Fast iterative solution of large undrained soil‐structure interaction problems

In view of rapid developments in iterative solvers, it is timely to re‐examine the merits of using mixed formulation for incompressible problems. This paper presents extensive numerical studies to compare the accuracy of undrained solutions resulting from the standard displacement formulation with a penalty term and the two‐field mixed formulation. The standard displacement and two‐field mixed formulations are solved using both direct and iterative approaches to assess if it is cost‐effective to achieve more accurate solutions. Numerical studies of a simple footing problem show that the mixed formulation is able to solve the incompressible problem ‘exactly’, does not create pressure and stress instabilities, and obviate the need for an ad hoc penalty number. In addition, for large‐scale problems where it is not possible to perform direct solutions entirely within available random access memory, it turns out that the larger system of equations from mixed formulation also can be solved much more efficiently than the smaller system of equations arising from standard formulation by using the symmetric quasi‐minimal residual (SQMR) method with the generalized Jacobi (GJ) preconditioner. Iterative solution by SQMR with GJ preconditioning also is more elegant, faster, and more accurate than the popular Uzawa method. Copyright © 2003 John Wiley & Sons, Ltd.     

Preconditioned IDR(s) iterative solver for non‐symmetric linear system associated with FEM analysis of shallow foundation

Non‐associated flow rule is essential when the popular Mohr–Coulomb model is used to model nonlinear behavior of soil. The global tangent stiffness matrix in nonlinear finite element analysis becomes non‐symmetric when this non‐associated flow rule is applied. Efficient solution of this large‐scale non‐symmetric linear system is of practical importance. The standard Krylov solver for a non‐symmetric solver is Bi‐CGSTAB. The Induced Dimension Reduction [IDR(s)] solver was proposed in the scientific computing literature relatively recently. Numerical studies of a drained strip footing problem on homogenous soil layer show that IDR(s = 6) is more efficient than Bi‐CGSTAB when the preconditioner is the incomplete factorization with zero fill‐in of global stiffness matrix K_ep (ILU(0)‐K_ep). Iteration time is reduced by 40% by using IDR(s = 6) with ILU(0)‐K_ep. To further reduce computational cost, the global stiffness matrix K_ep is divided into two parts. The first part is the linear elastic stiffness matrix K_e, which is formed only once at the beginning of solution step. The second part is a low‐rank matrix Δ, which is re‐formed at each Newton–Raphson iteration. Numerical studies show that IDR(s = 6) with this ILU(0)‐K_e preconditioner is more time effective than IDR(s = 6) with ILU(0)‐K_ep when the percentage of yielded Gauss points in the mesh is less than 15%. The total computation time is reduced by 60% when all the recommended optimizing methods are used. Copyright © 2013 John Wiley & Sons, Ltd.     

Effective block diagonal preconditioners for Biot's consolidation equations in piled‐raft foundations

The finite element (FE) simulation of large‐scale soil–structure interaction problems (e.g. piled‐raft, tunnelling, and excavation) typically involves structural and geomaterials with significant differences in stiffness and permeability. The symmetric quasi‐minimal residual solver coupled with recently developed generalized Jacobi, modified symmetric successive over‐relaxation (MSSOR), or standard incomplete LU factorization (ILU) preconditioners can be ineffective for this class of problems. Inexact block diagonal preconditioners that are inexpensive approximations of the theoretical form are systematically evaluated for mitigating the coupled adverse effects because of such heterogeneous material properties (stiffness and permeability) and because of the percentage of the structural component in the system in piled‐raft foundations. Such mitigation led the proposed preconditioners to offer a significant saving in runtime (up to more than 10 times faster) in comparison with generalized Jacobi, modified symmetric successive over‐relaxation, and ILU preconditioners in simulating piled‐raft foundations. Copyright © 2012 John Wiley & Sons, Ltd.     

Comparison between iterative solution of symmetric and non‐symmetric forms of Biot's FEM equations using the generalized Jacobi preconditioner

Finite element discretization of Biot's consolidation equations can produce a symmetric indefinite system (commonly used in geomechanics) or a non‐symmetric system. While this difference appears to be minor, however, it will require the selection of entirely different Krylov subspace solvers with potentially significant impact on solution efficiency. The former is solved using the symmetric quasi‐minimal residual whereas the latter is solved using the popular bi‐conjugate gradient stabilized. This paper presents an extensive comparison of the symmetric and non‐symmetric forms by varying the time step, size of the spatial domain, choice of physical units, and left versus left–right preconditioning. The generalized Jacobi (GJ) preconditioner is able to handle the non‐symmetric version of Biot's finite element method equation, although there are no practical incentives to do so. The convergence behaviour of GJ‐preconditioned systems and its relation to the spectral condition number or the complete spectrum are studied to clarify the concept of ill‐conditioning within the context of iteration solvers. Copyright © 2007 John Wiley & Sons, Ltd.     

An overview of electrical conductivity structures of the crust and upper mantle beneath the northwestern Pacific, the Japanese Islands, and continental East Asia

This article reviews the electrical conductivity structures of the oceanic upper mantle, subduction zones, and the mantle transition zone beneath the northwestern Pacific, the Japanese Islands, and continental East Asia, which have particularly large potential of water circulation in the global upper mantle. The oceanic upper mantle consists of an electrically resistive lid and a conductive layer underlying the lid. The depth of the top of the conductive layer is related to lithospheric cooling in the older mantle, whereas it is attributable to the difference in water distribution beneath the vicinity of the seafloor spreading-axis. The location of a lower crustal conductor in a subduction zone changes according to the subduction type. The difference can be explained by the characteristic dehydration from the subducting slab in each subduction zone and by advection from the backarc spreading. The latest one-dimensional electrical conductivity model of the mantle transition zone beneath the Pacific Ocean predicts values of 0.1–1.0 S/m. These values support a considerably dry oceanic mantle transition zone. However, one-dimensional electrical profiles may not be representative of the mantle transition zone there, since there exists a three-dimensional structure caused by the stagnant slab. Three-dimensional electromagnetic modeling should be made in future studies.     

