Partitioned Methods in Computational Modelling on Fluid-Structure Interactions of Concrete Gravity -

Fluid-structure interaction (FSI) has resulted in both complex applications and computing algorithmic improvements. The aim of this paper is to develop a better understanding of the fluid-structure interaction behaviour and the numerical coupling methods which can be used in analysing the FSI problem of a multi-physics nature computationally. There are two different systems in partitioned methods for coupling the fluid and structural domains which use strong and weak couple algorithms. Numerical results have been obtained on the hypothetical models for the close and open-spillways concrete gravity-dam. The two-way coupling partition method has been applied to the dynamic velocity flow and pressure using the ANSYS FEA software. A close comparison between the weak and strong coupled systems of two-way partitioned method has been made for the consideration of both close and open-spillways concrete gravity-dam.


Introduction
Fluid-structure interaction, FSI, can be described as the coupling of fluid mechanics and structure mechanics.FSI problems possess the classical multi-physics characteristics which occur in many engineering applications such as aerodynamics, wave-propagation, wind turbines, bio-engineering, offshore structures and bridges.In general, FSI or multi-physics problems can be solved with either experimental or numerical simulations.The advance on Computational Fluid Dynamics (CFD) and Computational Structure Dynamics (CSD) has allowed the numerical simulations of FSI to be conducted rapidly.The technique for the simulation of FSI has two distinctive approaches: the monolithic and partitioned approaches (Thomas, 2010).However, only the partitioned approach will be adopted in this paper for the FSI numerical examples.The partitioned approach in general can be categorised into weakly or strongly coupled problems.The coupling can be divided into one-way or two-way coupling cases.Although there are many existing methods and techniques in FSI applications (Friedrich-Karl et al., 2010;Joris et al., 2010;Bathe et al., 2009;Michler et al., 2003;Wulf & Djordje, 2008;Jo et al., 2005;Sandboge, 2010;Mitra, 2008;Akkose et al., 2008;Broderick & Leonard, 1989;Onate et al., 2007), the focus of this paper is to investigate 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 consideration of the Lagrangian and Arbitrary Lagrangian-Euler (ALE) methods.ALE formulations have been used 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 fluid and structural domains are set up separately and interacted with the coupled field methods of Multi-Field Single-Code Solver (MFS) and load transfer physics environment.

Fluid Structure Interaction Approach
The basic principle 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 widely known engineering problem such as FSI.The partitioned method is an approach in which the two distinctive solvers (fluid and structure) are activated separately for the fluid flow and the displacement of a structure.The fluid and structure equations are solved by integratation in time.Interface conditions are enforced asynchronously which means that the fluid's flow does not change while the solution of the structural equations is calculated and vice versa.This approach p determine (Thomas, 2 In general or weakly two-way c when both pressure ac transferred

Weak C
The weak solved sep as structur approach a can be use not consid (fluid and

Strong
The interfa allow para partitioned manner i.e 2. The bas

Computational Techniques
The analytical solutions of structural and fluid domains have both been conducted by ANSYS FLOTRAN-CFD and STRUCTURAL solution, respectively.Relevant element types used are element SOLID185 for the concrete dams and element FLUID142 for the reservoir fluid flow.Both SOLID185 and FLUID142 elements are compatible in relation to the coupling method of fluid interaction with solid structure in three dimensional models.The SOLID185 element is generally used for 3-D modelling of solid structures and 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 got mixed formulation capability for simulating deformations of nearly incompressible elasto-plastic materials, and fully incompressible hyperelastic materials (ANSYS Inc., 2009).
As for the FLUID142 element, it is defined by eight nodes and the material properties of the fluid density and viscosity.FLUID142 can model transient or steady state fluid/thermal systems that involve fluid and/or non-fluid region as well as the problem of fluid-solid interaction analysis with degrees of freedom: velocity and pressure.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 principles (ANSYS Inc., 2009) which are described in the governing equations below.

