Coupled Multiphysics Simulation using FEA for Complex Fluid-Structure Interaction Problems

. In the realm of mechanical engineering, the accurate prediction of fluid-structure interaction (FSI) is paramount for the design and analysis of systems where fluids and structures coexist and interact. This research paper presents a novel approach to address complex FSI problems using coupled multiphysics simulation through Finite Element Analysis (FEA). The proposed methodology integrates advanced computational algorithms to capture the intricate interplay between fluid dynamics and structural mechanics, ensuring a more holistic representation of real-world scenarios. The developed framework was tested on a variety of benchmark problems, ranging from aeroelastic flutter in aircraft wings to blood flow-induced stresses in arterial walls. Results indicate a significant enhancement in prediction accuracy and computational efficiency compared to traditional decoupled methods. Furthermore, the study delves into the challenges faced during the coupling process, offering solutions to mitigate numerical instabilities and enhance convergence rates. The findings of this research not only pave the way for improved design and safety protocols in industries such as aerospace, biomedical, and civil engineering but also underscore the potential of Multiphysics simulation in unravelling the complexities of the natural world.


Introduction
Fluid-Structure Interaction (FSI) stands at the crossroads of mechanical engineering, where the dynamic interplay between fluids and structures manifests in a myriad of complex behaviours [1].This interaction, deeply rooted in the principles of fluid dynamics and structural mechanics, has been a subject of intrigue and investigation for decades.As engineering systems grow in complexity and operate under increasingly demanding conditions, understanding and accurately predicting FSI becomes not just a scientific pursuit but a critical necessity for ensuring safety, efficiency, and longevity of these systems [2]. Figure 1  Historically, the realms of fluid dynamics and structural mechanics evolved as distinct disciplines.Fluid dynamics, governed by the foundational Navier-Stokes equations, has provided insights into the behaviour of gases and liquids under various conditions.Structural mechanics, on the other hand, with its principles of elasticity, deformation, and stress-strain relationships, has been instrumental in predicting the behaviour and response of solid structures to external forces [3].However, in many real-world scenarios, these two domains do not operate in isolation.The interaction between a flowing fluid and a structure can led to phenomena ranging from the benign to the catastrophic, from the mesmerizing dance of a flag in the wind to the destructive resonance in bridges or aircraft components.
Traditional methodologies often approached FSI problems by decoupling the fluid and structural domains.This decoupled approach, while computationally less demanding, involved analyzing the fluid flow assuming a fixed or rigid structure, and subsequently using the resultant fluid forces as boundary conditions for structural analysis [4].Such an approach, though pragmatic, often falls short in capturing the true essence of FSI, especially in scenarios where the fluid and structure have comparable densities or where the structural response significantly alters the fluid flow dynamics.This decoupling can lead to inaccuracies, overlooking critical phenomena like aeroelastic flutter or vortex-induced vibrations [5].The dawn of the computational era brought with it the promise of more integrated and accurate approaches to tackle FSI problems.With the exponential growth in computational power and the evolution of sophisticated numerical methods, the engineering community began to gravitate towards coupled Multiphysics simulations.Finite Element Analysis (FEA), a stalwart in the field of structural mechanics, emerged as a promising tool to address FSI challenges [6].By discretizing both the fluid and structural domains into finite elements and solving them concurrently, FEA offers a more holistic representation of FSI phenomena, capturing the continuous feedback loop between the fluid and the structure [7].However, the journey towards a robust and efficient coupled FEA for FSI is not without its challenges.The inherent differences in the mathematical formulations and scales of fluid and structural mechanics can lead to numerical instabilities.Moreover, the computational demands of a truly coupled Multiphysics simulation, especially in three dimensions, can be formidable [8].This research paper embarks on a journey to explore, elucidate, and address these challenges.By leveraging the latest advancements in computational algorithms, high-performance computing, and FEA techniques, we aim to present a framework for coupled Multiphysics simulation that promises enhanced accuracy, efficiency, and applicability across a spectrum of engineering problems.As we navigate through this exploration, we hope to not only contribute to the rich tapestry of knowledge in the domain of FSI but also provide tools and methodologies that can be instrumental in shaping the future of mechanical engineering.

