Hierarchical Divergence Conforming Bases for Edge Singularities in Quadrilateral Cells

First-order system least squares (FOSLS) is a methodology that offers an alternative to standard methods for solving partial differential equations. This paper studies the first-order system least-squares approach for scalar second-order elliptic boundary value problems with discontinuous coefficients. In a companion paper [M. Berndt, T. A. Manteuffel, S. F. McCormick, and G. Starke, Analysis of first-order system least squares ({FOSLS}) for elliptic problems with discontinuous coefficients: Part I, SIAM J. Numer. Anal., 43 (2005), pp. 386--408], ellipticity of an appropriately scaled least-squares bilinear form is established in a natural norm. For some geometries this ellipticity is independent of the size of the jumps in the coefficients. The occurrence of singularities at interface corners, cross-points, reentrant corners, and irregular boundary points is discussed, and a basis of singular functions with local support around singular points is established. This paper describes a method for including discrete versions of the singular basis functions together with standard finite element spaces in a least-squares format at little additional computational cost. The singular basis functions are constructed to match the jump conditions that arise at interfaces between regions of continuity of the diffusion coefficient. Because these basis functions must be approximated in practice, the resulting discretization is by nature nonconforming. This necessitates the establishment here of a general error estimate for FOSLS L2 minimization problems discretized by nonconforming finite elements. An advantage of the FOSLS formulation is that this estimate does not involve the consistency error term usually present in bounds for other methods. Based on this general estimate, error bounds are derived for the finite element space that includes singular basis functions. Numerical tests are included that confirm these discretization error bounds. Finally, a multilevel method is developed for solving the discrete system that uses singular basis functions on all levels, and its efficiency is demonstrated by the numerical results.

Inhomogeneous waveguiding structures containing corners, even dielectric corners, cannot be modeled efficiently with finite-element procedures based on conventional polynomial basis functions. Recently proposed singular vector basis functions for triangular cells are used in a formulation that combines transverse and longitudinal components for dielectric waveguide analysis. The additive hierarchical basis functions combine singular bases with a full set of existing hierarchical polynomial basis functions to form the representation. The improvement in the satisfaction of boundary conditions resulting from the use of the bases is illustrated.

Stable GFEM (SGFEM): Improved conditioning and accuracy of GFEM/XFEM for three-dimensional fracture mechanics

Spectral sensitivity, as one of the most important parameters of a digital camera, is playing a key role in many computer vision applications. In this paper, a confidence voting convolutional neural network (CVNet) is proposed to rebuild the spectral sensitivity function, modeled as the sum of weighted basis functions. By evaluating useful information supplied by different image segments, disparate confidence is calculated to automatically learn basis functions' weights, only using one image captured by the object camera. Three types of basis functions are made up and employed in the network, including Fourier basis function (FBF), singular value decomposition basis function (SVDBF), and radial basis function (RBF). Results show that the accuracy of the proposed method with FBF, SVDBF, and RBF is 97.92%, 98.69%, and 99.01%, respectively. We provide theory for network design, build a dataset, demonstrate training process, and present experimental results with high precision. Without bulky benchtop setups and strict experimental limitations, this proposed simple and effective method could be an alternative in the future for spectral sensitivity function estimation.

The Numerical Solution of Integral Equations on the Half-Line

The singular value decomposition characteristic basis function method (SVD-CBFM) is usually used to solve the electromagnetic scattering problems. When the number of unknowns increases, the memory and computation time consumed become insufferable. The parallel SVD-CBFM solution on graphics processing unit (GPU) is implemented by the compute unified device architecture (CUDA) technology in this paper. The acceleration ratio of parallel SVD-CBFM comparing with traditional central processing unit (CPU) is also analyzed. Numerical results show that the speed of impedance matrix filling and reduced matrix filling is greatly improved.

Numerial Methods for Volterra Integral Equations with Singular Kernels

The first section of this chapter collects results from [240] which gives a further contribution to the analysis of the hp-version of the boundary element method (BEM) by presenting a more general result for Dirichlet and Neumann problems than [21] allowing the use of a general geometric mesh refinement on the polygonal boundary Γ. Here as in [240] we prove the exponential convergence of the hp- version of the boundary element method by exploiting only features of the solutions of the boundary integral equations. The key result in this approach is an asymptotic expansion of the solution of the integral equations in singularity functions reflecting the singular behaviour of the solutions near corners of Γ. With such expansions we show that the solutions of the integral equations belong to countably normed spaces. Therefore these solutions can be approximated exponentially fast in the energy norm via the hp- Galerkin solutions of those integral equations. This result is not restricted to integral equations which stem from boundary value problems for the Laplacian but applies to Helmholtz problems as well. Further applications are 2D crack problems in linear elasticity. For numerical experiments with hp-version (BEM) see [165, 340].

