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Finite Element Method
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Finite Element Method
A numerical (approximate) method for the analysis ofcontinuum problems by:
reducing a mathematical model to a discreteidealization (meshing the domain)
assigning proper behavior to elements in thediscrete system (finite element formulation)
solving a set of linear algebra equations (linearsystem solver)
used extensively for the analysis of solids and
structures and for heat and fluid transfer
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Finite Difference Concept
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Finite Element Concept
Differential Equations : L u = F
General Technique: find an approximate solution that is a linear
combination of known (trial) functions
x
y
)y,x(c)y,x(* i
n
1ii==
u
Variational techniques can be used to reduce this problem to
the following linear algebra problems:
Solve the system K c = f
=
d)L(K jiij =
dFf ii
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FEM Programs
ALGOR
ANSYS
COSMOS/M
STARDYNE
IMAGES-3D
MSC/NASTRAN
SAP90
ADINA
NISA
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Sources of Error in the FEM
The three main sources of error in a typical FEM solution arediscretization errors, formulation errors and numerical errors.
Discretization error results from transforming the physical system(continuum) into a finite element model, and can be related tomodeling the boundary shape, the boundary conditions, etc.
Discretization error due to poor geometry
representation.
Discretization error effectively eliminated.
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Sources of Error in the FEM (cont.)
Formulation error results from the use of elements that don't preciselydescribe the behavior of the physical problem.
Elements which are used to model physical problems for which they are notsuited are sometimes referred to as ill-conditioned or mathematicallyunsuitable elements.
For example a particular finite element might be formulated on theassumption that displacements vary in a linear manner over the domain.
Such an element will produce no formulation error when it is used to model alinearly varying physical problem (linear varying displacement field in thisexample), but would create a significant formulation error if it used torepresent a quadratic or cubic varying displacement field.
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Sources of Error in the FEM (cont.)
Numerical error occurs as a result of numerical
calculation procedures, and includes truncation errorsand round off errors.
Numerical error is therefore a problem mainly concerningthe FEM vendors and developers.
The user can also contribute to the numerical accuracy,for example, by specifying a physical quantity, sayYoungs modulus, E, to an inadequate number of decimalplaces.
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Advantages of the Finite Element Method
Can readily handle complex geometry:
The heart and power of the FEM. Can handle complex analysis types:
Vibration
Transients
Nonlinear
Heat transfer
Fluids
Can handle complex loading:
Node-based loading (point loads).
Element-based loading (pressure, thermal, inertialforces).
Time or frequency dependent loading.
Can handle complex restraints:
Indeterminate structures can be analyzed.
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Advantages of the Finite Element Method (cont.)
Can handle bodies comprised of nonhomogeneous materials:
Every element in the model could be assigned a different set ofmaterial properties.
Can handle bodies comprised of nonisotropic materials:
Orthotropic
Anisotropic
Special material effects are handled:
Temperature dependent properties.
Plasticity
Creep
Swelling
Special geometric effects can be modeled:
Large displacements.
Large rotations.
Contact (gap) condition.
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Disadvantages of the Finite Element Method
A specific numerical result is obtained for a specific problem. A
general closed-form solution, which would permit one to examinesystem response to changes in various parameters, is notproduced.
The FEM is applied to an approximation of the mathematicalmodel of a system (the source of so-called inherited errors.)
Experience and judgment are needed in order to construct agood finite element model.
A powerful computer and reliable FEM software are essential.
Input and output data may be large and tedious to prepare andinterpret.
Di d f h Fi i El M h d ( )
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Disadvantages of the Finite Element Method (cont.)
Numerical problems:
Computers only carry a finite number of significant digits. Round off and error accumulation.
Can help the situation by not attaching stiff (small) elementsto flexible (large) elements.
Susceptible to user-introduced modeling errors:
Poor choice of element types.
Distorted elements.
Geometry not adequately modeled.
Certain effects not automatically included:
Buckling Large deflections and rotations.
Material nonlinearities .
Other nonlinearities.
