Chapter 41 - 38: Why is the Gap So Big?
Sun Yu let out a long sigh of relief. The tense muscles in his face finally relaxed, and he even managed a slight smile.
"Master, I don’t know if it’s just me, but this proposal seems a lot easier to understand than the last one."
He scratched his head and said, a little embarrassed, "For that last proposal, it took me two or three days of studying back at the company just to barely wrap my head around the math and methods involved."
Hearing this, Chen Lin chuckled to himself.
’Good heavens, President Peng is truly a meticulous man.’
The mathematical difficulty of this proposal was lower than the previous one on structural stability analysis.
This meant that President Peng could most likely have found someone within the design institute to perform this optimization. It would have just cost a bit more time and manpower.
But he still gave the project to him, and at the highest possible rate for an urgent consultation.
This five thousand yuan was less of a consultation fee and more of a sincere welcome gift.
Of course, Chen Lin also understood that President Peng’s actions likely had a deeper consideration behind them.
This was a final, on-the-spot confirmation of his abilities before they officially began a long-term partnership, with Sun Yu present as a witness.
’What if I just had a stroke of dumb luck last time and happened to run into a field I was good at?’
Far from being displeased by this cautious test, Chen Lin actually felt more reassured.
’Working with a boss who’s reliable, shrewd, and socially savvy is the most hassle-free way to go.’
At this thought, the smile on Chen Lin’s face became more relaxed. He looked up at Sun Yu and asked casually, "Shall I start explaining the new calculation method, then?"
Sun Yu was still reeling internally from Chen Lin’s freakish ability to fully comprehend everything after hearing it just once.
But having been through this once before, he quickly adjusted his mindset. He nodded vigorously, leaned his body forward slightly, and assumed a posture of rapt attention.
Chen Lin pulled a few blank sheets of scratch paper from the side pocket of his backpack and took a black gel pen from his pencil case.
He placed one of the sheets horizontally between them, hovered the tip of the pen over the paper, and began to speak:
"Your original method, whether it’s piecewise linear superposition or slicing the curve into finer micro-elements, is using discrete, linear thinking to approximate a continuous, non-linear problem. I’ll give you a new method based on Calculus. It can directly account for the continuous variation of tension along the curve."
Before he even finished speaking, the pen touched the paper. Lines of clear, neat formulas and symbols flowed onto the page, appearing before Sun Yu’s eyes accompanied by the steady sound of Chen Lin’s explanation:
"First, we establish two fundamental, precise differential equations."
[Precise differential equations for the elongation of a prestressed steel tendon, considering friction:]
[1. dP = -P * μ * |dθ| (Tension decay due to friction, where dθ is the incremental central angle for the infinitesimal arc segment)]
[2. d(ΔL) = (P * ds) / (E_s * A_p) (Elongation of the infinitesimal segment ds, based on Hooke’s Law)]
"Here, ds is the infinitesimal arc length along the tendon’s curve. The relationship between dθ and ds is determined by the radius of curvature at that point, R(s)."
[|dθ| = |ds| / R(s)]
"For the parts on your design drawings that are explicitly marked as standard circular arc segments—that is, where the radius of curvature R is constant—we can directly integrate to get an analytical solution for the elongation, taking friction into account."
Chen Lin’s wrist was steady as the pen glided across the paper, writing out the final integrated result:
[ΔL_arc = (P_start * L_arc) / (E_s * A_p * μ * θ) * (1 - e^(-μ*θ))]
[Where:]
[P_start: Tension at the start of the arc segment]
[L_arc: Length of the arc segment (L_arc = R * θ)]
[μ: Friction coefficient]
[θ: Enclosed angle of the arc segment (in radians)]
[E_s: Elastic modulus of the steel tendon]
[A_p: Cross-sectional area of the steel tendon]
After writing, Chen Lin gently tapped the formula with the body of his pen and emphasized to the dazed Sun Yu, "Note that this formula is the precise solution obtained by directly integrating the differential equations above. It fundamentally avoids the approximation errors introduced by your traditional ’average tension method’."
’This is the feeling! This is it!’
Sun Yu listened to Chen Lin, nodding mechanically while screaming wildly in his mind.
’It’s that completely emotionless tone, as if he’s reading from a pre-written instruction manual!’
’That’s the master’s signature style!’
Chen Lin, of course, was unaware of Sun Yu’s rich inner drama. He picked up another sheet of scratch paper and began to explain the specific calculation process:
"Based on this model, the overall calculation process can be divided into four steps."
