FVM for Nonlinear Soil Stress analysis involving …FVM for Nonlinear Soil Stress analysis involving Pore Pressure Coupling Tian Tang∗1, Ole Hededal1, Philip Cardi ﬀ2, Johan Roenby3

IntroductionSoil Solver Development using OpenFOAM

Collaborative works in UCD OpenFOAM groupSummary & Future works

FVM for Nonlinear Soil Stress analysisinvolving Pore Pressure Coupling

Tian Tang∗1, Ole Hededal1, Philip Cardiff2, Johan Roenby3

1Technical University of Denmark, 2University College Dublin, 3DHI

8th OpenFOAMrWorkshop, Jeju, 11th − 14th June 2013

IntroductionSystem Modeling

Simulation Results & AnalysisConclusion

1 / 31

1 / 24



Outline

1 IntroductionMotivationObjectives

2 Soil Solver Development using OpenFOAMMathematical modelDiscretization & Solution procedureTest cases

3 Collaborative works in UCD OpenFOAM groupMore soil featuresConvergence acceleration

4 Summary & Future worksAbbreviations

2 / 24



MotivationObjectives

Outline




4 Summary & Future works

3 / 24




Why we started with this research topic?

Offshore structure and foundation failuresdue to seabed instability (liquefaction) areobserved.

Integrated numerical modelling ofseabed-wave-structure interaction isdemanding.

OpenFOAM as an open source FVMlibrary facilitates the customer solverdevelopments.

Figure: An illustration ofwave-seabed-structure

interaction3 / 24




Our Goals

Current goal: Development of an efficient soil solver withplastic soil deformation and pore pressure coupling.Future goal: Multiphysics modeling by combining thedeveloped soil solver with existing fluid and structure solvers

Figure: The multiphysics solver structure using OpenFOAM4 / 24



Mathematical modelDiscretization & Solution procedureTest cases

Outline





5 / 24




Soil Mathematical ModelGoverning Equations:

Total momentum balance for the soil mixture (steady-state)

∇ · σ −∇p = 0

Storage equation for pore fluid flow

nK ′∂p∂t −

kγw∇2p + ∂

∂t (∇ · u) = 0

σ: soil effective stress, p: pore fluid pressure,u: soil skeleton displacement,n,K ′: soil porosity and pore fluid bulk modulus,k, γw : soil permeability and water specific weight.

5 / 24




Soil Mathematical ModelGoverning Equations:

Total momentum balance for the soil mixture (steady-state)

∇ · σ −∇p = 0

Storage equation for pore fluid flow

nK ′∂p∂t −

kγw∇2p + ∂

∂t (∇ · u) = 0

nonlinear consitutive relation & displacement-pressure coupling

σ: soil effective stress, p: pore fluid pressure,u: soil skeleton displacement,n,K ′: soil porosity and pore fluid bulk modulus,k, γw : soil permeability and water specific weight.

6 / 24




Soil Mathematical ModelConstitutive relations:

Linear elasticity

dσ = Ce : (dε− dεp) , dε = 12{∇(du) + [∇(du)]T

}Mohr-Coulomb perfect plasticity

Yield surface f = (σ1 − σ3) + (σ1 + σ3) sinϕ− 2c cosϕPlastic potential g = (σ1 − σ3) + (σ1 + σ3) sinψ

Flow rule dεp = dΛ · ∂g∂σ

, dΛ =

(∂f∂σ

)TCedε(

∂f∂σ

)TCe ∂g∂σ

σ1, σ3: maximum and minimum principal stress,ϕ, c, ψ:soil friction angle, cohesion, and dilation angle,C e : linear elastic stiffness tensor.

