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 Discretization 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, "Discretization 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.
B.F. Zalewski, “Uncertainties in the Solutions to Boundary Element Method: An Interval Approach”, Case Western Reserve University, Ph.D. Dissertation 2008.
B.F. Zalewski and R.L. Mullen, “Local Discretization Error Bounds Using Interval Boundary Element Method”, International Journal for Numerical Methods in Engineering, Volume 78, Issue 4, April 2009, Pages 403-428.
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
B.F. Zalewski, R.L. Mullena, and R.L. Muhanna, "Interval Boundary Element Method in the Presence of Uncertain Boundary Conditions, Integration Errors, and Truncation Errors", Engineering Analysis with Boundary Elements, Volume 33, Issue 4, April 2009, Pages 508-513. [1]
B.F. Zalewski, R.L. Mullen, and R.L. Muhanna, “Fuzzy Boundary Element Method for Geometric Uncertainty in Elasticity Problem”, SAE 2009 Reliability and Robust Design in Automotive Engineering, SP-2232, 2009.
B.F. Zalewski and R.L. Mullen, “Worst Case Bounds on the Point-wise Discretization Error in Boundary Element Method for the Elasticity Problem”, Computer Methods in Applied Mechanics and Engineering, In Press, 2009.

References

See also