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Home| Lecture notes| Exercises | Institute of Applied and Experimental Mechanics Boundary Element Methods
Lecture notes
Next:Introduction
Boundary Element Methods
Lothar Gaul and Matthias Fischer
IntroductionBE Formulation of Laplace's Equation
Weak formulation of the differential equationTransformation on the boundary
Fundamental solution as weighting functionBoundary integral equation of the 2-D problemPreparative example for the limit processCalculation of the limit
Discretisation of the boundaryThe collocation methodExample: Laplace problem of heat transfer
Numerical solution with the collocation methodAnalytical solution
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Computation of solution in the domainCalculation of Dirichlet variable in the domainCalculation of flux in the domain
BE formulation of Poisson's equationCalculation of domain integrals by integration of cellsCalculation of domain integrals by transformation into a boundary integralCalculation of the unknown boundary variables
Orthotropic constitutive behaviour in the domain
Indirect calculation of diagonal elements inConcentrated source termsSubstructure techniqueExample: Orthotropic heat transfer and subregion couplingFundamental solutions
Laplace equationFundamental solution of the 2D Laplace equationFundamental solution of the 3D Laplace equation
Helmholtz equationsFundamental solution of the 3D Helmholtz equation
About this document ...
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Boundary Element Methods
Introduction
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Introduction
This manuscript accompanies the lecture Boundary Element Methods in Statics and Dynamics. However, thematerial presented on the web cannot include all the aspects that are discussed in the class.
Focus point of the manuscript is the derivation of the standard boundary element method for Laplace's equation.Starting from the differential equation, the BEM is formulated step by step. Simple examples are calculated andcompared to analytical solutions. The handling of domain integrals in the BEM is discussed on the example ofPoisson's equation. Some advanced techniques and the derivation of selected fundamental solutions conclude
the manuscript.
The lecture covers additional important aspects of boundary elements. For example the application of themethod to elastostatics and elastodynamics as well as to acoustics. Furthermore advanced formulations such asthe Dual Reciprocity BEM and variational BEM are presented.
Page 1 of 2Introduction
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Next:BE Formulation of Laplace's EquationUp:Boundary Element MethodsPrevious:Boundary Element Methods
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Page 2 of 2BE Formulation of Laplace's Equation
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The collocation methodExample: Laplace problem of heat transfer
Numerical solution with the collocation method
Analytical solution
Computation of solution in the domainCalculation of Dirichlet variable in the domainCalculation of flux in the domain
Next:Weak formulation of the differential equationUp:Boundary Element MethodsPrevious:Introduction
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Page 1 of 2Weak formulation of the differential equation
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Boundary Element Methods
Weak formulation of the differential equation
Next:Transformation on the boundaryUp:BE Formulation of Laplace's EquationPrevious:BE Formulation of Laplace's Equation
Weak formulation of the differential equation
Starting point for the boundary element formulation is the weighted residual (or weak) statement of the differentialequation. For Laplace's equation , it is given by
with a test function.
Next:Transformation on the boundaryUp:BE Formulation of Laplace's EquationPrevious:BE Formulation of Laplace's Equation
(1)
Page 1 of 2Weak formulation of the differential equation
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Page 1 of 2Transformation on the boundary
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Boundary Element Methods
Transformation on the boundary
Next:Fundamental solution as weighting functionUp:BE Formulation of Laplace's EquationPrevious:Weak formulation of the differential equation
Transformation on the boundary
This step corresponds in 1-D to the partial integration of the differential operator. It requires the application ofspecial integral theorems depending on the problem dimension. These theorems reduce domain integrals inboundary integrals. This is different from the 1-D case where integrals reduce to scalar quantities. In the followingparagraphs the transformation on the boundary is treated for 2-D and 3-D by adopting Green's integral theoremin the plane and in space.
The transformation of the differential operator to the boundary is done by applying Green's theorem twice to theweighted residual statement. In index notation this reads
(2)
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Next:Fundamental solution as weighting functionUp:BE Formulation of Laplace's EquationPrevious:Weak formulation of the differential equation
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Boundary Element Methods
Fundamental solution as weighting function
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Fundamental solution as weighting function
The fundamental solution is the Green's function for the unbounded space and solves the differential equation
The minus sign of the Dirac distribution is introduced for convenience such that the obtained system matricesbecome positive. In 2-D the fundamental solution (s.f. Appendix 9) is given by:
with the common abbreviation of the Euclidean distance . The 3-D case leads to (s.f.
