30th Nordic Seminar on Computational Mechanics
J. Høgsberg. N.L. Pedersen (Eds.)
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
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.
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.
Jacob Oest and Erik Lund
Areference load vector ˆ
Pis applied to a structure, and a reference displacement ˆ
obtained by solving the static equilibrium state equation:
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:
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:
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:
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.
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.
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 0.24 m 0.24 m 0.24 m
(a) (b) (c)
0 100 200 300 400 500 600 700 800 900 1000
Mean of Load
(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).
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.
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.
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