Numerical modeling of elastic waves across imperfect contacts

A numerical method is described for studying how elastic waves interact with imperfect contacts such as fractures or glue layers existing between elastic solids. These contacts have been classicaly modeled by interfaces, using a simple rheological model consisting of a combination of normal and tangential linear springs and masses. The jump conditions satisfied by the elastic fields along the interfaces are called the "spring-mass conditions". By tuning the stiffness and mass values, it is possible to model various degrees of contact, from perfect bonding to stress-free surfaces. The conservation laws satisfied outside the interfaces are integrated using classical finite-difference schemes. The key problem arising here is how to discretize the spring-mass conditions, and how to insert them into a finite-difference scheme: this was the aim of the present paper. For this purpose, we adapted an interface method previously developed for use with perfect contacts [J. Comput. Phys. 195 (2004) 90-116]. This numerical method also describes closely the geometry of arbitrarily-shaped interfaces on a uniform Cartesian grid, at negligible extra computational cost. Comparisons with original analytical solutions show the efficiency of this approach.Comment: to be published in SIAM Journal of Scientific Computing (2006

Modeling 1-D elastic P-waves in a fractured rock with hyperbolic jump conditions

The propagation of elastic waves in a fractured rock is investigated, both theoretically and numerically. Outside the fractures, the propagation of compressional waves is described in the simple framework of one-dimensional linear elastodynamics. The focus here is on the interactions between the waves and fractures: for this purpose, the mechanical behavior of the fractures is modeled using nonlinear jump conditions deduced from the Bandis-Barton model classicaly used in geomechanics. Well-posedness of the initial-boundary value problem thus obtained is proved. Numerical modeling is performed by coupling a time-domain finite-difference scheme with an interface method accounting for the jump conditions. The numerical experiments show the effects of contact nonlinearities. The harmonics generated may provide a non-destructive means of evaluating the mechanical properties of fractures.Comment: accepted and to be published in the Journal of Computational and Applied Mathematic

How to incorporate the spring-mass conditions in finite-difference schemes

The spring-mass conditions are an efficient way to model imperfect contacts between elastic media. These conditions link together the limit values of the elastic stress and of the elastic displacement on both sides of interfaces. To insert these spring-mass conditions in classical finite-difference schemes, we use an interface method, the Explicit Simplified Interface Method (ESIM). This insertion is automatic for a wide class of schemes. The interfaces do not need to coincide with the uniform cartesian grid. The local truncation error analysis and numerical experiments show that the ESIM maintains, with interfaces, properties of the schemes in homogeneous medium

A new interface method for hyperbolic problems with discontinuous coefficients: one-dimensional acoustic example

International audienceA new numerical method, called the Explicit Simplified Interface Method (ESIM), is developed in the context of acoustic wave propagation in heterogeneous media. Equations of acoustics are written as a first-order linear hyperbolic system. Away from interfaces, a standard scheme (Lax-Wendroff, TVD, WENO...) is used in a classical way. Near interfaces, the same scheme is used, but it is applied on a set of modified values deduced from numerical values and from jump conditions at interfaces. It amounts to modify the scheme so that its order of accuracy is maintained at irregular points, despite the non-smoothness of the solution. This easy to implement interface method requires few additional computational resources and it can be applied to other partial differential equations

Numerical treatment of two-dimensional interfaces for acoustic and elastic waves

International audienceWe present a numerical method to take into account 2D arbitrary-shaped interfaces in classical finite-difference schemes, on a uniform Cartesian grid. This work extends the "Explicit Simplified Interface Method" (ESIM), previously proposed in 1D (2001, J. Comput. Phys., vol 168, pp. 227-248). The physical problem under study concerns the linear hyperbolic systems of acoustics and elastodynamics, with stationary interfaces. Our method maintains, near the interfaces, properties of the schemes in homogeneous medium, such as the order of accuracy and the stability limit. Moreover, it enforces the numerical solution to satisfy the exact interface conditions. Lastly, it provides subcell geometrical features of the interface inside the meshing. The ESIM can be coupled automatically with a wide class of numerical schemes (Lax-Wendroff, flux-limiter schemes,...) for a negligible additional computational cost. Throughout the paper, we focus on the challenging case of an interface between a fluid and an elastic solid. In numerical experiments, we provide comparisons between numerical solutions and original analytic solutions, showing the efficiency of the method

Propagation of compressional elastic waves through a 1-D medium with contact nonlinearities