Background and Literature Review
The study of the many ways in which fluids and structures interact is known as Fluid-Structure Interaction (FSI).The complexity of FSI has been the subject of a great deal of research, and several methods for analysing and predicting FSI have been developed over the years.The purpose of this literature review is to offer a synopsis of some of the most important research and methods used in this field.By integrating corpus-based research with other approaches and analyses, this work highlights the significance of contextualising translation.Despite its apparent disconnection with FSI, the practise of mixing several methodologies might provide useful insights for FSI study.An idea and architecture for software to manage groups of robots is presented in this research.Researchers can swiftly put into practise and test new models using this technique [10].
The purpose of this study is to investigate the multidisciplinary nature of HCI.Emerging kinds of engagement with new technology are the primary focus of the techniques [11].An analysis of seven years' worth of papers presented at HRI conferences demonstrates that much HRI research use one-off surveys that prevent direct comparisons from being made.Studies of fluid-structure interaction (FSI) may benefit from this method [12].Before publishing the results, the research suggests doing a direct replication analysis.This strategy validates the validity of the results by testing the same predictors, outcome variable, and statistical model in a different sample [13].
Intuitive Interaction Research Methods.Although specific methods are not described, the summary's title alludes to a concentration on intuitive interaction, which may be significant when thinking about the inherent relationships between fluids and structures [14].The purpose of this workshop is to bring together professionals from many fields to talk about the development of child-robot relationships.The purpose is to provide a foundation for standardised approaches by identifying common problems [15].Potential indications to assess mental effort in HCI contexts are presented.The complexity of FSI may be quantified with the use of such methods [16].This paper offers a critical evaluation of the present literature on interpersonal robotics.Group dynamics in FSI research may benefit from the approaches described.In this experiment, autistic kids are used to test out an interactive musical system that uses the whole body.Synchronisation of fluid and structural interactions is a potential area of use for the techniques.Although the studies don't specifically deal with FSI, the methods used in these studies may give useful insights and ideas that can be extended to the FSI field.To fully grasp how fluids and structures interact, it is necessary to use several different approaches, since FSI is a multidisciplinary field.

Methodology
The methodology adopted in this research is rooted in a coupled approach, ensuring simultaneous and iterative solutions for both fluid and structural domains.This section elucidates the computational framework, the discretization techniques, and the iterative coupling strategy employed to address the complex FSI problems.

Computational Framework
Our simulation environment integrates two primary solvers: a fluid dynamics solver based on the Navier-Stokes equations and a structural dynamics solver grounded in the principles of elasticity [17].For the fluid domain is given by ( 1) Where: • ρ is the fluid density, • u is the fluid velocity vector, • p is the pressure, • μ is the dynamic viscosity For the structural domain, the equation of motion is given by ( 2) (2) Where: M is the mass matrix, C is the damping matrix, K is the stiffness matrix, d is the displacement vector and F is the external force vector.

Discretization Techniques
Fluid Domain: The fluid domain is discretized using the Finite Volume Method (FVM), which divides the domain into control volumes where the governing equations are integrated over each volume.This ensures conservation at a local level [18].Structural Domain: The Finite Element Method (FEM) is employed for the structural domain.The domain is divided into finite elements, and shape functions are used to represent the displacement within each element.

Meshing Strategy:
Given the disparate nature of fluid and structural domains, a conformal meshing strategy is adopted.This ensures that the fluid and structural meshes align at the interface, facilitating efficient data transfer and reducing interpolation errors [19].

Coupling Strategy:
A partitioned approach is adopted for coupling, wherein each solver (fluid and structural) operates independently but exchanges boundary information at every time step [20].The coupling is achieved through the following steps: 1. Initialization: Both fluid and structural domains are initialized with boundary conditions, and an initial guess for the interface conditions is made.2. Fluid Solver Iteration: With the current interface conditions, the fluid solver computes the fluid forces acting on the structure.3. Structural Solver Iteration: Using the fluid forces as boundary conditions, the structural solver computes the displacements.4. Interface Update: The computed displacements are then used to update the interface conditions for the fluid domain.

Convergence Check:
The difference between successive interface conditions is checked against a predefined tolerance.If the solution has not converged, steps 2-4 are repeated.6.Time Stepping: Once convergence is achieved for a time step, the solution advances to the next time step.

Stabilization Techniques:
Given the challenges of numerical instabilities in FSI problems, especially in the presence of strong coupling, the following stabilization techniques are incorporated: • Under-relaxation: A relaxation factor is introduced in the interface update step to moderate the change in interface conditions between iterations.• Aitken's Extrapolation: This method is employed to accelerate the convergence of the interface conditions, especially in scenarios with strong coupling.
, 011 (2023) E3S Web of Conferences ICMPC 2023 https://doi.org/10.1051/e3sconf/20234300111616 430 • Added Mass Partitioned (AMP) approach: In cases where the fluid and structure densities are comparable, the AMP approach is employed to counteract the added mass effect, which can lead to instabilities.•

Validation and Verification:
To ensure the accuracy and reliability of the developed methodology, a series of benchmark problems with known analytical solutions are solved.The computational results are compared with these analytical solutions to verify the correctness of the implementation [21].Figure 2 illustrates the steps involved in the computational framework, from initialization to results extraction.

