III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et.al. (eds.) Lisbon, Portugal, 5–8 June 2006
AN ENRICHED SPACE-TIME FINITE ELEMENT METHOD FOR FLUID-STRUCTURE INTERACTION – PART I: PRESCRIBED STRUCTURAL DISPLACEMENT
A. Legay and A. K¨olke Structural Mechanics and Coupled Systems Laboratory Chaire de M´ecanique Conservatoire National des Arts et M´etiers, 2 rue Cont´e 75003 Paris, France e-mail:
[email protected]
Institut f¨ur Statik Technische Universit¨at Braunschweig, Beethovenstraße 51, 38106 Braunschweig, Germany e-mail:
[email protected]
Keywords: Fluid-structure interaction, thin immersed structures, large displacements, spacetime discretization, enriched finite elements. Abstract. This contribution introduces a new approach to treat fluid-structure interaction problems. This work (part one) focuses on applications with prescribed and a priori known displacement of thin structures. The extension of the presented numerical method to flexible structures enables the approach to handle fully coupled fluid-structure interaction situations (part two). A velocity-pressure-based weak formulation of the governing equations of viscous and incompressible fluid flow (Navier-Stokes-Equations) is discretized by finite space-time elements using a discontinuous Galerkin-scheme for time integration. The resulting space-time slabs are computed sequentially. The location of infinite thin structures in the fluid domain is represented by the zero level set of a space-time defined level set function. To capture the occuring moving discontinuities from embedding a thin solid body into the flow field, a locally enriched spacetime (EST) finite element method is applied to ensure a fluid mesh independent from the current configuration of the structure. Based on the concept of the extended finite element method, the space-time approximation of the pressure is enriched to represent strongly discontinuous solutions at the position of the structure. The velocity approximation is properly enriched to capture discontinuities in the gradient.
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A. Legay and A. K¨olke
1 INTRODUCTION Solving a fluid-structure interaction problem has always been a challenge for numerical engineers. The problem is strongly coupled and this coupling involves contrains in the numerical way of solving it. The most popular method Arbitray Lagrange Euler (so called ALE) uses both advantages of an Eulerian description for the fluid and a Lagrangian description for the solid. Combined with a finite element discretization for both domains, the coupling has to be ensure by a complete compatibility between the two meshes on the fluid-structure interface. This condition introduces severe fluid mesh distorsions when the structure has large displacements. Moreover, the method is almost unapplicable when several immersed structures have large relative displacements. In order to avoid these constrains, several methods have been developped. A class of methods are the meshless methods, for instance, SPH [1], EFG [2], NEM [3]. An other class of methods uses the idea of a fixed fluid mesh containing the structure: the immersed finite element method [4], the immersed boundary method [5] and the fictitious domain method [6] [7] [8]. The proposed method is in the second class. It is devoted for thin immersed structures with large displacements in a fluid domain. The fluid weak form is based on a velocity-pressure approach. The fluid is viscous and incompressible. It is discretized by finite space-time elements using a discontinuous Galerkin-scheme for time integration. The fluid approximation fields are enriched by appropriate functions in order to take into account the discontinuities involved by the structures. This leads to an Enriched Space-Time (EST) finite element method. The level-set concept is used to track the structures in the fluid. The present approach leads to a monotithic system, where each time slab in solved sequentially. This technique is based on the so-called extended finite element method [9]. The extended finite element method has been applied for fluid problems involving two-phase flow [10] [11] [12]. An enriched two-phase flow approach coupled with a flexible structure and using a space-time discretization has been developped in [13] and [14], following the work of [15]. The extended finite element method applied to thin immersed structures in a fluid has been applied in [16] for a compressible fluid, in [17] and [18] for an incompressible fluid using a fractionnal time scheme. Although the following ideas are applicable for a 3D space discretization as well as for flexible structures, the present work (part one) is focused on 2D space discretization and prescribed structure displacements. It is followed by the second part where the structures are flexibles [19]. 2 SPACE-TIME ELEMENT FORMULATION 2.1 Strong form of the equations The fluid is incompressible and viscous, with or with no flow. An Eulerian description is used in the fluid domain . The Navier-Stokes equations are in (1) where is the density, is the Eulerian velocity and is the gravity. The stress tensor is given by
!"$# %& ' )( *,+
(2)
where # is the pressure and * is the viscosity. The rate of strain tensor + is given by
+ -/. 0 1 2 3 54&6 ( 2
(3)
A. Legay and A. K¨olke
Enriched elements
Structures
Physical element
Reference element for velocity approximation
Figure 1: Space time fluid-structure interaction problem where several thin structures are immersed. The spacetime domain is cut in time direction by slabs. The reference element for the velocity approximation is a 18-node cube. The enriched fluid elements are the elements cut by the structures.
