Browsing by Author "Wang, Jenn-Nan"

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An efficient numerical algorithm for computing densely distributed positive interior transmission eigenvalues
(2017-03-01)
Li, Tiexiang
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Huang, Tsung-Ming
;
Lin, Wen-Wei
;
Wang, Jenn-Nan
;
應用數學系
We propose an efficient eigensolver for computing densely distributed spectra of the two-dimensional transmission eigenvalue problem (TEP), which is derived from Maxwell's equations with Tellegen media and the transverse magnetic mode. The governing equations, when discretized by the standard piecewise linear finite element method, give rise to a large-scale quadratic eigenvalue problem (QEP). Our numerical simulation shows that half of the positive eigenvalues of the QEP are densely distributed in some interval near the origin. The quadratic Jacobi-Davidson method with a so-called non-equivalence deflation technique is proposed to compute the dense spectrum of the QEP. Extensive numerical simulations show that our proposed method processes the convergence efficiently, even when it needs to compute more than 5000 desired eigenpairs. Numerical results also illustrate that the computed eigenvalue curves can be approximated by nonlinear functions, which can be applied to estimate the denseness of the eigenvalues for the TEP.
Efficient methods of computing interior transmission eigenvalues for the elastic waves
(2020-04-15)
Chang, Wei-Chen
;
Lin, Wen-Wei
;
Wang, Jenn-Nan
;
應用數學系
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Department of Applied Mathematics
We study the interior transmission eigenvalue problem for the elastic wave scattering in this paper. We aim to show the distribution of positive eigenvalues by efficient numerical algorithms. Here the elastic waves are scattered by the perturbations of medium parameters, which include the elasticity tensor C and the density rho. Let us denote (C-0, rho(0)) and (C-1, rho(1)) the background and the perturbed medium parameters, respectively. We consider two cases of perturbations, C-0 = C-1, rho 1 not equal rho(0) (case 1) and C-0 not equal C-1, rho(1) = rho(0) (case 2). After discretizing the associated PDEs by FEM, we are facing the computation of generalized eigenvalues problems (GEP) with matrices of large size. These GEPs contain huge number of nonphysical zeros (for case 1) or nonphysical infinities (for case 2). In order to locate several hundred positive eigenvalues effectively, we then convert GEPs to suitable quadratic eigenvalues problems (QEP). We then implement a quadratic JacobiDavidson method combining with partial locking or partial deflation techniques to compute 500 positive eigenvalues. (C) 2020 Elsevier Inc. All rights reserved.
ON THE TRANSMISSION EIGENVALUE PROBLEM FOR THE ACOUSTIC EQUATION WITH A NEGATIVE INDEX OF REFRACTION AND A PRACTICAL NUMERICAL RECONSTRUCTION METHOD
(2018-08-01)
Li, Tiexiang
;
Huang, Tsung-Ming
;
Lin, Wen-Wei
;
Wang, Jenn-Nan
;
應用數學系
In this paper, we consider the two-dimensional Maxwell's equations with the TM mode in pseudo-chiral media. The system can be reduced to the acoustic equation with a negative index of refraction. We first study the transmission eigenvalue problem (TEP) for this equation. By the continuous finite element method, we discretize the reduced equation and transform the study of TEP to a quadratic eigenvalue problem by deflating all nonphysical zeros. We then estimate half of the eigenvalues are negative with order of O(1) and the other half of eigenvalues are positive with order of O(10(2)). In the second part of the paper, we present a practical numerical method to reconstruct the support of the inhomogeneity by the near-field measurements, i.e., Cauchy data. Based on the linear sampling method, we propose the truncated singular value decomposition to solve the ill-posed near-field integral equation, at one wave number which is not a transmission eigenvalue. By carefully chosen an indicator function, this method produce different jumps for the sampling points inside and outside the support. Numerical results show that our method is able to reconstruct the support reliably.
Reconstruction of inclusions in an elastic body
(2009-06-01)
Uhlmann, Gunther
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Wang, Jenn-Nan
;
Wu, Chin-Tien
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應用數學系
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Department of Applied Mathematics
We consider the reconstruction of elastic inclusions embedded inside of a planar region, hounded or unbounded, with isotropic inhomogeneous elastic parameters by measuring displacements and tractions at the boundary. We probe the medium with complex geometrical Optics Solutions having polynomial-type phase functions. Using these solutions we develop all algorithm to reconstruct the exact shape of a large class of inclusions including star-shaped domains and we implement numerically this algorithm for some examples. (C) 2009 Elsevier Masson SAS. All rights reserved.