formulation of finite element method by variational principle€¦ · convergence of an area...
TRANSCRIPT
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.