An Adaptive Finite Element PML Method for the Open Cavity Scattering Problems

Consider the electromagnetic scattering of a time-harmonic plane wave byan open cavity which is embedded in a perfectly electrically conducting infinite groundplane. This paper is concerned with the numerical solutions of the transverse electricand magnetic polarizations of the open cavity scattering problems. In each polarization, the scattering problem is reduced equivalently into a boundary value problem ofthe two-dimensional Helmholtz equation in a bounded domain by using the transparent boundary condition (TBC). An a posteriori estimate based adaptive finite elementmethod with the perfectly matched layer (PML) technique is developed to solve thereduced problem. The estimate takes account ofboththe finite element approximationerror and the PML truncation error, where the latter is shown to decay exponentiallywith respect to the PML medium parameter and the thickness of the PML layer. Numerical experiments are presented and compared with the adaptive finite element TBCmethod for both polarizations to illustrate the competitive behavior of the proposedmethod.     

An Edge-Based Smoothed Finite Element Method with TBC for the Elastic Wave Scattering by an Obstacle

Elastic wave scattering has received ever-increasing attention in military and medical fields due to its high-precision solution. In this paper, an edge-based smoothed finite element method (ES-FEM) combined with the transparent boundary condition (TBC) is proposed to solve the elastic wave scattering problem by a rigid obstacle with smooth surface, which is embedded in an isotropic and homogeneous elastic medium in two dimensions. The elastic wave scattering problem satisfies Helmholtz equations with coupled boundary conditions obtained by Helmholtz decomposition. Firstly, the TBC of the elastic wave scattering is constructed by using the analytical solution to Helmholtz equations, which can truncate the boundary value problem (BVP) in an unbounded domain into the BVP in a bounded domain. Then the formulations of ES-FEM with the TBC are derived for Helmholtz equations with coupled boundary conditions. Finally, several numerical examples illustrate that the proposed ES-FEM with the TBC (ES-FEM-TBC) can work effectively and obtain more stable and accurate solution than the standard FEM with the TBC (FEM-TBC) for the elastic wave scattering problem.     

Finite Neuron Method and Convergence Analysis

We study a family of $H^m$-conforming piecewise polynomials based on theartificial neural network, referred to as the finite neuron method (FNM), for numericalsolution of $2m$-th-order partial differential equations in$mathbb{R}^d$ for any $m,d≥1$ and thenprovide convergence analysis for this method. Given a general domain Ω$⊂mathbb{R}^d$ and apartition$mathcal{T}_h$ of Ω, it is still an open problem in general how to construct a conforming finite element subspace of $H^m$(Ω) that has adequate approximation properties. By usingtechniques from artificial neural networks, we construct a family of $H^m$-conformingfunctions consisting of piecewise polynomials of degree $k$ for any $k≥m$ and we further obtain the error estimate when they are applied to solve the elliptic boundaryvalue problem of any order in any dimension. For example, the error estimates that $‖u−u_N‖_{H^m(rm{Ω})}=mathcal{O}(N^{−frac{1}{2}−frac{1}{d}})$ is obtained for the error between the exact solution $u$ andthe finite neuron approximation $u_N$. A discussion is also provided on the differenceand relationship between the finite neuron method and finite element methods (FEM).For example, for the finite neuron method, the underlying finite element grids are notgiven a priori and the discrete solution can be obtained by only solving a non-linearand non-convex optimization problem. Despite the many desirable theoretical properties of the finite neuron method analyzed in the paper, its practical value requiresfurther investigation as the aforementioned underlying non-linear and non-convex optimization problem can be expensive and challenging to solve. For completeness andthe convenience of the reader, some basic known results and their proofs are introduced.     

Acoustic Scattering Problems with Convolution Quadrature and the Method of Fundamental Solutions

Time-domain acoustic scattering problems in two dimensions are studied. The numerical scheme relies on the use of the Convolution Quadrature (CQ) method to reduce the time-domain problem to the solution of frequency-domain Helmholtz equations with complex wavenumbers. These equations are solved with the method of fundamental solutions (MFS), which approximates the solution by a linear combination of fundamental solutions defined at source points inside (outside) the scatterer for exterior (interior) problems. Numerical results show that the coupling of both methods works efficiently and accurately for multistep and multistage based CQ.     

