11iacmag_numericalmethods

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Application of Numerical Methods to the Analysis of

Liquefaction

Ahmed Elgamal, Jinchi Lu, and Zhaohui YangDepartment of Structural Engineering, University of California, San Diego, USA

Keywords: liquefaction, finite element, numerical analysis, cyclic mobility, plasticity

ABSTRACT: Liquefaction remains a topic that presents major challenges for numerical analysis. Anumber of these challenges are discussed in this paper. Salient ongoing research efforts are

presented and discussed, primarily in the area of cyclic mobility and accumulation of liquefaction-induced shear deformations. The significance of cyclic mobility during liquefaction is illustrated bydata sets from centrifuge model experiments. In the context of numerical simulation, the role of on-line computing, and scenario-specific user interfaces is highlighted, as a means for collaborationand associated technical advancements. Visualization is also viewed as an integral element withinthis framework. Directions for future research and the need for large-scale simulations are finallyaddressed.

1 Introduction

Liquefaction of soils and associated deformations remain among the main causes of damage duringearthquakes (Seed et al., 1990; Bardet et al., 1995; Sitar, 1995; JGS, 1996; Ansal et al., 1999).Indeed, dramatic unbounded deformations (flow failure) due to liquefaction in dams and other

structures (Seed et al., 1975; Seed et al., 1989; Davis and Bardet, 1996) have highlighted thesignificance of this problem in earthquake engineering. However, liquefaction often results in limited,albeit possibly high levels of deformation (Casagrande, 1975; Youd et al., 1999). The deformationprocess in such situations is mainly a consequence of limited-strain cyclic deformations (Seed,1979), commonly known as cyclic mobility (Castro and Poulos, 1977) or cyclic liquefaction(Casagrande, 1975).

A large number of computational models have been, and continue to be developed for simulation ofnonlinear soil response (e.g., Desai and Christian, 1977; Finn et al., 1977; Desai and Siriwardane,1984; Prevost, 1985; Pastor and Zienkiewicz, 1986; Prevost, 1989; Bardet et al., 1993; Manzari andDafalias, 1997; Borja et al., 1999a;b; Jeremic et al., 1999; Zienkiewicz et al., 1999; Desai, 2000; Liand Dafalias, 2000; Park and Desai, 2000; Shao and Desai, 2000; Arduino et al., 2001). Currently,liquefaction still remains a topic that presents major challenges for such numerical techniques. Theresearch presented in this paper addresses primarily the area of cyclic mobility and accumulation ofliquefaction induced shear deformations. Effort is dedicated to the analysis of liquefaction-induceddeformations in medium-dense cohesionless soils.

2 Mechanism of liquefaction-induced deformation

In saturated clean medium to dense sands (relative densities Dr of about 40% or above, Lambeand Whitman, 1969), the mechanism of liquefaction-induced cyclic mobility may be illustrated by the


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undrained cyclic response, schematically illustrated in Figure 1. In this figure, the following aspectsmay be observed: (i) as excess-pore pressure increases, cycle-by-cycle degradation in shear

strength is observed, manifested by the occurrence of increasingly larger shear strain excursions forthe same level of applied shear stress, and (ii) a regain in shear stiffness and strength at largeshear strain excursions, along with an increase in effective confinement (shear-induced dilativetendency).

In situation of an acting initial shear stress (e.g., in a slope or embankment), cycle-by-cycledeformation accumulates according to the schematic of Figure 2. Inspection of Figure 2 shows thata net finite increment of permanent shear strain occurs in a preferred down-slope direction on acycle-by-cycle basis. Realistic estimation of the magnitude of such increments is among the mostimportant considerations in assessments of liquefaction-induced hazards (Iai, 1998; Li and Dafalias,2000; Park and Desai, 2000).

Figure 1. Schematic stress-strain and stress path response for medium-to-dense sand in a stress-controlled, undrained cyclic shear loading (Parra, 1996).

Figure 2. Schematic stress-strain and stress path response for medium-to-dense sand in a stress-controlled, undrained cyclic shear loading with a static shear stress bias (Parra, 1996).

