Wave Function Expansion Research Articles

Active invisible metamaterial of elastic waves with double targets in a thick plate

Wave invisibility has received increasing attention due to its great theoretical value and application to aerospace, military and other fields. In this work, the invisible flexural cloak for double hole targets in an elastic thick plate is proposed and proportional integral derivative (PID) active control is applied. Based on Mindlin thick plate theory and wave function expansion method, the wave equation for the thick plate is derived and scattering mode coefficients are obtained. The influences of structural and material parameters on dynamic stress concentration, scattering amplitude and scattering cross section (SCS) for double targets are investigated. With the consideration of specific hole targets, the cloaking configuration includes cloaks 1 and 2 and consists of periodically arranged piezoelectric (PZT) patches. Every PZT patch is attached to the active circuit with PID control function. The theoretical and experimental results illustrate that the cloak can achieve an invisible function for double targets within multiple frequency regions. On the other hand, the cloak can reduce the dynamic stress concentration, scattering amplitude and SCS of the thick plate. Different from the original structure without control, the effective frequency range of the PID active cloak can be widely extended.

Resonant and Non-Resonant Impurity States Related to GaAs/AlGaAs Quantum Well Sub-Bands.

The energy positions and wave function shapes of the ground and excited states of impurities, including resonance states, are studied using the expansion of the impurity wave function in basis functions. The structures under study are rectangular GaAs/AlGaAs quantum wells with four different widths. In all cases, the impurity binding energy (relative to the corresponding sub-band) has a maximum at or near the center of the quantum well, decreases as the heterointerface is approached, and apparently has a limit of 0 if the impurity moves deeper into the barrier. If the impurity moves away from the center of the quantum well, then the "center of mass" of the electron charge of non-resonant impurity states follows the impurity atom, and the "center of mass" of the electron charge of the resonant impurity states moves away from it. The effect is more pronounced for the ground and first resonance states for wider quantum wells, and the shifts reach a maximum when the impurity atom is positioned near the midpoint of the path between the quantum well center and the heterointerface.

An analytical method for effective dynamic properties of SH wave in piezoelectric nanocomposites with coated nano-fibers

In this paper, a generalized self-consistent model of nanocoated fiber composite material considering interface effect is established by using the Gurtin-Murdoch surface/interface elasticity theory, elastic wave theory and micromechanics methods, and the propagation of anti-plane shear electroelastic wave in infinite nano-piezoelectric material is analyzed. Based on the wave function expansion method, the displacement potential and electric potential functions of each phase medium in composite materials are described, and the function expressions of displacement, electric potential, stress and electric displacement are obtained. The dynamic effective shear modulus of piezoelectric reinforced composites is obtained by satisfying the boundary conditions of the electroelastic coupling surface/interface theory and the multiple scattering formulas. The effects of coating thickness, coating material parameters, interfacial shear modulus and piezoelectric properties on dynamic effective shear modulus at different incident frequencies are discussed. The results show that the effective shear modulus fluctuates obviously with the increase of frequency. The effective shear modulus increases with the increase of interfacial shear modulus and piezoelectric constant, and is more sensitive to the change of interfacial shear modulus. The effective shear modulus is more affected by the interface effect on the outer surface of the coating than the inner interface, and the effect of interface mechanical properties decreases with the increase of coating stiffness. The larger the parameter of the coating material, the larger the dynamic effective shear modulus and the smaller the influence of the interface effect.

Analysis of the dynamic response and damage characteristic for the tunnel under near-field blasts and far-field earthquakes