The boundary conditions, in term of imposed velocity are
on
(4)
while in term of imposed stress vector are
on
(5)
where is the imposed velocity on and is the imposed stress vector on . Since there are immersed structures in the fluid, there are additionnal conditions on the fluidstructures interfaces :
on
(6)
where is the velocity of the structure and is the interface between the structure and the fluid. The previous condition is a no-slip contidion, for a slip interface condition it is written ! !
on
(7)
where " is a unit normal vector to the interface . Additionnaly to these equations, the incompressibility condition is enforced by the continuity equation
in
6
(8)
2.2 Weak form of the coupled system The basic idea of a space-time discretization is to include the time axis in the finite element discretization. This space-time domain is called # in the following (fig. 1). The initial condition are thus given straigthforward on the boundary of the space-time domain $# . 3
A. Legay and A. K¨olke
The weak form is obtained from the strong form (eq. 1 to 8). 0 0 Find # 4 such that % % # 4 % 3 # % # # % 3( * + #
%1 #
%3
%
% # #
(9)
The continuity of velocity on the interfaces is enforced by a penalty term in eq. 9 where is a large arbitrary scalar. In the case of a no-slip condition, this penalty term becomes
% ! " "
6
(10)
2.3 Construction of a space-time finite element In order to satisfy the LBB condition, the velocity approximation is one order higher than the pressure approximation. It is chosen quadratic in space for velocity and linear for pressure, while it is linear in the time direction. The reference finite element (fig. 1) is thus a 18-node 0! 4 cube element 0!for velocity ( shape functions) and the standard 8-node cube element for 4 shape functions). pressure ( 2.4 General computationnal strategy The previous weak form, combined with the space-time discretization, leads to the following "$discretized system: "$# "&+,
&%(' ,
+
-
%
)
* *
(11) * #
where , and are respectively the nodal velocity, pressure and external forces vectors. The matrix % corresponds to the velocity-pressure coupling term while the matrix corresponds to the other terms of the weak form, including the fluid-structure coupling penalty term. Since the structure position and velocity are known a priori in this work, there is no structure unknowns. For numerical efficiency the space-time domain is divided into a sequence of slabs from to which are solved successively (fig. 1). The velocity and pressure fields at the end of the previous time step / . is egal to the corresponding field at the beginning of the next time step . When the mesh does not move, which is the case in the present method, this condition can be enforced directly by imposing the nodal values at time to be equal to those at time . . This leads to a second-order accurate time scheme. The method can be improved by letting the velocity and pressure fields be unknowns at the time step . The velocity field continuity is enforced by an energy based term in the weak form: 0 1 1 7 6 % 3 0 3 3 84 9 (12) 2 1 354
This so called discontinuous Galerkin time scheme is third-order accurate.
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A. Legay and A. K¨olke
Figure 2: Level-sets function in the space-time domain to localize the structures. A structure is on the zero isocontour of , the two tips are on the zero iso-contour of and .
2.5 Geometrical representation of the structures For clarity reasons, it is assumed that there is only one immersed structure. The position of the structure in the space-time domain is given by the zero iso-contour of a level-set function !0 4 [20] (fig. 2). Such a function is for instance the distance function to the structure. Moreover, a sign is given to this function in such a way that it is positive on one side of the structure and negative on the other side. In the case of a 2-tip open structure, these 0!tips are located in the space-time domain by two 0 4 4 . The zero iso-contour of these two functions and additionnal level-set functions cut the interface on the structure tips. They are moreover positives on the part of the domain which contains the structure and negatives on the other part. These three level-set functions are 0 4. interpolated by using the space-time finite element shape functions Since the three level-sets are used only around the structure, in the case of several immersed structures, these three functions suffice to locate all the structures: they are discontinuous from a structure neighborhood to an other.
3 ENRICHED SPACE-TIME METHOD - The ”EST” METHOD 3.1 Partition of unity concept In the proposed method, the structure positions are independant of the fluid mesh, consequently, the discontinuities involved by the structures in the fluid have to be taken into account in the approximation. The velocity continuity is already enforced by a penalty term in the weak form (eq. 9), but the immersed structures involve a pressure discontinuity as well as a velocity gradient discontinuity from one side to the other side of the structure (fig. 3). In order to model these discontinuities, new functions are added to the approximation spaces through a partition of unity [21] [22]. The enriched domain is the set of all the space-time elements cut by the structures, this domain is called # . The set of nodes connected to these elements is called ! . The choosen partition of unity support for both velocity and pressure is the set of the 8 shape functions based on the 8-node cube element. 5
A. Legay and A. K¨olke
along
along
Figure 3: Discontinuity in the pressure field through the structure. Gradient discontinuity in the velocity field through the structure.
