Electromagnetic MUSIC-type imaging of perfectly conducting, arc-like cracks at single frequency

We propose a non-iterative MUSIC (MUltiple SIgnal Classification)-type algorithm for the time-harmonic electromagnetic imaging of one or more perfectly conducting, arc-like cracks found within a homogeneous space R²${\mathbb{R}}^2$. The algorithm is based on a factorization of the Multi-Static Response (MSR) matrix collected in the far-field at a single, nonzero frequency in either Transverse Magnetic (TM) mode (Dirichlet boundary condition) or Transverse Electric (TE) mode (Neumann boundary condition), followed by the calculation of a MUSIC cost functional expected to exhibit peaks along the crack curves each half a wavelength. Numerical experimentation from exact, noiseless and noisy data shows that this is indeed the case and that the proposed algorithm behaves in robust manner, with better results in the TM mode than in the TE mode for which one would have to estimate the normal to the crack to get the most optimal results.     

重心Lagrange插值配点法求解二维双曲电报方程

提出一种求解二维双曲电报方程的高精度重心Lagrange插值配点法.采用重心Lagrange插值构造包含时间和空间变量的近似函数.在给定Chebyshev-Gauss-Lobatto节点上,将多变量重心Lagrange插值近似函数代入双曲电报方程及其定解条件,得到离散代数方程组.包含狄里克雷和诺依曼边界条件的数值算例表明,本文方法程序实现方便并具有高精度,可应用于求解高维问题.     

A numerical scheme for the Stefan problem on adaptive Cartesian grids with supralinear convergence rate

We present a level set approach to the numerical simulation of the Stefan problem on non-graded adaptive Cartesian grids, i.e. grids for which the size ratio between adjacent cells is not constrained. We use the quadtree data structure to discretize the computational domain and a simple recursive algorithm to automatically generate the adaptive grids. We use the level set method on quadtree of Min and Gibou [C. Min, F. Gibou, A second order accurate level set method on non-graded adaptive Cartesian grids, J. Comput. Phys. 225 (2007) 300–321] to keep track of the moving front between the two phases, and the finite difference scheme of Chen et al. [H. Chen, C. Min, F. Gibou, A supra-convergent finite difference scheme for the poisson and heat equations on irregular domains and non-graded adaptive Cartesian grids, J. Sci. Comput. 31 (2007) 19–60] to solve the heat equations in each of the phases, with Dirichlet boundary conditions imposed on the interface. This scheme produces solutions that converge supralinearly (∼1.5)$(\sim 1.5)$ in both the L¹$L^1$ and the L^∞$L^{\infty }$ norms, which we demonstrate numerically for both the temperature field and the interface location. Numerical results also indicate that our method can simulate physical effects such as surface tension and crystalline anisotropy. We also present numerical data to quantify the saving in computational resources.     

Compact optimal quadratic spline collocation methods for the Helmholtz equation

Quadratic spline collocation methods are formulated for the numerical solution of the Helmholtz equation in the unit square subject to non-homogeneous Dirichlet, Neumann and mixed boundary conditions, and also periodic boundary conditions. The methods are constructed so that they are: (a) of optimal accuracy, and (b) compact; that is, the collocation equations can be solved using a matrix decomposition algorithm involving only tridiagonal linear systems. Using fast Fourier transforms, the computational cost of such an algorithm is O(N² log N) on an N × N uniform partition of the unit square. The results of numerical experiments demonstrate the optimal global accuracy of the methods as well as superconvergence phenomena. In particular, it is shown that the methods are fourth-order accurate at the nodes of the partition.     

Meshless analysis of three-dimensional steady-state heat conduction problems

Steady-state heat conduction problems arisen in connection with various physical and engineering problems where the functions satisfy a given partial differential equation and particular boundary conditions, have attracted much attention and research recently. These problems are independent of time and involve only space coordinates, as in Poisson’s equation or the Laplace equation with Dirichlet, Neuman, or mixed conditions. When the problems are too complex, it is difficult to find an analytical solution, the only choice left is an approximate numerical solution. This paper deals with the numerical solution of three-dimensional steady-state heat conduction problems using the meshless reproducing kernel particle method (RKPM). A variational method is used to obtain the discrete equations. The essential boundary conditions are enforced by the penalty method. The effectiveness of RKPM for three-dimensional steady-state heat conduction problems is investigated by two numerical examples.     

