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1.
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.  相似文献   

2.
This paper examines the performance of the Jacobi preconditioner when used with two Krylov subspace iterative methods. The number of iterations needed for convergence was shown to be different for drained, undrained and consolidation problems, even for similar condition number. The differences were due to differences in the eigenvalue distribution, which cannot be completely described by the condition number alone. For drained problems involving large stiffness ratios between different material zones, ill‐conditioning is caused by these large stiffness ratios. Since Jacobi preconditioning operates on degrees‐of‐freedom, it effectively homogenizes the different spatial sub‐domains. The undrained problem, modelled as a nearly incompressible problem, is much more resistant to Jacobi preconditioning, because its ill‐conditioning arises from the large stiffness ratios between volumetric and distortional deformational modes, many of which involve the similar spatial domains or sub‐domains. The consolidation problem has two sets of degrees‐of‐freedom, namely displacement and pore pressure. Some of the eigenvalues are displacement dominated whereas others are excess pore pressure dominated. Jacobi preconditioning compresses the displacement‐dominated eigenvalues in a similar manner as the drained problem, but pore‐pressure‐dominated eigenvalues are often over‐scaled. Convergence can be accelerated if this over‐scaling is recognized and corrected for. Copyright © 2002 John Wiley & Sons, Ltd.  相似文献   

3.
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.  相似文献   

4.
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 Kep (ILU(0)‐Kep). Iteration time is reduced by 40% by using IDR(s = 6) with ILU(0)‐Kep. To further reduce computational cost, the global stiffness matrix Kep is divided into two parts. The first part is the linear elastic stiffness matrix Ke, 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)‐Ke preconditioner is more time effective than IDR(s = 6) with ILU(0)‐Kep 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.  相似文献   

5.
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.  相似文献   

6.
The problem of finite element simulation of incompressible fluid flow in porous medium is considered. The porous medium is characterized by the X‐ray microtomography technique in three dimensions. The finite calculus‐based stabilization technique is reviewed to implement the equal order finite element interpolation functions for both velocity and pressure. A noble preconditioner, the nodal block diagonal preconditioner, is considered whose performance is thoroughly investigated. Combining this preconditioner with a standard iterative solver during the computational homogenization procedure, it is possible to carry out the large‐scale fluid flow simulation for estimating permeability of the porous medium with reasonable accuracy and reliability. Copyright © 2013 John Wiley & Sons, Ltd.  相似文献   

7.
Parallel computers are potentially very attractive for the implementation of large size geomechanical models. One of the main difficulties of parallelization, however, relies on the efficient solution of the frequently ill‐conditioned algebraic system arising from the linearization of the discretized equilibrium equations. While very efficient preconditioners have been developed for sequential computers, not much work has been devoted to parallel solution algorithms in geomechanics. The present study investigates the state‐of‐the‐art performance of the factorized sparse approximate inverse (FSAI) as a preconditioner for the iterative solution of ill‐conditioned geomechanical problems. Pre‐and post‐filtration strategies are experimented with to increase the FSAI efficiency. Numerical results show that FSAI exhibits a promising potential for parallel geomechanical models mainly because of its almost ideal scalability. With the present formulation, however, at least 4 or 8 processors are required in the selected test cases to outperform one of the most efficient sequential algorithms available for FE geomechanics, i.e. the multilevel incomplete factorization (MIF). Further research is needed to improve the FSAI efficiency with a more effective selection of the preconditioner non‐zero pattern. Copyright © 2011 John Wiley & Sons, Ltd.  相似文献   

8.
Sphalerite (ZnS) is an abundant ore mineral and an important carrier of elements such as Ge, Ga and In used in high‐technology applications. In situ measurements of trace elements in natural sphalerite samples using LA‐ICP‐MS are hampered by a lack of homogenous matrix‐matched sulfide reference materials available for calibration. The preparation of the MUL‐ZnS1 calibration material containing the trace elements V, Cr, Mn, Co, Ni, Cu, Ga, Ge, As, Se, Mo, Ag, Cd, In, Sn, Sb, Tl and Pb besides Zn, Fe and S is reported. Commercially available ZnS, FeS, CdS products were used as the major components, whereas the trace elements were added by doping with single‐element ICP‐MS standard solutions and natural mineral powders. The resulting powder mixture was pressed to pellets and sintered at 400 °C for 100 h using argon as an inert gas. To confirm the homogeneity of major and trace element distributions within the MUL‐ZnS1 calibration material, measurements were performed using EPMA, solution ICP‐MS, ICP‐OES and LA‐ICP‐MS. The results show that MUL‐ZnS‐1 is an appropriate material for calibrating trace element determination in sphalerite using LA‐ICP‐MS.  相似文献   

