Numerical simulation of a non linear coupled fluid-structure problem by explicit finite element-finite volume coupling

The present paper deals with the numerical s i m ul a ti o n of a coupled non linear fl u i d - s tru ct u re p rob l e m by e x p l i c i t coupling between a fi n it e element structure code and a f i n i t e volume fluid code. This numerical study is carried out i n order to develop robust and general coupling with FE and CFD commercial code for industrial app lic a t i o ns . A g eo m e tr i ca l l y simple non linear coupled problem is presented in order to validate the numerical approach. The structure non linear problem is solved with a finite element t ec hn i q u e , using a ite r at iv e implicit algorithm for time integration. The fluid problem i s solved using s ta n d ard numerical techniques (finite volume approach, implicit splitting operator scheme). The whole coupled problem is solved with a commercial CFD code: a d e di cat e d FE s tru c ture code is developed in the CFD code together with coupling (in time, in space) procedures. The proposed method is validated in the case of a i n c ompre ss i b l e inviscid fluid. for which t h e c oupl e d problem is solved with an analytical solution. The present study gives a reference test case f o r a full scale fluid-structure model. I n d u s tr ial applications can now be considered by c o u p ling commercial 1-'E and FV codes w i th g e n e r a l c ou pl i n g c o d e . c m t a i For coupled l n n fl a


INTRODUCTION
The present paper deals with the numerical simulation of coupled Ouid-structure problem, in which the coupling process is bases on mechanical exchanges between each sub-problem. ln the present study other fluid-structure coupling phenomena, such as   Nowadays, the numerical approach tends to propose general coupling algorithms with specific solvers for each sub-problems, with various coupling strategies [5] [8] [ 17) [20], depending on the phys ical coup ling phenomenon.
As the numerical design naval propulsion structures needs t ak i ng Ouidl:;tru�;tun:: irrtcm�;tiorr irrto a�;cuurrt, DCN Propul:;iurr launched, with Ecole Centrale Nantes GeM, a R&D study in order to apply numerical methods on industrial projects. From the academic point of view, the coupling process seems not to pose any particular difficulties. From the industrial point of view, an investigation of the vari ous aspecrs of the numerical fluid-strucrure coupling on a simple case is necessary before considering a full-scale coupling with existing commercial codes for industria l structure design.
The fluid-structure coupled problem studied in the present paper is described by Fig. 2'. The numerical resolution of the coupled 1 //is a11 ex/elision in the 3D case of the 11011linear 2D problem stud i ed in a previous publ i cation [21]. Coupling Subrouti ne C•lculabon of luld torcea and adjustm""t to Jhe structure "''"'" The structure problem is described by the following beam non linear equation of motion. taking into account geometrical non linearity [12] [26]. The coupled traction/bending equation of motion are formulated in the relative frame as:
: ! for the beam free end. {4) The numerical resolution of the structure problem is performed using the finite element method [1 ].   Using a GALERKIN method (i.e. writing an idc:.ntical approximation for unknown displacement 11 • v and virtual displacement with the: same shape functions). the: variationna l principle gi\·c:n by Eqs. (5) and (6) is written in the following discrete form: shape functio ns defined by Eqs. (8) and (9). these: matrices can be: analytically calculated. The: non linear tc:mlS arc:: In explicit techniques (7]. a three: points numerical scheme: is used to evaluate: the: acceleration at time: f 11• For instance:. the: centc:red finite: differc:nce scheme gives the: approximation: Using the: expression ofEq. (14) in the general equa ti on of motion leads to: (15) ID Eq. (15). all tc:nm in the right l>idc: arc: known. allowi ng a straightforward calculation of X n + 1 . The explicit technique can be: coupled '-Yith the finite volume approach of the fluid problc:nt in a staggered resolution of the coupled problem. Titough the implementation of the: coupling is rather c:asygoing in the: explicit approach. it suffers from stability limitations. This leads to the: use: of an implicit technique.
In implicit technique: (23]. a two points numerical scheme is used: the �EWMARK scheme gives for c:xantple: The: substitution of Eq. (16) in the: e quati on of motion leads to an implicit equation in terms of the: unknown di splacement X n + 1 . An iterat i ve: procedure: based on fixed point algorithm is performed to obtain X n+l. The algorithm starts with a linear calculation (with the linear part of K(X) ). which gives an estimation of the slmcture displacement X .. 1 ° . Iterations arc: performed with the following approximation: which gives: when a convergence criterion is satisfied: a simple criterion can be: written as: where c << 1 . This approach is equivalent to a NE\\ 'TO� procedure. and converges. under known assump tions. . The non linear scheme requires the calculation of the tangent stiffness matrix cK . In the ax present case. the matrix is expressed as: with: and: The non-linear terms of these matrices are computed using a GAUSS-LEGENDRE procedure (1]. that is using a numerical integration scheme based on the follow i ng approximation: (24) wi th (T Gauss integrati on point and Wf the associated integration weight. The numerical scheme (24) using L points is exact for the polynomial functions of degree 2L -1 : the numerical integration procedure used in the present case will be based on a 5 points scheme. Furthermore. the iterative scheme given by Eq. (18) needs with a LU fa�:torization.
The above finite c:lement procedure is implemented as FORTRA.!'\ subroutines. which will be integrated as a simple structural code in the commercial CFD code used for the numerical resolution of the fluid problem. Coupling procedures presented in §.3 w i ll also be implemented as FORTRAN subroutine of the CFD code.

