Certified Reduced Basis Method for the Electric Field Integral Equation

Parametric estimation of a stability factor for certified reduced basis methods via adaptive Gaussian process

A finite element method (FEM)/boundary element method (BEM)-hybrid approach is presented which allows electromagnetic field calculations in planar layered media including three-dimensional inhomogeneous regions. The fields within the inhomogeneities are modeled by the FEM in terms of the electric field strength Ē with edge elements on tetrahedral meshes. Based on Huygens’ principle the BEM is formulated with equivalent electric and magnetic surface currents on a surface completely enclosing the inhomogeneous regions. Additionally, ideal conducting objects can be included into the BEM-model. The electric field integral equation (EFIE) for the BEM is formulated as a mixed potential integral equation (MPIE). Applications of the method to field calculations in a microstrip resonator with finite metallization thickness and to the coupling of a cylindrical dielectric resonator to a microstrip line are presented.

A boundary element formulation, along with detailed solution procedure for determining the acoustic field inside a three-dimensional, rectangular duct is presented in this paper. The results of classical and boundary element solutions are compared for a typical rectangular duct by restricting the input frequency in such a way that only plane wave propagation is possible. The effect of changing the type and number of discrete boundary elements on the computed sound pressure levels inside the duct is also presented.

The integral equation (IE) method is one of the efficient approaches for solving electromagnetic problems, where dyadic Green’s function (DGF) plays an important role as the Kernel of the integrals. In general, a layered medium with planar, cylindrical, or spherical geometry can be used to model different biomedical media such as human skin, body, or head. Therefore, in this chapter, different approaches for the derivation of Green’s function for these structures will be introduced. Due to the recent great interest in two-dimensional (2D) materials, the chapter will also discuss the generalization of the technique to the same structures with interfaces made of isotropic and anisotropic surface impedances. To this end, general formulas for the dyadic Green’s function of the aforementioned structures are extracted based on the scattering superposition method by considering field and source points in the arbitrary locations. Apparently, by setting the surface conductivity of the interfaces equal to zero, the formulations will turn into the associated problem with dielectric boundaries. This section will also aid in the design of various biomedical devices such as sensors, cloaks, and spectrometers, with improved functionality. Finally, the Purcell factor of a dipole emitter in the presence of the layered structures will be discussed as another biomedical application of the formulation.

Forward electromagnetic scattering from buried dielectric objects is computed by the integral equation method in conjunction with efficient algorithm for half-space dyadic Green's functions. Concise representations of dyadic Green's functions are presented to establish the integral equation for buried dielectric objects. Through moving the spatial differential operator to the outside of the integral sign and computing the rest part by the discrete complex image method, closed forms of Green's functions are obtained. The bottleneck during calculating reaction matrix and scattering fields is overcome since it is not necessary to process the numerical evaluation of Sommerfeld integrals. The proposed method enhances the efficiency of the forward calculation of buried dielectric objects and enough accuracy can be guaranteed.

We present a reduced basis method for parametrized linear elasticity equations with two objectives: providing an error bound with respect to the exact weak solution of the PDE, as opposed to the typical finite-element “truth”, in the online stage; providing automatic adaptivity in both physical and parameter spaces in the offline stage. Our error bound builds on two ingredients: a minimum-residual mixed formulation with a built-in bound for the dual norm of the residual with respect to an infinite-dimensional function space; a combination of a minimum eigenvalue bound technique and the successive constraint method which provides a lower bound of the stability constant with respect to the infinite-dimensional function space. The automatic adaptivity combines spatial mesh adaptation and greedy parameter sampling for reduced bases and successive constraint method to yield a reliable online system in an efficient manner. We demonstrate the effectiveness of the approach for a parametrized linear elasticity problem with geometry transformations and parameter-dependent singularities induced by cracks.

An alternative set of Cartesian vector wave functions is presented to simplify the development of eigenfunction expansions of electromagnetic fields and dyadic Green's functions for bianisotropic media. With the eigenfields expanded using these functions, the determination of their expansion coefficients is facilitated via a concise 4×4 matrix method. Based on the complete dyadic discontinuity relations, the source consequents of dyadic Green's functions are also explicitly expressed in terms of these functions. By incorporating the modified reciprocity theorem into the discontinuity relations, more compact source consequents are found to be related directly to the eigenfields in a complementary medium. These complementary eigenfields, in turn, can be deduced readily from their original counterparts by performing certain simple transformations. This manner of incorporating reciprocity provides a clear insight into the significance of each discontinuity relation and reveals the connections between the singularities, discontinuities and the complementary eigenfield components. For illustration, the vector wave functions expansions of the eigenfields for isotropic and uniaxial bianisotropic media are discussed.

A boundary element method is presented for two-dimensional and steady-state analysis of the traveling magnetic field in conductive material. The boundary integral equation is derived from the Green's second identity in which the fundamental solution including an effect due to the motional induction is originated. By use of the constant boundary elements the boundary integral equation is transformed into a system of linear equations. Some numerical results are shown in order to demonstrate the validity of the boundary element method. It is proved that even at large magnetic Reynolds numbers the boundary element solution is stably calculated with the second-order accuracy.

