Analysis Of Wave Equation Research Articles

Solvability and microlocal analysis of the fractional Eringen wave equation

We discuss unique existence and microlocal regularity properties of Sobolev space solutions to the fractional Eringen wave equation, initially given in the form of a system of equations, in which the classical non-local Eringen constitutive equation is generalized by employing space fractional derivatives. Numerical examples illustrate the shape of solutions as a function of the order of the space fractional derivative.

Case studies to demonstrate challenges of driven piles on rock

In Wyoming, pile foundations for bridges are often driven on rock materials because of the state’s shallow bedrock stratigraphy. Unfortunately, no static analysis methods are currently available for estimating the resistance of these driven piles. In this paper, two recently completed bridge projects (the Owl Creek and Woods Wardell sites) on steel H-piles in Wyoming are explicitly presented, and data from three past projects are included to highlight the limited knowledge and challenges pertaining to the present design and construction practices. Static analysis methods were used to estimate the geotechnical resistances of these piles. The wave equation analysis program and the case pile wave analysis program were used to verify their performances during construction. Structural capacities of these piles were also calculated. The results of the studies show that the static analysis methods and structural analyses yield inconsistent pile resistance estimations. Recommendations in terms of pile bracing and embedded pile length are proposed to predict better the resistances of piles on soft rock.

AVBVICVII ferroelectrics as novel materials for phononic crystals

ABSTRACTIn the present work the acoustic band structure of a two-dimensional (2D) phononic crystal (PC) containing a semiconducting ferroelectric - AVBVICVII (A = Sb, Bi; B = S, Se, Te; C = I, Br, and Cl) was investigated theoretically and numerically by the plane-wave-expansion (PWE) method. Two-dimensional PC with square lattices composed of semiconducting ferroelectric cylindrical rods embedded in the organic/inorganic matrix is studied to find the existence of stop bands for the waves of certain energy. This phononic bandgap - forbidden frequency range - allows sound to be controlled in many useful ways in structures that can act as sonic filters, waveguides or resonant cavities. Phononic band diagram ω = ω(k) for a 2D PC was plotted versus the wavevector k along the Г-X-M-Г path in the square Brillouin zone (BZ). The band diagram shows four stop bands in the wide frequency range. The unusual properties of matrix and ferroelectric properties of AVBVICVII give us ability to control the wave propagation through the PC in over a wide frequency range. We study the 2D composites by solving the basic acoustic wave equation and use Bloch wave analysis to identify the band gaps.

A quadrature-based superconvergent isogeometric frequency analysis with macro-integration cells and quadratic splines

A quadrature-based superconvergent isogeometric method is presented for efficient frequency analysis with macro-integration cells and quadratic splines. The present development particularly refers to the isogeometric analysis of scalar wave equations. The single element, two-element and three-element macro-integration cells are proposed to develop a set of explicit superconvergent quadrature rules with a 6th order frequency accuracy, in contrast to the 4th order frequency accuracy associated with the standard isogeometric formulation with consistent mass matrix. The 1D quadrature rules with various macro-integration cells are derived from the condition of exact integration of the higher order mass matrix as well as the stiffness matrix. Consequently, 3-point, 5-point and 7-point quadrature rules with identical precision are established for the single element, two-element and three-element macro-integration cells, respectively. By construction these quadrature rules exactly recover the isogeometric higher order mass matrix formulation with frequency superconvergence in 1D case. The 2D and 3D superconvergent quadrature rules with versatile integration cells are directly constructed through the tensor product formulation of the 1D integration algorithms. It is shown that the multidimensional isogeometric analysis employing the proposed quadrature rules for both mass and stiffness matrices does produce the frequency superconvergence simultaneously without the wave propagation direction dependence problem, which needs special treatments for the multidimensional higher order mass matrix formulation. The proposed approach is featured by its simplicity for numerical implementation and efficiency using macro-integration cells. Numerical examples confirm the efficacy of the present methodology.

