SUMMARY Shape optimization strategies based on error indicators usually require strict...

SUMMARY Shape optimization strategies based on error indicators usually require strict...
SUMMARY
Shape optimization strategies based on error indicators usually require strict error control for every computed
design during the optimization run. The strict error control serves two purposes. Firstly, it allows for the
accurate computation of the structural response used to define the shape optimization problem itself. Secondly, it
reduces the discretization error, which in turn reduces the size of the step discontinuities in the objective
function that result from remeshing in the first place. These discontinuities may trap conventional optimization
algorithms, which rely on both function and gradient evaluations, in local minima. This has the drawback that
multiple analyses and error computations are often required per design to control the error.
In this study we propose a methodology that relaxes the requirements for strict error control for each design.
Instead, we rather control the error as the iterations progress. Our approach only requires a single analysis and
error computation per design. Consequently, large discontinuities may initially be accommodated; their
intensities reduce as the iterations progress. To circumvent the difficulties associated with local minima due to
remeshing, we rely on gradient-only optimization algorithms, which have previously been shown to be able to
robustly overcome these discontinuities.
274
D. N. WILKE, S. KOK AND A. A. GROENWOLD
Figure 1. Finite element-error indicator integration into optimization.
have to be repeated with the updated mesh topology [1]. This strategy may be efficient when design
changes are small and no error control is required between design updates.
Alternatively, the mesh topology may be updated to control the discretization error during each
optimization update, as depicted in Figure 1(b). Multiple FEAs may be required for each candidate
shape design to control the discretization error, that is, to limit the size of the jump discontinuities
resulting from different mesh topologies, allowing for the efficient use of classical gradient based
algorithms [1].
In this study, we implement a new approach that allows for relaxation of the requirement that
strict error control is done when using classical optimization algorithms that rely on both function
and gradient information. This new approach is depicted in Figure 1(c). We are able to relax the
requirement for strict discretization error control for each candidate shape design, because gradientonly optimization allows us to robustly and efficiently optimize the resulting discontinuous cost
functions, as we have previously demonstrated [5] (but at that time without any error indicators
whatsoever).
Instead of obtaining a converged error for each candidate shape design, we allow the error to
converge as the iterations progress. A loosely related strategy was implemented by Van Keulen
et al. [6]. The benefit of our proposed approach is enhanced computational efficiency. Note, however,
that the use of gradient-only optimization is an essential ingredient to enable relaxed error control.
Conventional gradient based algorithms are not able to solve these optimization problems robustly
[5,7], unless the magnitudes of the discontinuities are controlled to be sufficiently small by enforcing
strict error control. This is so because classical methods rely on function value information, which
contain numerical local minima in the presence of remeshing.
We demonstrate our approach by extending the previously proposed uniform remeshing shape
optimization strategy [7] on the basis of the efforts of Persson and Strang [8]. We extend the strategy by including the Zienkiewicz–Zhu (ZZ) error indicator [9], with the objective of improving the
accuracy of the global energy norm of the structure for a fixed number of DOFs. This ensures that
the computational cost for each design iteration remains more or less the same. In addition, (semi)
analytical sensitivities are made available for use in gradient-only optimization approaches. We can
indeed incorporate error indicators freely, because we are able to robustly and efficiently optimize
the discontinuous objective functions resulting from changes in the mesh topology. Although we
choose to keep the number of DOFs constant, the technique proposed herein is generic and can be
combined with any error estimator to control errors within user specified tolerances when allowing
for a varying number of DOFs.
In summary, the contribution of our proposed strategy is that it requires only a single FEA
for each candidate shape design. This is achieved by mapping the computed error indicator of a
Copyright © 2013 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2013; 94:273–289
DOI: 10.1002/nme
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275
given shape geometry to the geometry obtained after an iteration of our gradient-only optimization
algorithm. The mapping of the error indicator field between two shape geometries is merely a relocation of the nodal positions of the error indicator mesh from one shape geometry to the next using
radial basis functions (RBFs), as opposed to a linearization of the error indicator field between two
shape geometries [10]. This allows for a generic mapping of data between two geometries and may
be used to map additional or alternative information as opposed to the error indicator mapped in
this study. An additional advantage of this mapping is that the required number of computations are
fewer than that required to compute a linearized error indicator field, or the actual error indicator
field, which would require a full FEA.
We do not address convergence herein in a rigorous fashion. In engineering optimization, this is
not uncommon. Many efficient algorithms for engineering optimization simply rely on the accuracy
of the approximations used to drive the iteration sequence to termination; this is so because there
often is a computational penalty to be paid when convergence is enforced using, for example, conservatism, or trust region methods [11]. In turn, the motivation for this is that the simulations used
in engineering optimization, for example, FE or CFD simulations are expensive enough as they are.
Popular algorithms in this class include the convex linearization method (CONLIN) of Fleury and
Braibant [12] and its generalization, the method of moving asymptotes of Svanberg [13]. Having
said this, the use of unconstrained gradient-only optimization methods in certain classes of piecewise discontinuous functions have been proven to be convergent [14]. What is more, currently, we
at least have empirical evidence from our previous studies [5, 14] with a uniform remeshing strategy
that our approach seems promising. Nevertheless, an investigation into the convergence properties
of the methodologies we use herein is deserving of much attention.
