OPTIMIZATION ASSISTED STRUCTURAL DESIGN
OF THE REAR FUSELAGE OF THE A400M, A NEW
MILITARY TRANSPORT AIRCRAFT
Gruber, H.*; Schuhmacher, G.***; Förtsch, C.**; Rieder, E.*
*Altair Engineering, München
**Altair Engineering, Böblingen
***EADS Military Aircraft, München
Summary:
The following paper shows how topology optimization methods have been used in order to find new
concepts for a complete structure. The first part gives an overview of the functionality of the
optimization software Altair OptiStruct, and points out the key features used for this project. In the
second part, an example for the usage of topology optimization on system level, for the complete rear
fuselage of A400M transport aircraft, is shown.
Keywords:
topology optimization, system level, optimization software
NAFEMS Seminar:
„Optimization in Structural Mechanics“
1
April 27 - 28, 2005
Wiesbaden, Germany
1
Introduction
In the past years Topology optimization methods have been included into standard development
processes of the automotive and aerospace industry. Innovative CAE-Systems offer many possibilities
to establish weight optimized or load path optimum designs to reduce costly and time consuming
iterations in the development process. Examples of applications for these methods are mostly single
components, like wheel trunks.
In this paper topology optimization has been applied at system level, in order to determine a new
concept of the complete rear fuselage. The main goal of this project was to establish new ways in
designing the complete rear fuselage. Explained in simple words, the rear fuselage of the A400M is a
tube with a big cut which is loaded predominantly with torsion. Designing this part of the A400M as a
lightweight structure is a special challenge.
In a first step, the objective was to find the optimum material distribution with respect to a given design
space. The design space is bounded by the outer loft, driven by aerodynamic requirements and the
inner loft determined by the space required for passengers, cargo load, systems and other functional
requirements. The results were interpreted into an aerospace-like structure, consisting of stringer and
frames which was the input for another optimization step. As a result a detailed design proposal is
obtained which is important to make established design decisions. These optimization steps lead to a
significant weight reduction compared to the baseline design and showed new ways to fulfil different
requirements.
2
Optimization software
2.1
Formulation of the optimization problem
Most of the software used for structural optimization is based on finite element methods (FEM). First
used methods are size optimization, here for example shell thicknesses or beam parameter are the
design variables and shape optimization in which node coordinates of FEM meshes are parameterised
in order to vary the shape of an existing component. Both methods are based on a fix topology. At the
end of the 90’s, methods were introduced to find an optimum layout of a structure which was called
topology optimization. Based on a design space, modelled with finite elements and the relevant load
cases, an optimum material distribution can be found. All these methods work iterative and therefore
require a series of FE analyses to an optimal design (Fig. 2) [3].
In structural optimization the formulation of the optimization problem is important. It can be formulated
as follows:
Minimize :
f(x) = f ( x1, x2 ,K, xn )
Subject to :
g j (x) ≤ 0
j = 1,K, m
hk ( x) = 0
k = 1,K, mh
xiL ≤ xi ≤ xiU
i = 1,K, n
Fig. 1: General formulation of the optimization problem
The so called optimization model consists of the objective function f, the design constrains g and h
and the design variables x.
2.2
Mathematical approach
The optimization software Altair OptiStruct used in this project has a general mathematical approach.
The mechanical optimization problem is transferred into a mathematical one and then solved by the
optimizer. Compared to software working with optimum criteria there are following advantages [1]:
−
−
−
−
The user has maximum freedom in formulating the optimization problem
In principle all results of a FEM Analysis are available as a response for optimization
Responses of external servers could be used, e.g. damage values from fatigue
Different methods for optimization (chapter 2.1) could be combined in one run
NAFEMS Seminar:
„Optimization in Structural Mechanics“
2
April 27 - 28, 2005
Wiesbaden, Germany
First step in an optimization run is the finite element analysis which is necessary for the sensitivity
calculation. Based on this the search direction for the optimization algorithm is found, which leads to a
variation of the design variables to run this loop again with a better draft. This iteration process stops
when stop criterion is reached.
Fig. 2: Scheme of an optimization run
Altair OptiStruct uses an integrated FE-Solver and so sensitivities, the derivatives of a response g with
respect to the design variables x can be calculated very cheaply because the factorised stiffness
matrix K can be used and only a forward-backward division is necessary [5].
g j ( X ) = Q Tj U , KU = P
(I)
⇒
∂g j
=
∂x i
∂Q Tj
∂x i
U + Q Tj
∂U
∂U ∂P ∂K
Uj
, K
=
−
∂x i
∂x i ∂x i ∂x i
g j ( X ) = Q Tj U , KU = P
(II)
⇒
∂g j
=
∂x i
 ∂P ∂K 
U + U jT 
−
U  with KU j = Q j
∂x i
∂
x
∂
x
 i
i

∂Q Tj
Equation (I) shows the direct differences method which is efficient when less responses than design
variables are used, g << x (size and shape optimization). The adjoint method (equation II) is
preferable, if the number of responses is greater than the number of design variables, g >> x (topology
optimization).
