Download Structural Design Using Optimality Based Cellular

AIAA 2002–1676
AIAA 2002-1676
Structural Design Using Optimality Based
Cellular Automata
43rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Con
22-25 April 2002, Denver, Colorado
Mostafa M. Abdalla ∗
Department of Aerospace and Ocean Engineering,
Virginia Polytechnic institute and state university, Blacksburg, VA 24061
Zafer Gürdal †
Departments of Aerospace and Ocean Engineering,
and Engineering Science and Mechanics,
Virginia Polytechnic institute and state university, Blacksburg, VA 24061
Cellular Automata (CA) is an emerging paradigm for the combined analysis and design of complex systems using local update rules. An implementation of the paradigm
has recently been demonstrated successfully for the design of truss and beam structures.
In the present paper, CA is applied to two-dimensional continuum topology optimization problems. The topology problem is regularized using the popular SIMP approach.
The design rules are derived based on continuous optimality criteria interpreted as local
Kuhn-Tucker conditions. Both CA based and conventional finite element analyses are
considered. Numerical experiments with the proposed algorithm indicate that the approach is quite robust and does not suffer from checkerboard patterns, mesh-dependent
topologies, or numerical instabilities.
Introduction
U
SE of Cellular Automata (CA) paradigm for
modeling complex systems is finding widespread
applications in science and engineering. CA uses a
lattice of regularly spaced cells to model the physical
domain (e.g., a continuum structure). Each cell contains all the information needed to update its state.
This includes both field variables (e.g., displacements
or stresses) as well as local design variables (e.g., local
cross-sectional area or thickness). The only external information to the cell comes directly from the
adjacent cells, which along with the cell forms a neighborhood. The attraction of CA comes from two facts.
First, the application of simple local CA rules can be
used to generate global complex behavior. Second,
by limiting computations to neighborhoods and using
identical update rules for cell variables in the entire lattice, CA proves to be an inherently massively parallel
algorithm. For the solution of heat transfer equations,
Lowekamp et. al.1 showed that CA scales very well
with the number of available processors and concluded
that CA is ideally suited for distributed memory systems. For structural analysis, Hajela and Kim2 use
a genetic algorithm to find CA analysis rules for twodimensional isotropic continuum.
∗ Research
assistant. Member, AIAA.
Associate fellow, AIAA.
† Professor.
c 2002 by Mostafa M. Abdalla and Zafer Gürdal.
Copyright °
Published by the American Institute of Aeronautics and Astronautics, Inc. with permission.
Previous work on applying CA to structural design problems relied mainly on heuristic stress based
rules. Kita and Toyoda3 construct CA rules to obtain
two-dimensional topologies based on the Evolutionary Structural Optimization (ESO) method of Xie and
Steven,4 where the analysis of the entire structure is
still performed using the finite element method. The
possibility of simultaneously updating both the field
and the design variables, led to the application of the
CA paradigm to the combined analysis and design of
structural systems. Such an approach greatly reduces
the computational effort required to reach an improved
design over the current approaches, since most traditional design methods rely on repetitive analyses of
the entire structure as the design modifications are
performed. Gürdal and Tatting,5 and Tatting and
Gürdal6 were the first ones to use the CA paradigm to
perform the combined analysis and design, and used
Stress Ratio (SR) method for the design of trusses and
two dimensional continua modeled as an equivalent
truss. These formulations lack the more formal basis for the optimality, and might lead to significantly
suboptimal designs.7
In a previous paper,8 the present authors extended
CA to treat design problems for eigenvalue requirements. Buckling critical design of elastic EulerBernoulli columns under compressive load for weight
minimization was considered. The CA algorithm presented therein converges to the analytic optimal solu-
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American Institute of Aeronautics and Astronautics Paper 2002–1676
Copyright © 2002 by the author(s). Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.
tion for different boundary conditions and geometric
constraints. The design formulation is again heuristic,
and depends on the use of stress constraints for cross
sectional sizing. Lack of analytic justification of the
excellent results obtained using this formulation limits the confidence in the algorithm for more complex
problems.
Development of local design rules based on a sound
theoretical basis is deemed to be essential for the success of CA application to structural design. In this paper, the local design rules are formulated on the basis
of fundamental optimality criteria using the methods
of calculus of variations. The continuous optimality
conditions are interpreted as local Kuhn-Tucker conditions. This approach naturally leads to rigorous local
design rules that satisfy optimality.