The dynamic response and failure characteristics of tunnels vary significantly under various dynamic disturbances. These characteristics are crucial for assessing structural stability and designing effective support for surrounding rock. In this study, the theoretical solution for the dynamic stress concentration factor (DSCF) of a circular tunnel subjected to cylindrical and plane P-waves was derived using the wave function expansion method. The existing equivalent blast stress wave was optimized and the Ricker wavelet was introduced to represent the seismic stress waves. By combining Fourier transform and Duhamel’s integral, the transient response of the underground tunnel under near-field blasts and far-field earthquakes was determined in both the frequency and time domains. The theoretical results were validated by comparing them with those obtained from numerical simulations using ANSYS LS-DYNA software. Numerical simulations were conducted to further investigate the damage characteristics of the underground tunnel and evaluate the effect of initial stress on structural failure under both types of disturbances. The theoretical and numerical simulation results indicated that the differences in the dynamic response and damage characteristics of the underground tunnel were primarily due to the curvature of the stress waves and transient load waveform. The locations of the maximum DSCF values differed between near-field blasts and far-field earthquakes, whereas the minimum DSCF values occurred at the same positions. Without initial stress, the blast stress waves caused spalling damage to the rock mass on the wave-facing side. Shear failure occurred near the areas with maximum DSCF values, and tensile failure occurred near the areas with minimum DSCF values. In contrast, damage occurred only near the areas with maximum DSCF values under seismic stress waves. Furthermore, the initial stress exacerbated spalling and shear damage while suppressing tensile failure. Hence, the blast stress waves no longer induced tensile failure on the tunnel sidewalls under initial stress.

Study on deformation patterns of tunnel isolation layers and seismic response of a shield tunnel

Tunnel seismic isolation is an effective measure to reduce the seismic response of tunnels; however, the unclear mechanism limits the popularization and application of this measure. In this study, the deformation patterns of tunnel seismic isolation were investigated using theoretical analyses and shaking-table experiments. The analytical solution for the dynamic response of the seismic isolation tunnel under SH-wave incidence was derived using the wave-function expansion method. A novel loading device based on the reaction-displacement method was used to study the mechanism of the tunnel seismic isolation layer. It was found that the seismic isolation layer helped reduce and spread the deformation from the outer surface to the inner surface, effectively minimizing the deformation affecting the lining. The DSCF response of the lining is subsequently reduced. Moreover, the installation of seismic isolation did not result in a significant alteration in the acceleration response of the tunnel. These results provide a reference for the seismic isolation design of tunnels.

Про біквадратичний ангармонічний осцилятор - підхід у рамках розкладу по осциляторному базису. I. Дослідження та розрахунок енергій основного і збуджених станів

For the quantum quartic anharmonic oscillator with the Hamiltonian H=0.5(p2+x2)+λx4 which is one of the classic traditional quantum-mechanical and quantum-field-theory models, its main physical characteristics and properties are thoroughly studied and calculated based on the system's wave function expansion in a complete set of the harmonic oscillator eigenfunctions, i.e., in the basis of eigenfunctions {φ(0)n} of the unperturbed Hamiltonian H=0.5(p2+x2). Very good convergence of the calculated energy levels of the anharmonic oscillator is demonstrated with respect to the number of basis functions included in the expansion, across a wide range of variation of the parameter λ. Thus, we have computed the energies of the ground and the first six excited states of the system for an exceptionally wide range of the oscillator coupling constant λ. In general, the proposed method provides a very good and accurate way to calculate all system characteristics.

Analytical seismic model for a tunnel with segmental liner buried in the half-space soil

In this paper, by using the wave function expansion (WFE) and the expansion of the cylindrical wave into plane wave (ECPW) methods, an analytical method for the dynamic response of a circular tunnel with segmental liner buried in the half-space soil to harmonic elastic waves is developed. The liner of the tunnel is supposed to consist of several segments and joints. Both the segments and joints are treated as open cylindrical shells, and the segment and joint shells together thus form an equivalent continuous shell (ECS) liner, and the thin shell theory is utilized to describe its vibration. The scattered wave field due to the presence of the tunnel and boundary of the half-space soil is divided into two parts, namely, the direct scattered wave field and secondary scattered wave field. The expressions for the direct scattered cylindrical waves in the soil is determined via the WFE method, while the secondary scattered waves form the boundary of the half-space soil are obtained by the ECPW method together with the application of the boundary condition along the soil surface. Applying the cylindrical shell theory on the ECS liner and using the Fourier series expansions for the variables and parameters of the ECS liner along the azimuthal direction together with introducing the Fourier component constitutive relation for the ECS liner, a system of equations for the Fourier component ECS displacements are derived with the differential equations for the ECS liner displacements. By using the system of equations, the expressions for the free and scattered wave fields and the continuity conditions at the interface between the liner and soil, the system of equations for the ECS liner coupled with the soil are derived. By using the developed analytical method for the tunnel, some results of the ECS tunnel under different incident waves are given.

