finite-difference time-domain (fdtd) method · 2020. 9. 10. · 2 fdtd is a numerical method for...
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Finite-Difference
Time-Domain (FDTD)
Method
Chaiwoot Boonyasiriwat
September 3, 2020
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2
▪ FDTD is a numerical method for solving time-domain
wave equations using the finite difference method.
▪ Explicit FDTD schemes are easy to implement but only
suitable for problem domains that can be discretized as
structured grids, e.g., rectangle, circle, cube, cylinder,
and sphere.
▪ In addition, explicit FDTD schemes are conditionally
stable, i.e., a stability condition must be satisfied.
▪ Typically, the second-order centered FD formula is used
to approximate the time derivative while the spatial
derivatives can be approximated using arbitrary-order
FD formulas.
Introduction to FDTD
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Approximating the derivatives in the wave equation
by the second-order FD approximations
yields the finite-difference equation
where
Explicit FDTD in 1D
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Rearranging the finite difference equation yields the
explicit FDTD scheme:
where is called the Courant number.
Explicit FDTD in 1D
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Consider the wave equation in 3D
Applying the Fourier transform in space and time to the
wave equation yields the dispersion relation
Phase speed is frequency independent.
Hence, the wave equation governs nondispersive waves.
Dispersion Relation
Cohen (2002, p. 25-26)
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Let’s find a dispersion relation for the semi-discrete
wave equation
Inserting the plane wave solution of the form
into the wave equation and rearranging yields the
dispersion relation
Dispersion Relation
Cohen (2002, p. 65-66)
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Similarly, for the fully discrete wave equation
inserting the plane wave solution of the form
and rearranging yields the dispersion relation
Dispersion Relation
Cohen (2002, p. 66)
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In a homogeneous medium the wave speed is given by
Using an FD approximation, the numerical speed can be
computed by
Let’s define the numerical dispersion coefficient q as
When there is no dispersion error, q = 1.
Numerical Dispersion
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Numerical dispersion coefficient can be computed using
Phase velocity:
Group velocity:
Numerical Dispersion
Cohen (2002, p. 101-102)
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▪ “Numerical dispersion produces parasitic waves since
the numerical velocity is frequency dependent.”
▪ “These parasitic waves can leave the physical wave
and produce some ringing features in the waveform.”
Concept of Numerical Dispersion
Cohen (2002, p. 102-103)
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Semi-discrete wave equation
using the second-order scheme:
using a fourth-order scheme:
Note the leading error term in each case.
Order of Numerical Dispersion
Cohen (2002, p. 104)
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Fully-discrete wave equation using second-order in time
and second-order in space:
and fourth-order in space:
Order of Numerical Dispersion
Cohen (2002, p. 105)
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Dispersion curve for second-order scheme
Dispersion Curve
K=1/n (n = #points/wavelength)
Dis
per
sio
n C
oef
fici
ent
Cohen (2002, p. 114)
Order 2
14
Orders 2, 4, 6, 8, 10, 12, 14
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▪ To avoid a serious numerical dispersion issue, we need
to determine the number of grid points needed for
sampling a wavelength.
▪ This leads to a numerical dispersion condition as
shown, e.g., in the previous slide.
▪ This condition is used to determine the grid spacing
in terms of medium property, e.g. velocity c, and wave
frequency f:
where n is the number of points per wavelength
Numerical Dispersion Condition
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“Accuracy of approximation depends on the direction of
wave propagation.”
Isotropy Curve
Cohen (2002, p. 118-119)
Numerical dispersion
versus propagation angle
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16
▪ Von Neumann stability analysis is the most widely
used method for analyzing the stability of a finite
difference scheme for time-dependent problems, e.g.,
wave equations, diffusion equation.
▪ In wave propagation problems, plane wave solutions
are used for stability analysis.
▪ However, the method can only be used for linear PDEs.
▪ For nonlinear problems, Lyapunov stability analysis
has been widely used.
Linear Stability Analysis
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17
Consider an acoustic wave propagating in a 1D
homogeneous medium governed by
Assuming a plane wave solution given by
one can obtain the dispersion relation
Acoustic Plane Wave
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18
Consider the case when is complex
The plane wave solution becomes
When I > 0, we obtain an evanescent wave.
When I < 0, wave amplitude increases exponentially with time. This can happen when a numerical solution is
unstable.
Instability
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Dispersion relation for the second-order scheme is
We then obtain
This leads to the CFL condition (Courant et al., 1928)
Stability Analysis
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20
Consider the first-order coupled acoustic wave equations
where p is pressure, u particle velocity, K bulk modulus,
and mass density.
