比例边界有限元方法在波浪与开孔结构相互作用及电磁场问题中的研究
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摘要
本文发展了比例边界有限元方法(SBFEM)在波浪与开孔结构相互作用及电磁场问题中的研究。比例边界有限元方法是近年来提出和迅速发展的用来求解线性偏微分方程的一种半解析数值方法,它融合了有限元法和边界元法的优点,又有其特有的优点。该方法只需数值离散计算域边界,减少了一个空间维数；在没有离散的径向方向利用解析的方法求解,具有较高的计算精度。相对于边界元方法,它不需要基本解,也不存在奇异积分问题；对于无限域问题,也无需引入截断边界就能够自动满足无穷远处的辐射边界条件。比例边界有限元方法已成功地应用于弹性静、动力学、断裂力学、结构—无限地基动力相互作用、流固耦合、声波等领域,在许多领域有着非常大的应用前景。
根据国家自然科学基金重点项目以及中德合作研究项目的研究需要,本文的第一部分开展了SBFEM对波浪与圆弧型开孔柱结构相互作用问题的研究。开孔结构由于具有良好的减小波浪反射和自身所受波浪力的特性,越来越多地受到人们的重视。然而,大多研究工作者侧重于二维平面波浪与直立方沉箱开孔结构的相互作用问题的研究。迄今为止,更能反映实际海洋状况的三维短峰波与圆弧型开孔柱结构的相互作用问题的研究很少。为此,基于线形势流理论并采用改进的圆形比例边界有限元坐标变换系统,本文充分利用SBFEM优点将短峰波与圆弧型开孔柱结构相互作用的波动问题控制方程转换为贝塞尔方程,可以方便地通过贝塞尔或汉克尔函数进行解析求解。据此,重点研究了以下不同类型开孔结构的水动力相互作用：圆弧型贯底式开孔介质防波堤、双层开孔圆筒柱结构、双层圆弧型开孔柱结构、圆筒外接圆弧开孔柱结构、外壁局部开孔双筒柱结构、双层外壁开孔带内柱的圆筒结构、双层外壁开孔带内柱的圆筒结构和圆柱外接双层圆弧型开孔柱结构。当存在圆弧型结构时,本文巧妙地将圆弧延伸构建出了虚拟同心圆,其中圆弧段与圆弧延伸段的孔隙影响系数可由对角矩阵G统一进行表达,以便构建虚拟圆处的边界条件。针对不同结构类型,SBFEM将各个结构分成若干个有限域和一个无限域。无论结构多复杂,SBFEM只需对最外层圆边界进行离散,使空间维数降低一阶,并在圆的半径方向保持解析。本文首次利用变分原理方法推导了关于势函数的各个子域SBFEM方程。其中,有限域和无限域的包含未知展开系数势函数表达式可分别采用贝塞尔函数和汉克尔函数作为基底,并且通过开孔介质两侧的匹配边界条件可以求解待定的展开系数。针对每个结构,都通过数值算例验证了该方法是一种用很少单元便能得到精确结果的高效算法。进一步研究了诸如短峰波波浪参数、结构的几何参数以及孔隙影响系数G等因素对不同结构所受的波浪力和波浪绕射的影响。本文的研究成果为不同形式圆弧型开孔型结构的水动力分析和工程结构设计提供了有价值的参考。
由于SBFEM的特殊优越性,作者所在的工程抗震研究所开展了与SBFEM的创始人之一澳大利亚新南威尔士大学宋崇民教授的长期国际合作,聘请其为大连理工大学海天学者。宋教授经常来我校讲学和交流,根据我们与宋教授的合作研究的协议以及电磁场问题在人们的日常生活及工程具有重要的意义,本文的第二部分将开拓比例边界元有限元方法在电磁场问题中的研究。尽管论文的两部分物理现象关联性很小,但幸运的是,圆弧型开孔柱结构的相互作用和一些电磁场问题之间的数学表述有很多相似之处,通过同时使用SBFEM对两部分问题的求解,可以非常方便地将论文两部分有机结合起来,同时也为开展交叉学科研究提供了一条有效的途径。在电磁场问题的研究中,论文首先将SBFEM应用于电磁场问题中的静电场问题。从拉普拉斯方程出发,利用变分原理并通过比例坐标和笛卡尔坐标变换,推导和求解出了静电场分析的比例边界有限元方程、电位求解公式以及电场求解公式,提出了一种分析静电场问题半解析方法。与此同时,本文还引入了两种新型比例边界坐标,一类是含平行侧边面的比例坐标,另外一类是含圆形域比例坐标,并且也推导和求解了相应的比例边界有限元方程。在此基础上,文中还重点求解了非齐次、侧边面含有电位以及无穷远电位不为有限值的比例边界有限元方程。通过10个算例计算结果与解析解和其他数值方法比较,结果表明此方法在处理一些电磁工程含有奇异点、非均匀介质和无限域等复杂问题中能显著提高计算效率和计算精度。