Propagation of monochromatic elastic waves across cracks is investigated in 1D, both theoretically and numerically. Cracks are modeled by nonlinear jump conditions. The mean dilatation of a single crack and the generation of harmonics are estimated by a perturbation analysis, and computed by the harmonic balance method. With a periodic and finite network of cracks, direct numerical simulations are performed and compared with Bloch-Floquet's analysis.Comment: Article following the fifth meeting of GdR 2501 (CNRS

A 2D Time domain numerical method for the low frequency biot model

National audienceA numerical method is proposed to simulate the propagation of transient poroelastic waves across 2D heterogeneous media, in the low frequency range. A velocity-stress formulation of Biot's equations is followed, leading to a first-order system of partial differential equations. This system is splitted in two parts: a propagative one discretized by a fourth-order ADER scheme, and a diffusive one that is solved analytically. Near material interfaces, a space-time mesh refinement is implemented to capture the small spatial scales related to the slow compressional wave. Lastly, an immersed interface method is implemented to accurately model the jump conditions between the different media and the geometry of the interfaces. Numerical experiments and comparisons with exact solutions confirm the efficiency and the accuracy of the approach

Ultrasound characterization of red blood cells distribution: a wave scattering simulation study

International audienceUltrasonic backscattered signals from blood contain frequency-dependent information that can be used to obtain quantitative parameters describing the aggregation state of red blood cells (RBCs). However the relation between the parameters describing the aggregation level and the backscatterer coefficient needs to be better clarified. For that purpose, numerical wave simulations were performed to generate backscattered signals that mimic the response of two-dimensional (2D) RBC distributions to an ultrasound excitation. The simulated signals were computed with a time-domain method that has the advantages of requiring no physical approximations (within the framework of linear acoustics) and of limiting the numerical artefacts induced by the discretization of object interfaces. In the simple case of disaggregated RBCs, the relationship between the backscatter amplitude and scatterer concentration was studied. Backscatter coefficients (BSC) in the frequency range 10 to 20 MHz were calculated for weak scattering infinite cylinders (radius 2.8 $\mu$m) at concentrations ranging from 6 to 36$\%$. At low concentration, the BSC increased with scatterer concentrations; at higher concentrations, the BSC reached a maximum and then decreased with increasing concentration, as it was noted by previous authors in \emph{in vitro} blood experiments. In the case of aggregated RBCs, the relationship between the backscatter frequency dependence and level of aggregation at a concentration of 24$\%$ was studied for a larger frequency band (10 - 50 MHz). All these results were compared with a weak scattering model based on the analytical computing of the structure factor

Time-domain numerical simulations of multiple scattering to extract elastic effective wavenumbers

Elastic wave propagation is studied in a heterogeneous 2-D medium consisting of an elastic matrix containing randomly distributed circular elastic inclusions. The aim of this study is to determine the effective wavenumbers when the incident wavelength is similar to the radius of the inclusions. A purely numerical methodology is presented, with which the limitations usually associated with low scatterer concentrations can be avoided. The elastodynamic equations are integrated by a fourth-order time-domain numerical scheme. An immersed interface method is used to accurately discretize the interfaces on a Cartesian grid. The effective field is extracted from the simulated data, and signal-processing tools are used to obtain the complex effective wavenumbers. The numerical reference solution thus-obtained can be used to check the validity of multiple scattering analytical models. The method is applied to the case of concrete. A parametric study is performed on longitudinal and transverse incident plane waves at various scatterers concentrations. The phase velocities and attenuations determined numerically are compared with predictions obtained with multiple scattering models, such as the Independent Scattering Approximation model, the Waterman-Truell model, and the more recent Conoir-Norris model.Comment: Waves in Random and Complex Media (2012) XX

Numerical modeling of 1-D transient poroelastic waves in the low-frequency range

Propagation of transient mechanical waves in porous media is numerically investigated in 1D. The framework is the linear Biot's model with frequency-independant coefficients. The coexistence of a propagating fast wave and a diffusive slow wave makes numerical modeling tricky. A method combining three numerical tools is proposed: a fourth-order ADER scheme with time-splitting to deal with the time-marching, a space-time mesh refinement to account for the small-scale evolution of the slow wave, and an interface method to enforce the jump conditions at interfaces. Comparisons with analytical solutions confirm the validity of this approach.Comment: submitted to the Journal of Computational and Applied Mathematics (2008