Analysis of the 8th Shuwei Cup Mathematical Modeling Challenge in 2023

In order to better let everyone choose the topic of this Dimension Cup competition, I will briefly analyze the topic of this competition. In the topic selection of this competition, the postgraduate and undergraduate groups should choose one of A and B to complete the answer sheet, and the specialist group should choose one of B and C to complete the answer sheet. This also implies that the difficulty of this competition is A>B>C

The initial estimate of the number of candidates is also B>C>A

The following is a brief analysis of each question, so that you can choose the topic in advance.

Question A: Research on Water Pollution of River-Groundwater System

At first glance, this problem belongs to the category of physical equations, which is relatively difficult. We need to analyze and establish mathematical models of convection, dispersion and adsorption of organic pollutants in river-groundwater systems by consulting relevant literature and information. By simply looking up information (the formula is the input language of latex, if you don't understand, you can directly look at the picture)

The mathematical model of convection, dispersion and adsorption of organic pollutants in the river-groundwater system can refer to the following formula:

$$\frac{\partial C}{\partial t}+\nabla\cdot(\mathbf{v}C)=D\nabla^2C-\lambda C+R$$

Among them, $C$ is the concentration of organic pollutants, $\mathbf{v}$ is the velocity of groundwater, $D$ is the hydrodynamic dispersion coefficient, $\lambda$ is the degradation rate of organic pollutants, $R$ is the organic source or sink of pollutants.

For adsorption, a two-mode adsorption model can be used, namely:

$$S=\frac{K_dC}{1+bC}+\frac{S_0bC}{1+bC}$$

Among them, $S$ is the adsorption amount on the sediment, $K_d$ is the linear adsorption coefficient, $S_0$ is the maximum adsorption capacity, and $b$ is the adsorption surface affinity constant.

For the retardation effect, the retardation coefficient (R) can be used to express, namely:

$$R=\frac{1}{1+\rho_b\frac{dS}{dC}}$$

where $\rho_b$ is the density of the sediment.

On the whole, it is difficult and requires good mathematical or physical ability. Questions two and three will be explained in the follow-up question analysis.

Problem B Energy-saving train operation control optimization strategy

Question B is the same as question A. It is similar to the physical differential equation model. To solve this problem, we can use the method of numerical integration to combine the kinematic equation of the train during operation with traction force, braking force, and resistance. relationship combined. The following is a simple Python program (if you need it, I can also write the corresponding matlab code) to calculate the kinematic parameters and energy consumption of the train

import numpy as np
import matplotlib.pyplot as plt

# Parameters
m = 176.3 * 1000      # Mass of the train (kg)
p = 1.08              # Rotational mass factor
v_max = 100 / 3.6     # Maximum velocity (m/s)
f_davis = lambda v: 2.0895 + 0.0098*v + 0.006*v**2  # Davis resistance equation
F_max = 310 * 1000    # Maximum traction force (N)
B_max = 760 * 1000    # Maximum braking force (N)
L = 5144.7            # Distance between A and B (m)
delta_t = 0.01        # Time step (s)

# Initial conditions
x = 0                 # Initial position (m)
v = 0                 # Initial velocity (m/s)
t = 0                 # Initial time (s)
E_kin = 0             # Initial kinetic energy (J)
E_pot = 0             # Initial potential energy (J)
E_loss = 0            # Initial energy loss (J)

# Arrays for storing results
x_array = []
v_array = []
t_array = []
F_array = []
E_array = []

# Simulation loop
while x < L:
    # Calculate acceleration
    F_net = F_max if x < L/2 else -B_max  # Traction force or braking force
    F_resist = f_davis(v)                # Resistance force
    a = (F_net - F_resist) / (m + p*v*v)
    # Calculate velocity and position
    v += a * delta_t
    x += v * delta_t
    t += delta_t
    # Store results
    x_array.append(x)
    v_array.append(v)
    t_array.append(t)
    F_array.append(F_net if F_net > 0 else -F_resist)  # Store positive force (traction) or negative force (braking)
    # Calculate energy
    E_kin = 0.5 * m * v**2
    E_pot = m * 9.8 * x
    E_loss += abs(F_resist * v * delta_t)  # Accumulate energy loss
    E_array.append(E_kin + E_pot - E_loss)

