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M. Stupazzini, C. Zambelli, L. Massidda, L. Scandella R. Paolucci, F. Maggio, C. di Prisco

Politecnico di Milano Dep. of Structural Engineering. Center for Advanced Research and Studies in Sardinia. Ludwig Maximilians University Dep . of Earth and Environmental Sciences - Geophysics. THE SPECTRAL ELEMENT METHOD AS AN EFFECTIVE TOOL FOR SOLVING

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M. Stupazzini, C. Zambelli, L. Massidda, L. Scandella R. Paolucci, F. Maggio, C. di Prisco

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  1. Politecnico di Milano Dep. of Structural Engineering Center for Advanced Research and Studies in Sardinia Ludwig Maximilians University Dep. of Earth and Environmental Sciences - Geophysics THE SPECTRAL ELEMENT METHOD AS AN EFFECTIVE TOOL FOR SOLVING LARGE SCALE DYNAMIC SOIL-STRUCTURE INTERACTION PROBLEMS M. Stupazzini, C. Zambelli, L. Massidda, L. Scandella R. Paolucci, F. Maggio, C. di Prisco 20th ofApril 2006 San Francisco

  2. 3rd 1st Landslides Liquefaction Subsidence 2nd General problem

  3. DRM Study case Outlook GeoELSE

  4. GeoELSE(GEO-ELasticity by Spectral Elements) • GeoELSE is a Spectral Elements code for the study of wave propagation phenomena in 2D or 3D complex domain • Developers: • CRS4 (Center for Advanced, Research and Studies in Sardinia) • Politecnico di Milano, DIS (Department of Structural Engineering) • Native parallel implementation • Naturally oriented to large scale applications ( > at least 106 grid points)

  5. Formulation of the elastodynamic problem Dynamic equilibrium in the weak form: where ui= unknown displacement function vi= generic admissible displacement function (test function) ti= prescribed tractions at the boundary  fi= prescribed body force distribution in 

  6. Spatial discretization Spectral element method SEM (Faccioli et al., 1997) Time advancing scheme Finite difference 2nd order (LF2 – LF2) Courant-Friedrichs-Levy (CFL) stability condition

  7. Why using spectral elements ? • Suitable for modelling a variety of physical problems (acoustic and elastic wave propagation, thermo elasticity, fluid dynamics) • Accuracy of high-order methods • Suitable for implementation in parallel architectures • Great advantages from last generation of hexahedral mesh creation program (e.g.: CUBIT, Sandia Lab.)

  8. n=1 n=2 Acoustic wave propagation through an irregular domain. Simulation with spectral degree 1 (left) exhibits numerical dispersion due to poor accuracy. Simulation with spectral degree 2 (right) provides better results.Change of spectral degree is done at run time. Why using spectral elements ?acoustic problem

  9. Soil-Structure interaction EFFECTIVE NODAL FORCESP Pe(t) • A sub-structuring method : the Domain Reduction Method • (Bielak et al. 2003) Method for the simulation of seismic wave propagation from a half space containing the seismic source to a localized region of interest, characterized by strong geological and/or topographic heterogeneities or soil-structure interaction. Local geological feature Boundary region Inner region External region

  10. Pe(t) • DRM : 2 steps method Step I( AUXILIARY PROBLEM ) • The auxiliary problem simulates the seismic source and propagation path effects encompassing the source and a background structure from which the localized feature has been removed. • The free field displacement u0 may be calculated by different methods Analitical solutions (e.g.: Inclined incident waves) Numerical method (e.g.: FD, SEM, BEM, ADER-DG)

  11. Boundary region ub ui Inner region EFFECTIVE NODEL FORCES we External region • DRM : 2 steps method Step II( REDUCED PROBLEM ) • The reduced problem simulates the local site effects of the region of interest • The input is a set of equivalent localized forces derived from step I Inner region Boundary region External region • The effective forces act only within a single layer of elements adjacent to the interface between the external and internal regions where the coupled term of stiff matrix does not vanish

  12. Study case railway bridge

  13. Wave propagation in 2D “Source“ & “ Deep propagation“ zoom Fault “ Site effects “ & “ Soil Structures Interactions “ zoom

  14. 1.177 10-5 570 620 190.0 Computational comparison:

  15. Computational comparison:

  16. The computation with DRM is 2.8 times faster + Computational comparison:

  17. Dynamic rupture modelling (Festa G., IPGP) Interface behavior via friction Slip weakening law + Stress distribution Initial Principal stresses : 4.0 107 Pa s1 1.8 108 Pa s3 100° Orientation 0.67 Static friction 0.525 Dynamic friction 0.4 m DC 150-300m Cohesive zone thickness Kinematic source:Seismic moment tensor density (Aki and Richards, 1980):MW = 4.2, slip = 50 cm

  18. Comparison

  19. Comparison

  20. T = 0.5s T = 1.0s T = 1.5s T = 2.0s Wave propagation in 3D complex domain Fault 1 891 m Fault 2 1756 m 2160 m Snapshots of Displacement

  21. Conclusions • GeoELSE is capable to handle „source to structure“ wave propagation problem. • Thanks to DRM we acchieve: • reduced computational time • dialog between numerical codes oriented for different purposes • WEB SITE: • www.stru.polimi.it/Ccosmm/ccosmm.htm • www.spice-rtn.org

  22. Internal domain: External domain: • DRM : 2 steps method Internal domain Navier’s equation: Fault External domain

  23. Mass and stiffness matrices do not change because properties in + do not change External domain (0): Change of variables : • DRM : 2 steps method Internal domain (0) AUSILIARY PROBLEM (Step I) ujo= vector of nodal displacements j = i, b, e Pbo= forces from domain + to 0 Faglia External domain (0)

  24. Dominio interno: • DRM : 2 steps method External domain - External domain (0):

  25. REDUCED PROBLEM (Step II) (Step II) • DRM : 2 steps method • M and K matrices of the original problem • P localization within a single layer of elements in +adjacentto 

  26. The effectiveness of the method depend on the accuracy of the absorbing boundary conditions • DRM : 2 steps method Non linear properties in the internal domain

  27. DRM : 2D Validationsusing Spectral Elements (GeoELSE) Homogeneous valley in a layered half space

  28. Total displacements (u=w+uo) DRM : 2D Validationsusing Spectral Elements (GeoELSE) Homogeneous valley in a layered half space Relative displacements (w) Internal points External points

  29. DRM : 2D Validationsusing Spectral Elements (GeoELSE) Canyon in a homogeneous half space

  30. Total displacements (u=w+uo) DRM : 2D Validationsusing Spectral Elements (GeoELSE) Canyon in a homogeneous half space Relative displacements (w) Internal points External points

  31. I STEP Calculation of u0 for a homogeneous model • Analytical solution • Numerical methods (Ex. Hisada, 1994) • Same method used for step II (ex. SE) Interface elements Nodes b P Nodes e • DRM : 2 steps method ORIGINAL PROBLEM Oblique propagation of plane waves inside a valley II STEP Analysis of wave propagation inside the reduced model. Calculation of effective forces Pb and Pe

  32. Comparison

  33. Conclusions • Capabilities of DRM to handle „source to structure“ wave propagation problem with reduced CPU time • Dialog between numerical codes oriented for different purposes • Kinematic model are satisfactory to describe the low frequency bahaviour (e.g.: PGD and PGV) while PGA seems to be overestimated (nucleation, constant rupture velocity and instantaneous drop of the slip on the fault boundaries?).

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