Before proceeding, some clarification regarding the computation of gradients is provided. Given
a candidate shape design, this shape is discretized using some meshing algorithm. A structural
analysis is then performed, as well as a (semi) analytical sensitivity analysis. The sensitivity analysis is performed while disallowing any mesh topology changes. This computation can be performed
for every imaginable shape design. We denote the result of the sensitivity analysis as the gradient
of the objective function. It is this gradient that has to be provided to gradient-only optimization
algorithms [5, 14]. This usage of the term gradient is only inconsistent with the strict mathematical definition if the computation occurred exactly at the threshold of a change in mesh topology.
Because it is not necessary to identify such a case when using gradient-only optimization algorithms,
our definition of the gradient suffices for the purposes considered herein.
Our paper is arranged as follows. Firstly, we present the gradient-only shape optimization problem
in Section 2. Thereafter, we briefly outline the gradient-only optimization algorithm used in this
study in Section 3. We then discuss the structural analysis, including the a posteriori error indicator
and mesh refinement strategy, in Section 4. Our adaptive mesh generation strategy is presented in
Section 5, followed in Section 6 by a sensitivity analysis. Section 7 contains the numerical results,
which includes a convergence study and two example problems. Additional examples may be found
in Wilke [15]. Some conclusions are offered in Section 8.
2. SHAPE OPTIMIZATION PROBLEM
The problem under consideration is the equality constrained shape optimization problem, for which
the Lagrangian is given by
L.x, / D F ..x// C
m
X
j gj .x/, x 2 X Rn and 2 Rm ,
(1)
j D1
where the objective function F ..x// is a scalar function that depends on the geometry of the
structure, which in turn depends on the control variables x that describe the geometrical boundary
@. The equality constraints gj .x/ D 0, j D 1, 2, , m are scalar functions of the control variables x. For the sake of brevity, the cost function and the constraints will respectively be denoted by
F.x/ and g.x/; this notation will however imply dependency on .x/. We choose to represent the
Copyright © 2013 John Wiley & Sons, Ltd.
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D. N. WILKE, S. KOK AND A. A. GROENWOLD
geometrical boundary @ by a simple piecewise linear interpolation between the control variables.
However, Bezier curves or B-splines, and others, may of course also be used.
Normally, the saddle point of (1) is solved for using the dual formulation
±
°
max min L.x, / .
(2)
x
We however solve (1) using the gradient-only dual formulation [5]
² g
³
g
max min L.x, / ,
x
(3)
g
with max defined as follows: find , such that
rT L.x, C v v/v 6 0, 8 v 2 Rm .
(4)
g
Similarly, min is defined as follows: find x, such that
x
rxT L.x C ıu u, /u > 0, 8 u 2 Rn such that x C ıu u 2 X ,
(5)
with X as the convex set of all possible solutions, rx the partial derivatives w.r.t. x, r the partial
derivatives w.r.t. , and where ıu and v are real positive scalars. Note that we have only exploited
gradient information of L.x, /.
Following a similar approach, we may pose the problem as an inequality constrained problem by
imposing positive bounds on the Lagrangian multipliers. However, because our proposed approach
is currently limited to smooth constraint functions, we currently cannot consider displacements,
strains and stresses as constraints, but we hope to do so in the near future.
3. OPTIMIZATION ALGORITHM
We will use the gradient-only sequential spherical approximation algorithm presented by Wilke
et al. [5] to optimize the discontinuous shape optimization problem. For the sake of completeness
and brevity, we merely outline the algorithm here (for details on the algorithm, and a motivation for
using gradient-only optimization methods in the first place, the reader is referred to [5]):
¹0º
1. hInitialization:
i Select real constants > 0, ˛ > 1, initial curvature c > 0 and initial point
¹0º
. Set k WD 1, s WD 0.
x ¹0º h
i
2. Gradient evaluation: Compute rxT L x ¹kº ¹kº .
3. Approximate optimization: Construct the local gradient-only approximate subproblem
h
i
r fQ¹kº .x/ D rxT L x ¹kº ¹kº C H ¹kº x x ¹kº
(6)
at x ¹kº , using H ¹kº D c ¹kº I, where I is the identity matrix and
h
h
i
i
¹k1º
x
x ¹kº T rxT L x ¹k1º ¹kº rxT L x ¹kº ¹kº
.
c ¹kº D
x ¹k1º x ¹kº T x ¹k1º x ¹kº
(7)
(In an inner loop, use c ¹kº as calculated in step 6(b)). Solve this subproblem analytically, to
arrive at x ¹kº by setting (6)hequal to 0. i
4. Evaluation: Compute rxT L x ¹kº ¹kº .
Copyright © 2013 John Wiley & Sons, Ltd.
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5. Test if x ¹kº is acceptable: if
h
i rxT L x ¹kº ¹kº
x ¹kº x ¹kº 6 r T fQ x ¹kº x ¹kº x ¹kº D 0
277
(8)
go to step 7.
