183 Chapter VII Conclusion and Future Work This dissertation has presented work towards the development of partitioned fluid- structure interaction algorithm for thin flexible structures undergoing large displacements, where the flow is resolved using the remeshed Vortex Particle Method and the structure is computed using a FEM-based elastodynamics solver. Specifically, the main objectives were: to extend the development of remeshed- Vortex Particle Method to enhance its applicability for FSI simulations of flexible and relatively thin structures, both for 2D and 3D cases; to develop 2D and 3D structural solvers capable in simulating large displacements of highly flexible structures; and to couple the fluid and structural solver so that it can be applied to simulate complex dynamic interactions of FSI problems involving thin flexible structures undergoing large displacements. To address these objectives, a multiresolution remeshed-Vortex Particle Method combined with the iterative Brinkman Penalization method was developed and coupled to a newly developed consistent and approximately energy-conserving corotational beam formulation. The coupled FSI solver was then applied to investigate the physics of flapping dynamics of an inverted flag in a uniform flow. VII.1 Conclusions A partitioned fluid-structure interaction algorithm based on remeshed-Vortex Particle Method coupled with a consistent and approximately energy-conserving Corotational Beam Formulation was developed. This algorithm is intended to be applied for FSI simulations involving large displacements of thin flexible structures due to the increasing trend of the use of this type of structures in many engineering fields. These thin and flexible structures are susceptible to FSI phenomena and their interactions with the flows are of interest since they are crucial not only to help avoid potential catastrophic failures of important structures but also to develop flutter-based power harvesting system. 184 The Vortex Particle Method (VPM) combined with Brinkman penalization method was employed as the flow solver due to its ability to preserve rotational flow features and to precisely and naturally simulate flows dominated by the advection phenomenon, and its ability to handle boundary conditions for complex and/or moving geometries. In the present work, the remeshed version of VPM was adopted since it offers some advantages over the pure Lagrangian VPM. Firstly, the computation of the velocity field is more accurate and efficient thanks to the use of high order regularization kernels in solving the Poisson equation. Secondly, the approach is more attractive, easier to implement, and can save the computational expense since the positions and the vorticity of the particles are updated in Lagrangian manner whereas the particle’s velocities, the stretching term, and the diffusion term, in which pure Lagrangian vortex methods experience difficulties, are computed using the Eulerian framework. Based on comparative study using a two-dimensional flow model, it was clear that the remeshed-VPM is more computationally efficient than the pure particle-based VPM solver. In addition, the calculated forces are also stable and more accurate. The present work proposed a new force formulation that is capable to generate distributed body forces accurately, both for classical and iterative Brinkman penalization method. This force formulation is the first contribution of the present work. Not only accurate results can be obtained from the use of the iterative Brinkman penalization technique, but also force distributions can be generated across the immersed body, which is important for the FSI simulations. Hence, the computation of the pressure field and the velocity gradient field is not required since the distributed forces can be obtained from the accurate calculation of the distributed penalization forces, not from the surface forces. This new force formulation has been validated using 2D simulations of flow past an impulsively started cylinder at Re = 550 and normal to a thin flat plate at Re = 1000. Both simulation results confirmed that the new force formulation, which is based on the corrected penalization velocity, is highly accurate and the results are identical to those obtained from force formulation based on the penalization vorticity. 185 The flow solver was further developed by combining it with the multiresolution scheme to efficiently simulate flow around thin body. The 2D multiresolution scheme using different patches has been chosen due to its compatibility with the remeshed-VPM. This scheme was extended to 3D applications, which constituted the second contribution of the present work. In addition, this multiresolution scheme and the iterative Brinkman penalization method has not been combined before, both for 2D and 3D flow applications. It is this combination that constitutes one of the major contributions of the current work. This newly developed flow solver has been validated using a number of popular benchmark problems. For 2D cases, the current work simulated the flow past an impulsively started cylinder at Re = 550 and the impulsively started flow normal to a thin flat plate at Re = 1000. The mesh refinement ratio was varied up to 4, i.e. the region outside the penalization domain has resolution 4 times coarser than that on the penalization domain. Based on the simulation results, it can be concluded that the accuracy of the calculated drag coefficients is maintained even though high refinement ratio was employed providing that the penalization region has enough resolution