Two-dimensional minimum compliance topology design is considered as an application. Since the original
work of Bendsφe and Kikuchi,9 many numerical and
theoretical approaches to topology optimization were
attempted (see Rozvany10 and the references therein).
Many of these methods suffer from numerical instability problems in the form of checkerboard patterns
and mesh dependency.11 In the present paper, The
design rules are developed by adopting the Simple
Isotropic Microstructure with Penalization (SIMP)10
approach as a regularization of the topology problem. Closed form analytic expressions for the local
design rules are derived. The proposed technique of
deriving design rules can be easily extended to a wide
variety of elastic problems. The algorithm presented
in this paper avoids the need for sensitivity analysis
or the solution of large-scale mathematical programming problems. The CA design rules will be shown to
produce checkerboard-free mesh-independent optimal
topologies.
CA analysis update rules for two-dimensional linearly elastic continuum are derived from the principle
of minimum total potential energy. Alternatively, CA
rules can be used only for design while conventional
finite elements are used for analysis. Both CA analysis and finite elements are theoretically equivalent, but
differ in computational aspects. In applying CA analysis rules, different iteration schemes are considered,
and compared based on their numerical efficiency and
suitability for parallel implementation.
The formulation of local design rules is given first,
followed by the details of CA implementaion. CA design rule for FEM implementation is described, follwed
by numerical results.
topology design problem, artificial density variables
are introduced to determine the local contribution
of the material to stiffness. This approach converts
the discrete topology problem into a continuous sizing
problem. By using a suitable relation between local
structural stiffness and density variables (material interpolation scheme), the design is driven to what is
called a black/white topology, where the distribution
of the material is visualized graphically by black and
white regions in the domain. In the following first the
formulation of the continuous sizing problem for minimum compliance design is discussed, followed by its
specialization to the SIMP topology design.
The minimum compliance design problem can be
stated as:
Z
Minimize:
Φe dΩ
Ω
Z
ρ(d) dΩ = η VΩ , and g(d) ≤ 0
Subject to:
Ω
where Ω is the prescribed design domain, VΩ is its
volume, d is a vector of design functions, ρ is the local
density measure distribution, η is a prescribed volume
fraction, g are local side constraints, and Φe is the
strain energy density, given by:
Φe =
1
Γ · D(d) · Γ
2
(1)
where Γ is the strain measure vector, and D is the
elasticity tensor.
The local density measure ρ should ideally be either
1 or 0 at any given point. The functional dependence
of the density measure ρ and the elasticity tensor D on
the design variable vector d should be chosen so as to
penalize intermediate densities.
Following the standard procedure of calculus of variations we form the Lagrangian:
Z
£
¤
L=
Φe − γ (ρ − η) + λ · (g + s2 ) dΩ (2)
Ω
where λ is a vector of Lagrange multiplier functions,
s is a vector of slack variables functions, and γ is
a constant Lagrange multiplier associated with the
volume constraint. Setting the first variation of the
Lagrangian to zero, we obtain the following optimality conditions:
1. local stationarity
∂ Φe
∂ρ
∂g
−γ
+λ·
∂d
∂d
∂d
(3)
2. local side constraints
Formulation
Minimum compliance topology design attempts to
find the optimal distribution of material in a given domain to minimize the compliance of the structure under given loads with a prescribed volume constraint to
limit available material. Instead of solving the discrete
g(d) ≤ 0
(4)
3. switching conditions
λk sk = 0, no sum over k
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(5)
Objective function
Hence, the local optimization problem (9) at a specified point in the design domain reduces to:
Φ̂e
+γρ
ρp
Subject to: ² ≤ ρ ≤ 1
Minimize:
(13)
where ² is a small value selected to avoid numerical
ill-conditioning, and Φ̂e is given by:
ε
0
Fig. 1
ρ
1
Φ̂e =
Local optimization problem.
In addition to the volume constraint.
