INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICSInt. J. Numer. Anal. Meth. Geomech. (2015)Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nag.2361On finite volume method implementation of poro-elasto-plasticitysoil modelTian Tang1,*,,, Ole Hededal1,and Philip Cardiff2,1Department of Civil Engineering , Technical University of Denmark, 2800 Kongens Lyngby, Denmark2School of Mechanical and Materials Engineering, University College Dublin, Beleld, IrelandSUMMARYAccurate prediction of the interactions between the nonlinear soil skeleton and the pore uid under loadingplays a vital role in many geotechnical applications. It is therefore important to develop a numerical methodthat can effectively capture this nonlinear soil-pore uid coupling effect. This paper presents the implementa-tion of a new nite volume method code of poro-elasto-plasticity soil model. The model is formulated on thebasis of Biots consolidation theory and combined with a perfect plasticity Mohr-Coulomb constitutive rela-tion. The governing equation system is discretized in a segregated manner, namely, those conventional linearand uncoupled terms are treated implicitly, while those nonlinear and coupled terms are treated explicitly byusing any available values from previous time or iteration step. The implicitexplicit discretization leads toa linearized and decoupled algebraic system, which is solved using the xed-point iteration method. Uponthe convergence of the iterative method, fully nonlinear coupled solutions are obtained. Also explored in thispaper is the special way of treating traction boundary in nite volume method compared with FEM. Finally,three numerical test cases are simulated to verify the implementation procedure. It is shown in the simula-tion results that the implemented solver is capable of and efcient at predicting reasonable soil responseswith pore pressure coupling under different loading situations. Copyright 2015 John Wiley & Sons, Ltd.Received 8 November 2013; Revised 6 January 2015; Accepted 8 January 2015KEY WORDS: nite volume method; poro-elasto-plasticity; coupled analysis; soil model; segregatedsolution procedure; OpenFOAM1. INTRODUCTIONIt is commonlyrecognizedthat themechanical behavior of soils (andindeedother saturatedgeomaterials) is governed largely by the interaction of their solid skeleton with the uid present inthe pore structures [1, 2]. The fundamental mathematical framework describing the coupled effectsin porous soil was rst established by Biot in 1941, who by assuming a linear elastic behavior ofthe soil skeleton and a Darcian uid ow formulated a coupled model with the soil skeleton dis-placements and the pore uid pressure as the primary unknowns [3]. Extensive research works in thepast few decades have been devoted to analytical or numerical solutions of this linear poro-elasticitytheorybecauseof itssimplicityandbroadapplicabilityindifferent engineeringproblems[4].However, increasing focus on more complex applications has triggered research aimed at extendingtheoriginalBiottheorytoaccountfornonlinearmaterialbehaviors, thatis, thedevelopmentofporo-elasto-plasticity models [5]. Often, due to the complexity of those models (from the couplingandhighlynonlinear constitutiverelation), analytical solutionsareverydifcult toobtain, and*Correspondence to: T. Tang, Department of Civil Engineering, Technical University of Denmark, 2800 Kongens Lyngby,Denmark.E-mail: tiat@byg.dtu.dk