3 The constitutive model

A plasticity-based formulation is employed. Multi-surface kinematic plasticity allows for modellingthe desired hysteretic response. This soil constitutive model was developed with emphasis onsimulating the liquefaction-induced shear strain accumulation mechanism in clean medium-densesands (Elgamal et al., 2002a; Elgamal et al., 2002b; Yang and Elgamal, 2002; Elgamal et al., 2003;Yang et al., 2003). Special attention was given to the deviatoric-volumetric strain coupling (dilatancy)under cyclic loading (e.g., Figures 1 and 2), which causes increased shear stiffness and strength atlarge cyclic shear strain excursions (i.e., cyclic mobility). The main elements allowing for cyclicmobility response include:

The yield surface (Figure 3) is defined by the Lade and Duncan (1975) function:

013

3

1==

I

If (1)


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4 Finite element framework

For liquefaction problems, the saturated soil system is modeled as a two-phase material based onthe Biot (1962) theory for porus media. A simplified numerical formulation of this theory, known asu-p formulation (in which displacement of the soil skeleton u, and pore pressure p, are the primaryunknowns (Chan, 1988; Zienkiewicz et al., 1990), was implemented in a 3D Finite Element programCYCLIC (Parra, 1996; Yang, 2000; Yang and Elgamal, 2002; Lu et al., 2004a). Thisimplementation is based on the following assumptions: small deformation and rotation, constantdensity of the solid and fluid in both time and space, locally homogeneous porosity which isconstant with time, incompressibility of the soil grains, and equal accelerations for the solid and fluidphases.

The u-p formulation is defined by (Chan, 1988): 1) the equation of motion for the solid-fluid mixture,and 2) the equation of mass conservation for the fluid phase that incorporates equation of motionfor the fluid phase and Darcy's law. These two governing equations are expressed in the followingfinite element matrix form (Chan, 1988):

0fQpdBUMs

T =&& (2a)0fHppSUQ

pT =&& (2b)where M is the total mass matrix, U the displacement vector, B the strain-displacement matrix, the effective stress vector (determined by the soil constitutive model described above), Q the

discrete gradient operator coupling the solid and fluid phases, p the pore pressure vector, S the

compressibility matrix, and H the permeability matrix. The vectorss

f andp

f represent the effectsof body forces and prescribed boundary conditions for the solid-fluid mixture and the fluid phase,respectively. Equations 2a and 2b are integrated in the time domain using a single-step predictormulti-corrector scheme of the Newmark type (Chan, 1988; Parra, 1996). In the currentimplementation, the solution is obtained for each time step using the modified Newton-Raphsonapproach (Parra, 1996).

5 Model calibration

Data from centrifuge experimentation has been crucial in allowing for calibration. In particular,VELACS models 1 and 2 (Figure 5) and similar centrifuge (Figure 6) experiments conducted atRensselaer Polytechnic Institute (RPI) have been a key calibration component (Abdoun, 1997).

In the VELACS project, two centrifuge model tests (Figure 5) were conducted by Dobry andTaboada (1994b; 1994a) to simulate the dynamic response of level (Model 1) and mildly sloping(Model 2) sand sites. Results of these two tests were employed for calibration of model parameters,through finite element simulations. The main modeling parameters include typical dynamic soilproperties such as low-strain shear modulus and friction angle, as well as calibration constants tocontrol pore-pressure buildup rate, dilation tendency, and the level of liquefaction-induced cyclicshear strain. The computed surface lateral displacement histories for VELACS Model 2 and thecalibrated numerical response are shown in Figure 7 (sandy gravel k, where kis permeability).

In general terms, this type of centrifuge experimental data along with engineering judgmentcurrently suggest the post-liquefaction shear deformation levels shown in Figure 8 for medium,

medium-dense, and dense cohesionless soils.


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Figure 5. General configuration of RPI Models 1 and2 in laminar container (Dobry et al., 1995).

Figure 6. RPI 100-g ton geotechnicalcentrifuge (RPI, 2005).

1.3% /cycle

0.5% cycle

0.3% /cycle

1.3% /cycle

0.5% cycle

0.3% /cycle

Figure 7. Surface lateral displacement histories inuniform soil profile with different premeability

coefficients (Yang and Elgamal, 2002).

Figure 8. Post-liquefaction shear stress-strain response (deformation levels per

cycle are shown).

6 Role of permeability

A coupled solid-fluid framework such as the one described above is needed in order to account forexcess pore-pressure evolution and its distribution during and after seismic excitation. At anylocation, excess pore-pressure is dictated by the overall influence of shear loading throughout theentire ground domain under investigation. In this regard, permeability plays a critical role, locallyand globally. In simple terms, local effects might dictate the extent of dilation-induced regain ofshear stiffness and strength during a large shear strain excursion (and the resulting level of shearstrain accumulation). Globally, the distribution of pore-pressure with depth for instance can besignificantly affected by the natural layering of soil strata of different permeabilities, with thedramatic example being (Figure 9) the situation of alluvial deposits or man-made hydraulic fills(Scott and Zuckerman, 1972; Adalier, 1992).

