Finite element analysis
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Finite Element Analysis. Mathematical Background. http://www.kmitl.ac.th/~kpbumroo/12019801. Summary from Previous Lecture. Finite Element Computations Domain discretisation Element type and shape function Weak form and Galerkin WRM Element assemble Boundary conditions

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Finite Element Analysis

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Finite element analysis

Finite Element Analysis

Mathematical Background

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


Summary from previous lecture

Summary from Previous Lecture

  • 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


Summary from previous lecture1

Summary from Previous Lecture

local

global

nodal values

local

  • Finite Element Computations

    • Domain discretisation

    • Element type and shape function

12019801 Finite Element Analysis


Summary from previous lecture2

Summary from Previous Lecture

local

  • Finite Element Computations

    • Weak form and Galerkin WRM

12019801 Finite Element Analysis


Summary from previous lecture3

Summary from Previous Lecture

local

global

  • Finite Element Computations

    • Element assemble

12019801 Finite Element Analysis


Summary from previous lecture4

Summary from Previous Lecture

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

12019801 Finite Element Analysis


Numerical integration formulas

Numerical Integration Formulas

  • Newton-Cotes formulas

  • The Trapezoidal rule

    • The first of Newton-Cotes formulas

12019801 Finite Element Analysis


Numerical integration formulas1

Numerical Integration Formulas

The composite trapezoidal rule

12019801 Finite Element Analysis


Numerical integration formulas2

Numerical Integration Formulas

The composite trapezoidal rule – MatLab M-file

12019801 Finite Element Analysis


Numerical integration formulas3

Numerical Integration Formulas

  • Simpson’s rules

    • Simpson’s 1/3 rule

    • Simpson’s 3/8 rule

12019801 Finite Element Analysis


Numerical integration of functions

Numerical Integration of Functions

  • Romberg integration – based on Richardson’s extrapolation

    • Combine two numerical integrations from trapezoidal rule (with different h) for improving accuracy

12019801 Finite Element Analysis


Numerical integration of functions1

Numerical Integration of Functions

Romberg integration

12019801 Finite Element Analysis


Numerical integration of functions2

Numerical Integration of Functions

Romberg integration – MatLab M-file

12019801 Finite Element Analysis


Gaussian quadrature

Gaussian Quadrature

Gaussian Quadrature

12019801 Finite Element Analysis


Gaussian quadrature1

Gaussian Quadrature

Gauss-Legendre

12019801 Finite Element Analysis


Gaussian quadrature2

Gaussian Quadrature

Change of interval

12019801 Finite Element Analysis


Gaussian quadrature3

Gaussian Quadrature

Two-dimensional integrals

12019801 Finite Element Analysis


Partial differential equations

Partial Differential Equations

Partial derivative

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

12019801 Finite Element Analysis


Partial differential equations1

Partial Differential Equations

Linear

Non-Linear

Partial differential equation

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

12019801 Finite Element Analysis


Partial differential equations2

Partial Differential Equations

Linear second-order equations

Classification:

12019801 Finite Element Analysis


Partial differential equations3

Partial Differential Equations

  • Elliptic equations

    • Laplace equation

  • Parabolic equations

    • Heat conduction equation

  • Hyperbolic equations

    • Wave equation

12019801 Finite Element Analysis


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