Design Optimization of Subcavitating Hydrofoils for America's Cup Class Yachts

I am pleased to present this talk today. I would like to express my gratitude to our sponsor, the American Magic Team, from the recent America's Cup cycle. I also want to acknowledge my co-authors and collaborators: Eirikur Jonsson, Andrew Lamkin, Yingjian Liao, Galen Ng, Anil Yildirim, and Hal Youngren. All of them are students in my lab who contributed significantly to this work. Additionally, Hal Youngren played a crucial role on the American Magic side, ensuring our approach remained practical and grounded.

Reflecting on my childhood, I spent a considerable amount of time sailing. I began with an Optimist class boat, which is considerably slower than America's Cup vessels. The square bow design is particularly noticeable when it encounters waves, resulting in a distinct feeling of resistance. This experience exposed me to both aerodynamic and hydrodynamic forces, which I believe influenced my understanding of sailing dynamics.

It is a pleasure to be here today to present this talk. I would like to express my gratitude to the American Magic Team, the sponsor and founder of this work during the recent America's Cup cycle. I also want to acknowledge my co-authors and collaborators: Eirikur Jonsson, Andrew Lamkin, Yingjian Liao, Galen Ng, Anil Yildirim, and Hal Youngren. They are all students in my lab who contributed to this project. Hal Youngren, in particular, provided valuable insights from the American Magic side, ensuring our work remained grounded and practical.

As a child, I spent a significant amount of time sailing, particularly in an Optimist class boat. While much slower than America's Cup vessels, this boat's square bow illustrates the effects of wave drag; when it encounters waves, you can feel the deceleration. Both wave drag and friction drag significantly impede speed. My early experiences allowed me to appreciate the aerodynamic and hydrodynamic forces at play, which ultimately influenced my decision to pursue a career in aerospace engineering.

In my journey, I became increasingly interested in airplanes and the dynamics of flight. I discovered that when the wing design is robust enough, wave drag can be substantially reduced, leading to increased speed. However, challenges such as flow separation can arise, making this a complex and exciting area of study. Understanding the contributions of viscous drag and wave drag is crucial for improving performance in sailing and aviation.

In my journey into aerospace, I developed a strong interest in aircraft dynamics. I experienced firsthand how aerodynamic forces impact performance. For instance, when a wing is sufficiently strong, wave drag can be significantly reduced, allowing for increased speed. However, challenges such as separation can arise, which complicates the dynamics.

Now, let's examine a conventional sailboat. Consider the innovative project from Chalma's University, which involved lifting an optimist sailboat—a design intended for children and known for its slow speed—out of the water by utilizing an underwater wing. This project successfully demonstrated the concept.

The conventional sailboat design has evolved significantly with the introduction of hydrofoiling technology. By lifting the entire boat out of the water using an underwater wing, speed can be dramatically increased without a corresponding increase in wind speed. This innovative approach minimizes skin friction and wave drag, although there is a slight increase in induced drag due to the wing's presence.

In recent years, the hydrofoil revolution has gained momentum, driven by the pursuit of speed. A notable example is the wing foil design featured on the left, which represents a new class of recreational craft. The reduced forces acting on the wing due to lower drag allow for a lightweight structure that can be easily handled. Additionally, the market is seeing a rise in electric-powered hydrofoils, further expanding the possibilities for recreational use.

The introduction of hydrofoil technology has significantly increased speed while maintaining the same wind conditions. This advancement reduces skin friction and wave drag, although there is a slight increase in induced drag due to the wing being submerged. Consequently, hydrofoils have gained popularity in recent years, driven by the desire for higher speeds.

On the left, we see a new design known as a wing foil, which benefits from lower drag forces. This allows for a lightweight construction that can be easily held by hand, enhancing its appeal for recreational use. Additionally, the market has seen a rise in electric-powered hydrofoils, providing further options for enthusiasts.

Moreover, the potential applications of hydrofoil technology extend beyond recreation. The right side of the slide highlights the significant advantages in industry, including reduced fuel consumption and improved seakeeping characteristics. These benefits have led to the adoption of hydrofoils in competitive sailing, such as in the America's Cup.

To understand the latest design in the America's Cup, it is essential to recognize its innovative features. This design, utilized in the last two cycles of the America's Cup, represents a significant advancement in sailing technology. Previously, catamarans were favored for their speed, characterized by narrow hulls and foils that provided lift. However, these vessels were wide and lacked the elegance of the current design.

