Elastic Wave Problems Research Articles

Boundary integral equation methods for the elastic and thermoelastic waves in three dimensions

In this paper, we consider the boundary integral equation (BIE) methods for solving the exterior Neumann boundary value problems of elastic and thermoelastic waves in three dimensions based on the Fredholm integral equations of the first kind. The innovative contribution of this work lies in the proposal of the new regularized formulations for the hyper-singular boundary integral operators (BIO) associated with the time-harmonic elastic and thermoelastic wave equations. With the help of the new regularized formulations, we only need to compute the integrals with weak singularities at most in the corresponding variational forms of the boundary integral equations. The accuracy of the regularized formulations is demonstrated through numerical examples using the Galerkin boundary element method (BEM).

Near-Field Imaging of an Unbounded Elastic Rough Surface with a Direct Imaging Method

This paper is concerned with the inverse scattering problem of time-harmonic elastic waves by an unbounded rigid rough surface. A direct imaging method is developed to reconstruct the unbounded rou...

Numerical Solution of an Inverse Obstacle Scattering Problem for Elastic Waves via the Helmholtz Decomposition

Numerical Solution of an Inverse Obstacle Scattering Problem for Elastic Waves via the Helmholtz Decomposition

Root-finding absorbing boundary conditions for scalar and elastic waves in infinite media

Root-Finding Absorbing Boundary Conditions (RFABCs) for scalar and elastic waves in infinite media are developed. First, the existing RFABC formulation for scalar-wave propagation problems is refined. Specifically, a Fourier series expansion or an eigenfunction expansion, as in Sturm–Liouville problems, is applied to general scalar-wave fields and a consistent nodal flux is derived for eventual combination with the discrete representation of the problem under consideration. This newly-proposed approach for scalar waves is extended to elastic waves, governed by two scalar-wave equations for pressure and shear waves. Stability of the refined RFABCs can be proven at both the continuous and discrete levels. The newly-developed RFABCs for scalar and elastic waves are applied to various wave-propagation problems. It is verified that they produce accurate and stable results. Perfectly matched layers and high-order absorbing boundary conditions are the most popular of the available tools for wave-propagation problems. But, their stability properties have not been established in applications to problems of elastic waves in waveguides. It is expected that existing high-order absorbing boundaries will lead to stable results for elastic waves in waveguides, when implemented using the technique proposed in the present study.

Negative refraction in quasicrystalline multilayered metamaterials

Inspired by some recent results in elastodynamics of layered composites, we address here the problem of an antiplane elastic wave obliquely incident at the interface between a substrate and a periodic laminate with a quasicrystalline structure (generated by the Fibonacci substitution rule). The angles of refraction of the transmitted modes are computed by combining the transfer matrix method with the normal mode decomposition and evaluating the direction of the average Poynting vector. It is shown that, with respect to a periodic classical bilayer, on the one hand, beyond a certain frequency threshold, high order Fibonacci laminates can provide negative refraction for a wider range of angles of incidence, on the other, they allow negative wave refraction at lower frequencies. The outcome strongly relies on the Floquet–Bloch dynamic analysis of this class of laminates that is performed thoroughly. It is revealed that the corresponding spectra have a self-similar character linked to the specialisation of the Kohmoto’s invariant, a function of the frequency that was recently studied by the authors for periodic one-dimensional quasicrystalline-generated waveguides. This function is able to explain two types of scaling occurring in dispersion diagrams. The attained results represent an important advancement towards the realisation of multilayered quasicrystalline metamaterials with the aim to control negatively refracted elastic waves.

Grid-Characteristic Numerical Method for Low-Velocity Impact Testing of Fiber-Metal Laminates

