We present a mixed finite element formulation for the spatial discretization in dynamic analysis of non-isothermal variably saturated porous media using different order of approximating functions for solid displacements and fluid pressures/temperature. It is known in fact that there are limitations on the approximating functions N-u and N-p for displacements and pressures if the BabuskaBrezzi convergence conditions or their equivalent [5] are to be satisfied. Although this formulation complicates the numerical implementation compared to equal order interpolation, it provides competitive advantages e.g. in speed of computation, accuracy and convergence

Cao, Toan Duc

Sanavia, Lorenzo

Schrefler, B. A.

Wang, Yongqi

TUbiblio

Mixed finite element formulation for non-isothermal porous media in dynamics

Heaton, AH

Martin, SL

Littlefield, RS

Elsevier - Publisher Connector 

CAN3: BASELINE COST OF ILLNESS OF PARKINSON'S DISEASE IN A LARGE HEALTHCARE PLAN 

We present a mixed finite element formulation for the spatial discretization in dynamic analysis of non-isothermal variably saturated porous media using different order of approximating functions for solid displacements and fluid pressures/temperature. It is known in fact that there are limitations on the approximating functions Nu and Np for dis- placements and pressures if the Babuska-Brezzi convergence conditions or their equivalent [1] are to be satisfied. Although this formulation complicates the numerical implementa- tion compared to equal order of 

interpolation, it provides competitive advantages e.g. in speed of computation, accuracy and convergence

Cao, T.D.

Wang, Y.

Sanavia, L.

UPCommons

AH Heaton

SL Martin

RS Littlefield

Crossref

Value in Health

CAN3: BASELINE COST OF ILLNESS OF PARKINSON'S DISEASE IN A LARGE HEALTHCARE PLAN

We present a mixed finite element formulation for the spatial discretization in dynamic analysis of non-isothermal variably saturated porous media using different order of approximating functions for solid displacements and fluid pressures/temperature. It is known in fact that there are limitations on the approximating functions Nu and Np for dis- placements and pressures if the Babuska-Brezzi convergence conditions or their equivalent [1] are to be satisfied. Although this formulation complicates the numerical implementa- tion compared to equal order of 
interpolation, it provides competitive advantages e.g. in speed of computation, accuracy and convergence

