Wave Propagation Problems Research Articles

Space–time collocation meshfree method for modeling 3D wave propagation problems

Space–time collocation meshfree method for modeling 3D wave propagation problems

A direct time-domain FEM-SBFEM-SBPML method for soil-structure interaction analysis using quadtree mesh

In this paper, a novel direct time-domain numerical method, called the FEM-SBFEM-SBPML, is proposed to study the wave propagation problems in soil-structure interaction (SSI) system with complex geometry and geological condition. In the bounded domain, the structure is modeled as beam element based on the finite element method (FEM), and the surrounding truncated soil is discretized using the scaled boundary finite element method (SBFEM) combined with an efficient quadtree mesh generation technique, respectively. In addition, a solid-modified beam coupling model considering the eccentric effect of the tangential traction at the outer surface of the structures is implemented. The unbounded domain is modeled by a recently proposed scaled boundary perfectly matched layer (SBPML) method, which is a novel high-accuracy absorbing layer method satisfying the radiation condition. As a result, the combined FEM-SBFEM-SBPML approach is capable of modeling SSI with complex geometries and soil properties automatically. Some benchmark tests are conducted to illustrate the accuracy and efficiency of the proposed scheme, and a vibration analysis of a soil-tunnel system is further presented to demonstrate the applicability of the proposed method in engineering.

The explicit β1/β2-Bathe time integration method

In this paper we introduce the explicit β1/β2-Bathe method for solving dynamic problems, in particular wave propagations. Like for the implicit β1/β2-Bathe method, the proposed explicit scheme uses two sub-steps per time step and can be used directly as a first-order and a second-order method with the capability to suppress high spurious frequency response. In both sub-steps, standard Taylor series are employed resulting in an explicit solution scheme. The novelty is that we calculate the final displacements and velocities in each sub-step by applying correction terms using the generalized trapezoidal rule with control parameters β1 and β2. This approach makes the method a quite simple scheme. We consider the stability, accuracy and numerical dispersion and give recommendations on the parameter values β1 and β2 to be used in practice. We give the solutions of four problems, three of which are wave propagation problems, and compare the results with those obtained using other methods. While more experience in the use of the procedure is needed to understand its full solution capabilities, we can already conclude that the proposed method is effective in some wave propagation analyses.

Fundamental solution in a swelling porous medium containing a mixture of solid, liquid, and gas

This study considers the wave phenomena and fundamental solution in swelling porous elastic solids consisting of a mixture of solid, liquid, and gas. There exist three dilatational and two transversal waves which vibrate at distinct speeds. The basic characteristics of waves (phase velocity and attenuation coefficient) are computed numerically and displayed in the form of graphs. Also, a fundamental solution is explored for steady oscillation and some preliminary features are explored in this model. The obtained results for both waves are analyzed with and without swelling porous elastic solid. This new model enables us to improve the efficacy of swelling porous elastic solid which involves solid and fluid and find it applicable in exploration industries and to investigate various wave propagation problems.

Seismic wave propagation analysis in the framework of generalized continuum mechanics theory

For the problems of seismic wave propagation, scale effects will appear while considering the microstructures of medium. Scale effects mean that new components of wave fields appear in the seismograms due to the microstructure interactions in the medium and vary with the change of the characteristic length scale parameters. The generalized continuum mechanics (GCM) theory which introduces the intrinsic length scales of the microstructures is suitable for analyzing the scale effects of seismic wave propagation. We adopt the concept of multi-scale microstructure interactions, that is, the modified couple stress theory, and the one-parameter second strain gradient theory, as two branches of the GCM theory, have the same definition characteristic length scale parameters of medium, to describe the microstructure interactions at different scales. Through the quantitative relationship between the characteristic length scale parameters of the medium and the micropore, the layered model of the characteristic length scale parameters of the micropore is constructed for numerical modeling in a unified framework, and the scale effects of seismic wave propagation is analyzed. More specifically, the asymmetric elastic wave equations under a unified framework are derived, and the physical definition of the characteristic length scale parameters of medium is given. And the influence of the characteristic length scale parameter of the micropore on the seismic wave propagation is focused on. Finally, the scale effects of seismic waves excited by high-speed trains under the viaduct system are analyzed, and some conclusions are drawn.

An asymptotically exact first-order shear deformation theory for functionally graded plates

An asymptotically exact first-order shear deformation theory for functionally graded elastic plates is derived using the variational-asymptotic method. As an application, an analytical solution to the problem of wave propagation in a sandwich plate is found in accordance with this refined theory. Comparison between the dispersion curves obtained by 2-D plate theory and 3-D elasticity theory reveals that the former is accurate up to the order of h2/l2, where h is the plate thickness and l the wavelength.

