303 research outputs found
The Explicit Simplified Interface Method for compressible multicomponent flows
This paper concerns the numerical approximation of the Euler equations for
multicomponent flows. A numerical method is proposed to reduce spurious
oscillations that classically occur around material interfaces. It is based on
the "Explicit Simplified Interface Method" (ESIM), previously developed in the
linear case of acoustics with stationary interfaces (2001, J. Comput. Phys.
168, pp.~227-248). This technique amounts to a higher order extension of the
"Ghost Fluid Method" introduced in Euler multicomponent flows (1999, J. Comput.
Phys. 152, pp. 457-492). The ESIM is coupled to sophisticated shock-capturing
schemes for time-marching, and to level-sets for tracking material interfaces.
Jump conditions satisfied by the exact solution and by its spatial derivative
are incorporated in numerical schemes, ensuring a subcell resolution of
material interfaces inside the meshing. Numerical experiments show the
efficiency of the method for rich-structured flows.Comment: to be published in SIAM Journal of Scientific Computing (2005
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
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
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
Dilatation of a one-dimensional nonlinear crack impacted by a periodic elastic wave
The interactions between linear elastic waves and a nonlinear crack with
finite compressibility are studied in the one-dimensional context. Numerical
studies on a hyperbolic model of contact with sinusoidal forcing have shown
that the mean values of the scattered elastic displacements are discontinuous
across the crack. The mean dilatation of the crack also increases with the
amplitude of the forcing levels. The aim of the present theoretical study is to
analyse these nonlinear processes under a larger range of nonlinear jump
conditions. For this purpose, the problem is reduced to a nonlinear
differential equation. The dependence of the periodic solution on the forcing
amplitude is quantified under sinusoidal forcing conditions. Bounds for the
mean, maximum and minimum values of the solution are presented. Lastly,
periodic forcing with a null mean value is addressed. In that case, a result
about the mean dilatation of the crack is obtained.Comment: submitted to the SIAM J. App. Mat
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 investigation of acoustic solitons
Acoustic solitons can be obtained by considering the propagation of large
amplitude sound waves across a set of Helmholtz resonators. The model proposed
by Sugimoto and his coauthors has been validated experimentally in previous
works. Here we examine some of its theoretical properties: low-frequency
regime, balance of energy, stability. We propose also numerical experiments
illustrating typical features of solitary waves
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
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