9.
Preconditioned projection (or conjugate gradient like) methods are increasingly used for the accurate and efficient solution to finite element (FE) coupled consolidation equations. Theory indicates that preliminary row/column scaling does not affect the eigenspectrum of the iteration matrix controlling convergence as long as the preconditioner relies on the incomplete factorization of the FE coefficient matrix. However, computational experience with mid‐large size problems shows that the above inexpensive operation can significantly accelerate the solver convergence, and to a minor extent also improve the final accuracy, as a result of a better solver stability to the accumulation and propagation of floating point round‐off errors. This is demonstrated with the aid of the least square logarithm (LSL) scaling algorithm on FE consolidation problems of increasing size up to more than 100 000. It is shown that a major source of numerical instability rests with the sub‐matrix which couples the structural to the fluid part of the underlying mathematical model. It is concluded that for mid‐large size, possibly difficult, FE consolidation problems left/right LSL scaling is to be always recommended when the incomplete factorization is used as a preconditioning technique. Copyright © 2003 John Wiley & Sons, Ltd.  相似文献   

10.
Unbounded plane stress and plane strain domains subjected to static loading undergo infinite displacements, even when the zero displacement boundary condition at infinity is enforced. However, the stress and strain fields are well behaved, and are of practical interest. This causes significant difficulty when analysis is attempted using displacement‐based numerical methods, such as the finite‐element method. To circumvent this difficulty problems of this nature are often changed subtly before analysis to limit the displacements to finite values. Such a process is unsatisfactory, as it distorts the solution in some way, and may lead to a stiffness matrix that is nearly singular. In this paper, the semi‐analytical scaled boundary finite‐element method is extended to permit the analysis of such problems without requiring any modification of the problem itself. This is possible because the governing differential equations are solved analytically in the radial direction. The displacement solutions so obtained include an infinite component, but relative motion between any two points in the unbounded domain can be computed accurately. No small arbitrary constants are introduced, no arbitrary truncation of the domain is performed, and no ill‐conditioned matrices are inverted. Copyright © 2002 John Wiley & Sons, Ltd.  相似文献   

11.
A method for the simultaneous determination of Cd with In, Tl and Bi by isotope dilution‐internal standardisation (ID‐IS) ICP‐QMS using the same aliquot for rare earth element and other trace element determinations was developed. Samples mixed with an enriched 149Sm spike were decomposed using a HF‐HClO4 mixture, which was evaporated and then diluted with HNO3. After determination of Sm by ID‐ICP‐QMS and Cd, In, Tl and Bi concentrations were determined using the 149Sm intensity as an internal standard. The interference of MoO+ on Cd+ was corrected using the MoO+/Mo+ ratio separately measured using a Mo standard solution, and the validity of the externally determined oxide‐forming ratio correction was evaluated. The MoO+/Mo+ ratios measured using the standard solution and samples were ～ 0.0002 and < 0.002, respectively. Detection limits for Cd, In, Tl and Bi in silicate samples were at levels of < 1 ng g?1 with a total uncertainty of < 7%. Cadmium in the carbonaceous chondrites, Orgueil (CI1), Murchison (CM2) and Allende (CV3) as well as Cd, In, Tl and Bi in the reference materials, JB‐2, JB‐3, JA‐1, JA‐2, JA‐3, JP‐1 (GSJ), BHVO‐1, AGV‐1, PCC‐1 and DTS‐1 (USGS) and NIST SRM 610, 612, 614 and 616 were determined to show the applicability of this method.  相似文献   