Short-range expansion for the quantum many-body problem

In this work we derive a systematic short-range expansion of the many-body wave function. At leading order, the wave function is factorized to a zero-energy s-wave correlated pair and spectator particles, while terms that include energy derivatives and larger orbital angular momentum two-body functions appear at subleading orders. The validity of the expansion is tested for the two-body case, as well as the many-body case, where infinite neutron matter is considered. An accurate and consistent description of both coordinate-space two-body densities and the one-body momentum distribution is obtained. These results show the possibility to utilize such an expansion for describing different observables in strongly-interacting many-body systems, including nuclear, atomic and condensed-matter systems. This work also opens the path for a systematic description of large momentum transfer reactions in nuclear systems sensitive to short-range correlations, provides a link between such experiments and low-energy nuclear physics, and motivates measurement of new observables in these experiments.

Mechanical response and failure mechanism of circular inclusion embedded in brittle materials under dynamic impact

Inclusions are prevalent in both natural and synthetic materials. Gaining a comprehensive understanding of their dynamic response and interaction with the surrounding matrix is essential in fundamental mechanics and engineering applications. This study aims to achieve such understanding through theoretical analysis, experimental investigations, and numerical simulations, focusing on the dynamic destruction of inclusions and the underlying mechanisms. Firstly, the circumferential, radial, and shear stresses around elastic heterogeneous inclusions were derived by the wave function expansion and Duhamel integration methods. Subsequently, a set of experiments on sandstone specimens containing three types of inclusions (plaster, epoxy resin, cement) were conducted utilizing the Split Hopkinson Pressure Bar (SHPB) system in conjunction with high-speed photography and Digital Image Correlation (DIC) techniques to obtain the surface deformation of inclusion specimens. Eventually, a series of Finite Element Method (FEM) numerical simulations adopted suitable materials were also carried out to investigate the fracture process of inclusion and matrix under dynamic impact. The results demonstrate that the stress distributions and fracture mode of matrix and inclusion are highly dependent on the physical and mechanical properties of the inclusions and the surrounding matrix, specifically, density ρ, elastic modulus E, Poisson's ratio v, and strength. With an increase in the E and v, there is a reduction in the concentration of circumferential stress, while the radial and shear stresses experience an increase. The experimental and numerical results corroborated the theoretical findings and indicated that the localized dynamic stress concentrations induced by wave scattering around the inclusions directly dominated both local and overall specimen failure. Due to the dissimilarities in elastic parameters and strength between the inclusion and the matrix, the stress state of the inclusion and the interface is not homogeneous. Under dynamic loading, the weaker inclusion experiences tensile cracking at both ends of the loading, and as the mechanical properties (ρ, E, v, and strength) of the inclusion rise, a transition from tensile- hybrid-shear failure occurs within the inclusion. The findings of this study help us understand the dynamic failure mechanisms of dissimilar inclusions in brittle material.

Response of semicircular canyons and movable cylindrical cavities to SH waves in anisotropic half-space geology

The development of tunnels or the laying of underground pipelines are essential engineering projects in modern society, and in canyon tunnels and underground pipeline projects, the surface motion and cavity edge motion have been topics of concern in ground vibration problems. We investigate the wave scattering problem in an elastic half-space anisotropic medium containing a semicircular canyon and a subsurface movable cylindrical cavity by using the wave function expansion method, the complex function method, and the mirror method. By deriving the governing equation and transforming it into the standard form of the Helmholtz equation satisfying the zero-stress boundary condition, we solve the corresponding displacement functions. Introducing a position correction coefficient, the scattered wavefield in a half-space anisotropic medium is constructed by the mirror method, which improves the problem of scattered wave source singularity in anisotropic half-space media. Then, combining the free boundary conditions with a Fourier series expansion method, we solve the unknown coefficients in the equations. The correctness of the method is verified by degenerating it to a classical analytic solution. Finally, using frequency- and time-domain analysis, we investigate the effects of the relevant parameters on the surface motion [Formula: see text], the dynamic stress concentration factor, and the displacement amplitude [Formula: see text]. The results indicate that rock anisotropy and the presence of semicircular canyons have a significant effect on the dynamic response of subsurface structures. This not only provides a theoretical basis for practical unlined tunnels or pipeline projects but also can provide a basis for seismic design of underground structures.

