Transcript Lecture 1 - DigitalAddis

FEA Course Lecture I – Outline
10/02/03 - UCSD
A) Formal Definition of FEA:
An approximate mathematical analysis tool to study the behavior of a
continua (or a system) to an external influence such as stress, heat,
pressure, magnetic filed etc. This involves generating a mathematical
formulation of the physical process followed by a numerical solution of
the mathematics model.
History of FEA
1. Greek Mathematicians were the “first” to use the basic principles of FE to solve a
physical problem (i.e., finding the area of a circle, or find the value of Pi)
A=*R
C = 2R (More on this later).
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History of FEA (continued)
2. Archimedes had used a concept of ‘splitting a domain and reassembling it’ to calculate the volume of a wedge by breaking it
into a series of triangles.
3. Modern FEA – as we know it.
1941 – Hrenikoff – Framework method for Plane elastic
medium represented as collection of bars and
beams;
1943 – Courant solved a St. Venant’s Torsion Problem
through an assemblage of triangular elements;
1956 – Turner, Cough, Martin and Topp [UC
Berkeley/Aerospace];
1960 – Clough was the first to use the formal name of
“Finite Elements”.
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Basic Concept:
Division of a given domain into a set of simple sub-domains called finite
elements accompanied with polynomial approximations of solution over
each element in terms of nodal values. Assembly of element equation
with inter-element continuity of solution and balance of force
considered.
What are Finite Elements?
Any geometric shape that allows computation of solutions (with
approximation) or provides necessary relations among the values of
solution at selected points (called nodes) of the sub-domain.
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B) Basic Illustration: Approximation of Circumference of a Circle
Se
q
R
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1. FE Descitization:
• Each line segment is an element.
• Collection of these line segments is called a mesh.
• Elements are connected at nodes.
2. Element Equations
He = 2R sin (q/2)
3. Assembly of Equations and Solution
Pn = Sigma He (n=1, N)
For q = 2/n, He = 2R sin (/n), Pn = n2Rsin(/n)
4. Convergence
• As n approaches infinity, P = 2R
• if x = 1/n
Pn = 2Rsin(x)/x
• As n approaches infinity, x->0,
• Limit (2Rsin(x)/x) as x->0 = limit (2Rcos(x)/1) = 2
5. Error Estimation
Error, Ee = |Se – He|
= R[2/n – 2Sin(/n)]
Total Error = nEe
= 2R-Pn UCSD - FEA Course - Fall 2003
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C) Some Examples of the Second Order Equations in 1- Dimension, -d/dx(adu/dx) =
q for 0 < x < L
Primary
Variable u
Constant a
Source term q
Field
Secondary Variable
Qo
Transverse Deflection of
a Cable
Transverse
Deflection
Tension in Cable
Distributed
Transverse
Load
Axial Force
Axial Deformation of a
bar
Longitudinal
Displacement
EA (E= Young's
Modulus, A = Cross
Sectional Area)
Friction or
contact force
on surface of
bar
Axial Force
Heat Transfer
Temperature
Thermal
Conductivity
Heat Source
Heat
Flow Through Pipes
Hydrostatic
Pressure
D4/128m (D- Diameter, m
- viscosity)
Flow Source
(Generally
Zero)
Flow Rate
Laminar Incompressible
Flow through a Channel
under Constant Pressure
Gradient
Velocity
Viscosity
Pressure
Gradient
Pressure
Flow Through Porous
Media
Fluid Head
Coefficient of
Permeability
Fluid Flux
Flow (seepage)
Electrostatics
Dielectric Constant
Charge Density
Electrostatic
Potential UCSD - FEA Course - Fall 2003
Electric Flux
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D) Some Examples of the Poisson Equation – . (ku) = f
Field of Application
Primary Variable u
Material
Constant
K
Source Variable
f
Secondary
Variables d, du/dx,
du/dy
Heat Transfer
Temperature T
Conductivity k
Heat Source Q
Heat Flow q [comes
from conduction k
T/n and
convection h(T-T)
Irrotational Flow of
an Ideal Fluid
Stream Function y
Velocity Potential f
Density r
Density r
Mass Production
s (normally zero)
Mass Production
s (normally zero)
Velocities:
 y /x = -v;
 y /y = u
 f /x = -v;
 f /y = u
Groundwater Flow
Piezometric Head f
Permeability K
Recharge Q
Seepage: q = k f/dn
Velocities:
u = -k f/dx ,
v = -k f/dy
Torsion of Members
with Constant
Cross-Section
Stress Function Y
k=1
G = Shear
Modulus
f=2
q = angle of twist
per unit length
Gqdf/dx = -syz
Gqdf/dx = -sxz
Electrostatics
Scalar Potential f
Dielectric
Constant e
Charge Density r
Displacement Flux
density Dn
Magnetostatics
Magnetic Potential f
Permeability m
Charge density r
Magnetic Flux
density Bn
Transverse
Deflection of Elastic
Membranes
Transverse
deflection u
Tension T in
membrane
Transversely
distributed Load
Normal force q
[Both tables taken from J. N. Reddy's Book "Introduction to the Finite Element Method", J.N. Reddy, McGraw Hill Publishers, 2nd Edition, Page 71]
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E) Some Examples of Coupled Systems
1. Plane Elasticity
dsx/dx + dsxy/dy + fx = rd2u/dt2
dsxy/dx + dsy/dy + fy = rd2v/dt2
2. Flow of Viscous Incompressible Fluids [Navier Stokes Equations]
[Conservation of Linear Momentum]
rdu/dt - d/dx(2mdu/dx) - d/dy[m(du/dy + dv/dx ) ] + dP/dx – fx = 0
rdv/dt - d/dx[m(dv/dy + du/dx ) ] - d/dy(2mdu/dy) + dP/dy – fy = 0
[Conservation of Mass]
(du/dx + dv/dy) = 0
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System Level Modeling
System Level Modeling –
Reduced-order macro models are
converted into simulation templates
where the physically correct result
can be further optimized with
system level trade-offs.
Sample elector-mechanical library elements [Courtesy
Coventor]
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SOFTWARE-Specific Session:
1. Intro to ANSYS. Basic file operations. Simple plate problem.
2. Intro to FEMLAB. Fluid mechanics problem. Critical look at results.
3. Intro to software-specific issues. h-elements, p-Elements
Homework 1 and Reading Assignments.
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