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1 THE ACCURACY OF FEA

1 THE ACCURACY OF FEA. 2 REALITY MATHEMATICAL MODEL FEA MODEL RESULTS Discretization error Modeling error Solution error Discretization error is controlled

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Page 1: 1 THE ACCURACY OF FEA. 2 REALITY MATHEMATICAL MODEL FEA MODEL RESULTS Discretization error Modeling error Solution error Discretization error is controlled

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THE ACCURACY OF FEA

Page 2: 1 THE ACCURACY OF FEA. 2 REALITY MATHEMATICAL MODEL FEA MODEL RESULTS Discretization error Modeling error Solution error Discretization error is controlled

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REALITY

MATHEMATICAL

MODEL

FEA

MODEL

RESULTS

Discretization error

Modeling error

Solution error

Discretization error is

controlled in the

convergence process

Page 3: 1 THE ACCURACY OF FEA. 2 REALITY MATHEMATICAL MODEL FEA MODEL RESULTS Discretization error Modeling error Solution error Discretization error is controlled

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DISCRETIZATION OF STRESS DISTRIBUTION

Mesh built with first order triangular elements called constant stress triangles

First order element assumes linear distribution of displacements within each element. Strain, being derivative of

displacement, is constant within each element. Stress is also constant because it is calculated based on strain.

Discrete stress distribution in constant stress triangles

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Tensile hollow strip modelled with a coarse mesh of 2D plate elements.

An isometric view of von Mises effective stress distribution in

the upper right quarter of the model shown above. The

height of bars represents the magnitude of stress. Notice

that stresses are constant within each element.

DISCRETIZATION OF STRESS DISTRIBUTION

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CONVERGENCE ANALYSIS BY MESH REFINEMENT

The same tensile strip modelled three times with increasingly refined meshes. The

process of pprogressive mesh refinement is called h convergence

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CHARACTERISTIC ELEMENT SIZE

The process of pprogressive mesh refinement is called h convergence because

characteristic element size h is modified during this process

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Discretization errors

Discretization error is an inherent part of FEA. It is the price we pay for discretization of a continuous structure.

Discretization error can be defined either as solution error or convergence error.

Convergence error

Convergence error is the difference between two consecutive mesh refinements and/or element order upgrade.

Let’s say convergence error is 10%. If convergence takes place, then the next refinement and/or element order

upgrade will produce results that will be different from the current one by less than 10%.

Solution error

The solution error is the difference between the results produced by a discrete model with a finite number of

elements and the results that would be produced by a hypothetical model with an infinite number of infinitesimal

elements. To estimate the solution error, one has to assess the rate of convergence and predict changes in

results within the next few iterations as if they were performed.

CONVERGENCE CURVE

1 2 3

MESH REFINEMENT AND / OR ELEMENT ORDER UPGRADE NUMBER

CO

NV

ER

GE

NC

E C

RIT

ER

ION

SOLUTION OF THE HYPOTHETICAL “INFINITE” FINITE ELEMENT MODEL (UNKNOWN)

SOLUTION ERROR FOR MODEL # 3

CONVERGENCE ERROR

FOR MODEL # 3

# OF D.O.F. IN THE MODEL

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CONVERGENCE PROCESS WITH h ELEMENTS AND p ELEMENTS

The name h comes from characteristic element size usually denoted as h.

That characteristic element size is reduced during h convergence process.

The name p comes from polynomial function describing displacement field in the element.

The order of polynomial function is increased during p convergence process.

Both in h and p elements convergence process means adding degrees of freedom the the model. With h elements this is accomplished by mesh refinement. With p elements degrees of freedom and added by increasing elements order while mesh remains unchanged.

h - elements p - elements

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h - elements p - elements

Element shape: tetrahedral, wedge, hexahedralElement shape: tetrahedral, wedge, hexahedral

Mapping allows for only little deviation from the ideal shape.

Displacement field mapped by lower order polynomials (1st or 2nd), polynomial order does not change during solution

Mapping allows for higher deviation from the ideal shape but may introduce errors on highly curved edges and surfaces

Displacement field described by mapped higher order polynomials (up to 9th ), polynomial order adjusted automatically to meet user’s accuracy requirements.

COMPARISON BETWEEN h ELEMENTS AND p ELEMENTS

results are produced in the iterative process that continues until the known, user specified accuracy, has been obtained

results are produced in one single run with unknown accuracy

fewer large elements typically 500 - 10000many small elements typically 5000 - 500000

Note: Only tetrahedral elements elements can be reliably created with the available automeshers

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ec001 HOLLOW TENSILE STRIP

model file ec001

model type solid

material alloy steel

restraints built-in to left edge

load 100,000N tensile load to right edge in x direction

objectives

• meshing solid CAD geometry

• using solid elements

• demonstrating h convergence process

100,000 Ntensile load

Built-insupport

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ec001 HOLLOW TENSILE STRIP

Step 1 of h convergence process

Maximum von Mises stress 345.3 MPaSecond order (high quality) solid elements used

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ec001 HOLLOW TENSILE STRIP

Second order (high quality) solid elements used

Step 2 of h convergence process

Maximum von Mises stress 367.7 MPa

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Step 3 of h convergence process

Maximum von Mises stress 375.3 MPa

ec001 HOLLOW TENSILE STRIP

Second order (high quality) solid elements used

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ARTIFICIAL STIFNESS

ELEMENT STRESSES

NODAL STRESSES

H – ADAPTIVE SOLUTION

Page 15: 1 THE ACCURACY OF FEA. 2 REALITY MATHEMATICAL MODEL FEA MODEL RESULTS Discretization error Modeling error Solution error Discretization error is controlled

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0

100

200

300

400

0 1 2 3 4

von

Mis

es

stre

ss [

MP

a]

ec001 HOLLOW TENSILE STRIP

345 MPa 368 MPa 375 MPaMPa

TDW

P167

10)40100(

000,100

)(

216.2100

401212

33

W

DKn

MPaK nn 370216.2167max

368 3700.5%

368

COSMOS THEORYerror

COSMOS

Results of h convergence process presented in the form of a convergence curve

Verification of stress results:Haywood formula, Peterson p. 111

Ratio of global element size at iteration # 1to current global element size