Reshaping life starting from the Junior Class of HKUST

Chapter 54

After Wu Zhe came back from training, he brought breakfast for the three people in the dormitory as usual.

Huang Minghai and the others also got up today.Everyone went to bed early yesterday.Because the National Awards start today, I don't want to sleep well for the next three days. The 72-hour game time is still relatively tight.

At eight o'clock in the morning, the four people in Wu Zhe's dormitory left on time.The four of them also carried their laptops.

The computer room prepared by the school for the modeling competition is not in the east campus, but in the west campus.So you have to take the bus at the school gate.

The good thing is that after considering this situation, the school set up a few temporary sleeping spots next to the computer room.You go straight in, as long as there is no one, you can fall down and sleep.

Wu Zhe and the others arrived at the competition venue quite late, and it was full of people.After all, there are 1330 colleges and universities and more than 3 college students participating in the National Award.Guangke University has about 100 teams.

The competition is jointly organized by the Ministry of Education and the Chinese Mathematical Society to ensure authority and fairness.

Wu Zhe, Wang Chao, Huang Minghai, and Shen Zhiwen found seats and sat down.

At exactly nine o'clock, the bell rang.

Everyone quickly logged on to the website and downloaded the title.

The Higher Education Club Cup consists of four questions: A, B, C, and D.The undergraduate group chooses a short answer from two questions A and B. C and D are for the Opus Work group, the so-called junior college group.

Wu Zhe and the others don't need to consider questions C and D.As long as you do one of the two questions A and B, you can complete it within 72 hours and upload the MD5 code generated by the official website software. You also need to upload the corresponding pdf format paper on the end day.

Time is tight, and the topic selection will usually be completed within half an hour to an hour.The rest is the liver.

The topic download is complete.Wu Zhe took a look and found that this year's exams were relatively routine, and there were no particularly strange ones.

Question A: Optimizing the light source of the car light line.

Problem B: Math in the lottery.

The first question is about line generation, calculus, flexible use of geometry plus an optimization problem.The second problem is statistical nonlinear optimization and evaluation problems.Relatively speaking, it is relatively simple.

Wu Zhe estimated that more people would choose the second question. No.1 is gambling, no matter what the lottery says it is a game of gambling.What's more, now is the time to apply what you have learned.They will always think about whether the chances of winning are great or not, and whether it is possible to make a big fortune with a small one.

Sure enough, it didn't take long to start.There are many more people in the room shouting to choose B.Wu Zhe curled his lips, he didn't even need to think about it carefully, and the answer they came up with in the end probably wouldn't buy lottery tickets again.

Wu Zhe chose question A, and didn't bother Huang Minghai and the others to choose the topic.

Wu Zhe now has to solve three problems: (1) Under the condition of meeting the design specification, calculate the length of the line light source to minimize the power of the line light source.

(2) For the length of the light source obtained, draw the bright area of ​​the reflected light on the test screen in the coordinate system with a ruler.

(3) Discuss the rationality of the design specification.

Known: Its opening radius is 36 mm, and its depth is 21.6 mm.After the focal point of the car light, in the horizontal direction perpendicular to the axis of symmetry, a uniformly distributed light source of a certain length is placed symmetrically.

Then take point B and point C on the same side of point A on the straight line, so that AC=2AB=2.6 meters.It is required that the light intensity at point C is not less than a certain rated value (it can be taken as 1 unit), and the light intensity at point B is not less than twice the rated value (only one reflection needs to be considered).

Establish the coordinate system as follows, record the length of the line light source as l, the power as W, the light intensity at points B and C as W and W respectively, first sum the expression, and then establish the mathematical model of the whole problem.
The following are all in millimeters. From the given information, it is not difficult to find the equation of the reflective surface of the car lamp as , and the focal coordinates are (0, 0, 15).The energy reflected by a point light source with unit energy at point P(0, w, 15) to point C(0, 2600, 25015)

Let the coordinates of the reflection point be Q. Record the incident vector as , and the normal direction outside the reflection surface at this point is . It is not difficult to obtain the reflection vector, satisfying K as a constant.
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Where e is the light-up value, l is the maximum point, which is the maximum range of investigation, for example, it is taken as 20mm.Also has similar properties, and the bright value and the maximum point are correspondingly moved to the right. Numerical solution

Record its solution as, and then find out, it is not difficult to see that e falls in l.

This proves that e is indeed the minimum.Finally calculated e=3.294
The next step is to seek the problem of emitting bright areas.Respectively discretize the line light source and the reflective surface of the car light into a set of point light sources and surface elements, calculate the reflected light of each point light source on each light reflective surface element, and judge whether it intersects with the reflective surface of the car light, if it intersects, once The reflected light cannot reach the test screen, otherwise, find the intersection point of the reflected light and the plane of the reflective screen, which is the reflective bright spot.

The collection of all these bright spots is the reflective bright area.The upper part of the bright area is shown in the figure below (the abscissa is the x-axis, the ordinate is the y-axis, and the unit is mm), and the lower half and the upper part are symmetrical about the x-axis.
Another method of calculation is to establish a numerical model of the problem and solve it by numerical simulation. The specific method is: after obtaining the mathematical model of the reflected light and the energy reflected on the test screen, the line light source and the reflective surface of the car light are discrete Turn it into a collection of point light sources and surface elements, and take a small surface element near point B (or C) of the test screen. Calculate the reflected light of each point light source on each light reflection surface element, and combine all the light that can reach this surface element The reflected energy of the reflected light is superimposed and divided by the area of ​​the surface element is the reflected energy density of point B (or C).
Assume that the directional light source is transparent.It seems more realistic to assume that directional light sources are opaque.At this time, it is necessary to consider the occlusion of the reflected light by the line light source itself, the calculation will be more complicated, and the calculation results will be different.

At this point in the calculation, the problem has actually been solved by Wu Zhe.

Next, Wu Zhe only needs to program to convert the mathematical thinking just now into computer language.

Looking up at the sky, it was already time to eat.Many people are just getting the idea.

Wu Zhe is not in a hurry, his progress is obviously much faster than others.He glanced at Huang Minghai and the others.Wang Chao was doing calculations there, while Shen Zhiwen and Huang Minghai watched from behind, asking a few questions from time to time.It seems that they also divided the work, Wang Chao on the first day, Shen Zhiwen on the second day, and Huang Minghai on the final paper.

It looks like they don't want to eat anymore.Wu Zhe was about to go out for dinner, so he brought them here by the way.

After making a gesture with the boss Huang Minghai, Wu Zhe also stepped out of the computer room.

As soon as I went out, I heard someone calling myself.

"Wu Zhe, here." Qi Ming, secretary of the Communist Youth League, said while carrying the box.

"Have you come to the West Campus? Didn't you not participate in the National Award?" Wu Zhe was a little surprised.

"Here, I'm here to bring you lunch. This was decided by our whole class. Although we can't play, we have to do the logistics well." Qi Ming shook his hair and said.

"Thank you, I'm going out to eat? I'm not polite!" Wu Zhe took the bag from Qi Ming's hand.

"Well, the top one belongs to you, and the bottom one belongs to Wang Chao and the others. Don't forget." Qi Ming said with a smile.

"Okay, I can't forget, you go back first!" Wu Zhe raised the bag in his hand to Qi Ming.

"Okay, then I'm leaving, you don't have to go out to eat at night, I will deliver it to you on time in these three days."

After Qi Ming finished speaking, he turned and left with a smile. Now Wu Zhe and the others are racing against time, so don't delay them if you can.

(End of this chapter)