Comparativo FEM BEM FDM

7/21/2019 Comparativo FEM BEM FDM

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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

=

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q

q

T

T

LkA

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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:

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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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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

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