其次,本文发展了SBFEM在另外一类电磁场问题—波导本征值问题中的研究。波导截止频率的计算是一个富有挑战性的问题,各种形式的波导有一定的传输频率范围和传输特性,这对于波导的设计具有非常重要的意义。本文也利用了变分原理并通过比例边界坐标变换,推导了TE波和TM波波导的比例边界有限元频域方程以及波导动刚度方程,同时给出了波导动刚度矩阵的连分式解形式,通过引入辅助变量进一步得出波导特征值方程并求出波导本征值。以矩形、L形波导和叶型加载矩形波导的本征问题分析为例,通过与解析解和其他数值方法比较,结果表明此方法具有精度高、计算工作量小的优点,随着连分式阶数增加收敛速度快,而且SBFEM使用很少的单元数就能很好地解决了含有奇异点问题。脊波导具有较低的截止频率、较宽的工作带宽、低特性阻抗等优点,使得脊波导在微波和毫米波器件中被广泛应用。便随着高容量现代通信系统日益增长的需求,四脊波导也被广泛采用,尤其是在天线和雷达系统。对于四脊加载波导,往往在实际工程应用中,会对加载矩形进行剪切,而对于该类波导本征值研究较少。由于这类型波导多个角点处含有奇异性,使有限元计算中遭到困难,不得不采用网格加密或者采用高阶超级单元办法,增加了计算的复杂性,边界元方法在处理这类奇异问题也比较棘手,例如奇异积分的存在。为此,本文采用SBFEM的优越性可以顺利克服这些缺点,使计算效率和计算精度有很大程度的提高。这其中就包括分析了三类角切四脊加载(正方形、圆形和椭圆)波导的传输特性,并且给出了其中角切四脊加载正方形波导中的部分模式的截止波数计算经验公式,为工程设计提供一定理论依据。本文的解法也对计算电磁学发展作出了有意义的贡献,同时对工程应用也产生很好的效果。
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.
根据国家自然科学基金重点项目以及中德合作研究项目的研究需要,本文的第一部分开展了SBFEM对波浪与圆弧型开孔柱结构相互作用问题的研究。开孔结构由于具有良好的减小波浪反射和自身所受波浪力的特性,越来越多地受到人们的重视。然而,大多研究工作者侧重于二维平面波浪与直立方沉箱开孔结构的相互作用问题的研究。迄今为止,更能反映实际海洋状况的三维短峰波与圆弧型开孔柱结构的相互作用问题的研究很少。为此,基于线形势流理论并采用改进的圆形比例边界有限元坐标变换系统,本文充分利用SBFEM优点将短峰波与圆弧型开孔柱结构相互作用的波动问题控制方程转换为贝塞尔方程,可以方便地通过贝塞尔或汉克尔函数进行解析求解。据此,重点研究了以下不同类型开孔结构的水动力相互作用：圆弧型贯底式开孔介质防波堤、双层开孔圆筒柱结构、双层圆弧型开孔柱结构、圆筒外接圆弧开孔柱结构、外壁局部开孔双筒柱结构、双层外壁开孔带内柱的圆筒结构、双层外壁开孔带内柱的圆筒结构和圆柱外接双层圆弧型开孔柱结构。当存在圆弧型结构时,本文巧妙地将圆弧延伸构建出了虚拟同心圆,其中圆弧段与圆弧延伸段的孔隙影响系数可由对角矩阵G统一进行表达,以便构建虚拟圆处的边界条件。针对不同结构类型,SBFEM将各个结构分成若干个有限域和一个无限域。无论结构多复杂,SBFEM只需对最外层圆边界进行离散,使空间维数降低一阶,并在圆的半径方向保持解析。本文首次利用变分原理方法推导了关于势函数的各个子域SBFEM方程。其中,有限域和无限域的包含未知展开系数势函数表达式可分别采用贝塞尔函数和汉克尔函数作为基底,并且通过开孔介质两侧的匹配边界条件可以求解待定的展开系数。针对每个结构,都通过数值算例验证了该方法是一种用很少单元便能得到精确结果的高效算法。进一步研究了诸如短峰波波浪参数、结构的几何参数以及孔隙影响系数G等因素对不同结构所受的波浪力和波浪绕射的影响。本文的研究成果为不同形式圆弧型开孔型结构的水动力分析和工程结构设计提供了有价值的参考。
