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.     

Analysis of undrained cavity expansion in elasto‐plastic soils with non‐linear elasticity

A large strain analysis of undrained expansion of a spherical/cylindrical cavity in a soil modelled as non‐linear elastic modified Cam clay material is presented. The stress–strain response of the soil is assumed to obey non‐linear elasticity until yielding. A power‐law characteristic or a hyperbolic stress–strain curve is used to describe the gradual reduction of soil stiffness with shear strain. It is assumed that, after yielding, the elasto‐plastic behaviour of the soil can be described by the modified Cam clay model. Based on a closed‐form stress–strain response in undrained condition, a numerical solution is obtained with the aid of simple numerical integration technique. The results show that the stresses and the pore pressure in the soil around an expanded cavity are significantly affected by the non‐linear elasticity, especially if the soil is overconsolidated. The difference between large strain and small strain solutions in the elastic zone is not significant. The stresses and the pore pressure at the cavity wall can be expressed as an approximate closed‐form solution. Copyright © 2001 John Wiley & Sons, Ltd.     

A modified Jacobi preconditioner for solving ill‐conditioned Biot's consolidation equations using symmetric quasi‐minimal residual method

This paper identifies imbalanced columns (or rows) as a significant source of ill‐conditioning in the preconditioned coefficient matrix using the standard Jacobi preconditioner, for finite element solution of Biot's consolidation equations. A simple and heuristic preconditioner is proposed to reduce this source of ill‐conditioning. The proposed preconditioner modifies the standard Jacobi preconditioner by scaling the excess pore pressure degree‐of‐freedoms in the standard Jacobi preconditioner with appropriate factors. The performance of such preconditioner is examined using the symmetric quasi‐minimal residual method. To alleviate storage requirements, element‐by‐element iterative strategies are implemented. Numerical experiment results show that the proposed preconditioner reduces both the number of iteration and CPU execution time significantly as compared with the standard Jacobi preconditioner. Copyright © 2001 John Wiley & Sons, Ltd.     

Response evaluation for horizontally loaded fixed‐head pile groups using 3‐D non‐linear analysis

The response of laterally loaded pile foundations may be significantly important in the design of structures for such loads. A static horizontal pile load test is able to provide a load–deflection curve for a single free‐head pile, which significantly differs from that of a free‐ or fixed‐head pile group, depending on the particular group configuration. The aim of this paper is to evaluate the influence of the interaction between the piles of a group fixed in a rigid pile cap on both the lateral load capacity and the stiffness of the group. For this purpose, a parametric three‐dimensional non‐linear numerical analysis was carried out for different arrangements of pile groups. The response of the pile groups is compared to that of the single pile. The influence of the number of piles, the spacing and the deflection level to the group response is discussed. Furthermore, the contribution of the piles constituting the group to the total group resistance is examined. Finally, a relationship is proposed allowing a reasonable prediction of the response of fixed‐head pile groups at least for similar soil profile conditions. Copyright © 2005 John Wiley & Sons, Ltd.     

A bounding surface elasto‐viscoplastic constitutive model for non‐isothermal cyclic analysis of asphaltic materials

An elasto‐viscoplastic constitutive model for asphaltic materials is presented within the context of bounding surface plasticity theory, taking into account the effects of the stress state, void binder degree of saturation, temperature and strain rate on the material behaviour. A stress state dependent non‐linear elasticity model is introduced to represent time‐independent recoverable portion of the deformation. The consistent visco‐plasticity framework is utilised to capture the rate‐dependent, non‐recoverable strain components. The material parameters introduced in the model are identified, and their determination from conventional laboratory tests is discussed. The capability of the model to reproduce experimentally observed response of asphaltic materials is demonstrated through numerical simulations of several laboratory test data from the literature. Copyright © 2016 John Wiley & Sons, Ltd.     

