surrogate modeling for composite aircraft structure ... · the dimension estimation step in an...
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Surrogate Modeling for Composite Aircraft Structure Optimization
S. GRIHONa, S. ALESTRAb, E. BURNAEVc, P. PRIKHODKOc
aESAZO, Airbus Operations SAS
bApplied Mathematics Group, Airbus Group
cData Analysis Lab, DATADVANCE
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Contents
Aircraft Structural Components Optimization
Surrogate Modeling and Optimization: Motivation
Applications of Surrogate Modeling for StructureOptimization
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Contents
Aircraft Structural Components Optimization
Surrogate Modeling and Optimization: Motivation
Applications of Surrogate Modeling for StructureOptimization
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Introduction: Pre-sizing problem
Aeronautical structures
are mainly made of
stiffened panels enforced
with stiffeners
Define Super Stiffeners as the
theoretical union of a stringer
and two half panels
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Introduction: Pre-sizing problem
Super Stiffeners are
subject to highly non-linear
phenomena such as
buckling, collapse and
damage tolerance
To determine the optimal size of these super stiffeners,
static mechanical criteria must be computed using
dedicated software that is based on non-linear
calculation
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Introduction: Optimization
The analysis and the dimension estimation of the whole structure is currently computed by running a two-level study:
• global level: Finite
Element (FE) analysis run
on the whole FE model
provides internal loads,
applied to each S-Stiffener
• local level: these loads are
used to compute static
mechanical criteria
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Introduction: Optimization
Most of static mechanics criteria are formulated using Reserve Factors (RF): a structure is validated provided all its RFs are greater than one.
Optimization problem is to minimize weight under constraints on geometry and RFs
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Contents
Aircraft Structural Components Optimization
Surrogate Modeling and Optimization: Motivation
Applications of Surrogate Modeling for Structure Optimization
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Motivation: computational burden
The dimension estimation step in an aircraft developmentprogram is a repetitive, time-consuming process:
• Huge dimensionality of the problem (O(103) variables and O(105) constraints)
• Optimization methods require gradients of the constraint functions, which can only be obtained by finite differences
• Values of mechanical strength constraints are computed using dedicated software
• Each of numerous calls to this software takes up to 1 sec
The need for finite difference calculations in each of
numerous local optimizations greatly increases the time
between two update steps of optimization
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Motivation: computational burden
The dimension estimation step in an aircraft developmentprogram is a repetitive, time-consuming process:
• Huge dimensionality of the problem (O(103) variables and O(105) constraints)
• Optimization methods require gradients of the constraint functions, which can only be obtained by finite differences
• Values of mechanical strength constraints are computed using dedicated software
• Each of numerous calls to this software takes up to 1 sec
The need for finite difference calculations in each of
numerous local optimizations greatly increases the time
between two update steps of optimization
![Page 11: Surrogate Modeling for Composite Aircraft Structure ... · The dimension estimation step in an aircraft development program is a repetitive, time-consuming process: • Huge dimensionality](https://reader033.vdocuments.us/reader033/viewer/2022051806/5fff2660d287e31dbd1eb9c8/html5/thumbnails/11.jpg)
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Motivation: computational burden
The dimension estimation step in an aircraft developmentprogram is a repetitive, time-consuming process:
• Huge dimensionality of the problem (O(103) variables and O(105) constraints)
• Optimization methods require gradients of the constraint functions, which can only be obtained by finite differences
• Values of mechanical strength constraints are computed using dedicated software
• Each of numerous calls to this software takes up to 1 sec
The need for finite difference calculations in each of
numerous local optimizations greatly increases the time
between two update steps of optimization
![Page 12: Surrogate Modeling for Composite Aircraft Structure ... · The dimension estimation step in an aircraft development program is a repetitive, time-consuming process: • Huge dimensionality](https://reader033.vdocuments.us/reader033/viewer/2022051806/5fff2660d287e31dbd1eb9c8/html5/thumbnails/12.jpg)
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Motivation: computational burden
The dimension estimation step in an aircraft developmentprogram is a repetitive, time-consuming process:
• Huge dimensionality of the problem (O(103) variables and O(105) constraints)
• Optimization methods require gradients of the constraint functions, which can only be obtained by finite differences
• Values of mechanical strength constraints are computed using dedicated software
• Each of numerous calls to this software takes up to 1 sec