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140Mixed finite element formulation for non-isothermal porous media in dynamicsIV International Conference on Particle-based Methods – Fundamentals and ApplicationsPARTICLES 2015E. Oñate, M. Bischoff, D.R.J. Owen, P. Wriggers & T. Zohdi (Eds)MIXED FINITE ELEMENT FORMULATION FORNON-ISOTHERMAL POROUS MEDIA IN DYNAMICST.D. Cao1, Y. Wang1, L. Sanavia2 and B.A. Schrefler21 Department of Mechanical Engineering, Technische Universität DarmstadtOtto-Berndt-Strasse 2, 64287 Darmstadt, GermanyE-mail: cao@fdy.tu-darmstadt.de and wang@fdy.tu-darmstadt.de2 Dipartimento di Ingegneria Civile, Edile e Ambientale, Università degli Studi di Padovavia F. Marzolo 9, 35131 Padova, ItaliaE-mail: lorenzo.sanavia@unipd.it and bernhard.schrefler@dicea.unipd.itKey words: Hydro-Thermo-Mechanical processes; FEM; Mixed elements; MultiphasematerialsAbstract. We present a mixed finite element formulation for the spatial discretization indynamic analysis of non-isothermal variably saturated porous media using different orderof approximating functions for solid displacements and fluid pressures/temperature. It isknown in fact that there are limitations on the approximating functions Nu and Np for dis-placements and pressures if the Babuska-Brezzi convergence conditions or their equivalent[1] are to be satisfied. Although this formulation complicates the numerical implementa-tion compared to equal order of interpolation, it provides competitive advantages e.g. inspeed of computation, accuracy and convergence.1 INTRODUCTIONThe numerical modelling of the dynamic response of non-isothermal multiphase ge-omaterials is of great interest in many fields, including Environmental Geomechanics,Structural Mechanics and Biomechanics. Based on the improved efficiency and stabilityof appropriate numerical methods, and the increased performance of the computationalequipments, the efficient numerical treatment of coupled multi-field problems has becomepossible in the last decades. Effective numerical simulations require a mixed element (e.g.Taylor-Hood element [2]) with different order of approximating functions for solid dis-placements and fluid pressures/temperature. Consequently, numerical complexities arisein the solutions because the basis functions in the u-p (solid-displacements and pressure)formulation are of different order (e.g. second order for displacements, first order for pres-sure, to obtain the same kind of approximation for effective stress and fluid pressures).In the use of mixed elements it has been reported by several authors [3, 4, 5, 6] that in-accuracies in pressure estimates can result from spurious spatial fluctuations of estimated1141T.D. Cao, Y. Wang, L. Sanavia and B.A. Schreflerpore pressure between adjacent elements. This is especially likely to occur at perme-able boundaries in the initial time-marching steps, or when short time steps are used torepresent rapidly changing of fluid phases or transient applied loads. Moreover, Muradet al. [7] noted that, even with Taylor-Hood elements, potentially unstable results mayoccur when there are discontinuous initial conditions at the initiation of a time-marchingsolution.This work presents, as a novel contribution, a mixed finite element formulation for fullycoupled non-isothermal multiphase geomaterials in dynamics.The multiphase model is developed following Lewis and Schrefler [8]. The u-p-T formu-lation is obtained by neglecting the relative fluids acceleration and their convective terms,which is valid for low frequency problems as in earthquake engineering [9]. In the modeldevolvement, the dynamic seepage forcing terms in the mass balance equations and in theenthalpy balance equation and the compressibility of the grain at the microscopic levelare neglected. The Taylor-Hood approach is applied to discretize the governing equa-tions in space, while the generalized Newmark scheme is used for the time discretization.The final non-linear set of equations is solved by the Newton-Raphson method with amonolithic approach [8, 10]. The model has been implemented in the finite element codeCOMES-GEO, [8, 11, 12, 13, 14, 15, 16, 17, 18].The implemented model is validated through the comparison with analytical or finiteelement quasi-static or dynamic solutions.2 MACROSCOPIC BALANCE EQUATIONSThe full mathematical model necessary to simulate the thermo-hydro-mechanical be-haviour of partially saturated porous