(3)
(4)
(5)
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Appendix 9)
The functions and are denoted as single layer and double layer potentials, respectively. After selecting
, Eq. (2) and associated with the sifting property of the Dirac distribution lead to
where
The common notations for the field point or receiver point (marked by the vector ) and for the load point or
source point (marked by the vector ) have been used. It has to be noticed, that the definition (9) of deviates
from the physical definition of the heat flux vector
In an actual heat transfer problem, physical constants such as the heat conductivity need to be taken into
account.
(6)
(7)
(8)
(9)
(10)
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Other than in the 1-D case, higher dimensional problems lead to integrals over the boundary of the domain ,or - to be more precise - over the boundary consisting of all field points on .
By recalling all steps necessary to derive Eq. (8), one recognizes, that the weighted residual statement Eq. (1)does not lead to an approximation. The question arises, whether an exact solution of Eq. (8) is an exact solutionof Laplace's equation as well. This seems not to be the case, since the weighted residual statement allows forlocal errors in the domain but averages them to zero by domain integration. An exact solution of Eq. (8) whichfulfills the integral pointwise represents a weighting with infinitely many linearly independent test functions in theresidual statement (1). It follows that statement (1) is only fulfilled if the differential equation is satisfiedidentically. Thus, the exact solution of (8) is an exact solution of the corresponding differential equation as well.
Next:Boundary integral equation of the 2-D problemUp:BE Formulation of Laplace's EquationPrevious:Transformation on the boundary
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Page 1 of 2Boundar inte ral e uation of the 2-D roblem
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Boundary Element Methods
Boundary integral equation of the 2-D problem
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Boundary integral equation of the 2-D problemIf the load point in Eq. (8) moves on the boundary, only boundary data are present in Eq. (8). This equation is
called a boundary integral equation. Calculation of the integrals by boundary discretisation leads to algebraicequations for solving unknown boundary data in terms of known boundary data. The singularities of fundamentalsolutions requires careful analysis when the load point is accommodated on the boundary.
Subsections
Preparative example for the limit processCalculation of the limit
Discretisation of the boundary
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Next:Preparative example for the limit processUp:BE Formulation of Laplace's EquationPrevious:Fundamental solution as weighting function
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Preparative example for the limit process
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Preparative example for the limit process
If the integral
is solved as shown, the correct result is obtained by chance but the integration 'with closed eyes' incorporates
the singularity which is improper. The singularity at becomes obvious when the integral is
considered.
If one approaches the singularity in Eq. (11) from both sides by the small quantities and , one obtains
(11)
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Page 3 of 3Preparative example for the limit process
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In stress calculations and special BEM formulations such as the hybrid BEM, so called hyper singularities areencountered (s.f. Part III).
Next:Calculation of the limitUp:Boundary integral equation of the 2-D problemPrevious:Boundary integral equation of the 2-D problem
Table 1: Classification of singularities
Type Property Example
1. weak singularityIntegral is finite atsingularity
2. strong singularityInterpretation as Cauchy
Principal Value
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Calculation of the limit
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Calculation of the limit
To locate the load point on the boundary, we first adjust the boundary such that it contains the point inside acircle of radius according to Fig. 1
Thus the point is inside the domain and Eq. (8) is still valid.
(16)
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The integration along the -circle is parameterized by
(s.f. Fig. 2). Furthermore holds
Figure 1: Boundaryextension by a circle
(17)
(18)
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The limit value by taking Eq. (8) on the boundary can now be calculated. For we get by Eq. (4)
In the limit, the first integral is weakly singular. With Eqs (17, 18) and l'Hospital's rule, the last integral in Eq. (19)results in a vanishing contribution
Figure 2: Geometry for accommodating the load point on theboundary
(19)
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With Eq. (5), the integral leads to
The first integral in Eq. (21) is a strongly singular integral calculated by Cauchy's Principal Value. The secondintegral leads to
Summarizing these results and inserting in Eq. (8) leads to the integral equation
(20)
(21)
(22)
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respectively
The factor is called boundary factor and denotes the fraction of which is inside
Solving the integrals in Eq. (24) analytically is only possible for special cases. For a numerical integration theboundary is divided in segments with the interpolation of boundary data by piecewise continuous functions suchas polynomials. This approach is called discretisation.
