Overview of the finite element method.
The Finite Element Method (FEM) is a method for numerically analysing the behaviour and stress analysis of objects and structures, enabling detailed modelling of the effects of forces and loads on complex structures and objects and thereby calculating their physical behaviour, such as stress and displacement. The method can be used to calculate physical behaviour such as stresses and displacements. An overview of the finite element method is given below.
1. Principle: The finite element method divides a continuum into a finite number of elements and approximates the displacements and stresses within each element. This enables the differential equations of the continuum to be transformed into a finite number of algebraic equations and solved numerically. The main principles are as follows
Division: the analysis domain is divided into a finite number of elements. Typically, triangles and quadrilaterals (2D) and tetrahedra and hexahedra (3D) are used.
Approximation: physical quantities such as displacements and stresses within each element are expressed by approximate functions. Polynomials and trigonometric functions are usually used in this case.
Variational principle: Models the behaviour of a continuum by finding an approximate solution that minimises the energy variant within each element of the partition.
Construction of coupled equations: approximate solutions for each element are combined to construct a coupled equation (usually in weak form). The coupled equations are adjusted to satisfy the boundary conditions.
Solution: the physical quantities such as stresses and displacements of the continuum are calculated by solving the constructed coupled equations. As a typical solution method, finite element analysis (FEA) is used in the finite element method.
2. Features: the characteristics of the finite element method include
Flexibility: it can be applied flexibly to complex geometries and physical phenomena.
Accuracy: the accuracy of the analysis results can be improved by using finer element partitioning.
Computational efficiency: the computational load can be adjusted by setting the element geometry and size appropriately, resulting in efficient analysis.
Algorithms related to finite element methods.
The main algorithms associated with the finite element method (FEM) include.
1. calculation of the element stiffness matrix: the element stiffness matrix will be a matrix representing the relationship between displacements and stresses in the element. It is calculated based on the geometry and material properties of the element and typical algorithms include
Direct methods: these methods are based on the analytical determination of the element geometry and material properties. Specifically, the element stiffness matrix is obtained numerically by integration based on the defining equation of the stiffness matrix.
Numerical integration method: the element stiffness matrix is calculated by numerically approximating the integral. Typical methods include Gaussian integration and the Newton-Cotes method.
2. application of boundary conditions: boundary conditions define the displacement and stress conditions at the boundaries of the analysis domain. These conditions are applied as constraints to find the solution and the main algorithms for applying boundary conditions include
Dirichlet boundary conditions: specify boundary conditions with known displacements or stresses. These conditions are applied by directly setting the displacement or stress values.
Neumann boundary conditions: specify boundary conditions where external forces or loads are known. These conditions are applied by setting the values of forces and stresses on the boundary.
3. solution of coupled equations: coupled equations are equations that combine element-specific displacement and stress relationships to obtain an overall solution. The main algorithms for solving these equations include.
Direct methods: these involve directly solving the coupled equations, specifically by converting the coupled equations into matrix form and calculating the inverse matrix to obtain the solution.
Iterative methods: methods that solve the coupled equations iteratively, specifically by setting an initial estimate and iteratively updating the equations and modifying the solution until convergence. Typical methods include the Gauss-Zeidel method described in “Overview of the Gauss-Zeidel method, algorithms and implementation examples” and the conjugate gradient method described in “Overview of the conjugate gradient method, related algorithms and implementation examples“.
4. Combination of elements: The overall solution can be obtained by combining the elements that make up the entire analysis domain. Specifically, physical quantities such as displacements and stresses of the entire analysis domain are obtained by appropriately combining the boundary conditions and coupled equations between the elements.
Application examples of the finite element method.
The finite element method (FEM) is widely applied in various engineering and scientific disciplines. Typical applications of the finite element method are described below.
1. structural analysis: the finite element method is widely used in the analysis of structures such as buildings, bridges, aircraft and automobiles. The finite element method is used for stress analysis, displacement analysis, vibration analysis and fatigue analysis of these structures, e.g. to evaluate the load and earthquake resistance of bridges, strength analysis of aircraft structures and crash testing of automobiles.
2. fluid mechanics: the finite element method is also used in the analysis of liquid and gas flows. In the field of fluid mechanics, the finite element method is used to analyse fields such as pressure, velocity and temperature, for aerodynamic analysis of aircraft, wind tunnel testing of automobiles and hydraulic analysis of hydropower projects.
3. electromagnetics: finite element methods are also used to analyse electromagnetic fields and to design electrical equipment. In the field of electromagnetics, the finite element method is used to analyse the distribution of electric and magnetic fields, design electromagnetic equipment and evaluate the effects of electromagnetic fields, for example, in the design of motors and generators, evaluation of electromagnetic radiation and analysis of electromagnetic interference.
4. heat conduction analysis: in heat conduction analysis, the finite element method is used to analyse phenomena such as heat conduction, radiation and convection. This makes it possible to evaluate the temperature distribution of mechanical components and the efficiency of cooling systems, e.g. heat load analysis of engine components, cooling design of electronic equipment and evaluation of the thermal insulation performance of buildings and facilities.
5. biomechanics: In biomechanics, the finite element method is used to analyse the behaviour of the human body and animal tissue and skeleton. This has led to research into the distribution of loads in the human body, fracture mechanisms and the design of artificial joints, e.g. gait analysis of the human body, evaluation of the effectiveness of fracture treatment and the design of artificial heart valves.
