Chapter 2

Formulation of Finite Element Method by

Variational Principle

The Concept of Variation of FUNCTIONALS Variation Principle: Is to keep the DIFFERENCE between a REAL situation and an APPROXIMATE situation in MINIMUM

REAL SITUATION

APPROXIMATIE SITUATION = Minimum

Functional = “function of functions” In reality, we will find FUNCTIONALS to represent the “Real” and “Approximate” situations

Thus, derivation of appropriate FUNCTIONALS is an important step in FE formulations

Mathematical tool for determining MINIMUM of a function

Minimum/maximum values of function determined by calculus: Given function: f(x) Procedures in determining maximum value of f(x): 1) Solve the equation The solution xm can be either maximum or minimum value of function f(x) 2) Check:

0)(=

dxxdf

with solution x = xm

( ) 02

2

>= mxxdx

xfdIf Xm is the minimum value of f(x)

( ) 02

2

<= mxxdx

xfdIf Xm is the maximum value of f(x)

The Concept of Discretization

The essence of FEM is “Divide and Conquer” - meaning if the geometry/loading/boundary conditions of entire medium is too complicated to be solved by existing tools, one viable way is to divide the continuum into a finite number of sub-divisions (elements) inter-connected at nodes. This process is called DISCRETIZATION

Example of discretization : Estimate the land area of Antarctic: The land area of this continent is enclosed by complicated curved lines. One method for finding area is to use “squares” and “rectangles” enclosures over the entire area, because we know how to find the Enclosed areas of squares and rectangles. By super impose the surveyed land mass of the Antarctic on square grids with each square mesh 40,000 km2 counting 404.5 Square meshes leading to total land area = 16,180 million km2

This land area of Anarchic obtained by the above method of discretization obviously is an approximated value. Because the actual area is 13.6% less than this approximated value.

Antarctic

Convergence of an Area Computation by Discretization There are two ways one may compute the approximation of areas by discretization: Method 1: Enclose the individual areas within the actual curved boundaries, and Method 2: Enclose the individual areas outside the actual curved boundaries.

We observe that: The “exterior envelope” (Method 2) exceeds the actual values, and the “interior envelope” (Method 1) results in less than actual values. FE method often involves “interior envelopes” So, FE results are often less than the actual solutions.

Application of Discretization Principle in FEM

Manageable

by analytical solution method

NO available

analytical solution

Original geometry Discretized (approximate) geometry

Example on WHY Discretization is necessary in real-world stress analysis:

Original geometry Discretized (approximate) geometry

Variational Principle in FEM

= MINIMUM

for Close

Approximation

VARIATIONAL PROCESS

The Difference in Results ≠ 0 but can be made MINIMUM

Variational Process for General FE Formulation A continuum subject to ACTIONS with induced REACTIONS and boundary supports (conditions):

Stress analysis Heat transfer Fluid dynamics

ACTIONS: {P}: P1, P2, P3,……, Pn Forces {F}, pressures {p} Driving thermal forces: Q, q, etc.

Driving pressure

Induced REACTIONS: {Φ}: Φ1, Φ2, Φ3,……..,Φm

Local displacements {u}, Strains {ε}, stresses {σ}

Local temperatures T

Local velocities {V}

Actions and Reactions in Mechanical Engineering Analysis

Variational Process for General FE Formulation-Cont’d

Real situation: Original geometry + loading/boundary conditions

Discretized (approximate) geometry + loading/boundary conditions

= Minimum Difference expressed in FUNCTIONAL

Variational Process for General FE Formulation: Minimizing the FUNCTIONAL

Functionals for Variational Process for General FE Formulation A continuum subject to ACTIONS with induced REACTIONS and boundary supports (conditions):

Stress analysis Heat transfer Fluid dynamics

ACTIONS: {P}:P1, P2, P3,…,Pn Forces {F}, pressures {p} Driving thermal forces: Q, q, etc.

Driving pressure

Induced REACTIONS: {Φ}: Φ1, Φ2, Φ3,……..,Φm

Local displacements {u}, Strains {ε}, stresses {σ}

Local temperatures, T

Local velocities V

Actions, Reactions and Functionals in Mechanical Engineering Analysis

FUNCTIONAL Potential energy (P) Governing Diff. Equation plus boundary conditions

Governing Diff. Equation plus boundary conditions

Real Situation of solids Approximate Situation with elements

( )φχ

Real Situation on continuum

Mathematical Modeling of Variational Process in Finite Element Analysis Formulation

The FUNCTIONAL in the original continuum is: ( ) { } { } { } { }∫∫

∂∂

+

∂∂

=sv

sgdvf .,.........,..........,,rrφφφφφχ

Minimization of the functional will ensure the loaded continuum to be in equilibrium condition. Mathematically, This condition is satisfied by the relations:

( ){ } 02

1

=

•••∂∂∂∂

=∂∂ φ

χφχ

φφχ

where v is volume, s is surface (boundary), r denotes x,y,z coordinates

A number of equations for each induced reaction Φ:

( ) ( ) ( ) .......,.........0,0,0321

=∂∂

=∂∂

=∂∂

φφχ

φφχ

φφχ We will learn later that these are “element

equations” for the discretized FE model

Real Situation on solids Approximate Situation on elements

Mathematical Modeling of Variational Process in Finite Element Analysis Formulation

In FE Model with finite number of ELEMENTS interconnected at NODES: Actions {P}: p1, p2, p3,……Pn in discretized continuum Induced Reactions in elements: in ELEMENTS of discretized continuum

{ } em

eeee φφφφφ ......,,,,: 321

( ) ( )∑=m

e

1φχφχ

The functional of the discretized continuum is:

The Variational process required for the equilibrium of the discretized continuum becomes:

( ){ }

( ){ }

01

=∂∂

=∂∂ ∑

m

e

e

φφχ

φφχ

from which the “element equations” for every element in the discretized FE model is derived

where ( )φχ e is the functional in elements of discretized solid, and m is the total number of elements in the FE model

Element Equations in FE Model

The “element equations” derived from the above Variational process usually have the form of:

[ ]{ } { }qKe =φ

where [Ke] = coefficient matrix ( usually a square matrix) {Φ} = matrix of unknown quantities at the nodes {q} = the specified actions (or forces) at the nodes of the same element

The unknown quantities at ALL nodes in the FE mesh can be obtained by assembling all element equations in the FE model, and result in the following OVERALL equation in the form:

[ ]{ } { }QK =φ

in which [ ] [ ]∑=m

eKK1

= OVERALL coefficient matrix, and { } { }∑=n

qQ1

with n = total number of nodes in the FE model

( ) ( ) ( ) .......,.........0,0,0321

=∂∂

=∂∂

=∂∂

φφχ

φφχ

φφχ

SUMMARY OF VARIATIONAL PRINCIPLE

Variational principle is used to minimize the difference in the approximate solutions obtained by the FE method on Discretized situation corresponding to the Real situations.

Functionals are derived as the function to be minimized by the Variational process Functionals vary in the forms with the nature of the problems: ● functional for stress analysis of deformed solid structures is “Potential energy,” ● functional for heat conduction is the governing differential equation for heat conduction of solids ● functional for fluid dynamics is the differential equations called the Navier-Stoke’s equation Outcome of the Variational process of discretized media is the “element equations” for each element in the FE model Element equations are assembled to form the OVERALL stiffness equations, from which one may solve for all Primary unknown quantities at all the nodes in the discretized media Therefore, it is not an over statement to refer the Variational principle to be the basis of FE method.