A fast algorithm for the two-dimensional Helmholtz transmission problem with large multiple scattering configurationsa).

We develop an efficient three-stage algorithm for simulating multiple acoustic scattering by two-dimensional configurations comprising large numbers of penetrable scatterers. Our approach is based on a boundary integral equation reformulation of the Helmholtz transmission partial differential equation, and a reduction of the boundary integral system for computationally efficient evaluation of wave interactions between scatterers. A key ingredient of our algorithm is to represent the interactions between scatterers using expansions of cylindrical wavefunctions. For large numbers of scatterers, this approach facilitates the application of the fast multipole method, leading to linear complexity of the algorithm with respect to the number of scatterers. Numerical results demonstrate the efficiency of our algorithm for configurations containing a few hundred to hundreds of thousands of individual scatterers.

Analytical solution to 2D dynamic soil-tunnel-building interaction under seismic SH-wave excitation considering the influence of the stiffness of tunnel and building's foundation

This investigation examines the structure-soil-structure interaction (SSSI) problem between a tunnel and nearby aboveground building, considering the impact of the stiffness of tunnel and building's foundation. A two-dimensional (2D) model is utilized, which includes a flexible circular tunnel and a three-story building anchored by a flexible semi-circular foundation embedded in a homogeneous half-space. The response of the system to incident plane SH-wave excitation is solved analytically using the wave function expansion method. The analysis focuses on how the system response is affected by the stiffness of the tunnel and the building's foundation. Tunnel stiffness plays a leading role in tunnel response analysis. The model with a rigid tunnel and rigid foundation can offer a reasonable system response overall, but it fails to accurately represent tunnel deformation and stress distribution. Depending on the stiffness of the building's foundation, the maximum value of the inter-story displacement of the first floor can be overestimated by around 70 %, with even greater overestimations for higher floors. The results help understand observed seismic behaviors and provide insight into the effects of foundation and tunnel stiffness on the tunnel-building interaction system.

Propagation attenuation of elastic waves in multi-row infinitely periodic pile barriers: A closed-form analytical solution

Periodically arranged media exhibit attenuation zones for elastic waves, and the related periodic structures play a crucial role in the design of seismic isolation and vibration mitigation. From the perspective of wave theory, this paper presents a closed-form analytical solution for the propagation attenuation of elastic waves through infinitely periodic, multi-row pile barriers. The innovative approach begins by deriving the extended Graf’s addition theorem to transform wave potentials across arbitrary coordinate systems, overcoming the constraints of traditional formulas that limit pile centers to linear configurations. Subsequently, it completes the wave function expansion by utilizing the phase differences in the frequency domain of wave fields around different periodic elements, thus skillfully integrating the effects of incident and scattered wave fields. For the first time, this study delivers exact expressions for the scattering of various types of plane waves (SH, SV, and P) by multi-row pile barriers with an infinite periodic array, addressing the complexities of large pile numbers in previous theoretical studies. The proposed solution increases computational efficiency by tens of times and markedly reduces the majority of resource usage, improving the analysis of complex vibration isolation systems involving numerous piles. Building on the accuracy verification and convergence analysis, several numerical examples are conducted to explore the effects of pile row, pile spacing, and pile arrangement on the propagation attenuation. The findings elucidate the mechanisms driving vibration reduction design using periodic wave barrier, furthering its application in engineering structures.

Efficient Analysis of Unlined Elliptic Tunnels: An Approximate Method for Dynamic Response to Incident Plane SH Waves

ABSTRACT Dynamic analysis of underground structures, such as tunnels, is crucial to ensure their safety and stability during seismic loading. This paper proposes a more general method of large elliptic arc assumption-based wave function expansion, which is attributed to the important role of the large circular arc assumption-based wave function expansion method in the elastic wave scattering problem. The study is conducted within elliptic coordinate systems, employing the Mathieu function and Mathieu function addition theorem. The applicability and limitations of this method are analysed by comparing it with the existing exact analytical solutions obtained through the mirror-based wave function expansion method.

