International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP) (12th : 2015)

Sensitivity analysis of fluid-structure interaction using the PFEM Zhu, Minjie; Scott, Michael H. Jul 31, 2015

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Sensitivity Analysis of Fluid-Structure Interaction using the PFEMMinjie ZhuPost-Doctoral Researcher, School of Civil and Construction Engineering, Oregon StateUniv., Corvallis, OR, USAMichael H. ScottAssociate Professor, School of Civil and Construction Engineering, Oregon State Univ.,Corvallis, OR, USAABSTRACT: Sensitivity analysis of fluid-structure interaction (FSI) simulations provides an importanttool for assessing the reliability and performance of coastal infrastructure subjected to storm and tsunamihazards. As a preliminary step for gradient-based applications in reliability, optimization,system iden-tification, and performance-based engineering of coastal infrastructure, the direct differentiation method(DDM) is applied to FSI simulations using the particle finite element method (PFEM) to compute sensi-tivities of simulated FSI response with respect to uncertain parameters of the structural and fluid domainsthat are solved in a monolithic system via the PFEM. Due to geometric nonlinearity of the free-surfaceflow, geometric sensitivity of the fluid is considered in the governing equations of the DDM along withsensitivity of material nonlinear response in the structural domain. An example application shows sen-sitivity to load and resistance variables of a reinforced concrete frame subjected to tsunami loading withopen and closed first story design configurations.1. INTRODUCTIONWave loads induced by tsunami and storm surgeevents can cause significant damage to criticalcoastal infrastructure as observed in recent naturaldisasters such as the 2011 Great East Japan earth-quake and tsunami and the Superstorm Sandy hurri-cane of 2012. The modeling of wave loads as staticforces on a deformable body, or conversely, as hy-drodynamic forces on a rigid body, may not provideaccurate predictions of structural response. To ob-tain accurate response for structural displacementsand forces, fluid-structure interaction must be con-sidered accounting for the kinematics and deforma-tion of both the structural and fluid domains. It isalso imperative to assess the sensitivity of structuralresponse to stochastic wave loading and uncertainstructural properties. The sensitivity has importantimplications for the design of coastal infrastructureand in assessing the probability of failure of build-ings and bridges in tsunami and storm events as partof an over-arching performance-based engineeringframework.Fluid-structure interaction with incompressibleNewtonian fluid is one of the most challengingproblems in computational fluid mechanics be-cause the incompressibility condition leads to nu-merical instability of the computed solution. Alarge number of Finite Element Methods (FEM)have been developed for the computation of incom-pressible Navier-Stokes equations using the Eule-rian, Lagrangian or Arbitrary Lagrangian-Eulerian(ALE) formulations Girault and Raviart (1986);Gunzburger (1989); Baiges and Codina (2010);Radovitzky and Ortiz (1998); Tezduyar et al.(1992). The particle finite element method (PFEM)Oñate et al. (2004), has been shown to be an ef-fective Lagrangian approach to FSI because it usesthe same Lagrangian formulation as structures. Amonolithic system of equations is created for the si-multaneous solution of the response in the fluid andstructural domains via the fractional step method(FSM). This alleviates the need to couple disparatecomputational fluid and structural modules in orderto simulate FSI response, which can lead to uncon-112th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015servative estimates of structural response.While the PFEM solution of FSI simulations viaa monolithic system has computational advantagesin determining the fluid and structure response, thesensitivity of this response to uncertain modelingparameters is just as, if not more, important than theresponse itself. As a standalone product, sensitivityanalysis shows the effect of modeling assumptionsand uncertain properties on system response, butit also an important component to gradient-basedapplications. There are two methods for calculat-ing the sensitivity of a simulated response. Thefinite difference method (FDM) repeats the simu-lation with a perturbed value for each parameterand does not require additional implementation asperturbations and differencing can be handled withpre- and post-processing. The accuracy of the re-sulting finite difference approximation depends onthe size of the perturbation where the results arenot accurate for large perturbations and are prone tonumerical round-off error for very small perturba-tions. Due to the