media was developed within the Hybrid MixtureTheory by Lewis and Schrefler [8], using averaging theories according to Hassanizadehand Gray [19, 20]. After neglecting the acceleration of the fluids in the governing equa-tions of Lewis and Schrefler [8], a set of balance equations for the whole multiphasemedium is obtained and presented in this paper.Linear momentum balance equations of the mixturedivσ + ρg − ρas = 0 (1)Dry air mass balance equationdiv(ρga krgkgµg[−gradpg + ρgg])+ div(ρg MaMwM2gDgag grad(pgwpg))+ρgaSgdivvs + nSgρ̇ga − ρganṠw − ρgaβs [1− n]SgṪ = 0(2)Water species (liquid and vapour) mass balance equationdiv(ρw krwkwµw[−grad (pg − pc) + ρwg])+ div(ρgw krgkgµg[−gradpgw + ρgwg])−div(ρg MaMwM2gDgwg grad(pgwpg))+ [ρwSw + ρgwSg] divvs + n [ρw − ρgw] Ṡw−[ρwβsw + ρgwβs [1− n]Sg]Ṫ + nSgρ̇gw + ρw nSwKw [ṗg − ṗc] = 0(3)2142T.D. Cao, Y. Wang, L. Sanavia and B.A. SchreflerEnthalpy balance equation for the multiphase medium−div�ρw krwkµw[−grad (pg − pc) + ρwg]�∆Hvap − div (χeffgradT )− ρwSwdivvs∆Hvap+�Cwp ρw krwkµw[−grad (pg − pc) + ρwg] + Cgpρg krgkµg[−gradpg + ρgg]�· gradT+ (ρCp)eff Ṫ − ρw nSwKw�ṗg − ṗc�∆Hvap + βswṪ∆Hvap − n [ρw − ρgw] Ṡw∆Hvap = 0(4)The meaning of each variable of equations 1 to 4 is described in [8, 11, 15, 12].3 CONSTITUTIVE RELATIONSHIPSFor the gaseous mixture of dry air and water vapour, the ideal gas law is introduced.The equation of state of perfect gas (Clapeyrons equation) and Dalton’s law are appliedto dry air ( ga ), water vapor ( gw ) and moist air ( g ). In the partially saturatedzones, the equilibrium water vapor pressure pgw(x, t) can be obtained from the Kelvin-Laplace equation, where the water vapor saturation pressure pgws(x, t) depending onlyupon the temperature, can be calculated from the Clausius-Clapeyron equation or froman empirical correlation. The saturation degree Sπ(x, t) and the relative permeabilitykrπ(x, t) are experimentally determined functions. The solid skeleton is assumed elastic,homogeneous and isotropic in the numerical simulations described in Section 5.4 SPATIAL AND TIME DISCRETIZATIONThe Petrov-Galerkin method with different interpolation functions for the primaryvariables and the test functions is applied for the spatial discretization. Isoparametricmixed element (see. Figure 1) is used and the shape functions for each phase arepg = Ngp̄gpc = Ncp̄cT = NT T̄u = Nuū(5)After spatial discretization, the following non-symmetric, non-linear and coupled sys-tem of equations is obtainedMuu ¨̄u+�BTσ′dV −Kugp̄g +Kucp̄c = fuCgg ˙̄pg +Cgc ˙̄pc −CgT ˙̄T+Cgu ˙̄u+Kggp̄g −Kgcp̄c −KgT T̄ = fgCcg ˙̄pg +Ccc ˙̄pc +CcT˙̄T+Ccu ˙̄u−Kcgp̄g +Kccp̄c +KcT T̄ = fc−CTg ˙̄pg −CTc ˙̄pc +CTT ˙̄T−CTu ˙̄u−KTgp̄g +KTcp̄c +KTT T̄ = fT(6)The generalized Newmark method is applied to discretize in time the non-linear systemequations 6 and the dynamic response at step n+1 for the general non-linear systemis obtained, in which the unknowns are Xn+1 =�∆¨̄u,∆ ˙̄pg,∆ ˙̄pc,∆ ˙̄T�. The non-linear3143T.D. Cao, Y. Wang, L. Sanavia and B.A. SchreflerFigure 1: Isoparametric mixed element. According to the interpolation model (biquadratic for u, bilinearfor pg, pc and T ), the process variables are related to different nodes: (•) solid skeleton displacements,(◦) fluids pressure/temperaturesystem is solved by the Newton-Raphson method, written in concise form asJ(Xi)∣∣n+1[Xi+1 −Xi]n+1= −G(Xi)∣∣n+1(7)5 FINITE ELEMENT VALIDATION5.1 Force vibration tests under harmonic loadIn this example the response of an homogeneous and isotropic, water-saturated, poroe-lastic column defined in Figure 2 is analyzed under plane-strain confined compressionconditions, following [21] and [22]. The domain is surrounded by impermeable, fric-tionless but rigid boundaries except for the loaded top side which is perfectly drained.Furthermore, 2x2 Gauss integration points were used for numerical computation. Thevalue of the tolerance for the global iterative Newton-Rapson procedure is fixed to 10−4and the maximum number of iteration inside a time step is fixed to 30. The geomechanicalcharacteristics of the material are given in [21]. In particular, two scenarios are tested:an high permeability case, k=10−2 m/s, and a lower permeability case, k=10−5 m/s. Theobjective of this benchmark problem is to compare the dynamic finite element solutionobtained with standard and mixed elements with an existing analytical solution obtainedvia Laplace transform [22] as reported in [21].The vertical