The grid-characteristic numerical method (GCM) for hyperbolic equations systems is applied in many science fields—gas dynamics, hydrodynamics, plasma dynamics, etc. Its application for problems of dynamics of deformable solids is less popular, especially in comparison with finite elements methods. GCM shows good results and high performance for elastic wave problems in the approximation of small deformations—seismic survey and ultrasound non-destructive testing in medicine, aviation and railway industry. Low-velocity impacts (hail, dropped tool, bird strike, etc.) are one of the most dangerous load types for polymer composites. They cause barely visible impact damage (BVID) that can only be detected by a thorough ultrasound testing, but severely reduces the residual strength of the material, especially for a compression load along the surface. This testing increases the operating cost, and its necessity can be easily missed, which greatly reduces the reliability of polymer composites. Hybrid fiber-metal composites (GLARE, ARALL, titanium composite laminates) were developed to unify the advantageous properties of polymer composites and metal. The addition of a thin metal layer (1–2 mm) helps to reduce the impact vulnerability of polymer composites in case of a penetration or significant deformations of the material. The application of GCM for low-velocity impact problems can help to explain the damage pattern in fiber-metal composites in case of low-velocity strike, including delamination effects, by modelling elastic wave processes in the complex anisotropicmedium. This article contains the brief description of the GCM and numerical results that were obtained for model problems of a low-velocity impact on titanium composite laminates.

Uncertainty principles for inverse source problems for electromagnetic and elastic waves

In isotropic homogeneous media, far fields of time-harmonic electromagnetic waves radiated by compactly supported volume currents, and elastic waves radiated by compactly supported body force densities can be modelled in very similar fashions. Both are projected restricted Fourier transforms of vector-valued source terms. In this work we generalize two types of uncertainty principles recently developed for far fields of scalar-valued time-harmonic waves in Griesmaier and Sylvester (2017 SIAM J. Appl. Math. 77 154–80) to this vector-valued setting. These uncertainty principles yield stability criteria and algorithms for splitting far fields radiated by collections of well-separated sources into the far fields radiated by individual source components, and for the restoration of missing data segments. We discuss proper regularization strategies for these inverse problems, provide stability estimates based on the new uncertainty principles, and comment on reconstruction schemes. A numerical example illustrates our theoretical findings.

Analysis of axis symmetric circular crested elastic wave generated during crack propagation in a plate: A Helmholtz potential technique

This paper presents cylindrical coordinate solutions of axis symmetric circular crested elastic waves that appear due to sudden energy release during incremental crack propagation in a plate. Axis symmetric assumption decouples the elastic wave problem to Lamb (P+SV) and shear (SH) horizontal waves. Helmholtz decomposition principle was used to decompose displacement field in to unknown scalar and vector potentials; and body force vectors to known excitation scalar and vector potentials respectively. Therefore, Navier–Lame equations yield a set of four inhomogeneous wave equations of unknown potentials Φ, Hr, Hθ, Hz and known excitation potentials A*,Br*,Bθ*,Bz*. There are two types of potentials exist in a plate for axis symmetric circular crested Lamb wave: pressure potentials Φ, A* and shear potentials Hθ, Bθ*. Inhomogeneous wave equations for Φ and Hθ were solved due to generalized excitation potentials A* and Bθ* in a form, suitable for numerical calculation. The theoretical formulation shows that elastic waves generated in a plate using excitation potentials follow the Rayleigh-Lamb equations. The resulting solution is a series expansion containing the superposition of all the Lamb wave modes existing for the particular frequency-thickness combination under consideration. In addition, bulk wave solution is also recovered due to the effect of the excitation potentials. The numerical studies modeled the two-dimensional (2D) (circular crested) AE elastic wave propagation in order to simulate the out-of-plane displacement that would be recorded by an AE sensor placed on the plate surface at some distance away from the source. Parameter studies were performed to evaluate: (a) the effect of the pressure and shear potentials; (b) the effect of the thickness-wise location of the excitation potential sources varying from mid-plane to the top surface (source depth effect); (c) the effect of peak time (d) the effect of propagating distance away from the source. A Gaussian pulse is used to model the growth of the excitation potentials during the AE event; as a result, the actual excitation potential follows the error function variation in the time domain. The numerical studies show that the peak amplitude of A0 signal is higher than the peak amplitude of S0 signal and the peak amplitude of bulk wave is not significant compared to S0, A0 peak signals.

