Content uploaded by Jacob Oest

Author content

All content in this area was uploaded by Jacob Oest on Oct 31, 2017

Content may be subject to copyright.

30th Nordic Seminar on Computational Mechanics

NSCM-30

J. Høgsberg. N.L. Pedersen (Eds.)

2017

FINITE-LIFE FATIGUE CONSTRAINTS IN 2D TOPOLOGY

OPTIMIZATION OF CONTINUA

JACOB OEST∗AND ERIK LUND†

Materials and Production, Aalborg University

Fibigerstraede 16, 9220 Aalborg East, Denmark

e-mail: ∗oest@mp.aau.dk,†el@mp.aau.dk

Key words: Topology Optimization, Fatigue Constraints, Adjoint Method.

Summary. Topology optimization of 2D continua with the objective of minimizing

the mass while considering ﬁnite-life fatigue constraint is considered. The structure is

subjected to proportional variable-amplitude loading. The topology optimization problem

is solved using the density approach. The fractions of fatigue damage are estimated using

the stress-based Sines fatigue criterion and S−Ncurves, while the accumulated damage

is estimated using Palmgren-Miner’s rule. The method is a natural extension of classical

density-based topology optimization with static stress constraints, and thus utilizes many

of the same methods. A benchmark example is presented.

1 INTRODUCTION

Since the seminal work by Bendsøe and Kikuchi1, the topology method has been applied

to the optimal material distribution problem in a variety of ﬁelds. Most work has been

done on minimizing compliance subject to an overall volume constraint, but the method

has also been extended to e.g. stress-constrained optimization, ﬂuid-structure interaction

problems, and many complicated multi-physics problems. Limited research on fatigue-

constrained topology optimization has been published. However, fatigue failure is one of

the most common failure modes of structures subjected to repeated loading. A few works

on fatigue-constrained topology optimization have been published, where most work either

design for inﬁnite life, e.g. Collet et al.2, or reformulate the fatigue problem into a static

stress-constrained problem, see e.g. Holmberg, Torstenfeldt, and Klarbring 3.

In the recently published work by Oest and Lund 4a method is proposed where the

entire fatigue analysis is included directly in the optimization problem, including the entire

load-history. By utilizing an eﬀective adjoint formulation of the design sensitivities, the

computational cost of the ﬁnite-life fatigue-constrained problem is comparable to static

stress-constrained topology optimization. The method is currently limited to linear quasi-

static ﬁnite element analysis, linear elastic material behavior, and proportional loading.

The method can sometimes experience gray-scale issues, which in the current work is

addressed using the heaviside density ﬁlter.

149

Jacob Oest and Erik Lund

2 PHYSICS

Areference load vector ˆ

Pis applied to a structure, and a reference displacement ˆ

uis

obtained by solving the static equilibrium state equation:

K(˜

x(x))ˆ

u=ˆ

P(1)

Kis the interpolated global stiﬀness matrix, ˜

xis the vector containing the ﬁltered (phys-

ical) variables, and xis the design variables. The response for any other magnitude of the

reference load can then be determined eﬃciently by linear superposition of the reference

displacement vector. The global stiﬀness matrix is interpolated using the well-known

modiﬁed SIMP with a penalization factor p= 3. Consequently, the eﬀective Young’s

modulus Eein each element eis given by:

Ee(˜xe(x)) = Emin + ˜xe(x)p(E0−Emin ),x∈[0; 1] (2)

Here Emin << E0is a lower bound on the eﬀective modulus representing the stiﬀness

of a void region. The element reference stresses ˆσeare obtained using the reference

displacement and relaxed using the qp-stress relaxation method 5,6:

ˆσe= ˜xe(x)qEBˆue(3)

Here 0 ≤q < 1 is the stress interpolation exponent, Eis the constitutive matrix, Bis

the strain-displacement matrix, and ueis the vector of element displacements.

To estimate each fraction of damage caused by each load cycle, the stress-based Sines

criterion is applied. The Sines criterion includes all stress components and accounts for

amplitude and mean stresses, which are determined using traditional rainﬂow-counting. A

beneﬁcial property of the Sines criterion is that it reduces the stress state to an equivalent

uniaxial stress, ˜σei, for each stress cycle i. This allows for the use of S−Ncurves and

Palmgren-Miner’s rule. In the present work, a linear S−Ncurve has been applied.

Expressed in stress reversals, the S−Nrelation is:

˜σei=σ′

f(2Nei)b(4)

σ′

fand bare material parameters, and Neiis the expected amount of cycles to failure

in each element efor stress cycle i. The fractions of damage are accumulated using

Palmgren-Miner’s rule by:

De=cD

ni

X

i=1

ni

Nei≤η(5)

cD≥1 is a scaling factor to make the data representative of a lifetime, niis the amount

of cycles for a stress cycle, and ηis the allowable damage. The above equation constitutes

a fatigue constraint in every element. To reduce the computational costs, the P-norm

method is applied to reduce the large amount of constraints to a single global constraint.

