Awakening Elementary School System in University

"..."

A strange silence instantly fell over the entire conference room.

Everyone was stunned.

How long had it been since the problem was introduced? A minute? Two?

’He’s already come up with a solution?’

The faces of several young engineers were filled with shock and doubt. They couldn’t help but put their heads together and whisper.

"This fast? He’s not a con artist, is he?"

"I’ve never even heard of him. Where did President Peng find this prodigy?"

"This is ridiculous. Our team has pulled several all-nighters over this problem..."

In contrast, Engineer Pan and President Peng, who had already heard Sun Yu’s detailed report on Chen Lin’s first consultation, were the calmest of all.

"You can begin whenever you’re ready," President Peng said with an encouraging smile.

Chen Lin nodded, wasting no more words. He lowered his head, the tip of his pen touching the white paper. As he wrote with smooth, flowing strokes, he began to explain in a calm, steady voice.

"The alternative solution I’m proposing can be called an ’analytical-numerical hybrid method.’"

"The core idea is to bypass the current computational bottleneck you’re facing: the large-scale boundary element discretization and the solving of a large, dense linear system. By leveraging the geometric regularity of the caisson—namely, its rectangular shape—and the symmetry of the problem, we can seek a semi-analytical solution or perform a dimensional reduction for an analytical treatment."

His pen danced across the paper, and lines of mathematical formulas and symbols materialized, appearing clearly on the projection screen for all to see.

"First, the heave-pitch decoupling approximation. For your caisson’s length-to-width ratio and draft depth, while the heave and pitch motions are coupled at the target wave frequency, we can use the concept of two-dimensional strip theory on the caisson’s bottom for a preliminary decoupling estimation."

He fluently wrote out an analytical expression based on a two-dimensional source-sink distribution.

"This formula can be used to quickly estimate the main components of the heave added mass and pitch added mass—that is, the diagonal terms. The precision is sufficient to serve as a baseline."

"Second, handling the key coupling terms. This is the core of it." Chen Lin’s voice was unhurried, as if he were stating simple common sense. "The crux of the problem lies in the heave-pitch coupled added mass and the coupled radiation damping. Here, I propose an analytical transformation."

"We will map the fluid velocity potential perturbation, induced by the three-dimensional caisson surface during its pitching motion, onto an equivalent two-dimensional water channel problem with variable depth."

"Then, using complex functions and conformal mapping techniques—specifically, a variant of the Rukovski Transformation—we convert this complex boundary problem into an integral equation on a standard half-plane."

"This integral equation has a highly structured kernel function. Its form can be precisely expressed as a combination of complete elliptic integrals of the first and second kind."

"Third, a closed-form solution for the coupling coefficients."

At this point, Chen Lin’s writing took a sharp turn as he directly wrote out a long and extremely complex-looking formula on the paper.

"Through the aforementioned transformation and identification, we can directly provide the analytical closed-form expressions for the heave-pitch coupled added mass, which is (A_{35}), and the coupled radiation damping, (B_{35})! This expression depends only on the caisson’s geometric dimensions, fluid density, gravitational acceleration, and wave frequency. The calculation will be extremely fast and stable, completely avoiding the errors introduced by large-scale numerical discretization and solving."

"Fourth, the correction of the wave excitation force..."

"Fifth, solving the equations of motion. Take the precise analytical solution for the key coupling coefficients we’ve obtained, combine it with the other coefficients calculated by your software, and substitute them into the rigid-body equations of motion. Because the core uncertainty has been replaced by a precise analytical solution, solving the entire system of equations will become very stable and reliable."

As Chen Lin explained, the atmosphere in the conference room underwent a strange transformation.

At first, all the engineers were listening intently. A few of the more experienced senior engineers even nodded slightly in approval.

But gradually, as words like "Rukovski Transformation" and "complete elliptic integrals" came from Chen Lin’s mouth, the eyes of the younger engineers started to glaze over. Their brows furrowed, their expressions like they were listening to a foreign language.

By the end, when Chen Lin finished writing out that complex closed-form solution, it seemed the only ones in the entire conference room who could still barely follow his train of thought were Engineer Pan and the senior engineer who had spoken earlier.

Chen Lin finished the final step, put down his pen, and summarized, "An analytical solution is inherently precise. It can eliminate the errors and instability caused by numerical discretization, integration, and linear solving. The analytical formula I’ve provided can be easily implemented in code and independently verified."

"I was learning Python a while ago, and I remember a certain library has a ready-made toolkit for elliptic integrals that you can call directly. You can look it up later."

Engineer Pan took a deep breath. He turned to look at the subordinates behind him and asked in a deep voice, "Did everyone understand that?"

A few of the younger employees jolted as if waking from a dream.

In the conference room, everyone looked at each other, hesitant and embarrassed expressions appearing on their faces in unison. In the end, they all silently lowered their heads.

President Peng was not at all surprised by this. He chuckled, stood up, and clapped his hands. "Alright, let’s call it a day. Engineer Pan, take what Chen wrote back with you, organize a team to study it thoroughly, and then find the programmers in the department to code and verify it."

"Haaaaah..."

Chen Lin gave a great big stretch. He glanced at his phone and asked President Peng, "So, are we done for today? My return train ticket is for a departure in about two hours."