The current model is a monohull equipped with two large foil arms that adjust based on the wind direction. When the wind comes from one side, one foil is submerged while the other remains above water. The entire boat is supported by these foils, with the vertical force generated by the foils counterbalancing the boat's weight. While there is a rudder and a foil for trimming, their contribution to force is minimal compared to the main foils.

Each foil is constructed from solid steel, weighing 1.5 tons, which creates a substantial moment to counterbalance the force of the sail. This design allows the boat to maintain a straight course rather than leaning, significantly enhancing efficiency. Despite their high efficiency, these vessels are complex to operate and design, requiring extensive training in both aerodynamic and hydrodynamic forces.

To address these complexities, we focused on a small portion of the overall design, specifically the two foils. For this analysis, we employed the Mach framework, a tool developed within my team, which I previously discussed during the 85th anniversary of Professor Jason.

The latest design for the America's Cup, which occurs every four years, represents a significant advancement in sailing technology. This design features a monohull structure equipped with two large foil arms that adjust based on wind direction. When the wind originates from one side, one foil remains submerged while the other rises above the water. The lift generated by these foils counteracts the boat's weight, allowing for improved stability and efficiency.

In addition to the lift, the design incorporates a rudder and a fighting force to manage wind resistance. The foils, made of solid steel and weighing 1.5 tons each, create a substantial moment that balances the sail's force, enabling the boat to sail upright rather than leaning over. This design enhances overall efficiency, although it also adds complexity to both operation and design.

To address these complexities, we focus on a specific aspect of the design, utilizing the Mach framework developed in my group. This framework allows for gradient-based optimization, which we can compute efficiently. We employ SNOs, with contributions from experts like President Michael Saunders, to facilitate this process. Geometry parameterization is necessary to modify the foil's shape, and subsequent mesh adjustments must occur automatically with gradients.

We utilize a Computational Fluid Dynamics (CFD) solver based on the Reynolds-Averaged Navier-Stokes (RANS) equations, acknowledging its lower fidelity in steady-state analysis. The adjoint solver is crucial in this process, building on the legacy of Professor Jimmy. Our research has produced numerous publications detailing this framework and its components. For illustration, we can begin with a simple geometric shape, such as a cylinder, which typically exhibits high drag and minimal lift, demonstrating the potential for improvement through our methods.

The optimization process we employ is grade-based, allowing for efficient gradient computation. We utilize SNOs, and it's a privilege to have President Michael Saunders, who significantly contributes to SNO development, in the audience today.

To modify the geometry of the foil or a 3D shape, we require a geometry parameterization. Following this, it is essential to adapt the mesh automatically based on the gradients. The Computational Fluid Dynamics (CFD) solver we use is based on the Reynolds-Averaged Navier-Stokes (RANS) equations. While this approach offers low fidelity, it serves its purpose in our framework. A critical component of this process is the adjoint solver, which is a vital part of Professor Jimmy's legacy that I have built upon.

As an illustrative example, we can start with a basic shape, such as a circle or cylinder, which typically generates high drag and minimal lift. Our optimization process then seeks to minimize drag for a specified lift in transonic flow conditions. The optimization identifies necessary modifications, such as flattening the shape to reduce pressure drag, rounding the leading edge, and sharpening the trailing edge, ultimately leading to the creation of a supercritical airfoil design.

Additionally, we can extend our optimization to 3D shapes. A notable example is the "crappy wing," which is a random variation of a shape. Our optimization successfully achieves a smooth pressure distribution for this design, demonstrating the effectiveness of our approach over time.

We will now instruct the Optimizer to determine the lowest drag for a specified lift in transonic flow. The Optimizer identifies that it is essential to flatten the airfoil to reduce pressure drag. It concludes that rounding the leading edge and sharpening the trailing edge, along with flattening the top, are necessary to effectively redesign the supercritical airfoil, albeit with advancements made over the past few decades.

Additionally, our optimization process can be applied in three dimensions. Although the example presented is a few years old, it remains relevant. We refer to this design as the "crappy wing," which is a random variation of shape. Our optimization achieves a smooth pressure distribution, as illustrated by the straightened CP contours. This results in a shock-free, single-point optimized wing with an elliptical distribution.

Our research has extended beyond aircraft configurations to other applications, such as wind turbines and hydrofoils. This connection led us to collaborate with American Magic, who reached out after reviewing our research papers. We decided to begin with a straightforward approach, focusing on 2D sections of the T-foil, which is critical for providing lift and stability.