Non‐linear consolidation analysis of soil with variable compressibility and permeability under cyclic loadings

In this paper, a simple semi‐analytical method has been developed to solve the one‐dimensional non‐linear consolidation problems by considering the changes of compressibility and permeability of the soil layer, subjected to complicated time‐dependent cyclic loadings at the ground surface. The solution presented here takes into account e ～ lg k_v and e ～ lg σ′ linear responses. With c_k the slope of the e ～ lg k_v line and c_c as the slope of the e ～ lg σ′ line, the identified parameter c_c/c_k is found to control the rate of consolidation. Using the solutions obtained, some diagrams are prepared and the relevant behaviours of one‐dimensional non‐linear consolidation of saturated soft soil under cyclic loadings are discussed. The method in this paper does not require any special data; conventional oedometer data can be used. Therefore, the method is particularly efficient and convenient for engineering practice. Copyright © 2006 John Wiley & Sons, Ltd.     

The influence of a non‐associated flow rule on the calculation of the factor of safety of soil slopes

This paper presents a method that incorporates a non‐associated flow rule into the limit analysis to investigate the influence of the dilatancy angle on the factor of safety for the slope stability analysis. The proposed method retain's the advantage of the upper bound method, which is simple and has no stress involvement in the calculation of the energy dissipation and the factor of safety. Copyright © 2001 John Wiley & Sons, Ltd.     

Numerical simulations of simple shear with non‐coaxial soil models

This paper investigates the effects of a non‐coaxial model on simulated stress–strain behaviour of granular materials subject to simple shearing under various initial conditions. In most cases, a significant difference of predictions between coaxial and non‐coaxial modelling is found during the early stage in shearing. With the increase in shearing, non‐coaxial simulations approach and tend to coincide with coaxial simulations. It is also found that the roles of non‐coaxial modelling in simulating simple shear behaviour are considerably influenced by hardening rules, flow rules, initial static lateral pressure coefficients. In some cases, the non‐coaxial modelling gives a similar simulation as the coaxial modelling. In other cases, the non‐coaxial modelling decreases the hardening response or softening response of materials, compared with the coaxial modelling. Under certain conditions, the predicted peak strength of materials with non‐coaxial modelling is larger than that for coaxial modelling. Some of these observations can be attributed to the amount of principal stress rotation in various cases analysed. Others can be attributed to the difference between the directions of the non‐coaxial plastic flow and those for coaxial plastic flow. Copyright © 2005 John Wiley & Sons, Ltd.     

A finite strain solution for the elastoplastic ground response curve in tunnelling: rocks with non‐linear failure envelopes

This paper generalizes the finite strain Coulomb solution of Vrakas and Anagnostou (Int J Numer Anal Meth Geomech 2014; 38(11): 1131–1148) for the classic tunnel mechanics problem of the ground response curve to elastoplastic grounds satisfying a non‐linear Mohr's failure criterion. A linear (Coulomb‐type) plastic potential function is used, leading to a non‐associated flow law, and edge plastic flow is considered in the plastic zone. The solution for a general non‐linear Mohr's failure criterion is semi‐analytical in that it requires the evaluation of definite integrals. In the special case of the Hoek–Brown criterion, however, these integrals are calculated analytically, resulting in a rigorous closed‐form series solution. The applicability of the derived solution is illustrated through the example of the Yacambú‐Quibor tunnel, where very large deformations were observed when crossing of weak graphitic phyllites. Copyright © 2016 John Wiley & Sons, Ltd.     

Compressibility and permeability of sand–kaolin mixtures. Experiments versus non‐linear homogenization schemes

This study deals with the behaviour of mixtures of sand and saturated kaolin paste considered as composite materials made of permeable and deformable (with non‐linear behaviour) matrix (the kaolin paste) with rigid and impervious inclusions (the sand grains). Oedometric and permeability tests highlight the key role of the state of the clay paste, and show the existence of a threshold of sand grain concentration above which a structuring effect influences both compressibility and permeability. At the light of these experiments two homogenization schemes (with simplifying assumptions to make the problem manageable) are considered to model these two parameters. Qualitative and quantitative comparisons with experimental data point out their respective domain of interest and limitations: a tangent homogenization scheme is shown to be sufficient to describe the macroscopic properties for dilute sand concentration; above the concentration threshold, the structuring effect is captured by the new homogenization scheme developed in this paper. Copyright © 2010 John Wiley & Sons, Ltd.     

