263 research outputs found
Semi-analytical and numerical methods for computing transient waves in 2D acoustic / poroelastic stratified media
Wave propagation in a stratified fluid / porous medium is studied here using
analytical and numerical methods. The semi-analytical method is based on an
exact stiffness matrix method coupled with a matrix conditioning procedure,
preventing the occurrence of poorly conditioned numerical systems. Special
attention is paid to calculating the Fourier integrals. The numerical method is
based on a high order finite-difference time-domain scheme. Mesh refinement is
applied near the interfaces to discretize the slow compressional diffusive wave
predicted by Biot's theory. Lastly, an immersed interface method is used to
discretize the boundary conditions. The numerical benchmarks are based on
realistic soil parameters and on various degrees of hydraulic contact at the
fluid / porous boundary. The time evolution of the acoustic pressure and the
porous velocity is plotted in the case of one and four interfaces. The
excellent level of agreement found to exist between the two approaches confirms
the validity of both methods, which cross-checks them and provides useful tools
for future researches.Comment: Wave Motion (2012) XX
Wave propagation in a fractional viscoelastic Andrade medium: diffusive approximation and numerical modeling
This study focuses on the numerical modeling of wave propagation in
fractionally-dissipative media. These viscoelastic models are such that the
attenuation is frequency dependent and follows a power law with non-integer
exponent. As a prototypical example, the Andrade model is chosen for its
simplicity and its satisfactory fits of experimental flow laws in rocks and
metals. The corresponding constitutive equation features a fractional
derivative in time, a non-local term that can be expressed as a convolution
product which direct implementation bears substantial memory cost. To
circumvent this limitation, a diffusive representation approach is deployed,
replacing the convolution product by an integral of a function satisfying a
local time-domain ordinary differential equation. An associated quadrature
formula yields a local-in-time system of partial differential equations, which
is then proven to be well-posed. The properties of the resulting model are also
compared to those of the original Andrade model. The quadrature scheme
associated with the diffusive approximation, and constructed either from a
classical polynomial approach or from a constrained optimization method, is
investigated to finally highlight the benefits of using the latter approach.
Wave propagation simulations in homogeneous domains are performed within a
split formulation framework that yields an optimal stability condition and
which features a joint fourth-order time-marching scheme coupled with an exact
integration step. A set of numerical experiments is presented to assess the
efficiency of the diffusive approximation method for such wave propagation
problems.Comment: submitted to Wave Motio
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
Diffusive approximation of a time-fractional Burger's equation in nonlinear acoustics
A fractional time derivative is introduced into the Burger's equation to
model losses of nonlinear waves. This term amounts to a time convolution
product, which greatly penalizes the numerical modeling. A diffusive
representation of the fractional derivative is adopted here, replacing this
nonlocal operator by a continuum of memory variables that satisfy local-in-time
ordinary differential equations. Then a quadrature formula yields a system of
local partial differential equations, well-suited to numerical integration. The
determination of the quadrature coefficients is crucial to ensure both the
well-posedness of the system and the computational efficiency of the diffusive
approximation. For this purpose, optimization with constraint is shown to be a
very efficient strategy. Strang splitting is used to solve successively the
hyperbolic part by a shock-capturing scheme, and the diffusive part exactly.
Numerical experiments are proposed to assess the efficiency of the numerical
modeling, and to illustrate the effect of the fractional attenuation on the
wave propagation.Comment: submitted to Siam SIA
Wave propagation across acoustic / Biot's media: a finite-difference method
Numerical methods are developed to simulate the wave propagation in
heterogeneous 2D fluid / poroelastic media. Wave propagation is described by
the usual acoustics equations (in the fluid medium) and by the low-frequency
Biot's equations (in the porous medium). Interface conditions are introduced to
model various hydraulic contacts between the two media: open pores, sealed
pores, and imperfect pores. Well-possedness of the initial-boundary value
problem is proven. Cartesian grid numerical methods previously developed in
porous heterogeneous media are adapted to the present context: a fourth-order
ADER scheme with Strang splitting for time-marching; a space-time
mesh-refinement to capture the slow compressional wave predicted by Biot's
theory; and an immersed interface method to discretize the interface conditions
and to introduce a subcell resolution. Numerical experiments and comparisons
with exact solutions are proposed for the three types of interface conditions,
demonstrating the accuracy of the approach.Comment: Communications in Computational Physics (2012) XX
Time domain numerical modeling of wave propagation in 2D heterogeneous porous media
This paper deals with the numerical modeling of wave propagation in porous
media described by Biot's theory. The viscous efforts between the fluid and the
elastic skeleton are assumed to be a linear function of the relative velocity,
which is valid in the low-frequency range. The coexistence of propagating fast
compressional wave and shear wave, and of a diffusive slow compressional wave,
makes numerical modeling tricky. To avoid restrictions on the time step, the
Biot's system is splitted into two parts: the propagative part is discretized
by a fourth-order ADER scheme, while the diffusive part is solved analytically.