Next:Discretisation of the boundaryUp:Boundary integral equation of the 2-D problemPrevious:Preparative example for the limit process
(23)
(24)
(25)
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Discretisation of the boundary
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Discretisation of the boundary
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For an approximation of the geometry, the boundary of the domain is divided in boundary elements(Fig. 3). Every element has one or more nodes. At node of element the value of is and the
value of is . Shape functions describe the spatial distribution on the element. With nodes in element
, the shape of and are interpolated by
Figure 3: Discretisation of the
boundary
(26)
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respectively. Or in matrix notation by
where and are 1 x M row vectors and is a M x 1 column vector. The simplest shape functions areconstant and linear shape functions.
Constant shape function
Only one node exists per element, the values of and are constant throughout the element and have
the value at the node. This means and
and
and (27)
and (28)
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Linear shape function
Figure 4: Constant element shape function and localcoordinate
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Page 6 of 8Discretisation of the boundary
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with the local coordinate . The shape functions are depicted in Fig. 6.
Discretisation of Eq. (24) in 2-D leads to
(30)
Figure 6: Linear shape functions
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The nodal values and are constants and can be brought outside the integrals
If constant elements are used, the node is usually located in the middle of the perfectly flat element (s.f. Fig. 4).Therefore, and Eq. (25) lead to
The vector is perpendicular to if load point and field point are located on the same element, and
therefore
(31)
(32)
(33)
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This simplifies the calculation and makes the numerical implementation easier.
Next:The collocation methodUp:Boundary integral equation of the 2-D problemPrevious:Calculation of the limit
(34)
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The collocation method
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The collocation method
The collocation method allows to calculate the unknown boundary data from Eq. (32). The simplest approach isto establish a system of equations with as many unknowns as equations.
The principle of collocation means to locate the load point sequentially at all nodes of the discretisation such thatthe domain variable at the load point coincides with the nodal value. Because linear and higher order
polynomial shape functions lead to nodes which belong to more than one element, it is worthwhile to introduce aglobal node numbering ( ) which does not depend on the element.
If the load point is located on the first global node the first equation of the system reads
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The notation means the sum of integrals contributed from those elements which contain the global
node, where is the corresponding shape function. The Eq. (35) is given in matrix notation by
where ( ) denotes the element which contains the boundary term .
By collocating the load point with the nodes to the additional equations of the system (37) are obtained
(35)
(36)
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and read in matrix notation
The diagonal elements of the matrices and contain singular integrals because the distance
vanishes at the nodes. All other matrix elements contain regular integrals. Since both vectors and in
Eq. (38) contain known as well as unknown boundary data, it is necessary to rewrite the equations with allunknowns appearing in a vector on one side
A systematic way of doing this and solving the system of equations is demonstrated with a simple example whichcan be calculated by hand.
Next:Example: Laplace problem of heat transferUp:BE Formulation of Laplace's EquationPrevious:Discretisation of the boundary
(37)
(38)
(39)
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Boundary Element Methods
Example: Laplace problem of heat transfer
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Example: Laplace problem of heat transfer
Now, Laplace's equation of heat transfer is considered in a 2-D rectangular domain with aspect ratio 1:2 asdepicted in Fig. 7. At the horizontal boundary lines, the temperatures are prescribed. At the vertical
boundary lines the heat flux is given.
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The remaining boundary values , and , are unknown for a discretisation with four elements. The
numerical results are afterwards compared to the analytical solution.
Subsections
Figure 7: Example: Heat transfer in
rectangular domain
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Numerical solution with the collocation methodAnalytical solution
Next:Numerical solution with the collocation methodUp:BE Formulation of Laplace's EquationPrevious:The collocation method
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Numerical solution with the collocation method
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Numerical solution with the collocation method
For simplicity, constant elements are chosen in the example (i.e. , , . Each element has
only one node located in the middle. If Eq. (32) is written for the load point , one obtains
(40)
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Four equations for the four unknown boundary values are obtained if takes the values 1 to 4 and is located
at the four nodes sequentially. The elements of the matrices are the integrals in Eq. (40) for different values ofand . In matrix notation these equations are
Calculation of matrix elements and : ,
(41)
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Fig. 8 shows:
Figure 8: Calculation of matrix elementsand
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Inserting leads to
and
(42)
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Spatial isotropy of the problem at hand leads to
These symmetries do nothold in general for BEM.