FLUID PROBLEM
The fluid problem is described by the general XA\'lER-STOKES equation. The conservation equation are integrated over a moving control volume O(r) of boundary Bn(r). using the LIEBNITZ rule and the GAUSS theorem. This leads to the global conservation equation over an arbitrary control volume (25).in ALE formulation ( 19): In Eqs. (26) and (26). S P and Sv, stand respectively for mass and momentun1 sources: the fluid problem can then be formulated in the moving frame. The fluid unknown are the pressure and velocity fields p. v; . bu t as the conservation equation are integrated over a moving control volume. Eqs. (25) and (26) show another unknown fluid that is tl1e grid velocity v, • : another equation has to be us ed to close the fluid problem. This supplementary equation is the space law conservation [ 6] which is written: The fluid problen1 is then fully charact erized by Eqs. (25) to (27t Additional boundary conditions are to be taken into account   (Fig. 5 gives a typical representation of a computational grid for a 2D cartesian problem). for which a i ntegrated conservation equation is wrirten. A moving mesh teclu:tique is used. tl1e elementary grid velocity is deduced from known node displacement in order to satisfy Eq. (27) 4 In the case of a compress ible fluid, the fluid statll law p(p) allows a closure of thll problem. [6]. The grid velocity " i • appearing in the right side of Eq.   The iterative scheme is stopped \'.'hen a convergence criterion is satisfied or when a maximum number of inner iterations is reached.
The fluid problem is solved with the numerical principles expose before. using the commerc ial CFD code Star-CD [28). An elementary validation of the moving mesh technique and fluid force calculation is performed in the elementary case of an oscillating cylinder in a confined annular space . The fRITZ model [10) gives the added mass coefficient for small amplitude motions in the case of a perfec t fluid. Figure 6 gives a comparison of the analytical and computed added mass coefficient for several confinement ratios a 5.  which "-ill be used for the present study: the structural code w i ll be implemented in the STAR-CD code as FOTRA."J subroutines, together with coupling procedures.
The numerical solving strategy of the STAR-CD code does not allow numerical exchanges within a tinte iteration with PISO 5 -� numerical srudy currmu�v in progress focuses on the calculation of addtzd mass and added damping coeffi cient in the case of a l-iscotl$ fluid. PreliminaT)' results show a good agrell1 1l llll t l•dth the anal y tical model from CHEN and compari son ll-ith other numm-ical results [3] will be carritzd out. algorithm: coupling with a structure code will thus be possible only with an explicit technique as detailed bellow.

Coupling in time
As mentioned before. the time coupl i ng strategy will be based on a staggered. explicit coupling procedure. which is represented by the PISO algorithm (step [2]). Fluid forces on the flui<Lstru.cture 'Pn interface are then deduced and transferred on the structure problem (step [3]). The advance in time for the structural problem is then carried out. ,,;th inn er iterations for the implicit scheme (step (4]).
[21 This can induce numerical instabilities for the structure part. This problem is addressed by introducing a filtering process within the coupled procedure6. A first order filter is used. It is described by its transfer function: 6 From a numerical point ofl1ew, the «ristence of numerical oscillations in the computed fluid forces can also induce numerical struc/J/Te instabilities, enm in implicit coupling t(lChniques. Many authors use in this case a blending factor approach. The computed fluid forces are correcred u1th prlll 1ous where p is the L.u>L-\CE operator. and w. = 21T I r is the filtering pulsation. Using a bilinear transform of the LAPL�CE variable p . that 2 .:-1 .
is p = -· --. and usmg the z-transfonn [24]. one deduces the lil .:+1 follow i ng discrete relation: where rp and rp are the non-filtered and filtered discrete values. ol is the time step size-. In the 2D crue, tlus numencal rreatment pro1 ed to be effi cleJlt and precise, m · en when fluid forces ore dominam orer srruc1111·e inn er s ti ffness forces [ll].Tile major drawback of this rechnique lies on its emp irical approa ch Sll\'eral calculation u1th ,·arious filtering frequencies hO\·e to be performed in order to demonsn·are that numerical re.sulrs are not affeaed (11p to a cutain point) �·rh is numerical approach.
number of fluid finite volume are required to solve the fluid problem.
with local refinement to describe boundary layer). the space coupling procedure uses interpolation techniques in the displacement/force e�change� bem·een the nvo sub-problem� [2].
Since the fluid problem is re-meshed at each time step to take into account structure deformation. the fluid finite volume can undergo great shape deformation. Many re-meshing coupling techniques can be develop to preserve the fluid mesh quality [22]. As the fluid geometry is rather simple in the present case. it is possible to develop a re-meshing procedure that produces little deformati on of the fluid cells. based on a purely geometrical approach of the problem.
The fluid problem has an analytical solution that can be written in terms of pressure. from which fluid force on the elastic beam is deduced an \Yritten as: 8 For \'€1)' larg« duplaum#:tlt, 1.e. whm the inner C)lmd«r almost comf!S mto coma er with thll outer one, thll proposed re·meshing t«hnique nil/ foil (other ones too .' .. .) to presene mesh quality. I n this cost�, a mrmericol technique hosed on binh death fluid cells approach would be u.seful. Such a method is ami/able in the STAR-CD code {28] and is curnmtly im· estigated in the present coupl«i problem. Pa. The whole system is subjected to a sine wave acceleration: 1 0 if r< t Figure 11 shows the analytical solution and CFD calculation nith explicit fEIFV coupling in terms of beam free end displacement. The parameters of the imposed acceleration is are r =50 ws2• r = 100 ms. The numerical results sho w a good agreement between the two methods. which valid ates the explicit coupling procedure.