Power Series Expansion Technique Research Articles

A boundary point interpolation method for acoustic problems with particular emphasis on the calculation of Cauchy principal value integrals

A boundary point interpolation method (BPIM) is presented in this paper for numerical solving acoustic problems. The BPIM is a boundary-type meshless method that combines the point interpolation technique for constructing approximate functions with boundary integral equations for reformulating boundary value problems. Since effective numerical integration techniques are crucial in the BPIM and boundary integral equation methods for calculating regular and singular integrals, two Clenshaw-Curtis integration techniques (CCITs) for regular and weakly singular integrals are derived in a unified manner with the same integration points on quadratic integration cells. In addition, a unilateral CCIT, a power series expansion technique, and an improved CCIT are proposed to effectively calculate strongly singular integrals or Cauchy principal value integrals in the BPIM. These integration techniques provide direct and efficient formulas for calculating boundary integrals on quadratic integration cells, and can be directly applied to the calculation of regular and singular integrals in the boundary element method and other meshless boundary integral equation methods. Explicit expressions of integration weights in all CCITs are derived, and the values of integration points and weights are listed. Numerical results are finally provided to validate the effectiveness of the present meshless method and integration techniques.

New Bounds for the Sine Function and Tangent Function

Using the power series expansion technique, this paper established two new inequalities for the sine function and tangent function bounded by the functions x2sin(λx)/(λx)α and x2tan(μx)/(μx)β. These results are better than the ones in the previous literature.

Existence of series solutions for certain nonlinear systems of time fractional partial differential equations

This paper is concerned with the investigation of series solutions of some time fractional nonlinear systems of partial differential equations. The symmetries and reductions of the discussed time fractional nonlinear systems are presented in a recent work (J. Math. Phys. 57 (2016) 101504). But, their explicit solutions are investigated in the present study by using power series expansion technique. Moreover, the reported solutions are interpreted graphically to highlight the importance of present study.

Symmetries, Explicit Solutions and Conservation Laws for Some Time Space Fractional Nonlinear Systems

In this work, two space-time fractional nonlinear systems named Drinfeld–Sokolov–Satsuma–Hirota system and Broer Kaup system are considered with fractional derivatives in Riemann–Liouville type. The symmetry approach and power series expansion technique are applied to derive the explicit solutions of both the systems. In addition, the nontrivial conserved vectors are reported using the nonlinear self-adjointness method and Noether operators.

Analysis of the influence of boundary pressure and friction on determining fracture toughness of shale using cracked Brazilian disc test

As the exact distributions of the pressure and friction on the disc-jaw interface are not known in the cracked Brazilian disc test, we analyze their influence on determining mode I fracture toughness. The analytic solution of stress intensity factor is derived for the arbitrarily distributed pressure and friction by using the power series expansion technique and weight function method. This investigation shows that such influence should be considered for the case of large contact angle and long crack. The mode I fracture toughness of shale is determined indirectly from the test.

Influence of pressure distribution and friction on determining mechanical properties in the Brazilian test: Theory and experiment

The improved Brazilian test is a popular method for indirectly measuring rock mechanical parameters, in which the current assumption of a uniform contact pressure distribution on the disc-jaw interface would result in measurement deviations because the real distribution is non-uniform. This investigation examines the influence of pressure distribution and friction on the determination of mechanical properties such as tensile strength σt and Young's modulus E by both theoretical analyses and experiments. For a Brazilian disc under arbitrary distributions of normal and tangential loads, the power series expansion technique is used to find the analytical solutions of the full-field displacements and stresses. The related formulae for calculating the tensile strength σt and Young's modulus E are derived. Eight load distributions with identical resultant forces are considered. The results show that the series solution converges in the form of a negative power law. Based on the Griffith failure criterion, the minimum contact angles that make the crack initiate at the center of the disc are given for different load distributions. The analysis indicates that these distributions have a significant influence on Young's modulus E and, if the contact angle is smaller, on tensile strength σt. The deviation caused by ignoring friction is also discussed. The Brazilian test is implemented for shale disc samples. The tensile strength σt and Young's modulus E of shale are determined indirectly using the theoretical formulae discussed here and experimental data.