Using (1), the first term in the stationarity condition
evaluates to:
1
∂D
∂ Φe
= Γ·
·Γ
(6)
∂d
2
∂d
Furthermore, by defining the generalized stress measures to be:
∂ Φe
σ=
=D·Γ
(7)
∂Γ
the sensitivity of the strain energy density can be written, after algebraic manipulations, in terms of the
generalized stress measures as:
1
∂ D−1
∂ Φe
=− σ·
·σ
(8)
∂d
2
∂d
Thus, we can interpret the continuous optimality conditions given above as local Kuhn-Tucker conditions
for the problem:
Minimize: Φ∗e + γ ρ(d)
Subject to: g(d) ≤ 0
(9)
Φ∗e
where
is the complementary strain energy density
given by:
1
Φ∗e = σ · D−1 · σ
(10)
2
Thus, the minimum compliance problem is reduced
locally to a mathematical optimization problem. This
local optimization problem can be either solved analytically, or using standard numerical techniques.12
Specialization to SIMP
In the SIMP approach, the elasticity tensor is interpolated by:
D = ρp D̃
(11)
where D̃ is the elasticity tensor of the base material.
In the present approach the local density measure
distribution (0 < ρ(d) < 1) is used as the design
variable. To steer the design to a black and white
topology the penalization parameter p ≥ 1 is chosen high enough to penalize intermediate local density
measure without leading to numerical difficulties; typically p ≥ 3 is used. With this definition, the local
stress measure is given by:
σ = ρp D̃ · Γ
(12)
ρ2p
Γ · D̃ · Γ
2
(14)
Note that Φ̂e is calculated holding the generalized
stresses σ constant. This justifies the appearance of
ρ in (14), while Φ̂e is considered indepebdent of ρ in
(13). Figure 1 illustrates this one-dimensional problem for three different values of a parameter ρ̂ which
is given by:
1
à ! p+1
Φ̂e
ρ̂ =
(15)
γ̄
where γ̄ = γ/p is a modified Lagrange multiplier that
has units of energy density.
It is clear that the problem is convex, and depending
on the value of the parameter ρ̂ the solution of the local
optimization problem is given by:

ε < ρ̂ < 1
 ρ̂
ε
ρ̂ < ε
(16)
ρ=

1
ρ̂ > 1
The Lagrange multiplier γ̄ can be loosely interpreted
as an average strain energy density in the structure.
So, instead of pre-specifying the volume fraction η, γ̄
may be specified as input. This eliminates the need to
iteratively determine the Lagrange multiplier to satisfy
the volume constraint.
CA Implementation
In the previous section, optimality based local rules
for updating the material density were derived. We
now turn our attention to the discretization of a
two-dimensional structural domain using CA. Figure
2 depicts a generic two-dimensional topology design
problem. We use a lattice of regularly spaced cells
with the same spacing h in both x and y directions. Traditional Moore neighborhood is used to
define the connectivity of the lattice as shown in the
inset of Fig. 3. Each cell (C) has eight neighboring
cells (N, S, E, W, N W, N E, SW, SE), and the neighborhood is split into four quadrants. The state of a
cell i is denoted as φki where k is the iteration number.
For topology design in two dimensions we define a cell,
i, as:5
φi = {(ui , vi ) , (fxi , fyi ) , ρi }
(17)
where ui and vi are the cell displacements in x and y directions, respectively, and fxi and fyi are the external
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where ρi ’s are the density measures of the four cells
surrounding the quadrant, and Ncell is the number of
cells defining the quadrant. For the chosen neighborhood structure Ncell = 4.
This interpolation scheme is chosen so that any cell
with a density measure below the threshold value ²
(or zero) would turn-off all four quadrants in which
that cell participates. This makes the effect of cells
in white (void) regions to have a negligible (or no)
effect in the equilibrium equations of cells in the black
regions. Also, by smoothly interpolating cell densities,
checkerboard patterns are automatically suppressed.13
The strain energy in the cell neighborhood is calculated by summing the contribution of each quadrant.
Since the thickness of each quadrant is assumed to be
constant, the strain energy can be written in terms of
the strain energy of the base material as:
y
x
?
Fig. 2
NW
Sample domain for topology design.
N
Uq = ρ̄p Ũ
NE
where,
II
W
I
E
C
III
SW
Z
1
Ũ =
2
Γ · D̃ · Γ dxdy
(20)
quadrant
is expressed as a quadratic form in cell displacements.
The coefficients of this quadratic form are functions of
material parameters and base sheet thickness t.