need for repeated simulations, theFDM approach can become inefficient when thereis a large number of parameters.A more accurate approach is the direct differ-entiation method (DDM), where derivatives of thegoverning equations are implemented alongside theequations that govern the simulated response. Atthe one-time expense of derivation and implemen-tation, the DDM calculates the response sensitiv-ity efficiently for each parameter as the simulationproceeds rather than by repeating the analysis. TheDDM is generally more accurate than the FDM be-cause the sensitivity is computed using the same nu-merical algorithm as the response, making it subjectto only numerical precision rather than round off.Analytical approaches to sensitivity analysis basedon DDM of structural response have been well de-veloped in the literature Kleiber et al. (1997); Scottet al. (2004); Scott and Haukaas (2008), while sen-sitivity analysis of fluid-structure interaction (FSI)has not been addressed. This is partly due to thecomplexity of the computation for the FSI responseand the cumbersome nature of staggered computa-tional approaches.2. PFEM RESPONSE COMPUTATIONSThis section provides a brief review of the equa-tions that govern FSI response using the PFEM. Af-ter applying finite element techniques, discrete al-gebraic equations are formed from MINI elementsin the fluid domain and arbitrary line and solid ele-ments in the structural domain. The algebraic equa-tions will be differentiated in the following sectionfor sensitivity analysis via the DDM.In the combined equations of fluid and structures,particles connected to both domains are identifiedas interface particles, whose contributions appear inboth fluid and structural equations. Subscripts i, sand f are given to indicate interface, structural andfluid variables and equationsMssv˙s +Msiv˙i +Cssvs +Csivi (1)+Fints (us,ui) = FsMisv˙s +(Msii +M fii)v˙i +Cisvs +Ciivi (2)+Finti (us,ui)−Gip = Fsi +F fiM f f v˙ f −G fp = F f (3)GTf v f +GTi vi +Sp = Fp (4)Numerical time integration, such as the backwardEuler method, and nonlinear solution algorithm,such as fixed-point iteration, can be applied to thecombined equations in order to obtain a monolithicsystem of equations, which is solved by the frac-tional step method (FSM).3. DIRECT DIFFERENTIATION OF THE PFEMFor structural sensitivity analysis, even for non-linear materials, the configuration is often fixed.However, for large displacement applications suchas FSI, additional sensitivity terms that arise fromupdating the configuration at each iteration mustbe taken in to account in the derivation of sensi-tivity equations. The direct differentiation method(DDM) is used here to compute the sensitivity ofPFEM analysis with FSM. As in Kleiber et al.(1997), the DDM is applied on the combined FSIEqs. (1) to (4) to develop the combined FSI sensi-tivity equations for fluid and structure.By taking the derivative of the combined FSIEqs. (1) to (4) with respect to an uncertain param-212th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015eter θ , the combined sensitivity equations are ob-tained with assigned subscripts i, s and fMss∂ v˙s∂θ +Msi∂ v˙i∂θ +Css∂vs∂θ +Csi∂vi∂θ+Kss∂us∂θ +Ksi∂ui∂θ =∂Fs∂θ −∂Mss∂θ v˙s−∂Msi∂θ v˙i−∂Css∂θ vs−∂Csi∂θ vi−∂Fints∂θ(5)Mis∂ v˙s∂θ +(Msii +M fii)∂ v˙i∂θ +Cis∂vs∂θ+Cii∂vi∂θ +Kis∂us∂θ +(Kii +Hii)∂ui∂θ+Hi f∂u f∂θ −Gi∂p∂θ =∂Fsi∂θ +∂F fi∂θ−∂Mis∂θ v˙s−(∂Msii∂θ +∂M fii∂θ)v˙i−∂Cis∂θ vs−∂Cii∂θ vi−∂Finti∂θ +∂Gi∂θ p(6)M f f∂ v˙ f∂θ −G f∂p∂θ +H f f∂u f∂θ +H f i∂ui∂θ= ∂F f∂θ −∂M f f∂θ v˙ f +∂G f∂θ p(7)GTf∂v f∂θ +GTi∂vi∂θ +S∂p∂θ +T f∂u f∂θ +Ti∂ui∂θ= ∂Fp∂θ −∂GTf∂θ v f −∂GTi∂θ vi−∂S∂θ p(8)where, the matrices H and T account for geometricnonlinear of the fluid response and its influence onthe fluid response sensitivity.4. EXAMPLESThis example is of a tsunami bore impact-ing a three story reinforced concrete building.The structural model shown in Fig. 1 was de-veloped by Madurapperuma and Wijeyewickrema(2012) for the analysis of water-borne debris andwas further analyzed by Zhu and Scott (2014)to demonstrate fluid-structure interaction using thePFEM. To capture material and geometric non-linearity, each frame member is discretized in toten displacement-based beam-column finite ele-ments (dispBeamColumn in OpenSees) with fiber-discretized cross-sections at the element integrationpoints and the corotational geometric transforma-tion Crisfield (1991). As discussed above, the ge-ometric nonlinear sensitivity of the fluid is con-sidered through geometric tangent stiffness matri-ces. DDM sensitivity for the frame elements is de-scribed in Scott et al. (2004) while that for the coro-tational transformation is provided in Scott and Fil-ippou (2007).The cross-section dimensions, reinforcing de-tails, and concrete properties of the frame areshown in Fig. 2. Light transverse reinforcementprovides