Phd Student

Associate ProfessorPost Doctoral Research FellowCopyright 2015 John Wiley & Sons, Ltd.T. TANG, O. HEDEDAL AND P. CARDIFFthe numerical approaches, therefore, play an active role in nonlinear coupled analyses, such as indynamic earthquake problems [68] and in foundation bearing capacity problems [911].The aforementioned numerical poro-elasto-plasticity models adopted the traditional approach ofcontinuum solid mechanics, that is, nite element method (FEM) technique. An alternative numer-ical scheme is the nite volume method (FVM), which was originally applied to uid mechanicalproblems. Demirdzic and several other researchers [1223] established FVM as noteworthy alterna-tive to the FEM for linear and nonlinear continuum solid mechanics. It is pointed out that the drivingforce behind the development of segregated FVM stress analysis algorithms is the potential for deal-ing with nonlinear problems with only a marginal increase in computational cost [24]. Hence, it isinteresting to explore whether FVM is also capable of and efcient at treating poro-elasto-plasticitymodels containing both material nonlinearity and strong soil-uid coupling effects.This article provides background information and documentation for a new nite volume imple-mentationof aporo-elasto-plasticitysoil model. Theconstitutiverelationemployedisalinearelastic/plastic model based on a classical, nonassociated Mohr-Coulomb formulation. The selectedstress integration scheme is an explicit return mapping algorithmformulated in principal stress spaceby Clausen et al. [25]. Emphasis is given to the introduction of the implicitexplicit split method-ology, that is, the segregated method in FVM, to deal with the nonlinear and coupling effects inthe model. As a consequence of the special implicit and explicit treatment, the complicated non-linear coupled equations can be linearized and solved sequentially using a xed-point type iterativemethod, which usually provides a linear convergence. Additionally, the treatment of traction bound-ariesinthecurrentFVMimplementationdiffersmuchfromthatinFEMandthusisalsogivenspecial attention in this work.Thenumerical procedurehasbeenimplementedasacustomsolverinOpenFOAM(version1.6-ext)(ESI-OpenCFD,Bracknell,UK),whichisafree-to-useopensourcenumericalsoftwarethat has extensive computational uid dynamics and multiphysics capabilities [26]. This implemen-tation work, therefore, also opens a possibility of using the same numerical methodology, namelyFVM, for solving the wave-seabed-structure interaction problems in the future.The structure of this article is as follows: In Section 2, the mathematical (poro-elasto-plasticity)model for nonlinear porous soil is presented, and the key issues in modeling poro-elasto-plasticityas well as the corresponding solution strategies are proposed. In Section 3, the FVM discretizationtechniques including the discretization of the solution domain and the discretization of equationsaredescribedindetails. Section4outlinestheglobal iterativesolutionprocedureandthelocalreturn mapping stress update algorithm. A brief introduction of the relaxation method applied forthe improvement of convergence is also included. Finally, several test cases for which comparableanalytical solution or FEM simulation results exist are presented in Section 5 in order to assess thecapability and efciency of the implemented code.2. MATHEMATICAL MODELThe mathematical formulationof the poro-elasto-plasticitymodel employedinthis article isanextensionof the original Biots consolidationequations [3] toaccount for plasticity. Thederivationandhowtheequationsaremanipulatedtot FVMwill bebrieyexplainedinthefollowing subsections.2.1. Governing equationsThe behavior of the soil-pore uid(mainly water) mixture is governed by two types of differentialequations:thetotalmomentumbalanceofthesoilandtheconservationoftheowofwaterinthe pores.The momentum equation is formulated for static condition and without body force:V = V(0 I) = 0 (1)where V symbolizes the divergence operator,the total stress tensor, 0 the effective stress tensor,the pore uid pressure, and I the identity tensor. Here, tension is assumed positive as normallyCopyright 2015 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. (2015)DOI: 10.1002/nagFVM PORO-ELASTO-PLASTICITY SOIL MODELemployed in computational continuum mechanics. And no distinction is made between the initialand deformed congurations.The seepage equation is expressed as:k;wV2 =n10ddtddt(Vu) (2)where V2is the Laplacian operator, k is the permeability, ;w is the density of water, n is the porosityand u is the soil skeleton displacement vector. The effective bulk modulus of the pore uid, 10, maybe estimated assuming that air present in the pores remains close to atmospheric pressure:110 =Sr1w1 Sra(3)Here, Sris the degree of saturation, 1wthe pure water bulk modulus (~2 1091a), and a theatmospheric pressure (~ 1051a).2.2. Constitutive relationThe elasto-plastic constitutive relation of the soil skeleton is written in incremental form, applyingsmall-strain assumption and linear isotropic elasticity:J0 = 2j(J" J"p) ztr (J" J"p) (4)J" =12V(Ju) V(Ju)T| (5)whereJ"isthetotalincrementalstrain, J"ptheplasticincrementalstrain, Jutheincrementaldisplacement,and Vthegradientoperator.ThematerialconstantsjandzaretheLamcoef-cients, which can be dened by the more commonly used Youngs modulus 1 and Poissons ratio v[13, 27]. It is notable that the strain spilt in elastic and plastic strains although simple, is essentialfor the segregated nonlinear stress analysis in FVM. Determination of the incremental plastic strainJ"pis further done with a proper constitutive relation. The nonassociated Mohr-Coulomb perfectplasticity model formulated in principal stress space is employed: = (o01 o03) (o01o03) sin 2c cos (6)g = (o01 o03) (o01o03) sin [ (7)where istheyieldfunction, andgistheplasticpotential. Theprincipal stresses, o01ando03,canbeobtainedasthemaximumandminimumeigenvalueofthegeneraleffectivestresstensor