Yang and Elgamal (2002) attempted to shed light on the significance of permeability. For instance,

Figure 7 depicts the situation of a 10m-thick uniform soil profile, inclined by 4 degrees to simulatean infinite-slope response. This configuration is identical to that of the VELACS Model-2 centrifugeexperiment (Dobry et al., 1995; Taboada, 1995). Three numerical simulations were conducted, witha permeability coefficient kof 1.3 x 10

-2m/sec (gravel), 3.3 x 10

-3m/sec (VELACS Model-2 sandy

gravel calibration simulation), and 6.6 x 10-5

m/sec (clean sand) respectively. It is seen that: i) as


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mentioned earlier, computed lateral deformations with the sandy gravel k value are close to theexperimental response (part of the calibration process), and ii) the extent of lateral deformation in

this uniform profile is inversely proportional to soil permeability, i.e., a higherkresults in lower levelsof lateral deformation (the profile with the least kvalue had a lateral translation about 2.5 times thatwith the highest kvalue).

Figure 9. Natural layering of soil strata ofdifferent permeabilities (Adalier, 1992)

Figure 10. Excess pore-pressure profile anddeformed mesh for uniform sand profile with alow-permeability interlayer (deformations areexaggerated for clarity)(Yang and Elgamal,

2002).

Spatial variation of permeability in a soil profile is also potentially of primary significance in thedevelopment of liquefaction and associated deformations. Figure 10 shows an example of

liquefaction (excess pore pressure ratio ru=ue /v approaching and reaching 1.0, where ue=excess

pore pressure and v is effective vertical stress) with a low-permeability interlayer in a uniform soilprofile. Figure 10 and the observed deformations displayed (Yang and Elgamal, 2002):

1) A very high pore-pressure gradient within the silt-k layer. Below this layer, the post-shaking re-consolidation process eventually results in a constant distribution. This constant value is equal tothe initial effective confinement (overburden pressure) imposed by the thin layer and the layersabove. Dissipation of this trapped fluid through the low-permeability interlayer may take a very longtime in practical situations (if no sand boils develop).

2) After the shaking phase, void ratio continued to increase immediately beneath the silt-k layer,with large shear-strain concentration. Meanwhile, negligible additional shear strain was observed inthe rest of the profile.

7 Seismic response of deep foundation

The 1995 Kobe earthquake as well as earlier seismic events demonstrated the significant damageto piles due to liquefaction (Bardet et al., 1995; Sitar, 1995; JGS, 1996). Indeed muchexperimentation during the last 10 years has been focused on this problem (Abdoun, 1997).

Using the above described numerical framework, related computational efforts are underway(Figure 11 and Figure 12). Figure 11 shows a centrifuge experiment (RPI Model 3) conducted byAbdoun (1997) at RPI to simulate the response of a single pile subjected to lateral pressure of aliquefied soil due to lateral spreading. The experiment was conducted using the rectangular,flexible-wall laminar container shown in Figure 11. This centrifuge test was simulated using the


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above-described three-dimensional finite element framework. As shown in Figure 12, the mildinclination imposed a static shear stress component, causing the accumulated downslope lateral

deformation. The colormap in the final deformed mesh (Figure 12) clearly shows the influence of thepile on the ground deformation pattern.

Figure 11. Lateral spreading pile centrifugemodel in a two-layer soil profile (RPI Model

3, Abdoun, 1997).

Figure 12. Final deformed mesh (factor of 2) of RPIModel 3 simulation (Lu et al., 2004a).

In order to achieve more accurate results, the need for parallel computing quickly becomesapparent for 3D simulations (Figure 13). Figure 13 shows the partitioned mesh for a parallel run ofthe centrifuge test (Figure 11) using 16 processors (indicated by the different colors). Thesimulations were performed using a parallel nonlinear finite element program ParCYCLIC (Lu et al.,2004b), recently developed based on the existing serial code CYCLIC, conducted on the BlueHorizon machine at San Diego Supercomputer Center. Blue Horizon (Lu et al., 2004b) is an IBMScalable POWERparallel (SP) machine with 144 computer nodes, each with eight POWER3 RISC-based processors and 4 GBytes of memory. Each processor on the node has equal shared accessto the memory. Figure 14 displays the total execution times for performing this simulation on 8, 16,and 32 processors. Significant decrease in the total execution time can be observed, as alsoindicated by the speedup factor (relative to 8 processors) curve (Figure 14). Note that thecentrifuge model, with 363,492 degrees of freedom, cannot even fit into the memory of less than 8processors. A brief summary of ongoing parallel computing research is included in Appendix A.

