Bruggeman, Peter

Degroote, Joris

Vierendeels, Jan

English

Ghent University Academic Bibliography

Stability analysis of the coupling iterations influid-structure interaction simulationsJoris Degroote , Peter Bruggeman, Jan VierendeelsSupervisor(s): Jan VierendeelsAbstract—Fluid-structure interaction problems can be solved with sepa-rate solvers for the fluid and the structure. Equilibrium of the forces on thecommon boundary of the fluid and the structure is obtained by performingcoupling iterations between the two problems in every time step. This arti-cle gives insight into the parameters that influence the number of couplingiterations by means of Fourier analysis on a one-dimensional model of anelastic artery.Keywords— fluid-structure interaction, stability, coupling iterations,reduced-order model, arteryI. INTRODUCTIONTHE simulation of fluid-structure interaction and other cou-pled problems has gained in interest over the last decade.This due to the increase in computer power and the maturityof the codes to simulate fluid or structure problems separately.Interesting applications are parachute dynamics [1], biomedicalapplications [2] and flutter analysis [3], [4].Two main approaches exist to solve a coupled problem.• Monolithic approach: a new code that solves both problemssimultaneously is developed.• Partitioned approach: existing codes for both problems arecoupled by performing iterations in every time step to bring theforces of both problems on their common boundary into equi-librium.The former approach requires no coupling iterations everytime step but the development and maintenance of such a codeis very expensive. A new code has to be created for every com-bination of physical problems. The equations of two differentphysical problems can have diverse mathematical properties soit can be hard to solve them simultaneously.The partitioned approach reuses existing codes in variouscombinations, but both problems have to be solved multipletimes during the coupling iterations in a time step. However,a code for a single physical problem can be more optimized andthe partitioned approach makes it easier to use a different math-ematical formulation for each problem, e.g. finite volumes forthe fluid and finite elements for the structure.This article focuses on the partitioned approach to fluid-structure interaction for which several coupling algorithms exist.These algorithms can be classified according to their need to ac-cess the source code of the solvers. In this article, only Aitkenunderrelaxation and coupling with reduced-order models will bediscussed further. These algorithms treat the codes for the fluidand the structure as black boxes and thus do not require accessto the source code, enabling the use of commercial codes thatare common in industry.J. Degroote is with the Department of Flow, Heat and Combustion Mechanics,Ghent University (UGent), Gent, Belgium. E-mail: Joris.Degroote@UGent.be .However, the coupling iterations in a partitioned calculationare not automatically stable. The number of iterations must alsobe as low as possible to minimize the total calculation time.Therefore the parameters that influence the stability and thenumber of coupling iterations are investigated with Fourier anal-ysis on a linearized model of a one-dimensional elastic artery.The results of this analysis are verified with non-linear simula-tions, which indicate the shortcomings of the current couplingtechniques but also how they can be improved.II. PROBLEM DESCRIPTIONA. Fluid problemThe unsteady blood flow in an artery is analyzed with a one-dimensional model. The blood is represented by an incompress-ible and inviscid fluid and gravity is neglected. The governingequations are the conservation of mass and momentum whichcan be written in conservative form as∂a∂t+∂au∂x= 0 (1a)∂au∂t+∂au2∂x+1ρ(∂ap∗∂x− p∗ ∂a∂x)= 0 (1b)with a the cross sectional area of the artery, u the velocity alongthe axis of the artery and ∂∂t the time derivative. x is the spatialcoordinate, ρ is the density of the blood, p∗ the pressure andp = p∗/ρ the kinematic pressure.The Navier-Stokes equations are discretized on a one-dimensional equidistant mesh with N cells and mesh size ∆x.The time discretization scheme is implicit Euler and the timestep is indicated with ∆t. Solution of the fluid problem can besummarized as Y = F (X), with X a list with the coordinatesof the points on the fluid-structure interface and Y a list with thecomponents of the fluid load (pressure and viscous force) on theinterface.B. Structure problemThe behaviour of the elastic artery wall is described by aHookean constitutive relation without mass, as the inertia of theartery wall is neglected with regards to that of the fluid. An ax-isymmetric model is used in coordinate system (x,r,φ), with rthe inner radius of the artery and φ the angle in the cross sec-tional plane. σφφ is approximated asσφφ = Er − r0r0+ σ0 (2)with E Young’s modulus and r0 the radius where σ = σ0. Thismodel allows only radial motion of the nodes.Under the assumption that only pressure forces act on theboundary, the force balance reads p∗r = σφφh with h the thick-ness of the artery wall. Equation (2) and the definition of thekinematic pressure are substituted in the force balance, whichcan be rewritten asa = a0(p0 − 2c2mkp− 2c2mk)2(3)by using a = pir2 and by introducing the Moens-Korteweg wavespeed, c2mk =Eh2ρr0. Similar