Wave Function Expansion Research Articles

Dynamic analysis of piezoelectric materials with an elliptical hole under the action of shear horizontal waves

The dynamic mechanical properties of piezoelectric materials are of interest because of their brittle nature. Previous studies have mainly focused on cylindrical cavities or cracks. However, these are specific forms of elliptical cylindrical cavities. This study aims to develop a semi-analytical approach for analyzing the anti-plane dynamics of piezoelectric materials containing elliptical cavities by exploiting the geometrical generality of ellipses. The Helmholtz and Laplace equations were identified as governing equations by decoupling the displacement and electric potential functions. The wave function expansion method was employed to express the displacement and electric potential functions in the form of a Mathieu function series containing unknown coefficients in the elliptical coordinate system. The unknown coefficients were determined using the traction-free boundary and electric displacement continuous condition. Although the method of intercepting a finite-term series was used, the solutions are reliable because the wave function exhibits convergence. In addition, a parametric study was conducted. The results demonstrated the importance of using large axial ratios, small wave numbers, and large piezoelectric characteristic parameters in the dynamic analysis of piezoelectric materials with elliptical defects.

Electroelastic Coupled-Wave Scattering and Dynamic Stress Concentration of Triangular Defect Piezoceramics

In this paper, a method to calculate the dynamic stress concentration around the triangular defect of piezoelectric material under electroelastic coupling is studied and applied to the promising barium calcium zirconate titanate. Firstly, the electroelastic governing equation is decomposed by decoupling technique, and the analytical solutions of elastic wave field and electric field are obtained by wave function expansion method. Then, the conformal transformation is used to simplify the triangle boundary into a circular boundary, and the corresponding modal coefficients are determined according to the simplified boundary conditions. Finally, the analytical solution of the dynamic stress concentration factor can be obtained according to the constitutive equation. Substitute the relevant material parameters of (Ba0.85Ca0.15)(Zr0.1Ti0.9)O3 and set different temperatures, Ce doping amount, and incident wave number for numerical simulation. The numerical results show that the incident wave number, piezoelectric properties, and the shape parameters and deflection angle of the triangular defect have a great influence on the dynamic stress around the defect, and some meaningful laws are summarized through analysis.

Quantum flux densities for electronic-nuclear motion: exact versus Born-Oppenheimer dynamics.

We study the coupled electronic-nuclear dynamics in a model system to compare numerically exact calculations of electronic and nuclear flux densities with those obtained from the Born-Oppenheimer (BO) approximation. Within the adiabatic expansion of the total wave function, we identify the terms which contribute to the flux densities. It is found that only off-diagonal elements that involve the interaction between different electronic states contribute to the electronic flux whereas in the nuclear case the major contribution belongs to the BO electronic state. New flux densities are introduced where in both, the electronic and the nuclear case, the main contribution is contained in the component corresponding to the BO state. As a consequence, they can be determined within the BO approximation, and an excellent agreement with the exact results is found. This article is part of the theme issue 'Chemistry without the Born-Oppenheimer approximation'.

Transient responses of deep-buried unlined tunnels subjected to blasting P wave

Dynamic perturbation inevitably induces rock mass damage in deep engineering, especially for high-level in situ stress. The study presented in this paper focuses on the theoretical and numerical responses of deep twin circular tunnels subjected to transient disturbance. Based on wave function expansion method, Graf’s addition theorem, integral transform and its inversion, dynamic responses along the tunnel peripheries are analyzed theoretically. Moreover, with resort to a self-developed code, CASRock, the elasto-plastic responses triggered by coupled static geo-stress and blasting disturbance are investigated numerically. The analytical results indicate that the positions of maximum circumferential stress concentration are approximately perpendicular to the disturbance direction; the most remarkable radial vibrations occur at the incident sidewall and the peak hoop velocities are near the incident side. The numerical results demonstrate that the in situ stress field and characteristics of disturbance load affect the final failure region and damage mechanism around the tunnels. The theoretical and numerical results can provide guideline for the support and design of deep-buried tunnels.

Feasibility considerations for free-fall tests of gravitational decoherence

Space offers exciting opportunities to test the foundations of quantum physics using macroscopic quantum superpositions. It has been proposed to perform such tests in a dedicated space mission (MAQRO) using matter-wave interferometry with massive test particles or monitoring how the wave function of a test particle expands over time. Such experiments could test quantum physics with sufficiently high precision to resolve potential deviations from its unitary evolution due to gravitational decoherence. For example, such deviations have been predicted by the Diósi–Penrose (DP) model and the Károlyházy (K) model. The former predicts the collapse of massive or large superpositions due to a nonlinear modification of quantum evolution. The latter predicts decoherence because of an underlying uncertainty of space time. Potential advantages of a space environment are (1) long free-fall times, (2) low noise, and (3) taking a high number of data points over several years in a dedicated space mission. In contrast to interferometric tests, monitoring wave function expansion is less complex, but it does face some practical limitations. Here, we will discuss limitations of such non-interferometric experiments due to the limited number of data points achievable during a mission lifetime. Our results show that it will require an interferometric approach to conclusively test for gravitational decoherence as predicted by the DP or K models. In honor of the Nobel prize of Sir Roger Penrose, we will focus our discussion on the Diósi–Penrose model.