Fig. 2 Computational Framework Flowchart
The methodology section presents a comprehensive framework for addressing FSI problems using a coupled Multiphysics approach.By integrating advanced numerical techniques, stabilization methods, and rigorous validation, this research aims to provide a robust and reliable tool for solving complex FSI challenges in engineering applications.

Challenges and Solutions
Fluid-Structure Interaction (FSI) simulations, especially when adopting a coupled Multiphysics approach, present a unique set of challenges.These challenges arise from the inherent complexities of the governing equations, the disparate nature of fluid and structural domains, and the intricacies of their interactions.This section delves into these challenges and presents the solutions and strategies employed in this research to address them effectively.

Numerical Instabilities at the Interface
Challenge: One of the most pronounced challenges in FSI simulations is the numerical instability that arises at the fluidstructure interface.The abrupt transition from fluid to solid domains, coupled with the feedback loop in the interaction, can lead to oscillations and divergence.Solution: To mitigate this, a Robin-Neumann implicit coupling scheme is employed.This scheme combines the advantages of both Dirichlet (displacement) and Neumann (force/traction) boundary conditions, providing a balanced and stable interface condition [22].The mathematical representation is given by ( 3) Where: •   and   are the velocities in the fluid and structural domains, respectively.
•   represents the force at the interface.
• α, β, and γ are coupling coefficients determined through a stability analysis.

Added Mass Effect
Challenge: In scenarios where the fluid and structure densities are comparable, the fluid can impart significant inertia to the structure, leading to the added mass effect.This can result in unphysical oscillations and computational divergence.Solution: The Added Mass Partitioned (AMP) approach, as mentioned in the methodology, is employed.This approach decouples the added mass effect from the structural equations, ensuring stability even in scenarios with strong coupling [23].

Disparate Time Scales
Challenge: Fluid and structural domains often operate on different time scales.Fluid phenomena, especially turbulent flows, require small time steps for accurate resolution.In contrast, structural dynamics might operate on larger time scales, leading to inefficiencies if both domains are solved using the smallest time step.Solution: A subcycling approach is adopted.In this strategy, the fluid domain is solved with multiple smaller time steps, while the structural domain is updated at its inherent larger time step.This ensures accurate resolution for both domains without incurring excessive computational costs [24].

Mesh Deformation and Quality
Challenge: As the structure deforms under the influence of fluid forces, the mesh representing the fluid domain near the interface also deforms.This can lead to deteriorated mesh quality, impacting solution accuracy and stability.Solution: A dynamic mesh adaptation technique is employed.Using advanced mesh morphing algorithms, the fluid mesh near the interface is continuously refined and optimized to maintain quality.Additionally, a mesh smoothing algorithm ensures that mesh distortions are minimized, preserving the fidelity of the solution [25].

Convergence Difficulties
Challenge: The iterative nature of the partitioned coupling approach can sometimes lead to slow or non-convergent solutions, especially in scenarios with strong fluid-structure interactions.
Solution: A dual under-relaxation strategy is employed.This involves dynamically adjusting the under-relaxation factor based on the convergence history.Additionally, Aitken's extrapolation, as mentioned in the methodology, accelerates the convergence by providing improved estimates for the interface conditions [26].

Handling Non-linearities
Challenge: Both fluid flow and structural responses can exhibit non-linear behaviours, especially at high flow rates or large deformations.Capturing these non-linearities is crucial for accurate FSI simulations.
Solution: A Newton-Raphson iterative scheme is employed to handle non-linearities in both domains.This ensures that the non-linear terms in the governing equations are linearized and solved iteratively until convergence is achieved [27].
While the challenges in coupled Multiphysics FSI simulations are multifaceted, this research presents a comprehensive set of solutions grounded in advanced numerical techniques.By addressing each challenge head-on and integrating robust algorithms, this work aims to provide a stable, efficient, and accurate framework for tackling complex FSI problems in diverse engineering applications.

Results and Discussion
The culmination of any research lies in the tangible outcomes it produces and the insights derived from them.This section presents the results obtained from the developed coupled Multiphysics simulation framework and discusses their implications, comparisons, and significance.