For isotropic materials, the elasticity tensor can be
written in terms of Young’s modulus E, Poisson’s ratio
ν, and the sheet thickness as:


1 ν
0
0 
D̃ = t  ν 1
(21)
0 0 1−ν
IV
S
(19)
SE
Moore Neighborhood
and the small-strain tensor is given by:
Fig. 3
CA lattice and Moore neighborhood.
Γ = (εx , εy , εxy )
forces acting on the i-th cell in the respective directions. Note that each cell has its own density measure
ρi at the cell point independent of the thickness of the
quadrants that are used to define the neighborhood.
The optimization formulation described in the previous section for the derivation of the local optimum
material density (16) is valid for both linear and nonlinear analysis. For geometrically nonlinear analysis Γ
is the Green-Lagrange strain tensor. To handle material nonlinearity, the material elasticity tensor D̃ can
be related to the generalized stresses σ. In the present
work we only consider linear elastic behavior of an
isotropic material, thus the material elasticity tensor D̃
is considered to be the familiar Hooke’s law for plane
stress and infinitesimal strain-displacement relations
are used.
Each quadrant is assumed to have a constant density
ρ̄ given by:
1 X 1
1
=
(18)
p
ρ̄
Ncell
ρpi
cells
where
∂u
∂v
, εy =
,
∂x
∂y
µ
¶
1 ∂u ∂v
=
+
2 ∂y
∂x
εx =
εxy
(22)
(23)
Displacement update
On each quadrant (I to IV ), each displacement
component is expressed in terms of cell values using
bilinear interpolation. The approximate equilibrium
equations are found by minimizing the total potential
energy over the cell neighborhood with respect to cell
displacements, this gives the following form of equilibrium equations:
·
¸ ½
¾ ½
¾
fx + fxe
A B
uC
E
·
=
(24)
fy + fye
B A
vC
where A and B are parameters that depend on Poisson’s ratio and thicknesses. fxe and fye are elastic forces
that depend on thicknesses and material parameters
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and linearly on neighbor displacements. These forces
arise because neighbor displacements are assumed to
be fixed at their values in the previous iteration, while
cell displacements and/or forces are updated to restore
equilibrium during the new iteration. Closed form
expressions for these quantities are generated using
MATHEMATICATM . This 2 × 2 system of equations
is solved for cell displacements, cell forces or a mixture
of both according to the type of constraint or the lack
thereof.
Design update
Cell densities are updated using (16). To this end,
the cell strain energy density Φ̂e is calculated by averaging over cell neighborhood. Since the CA algorithm
should handle irregular domains, some cells will have
shadow neighbors that lie outside the computational
domain. Shadow cells are treated by setting their density ρ to zero. Due to the density interpolation scheme
(18), these cells automatically decouple from the solution.. The area of the quadrants corresponding to
shadow cells is not considered in averaging. In summary, we have:
Φ̂e =
IV
1 X 2p
ρ̄i Ũi
n h2
(25)
I
where n is the number of quadrants with nonzero density.
Fig. 4 Checkerboard ordering for Gauss-Seidel iteration.
checkerboard ordering is used for the analysis update,
and Jacobi mehod is used for the design update.
In previous work on CA,5, 6 the analysis and design
are nested. A fixed number of analysis update Na
is performed followed by a design update. The design changes were damped to prevent divergence of
the nonlinear iteration. Fixing the number of analysis updates is not completely satisfactory, since the
convergence rate of the Gauss-Seidel method depends
on the number of cells.14 In this paper, the analysis
and design are likewise nested, but instead of using
a fixed number of iterations, the analysis updates are
performed until the norm of the residual forces (unbalanced equilibrium) reaches a prespecified tolearnce.
Instead of damping design changes, a minimum value
of cell density ρmin is used:
ρmin = max {ε, (1 − α) ρk }
Iterative scheme
The CA analysis and design rules are applied at each
cell, keeping the values of other cells fixed. Cells can
be either updated simultaneously which corresponds
to a Jacobi iteration, in which case:
φk+1
= f (φkC , φkN , φkS , φkE , φkW , φkN W , φkN E , φkSW , φkSE )
C
(26)
or sequentially which corresponds to a Gauss-Seidel iteration. In order to preserve the symmetry of solutions
(when the domain and the loading are symmetric), a
variant of the Gauss-Seidel method is used where the
cells are updated in a checkerboard ordering (see fig.
