… within Epsilon

… within Epsilon

Twin Cities

ANSYS®

User Meeting

November 2011

Mesh Discretization Error

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ANSYS User Meeting

Mesh Discretization Error

1. Mesh Discretization: The “One-sided” Error Source

2. Tet Pregidous

3. Case Study A: Shape-Functions Effect– With and without mid-side nodes

– Stresses & Deflection

4. Case Study B: Mesh Convergence– Node vs. Element (Averaged vs Unaveraged)

– PRERR (SEPC/SMXB)

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ANSYS User Meeting

• Often small compared to load/material property error/scatter

• Ownership of error lands on analyst– Often linked to “credibility” of whole analysis

• True Error analysis would likely show Mesh Discretization is minor issue– And yet... Scrutiny continues

– And rules and criteria abound… (while other scatter goes unmentioned)

Mesh Discretization Error

“It can be measured? Well let’s fixate on it!”

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ANSYS User Meeting

• A “one-sided” error source*– Predictions are usually lower than actual (not higher)

• Excepting Singularities

– Nagging feeling because of non-conservative nature• Stress is usually underpredicted*

– Upper bound not determinable• Without employing knowledge of materials/loads/element

shape functions – discussed later

Mesh Discretization Error

*Powergraphics results (classic)

isn’t so one-sided – discussed later

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ANSYS User Meeting

• Bias Against Tetrahedrons (Tet’s)

– Source of grievance?• Low order Tets (a.k.a “T4”, a.k.a “non-midside noded Tet”)

– Too Stiff in bending / large error with 1 element through thickness

• 1st Tet’s (Berkely 1960’s) were high order

• You’d have to work at it to get ANSYS to create T4’s (structural)

– Tets (10 noded) are Less efficient per DOF• Longer solve times

• Shorter meshing times

• Added control allows refinement at location of interest– More efficient than Mapped meshing!

– Less pleasing to the eye (esp. higher aspect ratios)

– Stigmatism is receding over last decade

Tet-Pregidious

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ANSYS User Meeting

• Thick to thin rings with inner pressfit (radial expansion)– Stress gradient related to radius2

• Case Study A, Expansion of Thick/Thin Ring– Actually used 5° wedge

Case Study A

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ANSYS User Meeting

• Case Study A– Peak Stresses have similar convergence patterns/rate

Case Study A

15000

20000

25000

30000

35000

40000

45000

1.5 2 2.5 3

Pea

k S

tres

s

Ring ID

Low Order Elements

1/thick

4/Thick

5/Thick

15000

20000

25000

30000

35000

40000

45000

50000

1.5 2 2.5 3

Pea

k S

tres

s

Ring ID

High Order Elements

1/Thick

2/Thick

5/Thick

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ANSYS User Meeting

• Case Study A– Peak Stresses have similar convergence patterns/rate

Case Study A

15000

20000

25000

30000

35000

40000

45000

50000

1.5 1.7 1.9 2.1 2.3 2.5 2.7 2.9

Str

ess

Inner Radius

High & Low Order Elements

Low 1/Thick

Low 2/Thick

Low 5/Thick

High 1/Thick

High 2/Thick

High 5/Thick

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ANSYS User Meeting

• Case Study A– OD Deflections

Case Study A

0.0013

0.0014

0.0015

0.0016

0.0017

0.0018

0.0019

0.002

0.0021

1.5 1.7 1.9 2.1 2.3 2.5 2.7 2.9

Def

lect

ion

s

Inner Radius

High & Low Order Elements

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ANSYS User Meeting

• Case Study A– OD Deflections

Case Study A

-0.005

0.000

0.005

0.010

0.015

0.020

0.025

1.5 1.7 1.9 2.1 2.3 2.5 2.7 2.9

Def

lect

ion

Inner Radius

High & Low Order Elements

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ANSYS User Meeting

• Case Study A– Deflections Along Path

Case Study A

1.35E-03

1.45E-03

1.55E-03

1.65E-03

1.75E-03

1.85E-03

1.95E-03

2.05E-03

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Low 1/ThickLow 2/ThickLow 5/ThickHigh 1/ThickHigh 5/Thick

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ANSYS User Meeting

• Case Study A Conclusions

– Element Stress Gradient

• Linear for high or low order elements

– Element Displacement Gradient

• Linear for low order element

• 2nd order polynomial for high order element

– Thin Rings are well approximated with single element through the thickness

• This extends to beams as well

Case Study A

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ANSYS User Meeting

• Case Study B

– Stress along path

– Node vs. Element (Averaged vs Unaveraged)

– PRERR (SEPC/SMXB)

Mesh Discretization Error

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ANSYS User Meeting

• Stress along path

– Background stress of 180

– KT =2.0

Case Study B

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ANSYS User Meeting

• Stress along path

– Varying Mesh densities

Case Study B

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ANSYS User Meeting

• Peak Stress

– Varying Mesh densities

• WB ‘s adaptive mesh refinement automates this task refining only regions of interest (thanks, paul)

Case Study B

345

350

355

360

365

370

375

0 20 40 60 80 100 120

Str

ess

Mesh Density

Mesh Convergence

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ANSYS User Meeting

• Stress along path

Case Study B

180

200

220

240

260

280

300

320

340

360

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

Str

ess

Path Length

Very Fine

Fine

Medium

Coarse

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ANSYS User Meeting

• Stress along path: zoom

Case Study B

240

260

280

300

320

340

360

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

Str

ess

Path Length

Very Fine

Fine

Medium

Coarse

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ANSYS User Meeting

• Stress along path: Unaveraged Results

Case Study B

220

240

260

280

300

320

340

360

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

Str

ess

Path Length

Very Fine

Fine

Medium

Coarse

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ANSYS User Meeting

• Case Study B Conclusion:

– Discontinuity of stress element-to-element relates to degree of mesh discretization error

Case Study B

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ANSYS User Meeting

• Discontinuity at element boundaries is key

Error Assessment

Difference at boundary

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ANSYS User Meeting

• Discontinuity at element boundaries is key

Error Assessment

• Energy difference per element

• Considers volume/stiffness

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ANSYS User Meeting

• Discontinuity at element boundaries is key

Error Assessment

• Sum it over the model (selected region)

• Normalize it to the whole model energy

(includes load magnitude)

Yields a single number!(PRERR, or Percentage error in the energy norm)

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ANSYS User Meeting

• Percentage error in the energy norm (PRERR)

Error Assessment

0.797

4.0

MediumCoarse

1.59

8.95

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ANSYS User Meeting

• Percentage error in the energy norm (PRERR)

Error Assessment

0.32

0.06

Very FineFine

0.56

1.13

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ANSYS User Meeting

• SMXB– Checks all nodes (doesn’t necessarily correspond to the MX

location!)

– Only mentioned once in Help Manual!

– Training Classes refer to it as a “confidence band”…

Error Assessment

Root Mean Square of:

(avg. value – element value)

for each element sharing node

/EOF

Average stress from

contributing elements

(what’s plotted)