0 10 20 30 40 50 60 701000

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Time(second)

0 10 20 30 40 50 60 700.5

1

1.5

2

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Figure 13. Partitioned mesh of pile in liquefiablesoil model (Lu, 2005): RPI Model 3 (Abdoun,

1997)

Figure 14. Speed-up factor (relative to 8processors) and total execution times of the

single pile centrifuge simulation.


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8 User-Interfaces

Much time and effort is expended today in building an appropriate finite element mesh, particularlyfor 3D simulations. Preparation of data files is a step that requires careful attention to detail.Debugging can consume many weeks or even months. A minor oversight or misinterpretation mightgo undetected leading to erroneous results. Numerous opportunities for such errors abound. Theneed to address this challendge cannot be overstated (Elgamal et al., 2004).

Current efforts are underway to develop tools that simplify this process. Scenario-specific user-friendly interfaces, thought potentially restrictive, can significantly alleviate this problem allowing forhigh efficiency and much increased confidence. Such interfaces for analysis of plane-strain (withfoundation), earthdam, and pile response (including seismic applications and liquefaction) arecurrently under development (e.g., Figure 15). On-line remote computing versions (Figure 16) ofthese interfaces are also of potential value (Yang et al., 2004). Advantages of internet computinginclude: 1) ease of updating software, 2) creation of a collaborative environment, where a user canbuild-on work conducted and archived earlier by others.

Figure 15. User-interface for single pile in 3Dhalf-space simulation (e.g., Figures 11 and 12).

Figure 16. Internet-based simulation ofCyclic1D (Yang et al., 2004).

9 Visualization

Efficient visualization of the massive amounts of data from simulations (Yan et al., 2004) isbecoming increasingly of importance. Effort is underway to harness the latest 3D computer graphicsadvances in developing new stereoscopic scientific visualizations tools (e.g., Arduino et al., 1997;Czernuszenko et al., 1997; Hashash et al., 2002; Jeremic et al., 2002; Pape et al., 2002; Hashashet al., 2003; Schwarz et al., 2004).

Three-dimensional graphic models may be now constructed using the SGI Open Inventor toolkit(http://www.sgi.com/products/software/inventor/). Stereoscopic visualization is facilitated by Geowall(http://www.geowall.org; http://siogeowall.ucsd.edu), an efficient and low-cost system that exploitsthe power of PC-based commodity graphics hardware and passive polarization techniques.

Figure 17 shows the deformed mesh of a bridge-foundation-ground system. A 3D stereoscopicmodel of this system is displayed in Figure 18. This environment provides advanced interactiveoptions such as immersive viewing, which can significantly help researchers to understand data(close-up within the model body) from a new perspective.


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Figure 17. Deformed mesh of a bridge-foundation-ground finite element model (Yan et al., 2004).

Figure 18. 3D stereoscopic visualization of bridge-foundation-ground system (Yan et al., 2004).

10 Summary and conclusions

A numerical framework was presented for analysis of cyclic-mobility soil liquefaction scenarios. A

plasticity-based soil model within a finite element coupled solid-fluid formulation is employed.Among the current challenges are:1) Further understanding and calibration of soil response during liquefaction via experimentation.2) Simplification to the constitutive model logics.3) Necessity for high fidelity 3D simulation, and the implications of that on the need for parallel-computing environments.4) The vast time required for preparing a finite element model, and selection of soil modelparameters; an issue that can be simplified by appropriate user-interfaces.5) Management and display of 3D numerical response, where more efficient stereoscopicvisualization tools can be very helpful.

11 Acknowledgements

The reported research was supported by the Pacific Earthquake Engineering Research (PEER)Center, under the National Science Foundation Award Number EEC-9701568, and by the NationalScience Foundation (Grants No. CMS0084616 and CMS0200510). This support is mostappreciated. ParCYCLIC was developed based on the work and contributions of Professor KinchoLaw and Dr. Jun Peng of Stanford University.