displacement history at the top of the column for the two cases is plottedin Figure 3, where a period equal to 0.1 second, which is the period of the harmonic load,is shown. Figure 3a displays the comparison with the analytical solution presented in [21]4144T.D. Cao, Y. Wang, L. Sanavia and B.A. SchreflerFigure 2: Description of the test modelin case of hydraulic permeability of k=10−2 m/s and a good convergence is observed. Withthe lower permeability, the development of the vertical displacement of the top surfaceis lower than the case with higher permeability. Moreover, for permeability k=10−5 m/s,Figure 3b shows a diversity in the accuracy of the solution when the mixed elementformulation is used.Figure 3: Top displacement history for a) k=10−2 m/s, b) k=10−5 m/s5.2 Non-isothermal water saturated consolidation problemThe test problem concerning the fully saturated thermo-elastic consolidation definedin Figure 4 is now solved and its solution is compared with the solution of quasi-staticand dynamic problems (with standard elements) done in previous work [23, 18]. Theplane-strain medium is confined in a column of 7 m height and 2 m width and subjectedto a surface loading 10 kPa and a surface temperature jump of 50 K above the initialtemperature of 293.15 K.The geomechanical characteristics of the Aboustis material are given in [23, 13] withpermeability: k = 2.55 · 10−7 m/s. The liquid water and the solid grain are assumedincompressible for the static analysis, whereas the compressibility of the liquid water istaken into account in the dynamic analysis.The initial variables for whole node are the liquid water pressure in hydrostatic con-dition, the air pressure at atmospheric pressure and the temperature at 293.15 K. The5145T.D. Cao, Y. Wang, L. Sanavia and B.A. SchreflerFigure 4: Description of the Aboustit test modelboundary conditions of this problems are as follows: the bottom and lateral surface areimpervious to both water and air; the top boundary is drained. The normal and tangen-tial displacements are constrained at bottom; at lateral surface, the normal displacementis constrained. 2x2 Gauss point integration scheme was used. During the computation,the value of the tolerance for the global iterative Newton-Rapson procedure is fixed to10−4 and the maximum number of iteration inside a time step is fixed to 30.Figure 5 presents the comparison between the quasi-static solution with standard ele-ments, the dynamic solution with standard elements and the dynamic solution with mixedelements for the nodes 319 and 399 displayed in Figure 4. It is observed a difference inthe solution with mixed elements for vertical displacements and capillary pressure, whilethe solution of temperature is not affected by the element type used for the computations.6 CONCLUSIONSAmixed finite element formulation for the dynamic analysis of hydro-thermo-mechanicalbehavior of water saturated and partially saturated porous media has been presented. Themodel has been implemented in the finite element code Comes-Geo [8, 11, 12, 13, 14, 15,16, 17, 18] developed at the University of Padova, Department of Civil, Environmental andArchitectural Engineering. The comparison between the finite element solution obtainedwith standard elements and mixed elements in dynamics has been presented, showingthat the use of quadratic-linear Taylor-Hood elements for the monolithic treatment ofdynamics results less accurate with respect the case of applied equal-order approximationwhen low permeability is considered. This aspect deserves further investigations.AcknowledgmentsThe authors would like to thank the 7th Framework Programme of the European Union(ITN MuMoLaDe project 289911) for the financial support.6146T.D. Cao, Y. Wang, L. Sanavia and B.A. SchreflerFigure 5: Time-history variables at node references; a) vertical displacement at top sample, b) capillarypressure at bottom sample and c) temperature at bottom sampleREFERENCES[1] Zienkiewicz, O.C., Qu, S., Taylor, R.L. and Nakazawa, S. The patch test for mixedformulations.International Journal for Numerical Methods in Engineering (1986)23(10):1873–1883.[2] Falk, R.S. A fortin operator for two-dimensional taylor-hood elements.ESAIM:Mathematical Modelling and Numerical Analysis (2008) 42(14):411–424.doi:10.1051/m2an:2008008.[3] Ferronato, M., Gambolati, G. and Teatini, P. Ill-conditioning of finite element poroe-lasticity equations.International Journal of Solids and Structures (2001) 38(34-35):5995–6014.