6. Initiate an inner loop to effect conservatism:
(a) Set s WD s C 1.
(b) Set c ¹kº WD ˛c ¹kº .
(c) Go to step 3.
7. Move to the new iterate: Set x ¹kC1º WD x ¹kº .
¹kC1º
¹kC1º
¹kC1º
8. Update multiplier: Set
¹kº
the multiplier update step.
¹kº WD ¹kº
s
, with
C s
¹i
º
< , OR x < , 8 i D ¹k 4, k 3, : : : , kº,
9. Convergence test: if x OR k D kmax , stop‡ .
10. Initiate an additional outer loop: Set k WD k C 1 and go to step 2.
We elaborate as follows: in step 5, a step is deemed acceptable when the projection of the step onto
the directional derivative is negative, but this need not be optimal; superior strategies may well exist.
Again, we hope to investigate this in the near future.
4. STRUCTURAL ANALYSIS
In shape optimization, the cost function F.x/ D F .u.X .x/// is an explicit function of the
nodal displacements u, which in turn is a function of the discretized geometrical domain X .
The discretized geometrical domain X is described by the control variables x, which represent
the geometrical boundary @.
The nodal displacements u are obtained by solving the approximate FE equilibrium equations for
linear elasticity
Ku D f ,
(9)
where K represents the assembled structural stiffness matrix and f the consistent structural nodal
loads, from which the unknown displacements u can be computed. From u, we can then locally
compute elemental stress fields
O e D C B e ue ,
(10)
with constitutive relationship C , element kinematic relation B e and element displacement ue . By
combining the local stress fields O e of adjacent elements, we obtain a global discontinuous stress
field O over the entire structure, because inter-elemental stress continuity is not enforced. As the true
stress field is continuous, error indicators may be recovered from the discontinuous stress field [16].
As said, shape optimization and a posteriori adaptive FE mesh refinement may naturally
compliment each other, because both imply multiple FEAs. Instead of only conducting a FEA for
each candidate shape design, we also recover error indicators from these FEAs. The recovered error
from a given shape design is then used to discretize an updated shape design, causing the refinement
strategy to converge as the shape design converges.
4.1. Recovery-based global error indicator and refinement procedure
Although many recovery-based error indicators exist, which range from so-called global to local
indicators [17], we will herein opt for only the well-known ZZ global error indicator [9]. We do
so merely for the sake of brevity and simplicity—other indicators, be it global or local, may also
be used.
Using the computed error, we seek a refinement strategy to indicate spatial refinement (respectively de-refinement) of the mesh. For this, we modify the refinement procedure of Zienkiewicz and
‡
The notation used is ¹kº D ¹kC1º ¹kº .
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D. N. WILKE, S. KOK AND A. A. GROENWOLD
Zhu [9] to suit our r-refinement strategy: for the ith element at the kth iteration, we compute the
refinement ratio i¹kº as
¹kº e i i¹kº D ¹kº ,
(11)
eN
k
where e ¹kº
i represents the approximated energy norm and eN the average element error. In turn,
we use the refinement ratio i¹kº to compute the ideal element length§
¹k1º
h
hO ¹kº
D i 1=p ,
i
i¹kº
(12)
with h¹0º chosen as the ideal element length of a uniform mesh for the first iteration. This also
defines the initial number of nodes for our r-refinement strategy. Here, p is usually selected as the
polynomial order of the shape functions away from singularities, and adjusted near singularities [9].
Our mesh generator naturally generates linear strain triangle elements, for which we found experimentally that p < 4 may result in oscillatory behavior. Hence, we have somewhat arbitrarily selected
p D 5 herein [6]. We then smooth the discrete elemental scalar field hO ¹kº using nodal averaging to
obtain a piecewise continuous ideal element field hQ ¹kº described by a FE interpolation. Finally, we
normalize the continuous ideal element field hQ ¹kº to obtain
hQ ¹kº ¹0º
hM ¹kº D
h ,
hN ¹kº
(13)
with hN ¹kº the average continuous ideal element length and a scaling factor. We select as constant,
but allowing to vary (as some function of the initial area, current area, number of initial boundary
nodes and current boundary) may well be beneficial. Finally, we limit the minimum ideal element
length hM ¹kº as follows:
hM ¹kº D max hmin , hM ¹kº ,
(14)
with hmin prescribed.
5. ADAPTIVE MESH GENERATOR
Our mesh generator solves for the equilibrium position of a truss structure [8], which doubles as the
FE mesh, using the ideal element length field h¹kº , k D 1, 2, 3, : : : as the unloaded truss lengths, as
opposed to directly optimizing the mesh according to some optimality criterion [18, 19]. It has been
demonstrated [5, 7, 8] that this approach generates ‘good’ meshes.
We start with an initial uniform mesh X ¹0º at k D 0, using a uniform ideal element length field
h¹0º [7]. After each analysis, we compute the ideal element length field hM ¹kº using the refinement
strategy described in Section 4.1. Recall that we then merely relocate the nodal positions of the computed ideal element length field to the new candidate shape design obtained from the optimization
step, to avoid multiple analyses per candidate shape design, as illustrated in Figure 1(c). (We will
describe the details of the mapping of the error field in Section 5.2.)
