ESG2006, Grenoble, 30/08-01/09/2006 Third International Symposium on the Effects of Surface Geology on Seismic Motion Grenoble, France, 30 August - 1 September 2006 Paper Number: 43 WAVE2D: A COMPUTER PROGRAM FOR SH SEISMIC WAVE PROPAGATION IN HETEROGENEOUS MEDIA BY THE FOURIER PSEUDO-SPECTRAL METHOD Roberto PAOLUCCI 1 , Davide SPINELLI 2 1 Dept. of Structural Engineering, Politecnico di Milano, Italy. 2 Engineering consultant. Present address: SOIL s.r.l., Milano, Italy. ABSTRACT – A numerical code is introduced for SH seismic wave propagation in heterogeneous linear-viscoelastic media, based on the pseudo-spectral Fourier method. The program is freely accessible on the Internet and runs under Matlab environment. It enjoys a number of useful pre- and post-processing features, such as an on-line user- friendly construction of the numerical grid, plot of output time histories, convolution, filtering, calculation of Fourier and response spectra at selected receivers. 1. Introduction The pseudo-spectral Fourier method for numerical analysis of seismic wave propagation has been one of the major alternatives to more classical finite differences approaches. The first developments on this method date back to the early 80s (Kosloff and Baysal, 1982), yet its main feature of taking full advantage of fast Fourier transform algorithms to calculate spatial derivatives has made it attractive to a large number of researchers (Reshef et al., 1988; Paolucci, 1989; Witte 1989; Furumura et al., 1998). The main drawback of the Fourier method is that it involves periodicity of the computational domain, so that the local conditions involving either the free-surface or the transparent boundaries cannot be imposed in a straightforward way, but they require a special treatment that typically involves a decrease of accuracy of the numerical results. This is the main reason why, starting from the early 90s, other classes of spectral methods for elastic wave propagation analyses have emerged, firstly with the introduction of Chebyshev polynomials for the spatial decomposition of the wavefield (Kosloff et al., 1990) and subsequently of Lagrange polynomials coupled with Legendre-Gauss-Lobatto quadrature formulas (Faccioli et al., 1997; Komatitsch and Villotte, 1998). The latter spectral approach is probably the most suitable for seismic wave propagation problems, including arbitrary topographic profiles, since the free-surface boundary condition is included in an exact way in the variational formulation of the elastodynamic equations. This paper introduces a numerical code based on the Fourier method, WAVE2D, devised for SH wave propagation in heterogeneous media. The main features of the method, as well as the validations with independent solutions and details of implementation of boundary and initial conditions, are thoroughly described in Paolucci (1989) and in Faccioli (1991). Although the code was originally developed in the early 90s, it has recently undergone a significant update in the framework of the Sismovalp project (Seismic risk in Alpine valleys), sponsored by the EU within the Interreg IIIB Alpine Space research programme. 1
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ESG2006, Grenoble, 30/08-01/09/2006
Third International Symposium on the Effects of Surface Geology on Seismic Motion Grenoble, France, 30 August - 1 September 2006
Paper Number: 43
WAVE2D: A COMPUTER PROGRAM FOR SH SEISMIC WAVE PROPAGATION IN HETEROGENEOUS MEDIA BY THE FOURIER
PSEUDO-SPECTRAL METHOD
Roberto PAOLUCCI1, Davide SPINELLI
2
1 Dept. of Structural Engineering, Politecnico di Milano, Italy. 2 Engineering consultant. Present address: SOIL s.r.l., Milano, Italy.
ABSTRACT – A numerical code is introduced for SH seismic wave propagation in heterogeneous linear-viscoelastic media, based on the pseudo-spectral Fourier method. The program is freely accessible on the Internet and runs under Matlab environment. It enjoys a number of useful pre- and post-processing features, such as an on-line user-friendly construction of the numerical grid, plot of output time histories, convolution, filtering, calculation of Fourier and response spectra at selected receivers.