4). The update rule for black cells takes the form:
φk+1
= f (φkC , φkN , φkS , φkE , φkW , φkN W , φkN E , φkSW , φkSE )
C
(27)
and for white cells:
k+1
k+1
k+1
φk+1
= f (φkC , φkN , φkS , φkE , φkW , φk+1
C
N W , φN E , φSW , φSE )
(28)
The CA analysis updates correspond to block Jacobi or block Gauss-Seidel mehod applied to the finite element equations written for a mesh of bilinear
quadrilateral elements. Since the stiffness matrix of
the finite element method is symmetric positive definite, Gauss-Seidel method is proven to converge.14 On
the other hand Jacobi update may or may not converge. In the present paper, Gauss-Seidel method with
(29)
where α is a prescribed move limit. The overall iteration is terminated when the maximum change in cell
densities is less than a perspecified tolearnce.
CA Design Rules for FEM
Implementation
The local CA analysis rules for two-dimensional continuum are derived so as to be consistent with a finite
element mesh of square bilinear elements. Thus, the
analysis can be directly carried out using matrix techniques as well as the CA displacement update rules
derived earlier. In this section, we formulate the CA
design rule in a manner independent of mesh structure. In this fashion, the CA design rule can be used
as a post-processor for arbitrary finite element meshes.
First, the CA cells are identified with finite element
nodes, whereas in Kita and Toyoda3 CA cells were
identified with elements. Figure 5 depicts a general
cell (node) in an unstructured FEM mesh. Before
a FEM analysis, element material properties are assigned. Poisson’s ratio is kept fixed while the element
Young’s modulus Ē is altered according to:
Ē = ρ̄p E
(30)
where ρ̄ is evaluated from (18), and the summation is
extended over each element’s nodes.
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Fig. 6
Symmetric cantilever problem.
Symmetric cantilever (FEM analysis)
Fig. 5
mesh.
Typical node in an unstructured FEM
After the FEM analysis is performed, element strain
energy Ue and element volumes Ve are reported. The
density at each cell is updated according to (16), where
the cell strain energy density Φ̂e is calculated by:
P
ρ̄pe Ue
elements
P
Φ̂e =
(31)
Ve
elements
and the summation is extended over all elements having the cell as a node.
Updating the Lagrange Multiplier
When the volume fraction is pre-specified, the Lagrange multiplier γ̄ needs to be determined. Applying
approximate Newton method leads to very simple update rule for the Lagrange multiplier. The volume
constraint is approximated as:
X
X
ρ̄e Ve − η
Ve = 0
(32)
mesh
Geometry of a symmetric cantilever beam like domain is depicted in Figure 6. Poisson’s ratio is taken
as 0.3 for all subsequent examples. The aspect ratio of
the domain is 4, and the volume fraction is set to 0.5.
Since the analysis is linearly elastic, Young’s modulus,
sheet thickness and load value do not influence the optimal topology. Figure 7 shows the evolution of the
topology on a 161 × 41 cells mesh. The minimum density ε is set to 10−3 , and the penalization parameter
is chosen as p = 3. The value of the Lagrange multiplier was calculated at each re-design step to satisfy
the volume constraint within 10−3 . The CA design algorithm converges to an almost black/white topology
in 80 re-design steps. The figure clearly indicates that
no checkerboard patterns are encountered.
a) after 20 iterations.
mesh
We assume that the optimal density is determined by
ρ̂, and consider (32) as a function of γ̄. The derivative
of (32) with respect to γ̄ is obtained by applying the
chain rule to (18) and (16). The corresponding update
rule of the Lagrange multiplier is:
γ̄ k+1 = γ̄ k (1 + (p + 1) ξ)
where,
b) after 40 iterations.
(33)
P
Ve
mesh
P
ξ =1−η
ρ̄e Ve
(34)
c) after 60 iterations.
mesh
Results
The CA based iterative local analysis and design
update formulation described in the preceding sections was implemented using a Fortran90 code, while a
MATLABTM code was developed for the finite element
analysis based approach. Both approaches are tested
on a single processor Pentium III machine. Example
results for both analysis and design are provided to illustrate the possibilities of the CA combined analysis
and design, and the CA design rule for FEM implementation.
d) after 80 iterations.
Fig. 7
cells).