Stress and displacement fields around an arbitrary shape tunnel surrounded by a multilayered elastic medium subjected to harmonic waves under plane strain conditions

This study focuses on the propagation of harmonic mechanical waves around both circular and non-circular tunnels. The excavation disturbed zone is explicitly modeled with a multilayered description of the surroundings of the tunnel. The purpose of this paper is thus to establish closed-form solutions for stress and displacement fields for circular and non-circular geometries taking into account the mechanical heterogeneity of the area adjacent to the tunnel. Particularly, the multilayered non-circular shape configuration is a great step not treated previously. The classical approach based on the wave function expansion method is used to determine the expressions of stresses and displacements around circular geometries. To handle non-circular tunnel shapes, the complex variable method based on the Muskhelishvili approach and conformal mapping functions is introduced. Results indicate that the excavation damaged zone strongly modifies the distribution of stress and displacement fields around the tunnel. Geometrical tunnel shapes also affect stress and displacement polar diagrams.

Scattering of SH waves by an isosceles flat-topped hill with multistage slopes

This paper presents a flexible mechanical model, which can be used to study the dynamic response of irregular surfaces with complex shapes under the action of seismic waves. Take flat-topped hill as an example, an isosceles model with complex boundaries is established, and an appropriate multi-region-matching technique (MRMT) is proposed to simplify the complex boundary value problem and solve it analytically. Based on the wave function expansion method and the complex function method, the wave field expressions of each subregion, including the singular shape subregion, are constructed. By introducing multipolar coordinate transformation and combining the continuous conditions at three auxiliary boundaries, the infinite algebraic equations are established. To solve the unknown coefficients in the wave field expressions, the Fourier series expansion method in the complex domain is adopted. Finally, some typical numerical examples are calculated to analyze the influence of the shape parameters of the flat-topped hill, the incident angle and the wave number of seismic waves on surface displacement amplitudes.

Dynamic analysis of non‐confocal elliptical lined tunnel under the action of SH wave

The scattering characteristics of non‐confocal elliptical lined tunnels under SH‐wave disturbance are investigated. The problem is essentially a boundary value problem and its theoretical treatment is based on the wave function expansion method, which is an exact treatment of the problem. Multi‐elliptic co‐ordinate systems are employed and wave function translations between multi‐elliptic co‐ordinate systems are performed using the extra‐domain addition theorem of the Mathieu function. The method can be used to fit and handle scattering problems with a wide range of geometries by varying the minor‐and‐major axis ratio, lining thickness and relative position of the tunnel. Calculations have been performed on the truncated infinite linear equation established by considering the boundary conditions. Based on the calculation results a parametric study is carried out to analyze the effect of the dimensionless eccentricity distance, the minor‐and‐major axis ratio and the dimensionless wave number on the dynamic stresses around the tunnel. Two typical application examples of concrete lining in granite and steel lining in granite are presented.

Scattering of Magnetoacoustic Waves and Dynamic Stress Concentration around Double Openings in Piezomagnetic Composites

Based on the magnetoacoustic coupled dynamics theory, the wave function expansion method is used to solve the problem of acoustic wave scattering and dynamic stress concentration around the two openings in e-type piezomagnetic composites. To deal with the multiple scattering between openings, the local coordinate method is introduced. The general analytical solution to the problem and the expression of the dynamic stress concentration are derived. As an example, the numerical results of the dynamic stress distribution around two openings with equal diameters are given. The effects of the parameters, such as the incident wave number and the spacing between the openings, on the dynamic stress concentration factor are analyzed.

