higher-order finite-element analysis for fuzes subjected ... · higher-order elements used...

1

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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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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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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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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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


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Summary of errors at node in base (0 -12 µs)

order = 5

5x

10x

20x

80x40x

2010


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Convergence of top-node velocity with element order


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Convergence of top-node velocity with refinement of first-order quads

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27



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Convergence of top-node velocity with refinement of first-order triangles

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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

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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


40

2.93 GHz

15 GB RAM


data from 20th-

order elements

for t = 0-100 µs

(2) Intel Core i7:


order = 5

4

3

107

20x

15x

12x

8x

4x

3x

3x

5x

7x10x

15x

30x

20x

(1)

(2)


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order = 5

4

3

107

20

20x

15x

12x

8x

15x

40x

30x

20x

(1)


data from 20th-

order elements

for t = 0-100 µs


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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


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Convergence of top-node velocity with refinement of first-order hexes

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55



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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)


data from 7th-

order elements

for t = 0-100 µs


60

2.31 GHz

15.7 GB RAM


data from 7th-

order elements

for t = 0-100 µs

(2) AMD Opteron:


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order = 3

4

5

2

3x

4x

5x

7x

10x

12x

(1)

(2)

(1)


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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)


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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.

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