One-dimensional dynamic equations of a piezoelectric semiconductor beam with a rectangular cross section and their application in static and dynamic characteristic analysis

Within the framework of continuum mechanics, the double power series expansion technique is proposed, and a series of reduced one-dimensional (1D) equations for a piezoelectric semiconductor beam are obtained. These derived equations are universal, in which extension, flexure, and shear deformations are all included, and can be degenerated to a number of special cases, e.g., extensional motion, coupled extensional and flexural motion with shear deformations, and elementary flexural motion without shear deformations. As a typical application, the extensional motion of a ZnO beam is analyzed sequentially. It is revealed that semi-conduction has a great effect on the performance of the piezoelectric semiconductor beam, including static deformations and dynamic behaviors. A larger initial carrier density will evidently lead to a lower resonant frequency and a smaller displacement response, which is a little similar to the dissipative effect. Both the derived approximate equations and the corresponding qualitative analysis are general and widely applicable, which can clearly interpret the inner physical mechanism of the semiconductor in the piezoelectrics and provide theoretical guidance for further experimental design.

The investigation of trapped thickness shear modes in a contoured AT-cut quartz plate using the power series expansion technique

The dynamic model about the anti-plane vibration of a contoured quartz plate with thickness changing continuously is established by ignoring the effect of small elastic constant c56. The governing equation is solved using the power series expansion technique, and the trapped thickness shear modes caused by bulge thickness are revealed. Theoretically, the proposed method is more general, which can be capable of handling various thickness profiles defined mathematically. After the convergence of the series is demonstrated and the correctness is numerically validated with the aid of finite element method results, systematic parametric studies are subsequently carried out to quantify the effects of the geometry parameter upon the trapped modes, including resonant frequency and mode shape. After that, the band structures of thickness shear waves propagation in a periodically contoured quartz plate, as well as the power transmission spectra, are obtained based on the power series expansion technique. It is revealed that broad stop bands below cut-off frequency exist owing to the trapped modes excited by the geometry inhomogeneity, which has little relationship with the structural periodicity, and its physical mechanism is different from the Bragg scattering effect. The outcome is widely applicable, and can be utilized to provide theoretical and practical guidance for the design and manufacturing of quartz resonators and wave filters.

Propagation of thickness shear waves in a periodically corrugated quartz crystal plate and its application exploration in acoustic wave filters

The propagation of thickness shear waves in a periodically corrugated quartz crystal plate is investigated in the present paper using a power series expansion technique. In the proposed simulation model, an equivalent continuity of shear stress moment is introduced as an approximation to handle sectional interfaces with abrupt thickness changes. The Bloch theory is applied to simulate the band structures for three different thickness variation patterns. It is shown that the power series expansion method exhibits good convergence and accuracy, in agreement with results by finite element method (FEM). A broad stop band can be obtained in the power transmission spectra owing to the trapped thickness shear modes excited by the thickness variation, whose physical mechanism is totally different from the well-known Bragg scattering effect and is insensitive to the structural periodicity. Based on the observed energy trapping phenomenon, an acoustic wave filter is proposed in a quartz plate with sectional decreasing thickness, which inhibits wave propagation in different regions.

