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Finite Element Analysis

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Finite Element Analysis

Mathematical Background

http://www.kmitl.ac.th/~kpbumroo/12019801

- Finite Element Computations
- Domain discretisation
- Element type and shape function
- Weak form and Galerkin WRM
- Element assemble
- Boundary conditions
- Solving algebraic equations

12019801 Finite Element Analysis

local

global

nodal values

local

- Finite Element Computations
- Domain discretisation
- Element type and shape function

12019801 Finite Element Analysis

local

- Finite Element Computations
- Weak form and Galerkin WRM

12019801 Finite Element Analysis

local

global

- Finite Element Computations
- Element assemble

12019801 Finite Element Analysis

global

- Finite Element Computations
- Boundary conditions
- Neuman boundary conditions
- Dirichlet boundary conditions

- Algebraic Equation Solvers
- Direct solver
- Gaussian elimination with pivoting
- LU decomposition
- Choleski decomposition

- Iterative solver
- Gauss-Siedel
- Jacobi
- Conjugate gradient

- Direct solver

- Boundary conditions

12019801 Finite Element Analysis

- Newton-Cotes formulas
- The Trapezoidal rule
- The first of Newton-Cotes formulas

12019801 Finite Element Analysis

The composite trapezoidal rule

12019801 Finite Element Analysis

The composite trapezoidal rule – MatLab M-file

12019801 Finite Element Analysis

- Simpson’s rules
- Simpson’s 1/3 rule
- Simpson’s 3/8 rule

12019801 Finite Element Analysis

- Romberg integration – based on Richardson’s extrapolation
- Combine two numerical integrations from trapezoidal rule (with different h) for improving accuracy

12019801 Finite Element Analysis

Romberg integration

12019801 Finite Element Analysis

Romberg integration – MatLab M-file

12019801 Finite Element Analysis

Gaussian Quadrature

12019801 Finite Element Analysis

Gauss-Legendre

12019801 Finite Element Analysis

Change of interval

12019801 Finite Element Analysis

Two-dimensional integrals

12019801 Finite Element Analysis

Partial derivative

Partial differential equation = an equation involving partial derivatives of an unknown function of two or more independent variables

12019801 Finite Element Analysis

Linear

Non-Linear

Partial differential equation

Linear – if it is linear in the unknown function and all its derivatives

12019801 Finite Element Analysis

Linear second-order equations

Classification:

12019801 Finite Element Analysis

- Elliptic equations
- Laplace equation

- Parabolic equations
- Heat conduction equation

- Hyperbolic equations
- Wave equation

12019801 Finite Element Analysis