INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS Int. J. Numer. Anal. Meth. Geomech., 2003; 27:159–181 (DOI: 10.1002/nag.268)
Fast iterative solution of large undrained soil-structure interaction problems Kok-Kwang Phoon1,n,y, Swee-Huat Chan1, Kim-Chuan Toh2 and Fook-Hou Lee1 1
Department of Civil Engineering, National University of Singapore, Singapore 2 Department of Mathematics, National University of Singapore
SUMMARY 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. KEY WORDS:
undrained condition; Biot’s consolidation; generalized Jacobi (GJ) preconditioner; quasiminimal residual (QMR); element-by-element (EBE) iteration
1. INTRODUCTION The stability of structures supported on fine-grained soils is often most crucial in the short-term. This short-term ‘undrained’ design condition is typically idealized as an incompressible problem because volumetric strains are effectively zero over the period of interest. Undrained analysis may also be relevant for coarse-grained soils subjected to rapid rates of loading (e.g. blast and earthquake). It is well-known that the standard displacement finite element formulation of elastic problems fails when the material becomes incompressible or Poisson’s ratio ðnÞ reaches 0.5. A simple method to side-step this difficulty is to use a Poisson’s ratio close to 0.5 but not equal to it. This strategy is widely adopted in a number of commercial geotechnical finite element softwares (e.g. References [1–3]). n
y
Correspondence to: Kok-Kwang Phoon, Department of Civil Engineering, National University of Singapore, Block E1A #07-03, 1 Engineering Drive 2, Singapore 117576 E-mail:
[email protected]
Copyright # 2003 John Wiley & Sons, Ltd.
Received 8 July 2002 Revised 20 October 2002
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However, finite element solution of nearly incompressible problems is not entirely problem free [4]. The use of a Poisson’s ratio close to 0.5 is equivalent to the introduction of a penalty term to suppress volumetric compressibility arising from an arbitrary displacement field [5]. In principle, solutions of any desired accuracy can be determined using a sufficiently large penalty term. However, the penalized coefficient matrix becomes singular in the limit (zero volumetric strain), which implies that it is ill-conditioned for nearly incompressible problems. In practice, the solution of an ill-conditioned system of equations in finite precision arithmetic will eventually deteriorate as a result of roundoff errors when the condition number is high enough. This numerical difficulty is more severe for iterative solvers than direct solvers. It has also been recognized that discretization errors can be comparable to errors arising from approximating an incompressible problem by a nearly incompressible problem [6]. To ensure convergence to the correct limit solution, it is necessary to reduce the mesh size as the size of the penalty term increases. The main problem in the application of standard displacement formulation to incompressible or nearly incompressible problems lies in the computation of the mean stress or pressure. An alternate approach for incompressible problems is the two-field mixed formulation, which treats displacement and pressure as independent variables [4, 7]. However, this alternative approach is not widely adopted in practice for two main reasons. First, the number of unknowns ðN Þ increases resulting in longer solution time. For large 2-D and 3-D problems involving matrices with significant bandwidth, the number of operations required for direct matrix solvers will approach OðN 3 Þ [8]. Hence, it is computationally costly to improve accuracy by increasing N : Second, the augmented coefficient matrix will contain a zero diagonal block, which poses additional work for direct solvers that need to be suitably modified to handle zero pivots. Zienkiewicz et al. [9] demonstrated that the advantages of mixed formulation can be retained at a cost comparable to that of the irreducible form by using a global iterative procedure. The proposed Uzawa procedure involves iterating two decoupled systems of equations, one of which is complex and requires another iterative procedure for solution. A suitable choice of a convergence accelerator is needed to ensure convergence and to achieve an optimum balance between outer and inner iterations. It is tempting to ask if a single iterative procedure can be used to solve mixed finite element equations more efficiently. An obvious candidate is the preconditioned conjugate gradient (PCG) method, which has been applied to geomechanics with considerable success in recent years [10–14]. However, the coefficient matrix for mixed formulation is indefinite because of the zero (2, 2) block and PCG is not guaranteed to succeed for such matrices. Chan et al. [15] introduced the symmetric quasi-minimal residual (SQMR) method for solving the indefinite system of equations arising from Biot’s problem. The SQMR is also applicable to mixed finite element equations because it can be viewed as a special case of Biot’s formulation with zero flow matrix. However, it is well known that finite element discretization of Biot’s formulation will lead to a coefficient matrix that is both indefinite and ill-conditioned [14, 15]. The standard Jacobi (SJ) preconditioner becomes increasingly ineffective as the flow matrix tends to zero (decreasing soil permeability) and is not defined for the limiting case pertaining to mixed formulation. Even though SQMR is guaranteed to converge when the number of iterations exceeds the number of degrees-of-freedom (DOFs) in exact arithmetic, 3-D problems usually are so large that it is not practical to run even a small fraction of this theoretical maximum. To expedite convergence, preconditioning is crucial. Recently, Phoon et al. [16] demonstrated that a Copyright # 2003 John Wiley & Sons, Ltd.
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generalized Jacobi (GJ) preconditioner, which is an approximation to a theoretical block diagonal preconditioner proposed by Murphy et al. [17], could reduce the number of SQMR iterations significantly and practical 3-D consolidation analysis with modest desktop computers seems within reach. It must be acknowledged that Chan et al. [15] have observed heuristically that SJ could be modified to achieve fruitful results. This is a highly desirable result for largescale computing because the diagonal form is very easy to construct, trivial to invert, and incurs very little housekeeping overhead in ‘element-by-element’ or EBE strategies. In light of these recent 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. The main focus is on large-scale problems where it is not possible to perform direct solutions entirely within available random access memory.