A modulus‐multiplier approach for non‐linear analysis of laterally loaded pile groups

A modulus‐multiplier approach, which applies a reduction factor to the modulus of single pile p–y curves to account for the group effect, is presented for analysing the response of each individual pile in a laterally loaded pile group with any geometric arrangement based on non‐linear pile–soil–pile interaction. The pile–soil–pile interaction is conducted using a 3D non‐linear finite element approach. The interaction effect between piles under various loading directions is investigated in this paper. Group effects can be neglected at a pile spacing of 9 times the pile diameter for piles along the direction of the lateral load and at a pile spacing of 6 times the pile diameter for piles normal to the direction of loading. The modulus multipliers for a pair of piles are developed as a function of pile spacing for departure angle of 0, 90, and 180sup>/sup> with respect to the loading direction. The procedure proposed for computing the response of any individual pile within a pile group is verified using two well‐documented full‐scale pile load tests. Copyright © 2006 John Wiley & Sons, Ltd.     

A non‐coaxial constitutive model for sand deformation under rotation of principal stress axes

A constitutive model for the simulation of non‐coaxiality, an aspect of anisotropic behavior of sand subjected to the rotation of the principal stress axes, is presented in this paper. Experimental studies have shown that non‐coaxiality or non‐coincidence of principal plastic strain increments with principal stress axes under loadings involving the rotation of principal stress axes may be considerable. Besides, the rotation of the principal stress axes results in dramatic effects on stiffness and dilatant behavior of sand. Therefore, the consequences of principal stress axes rotation on deformational behavior, dilatancy and soil stiffness must be taken into account in theoretical and practical problems. To this aim, the following steps are taken: (1) A general relationship for flow direction with respect to possibility of non‐coaxial flow is developed. Moreover, special circumstances linking non‐coaxiality to instantaneous interaction between loading and soil fabric are proposed. (2) Proposing novel expressions for plastic modulus and dilatancy function, the model is enforced to provide realistic simulations when sand is subjected to the rotation of the principal stress axes. Finally, with numerous examples and comparisons, the model capabilities are shown under various stress paths and drainage conditions. Copyright © 2007 John Wiley & Sons, Ltd.     

An improved implicit numerical integration of a non‐associated,three‐invariant cap plasticity model with mixed isotropic–kinematic hardening for geomaterials

The Sandia GeoModel is a continuum elastoplastic constitutive model that captures many features of the mechanical response for geological materials over a wide range of porosities and strain rates. Among the specific features incorporated into the formulation are a smooth compression cap, isotropic/kinematic hardening, nonlinear pressure dependence, strength differential effect, and rate sensitivity. This study attempts to provide enhancements regarding computational tractability, domain of applicability, and robustness of the model. A new functional form is presented for the yield and plastic potential functions. This reformulation renders a more accurate, robust, and efficient model as it eliminates spurious solutions attributed to the original form. In addition, we achieve a high‐performance implementation, because the local iterative method is allowed to recast residual vectors with a uniform dimensionality. The model is also furnished with a smooth, elliptical tension cap to account for the tensile yielding. Moreover, an efficient algorithm is introduced, which decreases the computational cost by differentiating the updated shear yield surface from the cap surface based on the trial relative stress state. Finally, various numerical examples including a large‐scale boundary value problem are presented to demonstrate the fidelity of the modified model and to analyze its numerical performance. Copyright © 2015 John Wiley & Sons, Ltd.     