The optimization of CP contours results in a shock-free, single-point optimization with elliptical distribution. We conducted extensive research on various configurations, including full configurations for aircraft. This experience led us to explore applications in other areas, such as wind turbines and hydrofoils. Our collaboration with American Magic began when they reviewed our research papers and reached out to us for assistance. I emphasized the importance of approaching problems gradually, even when they appear straightforward. We initiated our work by analyzing 2D sections of the T-foil, which is crucial for generating lift.

In our design efforts, we also examined the rudder, although I will not discuss that aspect here. It is important to note that the design considerations for hydrofoils differ from those for aircraft wings. When evaluating the drag polar, it is essential to adopt a different perspective. Specifically, we aim to elevate the drag polar to achieve a lower takeoff speed.

This section is critical for the America's Cup. During races, a decrease in wind or a poor maneuver can result in a transition to displacement mode, severely impacting performance. In such scenarios, competitors may travel significantly slower, making it essential to maintain flight on the foils.

The upper range of the drag polar is particularly important for takeoff and maneuvering. A well-optimized polar can enhance either speed or range; for sailing vessels, the focus is primarily on speed. Conversely, for propelled craft, improvements can lead to increased range.

At the lower end of the drag polar, higher wind speeds can lead to cavitation, which limits top speed. We will discuss this limitation further. Our optimization process utilizes parameterized freeform deformation volumes, starting with a baseline foil shape from a previous year, which has already been fine-tuned and tested in competitive scenarios. Additionally, this foil design incorporates a flap that spans the entire length of the wing.

This analysis is crucial for the America's Cup races. When the wind decreases or a poor maneuver causes a boat to enter displacement mode, its performance significantly declines, while competitors may continue at much higher speeds. Therefore, it is essential to achieve and maintain flight on the foils for optimal performance, particularly during takeoff and maneuvering.

The mid-range performance is important for achieving either greater speed or range, depending on the type of vessel. For sailing boats, the focus is primarily on speed, while for propelled crafts, the emphasis can shift towards maximizing range. At the lower end of the performance spectrum, higher wind speeds can lead to limitations caused by cavitation, which restricts top speed.

Our optimization process utilizes parameterized freeform deformation volumes. We began with a baseline foil design from a previous year, which has been refined and tested in competitive conditions. Additionally, this foil features a flap that spans its entire length, which can be adjusted to an optimal setting based on varying conditions. This flap adjustment is a critical design variable that enhances performance across different scenarios.

Furthermore, we must consider cavitation, which occurs when pressure drops below vapor pressure, resulting in the formation of air bubbles. Understanding and managing this phenomenon is vital for maximizing speed and performance.

The flap is adjusted to an optimal setting for each condition, which we have included as a design variable. This setting can vary depending on the specific conditions.

Another important consideration is cavitation. Cavitation occurs when the pressure falls below the vapor pressure, resulting in the formation of air bubbles or cavities.

Cavitation presents various regimes, and it is a significant concern. While some of you may have experience with multi-phase flow simulations, I do not specialize in that area.

Currently, it is not feasible to incorporate cavitation analysis within an optimization loop. To address this, we implement a straightforward solution: we impose a pressure limit. We establish this limit based on the known vapor pressure, ensuring that the pressure at any point in the airfoil remains below this threshold.

In our analysis, we consider the critical pressure coefficient, Cpmin, which determines the onset of cavitation. The results can be visualized as a boundary; areas to the left of this boundary indicate conditions where cavitation occurs, while areas to the right indicate conditions without cavitation.

Currently, conducting an optimization loop for this process is not feasible. Instead, we limit the pressure in the airfoil to remain below the known vapor pressure. This approach prevents cavitation, which occurs when the pressure falls below a certain threshold.

The relationship between cavitation and pressure can be visualized through a graph. On the left side of the graph, we observe the minimum coefficient of pressure (Cpmin), which indicates the onset of cavitation. To the left of this boundary, cavitation occurs, while to the right, it does not.

Furthermore, we can translate pressure values into corresponding speeds. This allows us to plot the lift coefficient (CL) against cavitation speed. It is crucial to remain on the left side of the cavitation boundary, as the right side represents cavitation conditions.

Through optimization, we can modify the shape of the cavitation bucket, which illustrates the relationship between loading and wing size or foil design. This bucket has two critical components that define its characteristics.

Cavitation can occur during takeoff at an induced circulation lift (ICL) even at low speeds, making it essential to maintain sufficient speed. In high-speed conditions, it is important to shift the cavitation limit to the right to avoid being constrained by maximum speed.