As with our previous mesh generator [7], we partition the mesh X ¹kº along the interior nodes
¹kº
X
and boundary nodes X ¹kº@ , which allows for the independent treatment of the boundary
nodes @¹kº and interior nodes ¹kº . Superscript k denotes the iteration counter, which we will
omit for the sake of brevity, unless we explicitly want to highlight the dependency on k.
§
Ideal element length refers to the ‘unloaded’ truss lengths of the truss members in our truss structure analogous mesh
generator [7]—also see Section 5.
Copyright © 2013 John Wiley & Sons, Ltd.
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RELAXED ERROR CONTROL IN SHAPE OPTIMIZATION
The boundary nodes X @ are seeded according to the ideal element length field hM ¹kº along the
geometrical boundary @, with nodes explicitly placed on the control variable locations x. This
ensures accurate representation of the defined geometrical domain . Therefore, x X @ , with
@ described by a piecewise linear interpolation of x. The boundary nodes X @ remain fixed during the current iteration of the mesh generation process. We therefore only solve for X in finding
the equilibrium of the truss structure
(15)
F ±.X / D 0.
The equilibrium of the truss structure is related to the interior nodes X via the force function
(16)
±.X / D ± l .X /, l 0 hM ¹kº , X D K.l 0 l /,
which depends on the constant
K, the length of the truss members
l .X
spring stiffness
/ and the
undeformed truss lengths l 0 hM ¹kº , X . The undeformed truss lengths l 0 hM ¹kº , X depend on
the ideal element length field hM ¹kº , as well as the interior nodes X , because the ideal element
length field hM ¹kº is evaluated at the midpoint of each truss member.
However, the ideal element length field hM ¹kº .e/ is taken as a constant background field
[6, 10, 20, 21] during the mesh generation process, that is, we do not recompute the error field or
linearize the error field when the interior nodes X of the mesh vary. Consequently, the dependence of the ideal element length field hM ¹kº .e/ on the spatially varying a posteriori error field e
is constant and the sensitivity zero. Lastly, the displacement field u.X / depends on the interior
nodes X .
The reduced truss system in (15) is solved directly via the quadratically convergent Newton’s
method, which is given by
@F @X X D F .
(17)
The update of the nodal coordinates is given by
X
nC1 D X n C X ,
and for a constant ideal element length background field, the consistent tangent
@F @X
D
@F @l
@F @l 0
C
.
@l @X
@l 0 @X (18)
@F @X is given by
(19)
We obtain quadratic convergence using Newton’s method.
5.1. Boundary nodes
A first approach to accommodate a spatially varying ideal element length field may be to merely
change the ideal element length and boundary spacing, whereas the number of boundary nodes
and interior nodes follow from our previous uniform remeshing strategy [7]. However, limited
improvements are achieved using this naive strategy.
In this study, we let the number of boundary nodes follow from the spatially varying ideal element
length field by first placing nodes along the boundary X @ . We first compute the required number
of boundary nodes per piecewise linear boundary section by dividing the physical length of each
section by the ideal length of each section. To obtain an integer value for the required number of
nodes per section, we round the result down. For each section, the boundary nodes are then explicitly placed such that the integral of the error between the boundary nodes are the same. Because we
aim to keep the number of nodes constant, the remaining nodes are seeded in the interior domain
X . Although this strategy may seem simplistic, we found other strategies to determine the number
Copyright © 2013 John Wiley & Sons, Ltd.
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D. N. WILKE, S. KOK AND A. A. GROENWOLD
of boundary nodes, for example, using ratios of the circumference to interior area together with the
error along the circumference to the interior, susceptible to oscillations.
As stated previously, the boundary nodes remain fixed once they have been placed along the
boundary, which reduces the size of the Newton system when solving for the truss equilibrium.
Consequently, an increase in the number of boundary nodes X @ results in a decrease in the number of interior nodes X and vice versa, because we want to keep the total number of nodes fixed.
To reduce the computational effort, we use the nodes that describe the ideal element length field
hM ¹kº as an initial guess for our interior nodes. To keep the total number of nodes constant, we need
to either remove or add nodes to those nodes used in representing hM ¹kº . We do so by ranking the
nodes and elements according to their error densities. We remove nodes by starting with nodes with
smaller error densities, and add nodes by introducing nodes at the centroids of elements with higher
element error densities.
5.2. Mapping the error field between candidate shape designs
The ideal element length field hM ¹kº for the kth iteration is obtained by mapping the computed ideal
element length field after the analysis of the .k 1/th iteration to the candidate shape design for the
kth iteration. We achieve this by merely mapping the nodal positions, without requiring connectivity
information, using RBFs. We therefore have total flexibility on whether we want to use or discard
the nodal connectivity of the previous geometry. In this study, we pay the computational penalty
of retriangulation at every iteration to keep our strategy unsophisticated, while allowing for large
shape changes.
The details of the mapping strategy using RBFs are outlined in the Appendix.
6. SENSITIVITY ANALYSIS
Recall that the cost function F.u.X .x/// is an explicit function of the nodal displacements. Specifically, in all the examples herein, the cost function F is the nodal displacement at the point where a
point load F is applied, expressed as
F.u.x// D uF .x/.