[4] Ehlers, W., Karajan, N., Markert, B. An extended biphasic model for charged hy-drated tissues with application to the intervertebral disc.Biomechanics and Modelingin Mechanobiology (2009) 8(3):233–251. doi:10.1007/s10237-008-0129-y.[5] Görke, U.-J., Kaiser, S., Bucher, A. and Kreibig, R. A consistentmixed finite element formulation for hydro-mechanical processes in sat-urated porous media at large strains based on a generalized materialdescription.European Journal of Mechanics - A/Solids (2012) 32(0):88–102.doi:http://dx.doi.org/10.1016/j.euromechsol.2011.09.009.7147T.D. Cao, Y. Wang, L. Sanavia and B.A. Schrefler[6] Huang, M., Yue, Z.Q., Tham, L.G. and Zienkiewicz, O.C. On the stable finite elementprocedures for dynamic problems of saturated porous media.International Journal forNumerical Methods in Engineering (2004) 61(9):1421–1450.[7] Murad, M., Thomée, V. and Loula, A. Asymptotic behavior of semidiscrete finite-element approximations of biots consolidation problem. European Journal of Me-chanics - A/Solids (1996) 33(3):1065–1083.[8] Lewis, R.W. and Schrefler, B.A. The Finite Element Method in the Static and Dy-namic Deformation and Consolidation of Porous Media, 2nd ed. John Wiley & Sons:New York, (1998).[9] Zienkiewicz, O.C., Chan, A.H., Pastor, M., Schrefler, B.A. and Shiomi, T. Computa-tional geomechanics with special reference to earthquake engineering. Wiley, (1999).[10] Hauser, J.R. Numerical methods for nonlinear engineering models. Springer, (2009).[11] Gawin, D. and Schrefler, B.A. Thermo-hydro-mechanical analysis of partially satu-rated porous materials. Engineering Computations (1996) 13: 113–143.[12] Sanavia, L., Pesavento, F. and Schrefler, B.A. Finite element analysis of non-isothermal multiphase geomaterials with application to strain localization simulation.Computational Mechanics (2006) 37: 331–348.[13] Sanavia, L., Franois, B., Bortolotto, R., Luison, L. and Laloui, L. Finite elementmodelling of thermo-elasto-plastic water saturated porous materials. Journal of The-oretical and Applied Mechanics (2008) 38:7–24.[14] Sanavia, L. Numerical modelling of a slope stability test by means of porous mediamechanics. Engineering Computations (2009) 26: 245-266.[15] Gawin, D. and Sanavia, L. A unified approach to numerical modelling of fullyand partially saturated porous materials by considering air dissolved in water.CMES: Computer Modeling in Engineering & Sciences (2009) 53: 255-302. DOI10.3970/cmes.2009.053.255.[16] Gawin, D. and Sanavia, L. Simulation of cavitation in water saturated porous mediaconsidering effects of dissolved air. Transport in Porous Media (2010) 81: 141-160.DOI 10.1007/s11242-009-9391-4.[17] Sanavia, L., Luison, L., Passarotto, M. and Laloui, L. Finite element modellingof thermo-elasto-plastic multiphase porous materials. In: Bytes and Science, G.Zavarise and D.P. Boso (Eds.), Cimne, Barcelona, Spain (2012). ISBN: 978-84-940243-2-0. Pages: 99-118.8148T.D. Cao, Y. Wang, L. Sanavia and B.A. Schrefler[18] Sanavia, L., Cao, T.D., Passarotto, M., Schrefler, B.A. A model for non-isothermalvariably saturated porous media in dynamics. In Proceedings of the 11th WorldCongress of Computational Mechanics WCCM 2014, 20-25 July 2014, Barcelona;3473–3482.[19] Hassanizadeh, M. and Gray, W.G. General conservation equations for multi-phasesystem: 1. Averaging technique. Advances in Water Resources (1979) 2:131-144.[20] Hassanizadeh, M. and Gray, W.G. General conservation equations for multi-phasesystem: 2. mass, momenta, energy and entropy equations. Advances in Water Re-sources (1979) 2: 191-201.[21] Markert, B. Weak or strong on coupled problems in continuum mechanics. Habilita-tion thesis, Universitt Stuttgart, Germany, (2010).[22] de Boer, R., Ehlers, W. and Liu, Z. One-dimensional transient wave propagation influid-saturated incompressible porous media. Archive of Applied Mechanics (1993);63(1):59–72.[23] Aboustit, B.L., Advani, S.H., Lee J.K. Variational principles and finite element sim-ulations for thermo-elastic consolidation. International Journal for Numerical andAnalytical Methods in Geomechanics 1985; 9(1):49-69.9

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Cao Toan Duc

Wang Y.

Sanavia L.

Schrefler B. A.

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CAN3: BASELINE COST OF ILLNESS OF PARKINSON'S DISEASE IN A LARGE HEALTHCARE PLAN

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