The Generalization and Proof of “Square Root of 2 Is Not a Rational Number” on the Integral Domain

In this paper, the traditional proof of “square root of 2 is not a rational number” has been reviewed, and then the theory has been generalized to “if n is not a square, square root of n is not a rational number”. And then some conceptions of ring, integral domain, ideal, quotient ring in Advanced algebra, have been introduced. Integers can be regarded as an integral domain, the rational numbers can be regard as a fractional domain. Evens and odds are principal ideals in integral domain. The operations on evens and odds are operations on quotient ring. After introducing “the minimalist form” in fraction ring. The paper proves the main conclusion: in a integral domain, multiplicative subset S produces a fraction ring S−1R, and n is not a square element in R, then to every element a∈R, a2≠n.

Evaluation of strongly singular domain integrals for internal stresses in functionally graded materials analyses using RIBEM

An accurate evaluation of strongly singular domain integral appearing in the stress representation formula is a crucial problem in the stress analysis of functionally graded materials using boundary element method.To solve this problem,a singularity separation technique is presented in the paper to split the singular integral into regular and singular parts by subtracting and adding a singular term.The singular domain integral is transformed into a boundary integral using the radial integration method.Analytical expressions of the radial integrals are obtained for two commonly used shear moduli varying with spatial coordinates.The regular domain integral,after expressing the displacements in terms of the radial basis functions,is also transformed to the boundary using the radial integration method.Finally,a boundary element method without internal cells is established for computing the stresses at internal nodes of the functionally graded materials with varying shear modulus.

FAST SOLUTION OF TRANSIENT SCATTERING FROM ELECTRICALLY LARGE COMPLEX OBJECTS BASED ON TIME DOMAIN INTEGRAL EQUATION SOLVERS

A fast Time Domain Integral Equation(TDIE) solver is presented for analysis of transient scattering from electrically large conducting complex objects.The numerical process of Marching-On-in-Time(MOT) method based TDIE encounters high computational cost and exorbitant memory requirements.A group-style accelerated method-Plane Wave Time Domain(PWTD) algorithm,which permits rapid evaluation of transient wave field generated by temporally bandlimited sources,is employed to reduce the computational cost of MOT-based TDIE solvers.An efficient compressed storage technique for sparse matrix is adopted to decrease the enormous memory requirements of MOT.The scheme of the Multi-Level PWTD(MLPWTD)-enhanced MOT with compressed storage for sparse matrix is presented for analysis of transient scattering from electrically large complex objects in this paper.The numerical simulation results demonstrate the validity and efficiency of the presented scheme.

Time Domain Analysis of Transmit/Receive Dipole Pair Array

We introduce a new transmit/receive dipole pair array to obtain a compact quasi\|monostatic antenna structure for ground penetrating radar systems. And we analyze this transmit/receive dipole pair array in time domain. The numerical results show that if the distance between the transmit antenna and receive antenna is appropriate the array configuration is adoptable.

Time-Domain Analysis of a Wire Antenna Near Arbitrarily Shaped Conductor Bodies

A time domain electric al field integral equation (TDEFIE) is formulated for the problem of a thin wire antenna in the presence of conductor bodies, and this equation is solved by the method of time marching algorithm. The analysis is valid for any arbitrarily shaped, oriented and positioned wire antennas relative to arbitrarily shaped conductor bodies. Current at the excited point, input admittance and radiation pattern are given and agree with the results computed by the method in frequency domain.

An adaptive cell-based domain integration method for treatment of domain integrals in 3D boundary element method for potential and elasticity problems

An adaptive cell-based domain integration method（CDIM） is proposed for the treatment of domain integrals in 3D boundary element method（BEM）. The domain integrals are computed in background cells rather than volume elements. The cells are created from the boundary elements based on an adaptive oct-tree structure and no other discretization is needed. Cells containing the boundary elements are subdivided into smaller sub-cells adaptively according to the sizes and levels of the boundary elements; and the sub-cells outside the domain are deleted to obtain the desired accuracy. The method is applied in the 3D potential and elasticity problems in this paper.

Singular integral operators on product domains along twisted surfaces

We introduce a class of singular integral operators on product domains along twisted surfaces.We prove that the operators are bounded on L^(p) provided that the kernels satisfy weak conditions.

