Evolutionary Optimization Method for Thermal Protection System Design

1
Evolutionary Optimization Methodfor Thermal
Protection System Design
Department of Mechanical and Materials
Engineering Wright State University, Dayton, OH,
45435
Air Vehicles Directorate WPAFB, OH, 45433
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Thermal Protection System (TPS)

• Responsible for protecting the spacecrafts
  components from melting due to high re-entry
  temperatures.
• Key technology that enables a spacecraft to be
  lightweight, fully reusable, and easily
  maintained.
• Consists of different types of materials that
  are distributed all over the spacecraft, such as
  felt blankets, ceramic tiles, carbon-carbon
  leading edge, and metallic TPS.

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Metallic TPS

• The metallic TPS makes maintenance and
  replacement easier

• Its inherent ductility and design flexibility
  offers the potential for a more robust system
  with lower maintenance costs than competing TPS
  systems

lt ARMOR TPS model gt
lt X-33 metallic TPS panel gt
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Metallic TPS
PLATE (Unremovable region)
FRAME
z
x, y
SUPPORT
CHALLENGES IN DESIGN

• Weight is a key disadvantage in its use

• Thermal and acoustic loading conditions tend to
  be at odds with one another

It is difficult to develop an optimum TPS model
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Metallic TPS
REMEDIES

• The support height is increased to reduce the
  heat transfer to the fuselage by inserting more
  insulators.

• A frame for thin plate is attached to prevent
  flutter due to acoustic loading.

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Structural Optimization Method
Topology design is for determining the
distribution pattern of material and void.
This 0-1 discrete problem is transferred into a
continuous one

• Homogenization method (Bendsoe and Kikuchi, 1988)

The design region is assumed consisting of the
porous microstructure and material density
distribution is optimized by taking size and
direction of the hole of the micro-structure as
design variables.

• Density-based method (Yang and Chuang, 1999)

This method heuristically designs the material
properties such as Youngs modulus and density
for each finite element directly to find optimal
material distributions.

• Cellular automation generation method (Inou,
  1994)

Evolution of the organism is considered cell
automaton based on the local field rule and it is
the evolution makes the structure form that
adapted to the dynamics environment.
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Continuous approaches to topology optimization
Hard to derive sensitivity functions
(Mathematically difficult!!)
Make Final design to be different by a penalty
rule for gray regions
Hard to apply a sensitivity for stress
(Usually, strain energy is used)
The evolutionary structural optimization (ESO)
method
or
The bidirectional evolutionary structural
optimization (BESO) method
Discontinuous approach, which deals with only
0-1 distribution (No penalty function)
very simple concept !!
Stress, as well as strain energy can be easily
utilized
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Evolutionary Structural Optimization (ESO)
Methodfor Thermal Protection System Design
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Evolutionary Structural Optimization (ESO) Method

• Based on a simple concept that the residual
  structure evolves toward an optimum by gradually
  removing inefficient material.

STATIC OPTIMIZATION PROBLEM
low stress values low strain energy values
A number of elements with
are removed from structure
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CONVENTIONAL ESO FOR EIGENVALUE PROBLEM

• General eigenvalue problem

(1-1)
(1-2)

• Rayleigh quotient

• Change of the i th frequency

(1-3)
(1-4)
(1-5)
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CONVENTIONAL ESO FOR EIGENVALUE PROBLEM
Using a dynamic control parameter based on the
Rayleigh quotient,
(1-6)
A number of elements with high positive values
are eliminated to increase the natural frequency
of interest.
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Eliminating a large number of elements from a
structure through many iterative steps
Results in very weak modal stiffness !
Drastic alteration in the natural frequency of
interest !
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Derivation of New Dynamic Control Parameter
The response of the equation of motion (1) is
described in modal coordinates as
(1)
(2)
where ,
,
The displacement of each nodal point is computed
by implementing the concept of virtual static
displacement for each mode shape.
By assuming 0,
(3)
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The displacement of each nodal point is computed
by implementing the concept of virtual static
displacement for each mode shape.
By assuming ?0,
(3)
modal displacement (generalized displacement)
generated by external force F .
The natural mode can be treated as a
response (displacement) by applying the external
force whose modal displacement is 1 for the r th
mode and 0 for the other modes.
When Fr is the external force satisfying this
condition,
(4)
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• Force magnitude for each natural mode is
  different from others

