Benchmarking Multidisciplinary
Design Optimization Algorithms
16.842 Fundamentals of Systems Engineering
Alessandro Aliakbargolkar, Rhea Liem, Brian Yutko
October 16, 2009
Tedford, Nathan, and Joaquim R. R. A. Martins. "Benchmarking Multidisciplinary Design
Optimization Algorithms." (2007): 1-30.
1
Intro/Objective
• Traditional sequential optimization
– Can’t find true optimum in MD systems
– Consider interdisciplinary interactions  true optimum
• Various MDO architectures developed
– Monolithic (MDF, SAND)
• Good for small systems – scale poorly
• Poor for industrial settings where disciplines act largely indep
– System-level (CO, BLISS)
• Method selection is typically done ad hoc
– Performance dependent on implementation
– Comparison of results between studies is difficult
πMDO
• Provide a framework to facilitate benchmarking architectures
• Describe the problem once and implement MDO methods
automatically
Sequential Optimization
Image by MIT OpenCourseWare.
• Wing sweep is a global
variable
• Wing thickness is local
to structures
• Wing twist is local to
aerodynamics
Sequential Optimization (cont)
Aerodynamic
Optimization
• Max Range
• w.r.t twist
• s.t. lift = weight
Forces
Drag
Structural
Optimization
Displacements
Weight
• Max Range
• w.r.t thickness
• s.t. stress constraints
Always results in an elliptical lift distribution.
Non-hierarchical vs Hierarchical MDO Architectures
Distributed Analysis – Non-hierarchical
• analysis performed in disciplinary models
• “centralized” optimization  system level
Distributed Design – Hierarchical
• disciplinary model performs design tasks
• bilevel optimization
System Level Optimizer
System Level Optimizer
Analysis 1
Analysis 2
e.g., MDF, IDF, SAND
Opt. 1
Opt. 2
Opt. 3
An. 1
An. 2
An. 3
Analysis 3
e.g., CO, CSSO
Multidisciplinary Feasible (MDF)
x: Local Design Variable
z: Global Design Variable
y: Discipline states and coupling variables
Optimizer
f, c
z2, x2
z1, x1
y1
Discipline 1
z3, x3
MDA
y1
y2
Discipline 2
y2
y3
Discipline 3
• MDA solves all governing eqns using design variables until coupling
variables converge
• Objectives and constraints then computed
• Requiring MDA solution at each design point  iteration is MD Feasible
Aero/Structural MDF
6
x 104
5
Lift (N)
4
Sequential
MDF
3
Elliptical Distribution
2
1
0
2
4
6
8
10
12
14
16
18
20
Spanwise distance (m) - [Root at left, Tip at right]
Image by MIT OpenCourseWare.
• MDF method differs up to 33% with non-MDO analysis
• Development of MDA is costly
– Sensitivity analysis is especially expensive
• No parallel computation outside MDA module
Individual Discipline Feasible (IDF)
Optimizer
f, c
x: Local Design Variable
z: Global Design Variable
yt: Coupling variable estimates (targets)
yi: Coupling variable outputs
z, x, yt
z1, x1, y2t, y3t
y1
z2, x2, y1t, y3t
Discipline 1
y2
Discipline 2
y3
•
z3, x3, y1t, y2t
Discipline 3
Decoupled version of MDF
– Coupling variables are passed with design variables. Optimizer provides estimates.
– Enforces discipline feasibility rather than multidiscipline feasibility
•
To ensure multidiscipline feasibility one constraint added for each coupling variable
– At optimum, coupling estimate = coupling variable computed at the discipline
•
Requiring MDA solution at each design point  iteration is MD Feasible
Simultaneous Analysis and Design (SAND)
z: common variable
x: local variable
y: coupling variable
u: State
System Level
Optimizer
• governing equations
As equality
constraints
z, x1, u
z, x2, u
z, x3, u
• optimization
Compute:
• objective function
• global constraint
• discipline constraint
y1
Discipline 1
y2
y3
•
•
•
Discipline 2
Discipline 3
Not always easy to compute the residuals
Discipline feasibility is not generally attained at intermediate design points
Residual constraints  increase dimensionality
– Sensitivity analyses can be very expensive
Collaborative Optimization (CO)
System Level
Optimizer
Global & compatibility
constraints
J1*
z, y2t, y3t
z, x1,
y2t, y3t
Optimizer 1
y1
z, y1t, y3t
Independent
optimization
subproblems
Discipline 1
J2*
Optimizer 2
y2
J3*
•
z, x2,
y1t, y3t
Discipline 2
Optimizer 3
y3
Ji* : interdisciplinary compatibility
–
•
•
•
z, y1t, y2t
Minimize Ji = [discipline variable – target
variable]2
z, x3,
y1t, y2t
Discipline 3
(Alexandrov and Lewis, 1999)
Discipline feasibility is always maintained
Multidisciplinary feasibility is achieved when converged compatibility constraints
Typical industry practice, parallelizable
Concurrent Subspace Optimization (CSSO) (1/2)
x : local design variables
z : global design variables
System level optimizer
ỹ : coupling variables
Responsible for satisfying
global and local constraints
System
level optimizer
z, x
f,c
z i : global variables assigned to discipline i
ỹ j : coupling variables assigned to discipline i
z, x
f,c
~y
x i : local variables assigned to discipline i
0 : variable held constant during optimization
RS
minimize : f(z,x, y˜ )
w.r.t.:
z,x
s.t.:
c(z,x, y˜ )  0
Subspace level optimizer
Optimizer 1
z1, x1
f,c
f,c
~y23
z1, x1
z1, x1
~y23
RS
Efficiency depends on
the cost of producing
response
y1
Discipline 1
the response surface
surface
(MDA calls)
minimize : f(z i ,z 0,x i ,x0 ,y i (z i ,z 0 ,xi ,x 0, y˜ j ), y˜ j )
w.r.t.:
s.t.:
zi ,xi
c(zi ,z 0 ,x i ,x 0 ,y i (z i ,z 0 ,xi ,x 0 , y˜ j ), y˜ j )  0
(CSSO diagram from J. Martins, ESD.77 lecture, April 2008)
• Non-local variables are calculated
through a quadratic response surface
(N^2/2 function calls, N=number of
variables);
•Response surface is created by
completing an MDA through a
number of design points.