12.
There is currently a lack of well‐characterised matrix‐matched reference materials (RMs) for forensic analysis of U‐rich materials at high spatial resolution. This study reports a detailed characterisation of uraninite (nominally UO2+x) from the Happy Jack Mine (UT, USA). The Happy Jack uraninite can be used as a RM for the determination of rare earth element (REE) mass fractions in nuclear materials, which provide critical information for source attribution purposes. This investigation includes powder X‐ray diffraction (pXRD) data, as well as major, minor and trace element abundances determined using a variety of micro‐analytical techniques. The chemical signature of the uraninite was investigated at the macro (cm)‐scale with micro‐X‐ray fluorescence (µXRF) mapping and at high spatial resolution (tens of micrometre scale) using electron probe microanalysis (EPMA) and laser ablation‐inductively coupled plasma‐mass spectrometry (LA‐ICP‐MS) analyses. Based on EPMA results, the uraninite is characterised by homogeneous UO2 and CaO contents of 91.57 ± 1.49% m/m (2s uncertainty) and 2.70 ± 0.38% m/m (2s), respectively. Therefore, CaO abundances were used as the internal standard when conducting LA‐ICP‐MS analyses. Overall, the major element and REE compositions are homogeneous at both the centimetre and micrometre scales, allowing this material to be used as a RM for high spatial resolution analysis of U‐rich samples.  相似文献   

13.
The anomaly response characteristics of the vertical line‐source 3D borehole‐to‐surface model are simulated by an adaptive finite element method. The calculation shows that the anomaly in the radial direction is pressed and the closer to the source, the more pronounced, the anomaly is stretched in the direction perpendicular to the radius. Adding a B pole in the Y direction can offset the effect of stretching to some extent. The anomaly that is closer to the source of the profile is clearer than the anomaly that is far from the source. The research results are of great significance for guiding the practical application of the borehole‐to‐surface electrical method.  相似文献   

14.
Strain‐softening in geomaterials often leads to ill‐posed boundary‐valued problems (BVP), which cannot be solved with finite element methods without introducing some kind of regularization such as nonlocal plasticity. Hereafter we propose to apply spectral analysis for testing the performance of nonlocal plasticity in regularizing ill‐posed BVP and producing mesh‐independent solutions when local plasticity usually fails. The spectral analysis consists of examining the eigenvalues and eigenvectors of the global tangential stiffness matrix of the incremental equilibrium equations. Based on spectral analysis, we propose a criterion for passing or failing the test of constitutive regularization in the context of BVP. If the eigenvalues of the tangential operator are all positive then the regularization succeeds, otherwise it fails and may not prevent artificial mesh‐dependent solutions from appearing. The approach is illustrated in the particular case of a biaxial compression with strain‐softening plasticity. In this particular case, local softening plasticity is found to produce negative eigenvalues in the tangential stiffness matrix, which indicates ill‐posed BVP. In contrast, nonlocal softening plasticity always produces positive eigenvalues, which regularizes ill‐posed BVP. The dominant eigenvectors, which generate localized deformation patterns, have a bandwidth independent of mesh size, provided that the mesh is fine enough to capture localization. These mesh‐independent eigenmodes explain why nonlocal plasticity produces numerical solutions that are mesh‐independent. Copyright © 2011 John Wiley & Sons, Ltd.  相似文献   

