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Introduction to Finite Element Method

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Introduction to Finite Element Method
Mathematic Model
Finite Element Method
Historical Background
Analytical Process of FEM
Applications of FEM
Computer Programs for FEM

1. Mathematical Model
(1) Modeling
Physical
Problems

Mathematica l Model

Solution

Identify control variables
Assumptions (empirical law)

(2) Types of solution
Sol.

Eq.

Exact Sol.
Approx. Sol.

Exact Eq.

Approx. Eq.







(3) Methods of Solution

(3) Method of Solution
A. Classical methods
They offer a high degree of insight, but the problems are difficult or impossible to solve for anything but simple geometries and loadings.
B. Numerical methods
(I) Energy: Minimize an expression for the potential energy of the structure over the whole domain.
(II) Boundary element: Approximates functions satisfying the governing differential equations not the boundary conditions. (III) Finite difference: Replaces governing differential equations and boundary conditions with algebraic finite difference equations. (IV) Finite element: Approximates the behavior of an irregular, continuous structure under general loadings and constraints with an assembly of discrete elements.

2. Finite Element Method
(1) Definition
FEM is a numerical method for solving a system of governing equations over the domain of a continuous physical system, which is discretized into simple geometric shapes called finite element.
Continuous system
Time-independent PDE
Time-dependent PDE

Discrete system
Linear algebraic eq.
ODE

(2) Discretization
Modeling a body by dividing it into an equivalent system of finite elements interconnected at a finite number of points on each element called nodes.

3. Historical Background

Chronicle of Finite Element Method
Year

Scholar

Theory

1941

Hrennikoff

Presented a solution of elasticity problem using one-dimensional elements.

1943

McHenry

Same as above.

1943

Courant

1947

Levy

Introduced shape functions over triangular subregions to model the whole region. Developed the force (flexibility) method for structure problem.

1953

Levy

Developed the displacement (stiffness) method for structure problem.

1954

Argyris & Kelsey

Developed matrix structural analysis methods using energy principles.

1956

Turner,
Clough,
Martin, Topp

Derived stiffness matrices for truss, beam and 2D plane stress elements. Direct stiffness method.

1960

Clough

Introduced the phrase finite element .

1960

Turner et. al

Large deflection and thermal analysis.

1961

Melosh

Developed plate bending element stiffness matrix.

1961

Martin

Developed the tetrahedral stiffness matrix for 3D problems.

1962

Gallagher et al

Material nonlinearity.

Chronicle of Finite Element Method
Year

Scholar

Theory

1963

Grafton, Strome

Developed curved-shell bending element stiffness matrix.

1963

Melosh

Applied variational formulation to solve nonstructural problems.

1965

Clough et. al

3D elements of axisymmetric solids.

1967

Zienkiewicz et.

Published the first book on finite element.

1968

Zienkiewicz et.

Visco-elasticity problems.

1969

Szabo & Lee

Adapted weighted residual methods in structural analysis.

1972

Oden

Book on nonlinear continua.

1976

Belytschko

Large-displacement nonlinear dynamic behavior.

~1997

New element development, convergence studies, the developments of supercomputers, the availability of powerful microcomputers, the development of user-friendly general-purpose finite element software packages.

4. Analytical Processes of Finite Element Method
(1) Structural stress analysis problem
A. Conditions that solution must satisfy
a. Equilibrium
b. Compatibility
c. Constitutive law
d. Boundary conditions
Above conditions are used to generate a system of equations representing system behavior.
B. Approach
a. Force (flexibility) method: internal forces as unknowns.
b. Displacement (stiffness) method: nodal disp. As unknowns.
For computational purpose, the displacement method is more desirable because its formulation is simple. A vast majority of general purpose FE softwares have incorporated the displacement method for solving structural problems.

(2) Analysis procedures of linear static structural analysis

A. Build up geometric model
a. 1D problem line b. 2D problem surface c. 3D problem solid B. Construct the finite element model
a. Discretize and select the element types
(a) element type
1D line element
2D element
3D brick element

(b) total number of element (mesh)
1D:
2D:
3D:

b. Select a shape function
1D line element: u=ax+b
c. Define the compatibility and constitutive law

d. Form the element stiffness matrix and equations
(a) Direct equilibrium method
(b) Work or energy method
(c) Method of weight Residuals

e. Form the system equation
Assemble the element equations to obtain global system equation and introduce boundary conditions

C. Solve the system equations
a. elimination method
Gauss’s method (Nastran)
b. iteration method
Gauss Seidel’s method
Displacement field

strain field

stress field

D. Interpret the results (postprocessing)
a. deformation plot

b. stress contour

5. Applications of Finite Element Method
Structural Problem

Non-structural Problem

Stress Analysis
- truss & frame analysis
- stress concentrated problem
Buckling problem
Vibration Analysis
Impact Problem

Heat Transfer
Fluid Mechanics
Electric or Magnetic
Potential

6. Computer Programs for Finite Element Method
結構

非線


最佳


複合
材料

ANSYS









NASTRAN









ABAQUS





MARC





碰撞
分析

噪音
分析

流力
分析

























LS-DYNA3D

機構
分析

模流
分析

熱傳
分析




MSC/DYNA

疲勞
分析




ADAMS/
DADS
COSMOS















MOLDFLOW



C-FLOW



PHOENICS



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