6 / 24




Discretization & Solution procedure

Cell-centered finite volume discretizationGlobal solution procedure:1. Partitioned (segregated) approach

fvScalarMatrix pEqn(

fvm::ddt(p) == fvm::laplacian(Dp, p) - fvc::div(fvc::ddt(Dp2,U)));

fvVectorMatrix dUEqn(

fvm::laplacian(2.0*mu + lambda, dU, "laplacian(dU)")==

- divDsigmaExp+ fvc::div(2.0*mu*(mesh.Sf() & fvc::interpolate(dEpsP)))+ fvc::div(lambda*(mesh.Sf() & I*fvc::interpolate(tr(dEpsP))))+ fvc::grad(dp)

);

7 / 24




Discretization & Solution procedure2. Fixed Point iteration + Underrelaxation

Figure: The iterative solution strategy of nonLinearBiotFoam in OpenFOAM

8 / 24




Discretization & Solution procedure

Local stress update

Figure: Illustration ofreturn mapping

Stress return algorithmINPUT: du, displacement increments

σA, initial/old-time stress

1. Compute the elastic trial stress σB by:σB = σA + {µ∇(du) + µ[∇(du)]T + λItr[(du)]}

2. Transform σB into principal space as σBprin . Store the principal directions.3. Evaluate the yield function f (σBprin):

if f < 0, EXIT, σC = σB , dεp = 0if f ≥ 0, CONTINUE

4. Determine the right stress return type.Obtain the principal plastic corrector stress σCprin .

5. Reuse the preserved principal directions and transform σCprin back to σC

6. Calculate the plastic strain increment dεp

OUTPUT: σC , dεp

9 / 24




Test cases

Elastic consolidation test

Drained triaxial compression test

Elasto-plastic consolidation test

10 / 24




Test cases: Elastic consolidation testA saturated soil column subjected to a surface step loading:

Figure: Case definitionFigure: Pore pressure distribution

along the depth after differentconsolidation time

11 / 24




Test cases: Drained triaxial compression testA drained soil cube compressed by constant strain rate:

Figure: Simulated elastic perfect-plastic soil response

12 / 24




Test cases: Elasto-plastic consolidation testA layer of saturated soil loaded by a strip footing with differentloading rate:

Figure: A sketch of the case geometry (left); OpenFOAM mesh and boundaryconditions (right)

13 / 24




Test cases: Elasto-plastic consolidation testA layer of saturated soil loaded by a strip footing with differentloading rate:

Figure: FEM results by Small et al. (left); FVM results by nonLinearBiotFoam (right).

13 / 24




Test cases: Elasto-plastic consolidation test

Animations: Pore pressure variation + Yielding zone

(Loading Video...) (Loading Video...)

(displacement exaggerated by a factor of 10)

14 / 24

porepressure.aviMedia File (video/avi)

yielding.aviMedia File (video/avi)



More soil featuresConvergence acceleration

Outline





15 / 24




Modeling more soil features: Large deformation

Linear elastic stress-strain + Nonlinear strain-displacementrelation (Total Lagrangian format):

All the nonlinear terms treated explicit + Fixed point iteration:

Extend to small strain large displacement elasto-plastic solver(similar solution procedure to small strain small displacementEP solver)

15 / 24




Modeling more soil features: Large deformation

Verification: a simple tension test

(a) (b)

Figure: a) Resulting stress-displacement relationships from soilEpTLFoam prediction;b) Resulting force-displacement relationships from soilEpTLFoam prediction

16 / 24




Modeling more soil features: Anisotropy

Soil deposits are inherently anisotropic due to the process ofsedimentaton followed by predominantly one-dimensionalconsolidation. Soil anisotropy is used in reference to soil structure,soil strength, and soil permeability changes with direction ofmeasurement.

Assumption: cross-anisotropic soil

Ce =

C11 C12 C13 0 0 0C12 C11 C13 0 0 0C13 C13 C33 0 0 00 0 0 C44 0 00 0 0 0 C44 00 0 0 0 0 (C11 − C12)/2

k = k1 0 00 k1 0

0 0 k3

Ce : cross-anisotropic elastic stiffness tensor, k: permeability coefficienttensor.

17 / 24





Implementation:

∇ · (K · ∇(du))︸︷︷︸Implicit

+∇ · (C : dε)−∇ · (K · ∇(du))−∇p︸︷︷︸explicit

= 0

nK ′∂p∂t −

1γw∇ · (k · ∇p)︸︷︷︸

Implicit

+∂

∂t (∇ · u)︸︷︷︸explicit

= 0

Tested by a case of standing wave induced cross-anisotropicseabed response.