I f ti A il bl f V i T f FEM A l i
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Information Available from Various Types of FEM Analysis
Static analysis
Deflection Stresses
Strains
Forces
Energies
Dynamic analysis
Frequencies
Deflection (modeshape)
Stresses Strains
Forces
Energies
Heat transfer analysis
Temperature Heat fluxes
Thermal gradients
Heat flow fromconvection faces
Fluid analysis
Pressures
Gas temperatures
Convection coefficients Velocities
V i t f FEM S l ti i Wid d G i Wid
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Variety of FEM Solutions is Wide and Growing Wider
The FEM has been applied to a richly diverse array of scientificand technological problems.
The next few slides present some examples of the FEM appliedto a variety of real-world design and analysis problems.
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Six Steps in the Finite Element Method
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Six Steps in the Finite Element Method
Step 1 - Discretization: The problem domain is discretizedinto a collection of simple shapes, or elements.
Step 2 - Develop Element Equations: Developed using thephysics of the problem, and typically Galerkins Method orvariational principles.
Step 3 - Assembly: The element equations for each elementin the FEM mesh are assembled into a set of global equationsthat model the properties of the entire system.
Step 4 - Application of Boundary Conditions : Solutioncannot be obtained unless boundary conditions are applied.They reflect the known values for certain primary unknowns.Imposing the boundary conditions modifies the globalequations.
Step 5 - Solve for Primary Unknowns: The modified globalequations are solved for the primary unknowns at the nodes.
Step 6 - Calculate Derived Variables: Calculated using thenodal values of the primary variables.
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Thermal Analysis - Introduction:
Thermal Analysis involves calculating:
1. Temperature distributions
2. Amount of Heat lost or gained
3. Thermal gradients
4. Thermal fluxes.
There are two types of thermal analysis:
Steady-state analysis
Transient thermal analysis
Two types of Thermal Analysis:
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Two types of Thermal Analysis:
Steady-state Thermal Analysis . It involves determining the temperature
distribution and other thermal quantities under steady-state loading
conditions. A steady-state loading condition is a situation where heatstorage effects varying over a period of time can be ignored.
Some examples of thermal loads are:
1. Convections
2. Radiation
3. Heat Flow Rates
4. Heat Fluxes (Heat Flow/unit area)
5. Heat Generation Rates (heat flow/unit volume)
6. Constant Temperature Boundaries
Steady State thermal analysis may be linear or nonlinear (due to material
properties not geometry). Radiation is a nonlinear problem.
Transient Thermal Analysis. It involves determining the temperature
distribution and other thermal quantities under conditions that vary over a
period of time.
Theoretical Basis for Thermal Analysis
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Theoretical Basis for Thermal Analysis
[KT] {T} = {Q} where [KT] = f (conductivity of material).T = vector of nodal temperatures
Q = vector of thermal loads.
[KT] is nonlinear when radiation heat transfer is present. Note that convection andradiation BCs contribute terms to both [KT] and {Q}.
Heat is transferred to or from a body by convection and radiation.
Prescribed rate of heat flow across
boundary (in or out)
Prescribed temperature (BC)
insulated for example.
Heat generated internally (eg.,
Joule heating)
Heat Flow across boundary due to radiation (in-out)
Heat Flow across boundary
due to radiation (in-out)
Equation of Heat Flow (1D Systems)
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Equation of Heat Flow (1D Systems)fx = -k dT/dx [Fourier Heat Conduction Equation]. Heat flows from high
temperature region to low temperature region.
Q = -kA dT/dx Q heat flow, fx = Q/Awhere fx = heat flux/unit area, k = thermal conduc tivity, A = area of cross-section, dT/dx = temperature gradient
In general, {fx, fy, fz} = -k{dT/dX, dT/dY, dT/dZ} T
For an elemental area of length dx, the balance of energy is given as:
-KA dT/dx +qAdx = ca dT/dt dx [KA dT/dx + d/dx(KA dT/dx)dx]d/dx(KA dT/dx) + Aq =
ca dT/dtrate in rate out = rate of increase within
For generally anisotropic material
-[d/dx d/dy d/dz] {fx fy fz}T +qv = c dT/dtwhere c is specific heat, t is ti me, mass density and q v rate of in ternal heat generat ion / uni t volume.
