november 2011 mesh discretization error - epsilon fea, llc › wp-content › uploads › 2015 ›...
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… within Epsilon
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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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• 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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• 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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• 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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• 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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• 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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• 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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• 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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• 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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• 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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• Case Study B
– Stress along path
– Node vs. Element (Averaged vs Unaveraged)
– PRERR (SEPC/SMXB)
Mesh Discretization Error
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• Stress along path
– Background stress of 180
– KT =2.0
Case Study B
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• Stress along path
– Varying Mesh densities
Case Study B
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• 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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• 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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• Case Study B Conclusion:
– Discontinuity of stress element-to-element relates to degree of mesh discretization error
Case Study B
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• Discontinuity at element boundaries is key
Error Assessment
Difference at boundary
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• Discontinuity at element boundaries is key
Error Assessment
• Energy difference per element
• Considers volume/stiffness
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• 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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• Percentage error in the energy norm (PRERR)
Error Assessment
0.797
4.0
MediumCoarse
1.59
8.95
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• 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)