15.
Apatite incorporates variable and significant amounts of halogens (mainly F and Cl) in its crystal structure, which can be used to determine the initial F and Cl concentrations of magmas. The amount of chlorine in the apatite lattice also exerts an important compositional control on the degree of fission‐track annealing. Chlorine measurements in apatite have conventionally required electron probe microanalysis (EPMA). Laser ablation inductively coupled plasma‐mass spectrometry (LA‐ICP‐MS) is increasingly used in apatite fission‐track dating to determine U concentrations and also in simultaneous U‐Pb dating and trace element measurements of apatite. Apatite Cl measurements by ICP‐MS would remove the need for EPMA but the high (12.97 eV) first ionisation potential makes analysis challenging. Apatite Cl data were acquired using two analytical set‐ups: a Resonetics M‐50 193 nm ArF Excimer laser coupled to an Agilent 7700× quadrupole ICP‐MS (using a 26 μm spot with an 8 Hz repetition rate) and a Photon Machines Analyte Excite 193 nm ArF Excimer laser coupled to a Thermo Scientific iCAP Qc (using a 30 μm spot with a 4 Hz repetition rate). Chlorine concentrations were determined by LA‐ICP‐MS (1140 analyses in total) for nineteen apatite occurrences, and there is a comprehensive EPMA Cl and F data set for 13 of the apatite samples. The apatite sample suite includes different compositions representative of the range likely to be encountered in natural apatites, along with extreme variants including two end‐member chlorapatites. Between twenty‐six and thirty‐nine isotopes were determined in each apatite sample corresponding to a typical analytical protocol for integrated apatite fission track (U and Cl contents) and U‐Pb dating, along with REE and trace element measurements. 35Cl backgrounds (present mainly in the argon gas) were ~ 45–65 kcps in the first set‐up and ~ 4 kcps in the second set‐up. 35Cl background‐corrected signals ranged from ~ 0 cps in end‐member fluorapatite to up to ~ 90 kcps in end‐member chlorapatite. Use of a collision cell in both analytical set‐ups decreased the low mass sensitivity by approximately an order of magnitude without improving the 35Cl signal‐to‐background ratio. A minor Ca isotope was used as the internal standard to correct for drift in instrument sensitivity and variations in ablation volume during sessions. The 35Cl/43Ca values for each apatite (10–20 analyses each) when plotted against the EPMA Cl concentrations yield excellently constrained calibration relationships, demonstrating the suitability of the analytical protocol and that routine apatite Cl measurements by ICP‐MS are achievable.  相似文献   

16.
A novel three‐dimensional particle‐based technique utilizing the discrete element method is proposed to analyze the seismic response of soil‐foundation‐structure systems. The proposed approach is employed to investigate the response of a single‐degree‐of‐freedom structure on a square spread footing founded on a dry granular deposit. The soil is idealized as a collection of spherical particles using discrete element method. The spread footing is modeled as a rigid block composed of clumped particles, and its motion is described by the resultant forces and moments acting upon it. The structure is modeled as a column made of particles that are either clumped to idealize a rigid structure or bonded to simulate a flexible structure of prescribed stiffness. Analysis is done in a fully coupled scheme in time domain while taking into account the effects of soil nonlinear behavior, the possible separation between foundation base and soil caused by rocking, the possible sliding of the footing, and the dynamic soil‐foundation interaction as well as the dynamic characteristics of the superstructure. High fidelity computational simulations comprising about half a million particles were conducted to examine the ability of the proposed technique to model the response of soil‐foundation‐structure systems. The computational approach is able to capture essential dynamic response patterns. The cyclic moment–rotation relationships at the base center point of the footing showed degradation of rotational stiffness by increasing the level of strain. Permanent deformations under the foundation continued to accumulate with the increase in number of loading cycles. Copyright © 2011 John Wiley & Sons, Ltd.  相似文献   

17.
Lithium separation technique for three reference materials has been established together with precise determination of lithium isotope using a Neptune multi collector-inductively coupled plasma mass spectrometry (MC-ICP-MS). The solutions of lithium element standard reference materials, potassium, calcium, sodium, magnesium and iron single element, were used to evaluate analytical methods applied. Three separate stages of ion-exchange chromatography were carried out using organic cation-exchange resin (AG 50W-X8). Lithium was enriched for the three stages using different eluants, which are 2.8 M HCl, 0.15 M HCl and 0.5 M HCl in 30% ethanol, respectively. The columns for the first and second stages are made of polypropylene, and those for the third stage are made of quartz. Total reagent volume for the entire chemical process was 35 mL for three reference materials. The recovery yielded for the three stages is 98.9–101.2% with an average of 100.0%, 97.6–101.9% with an average of 99.9%, and 99.8–103.3% with an average of 100.6%, respectively. The precision of this technique is conservatively estimated to be ±0.72–1.04‰ (2σ population), which is similar to the precision obtained by different authors in different laboratories with MC-ICP-MS. The δ7Li values (7Li/6Li relative to the IRMM-016 standard) determined for andesite (AGV-2) and basalt (BHVO-2) are 5.68‰ (n=18), 4.33‰ (n=18), respectively. The δ7Li value (7Li/6Li relative to the L-SVEC standard) determined for IRMM-016 is –0.01‰ (n=15). All these analytical results are in good agreement with those previously reported. In addition, the results for the same kinds of samples analyzed at the MLR Key Laboratory of Metallogeny and Mineral Assessment, Institute of Mineral Resources, Chinese Academy of Geological Sciences, are consistent with those obtained at the Plasma Laboratory, University of Maryland, within analytical uncertainty. According to these experiment results, it is concluded that this proposed procedure is a suitable method for determining the lithium isotopic composition of natural samples.  相似文献   