2. THEORETICAL FORMULATIONS Finite element discretization of an elastic soil continuum leads to the well-known symmetric positive definite linear system of equations: Ku ¼ f
ð1Þ
where K is the effective stress stiffness matrix, f the load vector and u the unknown displacement vector. This standard linear form can be viewed as the stationary solution of a quadratic functional P: P ¼ 12 uT Ku uT f
ð2Þ
For undrained problems, it is necessary to impose the condition of zero volumetric strain on u over the whole domain. This constraint can be introduced into the quadratic functional via the standard penalty function approach or the less common Lagrange multiplier approach as briefly described below. 2.1. Penalty function The volumetric strain ðev Þ in each element can be expressed using standard finite element notations as: ev ¼ mT Bue
ð3Þ
where B is the strain shape function, ue the element displacement vector and mT the matrix equivalent of Kronecker delta}mT ¼ ½1 for 1-D analyses, [1, 1, 0] for 2-D analyses and [1, 1, 1, 0, 0, 0] for 3-D analyses. The solution obtained by stationarity of the quadratic functional will satisfy the zero volumetric strain condition approximately if a product term is appended as follows (e.g. Reference [4]) X Z % ¼ 1 uT Ku uT f þ a P eTv ev dV ð4Þ 2 V e Copyright # 2003 John Wiley & Sons, Ltd.
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where a is a ‘penalty number’. Note that Pintegration is performed over the volume domain (denoted as V ) of each finite element and e denotes summation over all elements. Clearly, the product reaches a minimum value of zero when the constraint is satisfied and for a sufficiently % will approximately produce the desired undrained solution. large positive a, stationarity of P The procedure of introducing a constraint as a penalty term in the functional is general. When applied to undrained problems, it is quite simple to demonstrate that the procedure is also physically meaningful by expanding the effective stress stiffness matrix K in the usual way and substituting Equation (3) into (4): Z X Z 1 T X T T T e T T e % P¼ u B DB dV u u f þ a ðm Bu Þ ðm Bu Þ dV 2 V V e e Z 1 T X T T ¼ u B ðD þ 2amm ÞB dV u uT f ð5Þ 2 V e where D is the element effective stress–strain matrix. The stationary solution for Equation (5) is X Z T T B ðD þ 2amm ÞB dV u ¼ f ð6Þ e
V
However, the total stress–strain matrix is given by [18] 2 4 2 2 0 0 0 6K þ 3G K 3G K 3G 6 6 4 2 6 0 2 6 K G K0 þ G K0 G 6 3 3 3 6 6 2 2 4 0 0 0 DT ¼ 6 6K 3G K 3G K þ 3G 6 6 6 0 0 0 6 6 6 0 0 0 4 0 0 0 3 2 Kw Kw Kw 0 0 07 6 n n n 7 6 7 6K 7 6 w Kw Kw 0 0 07 6 7 6 n n n 7 6 7 6 Kw Kw Kw 7 þ6 0 0 0 7 6 n n n 7 6 7 6 6 0 0 0 0 0 07 7 6 7 6 7 6 0 0 0 0 0 0 5 4 0 0 0 0 0 0
0
0
0
0
0
0
G
0
0
G
0
0
3 07 7 7 7 07 7 7 7 07 7 7 7 07 7 7 07 5 G
Kw mmT ð7Þ n where K 0 is the effective bulk modulus of the soil skeleton, G the shear modules, Kw the bulk modulus of water and n the porosity of soil. Comparing Equations (6) and (7), it is clear that the ¼D þ
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penalty number a is proportional to the bulk modulus of water Kw and the solution is nearly incompressible when Kw is sufficiently large. Equation (6) is the standard penalty approach implemented in many geotechnical finite element softwares [1, 2, 18, 19], although it is usually derived using soil mechanics in the form of Equation (7) rather than using a more general constrained variational principle. Note that DT becomes singular in the limit as Kw =n tends to infinity because mmT is singular. 2.2. Lagrange multiplier The constrained functional of the Lagrange multiplier method has the form (e.g. Reference [4]): X Z % ¼ 1 uT Ku uT f þ P eTv p dV ð8Þ 2 V e where p is a Lagrange multiplier field that can be approximated in each element using shape functions N% and nodal values pe : ð9Þ p ¼ N% pe Equation (8) can be expanded thus: % ¼ 1 uT Ku uT f þ P 2 1 ¼ uT Ku þ uT 2
X Z
ðm Bu Þ ðN% pe Þ dV T
e T
V
e
X Z
BT mN% dV p uT f
ð10Þ
V
e
% ¼ 0 for all where p is the global vector of nodal Lagrange multipliers. It is easy to show that dP variations du and dp when: X Z BT mN% dV p ¼ f ð11Þ Ku þ V
e
and "
X Z e
B mN% dV T
#T
u ¼ LT u ¼ 0
ð12Þ
V
Equations (11) and (12) can be combined in a 2 2 matrix block form as " #( ) ( ) K L u f ¼ LT 0 p 0
ð13Þ
The procedure of introducing a constraint by using a Lagrange multiplier field in the functional also is general. When applied to undrained problems, it is quite easy to see that Equation (13) is a special case of the Biot’s formulation with zero flow matrix and p can be interpreted in a physically meaningful way as the vector of nodal excess pore water pressures. The 2 2 matrix block form shown in Equation (13) can also be derived using the two-field mixed formulation that treats displacement and pressure as independent variables [4, 7]. In principle, the Lagrange multiplier or mixed formulation approach can enforce zero volumetric strain over the whole domain ‘exactly’. This undrained state can only be achieved as the limit of a sequence of nearly incompressible states with increasing Kw if the penalty function approach is Copyright # 2003 John Wiley & Sons, Ltd.
