A Simple and Efficient Higher Order Finite Element Scheme for Helmholtz Waveguides
This paper presents a simple and efficient finite element scheme for computing the cutoff wave numbers of arbitrary-shaped waveguides using higher order triangular elements. The waveguide geometry is divided into a set of triangular elements and each of t
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Abstract This paper presents a simple and efficient finite element scheme for computing the cutoff wave numbers of arbitrary-shaped waveguides using higher order triangular elements. The waveguide geometry is divided into a set of triangular elements and each of these elements is mapped to a standard isosceles triangle by discritizing with subparametric finite elements. For waveguides containing arbitrary cross sections, the transformation is done using a series of higher order parabolic arcs. In this case, the curve boundaries are approximated by curved triangular finite elements and then transformed to an isosceles triangle. Numerical results are illustrated to validate the present approach. The obtained results have converged very well with the existing literature with minimum number of triangular elements, degree of freedoms, order of computational matrix, etc.
Keywords Helmholtz equation Finite element method Eigenvalue problem Subparametric transformation Waveguides Parabolic arcs
1 Introduction There are many numerical methods developed for computing accurate and optimal numerical solutions to waveguide eigenvalue problem. Sensale et al. have presented a unique approach for microwave waveguides based on hypersingular boundary T.D. Panda (&) Department of Electronics and Instrumentation Engineering, Amrita School of Engineering, Amrita Vishwa Vidyapeetham, Amrita University, Bangalore 560035, India e-mail: [email protected] K.V. Nagaraja V. Kesavulu Naidu Department of Mathematics, Amrita School of Engineering, Amrita Vishwa Vidyapeetham, Amrita University, Bangalore 560035, India e-mail: [email protected] V. Kesavulu Naidu e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2018 A. Kalam et al. (eds.), Advances in Electronics, Communication and Computing, Lecture Notes in Electrical Engineering 443, https://doi.org/10.1007/978-981-10-4765-7_45
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element formation and can be found [1]. Dong et al. [2] have derived some solutions to waveguide eigenvalue problem based on generalized differential quadrature method. Numerical approximation to the two-dimensional Helmholtz equation by a meshfree approach can be found in [3, 4]. This paper is concerned with higher order finite element solution to the Helmholtz equation in waveguides using straight-sided and curved triangular elements. In the present case, solutions are determined for TM modes. For waveguides with rectangular boundaries, the geometry is discretized with a series of higher order finite elements and then suitably transformed to a standard isosceles right-angled triangle. Certain point transformation formulae are used for ensuring highest level of mapping. For waveguides with irregular cross sections, a series of higher order parabolic arcs are been used to discretize the geometry. Rathod et al. [5] have derived a mapping procedure of mapping the curved boundary by a series of parabolic arcs. The point transformation formula in this case gives an interpolating polynomial that ensures
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