Let’s use the second-order FD
In both space and time on the
staggered grid to solve the
acoustic wave equation
1D Acoustic Wave Equations
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21
We then obtain the finite difference equations
1D Acoustic Wave Equations
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22
Assuming plane wave solutions
we can obtain
where
Stability Analysis: Acoustics
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23
▪ The coefficient matrix is called the amplification
matrix whose eigenvalues determine the stability of
the FD scheme.
▪ If , wave amplitudes are amplified every time
step leading to a blow up and the scheme is unstable.
▪ Otherwise the scheme is stable as long as a stability
condition is satisfied.
▪ In this case, the characteristic equation is
▪ The eigenvalues are
Stability Analysis: Acoustics
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▪ If , “the eigenvalues are real and of magnitude
greater than 1, so giving instability (Woolfson and Pert,
1999, p. 163).
▪ If , the eigenvalues are complex.
▪ If , both eigenvalues are equal to -1.
▪ In the last two cases, the scheme is stable.
▪ The stability conditions are satisfied when the CFL
condition is satisfied
▪ This is in agreement with the stability analysis of the
second-order wave equation using the second-order
FDTD scheme.
Stability Analysis: Acoustics
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25
▪ Recall the plane wave solution
▪ At the next time step the wave field will become
▪ Phase shift due to wave propagation is
▪ Recall the dispersion relation of the numerical solution
▪ Using the identity , the
dispersion relation becomes
Phase Shift
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26
▪ Consequently, we obtain the numerical phase error
▪ When C = 1, and there is no phase error, i.e.,
no numerical dispersion – the FDTD solution is equal to
the exact solution.
Phase Shift (continued)
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27
In practice, there usually exists an energy source and the
governing equation will have an additional source term
FD implementation becomes
for i = 2,n-1
Source Term
where is is the index of source position.
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28
▪ In many situations we want to model wave propagation in unbounded domains.
▪ It is impossible to construct a physical model representing an unbounded domain.
▪ Only part of the domain can be represented.▪ The region of interest (ROI) is only used in a
computational study.▪ Reflection from ROI boundaries is undesirable
and must be reduced as much as possible to avoid the interference of unwanted reflected waves with other waves.
Wave in Unbounded Domains
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29
Consider the 1D wave equation
where L is the two-way propagation operator which can be decomposed into a concatenation of two one-way (paraxial) operators:
Clayton-Engquist ABC
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Applying each one-way operator to a wave field yields one-way wave equations
which govern waves that propagate only in one direction, i.e., left or right.
One-Way Wave Equations
Left-going wave
Right-going wave
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The paraxial operators can be used as an absorbing boundary condition to avoid boundary reflection as follows.
This method can perfectly absorbs only normally incident waves.
Clayton-Engquist ABC
Allow left-going
wave only
Allow right-going
wave only
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▪ Clayton-Engquist (1977) proposed to use paraxial wave equations as absorbing boundary conditions for 2D acoustic and elastic wave propagation.
▪ Consider the dispersion relation in 2D
▪ Rearranging the last equation yields
Paraxial Approximation
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Padé or rational approximation is usually used to expand the square root term as a rational function
where the derivatives of the rational function are the same as the original function up to derivative of order m+n.
Padé Approximation
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This is equivalent to
where is the truncation error.Suppose is an analytic function and can be expanded as a Maclaurin series
Padé Approximation
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We then have
This leads to a system of n+m+1 linear equations whose solution is the coefficients of the rational function.
Padé Approximation
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36
▪ Padé approximation has been widely used in computational science research because it usually provides a more accurate function approximation than Taylor’s expansion of the same order.
▪ When Taylor series is divergent, Padé series is usually convergent.
Padé Approximation
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37
Using rational approximation, we can obtain various approximations
Paraxial Approximation
0 degree
15 degree
45 degree
Reference: Clayton and Engquist (1977, 1980)
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Paraxial wave equations corresponding to the dispersion relations are
Paraxial Wave Equations
0 degree
15 degree
45 degree
Reference: Clayton and Engquist (1977)
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39
Paraxial Approximation
Image Source: Clayton and Engquist (1977)
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▪ Applying these paraxial wave equations at top and bottom boundaries will reduce spurious reflections from the boundaries.
▪ The boundary conditions for left and right boundaries can be obtained by switching the locations of x and z in the previous equations.
▪ Higher-order ABCs are too complicated and only the lower-order (2 and 4) ABCs are used.
▪ Many ABCs have been proposed later including that of Keys (1985)
Clayton-Engquist ABC
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Derive the paraxial dispersion relations
Exercise
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where are locations of the left and
right boundaries, respectively.42
▪ Sponge ABL is an artificial layer attached to the physical domain at the boundaries.
