Application of the Finite Element Method in Numerical Computation

# 1. Introduction to the Finite Element Method

### 1.1 Origin and Development of the Finite Element Method

The Finite Element Method (FEM) originated in the 1950s, initially applied to the field of structural mechanics. With the development of computer technology and continuous improvement in numerical computing methods, the finite element method has gradually become a widely used method in numerical computations. The finite element method divides complex continuous bodies into a finite number of small elements, uses basis functions for interpolation, and employs numerical methods to solve algebraic equations, thus obtaining an approximate solution to the continuous domain.

### 1.2 Basic Principles of the Finite Element Method

The basic principles of the finite element method include discretization, variational methods, and interpolation. First, the continuous problem is transformed into a discrete one through discretization, dividing the continuous domain into a finite number of small elements and defining appropriate function spaces within each element. Then, the original problem is transformed into a variational problem using variational methods, where a test function is introduced and multiplied with the original equation and integrated over the entire discrete domain. Finally, the interpolation method matches the discrete approximations of known functions with the form of test functions, allowing the variational problem to be solved through the solution of a system of algebraic equations.

### 1.3 Advantages and Limitations of the Finite Element Method

The finite element method has the following advantages:
- Wide applicability: The finite element method can be used to solve various types of physical problems, such as structural mechanics, fluid dynamics, and heat conduction.
- Ability to handle complex geometries: Because the finite element method discretizes the continuous domain into small elements, it can handle complex geometries.
- High precision and flexibility: Higher accuracy of approximate solutions can be achieved by increasing the number of elements or by improving the order of interpolation functions.

However, the finite element method also has some limitations:
- Discretization errors: The results of the finite element method are influenced by discretization, and when there are fewer elements or lower-order interpolation functions, this may lead to results with significant errors.
- Mesh dependence: The results of the finite element method are influenced by the meshing, and poor mesh quality or significant mesh deformation may cause distorted results.
- Time-consuming: Due to the finite element method involving the solution of large-scale algebraic equations, it requires high computational resources and time.

In summary, as a numerical computing method, the finite element method has a wide range of applications and flexibility, but it is also necessary to select appropriate models and parameters based on the nature and requirements of specific problems.

# 2. Applications of the Finite Element Method in Structural Mechanics

Structural mechanics is the study of the mechanical behavior of structural objects under external forces. The finite element method has been widely applied in structural mechanics. This chapter will introduce the application of the finite element method in structural mechanics and is divided into the following three sections for detailed introduction.

### 2.1 Basic Equations of Structural Mechanics

Structural mechanics includes statics and dynamics. Statics studies the mechanical properties of objects in equilibrium states, mainly involving equilibrium equations and stress-strain relationships; dynamics studies the vibration and wave phenomena of objects under external forces, mainly including elastic wave equations and vibration equations.

### 2.2 Applications of the Finite Element Method in Static Analysis

In structural static analysis, the finite element method can be used to solve the distribution of forces and displacements of structural objects. By discretizing structural objects and establishing a finite element model, structural displacements, stress distributions, and other important parameters can be obtained using the finite element method. In the engineering field, the finite element method has been widely used in static analysis.

# Python code example
import numpy as np
from scipy.sparse import csr_matrix
from scipy.sparse.linalg import spsolve

# Building a finite element model

# Define node coordinates
nodes = np.array([[0, 0], [1, 0], [0, 1], [1, 1]])

# Define elements
elements = np.array([[0, 1, 3], [0, 3, 2]])

# Define boundary conditions and external loads
boundary_nodes = [0, 1, 2]   # Assume nodes on the boundary
boundary_values = np.array([0, 0, 0])   # Displacements at boundary nodes
external_force = np.array([0, 0, -1])   # External load

# Assemble stiffness matrix and load vector

# Calculate displacements

# Output results
print("Node displacements:", displacement)

Code summary: The above Python code demonstrates how to use the finite element method for structural static analysis, including the establishment of a finite element model, definition of boundary conditions, assembly of the stiffness matrix and load vector, and final displacement calculation.

### 2.3 Applications of the Finite Element Method in Dynamic Analysis

In structural dynamic analysis, the finite element method can be used to solve the vibration response and wave propagation of structural objects under external forces. By establishing the corresponding dynamic finite element model, the vibration modes, natural frequencies, and dynamic responses of structural objects can be analyzed.

// Java code example
// Building a finite element model

int[] boundaryNodes = {0, 1, 2};   // Assume nodes on the boundary
double[] boundaryValues = {0, 0, 0};   // Displacements at boundary nodes
double[] externalForce = {0, 0, -1};   // External load

// Assemble stiffness matrix and load vector

// Calculate dynamic response

// Output results
System.out.println("Node vibration modes:");

Code summary: The above Java code demonstrates how to use the finite element method for structural dynamic analysis, including the establishment of a dynamic finite element model, definition of boundary conditions, assembly of the stiffness matrix and load vector, and calculation of dynamic responses.

This section introduces the application of the finite element method in structural mechanics, including static analysis and dynamic analysis. The finite element method provides effective numerical comp