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used. In practice, undrained analysis is carried out using one penalty number that is selected on an ad hoc basis because practitioners could ill-afford to perform limit analysis using a sequence of increasing penalty numbers. It is also not possible to increase the penalty number arbitrarily in finite precision arithmetic because of ill-conditioning problems. However, the Lagrange multiplier approach will produce a zero (2, 2) block and increase the number of unknowns ðN Þ as illustrated in Equation (13) for the undrained problem. For direct equation solvers, the former poses problems because of zero pivots while the latter is quite unappealing because matrix inversion is a N 3 process. For iterative solvers, the latter is not as onerous but the former results in an indefinite matrix [20] that cannot be handled by PCG (cheapest Krylov subspace method) and SJ preconditioning (cheapest preconditioner). Detailed comparisons using numerical experiments are presented next.
3. NUMERICAL SCHEMES The penalty function and the Lagrange multiplier methods result in a symmetric positive definite 1 1 block system of the form shown in Equation (6) and a symmetric indefinite 2 2 block system of the form shown in Equation (13), respectively. Direct and iterative solution methods will be used for solving both block systems. For the direct approach, the well-known frontal method [18, 21] is used. To avoid zero pivots in the 2 2 block system, Equation (13) is approximated using Biot’s finite element system of equations with a very small flow matrix. This can be achieved by setting the time step ðDtÞ and/or hydraulic permeability ðkÞ to a very small value, e.g. kDt ¼ 1 1014 m: Note that this approach is very similar to the perturbed Lagrangian form (e.g. Reference [4]) " #( ) ( ) K L u f ¼ ð14Þ T L eI p 0 Here e is a small positive number that can be reduced to make Equations (13) and (14) nearly equivalent and I denotes the identity matrix. The presence of a zero (2, 2) block does not introduce numerical difficulties when K is non-singular and L has full column rank. Thus, the perturbed Lagrangian form is not necessary but useful to adopt in this paper in order to apply an identical direct solver to both block systems for fair comparison of runtime. It is also interesting to note that the perturbed Lagrangian form can be re-written in a 1 1 block penalty form: 1 T K þ LL u ¼ f ð15Þ e The similarity between Equation (15) and the standard penalty approach (Equation (6)) is striking but a closer inspection will show that they are not identical. To our knowledge, this alternate penalty approach has not been applied in the literature to solve undrained problems. For the iterative approach, three methods are studied, namely: (a) preconditioned conjugate gradient (PCG), (b) symmetric quasi-minimal residual (SQMR), and (c) Uzawa method. The former is used for solving the 1 1 block system (Equation (6)), while the latter two are used for the 2 2 block system (Equation (13)). The following subsections briefly describe these iterative solvers. Copyright # 2003 John Wiley & Sons, Ltd.
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3.1. PCG and SJ preconditioner The PCG method [22, 23] is the best known and probably most efficient, Krylov subspace method (e.g. References [24–27]) for solving symmetric positive definite linear systems. Coupled with element-by-element (EBE) iterative strategies that obviate the assembly of the global coefficient matrix [28, 29], the PCG method should allow large 3-D problems to be solved on relatively modest computing platforms, e.g. PCs, owing to its lower memory demand (e.g. References [14, 19, 30]). The standard Jacobi (SJ) preconditioner, a diagonal matrix whose diagonal entries are identical to those of the global coefficient matrix, is used in the PCG solver to accelerate the convergence rate. It should be noted that the SJ preconditioner is usually not an efficient preconditioner in terms of iteration count, in comparison with other more complicated EBE preconditioners, e.g. References [30–35]. However, SJ is very competitive when the overall runtime is considered because of its simplicity and economy in construction, storage and implementation (e.g. References [10, 36]. This is particularly true for large-scale simulation of construction processes, which often involve complex evolving geometry and boundaries, multiple material zones, and irregular soil stratification. Hence, the SJ preconditioner is still commonly used in many applications (e.g. References [13, 14, 19]). 3.2. Symmetric QMR and GJ preconditioner The SQMR method was developed by Freund and Nachtigal [37] for solving symmetric indefinite linear systems to overcome limitations in existing methods such as the minimal residual (MINRES) and symmetric LQ (SYMMLQ) methods proposed by Paige and Saunders [38]. The SQMR is chosen in this study because it can be combined with symmetric indefinite preconditioners while the latter can only admit positive definite preconditioners. Moreover, it usually converges considerably faster than the latter when appropriate indefinite preconditioners are incorporated [37]. To reduce storage requirement, EBE iterative strategies are implemented as well. The generalized Jacobi (GJ) preconditioner developed by Phoon et al. [16] for Biot’s finite element system of equations is used in SQMR to accelerate the convergence rate. The preconditioning overheads for GJ and SJ are comparable, but the former results in significantly smaller number of iterations than the latter for Biot’s indefinite system of linear equations [16]. The GJ preconditioner is given by " # diagðKÞ 0 PGJ ¼ ð16Þ 0 4diagfLT ½diagðKÞ1 L þ Cg where diagðÞ denotes the diagonal of the matrix and C is the flow matrix. Note that Equation (16) is identical to SJ insofar as the displacement DOFs are concerned, i.e. for i ¼ 1 to m: pii ¼ kii For the excess pore water pressure DOFs, i ¼ m þ 1 to n: " ! # m l2 X ji pii ¼ 4 þ cii k j¼1 jj
ð17aÞ
ð17bÞ
where lij ; kjj and cii are the entries in L; K and C referenced by the global DOF indexing system, respectively. To store PGJ efficiently using only one n-dimensional vector in an EBE Copyright # 2003 John Wiley & Sons, Ltd.