and : ,
(43)
and (44)
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Fig. 9 shows:
Figure 9: Calculation of matrix
elements and
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Inserting leads to
and
(45)
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Spatial isotropy leads to
and : and
The geometry for calculating and is obtained the same way
(46)
and (47)
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Inserting leads to
and
(48)
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and by virtue of symmetry
and : and
The geometry for calculating and is obtained the same way
(49)
and (50)
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Inserting leads to
and
(51)
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and by virtue of symmetry
Diagonal terms:According to Eq. (34) the main diagonal of matrix vanishes
(52)
and (53)
(54)
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For one obtains
And by inserting
The integrand in Eq. (55) is weakly singular, but the integral exists. With (s.f. Eq. (15))
(55)
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is obtained and because of symmetry
holds. The same way follows
and
(56)
(57)
(58)
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(59)
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Assembling and solving the system of equations
Rewriting of Eq. (41) in index notation and summation of the terms leads to
Plugging the matrix elements and the known boundary data leads to
(59)
(60)
(61)
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and after rewriting the equations with all unknown boundary data appearing on the left side
Solving the equations leads to
In terms of physics, the flux has to be multiplied by in order to obtain the heat flux (s.f. Eq. (8) and
(62)
(63)
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Eq. (10)). After multiplication the nodal value is positive and is negative. This means that the heat flux
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through element 1 has the direction of the outward normal vector and the heat flux through element 3 has thedirection opposite to the outward normal vector.
After having used the crudest form of discretisation, a finer boundary mesh with six constant elements of length 1is used. This discretisation still allows the calculation by hand. The matrix entries are calculated as shown for thefour element mesh. The system of equations is obtained as
Rearranging of known and unknown nodal data in the equations and solving the system of equations leads to thesolution
(64)
(65)
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where the nodes 1 and 4 of the six element discretisation coincide with the nodes 1 and 3 of the four element
discretisation.
Next:Analytical solutionUp:Example: Laplace problem of heat transferPrevious:Example: Laplace problem of heat transfer
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Boundary Element Methods
Analytical solution
Next:Computation of solution in the domainUp:Example: Laplace problem of heat transferPrevious:Numerical solution with the collocation method
Analytical solution
The 2-D Laplacian is the field equation of the heat flux problem
The flux in direction vanishes on the boundaries and , . On the boundaries and
the gradient of vanishes in direction. This leads to the conclusion that the seeked solution is
independent of and a trial function with unknown coefficients and depends linearly on
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Fitting the boundary conditions
leads to the constants
The analytical solution is thus given by
The comparison between analytical and numerical solution shows, that for the Dirichlet variable even a coarse
(67)
(68)
and (69)
and (70)
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discretisation leads to good accuracy. The larger error of the Neumann variable for the four element
discretisation can be explained by the fact that the differentiated quantity requires finer discretisation because
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discretisation can be explained by the fact that the differentiated quantity requires finer discretisation becauseintegration smoothes while differentiation creates roughness. The finer discretisation by six elements already
shows a considerable improvement of accuracy.
Next:Computation of solution in the domainUp:Example: Laplace problem of heat transferPrevious:Numerical solution with the collocation method
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Boundary Element Methods
Computation of solution in the domain
Next:Calculation of Dirichlet variable in the domainUp:BE Formulation of Laplace's EquationPrevious:Analytical solution
Computation of solution in the domainThe solution of unknown data in the domain can only be obtained after the data on the boundary have been
calculated. The load point is placed where the domain data shall be calculated. Integration of Eq. (32) along
the boundary with a vector connecting each boundary point with the interior load point gives the value of the
field variables. The boundary factor is chosen according to Eq. (25). The boundary data is completelyknown and consist of given boundary conditions and the values that were calculated using the collocationmethod.
Subsections
Calculation of Dirichlet variable in the domainCalculation of flux in the domain
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Next: Calculation of Dirichlet variable in the domain
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Next:Calculation of Dirichlet variable in the domainUp:BE Formulation of Laplace's EquationPrevious:Analytical solution
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Boundary Element Methods
Calculation of Dirichlet variable in the domain
Next:Calculation of flux in the domainUp:Computation of solution in the domainPrevious:Computation of solution in the domain
Calculation of Dirichlet variable in the domain
The calculation of in the domain is demonstrated on the 2-D example. Rewriting Eq. (40) for the load point in
the domain leads to
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with the boundary factor according to Eq. (25) and the nodal data and . For the solution of the
domain variable all boundary integrals need to be evaluated
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domain variable all boundary integrals need to be evaluated.