1. Introduction
The pseudo-spectral Fourier method for numerical analysis of seismic wave propagation has been one of the major alternatives to more classical finite differences approaches. The first developments on this method date back to the early 80s (Kosloff and Baysal, 1982), yet its main feature of taking full advantage of fast Fourier transform algorithms to calculate spatial derivatives has made it attractive to a large number of researchers (Reshef et al., 1988; Paolucci, 1989; Witte 1989; Furumura et al., 1998). The main drawback of the Fourier method is that it involves periodicity of the computational domain, so that the local conditions involving either the free-surface or the transparent boundaries cannot be imposed in a straightforward way, but they require a special treatment that typically involves a decrease of accuracy of the numerical results. This is the main reason why, starting from the early 90s, other classes of spectral methods for elastic wave propagation analyses have emerged, firstly with the introduction of Chebyshev polynomials for the spatial decomposition of the wavefield (Kosloff et al., 1990) and subsequently of Lagrange polynomials coupled with Legendre-Gauss-Lobatto quadrature formulas (Faccioli et al., 1997; Komatitsch and Villotte, 1998). The latter spectral approach is probably the most suitable for seismic wave propagation problems, including arbitrary topographic profiles, since the free-surface boundary condition is included in an exact way in the variational formulation of the elastodynamic equations. This paper introduces a numerical code based on the Fourier method, WAVE2D, devised for SH wave propagation in heterogeneous media. The main features of the method, as well as the validations with independent solutions and details of implementation of boundary and initial conditions, are thoroughly described in Paolucci (1989) and in Faccioli (1991). Although the code was originally developed in the early 90s, it has recently undergone a significant update in the framework of the Sismovalp project (Seismic risk in Alpine valleys), sponsored by the EU within the Interreg IIIB Alpine Space research programme.
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Our aim is to make freely available an easy-to-use computer code for seismic wave propagation analyses in Alpine valleys. The release of WAVE2D, the main features of which will be described in detail in this paper, is in parallel with the one of REFORM, based on the method devised by Paolucci (1999), for the fast evaluation of the resonance frequencies of alluvial valleys by Rayleigh’s method, including 2D and 3D simplified geometries,.
2. Main features of WAVE2D WAVE2D is based on a series of Fortran routines that have been assembled into a single pre-post processing graphical interface, based on the Matlab programming language (versions tested: R13 and R14). A sketch of the introductory window is plotted in Fig. 1.
Figure 1.Introductory window of WAVE2D The main features of WAVE2D include: • Vertical and oblique SH wave propagation analysis in heterogeneous media. • Possibility of including a topographic irregularity. • Linear visco-elastic soil behaviour, where the quality factor Q (Q=1/2ζ, ζ being the soil damping ratio) is proportional to frequency: Q=q0f. For this purpose, the equation of SH motion is modified as follows (Kosloff and Kosloff, 1986):
where v(x,z,t) is the unknown displacement field, G(x,z) and ρ(x,z) are the shear modulus and mass density, respectively, and γ(x,z)=π/q0 is a damping coefficient. • Absorbing strips at the lateral and bottom boundaries of the numerical model, where a suitable spatial variation of the γ(x,z) damping parameter in eq. (1) can be defined, based on a pre-processing tool designed for its optimum calibration depending on a specified error tolerance. • Analysis in the time domain using simple input signals and possibility of convolving the response with arbitrary accelerograms. • Band pass, low pass and high pass filter design. • Output in terms of time history, Fourier spectrum, transfer function and response spectrum at selected receivers. • Saving of input and output data into text files and restart option.
