to understand that current boundary element codes are not perfect. They probably do not take care of the singularities in the KHIE such that the acoustic field is strict- ly non-singular. Nonetheless, it has been well shown over the years that even simple approximations for the pressure and
easier to understand.
normal surface velocity weighting functions yield adequate 2 22
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