Next:Calculation of flux in the domainUp:Computation of solution in the domainPrevious:Computation of solution in the domain
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Home| Lecture notes| Exercises | Institute of Applied and Experimental Mechanics
Boundary Element Methods
Calculation of flux in the domain
Next:BE formulation of Poisson's equationUp:Computation of solution in the domainPrevious:Calculation of Dirichlet variable in the domain
Calculation of flux in the domain
For the calculation of the flux, it is necessary to calculate the gradient of at the load point . This
leads to both flux coordinates
With a matrix notation of scalar products, Eq. (71) leads to
and
and (72)
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Exchanging differentiation and integration, the derivatives with respect to are obtained as
and
The derivatives of the matrix entries of and with respect to are
and
After this, the corresponding integrals need to be solved and lead to the flux at the load point in the domain.
(75)
(76)
(77)
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N t BE f l ti f P i ' ti
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Next:BE formulation of Poisson's equationUp:Computation of solution in the domain
Previous:Calculation of Dirichlet variable in the domain
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H | L t t | E i | I tit t f A li d d E i t l M h i
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Boundary Element Methods
BE formulation of Poisson's equation
Next:Calculation of domain integrals by integration of cellsUp:Boundary Element MethodsPrevious:Calculation of flux in the domain
Boundary element formulation ofPoisson's equation
Poisson's equation with a non-homogeneous term
describes for example the local heat conduction with sources in the domain or torsion of non-circular crosssections. The weighted residue statement
in (79)
(80)
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and the inverse form with Green's theorem lead to the presence of a domain integral
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With the domain integral is given in 2-D by
A mean to calculate the domain integral is to discretise the domain into integration cells and then
using subsequent numerical integrations. The cells look like a finite element mesh. However, the procedure hasan essential difference because there are no unknowns in the domain. The cells are used as integration regionsover which analytical or Gaussian quadrature is performed.
Discretisation of the boundary with elements and of the domain with cells leads to (s.f. Eq. (32))
(81)
(82)
(83)
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The result is a system of equations
and after separating given boundary variables from unknown boundary variables
By adding the vectors and , the system of equations allows to calculate the vector containing the
unknown boundary variables.
Subsections
Calculation of domain integrals by integration of cellsCalculation of domain integrals by transformation into a boundary integral
Calculation of the unknown boundary variables
Next:Calculation of domain integrals by integration of cellsUp:Boundary Element MethodsPrevious:Calculation of flux in the domain
(84)
(85)
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o e| ectu e otes| e c ses | st tute o pp ed a d pe e ta ec a cs
Boundary Element Methods
Calculation of domain integrals by integration of cells
Next:Calculation of domain integrals by transformation into a boundary integralUp:BE formulation of Poisson's equationPrevious:BE formulation of Poisson's equation
Calculation of domain integrals by integration of cells
The example of Laplace's equation in a rectangular domain is now modified such that Poisson's equation (79)holds with const. If the complete domain is taken as integration cell and the boundary is dicretised
with four constant elements, Eq. (83) leads to
with
(86)
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The integration is carried out
Symmetry results in
The result for is obtained by
(87)
(88)
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Symmetry results in
Next:Calculation of domain integrals by transformation into a boundary integralUp:BE formulation of Poisson's equationPrevious:BE formulation of Poisson's equation
(89)
(90)
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B d El M h d
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Boundary Element Methods
Calculation of domain integrals by transformation into a boundary integral
Next:Calculation of the unknown boundary variablesUp:BE formulation of Poisson's equationPrevious:Calculation of domain integrals by integration of cells
Calculation of domain integrals by transformation into a boundary integral
If is a harmonic function, that is, if it satisfies , the domain integral may be transformed into a
boundary integral.
After introducing a function defined by
and using Green's theorem
along with , one arrives at
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with the fundamental solution of Laplace's equation. A so called higher order fundamental solution can becalculated from Eq. (91). With its directional derivative all terms in the boundary representation Eq. (93)
are known.
Determination of
With the 2-D fundamental solution, Eq. (91) is given by
In polar coordinates , Eq. (94) is expressed by
If is assumed to have no dependence, Eq. (96) remains
A first integration
where (94)
(95)
(96)
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(97)
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and a second integration lead to
The choice of results in
Determination of
The directional derivative is executed by
(98)
(99)
(100)
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With Eq. (93), can be obtained from
The simple example where const leads to
The boundary integrals from to can be split into sub-integrals corresponding to each boundary element.