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implementation, Equation (17b) can be further approximated as " # " # P e 2! P e 2! m ð m X X e lji Þ e ðlji Þ pii ¼ 4 þ cii 4 þ cii kjj kjj j¼1 j¼1
ð18Þ
where Iije are the entries in the L-block of the eth finite element referenced globally. Note that C ¼ 0 for the undrained problem. Only the EBE version of GJ is implemented in this paper. 3.3. Uzawa method The Uzawa method [39] is a popular iterative method for solving the 2 2 block system by recursively solving two decoupled systems [4, 9, 25]. The standard implementation of the Uzawa’s method for solving a system of the form shown by Equation (13) can be described by the following simple pseudocode: Choose a positive parameter o; and initial vectors u0 and p0 : DO for i ¼ 0; 1; 2; . . . ; Solve Kuiþ1 ¼ f Lpi Compute piþ1 ¼ pi þ oLT uiþ1 END DO Let S ¼ LT K 1 L be the Schur complement matrix of Equation (13) and lmax ðSÞ; lmin ðSÞ be the maximum and minimum eigenvalues of S; respectively. To ensure convergence, the parameter o must satisfy 05o51=lmax ðSÞ [25]. The optimal convergence rate is given by: kðSÞ 1 kðSÞ þ 1
ð19Þ
2 lmin ðSÞ þ lmax ðSÞ
ð20Þ
ropt ¼ and this is achieved when: oopt ¼
where kðSÞ ¼ lmax ðSÞ=lmin ðSÞ is the condition number of S: Notice that in each step of the Uzawa method, a symmetric positive definite linear system involving the coefficient matrix K has to be solved. If an iterative method, e.g. PCG (as in Reference [40]), is used to solve this linear system, then the computational time of the Uzawa method is determined by the total number of Uzawa steps taken and the number of PCG iterations needed in each step. In such circumstances, the conditioning of K will greatly affect the time needed to solve the symmetric positive definite linear system in each Uzawa step. If K is relatively well-conditioned but S is highly ill-conditioned, each Uzawa step will require a relatively small number of PCG iterations but a large number of Uzawa steps. The reverse can happen as well. It is possible to modify the 2 2 block system to an equivalent system so that when the Uzawa method is applied to the equivalent system, a balance between the computational time needed in each step and the total number of Uzawa steps needed for convergence is achieved. Copyright # 2003 John Wiley & Sons, Ltd.
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This was observed in Reference [9], and the equivalent system is as follows: " #( ) ( ) f u K þ gLLT L ¼ T 0 p L 0
167
ð21Þ
where g is an arbitrary positive number. It is easy to see algebraically that Equation (21) gives the same solution as Equation (13) since LT u ¼ 0: Now, let Kg ¼ K þ gLLT and Sg ¼ LT Kg1 L: Then the optimal convergence rate for the Uzawa method when applied to Equation (21) is given by kðSg Þ 1 kðSg Þ þ 1
ð22Þ
2 lmin ðSg Þ þ lmax ðSg Þ
ð23Þ
ropt ðgÞ ¼ and this is achieved when: oopt ðgÞ ¼
The eigenvalues of Sg are generally too expensive to be estimated and thus the optimal parameter is generally not known. However, in the case when g is large, an asymptotic estimates of the largest and smallest eigenvalues of Sg can be obtained, and thus a nearly optimal value for o is known. It turns out that o ¼ g is a nearly optimal choice and the corresponding convergence rate for the Uzawa method is given by rnear
opt ðgÞ
¼
1 ½lmin ðT Þ þ lmax ðT Þ 2g
ð24Þ
where T is a matrix that depends only on K and L; but not g: Thus, if g is sufficiently large, the Uzawa method can converge in one step. But there is a hidden cost in achieving such a fast convergence in that the matrix Kg can be extremely ill-conditioned when g is large and the time spent in the PCG iteration can therefore be tremendous. Moreover, accuracy of the solution can also be seriously degraded as a result of amplification of roundoff errors. In general, the condition number of Kg increases with g because LLT is singular. Thus the time spent in the PCG iterations for each Uzawa step is expected to increase with g: On the other hand, Equation (23) shows that the number of Uzawa steps generally decrease with g for reasonably large values of g: Hence, g should be chosen so that the time spent for PCG iterations in each Uzawa step and the total number of Uzawa steps needed to achieve convergence are moderate. Unfortunately, this moderate choice of g can only be gauged from numerical experiments and it is problem dependent.
4. NUMERICAL RESULTS 4.1. Problem configuration Figure 1 shows the finite element meshes of a flexible square footing resting on homogeneous soil subjected to a uniform vertical pressure of 0:1 MPa: Symmetry considerations allow a quadrant of the footing to be analysed. The fine mesh (Figure 1(a)) comprises 1728 20-noded brick elements, with a total of 24843 displacement DOFs and 2197 excess pore water pressure DOFs. The coarse mesh (Figure 1(b)) comprises 64 20-noded brick elements, with a total of Copyright # 2003 John Wiley & Sons, Ltd.