• A higher natural mode requires a larger external
  force due to the complexity of the mode shape

The magnitude of Fr is scaled to be that of the
external force of the interested natural mode.
(5)
The scaled external force vector can estimate the
displacement of the r th natural mode by the
force magnitude of the interested natural mode
(6)
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Dynamic Control Parameter by von Mises Stress
(7)
Eq. (7) considers not only the i th natural
frequency, but also its neighboring natural
frequencies by choosing maximum stress of r th
natural mode
If , , as well as
are selected because
approaches to that of the natural mode of
interest.
A number of elements with the smallest stresses,
that is, the most inefficient elements, are
removed from the structure.
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Cantilever Structure

• Fixed Region

• Removable region

• 2 of the total number are eliminated in each
  iteration.

• Unremovable region

Weighting Factor Change
Evolutionary History of Frequencies
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Comparison between conventional ESO and new ESO
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TPS Design with Heat Transfer Problem

• Minimize TPS weight

subject to

• Using weighting objectives method

(8)

• Transient heat analysis

Four corners are simply supported
Initial Metallic TPS Model
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Optimization Challenge for TPS Model with a Thin
Plate
Plate (Unremovable)

• A local mode can be observed at the plate region
  due to its thinness

Frame
Z
Support
X
To avoid the local mode,
Fixed region
Modified
- Dense-meshed structure
- Few elements removal in one iteration
Too much computational cost !
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TPS Design Process
Design optimization of the support and frame
regions are conducted separately
TPS initial model with dense mesh
TPS support design by topology optimization
Modified TPS initial model based on the topology
result
TPS support design by shape optimization
TPS frame design
Optimum TPS model
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Research Approach
Research Approach

• The support region is designed by topology
  optimization using ESO algorithm

Initial structure modified from topology result
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TPS Support Design

• The initial TPS model is set up as a full-meshed
  structure.

• Plate and frame region are set as unremovable.

• Topology optimization is applied to the
  conventional ESO and the new ESO with the
  proposed control parameter.

Unremovable
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Change of the Fundamental Natural Frequency

• Fundamental natural frequency is improved up to
  certain percentage of material removal.

• As the removed volume percentage increases, the
  conventional method decreases the fundamental
  natural frequency very quickly because there are
  no direct considerations of mode-switching
  phenomenon and modal stiffness.

• The proposed method keeps the fundamental
  natural frequency higher.

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Change in Maximum Thermal Stress

• In the conventional method, the maximum thermal
  stress increases at an early stage because the
  control parameter doesnt consider thermal stress.

• As the removed volume percentage increases, the
  maximum thermal stress decreases because the heat
  transfer to the bottom side is decreased by
  eliminating elements.
• ( Elements that connect between the plate and
  the support region )

• The proposed method restrains the increase in
  maximum thermal stress from increasing at an
  early stage, as well as at increased volume
  reduction.

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Resultant Models with the Fundamental Frequency
at 900 Hz
Conventional Method
Proposed Method
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TPS Support Redesign by Shape Optimization

• The initial TPS model with hollow cube is chosen
  from the previous result.

• Shape optimization (Called Nibbling ESO) is
  applied.

• Plate and frame region are set as unremovable.

• For the fixed region,

- In dynamic analysis, additional stiffness kX,
kY, kZ 0, 0, 108 (N/m) are set up at four
edges of the plate.
The local mode will have a higher natural
frequency when compared with the fundamental
natural frequency of the supports first bending
mode.
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Modified TPS Support Model

• Fundamental frequency at 77.7 901.5 Hz

• Maximum stress at 77.7 0.236 GPa

• When the additional stiffness at the plate edge
  is removed from the structure, the fundamental
  frequency becomes 871.9 Hz due to the increase in
  modal mass.