Meshless analysis of two-dimensional two-sided space-fractional wave equation based on improved moving least-squares approximation

ABSTRACTThe Space-fractional wave equations (SFWE) have been found to be very adequate in describing anomalous transport and dispersion phenomena. Due to the non-local property of integro-differential operator of space-fractional derivative, it is very challenging to deal with fractional model. In this paper, a meshless analysis of two-dimensional two-sided SFWE is proposed based on the improved moving least-squares (IMLS) approximation. The trial function for the SFWE is constructed by the IMLS approximation, where the resulting algebraic equation system to obtain the shape functions is no more ill conditioned and has high computational efficiency. The Riemann–Liouville operator is discretized by the Grünwald formula. The centre difference method and the strong-forms of the SFWE are used to obtain the final fully discrete algebraic equation. And the essential boundary conditions can be directly and easily imposed on as a finite element method. Due to the adoption of IMLS approximation and strong-forms, this method will be highly accurate and efficient. Numerical results demonstrate that this method is highly accurate and computationally efficient for SFWE. Moreover, the convergence and error estimate have been analysed in our study.

Analysis of wave equation in electromagnetic field by Proca equation

This research is aimed to analyze wave equation for the electric and magnetic field, vector and scalar potential, and continuity equation using Proca equation. Then, also analyze comparison of the solution on Maxwell and Proca equation for scalar potential and electric field, both as a function of distance and constant wave number.

Low-frequency AC electric field induced thermoacoustic oscillation of a premixed stagnation flame

We present a novel phenomenon of low-frequency AC electric field induced thermoacoustic oscillations by employing a wall-jet premixed stagnation flame configuration. A high speed camera and a microphone are used to record the instantaneous flame topographies and the acoustic oscillations, respectively. We find that, under manipulations of the AC electric field, the acoustic oscillation occurs after the flame transits from a ‘disc’ mode to an ‘envelope’ mode. The observed correlation between the changing rates of the flame surface area and the pressure fluctuations in the time domain indicates a significant contribution from the unsteady heat release to the acoustic oscillation; meanwhile, the other possible mechanism, i.e. the direct impact of oscillating electric body force on the generated pressure wave, can be ruled out based on a theoretical analysis of a pressure wave equation. The oscillating behaviors of the flame front show that the electric field drives the movement of the flame cusp with a large stretch rate. Assessment of the current density distribution of the weakly-ionized flame plasma is conducted by non-dimensionalizing the convective–diffusive–reactive equation, estimating the electric conductivity, and solving Ohm's Law. This simplified model divulges that the non-uniform current density induces large perturbations at the corner positions of the flame front. The unsteady flame fronts strongly influence the current response of the flame, which indicates a self-looping process among the flame current, the electric body force and the flame reaction rate.

Second-order two-scale analysis and numerical algorithm for the damped wave equations of composite materials with quasi-periodic structures

In this paper, we perform a second-order two-scale analysis and introduce a numerical algorithm for the damped wave equations of composite materials with a quasi-periodic structure. Firstly, second-order two-scale asymptotic expansion solutions for these problems are constructed by a multiscale asymptotic analysis. In addition, we explain the importance of the second-order two-scale solutions by the error analysis in the pointwise sense. Moreover, explicit convergence rates of these second-order two-scale solutions are obtained in the integral sense. Then a second-order two-scale numerical method based on a Newmark scheme is presented to solve these multiscale problems. Finally, some numerical examples show the effectiveness and efficiency of the multiscale numerical method we proposed.

Region-specific calibration of resistance factors for use with static and wave equation analyses of driven piles

In this study, resistance factors for the wave equation analysis of piles (WEAP) and a commonly used static analysis (SA) method are calibrated for use with piles driven in the Puget Sound Lowlands...