0. The soil properties . c. [are the friction angle, cohesion, and dilation angle, respectively. Anexplicit returning mapping approach developed by Clausen et al. [25] is adopted for the plastic straincomputation. Readers of interest are suggested to refer to the original paper for details.Otherplasticityconstitutivemodels, ifneeded, canalsobeusedaslongasanefcientstressupdate procedure has been carefully selected.2.3. Poro-elasto-plasticity modelBysubstitutingEquations(4)and(5)intoEquation(1)andrearrangingthelineartermsontheleft side and the nonlinear or coupling terms on the right side, the completed poro-elasto-plasticitymodel is achieved and consists of the following equations:- Displacement equationV _jV(Ju) jV(Ju)TzItr(V(Ju))_ = V2j(J"p) zItr (J"p)| nonlinearityV(J)coupling(8)Copyright 2015 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. (2015)DOI: 10.1002/nagT. TANG, O. HEDEDAL AND P. CARDIFF- Pore pressure equationk;V2 n10ddt=ddt(Vu) coupling(9)As shown in Equations (8) and (9), the poro-elasto-plasticity soil model has two important fea-tures: the nonlinearity, that is, the plastic strain term J"pdepends on the unknown displacement andstress level; and the strong coupling, that is, having pore pressure gradient term present in the dis-placement equation and velocity divergence term present in the pore pressure equation. In order todeal with these two issues, an implicitexplicit split strategy from the FVM segregated method willbe used: namely, the conventional linear, uncoupled terms categorized on the left hand side of theequations will be treated implicitly, while the nonlinear, coupled terms on the right hand side will betreated explicitly by using any available values from previous time or iteration step. This splitting ofthe nonlinearity and coupling in general guarantees the linear convergence of the employed globaliterative solution procedure, that is, the Fixed Point Iteration method (more details can be found inthe Appendix A). In general, the iterations are performed over the systems of equations until all theexplicit terms have converged.2.4. Initial and boundary conditionsThe poro-elasto-plasticity shall be further complemented with the denition of the initial and bound-ary conditions. The initial conditions consist of the displacements and pore pressure in the wholesolution domain at the initial instant of time. The boundary conditions can be specied as eitherconstant or time varying of the types summarized in Table I.It should be mentioned that the current implementation of traction boundaries in FVM is not astrivial as that in FEM. In FEM, the traction is straightforwardly accounted for as an external forcecontribution into the source term. Whereas in our implementation, we must ensure fulllment ofthe momentum balance on boundary surfaces and it thus involves some computations in solving theCauchys stress theorem equation on the boundary:Jt = n(J)= n(J0 JI)= jn2jJ" ztrJ" (2jJ"pztr (J"p)) JI nonlinearity and coupling on BC|(10)where t stands for the traction force vector acting on the boundary,is the total stress in the soiland n is the surface normal vector of the boundary.It is clear from Equation (10) that the same nonlinearity and coupling effects exist in the tractionboundary condition as the inner solution domain. It is therefore vital to apply the same implicitexplicit split strategy and iteration method on the traction boundary as for the inner domain, whichwill be described later.Table I. Boundary conditions.Ju : - prescribed displacement- prescribed traction- xedDisplZeroShear (zero value on normal direction, zero traction on tangent directions)- plane of symmetry : - prescribed pressure- zero gradient (impermeable)- plane of symmetryCopyright 2015 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. (2015)DOI: 10.1002/nagFVM PORO-ELASTO-PLASTICITY SOIL MODELFigure 1. Polyhedral control volume (Adopted from [28]).3. FINITE VOLUME DISCRETIZATIONIn this section, cell-centered FVM technique is applied to discretize the poro-elasto-plasticity modelderived in the previous section. FVM discretization is often second order accurate by assuming alinear variation of the variables over the control volume (CV). The standard discretization procedureconsists of two parts: the discretization of solution domain and the discretization of equations.3.1. Discretization of solution domainDiscretization of the solution domain consists of discretization of the time interval and discretizationof space. The time interval is split into a nite number of time steps ^tand the equations are thensolved in a time-marching manner using rst-order accurate implicit method. The computationalspace domain is subdivided into a nite number of convex polyhedral CVs, which are bounded bypolygonalfaces. Figure 1 illustrates a typicalCVVPwiththe computationalpoint 1locatedinits centroid, the face , the face areaSf , the face unit normal vector nf , the centroidNof theneighboringCVsharingtheface ,thedistancevector dfjoining1andN,andtheskewnessvector mfpointing from the intersection points between dfand the face to the face center [28].Notice,ifthevectors nfand dfarenotparallel,themeshisdenotedasnonorthogonal.Fur-thermore, if the vector mfis nonzero, the mesh is skewed. Both features have a large effect on theaccuracy of the FVM technique and thus requires special corrections in the discretization procedure[29]. Because the soil geometry is often simple and has a regular mesh, nonorthogonal and skewnesscorrections will not be discussed further.3.2. Discretization of equationsFor each time step, the discretization of equations uses the integral form of Equations (8) and (9)over VPand then applies the Gauss theorem to convert the volume integrals into surface integrals:implicit _@VPJs(2j z)V(Ju)| = inter-component coupling, explicit _@VPJs _jV(Ju)TzItrV(Ju)| (j z)V(Ju)_nonlinearity, explicit _@VPJs2j(J"p) zItr (J"p)|pressure coupling, explicit _@VPJs(J)(11)Copyright 2015 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. (2015)DOI: 10.1002/nagT. TANG, O. HEDEDAL AND P. CARDIFFimplicit _@VPJs _k;wV__VPn10ddtJV =displacement coupling, explicit _@VPJs _dudt_(12)The integral equations have been split into Implicit and Explicit discretization parts in order toallow the system being solved in the segregated manner as mentioned in Section 2.3. Iterations willbe applied until the convergence of the whole system is attained. It is worth mentioning that besidesfrom the nonlinear and pressuredisplacement coupling terms, an extra intercomponent couplingpart is also treated explicitly to promote faster convergence, see Jasak & Weller [27].Let us now describe the implicit and explicit discretization parts on a term-by-term basis.3.2.1. Implicit discretization. The implicit surface diffusion term(Laplacian terms) fromthemomentum equation is discretized as:_@VPJs(2j z) V(Ju)| =F

f D1_2jf zf_nfV(Ju)|fSf(13)where Jis for the number of faces of VP. By assuming a linear variation across face , the face-center gradient V(Ju)|fis evaluated as:nfV(Ju)|f =nf (Ju)N (Ju)Pdf(14)Similarly, the pore pressure diffusion term is discretized as follows:_@VPJs _k;wV_ =F

f D1kf;wfnf(V)fSf =F

f D1kf;wf_nf N Pdf_Sf(15)For constant soil properties, jf , zf , kf , and ;wfare simply equal to j ,z, k, and ;w.The volume integral of the time derivative of is calculated using the midpoint rule and a rst-order implicit Euler method:_VPn10ddtJV =_VPn10 o^tJV =n10P oP^tVP(16)where the upper index o is representing the old-time step value.3.2.2. Explicit discretization. Theexplicitsurfacediffusionterms(theintercomponentcouplingterms) in Equation (11) are rstly approximated the same way as in Equation (13):_@VPJs _jV(Ju)TzItr V(Ju)| (j z) V(Ju)_=F

f D1nf _jf_V(Ju)T_f zf Itr V(Ju)|f (jf zf ) V(Ju)|f_Sf(17)But unlike the implicit discretization, the face-center gradient V(Ju)|fwill now be calculated fromlinear interpolation of the cell center gradients (available from previous iteration) as follows:V(Ju)|f = xV(Ju)|P (1 x) V(Ju)|N(18)wherex=nf,dfistheinterpolationfactor, andthecell centergradient values, that is,V(Ju)|Pand V(Ju)|Ncan be evaluated using the least square t approach based on the availabledistribution of Ju [27].Copyright 2015 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. (2015)DOI: 10.1002/nagFVM PORO-ELASTO-PLASTICITY SOIL MODELThe explicit gradient terms are relatedtothe nonlinearityandpressure couplingterms inEquation(11). Theywill bediscretizedassuminglinearvariationofthevaluesacrosstheface(i.e., linear interpolation approach):_@VPJs2j(J"p) zItr (J"p)| =F