The need for finite difference calculations in each of
numerous local optimizations greatly increases the time
between two update steps of optimization
![Page 13: Surrogate Modeling for Composite Aircraft Structure ... · The dimension estimation step in an aircraft development program is a repetitive, time-consuming process: • Huge dimensionality](https://reader033.vdocuments.us/reader033/viewer/2022051806/5fff2660d287e31dbd1eb9c8/html5/thumbnails/13.jpg)
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Motivation: computational burden
The dimension estimation step in an aircraft developmentprogram is a repetitive, time-consuming process:
• Huge dimensionality of the problem (O(103) variables and O(105) constraints)
• Optimization methods require gradients of the constraint functions, which can only be obtained by finite differences
• Values of mechanical strength constraints are computed using dedicated software
• Each of numerous calls to this software takes up to 1 sec
The need for finite difference calculations in each of
numerous local optimizations greatly increases the time
between two update steps of optimization
![Page 14: Surrogate Modeling for Composite Aircraft Structure ... · The dimension estimation step in an aircraft development program is a repetitive, time-consuming process: • Huge dimensionality](https://reader033.vdocuments.us/reader033/viewer/2022051806/5fff2660d287e31dbd1eb9c8/html5/thumbnails/14.jpg)
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Motivation: computational burden
The dimension estimation step in an aircraft developmentprogram is a repetitive, time-consuming process:
• Huge dimensionality of the problem (O(103) variables and O(105) constraints)
• Optimization methods require gradients of the constraint functions, which can only be obtained by finite differences
• Values of mechanical strength constraints are computed using dedicated software
• Each of numerous calls to this software takes up to 1 sec
The need for finite difference calculations in each of
numerous local optimizations greatly increases the time
between two update steps of optimization
![Page 15: Surrogate Modeling for Composite Aircraft Structure ... · The dimension estimation step in an aircraft development program is a repetitive, time-consuming process: • Huge dimensionality](https://reader033.vdocuments.us/reader033/viewer/2022051806/5fff2660d287e31dbd1eb9c8/html5/thumbnails/15.jpg)
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Solution: Use of Surrogate Models (Response Surface Models)
Expected benefits of surrogate models use:
• time saving in pre-sizing processes
• response smoothingRFs are sometimes are not themselves continuous (as
often for semi-empirical approaches)
BUT
Surrogate Models (SMs) may provide a continuous and
differentiable approximation
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Solution: Use of Surrogate Models (Response Surface Models)
Expected benefits of surrogate models use:
• time saving in pre-sizing processes
• response smoothingRFs are sometimes are not themselves continuous (as
often for semi-empirical approaches)
BUT
Surrogate Models (SMs) may provide a continuous and
differentiable approximation
=> calculation of gradients directly instead of using
costly finite differences
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Contents
Aircraft Structural Components Optimization
Surrogate Modeling and Optimization: Motivation
Applications of Surrogate Modeling for StructureOptimization
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Examples of Surrogate Modeling Applications
Problem 1: Surrogate Optimization of Wing Covers
Problem 2: Surrogate Modeling of Thin Composite Plates
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Examples of Surrogate Modeling Applications
Problem 1: Surrogate Optimization of Wing Covers
Problem 2: Surrogate Modeling of Thin Composite Plates
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Design Variables and Criteria
Design variables
h
ba
tatw
t2
t1 %0, %45%0, %45
h
ba
tatw
t2
t1 %0, %45%0, %45
h
ba
tatw
t2
t1
h
ba
tatw
t2
t1
h
ba
tatw
t2
t1 %0, %45%0, %45
Different
Stringer
Geometries
Skin angle thicknesses / stringer section parameters
Design criteria (calculated by Airbus in-house software)- stability: Rayleigh Ritz approach & Karman theory for post-buckling- damage tolerance- reparability: bearing, by-pass
All composite design variables and criteria are considered
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Construct such Y = Fapprox(X)
that
Fapprox(X) ≈ F(X)
Y = F(X) – simulates behaviour
relationship between X and Y
Original model
DoE Data Base
(Xi, Yi=F(Xi)), i = 1,…,N –
training data set
In the considered case:
X is composed of • Skin thickness,
• Percentages of standard
draping angles 0%, 45%, 90%,
• T-stringer core and web
percentages 0%, 45%, 90%,
• etc. (> 20 parameters)
F(X) is realized by
Airbus in-house skill-tool
Y = F(X) is composed
of various Reserve Factors
(>20 RFs)
Surrogate Models (Response Surface Models)
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• nonlinear multivariate functions (> 20 input parameters)
• discontinuities and large gradients
• strict requirements on model accuracy (for 95% of errors are less than 5%)
Challenges
Example of two-dimensional slice of function to approximate
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MACROS is a software toolkit for
• intelligent data analysis and
• multi-disciplinary optimization
developed by DATADVANCE llc.