(20)
The displacement field u.x/ depends on the discretized geometrical domain X , which is obtained
by solving for the nodal positions of a truss structure at equilibrium. The ideal element lengths
hM ¹kº of the truss structure are obtained from the error indicator discussed in Section 4. Then, the
sensitivity of the displacement uF w.r.t. the control variables x is obtained by computing
duF dX
duF
D
.
dx
dX dx
The computation of
K u D f , given by
d uF
dX
(21)
is obtained by direct differentiation of the FE equilibrium equations
dK
du
df
uCK
D
.
dX
dX
dX
For the constant external loads f we restrict ourselves to in this study,
K
du
dK
D
u.
dX
dX
(22)
df
dX
D 0 and (22) reduces to
(23)
dK
by direct differentiation of the analytical stiffness matrices for the linear strain
We compute dX
du
F
triangle elements [7, 22], which allows to solve for dX
, to then obtain du
.
dX
, are obtained by difThe sensitivities of the nodal coordinates X w.r.t. the control variables x, dX
dx
ferentiating the truss structure equilibrium equations from the mesh generation in Section 5. Recall
that we partitioned X along the interior nodes X and boundary nodes X @ . Also, recollect that
Copyright © 2013 John Wiley & Sons, Ltd.
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DOI: 10.1002/nme
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RELAXED ERROR CONTROL IN SHAPE OPTIMIZATION
the boundary nodes X @ are seeded according to the ideal element length field hM ¹kº along the geometrical boundary @, and that they remain fixed during the mesh generation process. Hence, the
equilibrium of the truss structure F only depends implicitly on the interior nodes X .
Consider the dependency of the equilibrium equations F on x:
(24)
F l .X .x/, X @ .hM ¹kº .x///, l 0 .hM ¹kº .x/, X .x/, X @ .x// D 0.
The equilibrium equations
F depend on the deformed lengths l .X , X @ / and unde
formed lengths l 0 hM ¹kº .x/, X .x/, X @ .x/ of the truss members. The truss member lengths
l .X , X @
on the interior nodes X and boundary nodes X @ . The undeformed
/ in turn depend
lengths l 0 hM ¹kº , X @ are evaluated at the midpoints of the truss members, which depend on the
interior nodes X and the boundary nodes X @ . In addition, the ideal element length field hM ¹kº .x/
changes as a function of x because of the RBF mapping. By taking the derivative of (24) w.r.t. to
the control variables x we obtain
dF @F @l @X @F @l @X @ @hM ¹kº
D
C
dx
@l @X @x
@l @X @ @hM ¹kº @x
@F @l 0 @hM ¹kº
@F @l 0 @X @F @l 0 @X @
C
C
C
D 0,
@l 0 @hM ¹kº @x
@l 0 @X @x
@l 0 @X @ @x
(25)
to give
@F @l 0
@F @l
C
@l @X
@l 0 @X @F @l 0
@F @l @X @
@X C
D
@
¹kº
@x
@l @X
@l 0 @hM ¹kº
@hM
!
@hM ¹kº
@x
@F @l 0 @X @
.
(26)
@l 0 @X @ @x
@F @l
@F @l 0
,
when
C
From (26), we may solve for @X
and the right-hand side of (26)
@x
@l @X @l 0 @X are
known. We compute the right-hand side with a finite difference perturbation, and recall that
@F @l
@l 0
C @F
is available from the Newton update, as noted in Section 5. Once we have
@l @X @l 0 @X @
@
solved for @X
, we obtain @X
as the union between @[email protected] and @X
, where @[email protected] is obtained
@x
@x
@x
numerically. We have verified that the (semi) analytical sensitivities are computed correctly by
comparing them with numerical forward finite difference sensitivities obtained with a range of perturbations. Calculation of the finite difference sensitivities is done without Delaunay triangulation
steps, to avoid the introduction of discontinuities (due to the addition or removal of nodes) in the
finite difference calculations.
7. NUMERICAL STUDY
We will consider two remeshing strategies. Firstly, we use a remeshing strategy, which we will refer
to as uniform, which uses an ideal element length field that is spatially uniform, and kept constant as
the optimization iterations progress [7]. Secondly, we will use the newly developed remeshing strategy presented herein, which is characterized by a spatially varying ideal element field that changes
as the optimization iterations progress, to which we will refer as adaptive.
The parameters used in the remeshing strategies, the FEAs and the optimization algorithm are
as follows. For the adaptive remeshing strategy, we set the mesh refinement parameter p in (12) to
5, whereas the minimum element length hmin is selected as 0.1h¹0º , unless otherwise stated. The
material considered is an orthotropic Boron-Epoxy in a tape lay-out, that is, the fibers are all aligned
in a single direction, aligned with the global x-axis. In other words, we do not determine an optimal
fiber orientation. We assume plane stress conditions and use classical laminate theory. The material
Copyright © 2013 John Wiley & Sons, Ltd.