to properly enforce the boundary conditions via penalization technique. The use of this multiresolution scheme, however, reduces the quality of the flow visualization outside the penalization region and requires further interpolation to extract the flow structures. Regarding the structural solver, the corotational beam formulation was employed since it is simple, accurate, and efficient to simulate the geometric nonlinear effects resulted from the large displacements of the thin flexible structures. A consistent and approximately energy-preserving algorithm was developed. The novelty is the extension of this algorithm from a 2D formulation to a 3D formulation, constituting the third contribution of the present work. The proposed structural solver was validated using a number of challenging benchmark problems both in 2D and 3D cases. The results have shown that the algorithm can approximately conserve the total energy and momentum of the systems, leading to accurate and stable solutions even for long-term simulations. 186 The FSI solver was then constructed by coupling the multiresolution remeshed- VPM with the corotational beam formulation. This constitutes the fourth contribution of the present work. Indeed, it is the main contribution of the present study since it is here for the first time that the iterative Brinkman penalization method for remeshed-VPM was applied to simulate two-way FSI problems. Both loosely and strongly coupled schemes were implemented to handle various fluid- to-solid mass ratios. However, the loosely coupled scheme was used in the present work since the selected test cases has low fluid-to-solid mass ratios. The coupled solver was validated using some challenging FSI problems involving large structural displacements such as cantilever oscillation under vortex shedding from a square cylinder and large amplitude flapping of an inverted flag in a uniform flow. Comparisons between the simulation results with experimental and other numerical data have shown very good agreement. The simulations were firstly conducted using 2D FSI solver since some important flow features and structural responses in 3D are similarly reproduced in 2D simulations. The 2D simulation results have shown very good agreement with other 2D numerical observations and were able to resolve the physics of some main modes of the inverted flag systems. The 3D FSI solver was then applied to verify the 2D simulation results and to investigate the ’three-dimensional effect’ on the large amplitude flapping of the inverted flag. The 3D simulations have confirmed this ’three-dimensional effects’ since the results were in a good agreement with the experimental data. The 3D FSI simulations of the inverted flag model constitute the main contribution of the present work since it is here for the first time that the inverted flag model was simulated using fully 3D remeshed-VPM coupled with 3D consistent and energy-conserving corotational beam formulation. Due to the rich set of complex interactions of the generated wake dynamics with the flexible inverted flag, the FSI solver was further employed to a series of FSI simulations using the inverted flag model to understand the physics of these complex interactions. The simulations were performed over a range of nondimensional bending rigidity and flow conditions to investigate the dynamics 187 of the fluid–structure interaction of this flag configuration. Some key behaviour mechanisms of the flapping regimes of the inverted flag system especially in the LAF regime and the deflected regime have been investigated. VII.2 Future works During the FSI simulations, some numerical instabilities were encountered due to the use of certain convergence parameter for the iterative Brinkman penalization method and could not be solved by reducing the time step size, requiring further convergence tolerance adjustment. These instabilities also occurred when the FSI solver evaluated the convergence of the structural responses. It is assumed that this problem was caused by the use of linear interpolation when projecting the nodal displacements and velocities to the fluid cells inside the segmented body. In future, proper nodals-to-cells projection needs to studied to eliminate these instabilities. In addition, the flow simulations and FSI simulations for 3D objects were still expensive even though the multiresolution scheme has been implemented. To improve its computational efficiency, parallel computing can be implemented. Moreover, the flow visualization has poor quality due to the use of semi-adaptive multiresolution scheme. The fully adaptive multiresolution scheme can be developed in future by automatically recognizing the region in which high velocity gradient occurs. Furthermore, the proposed FSI algorithm has not demonstrated its capability in simulating FSI problems with high fluid-to-solid mass ratio. Hence, some numerical tests for this kind of problems can be performed in future work. Even though the results of FSI simulations in the present work are considered accurate, they were conducted without initially performing the mesh convergence studies. The use of the converged mesh / resolution in future work will certainly improve the validity the proposed FSI solver. In addition, the Flow and FSI simulations have not included the gravitational effect, which obviously can induce natural asymmetry to the system. The structural model can also be replaced by another finite element model, such as nonlinear shell model, to enhance the capability of the FSI solver in simulating more complex thin deformable bodies. 188.