3.2 Fluid field enrichment 3.2.1 Pressure field The pressure discontinuity is taken into account by adding a 0 Heaviside like function in the 4 . The enriched pressure approximation. Such a function is the sign of the level-set , approximation is then 0 # !0 4$ 0 4 4 0 0! 4 4 , (13)
3
where is the pressure on the node and is a new unknown associated to the enriched node. Since the enriched function is constant piecewise, it does not introduce spurious terms in the partial enriched elements or blending elements [23]. 3.2.2 Velocity field The gradient discontinuity in the velocity is taken into account by adding a0 ramp like function 4 . The enriched in the approximation. Such a function is the absolute value of the level-set velocity approximation is then + 0 !0 4' !0 4 4 !0 4 (14) +
3
node and is a new unknown where is the componant of the velocity on the associated to the enriched node. It has been shown in [24] that this choice of partition of unity one order lower than the standard part of the approximation associated with a ramp like function enrichment does not introduced spurious terms in the partial enriched elements. 4 APPLICATIONS The discontinuous Galerkin time scheme is used in the two following applications. It is assumed that there is no gravity and a no-slip interface condition is taken into account. 4.1 Driven cavity The driven cavity problem is proposed as an application to show the ability of the method to decouple 2 physical independant problems. The discretized fluid domain is a rectangular cut 6
A. Legay and A. K¨olke
1m.s =1kg.m =0.135 .
.
1m, 12 elements
Fixed structure Driven cavity
1m 1.35m, 17 elements Figure 4: Driven cavity
Pressure profile along :
Streamlines with enrichment
o: with enrichment +: with no enrichment
Figure 5: Streamlines for the driven cavity problem when the enrichment is used. Pressure profile along .
by a fixed structure which divides the fluid into a square part an a rectangular part (fig. 4). The square part is the so-called driven cavity and the rectangular part is an isolated domain where no velocity as well as a zero pressure are expected. The streamlines are shown on figure 5 in the case where the enrichment is used. The enrichment produces no spurious flow in the right part as it can also be seen on figure 6. If no enrichment is used, the enforcement of the velocity continuity on the interface is well done, but the velocity is not equal to zero in the right part. The pressure profile shown on figure 5 gives the same results; the jump in pressure is well catched by the enrichment but is not with no enrichment. 4.2 Translating structure in a cavity A straight structure is immersed in a closed cavity (fig. 7). The structure is initially placed vertically in the left side of the cavity, it has an horizontal velocity. Figure 8 shows the streamlines as well as the pressure field for several time steps. The pressure jump accross the structure 7
A. Legay and A. K¨olke
1m.s .
along
Zoom in the right area
o: with enrichment +: with no enrichment Figure 6: Velocity profile for the driven cavity problem. The scale coefficient to represent the velocity in the zoom-in picture is 20.
1.5 m, 17 elements . structure velocity 0.42 m at t=0s =1 kg.m =0.15 0.5 m time 0.5s
1m.s .
1 m, 10 elements
Figure 7: Translating structure in a cavity.
is well catched. The streamlines, represented for the relative velocity between fluid and structure, are correct. 5 CONCLUSIONS The proposed method is developped to deal with a fluid-structure interaction problem where thin structures are immersed. It is based on the three following main points : 1. the fluid mesh is independant of the structure positions, 2. the fluid fields (velocity and pressure) are enriched by additionnal functions around the fluid-structure interfaces through a partition of unity, 3. a space-time discretization is used, combined with a Discontinuous time scheme. 8
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(a) Pressure field at time 0.13s
(b) Streamlines at time 0.13s
(c) Pressure field at time 0.40s
(d) Streamlines at time 0.40s
(e) Pressure field at time 0.66s
(f) Streamlines at time 0.66s
(g) Pressure field at time 0.93s
(h) Streamlines at time 0.93s
Figure 8: Pressure field and streamlines for the translating structure.