Fundamental frequency may be increased by
reducing the frame weight.
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TPS Frame Design

• Shape optimization is applied by eliminating
  elements from the bottom surface of the frame
  region.

• Plate and support regions are set up as
  unremovable regions.

• No thermal stress analysis is used.

• TPS model is designed to be lightweight with the
  fundamental frequency greater than 900 Hz.

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Evolutionary Process of the TPS Frame Model

• Up to the 41st iteration, fundamental frequency
  increases.

• In the end iterations, the frequency drastically
  decreases.

• Modified TPS model satisfies the frequency
  constraint at the 45th iteration.

• A transient heat transfer analysis is conducted
  on the TPS model of 45th iteration.

View from the Bottom Side of the Frame
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Optimized TPS Model

• Fundamental natural frequency 919.8 Hz (gt900
  Hz)
• Maximum thermal stress0.228 GPa (lt0.3 GPa)
• TPS mass 76.50 kg (cf. For the full meshed
  structure, 470 kg)

This model reduced its weight until 84 in
comparison with the full-meshed structure
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Summary
1. A multi-objective optimization problem for
thermal stress and fundamental natural frequency
was conducted to make a lightweight TPS model by
using Evolutionary Structural Optimization (ESO)
algorithm.
2. New control parameter based on static analysis
was proposed for the TPS design concerned with
dynamic analysis.
3. An efficient way to obtain a metallic TPS
model was shown by designing the support and the
frame region separately using the ESO method with
the proposed control parameter.
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Bidirectional ESO Methodfor Thermal Protection
System Design
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Conventional Bidirectional Evolutionary
Structural Optimization Method
applies element addition, as well as element
removal (e.g. New elements are attached around
the elements with overly stress to reduce
localized high stress regions in static
optimization problem)
starts from a simple structure satisfying
boundary conditions, not a full-meshed one
uses control parameters based on Rayleigh
quotient for both element addition and removal in
eigenvalue optimization problem
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Control Parameter Based on Rayleigh Quotient
IN THE ADDITION PROCESS
(A)
This equation expresses the change of the i-th
eigenvalue due to attaching the l-th virtual
element to the original structure.
Some elements whose are most
positive among all virtual elements are
converted into real elements !
Eq. (A) is also applied to following situation

• The i-th natural frequency increases by adding
  the l-th element of 0
• The i-th natural frequency decreases by removing
  the l-th element of 0

There is no drastic change of natural frequency
of interest even if mode-switching occurs in
iterative steps because there is no sudden
changes of stiffness of structure.
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Control Parameter Based on Rayleigh Quotient
IN THE REMOVAL PROCESS
(B)
This equation expresses the change of the i-th
eigenvalue due to removing the l-th element from
the original structure.

• Only the changes in natural frequencies are
  evaluated. Nothing is considered about the
  stiffness of any part of the structure.
• The sensitivities of higher-order natural modes
  are incorrect due to the inaccuracy of the
  natural frequency calculations.
• In the case of mode-tracking, comparison between
  the current and previous modes is impossible if a
  new mode appears at the current step, or if a
  previous mode disappears at the current step.
• The change of the natural mode of interest due to
  the mode-switching phenomenon induces a
  significant alteration of the structural
  stiffness in the next evolutionary process.

1 and 2 are the main challenges for the
evolutionary method with the fundamental natural
frequency optimization.
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Challenge in Conventional BESO Method
ADDITION PROCESS
REMOVAL PROCESS
(A)
(B)
No significant decrease in structural stiffness
Significant decrease in structural stiffness due
to mode-switching
Sudden drop of the natural frequency of interest
Unstable eigenvalue optimization process

• Static control parameter using strain energy or
  von Mises stress is applied to consider the modal
  stiffness

• The static control parameter is expanded to
  include the neighboring natural modes, as well as
  the natural mode of interest

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Static Control Parameter Using Strain Energy
Modified natural mode
(6)
Strain energy
Static control parameter in the removal process
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Proposed Bidirectional Evolutionary Structural
Optimization Process

• Virtual element is newly applied.