Pile driving formulas based on pile wave equation analyses

Pile driving formulas, which directly relate the pile set resulting from a hammer blow to the static load capacity of the pile, are often used to decide whether a pile will have the required design capacity. However, existing formulas do not consider soil or pile type, and do not, in general, reliably predict pile capacity. In this paper, an advanced model for dynamic pile driving analysis was used to develop accurate pile driving formulas. The proposed driving formulas are validated through well-documented case histories. Comparisons of predictions from the proposed formulas with results from static and dynamic load tests show that they produce reasonably accurate predictions of pile capacity based on pile set observations.

Long-term analysis of semilinear wave equations with slowly varying wave speed

ABSTRACTA semilinear wave equation with slowly varying wave speed is considered in one to three space dimensions on a bounded interval, a rectangle or a box, respectively. It is shown that the action, which is the harmonic energy divided by the wave speed and multiplied with the diameter of the spatial domain, is an adiabatic invariant: it remains nearly conserved over long times, longer than any fixed power of the time scale of changes in the wave speed in the case of one space dimension, and longer than can be attained by standard perturbation arguments in the two- and three-dimensional cases. The long-time near-conservation of the action yields long-time existence of the solution. The proofs use modulated Fourier expansions in time.

Microlocal analysis of fractional wave equations

We determine the wave front sets of solutions to two special cases of the Cauchy problem for the space‐time fractional Zener wave equation, one being fractional in space, the other being fractional in time. For the case of the space fractional wave equation, we show that no spatial propagation of singularities occurs. For the time fractional Zener wave equation, we show an analogue of non‐characteristic regularity.

Symmetry computation and reduction of a wave model in $$\mathbf{2}+\mathbf{1}$$ 2 + 1 dimensions

We present the iterative classical point symmetry analysis of a shallow water wave equation in $$2+1$$ dimensions and that of its corresponding nonisospectral, two-component Lax pair. A few reductions arise and are identified with celebrate equations in the Physics and Mathematics literature of nonlinear waves. We pay particular attention to the isospectral or nonisospectral nature of the reduced spectral problems.

Efficient 2-D Finite-Element Solution of Vector Wave Equation in a Class of Curved Polygons

We propose an efficient finite-element analysis of the vector wave equation in a class of relatively general curved polygons. The proposed method is suitable for an accurate and efficient calculation of the propagation constants of waveguides filled with pieces of homogeneous materials. To apply the method, we first decompose the 2-D problem domain into a set of curved polygons of a specific characteristic. Then, we divide every polygon into a set of triangular elements with two straight edges. Finally, we introduce a set of hierarchical mixed-order curl-conforming vector basis functions inside every triangular element to discretize the vector wave equation. The advantages of the method are as follows. The curved boundaries of the elements are modeled exactly, and hence, there is no approximation in the geometrical modeling. The 2-D integrals of the matrix elements are reduced to 1-D integrals. Therefore, the matrix filling can be performed very fast. The total number of elements due to the discretization of a given domain is very small, and hence, the discretization of the problem domain takes up a very small percentage of the total computational time. We validate the method by comparing the results with those of the Ansoft High Frequency Structure Simulator simulator and investigate the accuracy and efficiency of the method through a numerical example.

The Numerical Analysis of Two-Sided Space-Fractional Wave Equation with Improved Moving Least-Square Ritz Method

A numerical analysis of the space-fractional wave equation is carried out by the improved moving least-square Ritz (IMLS-Ritz) method. The trial functions for the space-fractional wave equation are constructed by the IMLS approximation. By the Galerkin weak form, the energy functional is formulated. Employing the Ritz minimization procedure, the final algebraic equations system is obtained. In this numerical analysis, the applicability and efficiency of the IMLS-Ritz method are examined by some example problems. Comparing the numerical results with the analytical solutions, the stability and accuracy of the IMLS-Ritz method are also presented.