An implicit high-order hybridizable discontinuous Galerkin method for nonlinear convection–diffusion equations

In this paper, we present hybridizable discontinuous Galerkin methods for the numerical solution of steady and time-dependent nonlinear convection–diffusion equations. The methods are devised by expressing the approximate scalar variable and corresponding flux in terms of an approximate trace of the scalar variable and then explicitly enforcing the jump condition of the numerical fluxes across the element boundary. Applying the Newton–Raphson procedure and the hybridization technique, we obtain a global equation system solely in terms of the approximate trace of the scalar variable at every Newton iteration. The high number of globally coupled degrees of freedom in the discontinuous Galerkin approximation is therefore significantly reduced. We then extend the method to time-dependent problems by approximating the time derivative by means of backward difference formulae. When the time-marching method is (p+1)$(p+1)$th order accurate and when polynomials of degree p?0$p?0$ are used to represent the scalar variable, each component of the flux and the approximate trace, we observe that the approximations for the scalar variable and the flux converge with the optimal order of p+1$p+1$ in the L²$L^2$-norm. Finally, we apply element-by-element postprocessing schemes to obtain new approximations of the flux and the scalar variable. The new approximate flux, which has a continuous interelement normal component, is shown to converge with order p+1$p+1$ in the L²$L^2$-norm. The new approximate scalar variable is shown to converge with order p+2$p+2$ in the L²$L^2$-norm. The postprocessing is performed at the element level and is thus much less expensive than the solution procedure. For the time-dependent case, the postprocessing does not need to be applied at each time step but only at the times for which an enhanced solution is required. Extensive numerical results are provided to demonstrate the performance of the present method.     

Black rings in six dimensions

We propose a general framework for the numerical study of balanced black rings for any spacetime dimensions d?5$d?5$. Numerical solutions are constructed in a systematic way for d=6$d=6$, by solving the Einstein field equations with suitable boundary conditions. These black rings have a regular event horizon with S¹×S³$S^1\times S^3$ topology, and they approach the Minkowski background asymptotically. We analyze their global and horizon properties.     

The Glimm-Jaffe-Spencer expansion for the classical boundary conditions and coexistence of phases in the λφ24 Euclidean (quantum) field theory

The λφ₂⁴ Euclidean (quantum) field theory is studied in the multiphase region, and the following results are proven: (1) The “low temperature” expansion converges for Dirichlet (D), free (F), Neumann (N), and periodic (P), boundary conditions, and the even-point Schwinger functions for these boundary conditions have a mass gap; (2) $\text{〈}\text{o}\text{〉}{}^{\mathrm{b}}\text{=}\text{1}\text{2}\text{〈o〉}{}^{\mathrm{+}}\text{+}\text{1}\text{2}\text{〈o〉}{}^{\mathrm{?}}$, where b = D, F, N, P, and 〈o〉^± are the pure states of Glimm, Jaffe, and Spencer; (3) 〈o〉^±ξ = 〈o〉^± for all ξ > 0, where ξ is the buondary field; (4) alternative characterizations of the pure states 〈·〉^± are given.     

Noncommutative deformation of instantons

We study instanton solutions on noncommutative Euclidean 4-space which are deformations of instanton solutions on commutative Euclidean 4-space. We show that the instanton numbers of these noncommutative instanton solutions coincide with the commutative solutions and conjecture that the instanton number in R⁴${\mathbb{R}}^4$ is preserved for general noncommutative deformations. We also study noncommutative deformation of instanton solutions on a T⁴$T^4$ with twisted boundary conditions.     

Vector NLS solitons interacting with a boundary

We construct multi-soliton solutions of the n-component vector nonlinear Schrödinger equation on the half-line subject to two classes of integrable boundary conditions (BCs): the homogeneous Robin BCs and the mixed Neumann/Dirichlet BCs. The construction is based on the so-called dressing the boundary, which generates soliton solutions by preserving the integrable BCs at each step of the Darboux-dressing process. Under the Robin BCs, examples, including boundary-bound solitons, are explicitly derived; under the mixed Neumann/Dirichlet BCs, the boundary can act as a polarizer that tunes different components of the vector solitons. Connection of our construction to the inverse scattering transform is also provided.     

Boundary from Bulk Integrability in Three Dimensions: 3D Reflection Maps from Tetrahedron Maps

We derive the macroscopic laws that govern the evolution of the density of particles in the exclusion process on the Sierpinski gasket in the presence of a variable speed boundary. We obtain, at the hydrodynamics level, the heat equation evolving on the Sierpinski gasket with either Dirichlet or Neumann boundary conditions, depending on whether the reservoirs are fast or slow. For a particular strength of the boundary dynamics we obtain linear Robin boundary conditions. As for the fluctuations, we prove that, when starting from the stationary measure, namely the product Bernoulli measure in the equilibrium setting, they are governed by Ornstein-Uhlenbeck processes with the respective boundary conditions.

     

Three Boundary Meshless Methods for Heat Conduction Analysis in Nonlinear FGMs with Kirchhoff and Laplace Transformation

This paper presents three boundary meshless methods for solving problems of steady-state and transient heat conduction in nonlinear functionally graded materials (FGMs). The three methods are, respectively, the method of fundamental solution (MFS), the boundary knot method (BKM), and the collocation Trefftz method (CTM) in conjunction with Kirchhoff transformation and various variable transformations. In the analysis, Laplace transform technique is employed to handle the time variable in transient heat conduction problem and the Stehfest numerical Laplace inversion is applied to retrieve the corresponding time-dependent solutions. The proposed MFS, BKM and CTM are mathematically simple, easy-to-programming, meshless, highly accurate and integration-free. Three numerical examples of steady state and transient heat conduction in nonlinear FGMs are considered, and the results are compared with those from meshless local boundary integral equation method (LBIEM) and analytical solutions to demonstrate the efficiency of the present schemes.     