18.
A method is presented for coupling cubic‐order quadrilateral finite elements with the finite side of a new coordinate ascent hierarchical infinite element. At a common side shared by a hierarchical infinite element and an arbitrary number of finite elements, the displacements are minimized in the least square sense with respect to the degrees‐of‐freedom of the finite elements. This leads to a set of equations that relate the degrees‐of‐freedom of the finite and hierarchical infinite elements on the shared side. The method is applied to a non‐homogeneous cross‐anisotropic half‐space subjected to a non‐uniform circular loading with Young's and shear moduli varying with depth according to the power law. A constant mesh constructed from coupled finite and hierarchical infinite elements is used and convergence is sought simply by increasing the degree of the interpolating polynomial. The displacements and stresses produced by conical and parabolic circular loads applied on the surface are obtained. The efficiency of the proposed method is demonstrated through convergence and comparison studies. New results produced by a frusto‐conical circular load applied on the surface of a half‐space made up of heavily consolidated London clay are provided. The non‐homogeneity parameter and degree of anisotropy are shown to influence the soil response. Copyright © 2012 John Wiley & Sons, Ltd.  相似文献   

19.
Artificial ground freezing (AGF) is a commonly used technique in geotechnical engineering for ground improvement such as ground water control and temporary excavation support during tunnel construction in soft soils. The main potential problem connected with this technique is that it may produce heave and settlement at the ground surface, which may cause damage to the surface infrastructure. Additionally, the freezing process and the energy needed to obtain a stable frozen ground may be significantly influenced by seepage flow. Evidently, safe design and execution of AGF require a reliable prediction of the coupled thermo‐hydro‐mechanical behavior of freezing soils. With the theory of poromechanics, a three‐phase finite element soil model is proposed, considering solid particles, liquid water, and crystal ice as separate phases and mixture temperature, liquid pressure, and solid displacement as the primary field variables. In addition to the volume expansion of water transforming into ice, the contribution of the micro‐cryo‐suction mechanism to the frost heave phenomenon is described in the model using the theory of premelting dynamics. Through fundamental physical laws and corresponding state relations, the model captures various couplings among the phase transition, the liquid transport within the pore space, and the accompanying mechanical deformation. The verification and validation of the model are accomplished by means of selected analyses. An application example is related to AGF during tunnel excavation, investigating the influence of seepage flow on the freezing process and the time required to establish a closed supporting frozen arch. Copyright © 2013 John Wiley & Sons, Ltd.  相似文献   

20.
Three synthetic reference glasses were prepared by directly fusing and stirring 3.8 kg of high‐purity oxide powders to provide reference materials for microanalytical work. These glasses have andesitic major compositions and are doped with fifty‐four trace elements in nearly identical abundance (500, 50, 5 µg g?1) using oxide powders or element solutions, and are named ARM‐1, 2 and 3, respectively. We further document that sector‐field (SF) ICP‐MS (Element 2 or Element XR) is capable of sweeping seventy‐seven isotopes (from 7Li to 238U, a total of sixty‐eight elements) in 1 s and, thus, is able to quantify up to sixty‐eight elements by laser sampling. Micro‐ and bulk analyses indicate that the glasses are homogeneous with respect to major and trace elements. This paper provides preliminary data for the ARM glasses using a variety of analytical techniques (EPMA, XRF, ICP‐OES, ICP‐MS, LA‐Q‐ICP‐MS and LA‐SF‐ICP‐MS) performed in ten laboratories. Discrepancies in the data of V, Cr, Ni and Tl exist, mainly caused by analytical limitations. Preliminary reference and information values for fifty‐six elements were calculated with uncertainties [2 relative standard error (RSE)] estimated in the range of 1–20%.  相似文献   

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