Optimizer 2
z2, x2
f,c
f,c
~y23
z2, x2
RS
y2
z2, x2
~y23
Discipline 2
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Concurrent Subspace Optimization (CSSO) (2/2)
System Analysis
• Non hierarchical structure
• Bi-level
• Concurrent architecture
Model Update
Approximation
System 1
Approximation
System 2
Approximation
System 3
Approximation
Optimizer
Optimizer
Optimizer
System 1
Analysis
System 2
Analysis
System 3
Analysis
Model Update
System
Approximation
System
Optimizer
Image by MIT OpenCourseWare.
(CSSO diagram from J. Martins, ESD.77 lecture, April 2008)
Method Comparison
MDF
IDF
SAND
CO
CSSO
Parallelizability
Discipline
feasibility
Not always
Multidisciplinary
feasibility
Computational
cost
≈ MDA
module cost
Not always
Not always
Ensured at
convergence
≈ # of coupling
variables
≈ residual
comp. cost
≈ # of coupling
variables
 Additional
feasibility
constraint
Typically the
best
≈ response
surface cost
MDA at
initialization:
limit N ≤ 20
Sensitivity
analysis
Semi-analytic
Semi-analytic
Expensive (SA
of residuals)
Semi-analytic
Does not
really matter
(using
response
surface)
Convergence
OK
OK
OK
Slow
Slow
Application 1: Combustion of Propane
• Objective: optimize the combustion of propane in air;
• Propane + Air  10 combustion products
• Traditional approach: nonlinear system of 11 equations
– 1 equation per combustion product
– 1 equation for chemical equilibrium (sum of products)
– Design constants (8):
• pressure (p)
• air to fuel ratio (R)
• empirical constants (6)
• MDO approach:
– Equations partitioned in 3 disciplines
• Partitioned so to couple disciplines and system-level objective
Application 1: Computational performance comparison
the table shows
the number of
required residual
evaluations per
method per disc.
102
Notice the difference in time
required. Take away message:
Optimal MDO architecture selection
depends on problem formulation
Relative error
100
10-2
CSSO
MDF
10-4
CO
10-6
SAND
IDF
10-8
0
1
2
3
Time (s)
4
5
6
Image by MIT OpenCourseWare.
Application 2: Scalable problem
• Objective
– A sensitivity analysis to increasing dimensionality while managing the
computational requirements;
• Parameters of the problem
1.
2.
3.
4.
5.
Number of disciplines
Number of output coupling variables associated with each discipline
Number of local design variables associated with each discipline
Number of global design variables
Strength of coupling between the disciplines
• Objective Function: Quadratic
• Disciplines Dependence: Linear
• Disciplines Equations: Linear systems dependent on global/local design
variables and non-local coupling variables;
• Constraints: Local on each coupling variable
Application 2: Formulation
N
minimize : z z +  y Ti y i
T
i=1
w.r.t.: z,x
s.t.: 1-
yi
 0, i  1,...,N
Ci
discipline i equation :
y i (z, x i , y j )  
1
Cz z  Cx i x i  C y j y j
Cy i


(C are matrices of random positive
coefficients generated before the optimization)
Application 2: Sensitivity to the # of design variables
Scalable problem - effect of number of design variables
Optimization Time (s)
103
MDF
102
IDF
101
SAND
100
100
101
102
103
Number of Local Design Variables
Design variables per discipline
2
20
200
MDF
10348
305145
515264
**
IDF
1493
12202
30918
350782
**
**
25637
**
SAND
Image by MIT OpenCourseWare.
2000
Application 2: Sensitivity to the # of coupling variables
Scalable problem - effect of number of coupling variables
Optimization Time (s)
104
103
MDF
102
IDF
101
SAND
100
100
101
102
Number of Coupling Variables
Coupling variables per discipline
103
2
20
200
MDF
55722
134653
136262
IDF
2724
9421
33934
SAND
1921
5351
**
Image by MIT OpenCourseWare.
Conclusion
• MDO platform provides ease in the development,
benchmarking, and use of MDO architectures
• The performance of each architecture is dependent on:
– Characteristics of the problem
– Optimization algorithm
– Sensitivity analysis method
• Scalable problem  enables studies on the effect of number
of local and coupling variables on the optimization time
Discussion points
• How extensive are the MDO applications (industrial, design projects, …)?
And if MDO is still not extensively used at a design stage:
– What are the limiting factors?
– Too complex, too expensive, too mathematical, …?
• How applicable is MDO in real-world problems?
– How do the nice and elegant MDO algorithms behave under the complexities
of real-world problems?
– Presence of multiple local minima, islands of feasibility, combination of
continuous and discrete variables, …
• What is the best way to define the boundary between systems
architecting (SA) and MDO?
• How to improve the interactions between SA and MDO in the early
stages of the design?
21
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16.842 Fundamentals of Systems Engineering
Fall 2009
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