f D1nf _2jf(J"p)f zf Itr (J"p)f_Sf=F

f D1nf _2jf x (J"p)P (1 x) (J"p)N|_SfF

f D1nf _zf I xtr (J"p)P (1 x)tr (J"p)N|_Sf(19)_@VPJs(J) =F

f D1nf(J)f Sf =F

f D1nfx(J)P (1 x)(J)N| Sf(20)wherein above all the cell center values:(J"p)P. (J"p)N. (J)P. (J)Nare evaluated from theprevious iterative values.The explicit displacement coupling term is differentiated with respect to both time and space, andit can be approximated as follows:_@VPJs _dudt_ =_@VPJs _u uo^t_ =1^tF

f D1nf _uf uof_Sf=1^tF

f D1nfxuP (1 x)uN| Sf1^tF

f D1nf _xuoP (1 x)uoN_Sf(21)where uP. uNwill be evaluated from the current available iterative value.3.2.3. Traction boundary conditions. The traction boundary condition in FVM is discretized intodisplacement gradients of the CVs on the boundary patches based on Cauchys stress theorem. It hasbeen discussed previously in Section 2.4 that the same ImplicitExplicit split and iteration methodshall be applied on the boundary as the inner solution domain, expressed by the following formula:(2j z)nbV(Ju)|b implicit= (Jt)b {jV(Ju)|bnb ztr (V(Ju)|b) nb (j z)nbV(Ju)|b} explicit inter-component coupling{2jnb(J"p)b ztr(J"p)bnb} (J)bnb explicit nonlinear and pressure coupling(22)wherenbistheoutward-pointingboundaryfaceareavector, (Jt)bisthetractionincrementonthe boundary (be either directly specied or computed using the specied total traction minus theprevious time total traction), and the variables with lower index b are values on boundary face.In this way, the traction boundary will be iteratively updated until both the convergence of theinner solution domain and the convergence of the boundaries have been obtained. In general, thisextracomputationeffortonthetractionboundaryconditions(BC)ismarginalasthesizeoftheinternal domain is often far larger than that of the boundary patches.Copyright 2015 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. (2015)DOI: 10.1002/nagT. TANG, O. HEDEDAL AND P. CARDIFF4. SOLUTION PROCEDUREAsaconsequenceoftheearlierdiscretization,thediscretizedformofEquations(8)and(9)foreach CV can nally be arranged as four linearized algebraic equations (one vector equation and onescalar equation):a.du/P(Ju)P

Fa.du/N(Ju)N = b.du/P(23)a.p/PP

Fa.p/NN = b.p/P(24)where Jis the number of CV faces.The discretized coefcients a.du/P, a.du/N, a.p/P, a.p/N, and source terms b.du/P, b.p/Pare summarizedas:a.du/P=

FaduN. a.du/N= _2jf zf_ nfdfSf(25)b.du/P=

Fnf _jfV(Ju)|Tf zf Itr V(Ju)|f _jf zf_V(Ju)|f_Sf

Fnf _2jf (J"p)f zf Itr (J"p)f_Sf

Fnf(J)f Sf(26)a.p/P=

FapN nVP10^t. a.p/N=kf;fnfdfSf(27)b.p/P= n10oP^tVP 1^t

Fnf _uf uof_Sf(28)In the preceding text, the source terms b.du/Pand b.p/Pcover the contribution from explicit nonlinearterms, explicit couplings, and boundary conditions.Assembling Equations (23) and (24) for each CV in the mesh results in four sets of algebraicequations as follows:_.du/_Ju| =_b.du/_(29)_.p/_| =_b.p/_(30)where _.du/_._.p/_arethesparseN Nmatrices(NstandsforthetotalnumberofCVs),with coefcients a.du/P. a.p/Pon the diagonal and Jnonzero neighbor coefcients a.du/N. a.p/Noff thediagonal (recall Jas the total internal face numbers of each CV), Ju| the displacement incrementvector consisting of Ju at the cell center of all CVs, | the vector of pore pressure s at each CVcenter, and b.du/|. b.p/| the assembled source vectors.Having solved the incremental system for the four primary unknown variables,Ju and, it ispossible to compute the dependent unknown variables, that is, the total displacement vector u, theplastic strain incrementJ"p, the effective stresso0, and the pore pressure incrementJ. Overall,the segregated solution procedure with xed point iterations is summarized in Table II.Two levels of iteration are involved in the global solution procedure. The outer iterations, namely,the repeated steps 26, are performed to account for the explicitly lagged nonlinear and couplingsterms. While the inner iterations are performed during the iterative solution of the symmetric andCopyright 2015 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. (2015)DOI: 10.1002/nagFVM PORO-ELASTO-PLASTICITY SOIL MODELTable II. Global solution procedure.TIME STEP: t , t = nITERATION: i , i = 01. Initialize variables with the value from previous time(load) step:(Ju)0n = (Ju)n1, (u)0n = (u)n1, ()0n = ()n1, (J"p)0n = (J"p)n12. Assemble and solve the pore pressure equation for ()in by:fvm::laplacian(k,;. ) fvm::ddt(n,10. ) == fvc::div(fvc::ddt(u))13. Calculate the pore pressure increment:J = in n1:4. Assemble and solve the displacement equation for (Ju)in by:fvm::laplacian(2j z. Ju) == fvc::div(dsigmaExp)fvc::div(2jJ"pzItr(J"p))fvc::div(dp)withdsigmaExp = j + fvc::grad(Ju).T() z + I + tr(fvc::grad(Ju)) (j z) + fvc::grad(Ju)5. Obtain plastic corrections (J"p)in through local stress update algorithm in Table III.6. Update the total displacement:uin = un1 (Ju)in7. Return to step 2 until converged solutions.i = i 1EXIT: n = n 11Thefvm::operatorindicatesanimplicitdiscretizationterminOpenFOAM,whereasthefvc::operatorindicates an explicit calculus.Table III. Local return mapping stress update.INPUT: Ju, displacement increments