Permanent Charge Effects on Ionic Flow: A Numerical Study of Flux Ratios and Their Bifurcation

Ionic flow carries electrical signals for cells to communicate with each other.The permanent charge of an ion channel is a crucial protein structure for flow properties while boundary conditions play a role of the driving force. Their effects on flowproperties have been analyzed via a quasi-one-dimensional Poisson-Nernst-Planckmodel for small and relatively large permanent charges. The analytical studies haveled to the introduction of flux ratios that reflect permanent charge effects and have auniversal property. The studies also show that the flux ratios have different behaviorsfor small and large permanent charges. However, the existing analytical techniquescan reveal neither behaviors of flux ratios nor transitions between small and large permanent charges. In this work we present a numerical investigation on flux ratios tobridge between small and large permanent charges. Numerical results verify the analytical predictions for the two extremal regions. More significantly, emergence of non-trivial behaviors is detected as the permanent charge varies from small to large. Inparticular, saddle-node bifurcations of flux ratios are revealed, showing rich phenomena of permanent charge effects by the power of combining analytical and numericaltechniques. An adaptive moving mesh finite element method is used in the numericalstudies.     

One-Dimensional Blood Flow with Discontinuous Properties and Transport: Mathematical Analysis and Numerical Schemes

In this paper we consider the one-dimensional blood flow model with discontinuous mechanical and geometrical properties, as well as passive scalar transport, proposed in [E.F. Toro and A. Siviglia. Flow in collapsible tubes with discontinuous mechanical properties: mathematical model and exact solutions. Communications in Computational Physics. 13(2), 361-385, 2013], completing the mathematical analysis by providing new propositions and new proofs of relations valid across different waves. Next we consider a first order DOT Riemann solver, proposing an integration path that incorporates the passive scalar and proving the well-balanced properties of the resulting numerical scheme for stationary solutions. Finally we describe a novel and simple well-balanced, second order, non-linear numerical scheme to solve the equations under study; by using suitable test problems for which exact solutions are available, we assess the well-balanced properties of the scheme, its capacity to provide accurate solutions in challenging flow conditions and its accuracy.     

Detection of heterogeneities embedded within a turbid slab media using time- and frequency-domain methods: application to the mammography

During the last decade, several methods have been devoted to the detection and imaging of tumor-like objects embedded in turbid slab media. Optical methods are broadly investigated as potential non-invasive medical diagnosis used for the detection of tumors. In this paper, we model the photon migration due to a pulsed source laser, through a multiple scattering slab to locate and characterize heterogeneities of different optical properties. The time-dependent diffusion equation is used and solved by means of a finite element model, taking into account air–tissue boundary conditions. The transmitted time-spectra associated to their Fast Fourier Transforms are used to detect embedded objects within diffusive slab media. We show that for an inclusion of identical scattering coefficient to the surrounding medium, the phase shift increases as the absorption coefficient of the inclusion is increased. For a homogeneous absorption, the phase shift is very sensitive to local variations in scattering properties. We then compare these results with those reported by other workers and conclude that the computational model allows the lateral detection of these inclusions, so it should be possible to enhance the detection of a malignant tumor surrounded by the healthy breast tissue.This paper was first presented at the Laser Florence 2005 International Congress, IALMS Florence, 10–12 November 2005     

Analysis and Numerical Simulation of Hyperbolic Shallow Water Moment Equations

Shallow Water Moment Equations allow for vertical changes in the horizontal velocity, so that complex shallow flows can be described accurately. However, weshow that these models lack global hyperbolicity and the loss of hyperbolicity alreadyoccurs for small deviations from equilibrium. This leads to instabilities in a numericaltest case. We then derive new Hyperbolic Shallow Water Moment Equations based ona modification of the system matrix. The model can be written in analytical form andhyperbolicity can be proven for a large number of equations. A second variant of thismodel is obtained by generalizing the modification with the help of additional parameters. Numerical tests of a smooth periodic problem and a dam break problem usingthe new models yield accurate and fast solutions while guaranteeing hyperbolicity.     

An Efficient Finite Element Method with Exponential Mesh Refinement for the Solution of the Allen-Cahn Equation in Non-Convex Polygons

In this paper we consider the numerical solution of the Allen-Cahn typediffuse interface model in a polygonal domain. The intersection of the interface withthe re-entrant corners of the polygon causes strong corner singularities in the solution.To overcome the effect of these singularities on the accuracy of the approximate solution, for the spatial discretization we develop an efficient finite element method withexponential mesh refinement in the vicinity of the singular corners, that is based on($k$−1)-th order Lagrange elements, $k$≥2 an integer. The problem is fully discretized byemploying a first-order, semi-implicit time stepping scheme with the Invariant EnergyQuadratization approach in time, which is an unconditionally energy stable method.It is shown that for the error between the exact and the approximate solution, an accuracy of $mathcal{O}$($h^k$+$τ$) is attained in the $L^2$-norm for the number of $mathcal{O}$($h^{−2}$ln$h^{−1}$) spatialelements, where $h$ and $τ$ are the mesh and time steps, respectively. The numericalresults obtained support the analysis made.