Research on the Problems of Wave Interaction with Porous Structures and Electromagnetic Field Based on Scaled Boundary Finite Element Method
|School||Dalian University of Technology|
|Course||Hydraulic Structure Engineering|
|Keywords||Scaled Boundary Finite Element Method Arc-Shaped Porous CylindricalStructure Wave Force Wave Runup Short-Crested Wave Electrostatic Field Eigenvalues Waveguide|
In this paper, the scaled boundary finite element method (SBFEM) has applied to the wave interaction with porous structures and electromagnetic field problems. The scaled boundary finite element method is a newly developed semi-analytical technique to solve systems of partial differential equations. It works by employing a special local coordinate system, called scaled boundary coordinate system, to define the computational field, and then weakening the partial differential equation in the circumferential direction with the standard finite element whilst keeping the equation strong in the radial direction, finally analytically solving the resulting system of equations, termed the scaled boundary finite element equation. This unique feature of the scaled boundary finite element method enables it to combine mangy of advantages of the finite element method (FEM) and the boundary element method (BEM) with the features of its own. For instance, since only the boundaries of computational fields are discretized, the spatial dimensions can be reduced by one. Consequently the data preparation effort can be significantly decreased. Due to its analytical nature in the radial direction, the singularity of field gradients near sharp re-entrant corenes can be modlled with ease and the radiation condition at infinity can be satisfied rigorously. The scaled boundary finite element method was originally developed for solving problems of elasto-statics and elasto-dynamics in solid mechanics, and recently extended to fluid dynamics, fracture mechanics, structure-infinite foundation interaction, acoustic and fluid mechanics, etc. It has been employed successfully for solving problems with singularities and unbounded domains, and has very large application prospect in many fields.According to the projects supported by the State Key Program of National Natural Science of China and China-Germany joint research, the scaled boundary finite element method has firstly applied to the wave interaction with arc-shaped porous cylindrical structures. The porous structures have been considered for the sake of good effect on reduction of wave force and wave run-up around the outside of the structure. However, most researchers have focused on the two-dimensional plane wave interaction with upright porous caisson structure, and there is little literature has been report of its applications to the three-dimensional short-crested wave interaction with the arc-shaped porous cylindrical structures. Based on the linear wave theory and modified scaled boundary finite element method with circular shape, the scaled boundary finite element method can easily transform the governing wave equation of the problem into Bessel equation, so the problem can be solved analytically by using Bessel or Hankel functions. Based on the above-mentioned theories, the scaled boundary finite element method has been applied to the short-crested wave interaction with several types of circular or arc-shaped porous structures including arc-shaped bottom-mounted porous breakwater, double-layered porous cylindrical columns, double-layered arc-shaped bottom-mounted porous breakwater, circular cylinder circumscribed arc-shaped porous cylindrical structure, concentric porous cylinder system with partially porous outer cylinder, concentric cylindrical structure with double-layered perforated walls and combined cylinders structure with dual arc-shaped porous outer walls. A central feature of the newly extended method is that, when the porous structures includes arc-shaped porous cylinder, virtual outer cylinder extending the arc-shaped porous cylinder with the same centre is introduced and variable porous-effect parameters is also introduced for the virtual cylinders, so that the final SBFEM equation still can be handled in a closed-form analytical manner in the radial direction and by a finite element approximation in the circumferential direction. For those seven types of porous structures, the entire computational domain for each type is divided into several bounded domains and one unbounded domain, and a variational principle formulation is used to derive the SBFEM equation in each sub-domain. The velocity potential in bounded and unbounded domains are formulated using a sets of Bessel and Hankel functions respectively, and the unknown coefficients are determined from the matching conditions. The results of numerical verification for each type’s structure show that the approach discretises only the outermost virtual cylinder with surface finite-elements and fewer elements are required to obtain very accurate results. The influences of the wave parameters, the configuration of the structures and porous-effect parameters on the systems hydrodynamics, including the wave force, wave and diffracted wave contour are extensively examined. The present results are of practical significance to the hydrodynamic analysis and design for the porous structures.Thinking about the special advantages of the scaled boundary finite element, the author’s Institute of Earthquake Engineering has developed a long-term international cooperation with Professor Song Chongmin of University of New South Wales in Australia, which is co-founder of the scaled boundary finite element and also the Haitian scholars of Dalian University of Technology. Professor Song often gives lectures and exchanges with us. According to the agreement with the cooperation of Professor Song and great