0A, initial/old-time stress1. Compute the elastic trial stress 0Bby:

0B= 0A{jV(Ju) jV(Ju)|TzItr(Ju)|}2. Transform Binto principal space as 0Bpri n (eigenvalues).Store the principal directions (eigenvectors).3. Evaluate the yield function _

0Bpri n_:if < 0, EXIT, 0C= B, J"p= 0if _ 0, CONTINUE4. Determine the right stress return type.Obtain the principal plastic corrector stress 0Cpri n.5. Reuse the preserved principal directions.Transform 0Cpri n back to the general space as 0C6. Compute the correct elastic strain increment J"eby:J"e=iso_

0C 0A_3z 2jdev_

0C 0A_2j7. Calculate the plastic strain increment J"pby:J"p=12V(Ju) V(Ju)T| J"eOUTPUT: 0C, J"pdiagonally dominant linear sparse systems in steps 2 and 4, typically using the incomplete Choleskypreconditionedconjugategradient method. Duetotheappliedlinear Mohr-Coulombplasticitymodel, the local return mapping process to a linear yield plane with a linear plastic potential can beperformed in one step.Copyright 2015 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. (2015)DOI: 10.1002/nagT. TANG, O. HEDEDAL AND P. CARDIFFInprinciple, it isnot necessarytosolvetheinneriterationstoaverytight tolerance, asthesolutions will only be used to update the explicit terms in the outer iterations. The entire system isconsidered to be solved when the solution changes less than a predened tolerance.In addition, in order to promote the stability of the iteration method, an under-relaxation methodis often necessary. In this work, we employ two levels of under-relaxation for the outer iteration andinner iteration, respectively:1. Under-relaxation of the variables in outer iterationThe variable under-relaxation can be explained briey by the form below: .i= (1 0).i10.i(31)where .istands for the ith under-relaxed iterative value of a generalized unknown variable .and0the under-relaxation factor with a value between 0 to 1. The variable under-relaxationslows down the change of two successive iterations in the outer iteration and thus promotesthe stabilization of the solution method. The relaxation factor 0 can be specied either as con-stant value (xed under-relaxation) or more efciently as adaptive value based on the iterationsequences (e.g., Aitkens method and Anderson acceleration). In this paper, the simple xedunder-relaxation has been adopted.2. Under-relaxation of equations in inner iterationThe equation under-relaxation is dealing with the linear algebraic system from steps 2 and4 in the global solution procedure. If we take Equation 23 as an example for explanation, itmay be written as:a.du/P(Ju)iP =