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These three points are linked and lead to an original and advantageous method. The first point makes the method flexible and allows the structures to have large displacements and relative displacements. The second point makes the first point to be possible by taking into account the discontinuities introduced by the structures in the fluid fields. The third point makes the second point efficient and accurate since the enrichment is continuous in time. The method may be applicable for more general problems involving moving discontinuies and leads to the Enriched Space-Time (EST) finite element method. The EST-method is applied for prescribed structure displacements in this first part, it is extended to flexible strutures the second part [19]. REFERENCES [1] J.J Monaghan and R.A. Gingold, Smoothed particles hydrodynamics: theory and application to non-spherical stars. Monthly Notices of the Royal Astronomical Society, 181, 375-389, 1977. [2] T. Belytschko, Y. Y. Lu and L. Gu, Element-free Galerkin methods. International Journal for Numerical Methods in Engineering, 37(2), 229–256, 1994. [3] N. Sukumar, B. Moran and T. Beltyschko, The natural element method in solid mechanics. International Journal for Numerical Methods in Engineering, 43(5), 839–887, 1998. [4] L. Zhang, A. Gerstenberger, X. Wang and W.K. Liu, Immersed finite element method. Computer Methods in Applied Mechanics and Engineering, 193(21-22), 2051–2067, 2004. [5] C.S. Peskin, The immersed boundary method. Acta Numerica, 11, 479–517, 2002. [6] R. Glowinski, T.W. Pan, T.I. Hesla, D.D. Joseph and J. P´eriaux, A fictitious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies: Application to particulate Flow. Journal of Computational Physics, 169(2), 363–426, 2001. [7] F.P.T. Baaijens, A fictitious domain/mortar element method for fluid-structure interaction. International Journal for Numerical Methods in Fluids, 35(7), 743–761, 2001. [8] F. Bertrand, P.A. Tanguy and F. Thibault, A three-dimensional fictitious domain method for incompressible fluid flow problem. International Journal for Numerical Methods in Fluids, 25(6), 719–736, 1997. [9] T. Belytschko, T.N. Mo¨es, S. Usui and C. Parimi. Arbitrary Discontinuities in Finite Elements. International Journal of Numerical Methods in Engineering, 50(4), 993–1013, 2001. [10] J. Chessa and T. Belytschko, An enriched finite element method and level sets for axisymmetric two-phase flow with surface tension. International Journal for Numerical Methods in Engineering, 58(13), 2041–2064, 2003. [11] A. K¨olke and D. Dinkler, Extended space-time finite elements for two-fluid flow. Computational Fluid and Solid Mechanics 2005, Proceedings of Third M.I.T. Conference on Computational Fluid and Solid Mechanics, 2005. 10
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[12] A. K¨olke and D. Dinkler, Extended Space-Time Finite Elements for Boundary-Coupled Multi-Field Problems on Fixed Grids. Proceedings of International Conference on Computational Methods for Coupled Problems in Science and Engineering, Greece, 2005 [13] E. Walhorn, A. K¨olke, B. H¨ubner and D. Dinkler, Fluid-structure coupling within a monolithic model involving free surface flows. Computers and Structures, 83(25-26), 2100– 2111, 2005. [14] A. K¨olke and D. Dinkler, Extended Space-Time Finite Elements for Two-Fluid Flows in Fluid-Structure Interaction. Proceedings of Sixth World Congress on Computational Mechanics, China, 2004 [15] B. H¨ubner, E. Walhorn and D. Dinkler, A Monolithic Approach to Fluid-structure Interaction using Space-time Finite Elements. Computer Methods in Applied Mechanics and Engineering, 193(23-26), 2069–2086, 2004. [16] A. Legay, J. Chessa and T. Belytschko, An Eulerian-Lagrangian Method for FluidStructure Interaction Based on Level Sets. Computer Methods in Applied Mechanics and Engineering, 195(17-18), 2070–2087, 2006. [17] A. Legay et A. Tralli, Une approche e´ l´ements finis enrichis Euler-Lagrange pour l’interaction fluide-structure. Premier colloque du GDR fluide-structure, SophiaAntipolis, France, 26-27 septembre 2006. [18] A. Legay and A. Tralli, An Euler-Lagrange enriched finite element approach for fluidstructure interaction. European Journal of Computational Mechanics, submitted, 2006. [19] A. K¨olke and A. Legay, An enriched space-time finite element method for fluid-structure interaction - Part II: Thin flexible structures. Proceedings of the third European Conference on Computational Mechanics, Lisbon, 5-8 june 2006. [20] J.A. Sethian, Level Set Methods and Fast Marching Methods Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science. Cambridge University Press, 1999. [21] J.M. Melenk and I. Babu˘ska, The partition of unity finite element method: Basic theory and applications. Computer Methods in Applied Mechanics and Engineering, 139(1-4), 289–314, 1996. [22] I. Babu˘ska and J.M. Melenk, The partition of unity method. International Journal for Numerical Methods in Engineering, 40(4), 727–758, 1997. [23] J. Chessa, H. Wang and T. Belytschko, On the construction of blending elements for local partition of unity enriched finite elements. International Journal for Numerical Methods in Engineering, 57(7), 1015–1038, 2003. [24] A. Legay, H.W. Wang and T. Belytschko, Strong and weak arbitrary discontinuities in spectral finite elements. International Journal for Numerical Methods in Engineering, 64(8), 991–1008, 2005.
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