1. Boundary Element Analysis of Systems Using Interval Methods
Bart F. Zalewski and Robert L. Mullen
Department of Civil Engineering
Case Western Reserve University
Rafi L. Muhanna
Department of Civil and Environmental Engineering
Georgia Institute of Technology

2. BEA
Boundary Element Analysis (BEA) is a method for obtaining approximate solutions of partial differential equations.
BEA is performed using transformation of the system’s domain variables to the variables on the boundaries of the system.
BEA requires less meshing than finite element analysis and thus it is comparatively faster in generating or refining the mesh.

3. BEA (Cont.)
The variable transformation is constructed using singular solutions of the governing partial differential equation.
The transformed boundary integral equations are solved using collocation methods.
Source points are located sequentially at all boundary nodes that map the domain variables such that they coincide to their nodal values.

4. BEA (Cont.) The following sources are a cause of error in BEA:
Uncertainty in boundary conditions
Uncertainty in the parameters of the system
Errors in integration
Errors in the solution of the resulting linear system of equations
Discretization errors

5. Objective
To explore the applications of interval concepts to boundary element methods
Procedure
To perform the boundary element analysis of Laplace equation considering interval uncertainties

6. Presentation Outline Deterministic BEA of Laplace equation
Interval BEA
Truncation error
Integration error
Uncertain boundary conditions
Discretization error
Examples
Conclusion

7. Boundary Element Analysis of Laplace Equation
The Laplace equation is:
is the domain of the system
is the boundary of the system
, are the values at the boundary

8. Boundary Element Analysis of Laplace Equation (Cont.)
Orthogonalization of Laplace equation with respect to a test function is performed to minimize the error due to approximation of the the exact solution of and :

9. Boundary Element Analysis of Laplace Equation (Cont.)
Twice integrating by parts on the left side and considering and yields:
is a source point
(Brebbia 1992)

10. Boundary Element Analysis of Laplace Equation (Cont.)
Before the application of boundary conditions the boundary is taken along
The boundary integral equation is integrated such that the source point is included on the circular boundary of radius , as :

11. Constant Boundary Element Discretization
Any boundary can be discretized into boundary elements consisting of nodes, at which a value of either or is known, and also consisting of assumed polynomial shape functions between nodes.

12. Constant Boundary Element Discretization (Cont.)
In this work one-noded boundary elements with constant shape functions are used, leading to the following discretization:
and are vectors of nodal values
is a vector of constant shape functions

13. Constant Boundary Element Discretization (Cont.)
The discretized integral equation is written as:
or in matrix form:
Applying the boundary conditions, the system of linear equations is rearranged as:

14. Boundary Element Formulation
and are fully populated non-symmetric matrices.
The diagonal terms of the matrix are computed such that the matrix is singular.
The diagonal terms of the matrix require special consideration since they contain singular integrals.

15. Interval Boundary Element Formulation
The following are considered as interval quantities in the interval BEA analysis:
Truncation error
Integration error
Uncertainty in boundary conditions
Discretization error

16. Interval Boundary Element Formulation (Cont.)
Interval boundary element analysis is performed as:
The equation is rearranged as:
The interval linear system of equation can be solved by Matlab Interval Toolbox [MATLAB 6.5.1], which uses Newton-Krawczyk iteration.

17. Interval Boundary Element Formulation (Cont.)
The approximate value of the diagonal terms of the
matrix is computed using the Taylor series expansion.
In order to bound the integration error, the closed form solution of the improper integral of the diagonal terms is found, which is not necessarily in the domain of the actual problem.

18. Interval Boundary Element Formulation (Cont.)
If the domain of the improper integral is different than that of the problem, the remaining domain is integrated numerically using Taylor series expansion and the integration error found using the given error bound.
If the domain of the improper integral is that of the problem, the difference between the closed form solution and the numerical integration is considered as integration error.

19. Taylor Series Expansion
A function can be expressed as a polynomial using Taylor series expansion [Taylor, 1715]:
where
Any function can be integrated exactly as:
where corresponds to the derivative of the function

20. Error Analysis on Taylor Series Expansion
is the series remainder as:
Integrating the remainder yields:

21. Error Analysis on Taylor Series Expansion (Cont.)
The maximum truncation error is found:
The approximate terms of the and matrices are computed as:

22. Integration Error
The integration error can be represented by an interval variable. The real interval number is a closed set as:
Then the bounds on the integration error are computed:
Alternative series may yield sharper bounds depending on the partial differential equation considered.