Fa.du/N(Ju)iN b.du/P(1 )a.du/P(Ju)i1P(32)whereis another under-relaxation factor between 0 and 1. It enhances the diagonal domi-nance of the linear system and thus improves the convergence of most iterative linear algebraicsolvers [13, 30].Thesegregatedxedpoint iterationalgorithmcombinedwiththeunder-relaxationhasbeenimplemented as a customsolver nonlinearBiotFoam in the open-source FVMcodeOpenFOAM-1.6-ext.5. NUMERICAL EXAMPLESIn order to assess the performance of the nonlinearBiotFoam solver, three geotechnical prob-lems with comparable solutions in either analytical or numerical (FEM) forms have been simulated:1-D consolidation of a poro-elastic soil column, the bearing capacity of a strip footing, and the upliftcapacity of a circular suction caisson foundation. Additionally, the computational cost for each caseis presented to demonstrate the efciency of the solver.5.1. Consolidation of poro-elastic soil columnThe rst simple case is aimed to verify the elastic soil-pore uid coupling. Figure 2 outlines thecase denition: a saturated soil column with heighth is subjected to a surface step loading (T ) of1kN,m2applied over a time of0.1 s. The boundary conditions and soil material properties havealso been shown in the gure itself.A comparison between FVM prediction, the analytical solution of Terzaghis 1-D consolidationtheory[31], andFEMsimulationinAbaqusisshowninFigure3. Thehorizontal axisrepre-sents the pore pressure normalized by an initial pore pressure built up immediately after loading,andtheverticalaxisstandsforthedepthofthesoilcolumnnormalizedbythetotalheight.Thevedataseriescorrespondtodifferentdimensionlessconsolidationtime, _cvt ,h2_, wherecvisthe consolidation coefcient. Overall, the FVM results agree very well to both the analytical andAbaqus solutions.Copyright 2015 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. (2015)DOI: 10.1002/nagFVM PORO-ELASTO-PLASTICITY SOIL MODELFigure 2. A poro-elastic soil column subjected to a surface step loading.Figure 3. Numerical and analytical excess pore pressure dissipation isochrones (sampled along the centerline of soil column).Table IV. Finite volume method computation cost of the poro-elastic consolidation case.No. control volume No. time step No. outer iterations per time step Total CPU time (s)40 110 12 10.59TheOpenFOAMFVMsimulationswerecarriedoutonalaptopcomputerwithIntelCorei71.5 GHz processor and 8GB RAM, and the total computation cost is listed in Table IV.5.2. Bearing capacity of strip footingThesecondtest aimsat demonstratingtheelasto-plasticsoil-poreuidcoupling. Followingthework done by Small et al. [11], a similar elasto-plastic consolidation validation case is examinedhere: a smooth, perfectly exible, uniformly loaded, and permeable strip footing acting on a layerof soil resting on a smooth rigid base. In order to completely dene the problem, it is also assumedCopyright 2015 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. (2015)DOI: 10.1002/nagT. TANG, O. HEDEDAL AND P. CARDIFFFigure 4. Illustration of case geometry (adopted from [11]).Figure 5. Finite volume mesh used for footing (half domain).Table V. The boundary conditions used in OpenFOAM.soilStructureInterface Ground Left Right soilDomainBottomJu timeVaryingTraction zeroTraction symmetryPlane zeroTraction xedDispZeroShear(Ju = 0. tx = 0) xedValue xedValue symmetryPlane zeroGradient zeroGradient( = 0) ( = 0)that there is no horizontal force on any vertical section. The original case geometry is sketched inFigure 4 and the OpenFOAM case setup with regular mesh and soil properties is shown in Figure 5.Plain strain condition has been considered. To demonstrate the computational efciency of the pro-posedFVMtechnique,acomparablecoupledMohr-CoulombFEMsimulationisruninAbaqus(version 6.11-1).Theboundaryconditionsforthedisplacement increment (Ju)andporeuidpressure()inOpenFOAM are listed in Table V. Correspondingly, the Abaqus model input of the same boundaryand loading conditions is summed in Table VI.In the beginning, it is assumed that there is no initial stress and pore pressure in the domain. Aload rate parameter o is adopted from [11]:o =J(1,c)J(Tv). Tv =cvta2(33)Copyright 2015 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. (2015)DOI: 10.1002/nagFVM PORO-ELASTO-PLASTICITY SOIL MODELTable VI. The prescribed boundary and load conditions in Abaqus.soilStructureInterface Ground Left Right soilDomainBottomDisplacement BC XSYMM YSYMMPore pressure BC = 0 = 0 Load condition Uniform pressure Figure 6. Bearing capacities under different loading rates (a) FEM results of Small et al., (b) FEM results inAbaqus, and (c) nonlinearBiotFoam simulations.Here, 1,c represents the external load pressure (normalized by soil cohesion), Tv the dimensionlesstime dened bycvwhich is the aforementioned one-dimensional consolidation coefcient, andathe width of strip footing.In general,thesmaller theloadparametero,theslowertheloadrateactingonthesoillayer.It is suggested in [11] thato=143 for fullyundrained condition and the two intermediate rates of o =1.43 and o =14.3 for a slow load rateand a fast load rate in partially drained conditions, respectively. For comparison, Figure 6a showsthe original results of Small et al., Figure 6b presents the Abaqus predictions, while Figure 6c showsthe simulation results of nonlinearBiotFoam in OpenFOAM.It can be clearly seen that the nonLinearBiotFoam solver successfully predicts the differentfailure load levels under various load rates (i.e., stronger response at slower load rate) consistentwiththeresultsofSmalletal. andAbaqus. Moreover, Figures7and8showtheporepressuredistributions along the footing center line at different load levels, for the slow load rateo =1.43and for the fast load rateo =14.3, respectively. The numbers next to the lines are indicating theloadlevel(1,c).Atlowloadlevelsinthebeginning,thewholesoildomainbehaveselastically(solid lines). Whereas, when load pressure increases to high levels, soil reaches failure (dotted line).It is interesting to observe from the simulation results that the excess pore pressure in the soil startsCopyright 2015 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. (2015)DOI: 10.1002/nagT. TANG, O. HEDEDAL AND P. CARDIFFFigure 7. Pore pressure distributions along the footing center line with slow load rate o = 1.43.Figure 8. Pore pressure distributions along the footing center line with fast load rate o = 14.3.to dissipate fast as long as plastic deformation occurs and results in dilation. This demonstrates wellthe capability ofnonlinearBiotFoam to effectively model the interactions between the porepressure development and the nonlinear soil behavior.TheOpenFOAMsimulationwasperformedunder parallel computationusing8coresonanIBM HPC cluster (2.66 GHz Intel Xeon E5430 cores). Comparably, the Abaqus simulations wereeachranon6coresinparallel onanSGIICEXcluster(2.4 GHzIntel XeonE5-2695v2IvyBridge