▪ Wave fields in the sponge zones are gradually damped to reduce boundary reflection.
Cerjan Sponge ABL
ABL 1D Physical Domain ABL
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43
▪ ABL proposed by Cerjan et al. (1985) can be used for modeling in both time-space and time-wavenumber (Fourier) domains.
▪ They argued that previously proposed ABCs cannot be used for modeling in the Fourier domain.
▪ Later Clayton and Engquist (1980) extended their method to the Fourier domain.
▪ Sponge ABL is simpler but more expensive than ABC.
Cerjan Sponge ABL
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44
▪ Perfectly matched layer (PML) proposed by Bérenger in 1994 for modeling EM wave propagation is the most widely used method for simulation of wave propagation in unbounded domains.
▪ PML is also an artificial layer to absorb wave energy.
Bérenger PML
PML 1D Physical Domain PML
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45
▪ Bérenger’s PML for EM is an artificial medium whose impedance is equal to that of free space, i.e.,
▪ In 2D TE case, EM waves in the PML satisfy
Bérenger PML
= electrical
conductivity
* = magnetic
conductivity
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46
▪ To match the impedance of free space, Bérenger’s PML must satisfy the condition
▪ An additional complication in Bérenger work is that the magnetic field must be split into two components
▪ The resulting method is then called split-field PML (SPML) which is conditionally stable.
Bérenger PML
Reference: Abarbanel and Gottlieb (1998)
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47
▪ Many efforts have been done to improve the stability of PML leading to nonsplit PML formulations (See Zhou (2003)).
▪ Modern PML formulations can be directly derived by using coordinate stretching (Chew and Liu, 1996) or analytic continuation (Johnson, 2010), namely
where vanishes in the physical region.
Nonsplit PML
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48
▪ Inserting the stretched coordinate into a plane wave yields an evanescent wave
▪ Using the stretched coordinate, the partial derivative becomes
Coordinate Stretching
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49
▪ Let’s apply PML to the acoustic wave equations
▪ The first step is to temporal Fourier transform to the equations and we obtain
▪ Then apply the stretched coordinate to the spatial derivatives.
PML for Acoustic Wave
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50
▪ We then obtain
▪ Taking inverse Fourier transform yields the PML formulation of acoustic wave equations
PML for Acoustic Wave
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51
▪ In EM problems, the damping function has the physical meaning of conductivity.
▪ In acoustic and elastic cases, there is no direct analogy to any physical quantity but it could be related to viscosity of the medium.
▪ Bérenger (1994, p. 191) used
where is the PML thickness.
PML Damping Function
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52
Collino and Tsogka (2001, p. 301) used
where
and R is reflection coefficient.
PML Damping Function
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53
▪ Many researchers have combined various methods to develop hybrid methods.
▪ Recently, Liu and Sen (2010) proposed a hybrid method to combine Clayton-Engquist ABC with an absorbing layer.
▪ In the absorbing layer, wave fields are computed by the two-way wave equationand one-way wave equation
▪ Total wave field is a linear combination
Hybrid Methods
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▪ Abarbanel, S., and D. Gottlieb, 1988, On the construction and analysis of absorbing layers in CEM, Applied Numerical Mathematics, 27, no. 4, 331-340.
▪ Bérenger, J. P., 1994, A perfectly matched layer for the absorption of electromagnetic waves, Journal of Computational Physics, 114, 185-200.
▪ Cerjan, C., D. Kosloff, R. Kosloff, M. Reshef, 1985, A nonreflecting boundary condition for discrete acoustic and elastic wave equations, Geophysics, 50, no. 4, 705-708.
▪ Clayton, R., and B. Engquist, 1977, Absorbing boundary conditions for acoustic and elastic wave equation, Bulletin of the Seismological Society of America, 67, no. 6, 1529-1540.
▪ Clayton, R., and B. Engquist, 1980, Absorbing boundary conditions for wave-equation migration, Geophysics, 45, no. 5, 895-904.
▪ Cohen, G. C., 2002, Higher-Order Numerical Methods for Transient Wave Equations, Springer.
▪ Collino, F., and C. Tsogka, 2001, Application of the perfectly matched absorbing layer model to the linear elastodynamic problem in anisotropic heterogeneous media, Geophysics, 66, no. 1, 294-307.
▪ Courant, R., K. Friedrichs, and H. Lewy, 1928, On the partial differential equations of mathematical physics, Physik. Math. Ann., 100, 32-74.
▪ Johnson, S. G., 2010, Notes on perfectly matched layers, Online MIT Course Notes.▪ Woolfson, M. M., and G. J. Pert, 1999, An Introduction to Computer Simulation, Oxford
University Press.
References
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