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D. N. WILKE, S. KOK AND A. A. GROENWOLD
properties used are a longitudinal Young’s modulus of E1 D 228 GPa, a transverse Young’s modulus of E2 D 145 GPa, and a shear modulus of G12 D 48 GPa. The last independent parameter
in classical laminate theory is Poisson ratio 12 D 0.23, because 21 follows from the symmetry
relation E1 21 D E2 12 .
The selected parameters for the gradient-only conservative sequential spherical approximation
algorithm [5] are the curvature factor ˛ D 2, initial curvature c ¹0º D 1, convergence tolerance
D 104 , and a maximum number of
h outer iterations
i kmax D 300. The Lagrange multiplier update
¹kC1º
¹kº
T
¹kC1º
D r L x
. We also limit the maximum step size to 1,
step is selected as s
which was experimentally found to result in good convergence behavior, but this value will, in general, of course, strongly depend on the scaling of the problem. Before we proceed, we study the
convergence behavior of the meshing strategies.
7.1. Convergence behavior
The convergence characteristics of the ZZ error indicator with established refinement strategies are
well known, but we are herein interested in an indication of the effects of our own remeshing strategy in a shape optimization setting on convergence, if only qualitatively. We start the demonstration
by considering a fixed boundary, being indicative of a converged shape design.
10
1
3
4
5
15
2
6
F, uF
5
7
8
20
Figure 2. Bow-tie structure used to study the convergence behavior of the remeshing strategies.
0.8
1
1.5
3500
0.32
2500
{k}
SDOF
3000
0.8
1
1.5
0.34
2000
0.3
1500
0.28
1000
5
10
15
5
k
(a)
10
15
k
(b)
Figure 3. (a) System degrees of freedom (SDOF) and (b) global error ¹kº for the mesh convergence study
on the bow-tie structure for initial uniform element lengths h0 D ¹1.5, 1, 0.8º.
Copyright © 2013 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2013; 94:273–289
DOI: 10.1002/nme
283
RELAXED ERROR CONTROL IN SHAPE OPTIMIZATION
Consider the bow-tie structure depicted in Figure 2. The structure is meshed using three initial
undeformed truss lengths h0 D ¹1.5, 1, 0.8º. The convergence criterion for the error indicator is
given by
¹kº ¹k1º
< 106 .
¹kº
(27)
Results are presented in Figure 3, with the system DOF (SDOF) depicted in Figure 3(a) and the
global error indicator depicted in Figure 3(b). From Figure 3(a), it is clear that the SDOFs remain
practically constant as the iterations progress. Figure 3(b) reveals that the final global error ¹max.k/º
is lower than the global error of the initial uniform mesh ¹1º , for each of the three initial undeformed
truss length choices. We also observe that the required number of refinement iterations reduces as
the SDOFs increase. Note that the global errors ¹kº for the various undeformed truss lengths cannot
be compared with each other, because each uses a different M field. For each of the initial uniform
element lengths h0 D ¹1.5, 1, 0.8º, we depict the initial meshes in Figures 4(a)–(c), the final meshes
in Figures 4(d)–(f) and the ideal element length fields in Figures 4(g)–(i).
We depict the convergence of the displacements of the bow-tie structure in Figure 5. To do so,
we approximate the analytical solution uF using Richardson’s extrapolation method [23], because
an analytical solution is not available for this problem. For Richardson’s extrapolation method, we
have used the initial uniform element lengths h0 D ¹1.6, 0.8, 0.4º to compute uniform meshes, with
h0 representative of the average element length. To approximate the asymptotic convergence rate,
we fit a straight line in a least squares sense through the four data points with the highest SDOFs.
15
15
15
10
10
10
5
5
5
0
0
5
10
15
20
0
0
5
10
15
20
(d)
(a)
5
10
15
20
15
10
10
0
(g)
15
15
0
3.5
3
2.5
2
1.5
1
0.5
2.5
2
10
1.5
5
5
0
0
0
0
1
5
0.5
5
10
15
20
5
10
15
20
(e)
(b)
0
5
10
15
20
0
(h)
15
15
0
15
2
10
10
10
5
5
5
1.5
1
0.5
0
0
(c)
5
10
15
20
0
0
(f)
5
10
15
20
0
0
5
10
15
20
0
(i)
Figure 4. Convergence study showing (a)–(c) the initial mesh, (d)–(f) the final mesh and (g)–(i) the
final ideal element length field of the bow-tie structure for various initial uniform element lengths
h0 D ¹1.5, 1, 0.8º.
Copyright © 2013 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2013; 94:273–289
DOI: 10.1002/nme
284
D. N. WILKE, S. KOK AND A. A. GROENWOLD
Figure 5. Approximated displacement convergence rate for the bow-tie structure problem using the uniform
and adaptive mesh generators.
x
30
F, u
F
Figure 6. Initial geometry of the cantilever beam using 13 control points x.
As shown, the error of the adaptive mesh is less than that of the uniform mesh for a given number
of DOFs, whereas the convergence rate is superior.
Having offered some numerical evidence that our remeshing strategy seemingly converges for a
fixed boundary, we now proceed to illustrate the same when both the boundary and error field are
allowed to change simultaneously. We consider equality constrained example problems using both
uniform and adaptive strategies. Note that the initial design is not required to satisfy the equality
constraint (and indeed does not).