cores).Theoverall computational costsfor FVMandFEManalysisaresummarizedinTablesVIIand VIII, respectively.The proposed segregated FVM simulations use constant time steps with large amounts of xed-point iterations, while the implicit FEM predictions in Abaqus employ automatic (small) time stepwithNewton-typeiterations.Thetwomethodsencounterthesamechallengeconvergenceratedecreases during plastic steps and requires special algorithm handling. We, therefore, applied under-relaxation methods in OpenFOAM and automatic stabilization with xed damping factor in Abaqusto help for convergence.The Abaqus soil analysis only assumes fully saturated condition (i.e., incompressible pore uid)using standard nite element coupled formulation (e.g., [32, sec. 6.8]). Element type with hybridCopyright 2015 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. (2015)DOI: 10.1002/nagFVM PORO-ELASTO-PLASTICITY SOIL MODELTable VII. Finite volume method OpenFOAM computation cost of the strip footing case.Case No. control volume No. time steps/ Max. no. outer iteration. Total wall timeplasticity occurs at per plastic time step (h)Drained 3200 53/38 3.4 10400:17:23Slow load rate 3200 46/34 1.0 10500:42:43Fast load rate 3200 38/27 1.2 10400:18:58Undrained 3200 33/25 1.0 10500:33:04Table VIII. FEM Abaqus computation cost of the strip footing case.Case No. elements/ No. time steps/ Max. no. equil iter. Total wall timeno. nodes per element plasticity occurs at per plastic time step (h)Drained 4800/8 1.150 104/396 16 14:50:00Slow load rate 4800/8 6.142 103/358 16 08:07:00Fast load rate 4800/8 9.558 103/263 16 13:21:00Undrained 4800/8 1.752 104/257 16 06:50:00formulation or reduced integration has been used to alleviate the volumetric locking problem (inthis case, we applied the 8-node quadratic elements with hybrid formulation CPE8PH). Further-more, thefullysaturatedassumptionyieldssingularitiesinthestiffnessmatrixandsignicantlyslows down the convergence. The solution of the large implicit matrix system in Abaqus also takesmuchmoreRAMandlongercomputational time. Whereas, theFVMnonlinearBiotFoamsolver is formulated on the basis of unsaturated condition, in which true bulk modulus of pore uid(comparable to that of soil skeleton) is considered and the convergence is improved. The staggeredsolutionprocedurealsoenablesefcientiterativesolvertobeusedforsolvingthesmallmatrixsystem. Alternatives to the Abaqus implicit solver are explicit-FEM/hybrid-FEM solvers (with opti-mizedalgorithmsasfortheproposedFVM); however, theyarenot availableforfullycoupledporo-elasto-plastic stress analysis in the commercial software packages.It is necessary to note that the Abaqus FEM and our OpenFOAM FVM implementations are notexactlythesameduetotheassumed(in)compressibilityoftheporeuid, sothecomparisonisnot entirely fair but it does give a good indication of the efciency of the developed method as aviable alternative.Additionally, if comparing the two methods in terms of traction boundary condition treatment:the FEM Abaqus analysis only requires a simple external load input, while the proposed FVM usesuser-dened boundaries which do iterative computation from the external traction to displacementon the patches. This drawback of FVM may be acceptable as long as a consistent staggered iterativemethod has been applied on the boundary and the inner solution domain, which provides an overallefciency of the algorithm.5.3. Uplift capacity of suction caissonThethirdtest caseisrepresentingaproblemthat drawsincreasingapplicationsintheoffshoregeotechnicaleld:namely, theverticalupliftcapacityofsuctioncaissonfoundations. Asuctioncaisson, open at the bottom and closed at the top, is designed to penetrate to the sea oor by its ownweight and also by creating an internal under-pressure relative to the external water pressure [33].The developed passive suction inside the soil plug signicantly enhances the pullout capacity of thesuction caisson.Figure9illustratesacylindricalsuctioncaissonwithanoveralldiameterJandanembeddedlength 1, subjected to an uplift load at the top of the caisson.CPE8PH:an8-nodeplanestrainquadratic,biquadraticdisplacement,bilinearporepressure,hybrid,linearpressurestressCopyright 2015 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. (2015)DOI: 10.1002/nagT. TANG, O. HEDEDAL AND P. CARDIFFFigure 9. Suction caisson foundation under vertical uplift loading (adopted from [33]).Table IX. Parameters employed in the suction caisson case.Component Property Symbol Value(s)Caisson geometry Embedment depth 1 1 mDiameter J 2 mYoungs modulus 1 2 107PaPoissons ratio v 0.3Permeability k 0.001 m/sPorosity n 0.2Soil properties Saturation factor Sr 0.98Friction angle 30Cohesion c 1 105PaDilation angle [ 0Bulk modulus of water 1w2.1109PaSpecic weight of water ;w1 104PaLoading rate Pullout velocity 2 104to 2 101m/s 0 100 200 300 400 500 600 00.020.040.060.080.1Pullout traction (in kPa)Displacement (in m)Comparison of pullout capacity under different loading rates0.2m/s0.02m/s0.002m/s0.0002m/sFigure 10.Pullout load-displacement behavior under various loading rate.The above geometry are then set up in OpenFOAM using the same rectangular FVM mesh typeasinFigure5.AwedgetypeboundaryinOpenFOAMisalsoappliedtosimulatethecaseas2-Daxisymmetric.Ithastobementionedthattheentiresuctioncaissonisassumedtobelifteduniformly under constant rate (as a prescribed displacement boundary). All the parameters involvedin the simulations have been listed in Table IX.Our focus is to evaluate the effects of different loading rate on the uplift capacity. It is revealedin Figure 10 that faster load rates give greater pullout capacities, which comes from the ability tobuild up and sustain a signicant negative pore pressure inside the bucket, Figure 11. This capturedCopyright 2015 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. (2015)DOI: 10.1002/nagFVM PORO-ELASTO-PLASTICITY SOIL MODELFigure 11. Passive suction development under various loading rate.Table X. Computation cost of the suction caisson.Case (m/s) No. control volume No. time steps Max. no. outer iterations Total used wall time (h)per time step0.2 1840 10 1 10400:18:580.02 1840 10 1 10400:20:430.002 1840 10 1 10400:20:550.0002 1840 10 1 10400:23:31phenomenon agrees well to those simulation results in the references [33, 34] with FEM analysis,although they are dealing with less permeable clay.The simulation was carried out under parallel computation using 8 cores of a single computationnode that has 2 XEON 5550 2.66 GHZ quad core CPUs and 24 GB RAM. The overall cost is listedin Table X.6. CONCLUSIONA new numerical method for poro-elasto-plastic soil is presented, based on a nite volume methoddiscretization and segregated solution procedure. The main features of the method are:(i) It is computationally efcient in terms of both memory and CPU time-wise, as the implicitexplicit split of discretizationandlinearizationresults ina decoupledsystemof linearalgebraic system with sparse diagonally dominant coefcient matrices, solvable by efcientiterative solvers.