In this paper, we present a time domain combined field integral equation formulation (TD-CFIE) to analyze the transient electromagnetic response from dielectric objects. The solution method is based on the method of moments which involves separate spatial and temporal testing procedures. A set of the RWG functions is used for spatial expansion of the equivalent electric and magnetic current densities, and a combination of RWG and its orthogonal component is used for spatial testing. The time domain unknowns are approximated by a set of orthonormal basis functions derived from the Laguerre polynomials. These basis functions are also used for temporal testing. Use of this temporal expansion function characterizing the time variable makes it possible to handle the time derivative terms in the integral equation and decouples the space-time continuum in an analytic fashion. Numerical results computed by the proposed formulation are compared with the solutions of the frequency domain combined field integral equation.

An efficient method of moments (MoM) solution of the volume-surface integral equation (VSIE) is presented for the analysis of electromagnetic (EM) scattering and radiation of composite objects comprising arbitrarily contacted dielectrics and conductors. In contrast to the conventional Schaubert-Wilton-Glisson (SWG) and Rao-Wilton-Glisson (RWG) basis functions, which were widely used in the MoM solutions of the VSIE, a new hybrid pulse-rooftop basis function is proposed to explicitly enforce the boundary condition of vanishing tangential electric field at the dielectric-conductor interface. As a consequence, the displacement currents in the dielectrics can be efficiently modeled by a reduced number of volume bases on the same set of meshes, particularly when thin dielectric slabs are involved in the composite structures. The numerical results of EM scattering and radiation of several composite objects are shown to illustrate the accuracy and efficiency of the proposed scheme. Since the new hybrid basis function can be easily implemented into the existing fast algorithms, the VSIE solution presented in this paper is promising for EM problems of composite targets with large size.

In this work, we propose a new decomposed subspace reduction (DSR) method for reduced-order modeling of fracture mechanics based on the integrated singular basis function method (ISBFM) with reproducing kernel approximation enriched by crack-tip basis functions. It is shown that the standard MOR approach based on modal analysis (ISBFM-MA) with a direct employment of the crack-tip enrichment functions yields an inappropriate scaling effect to the stiffness matrix, and results in the loss of essential crack features and the erroneous representation of inhomogeneous Dirichlet boundary conditions in the reduced subspace. On the other hand, the solution of ISBFM-DSR is not affected by the arbitrary scaling of the enrichment functions, and it properly captures the singularity and discontinuity properties of fracture problems in its low-dimensional reduced-order approximation. It is also shown that the inhomogeneous boundary conditions can be accurately represented in the ISBFM-DSR solution. Validations are given in the numerical examples.

Recently, the authors have developed a number of new hierarchical and singular basis functions for field representation in two- and three-dimensional cells. Scalar and vector basis functions of both interpolatory and hierarchical kind together with some singular functions of the additive kind are discussed in detail, with several results, in a recent SciTech-IET book (R.D. Graglia and A.F. Peterson: Higher-order Techniques in Computational Electromagnetics), as well as in several other publications (for example, Graglia et al., IEEE TAP, vol. 61, pp. 3674–3692, Jul. 2013; IEEE TAP, vol. 62, pp 3632–3644, Jul. 2014; IEEE AWPL, vol. 13, pp 1701–1704, 2014, and in the special issue on "Finite Elements for Microwave Engineering," Electromagnetics, vol. 34, pp. 171–198, 2014).

The different methods of solving problems of viscoelasticity for hereditary media that employ Laplace transforms or exact solutions of integral equations for inaccurately approximated kernels inevitably introduce errors associated with the approximation and the inverse transformations. Accordingly, it is necessary to estimate the accuracy of these methods. It is shown that the kernels of the integral equations of viscoelasticity permit the estimation, with a certain accuracy, of upper and lower bounds directly for the solutions of these integral equations. In cases when the accuracy of the estimate is sufficient, there is no need to employ other methods of solution.

By integrating the electric field integral equation and the transmission line equation, an iterative solution for the electromagnetic field coupling to buried wires is obtained. At first we establish the integral equation which the difference between solutions of the integral equation and the telegraph equation satisfies. Then the solution of the telegraph equation is used to approximate the solution of this integral equation. Every following step of iteration is an improvement on the transmission line solution, and with several iterations, a well approximation to the solution of electric field integral equation can be obtained.

C15 - ON THE INTERACTION BETWEEN CUBIC ANHARMONICITY AND A SECOND-ORDER ELECTRIC MOMENT IN THE OPTICAL ABSORPTION OF CRYSTALS