residual concrete compressive strength inthe core regions of the members. Zero tensilestrength is assumed for the concrete (Concrete01in OpenSees) and the longitudinal reinforcing steelis assumed bilinear with elastic modulus 200 GPa,yield strength 420 MPa, and 1% kinematic strainhardening (Steel01). Gravity loads and nodal masswere calculated assuming uniform pressure of 4.8kPa on floor slabs and 1.0 kPa on the roof with trib-utary width of 6 m.A refined mesh of beam-column elements is usedfor the first floor column members in order to rep-resent a design configuration with a closed firstfloor that resists hydrodynamic loading. A com-monly proposed tsunami mitigation strategy is todesign buildings with an open first floor config-uration that allows fluid to pass through withoutdeveloping significant impact and drag forces onthe structure. This is accomplished by using onlyone beam-column element for the first floor columnmembers, i.e., a coarse mesh that will not develop afluid-structure interface.The tsunami bore has height 4.5 m, initial veloc-ity 2 m/s, and out-of-plane thickness 6 m. The sim-ulation begins at impending impact on the frame.Snap shots of the simulation are shown in Fig. 3for the case of closed first story with a refined meshof first floor beam-column elements. The simula-tion is repeated and shown in Fig. 4 using a singleelement coarse mesh for an open first story, whichallows the fluid passing through the structure.The roof displacements for the Fig. 3 and Fig. 4are compared in Fig. 5, where the closed floorhas much larger displacement than the open floor.Whereas, in Fig. 6, the axial forces of right col-312th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 20156 kN/m28.8 kN/m28.8 kN/m6 m 6 m4 m3.6 m3.6 mFigure 1: Geometry and floor loads of reinforced con-crete frame example.600 mm300 mm550 mm 600 mmσεcover0.002 0.005 0.054 No. 6 Bars12 No. 8 Bars core27.6 MPa28.5 MPa5.7 MPaBeam Cross Section Column Cross Section Concrete Stress-StrainFigure 2: Beam and column cross-sections of rein-forced concrete frame.Time = 0.0 s Time = 1.0 sTime = 2.0 s Time = 5.0 sFigure 3: Snap shots of the tsunami waves runup oncoastal structure.Time = 0.0 s Time = 0.5 sTime = 1.0 s Time = 5.0 sFigure 4: Snap shots of the tsunami waves passingthrough coastal structure.0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50100200Time (sec)Displacement(mm) Closed floor Open floorFigure 5: Roof displacements for two cases.0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−400−2000200Time (sec)Axialforce(kN)Closed floor Open floorFigure 6: Axial forces at the base of right column fortwo cases.0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−1,0000Time (sec)Bendingmoment(kN·m)Closed floor Open floorFigure 7: Bending moments at the base of right columnfor two cases.412th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015umn don’t shown much difference between the twocases. This is due to the fact that axial forces ofcolumns are primarily determined by vertical loadsrather than the lateral wave loading. The bendingmoments in Fig. 7, which are greatly influenced bythe wave loading, show the difference between twocases.0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−40−20020Time (sec)(∂U/∂fc c)fc c(mm)Closed floor Open floorFigure 8: Disp. sens. to f cc for two cases.0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−40−200Time (sec)(∂U/∂fc c)fc c(mm)DDM FDMFigure 9: Disp. sens. to f cc for closed floor using DDMand FDM.0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−20020Time (sec)(∂U/∂fc c)fc c(mm) DDM FDMFigure 10: Disp. sens. to f cc for open floor using DDMand FDM.The floor displacement sensitivity to uncer-tain parameters, the column concrete compressivestrength f cc and the fluid density ρ f are shown inFig. 8 to Fig. 13. The sensitivity to f cc are similarfor two cases since the similar structures are used.But the sensitivity to ρ f are very different since thewaves are running in different directions.The bending moment sensitivity to uncertain pa-rameters, the column concrete compressive strength0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−2000200400Time (sec)(∂U/∂ρf)ρf(mm)Closed floor Open floorFigure 11: Disp. sens. to ρ f for two cases.0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50200400Time (sec)(∂U/∂ρf)ρf(mm)DDM FDMFigure 12: Disp. sens. to ρ f for closed floor usingDDM and FDM.0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−100−50050100Time (sec)(∂U/∂ρf)ρf(mm)DDM FDMFigure 13: Disp. sens. to ρ f for open floor using DDMand FDM.0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−4,000−2,00002,000Time (sec)(∂M/∂fc c)fc c(kN·m)Closed floor Open floorFigure 14: Bending moment sens. to f cc for two cases.0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−4,000−2,00002,000Time (sec)(∂M/∂fc c)fc