7.2. Simple cantilever beam
Next, we progress to the equality constrained design of the orthotropic cantilever beam depicted in
Figure 6. The structure has a predefined length of 30 mm and a thickness of 1 mm. A point load
F of 10 N acts at the bottom right corner of the structure. The boundary of the structure is controlled by the 13 control points or design variables x that can only move vertically. The boundary
is interpolated linearly between the control points. We minimize the displacement uF at the point
of load application, subject to an equality constraint on volume, expressed as V .x/ D V0 , with
V0 D 150 mm3 , the prescribed volume of the structure.
histories for the value of the Lagrangian L x ¹kº , ¹kº , the constraint function
ˇ Convergence
ˇ
ˇg.x ¹kº /ˇ, the Lagrange multiplier ¹kº and the SDOFs are depicted in Figure 7(a)–(d) for the uniform and adaptive mesh generators. (We have used an initial ideal element length of 1.05 for the
uniform mesh generator and one for the adaptive mesh generator to get a comparable number of
SDOFs for the converged shapes.)
Copyright © 2013 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2013; 94:273–289
DOI: 10.1002/nme
285
RELAXED ERROR CONTROL IN SHAPE OPTIMIZATION
Uniform
Adapted
0.38
0.04
0.34
|g(x{k})|
{k})
0.36
L(x{k},
Uniform
Adapted
0.05
0.32
0.03
0.02
0.3
0.01
0.28
0
0.26
10
20
30
40
10
20
k
30
40
k
(a)
(b)
1.6
1.5
1400
Uniform
Adapted
1.3
1.2
SDOF
{k}
1.4
1350
1300
1.1
1250
Uniform
Adapted
1
10
20
30
40
10
20
k
30
40
k
(c)
(d)
Figure 7. The cantilever beam convergence histories of (a) the Lagrangian L x ¹kº , ¹kº , (b) absolute
¹kº value of the constraint function g x
, (c) Lagrange multiplier ¹kº , and (d) system degrees of freedom
(SDOF) for a uniform and adapted mesh using initial ideal element lengths h0 of respectively 1.05 and 1.
The required number of iterations and final designs are comparable. The SDOFs of the uniform
remeshing strategy changes as the geometry varies, because the defined geometrical domain
changes. The number of SDOFs of our adaptive remeshing strategy is roughly constant; the small
variations present being the result of nodes being eliminated during convergence of the mesh
generator. The interesting aspects of this example are depicted in Figure 8. The initial and final
designs are depicted in Figure 8(a) and (b), and Figure 8(c) and (d), respectively, with the final ideal
element length fields depicted in Figure 8(e) and (f), for the uniform and adaptive mesh generators,
respectively—note the superiority of the latter mesh. The converged designs depicted in Figure 8(c)
and (d) reflect the parabolic shape known as the analytical solution of the equivalent beam problem.
10
10
10
5
5
5
2
0
0
10
20
30
0
0
10
20
30
0
0
(a)
(c)
(e)
10
10
10
5
5
5
1
10
20
30
0
2
0
0
(b)
10
20
30
0
0
(d)
10
20
30
0
0
1
10
20
30
0
(f)
Figure 8. Initial (a) and (b) and final (c) and (d) designs of the cantilever beam with the associated final
ideal element length field (e) and (f), for a uniform and adapted mesh using initial ideal element lengths h0
of respectively 1.05 and 1.
Copyright © 2013 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2013; 94:273–289
DOI: 10.1002/nme
286
D. N. WILKE, S. KOK AND A. A. GROENWOLD
7.3. Spanner design
Finally, we consider the shape design of the full spanner problem presented in Figure 9, which is
subjected to multiple load cases. The objective is to minimize 12 .uFA uFB /, with uFA and uFB
the vertical displacements at the point of load application, for the two independent load cases FA
and FB , respectively. The spanner is subjected to an equality constraint on volume, expressed as
V .x/ D V0 , with V0 D 70 mm3 , the prescribed volume of the structure.
The upper and lower boundaries of the geometry are described using 11 control points each. The
structure has a predefined length of 24 mm and thickness of 1 mm. The magnitude of the point loads
FA and FB is 1 N each. Symmetry is not enforced; deviations from symmetry may be used to qualitatively evaluate the obtained designs because this problem should result in a symmetric geometry.
The ideal element length field hM ¹kº for the mesh is obtained by nodal averaging of the ideal element
length fields obtained from the different load cases.
24
1
FB , u FB
A
10
B
2
2
x
A
1
B
FA , u FA
Figure 9. Initial geometry and loads of the full spanner problem using 22 control points x.
Uniform
Adapted
0.25
0.6
0.2
|g(x{k})|
{k})
L(x{k},
0.3
Uniform
Adapted
0.7
0.5
0.15
0.1
0.05
0.4
0
50
100
150
200
50
250
100
150
200
250
k
k
(a)
(b)
3
2500
2000
{k}
Uniform
Adapted
2
SDOF
2.5
1500
1000
1.5
Uniform
Adapted
500
50
100
150
200
50
250
(c)
100
150
200
250
k
k
(d)
Figure 10. The full spanner convergence histories of (a) the Lagrangian L x ¹kº , ¹kº , (b) absolute value
of the constraint function g x ¹kº , (c) Lagrange multiplier ¹kº , and (d) system degrees of freedom for a
uniform and adaptive mesh using initial ideal element lengths h0 of respectively 0.7 and 1.
