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# Solution of Inverse Boundary Optimization Problem by Trefftz Method and Exponentially Convergent Scalar Homotopy Algorithm

Hsin-Fang Chan1, Chia-Ming Fan1,2, Weichung Yeih1

Department of Harbor and River Engineering & Computation and Simulation Center, NationalTaiwan Ocean University, Keelung 20224, Taiwan
Corresponding Author, E-mail: cmfan@ntou.edu.tw

Computers, Materials & Continua 2011, 24(2), 125-142. https://doi.org/10.3970/cmc.2011.024.125

## Abstract

The inverse boundary optimization problem, governed by the Helmholtz equation, is analyzed by the Trefftz method (TM) and the exponentially convergent scalar homotopy algorithm (ECSHA). In the inverse boundary optimization problem, the position for part of boundary with given boundary condition is unknown, and the position for the rest of boundary with additionally specified boundary conditions is given. Therefore, it is very difficult to handle the boundary optimization problem by any numerical scheme. In order to stably solve the boundary optimization problem, the TM, one kind of boundary-type meshless methods, is adopted in this study, since it can avoid the generation of mesh grid and numerical integration. In the boundary optimization problem governed by the Helmholtz equation, the numerical solution of TM is expressed as linear combination of the T-complete functions. When this problem is considered by TM, a system of nonlinear algebraic equations will be formed and solved by ECSHA which will converge exponentially. The evolutionary process of ECSHA can acquire the unknown coefficients in TM and the spatial position of the unknown boundary simultaneously. Some numerical examples will be provided to demonstrate the ability and accuracy of the proposed scheme. Besides, the stability of the proposed meshless method will be validated by adding some noise into the boundary conditions.

## Keywords

H. . Chan, C. . Fan and W. . Yeih, "Solution of inverse boundary optimization problem by trefftz method and exponentially convergent scalar homotopy algorithm," Computers, Materials & Continua, vol. 24, no.2, pp. 125–142, 2011. https://doi.org/10.3970/cmc.2011.024.125

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