18 / 24





Figure: Standing-wave induced anisotropic and isotropic soil response. Isotropy(right):E = 107Pa, ν = 0.3, cross-anisotropy(left):

Ez = 107Pa, νxx = νzx = 0.3, n = m = 0.6, kx = kz = 10−4m/s.19 / 24




Convergence consideration

Fixed point iteration method in FVM has both advantages anddrawbacks:

No need to form and update the Jacobian matrix. ,Create diagonally dominant sparse matrices ideally suited

for iterative solver. ,No convergence for some highly nonlinear and strongcoupling problems. If combined with fixedunderrelaxation, slow convergence./Seek for adaptive underrelaxation → Aitken’s method

20 / 24




Convergence consideration: Aitken’s methodAitken’s method is most useful for accelerating a linear convergentsequence:

x i+1 = x̃ i+1 + θi+1r i

r i = x̃ i+1 − x i

θi+1 = −θi ri−1 (r i − r i−1)

(r i − r i−1) (r i − r i−1)where, x is the solving variable (or variable vector). The tilde signdenotes the solved value before underrelaxation.

Apply to a large deformation elastoplastic simple tension test case:

Number of outer iterations total CPU time (s)(plastic step)

Fixed under-relaxation 618 38.34Aitken’s relaxation 264 15.79

21 / 24



Outline





22 / 24



Summary & Future worksSummary:

FVM soil solver has been developed using OpenFOAM, it hasthe following features:

elasto-plastic soil deformationspore pressure coupling

The nonlinearity and coupling in the equations are tackled bypartitioned approach and fixed point iteration.Large deformation and soil anisotropy added to the soil solver.Aitken’s method is applied for convergence acceleration.

Next step:Implementaion of advanced soil solver based on critical stateand cyclic plasticity.Further convergence improvement.

22 / 24



References:D.S. Jeng, 1997, Soil response in cross-anisotropic seabed due to standing waves, Journal of Geotechnicaland Geoenvironment Engineering, Vol. 123, 9-19

H. Jasak, 1996, Error analysis and estimation for the Finite Volume method with applications to fluid flows,PhD. Thesis, Imperial College, University of London

H. Jasak and H.G. Weller, 2000, Application of the finite volume method and unstructured meshes to linearelasticity, International Journal for Numerical Methods in Engineering, Vol.48, No.2, 267-287

I. Demirdzic and D. Martinovic 1993, Finite volume method for thermo-elasto-plastic stress analysis,Computer Methods in Applied Mechanics and Engineering, Vol.109, No.3-4, 331-349

I. Demirdzic, I. Horman and D. Martinovic 2000, Finite volume analysis of stress and deformation inhygro-thermo-elastic orthotropic body, Computer Methods in Applied Mechanics and Engineering, Vol.190,1221-1232

J. Clausen and L. Damkilde and L. Andersen, 2007, An efficient return algorithm for non-associatedplasticity with linear yield criteria in principal stress space, Computers and Structures, Vol.85, No.23-24,1795-1807

J.C. Small, J.R. Booker and E.H. Davis, 1976, Elasto-plastic consolidation of soil, International Journal ofSolids Structures, Vol.12, 431-448

P. Cardiff, 2012, Development of the Finite Volume Method for Hip Joint Stress Analysis, PhD Thesis,University College Dublin

P. Cardiff, A. Karac, and A. Ivankovic 2013, A large strain finite volume method for orthotropic bodies withgeneral material orientations, Computer Methods in Applied Mechanics and Engineering, submitted

23 / 24



Thank you for your attention!

24 / 24

IntroductionMotivationObjectives

Soil Solver Development using OpenFOAMMathematical modelDiscretization & Solution procedureTest cases

Collaborative works in UCD OpenFOAM groupMore soil featuresConvergence acceleration

Summary & Future works

Documents

FVM for Nonlinear Soil Stress analysis involving …FVM for Nonlinear Soil Stress analysis involving Pore Pressure Coupling Tian Tang∗1, Ole Hededal1, Philip Cardi ﬀ2, Johan Roenby3