Copyright © 2013 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2013; 94:273–289
DOI: 10.1002/nme
287
RELAXED ERROR CONTROL IN SHAPE OPTIMIZATION
10
10
10
1.5
5
5
5
0
0
0
0
0
1
0.5
0
5
10
15
20
5
10
15
20
(a)
(c)
(e)
10
10
10
5
5
5
0
0
(b)
5
10
15
20
0
0
5
10
15
20
(d)
0
0
5
10
15
20
0
2.5
2
1.5
1
0.5
5
10
15
20
(f)
Figure 11. Initial (a) and (b) and final (c) and (d) designs of the full spanner with the associated final ideal
element length field (e) and (f), for a uniform and adapted mesh using initial ideal element lengths h0 of
respectively 0.7 and 1.
histories for the value of the Lagrangian L x ¹kº , ¹kº , the constraint function
ˇ Convergence
¹kº ˇ
ˇg x
ˇ, the Lagrange multiplier ¹kº and the SDOFs are depicted in Figure 10(a)–(d) for the
uniform and adaptive mesh generators. This time, the required number of iterations for the uniform mesh generator is slightly more than that required for the adaptive mesh generator. Again, the
SDOFs of the uniform remeshing strategy changes as the geometry varies, whereas the SDOFs of
our adaptive remeshing strategy remains roughly constant after an initial unstable 80 iterations. The
results depicted in Figure 11 compare well with results obtained in previous studies [24, 25]. Due
to changes in the
SDOF, as depicted in Figure 10(d), some oscillatory behavior is observed in the
Lagrangian L x ¹kº , ¹kº within the first 70 iterations, as seen in Figure 10(a).
8. CONCLUSIONS
We have proposed a remeshing shape optimization methodology based on error estimates that
only requires a single analysis and error calculation per iteration or shape design. This is in contrast to conventional approaches that often require multiple analyses per shape design. We have
exploited gradient-only optimization to efficiently incorporate error indicators and refinement strategies, because we only require a single FEA followed by a posteriori error computation for each
candidate shape design, without sacrificing optimization robustness.
We have demonstrated the proposed methodology by extending an existing uniform remeshing strategy [7] by incorporation of the very well known ZZ global error indicator. We improve
significantly on the accuracy of the results obtained with uniform meshes. In addition, we have
demonstrated convergence and efficiency of our strategy on two equality constrained example problems, albeit that we have not developed a theoretical framework that guarantees convergence at
this stage.
In a follow-up study, we plan to linearize the error indicator field and investigate whether this
improves the convergence of the gradient-only optimization strategies, and the implications for the
development of theoretical frameworks for convergence.
APPENDIX A
The radial basis function s.´/ with ´ 2 R2 for the two dimensional case, is given by
s.´/ D
nb
X
˛j ´ X @
j C p.´/,
(A.1)
j D1
Copyright © 2013 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2013; 94:273–289
DOI: 10.1002/nme
288
D. N. WILKE, S. KOK AND A. A. GROENWOLD
with p a polynomial, nb the number of boundary nodes and a given basis function with respect
to the norm k´k. The coefficients ˛j and the polynomial p are determined by the interpolation
conditions
s X @
(A.2)
D d @
i , i D 1, 2, : : : , nb
i
with d @
the displacement of the ith boundary node. In addition, it is required that
i
nb
X
˛j q X @
D 0,
j
(A.3)
j D1
for all polynomials q with a degree less or equal than that of polynomial p. We rewrite (A.2) into a
matrix form as
d @ D M ˛ C Pˇ,
(A.4)
with
where M is an n
b nb matrix the ith row and j th column containing the evaluation of the
@
@ basis function X i X j . For two dimensional interpolations P is an nb 3 matrix with
h
i
@
@
@
X
the ith row given by 1 X @
ix
iy , where X ix and X iy are respectively the x and y coordinates
of the ith boundary node. Similarly, (A.3) can be written in a matrix form
P T ˛ D 0.
(A.5)
We therefore need to solve the two systems of linear equations (A.4) and (A.5). We start by rewriting
(A.4) to obtain
˛ D M 1 d @ M 1 Pˇ,
(A.6)
which we substitute into (A.5) to obtain
P T M 1 d @ M 1 Pˇ D 0.
We then solve for ˇ from
P T M 1 Pˇ D P T M 1 d @ ,
(A.7)
and ˛ from (A.6). After solving for ˛ and ˇ, the RBF s.´/ is defined and can be used to update the
interior nodes X as the geometry changes. The boundary displacements d @ are obtained from
the control variables x and piecewise linear boundary interpolation.
ACKNOWLEDGEMENT
This study was funded by the South African National Research Foundation (NRF) with contract/grant
number 2769.
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DOI: 10.1002/nme
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