This means e.g. for
For an element which contains the load point as well as the field point, orthogonality leads to
(101)
(102)
(103)
(104)
(105)
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The second integral leads to
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and because of symmetry follows
The result for is
so that at the end is obtained as
(106)
(107)
(108)
(109)
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The result for contains
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The integral is now given by
and because of the problem, symmetry holds
The integral leads to
(110)
(111)
(112)
(113)
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and finally
(114)
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is obtained. One realizes that this result is identical with the one calculated by domain integration. The presentedapproach allows to transform the domain integral onto the boundary But it has to be noticed that the approachonly applies for special functions of . For more general distributions other methods are available such as the
Multiple Reciprocity Method[#!nowak!#] which represents an extension of the approach presented in thischapter, or the Dual Reciprocity Method[#!drm!#] with a slightly different approach.
Next:Calculation of the unknown boundary variablesUp:BE formulation of Poisson's equationPrevious:Calculation of domain integrals by integration of cells
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Boundary Element Methods
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y
Calculation of the unknown boundary variables
Next:Orthotropic constitutive behaviour in the domainUp:BE formulation of Poisson's equationPrevious:Calculation of domain integrals by transformation into a boundary integral
Calculation of the unknown boundary variables
Inserting the results for in Eq. (85) leads to
As compared to the inhomogeneous set of equations (62), another known vector is added on the right hand side.Solving Eq. (115) for a fixed value of leads to the unknown boundary variables. The calculation of the domainvariables proceeds analogue as shown for Laplace's equation.
Next:Orthotropic constitutive behaviour in the domain
(115)
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Orthotropic constitutive behaviour in the domain
Next:Indirect calculation of diagonal elements inUp:Boundary Element MethodsPrevious:Calculation of the unknown boundary variables
Orthotropic constitutive behaviour
in the domainIn an anisotropic domain the constitutive parameters depend on the direction. Orthotropic heat transfer, e.g. withcoordinates and in the direction of orthotropy, is associated with conduction coefficients and .
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The modified Fourier heat conduction equation in an orthotropic domain reads in index notation
where the brackets around the index exclude summation. In 2-D, this leads to
Figure 10: Direction of orthotropy
(116)
(117)
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and
Eq. (116) leads to the orthotropic heat transfer equation
Stationary heat transfer along with homogeneous orthotropic constants and lead to
(118)
(119)
(120)
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The associated fundamental solution is obtained from
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by invoking a coordinate transformation
such that the left hand side leads to the ordinary Laplacian operator
(121)
and (122)
(123)
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Theorem 2 The -distribution has the property
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With this property, Eq. (123) is given by
and leads to the already known 2D fundamental solution (s.f. Eq. (4))
(124)
(125)
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where
d (126)
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can be transformed back to the physical space using Eq. (122). The following calculation is now handled in amanner analogous to the case const.
Next:Indirect calculation of diagonal elements inUp:Boundary Element MethodsPrevious:Calculation of the unknown boundary variables
and (126)
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Page 1 of 3Indirect calculation of diagonal elements in
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Indirect calculation of diagonal elements in
Next:Concentrated source termsUp:Boundary Element MethodsPrevious:Orthotropic constitutive behaviour in the domain
Indirect calculation of diagonal elements in matrix
from physical considerations
Calculation of diagonal elements of matrix requires to determine the fractional boundary coefficients by
integration. Different from the coefficient for a boundary point on a constant element, the determination is more
complex for more complex elements. In the following, a simple way is discussed in which the diagonal elements
can be computed regardless of the element complexity.
The most simple solution to be described by the system matrices is a uniformly constant temperature on theboundary. In this homogeneous case, there is no flux in the domain or on the boundary. With these boundaryconditions and an arbitrary constant , the vectors and are given by
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and (127)
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Eq. (60) leads to
with the singular matrix . The sum of terms in any row of must vanish. This leads to the diagonal elements
of by the negative sum of the off diagonal elements
As the matrix entries of in Eq. (61) show, the sum of the entries in a row does as well vanish when theboundary are determined explicitly. Both procedures lead to the same result.
(128)
(129)
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Next:Concentrated source termsUp:Boundary Element MethodsPrevious:Orthotropic constitutive behaviour in the domain
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Concentrated source terms
Next:Substructure techniqueUp:Boundary Element MethodsPrevious:Indirect calculation of diagonal elements in
Concentrated source terms
In the presence of concentrated source terms in the domain, the volume integral
can be simplified for
(130)
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(131)
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By inserting in Eq. (130):
Next:Substructure techniqueUp:Boundary Element MethodsPrevious:Indirect calculation of diagonal elements in
(132)
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Boundary Element Methods
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Substructure technique
Next:Example: Orthotropic heat transfer and subregion couplingUp:Boundary Element MethodsPrevious:Concentrated source terms
Substructure technique
So far, only homogeneous domains have been treated in which the constitutive properties do not vary. Domainswith piecewise non-homogeneity are now subdivided into homogeneous separate subregions. Afterwards theformulations of the distinct regions are coupled by a substructure technique.