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Figure 1. Finite element meshes for footing problem (quadrant symmetry): (a) fine mesh and (b) coarse mesh.
1275 displacement DOFs and 125 excess pore water pressure DOFs. Note that the excess pore water pressure DOFs only exist in the 2 2 block system. The ground water table is assumed to be at the ground surface and is in hydrostatic condition at the initial stage. The material used in the analysis is assumed to behave in a linear elastic manner, with an effective Young’s modulus ðE0 Þ of 10 MPa and effective Poisson’s ratio ðn0 Þ of 0.3. Footing load is applied in one increment. For all iterative methods, a zero vector is adopted Copyright # 2003 John Wiley & Sons, Ltd.
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as the initial guess for the solution. Convergence is achieved when the relative residual norm is less than a prescribed tolerance ðdÞ [14, 15, 37]: jjb Axi jj2 5d ð25Þ jjb Ax0 jj2 where A denotes the coefficient matrix, b the right-hand side vector, xi the solution at the current iteration, x0 the initial guess for the solution and jj:jj2 the vector 2-norm function. 4.2. Verification of accuracy Figures 2 and 3 show the undrained results from the 1 1 block system (Equation (6)) for Kw =n ¼ 50K 0 and 500K 0 ; respectively. Note that Kw =n is the penalty number as shown in Equation (7). The use of Kw =n ¼ 50K 0 and 500K 0 are equivalent to the use of Poisson’s ratios n ¼ 0:49 and 0.499, respectively [1, 41]. Results for surface settlement, total horizontal stress, total vertical stress, total pore water pressure and volumetric strain are compared with analytical solutions. For surface settlement, the analytical solution for a loaded area on an elastic halfspace of finite depth overlying a rigid base can be found in NAVFAC [42]. Analytical solutions for the total horizontal and vertical stresses are obtained using the tables given by Poulos and Davis [43] (citing Milovic and Tournier [44]). The total pore water pressure is given as the summation of initial hydrostatic water pressure and excess pore water pressure. Analytical solution for excess pore water pressure is obtained using the following formula for an isotropic and elastic soil mass saturated with an incompressible pore fluid (e.g. Reference [45]) Dp ¼ Ds3 þ 13 ðDs1 Ds3 Þ
ð26Þ
where Dp denotes the pore water pressure increment, Ds3 the horizontal stress increment and Ds1 the vertical stress increment. It can be observed from Figures 2 and 3 that the settlement profile fits the analytical solution well for both penalty values. However, significant deviations are observed in the total horizontal stress, total vertical stress and total pore water pressure profiles, when a larger penalty number is adopted. The stress and pressure fields are thus more sensitive to the value of penalty number than the displacement field. As expected, the use of a large penalty number results in smaller volumetric strains. The inaccuracy in the stress and pressure fields are exacerbated when a coarse mesh is used as shown in Figures 4 and 5. The difficulty of computing the mean stress or pressure in a penalty function approach is quite well-known and has been reported by Zienkiewicz and Taylor [4] and Reddy [46]. Although the undrained state can be theoretically determined as the limit of a sequence of nearly incompressible states with increasing penalty numbers, this is practically unachievable because oscillations in the pressure will be amplified. Hence, a moderate penalty number is necessary for a given mesh density but there is no systematic way of determining the optimal value for such an ad hoc number. Figures 6 and 7 show the undrained results from the 2 2 block system (Equation (13)) for the fine and coarse mesh problem, respectively. The surface settlement, total horizontal stress, total vertical stress, and total pore water pressure are in good agreement with the respective analytical solutions, even when the mesh is coarse. The volumetric strains are also relatively small. It appears to be possible to solve undrained problems ‘exactly’ and accurately without numerical difficulties using the Lagrangian 2 2 block system. In contrast, the 1 1 block system resulting from the penalty function approach only solves the incompressible approximately, is susceptible to oscillations in the mean stress, and requires selection of an ad Copyright # 2003 John Wiley & Sons, Ltd.
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Settlement (m)
0. 005 0.000 -0.005 -0.010
(a) -0.015 0
1
2
3
4
5
6
7
8
9
10
X-direction (m)
Depth (m)
0
5
(b) 10 -0.05
0.00
0.05
0.10
0.15
0.20
0.25
Total horizontal stress (MPa)
Depth (m)
0
5
(c) 10 -0.05
0.00
0.05
0.10
0.15
0.20
0.25
Total vertical stress (MPa)
Depth (m)
0
5
(d)
10 -0.05
0.00
0.05
0.10
0.15
0.20
0.25
Total pore water pressure (MPa)
Depth (m)
0
5 Kw /n = 50K'
Analytical solution
(e)
Initial value 10 -5.0E-4
-4.0E-4
-3.0E-4
-2.0E-4
-1.0E-4
0.0E+0
1.0E-4
2.0E-4
3.0E-4
4.0E-4
5.0E-4
Volumetric strain
Figure 2. Undrained results from 1 1 block system (Equation (6)) with Kw =n ¼ 50K 0 for fine mesh: (a) settlement; (b) total vertical stress; (c) total horizontal stress; (d) total pore water pressure; and (e) volumetric strain. Copyright # 2003 John Wiley & Sons, Ltd.
