A COMPARISON OF FLUID-STRUCTURE INTERACTION METHODS FOR A SIMPLE NUMERICAL ANALYSIS OF CONCRETE GRAVITY-DAM *W.Z Lim1, R.Y Xiao1 and C.S Chin1 1
Department of Urban Engineering, London South Bank University, SE1 0AA, London, UK *
[email protected]
Key Words: partitioned method; fluid-structure interaction; concrete gravity-dam
ABSTRACT Fluid-structure interaction (FSI) has resulted in both complex applications and computing algorithmic improvements. This multi-physics problem consists of a fluid domain and a structural domain with the interaction in between which has been developed into a very attractive method for researchers and industrial applications. The aim of this paper is to develop a thorough understanding of the fluidstructure interaction behaviour and the numerical coupling methods apply in analyzing the problem of FSI. There are two distinctive methods for coupling the fluid and structural domains which are the monolithic and partition methods. In this paper, numerical cases were produced on the hypothetical models of the rigid and flexible (spillways) concrete gravity-dam. A dynamic velocity flow and pressure have been considered with the approach of two-way coupling partitioned method for the weak and strong coupled systems in ANSYS FEA software. A close comparison between the weak and strong coupled systems of two-way partitioned method has been observed for the consideration of both rigid and flexible (spillways) concrete gravity-dam.
1 INTRODUCTION Fluid-structure interaction, FSI can be described as the coupling of fluid mechanics and structure mechanics with interaction surrounding it. FSI problems have got the classical multi-physics characteristics which cover many engineering applications such as aerodynamics, wave-propagation, wind turbine, bio-engineering, offshore structures and bridges. In general, FSI or multi-physics
problems can be solved analytically with experiments or numerical simulations. The advancement of Computational Fluid Dynamics, CFD and Computational Structure Dynamics has allowed the numerical simulations of FSI to be expanded with availability of these techniques. The technique for the simulation of FSI has two distinctive approaches; the monolithic and partitioned approaches [1]. In this paper, the partitioned approach will be considered for the FSI problems. The partitioned approach in general can be categorized into weakly or strongly coupled problem. FSI problems can be divided into one-way or two-way coupling case and each coupling problem can be either weakly or strongly coupled system in the partitioned approach. Although there are many existing methods and techniques in FSI applications [2-13], the focus of this paper is to study the differences of the partitioned two-way coupling method for the weakly and strongly coupled system. The finite element method, FEM has been adopted with the consideration of Lagrangian and Arbitrary Lagrangian-Euler, ALE formulations as the numerical technique in investigating and analysing the FSI problem. The partitioned method of the FSI problems has been used in ANSYS software where both the fluid and structural domain were constructed separately and interacts with the coupled field method of MFS and Physics Environment within ANSYS.
2 PARTITIONED METHOD The basic understanding of couple-field analysis or multi-physics analysis is the combination of analysis from different engineering disciplines or physics that interact with each other to solve a wide existing engineering problem such as FSI. The partitioned method is an approach of which the two distinctive solvers (fluid and structure) are solved separately for the fluid flow and the displacement of a structure. The fluid and structure equations are integrated in time and the interface conditions are enforced asynchronously which means that the fluid flow does not change while the solution of the structural equations is calculated and vice versa. This approach preserves the software modularity and requires a coupling algorithm to allow for the interaction and to determine the solution of the coupled problem where information can be transferred between the two solvers [1].
In general, partitioned method can be categorized into two different kinds of coupling algorithms; staggered or weakly coupled and iterative staggered or strongly coupled. There are two types of coupling systems in the category of weak coupling; the one-way and two-way coupling system. However, the focus will be entire on the two-way coupling system when both, fluid and structure are able to interact fully with each other. In the two-way coupling calculations, fluid pressure acting on the structure is transferred to the structure solver and the displacement of the structure is also transferred to the fluid solver.
2.1 Weak Coupling The weak or staggered coupling can best be described from Fig. 1. In every time-step tn to tn + 1, both problems Ωf and Ωs are solved separately. In that case, the flow problem, Ωf at time tn is fully dependent on the flow and as structure problem, Ωs at time tn. However, the interaction at time tn + 1 is not taken into account and the same approach apply for the structure problem. The convergence at the boundary between structure and fluid is not considered and a new time step is launched directly. The weakly coupled approach solve every subproblem (fluid and structure) only once per time-step and such approach is generic explicit approach.