Time‐domain scaled boundary perfectly matched layer for elastic wave propagation

AbstractA novel artificial boundary method called the scaled boundary perfectly matched layer (SBPML) is proposed for the transient analysis of elastic wave propagation in unbounded domains. This method constructs a generalized perfectly matched layer (PML) around the finite interior subdomain based on a coordinate transformation inspired by the scaled boundary finite element method (SBFEM). It can model the infinite exterior subdomains of heterogeneous materials, featuring both parallel and radial straight‐line physical surfaces and interfaces. In addition, generally‐shaped (even nonconvex‐shape) artificial boundaries can be used to provide high flexibility in choosing the geometry of the finite interior subdomain. Two coordinate mapping technologies, namely the modified scaled boundary coordinate transformation from the SBFEM and the complex coordinate stretching from the PML, are successively utilized to construct the proposed method. By introducing the integral of stress as an auxiliary variable, the resulting SBPML subdomain is represented in the time domain using a mixed displacement‐stress unsplit‐field formulation. This has the added benefit of allowing for easy incorporation into standard dynamic finite element methods. The performance of the SBPML is demonstrated through several numerical examples, including P‐SV and SH wave propagation problems occurring in unbounded domains with complex geometries and heterogeneous material properties. The numerical results are evaluated in comparison to both reference solution and existing artificial boundary methods, ultimately demonstrating the flexibility, robustness, accuracy, and stability of the proposed approach.

Shear Wave-Induced Friction at Periodic Interfaces for Programmable Mechanical Responses

Abstract Nonlinear phononic materials enable superior wave responses by combining nonlinearity with their inherent periodicity, creating opportunities for the development of novel acoustic devices. However, the field has largely focused on reversible nonlinearities, whereas the role of hysteretic nonlinearity remains unexplored. In this work, we investigate nonlinear shear wave responses arising from the hysteretic nonlinearity of frictional rough contacts, and harness these responses to enable programmable functions. By using a numerical approach, we solve the strongly nonlinear problem of shear wave propagation through a single contact and a periodic array of contacts, accounting for frictional effects. Specifically, the Jenkin friction model with experimentally obtained properties is used to capture the effects of stick–slip transition at the contacts. Results show that friction gives rise to shear-polarized eigenstrains, which are residual static deformations within the system. We then demonstrate how eigenstrain generation in multiple contacts can enable programmable functionalities such as an acoustically controlled mechanical switch, precision position control, and surface reconfigurability. Overall, our findings open new avenues for designing smart materials and devices with advanced functionalities via acoustic waves using the hysteretic nonlinearity of frictional contacts.

Study of stress wave propagation across a non-persistent joint based on a boundary integral equation method

Studies on stress wave propagation across persistent joints have been conducted extensively. Nevertheless, there exists a consensus that non-persistent joints are widely and densely distributed, which have a profound impact on wave propagation in jointed rock masses. A boundary integral equation method is suggested in this paper to investigate the characteristics of transmitted wave field for the case of stress wave propagation across a single non-persistent joint. The displacement continuity and discontinuity boundaries are combined in the method. The method presented in the current study is applicable to the analysis of wave propagation across non-persistent joints with arbitrary incident angles. Then, taking a single non-persistent joint arranged with only one joint segment as an example, the applicability of the method in dealing with the problems of wave propagation is verified by comparing the results with those from the discrete element method and analytical methods. Subsequently, parametric studies are carried out, including the effects of joint-segment length, rock-bridge length, wave frequency and incident angle on the transmitted wave. The result indicates that the existence of non-persistent joint makes the transmitted displacement field different from that of persistent joint, because the scattered wave is produced during the process of wave propagation. The displacement amplitude may be amplified evidently in some regions and the spatial distribution pattern of the transmission coefficient is closely related to the joint-segment length, rock-bridge length and incident wavelength.

Analyzing the acoustic wave propagation characteristics in discontinuous bifurcated waveguide

This article examines the dispersion characteristics and energy distribution in different regions of the bifurcated waveguide with vertical step discontinuities. In this research study, we have considered mathematical model of noise reduction devices, specifically analyzing the acoustic wave propagation problem in a bifurcated waveguide with elastic membrane outer plates and rigid inner plates. Mode-matching (MM) technique is employed to match the eigenfunctions at the interface and analyzed the reflection and transmission amplitudes in all regions. Through MM scheme an infinite system of equations has been obtained for these boundary conditions by dividing the structure into three regions and finding dispersion relations and eigenfunctions using separation of variables and the corresponding orthogonality relations. Because of the flexible nature of the boundary surfaces, the acquired eigenfunctions and the dispersion relation do not satisfy the standard orthogonality relation. To resolve this difficulty, we have established a modified orthogonality relation to take into account the flexible nature of the boundary surfaces. With the implication of physical connections, the uniqueness of the problem has been ensured, and zero displacement and zero gradient edge conditions on bending surfaces have been considered to present results. Energy expressions are obtained showing how the energy is distributed in different sections of the waveguide, which is consistent with the energy conservation law. Finally, the numerical discussion illustrates the quantitative behavior of the physical parameters and summarizes the significant contribution of the study by presenting graphical data.

