2025 in Review

2025 was a unique year. I lived in four different states and was a resident of those states. I moved across the country three times, with professional movers, packing and unpacking, registering my car, updating bank accounts, IDs, insurance, and permanent addresses. I leased apartments, sold my beloved house in Florida, and navigated multiple career changes quickly.

Along the way, I said goodbye to long term friends, familiar rhythms, favorite restaurants, coffee shops, and the commutes that had become quiet anchors, the route to the university, the small towns, the streams, and the art museums that made a place feel peaceful. In the middle of so much change, I found myself thinking of what I have done, where I have been, and what I still carry. When I look back at the people I knew, and the work, and the ordinary days that now feel distant, I do not see loss. I see what I learned, what it built in me, and the good that came from it.

Socially Sanctioned Incompetence

In America, there is a funny, and revealing, version of mathematics.

One popular joke is that, at a dinner party, the only socially acceptable thing to be bad at is mathematics. Someone recalls high school and says they are terrible at math, another says, me too, and everybody laughs. This is not so common in other countries. America is not unique, but this is a recognizable piece of day to day American culture. Americans are rarely proud of being bad at anything, yet mathematics is the one domain where the common American can sound almost pleased to be incompetent.

Americans often connect academic effort with outcome. A scholarship, a job, a credential, a title, a salary. So how does one convince American families that mathematics matters for a child’s education if the payoff is not immediate, or even legible? This is part of the motivation behind the Clay Mathematics Institute putting serious money behind difficult, century class problems. There is an old story, I cannot remember whether it comes from von Neumann or someone adjacent. The idea is simple: attach a financial reward to hard mathematics so American parents can tell their kids to study it because there is a concrete outcome.

This is very different from what drives professional mathematicians at the top of the field, like Andrew Wiles, who solved one of the great century class problems. I recall watching a documentary on him, and in one interview he cried while talking about the moment it finally worked. He was not crying because of money, status, or career positioning. He was crying because he had touched mathematical beauty. Many engineers, and many students in engineering courses, do not fully register how powerful that can be, the lived experience of beauty in a symbolic language that does not need a practical pretext.

Note on Early Medieval Universities, the Scholastic Method, and the Formation of Critical Thought

Early medieval universities formed a distinct intellectual system built around structure, discipline, and the controlled expansion of reasoning. These institutions emerged from cathedral schools and monastic centers and evolved into formal environments where knowledge was not only preserved but interrogated. Their core mechanism was the scholastic method, a systematic approach that treated ideas as analytical objects. The process was not ornamental. It was engineered to extract clarity from ambiguity through ordered reasoning.

The foundation of the scholastic method began with the authoritative text. Masters performed the lectio, reading the text line by line, isolating definitions, identifying structure, and parsing each conceptual element. This was not a passive reading but an initial decomposition step. The goal was to understand the internal architecture of the argument and to expose points where logical tension might develop.

From this, the quaestio emerged. The method treated contradictions as formal problems to be resolved. A question was constructed with precise logical framing that forced the students to examine the boundary where an argument failed to align with another authority or with its own implications. The university classroom became a controlled environment for isolating assumptions, identifying conflicts, and preparing for systematic resolution.

Disputation was the operational phase. In this setting, objections were raised, defended, and countered with ordered rigor. The disputation structure required the student to articulate objections with precision, evaluate their strength, and identify where an argument carried or collapsed. The master responded with the determinatio, a final synthesis that reconciled the competing claims and provided a coherent resolution grounded in logic rather than force of assertion. This served as a model for integrative reasoning.

The outcome was a unique form of critical thinking. Students were trained to interrogate any claim, including those from revered authorities, through systematic analysis. The method normalized structured critique. The reliance on logical form eliminated vague reasoning and required explicit premises. Students learned to perform rapid evaluation during oral defense, to track argument flow, and to maintain internal consistency. The result was a disciplined intellectual framework that enabled scholars to construct large, stable bodies of knowledge, such as the theological and legal systems that later defined medieval and early modern thought.

The scholastic method represents an early creation of cognition. It imposed constraints that produced clarity and forced internal coherence. While the topics were often theological, the underlying process was analytical, and it shaped the evolution of Western intellectual life. The early universities did not invent critical thinking, but they built the first institutional architecture that required it, refined it, and transmitted it across generations.