A systematic approach to derive dynamic equations for homogeneous and functionally graded micropolar plates

This work considers a systematic derivation process to obtain hierarchies of dynamical equations for micropolar plates being either homogeneous or with a functionally graded (FG) material variation over the thickness. Based on the three dimensional micropolar continuum theory, a power series expansion technique of the displacement and micro-rotation fields in the thickness coordinate of the plate is adopted. The construction of the sets of plate equations is systematized by the introduction of recursion relations which relates higher order powers of displacement and micro-rotation terms with the lower order terms. This results in variationally consistent partial differential plate equations of motion and pertinent boundary conditions. Such plate equations can be constructed in a systematic fashion to any desired truncation order, where each equation order is hyperbolic and asymptotically correct. The resulting lowest order flexural plate equation is seen to be of a generalized Mindlin type. The numerical results illustrate that the present approach may render accurate solutions of benchmark type for both homogeneous and functionally graded micropolar plates provided higher order truncations are used. Moreover, low order truncations render new sets of plate equations that can act as engineering plate equations, e.g. of a generalized Mindlin type.

A three-dimensional solution for laminated orthotropic rectangular plates with viscoelastic interfaces

When a body consists completely or even partly of viscoelastic materials, its response under static loading will be time-dependent. The adhesives used to glue together single plies in laminates usually exhibit a certain viscoelastic characteristic in a high temperature environment. In this paper, a laminated orthotropic rectangular plate with viscoelastic interfaces, described by the Kelvin-Voigt model, is considered. A power series expansion technique is adopted to approximate the time-variation of various field quantities. Results indicate that the response of the laminated plate with viscoelastic interfaces changes remarkably with time, and is much different from that of a plate with spring-like or viscous interfaces.

On time-dependent behavior of cross-ply laminated strips with viscoelastic interfaces

The two-dimensional problem of a simply supported laminated orthotropic strip with viscoelastic interfaces under static loading is studied. State-space formulations are developed based on the exact elasticity equations governing orthotropic media and the Kelvin–Voigt constitutive relation of interfaces. Since the response of the strip is time-dependent, the power series expansion technique is adopted to model the variations of elastic fields with time. Results show that the response of the laminated strip with viscoelastic interfaces changes remarkably with time, which is also significantly different from that of a plate with perfect interfaces or with viscous interfaces. Note that from the present analysis, the response for a laminated plate with spring-like interfaces or with viscous interfaces can be easily obtained because they are just two particular cases of the present Kelvin–Voigt model.

The behavior of angle-ply laminated cylindrical shells with viscoelastic interfaces in cylindrical bending

The behavior of a simply supported angle-ply laminated cylindrical shell in cylindrical bending with viscoelastic interfaces is studied. The Kelvin–Voigt model is adopted to represent the character of interfaces. State-space formulations are developed based on the exact elasticity equations, and in particular, a variable substitution technique is used to derive the state equations with constant coefficients. Since the behavior of this structure under static loading is time-dependent, the power series expansion technique is used to approximate the variations of physical variables with time. The response for a laminated shell with viscous interfaces is also investigated as a particular case. Results show that the displacements as well as the maximum stresses in the panel increase rapidly with time. Thus, the imperfect bonding properties should be considered carefully in the design of laminated structures.

Mutual Coupling Mechanisms Within Arrays of Nonlinear Antennas

In this paper, mathematical analyses and physical interpretations for the mutual coupling mechanisms within a finite nonlinearly loaded antenna array are presented. These analyses are based on the method of nonlinear currents with the array mutual coupling effects treated by a power series expansion technique. Initially, the antenna input terminals in an antenna array are related by an equivalent multiport microwave circuit. Each port of this equivalent circuit network is then treated by the method of nonlinear currents. The antenna array mutual coupling effect is considered in this linear network and treated by a power series expansion technique. This study gives not only full analyses, but also physical insight into the mutual coupling mechanisms of nonlinearly loaded antenna arrays.

Response of laminated adaptive composite beams with viscoelastic interfaces

The time-dependent behavior of a simply supported laminated beam bonded with two piezoelectric layers (one is bonded onto the uppermost surface of the laminated elastic beam, another onto the lowermost surface), which may be sensor or actuator, is studied using the state-space method. The Kelvin–Voigt viscoelastic constitutive relation of interfaces is introduced to model the creep behavior of bonding adhesives. The response for a laminated beam with viscous interfaces is also investigated as a particular case of the present model. Since the behavior of this smart structure under static loading is time-dependent, the power series expansion technique is adopted to approximate the variations of variable fields with time. It is shown that the function of actuator or sensor will be counteracted partly by the effect of imperfect interfaces with time elapsing. The study should open a door for the failure analysis in health monitoring of engineering structures when bonded piezoelectric sensors and actuators are used.