A Series Solution for 2D Scattering of Cylindrical SH-Waves by Surrounding Loose Rock Zone of Underground Tunnel Lining

This paper presents a closed-form series solution of cylindrical SH-wave scattering by the surrounding loose rock zone of underground tunnel lining in a uniform half-space based on the wave function expansion method and the mirror image method. The correctness of the series solution is verified through residual convergence and comparison with the published results. The influence of the frequency of the incident cylindrical SH-wave, the distance between the wave source and the lining, the lining buried depth, and the properties of the surrounding loose rock zone on the dynamic stress concentration of the tunnel lining is investigated. The results show that the incident wave with high frequency always makes the dynamic stress concentration of the tunnel lining obvious. With the increase of the distance between the wave source and the tunnel lining, the stress around the tunnel lining decreases, but the dynamic stress concentration factor around the tunnel lining does not decrease significantly but occasionally increases. The ground surface has a great influence on the stress concentration of the tunnel lining. The amplitude of the stress concentration factor of tunnel lining is highly related to the shear wave velocity of the surrounding loose rock zone. When the property of the surrounding rock (shear wave velocity) changes more, the amplitude of the stress concentration factor is larger, that is, the stress concentration is more significant.

Dynamic stress responses of defects in periodic heterogeneous media under SH waves

Due to the complexity of heterogeneous medium models, the corresponding research methods still have great limitations, and hence, there is a dearth of research findings. To study the problem of dynamic stress concentration in heterogeneous media, the scattering of SH waves by circular holes in a full-space periodic heterogeneous continuum medium is investigated based on the propagation characteristics of elastic waves in complex heterogeneous media. The wave equation of SH waves propagating in heterogeneous media with the elastic shear modulus and density varying in the form of trigonometric functions is solved by using the complex function method, the wave function expansion method, and the auxiliary function method. On this basis, the wave field and stress field are derived, and the expression of the dynamic stress concentration factor (DSCF) on the boundary of a circular hole is given. The effects of the reference wavenumber and the variation period of the heterogeneous medium on the distribution of the DSCF on the boundary of a circular hole are analyzed through numerical examples. The results show that when the circular hole defect is within one heterogeneous period of the medium, shortening of the heterogeneous period of the medium reduces the peak of the DSCF due to high-frequency waves on the boundary of the circular hole.

An Analytical Solution for 2D Dynamic Structure-Soil-Structure Interaction for Twin Flexible Tunnels Embedded in a Homogeneous Half-Space

The interaction between subway tunnels is investigated by using a 2D analytic model of a twin tunnels system embedded in a homogenous half-space. The closed-form analytical solution for tunnel displacement response is derived through the wave function expansion method and the mirror method, and the correctness of the solution is verified through residuals convergence and comparison with the published results. The analysis focuses on the effects of tunnel relative stiffness on tunnel–soil–tunnel interaction. Tunnel relative stiffness has a great influence on tunnel displacement response. For small tunnel relative stiffness, tunnel displacement amplitude can be enlarged by 3.3 times that of single rigid tunnel model. The response of the tunnel–soil–tunnel interaction system depends not only on the distances between tunnels but also on the frequency of the incident wave and the incident angle. The strength of the interaction between the tunnels is highly related to the tunnel spacing distance. The smaller the distance between tunnels, the stronger the interaction between them. When the distance between tunnels reaches s/a = 20, the interaction between tunnels can be ignored. It is worth pointing out that the seismic design of underground tunnels should consider the interaction between tunnels when the tunnel distance is small.

On propagation characteristics of Rayleigh wave in saturated porous media based on the strain gradient nonlocal Biot theory

Based on the strain gradient nonlocal Biot theory, the semi-analytical solution of Rayleigh wave fields is obtained using the wave function expansion method. The influence of nonlocal pore size (characterized by nonlocal parameter) and heterogeneity structure effect (characterized by scale factor) on the propagation of Rayleigh wave are numerically implemented and then discussed in detail. The results show that the nonlocal parameter and scale factor have an ignorable influence on the propagation of Rayleigh wave in low frequency region while a significant influence within high frequency range by comparing with the results obtained from the classical Biot theory. A critical ratio of scale factor to nonlocal parameter defining the transition of negative dispersion to positive dispersion is observed. The Rayleigh wave velocity in saturated porous media increases with increasing nonlocal parameter and decreases with the increase of the scale factor. The length of the principal axis and minor axis of the ellipse formed by the displacement fields strongly depends on the ratio of nonlocal parameter to scale factor and the depth of the observation point. Furthermore, the displacement amplitude is intensified by increasing the nonlocal parameter and weakened by increasing the scale factor.

Surface motion of an arbitrary number of semi-circular viscoelastic alluvial valleys for incident planeSHwaves

SUMMARYA semi-analytical treatment of the seismic response of an arbitrary number of semi-circular viscoelastic alluvial valleys (including an infinite number of periodically distributed ones) in an elastic half-space under obliquely incident plane SH waves is presented. In terms of a radial wave function expansion and a region matching approach, a rigorous solution is derived for general computation for the anti-plane shear wave scattering problem. Upon confirmation of its accuracy with past solutions for single and twin elastic alluvial valleys, the proposed solution is used to present a comprehensive set of numerical examples to illustrate the sensitivity of the ground motion to the size, distribution, modulus, damping, and number of multiple alluvial valleys and frequency and direction of incident waves. It is found that the magnitudes of ground motions of a central alluvial valley are intensified and reduced by the surrounding valleys for vertical and grazing wave incidences, respectively. In addition, the duration of ground motions of the central alluvial valley tends to be prolonged due to the incoming secondary waves generated by surrounding valleys.

