A stabilization method of F-barES-FEM-T4 for dynamic...
Transcript of A stabilization method of F-barES-FEM-T4 for dynamic...
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ICCM2017
A stabilization method ofF-barES-FEM-T4
for dynamic explicit analysisof nearly incompressible solids
Ryoya IIDA, Yuki ONISHI, Kenji AMAYA
Tokyo Institute of Technology, Japan
P. 1
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ICCM2017
BackgroundOur group has proposed new FEM formulation named
F-bar aided edge-based smoothed finite element
method with tetrahedral elements (F-barES-FEM-T4).
P. 2
ABAQUS C3D4H✗ pressure oscillation
F-barES-FEM-T4(3)
# of cyclic smoothings
Our method shows excellent accuracy in static problems!
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Drawbacks in explicit dynamics
Highly accurate results are restricted to short-term analysis.
F-barES-FEM-T4 causes energy divergence.
P. 3
time
ABAQUS/Explicit
C3D8
Reference
F-barES-FEM
-T4(2)
Cantilever bending analysis
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ICCM2017
Cause of energy divergenceDue to the adoption of F-bar method,
the stiffness matrix 𝐾 becomes asymmetric.
Equation of Motion: 𝑀 ሷ𝑥 + 𝐾 𝑥 = {𝑓ext}
Asymmetric stiffness matrix gives rise to imaginary part of
natural frequencies and thus causes energy divergence
(instability) in dynamic problem.
P. 4
asymmetric
To realize long-term analysis,
stiffness matrix must be symmetrized.
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Objective
Objective
To stabilize F-barES-FEM-T4 in explicit dynamics
for nearly incompressible materials.
Table of Body Contents Methods: Quick introductions of F-barES-FEM-T4 and
stabilized method
Results & Discussion: A few verification analyses
Summary
P. 5
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ICCM2017
Methods
P. 6
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ICCM2017
Procedure of F-barES-FEM (1 of 2)
P. 7
ES-FEM
ഥ𝑭 = ෩𝑭iso ∙ ഥ𝑭vol
Deformation gradient of each edge, ഥ𝑭 is
derived as
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ICCM2017
Procedure of F-barES-FEM (2 of 2)
P. 8
Each part of ഥ𝑭 is calculated as
(1)
Isovolumetric part
Smoothing the value of
adjacent elements.
↓
The same manner as
ES-FEM
(1)Calculating node’s value by smoothing
the value of adjacent elements
(2)Calculating elements’ value by smoothing
the value of adjacent nodes
(3)Repeating (1) and (2) a few times
(2)
Volumetric part
ഥ𝑭 = ෩𝑭iso ∙ ഥ𝑭vol
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Advantages of F-barES-FEM
P. 9
Like a ES-FEM
1. Shear locking free
Like a NS-FEM
2. Little pressure oscillation
3. Volumetric locking free
with the aid of F-bar method
Isovolumetric part Volumetric part
This formulation is designed to have 3 advantages.
ഥ𝑭 = ෩𝑭iso ∙ ഥ𝑭vol
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Construction of internal force vectorInternal force vector of F-barES-FEM-T4 is calculated
as followings:
𝑓int = ෨𝐵 ത𝑇 ෨𝑉
Combination of ෨𝐵 , {ത𝑇} and ෨𝑉 causes asymmetric
stiffness matrix and thus causes energy divergence.
P. 10
𝐵-matrix of ES-FEM Stress derived from ഥ𝑭
Defenition of 𝑓int
in the same fashion
as F-bar method
𝑉 of ES-FEM
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𝑓int = ത𝐵 ത𝑇 ത𝑉
Proposed method: SymF-barES-FEM-T4
P. 11
We modify the internal force vector in order to realize
stabilization as followings:
𝐵-matrix
Derived from ഥ𝑭
Stress
derived from ഥ𝑭
volume
derived as
det ഥ𝑭 𝑉ini
𝑓int = ෨𝐵 ത𝑇 ෨𝑉F-barES-FEM-T4
SymF-barES-FEM-T4
Proposed
We named this formulation SymF-barES-FEM-T4.
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Concept of SymF-barES-FEM-T4
P. 12
The replacement ෨𝐵 to [ ത𝐵] means to symmetrize
stiffness matrix as standard FEM.
The replacement ෨𝑉 to ത𝑉 is natural extension of
volume term and introduced empirically.
𝑓int = ത𝐵 ത𝑇 ത𝑉
𝐵-matrix
Derived from ഥ𝑭
Stress
derived from ഥ𝑭
volume
derived as
det ഥ𝑭 𝑉iniSymF-barES-FEM-T4
Proposed
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Result & Discussion
P. 13
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ICCM2017
#1 Bending of a cantilever
P. 14
Dynamic explicit analysis.
Neo-Hookean material
Initial Young’s modulus: 6.0 MPa,
Initial Poisson’s ratio: 0.49,
Density: 920 kg/m3.
Compare the results of SymF-barES-FEM-T4,
F-barES-FEM-T4 and Selective H8 (ABAQUS/Explicit
C3D8) elements.