Remember a Perfect Day

On a spring day at University of Florida, I walked from Professor Haftka’s office, now my own. Up the small rise from aerospace, near the student union and under the trees, I made my way to class. Years of developing the Navier–Stokes equations for aerospace students, joined by mathematics students. The room held thirty, yet forty arrived, ten seated on the floor. A failed attempt at seeing mathematical beauty, but a perfect walk. Then Covid arrived.

Leaves

In the late summer of that year we lived in a house in a village that looked across the river and the plain to the mountains. In the bed of the river there were pebbles and boulders, dry and white in the sun, and the water was clear and swiftly moving and blue in the channels. Troops went by the house and down the road and the dust they raised powdered the leaves of the trees. The trunks of the trees too were dusty and the leaves fell early that year and we saw the troops marching along the road and the dust rising and leaves, stirred by the breeze, falling and the soldiers marching and afterward the road bare and white except for the leaves.

Ernest Hemingway, A Farewell to Arms

Remembering Dennis Bushnell

I received word via the NASA Alumni Association that Dennis Bushnell passed away. Dennis Bushnell was a long-serving Chief Scientist of NASA Langley Research Center. I remember meeting him for the first time when there was internal research funding available at NASA for researchers. I wrote a proposal on fluid thrust factoring of a particular kind. He was patient and read the proposal, but he told me that the Israeli Air Force had already accomplished this research and put it into production.

Perhaps more importantly than just another one-off proposal-or one of the first in my career that was rejected-was that he taught me one of the most invaluable lessons for any researcher: all good research gets funded. Perhaps if research is not funded, it is not so good after all.

Sometimes the long-time Chief Scientist might have been made fun of for his perhaps quizzical delivery of futurism in center addresses, but he succeeded in generating discussions among researchers. Perhaps this was one of the points of a center speech.

I recall an article describing how he would go to the top of the Chamberlin Hotel at Fort Monroe in Virginia-where part of the Civil War was fought and Jefferson Davis was held prisoner-and look out at the water each year to decide if he would spend another ten years at NASA Langley. Of course, he did. He decided to stay for a long time.

Upon receiving the news of his death, I think of another type of death-or rebirth-and that is of our beloved agency, the National Aeronautics and Space Administration. It was something that we were all proud to be a part of-a special agency. What makes me sad, perhaps, is that Dennis saw the great changes, and perhaps sadness, in the research core he served, and how NASA is forever changed today due to political interference.

Wiki: https://en.wikipedia.org/wiki/Dennis_M._Bushnell

High-Order CFD for Validating Analytical Solution of the Navier-Stokes Equations – ‘BlackJack’

The Computational Fluid Dynamics (CFD) code, ‘BlackJack,’ was created with a singular purpose: to generate extremely accurate numerical solutions of the Navier-Stokes equations in support of a broader program to develop new analytical solutions to those same equations. Unlike general-purpose CFD tools designed for industrial applications or engineering approximations, BlackJack CFD is a research code, engineered from the ground up to deliver benchmark-level fidelity for theoretical validation. Its development was motivated by the realization that no existing solver provided the order of accuracy, control over discretization, or modular extensibility required to confirm and refine exact or semi-exact solutions to the full compressible Navier-Stokes equations. Every algorithmic choice in BlackJack from the finite difference stencils to the boundary condition handling, damping strategies, and grid metrics was selected to eliminate numerical artifacts and ensure transparent, reproducible correspondence with analytical theory. It allows one to see what the analytical solutions will be, unlike production CFD codes that yield only relative insights into solutions.

I created the code BlackJack as an extension of my earlier DARPA supported program focused on developing new analytical solutions to the Navier-Stokes equations in high-speed flows. As I began deriving exact and semi-empirical solutions, it became clear that the engineering and applied communities no longer accept mathematical correctness based solely on symbolic insertion into the governing equations. Instead, they now demand that analytical solutions be compared directly to numerical simulations or experimental data, as if analytical solutions require the same form of validation as CFD. This mindset is fundamentally backwards, but it is the reality faced today.

Just as CFD is validated against wind tunnel measurements, engineers now insist that analytical solutions be benchmarked against numerical simulations, regardless of the fact that the solution may already satisfy the governing equations exactly. To confront this, I needed a numerical tool that was capable of producing reference-quality data free of dissipation, dispersion, and algorithmic ambiguity so that it could serve as a baseline for validating these theoretical results. Existing CFD codes, including commercial and open-source tools, were simply not built for this purpose.