State Expansion of a Levitated Nanoparticle in a Dark Harmonic Potential.

We spatially expand and subsequently contract the motional thermal state of a levitated nanoparticle using a hybrid trapping scheme. The particle's center-of-mass motion is initialized in a thermal state (temperature 155mK) in an optical trap and then expanded by subsequent evolution in a much softer Paul trap in the absence of optical fields. We demonstrate expansion of the motional state's standard deviation in position by a factor of 24. In our system, state expansion occurs devoid of backaction from photon recoil, making this approach suitable for coherent wave function expansion.

Kinematic response of a pile within a soil slope to SH wave excitation

It is widely recognized that the topography of a slope can significantly enhance the magnitude of seismic waves. Nevertheless, the mechanism behind the topographic amplification on the kinematic response of a pile within a soil slope remains unclear. Consequently, a newly rigorous method is employed to investigate the kinematic response of a pile within a soil slope. The proposed method which combines the Wave Function Expansion (WFE) method and the Beam on Dynamic Winkler Foundation (BDWF) model advances the rigorous theoretical study for the kinematic response of a pile on slope from the pseudo-static analysis to the frequency-domain analysis. The proposed solution well captures the kinematic response of a piles within a slope. The parametric analysis focuses on investigating the influence of slope topographic amplification effects on pile vibration under different incident frequencies, slope gradients, and pile-to-soil modulus ratios. The results indicate that steep slopes and high pile-to-soil modulus ratios lead to higher topographic amplification effects, with specific frequencies significantly amplifying the kinematic response of pile within a slope. Nonetheless, as the pile located beyond twice the length of the slope, the amplification effect notably diminishes.

Influence of Incident Orientation on the Dynamic Response of Deep U-Shaped Cavern Subjected to Transient Loading

In deep rock engineering, caverns are often disturbed by engineering loads from different directions. To investigate the dynamic response of deep U-shaped caverns under different incident orientations, a theoretical solution of the dynamic stress concentration factor along the cavern boundary was derived based on the wave function expansion and conformal mapping method, and the failure characteristics around the cavern were further investigated by PFC2D (Particle Flow Code in two dimensions). As the incident orientation increases from 0° to 90°, the dynamic compressive stress concentration area transforms from both the roof and the floor to the sidewalls, and the peak dynamic stress concentration factor of the roof decreases from 2.98 to −0.20. The failure of the floor converts from dynamic compression shear failure to dynamic tensile failure. Compared to a stress wave incident from the curved boundary, a stress wave incident from the flat boundary causes severer damage. When the stress wave is incident from the sidewall, the cavern with a larger height-to-width (h/w) ratio exhibits severer damage. Conversely, the cavern with a smaller h/w ratio tends to fail as the stress wave is incident from the floor. This paper provides a basic understanding of dynamic responses of the deep U-shaped cavern.

Energy levels and characteristic features in photoionization of Cl III using the R-matrix method

We report study of Cl III for its large number of fine structure bound levels, 890, with n ≤ 10 and l ≤ 9, and 1/2 ≤ j ≤ 19/2 of even and odd parities with spectroscopic designation and photoionization cross sections (σPI) of the levels revealing various characteristic features. Various resonant structures and the shapes of the background, and their interference in σPI are illustrated for the ground, excited equivalent electron, low and high lying excited levels with single valence electron, and effects of fine structure couplings. σPI of the ground level shows Rydberg series of resonances on smooth background, and of equivalent electron levels producing strong closely spaced Rydberg series of resonances belonging to the low lying core ion excitation thresholds. They will impact applications in low temperature plasma. For the single valence electron excited levels, we find that σPI of low lying excited states are dominated by Rydberg series of resonances and of high lying excited states exhibit prominent broad feature of Seaton resonances. Partial photoionization cross sections of the ground level for leaving the core ion in the ground and various excited levels are also presented for applications in plasma modeling. The study was carried out in relativistic Breit–Pauli R-matrix method using a large close coupling wave function expansion of 45 levels of the core ion configurations 3 s23 p2, 3 s3 p3, 3 s23 p3 d, 3 s23 p4 s, 3 s3 p23 d, and 3 p4 with closed 1s, 2s, 2p orbitals. They belong to the optimized set of 13 configurations of Cl IV, 3 s23 p2, 3 s3 p3, 3 s23 p3 d, 3 s23 p4 s, 3 s23 p4 p, 3 s23 p4 d, 3 s3 p23 d, 3 s3 p24 s, 3 s3 p24 p, 3 p4, 3 p33 d, 3 p34 s, and 3 p34 p. Cl III energies are in good agreement with measured values. σPI features of low lying levels were benchmark with observation carried out at Advanced Light Source at LBNL with very good agreement. The present set of high accuracy data should complete for any practical applications.

