Interval boundary element method

Interval boundary element method is classical boundary element method with the interval parameters.
Boundary element method is based on the following integral equation

$c\cdot u=\int \limits _{\partial \Omega }\left(G{\frac {\partial u}{\partial n}}-{\frac {\partial G}{\partial n}}u\right)dS$

The exact interval solution on the boundary can be defined in the following way:

${\tilde {u}}(x)=\{u(x,p):c(p)\cdot u(p)=\int \limits _{\partial \Omega }\left(G(p){\frac {\partial u(p)}{\partial n}}-{\frac {\partial G(p)}{\partial n}}u(p)\right)dS,p\in {\hat {p}}\}$

In practice we are interested in the smallest interval which contain the exact solution set

${\hat {u}}(x)=hull\ {\tilde {u}}(x)=hull\{u(x,p):c(p)\cdot u(p)=\int \limits _{\partial \Omega }\left(G(p){\frac {\partial u(p)}{\partial n}}-{\frac {\partial G(p)}{\partial n}}u(p)\right)dS,p\in {\hat {p}}\}$

In similar way it is possible to calculate the interval solution inside the boundary $\Omega$ .

References

T. Burczynski, J. Skrzypczyk J. The fuzzy boundary element method: a new methodology. Series Civil Eng, Vol. 83. Gliwice: Sci Fasc of Silesian Tech Univ; 1995, pp. 25–42.
J. Skrzypczyk, A Note on Interval Fredholm Integral Equations. Zeszyty Naukowe Politechniki Śląskiej, Seria Budownictwo, Z.85, pp.75-83, 1998
T. Burczynski, J. Skrzypczyk, Fuzzy aspects of the boundary element method, Engineering Analysis with Boundary Elements, Vol.19, No.3, pp.209-216, 1997
H. Witek, Boundary element method in analysis of civil engineering structures with uncertain parameters. Ph.D. Dissertation, Silesian University of Technology, Faculty of Civil Engineering, Poland, 2005
B.F. Zalewski, R.L. Mullen, R.L. Muhanna, “Boundary Element Analysis of Systems Using Interval Methods”, Proceedings of the NSF Workshop on Modeling Errors and Uncertainty in Engineering Computations, Georgia Tech Savannah, February 2006.
B.F. Zalewski and R.L. Mullen, "Interval Bounds on the Local Discretizaton Error in Boundary Element Analysis for Domains with Singular Flux", SAE 2008 Reliability and Robust Design in Automotive Engineering, SP-2170, pp. 237-246, 2008.
B.F. Zalewski and R.L. Mullen, "Discretizaton Error in Boundary Element Analysis using Interval Methods", SAE 2007 Transactions Journal of Passenger Cars: Mechanical Systems, V116-6, pp. 1353-1361, 2008.
B.F. Zalewski and R.L. Mullen, "Point-wise Discretization Errors in Boundary Element Method for Elasticity Problem", Third NSF Workshop on Reliable Engineering Computing, pp. 429-457, February 2008.
Alicja Piasecka Belkhayat, Interval boundary element method for 2D transient diffusion problem, Engineering Analysis with Boundary Elements, Volume 32, Issue 5, May 2008, Pages 424-430

References

See also