Numerical Models and Algorithms for Multidisciplinary Design

Transcription

Introduction
Numerical Models for Engineering Optimization
Optimization Algorithms
Conclusions and Further Developments
Numerical Models and Algorithms for
Multidisciplinary Design Optimization
José Herskovits
[email protected]
OptimizE - Interdisciplinary Lab for Engineering Optimization
Mechanical Engineering Program
COPPE - Federal University of Rio de Janeiro
Ter@tec FORUM
Ecole Polytechnique
France, June 2010
José Herskovits
Introduction
Outline
1
Introduction
2
The Classical Model for Engineering Optimization
SAND Optimization
3
FAIPA
FAIPA-SAND
FAIPA-MDO
4
José Herskovits
Introduction
Engineering Design
Engineering Design is done in two stages:
1
2
CONCEPTION: the basic ideas are defined.
SIZING:
sizes
shapes,
Materials,
controls, that verify Design requirements are obtained.
3
OPTIMIZATION: Looks for the best SET OF SIZES that
satisfy DESIGN REQUIREMENTS
José Herskovits
Introduction
The Truss Optimization Example
Find the CROSS AREAS of the bars ⇒ DESIGN VARIABLES
that minimize the STRUCTURAL WEIGHT ⇒ OBJECTIVE
FUNCTION
and such that the STRESSES are allowable ⇒
CONSTRAINTS
José Herskovits
Introduction
SAND Optimization
Outline
1
Introduction
2
SAND Optimization
3
FAIPA
FAIPA-SAND
FAIPA-MDO
4
José Herskovits
Introduction
SAND Optimization
Engineering Systems Design
State Equations and State Variables
We consider the optimal design of engineering systems described
by the state equation
e(x, u) = 0
e ∈ Rr
where x ∈ Rn and u ∈ Rr are the design variables and the state
variables respectively.
Example
The displacements vector and the equilibrium equation, in
structures.
José Herskovits
Introduction
SAND Optimization
Engineering Optimization
The objective function is
f (x, u)
we have m inequality constraints
g(x, u) ≤ 0; g ∈ R m
and p equality constraints
h(x, u) = 0; h ∈ R p
José Herskovits
Introduction
SAND Optimization
The Optimization Problem Formulation
The optimization problem can be represented by the following
mathematical programming problem:

Minimize
f (x, u(x))





 Submitted to:
where






g(x, u(x)) ≤ 0; g ∈ Rm
h(x, u(x)) = 0; h ∈ Rp
x ∈ Rn
u(x) must verify e(x, u(x)) = 0
Then, for each design, the state equation must be solved.
José Herskovits
Introduction
SAND Optimization
Outline
1
Introduction
2
SAND Optimization
3
FAIPA
FAIPA-SAND
FAIPA-MDO
4
José Herskovits
Introduction
SAND Optimization
SAND - Simultaneous Analysis and Design
SAND
Also called “ONE SHOT OPTIMIZATION”
The State Variables (i.e. nodal displacements) are included in
the Mathematical Program within the design variables.
The State Equations (i. e. the equilibrium eq.) are included
as equality constraints.
José Herskovits
Introduction
SAND Optimization
The SAND Mathematical program
The optimization problem can be represented by the following
optimization problem:

Minimize x,u
f (x, u)