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Inability of standard T4 element
P. 15
ABAQUS/Explicit C3D4Standard T4 element
✗Pressure oscillation and locking
Standard T4 element is useless!
at 𝑡 = 0.75 s (pressure distribution)
ABAQUS/Explicit C3D8(Selective H8 element)
Reference
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Time history of deformed shapes
P. 16
SymF-barES-FEM-T4(2)
(Proposed method)
✔ Less pressure oscillation
✔ No locking
✔ No energy divergence
ABAQUS/Explicit C3D8(Selective H8 element)
Reference
F-barES-FEM-T4(2)
✔ No pressure oscillation
✔ No locking
✗ Energy divergence
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ICCM2017
Time history of total energy
P. 17
SymF-barES-FEM-T4 can suppress energy divergence!
SymF-barES-
FEM-T4
(Proposed)
F-barES-
FEM-T4
150,000 time steps
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ICCM2017
Deformed shapes and pressure distributions
P. 18
at 𝑡 = 0.75 s
Proposed method can give acceptable pressure distribution.
F-barES-FEM-T4(2)
✔ No pressure oscillation
✔ No locking
✗ Energy divergence
ABAQUS/Explicit C3D8(Selective H8 element)
Reference
SymF-barES-FEM-T4(2)
(Proposed method)
✔ Less pressure oscillation
✔ No locking
✔ No energy divergence
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ICCM2017
Time history of displacement
P. 19
Proposed method shows good result without locking.
The accuracy of displacement does not depend on the number of cyclic
smoothings as well as F-baES-FEM-T4.
Standard T4
Proposed &
Reference
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#2 Cantilever twisting analysis
P. 20
Dynamic explicit analysis.
Twisting initial velocity fields:
𝒗0 𝑥, 𝑦, 𝑧 = 100 sin𝑦𝜋
12𝑧, 0, −𝑥 𝑇 .
Neo-Hookean material
Initial Young’s modulus: 17.0 MPa,
Initial Poisson’s ratio: 0.49,
Density: 1100 kg/m3.
Compare the results of
SymF-barES-FEM-T4,
F-barES-FEM-T4,
and Selective H8
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ICCM2017
Time history of deformed shapes
P. 21
F-barES-FEM-T4(2)
✔ No pressure oscillation
✔ No locking
✗ Energy divergence
ABAQUS/Explicit C3D8(Selective H8 element)
Reference
SymF-barES-FEM-T4(2)
(Proposed method)
✔ Less pressure oscillation
✔ No locking
✔ No energy divergence
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Time history of total energy
P. 22
SymF-barES-FEM-T4 can suppress energy divergence!
F-barES-
FEM-T4SymF-barES-
FEM-T4
(Proposed)
60,000 time steps
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Effect of cyclic smoothings
P. 23
F-barES-FEM-T4 SymF-barES-FEM-T4
(Proposed method)
#cyclic
smoothings1 2 3 1 2 3
The increase in cyclic smoothings no longer improves distribution
unlike F-barES-FEM-T4
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Time history of displacement
P. 24
Standard T4
Proposed
Proposed methods can show as good result as F-barES-FEM-T4
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#3 Swinging of Bunny EarsOutline
Iron ears: 𝐸ini = 200 GPa, 𝜈ini = 𝟎. 𝟑, 𝜌 = 7800 kg/m3,
Neo-Hookean, No cyclic smoothing.
Rubber body: 𝐸ini = 6 MPa, 𝜈ini = 𝟎. 𝟒𝟗, 𝜌 = 920 kg/m3,
Neo-Hookean, 1 cycle of smoothing.
Compared to ABAQUS/Explicit C3D4. No Hex mesh available!
P. 25
Iron Ears
Rubber
Body
Fixed
Initial Velocity
of Iron Ears
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ICCM2017
Time histories of deformed shapes
P. 26
SymF-barES-FEM-T4(1)
(Proposed method)ABAQUS/Explicit C3D4
(Standard T4 element)
✗ Pressure oscillation
✗ Locking
✔ No energy divergence
✔ Less pressure oscillation
✔ No locking
✔ No energy divergence
F-barES-FEM-T4(1)
✔ No pressure oscillation
✔ No locking
✗ Energy divergence
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ICCM2017
Deformed shapes and sign of pressure
Our method represents pressure waves appropriately!
P. 27
SymF-barES-FEM-T4(1)
(Proposed method)
✔ Pressure
waves
ABAQUS/Explicit C3D4
(Standard T4 element)
In an early stage
✗ Pressure
oscillations
F-barES-FEM-T4(1)
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Summary
P. 28
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ICCM2017
Summary SymF-barES-FEM-T4 was proposed in order to
realize accurate and stable dynamic explicit analysis.
Proposed method realizes
✔ Less pressure oscillation
✔ No locking
✔ No energy divergence
Further improvement for perfect suppression of
pressure oscillation is our future work.
P. 29
Thank you for your kind attention.