150

Jacob Oest and Erik Lund

Due to the cumulative nature of Palmgren-Miner’s rule, it is possible to solve just one

adjoint equation per load case4. Thus, for large load series, the computational costs are

only increased slightly, as the amount of adjoint equations is independent of the size of

the load cases.

In the original work, it was shown that grayscale issues may occur. To address this the

regularization scheme has in this work been changed from the consistent density ﬁlter to

the heaviside density ﬁlter7. The heaviside ﬁlter is given by:

˜xe= 1 −e−β¯xe+¯xee−β(6)

¯xeis the density as obtained using the classical density ﬁlter, and βis a ﬁlter parameter.

For a βvalue of zero, the physical variables ˜

xare similar to those obtained using the

classical density ﬁlter. When βgoes to inﬁnity the design variables are all 0 −1. Note

that the implementation is using the standard element density, and not the mapped nodal

design variables which reportedly improves the problem numerically.

3 EXAMPLES

An optimized cantilever beam with two holes added to introduce stress concentrations

is shown in Figure 1. The asymmetric design is a direct consequence of the mean stress

contributions caused by the loading condition. The example is obtained using careful

optimizer settings, a very slowly increasing βparameter, and with a P-norm factor of

P= 8. For comparison, the same problem is solved using the original formulation, i.e.

with a density ﬁlter and a P-norm factor of P= 12. However, as compared with the

original publication, more iterations are allowed in this framework in both examples. The

0.24 m

0.10 m

0.24 m 0.24 m 0.24 m 0.24 m

0.10 m

0.20 m

?

P

^

(a) (b) (c)

0 100 200 300 400 500 600 700 800 900 1000

Timestep [-]

-2

0

2

4

Force [N]

104

Load

Mean of Load

Zero Mean

(d) (e) (f)

Figure 1: Minimization of mass problem with fatigue constraints demonstrated on a

cantilever beam, sketched in (a). The time-varying load is shown in (d). The original

formulation from Oest and Lund4is shown in (b) and (e), and the results obtained using

Heaviside ﬁltering is shown in (c) and (f).

151

Jacob Oest and Erik Lund

design obtained using the heaviside is with β=20, presents a slightly more 0 −1 design.

However, the formulation adds an additional tuning parameter and is more diﬃcult to

solve. Thus, a lower P-norm factor had to be applied, which resulted in a less well-

distributed damage as compared with the original formulation.

3.1 CONCLUSION

A minimization of mass topology optimization problem constrained by ﬁnite-life fa-

tigue is demonstrated using the heaviside ﬁlter. However, the added non-linearity makes

the already very non-linear problem even more diﬃcult to solve. Thus, the example

shown here is solved with a P-norm parameter lower than in the original work. It is not

necessarily worth the extra eﬀort to use the heaviside ﬁlter for this speciﬁc purpose as

compared with the density ﬁlter. However, if the optimization can be made more stable

by other means, the heaviside ﬁlter may prove a good regularization approach for fatigue

constrained topology optimization using the density method.

Acknowledgements This research is part of the ABYSS project, sponsored by the Dan-

ish Council for Strategic Research, Grant no. 1305-00020B. The support is acknowledged.

REFERENCES

[1] Bendsøe, M. P. & Kikuchi, N. Generating optimal topologies in structural design using

a homogenization method. Computer Methods in Applied Mechanics and Engineering

71, 197–224 (1988).

[2] Collet, M., Bruggi, M. & Duysinx, P. Topology optimization for minimum weight with

compliance and simpliﬁed nominal stress constraints for fatigue resistance. Structural

and Multidisciplinary Optimization 55, 839–855 (2017).

[3] Holmberg, E., Torstenfelt, B. & Klarbring, A. Fatigue constrained topology optimiza-

tion. Structural and Multidisciplinary Optimization 50, 207–219 (2014).

[4] Oest, J. & Lund, E. Topology optimization with ﬁnite-life fatigue constraints. Struc-

tural and Multidisciplinary Optimization 1–15 (2017).

[5] Bruggi, M. On an alternative approach to stress constraints relaxation in topology

optimization. Structural and Multidisciplinary Optimization 36, 125–141 (2008).

[6] Le, C., Norato, J., Bruns, T., Ha, C. & Tortorelli, D. Stress-based topology op-

timization for continua. Structural and Multidisciplinary Optimization 41, 605–620

(2010).

[7] Guest, J. K., Pr´evost, J. H. & Belytschko, T. Achieving minimum length scale in

topology optimization using nodal design variables and projection functions. Interna-

tional Journal for Numerical Methods in Engineering 61, 238–254 (2004).

152