A non‐linear coupled finite element–boundary element model for the prediction of vibrations due to vibratory and impact pile driving

This paper presents a non‐linear coupled finite element–boundary element approach for the prediction of free field vibrations due to vibratory and impact pile driving. Both the non‐linear constitutive behavior of the soil in the vicinity of the pile and the dynamic interaction between the pile and the soil are accounted for. A subdomain approach is used, defining a generalized structure consisting of the pile and a bounded region of soil around the pile, and an unbounded exterior linear soil domain. The soil around the pile may exhibit non‐linear constitutive behavior and is modelled with a time‐domain finite element method. The dynamic stiffness matrix of the exterior unbounded soil domain is calculated using a boundary element formulation in the frequency domain based on a limited number of modes defined on the interface between the generalized structure and the unbounded soil. The soil–structure interaction forces are evaluated as a convolution of the displacement history and the soil flexibility matrices, which are obtained by an inverse Fourier transformation from the frequency to the time domain. This results in a hybrid frequency–time domain formulation of the non‐linear dynamic soil–structure interaction problem, which is solved in the time domain using Newmark's time integration method; the interaction force time history is evaluated using the θ‐scheme in order to obtain stable solutions. The proposed hybrid formulation is validated for linear problems of vibratory and impact pile driving, showing very good agreement with the results obtained with a frequency‐domain solution. Linear predictions, however, overestimate the free field peak particle velocities as observed in reported field experiments during vibratory and impact pile driving at comparable levels of the transferred energy. This is mainly due to energy dissipation related to plastic deformations in the soil around the pile. Ground vibrations due to vibratory and impact pile driving are, therefore, also computed with a non‐linear model where the soil is modelled as an isotropic elastic, perfectly plastic solid, which yields according to the Drucker–Prager failure criterion. This results in lower predicted free field vibrations with respect to linear predictions, which are also in much better agreement with experimental results recorded during vibratory and impact pile driving. Copyright © 2008 John Wiley & Sons, Ltd.     

A new parabolic variational inequality formulation of Signorini's condition for non‐steady seepage problems with complex seepage control systems

By extending Darcy's law to the dry domain above the free surface and specifying the boundary condition on the potential seepage surfaces as Signorini's type, a partial differential equation (PDE) defined in the entire domain of interest is formulated for non‐steady seepage flow problems with free surfaces. A new parabolic variational inequality (PVI) formulation equivalent to the PDE formulation is then proposed, in which the flux part of the complementary condition of Signorini's type in the PDE formulation is transformed into natural boundary condition. Consequently, the singularity at the seepage points is eliminated and the difficulty in selecting the trial functions is significantly reduced. By introducing an adaptive penalized Heaviside function in the finite element analysis, the numerical stability of the discrete PVI formulation is well guaranteed. The proposed approach is validated by the existing laboratory tests with sudden rise and dropdown of water heads, and then applied to capture the non‐steady seepage flow behaviors in a homogeneous rectangular dam with five drainage tunnels during a linear dropdown of upstream water head. The non‐steady seepage flow in the surrounding rocks of the underground powerhouse in the Shuibuya Hydropower Project is further modeled, in which a complex seepage control system is involved. Comparisons with the in situ monitoring data show that the calculation results well illustrate the non‐steady seepage flow process during impounding and the operation of the reservoir as well as the seepage control effects of the drainage hole arrays and drainage tunnels. Copyright © 2010 John Wiley & Sons, Ltd.     

High‐pressure metamorphic evolution of eclogite and associated metapelite from the Chuacús complex (Guatemala Suture Zone): Constraints from phase equilibria modelling coupled with Lu‐Hf and U‐Pb geochronology