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12 Appendix A: ParCYCLIC

Recently, a parallel nonlinear finite element program, ParCYCLIC (Lu, 2005), was developed(based on the serial code CYCLIC (Parra, 1996; Yang, 2000)) for modeling earthquake groundresponse and liquefaction. Key elements of the computational strategy employed in ParCYCLIC,designed for distributed-memory message-passing parallel computer systems, include (Lu et al.,2004b; Peng et al., 2004): (a) a parallel sparse direct solver (Law and Mackay, 1993), in which LDL

T

factorization is performed (where L is a unit lower triangular matrix and D is a diagonal matrix); (b)nodal ordering strategies to minimize storage space for the matrix coefficients; (c) an efficientscheme for the allocation of sparse matrix coefficients among the processors; and (d) an automaticdomain decomposer, where METIS (Karypis and Kumar, 1997) is used to partition the finite elementmesh so that the workload on each processor is more or less evenly distributed and thecommunication among processors is minimized.

ParCYCLIC employes the single-program-multiple-data (SPMD) programming paradigm, a commonapproach in developing application software for distributed memory parallel computers.Communication in ParCYCLIC is written in MPI (Snir and Gropp, 1998), making ParCYCLICportable, capable of running on a wide range of parallel computers and workstation clusters.

ParCYCLIC has been successfully ported on IBM SP machines, SUN super computers, and Linuxworkstation clusters.

Large-scale experimental results for 3D geotechnical simulations have been conducted todemonstrate the capability and performance of ParCYCLIC (Lu et al., 2004b). Simulation resultsdemonstrated that ParCYCLIC is suitable for large-scale geotechnical simulations, and goodagreement has been achieved between the computed and the recorded acceleration, displacement,and pore pressure responses. Remarkable parallel speedup has also been obtained from thesimulation results. Last but not least, it was shown that ParCYCLIC, which employs a directsolution scheme, remains scalable to a large number of processors, e.g., 64 or more. The parallelcomputational strategies employed in ParCYCLIC are general and can be adapted to other similiarapplications without difficulties.

ParCYCLIC handles symmetric systems of linear equations (resulting from the employed implicittime integration scheme) using the parallel sparse solver (Law and Mackay, 1993). This solver isbased on a row-oriented storage scheme that takes full advantage of the sparsity of the stiffnessmatrix. The concept of the sparse solver is briefly described below (Lu et al., 2004b; Peng et al.,2004; Lu, 2005).

It is well known that the nonzero entries in the numerical factor L can be determined by the originalnonzero entries of the stiffness matrix K(Law and Fenves, 1986; Liu, 1991)and a list vector, whichis defined as:

}0|min{)( = ijLijPARENT (3)

The array PARENTrepresents the row subscript of the first nonzero entry in each column of thelower triangular matrix factorL. The definition of the list array PARENTresults in a monotonicallyordered elimination tree (Liu, 1990) of which each node has its numbering higher than itsdescendants. By topologically post-ordering the elimination tree, the nodes in any subtree can benumbered consecutively. The resulting sparse matrix factor is partitioned into block submatriceswhere the columns/row of each block corresponds to the node set of a branch in the elimination tree.Figure 19 shows a simple square finite element grid and its post-ordered elimination tree

representation.

The coefficients of a sparse matrix factor are distributively stored among the processors accordingto the column blocks. The strategy is to assign the rows corresponding to the nodes along eachbranch of the elimination tree (column block) to a processor or a group of processors. Beginning at


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the root of the elimination tree, the nodes belonging to this branch of the tree are assigned amongthe available processors in a rotating round robin fashion (Golub and Van Loan, 1989). As we

traverse down the elimination tree, at each fork of the elimination tree, the group of processors isdivided to match the number and size of the subtrees below the current branch. A separate groupof processors is assigned to each branch at the fork and the process is repeated for each subtree.The process of assigning groups of processors to each branch of the elimination tree continues untilonly one processor remains for the subtree. At this stage, all remaining nodes in the subtree areassigned to this single processor (Figure 19).

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Figure 19. A finite element grid and its elimination tree representation (Peng et al., 2004)

The parallel numerical factorization procedure is divided into two phases (Law and Mackay, 1993).In the first phase, each processor independently factorizes certain portions of the matrix assigned toa single processor. In the second phase, other portions of the matrix shared by more than oneprocessor are factored. Following the parallel factorization, the parallel forward and backwardsolution phases proceed to compute the solution to the global system of equations (Lu et al., 2004b;Peng et al., 2004).

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