Fluid Flow Governing Equations
The fluid flow is defined by the laws of conservation of mass, momentum, and energy.Such laws are expressed in terms of partial differential equations which are discretised with a finite element based technique.The fluid flow equations are governed by Navier-Stokes equations of incompressible flow.
The continuity equation of the fluid flow is shown in (ANSYS Inc., 2009) as the following: where , and are the components of the velocity vector in the x, y and z direction, respectively.is the density of the fluid and t is the time shown in the equation above.The rate of change of density can be replaced by the rate of change of pressure: As for the incompressible solution: where P and β are the pressure and bulk modulus of the fluid flow, respectively.
In a Newtonian fluid, the relationship between the stress and rate of deformation of the fluid is shown in (ANSYS Inc., 2009) as: where , P, , μ and λ represent the stress tensor, the fluid pressure, orthogonal velocities ( = , = =, = ), dynamic viscosity and second coefficient of viscosity, respectively.The product of the second coefficient of viscosity and the divergence of the velocity is zero for a constant density fluid.Equation (4) transforms the momentum equations to the Navier-Stokes equations as follows (ANSYS Inc., 2009): where , R and T represent the acceleration due to gravity, distributed resistances and viscous loss terms, respectively with subscript x, y and z as the coordinates.The density of the fluid properties and effective viscosity are presented as ρ and , respectively.The viscous loss term, T for all coordinate directions is eliminated in the incompressible, constant property case.The order of the differentiation is reversed in each term, reducing the term to a derivative of the continuity equation, where it is zero.
The energy equation for the incompressible is shown in (ANSYS Inc., 2009) as: where the specific heat, thermal conductivity, volumetric heat source are represented by , K and , respectively.The static temperature, T is calculated from the total temperature with v as the magnitude of the fluid velocity vector specified below in the absence state of heat transfer, adiabatic incompressible case: In the incompressible fluid flow state, the viscous work, pressure work, viscous dissipation and kinetic energy are neglected in the compression case which can best be referred to (ANSYS Inc., 2009).
For the calculation of the pressure, the defining expression for the relative pressure is: Combining the momentum equations into vector form, the result is changed to: where , , , , , , , , , and are the reference density, reference pressure, gravity vector, absolute pressure, relative pressure, position vector of fluid particle relating to rotating coordinate system, angular velocity vector, velocity vector in global coordinate system, fluid viscosity and fluid density 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 needs to satisfy the boundary conditions at the moving interfaces.Arbitrary Lagrangian-Eulerian (ALE) formulation (Thomas, 2010;ANSYS Inc., 2009;Donea et al., 2003) has been applied in solving such a problem, this can be found in (Joris et al., 2010;Bathe et al., 2009).

Arbitrary Lagrangian Eulerian, ALE Formulation
ALE algorithms are particularly useful in flow problem solutions which involve large distortions in the case of mobile and deforming boundaries available within the interaction between a fluid and a flexible structure such as FSI (ANSYS Inc., 2009).Fluid flow problems often involve moving interfaces which include moving internal walls (for example, a solid moving through a fluid), external walls or free surfaces.ALE formulations will become useful to solve such a problem as shown in (Donea et al., 2003).Eulerian equations of motion need to be modified to reflect the moving frame of reference.The time derivative terms are essentially rewritten in terms of the moving frame of reference where ϕ and are the degree of freedom and velocity of the moving frame of reference, respectively as shown below: With the robust and versatile software, ANSYS, three-dimensional models of FSI can be applied and solved in some numerical examples.However, various research has been conducted in the applications of ALE which can be referred to in Bathe (2009) and Wulf & Djordje (2008).Such algorithms are applicable in the monolithic or partitioned manner in mitigating FSI problems.The functions of FLOTRAN-CFD and Structural in ANSYS allow the analysis of fluid and structure domains in conjunction with different coupling analysis solution methods.

Segregation Solution Algorithm
In the case of coupling algorithms, the pressure and momentum equations are coupled with the SIMPLEF algorithm originally belonging to a general class referred to as the Semi-Implicit Method for Pressure Linked Equations (SIMPLE).SIMPLEF is the sole pressure-velocity coupling algorithm developed by Schnipke and Rice (1985) with the further improved SIMPLE-Consistent algorithm by Van Doormaal and Raithby (1984).An approximation of the velocity field is obtained by solving the momentum equation.The pressure gradient term is calculated by using the pressure distribution from the previous iteration or an initial guess.The pressure equation is then formulated and solved in order to obtain the new pressure distribution.Velocities are corrected and a new set of conservative fluxes is calculated.The implementation of the SIMPLEF algorithm can best be referred to in (ANSYS Inc., 2009).