Typically, the middle section of the cavitation bucket is managed effectively, as indicated by the constant loading curve falling below this region, suggesting it is not a limiting factor. To control cavitation, we impose constraints on the maximum mean coefficient of pressure (CP). This is achieved using an aggregation function known as the Chryselmeyer-Steinhauser function.

Cavitation can occur during takeoff at low speeds, making it essential to maintain an adequate speed threshold. In high-speed conditions, it is also important to push the limits to avoid constraints imposed by maximum speed. Typically, addressing these aspects resolves the central portion of the performance curve, which is generally not a limiting factor.

To manage cavitation effectively, we constrain the maximum mean coefficient of pressure (CP) using the Chryselmeyer-Steinhauser (KS) function. This aggregation function has been utilized by my team for various applications over the years due to its versatility.

The accompanying contour plot illustrates this aggregation process. The two red lines represent constraints, and their intersection indicates a discontinuity. While this poses challenges for Sequential Quadratic Programming (SQP), the KS function smooths out this discontinuity, providing a conservative yet effective solution. By tuning the parameter ρ in the KS function, we can achieve results that closely align with our optimization goals.

In a sample optimization scenario, the baseline exceeded the cavitation limit, but after applying the KS constraint, the results remained below this threshold. In this case, the tuning of the ρ parameter ensured that the optimization was not overly conservative.

For the AC 75 class boats, it is noteworthy that they can achieve speeds up to four times that of the wind.

This method has been utilized by my group for many years across various applications due to its versatility. As illustrated in the contour plot from my book, which explains the aggregation process, two constraints are represented by the red lines. At their intersection, a discontinuity occurs; however, this poses no issue for Sequential Quadratic Programming (SQP). If one were to aggregate the constraints using the maximum function, this discontinuity would present a problem. The KS function effectively smooths out this discontinuity. Although it is conservative, careful tuning of the row parameter (row KS) allows for closer approximation to the desired solution.

In a sample optimization scenario, the baseline exceeded the cavitation constraint. After applying the optimization with the constraint, the results remained below the threshold. In this instance, the approach was not conservative due to the tuning of the row parameter.

Now, focusing on the conditions for AC 75 class boats, it is remarkable that these vessels can achieve speeds four times that of the wind. Depending on the angle to the wind, the speed can range from two to four times the wind speed. The performance polar demonstrates this relationship, with the vertical axis representing boat speed and the horizontal axis representing wind speed. The lines indicate different wind speeds, and it is important to note that these boats cannot sail directly into the wind; they require an angle to achieve optimal speed.

When sailing downwind, the maximum speed is limited to the wind speed, which is often insufficient. Interestingly, during upwind and downwind legs, the boats maintain a similar angle, and the sails are consistently trimmed in. Unlike smaller boats, where sails may be fully out, this is not the case for AC 75 class boats. For optimization, American Magic identified critical points that are essential for performance evaluation.

The optimization process aims to minimize cumulative drag across 67 varying flight conditions. Each condition involves adjustments to the angle of attack and flap angle. The shape remains constant, as it is fixed in the design. For each condition, we establish a target lift contribution and impose constraints on cavitation, thickness for structural integrity, and weight of the foam. Additionally, we include a volume constraint and various geometric constraints, resulting in a total of 237 constraints and 52 design variables.

American Magic provided us with critical data points, consisting of three upwind and three downwind points, which correspond to different wind speeds. Each point is associated with a specific Reynolds number and target lift coefficient. The parameter σ represents the cavitation coefficient (CP), which must be constrained based on speed.

The last condition we are examining is a maneuver condition. In this scenario, the aircraft slows down while also requiring high lift to facilitate turning, necessitating an increased lift coefficient (CL) of approximately 1.3.

Now, let’s focus on the optimization process. We are plotting the shape of the airfoil alongside the flap angle. It is important to note that while the airfoil shape remains constant across all conditions, the flap angle varies for each specific condition.

The final condition we are examining pertains to maneuvering. This scenario involves a reduction in speed while simultaneously requiring a high lift for turning, particularly when additional lift is necessary to counteract side forces. To achieve this, a relatively high lift coefficient (CL) of approximately 1.3 is required.

Now, let us focus on the optimization process. We are plotting the airfoil shape alongside varying flap angles. While the airfoil shapes remain consistent, the flap angles differ across each condition.