Based on the sensitivities the optimization problem is approximated. The behaviour of the design
constrains and objective function are considered in a small proximity around the current draft. Thus
the total number of iterations and expensive FE-Analysis runs are reduced [3].
NAFEMS Seminar:
„Optimization in Structural Mechanics“
3
April 27 - 28, 2005
Wiesbaden, Germany
∂g
g~ j (x) = g j 0 + ∑ j ( xi − xi 0 )
i ∂x i
(III)
Equation III shows a linear approximation of a response g against the gradient. In addition reciprocal
and convex approximation types are also used in OptiStruct.
2.3
Topology optimization
2.3.1 Density method (SIMP)
That approach for topology optimization (solid isotropic material with penalty) is based on a design
space meshed of finite elements. Every element represents one design variable in the optimization
problem which leads to a variation of the stiffness and the density. Due to exponent p, the coherence
between a normalized density ρ/ρ0 and the young modulus E shows a hyperbolic behaviour. Resultant
from this approach, semi dense elements should be penalized to decide whether to be full material or
a hole. At the end the design result should be discrete which is easier to be interpreted. [2].
Fig. 3: Approach of the density method
Figure 3 shows a result of this approach. On the black contoured elements material is needed and for
the white elements material can be dismissed for an optimum structure.
2.3.2 Integration of manufacturing constrains
Not in every case is the optimum material distribution the best in developing a component.
Manufacturing aspects have also to be considered in topology optimization, to ensure that design
proposals from the optimizer can be transferred into a construction without fundamental changes. To
include these aspects into the optimization process, special features can be activated by the user in
order to control the design result. Fig. 4 shows such a manufacturing constrain (minimum member size
control), which suppresses small design features and create bigger structures – not smaller than a
diameter d - instead. Switching on a manufacturing constrain has more positive affects:
−
−
−
Reaching a discrete material distribution
Checkerboard control
Avoiding mesh dependency [5]
NAFEMS Seminar:
„Optimization in Structural Mechanics“
4
April 27 - 28, 2005
Wiesbaden, Germany
Fig. 4: Effect of manufacturing constrains on the design result
More manufacturing constrains which are available in Altair OptiStruct are:
−
−
−
−
Maximum member size (spitting up material accumulations)
Draw direction
Symmetry
Pattern repetition [1]
3
Concept studies using topology optimization considering the A400M airfreighter
tail structure as an example
3.1
Requirements on an ideal tail structure
The A400M’s tail structure consists of an outer skin, supported by stringers and vertical frames. This
tube has a big tailboard for loading and unloading goods or even vehicles. It can be closed with a
ramp and a door both outwards and inwards.
Section C
Section B
Section A
Cargo Door
Ramp
Longerons
Floor
Fig. 5: Baseline design (half model)
The objective of the optimization is to increase the stiffness while keeping the mass bounded above.
As the door stands slant under tensional load, the relative displacements shouldn’t exceed a certain
value to allow closing the door securely during the flight after launching cargo or paratroops.
Particularly critical are load cases caused by extreme manoeuvres or squalls which load the fuselage
by torsion or bending.
NAFEMS Seminar:
„Optimization in Structural Mechanics“
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April 27 - 28, 2005
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3.2
Design space and optimization model
In this case the maximum allowable design space is limited by the outer skin of the fuselage and
inside by system requirements, e.g. space for opening and closing door and ramp, loading space, etc.
. Within this area the material can be distributed arbitrarily during the optimization to replace the
stringers and vertical frames, which were removed from the original model, in order to increase the
total stiffness of the afterbody. Various other components, e.g. door and ramp, serve during the
calculation as non design, they are considered to be unchangeable. Beneath the topological design
space the thicknesses of the main fields were defined as variables to enable the Optimizer to decide
weather the load path should be carried by the outer skin (via increasing the thickness) or rather by
using additional braces inwards.
Sec. B: Frames
removed
Sec. C: Frames
removed
Sec. A: Frames
removed
Design Space
modelled by Solids
+
Floor & longerons removed
Fig. 6: Non design and design space
The optimization task includes the minimization of the linear combination of the Compliance, an
equivalent of the extension energy for all relevant loading cases (k). As design constrains for the
calculation, the exceeding of the mass – in comparison to the original model – and the relative
displacement between door and fuselage structure are limited. This conforms to a modified matter of
formulating the traditional topology optimization task.