EMRIDGE: The electromagnetic investigation of the Juan de Fuca Ridge

From July to November 1988, a major electromagnetic (EM) experiment, known as EMRIDGE, took place over the southern end of the Juan de Fuca Ridge in the northeast Pacific. It was designed to complement the previous EMSLAB experiment which covered the entire Juan de Fuca Plate, from the spreading ridge to subduction zone. The principal objective of EMRIDGE was to use natural sources of EM induction to investigate the processes of ridge accretion. Magnetotelluric (MT) sounding and Geomagnetic Depth Sounding (GDS) are well suited to the study of the migration and accumulation of melt, hydrothermal circulation, and the thermal evolution of dry lithosphere. Eleven magnetometers and two electrometers were deployed on the seafloor for a period of three months. Simultaneous land-based data were made available from the Victoria Magnetic Observatory, B.C., Canada and from a magnetometer sited in Oregon, U.S.A.Changes in seafloor bathymetry have a major influence on seafloor EM observations as shown by the orientation of the real GDS induction arrows away from the ridge axis and towards the deep ocean. Three-dimensional (3D) modelling, using a thin-sheet algorithm, shows that the observed EM signature of the Juan de Fuca Ridge and Blanco Fracture Zone is primarily due to nonuniform EM induction within the ocean, associated with changes in ocean depth. Furthermore, if the influence of the bathymetry is removed from the observations, then no significant conductivity anomaly is required at the ridge axis. The lack of a major anomaly is significant in the light of evidence for almost continuous hydrothermal venting along the neo-volcanic zone of the southern Juan de Fuca Ridge: such magmatic activity may be expected to have a distinct electrical conductivity signature, from high temperatures, hydrothermal fluids and possible melt accumulation in the crust.Estimates of seafloor electrical conductivity are made by the MT method, using electric field records at a site 35 km east of the ridge axis, on lithosphere of age 1.2 Ma, and magnetic field records at other seafloor sites. On rotating the MT impedance tensor to the principal axis orientation, significant anisotropy between the major (TE) and minor (TM) apparent resistivities is evident. Phase angles also differ between the principal axis polarisations, and TM phase are greater than 90° at short periods. Thin-sheet modelling suggests that bathymetric changes accounts for some of the observed 3D induction, but two-dimensional (2D) electrical conductivity structure in the crust and upper mantle, aligned with the ridge axis, may also be present. A one-dimensional (1D) inversion of the MT data suggests that the top 50 km of Earth is electrically resistive, and that there is a rise in conductivity at approximately 300 km. A high conductivity layer at 100 km depth is also a feature of the 1D inversion, but its presence is less well constrained.     

Estimation of sand transport by use of tracers along a reclaimed shoreline at Singapore Changi Airport

This paper describes a probabilistic approach in the estimation of the sediment transport along the reclaimed shoreline at the Singapore Changi Airport through a detailed analysis of the data obtained from a radio-active tracer experiment conducted on site by the Port of Singapore Authority (PSA).The probabilistic approach showed that the transport rate per unit transverse width of the beach is proportional to the effective depth of transport and the speed of the centroid of the probability function describing the dispersion of the cloud of tracers.     

The Earth-Moon system during the late heavy bombardment period - Geochemical support for impacts dominated by comets

The solid planets assembled 4.57 Gyr ago during a period of less than 100 Myr, but the bulk of the impact craters we see on the inner planets formed much later, in a narrow time interval between 3.8 and 3.9 Gyr ago, during the so-called late heavy bombardment (LHB). It is not certain what caused the LHB, and it has not been well known whether the impactors were comets or asteroids, but our present study lend support to the idea that it was comets. Due to the Earth’s higher gravity, the impactors will have hit the Earth with ∼twice the energy density that they hit the Moon, and the bombardment will have continued on Earth longer than on the Moon. All solid surface of the Earth will have been completely covered with craters by the end of the LHB.However, almost nothing of the Earth’s crust from even the end of this epoch, is preserved today. One of the very few remnants, though, is exposed as the Isua greenstone belt (IGB) and nearby areas in Western Greenland. During a field expedition to Isua, we sampled three types of metasedimentary rocks, deposited ∼3.8 billion years ago, that contain information about the sedimentary river load from larger areas of surrounding land surfaces (mica-schist and turbidites) and of the contemporaneous seawater (BIF). Our samples show evidence of the LHB impacts that took place on Earth, by an average of a seven times enrichment (150 ppt) in iridium compared to present-day ocean crust (20 ppt). The clastic sediments show slightly higher enrichment than the chemical sediments, which may be due to contamination from admixtures of mafic (proto-crustal) sources.We show that this enrichment is in agreement with the lunar cratering rate and a corresponding extraterrestrial LHB contribution to the Earth’s Hadean-Eoarchean crust, provided the bulk of the influx was cometary (i.e., of high velocity and low in CI abundance), but not if the impactors were meteorites (i.e. had velocities and abundances similar to present-day Earth-crossing asteroids). Our study is a first direct indication of the nature of the LHB impactors, and the first to find an agreement between the LHB lunar cratering rate and the Earth’s early geochemical record (and the corresponding lunar record). The LHB comets that delivered the iridium we see at Isua will at the same time have delivered the equivalent of a ∼1 km deep ocean, and we explain why one should expect a cometary ocean to become roughly the size of the Earth’s present-day ocean, not only in terms of depth but also in terms of the surface area it covers. The total impacting mass on the Earth during the LHB will have been ∼1000 tons/m².