Fig. 1. Two-way partitioned solution of the coupled system
2.2 Strong Coupling The interface between both domains is crucial and in most applications, a strong coupling system is approachable and this decoupling system allows parallel solution of the fluid and the structure domains. A strongly coupled system is a further development of the partitioned approach of which two domains are solved independently in a decoupled way with an iterative manner with interaction loop between each time-step [1]. The coupled system is shown in Fig. 2. The basic iterative algorithm can best be referred to [1].
Fig. 2. Strong partitioned solution of the coupled system
3 COMPUTATIONAL TECHNIQUES The analytical solutions of structural and fluid domain were both conducted by ANSYS FLOTRANCFD and STRUCTURAL disciplines respectively. The corresponding element types that were used are element SOLID185 [14] for the concrete dams and element FLUID142 [15] for the reservoir fluid flow are shown in Fig. 3 and 4. The SOLID185 element is used for 3-D modeling of solid structures and Fig. 3 shows the geometry and node locations for this element. It is defined by eight nodes having three degrees of freedom at each node: translations in the nodal x, y, and z directions. The element has plasticity, hyperelasticity, stress stiffening, creep, large deflection, and large strain capabilities. It also
has mixed formulation capability for simulating deformations of nearly incompressible elastoplastic materials, and fully incompressible hyperelastic materials [14].
Fig. 3. SOLID185 Structural Solid Geometry [14]
As for the FLUID142 element, the Fig. 4 shows the geometry, node locations, and the coordinate system for this element that is defined by eight nodes and the material properties. FLUID142 can model transient or steady state fluid/thermal systems that involve fluid and/or non-fluid region as well as in the problem of fluid-solid interaction analysis with the degree of freedom; velocities, pressure, and temperature. By using the FLUID142 element, the velocities are obtained from the conservation of momentum principle, and the pressure is obtained from the conservation of mass principle [15].
Fig. 4. FLUID142 Geometry [15]
3.1 Fluid Flow In a Newtonian fluid, the relationship between the stress and rate of deformation of the fluid is shown as [16]: (1) where
, P,
, μ and λ represent the stress tensor, the fluid pressure, orthogonal velocities (u1= vx,
u2 = vy, u3 = vz), dynamic viscosity and second coefficient of viscosity respectively. For the case of two-way coupling in fluid flow, moving interfaces are included with the effect on the structural deformation that will deform the fluid mesh. Such phenomenon changes with time and to satisfy the boundary conditions at the moving interfaces, Arbitrary Lagrangian-Eulerian (ALE) formulation [1,17, 18] has been applied in solving such problems that can be found in [3,4].
3.2 Solid Structure The solid structure equation is based on the impulse conservation that is solved by using a finite element approach as shown below where M, C, K, ,
and
are the mass, damping coefficient,
stiffness, acceleration, velocity, and displacement vectors, respectively: (2) For the computed equivalent strain shown as: (3) The equivalent stress (von Mises) related to the principal stress can be obtained from
(4) where
is the equivalent stress of any arbitrary three-dimensional stress state to be represented as a
single positive stress values. The equivalent stress is part of the maximum equivalent stress failure theory known as yield functions which can be referred to [19].
3.3 Coupled-Field Analysis Methods There are several coupling field methods available in ANSYS and they are categorized as Direct Coupling-Field (Monolithic) or Load Transfer Coupling Methods (Partitioned) which includes MultiField Analysis with Single-Code Coupling (MFS) [20], Multi-Field Analysis with Multiple-Code Coupling (MFX), Load Transfer Coupled Physics Analysis [21] and Unidirectional Load Transfer. However, only the partitioned approach of load transfer coupling methods, MFS and Physics Environment Coupling are considered. The MFS coupling solver is consider the strongly coupled system as shown in Fig. 6 whereas the load transfer physics environment is consider the weakly coupled system as shown in Fig. 5. Both methods are categorized in the load transfer coupling that involve two or more analysis where each belong to a different field with interaction in between which allow load transfer from a result of one analysis to another analysis.