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Settlement (m)
0.005 0.000 -0.005 -0.010
(a) -0.015 0
1
2
3
4
5
6
7
8
9
10
X-direction (m)
Depth (m)
0
5
(b)
10 -0.05
0.00
0.05
0.10
0.15
0.20
0.25
Total horizontal stress (MPa)
Depth (m)
0
5
(c) 10 -0.05
0.00
0.05
0.10
0.15
0.20
0.25
Total vertical stress (MPa)
Depth (m)
0
5
(d)
10 -0.05
0.00
0.05
0.10
0.15
0.20
0.25
Total pore water pressure(MPa)
Depth (m)
0
5
10 -5.0E-4
Kw/n = 500K'
Analytical solution
(e)
In itial v alue -4.0E-4
-3.0E-4
-2.0E-4
-1.0E-4
0.0E+0
1.0E-4
2.0E-4
3.0E-4
4.0E-4
5.0E-4
Volumetric strain
Figure 3. Undrained results from 1 1 block system (Equation (6)) with Kw =n ¼ 500K 0 for fine mesh: (a) settlement; (b) total vertical stress; (c) total horizontal stress; (d) total pore water pressure; and (e) volumetric strain. Copyright # 2003 John Wiley & Sons, Ltd.
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Settlement (m)
0.005 0.000 -0.005 -0.010
(a) -0.015 0
1
2
3
4
5
6
7
8
9
10
X-direction (m)
Depth (m)
0
5
(b) 10 -0.05
0.00
0.05
0.10
0.15
0.20
0.25
Total horizontal stress (MPa)
Depth (m)
0
5
(c) 10 -0.05
0.00
0.05
0.10
0.15
0.20
0.25
Total vertical stress (MPa)
Depth (m)
0
5
(d) 10 -0.05
0.00
0.05
0.10
0.15
0.20
0.25
Total pore pressure (MPa)
Depth (m)
0
5
Kw /n = 50K'
Analytical solution 10 -5.0E-4
(e)
Initial value -4.0E-4
-3.0E-4
-2.0E-4
-1.0E-4
0.0E+0
1.0E-4
2.0E-4
3.0E-4
4.0E-4
5.0E-4
Volumetric strain
Figure 4. Undrained results from 1 1 block system (Equation (6)) with Kw =n ¼ 50K 0 for coarse mesh: (a) settlement; (b) total vertical stress; (c) total horizontal stress; (d) total pore water pressure; and (e) volumetric strain. Copyright # 2003 John Wiley & Sons, Ltd.
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FAST ITERATIVE SOLUTION
Settlement (m)
0.005 0.000 -0.005 -0.010
(a) -0.015 0
1
2
3
4
5
6
7
8
9
10
X-direction (m)
Depth (m)
0
5
(b) 10 -0.05
0.00
0.05
0.10
0.15
0.20
0.25
Total horizontalstress (MPa)
Depth (m)
0
5
(c) 10 -0.05
0.00
0.05
0.10
0.15
0.20
0.25
Total vertical stress (MPa)
Depth (m)
0
5
(d) 10 -0.05
0.00
0.05
0.10
0.15
0.20
0.25
Total pore pressure (MPa)
Depth (m)
0
5
Kw /n = 500K'
Analytical solution 10 -5.0E-4
(e)
Initial value -4.0E-4
-3.0E-4
-2.0E-4
-1.0E-4
0.0E+0
1.0E-4
2.0E-4
3.0E-4
4.0E-4
5.0E-4
Volumetric strain
Figure 5. Undrained results from 1 1 block system (Equation (6)) with Kw =n ¼ 500K 0 for coarse mesh: (a) settlement; (b) total vertical stress; (c) total horizontal stress; (d) total pore water pressure; and (e) volumetric strain. Copyright # 2003 John Wiley & Sons, Ltd.
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Settlement (m)
0.005 0.000 -0.005 -0.010
(a) -0.015 0
1
2
3
4
5
6
7
8
9
10
X-direction (m)
Depth (m)
0
5
(b) 10 -0.05
0.00
0.05
0.10
0.15
0.20
0.25
Total horizontal stress (MPa)
Depth (m)
0
5
(c) 10 -0.05
0.00
0.05
0.10
0.15
0.20
0.25
Total vertical stress (MPa)
Depth (m)
0
5
(d) 10 -0.05
0.00
0.05
0.10
0.15
0.20
0.25
Total pore water pressure (MPa)
Depth (m)
0
5
10 -5.0E-4
Biot's formulation (∆t = 0)
Analytical solution
(e)
Initial value -4.0E-4
-3.0E-4
-2.0E-4
-1.0E-4
0.0E+0
1.0E-4
2.0E-4
3.0E-4
4.0E-4
5.0E-4
Volumetric strain
Figure 6. Undrained results from 2 2 block system (Equation (13)) for fine mesh: (a) settlement; (b) total vertical stress; (c) total horizontal stress; (d) total pore water pressure; and (e) volumetric strain.
Copyright # 2003 John Wiley & Sons, Ltd.
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FAST ITERATIVE SOLUTION
Settlement (m)
0.005 0.000 -0.005 -0.010
(a) -0.015 0
1
2
3
4
5
6
7
8
9
10
X-direction (m)
Depth (m)
0
5
(b) 10 -0.05
0.00
0.05
0.10
0.15
0.20
0.25
Total horizontal stress (MPa)
Depth (m)
0
5
(c) 10 -0.05
0.00
0.05
0.10
0.15
0.20
0.25
Total vertical stress (MPa)
Depth (m)
0
5
(d) 10 -0.05
0.00
0.05
0.10
0.15
0.20
0.25
Total pore pressure (MPa)
Depth (m)
0
5
Biot's formulation (∆ t = 0)
Analytical solution 10 -5.0E-4
(e)
Initial value -4.0E-4
-3.0E-4
-2.0E-4
-1.0E-4
0.0E+0
1.0E-4
2.0E-4
3.0E-4
4.0E-4
5.0E-4
Volumetric strain
Figure 7. Undrained results from 2 2 block system (Equation (13)) for coarse mesh: (a) settlement; (b) total vertical stress; (c) total horizontal stress; (d) total pore water pressure; and (e) volumetric strain.
