Heat kernel asymptotics with mixed boundary conditions

We calculate the coefficient a₅ of the heat kernel asymptotics for an operator of Laplace type with mixed boundary conditions on a general compact manifold.     

Ising model with mixed boundary conditions: Universal amplitude ratios

In the vicinity of boundaries the bulk universality class of critical phenomena splits into several boundary universality classes, depending upon whether the tendency to order in the boundary is smaller or larger than in the bulk. For Ising universality class there are five different boundary universality classes: periodic, antiperiodic, free, fixed and mixed (mixture of the last two). In this paper we present the new set of the universal amplitude ratios for the mixed boundary universality class. The results are in perfect agreement with a perturbated conformal field theory scenario proposed by Cardy [J. Cardy, Nucl. Phys. B 270 (1986) 186].     

Integral equation methods for elliptic problems with boundary conditions of mixed type

Laplace’s equation with mixed boundary conditions, that is, Dirichlet conditions on parts of the boundary and Neumann conditions on the remaining contiguous parts, is solved on an interior planar domain using an integral equation method. Rapid execution and high accuracy is obtained by combining equations which are of Fredholm’s second kind with compact operators on almost the entire boundary with a recursive compressed inverse preconditioning technique. Then an elastic problem with mixed boundary conditions is formulated and solved in an analogous manner and with similar results. This opens up for the rapid and accurate solution of several elliptic problems of mixed type.     

Two transparent boundary conditions for the electromagnetic scattering from two-dimensional overfilled cavities

We consider electromagnetic scattering from two-dimensional (2D) overfilled cavities embedded in an infinite ground plane. The unbounded computational domain is truncated to a bounded one by using a transparent boundary condition (TBC) proposed on a semi-ellipse. For overfilled rectangular cavities with homogeneous media, another TBC is introduced on the cavity apertures, which produces a smaller computational domain. The existence and uniqueness of the solutions of the variational formulations for the transverse magnetic and transverse electric polarizations are established. In the exterior domain, the 2D scattering problem is solved in the elliptic coordinate system using the Mathieu functions. In the interior domain, the problem is solved by a finite element method. Numerical experiments show the efficiency and accuracy of the new boundary conditions.     

Vibration analysis of functionally graded annular plates with mixed boundary conditions in thermal environment

The free vibration analysis of functionally graded annular plates with mixed boundary conditions in thermal environment is carried out by the 3D elasticity theory and the Chebyshev–Ritz method. The material properties are assumed to be temperature dependent and graded in the thickness direction. The mixed boundary conditions which include upper and lower surfaces partially fixed, inner side partially fixed and outer side partially fixed are considered, respectively. The accuracy of the present approach for solving the free vibration of the plates with different boundary conditions is validated by comparing the present numerical results with the results available. The effects of the different mixed boundary conditions, the temperature rise, the material graded index and the geometrical parameters on the eigen-frequencies are studied.     

Stress and mixed boundary conditions for two-dimensional dodecagonal quasi-crystal plates

For plate bending and stretching problems in two-dimensional (2D) dodecagonal quasi-crystal (QC) media, the reciprocal theorem and the general solution for QCs are applied in a novel way to obtain the appropriate stress and mixed boundary conditions accurate to all order. The method developed by Gregory and Wan is used to generate necessary conditions which the prescribed data on the edge of the plate must satisfy in order that it should generate a decaying state within the plate; these decaying state conditions are obtained explicitly for axisymmetric bending and stretching of a circular plate when stress or mixed conditions are imposed on the plate edge. They are then used for the correct formulation of boundary conditions for the interior solution. For the stress data, our boundary conditions coincide with those obtained in conventional forms of plate theories. More importantly, appropriate boundary conditions with a set of mixed edge-data are obtained for the first time. Furthermore, the corresponding necessary conditions for transversely isotropic elastic plate are obtained directly, and their isotropic elastic counterparts are also obtained.      

Stability conditions for laser cavities with two foci

Laser cavities with two foci are important in applications such as the frequency doubling of continuous-wave dye lasers. An analysis is presented that predicts stability regions, spot sizes and positions of the foci for such cavities. Requirements for minimizing the interaction between the foci are examined, and a practical method of setting up such a cavity described.     

Performance of noise barriers with various edge shapes and acoustical conditions

The performances of barriers having different shapes and surface conditions were tested using the boundary element method in a well-controlled environment. The heights and widths of the barriers were standardized and the insertion losses for six receiver positions were averaged and compared. Figures displaying the results allow for straightforward barrier performance estimation. It was shown that absorbing and soft edges significantly improve the efficiency of the barrier, but configuration modifications provide only a slight improvement. The soft T-shaped barrier produces the highest performance. A 3 m high T-shaped barrier provides the same performance as a 10 m high plain barrier. The spectral efficiency was also investigated. The insertion loss spectra for the absorbing and the soft barriers exhibit a similar shape, but the rigid barrier differs from these two.     