"First, data preprocessing. Take the centerline of the prestressed steel tendon from the design drawings—whether it’s a straight line, a circular arc, or a transition curve—and accurately discretize it into a series of sufficiently dense coordinate points (x_i, y_i, z_i)."
"Second, calculate the geometric properties of the micro-segments. For any micro-segment formed by two adjacent points, we treat it as a tiny straight line segment. You need to calculate two things: one is the length of this micro-segment, ds_i, which is easy; the other is the instantaneous curvature at the location of this micro-segment, κ_i, and its corresponding radius of curvature, R_i = 1/κ_i. This is the key part. It can be achieved using the three-point circle method or more advanced Differential Geometry methods."
"Third, the core step: tension transfer calculation. We start from the tensioning end and set the initial tension as P₀. Then, we iterate through each micro-segment i:"
Chen Lin’s pen moved without slowing as he wrote down the different cases with clear logic:
[If the micro-segment is on a straight section (κ_i ≈ 0):]
[P_end_i = P_start_i (ignoring minor friction loss)]
[ΔL_i = (P_start_i * ds_i) / (E_s * A_p)]
[If the micro-segment is on a curved section (κ_i > 0):]
[1. Calculate the incremental central angle for the micro-segment: dθ_i ≈ κ_i * ds_i (since dθ = ds / R = κ * ds)]
[2. Apply the precise friction model to calculate tension decay: P_end_i = P_start_i * e^(-μ*|dθ_i|)]
[3. Apply the precise elongation formula (micro-segment version): ΔL_i = (P_start_i * ds_i) / (E_s * A_p * μ * |dθ_i|) * (1 - e^(-μ*|dθ_i|))]
[Then, the starting tension for the next micro-segment is P_start_{i+1} = P_end_i. Repeat this process.]
"Fourth, and the final step, is to sum up the total elongation. ΔL_total = ΣΔL_i. Add up the elongations of all micro-segments, and that’s your final result."
After explaining the four steps, Chen Lin paused, picked up the lemonade from the table for a sip to wet his throat, and continued, "The benefit of this method is that it not only handles standard straight lines and circular arcs, but it can also provide an integrated solution for any oddly shaped spatial curves you might encounter in the future, like transition curve segments. All you need is to provide the curvature κ(s) at each point on the curve. Moreover, its core calculation process is also a single pass, so the computational load should be comparable to your traditional method."
He paused for a moment, then added, "In theory, this algorithm can be implemented very quickly in Python. However, I’ve only just started teaching myself programming and I’m not very proficient yet. So this time, I’ll probably have to trouble your company’s programmers to implement the code based on the process and formulas I’ve provided."
When he finished, he noticed that Sun Yu beside him had once again fallen into a daze, his eyes fixed motionlessly on the scratch paper.
Chen Lin reached out and waved a hand in front of his face.
Sun Yu jolted and snapped back to his senses.
He hadn’t zoned out because he didn’t understand Chen Lin’s proposal. Ever since the last consultation and his conversation with President Peng, he had developed an almost blind faith in Chen Lin’s mathematical abilities.
The reason he was so shocked again was that during the explanation, he had caught two details that were terrifying to think about.
First, from beginning to end, Chen Lin never once smudged, paused, or corrected anything—not a single formula or symbol he wrote on the scratch paper, nor any technical term he spoke. The entire process was as precise and fluid as a pre-programmed supercomputer outputting results.
Sun Yu thought back carefully and realized it had been the same during the first consultation. It was just that back then, his entire mind had been captivated by the indecipherable mathematics, causing him to completely overlook Chen Lin’s inhuman performance.
Second, and most terrifyingly, throughout the entire explanation, Chen Lin hadn’t looked at a computer, touched his phone, or consulted any materials. He didn’t even hesitate for a second to confirm anything. It was as if all the profound mathematical knowledge and sophisticated algorithmic processes were already seared into his brain, ready to be called upon effortlessly and perfectly at any moment.
Sun Yu looked at Chen Lin’s handsome face, then down at the sheets of scratch paper filled to the brim with writing. A thought uncontrollably bubbled up from the bottom of his heart.
’This must be what they call a protagonist in web novels or games.’
’And a nobody like me, with my average background, intelligence, and looks, can only be content with being a roadside NPC.’
’Sigh, how can the gap between people who studied math be so huge? Oh, wait, Chen Lin didn’t even study math. He’s a liberal arts major!’