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Higher-Order Finite-Element
Analysis for Fuzes Subjected to
High-Frequency Environments
Presented at:
The 60th Annual Fuze Conference9-11 May, 2017
Cincinnati, OH
Stephen Beissel
Southwest Research Institute
[email protected]; (210)522-5631
Approved for Public Release
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Outline
Background
Comparison of first-order and higher-order elements in
explicit solid dynamics
Finite-deformation plasticity
Wave propagation
Summary and Conclusions
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Background
Impact
generates
wave front
target
steel case
electronics
fuze
Al housing
potting
circuit
boards
Reflections
disrupt
wave front
Fuze components subjected
to high-frequency waves
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Background
Lagrangian finite-element codes are industry standard for
analysis of wave propagation
Explicit time integration by central differences
First-order elements
Computations often can’t resolve high-frequency modes,
resulting in spurious oscillations (Gibbs’ phenomenon)
Artificial viscosity damps oscillations and high-frequency modes
Objective is to improve the accuracy of Lagrangian
computations of wave propagation
Systematic survey of numerical methods uncovered advantages
of higher-order ( > 2nd order) elements
Higher-Order elements formulated and added to the EPIC code
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Background
Higher-order elements used successfully for years in CFD
Higher-order elements not used for solid mechanics because:
Computational efficiency of explicit schemes historically equated
to minimizing the floating-point operations (FLOPS) in evaluation
of internal-force term, and FLOPs increase with element order.
Greater complexity of curved-surface contact algorithms
Decades of research invested in various formulaic tradeoffs
between locking and zero-energy modes of first-order elements
Mass lumping of 2nd-order serendipity elements yields vertex
nodes with zero or negative:
Masses
Nodal forces due to uniform external traction
Lack of meshing and visualization software for higher orders
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Finite-deformation plasticity
order-1 hexahedra
(Flanagan-Belytschko)
non-symmetric
order-1 tetrahedra
symmetric
order-1 tetrahedra
plastic
strain
Square copper rod impacting a rigid surface at 200 m/s
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Finite-deformation plasticity
order-2 hexahedra
plastic
strain
Square copper rod impacting a rigid surface at 200 m/s
order-3 hexahedra order-4 hexahedra
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Finite-deformation plasticity
No volumetric
locking
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Wave propagation in 2-D axisymmetry
Baseline mesh of simple part loaded by a pulse
𝑣𝑧 𝑡
𝑡
5 m/s
2 ms
pulse
16 cm
7 cm
monitored node
near top
monitored node
in base
element size: 1x1 cm
4340 steel:
c1 = 5845 m/s
c2 = 4451 m/s
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Wave propagation in 2-D axisymmetry
t = 5 µs 10 µs 15 µs 20 µs 25 µs 30 µs 35 µs
40 µs 45 µs 50 µs 55 µs 60 µs 65 µs 70 µs
75 µs 80 µs 85 µs 90 µs 95 µs 100 µs
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Velocities of node in base at equal mesh refinement
Wave propagation in 2-D axisymmetry
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Velocities of node in base at equal mesh refinement
Wave propagation in 2-D axisymmetry
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Velocities of node in base at equal mesh refinement
Wave propagation in 2-D axisymmetry
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Summary of errors at node in base (0 -12 µs)
order = 5
5x
10x
20x
80x40x
2010
Wave propagation in 2-D axisymmetry
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Convergence of top-node velocity with element order
Wave propagation in 2-D axisymmetry
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Wave propagation in 2-D axisymmetry
Convergence of top-node velocity with element order
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Wave propagation in 2-D axisymmetry
Convergence of top-node velocity with element order
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Wave propagation in 2-D axisymmetry
Convergence of top-node velocity with element order
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Wave propagation in 2-D axisymmetry
Convergence of top-node velocity with element order
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Wave propagation in 2-D axisymmetry
Convergence of top-node velocity with refinement of first-order quads
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Wave propagation in 2-D axisymmetry
Convergence of top-node velocity with refinement of first-order quads
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Wave propagation in 2-D axisymmetry
Convergence of top-node velocity with refinement of first-order quads
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Wave propagation in 2-D axisymmetry
Convergence of top-node velocity with refinement of first-order quads
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Wave propagation in 2-D axisymmetry
Convergence of top-node velocity with refinement of first-order quads
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Wave propagation in 2-D axisymmetry
Convergence of top-node velocity with refinement of first-order quads
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Wave propagation in 2-D axisymmetry
Convergence of top-node velocity with refinement of first-order quads
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Wave propagation in 2-D axisymmetry
Convergence of top-node velocity with refinement of first-order quads
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Wave propagation in 2-D axisymmetry
Convergence of top-node velocity with refinement of first-order triangles
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Wave propagation in 2-D axisymmetry
Convergence of top-node velocity with refinement of first-order triangles
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Wave propagation in 2-D axisymmetry