23. Uncertainty in Boundary Conditions
Considering uncertainty in boundary conditions results in interval vectors and
Parameterization of the rows of and matrices is considered to obtain sharp results and the system is solved as:

24. Discretization Error
In the analysis of the discretiztion error, we will look for interval bounded unknown functions that will satisfy the continuous problem.

25. Discretization Error (Cont.)
The boundary is subdivided into elements. For each element, we will seek the interval values and that bound the functions and over an element such that:

26. Discretization Error (Cont.)
The integral of the product will be expanded to the product of the interval value of or and the interval bounds of the integral of the singular solution over the element for all values of :
if has the same sign over the element.

27. Discretization Error (Cont.)
If not, the integration domain is subdivided into portions that have the same sign for Then the integral is replaced by interval bounds:
Thus, the interval bounds on the solution can be expressed as a generalized interval system of linear equations.
For sharp bounds, the parametric dependence of each row of the or matrices on must be included in the solution of the interval system.

28. Examples
The first example uses interval BEA to bound the solution to Laplace equation in the presence of truncation and integration errors. Four point integration method based on a Taylor series is used to develop interval terms in matrices.
Boundary Conditions: u1=0, q2=0, q3=0, u4=50, q5=0, q6=0
1:2 ratio

29. Examples (Cont.) Results: Node Value Lower Bound Solution with exact
integration
Upper Bound q1 u2 u3 q4 u5 u6

30. Examples (Cont.)
The second example is a demonstration of the interval treatment of uncertain boundary conditions.
Boundary Conditions: u1=[0,1], q2=0, q3=0, u4=[1,2], q5=0, q6=0
1:1 ratio

31. Examples (Cont.) Results: Node Value Lower Bound Lower Bound with
Parame-terization
Combinatorial
Upper Bound
Upper Bound with q1
0.0000
0.5007 u2
0.0922
0.2451
1.2451
1.3981 u3
0.6019
0.7549
1.7549
1.9078 q4
2.0763
2.5770 u5 u6

32. Examples (Cont.)
The third example obtains the bounds on discretization error for the BEA of the Laplace equation. Interval bounds of the and matrices are constructed by moving the point over the domain of an element.
Boundary Conditions: Boundary Conditions: Boundary Conditions:
u1= u1= u1= u5=1
q2= q2= q2= q6=0
u3= q3= q3= q7=0
q4= u4= q4= q8=0
q5=0
q6=0 ,
1:1 ratio
1:1 ratio
1:1 ratio

33. Mid-point Node Solution
Examples (Cont.)
Results:
Node Value
Lower Bound
100,000 Simulations
Middle Value
Upper Bound
Width
Mid-point Node Solution q1 u2
0.5000 q3
1.1746 u4

34. Mid-point Node Solution
Examples (Cont.)
Results:
Node Value
Lower Bound
Middle Value
Upper Bound
Width
Mid-point Node Solution q1
1.4768 u2
0.0000
0.5000
1.0000 q3
0.5129
1.2512
1.1746 u4

35. Examples (Cont.) Results: Node Value Lower Bound Central Value
Upper Bound
Width
Mid-point
Node Solution q1
0.7131 u2
0.2431
0.5655
0.6448
0.2451 u3
0.4345
0.7569
1.0793
0.7549 q4
0.7258
1.0823
1.4389
1.0382 u5 u6

36. Examples (Cont.) Results: Node Value Lower Bound Central Value
Upper Bound
Width
Mid-point
Node Solution q1
0.4680 u2
0.1539
0.3808
0.4539
0.1639 u3
0.2856
0.5000
0.7144
0.4288 u4
0.6192
0.8461
1.0731
0.8361 q5
0.8057
1.0397
1.2737
1.0161 u6 u7 u8

37. Conclusion
In this work, new methods are presented to perform boundary element analysis in the presence of the truncation, integration and discretization errors as well as uncertain boundary conditions using interval methods.
The examples presented demonstrate the potential of interval based boundary element methods to provide reliable engineering computations.
Further work is needed to optimally solve the parametric form of the interval equations to advance interval based BEA to a truly reliable and efficient engineering analysis tool.

38. Acknowledgment The authors thank Dr. Mehdi Modares for his
tremendous help throughout the present work.

39. Thank you.