 Submitted to:
where
g(x, u) ≤ 0
h(x, u) = 0
e(x, u) = 0






u is a vector of State Variables,
e(x, u) is a vector of State Equations.
José Herskovits
Introduction
SAND Optimization
The main advantage
Solves simultaneously Analysis and Optimization Problems
Even for nonlinear analysis!
The main difficulty
The size of the Mathematical Program is greatly increased.
Sensitivity Analysis for SAND Optimization is simpler
José Herskovits
Introduction
SAND Optimization
Outline
1
Introduction
2
SAND Optimization
3
FAIPA
FAIPA-SAND
FAIPA-MDO
4
José Herskovits
Introduction
SAND Optimization
MDO
Optimal Design of complex engineering systems:
that are governed by mutually interacting physical phenomena
Made up of distinct interacting subsystems.
José Herskovits
Introduction
SAND Optimization
MDO
Modern design techniques require numerical models of each of
the parts of the system and each of the interacting physical
phenomena.
These models and the simulation codes were generally
developed independently.
To be successful, MDO must be based on existing analysis
codes, as they are.
Is not reasonable to ask engineers to modify their
numerical models and computer codes to adapt to MDO.
Due to the complexity of the problem, in general, MDO
techniques work with approximated problems and/or looks for
approximated solutions of the problem.
José Herskovits
Introduction
SAND Optimization
MDO
We propose a model for MDO that that considers the
complete problem without reductions, decompositions or
simplifications.
Low fidelity models or surrogates can also be employed
The present model uses existing numerical techniques and
analysis codes for each discipline
This goal is very ambitious due to the size and complexity of
the problems, but it can be a way to obtain strong and
efficient tools for MDO.
José Herskovits
Introduction
SAND Optimization
MDO
We have r Engineering Systems.
u = (u1 , u2 , ...ur ) are the State Variables of sub-systems.
e1 (x , z, u1 ), e2 (x , z, u2 ), ..., er (x , z, ur ) are the State Equations
of the sub-systems.
z are auxiliary variables that represent interactions between
the sub-systems, for example:
Aerodynamic Forces acting on the structure.
Multibody.
h(x , z, u) = 0 Impose compatibility of sub-systems.
Sub-Systems can be also included implicitly, as in the classic
approach
José Herskovits
Introduction
SAND Optimization
MDO
A Mathematical programming for MDO:


Minimize x,z,u f (x, z, u)




Submitted
to:





g(x, z, u) ≤ 0




h(x, z, u) = 0
e1 (x, z, u1 ) = 0
e2 (x, z, u2 ) = 0
..
.














er (x, z, ur ) = 0
José Herskovits
Introduction
FAIPA
FAIPA-SAND
FAIPA-MDO
Outline
1
Introduction
2
SAND Optimization
3
FAIPA
FAIPA-SAND
FAIPA-MDO
4
José Herskovits
Introduction
FAIPA
FAIPA-SAND
FAIPA-MDO
The nonlinear optimization problem

Minimize x
f (x)





Submitted
to:

g(x) ≤ 0 ; g ∈ Rm
h(x) = 0 ; h ∈ Rp
x ∈ Rn






José Herskovits
Introduction
FAIPA
FAIPA-SAND
FAIPA-MDO
Nonlinear optimization algorithms:
Are iterative with asymptotic convergence.
Given an initial point, generate a sequence of points
converging to the solution of the problem.
At each point, the modern methods:
Generate a direction that points to the solution.
Determine a step, along this direction, in order to go to a new
point closer of the solution.
José Herskovits
Introduction
FAIPA
FAIPA-SAND
FAIPA-MDO
Feasible Point Algorithms
Require an initial point that verifies the inequalities and generates
a descent sequence also verifying them.
Advantage
Since all the points are feasible, any iterate can be employed. This
is an advantage in:
Engineering Design.
Optimal control in real time.
José Herskovits
Introduction
FAIPA
FAIPA-SAND
FAIPA-MDO
FAIPA - Feasible Arc Interior Point Algorithm
Introduction
Developed at OptimizE, COPPE/UFRJ
MATLAB and FORTRAN codes where implemented with the
support of RENAULT, France, 2000.
Several improvements have been done since that.
Problems up to 270,000 design variables and constraints are
solved in a PC.
At each iteration:
Defines a Feasible Descent Arc.
Then, makes an inexact line search along this arc looking for a
step-length t step that:
Makes the function smaller,
Verifies the inequality constraints.
José Herskovits
Introduction
FAIPA
FAIPA-SAND
FAIPA-MDO
At a point on the boundary:
d k = d0k + ρd1k is a feasible descent direction.
x k+1 = x k + t d k + t 2 d̃ k is a feasible descent arc.
José Herskovits
Introduction
FAIPA
FAIPA-SAND
FAIPA-MDO
System 1
At each iteration FAIPA computes a Descent Direction d0 , by
solving:
 




∇f (x)
d0
B
∇g(x) ∇h(x)



 