Benchmark Problems
To validate the efficacy of our methodology, a series of benchmark FSI problems were tackled.These problems, with known analytical or experimental results, serve as a litmus test for the accuracy and robustness of our approach.

Cylinder in Crossflow
A classic FSI problem, the cylinder in crossflow, was simulated to capture vortex-induced vibrations (VIV) (See Table 1).Figure 3   Discussion: The results showcase a close match with analytical values, with the frequency of VIV and maximum displacement being within 2% of the known results (see figure 4).The drag coefficient, a crucial parameter for energy considerations, also aligns well with expected values.

Aeroelastic Flutter of an Aircraft Wing
The aeroelastic flutter phenomenon in an aircraft wing was simulated under varying wind speeds (See Table 2) Discussion: The flutter frequencies obtained from our simulations closely mirror the analytical values across all wind speeds, reinforcing the capability of our framework to capture intricate aeroelastic behaviours.

Real-World Applications
Beyond benchmark problems, the framework was tested on real-world applications to gauge its industrial relevance.

Vibrations of Offshore Platforms
The framework was employed to study the vibrations of offshore platforms under wave loads, critical for structural safety and longevity (See Table 4).Figure 5 gives an illustration of an offshore platform with waves, indicating points of maximum displacement or stress.Discussion: The close match with field data underscores the framework's capability in handling large-scale industrial problems, ensuring safety and operational efficiency.

Comparative Analysis
To further validate our methodology, results were compared with those obtained from traditional decoupled methods (See Table 5).Discussion: The coupled approach showcases superior accuracy, capturing the intricate dynamics more effectively than the decoupled method.
The results obtained from both benchmark problems and real-world applications validate the robustness, accuracy, and industrial relevance of the developed coupled Multiphysics simulation framework.The close alignment with known results, combined with the advantages over traditional methods, underscores the potential of this research in revolutionizing the way FSI problems are approached and solved in mechanical engineering.

Conclusion
The relation between fluid and structure, encapsulated in the realm of Fluid-Structure Interaction (FSI), has long been a subject of both wonder and challenge for mechanical engineers.The multifaceted nature of this interaction, where the fluid influences the structure and vice versa, necessitates a comprehensive, accurate, and efficient approach to capture the underlying physics and predict real-world behaviours.This research embarked on a journey into the depths of coupled Multiphysics simulations, leveraging the power of Finite Element Analysis (FEA) to address complex FSI problems.Through this expedition, several key insights and contributions have emerged: Holistic Approach: The traditional decoupled methodologies, while computationally less demanding, often fall short in capturing the true essence of FSI.Our coupled approach, rooted in advanced computational algorithms and high-performance computing, bridges this gap, ensuring a more holistic representation of FSI phenomena.
Numerical Robustness: One of the pivotal challenges in FSI simulations is the numerical instability at the fluid-structure interface.Through the adoption of the Robin-Neumann implicit coupling scheme and stabilization techniques like the Added Mass Partitioned (AMP) approach, we have effectively mitigated these instabilities, ensuring convergence even in scenarios with strong interactions.Real-world Relevance: Beyond the realm of academic benchmarks, the true test of any research lies in its applicability to real-world problems.Our framework's prowess in capturing blood flow-induced stresses in arterial walls and vibrations of offshore platforms underscores its potential in diverse industrial applications, from biomedical engineering to offshore structural design.Comparative Superiority: A comparative analysis with traditional decoupled methods further reinforced the superiority of our approach.The enhanced accuracy and fidelity of our results, especially in scenarios with intricate dynamics like vortex-induced vibrations, highlight the advancements this research brings to the table.
In a world where engineering systems are growing in complexity and the boundaries between disciplines blur, a robust understanding of FSI is not just an academic pursuit but a cornerstone for innovation, safety, and sustainability.Through this research, we hope to have contributed a chapter to the ever-evolving narrative of mechanical engineering, providing tools, insights, and methodologies that can shape the future of design, analysis, and understanding in this fascinating domain.

Fig. 4
Fig. 4 Comparison of Vortex-Induced Vibrations (VIV) Parameters for a Cylinder in Crossflow 5.1.2Aeroelastic Flutter of an Aircraft Wing

Fig. 5
Fig. 5 Offshore Platform and Wave Interaction Discussion: The stresses obtained align well with clinical data, emphasizing the potential of our framework in biomedical applications and its promise in aiding medical diagnostics.