Implementation and computational aspects of a 3D elastic wave modelling by PUFEM

This paper presents an enriched finite element model for three dimensional elastic wave problems, in the frequency domain, capable of containing many wavelengths per nodal spacing. This is achieved by applying the plane wave basis decomposition to the three-dimensional (3D) elastic wave equation and expressing the displacement field as a sum of both pressure (P) and shear (S) plane waves. The implementation of this model in 3D presents a number of issues in comparison to its 2D counterpart, especially regarding how S-waves are used in the basis at each node and how to choose the balance between P and S-waves in the approximation space. Various proposed techniques that could be used for the selection of wave directions in 3D are also summarised and used. The developed elements allow us to relax the traditional requirement which consists to consider many nodal points per wavelength, used with low order polynomial based finite elements, and therefore solve elastic wave problems without refining the mesh of the computational domain at each frequency. The effectiveness of the proposed technique is determined by comparing solutions for selected problems with available analytical models or to high resolution numerical results using conventional finite elements, by considering the effect of the mesh size and the number of enriching 3D plane waves. Both balanced and unbalanced choices of plane wave directions in space on structured mesh grids are investigated for assessing the accuracy and conditioning of this 3D PUFEM model for elastic waves.

Uniqueness and Lipschitz stability of an inverse boundary value problem for time-harmonic elastic waves

We consider the inverse problem of determining the Lamé parameters and the density of a three-dimensional elastic body from the local time-harmonic Dirichlet-to-Neumann map. We prove uniqueness and Lipschitz stability of this inverse problem when the Lamé parameters and the density are assumed to be piecewise constant on a given domain partition.

Inverse Random Source Scattering for Elastic Waves

This paper is concerned with the direct and inverse random source scattering problems for elastic waves, where the source is assumed to be driven by an additive white noise. Given the source, the direct problem is to determine the displacement of the random wave field. The inverse problem is to reconstruct the mean and variance of the random source from the boundary measurement of the wave field at multiple frequencies. The direct problem is shown to have a unique mild solution by using a constructive proof. Based on the explicit mild solution, Fredholm integral equations of the first kind are deduced for the inverse problem. The regularized block Kaczmarz method is developed to solve the ill-posed integral equations. The convergence is proved for the method. Numerical experiments are shown to demonstrate the effectiveness of the proposed method.

A numerical approximation of the two-dimensional elastic wave scattering problem via integral equation method

In this paper, a numerical method is proposed to approximate the solution of a two-dimensional scattering problem of time-harmonic elastic wave from a rigid obstacle. By Helmholtz decomposition, the scattering problem is reduced to a system of Helmholtz equations with coupled boundary conditions. Then, we prove that the system of Helmholtz equations has only one solution under certain conditions, and propose an integral equation method to solve it numerically based on Tikhonov regularization method. Finally, numerical examples are presented to show the feasibility and effectiveness of the proposed method.

Global existence for nonlinear elastic waves in high space dimensions

In Georgiev et al. (2005), the authors proved the global existence of solutions to the Cauchy problem of nonlinear elastic waves with memory for space dimensions n≥3. In this manuscript, we show that in high space dimensions n≥4, even without such memory effect, global solutions can be also constructed. The main tool we used is the vector fields method adapted to elastic waves in Sideris’s work (Sideris, 2000).

English

In this article, gauge condition in elastodynamics is explored more to revive its potential capability of simplifying wave propagation problems in elastic medium. The inception of gauge condition in elastodynamics happens from the Navier-Lame equations upon application of Helmholtz theorem. In order to solve the elastic wave problems by potential function approach, the gauge condition provides the necessary conditions for the potential functions. The gauge condition may be considered as the superposition of the separate gauge conditions of Lamb waves and shear horizontal (SH) guided waves respectively, and thus, it may be resolved into corresponding gauges of Lamb waves and SH waves. The manipulation and proper choice of the gauge condition does not violate the classical solutions of elastic waves in plates; rather, it simplifies the problems. The gauge condition allows to obtain the analytical solution of complicated problems in a simplified manner.