Casimir forces in models with non-trivial topology

The Casimir forces, acting on the parallel plates in models with the compact subspace are investigated for the case of a scalar field. The field obeys the Robin boundary conditions on the plates. Depending on the values of the coefficients in the boundary conditions, the forces can be either attractive or repulsive. In models with a homogeneous compact subspace, they are the same for both the plates. In special cases of the Dirichlet and Neumann boundary conditions, the Casimir forces are attractive. Proceeding from general results, two particular cases with the subspaces S¹ and S² are considered.     

Waveguides with Combined Dirichlet and Robin Boundary Conditions

We consider the Laplacian in a curved two-dimensional strip of constant width squeezed between two curves, subject to Dirichlet boundary conditions on one of the curves and variable Robin boundary conditions on the other. We prove that, for certain types of Robin boundary conditions, the spectral threshold of the Laplacian is estimated from below by the lowest eigenvalue of the Laplacian in a Dirichlet-Robin annulus determined by the geometry of the strip. Moreover, we show that an appropriate combination of the geometric setting and boundary conditions leads to a Hardy-type inequality in infinite strips. As an application, we derive certain stability of the spectrum for the Laplacian in Dirichlet–Neumann strips along a class of curves of sign-changing curvature, improving in this way an initial result of Dittrich and Kříž (J. Phys. A, 35:L269–275, 2002).      

Performance of a parallel algebraic multilevel preconditioner for stabilized finite element semiconductor device modeling

In this study results are presented for the large-scale parallel performance of an algebraic multilevel preconditioner for solution of the drift-diffusion model for semiconductor devices. The preconditioner is the key numerical procedure determining the robustness, efficiency and scalability of the fully-coupled Newton–Krylov based, nonlinear solution method that is employed for this system of equations. The coupled system is comprised of a source term dominated Poisson equation for the electric potential, and two convection–diffusion-reaction type equations for the electron and hole concentration. The governing PDEs are discretized in space by a stabilized finite element method. Solution of the discrete system is obtained through a fully-implicit time integrator, a fully-coupled Newton-based nonlinear solver, and a restarted GMRES Krylov linear system solver. The algebraic multilevel preconditioner is based on an aggressive coarsening graph partitioning of the nonzero block structure of the Jacobian matrix. Representative performance results are presented for various choices of multigrid V-cycles and W-cycles and parameter variations for smoothers based on incomplete factorizations. Parallel scalability results are presented for solution of up to 10⁸${10}^8$ unknowns on 4096 processors of a Cray XT3/4 and an IBM POWER eServer system.     

A holographic model for hall viscosity

We have modified the holographic model of Saremi and Son [12] by using a charged black brane, instead of a neutral one, such that when the bulk pseudo scalar (θ  ) potential is made of θ²${\theta }^2$ and θ⁴${\theta }^4$ terms, parity can still be broken spontaneously in the boundary theory. In our model, the 3+1$3+1$ dimensional bulk has a pseudo scalar coupled to the gravitational Chern–Simons term in the anti de Sitter charged black brane back ground. Parity could be broken spontaneously in the bulk by the pseudo scalar hairy solution and give rise to non-zero Hall viscosity at the boundary theory.     

Three-phase compressible flow in porous media: Total Differential Compatible interpolation of relative permeabilities

We describe the construction of Total Differential (TD) three-phase data for the implementation of the exact global pressure formulation for the modeling of three-phase compressible flow in porous media. This global formulation is preferred since it reduces the coupling between the pressure and saturation equations, compared to phase or weighted formulations. It simplifies the numerical analysis of the problem and boosts its computational efficiency. However, this global pressure approach exists only for three-phase data (relative permeabilities, capillary pressures) which satisfy a TD condition. Such TD three-phase data are determined by the choice of a global capillary pressure function and a global mobility function, which take both saturations and global pressure level as argument. Boundary conditions for global capillary pressure and global mobility are given such that the corresponding three-phase data are consistent with a given set of three two-phase data. The numerical construction of global capillary pressure and global mobility functions by C¹${\text{C}}^1$ and C⁰${\text{C}}^0$ finite element is then performed using bi-Laplacian and Laplacian interpolation. Examples of the corresponding TD three-phase data are given for a compressible and an incompressible case.     

Integrable boundary interaction in 3D target space: The “pillow-brane” model

We propose a model of boundary interaction, with three-dimensional target space, and the boundary values of the field X∈R³$\mathbf{X}\in {\mathbb{R}}^3$ constrained to lay on a two-dimensional surface of the “pillow” shape. We argue that the model is integrable, and suggest that its exact solution is described in terms of certain linear ordinary differential equation.