to the fluid problem, X = S(Y )represents the solution of the structure problem.III. COUPLING TECHNIQUESA. Coupling iterationsCoupling iterations between the fluid and structure in a timestep can be performed by the following scheme where a sub-script k indicates the coupling iteration.1. Calculate the fluid load on the interface by solving the fluidproblem, Yk = F (Xk).2. Calculate the position of the interface by solving the structureproblem, Xk+1 = S(Yk)3. Increase k and go to 1 if the L1-norm ofXk−Xk+1 is biggerthan the convergence criterium.B. Aitken underrelaxationAitken’s method [5] applies a variable underrelaxation factoron the difference between the current interface position (Xk+1)and the previous one (Xk) after step 2. This factor is calculatedfrom the differences in the position of the interface during thelast two coupling iterations.C. Reduced-order modelsAfter the solution of the fluid or structure problem, an inputand output of that code is known. With this new input/outputpair and all the ones from the preceding iterations, a linearreduced-order model of the solver is created which is an ap-proximation for the real solver. The coupled solution of thosesimplified models is calculated and given to the other solver [2].This procedure is iterated until convergence is obtained.IV. FOURIER STABILITY ANALYSISBy linearizing the discretized fluid and structure equations,one can perform a Fourier stability analysis of the error betweenthe interface’s current position and load and the exact solution ina coupling iteration. The amplification factor of the error mustbe smaller than one for all spatial frequencies for an iterationscheme to be stable. The influence of the structure’s stiffnessand the time step on the stability is shown in Figure 1 for cou-pling iterations without underrelaxation or reduced-order mod-els. Perturbations with a low spatial frequency cause instabilityand a decreasing stiffness or decreasing time step further desta-bilizes the coupling iterations. A smaller time step or lowerstructural stiffness thus increase the number of coupling itera-tions. The difficulties arising from the decrease in time step ishighly unwanted, but this result gives a theoretical basis for thefindings in previous work [2]. None of the simulations shown   0120 π/2 π  E=3E6N/m2E=12E6N/m2E=75E6N/m2Decreasingstiffness(a)   0120 π/2 π  Δ t=0.001sΔ t=0.025sΔ t=0.625sDecreasingtime step(b)Fig. 1. Error amplification from Fourier analysis (a) for different values of thestructural stiffness and (b) for different values of the time step.in Figure 1 is stable without underrelaxation as the error am-plification is bigger than one for the lower spatial frequencies.V. NON-LINEAR RESULTSThe configurations studied with the Fourier analysis havebeen simulated with the non-linear fluid and structure equationsusing Aitken underrelaxation and coupling with reduced-ordermodels. Both techniques require more coupling iterations if thetime step or the structural stiffness decreases, which extends theconclusions from the Fourier analysis to non-linear problems.Coupling with reduced-order models outperforms Aitken under-relaxation in every configuration.VI. CONCLUSIONSThe partitioned approach to fluid-structure interaction enablesthe reuse of existing codes but requires a coupling algorithm toobtain force equilibrium on the fluid-structure interface. If themass of the structure is small with respect to the mass of thefluid then more coupling iterations are required when the timestep or structural stiffness decreases. New coupling techniqueswithout this unwanted behaviour should be developed by usingthe information of the previous time steps.ACKNOWLEDGMENTSThe author gratefully acknowledges his funding by a Ph.D.fellowship of the Research Foundation - Flanders (FWO).REFERENCES[1] K.R. Stein, T.E. Tezduyar, V. Kumar, S.V. Sathe, R.J. Benney, and R.D.Charles, “Numerical simulation of soft landing for clusters of cargoparachutes,” in European congres on computational methods in applied sci-ences and engineering ECCOMAS 2004, P. Niettaanmki, T. Rossi, K. Ma-java, and O. Pieronneau, Eds., Jyvskyl, July 2004.[2] J. Vierendeels, L. Lanoye, J. Degroote, and P. Verdonck, “Implicit cou-pling of partitioned fluidstructure interaction problems with reduced ordermodels,” Computers and Structures, vol. 85, pp. 970–976, 2007.[3] E.H. Dowell, J.P. Thomas, and K.C. Hall, “Transonic limit cycle oscillationanalysis using reduced order aerodynamic models,” in 42nd AIAA ASMEASCE AHS ASC Structures, Structural Dynamics and Materials Conferenceand Exhibit, Seattle, WA, April 2001, AIAA ASME ASCE AHS ASC.[4] K. Willcox, J.D. Paduano, and J. Peraire, “Low order aerodynamic modelsfor aeroelastic control of turbomachines,” in 40th AIAA ASME ASCE AHSASC Structures, Structural Dynamics and Materials Conference, St Louis,MO, april 1999, AIAA ASME ASCE AHS ASC.[5] D.P. Mok, W.A. Wall, and E. Ramm, “Accelerated iterative substructur-ing schemes for instationary fluid-structure interaction,” in ComputationalFluid and Solid Mechanics, K.J. Bathe, Ed. 2001, pp. 1325–1328, Elsevier.

Stability analysis of the coupling iterations in fluid-structure interaction simulations

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Stability analysis of the coupling iterations in fluid-structure interaction simulations

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