# Plot results
fig, axs = plt.subplots(2, 2, figsize=(12, 8))
axs[0, 0].plot(x_array, v_array)
axs[0, 0].set_xlabel("Distance (m)")
axs[0, 0].set_ylabel("Velocity (m/s)")
axs[0, 1].plot(x_array, F_array)
axs[0, 1].set_xlabel("Distance (m)")
axs[0, 1].set_ylabel("Force (N)")
axs[1, 0].plot(x_array, t_array)
axs[1, 0].set_xlabel("Distance (m)")
axs[1, 0].set_ylabel("Time (s)")
axs[1, 1].plot(x_array, E_array)
axs[1, 1].set

In this case, it is necessary to further consider various factors in the train running process, such as the speed limit and slope of different road sections, the dynamic characteristics of the train motor, the use of energy storage devices, etc., to build a more accurate model. The specific modeling process needs to comprehensively consider the knowledge of physics, mathematics, mechanics and electricity, and use appropriate software tools for simulation and analysis.

There are many factors to be considered in the modeling process, such as the dynamic characteristics of the train, the control strategy of the traction system and the braking system, and the energy management system. In specific modeling, it is necessary to determine the input and output of the model, as well as the relationship between various subsystems, and then determine the structure and parameters of the model. At the same time, the validity and reliability of the model also needs to be considered, and the model should be verified and tested to ensure that it can accurately reflect the actual situation.

In general, establishing a train operation model is a complex process, which requires full consideration of various factors, and the use of appropriate modeling methods and software tools for simulation and analysis. At the same time, the model needs to be verified and tested to ensure that it can accurately reflect the actual situation and provide effective support for practical applications.

Alternatively, we can also use

According to the search results, the train operation modeling method can be divided into the following steps: use Simulink or other tools to build a train operation simulation model, you can refer to the Powertrain Blockset and Vehicle Dynamics Blockset toolboxes provided by MATLAB, or use professional software such as CarMaker, CarSim and others.

Design the speed control algorithm for train operation, and automatically adjust the traction/braking force according to the train's operating status, line parameters, speed limit conditions and other information, so that the train runs along the optimal speed-distance curve.

Verify the speed control effect of train operation through simulation experiments, draw speed-distance curves, traction braking force-distance curves, time-distance curves and energy consumption-distance curves, and analyze the safety, punctuality, comfort and performance of train operations Economical¹⁴.

The running time of the program depends on the complexity of the simulation model, the efficiency of the control algorithm, the performance of the computer and other factors, which cannot be generalized. When it is necessary to obtain curves with different arrival times, it can be realized by adjusting the parameters or the objective function in the control algorithm.

 

Question C Production of IUDs

Question C is the easiest question in this competition, but unfortunately the choice of questions is limited. The following is a brief explanation of the problem-solving ideas.

Question 1: To analyze whether there is a significant difference in the clinical data of the two hospitals, that is, to perform a significant analysis, the corresponding software of SPSS, matlab, and python can be realized.

import pandas as pd
import numpy as np
import scipy.stats as stats

# 读取数据
data = pd.read_csv('data.csv')

# 计算相关系数和P值
corr_matrix = data.corr()
p_matrix = np.zeros(corr_matrix.shape)
for i in range(corr_matrix.shape[0]):
    for j in range(corr_matrix.shape[1]):
        pearson_coef, p_value = stats.pearsonr(data.iloc[:, i], data.iloc[:, j])
        p_matrix[i, j] = p_value

# 显示相关系数矩阵和P值矩阵
print("Correlation Matrix:")
print(corr_matrix)
print("P-Value Matrix:")
print(p_matrix)

# 进行显著性检验
threshold = 0.05  # 设置显著性水平
sig_matrix = p_matrix < threshold  # 判断P值是否小于显著性水平
print("Significant Matrix:")
print(sig_matrix)

Only the significance test of correlation is considered here. In fact, significance analysis can also involve multiple regression analysis, variance analysis, chi-square test and other methods. In practical application, we need to choose the appropriate significance test method according to the specific problem, and use the corresponding tool library for analysis.

Question 2. Analyze the relationship between the physical indicators of the subjects and the follow-up complaints, typical correlation analysis. Choose the appropriate method according to the situation.

Question 3: Analyze the quality of VCu260 and VCu380 memory type IUD which is better. It can be understood as an optimization model, and the 0-1 variable is set as the decision variable to solve the optimal IUD, or a comprehensive evaluation model can be established, and the specific choice varies from pair to pair.