A new series solution method for two-dimensional elastic wave scattering along a canyon in half-space

This paper presents a semi-analytical method for studying the two-dimensional problem of elastic wave scattering by surface irregularities in a half-space. The new method makes use of the member of a c-completeness family of wave functions to construct the scattering fields, and then applies equal but opposite tractions to those of the foregoing constructed scattering fields on the horizontal surface of the half-space to produce additional scattering fields. These additional scattering fields are a series of Lamb's solutions. Thus the whole scattering field constructed in the series automatically satisfies the Navier equations, the condition of zero traction on the half-space surface, and the radiation boundary conditions at infinity. Using the traction-free conditions along the canyon surface, the coefficients of the series solutions are determined via a least-squares method. For incident P, SV, and Rayleigh waves, the numerical results are presented for the scattering displacements in the vicinity of a semi-circular canyon in the half-space.

Dynamic Stress Concentrations Due to Scattering of Elastic SV Waves from a Coated Nanoinclusion with Considerations in the Interfacial Region

AbstractThe problem of plane elastic shear waves (SV waves) scattering from a circular nanoinclusion surrounded by an inhomogeneous interphase embedded in an elastic matrix is investigated analytically in this paper. An approach is introduced to account for the simultaneous effects of a graded interphase and surface/interface energy based on Gurtin-Murdoch's model of surface elasticity. Using the wave function expansion method, the Navier equation is solved for all three phases (nanofiber-interphase- matrix). Presenting the results in dimensionless manner, Dynamic Stress Concentration Factors (DSCF) for the present problem are obtained and the effects of several parameters on the results are studied in detail. It is understood that taking the effects of both surface/interface and interphase inhomogeneity into account leads to a significant influence on the DSCF results and consequently on the overall dynamic behavior of the nanocomposites.

Extending the method of fundamental solutions to non-homogeneous elastic wave problems

Two meshfree methods are developed for the numerical solution of the non-homogeneous Cauchy–Navier equations of elastodynamics in an isotropic material. The two approaches differ upon the choice of the basis functions used for the approximation of the unknown wave amplitude. In the first case, the solution is approximated in terms of a linear combination of fundamental solutions of the Navier differential operator with different source points and test frequencies. In the second method the solution is approximated by superposition of acoustic waves, i.e. fundamental solutions of the Helmholtz operator, with different source points and test frequencies. The applicability of the two methods is justified in terms of density results and a convergence result is proven. The accuracy of the methods is illustrated through 2D numerical examples. Applications to interior elastic wave scattering problems are also presented.

Equivalent Robin boundary conditions for acoustic and elastic media

We present equivalent conditions and asymptotic models for a diffraction problem of acoustic and elastic waves. The mathematical problem is set with a Robin boundary condition. Elastic and acoustic waves propagate in a solid medium surrounded by a thin layer of fluid medium. Due to the thinness of the layer with respect to the wavelength, this problem is well suited for the notion of equivalent conditions and the effect of the fluid medium on the solid is as a first approximation local. This approach leads to solve only elastic equations. We derive and validate equivalent conditions up to the third order for the elastic displacement. The construction of equivalent conditions is based on a multiscale expansion in power series of the thickness of the layer for the solution of the transmission problem.

Designing meta material slabs exhibiting negative refraction using topology optimization

This paper proposes a topology optimization based approach for designing meta materials exhibiting a desired negative refraction with high transmission at a given angle of incidence and frequency. The approach considers a finite slab of meta material consisting of axis-symmetric designable unit cells subjected to an exterior field. The unit cell is designed to achieve the desired properties based on tailoring the response of the meta material slab under the exterior field. The approach is directly applicable to physical problems modeled by the Helmholtz equation, such as acoustic, elastic and electromagnetic wave problems. Acoustic meta materials with unit cell size on the order of half the wave length are considered as examples. Optimized designs are presented and their performance under varying frequency and angle of incidence is investigated.

Ballistic electron transport in wrinkled superlattices

Inspired by the problem of elastic wave scattering on wrinkled interfaces, we studied the scattering of ballistic electrons on a wrinkled potential energy region. The electron transmission coefficient depends on both wrinkle amplitude and periodicity, having different behaviors for positive and negative scattering potential energies. For scattering on potential barriers, minibands appear in the electron transmission, as in superlattices, whereas for scattering on periodic potential wells the transmission coefficient has a more complex form. Besides suggesting that tuning of electron transmission is possible by modifying the scattering potential via voltages on wrinkled gate electrodes, our results emphasize the analogies between ballistic electrons and elastic waves even in scattering problems on non-typical configurations.