In the plot, the line representing the maximum coefficient of pressure (CP) is currently being exceeded in almost all conditions, particularly in the maneuvering scenario. An accompanying animation illustrates the CP behavior, which fluctuates before stabilizing beneath the maximum CP line. Although there is a slight increase in drag, the initial design failed to meet the cavitation constraints, resulting in a net gain in performance.

The graph illustrates the maximum cavitation pressure (CP) line, which is currently being violated under nearly all conditions. The last data point represents a maneuver scenario. An animation will demonstrate how the CP fluctuates but ultimately stabilizes below the maximum CP line. Although drag increases slightly, the initial design did not meet cavitation constraints, resulting in a net gain.

In our analysis, we focused on significant iterations within the SNOt framework. During the line search, substantial adjustments were made, particularly at iteration 4, where a notable step was taken. It is crucial for the computational fluid dynamics (CFD) model and the mesh to converge successfully. While I acknowledge there were instances of non-convergence, I am only presenting the successful outcomes. For specific geometries with which we have extensive experience, achieving convergence becomes routine after fine-tuning several parameters.

This optimization process typically takes less than an hour, often performed on a MacBook Pro with a 12-core M1 chip, and will likely be even faster with the M4 chip. The optimization considers numerous constraints, each associated with an adjoint, making the effort worthwhile.

In terms of the cavitation bucket, we have successfully shifted it to the right. The dashed green line indicates the requested position of the solid line, while the black points represent our test conditions. This adjustment ensures that all requested speeds for the cavitation bucket are feasible.

The optimization process did not involve any drastic measures; however, we focused on significant iterations in the SNOt framework. During the line search, we explored various options, including substantial steps. For instance, at iteration 4, a notable step was taken, as illustrated. The challenge lies in ensuring that the Computational Fluid Dynamics (CFD) converges successfully, meaning that both the N CFD and the mesh must operate without failure. Although there may be questions regarding instances of NaNs and failures, I will only present the successful outcomes.

For a specific geometry and case where we have extensive experience, the process becomes routine and reliable after fine-tuning a few parameters. Notably, this optimization typically requires less than an hour to complete, often performed on a MacBook Pro with a 12-core M1 chip, and likely even faster on the M4. This optimization encompasses multiple points and involves numerous constraints, each associated with an adjoint, which makes the effort worthwhile.

Regarding the cavitation bucket, we successfully shifted it significantly to the right. The dashed green line indicates this adjustment, moving the solid line accordingly. The black points represent our specified conditions, which demonstrate that we achieved the desired placement for the bucket. Importantly, all configurations remain feasible concerning the requested speeds for the cavitation bucket.

Overall, this optimization was accomplished with minimal drag penalty. As previously mentioned, there was a slight drag penalty observed, but the overall drag polar remains favorable.

In a subsequent study, we examined various flap sizes. While we did not vary the sizes continuously due to the complexity of mesh generation, we conducted several quick assessments. The findings indicate that under high lift conditions, larger flaps provide significant benefits. However, this advantage comes with trade-offs related to hinging moments, actuation, and other engineering factors beyond aerodynamics.

In our recent study, we examined various flap sizes. While we did not vary the flap sizes continuously due to the complexities involved with different meshes, we were able to conduct a few rapid assessments. The results indicate that under high lift conditions, larger flaps provide significant advantages. However, this benefit comes with trade-offs related to hinging moments, actuation, and other engineering considerations beyond aerodynamics.

The accompanying plot illustrates the performance of the different flaps, all of which meet the cavitation constraint. Additionally, we briefly explored 3D optimization; however, this presents a more complex challenge.

The current situation involves analyzing a non-symmetric condition, as opposed to the typical symmetric conditions encountered in airplane cruise scenarios. The angle of this configuration complicates the meshing process, but it is manageable. Additionally, the presence of the free surface influences the overall dynamics.

A more significant challenge arises from the piercing phenomenon. Under normal conditions, the tip of the foil often pierces the surface, leading to ventilation. This aspect is crucial, as ventilation can significantly reduce lift. Due to these complexities, we were unable to fully capture all the relevant physics in our current model, but there are promising avenues for future exploration.

Transitioning to the academic side, the work I will present predates our collaboration with American Magic. This research, although at a lower Technology Readiness Level (TRL), incorporates three-dimensional modeling and structural analysis. Unlike the previous Mac framework, which focused solely on Computational Fluid Dynamics (CFD), our current approach couples CFD with a structural solver. This integration allows for Fluid-Structure Interaction (FSI) analysis, ensuring that optimization considers the interactions between fluid dynamics and structural responses for each flow condition.