Erratum to: ”On a time discrete convolution - space collocation BEM for the numerical solution of two-dimensional wave propagation problems in unbounded domains”

Abstract

Formulation for wave propagation in dissipative media and its application to absorbing layers in elastoplastic analysis using mathematical programming

AbstractIn this article, we propose a new solution scheme for modeling elastoplastic problems with stress wave propagation in dissipative media. The scheme is founded on a generalized Hellinger–Reissner (HR) variational principle. The principle renders the discretized boundary‐value problem into an equivalent second‐order cone programming (SOCP) problem that can be resolved in mathematical programming using the advanced optimization algorithm—the interior point method. In such a way, the developed method not only inherits admirable features of the SOCP‐based finite element method in solving elastoplastic problems but also enables the enforcement of absorbing layers (i.e., Caughey absorbing layer), which is essential in modeling stress wave propagation problems, to absorb wave energy. The proposed scheme is validated via the comparison between analytical and numerical results for seismic wave propagation in dissipative media. Its application to elastoplastic dynamic problems with stress wave propagation is also illustrated to demonstrate its efficiency.

The Nonlinear Eigenvalue Problem of Electromagnetic Wave Propagation in a Dielectric Layer Covered with Graphene

The paper focuses on the problem of a monochromatic terahertz TE-polarized wave propagation in a plane dielectric layer filled with a homogeneous isotropic medium; one of the boundaries of the waveguide is covered with a layer of graphene. In fact, the paper aims to find the eigenwaves of the described waveguiding structure. On the one hand, in the study, energy losses both in the dielectric layer and in the graphene layer are neglected; the latter assumption is reasonable in the terahertz range of electromagnetic radiation (on which the paper focuses), where graphene has a strong plasmonic response and much less loss. On the other hand, this study takes into account the significant third-order nonlinearity resulting from the interaction of the electromagnetic wave with the charge carriers in the graphene layer. The paper aims to study the guiding properties of the above structure using primarily an analytical approach. The wave propagation problem is reduced to an eigenvalue problem, where one of the boundary conditions is nonlinear with respect to the sought-for function. The main result of the paper is a dispersion equation allowing for a waveguide of a given thickness to determine a set of its propagation constants and, consequently, a set of its eigenwaves. It is worth noting that the dispersion equation being written in an explicit form can be used to obtain deep qualitative results related to the solvability of the problem and the properties of its solutions. For example, in the paper, the existence of several propagation constants (and, consequently, the eigenwaves) of the studied waveguiding structure is proved under some conditions. Besides studying the problem analytically, the paper presents some numerical results as well. In particular, the presented figures demonstrate how the nonlinearity in graphene affects the propagation constants and eigenwaves, providing the dispersion curves and eigenwaves for nonlinear graphene as well as for the linear one.

A combined compact finite difference scheme for solving the acoustic wave equation in heterogeneous media

AbstractIn this paper, we consider the development and analysis of a new explicit compact high‐order finite difference scheme for acoustic wave equation formulated in divergence form, which is widely used to describe seismic wave propagation through a heterogeneous media with variable media density and acoustic velocity. The new scheme is compact and of fourth‐order accuracy in space and second‐order accuracy in time. The compactness of the scheme is obtained by the so‐called combined finite difference method, which utilizes the boundary values of the spatial derivatives and those boundary values are obtained by one‐sided finite difference approximation. An empirical stability analysis has been conducted to obtain the Courant‐Friedrichs‐Levy (CFL) condition, which confirmed the conditional stability of the new scheme. Four numerical examples have been conducted to validate the convergence and effectiveness of the new scheme. The application of the new scheme to a realistic wave propagation problem with a Perfect Matched Layer is validated in this paper as well.