As is common in suture zones, widespread high‐pressure rocks in the Caribbean region reached eclogite facies conditions close to ultrahigh‐pressure metamorphism. Besides eclogite lenses, abundant metapelitic rocks in the Chuacús complex (Guatemala Suture Zone) also preserve evidence for high‐pressure metamorphism. A comprehensive petrological and geochronological study was undertaken to constrain the tectonometamorphic evolution of eclogite and associated metapelite from this area in central Guatemala. The integration of field and petrological data allows the reconstruction of a previously unknown segment of the prograde P–T path and shows that these contrasting rock types share a common high‐pressure evolution. An early stage of high‐pressure/low‐temperature metamorphism at 18–20 kbar and 530–580°C is indicated by garnet core compositions as well as the nature and composition of mineral inclusions in garnet, including kyanite–jadeite–paragonite in an eclogite, and chloritoid–paragonite–rutile in a pelitic schist. Peak high‐pressure conditions are constrained at 23–25 kbar and 620–690°C by combining mineral assemblages, isopleth thermobarometry and Zr‐in‐rutile thermometry. A garnet/whole‐rock Lu‐Hf date of 101.8 ± 3.1 Ma in the kyanite‐bearing eclogite indicates the timing of final garnet growth at eclogite facies conditions, while a Lu‐Hf date of 95.5 ± 2.1 Ma in the pelitic schist reflects the average age of garnet growth spanning from an early eclogite facies evolution to a final amphibolite facies stage. Concordant U‐Pb LA‐ICP‐MS zircon data from the pelitic schist, in contrast, yield a mean age of 74.0 ± 0.5 Ma, which is equivalent to a U‐Pb monazite lower‐intercept age of 73.6 ± 2.0 Ma in the same sample, and comparable within errors with a less precise U‐Pb lower‐intercept age of 80 ± 13 Ma obtained in post‐eclogitic titanite from the kyanite‐bearing eclogite. These U‐Pb metamorphic ages are interpreted as dating an amphibolite facies overprint. Protolith U‐Pb zircon ages of 167.1 ± 4.2 Ma and 424.6 ± 5.0 Ma from two eclogite samples reveal that mafic precursors in the Chuacús complex originated in multiple tectonotemporal settings from the Silurian to Jurassic. The integration of petrological and geochronological data suggests that subduction of the continental margin of the North American plate (Chuacús complex) beneath the Greater Antilles arc occurred during an Albian‐Cenomanian pre‐collisional stage, and that a subsequent Campanian collisional stage is probably responsible of the amphibolite facies overprint and late syncollisional exhumation.     

Conceptual models for short‐eccentricity‐scale climate control on peat formation in a lower Palaeocene fluvial system,north‐eastern Montana (USA)

Fluvial systems in which peat formation occurs are typified by autogenic processes such as river meandering, crevasse splaying and channel avulsion. Nevertheless, autogenic processes cannot satisfactorily explain the repetitive nature and lateral continuity of many coal seams (compacted peats). The fluvial lower Palaeocene Tullock Member of the Fort Union Formation (Western Interior Williston Basin; Montana, USA ) contains lignite rank coal seams that are traceable over distances of several kilometres. This sequence is used to test the hypothesis that peat formation in the fluvial system was controlled by orbitally forced climate change interacting with autogenic processes. Major successions are documented with an average thickness of 6·8 m consisting of ca 6 m thick intervals of channel and overbank deposits overlain by ca 1 m thick coal seam units. These major coal seams locally split and merge. Time‐stratigraphic correlation, using a Cretaceous–Palaeogene boundary event horizon, several distinctive volcanic ash‐fall layers, and the C29r/C29n magnetic polarity reversal, shows consistent lateral recurrence of seven successive major successions along a 10 km wide fence panel perpendicular to east/south‐east palaeo‐flow. The stratigraphic pattern, complemented by stratigraphic age control and cyclostratigraphic tests, suggests that the major peat‐forming phases, resulting in major coal seams, were driven by 100 kyr eccentricity‐related climate cycles. Two distinct conceptual models were developed, both based on the hypothesis that the major peat‐forming phases ended when enhanced seasonal contrast, at times of minimum precession during increasing eccentricity, intensified mire degradation and flooding. In model 1, orbitally forced climate change controls the timing of peat compaction, leading to enhancement of autogenic channel avulsions. In model 2, orbitally forced climate change controls upstream sediment supply and clastic influx determining the persistence of peat‐forming conditions. At the scale of the major successions, model 2 is supported because interfingering channel sandstones do not interrupt lateral continuity of major coal seams.     