Fig. 5. Weak Two-Way Coupling System (Load Transfer Physics Environment)
Fig. 6. Strong Two-Way Coupling System (Multi-Field Solver, MFS)
4 CONCRETE GRAVITY-DAMS RESERVOIR ANALYSIS A hypothetical Oued-Fodda concrete gravity-dam water reservoir [22] has been used for FSI analysis truncated to 300.0 m in length with the assumed depth of 300.0 m in the consideration of a rigid and flexible(open spillways) concrete gravity-dam scenario. A turbulence current inflow from the dynamic effects of the recent Tohoku earthquake in Japan, 2011 was applied which induced a high velocity inflow. The actual recorded acceleration of 26.49 m/s2 at t=1.10 second [23] was taken to be the peak horizontal inflow acceleration under that dynamic condition. All the materials properties were taken with assumption shown in Table 1.
Table 1 Materials Properties for the Concrete Gravity-Dams and Water-Reservoirs Elastic Modulus
Density
Viscosity
(kg/m3)
(x 10-4 Pa.s)
Poisson Ratio’s
Materials (GPa) Concrete
30
0.2
2400
-
Water
-
-
1000
8.9
4.1 Comparison of Numerical Methods In comparison of the MFS method and the physics environment method, the results obtained for the rigid concrete gravity-dam structure of the hydrodynamics pressure and von Mises stress are fairly reasonable and agreeable in terms of their distribution patterns. Five different node locations were selected for the rigid concrete gravity dam structure and water-reservoir on von Mises and hydrodynamics pressure results respectively in purpose of verification in comparison of the both methods. Comparison of the average hydrodynamics pressure and von Mises stresses for the different node locations are demonstrated in Fig. 7(a) and Fig. 7(b). The distribution patterns for the pressures and stresses of both methods in comparison were quite similar for all the locations. The recorded maximum average ratio values between them are 1.147 (Location Case 5) for pressure and 2.600 (Location Case 5) for stress. In overall, it has an overall average ratio of 1.031 for the pressure and 2.321 for the stress, respectively. The higher ratio in the stress may be caused by strong two-way MFS analysis solution that requires longer staggering iterations in achieving a full convergence at the strong coupling of interface(s). This is compared with load transfer physics environment method of weak two-way coupling which could converge easily in a weak coupling system. The trends of both graphs oscillate with almost similar patterns and distributions.
(a) Average hydrodynamics pressure
(b) Average von Mises stress Fig. 7. Comparison in the average; (a) hydrodynamics pressure and (b) von Mises stress of the rigid concrete gravity-dam reservoir for the multi-field solver, MFS with load transfer physics environment.
(a) Average hydrodynamics pressure
(b) Average von Mises stress Fig. 8. Comparison in the average; (a) hydrodynamics pressure and (b) von Mises stress of the flexible concrete gravity-dam reservoir for the multi-field solver, MFS with load transfer physics environment.
In further applications in both couple-field methods, results were compared for the flexible (open spillway) concrete gravity-dam reservoir analysis. Five different node locations were selected as shown in Fig. 8(a) and Fig. 8(b) for each case of stress and pressure, respectively. The distribution patterns of the hydrodynamics pressures and von Mises stresses are agreeable in all cases (node locations) as shown in Fig. 8(a) and Fig. 8(b). With the recorded maximum average ratio value of 1.412 for pressure (Location Case 2) and 1.816 for stress (Location Case 5) is achieved. For the computed results of the flexible concrete gravity-dam structure, it shows that the overall average ratio of 1.260 and 1.474 for both pressure and stress, respectively. This is similar to the rigid case. The difference in results could be the reason of the strong coupling system between the interactions in MFS method and its staggering iterations between fields analysis that caused the computational time duration longer than that of the load transfer physic environment method. Hence, in these FSI computing applications, the results obtained have justified the use of both methods in solving the analysis of FSI problems.
5 CONCLUSIONS Comparison between the coupled-field analysis solution approach for the MFS and load transfer physics environment method was carried out on a concrete gravity-dam structure and with the Lagrangian and ALE with FEM techniques adopted, used in the ANSYS FEA software. The results obtained have proved that both weak and strong coupled field methods are suitable for the solution of the FSI problems with close average ratios for stress and pressure in comparison for both cases. In both scenarios, the dam structures were responsive to the pressure impact through the interaction surface or region although the values of the results do not differ much in comparison. From the comparisons for MFS and load transfer physics coupling methods, it can be seen that strong coupled system, MFS is capable of producing more accurate and realistic results. But the computational time of such method is far longer than that of load transfer physics coupling owing to its staggering iteration. Furthermore, the results of the load transfer physics coupling do not differ much from the results of MFS coupling method.
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