c(kN·m)DDM FDMFigure 15: Bending moment sens. to f cc for closed floorusing DDM and FDM.512th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 20150 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−400−2000200Time (sec)(∂M/∂fc c)fc c(kN·m)DDM FDMFigure 16: Bending moment sens. to f cc for open floorusing DDM and FDM.0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 501·104Time (sec)(∂M/∂ρf)ρf(kN·m)Closed floor Open floorFigure 17: Bending moment sens. to ρ f for two cases.0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 501·104Time (sec)(∂M/∂ρf)ρf(kN·m) DDM FDMFigure 18: Bending moment sens. to ρ f for closed floorusing DDM and FDM.0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−1,00001,000Time (sec)(∂M/∂ρf)ρf(kN·m)DDM FDMFigure 19: Bending moment sens. to ρ f for open floorusing DDM and FDM.f cc and the fluid density ρ f are shown in Fig. 14 toFig. 19.For all sensitivities, the DDM results are match-ing to FDM results with a small perturbation be-tween 10−8 and 10−10.5. CONCLUSIONSThe PFEM is an effective approach to simulat-ing FSI because it uses a Lagrangian formulationfor the fluid domain, which is the same formula-tion typically employed for finite element analy-sis of structures. Development of DDM sensitivityequations for the PFEM broaden its application togradient-based applications in structural reliability,optimization, and system identification. Due to ge-ometric nonlinearity of the fluid domain, additionalterms were required to derive and implement theDDM equations for the PFEM. Following the sameanalysis procedure as for the response itself, thesensitivity equations are solved using the fractionalstep method (FSM). Tsunami loading on a rein-forced concrete frame verifies the DDM implemen-tation for PFEM with sensitivity results by match-ing finite difference solutions with small parame-ter perturbations. Two cases with first floor openand closed are compared in structural responses andsensitivities to various uncertain parameters. Fu-ture applications of DDM sensitivity for the PFEMinclude time variable reliability analysis of fluid-structure interaction, which is an important con-sideration for multi-hazard analysis involving windloading concurrent with storm surge and tsunamifollowing an earthquake.6. ACKNOWLEDGMENTSThis material is based on work supported bythe National Science Foundation under Grant No.0847055. Any opinions, findings, and conclusionsor recommendations expressed in this material arethose of the authors and do not necessarily reflectthe views of the National Science Foundation.7. REFERENCESBaiges, J. and Codina, R. (2010). “The fixed-meshale approach applied to solid mechanics and fluid-structure interaction problems.” International Journalfor Numerical Methods in Engineering, 81, 1529–1557.612th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Crisfield, M. A. (1991). Non-linear Finite Element Anal-ysis of Solids and Structures, Vol. 1. John Wiley &Sons.Girault, V. and Raviart, P. (1986). Finite Element Meth-ods for Navier-Stokes Equations. Springer-Verlag.Gunzburger, M. (1989). Finite Element Methods for Vis-cous Incompressible Flows. Academic Press.Kleiber, M., Antunez, H., Hien, T., and Kowalczyk, P.(1997). Parameter Sensitivity in Nonlinear Mechan-ics. John Wiley & Sons.Madurapperuma, M. A. K. M. and Wijeyewickrema,A. C. (2012). “Inelastic dynamic analysis of an RCbuilding impacted by a tsunami water-borne shippingcontainer.” Journal of Earthquake and Tsunami, 6(1),1250001 1–17.Oñate, E., Idelsohn, S., Pin, F. D., and Aubry, R. (2004).“The particle finite element method. an overview.” In-ternational Journal of Computational Methods, 1(2),267–307.Radovitzky, R. and Ortiz, M. (1998). “Lagrangian finiteelement analysis of newtonian fluid flows.” Interna-tional Journal for Numerical Methods in Engineer-ing, 43, 607–619.Scott, M. H. and Filippou, F. C. (2007). “Exact re-sponse gradients for large displacement nonlinearbeam-column elements.” Journal of Structural Engi-neering, 133(2), 155–165.Scott, M. H., Franchin, P., Fenves, G. L., and Filip-pou, F. C. (2004). “Response sensitivity for nonlinearbeam-column elements.” Journal of Structural Engi-neering, 130(9), 1281–1288.Scott, M. H. and Haukaas, T. (2008). “Software frame-work for parameter updating and finite-element re-sponse sensitivity analysis.” Journal of Computing inCivil Engineering, 22, 281–291.Tezduyar, T., Mittal, S., Ray, S., and Shih, R. (1992).“Incompressible flow computations with stabilizedbilinear and linear equal-order-interpolation velocity-pressure elements.” Computer Methods in AppliedMechanics and Engineering, 95, 221–242.Zhu, M. and Scott, M. H. (2014). “Modeling fluid-structure interaction by the Particle Finite ElementMethod in OpenSees.” Computers & Structures, 132,12–21.7

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