High-order numerical methods for 2D parabolic problems in single and composite domains

In this work, we discuss and compare three methods for the numerical approximation of constant- and variable-coefficient diffusion equations in both single and composite domains with possible discontinuity in the solution/flux at interfaces, considering (i) the Cut Finite Element Method; (ii) the Difference Potentials Method; and (iii) the summation-by-parts Finite Difference Method. First we give a brief introduction for each of the three methods. Next, we propose benchmark problems, and consider numerical tests-with respect to accuracy and convergence-for linear parabolic problems on a single domain, and continue with similar tests for linear parabolic problems on a composite domain (with the interface defined either explicitly or implicitly). Lastly, a comparative discussion of the methods and numerical results will be given.Comment: 45 pages, 12 figures, in revision for Journal of Scientific Computin

WAVE PROPAGATION IN ELASTIC MEDIA WITH INTERNAL STRUCTURE. PERIODIC TRANSFORMATIONS AND CURVED BEAMS

In this thesis we will focus on the linear elastodynamic properties of complex materials. In Chapter 2, we review the linear theory of elasticity. This chapter provides a brief overview of the basic laws of elasticity theory for di_erent coordinate systems, that will facilitate the further development of our research. Dispersion properties of 2D periodic systems for di_erent lattices are reported in Chapter 3. The governing equations of out of plane wave and examples of coordinate transformations in Cartesian and cylindrical are considered in Chapter 4. Also the scattering problems by square and cylindrical uncloaked and cloaked holes are investigated analytically and numerically. In Chapter 5 a periodic transformation approach has been applied to the problem of out of plane shear wave propagation in an isotropic linear elastic material. The Chapter is organized as follows. In Section 5.1 we present initial and transformed equations of motion and corresponding boundary conditions, describing the periodic locally radial geometric transformation. In Section 5.2 we report the comparative analysis of dispersion properties and briey describe the applied multipole expansion method. In particular, we focus our attention on classical, overlapping and unfolding transformations by also performing a low-frequency, long wavelength homogenisation. In Section 5.3 we show several application including a transmission problems in a continuum and in a waveguide, the detection of defect modes and the design of the transformation for the existence of Dirac points. In Chapter 6, the mathematical model of a curved beam that is connected to two semi-in_nite straight beams is developed. Dispersion properties of curved beams are derived, characterized by three di_erent propagating regimes. By implementing the Transfer matrix approach, the reection and transmission coe_cients that depend on the curvature, frequency and total angle of the curved beam are determined. By analysing the e_ect of the curvature, frequency and total angle on energy ux, separation between high frequency/low curvature regime, where the incident wave is practically totally transmitted, and low frequency/high curvature regime where, in addition to reection there is a strong coupling between longitudinal and exural waves, are de_ned. Finally, general conclusions are given in the last chapter

Surface-plasmon resonances of arbitrarily shaped nanometallic structures in the small-screening-length limit

According to the hydrodynamic Drude model, surface-plasmon resonances of metallic nanostructures blueshift owing to the nonlocal response of the metal's electron gas. The screening length characterising the nonlocal effect is often small relative to the overall dimensions of the metallic structure, which enables us to derive a coarse-grained nonlocal description using matched asymptotic expansions; a perturbation theory for the blueshifts of arbitrary shaped nanometallic structures is then developed. The effect of nonlocality is not always a perturbation and we present a detailed analysis of the "bonding" modes of a dimer of nearly touching nanowires where the leading-order eigenfrequencies and eigenmode distributions are shown to be a renormalisation of those predicted assuming a local metal permittivity

Ultrafast Analog Fourier Transform Using 2-D LC Lattice

We describe how a 2-D rectangular lattice of inductors and capacitors can serve as an analog Fourier transform device, generating an approximate discrete Fourier transform (DFT) of an arbitrary input vector of fixed length. The lattice displays diffractive and refractive effects and mimics the combined optical effects of a thin-slit aperture and lens. Diffraction theories in optics are usually derived for 3-D media, whereas our derivations proceed in 2-D. Analytical and numerical results show agreement between lattice output and the true DFT. Potentially, this lattice can be used for an extremely low latency and high throughput analog signal processing device. The lattice can be fabricated on-chip with frequency of operation of more than 10 GHz