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Fig. 11 illustrates homogeneous subregions 1 and 2 with different constitutive parameters. According to Eq. (84),the formulation for subdomain 1 is
Figure 11: Substructure technique: Division of non-homogeneousdomain in piecewise homogeneous subregions
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(133)
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and for subdomain 2 is
Compatibility of at the interface ( compatibility)
as well as compatibility of ( compatibility)
(134)
(135)
(136)
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lead to a coupled system of equations which can be solved by two methods.
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Method 1
After rearranging Eq. (133) for subregion 1
and Eq. (134) for subregion 2
(137)
(138)
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the coupling with and compatibility according to the constraints in Eq. (135) and Eq. (136), respectively,leads to
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No inversion is necessary for setting up the coupled equations and the system matrix is banded which is anadvantage for the numerical treatment. Another advantage is that all unknowns in the interface, and , are
obtained at once.
Method 2
Multiplication of Eq. (84) with the inverse gives
(139)
(140)
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with matrix and vector .
The application to subregion 1 is
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and correspondingly to subregion 2
Coupling with the constraints in Eq. (135) and Eq. (136) leads to
(141)
(142)
(143)
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As compared to Eq. (139), the smaller set of equations is an advantage. But this is obtained at the cost of aninversion of and the necessity to calculate the flux in the interface from Eq. (141) or Eq. (142) after Eq. (143)has been solved.
Next:Example: Orthotropic heat transfer and subregion couplingUp:Boundary Element MethodsPrevious:Concentrated source terms
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E l O th t i h t t f d b i li
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Example: Orthotropic heat transfer and subregion coupling
Next:Fundamental solutions
Up:Boundary Element MethodsPrevious:Substructure technique
Coupling of an orthotropic and
an isotropic subregionFor illustrating the methods outlined in the preceding chapters, the coupling of an orthotropic and an isotropicsubregion is treated. The matrices from the example in Chapter 2.6 for the isotropic subregion are used. Fig. 12shows the rectangular subdomains with aspect ratio 2:1 which are discretised by 6 constant elements,respectively. Subregion 1 is isotropic and subregion 2 is orthotropic with conduction coefficients and
.
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The elements of matrices and of Eq. (64) in Chapter 2.6.1 define the system of equations for subregion 1.After rearranging, one obtains
Figure 12: Example: Coupling of isotropic and orthotropic subregions
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The calculation of the system matrices for subregion 2 is demonstrated for an example of the matrix and anelement of the matrix . According to Eq. (122), new variables are introduced
Calculation of matrix elements and ( )
(144)
and (145)
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According to Fig. 13 the following relations hold. The variables and are replaced in the correspondingexpressions
Figure 13: Calculation of matrix elements H78 and G78
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Inserting leads to
(146)
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and
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The remaining elements can be calculated the same way. The following system of equations is obtained
(147)
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Coupling of Eq. (144) and Eq. (148) according to either method 1 in Eq. (139) or method 2 in Eq. (143) along withEq. (141) or Eq. (142) leads to sets of equations from which the unknown boundary variables and the interfacevariables can be solved.
The solution is given by:
(148)
(149)
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Analytical solution
The field equations for the analytical solution are
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where , in subregion 1 and , in subregion 2. According to the boundary and
compatibility conditions, the variables at the nodes of the discretisation are obtained as
(150)
(151)
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The crude discretisation of Fig. 12 should be taken into consideration when the numerical results in Eq. (149) arecompared to the analytical results in Eq. (151). The Dirichlet data lead to very good accuracy while the
Neumann data or fluxes show reasonable approximations.
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Next:Fundamental solutionsUp:Boundary Element MethodsPrevious:Substructure technique
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Fundamental solutions
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Fundamental solutions
In this section fundamental solutions are derived for potential problems. Potential problems are scalar fieldproblems, thus the fundamental solution consists of a scalar function relating the effect of a source term at theload-point to its influence point . This point is usually called field-point.
Subsections
Laplace equation
Fundamental solution of the 2D Laplace equationFundamental solution of the 3D Laplace equation
Helmholtz equationsFundamental solution of the 3D Helmholtz equation
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Laplace equationThe fundamental solution of the Laplace equation is a solution of the equation
Note that is the distance between the load- and field-point. This implies that a fundamental solution
is a symmetric function
(152)
(153)
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The derivation of the fundamental solution of the Laplace equation in 2D and 3D is carried out here as anexample for the general appraoch.