The current analysis involves a coupled adjoint approach that accounts for the interaction between aerodynamic and structural elements. This framework includes both a structural adjoint and an aerodynamic adjoint, along with off-diagonal terms to facilitate full coupling. A prototype of this methodology was first developed during my PhD thesis at Stanford, in collaboration with James Reuter and Juan. This project marked the initial integration of Computational Fluid Dynamics (CFD) within the optimization loop.

In hydrostatic applications, the scenario differs from traditional wing box configurations, as we are dealing with solid structures. Consequently, the elements utilized are solid elements. In this context, the outer mold line serves as a structural variable; adjustments to the outer mold line directly impact the structure. This relationship is akin to that of a wing box but exhibits even greater influence due to the absence of thickness variables.

The adjoint method we are discussing is a coupled adjoint, which considers the interactions between error structures. It includes a structural adjoint, an aerodynamic adjoint, and off-diagonal terms to form a complete coupled adjoint. This approach was first developed during my PhD thesis at Stanford, in collaboration with James Reuter and Juan. We successfully integrated Computational Fluid Dynamics (CFD) into this process.

In the context of hydroelastic structures, the dynamics differ from those of a wing box due to the solid nature of the structure. Here, the elements are solid rather than aerodynamic, and the outer mold line acts as a structural variable. Changes to the outer mold line directly affect the structure, which is similar to a Libox, but with a greater influence since there are no thickness variables involved.

This work was conducted with Professor Julie Young, formerly of the University of Michigan, who introduced me to this field. Together, we co-advised three students sequentially. The first student, Nitin Garg, has since joined the Ios Britannia Team for the current America's Cup cycle. Notably, this research represents the first validation of an optimization process conducted without a framework, although it focused on the hydronaval side rather than the aerodynamic side.

We conducted an optimization study on a solid aluminum foil, beginning with a NACA 009 profile. Our analysis of the NACA 009 and the optimized result closely matched experimental data obtained from a water channel. The tested foil demonstrated cavitation, but we successfully constrained it during the optimization process.

Subsequently, we advanced to composite hydrofoils, incorporating fiber tailoring. The resulting foil features a significant sweep and a smoothly curved sweep line. This design introduced hydroelastic washout, which effectively reduced the pressure peak and, consequently, prevented cavitation. It is important to note that we maintained cavitation constraints throughout this study.

This slide presents a comparison between the NACA 39 hydrofoil, which is limited by its lack of camber, and the Aer hydrofoil, a classic design known for its superior performance. Our enhancements have significantly improved the Aer hydrofoil's efficiency.

The optimization points we identified have shifted the performance metrics notably to the left. We also employed TIFA optimization, which is relevant for both rudders and hydrofoils, such as those used in the Optimist class.

In practice, many modern hydrofoils share similar designs. The key challenge lies in optimizing both the foil and the strut simultaneously. Careful attention must be paid to their intersection and shape to prevent cavitation and flow separation.

In this analysis, we examine both cavitation and separation constraints, drawing on our previous work in this area. Our optimization process effectively eliminates cavitation and separation issues. Initially, we identified a hotspot at the intersection, but through careful shaping of the airfoil in this region, we successfully mitigated the problem.

To conclude, I want to emphasize that the advancements we achieved would not have been possible without the pioneering adjoint method developed by President Anthony Jameson. I am fortunate to have learned from his work. Throughout his career, he demonstrated a commitment to integrating theory with practical implementation, developing code that is applicable in industrial contexts and making a significant impact on the industry.

In this presentation, we address both cavitation and separation constraints, building on our previous work in this area. The analysis demonstrates a rapid elimination of cavitation separation. Initially, a minor hotspot appears at the intersection; however, by appropriately shaping the airfoil at this location, we can effectively resolve the issue.

I would like to conclude by emphasizing that this work would not have been possible without the pioneering adjoint method developed by President Anthony Jameson. I am fortunate to have learned from his expertise. Throughout his career, he consistently integrated theoretical knowledge with practical implementation, creating usable codes for industrial applications that have had a direct impact on the industry.

I want to express my admiration for this approach and aspire to embody it in my own career. Additionally, I would like to wish a happy birthday to all three of you. Thank you.

It is essential to integrate both theory and practical implementation. The implementation involves coding that can be directly applied in industrial applications, resulting in a significant impact on the industry.