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Table I. Comparison of computer runtime between 1 1 and 2 2 block systems using Pentium II, 450 MHz PC for fine mesh. Direct approach
Iterative approach (tolerance d ¼ 1 106 )
1 1 block system (24843 DOFs) Frontal Runtime (s) 3786 Kw =n ¼ 50K 0 Kw =n ¼ 500K 0 3786
PCG-SJ Kw =n ¼ 50K 0 Kw =n ¼ 500K 0
Runtime (s) 851 2719
Iteration 1779 5723
2 2 block system (27040 DOFs) Frontal Runtime (s) 4888 kDt ¼ 1014 m
SQMR-GJ Dt ¼ 0
Runtime (s) 452
Iteration 700
hoc number. However, it is not necessarily worthwhile to achieve this accuracy in practice if the computation costs are significantly higher. This important issue is addressed next. 4.3. Efficiency Table I compares the computer runtimes between the 1 1 and 2 2 block systems for both the direct and iterative solution methods. For the 1 1 block system consisting of 24843 equations, the frontal method requires 336 MB of memory for double precision solutions, whereas the PCG method with the SJ preconditioner only requires 39 MB: For the 2 2 block system consisting of 27040 equations, the memory requirement is 414 MB for the frontal method but is only 47 MB for the SQMR method with the GJ preconditioner. Note that the mesh for 1 1 and 2 2 block systems is the same, but the latter requires solution of a large system of equations as noted above. In this study, the frontal solution is obtained out-of-core, where a main memory (RAM) of 256 MB and a secondary memory (disk) of 80 MB is used for the 1 1 block system, and a main memory of 256 MB and a secondary memory of 158 MB is used for the 2 2 block system. The tolerance ðdÞ used for convergence checks in iterative solution methods is 1 106 : For the direct approach, the increase in computer runtime from the 1 1 to 2 2 block system is primarily due to the increase in DOFs and frontwidth (or number of equations). As to be expected, the value of the penalty number used only affects accuracy rather than efficiency. Comparison of direct solution runtimes for the 1 1 and 2 2 block systems shows that the runtime is indeed approximately proportional to N 3 ; as expected. It is probably not worthwhile to overcome the limitations of the existing standard approach at the expense of consuming more memory and time to solve a larger system of equations. However, this is not true for iterative methods because their convergence rates depend more crucially on the spectral properties of the coefficient matrix than the number of DOFs. This is clearly shown in Table I. In fact, the runtime required to solve a larger 2 2 block system is actually less than that required to solve the smaller 1 1 block system. The reason is that the GJ preconditioner is optimized for the 2 2 block system and is extremely efficient [16]. The runtime required to solve the 2 2 block system using the SQMR method with GJ ð452 sÞ is roughly 50% faster than the PCG with SJ 1 1 block system solver with Kw =n ¼ 50K 0 ð851 sÞ and 80% faster than the one with Kw =n ¼ 500K 0 ð2719 sÞ: In contrast to direct solution, the runtime required for the iterative solution of the 1 1 block system is a function of the penalty term Kw =n because the number of iterations is sensitive to the conditioning of the coefficient Copyright # 2003 John Wiley & Sons, Ltd.
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matrix. It is known that the penalized matrix with Kw =n ¼ 500K 0 is much more ill-conditioned than the one with Kw =n ¼ 50K 0 [14]. Hence, more iterations and more time are needed to solve the former. Although the runtimes between the direct and iterative approaches are not directly comparable because the former is partially solved out-of-core, it is nonetheless illuminating that iterative approaches are significantly more efficient for large systems. The situation portrayed in Table I is realistic because 3-D soil–structure interaction problems are commonly very large but generally cannot justify expenditure of high-end computing resources.