The finite element method plays an important role as an advanced numerical analysis method in the analysis and design of objects and structures.
Examples of finite element method implementations.
An example implementation of the finite element method (FEM) is given. Here, a Python code is presented that performs a static stress analysis of a one-dimensional elastic body. The code implements the calculation of the stiffness matrix of the element, the application of boundary conditions and the solution of the coupled equations.
import numpy as np
# g
E = 200e9 # Young's modulus [Pa]
A = 0.01 # cross-sectional area [m^2]
L = 1.0 # Length of stick [m]
F = 1000 # external force [N]
n_elements = 5 # Number of elements
# Calculation of element stiffness matrices.
def calculate_stiffness_matrix(element_length):
k = (E * A / element_length) * np.array([[1, -1], [-1, 1]])
return k
# Construction of a global stiffness matrix.
def assemble_global_stiffness_matrix(n_elements, element_length):
K_global = np.zeros((n_elements + 1, n_elements + 1))
for i in range(n_elements):
k = calculate_stiffness_matrix(element_length)
K_global[i:i+2, i:i+2] += k
return K_global
# Application of boundary conditions
def apply_boundary_conditions(K_global):
K_global[0, :] = 0
K_global[0, 0] = 1
return K_global
# Creation of external force vectors
def create_load_vector(n_elements, F):
f = np.zeros((n_elements + 1, 1))
f[-1] = F
return f
# Solution of coupled equations
def solve_system(K_global, f):
u = np.linalg.solve(K_global, f)
return u
# main function
def main():
# Element length
element_length = L / n_elements
# Construction of a global stiffness matrix.
K_global = assemble_global_stiffness_matrix(n_elements, element_length)
# Application of boundary conditions
K_global = apply_boundary_conditions(K_global)
# Creation of external force vectors
f = create_load_vector(n_elements, F)
# Solution of coupled equations
u = solve_system(K_global, f)
print("Displacements (m):")
print(u)
if __name__ == "__main__":
main()The code divides a one-dimensional elastic body into five elements and calculates the stiffness matrix of each element. Boundary conditions are then applied, external force vectors are created and the coupled equations are solved to obtain the displacements. The results of the run show the displacements at each node. In this way, a simple static stress analysis of elastic bodies can be implemented using the finite element method.
Challenges and remedies for finite element methods.
The finite element method (FEM) is a powerful numerical analysis method, but it has several challenges. The main challenges of the finite element method and the measures taken to address them are described below.
1. computational cost and memory usage:
Challenge: As the complexity of the analysis object increases, the number of elements and the amount of calculations increases rapidly, leading to increased computational costs and memory usage.
Solution:
High-performance computing: use high-performance computing resources to analyse large problems.
Parallel computing: implement parallelisation of computations and use multiple processes and threads to increase computation speed.
Memory efficiency: use memory-efficient data structures, such as sparse matrix formats, to reduce memory usage.
2. non-linearity of geometry and materials:
Challenge: when considering non-linear behaviour of objects and large deformations of geometry, the analysis becomes more complex and computationally difficult.
Solution:
Non-linear analysis methods: select appropriate analysis methods to account for non-linear material behaviour and geometry deformations. Examples include finite deformation analysis and contact analysis.
Iterative solution methods: use methods for solving non-linear problems iteratively to achieve convergence. Newton-Raphson and modified Newton methods are commonly used.
3. element quality:
Challenge: poor element geometry and quality may reduce the accuracy of the analysis results.
Solution:
Element optimisation: use optimisation methods to obtain good element geometry and apply mesh generation algorithms and element geometry optimisation methods.
Admissibility conditions: define evaluation criteria for element geometry and quality to ensure that the admisibility conditions are met. This improves the accuracy of the analysis results.
4. linearity approximations:
Challenge: for some problems, the results of the analysis may differ from the actual physical phenomena due to inadequate linearity approximations.
Solution:
Advanced analytical methods: apply advanced analytical methods and use extended or integrated approaches to finite element methods to accurately model complex phenomena such as non-linear, thermal and contact.
Comparison with experiment: compare the analytical results with experimental results to check the validity of the results and adjust the model if necessary.
5. setting boundary conditions:
Challenge: setting appropriate boundary conditions can be difficult and may affect the results of the analysis.
Solution:
Understand the physics: set appropriate boundary conditions based on a good understanding of the physics.
Verification and review: improve the reliability of the analysis results by verifying and reviewing the effects of the boundary conditions.
Reference Information and Reference Books
More information on the combination of simulation and machine learning can be found in “Simulation, Data Science, and Artificial Intelligence” and “Artificial Life and Agent Technology. See also “Simulation, Data Science, and Artificial Intelligence” and “Artificial Life and Agent Technology. Reinforcement learning techniques that take a simulation approach are described in “Theory and Algorithms of Various Reinforcement Learning Techniques and Their Implementation in Python” and the stochastic generative model approach described in “About Stochastic Generative Models“.
Practical Simulations for Machine Learning” as a reference book.
“Reservoir Simulations: Machine Learning and Modeling“
“Machine Learning in Modeling and Simulation: Methods and Applications“
“Statistical Modeling and Simulation for Experimental Design and Machine Learning Applications“
コメント