Solid Structure Governing Equations
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 as described in (Zienkiewicz & Taylor, 2005): (13) The computed equivalent strain is shown as: The equivalent stress (von Mises) related to the principal stress can be obtained from where is the equivalent stress of any arbitrary three-dimensional stress state to be represented as a single positive stress value.The equivalent stress is part of the maximum equivalent stress failure theory known as yield functions which can be referred to in Zienkiewicz and Taylor (2005).
The general element formulation used for SOLID185 is the fundamental equation applicable for general finite strain deformation where the updated Lagrangian method is applied to simulate geometric nonlinearities.Convention of index notation are used in the equations below and all variables such as coordinates , displacements , strains ij , stresses , velocities , volume V and other material variables have been assumed to be solved at time t.The equations are derived from the element formulations which are based on the principle of virtual work as in ANSYS Inc. ( 2009): where , , , , f and f are the Cauchy stress component, deformation tensor, displacement, current coordinates (x, y and z), component of body force and component of surface traction, respectively.The volume of deformed body, V, and surface traction of deformed body, S, were represented in Equation ( 16) as well.The internal virtual work, W, can be described as: The element formulations are obtained by differentiating the virtual work and in the derivation, only the linear differential terms are kept and all higher order terms are ignored just to obtain a linear set of equations.The Mmaterial constitutive law was used to create the relation between stress increment and strain increment which reflects the stress increment due to straining.The Jaumann rate of Cauchy stress expressed by McMeeking and Rice (1975) was applied in the constitutive law because the Cauchy stress is affected by the rigid body rotation which is not a frame invariant (ANSYS Inc., 2009).
where σ is the Jaumann rate of Cauchy stress and is the time rate of Cauchy stress.
The spin tensor ω can be expressed as: Thus, the Cauchy stress rate is: The rate of deformation tensor d can be expressed as: The stress change due to straining based on the constitutive law is shown as: And the Cauchy stress rate can now be expressed as: where c and v are the material constitutive tensor and velocity, respectively.

Coupled-Field Analysis Methods
Multi-Field Analysis with Single-Code Coupling (MFS) (ANSYS Inc., 2009) andLoad Transfer Coupled Physics Analysis (ANSYS Inc., 2009) are the two sequential or partitioned coupling solvers in the ANSYS Mechanical APDL.The MFS coupling solver is considered as a strongly coupled system as shown in Figure 4 whereas the weakly coupled system shown in Figure 3.Both methods are categorised by the load transfer coupling that involves two or more calculations where each belongs to a different field with interaction in between.This will allow load transfer from one result of an analysis to another.

Conclusions
Numerical examples have been used in this paper to illustrate the differences between the coupling algorithms for both strongly and weakly coupled partitioned methods with techniques adopted such as the Lagrangian and ALE with FEM techniques.The feasibility and capability of both methods have been tested and compared on a large scale three dimensional concrete gravity reservoir dam.From the numerical results obtained, it has been proved that both weak and strong coupled field methods are oscillating with the same pressure and stress distributions which both justify their capabilities of transferring load between the surfaces of interaction.In both examples, the dam structures have responded to the pressure impact through the interaction surface or region with the distribution patterns being similar.However, the differences of average ratio in the stress value could be caused by the stringent convergence in the strong coupling algorithm due to the large surface interactions of the numerical model whereas the weak coupling algorithm has loose convergence within the surface of interaction.Therefore, the strong coupling algorithm could be justified to obtain more realistic results, however its strict convergence requirement for the large interaction surface may need longer computational time and initiate instability for computing.Hence, the weak two-way coupling algorithm could be an ideal method in solving both small and large scale numerical models without any convergence problems within the interaction surfaces. Figure

Table 3 .
Average ratio of hydrodynamic pressure and von Mises stress of both MFS and load transfer physics environment methods for the open-spillway concrete gravity-dam reservoir