Mesh Discretization Error

… within Epsilon
Twin Cities
ANSYS® User Meeting
November 2011
Mesh Discretization Error
… within Epsilon
… within Epsilon
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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Mesh Discretization Error
… within Epsilon
• 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)
“It can be measured? Well let’s fixate on it!”
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Mesh Discretization Error
… within Epsilon
• 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
*Powergraphics results (classic)
isn’t so one-sided – discussed later
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Tet-Pregidious
… within Epsilon
• 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
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Case Study A
… within Epsilon
• 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
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Case Study A
… within Epsilon
• Case Study A
– Peak Stresses have similar convergence patterns/rate
Low Order Elements
High Order Elements
50000
45000
1/thick
45000
40000
4/Thick
5/Thick
Peak Stress
Peak Stress
35000
30000
1/Thick
40000
2/Thick
35000
5/Thick
30000
25000
25000
20000
20000
15000
15000
1.5
1.5
2
2.5
3
2
2.5
3
Ring ID
Ring ID
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Case Study A
… within Epsilon
• Case Study A
– Peak Stresses have similar convergence patterns/rate
High & Low Order Elements
50000
Low 1/Thick
45000
Low 2/Thick
Low 5/Thick
Stress
40000
High 1/Thick
High 2/Thick
35000
High 5/Thick
30000
25000
20000
15000
1.5
1.7
1.9
2.1
2.3
2.5
2.7
2.9
Inner Radius
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Case Study A
… within Epsilon
• Case Study A
– OD Deflections
High & Low Order Elements
0.0021
0.002
0.0019
Deflections
0.0018
0.0017
0.0016
0.0015
0.0014
0.0013
1.5
1.7
1.9
2.1
2.3
2.5
2.7
2.9
Inner Radius
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Case Study A
… within Epsilon
• Case Study A
– OD Deflections
High & Low Order Elements
0.025
0.020
Deflection
0.015
0.010
0.005
0.000
1.5
-0.005
1.7
1.9
2.1
2.3
2.5
2.7
2.9
Inner Radius
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Case Study A
… within Epsilon
• Case Study A
– Deflections Along Path
2.05E-03
Low 1/Thick
Low 2/Thick
Low 5/Thick
High 1/Thick
High 5/Thick
1.95E-03
1.85E-03
1.75E-03
1.65E-03
1.55E-03
1.45E-03
1.35E-03
0
0.2
0.4
0.6
0.8
1
1.2
1.4
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Case Study A
… within Epsilon
• 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
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Mesh Discretization Error
… within Epsilon
• Case Study B
– Stress along path
– Node vs. Element (Averaged vs Unaveraged)
– PRERR (SEPC/SMXB)
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Case Study B
… within Epsilon
• Stress along path
– Background stress of 180
– KT =2.0
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Case Study B
… within Epsilon
• Stress along path
– Varying Mesh densities
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Case Study B
… within Epsilon
• Peak Stress
– Varying Mesh densities
• WB ‘s adaptive mesh refinement automates this
task refining only regions of interest (thanks, paul)
Mesh Convergence
375
370
Stress
365
360
355
350
345
0
20
40
60
80
100
120
Mesh Density
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Case Study B
… within Epsilon
• Stress along path
360
340
Very Fine
Fine
Medium
Coarse
320
Stress
300
280
260
240
220
200
180
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
Path Length
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Case Study B
… within Epsilon
• Stress along path: zoom
360
Very Fine
Fine
Medium
Coarse
340
Stress
320
300
280
260
240
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
Path Length
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Case Study B
… within Epsilon
• Stress along path: Unaveraged Results
360
340
Very Fine
320
Fine
Medium
Stress
300
Coarse
280
260
240
220
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Path Length
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Case Study B
… within Epsilon
• Case Study B Conclusion:
– Discontinuity of stress element-to-element
relates to degree of mesh discretization error
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Error Assessment
… within Epsilon
• Discontinuity at element boundaries is key
Difference at boundary
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Error Assessment
… within Epsilon
• Discontinuity at element boundaries is key
• Energy difference per element
• Considers volume/stiffness
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Error Assessment
… within Epsilon
• Discontinuity at element boundaries is key
• 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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Error Assessment
… within Epsilon
• Percentage error in the energy norm (PRERR)
Coarse
Medium
1.59
0.797
8.95
4.0
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Error Assessment
… within Epsilon
• Percentage error in the energy norm (PRERR)
Fine
Very Fine
0.56
0.32
0.06
1.13
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Error Assessment
… within Epsilon
• 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”…
Root Mean Square of:
(avg. value – element value)
for each element sharing node
Average stress from
contributing elements
(what’s plotted)
/EOF
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