Convergence of top-node velocity with refinement of first-order triangles
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Wave propagation in 2-D axisymmetry
Convergence of top-node velocity with refinement of first-order triangles
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Wave propagation in 2-D axisymmetry
Convergence of top-node velocity with refinement of first-order triangles
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Wave propagation in 2-D axisymmetry
Convergence of top-node velocity with refinement of first-order triangles
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Wave propagation in 2-D axisymmetry
Convergence of top-node velocity with refinement of first-order triangles
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Wave propagation in 2-D axisymmetry
Comparison of velocity convergence with order and refinement
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Wave propagation in 2-D axisymmetry
Comparison of velocity convergence with order and refinement
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Wave propagation in 2-D axisymmetry
Comparison of velocity convergence with order and refinement
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Wave propagation in 2-D axisymmetry
Comparison of velocity convergence with order and refinement
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Summary of errors in velocity at node near top
order = 3
4
5
10 7
3x
5x
7x10x
15x
40x
first-order
triangles
higher-order
elements
20
first-order
quads
20x
15x
12x
8x
4x
3x
40x
80x
30x
20x
(1)
(1) Errors relative to
data from 20th-
order elements
for t = 0-100 µs
Wave propagation in 2-D axisymmetry
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2.93 GHz
15 GB RAM
(1) Errors relative to
data from 20th-
order elements
for t = 0-100 µs
(2) Intel Core i7:
Summary of errors in velocity at node near top
order = 5
4
3
107
20x
15x
12x
8x
4x
3x
3x
5x
7x10x
15x
30x
20x
(1)
(2)
Wave propagation in 2-D axisymmetry
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Summary of errors in velocity at node near top
order = 5
4
3
107
20
20x
15x
12x
8x
15x
40x
30x
20x
(1)
(1) Errors relative to
data from 20th-
order elements
for t = 0-100 µs
Wave propagation in 2-D axisymmetry
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monitored node
(same location as 2D)
mesh on
plane of
symmetry
identical to
2-D mesh
model differs from 2D due to
facets along hoop direction
𝑣𝑧 𝑡
𝑡
5 m/s
2 ms𝑣𝑧 𝑡
Baseline mesh of simple part loaded by a pulse
Wave propagation in 3D
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Comparison of 2-D and 3-D node velocities
Wave propagation in 3D
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Convergence of top-node velocity with element order
Wave propagation in 3D
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Wave propagation in 3D
Convergence of top-node velocity with element order
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Wave propagation in 3D
Convergence of top-node velocity with element order
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Wave propagation in 3D
Convergence of top-node velocity with element order
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Wave propagation in 3D
Convergence of top-node velocity with refinement of first-order hexes
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Wave propagation in 3D
Convergence of top-node velocity with refinement of first-order hexes
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Wave propagation in 3D
Convergence of top-node velocity with refinement of first-order hexes
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Wave propagation in 3D
Convergence of top-node velocity with refinement of first-order hexes
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Wave propagation in 3D
Convergence of top-node velocity with refinement of first-order hexes
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Wave propagation in 3D
Convergence of top-node velocity with refinement of first-order hexes
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Wave propagation in 3D
Convergence of top-node velocity with refinement of first-order hexes
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Wave propagation in 3D
Convergence of top-node velocity with refinement of first-order hexes
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Comparison of convergence with order and refinement
Wave propagation in 3D
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Wave propagation in 3D
Comparison of convergence with order and refinement
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Wave propagation in 3D
Comparison of convergence with order and refinement
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Summary of velocity errors at monitored node
order = 3
4
5
6
7
3x
4x
5x
7x
10x
12x
15x
20x
first-order hexahedra
3-D higher-order
elements
first-order quads
2-D higher-order
elements
(1)
(1) Errors relative to
data from 7th-
order elements
for t = 0-100 µs
Wave propagation in 3D
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2.31 GHz
15.7 GB RAM
(1) Errors relative to
data from 7th-
order elements
for t = 0-100 µs
(2) AMD Opteron:
Summary of velocity errors at monitored node
6
order = 3
4
5
2
3x
4x
5x
7x
10x
12x
(1)
(2)
(1)
Wave propagation in 3D
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Summary of velocity errors at monitored node
(1) Errors relative to
data from 7th-
order elements
for t = 0-100 µs
order = 3
4
5
6
7
3x
4x
5x
7x
10x
12x
15x
2
(1)
Wave propagation in 3D
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Summary and conclusions
Analysis of wave propagation is essential to fuze design
1D, 2D and 3D higher-order elements have been formulated
and implemented in EPIC
The higher-order elements show no signs of volumetric locking
Accuracy of higher-order elements is compared to standard
first-order elements in simulations of wave propagation.
Higher-order elements provide much greater accuracy at equal:
Mesh refinement
Computing time
Allocated memory
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Acknowledgment
This work was funded by the DoD Joint Fuze
Technology Program.