Two-dimensional Scalar Model Research Articles

Spectral analysis of polygonal cavities containing a negative-index material

The purpose of this paper is to investigate the spectral effects of an interface between vacuum and a negative-index material (NIM), that is, a dispersive material whose electric permittivity and magnetic permeability become negative in some frequency range. We consider here an elementary situation, namely, 1) the simplest existing model of NIM : the non dissipative Drude model, for which negativity occurs at low frequencies; 2) a two-dimensional scalar model derived from the complete Maxwell’s equations; 3) the case of a simple bounded cavity: a polygonal domain partially filled with a portion of Drude material. Because of the frequency dispersion (the permittivity and permeability depend on the frequency), the spectral analysis of such a cavity is unusual since it yields a nonlinear eigenvalue problem. Thanks to the use of an additional unknown, we linearize the problem and we present a complete description of the spectrum. We show in particular that the interface between the NIM and vacuum is responsible for various resonance phenomena related to various components of an essential spectrum.

Striped Periodic Minimizers of a Two-Dimensional Model for Martensitic Phase Transitions

In this paper we consider a simplified two-dimensional scalar model for the formation of mesoscopic domain patterns in martensitic shape-memory alloys at the interface between a region occupied by the parent (austenite) phase and a region occupied by the product (martensite) phase, which can occur in two variants (twins). The model, first proposed by Kohn and Mueller, is defined by the following functional: $${\cal E}(u)=\beta||u(0,\cdot)||^2_{H^{1/2}([0,h])}+ \int_{0}^{L} dx \int_0^h dy \big(|u_x|^2 + \epsilon |u_{yy}| \big)$$ where $u:[0,L]\times[0,h]\to R$ is periodic in $y$ and $u_y=\pm 1$ almost everywhere. Conti proved that if $\beta\gtrsim\epsilon L/h^2$ then the minimal specific energy scales like $\sim \min\{(\epsilon\beta/L)^{1/2}, (\epsilon/L)^{2/3}\}$, as $(\epsilon/L)\to 0$. In the regime $(\epsilon\beta/L)^{1/2}\ll (\epsilon/L)^{2/3}$, we improve Conti's results, by computing exactly the minimal energy and by proving that minimizers are periodic one-dimensional sawtooth functions.

A multimodal method for non-uniform open waveguides

We present an original method for the simulation of ultrasonic waves in a varying cross section elastic waveguide embedded in a homogeneous matrix. The so-called multimodal methods consist in expanding the acoustic field in each transverse section as a series which involves the modes of a perfectly cylindrical waveguide with this particular section. For such a cylindrical waveguide, the modal components of the acoustic field are uncoupled. But a varying cross section yields a coupling of these components. A multimodal approach consists in solving the system of differential equations which couples these components instead of the propagation equations.Such methods have been extensively studied for closed waveguides, that is, when no wave can radiate in the transverse direction. In our case, radiation waves occur, because of the presence of the matrix. Actually, such an open waveguide is characterized by a continuum of radiation modes together with a possible finite set of guided modes. The multimodal method proposed here does not take into account this continuum, but uses instead the concept of leaky modes. These modes appear naturally when the computational domain is artificially limited by Perfectly Matched Layers (PMLs). The main advantage of using leaky modes instead of radiation modes is that the formers compose a discrete set, which simplifies considerably the implementation of a multimodal method. In the case of a simple two-dimensional scalar model, the basic ideas of such a method are presented and illustrated by numerical results.

Reconstruction of thin electromagnetic inclusions by a level-set method

In this contribution, we consider a technique of electromagnetic imaging (at a single, non-zero frequency) which uses the level-set evolution method for reconstructing a thin inclusion (possibly made of disconnected parts) with either dielectric or magnetic contrast with respect to the embedding homogeneous medium. Emphasis is on the proof of the concept, the scattering problem at hand being so far based on a two-dimensional scalar model. To do so, two level-set functions are employed; the first one describes location and shape, and the other one describes connectivity and length. Speeds of evolution of the level-set functions are calculated via the introduction of Fréchet derivatives of a least-square cost functional. Several numerical experiments on noiseless and noisy data as well illustrate how the proposed method behaves.

Langevin simulations of a model for ultrathin magnetic films

We show results from simulations of the Langevin dynamics of a two-dimensional scalar model with competing interactions for ultrathin magnetic films. We find a phase transition from a high temperature disordered phase to a low temperature phase with both positional and orientational orders. Both kinds of order emerge at the same temperature, probably due to the isotropy of the model Hamiltonian. In the low temperature phase orientational correlations show long range order while positional correlations show only quasi-long-range order in a wide temperature range. The orientational correlation length and the associated susceptibility seem to diverge with power laws at the transition. While at zero temperature the system exhibits stripe long range order, as temperature grows we observe the proliferation of different kinds of topological defects that ultimately drive the system to the disordered phase. The magnetic structures observed are similar to experimental results on ultrathin ferromagnetic films.

Wormholes and effective actions in two dimensions

We discuss the replacement of wormholes by effective interactions in a simple two-dimensional scalar model. In the region of moduli space where the wormholes are small this leads to a bilocal effective addition to the action on the complex plane. The connection with string theory is described.

Universal configurational structure in two-dimensional scalar models

Monte Carlo methods are used to explore the universal configurational structure of two-dimensional spin- 1/2, spin-1 and border- phi 4 models. Comparison of spin- 1/2 and spin-1 data provides evidence that the magnetisation distribution (effectively the Helmholtz free-energy function) and its coupling derivative (effectively the internal-energy function) constitute readily accessible signatures of a universality class. It is shown that, when allowance is made for relatively large corrections-to-scaling effects, the behaviour of the border- phi 4 model may be satisfactorily matched to that of the other two models, substantiating the view that the border model does indeed belong to the Ising universality class.

Time coarse graining near critical points

The authors investigate the statistical properties of local variables, coarse grained in time, in systems undergoing continuous phase transitions. A general theory is developed which suggests universal scaling behaviour in the limit of large coarse-graining times. The predictions are substantiated with Monte Carlo studies of two-dimensional scalar models. A simple fractal model of the time profile of the ordering variable is developed which captures and illuminates its timescale-invariant character. The theory is used to study the critical behaviour of local resonance lineshapes. In the two-dimensional case studied explicitly the critical slowing down is shown to drive a crossover to a split slow-motion lineshape.

Universal local dynamics

The authors suggest that, near a critical point, the statistical behaviour of local variables averaged over times long on microscopic scales is universal, reflecting large scale partial structure. The contention is supported with Monte Carlo studies of two-dimensional scalar models.

Nonperturbative vacuum energy density in two-dimensional scalar models

Nonperturbative vacuum energy density in two-dimensional scalar models