由于SBFEM的特殊优越性,作者所在的工程抗震研究所开展了与SBFEM的创始人之一澳大利亚新南威尔士大学宋崇民教授的长期国际合作,聘请其为大连理工大学海天学者。宋教授经常来我校讲学和交流,根据我们与宋教授的合作研究的协议以及电磁场问题在人们的日常生活及工程具有重要的意义,本文的第二部分将开拓比例边界元有限元方法在电磁场问题中的研究。尽管论文的两部分物理现象关联性很小,但幸运的是,圆弧型开孔柱结构的相互作用和一些电磁场问题之间的数学表述有很多相似之处,通过同时使用SBFEM对两部分问题的求解,可以非常方便地将论文两部分有机结合起来,同时也为开展交叉学科研究提供了一条有效的途径。在电磁场问题的研究中,论文首先将SBFEM应用于电磁场问题中的静电场问题。从拉普拉斯方程出发,利用变分原理并通过比例坐标和笛卡尔坐标变换,推导和求解出了静电场分析的比例边界有限元方程、电位求解公式以及电场求解公式,提出了一种分析静电场问题半解析方法。与此同时,本文还引入了两种新型比例边界坐标,一类是含平行侧边面的比例坐标,另外一类是含圆形域比例坐标,并且也推导和求解了相应的比例边界有限元方程。在此基础上,文中还重点求解了非齐次、侧边面含有电位以及无穷远电位不为有限值的比例边界有限元方程。通过10个算例计算结果与解析解和其他数值方法比较,结果表明此方法在处理一些电磁工程含有奇异点、非均匀介质和无限域等复杂问题中能显著提高计算效率和计算精度。
其次,本文发展了SBFEM在另外一类电磁场问题—波导本征值问题中的研究。波导截止频率的计算是一个富有挑战性的问题,各种形式的波导有一定的传输频率范围和传输特性,这对于波导的设计具有非常重要的意义。本文也利用了变分原理并通过比例边界坐标变换,推导了TE波和TM波波导的比例边界有限元频域方程以及波导动刚度方程,同时给出了波导动刚度矩阵的连分式解形式,通过引入辅助变量进一步得出波导特征值方程并求出波导本征值。以矩形、L形波导和叶型加载矩形波导的本征问题分析为例,通过与解析解和其他数值方法比较,结果表明此方法具有精度高、计算工作量小的优点,随着连分式阶数增加收敛速度快,而且SBFEM使用很少的单元数就能很好地解决了含有奇异点问题。脊波导具有较低的截止频率、较宽的工作带宽、低特性阻抗等优点,使得脊波导在微波和毫米波器件中被广泛应用。便随着高容量现代通信系统日益增长的需求,四脊波导也被广泛采用,尤其是在天线和雷达系统。对于四脊加载波导,往往在实际工程应用中,会对加载矩形进行剪切,而对于该类波导本征值研究较少。由于这类型波导多个角点处含有奇异性,使有限元计算中遭到困难,不得不采用网格加密或者采用高阶超级单元办法,增加了计算的复杂性,边界元方法在处理这类奇异问题也比较棘手,例如奇异积分的存在。为此,本文采用SBFEM的优越性可以顺利克服这些缺点,使计算效率和计算精度有很大程度的提高。这其中就包括分析了三类角切四脊加载(正方形、圆形和椭圆)波导的传输特性,并且给出了其中角切四脊加载正方形波导中的部分模式的截止波数计算经验公式,为工程设计提供一定理论依据。本文的解法也对计算电磁学发展作出了有意义的贡献,同时对工程应用也产生很好的效果。
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.
引文
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