Boundary Element Method-CSA

Acoustics Research Centre

Boundary Element Method(BEM)Brief

Prof. Y. W. Lam


Basics• BEM is a numerical technique to solve partial

differential equations (PDEs) of a variety of physical problems with well defined boundary conditions.

• The PDE over a problem domain is transformed into a surface integral equation over the surfaces that enclosed the domain.

• The integral equation can be solved by discretisingthe surfaces into small patches - boundary elements.

• Only surfaces need to be discretised - resulting in a lot less elements than FEM.

• Particularly useful for acoustics - problem domain often involves the entire 3D space in free field


Acoustic ProblemThe (harmonic) acoustic problem is defined by the wave equation (with or without sources)

and the boundary conditions

special cases:


Transform to Integral Equation

Since the problem is defined by the bc, we can re-formulate the problem on the boundary itself - mathematically transformed into an integral equation(no source exterior problem):

Unknown 1:Dipole sourcedistribution

Unknown 2:Monopole sourcedistribution

In full 3Dspace

In half 3Dspace

Inside solid

Huygens principle - wave propagation as originates from sources on a wave front (in our case the boundary)


Interior Problem:No source Interior problem:

Unknown 1:Dipole sourcedistribution

Unknown 2:Monopole sourcedistribution

In full 3Dspace

In half 3Dspace

Outside Volumn


Application of Boundary ConditionsThe boundary condition relates surface pressure to surface velocity, reduces the number of unknown fields from 2 to 1 on the surface:


Adding Sources:Exterior Problem:

Interior Problem:


Numerical SolutionBoundary divided up into small boundary elements - small enough so that approximations can be made to facilitate a solution.

Unknowns out of integral

Simplest approximation - surface pressure and velocity constant over each element, andShape function [N]=1 (zero order)


Linear EquationsA set of linear equations that can be solved numerically:

Coefficient matrices

Known variablesUnknown variable


Characteristic Frequencies

• Non-unique solution (failure) when ~ interior region resonates.

• Overcome by either:– Combined with the equation for r in Vint in the case of exterior

problems (or r in Vext in the case of interior problems). An over-determined matrix equation is formed and solved by a least square method.

– A combination of the integral equation and its normal derivative formulation, provided that the coefficient of the combination satisfies certain requirements.

• Automatically taken care of by the computer program.


Thin Panel Approximation

When a panel is thin, the surface pressure and velocity on either side of the panel is related and an analytical equation can be used to reduce the modelling to just one side of the panel

Automatically done in DIFTHINI


DIFTHINI

• Purely for number crunching – calculation• Console program – run in a Windows Command

Prompt Box• Problem/model specified entirely by the Input Data

File• Results in 2 Output Files

– .out – mainly for debugging– .lst – main results in dB


Example Input File/*/* Example data input file for DIFTHINI./*/* These are comment lines/*/* Modelling a simple panel (0.3mx0.2m) with uniform surface velocity./* Symmetry modelled in the x and y directions./* Polar radiation pattern at 1kHz calculated in the x-z principle plane/* from front to back at a distance of 1m./*/* Number of nodes used to form the BE model

12/* Node X Y Z 1 .0000 .0000 .0002 .0500 .0000 .0003 .1000 .0000 .0004 .1500 .0000 .0005 .0000 .0500 .0006 .0500 .0500 .0007 .1000 .0500 .0008 .1500 .0500 .0009 .0000 .1000 .00010 .0500 .1000 .00011 .1000 .1000 .00012 .1500 .1000 .000/*/* Specification of sound field calculation option/* -1 to calculate SPL at points on spherical surfaces /* (NB. a positive number indicates a list of specific points)/*

-1/*/* Radius of spherical surface, Min Max and Interval of calculation

1. 1. 1.0/*/* Angle (degree) from z-axis, Min Max and Interval of calculation

0 360 2/*/* Angle (degree) from x-axis in xy plane, Min Max and Interval

0. 0. 1/*

/* Specification of Boundary Elements/* /* Elements should be square Longest side <1/4 (1/6 if possible) wavelength/* Number of 4-noded Boundary Elements

6/* Element Node1 Node2 Node3 Node4 0 (to indicate Rectangular elements)1 1 2 6 5 02 2 3 7 6 03 3 4 8 7 04 5 6 10 9 05 6 7 11 10 06 7 8 12 11 0/*/* Symmetry option (1 for none, 2 for x, 4 for x & y)

4/*/* frequency

10000.000000/* /* List of complex normal velocity values v on each boundary element./* The velocity over each element is assumed to be constant./* Element Real(v) Imaginary(v)/*Velocity1 1 02 1 03 1 04 1 05 1 06 1 0/*/* use -1 -1 -1 to indicate end of this list of velocity values -1 -1 -1/* /* 0 to indicate no external monopole sources 0

/*/* -1 to indicate end of section -1

/*/* -1 to indicate end of data-1

Commentlines

Nodecoordinates

CalculationPoints spec

Elementdefinitions

Surfacevelocity spec


Model of Example Input


Output .OUT file

• The .OUT file contains detailed sound pressure (surface and external) data, an image of the interpreted input model data, and a diagnostic message:

• INTEGRAL FAILED TO CONVERGE X TIMES OUT OF Y

• Y is the total number of integration done by the program. Needless to say the number of failures X should be 0


Example Output .LST File2 0

0.350000 0.350000 1.000000.000000E+00 360.000 2.000000.000000E+00 0.000000E+00 1.00000

PRESSURE DB-P/PIN DB-PS/PIN1000.00

10.000000E+00 0.000000E+00

( 0.350 0.000 0.000 ) 73.39 -4.72 -6.52( 0.350 2.000 0.000 ) 73.34 -4.75 -6.53( 0.350 4.000 0.000 ) 73.22 -4.84 -6.54( 0.350 6.000 0.000 ) 73.03 -4.99 -6.57( 0.350 8.000 0.000 ) 72.79 -5.15 -6.60( 0.350 10.000 0.000 ) 72.56 -5.29 -6.65...

( 0.350 360.000 0.000 ) 73.39 -4.72 -6.52-1 -1 -1

-1.00000

Echo of calculationPoint spec

Calculation Point Coord,(r,θ,Φ) in this example

SPL IL

Documents

Boundary Element Method-CSA