Biomathematical Analysis on Mechanical Engineering during COVID-19 Pandemic

**Discipline Formula**

In mechanical engineering, we first contact the mathematical content and the most direct contact with mathematics is the formula. In the research process of mechanical engineering, each specific discipline has its own natural laws, and the intuitive expression of these laws is the mathematical formula. At the same time, most of our research is to find out the mathematical relationship between multiple physical quantities. Such as:

In engineering mechanics, it is often necessary to consider the bending of the beam. Under the action of external force, the beam will bend, bend to a certain extent, and the beam may break. If the highway bend to what extent, will affect the high-speed vehicle? To what extent does the work piece cavity bend to avoid excessive milling? The curve of the beam after bending becomes the deflection curve. The equation of the deflection curve is as follows:

$$\frac{d^2 \omega}{dx^2} - \frac{M(x)}{EI} (1)$$

The mechanical characteristic $n = f(TM)$ of DC motor reflects the relationship between electromagnetic torque and speed, and the corresponding curve is a straight line. Let the load be a constant torque load, that is, the torque TL is constant. The mechanical characteristic expression of DC excitation motor can be written as:

$$TM/Tst + n/n0 = 1$$

There are a lot of mathematical formulas in every class and not one by one here.

**Mathematic Model**

In solving practical engineering problems or research process, often establish a mathematical model in advance, the actual problem into mathematical problems. The mathematical model is also used to describe a certain scientific law.

Taking solid heat transfer as an example, three-dimensional objects in space will produce temperature distribution in space when subjected to heat sources. The distribution of temperature is related to heat source and material. In the process of research, we data the actual material and heat source: the heat source will be transformed into a function that changes with time and space position. For materials, it is often regarded as parameter geometry, and the relevant parameters are extracted and calculated in the research process.

The simplest model of solid heat conduction is the Fourier heat transfer model. The differential equation is used to describe a heat source acting on a solid surface. The solid heat conduction in the direction perpendicular to the surface is in the temperature distribution. The expression is:

$$ \frac {d Q}{d r} = - \lambda A \frac {d t}{d \delta} \tag {2} $$

Where Q is capacity of heat transmission, 𝑟is time, 𝜆is thermal coefficient, 𝐴is area, 𝑡is temperature, 𝛿is depth.

The term on the left of the equation represents the heat per unit time acting on the surface of the object, an item on the right representing the temperature distribution in the inner depth direction of the heat conduction area.

However, the Fourier heat conduction model has certain limitations. For example, when high-speed laser acts on metal, we often use the two-temperature model to study it. The expression of the two-temperature model is as follows:

$$ C e \frac {\partial T e}{\partial t} = - \frac {\partial Q z}{\partial z} - \gamma (T e - T i) + S $$ (3) $$ C i \frac {\partial T i}{\partial t} = \gamma (T e - T i) $$ (4) $$ Q z = - k _ {e} \frac {\partial T e}{\partial z} $$ (5) $$ S = I (t) A \alpha \exp (- \alpha z) \tag {6} $$ Where 𝑧 is direction perpendicular to the surface, 𝑄𝑧is heat flux, 𝐼(𝑡) is laser intensity, 𝑇 is temperature, 𝐶 is isobaric specific heat capacity, 𝑒 is electron, 𝑖 is crystal lattice, 𝛾is lattice and electron coupling coefficient, A is surface transmittance, 𝑘𝑒 is thermal conductivity of electrons, 𝛼is absorption coefficient.

In this model, the object is regarded as the composition of electrons and lattices, and it is also transformed into a set of numerical values as parameters in the model. Each item in the model has its actual physical significance [3]. The physical meaning of each term from left to right is: the temperature change of electrons, the conduction of heat in the vertical surface direction, the coupling of energy between electrons and lattices and the heat source. In Equation [4], the temperature change of lattice and the coupling energy between lattice and electron are respectively. In Equation [5], is the heat conduction of heat flux and heat in the direction perpendicular to the surface [6]. Formula is the expression of heat source.

It can be seen that the above mathematical model is very complex, but when each parameter is determined, it is still difficult or even impossible to obtain its analytical solution. At this time, we need to use the knowledge of another mathematics course, namely, numerical calculation method. Our modeling of actual phenomena and problems is often very complex, and the equations in the model are often unable to calculate the analytical solution. However, we solve the model through iteration, fitting and other mathematical ideas to obtain the numerical solution of the model and the accuracy of the solution. However, a large number of calculations are needed to obtain the solution with high accuracy through this solution. Manual calculation is not only slow, but also there may be artificial calculation errors, even if the birth of computers perfectly solves this problem. In summary, the application of mathematics can quantitatively analyze the problems and phenomena in mechanical engineering, which is more helpful for us to understand and share the research content. The emergence of computers makes the application of mathematics in mechanical engineering more in-depth, and also makes mathematics play a greater role in mechanical engineering, helping to solve more problems [7, 8].

Finite Element Analysis

In today’s research on mechanical engineering problems, people often use computer simulation to replace the actual experiment, which greatly saves the time and cost of research. At present, many researches on mechanical engineering are about the establishment of models. When we establish a model that can better reflect the data required for the experiment, the related research on similar experiments can replace the real experiment by simulation of the model.

In the current field of mechanical engineering, the operation of a model often relies on the finite element method and its related software. The following will take a specific finite element simulation as an example to illustrate the important role of finite element method and mathematics in mechanical engineering. In mechanical engineering finite element simulation is often to establish a geometric model, and then give the parameters, determine the boundary conditions, mesh, set the numerical solver, and finally solve the data.

Taking laser ablation of materials as an example, the finite element simulation of materials begin and after laser ablation is shown in (Figure 1).

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The pictures vividly show the temperature distribution of the solid after laser ablation, which is helpful to study the indexes and states of the materials during the laser action.

An important performance of finite element analysis is grid division, in the simulation process, the more refined grid division will make us get more accurate results, as a cost, will also spend more time. Therefore, the grid division is very important, and fine grids are needed where the model is greatly affected or important. In order to ensure the simulation efficiency, the grid size of other parts can be slightly larger. For example, the meshing of the above finite element simulation is shown in (Figure 2). The upper surface grid in the figure is obviously denser than elsewhere, because the laser part is on the upper surface of the material and we pay more attention to the region.

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The above simulation examples do not have the same phenomenon as the real experiment. To make the simulation results closer to the real situation, it is necessary to improve the parameters and settings. However, we can still realize from the above simulation that the mathematical calculation method can simulate the phenomena in reality. The power of mathematics is fully demonstrated by finite element simulation in mechanical engineering.