Linear analytical model for the seismic response of a straight‐jointed shield tunnel

AbstractIn this study, the segment joint and ring joint of the lining of the straight‐jointed tunnel are simplified as an equivalent open cylindrical shell with a small central angle and an equivalent closed cylindrical shell with a small width, and the lining of the straight‐jointed tunnel is thus simplified as an equivalent continuous periodic shell (ECPS) described by the cylindrical shell theory. Since the ECPS lining is periodic along the longitudinal direction, the tunnel‐soil system can thus be treated as a periodic system, which is referred to as the periodic tunnel‐soil system (PTSS) in this study. Based on the proposed ECPS lining model, an analytical method for the tunnel with the ECPS lining (ECPS tunnel) under seismic waves is established in this study. By employing the periodicity condition for the PTSS as well as the wave function expansion method, the representation for the wave field in the soil is established. With the aforementioned cylindrical shell theory and Fourier series expansion method, the convolution type constitutive relation for the ECPS lining and the Fourier space equations of motion are derived. By using the soil‐lining continuity condition and aforementioned formulations for the ECPS lining and soil, the coupled Fourier space equations of motion for the PTSS are established, with which the response of the ECPS lining and scattered waves in the soil can be determined.

R-Matrix calculations for opacities: II. Photoionization and oscillator strengths of iron ions Fe xvii, Fe xviii and Fe xix

Iron is the dominant heavy element that plays an important role in radiation transport in stellar interiors. Owing to its abundance and large number of bound levels and transitions, iron ions determine the opacity more than any other astrophysically abundant element. A few iron ions constitute the abundance and opacity of iron at the base of the convection zone (BCZ) at the boundary between the solar convection and radiative zones and are the focus of the present study. Together, Fe xvii, Fe xviii and Fe xix represent 85% of iron ion fractions, 20%, 39% and 26% respectively, at the BCZ physical conditions of temperature T ∼ 2.11×106 K and electron density Ne = 3.1×1022 c.c. We report the most extensive R-matrix atomic calculations for these ions for bound–bound and bound–free transitions, the two main processes of radiation absorption. We consider wavefunction expansions with 218 target or core ion fine structure levels of Fe xviii for Fe xvii, 276 levels of Fe xix for Fe xviii, in the Breit–Pauli R-matrix (BPRM) approximation, and 180 LS terms (equivalent to 415 fine structure levels) of Fe xx for Fe xix calculations. These large target expansions, which include core ion excitations to n = 2,3,4 complexes, enable accuracy and convergence of photoionization cross sections, as well as the inclusion of high lying resonances. The resulting R-matrix datasets include 454 bound levels for Fe xvii, 1,174 levels for Fe xviii, and 1,626 for Fe xix up to n⩽ 10 and l = 0–9. Corresponding datasets of oscillator strengths for photoabsorption are: 20 951 transitions for Fe xvii, 141 869 for Fe xviii, and 289 291 for Fe xix. Photoionization cross sections have been obtained for all bound fine structure levels of Fe xvii and Fe xviii, and for 900 bound LS states of Fe xix. Selected results demonstrating prominent characteristic features of photoionization are presented, particularly the strong Seaton photoexcitation-of-core resonances formed via high-lying core excitations with Δn=1 that significantly impact bound–free opacity.