Above equat ion can be re-writ ten as:
Steady state if dT/dt = 0
/
xfx
fy
vqTk = ).(
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For heat transfer problem in 1-dimensional, we have:
fx = -Kdt/dx [Fourier Heat Conduction Equation]
Q = -KAdt/dx (where Q=A fx)[KT}{T} = {Q} [applicable for steady-state heat transfer problems]
1
5
Tbase=100oC Tamb=20
oC
=
2
1
2
1
11
11
q
q
T
T
LkA
5
Some Notes
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Some Notes
If the body is plane and there is convection and or radiation heat transfer across its f lat lateralsurfaces, additional equations for flux terms are needed:
Convection BCf = h(Tf T) (Newtons Law of cooling)
[K] += f(h)
{Q} = f(h,Tf)
where f = flux normal to the surface; Tf temperature of surrounding fluid; h heat transfercoefficients (which may depend on many factors li ke velocity of f luid, roughness/geometry ofsurface, etc) and T- temperature of sur face.
Radiation BC
f = h r(Tr T)
[K] += f(hr)
{Q} = f(hr,T)
where, Tr temperature of the surface; h r temperature dependent heat t ransfer coeff ic ients.
hr= F(Tr2
+T2
)(Tr+T).Where F is a factor that accounts for geometries of radiating surfaces.s is Stefan-Boltzmannconstant.
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Element Types
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Element Types
Nodal Numbering Schemes:
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Mesh Generation
Mesh Generation
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Mesh Generation
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Mesh Generation
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Smoothing/Rafinement
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Introduction toNonlinear Problems
Types of Nonlinear Problems
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yp
1. Material nonlinearitya. Conductivity depends on
temperature
b. BC depends on temperature
2. Geometric nonlinearity
a. Change in solution domain
Linear Problem
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[ ]{ } { }
[ ] { }( )[ ]{ } { }( ){ }DRRDKK
RDK
=
Stiffness [K] and Forces [R] are notfunctions of displacements.
Nonlinear Problem
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[ ]{ } { }
[ ] { }( )[ ]{ } { }( ){ }DRRDKK
RDK
=
==
Stiffness and Forces arefunctions of displacements.
Newton-Raphson Approach
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p pp
( )
11
0
udu
dP)u(f)uu(f
)u(fk
Pukk
AAA
ANA
AANA
+=+
=
=+
:SeriesTaylorTermOne
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Visualization TechniquesTwo Dimensional Scalar Data
2D Interpolation - RectangularGrid
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Grid
Suppose we are given data onrectangular grid:
f given at eachgrid point;data enrichmentfills out the emptyspaces byinterpolating valueswithin each cell
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Bilinear Interpolation
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Bilinear Interpolation
f00 f10
f01
f11
(i) interpolate in x-direction between f00,f10; and f01,f11(ii) interpolate in y-direction
We carry out three 1D interpolations:
Exercise: Show this is equivalent to calculating -f(x,y) = (1-x)(1-y)f00+x(1-y)f10+(1-x)yf01+ xyf11
(x,y)
Piecewise Bil inear Interpolation
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Apply within each grid rectangle Fast
Continuity of value, not slope (C0)
Bounds fixed at data extremes
Contour Drawing
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Contouring is verycommon technique for2D scalar data
Isolines join points ofequal value
sometimes w ith shadingadded
How can we quickly andaccurately draw theseisolines?
An Example
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As an example, consider this data:
10 -5
1 -2
Where does the zero level contour go?
Intersections w ith sides
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The bilinear interpolant is l inear along any edge -thus we can predict where the contour will cut theedges (just by simple proportions)
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-21
10
-5
c ro ss-se c t ion view
a long top edg e
Simple Approach
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A simple approach to get the contour inside the gridrectangle is just to join up the intersection points
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Question:Does this always work?
Try an example where
one pair of oppositecorners are positive,other pair negative
Contouring from Triangulated Data
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The final step is to contourfrom the triangulated data
Easy because contours oflinear interpolant are straightlines see earlier
http://www.tecplot.com
Examples - with added contours
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www.tecplot.com