Thermal boundary layer effects on the acoustical impedance of enclosures and consequences for acoustical sensing devices

Expressions are derived for the acoustical impedance of a rectangular enclosure and of a finite annular cylindrical enclosure. The derivation is valid throughout the frequency range in which all dimensions of the enclosure are much less than the wavelength. The results are valid throughout the range from adiabatic to isothermal conditions in the enclosure. The effect is equivalent to placing an additional, frequency-dependent complex impedance in parallel with the adiabatic compliance. As the thermal boundary layer grows to fill the cavity, the reactive part of the impedance varies smoothly from the adiabatic value to the isothermal value. In some microphones, this change in cavity stiffness is sufficient to modify the sensitivity. The resistive part of the additional cavity impedance varies as the inverse square root of frequency at high frequencies where the boundary layer has not grown to fill the enclosure. The thermal modification gives rise to a thermal noise whose spectral density varies asymptotically as l/f(3/2) above the isothermal transition frequency.     

An exact analytical approach to the free vibration analysis of rectangular plates with mixed boundary conditions

A comprehensive analytical technique is developed for the free vibration analysis of rectangular plates with discontinuities along the boundaries. For illustrative purposes a solution is obtained for plates with edges partially clamped and partially simply supported and plates with edges partially and partially simply supported. A vast array of first mode eigenvalues is provided for these families of plates. Solutions to the equations are obtained by exploiting a mathematical technique described by the author during an earlier publication. It is shown that eigenvalue matrices are easily generated for a wide range of plates with discontinuities in boundary conditions.     

Eigenmodes of index-modulated layers with lateral PMLs

Maxwell equations are solved in a layer comprising a finite number of homogeneous isotropic dielectric regions ended by anisotropic perfectly matched layers (PMLs). The boundary-value problem is solved and the dispersion relation inside the PML is derived. The general expression of the eigenvalues equation for an arbitrary number of regions in each layer is obtained, and both polarization modes are considered. The modal functions of a single layer ended by PMLs are found, and their orthogonality relation is derived. The present method is useful to simulate scattering problems from dielectric objects as well as propagation in planar slab waveguides. Its potential to deal with more complex problems such as the scattering from an object with arbitrary cross section in open space using the multilayer modal method is briefly discussed.     

Effect of occupancy on acoustical conditions in university classrooms

Acoustical measurements were made in 12 university classrooms with and without occupants at Kangwon National University (KNU), Korea to investigate how the occupants influence the acoustical conditions of the classrooms. The mean sound absorption per occupant was calculated from the measured T₃₀ values and compared in two different groups of classrooms (6 reflective and 6 absorptive classrooms). At 250 and 500 Hz, the mean sound absorption per occupant was nearly the same for both groups of classrooms, but not in the other octave bands. The results showed that the effect of the added absorption of occupants is dependent on the acoustical conditions of the classroom. The changes in acoustical parameter values, due to added occupants in the classrooms, tended to be largest for the more reflective classrooms. The occupants may contribute to achieving more ideal reverberation times for speech (typically 0.4–0.7 s in classrooms) in the more reflective classrooms, but not in the more absorptive classrooms. A simple process is described to predict the expected effects of adding occupants to other classrooms based on the total sound absorption of unoccupied rooms.     

Ramond-Neveu-Schwarz string with new boundary conditions

For the Ramond-Neveu-Schwarz string in D space-time dimensions we seek boundary conditions which preserve Poincaré invariance in d dimensions, d<D. We obtain twisted closed and twisted open strings preserving Gervais-Sakita supersymmetry. Covariant BRST quantization yields D=10. For some boundary conditions, partition functions exhibit space-time supersymmetry.     

Six-vertex model with rotated boundary conditions

We investigate the six-vertex model on a square lattice rotated through an arbitrary angle with respect to the coordinate axes, a model recently introduced by Litvin and Priezzhev. Auxiliary vertices are used to define an inhomogeneous system which leads to a one-parameter family of commuting transfer matrices. A product of commuting transfer matrices can be interpreted as a transfer matrix acting on zigzag walls in the rotated system. Using an equation for commuting transfer matrices, we calculate their eigenvalues. Finite-size properties of the model are discussed from the viewpoint of the conformal field theory.