Joël Piraux

Free and smooth boundaries in 2-D finite-difference schemes for transient elastic waves

A method is proposed for accurately describing arbitrary-shaped free boundaries in single-grid finite-difference schemes for elastodynamics, in a time-domain velocity-stress framework. The basic idea is as follows: fictitious values of the solution are built in vacuum, and injected into the numerical integration scheme near boundaries. The most original feature of this method is the way in which these fictitious values are calculated. They are based on boundary conditions and compatibility conditions satisfied by the successive spatial derivatives of the solution, up to a given order that depends on the spatial accuracy of the integration scheme adopted. Since the work is mostly done during the preprocessing step, the extra computational cost is negligible. Stress-free conditions can be designed at any arbitrary order without any numerical instability, as numerically checked. Using 10 grid nodes per minimal S-wavelength with a propagation distance of 50 wavelengths yields highly accurate results. With 5 grid nodes per minimal S-wavelength, the solution is less accurate but still acceptable. A subcell resolution of the boundary inside the Cartesian meshing is obtained, and the spurious diffractions induced by staircase descriptions of boundaries are avoided. Contrary to what occurs with the vacuum method, the quality of the numerical solution obtained with this method is almost independent of the angle between the free boundary and the Cartesian meshing.

Numerical modeling of transient two-dimensional viscoelastic waves

This paper deals with the numerical modeling of transient mechanical waves in linear viscoelastic solids. Dissipation mechanisms are described using the generalized Zener model. No time convolutions are required thanks to the introduction of memory variables that satisfy local-in-time differential equations. By appropriately choosing the relaxation parameters, it is possible to accurately describe a large range of materials, such as solids with constant quality factors. The evolution equations satisfied by the velocity, the stress, and the memory variables are written in the form of a first-order system of PDEs with a source term. This system is solved by splitting it into two parts: the propagative part is discretized explicitly, using a fourth-order ADER scheme on a Cartesian grid, and the diffusive part is then solved exactly. Jump conditions along the interfaces are discretized by applying an immersed interface method. Numerical experiments of wave propagation in viscoelastic and fluid media show the efficiency of this numerical modeling for dealing with challenging problems, such as multiple scattering configurations.

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.

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

Elastic wave propagation is studied in a heterogeneous two-dimensional 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 scatterer 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.

Numerical modeling of 1D 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 model with frequency-independent 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.

Comparison between a multiple scattering method and direct numerical simulations for elastic wave propagation in concrete

Numerical simulations are performed to study the propagation of elastic waves in a 2-D random heterogeneous medium such as concrete. To reduce spurious numerical artefacts to a negligible level, a fourth-order time-domain numerical scheme and an immersed interface method are used together. Effective properties of the equivalent homogeneous medium are extracted and compared to the predictions of a multiple scattering method (ISA), to evaluate the validity of this latter.

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

Ultrasonic 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 μm) 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 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 – 40 MHz). All these results were compared with a weak scattering model based on the analytical computing of the structure factor.

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.

Comparisons between multiple scattering methods and time‐domain numerical simulations for elastic waves

Propagation of elastic waves in heterogeneous medium composed of scatterers embedded in a homogeneous matrix is considered. Both matrix and scatterers are isotropic elastic media. The multiple scattering regime is assumed, and the focus is put on the coherent field obtained by averaging several equivalent realizations of disorder. Classical methods, such as Independent Scattering Approximation, Foldy or Waterman‐Truells model, provide expressions of the complex effective wave number of the coherent field, leading to an effective phase velocity and effective damping factor. Two‐dimensional time‐domain numerical simulations are performed for studying the validity of these analytical or semianalytical methods. To reduce spurious effects, such as numerical diffraction, to a negligible level, a high‐order numerical scheme and an immersed interface method are used together. Comparisons between theoretical and numerical values of the effective phase velocity and damping factor are proposed and analyzed in terms...

Numerical modeling of transient poroelastic waves in the low frequency range

A numerical method is proposed to simulate the propagation of transient poroelastic waves across heterogeneous media, in the low frequency range. A velocity‐stress formulation of Biots equations is followed, leading to a first‐order differential system. The latter is splitted in two parts: a propagative one discretized by a fourth‐order ADER scheme, and a diffusive one solved analytically. Near sources and interfaces, a space‐time mesh refinement is implemented to capture the small scales of evolution of the diffusive slow compressional wave. Lastly, an immersed interface method is implemented to accurately model the jump conditions at interfaces between the different media. Numerical experiments in one and two dimensions are shown, with porous/porous or fluid/porous interfaces. Comparisons with analytical solutions confirm the efficiency of the approach. [1] G. Chiavassa, B. Lombard, J. Piraux, Numerical modeling of 1‐D transient poroelastic waves in the low‐frequency range, soumis au J. Comput. Appl. M...