significance of electromagnetic field problems in people’s daily life and works, the second part of the paper is that the scaled boundary finite element method has also been firstly and successfully applied to electromagnetic field. Although the physical phenomena of the two parts in the paper has little correlation, fortunately, it is well known that there many mathematical similarities between fluid mechanics and electromagnetic field, the scaled boundary finite element can easily combine the two parts, and also provides an effective way to carry out interdisciplinary research. As to the electromagnetic field problems, the scaled boundary finite element method is firstly successfully extended to solve one type of electromagnetic field problems-electrostatic field problems. Based on Laplace equation of electrostatic field problems and a variational principle, the derivations and solutions of SBFEM equations for bounded domain and unbounded domain problems are expressed in details, and the solution for the inclusion of prescribed potential along the side-faces of bounded domains is also presented in details, then the total charges on the side-faces can be semi-analytical solved. Meanwhile, modified scaled boundary finite element method for problems with parallel side-faces and circular shape are introduced, and the SBFE equations for those problems are also derivated and solved in detail. Furthermore, The SBFE non-homogeneous equation, the SBFE equation with prescribed side-face electric potential and the SBFE equation with infinity electric potential at infinity are also derived in detail. The accuracy and efficiency of the method are illustrated by ten numerical examples of electromagnetic field problems with complicated field domain, potential singularity, inhomogeneous and open boundary. In comparison with analytic solution method and other numerical methods, the results show that the present method has a strong ability to resolve potential field singularities analytically by choosing the scaling centre at the singular point, has the inherent advantage of solving the open boundary problems without truncation boundary condition, has efficient application to the problems with inhomogeneous media by placing the scaling centre in the bi-material interfaces, and produces more accurate solution than conventional numerical methods with far less number of degrees of freedom.Then, the scaled boundary finite element method is developed for the solution of waveguide eigenvalue problems. The calculation of the waveguide cutoff frequency is a challenging problem, and various types of waveguides have different transmission frequency range and transmission characteristics, and this has very important significance for the design of the waveguide. This paper develops a new variational principle formulation to derive the SBFEM equations for waveguide eigenvalue problems. And an equation of the dynamic stiffness matrix for waveguide representing between the’flux’and the longitudinal field components relationship at the discretized boundary is established. A continued fraction solution in terms of eigenvalue is obtained. By using the continued fraction solution and introducing auxiliary variables, the flux-longitudinal field relationship is formulated as a system of linear equations in eigenvalue then a generalized eigenvalue equation is obtained. The eigenvalues of rectangular, L-shaped, vaned rectangular are calculated and compared with analytical solution or other numerical methods. The results show that the present method yields excellent results, high precision and less amount of computation time and rapid convergence is observed, Moreover, the problem with the singular point has been successfully solved with few elements. Meanwhile, ridged waveguides have been widely used in microwave and millimeter-wave devices because of their unique characteristics such as low cutoff frequency, wide bandwidth and low impedance characteristics. Among them, as the ever-growing needs of the modern communication systems working at higher and higher capacity, quadruple-ridge waveguides find wide applications, especially in antenna and radar systems. In practical applications, the quadruple ridges in a square waveguide are usually cut at their corners, which contain several reentrant corners. However, the standard FEM yields comparatively poor results when applied to the waveguide whose domain contains re-entrant corners, owing to the singular nature of the solution. The method used to circumvent this difficulty is to refine the mesh locally in the region of the singularity or using higher order basis functions which bring out time-consuming task. The BEM is an attractive technique for solving the waveguide problems. However, fundamental solutions are required and singular integrals exist. Furthermore, it may suffer from the problems caused by sharp corners. In this paper, the scaled boundary can easily overcome these difficulties and make a great improvement for the computational efficiency and computational accuracy. Three types quadruple corner-cut ridged (square, circular and elliptical) waveguides are taken as examples. Variations of the cutoff wave numbers of the dominant and higher-order modes for both TE and TM cases with the corner-cut ridge dimensions are investigated in details. Simple approximate equations are found to accurately predict the cutoff wave number of several modes for the quadruple corner-cut ridged square waveguides. The single mode bandwidths of the waveguides are also calculated. Therefore, these results provide an extension to the existing design data for ridge waveguide and are considered helpful in practical applications. The solution of this paper makes a meaningful contribution on Computational electromagnetics and also produces good results for engineering applications.