The integral equations encountered in boundary element methods are frequently solved numerically using collocation with spline trial functions. Convergence proofs and error estimates for these approximation methods have been only available in the following cases: Fredholm integral equations of the second kind [4], [7], one-dimensional pseudodifferential equations and singular integral equations with piecewise smooth coefficients on smooth curves [2], [3], [17], [26]—[29], and some special results on the classical Neumann integral equation of potential theory for polygonal plane domains [5], [8], [9]. Here we give convergence proofs for collocation with piecewise linear trial functions for Neumann’s integral equation and Symm’s integral equation on plane curves with corners. We derive asymptotic error estimates in Sobolev norms and analyze the effect of graded meshes.

A full-wave, integral equation method is presented which generalizes the treatment of multi-layer substrates and superstrates for shielded circuits so that any combination and number of layers can be handled without reformulation. A vector wave function approach with dyadic analysis is used to expand the fields, producing all components of the dyadic Green's functions in one solution. The approach makes application to the analysis of shielded transmission lines very straightforward including the calculation of the fields in all layers. Application to the determination of fundamental properties of strip transmission lines are emphasized in the examples provided.

Most recently, we have developed a novel multilevel boundary element method (MLBEM) for steady Stokes flows in irregular two-dimensional domains (Grigoriev, M.M., and Dargush, G.F., Comput. Methods. Appl. Mech. Eng., 2005). The multilevel algorithm permitted boundary element solutions with slightly over 16,000 degrees of freedom, for which approximately 40-fold speedups were demonstrated for the fast MLBEM algorithm compared to a conventional Gauss elimination approach. Meanwhile, the sevenfold memory savings were attained for the fast algorithm. This paper extends the MLBEM methodology to dramatically improve the performance of the original multilevel formulation for the steady Stokes flows. For a model problem in an irregular pentagon, we demonstrate that the new MLBEM formulation reduces the CPU times by a factor of nearly 700,000. Meanwhile, the memory requirements are reduced more than 16,000 times. These superior run-time and memory reductions compared to regular boundary element methods are achieved while preserving the accuracy of the boundary element solution.

The interaction between acoustic wave propagation and a fluid in motion can be described by a system of coupled equations. For many applications like ultrasound flowmeters, medical ultrasound diagnostics, environmental acoustics or air conditioning systems, the influence of the acoustic waves upon the flow can be neglected. This allows a separation and therefore a successive solution of the problem. The results of finite element (FEM) simulations based on a modified wave equation with the flow profile as a boundary condition have been previously presented. In this paper, a new boundary element (BEM) technique is reported to solve this problem. One major advantage over FEM is the reduced dimension of the problem, leading to easier meshing and, especially for unbounded domains, to reduced computational effort. For the new BEM approach the Green's function had to be modified to consider flow. Based on an approximate Green's function a boundary element solution was developed. This new approach is compared with the previously implemented FEM scheme as well as with experiments measuring the sound pressure in a flow channel for various geometries and flow profiles. The results show good agreement between experiment and simulation. The limitations of this approach will be discussed.

Special boundary element methods for the solution of plate bending problems are presented in this chapter. Some of these methods have been developed during the first efforts to apply the boundary integral equation method to the plate bending problem and before the appearance of general direct BEM formulations in the literature. Other special BEM’s have been developed later in order to overcome certain shortcomings and computational difficulties in the general direct BEM formulations or in order to solve problems for which the fundamental solution can not be established or is difficult to treat numerically. On the basis of common characteristics the special methods are classified in five groups: (a) Methods based on the biharmonic analysis, (b) Indirect boundary element methods (IBEM’s) (c) Boundary differential — integral equation methods (BDIEM’s) (d) Green’s function methods (GFM’s), (e) Other methods which lack common characteristics. For each group of special methods a separate section is devoted. The special methods are described by the most representative methods of each group. The description of each special method is supplemented by several numerical results which are compared with those obtained by analytical or other numerical methods in order to illustrate the versatility, the effectiveness and the accuracy of the special BEM’s. Concluding remarks are presented in the last section.

It is well known that electric field integral equation (EFIE) has the low frequency breakdown problem due to the decoupling of electric and magnetic field. Augmented electric field integral equation (AEFIE) method overcomes the low frequency breakdown. However, its performance degrades with the increasing frequency. To obtain a unified method that works for both low and high frequencies, a novel method using the EFIE complement process inside AEFIE is proposed. Numerical results demonstrated that this alternative AEFIE‐EFIE method is able to extend integral equations to wide band applications. © 2012 Wiley Periodicals, Inc. Microwave Opt Technol Lett 54:1737–1740, 2012; View this article online at wileyonlinelibrary.com. DOI 10.1002/mop.26919

We analyze adaptive mesh-refining algorithms in the frame of boundary element methods (BEM) and the coupling of finite elements and boundary elements (FEM-BEM). Adaptivity is driven by the two-level error estimator proposed by Ernst P. Stephan, Norbert Heuer, and coworkers in the frame of BEM and FEM-BEM or by the residual error estimator introduced by Birgit Faermann for BEM for weakly-singular integral equations. We prove that in either case the usual adaptive algorithm drives the associated error estimator to zero. Emphasis is put on the fact that the error estimators considered are not even globally equivalent to weighted-residual error estimators for which recently convergence with quasi-optimal algebraic rates has been derived.