Minimize { f ( ρ,t ) = ∑ f kT u k }
with
k
p = topology variable
Subject to m ( ρ,t ) ≤ m baseline
t = thickness variable
d 1 ( ρ,t ) − d 2 ( ρ,t ) ≤ d max
′ ≤ ρi ≤ 1, i = 1,...,n
ρmin
fTu = compliance
m = mass
d = displacement
t min ≤ t j ≤ t max , j = 1,...,m
Fig. 7: Formulation of the optimization task
3.3
Design result of the 3-D Topology optimization
The result of the first optimization step is a qualitative material distribution within the design space
idealized of 3-D volume elements, as well as a set of thickness variables for the main fields. Figure 8
shows this distribution, where all non design areas and the outer skin are blanked. It shows the
accumulation of material in the area of the load introduction underneath the tail unit, along the original
edge beams (longerons) and confirms the importance of the original model’s details. Nevertheless the
result shows new insights and design elements, e.g. an extended inner skin in the area of the door as
well as an intelligent connection of the door and the fuselage structure.
NAFEMS Seminar:
„Optimization in Structural Mechanics“
6
April 27 - 28, 2005
Wiesbaden, Germany
Fig. 8: Material distribution regarding topology optimization on volume meshed design space
A kind of kinematics like scissors takes care that the side conditions limitation of the relative
displacement between door and fuselage structure can be fulfilled. This is an example that a complex
formulation of an optimization problem can lead to smart ideas and solution, which are hard or even
impossible to obtain only by intuition.
Fig. 9: Smart design proposal for a connection cargo door to the substructure
3.4
Results of the continuative topology optimization on shell base
Starting from the abovementioned 3-D material distribution, another design space model meshed with
shell elements is originated. This continuative step is performed to make a detailed statement about
the designs of the single components and determine the ideal stiffness distribution for the complete tail
structure. The optimization model is not changed in comparison to step one. For the visualization of
the design results for this case, a contour display of the element density suited, whereas all ranges of
minimal wall thicknesses (e.g. first frame) and areas in which material accumulations are necessary
(e.g. inner skin under the tail unit) can be distinguished by colour.
NAFEMS Seminar:
„Optimization in Structural Mechanics“
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April 27 - 28, 2005
Wiesbaden, Germany
Fig. 10: Density distribution of the topology optimization based on a shell meshed design space
This second step was performed based on the complete model, to allow a stiffness rearrangement
e.g. from frame to frame. By conducting the optimization of the single component at this point, it can
lead to changed loads and as a consequence to problems in other areas of the model
3.5
Following steps in the development process
Another step in the development of the tail structure consists of the quantification of the solution. The
topology optimization is first of all supposed to acquire qualitative material distributions, e.g. where are
the best spots to place frames or connection elements. The question of ideal fin thicknesses can only
be answered by using size- , shape optimization or a combination of both. The optimization model can
be upgraded by regarding stress constrains, damage values of lifetime calculations or buckle safety
factors [1]. A detailed description of the tail structure’s development process can be found in a
separate publication [5].
4
Conclusion
The topology optimization with Altair OptiStruct was an important tool in the concept phase of the
A400M airfreighter’s development process. Already in the early development stage, statements about
the material distribution based on the given loads can be made in order to decide design questions
based on fact.
Further on tools around the optimization software take care that optimizations can be performed with
complex FE-models and accordingly formulate the task deserved on demand, if it is about the
definition of design variables, boundary conditions of the optimization and objective function. For this
reason optimization tools are more and more integrated into the development process, in order to
save cost-intensive and time consuming iteration loops.
NAFEMS Seminar:
„Optimization in Structural Mechanics“
8
April 27 - 28, 2005
Wiesbaden, Germany
5
References
[1]
Altair OptiStruct, Users Manual v7.0, (2004), Altair Engineering inc., Troy MI.
[2]
Bendsoe, M.P.; Sigmund, O.; Topology Optimization – Theory, Methods and Applications,
Springer Verlag, Heidelberg, 2003.
[3]
Schramm, U.; Structural Optimization – An efficient Tool in Automotive Design. ATZ –
Automobiltechnische Zeitschrift, 100 (1998) Part 1: 456-462, Part 2: 566-572. (In German,
English in ATZ worldwide).
[4]
Schramm, U.; Thomas, H. L.; Zou M.; Manufacturing Considerations and Structural
Optimization for Automotive Components, (2002), Society of Automotive Engineers, Inc.
[5]
Schuhmacher, G.; Stettner, M.; Zotemantel, R.; O’Leary, O.; Wagner, M.; optimization
assisted structural design of a new military transport aircraft, (2004), American Institute of
Aeronautics and Astronautics.
[6]
Zou, M.; Shyy, Y.K.; Thomas, H. L.; Checkerboard and minimum member size control in
topology optimization, (1999), Proceedings of the 3rd World Congress of Structural and
Multidisciplinary Optimization, Buffalo, New York.
NAFEMS Seminar:
„Optimization in Structural Mechanics“
9
April 27 - 28, 2005
Wiesbaden, Germany