3. Example of application To illustrate the features of WAVE2D in a practical example, we have selected one of the geological cross-sections of Ashigara Valley, Japan, that has been the object of an international experiment of “blind” ground motion prediction in the early 90s (Kudo and Sawada, 1992; Faccioli and Paolucci, 1992). The front page of the graphical user interface for this case is shown in Fig. 2. For a detailed description and explanation of all the items included in this window, the reader is referred to the User’s manual, available together with the software. Herein, we will just show how the numerical model can be simply constructed as a series of layers, with arbitrary interfaces defined by polygonal lines, as illustrated on the right side of Fig. 3, with reference to the first (upper) of the three layers of the cross-section. The corresponding dynamic properties are illustrated on the left side of the same figure. The resulting numerical model is shown in Fig. 2, where the lateral and bottom absorbing regions are emphasized as well as the extension of the fictitious region above the free surface, that is necessary to accurately impose the stress-free boundary condition. The seismic response of the Ashigara Valley geological cross-section has been calculated considering the vertical incidence of a plane SH wave with a “double-impulse” time-dependence (first derivative of a Gaussian function), having a peak frequency of 5 Hz (maximum frequency propagated 15 Hz). The spatial time step was chosen in order to comply with the limit of 3÷4 points required to accurately propagate the minimum wavelength in a highly heterogeneous medium (Paolucci, 1989). The total number of nodes of the numerical mesh is 24300, including the absorbing and free-surface strips.
The analysis has required about 5 min of CPU time on an Intel Pentium IV (3.00 GHz) personal computer, for a total of 6 s of wave propagation, considering a time step ∆t=0.001 s (6000 iterations). The output is given in terms of time history, Fourier spectrum and transfer function at selected receivers, as shown in Fig. 4.
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Figure 2. Front page of the graphical user interface for the mesh generation of the Ashigara Valley case. The bottom and lateral boundaries include the absorbing strips to reduce the spurious
reflections from the edges of the numerical model, while the strip above the model includes the domain to imposing the free-surface condition.
Figure 3. WAVE2D windows to input the soil layer properties and the coordinates of the polygonal line so that to define the layer.
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Figure 4. Example of output of the analysis at a generic receiver, consisting of the displacement time history (top), of the corresponding Fourier spectrum (centre, the red line indicates the
maximum frequency of the input motion) and (bottom) of the numerical transfer function calculated as the spectral ratio of output and input motions.
After the response to the prescribed input motion has been calculated, WAVE2D allows
the user to calculate the response at the generic receiver when the input motion consists of a generic acceleration, velocity or displacement time history, by simply convolving the numerical transfer function at a selected receiver, calculated by the spectral ratio of output vs. input motion, with the input signal. Therefore, this operation does not require any further application of the Fourier method. In this case, the convolution can be computed with the graphical interface shown in Fig. 5 and the output convolved signals are represented in terms of time history, Fourier spectrum and response spectrum at selected receivers as Fig. 6 shows.
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Figure 5. Window of WAVE2D to read the convolution signal and calculate the response spectra of the output.
Figure 6. Acceleration time history, corresponding Fourier spectrum and acceleration response spectra of the output signal from the convolution operation.
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5. Conclusions
Although more accurate and more powerful techniques of the Spectral Element family have emerged in the last 15 years, the Fourier pseudo-spectral method, the software WAVE2D is based upon, is still to be considered a powerful approach to seismic wave propagation problems. WAVE2D is one of the few numerical tools, devised for SH seismic site effects analyses in heterogeneous media with arbitrary topographic profile, with a user-friendly graphical interface in the Matlab environment, and a set of post-processing tools including the convolution with an accelerogram and calculation of response spectra. The software can be freely obtained upon request (email: [email protected]) and it is distributed as a deliverable part of the project Sismovalp. Its use is allowed, with proper acknowledgements, for teaching and research purposes,.
Acknowledgements
The preparation of the Matlab Graphical User interface for WAVE2D was carried out in the framework of the EC funded project Sismovalp (Seismic risk of Alpine valleys), within the Interreg IIIB research programme Alpine Space. Ezio Faccioli from Politecnico di Milano and Maurizio Fontana from Studio Geotecnico Italiano, Milano, deeply contributed to the early developments of this numerical code.
References
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