Boosting the Rate Performance of Li–S Batteries via Highly Dispersed Cobalt Nanoparticles Embedded into Nitrogen-Doped Hierarchical Porous Carbon

Boosting the Rate Performance of Li–S Batteries via Highly Dispersed Cobalt Nanoparticles Embedded into Nitrogen-Doped Hierarchical Porous Carbon

An Analytical Solution for Seismic Interaction between Urban River-Canyon Topography and Nearby Buildings

The interaction between urban river-canyon topography and the river-side building is investigated by using a whole analytic model of a semicircle river-canyon and a shear wall supported by a semicircle rigid foundation embedded in a homogenous half-space. The closed-form analytical solution for system response is presented based on the wave function expansion method. The analysis focuses on the effects of the canyon-building interaction on system response. The strength of the interaction between the river-canyon topography and the building changes periodically as the distance between the canyon and the structure increases, leading to the interaction having beneficial or harmful effects on the building’s seismic response. The foundation peak response of the building can be amplified by about 10%, and the peak of the building relative response can be amplified by about 40%. The distribution of canyon-structure spacing with strong or weak interaction is closely related to the dynamic characteristics of the building and the incident angle of the wave. When designing buildings along the river, the building and canyon should be analyzed as a whole model to determine whether the location of the building is in a position with strong interaction with the river-canyon. The model in this paper may be useful for obtaining insight into the effects of canyon-structure interaction and interpreting the observed response in buildings and seismic response estimation in general.

Low-rank matrix decompositions for ab initio nuclear structure

The extension of ab initio quantum many-body theory to higher accuracy and larger systems is intrinsically limited by the handling of large data objects in form of wave-function expansions and/or many-body operators. In this work we present matrix factorization techniques as a systematically improvable and robust tool to significantly reduce the computational cost in many-body applications at the price of introducing a moderate decomposition error. We demonstrate the power of this approach for the nuclear two-body systems, for many-body perturbation theory calculations of symmetric nuclear matter, and for non-perturbative in-medium similarity renormalization group simulations of finite nuclei. Establishing low-rank expansions of chiral nuclear interactions offers possibilities to reformulate many-body methods in ways that take advantage of tensor factorization strategies.

Theoretical responses of shallow-buried circular cavity subjected to transient P wave

Shallow-buried structures are inevitably subjected to time-varying disturbances such as drilling and blasting, earthquake and tectonic activities. The work presented in this paper focuses on the transient responses of a shallow-buried cavity. Based on the wave function expansion method, the large arc approximation and the integral transform, the time-domain responses around a cylindrical cavity subjected to transient P wave in an elastic half space are investigated analytically. The analytical results reveal that the wavelength-to-diameter ratio, the direction of disturbance and the embedment depth have a significant effect on the velocity vibrations and dynamic stress concentration around the tunnel. The theoretical investigations can provide practical value for the support of shallow tunnels.

Electromagnetic wave beam propagation through a chiral slab

A general scheme for investigating the electromagnetic wave beam reflection by and transmission through a chiral slab is proposed. The electromagnetic wave beams within different regions are described by appropriate cylindrical vector wave function expansions, and their expansion coefficients are determined by using the boundary conditions and projection method. Under illumination by a fundamental Gaussian beam, a Hermite–Gaussian beam and a doughnut mode beam, numerical results of the normalized field intensity distributions are provided and discussed briefly.

Real space electron delocalization, resonance, and aromaticity in chemistry

Chemists explaining a molecule’s stability and reactivity often refer to the concepts of delocalization, resonance, and aromaticity. Resonance is commonly discussed within valence bond theory as the stabilizing effect of mixing different Lewis structures. Yet, most computational chemists work with delocalized molecular orbitals, which are also usually employed to explain the concept of aromaticity, a ring delocalization in cyclic planar systems which abide certain number rules. However, all three concepts lack a real space definition, that is not reliant on orbitals or specific wave function expansions. Here, we outline a redefinition from first principles: delocalization means that likely electron arrangements are connected via paths of high probability density in the many-electron real space. In this picture, resonance is the consideration of additional electron arrangements, which offer alternative paths. Most notably, the famous 4n + 2 Hückel rule is generalized and derived from nothing but the antisymmetry of fermionic wave functions.