Nonclassical perturbation approach to a nonlinear multiparameter eigenvalue problem arising in electromagnetics

The paper focuses on a problem of nonlinear electromagnetic wave propagation in a plane shielded waveguide filled with nonlinear medium. We look for nonlinear eigenvalues that correspond to propagation constants of the waveguide. One of the main feature of the problem is that it has nonperturbative solutions and therefore the corresponding linearized problem cannot be used in any perturbation approach to find such eigenvalues (moreover, it turns out that the linearized problem is solvable in exceptional cases only). To solve the main problem, we develop a nonlinear perturbation approach based on usage of a certain simpler nonlinear problem that also has nonlinearizable solutions. Using this approach, we prove existence of nonlinearizable solutions to the main problem. Numerical results are also presented. Briefly, we present electromagnetic background for the main problem.

Research on Laplace transform of stress wave propagation relaxation function in multi-layer media

There are some difficulties in solving the theory of stress wave propagation in multi-layer media. To solve the problem of stress wave propagation in multi-layer media under the impact load, this paper theoretically makes further derivation and discussion based on viscoelastic analogy method. To further verify the feasibility of the theoretical solution, LS-DYNA is used to simulate the three-layer medium. The simulation results show that comparing the peak stress obtained from the numerical inversion of the Laplace function after the Laplace transform with the simulation results, the maximum error of the peak stress is about 9.5%, and the minimum error is about 1.7%. The results show that the theoretical solution agree with the simulation results, and the theoretical solution has specific feasibility. The theoretical solution method deduced in this paper further develops the viscoelastic analogy method, which has a specific reference value for the rapid calculation of the stress wave propagation law in multi-layer media.

A novel three sub‐step explicit time integration method for wave propagation and dynamic problems

AbstractA novel explicit time integration method is formulated with sub‐step strategy and Cubic B‐spline interpolation method. Theoretical and numerical analysis are conducted to obtain optimized algorithm properties including algorithm accuracy, spectral stability and numerical dissipation/dispersion. A demonstrative dispersion analysis for wave propagation is presented to acquire optimal algorithm parameter value for finite element analysis of wave propagation problems. Numerical tests demonstrate that new method show desirable algorithm accuracy and convergence for dynamic problems. Especially, for highly nonlinear problems, the new method can provide very accurate and stable solutions.

Generalized finite difference method applied to solve seismic wave propagation problems. Examples of 3D simulations

The simulation of seismic wave propagation generally requires dealing with complex tridimensional geometries that are irregular in shape and have non‐uniform properties, features that make the application of the generalized finite difference method in this field interesting. This work continues the extensive developments by the research team focused on the simulation of seismic wave propagation in two‐dimensional domains. In this new contribution, the general formulation and the treatment of free surface boundary conditions are extended for the three‐dimensional case, and the results obtained from different examples are analyzed.

Wave packet enriched finite elements for thermo-electro-elastic wave propagation problems with discontinuous wavefronts

In this article, we study the numerical properties of the recently developed local (element-domain) harmonic basis function enriched finite element (FE) for wave propagation problems. The study includes the effect of the enrichment harmonics on the conditioning of system matrices and how the diagonal scaling mass lumping technique influences the possible ill-conditioning and the solution accuracy. Subsequently, a wave packet enriched thermoelectromechanical FE formulation is presented for axisymmetric and planar wave propagation problems in piezoelastic media. The conventional Lagrange interpolations for the displacement, temperature, and electric potential fields are enriched with the element-domain sinusoidal functions which satisfy the partition of unity condition. The extended Hamilton’s principle and constitutive relations of the Lord–Shulman and Green–Lindsay generalized piezothermoelasticity are employed to derive the coupled system of equations of motion which is solved using Newmark- direct time integration scheme. The performance of the proposed elements is assessed and wave characteristics are studied for the problems of thermoelastic shock waves in an elastic annular disk and thermoelectric shock waves in a piezoelectric hollow cylinder. The element shows significant improvement in the computational efficiency and accuracy over the conventional FE for these problems involving sharp discontinuities in the fields at the wavefronts.

Topology optimization of microwave frequency dividing multiplexers

We use material-distribution-based topology optimization to design a three-port frequency dividing multiplexer at microwave frequencies. That is, by placing a good electric conductor inside the design domain, we aim to design a passive device that splits the incoming signal’s frequencies into two frequency bands and transmits them to their respective output ports. The Helmholtz equation models the time-harmonic wave propagation problem. We solve the governing equation using the finite element method. The adjoint variable method provides the required gradients, and we solve the topology optimization problem using Svanberg’s MMA algorithm. In this study, we present a technique for modeling the distribution of a good electric conductor within the design domain. In addition, we derive a power balance expression, which aids in formulating a series of three objective functions. In each successive objective function, we add more information and evaluate its impact on the results. The results show that by selecting a suitable objective function, we achieve more than 93.7,% transmission for both the frequency bands. Moreover, the numerical experiments suggest that the optimization problem is self penalized and is sensitive to the initial design.