Numerical analysis methods in engineering analysis and basic knowledge of computer-aided engineering analysis.
The weak form of the equivalent integral of the differential equation.
The component modules and calculation process of the finite element method.
The role of weighted residual method and virtual work principle in constructing finite element determinants.
1.1 Numerical analysis methods in engineering analysis
In order to design, construct and operate an engineering system, engineers must first fully understand the behavior (structural characteristics, internal links and subsystems) of these engineering systems
This process is engineering analysis.
Figure 1-1 shows the traditional engineering analysis steps.The squares represent the models at different analysis stages.
The circles represent the four analysis steps.
These five models are actually different representations of the same thing:
Engineering system: real objects in the real world,
Analysis model: A simplified abstract model.
Mathematical equations often have the form of differential equations,
But analytical or numerical solutions
If the numerical solution is then the analytical model is called the numerical model.
When engineering analysis, engineers use (CAE) to complete many of the steps in Figure 1-2.
If you are unfamiliar with mathematical methods or computer-aided engineering analysis
The "building equations" and "solving equations" and their related models (analysis models, mathematical equations and numerical solutions) can be packed in a "black box".
The black box represents a closed computer processing core:
The computer automatically converts an analytical model into a mathematical equation
And easily solve
Input data is injected outside the black box
The final numerical result is also represented graphically outside the black box
Figure 1-3 shows the steps of computer-aided engineering analysis. The black box represents a computer program for solving engineering problems:
An analysis model is used as input, and a numerical solution is used as output.
Mechanics Mathematics and Computerists
Developed a method for solving differential equations in engineering problems
FDM, FEM, boundary element method
In essence, the solution grid is discretized, and then the numerical results are obtained by solving the equations.
The most mature and widely used is the finite element method.
Change the computer-aided engineering analysis from Figure 1-2 to Figure 1-3.
When the finite element analysis method is used, Figure 1-4 is special for Figure 1-2.
In Figure 1-4, the analytical model (or finite element model) and numerical solution are still in the black box.
The two steps of "building model equations" and "solving equations" are replaced by "finite element analysis". The step of "building a system mathematical model" is often called "preprocessing"
"Data processing and display" is called "post-processing"
Figure 1-4 divides computer-aided engineering analysis into three main steps
Pre-treatment
Finite element analysis
Post-processing
1.2 The equivalent integral form of the differential equation
1.2.1 Controlling differential equations
The nature of many engineering problems is a physical problem, and the differential process of control can be used to describe the essential process of material changes .
Consider the deterministic conditions of space, time, and time delay, then form the boundary value and initial conditions that can be solved
It is a differential operator for basic unknown functions in differential equations.