Analysis and computation of the elastic wave equation with random coefficients

We consider the stochastic initial–boundary value problem for the elastic wave equation with random coefficients and deterministic data. We propose a stochastic collocation method for computing statistical moments of the solution or statistics of some given quantities of interest. We study the convergence rate of the error in the stochastic collocation method. In particular, we show that, the rate of convergence depends on the regularity of the solution or the quantity of interest in the stochastic space, which is in turn related to the regularity of the deterministic data in the physical space and the type of the quantity of interest. We demonstrate that a fast rate of convergence is possible in two cases: for the elastic wave solutions with high regular data; and for some high regular quantities of interest even in the presence of low regular data. We perform numerical examples, including a simplified earthquake, which confirm the analysis and show that the collocation method is a valid alternative to the more traditional Monte Carlo sampling method for approximating quantities with high stochastic regularity.

An application of wave equation analysis program to pile dynamic formulae

Wave equation analysis programs (WEAP) such as GRLWEAP and TNOWave were primarily developed for pre-driving analysis. They can also be used for post-driving measurement applications with some refinements. In the case of pre-driving analysis, the programs are used for the purpose of selecting the right equipment for a given ground condition and controlling stresses during pile driving processes. Recently, the program is increasingly used for the post-driving measurement application, where an assessment based on a variety of input parameters such as hammer, driving system and dynamic behaviour of soil is carried out. The process of this type of analysis is quite simple and it is performed by matching accurately known parameters, such as from CAPWAP analysis, to the parameters used in GRLWEAP analysis. The parameters that are refined in the typical analysis are pile stresses, hammer energy, capacity, damping and quakes. Matching of these known quantities by adjusting hammer, cushion and soil parameters in the wave equation program results in blow counts or sets and stresses for other hammer energies and capacities and cushion configuration. The result of this analysis is output on a Bearing Graph that establishes a relationship between ultimate capacity and net set per blow. A further application of this refinement method can be applied to the assessment of dynamic formulae, which are extensively used in pile capacity calculation during pile driving process. In this paper, WEAP analysis is carried out to establish the relationship between the ultimate capacities and sets using the various parameters and using this relationship to recalibrate the dynamic formula. The results of this analysis presented show that some of the shortcoming of the dynamic formula can be overcome and the results can be improved by the introduction of a correction factor.

Erratum: Wave equation analyses of fiber reinforced polymer composite piling (Journal of Composites for Construction (May 2002) 6:2 (88-96))

Erratum: Wave equation analyses of fiber reinforced polymer composite piling (Journal of Composites for Construction (May 2002) 6:2 (88-96))

GFEM for modal analysis of 2D wave equation

Purpose – The purpose of this paper is to present an application of the Generalized Finite Element Method (GFEM) for modal analysis of 2D wave equation. Design/methodology/approach – The GFEM can be viewed as an extension of the standard Finite Element Method (FEM) that allows non-polynomial enrichment of the approximation space. In this paper the authors enrich the approximation space with sine e cosine functions, since these functions frequently appear in the analytical solution of the problem under study. The results are compared with the ones obtained with the polynomial FEM using higher order elements. Findings – The results indicate that the proposed approach is able to obtain more accurate results for higher vibration modes than standard polynomial FEM. Originality/value – The examples studied in this paper indicate a strong potential of the GFEM for the approximation of higher vibration modes of structures, analysis of structures subject to high frequency excitations and other problems that concern high frequency oscillatory phenomena.

A new approach for time effect analysis in the settlement of single pile in nonlinear viscoelastic soil deposits

A new and simple approach is presented to analyze the time effect in the settlement of single pile and the distributions of pile shaft resistance and pile axial force. First, the viscosity of soil is considered by using a linear damper, and the nonlinear elasticity of pile lateral soil and pile end soil are simulated by using a hyperbola model and idealized elastoplastic model, respectively. Then, the settlement of the pile head, shaft resistance, and axial force along the pile are derived by virtue of a wave equation analysis program based on traveling wave decomposition. Based on the solutions, a parametric study has been undertaken to investigate the influences of the parameters of a pile-soil system on the settlement behavior of a single pile. Finally, the calculated results are compared with the measured results to demonstrate the effectiveness and accuracy of the proposed approach. Note that the presented solution allows for a good prediction of the settlement behavior of a single pile and can provide a reference for the preliminary design of a pile foundation.