Provides proprietary and state-of-the-art data analysis and optimization
techniques and consists of Generic Tools for
• Dimension reduction
• Sensitivity Analysis
• Design of Experiments
• Construction of Surrogate Models
• Variable Fidelity Data Modeling
• Optimization
MACROS: a surrogate modeling and optimization software toolkit
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Results: Wing stress model
Optimization of the wing
lower and upper covers
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Results: Heavy Point optimization run
Smoother convergence with MACROS SM
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A check of Reserve Factors was performed with the optimum based onMACROS SM
Satisfactory accuracy for a pre-sizing result, according to AIRBUS experts and considering that a pre-sizing is always to be re-engineered including, for example, manufacturing constraints
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Conclusions
MACROS surrogate model gives high accuracy ofapproximation
MACROS surrogate model allows obtaining smootherconvergence in less iterations with a smoother distribution ofthickness/stringer dimensions and a small violation of constraintswhich then could be easily repaired at the detailed design phase
MACROS surrogate model provides expected reduction ofstructure optimization computational time from several days toa few hours.
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Examples of Surrogate Modeling Applications
Problem 1: Surrogate Optimization of Wing Covers
Problem 2: Surrogate Modeling of Thin Composite Plates
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Objective:
Approximate Reserve Factors (RFs) of thin composite plates as a functions of loads with smoothand fast surrogate model
PRESTO Data Base:
• ~ 900 Gigabytes divided into catalogues
• Each catalogue is defined by a
specific stacking sequence
• Each catalogue contains for
different pairs of thickness
and area values of different
RFs depending on applied
forces
• For any new load RFs are evaluated using piecewise-linear interpolation
• too huge size of the PRESTO Data Base,
• non-smooth approximation
Problem statement
PROBLEM:
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Requirements to Surrogate Models:
• relative prediction error is smaller than 5% for 95% of observations
• model size is at least 9-10 times smaller than the size of data base
• It should take no more than a few seconds to construct a new model
Problem statement
Test Case:• 7 RFs
• 47 pairs of t, A
That is 329 models in total
For each (pair t, A + failure mode) training sample
is defined as
• inputs (DoE) = 2D grid of 23 x 45 = 1035 points
• outputs = values of the corresponding RF
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t. • nonlinear 2D/3D function
• discontinuities and large gradients
• strict limitations on model size,model construction time and accuracy
Challenges
Examples of functions to approximate
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Algorithm
Steps of model construction:
1. Construct classifier for “nan” points
2. Transform data (box-cox transformation of y gives better accuracy)
3. Build separate model on filtered sample (“nan” points with y>100 removed) for each output using MACROS techniques TA/iTA
4. Merge separate models and classifier into one model for catalog (in form of C99 code)
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TCDB size
Comparison of TCDB size:
Size RFDB: 11553792 octets
Size MACROS surrogate: 1284480 octets
Ratio = RFDB / Surrogate ~ 9
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Comparison of RF values
Comparison of critical RF values
MACROS:
RFDB:
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Comparison of RF values
Comparison of critical RF types (failure modes)
MACROS:
RFDB:
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Conclusions
Following activities are planned:
• Integration of MACROS surrogate modeling techniques into
PRESTO (to allow construction of surrogate models on the side
of LMS-SAMTECH)
• Research on approaches to reduction of design of experiments
(to further reduce space storage size for the models)
Results of tests of MACROS surrogate model for CFRP fuselage
stiffened panels (LMS-SAMTECH) showed that MACROS surrogate
models give good accuracy with respect to the RF databases with a
reduction factor of about 10 for the data storage.
• Thus the use of MACROS is beneficial for data compression
within PRESTO