Near the material interfaces, a space-time mesh refinement is implemented to
capture the small spatial scales related to the slow compressional wave. The
jump conditions along the interfaces are discretized by an immersed interface
method. Numerical experiments and comparisons with exact solutions confirm the
accuracy of the numerical modeling. The efficiency of the approach is
illustrated by simulations of multiple scattering.Comment: Journal of Computational Physics (March 2011
Aerodynamic noise from rigid trailing edges with finite porous extensions
This paper investigates the effects of finite flat porous extensions to
semi-infinite impermeable flat plates in an attempt to control trailing-edge
noise through bio-inspired adaptations. Specifically the problem of sound
generated by a gust convecting in uniform mean steady flow scattering off the
trailing edge and permeable-impermeable junction is considered. This setup
supposes that any realistic trailing-edge adaptation to a blade would be
sufficiently small so that the turbulent boundary layer encapsulates both the
porous edge and the permeable-impermeable junction, and therefore the
interaction of acoustics generated at these two discontinuous boundaries is
important. The acoustic problem is tackled analytically through use of the
Wiener-Hopf method. A two-dimensional matrix Wiener-Hopf problem arises due to
the two interaction points (the trailing edge and the permeable-impermeable
junction). This paper discusses a new iterative method for solving this matrix
Wiener-Hopf equation which extends to further two-dimensional problems in
particular those involving analytic terms that exponentially grow in the upper
or lower half planes. This method is an extension of the commonly used "pole
removal" technique and avoids the needs for full matrix factorisation.
Convergence of this iterative method to an exact solution is shown to be
particularly fast when terms neglected in the second step are formally smaller
than all other terms retained. The final acoustic solution highlights the
effects of the permeable-impermeable junction on the generated noise, in
particular how this junction affects the far-field noise generated by
high-frequency gusts by creating an interference to typical trailing-edge
scattering. This effect results in partially porous plates predicting a lower
noise reduction than fully porous plates when compared to fully impermeable
plates.Comment: LaTeX, 20 pp., 19 graphics in 6 figure
Seismic wave propagation, attenuation and scattering in porous media across various scales
The theory of poroelasticity describes the mechanics of fluid saturated deformable porous solids. Poroelastic theory accurately model the seismic wave propagation in oil and gas reservoirs with an extension to other porous medium e.g. living bones and soft tissues. The poroelastic theory also incorporates the attenuation of energy caused by the relative motion between solid and fluid excited by a point source perturbation.First and foremost, this thesis presents a comprehensive description of seismic attenuation in a viscoacoustic medium by using a fractional derivative approach. Representation of attenuation by using a fractional derivative avoids the solution of augmented system represented by memory process i.e. convolution. In this thesis, a highly accurate time integration scheme, known as rapid expansion method, with a spectral accuracy is implemented to solve the viscoacoustic wave equation.Next, this thesis describes the implementation of a high-order weight-adjusted discontinuous Galerkin (WADG) scheme for the numerical solution of two and three-dimensional (3D) wave propagation problems in anisotropic porous media. The use of a penalty-based numerical flux avoids the diagonalization of Jacobian matrices into polarized wave constituents necessary when solving element-wise Riemann problems.Additionally, a system of hyperbolic partial differential equations describing Biot's poroelastic wave equation for quasi-static Poisseulle and potential flow is also introduced. To incorporate effects from micro-heterogeneities due to pores, we have used the Johnson-Koplik-Dashen (JKD) model of dynamic permeability, which also account for frequency-dependent viscous dissipation caused by wave-induced pore fluids.Next, this thesis uses a model to quantify capillary effects on velocity and attenuation. Studies that have attempted to extend Biot's poroelasticity to include capillary effects found changes in fast P-wave velocity of up to 5 % between the sonic and ultrasonic frequency ranges
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