(ii) The solution algorithm is neat and economic, because any nonlinearity and coupling can behandled with one single outer iteration process (i.e., the xed point iteration method), thus,requiring no or little additional cost. Moreover, it has been demonstrated in the convergencestudy of the iteration method (Appendices A.1 and A.2): Careful splitting of the nonlinearand coupling terms guarantees a linear convergence rate for unsaturated soil problems. Forfullysaturatedsituations, theselectionoftimestepandmeshdensityneedstofulll theconvergence criteria, namely, Equation A.11.(iii) The implementation work is time-saving under the help of OpenFOAM C++ library, whererich FVM classes and functions are freely available.(iv) The model itself is easily extendable, as other nonlinear constitutive relations can be integratedinto the model with the same global solution procedure.(v) It has a lot of application potentials to model many geotechnical problems in a simple andeffective manner, as illustrated by the test cases.Copyright 2015 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. (2015)DOI: 10.1002/nagT. TANG, O. HEDEDAL AND P. CARDIFFAPPENDIX: REMARKS ON FIXED POINT ITERATION METHODThe xed-point (FP) iteration method in FVM simply uses the following recursive form:.iC1= G_.i_. i = 0. 1. 2.(A.1)The FP method usually results in linear convergence [35], slightly inferior to the commonly appliedNewton-typeiterationmethodin FEM that givesquadraticconvergence.However,the importantfeatureofFPmethodliesinitsavoidancetotheformationofthelargenonsymmetricJacobianmatrix, and instead, it produces several small symmetric matrices that can be very efciently solvedby standard fast iterative algebraic solvers. The main problem of applying FP method comes outthat the convergence condition is dependent on the choice of function G(.), which shall in principlebe differentiable and fulll [G0(.)[ _1 in some operator norm [36]. In our case, the denition offunction G(.) is closely related to the method of partitioning the nonlinear terms in the momentumequation and the soil-pore uid coupling operators. The following paragraphs will then discuss thesetwo aspects separately.A1. THE PARTITIONING OF NONLINEAR TERMS IN MOMENTUM EQUATIONIf we are solving a nonlinear system equation (.). = b, the splitting of linear and nonlinear termsby writing (.) = l nl(.), results in:l. = nl(.). b (A.2)The FP iteration method is then:.iC1= 1l_nl_.i_.ib_(A.3)where,.symbolizes the displacement incrementJu in the momentum equation Equation 11,lrelates to the linear (implicit) elastic stiffness matrix, and nl(.) the nonlinear (explicit) plastic stiff-ness matrix. In this way, the function G(.) can simply be written out as G(.) = 1l(nl(.). b)and its differential is:G0(.) = 1lnl(.) 1l0nl(.). (A.4)Particularly, as we used a perfect plasticity model without any hardening or softening effect in theconstitutive relation, the derivative of plastic stiffness 0nl(.) is equivalent to zero and, therefore, thesecond term on the right hand side of Equation A.4 simply vanishes. In the end, the sufcient con-vergence condition for the partitioning scheme turns to be [G0(.)[ =1lnl(.) _1. We mayprove such convergence condition in a qualitative way instead of complicated matrix norm manip-ulation. The elastic stiffness contribution lis the maximum stiffness that the system can provide,in other words the plastic stiffnessnlis always weaker than the preyielding elastic stiffnessl,which means that 1lnl(.) _1 always holds. In general, as long as the implicit stiffness partis stronger than its counterpart explicit stiffness part, the system of momentum equation will con-verge. While for other advanced hardening/softening plastic models, more terms will be involved inEquation A.4, and, hence, careful evaluation might be needed.A2. THE PARTITIONING OF SOIL-PORE FLUID COUPLINGConsider a simple linearly coupled equation system in the following form:an1 = n2b n2cn2 = n1(A.5)The aforementioned system has the same coupling feature as our poro-elasto-plasticity model, thatis, Equations 8 and 9, but is formed in a much simpler form for demonstration purpose. ReferenceCopyright 2015 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. (2015)DOI: 10.1002/nagFVM PORO-ELASTO-PLASTICITY SOIL MODEL[37] has pointed out that the sufcient condition for convergence of a coupled system is related to thespectral radius of the coupled operator, which is indeed equivalent to the value of aforementioned[G0(.)[. We here adopted the same procedure of dening the coupled spectral radius:Firstly, use the backward Euler scheme and the coupled system Equation A.5 is discretized intime as,_1 1,a1,(b c^t ) 1_ _nnC1;iC11nnC1;iC12_ =_0bnn2 nn1_(A.6)where the superscript n stands for the time step counter and ithe iterative counter.Then apply the FP splitting, Equation A.7 can be written as,_1001_ _nnC1;iC11nnC1;iC12_ =_0bnn2 nn1__(1,a)nnC1;i2(1,(b c^t ))nnC1;i1_(A.7)Further rewrite Equation A.7 into the following form as,_nnC1;iC11nnC1;iC12_ =_0 1,a1,(b c^t ) 0_ G_nnC1;i1nnC1;i2__0bnn2 nn1_(A.8)The spectra radius of the coupled operator is the eigenvalues of the matrix G,j(G) = eig_0 1,a1,(b c^t ) 0_ = _1,(a(b c^t )) (A.9)The convergence condition requires that j(G) shall be less than 1. Let us now examine whether thiscriteria could be fullled in our case. Recall the model Equations 8 and 9 and ignore the nonlinearterms in this stage, we could gure out the expression of the coefcients a. b. c in terms of the soilproperties and the mesh and time step sizes:a =2j zh. b =nh10. c =k^t;h(A.10)whereh could represent the generalized mesh size. The convergence criteria now can be writtenout as:a(b c^t ) =2j z10,n(2j z)k;w

^th2 _ 1 (A.11)the previously mentioned criteria imply that once the bulk modulus of the soil skeleton is strongerthan the effective bulk modulus of the uid normalized by the porosity, that is,2j z _10,n,theconvergencecanbeguaranteedwithoutlimitationontimeandmeshconditions. Itthereforereveals that the poro-elasto-plastic model might fail to simulate the case when pure water presentsin the soil pore, as pure water is often considered as incompressible compared with the soil skeleton.However, favorably, for natural soil that even only contains one percent of air in the pore, the uidbulk modulus is reduced dramatically by a factor of 200 to 1071a, which often turns out to be smallenough compared with the solid bulk modulus and thus fullls Equation A.11.ACKNOWLEDGEMENTSWe would like to thank University College Dublin OpenFOAM Simulation Group, especially Pro-fessor Alojz Ivankovic, for all the support and helpful discussions during the rst authors 3-monthexternal research stay there. 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