Three‐dimensional wave polynomials

We demonstrate a specific power series expansion technique to solve the three‐dimensional homogeneous and inhomogeneous wave equations. As solving functions, so‐called wave polynomials are used. The presented method is useful for a finite body of certain shape. Recurrent formulas to improve efficiency are obtained for the wave polynomials and their derivatives in a Cartesian, spherical, and cylindrical coordinate system. Formulas for a particular solution of the inhomogeneous wave equation are derived. The accuracy of the method is discussed and some typical examples are shown.

Solution of the three‐dimension wave equation by using wave polynomials

AbstractThe paper demonstrates a specific power series expansion technique to solve the three‐dimensional wave equation. As solving functions, so‐called wave polynomials are used. The presented method is useful for a finite body of certain shape geometry without strength restrictions. The wave polynomials are defined. Recurrent formulas for the wave polynomials and their derivatives are obtained. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Helmholtz equation in parabolic rotational coordinates: application to wave problems in quantum mechanics and acoustics

A method for solving exactly the Helmholtz equation in a domain bounded by two paraboloids and subject to Dirichlet, Neumann, and Dirichlet-Neumann boundary conditions is presented. It is based upon the separability of the eigenfunctions and the Frobenius power series expansion technique. Two examples of interest in wave physics are considered and analyzed quasi-analytically: the energy spectrum and wave functions of an electron in a quantum dot, and the acoustic eigenfrequencies and eigenmodes of the pressure field inside an acoustic cavity. Eigensolutions are calculated and the shape dependence of the first solution is examined. Dimensional scaling is used to factor out the size dependence of the solutions.

Acoustic cavity modes in lens-shaped structures

A method for solving exactly the Helmholtz equation in parabolic rotational coordinates is presented using separability of the eigenfunctions and the Frobenius power series expansion technique. Two examples of interest in acoustics are considered and analyzed quasianalytically: The acoustic pressure in a cavity defined by two paraboloids (forming a lens-shaped structure) with (I) rigid wall boundary conditions and (II) pressure-release boundaries. The rigid-wall (pressure-release) acoustic enclosure problem is a Neumann (Dirichlet) boundary condition problem. In both cases, eigenfunctions and eigenmodes are calculated and the shape dependence of the eigenvalue for the ground state is examined.

Numerical test of approximate single-step propagators: Harmonic power series expansions versus system-specific split operator representations

This paper compares the efficiency and accuracy of different second-order approximations for the single-step propagator obtained by means of the power series expansion formalism and the split operator method. The former is based on a harmonic reference system, while the latter allows for the use of physically motivated (anharmonic) zeroth-order representations. Three typical examples---a system-bath Hamiltonian, a H\'enon-Heiles anharmonic resonating system, and a Fokker-Planck chaotic model---are considered in the present testing. The examples cover a variety of situations with strongly anharmonic coupling. Although no system-specific reference system is involved in the power series representation of the propagator, it quite accurately describes the dynamics of very anharmonic processes in the entire time domain even though the coupling is strong. Another appealing feature of the approach is that it is essentially analytical and therefore does not require any computational effort to implement. This makes the power series expansion technique particularly attractive for efficiently treating many-body problems. In contrast, numerical implementation of the split operator method can be arduous as a general multidimensional calculation, while its utility is in general restricted to the separable limit, when the coupling is almost turned off. In that limit the method indeed provides accurate results over a broad range of t. With increasing coupling, however, the efficiency of the method deteriorates very rapidly regardless of the particular choice of the zeroth-order representation.