A NEW SINGULARITY EXTRACTION TECHNIQUE FOR TDEFIE

A new singularity extraction technique is presented to calculate accurately the singular integrals in Time Domain Electric Field Integral Equation (TDEFIE).In singularity extraction pro- cedure,through the aid of the first order Taylor series of time base function including time-retardation,the singularity of the integrand can be removed.The surface current density and backscattered far-field response of a conducting cube illuminated by a Gaussian plane wave is com- puted using the presented technique.Comparisons are made with the results obtained by the Inverse Discrete Fourier Transform (IDFT) of the frequency domain and the results obtained by using Ve- chinski's time averaging technique,which demonstrate that the presented method with this new time domain singularity extraction technique to solve TDEFIE is very accurate and stable.

Stability conditions of explicit integration algorithms when using 3D viscoelastic artificial boundaries

Viscoelastic artificial boundaries are widely adopted in numerical simulations of wave propagation problems.When explicit time-domain integration algorithms are used,the stability condition of the boundary domain is stricter than that of the internal region due to the influence of the damping and stiffness of an viscoelastic artificial boundary.The lack of a clear and practical stability criterion for this problem,however,affects the reasonable selection of an integral time step when using viscoelastic artificial boundaries.In this study,we investigate the stability conditions of explicit integration algorithms when using three-dimensional(3D)viscoelastic artificial boundaries through an analysis method based on a local subsystem.Several boundary subsystems that can represent localized characteristics of a complete numerical model are established,and their analytical stability conditions are derived from and further compared to one another.The stability of the complete model is controlled by the corner regions,and thus,the global stability criterion for the numerical model with viscoelastic artificial boundaries is obtained.Next,by analyzing the impact of different factors on stability conditions,we recommend a stability coefficient for practically estimating the maximum stable integral time step in the dynamic analysis when using 3D viscoelastic artificial boundaries.

On hybrid temporal basis functions for stable numerical solution of time domain boundary integral equations

Problems in unsteady aerodynamics and aeroacoustics can sometimes be formulated as integral equations,such as the boundary integral equations.Numerical discretization of integral equations in the time domain often leads to so-called March-On-in-Time(MOT)schemes.In the literature,the temporal basis functions used in MOT schemes have been largely limited to low-order shifted Lagrange basis functions.In order to evaluate the accuracy and effectiveness of the temporal basis functions,a Fourier analysis of the temporal interpolation schemes is carried out.Based on the Fourier analysis,the spectral resolutions of various temporal basis functions are quantified.It is argued that hybrid temporal basis functions be used for interpolation of the numerical solution and its derivatives with respect to time.Stability of the proposed hybrid schemes is studied by a matrix eigenvalue method.Substantial improvement in accuracy and efficiency by using the hybrid temporal basis functions for time domain integral equations is demonstrated by numerical examples.Compared with the traditional temporal basis functions,the use of hybrid basis functions keeps numerical errors low for a larger frequency range given the same time step size.Conversely,for a given range of frequency of interest,a larger time step can be used with the hybrid temporal basis functions,resulting in an increase in computational efficiency and,at the same time,a reduction in memory requirement.

A Simplified Full-Wave Analysis for Microstrip Patch Antennas

The spectral domain integral equation(SDIE) provides an accurate and efficient method for computing the resonant frequency, radiation patterns, etc . Using continuous Fourier transform, the formulation utilizes the singular integral equations via the Glerkin's method to derive the deterministic equation with fewer mathematical manipulations. In contrast, discrete Fourier transform(DFT) requires intricate mathematical labor. The present scheme requires a small size, i.e ., (2×2) matrix, and it is possible to extract higher order modal solutions conveniently. Moreover, computation is reduced with the same convergence properties. Based on the present scheme, some results for resonant frequency and radiation patterns compared with available data and computed current distribution on the patch are presented.

Microball lens integrated fiber probe for optical frequency domain imaging

An integrated microball lens fiber catheter probe is demonstrated, which has better lateral resolution and longer working distance than a corresponding bare fiber probe with diverging beam for Fourier domain optical coherence tomography （FDOCT）. Simulation results are shown to gain the effect of the distance between the mieroball lens and the bare fiber to the focusing plane and beam width. The freedom of modifying the working distance and lateral resolution is shown. This is achieved by changing the gap distance between the single-mode fiber and the microball lens within the packaged surgical needle catheter without using an additional beam expander having a fixed length. The probe successfully acquired crosssectional images of ocular tissues from an animal sample with the proposed miniaturized imaging probe.