This became the motivation for the new code. The goal was to construct a solver from the ground up that prioritizes fidelity over generality, with numerical methods capable of matching analytical solutions point for point in both steady and unsteady flow regimes. I began development nearly two years ago, writing the entire codebase by myself in Fortran. The architecture reflects the demands of research rather than industry: everything is explicitly structured, highest-order accurate, and open to modification.

BlackJack is a high-order structured CFD solver built on a multi-block, sub-block framework. It supports fully three-dimensional simulations on structured grids with arbitrary block decomposition, allowing exact enforcement of interface conditions and strict memory locality across parallel threads. Discretization schemes are user-specified and include a wide range of central and biased finite difference operators, including dispersion-optimized methods suitable for acoustics, shocks, and nonlinear wave propagation. Damping and filtering routines, including high-order sponge layers and localized artificial diffusivity, are tightly integrated into the residual formulation to stabilize underresolved gradients while preserving accuracy in smooth regions. Boundary conditions allow for subsonic and supersonic inflow and outflow, slip and no-slip walls, moving boundaries, periodic conditions, and radiation conditions. Each of these can be independently specified in each spatial direction, offering full control over problem setup and experimental design.

The solver operates on conservative variables, advancing them in time using a suite of advanced integrators, including low-storage and error-controlled methods up to 21st (and higher) order in accuracy. Grid metrics are computed analytically using the full inverse Jacobian formulation, enabling consistent representation of curvilinear geometry and ensuring the fidelity of derivative operations. All numerical routines-including finite difference, damping, boundary enforcement, and thermodynamic evaluation are modularized and can be extended with minimal disruption to the core solver. This modularity is essential, as BlackJack is designed not only to run known problems but to serve as a platform for exploring new equations, new models, and new forms of physical boundary data. The code can easily add additional equations, such as those for electromagnetics with minimal changes.

The code continues to evolve. Current development focuses on enhanced GPU parallelism, support for structured GMSH grid input, and expansion of the analytical validation suite with canonical solutions ranging from nonlinear acoustics to transitional flows and shock interactions. The code is already capable of resolving shock formation in N-wave propagation, reproducing linear and nonlinear acoustic fields, and achieving global accuracies on the order of 21st order or higher with only a handful of points per wavelength.

The name of the code originated as a joke I made with coworkers while I was a professor at the University of Florida. My original goal was to develop a high-order CFD code with spatial accuracy up to 21 points. Since 21 is the critical number in the card game blackjack, I began informally referring to the code as “BlackJack.” I also considered naming it “Gambit,” as a nod to the X-Men series, but BlackJack (or informally Gambit) ultimately stuck. Achieving practical CFD with shock capturing at 21st-order accuracy is not trivial, and the code continues to be pushed toward arbitrary order in both space and time for more complicated problems.

Olaf O. Storaasli

Recently I lost a friend, Olaf O. Storaasli, Ph.D., due to medical complications. I first met him through the NASA Langley Alumni Association after giving a lecture titled Life After Academics. Although we did not overlap during my time at NASA, we connected through shared experiences. Later, he and his wife visited Florida, and I hosted them at the University of Florida. During that visit, I introduced him to several senior design groups, invited him to speak to my class, and arranged a seminar with my research group.

Olaf had spent many years at NASA, primarily in Tennessee and at Oak Ridge, where he was a pioneer in early high performance computing. I believe he was at NASA during one of the space shuttle disasters. Many people from that era carry a lasting sense of responsibility for those events. It raises an important question—whether the hard-won lessons of safety from that time have truly endured, or whether we have started to drift back toward the cowboy mindset of the so-called Golden Age of Aerospace in the 1950s and 60s.

His visit to the university left a strong impression. I remember a dinner I hosted for him and his wife in Gainesville, a moment of quiet conversation and reflection. The students were especially engaged. In one senior design course, he stayed after class answering questions for over an hour, until I finally had to pull him away. Experiences like these remind me how valuable it is for students to engage with experienced professionals. Despite the noise of modern academic life, students remain eager to learn, listen, and absorb the lessons of those who came before them. That, I think, is a good sign.

Univ. Mich Art Exhibit

Today I attended the University of Michigan Art Exhibit on North Campus, which featured thousands of works by prisoners in the state of Michigan. The event serves both as a charity and as a program designed to provide the incarcerated with a creative outlet and a means of connecting with the broader community. The artwork is filled with pain and emotion. Some pieces are remarkably well executed.

https://dcc.carceralstateproject.lsa.umich.edu/s/pcapexhibition29/page/home