Subsections
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Fundamental solution of the 2D Laplace equationFundamental solution of the 3D Laplace equation
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Boundary Element Methods
Fundamental solution of the 2D Laplace equation
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Fundamental solution of the 2D Laplace equation
For simplicity the load-point is shifted in the origin. The derivation starts out by transforming the Laplace operatorto polar coordinates
The excitation with the Dirac impulse is radial-symmetric and, since we are dealing with an infinite problem, thereare no disturbances from the boundary, it is implied that the fundamental solution is radial-symmetric, too. Thusthe last term in Eq. (154) vanishes. The Dirac impulse in polar coordinates is stated as .
Hence, a way to solve for is to integrate Eq. (154). This yields
(154)
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with the integration constants and . It will be shown in the next section that for Eq. (155) being a
valid solution of Eq. (152). The constant introduces the notion of a constant potential. It is arbitrary and is
generally set to zero.
Verification of the impulse condition
The validity of a fundamental solution can be verified by evaluating the impulse condition. This condition carriesout the integral over the partial differential equation over an arbitrary volume enclosing the Dirac impulse
Application of Gauss' theorem transforms the volume integral on the left to a surface integral
(155)
(156)
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(157)
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Since is radial-symmetric, the gradient is also a pure function of the radius. In polar coordinates this reads as
Moreover, the outward normal on a circle is defined in polar coordinates as
Choosing the surface as a circle of arbitrary radius leads to the impulse condition
(158)
(159)
(160)
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Since depends only on and is constant on a specific circle , the impulse condition is reformulated as
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which proves that Eq. (155) is indeed a valid fundamental solution of Eq. (152). Note that this condition alsoimplies the must be zero as stated before because otherwise the terms would not cancel to -1.
Next:Fundamental solution of the 3D Laplace equationUp:Laplace equationPrevious:Laplace equation
(161)
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Fundamental solution of the 3D Laplace equation
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Fundamental solution of the 3D Laplace equation
In this case a transformation on spherical coordinates is carried out
Assuming radial symmetry yields
(162)
(163)
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The Dirac impulse in spherical coordinates is
(164)
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since
and by equivalence
(164)
(165)
(166)
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The integration of the Laplace equation yields for the 3D case
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Again, the impulse condition will show that . As before is arbitrary and is set to zero for convenience.
Verification of the impulse condition
The derivation is analogous to the 2D case. The integral over the partial differential equation is transformed tothe boundary. Since the solution is radial symmetric the gradient has only a component in the radial direction.
A sphere is chosen as arbitrary enclosing surface in the 3D case. The normal vector is a unit vector in sphericalcoordinates
(167)
(168)
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With this the impulse condition is
(169)
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p
Again, only depends on and thus is constant on a sphere of constant radius. It follows
As in the 2D case, this shows that must be set to zero so that fulfills this equation.
The solutions for the potential and the flux as the normal derivative of the potential in 2D and 3D are
(170)
(171)
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summarized in Table 2.
Table 2: Fundamental solutions of
the Laplace equation
2D 3D
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Helmholtz equationsA fundamental solution for the Helmholtz equation is derived by solving
As in the case of the Laplace equation, the function is scalar.
Subsections
Fundamental solution of the 3D Helmholtz equation
(172)
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Fundamental solution of the 3D Helmholtz equation
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Fundamental solution of the 3D Helmholtz equation
Transformation on spherical coordinates and taking radial symmetry into account yields
A solution is obtained by choosing
(173)
(174)
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where the term with the function is singular for while the term with the function remains regular.To verify the impulse condition the behavior for small is considered. A series expansion of the cosine-function
shows that the singularity behavior is , which after comparison to Eq. (167) yields
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g y p q ( ) y
The sine-term is again a homogeneous solution and does not contribute to the Dirac impulse. Hence, this second
constant is free and is adjusted such that the ansatz in Eq. (174) fulfills the Sommerfeld condition Eq. ( ) for the3D case.
This yields
(175)
(176)
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and the complete solution is
(177)
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For small the behavior is
This implies that also in 3D the fundamental solution behaves for small or like the fundamental solution of
the Laplace equation.
The flux of the 3D solution is
(178)
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The solutions for the potential and the flux as the normal derivative of the potential in 2D and 3D are
summarized in Table 3.
(180)
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Table 3: Fundamental solutions of the Helmholtz equation
2D 3D
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Boundary Element Methods
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