Interaction of an antecedent fluvial system with early normal fault growth: Implications for syn‐rift stratigraphy,western Corinth rift (Greece)

Continental ‘overfilled’ conditions during rift initiation are conventionally explained as due to low creation of accommodation compared with sediment supply. Alternatively, sediment supply can be relatively high from the onset of rifting due to an antecedent drainage system. The alluvial Lower Group of the western Plio–Pleistocene Corinth rift is used to investigate the interaction of fluvial sedimentation with early rifting. This rift was obliquely superimposed on the Hellenide mountain belt from which it inherited a significant palaeorelief. Detailed sedimentary logging and mapping of the well‐exposed syn‐rift succession document the facies distributions, palaeocurrents and stratigraphic architecture. Magnetostratigraphy and biostratigraphy are used to date and correlate the alluvial succession across and between fault blocks. From 3·2 to 1·8 Ma, a transverse low sinuosity braided river system flowed north/north‐east to east across east–west‐striking active fault blocks (4 to 7 km in width). Deposits evolved downstream from coarse alluvial conglomerates to fine‐grained lacustrine deposits over 15 to 30 km. The length scale of facies belts is much greater than, and thus not directly controlled by, the width of the fault blocks. At its termination, the distributive river system built small, stacked deltas into a shallow lake margin. The presence of a major antecedent drainage system is supported by: (i) a single major sediment entry point; (ii) persistence of a main channel belt axis; (iii) downstream fining at the scale of the rift basin. The zones of maximum subsidence on individual faults are aligned with the persistent fluvial axis, suggesting that sediment supply influenced normal fault growth. Instead of low accommodation rate during the early rift phase, this study proposes that facies progradation can be controlled by continuous and high sediment supply from antecedent rivers.     

Setting,sulfur isotope variations,and metamorphism of Jurassic massive Zn‐Pb‐Ag sulfide mineralization associated with arc‐type volcanism (Skra,Vardar zone, Νorthern Greece)

Massive Zn‐Pb‐Ag sulfide mineralization appears conformable with felsic volcanism, developed in an Upper Jurassic volcanic arc to the Southwest (SW) of the Serbo‐Macedonian continent in Northern Greece. The host volcanic sequence of the mineralization comprises mylonitized rhyolitic to rhyodacitic lavas, pyroclastics, quartz‐feldspar porphyries, and cherty tuffs. A “white mica—quartz—pyrite” mineral assemblage characterizes the volcanic rocks in the footwall and hanging‐wall of massive sulfide ore layers, formed as a result of greenschist‐grade regional metamorphism on “clay‐quartz‐pyrite” hydrothermal alteration haloes. Massive ore lenses are usually underlain by deformed Cu‐pyrite and quartz‐pyrite stockworks. Most of the sulfide ore bodies have proximal‐type features. Ductile deformation and regional metamorphism have transformed many of the stockwork structures. The mineralization is characterized by high Zn, Pb, and Ag contents, while Cu and critical metals are low. Primary depositional textures, for example, layering, clastic pyrite, colloform, and atoll textures were identified. The overall textural features of the mineralization indicate it has undergone mechanical deformation. The most prominent features of the effects of metamorphism, folding and shearing, are modification of the ore body morphology toward flattened and boudinage structures and transformation of the ore textures toward the dominance of planar fabrics. Sulfur isotope analyses of sulfides along with textural observations are consistent with a dual source of sulfide sulfur. Sulfur isotope values for sphalerite, non‐colloform pyrite, galena, and chalcopyrite fall in a limited range from ?1.6 to +4.8‰ (mean δ³⁴S + 2‰), indicating a hydrothermal source derived from the reduction of coeval seawater sulfate in the convective system. Pyrites with colloform and atoll textures are characterized by a ³⁴S depletion, indicating a bacterial reduction of coeval seawater sulfate. The morphology of ore beds, the mineralogy, sulfide textures, and ore chemistry along with the petrology and tectonic setting of the host rocks can be attributed to typical of a bimodal‐felsic metallogenesis. Although similar in many respects to classic Kuroko‐type volcanogenic massive sulfide mineralization, it has some atypical features, like the absence of barite ore, which is possibly a result of significant temporal depletion in sulfate due to bacterial reduction, a conclusion supported by the widespread occurrence of colloidal and atoll textures of pyrite.