4.4. Comparison with Uzawa method The coarse mesh is used in this section because the solution time for the Uzawa method with extreme values of the convergence accelerator ðgÞ would otherwise be excessive. The undrained results from the 2 2 block system for the coarse mesh problem were already shown to be good in Figure 7. Hence, it is valid to compare the performance between the SQMR-GJ and Uzawa approach using a coarse mesh. Figure 8 shows the computer runtime and relative residual norm for the Uzawa and SQMRGJ approach over a wide range of g values. Note that SQMR-GJ solves Equation (13) directly, which can be considered as a special case of Equation (21) with g ¼ 0: Each Uzawa step requires the solution of two equations}the most expensive being ðK þ gLLT Þuiþ1 ¼ f Lpi : Because the coefficient matrix is large, an inner loop consisting of PCG iterations is used for solution in each Uzawa step. This inner loop is terminated when the relative residual norm is less than d ¼ 1 1014 : The same value of d is used for the SQMR method. The outer Uzawa loop is stopped when the relative residual norm is no longer decreasing. As noted above, g should be chosen so that the time spent for PCG iterations in each Uzawa step and the total number of Uzawa steps needed to achieve convergence are moderate. In this example, the minimum computer runtime for the Uzawa method is about 1700 s; corresponding to an optimum g value of 104 and a relative residual norm of about 1010 : As shown in Table II, it is possible to reduce the number of Uzawa steps by increasing g only at the expense of increasing the number of PCG iterations in the inner loop because the matrix Kg ¼ ðK þ gLLT Þ becomes ill-conditioned. The converse is true, which explains the U-shape curve for the runtime (Figure 8(a)). However, the minimum relative residual norm achievable seems to increase with g: This observation is consistent with the result in Greenbaum [47] where it is shown that the ultimate attainable accuracy in the residual norm is proportional to the norm of the coefficient matrix. It may be noted in passing that the alternate penalty function approach given in Equation (15) is identical to the one-step Uzawa approach, with g ¼ 1=e: The undrained solutions are of comparable accuracy with those shown in Figure 7 (results not shown herein) when g is very large (say 108 or larger) because the actual Uzawa approach only requires one additional step in such cases as shown in Table II. However, the penalty term of 108 is very large compared to that used in the standard penalty approach (Equation (6)). Hence, the number of iterations is much larger (> 13; 000 for g ¼ 108 compared to 5723 for Kw =n ¼ 500K 0 ). This alternate penalty function approach is therefore not practical. For the GJ approach, it only takes 8 s to achieve a relative residual norm of 1 1014 : Hence, it is more than two orders of magnitude faster and four orders of magnitude more accurate than the Uzawa solution based on an optimum g ¼ 104 : In addition, it does away with the need of an arbitrary g parameter, which cannot be determined without costly numerical experiments. Copyright # 2003 John Wiley & Sons, Ltd.
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4500 4000
Computer runtime (s)
3500 3000 2500 2000 1500 1000 500
(a)
0 0
1E+2
1E+3
1E+4
1E+5
1E+6
1E+7
1E+8
1E+9 1E+10 1E+11 1E+12
1E-1 1E-2
Uzawa
1E-3
SQMR-GJ
1E-4
Relative residual norm
1E-5 1E-6 1E-7 1E-8 1E-9 1E-10 1E-11 1E-12 1E-13 1E-14
(b)
1E-15 0
1E+2
1E+3
1E+4
1E+5
1E+6
1E+7
1E+8
1E+9 1E+10 1E+11 1E+12
Gamma, % Note: Runtime is monitored using a Pentium II, 450 MHz PC.
Figure 8. Comparison between Uzawa method and SQMR with GJ preconditioner for coarse mesh: (a) computer runtime and (b) relative residual norm.
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FAST ITERATIVE SOLUTION
Table II. Variation in iteration count with convergence accelerator for Uzawa method (coarse mesh). g
Uzawa method
0 102 103 104 105 106 107 108 109 1012 y
Outer Uzawa steps
Inner PCGy iterations
} 81 15 7 4 3 3 2 2 2
} 830–843 2303–2337 4911–4973 7912–7950 10796–10904 13914–14115 17246–17313 20627–21019 33252–33555
SQMR-GJ method
315 } } } } } } } } }
A range is reported because PCG iterations vary from step to step.
5. CONCLUSION For undrained problems, it is necessary to impose the condition of zero volumetric strain on over the whole domain. The standard penalty function approach using a large bulk modulus of water can be derived from a more general constrained variational principle consisting of the standard quadratic functional with a penalty applied on the square of the volumetric strains. The alternate approach of using Biot’s equations with a zero flow matrix can be derived using the two-field mixed formulation which arises from the use of Lagrangian multipliers in the functional. The former results in a symmetric positive definite 1 1 block system while the latter results in a symmetric indefinite 2 2 block system. Both system of equations are solved using the direct frontal method and three iterative methods, namely the: (a) preconditioned conjugate gradient (PCG) with standard Jacobi (SJ), (b) symmetric quasi-minimal residual (SQMR) with generalized Jacobi (GJ), and (c) Uzawa method. Numerical studies of a simple footing problem show that both 1 1 and 2 2 block systems are able to produce accurate solutions for surface settlement. However, the stress and pressure fields exhibit signs of instability for the 1 1 block system when the penalty term is large. In contrast, the 2 2 block system is able to solve the incompressible problem ‘exactly’ in principle, does not create pressure and stress instabilities, and obviate the need for an ad hoc penalty number. The 2 2 block system clearly is preferable from an accuracy point of view. For large-scale problems where it is not possible to perform direct solutions entirely within available random access memory, iterative methods are more efficient for both 1 1 and 2 2 block systems. Surprisingly, it turns out that the smaller 1 1 block system that could utilize the highly efficient PCG method incurs a significantly longer solution time than the larger 2 2 block system that utilizes a more costly SQMR method. The reason is that the 1 1 block system becomes very ill-conditioned when a large penalty term is used and SJ is ineffective in this situation. On the other hand, GJ is very effective in preconditioning the indefinite 2 2 block system, even though it carries a zero (2,2) block. The Uzawa method has been proposed as an efficient iterative solver for the 2 2 block system in previous studies. However, it requires judicious selection of an ad hoc convergence accelerator parameter to maintain an optimal balance between time spent for PCG iterations in each Uzawa step and the total number of Copyright # 2003 John Wiley & Sons, Ltd.
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Uzawa steps needed to achieve convergence. Even after tuning the convergence accelerator to an optimum value using costly parametric studies, the GJ is able to achieve more accurate results in significantly less time. For large undrained problems, it appears that solving the 2 2 block system using SQMR preconditioned by GJ is currently the top contender in terms of speed and accuracy. Although the examples shown are restricted to undrained problems in the context of geomechanics, its general applicability to other examples such as incompressible flow and metal forming problems studied by Zienkiewicz et al. [9] is clear. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.
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