Evolution of cantilever topology (161 × 41
To check whether mesh independent results are obtained, the same topology problem with the same pa-
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rameters as above is solved on three successively finer
meshes. Figure 8 shows the converged optimal topologies for the three meshes. We note that the design
features are the same in all three solutions. As the
number of cells is increased, the grey areas encountered on coarser meshes disappear giving a more crisp
solution. The run time for the MATLABTM implementation ranges from 226 seconds for the coarsest
mesh to about 2000 seconds for the finest mesh.
89
617
05
34 2
/10
!#"%$'&)(!*)+&,$-.
Fig. 10
Converged optimal topology using CA
combined analysis and design algorithm.
a) 81 × 21 cells
b) 121 × 31 cells
c) 161 × 41 cells
Fig. 8
(p = 3).
Cantilever topology after 100 re-designs
The effect of changing the value of the penalization
parameter p is shown in fig. 9. The converged designs
are similar for the cases p = 4 and p = 5 to the design
for the case p = 3.
a) 161 × 41 cells, p = 4
the convergence behavior. The convergence history
is shown in fig. 10. It is clear that the Gauss-Seidel
iteration achieves considerable reduction in residue initially and then, the convergence rate deteriorates. The
combined analysis and design algorithm was run with
p = 3 and move limit α = 5%. The Lagrange multiplier was adjusted to produce a volume fraction of
approximately 0.5. The algorithm converged in a total of 294,414 analysis updates. The run time on
a Pentium III machine is about 908 seconds. The
predicted minimum compliance (non-dimensionalized
by P 2 /E L, where P is the applied load) is 440.5 as
compared to 410.6 predicted by the FEM solution of
the previous section. This discrepancy (≈ 7%) is attributed to the way the Lagrange multiplier is handled.
While in the FEM based implementation the Lagrange
multiplier is updated after each analysis iteration, the
CA algorithm uses a user-supplied value of the Lagrange multiplier. Notwithstanding this discrepancy,
the converged topology as shown in fig. 11 compares
very well with FEM results as shown in fig. 8-a.
Fig. 11
Converged optimal topology using CA
combined analysis and design algorithm.
Unsymmetric cantilever (FEM analysis)
b) 161 × 41 cells, p = 5
Fig. 9 Effect of penalization parameter on converged cantilever topology.
To further illustrate the robustness of the CA design
rule, we consider an unsymmetric cantilver of aspect
ratio 4 as shown in fig. 12. The algorithm was run on
a 161 × 41 cells mesh and p = 3 for a volume fraction
η = 0.5. The converged topology as shown in fig. 13
is close to the optimal layout calculated by.13
Symmetric Cantilever (CA analysis)
Michell truss (FEM analysis)
The symmetric cantilever problem analyzed previously is re-solved using the CA combined analysis and
design algorithm. First the problem was run on analysis mode only on a 81 × 21 cells lattice to study
The problem under study is a rectangular ground
structure supported at the lower corners, with a downward force applied at the center of the bottom edge
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American Institute of Aeronautics and Astronautics Paper 2002–1676
Fig. 12
Fig. 13
terpreted as local Kuhn-Tucker condition and using
the SIMP approach. The effectiveness of the design
algorithm in suppressing numerical instabilities such
as checkerboard patterns or mesh-dependency comes
from using continuous interpolation of density. The
analysis is carried out using either traditional finite
element analysis or using CA analysis rules. Initial
results indicate that the CA combined analysis and design algorithm performs satisfactorily even on a serial
machine. Further numerical experiments on parallel
architectures are required to determine the relative
merits of using matrix finite element techniques versus CA local analysis rules.
Unsymmetric cantilever problem.
Unsymmetric cantilever optimal topology.
Fig. 14
Michell truss, domain and optimal layout.
(see fig. 14). This problem roughly corresponds to the
Mitchell truss, a classical topology optimization problem. The ideal solution is shown superimposed on the
ground structure.
The CA design algorithm is run on a 81 × 81 cells
modeling half the domain by using symmetry. The
converged design for p = 3 and a volume fraction
η = 0.3 is shown in fig. 15. The converged topology corresponds reasonably well to the exact solution.
It is noteworthy that the Michell truss solution can
be approached only in the limit of vanishingly small
volume fraction.
Fig. 15
Michell truss optimal topology.
Conclusion
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