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The aim of this test is to show that MSEM has the same property of convergence (both in h and N) and stability compared with SEM method:
In the first an the second column (convergence in N and h) we prove that MSEM has spectral convergence in N and algebraic convergence in h .
In the third column we show that our method is obviously stable until CFL condition is verified.
Finally the last graphic displays that we use second order time integration scheme.
n
Level 1Level 1
MSEMGeoELSEMSEMGeoELSE
PGVX
The mortar element method as an effective tool for solving large scale dynamic soilstructure interaction problemsThe mortar element method as an effective tool for solving large scale dynamic soilstructure interaction problemsI. Mazzieri I. Mazzieri (1)(1), C. Smerzini , C. Smerzini (2)(2), M. Stupazzini , M. Stupazzini (3) (3) and F. Rapetti and F. Rapetti (4)(4)
(1) Dipartimento di Matematica MOX Politecnico di Milano, Italy (1) Dipartimento di Matematica MOX Politecnico di Milano, Italy (2) ROSE School, Pavia, Italy (2) ROSE School, Pavia, Italy (3) Munich RE, Munich, Germany(3) Munich RE, Munich, Germany
(4) Université de Nice SophiaAntipolis, Laboratoire de Mathématiques J.A. Dieudonné, Nice,France (4) Université de Nice SophiaAntipolis, Laboratoire de Mathématiques J.A. Dieudonné, Nice,France
2D seismic response of a railway viaduct in Italy2D seismic response of a railway viaduct in Italy
Numerical Numerical validationvalidation
Convergence in hConvergence in hConvergence in NConvergence in N Convergence in timeConvergence in time
Error estimate in LError estimate in L22 norm norm
Analyitic solutionAnalyitic solution
Mathematical FrameworkMathematical Framework
Variational FormulationVariational Formulation
Discrete Galerkin FormulationDiscrete Galerkin Formulation
MSEM FrameworkMSEM Framework
Projection operator QProjection operator Q
Solution schemeSolution scheme
Master unknonwnsSlave unknowns
Conclusion Conclusion
Level 2Level 2 Level 3Level 3
Some remarksSome remarks
Anderson CriteriaAnderson Criteria
Legend
1 Arias Duration
2 Energy Duration
3 Arias Intensity
4 Energy Integral
5 Peak Acceleration
6 Peak Velocity
7 Peak Displacement
8 Response Spectra
9 Fourier Spectra
10 Cross Correlation
1 3 5 7 9
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Score8 – 10 Excellent Fit6 – 8 Good Fit4 – 6 Fair Fit < 4 Poor Fit
(J.Anderson, Quantitative measure of the goodnessoffit of synthetic seismograms, 13th World Conference on Earthquake Engineering Vancouver, B.C., Canada,August 16, 2004
Paper No. 243)
Band Frequency Limits[Hz]B1 0.050.1B2 0.10.2B3 0.20.5B4 0.51.0B5 1.02.0B6 2.05.0
(Miriam Kristekova, Jozef Kristek, Peter Moczo, and Steven M. Day, Misfit Criteria for Quantitative Comparison of Seismograms, Bulletin of the Seismological Society of America, Vol. 96, No. 5, pp. 1836–1850, October 2006, doi: 10.1785/0120060012)All results are obtained with the code downloadable from http://www.nuquake.sk/Computer_Codes (Authors:Miriam Kristekova, Jozef Kristek & Peter Moczo )
Deep Structure: Mechanical ParametersLayer V
p[m/s] V
s[m/s] [Kg/m3] Q
s
B1 2300 1130,0 2800 ? B2 2300 1130,0 2800 ? B3 1700 982,5 2600 ? B4 1100 635,0 2400 ? B5 1100 635,0 2400 ?
B6 1218 716,7 1750 ?
For the spatial approximation spectral degrees SDs are choosen in this way: (h = mesh size and Fmax = max value of signal frequency)
if h is approximately Vs/Fmax we choose SD greater or equal 4if h is approximately 0.1Vs/Fmax we choose SD less than 4.
This lead the choiceGeoELSE :SD4 in B1B6MSEMGeoELSE: SD 4 in B1B4 and SD2 in B5B6.
R1R2
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The meshes has been designed with the aid of CUBIT software (http://cubit.sandia.gov/) while all the computation have been performed with GeoELSE and its new version MSEMGeoELSE.
Conforming( Spectral nodes 51320 )
NonConforming( Spectral nodes 20661 )
S1S1
Source term
Choice of Spectral Degrees
StabilityStability
All informations about GeoElse are available on the website http://geoelse.stru.polimi.it/
GeoELSE is a Spectral Elements code for the study of wave propagation phenomena in 2D or 3D complex domain, developed by CRS4 (Center for Advanced, Research and Studies in Sardinia) and the Politecnico di Milano, DIS (Department of Structural Engineering).
The main features of the code are: 1. Naturally oriented to large scale applications (millions of grid points);
2. Dealing with externally created 3D unstructured meshes (e.g.: CUBIT);
3. Native parallel implementation;
4. Handling the partitioning and load balancing of the computational domain by incorporating the METIS software library;
5. Implementing of complex constitutive behavior like viscoplasticity or non linear elasticity;
6. Communicating with other codes through a substructuring interface based upon a domain reduction method;
7. Postprocessing output in GID and VTK format.
The MSEM formulation proposed, is designed for the general case: a geometrically nonconforming domain partition where local meshes are independently generated from the neighbouring ones and associated with different spectral approximation degrees.
In this contribution we have present a first set of benchmark assessing the accuracy and flexibility of the MSEM method, implemented in the well known spectral elements based code GeoELSE (Faccioli et al., 1997 and Stupazzini et al., 2009).
We have also illustrate how this approach implemented in the code GeoELSE can be effectively used also for the numerical analysis of DSSI problems, with reference to the 2D seismic response of a railway viaduct in Italy.
Functional SpacesFunctional Spaces
Misfits between the signal obtained with GeoELSE ( ) and the signal obtained with MSEMGeoELSElse (). (Middle) GeoELSE and MSEMGeoELSE signals, values of the singlevalued envelope misfit EM and phase misfit PM. (Top) Timefrequency envelope misfits TFEM(t, f), time envelope misfits TEM (t), and frequency envelope misfits FEM (f). (Bottom) Timefrequency phase misfits TFPM (t, f), time phase misfits TPM (t), and frequency phase misfits FPM (f).
Misfit CriteriaMisfit Criteria
Future WorkFuture WorkIn the immediate future we will extend MSEMformulation to overlapping decomposition of the computational domain in order to treat with completly unstructured mesh
Analyze the behaviour of this last formulation compared with other formulation known in literature (like honoring or nothonoring SEM formulations)
Extend MSEMformulation to 3D problem and implement it in the parallel version of GeoELSE
R1
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EM = 2.63PM = 2.07
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EM = 11.9PM = 7.1
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EM = 10.7PM = 3.4
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EM = 6.23PM = 2.36
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IntroductionIntroduction
