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

1 st 2 nd 3 rd Subsidence Liquefaction Landslides General problem

Outlook DRM Study case GeoELSE

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 10 6 grid points)

Formulation of the elastodynamic problem Dynamic equilibrium in the weak form: where u i = unknown displacement function v i = generic admissible displacement function (test function) t i = prescribed tractions at the boundary  f i = prescribed body force distribution in 

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

 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.) Why using spectral elements ?

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

A sub-structuring method : the Domain Reduction Method (Bielak et al. 2003) Local geological feature P e (t) Soil-Structure interaction Inner region External region EFFECTIVE NODAL FORCES P Boundary region 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.

The free field displacement u 0 may be calculated by different methods 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. P e (t) Analitical solutions (e.g.: Inclined incident waves) Numerical method (e.g.: FD, SEM, BEM, ADER-DG) DRM : 2 steps method

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 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 EFFECTIVE NODEL FORCES uiui wewe ubub Inner region External region Boundary region Inner region Boundary region External region Step II ( REDUCED PROBLEM ) DRM : 2 steps method

Study case railway bridge

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

Computational comparison:Simulation # elem. Memory[Mb]  t simulation [sec.] # time steps Tot. CPU time [min.] Single model

Computational comparison:Simulation # elem. Memory[Mb]  t simulation [sec.] # time steps Tot. CPU time [min.] Single model DRM 1 st step

Computational comparison:Simulation # elem. Memory[Mb]  t simulation [sec.] # time steps Tot. CPU time [min.] Single model DRM 1 st step DRM 2 nd step The computation with DRM is 2.8 times faster

Kinematic source: Kinematic source: Seismic moment tensor density (Aki and Richards, 1980): M W = 4.2, slip = 50 cm Dynamic rupture modelling (Festa G., IPGP) Interface behavior via friction Slip weakening law + Stress distribution Initial Principal stresses : Pa  Pa  3 100° Orientation 0.67 Static friction Dynamic friction 0.4 m D C m Cohesive zone thickness

Comparison

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

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:

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

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

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

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

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

DRM : 2D Validations using Spectral Elements (GeoELSE) Homogeneous valley in a layered half space Mechanical properties V S [m/s]V P [m/s]  [m/s] Valley Half space

DRM : 2D Validations using Spectral Elements (GeoELSE) Relative displacements (w) Total displacements (u=w+u o ) Homogeneous valley in a layered half space Internal points External points

Canyon in a homogeneous half space Mechanical properties V S [m/s]V P [m/s]  [m/s] Canyon Half space DRM : 2D Validations using Spectral Elements (GeoELSE)

Relative displacements (w) Total displacements (u=w+u o ) Internal points External points Canyon in a homogeneous half space DRM : 2D Validations using Spectral Elements (GeoELSE)

Calculation of effective forces P b and P e ORIGINAL PROBLEM II STEP Analysis of wave propagation inside the reduced model. Interface elements Nodes e Nodes b P Calculation of u 0 for a homogeneous model I STEP Analytical solution Numerical methods (Ex. Hisada, 1994) Same method used for step II (ex. SE) Oblique propagation of plane waves inside a valley DRM : 2 steps method

Comparison

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?).