• In element removal, structural stiffness is only
  considered to avoid sudden drop of the stiffness.

• In element removal, new control parameter is
  applied to consider mode-switching phenomenon.

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Algorithm
The objective of this method is
to shift the fundamental natural frequency of a
three-dimensional structure to a target
frequency as closely as possible.
Because of starting from a simply small structure,
The number of FE to be added gt The number of FE
to be removed
Stage 1
the growth stage to grow the structure until
raising the fundamental natural frequency to a
little bit higher than the target frequency (
upper limit).
( upper limit target frequency a, where a
1.051.50)
The number of FE to be removed gt The number of FE
to be added
Stage 2
the weight reduction stage to control the
fundamental natural frequency until it reaches to
lower limit ( target freq ß, where ß1.001.02).
Stage 3
The number of FE to be removed The number of FE
to be added
the alternation stage to change the positions
of the elements
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Algorithm
The evolutionary iterative process is continued
until the i th natural frequency ( fundamental
natural frequency) converges after the
application of the three kinds of stages.
The number of finite elements, which are added
and removed in an iterative process, is set
within 12 of the number of elements of the
structure .
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A Bridge Model
1st natural freq. 2349 Hz
Upper limit frequency 4250 Hz (target
frequency 1.05)
1st mode (2349Hz)
2nd mode (2506Hz)
3rd mode (3658Hz)
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A Bridge Model
V shape
Z
X
1st mode (2349Hz)
2nd mode (2506Hz)
Reverse-V shape
Z
Y
3rd mode (3658Hz)
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Availability of Proposed Method
ltlt ADDITION gtgt
Control parameter using Rayleigh quotient
ltlt REMOVAL gtgt
ltlt REMOVAL gtgt
Control parameter using strain energy
Control parameter using Rayleigh quotient
METHOD 1 (Conventional method)
METHOD 2 (Proposed method)
COMPARISON
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A Bridge Model
In the case of METHOD 1,
In the case of METHOD 2,
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A Bridge Model
1st mode (2349Hz)
2nd mode (2506Hz)
3rd mode (3658Hz)
1st mode (4028Hz)
2nd mode (4176Hz)
3rd mode (4299Hz)
METHOD 2 can select appropriate finite elements
to be removed, and can evolve the bridge model
to the optimum design stably under the condition
that Mode-switching happens in.
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A Thermal Protection System Model
Adaptable, Robust, Metallic, Operable, Reusable
(ARMOR) TPS panel
X-33's Innovative Metallic Thermal Shield
1st natural freq. 104.8 Hz
ltlt Inconel 693 alloy gtgt
E 196 GPa Density 7770 kg/m3 Poisson 0.32
Target freq. 1000 Hz
Initial model for designing TPS
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A Thermal Protection System Model
At the 100th iteration, the fundamental natural
frequency is 1011.2 Hz
Although "mode-switching" occurs irregularly in
the iterative processes between the bending mode
in X-axis and the bending mode of Y-axis, which
are mutually orthogonal directions, the evolution
process is executed very smoothly.
Evolutionary history of the first three natural
frequencies
49
A Thermal Protection System Model
Initial model
Modified model
ltThe fundamental natural mode for the initial
model and the modified model gt
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Summary
1. Two kinds of control parameters are introduced
in the bidirectional evolutionary structural
optimization.
2. It is shown that the static control parameter
can make a natural frequency of interest
convergence stable or improved by removing
elements with low values.
3. Static control parameter is modified to
consider adjacent natural modes, as well as the
targeted mode to alleviate the problem of the
mode-switching phenomenon.
4. Using the proposed control parameter and new
addition process, two representative models are
analyzed.