0
G
0

  λ0  = − 
 Λ ∇g(x)
h(x)
µ0
∇h(x)
0
0
This Linear System comes from a Newton like Primal-Dual
iteration to solve the equalities in Karush-Kuhn-Tucker
condition.
José Herskovits
Introduction
FAIPA
FAIPA-SAND
FAIPA-MDO
System 2
The following system with the same matrix gives a Centering
Direction d1 :
 




0
d1
B
∇g(x) ∇h(x)



 

G
0
  λ1  = −  λ 
 Λ ∇g(x)
µ
µ1
∇h(x)
0
0
Taking an appropriate ρ > 0, we have that
d = d0 + ρ d1
is a Descent Direction of an exact penalty function of the
inequalities and a Feasible Direction of the problem.
José Herskovits
Introduction
FAIPA
FAIPA-SAND
FAIPA-MDO
System 3
An additional Centering Direction compensates the curvature of
the constraints, is computed:
 

where



0
d̄
B
∇g(x) ∇h(x)



 

G
0
  λ̄  = −  λω̃ I 
 Λ ∇g(x)
E
µ̄
∇h(x)
0
0
λω̃
ω̃iI = gi (x + d) − gi (x) − ∇git (x)d
ω̃iE = hi (x + d) − hi (x) − ∇hit (x)d
are estimates of the 2nd derivatives of the constraints along d.
José Herskovits
Introduction
FAIPA
FAIPA-SAND
FAIPA-MDO
Line Search
Finally a line search along the Feasible Descent Arc
x k+1 = x k + t d k + t 2 d̃ k
is performed to get a new feasible point with a reduction of the
penalty function.
José Herskovits
Introduction
FAIPA
FAIPA-SAND
FAIPA-MDO
At a point on the boundary:
d k = d0k + ρd1k is a feasible descent direction.
x k+1 = x k + t d k + t 2 d̃ k is a feasible descent arc.
José Herskovits
Introduction
FAIPA
FAIPA-SAND
FAIPA-MDO
Versions
Different versions of FAIPA:
First Order.
Newton.
Quasi-Newton.
Theoretical results
It was proved:
Global convergence to a KKT point.
Superlinear rate of convergence.
Maratos’ Effect is avoid.
José Herskovits
Introduction
FAIPA
FAIPA-SAND
FAIPA-MDO
Quasi-Newton method
Modern Nonlinear Programming works with an estimate of
the 2nd derivative, called quasi - Newton Matrix.
The quasi-Newton matrix is full.
Limited Memory quasi-Newton
Is a technique for unconstrained optimization that avoids the
storage of this matrix.
Economizes a lot of memory in large problems.
José Herskovits
Introduction
FAIPA
FAIPA-SAND
FAIPA-MDO
Limited Memory FAIPA
FAIPA-LM is the only existing Limited Memory Algorithm for
nonlinear constraints.
Solving iteratively the internal linear systems:
Versions of FAIPA-LM without memory are obtained: The
quasi-Newton Matrix and the System Matrix are not stored.
Very large problems can be treated.
José Herskovits
Introduction
FAIPA
FAIPA-SAND
FAIPA-MDO
FAIPA-Low Memory
When there are inequality constraints only, we solve:
The Primal-Dual System
B
Λ ∇g(x)
∇g(x)
G
d0
λ0
d1
λ1
d̃
λ̃
=−
∇f (x)
0
=
0
−λ
0
−λω̃ I
Or the Dual system
∇t g(x) B −1 ∇g(x) − Λ−1 G(x)
λ0 λ1 λ̃
which is symmetric and positive definite.
José Herskovits
−∇t g(x) B −1 ∇f (x)
Introduction
FAIPA
FAIPA-SAND
FAIPA-MDO
FAIPA-Low Memory
In order to solve the system
∇t g(x) B −1 ∇g(x) − Λ−1 G(x)
We have to compute and store:
λ0 λ1 λ̃
=
−∇t g(x) B −1 ∇f (x)
The constraints derivative matrix ∇g(x)
The quasi-newton matrix B
The Dual-system matrix
h
José Herskovits
∇t g(x) B −1 ∇g(x) − Λ−1 G(x)
i
Introduction
FAIPA
FAIPA-SAND
FAIPA-MDO
FAIPA-Low Memory
To solve the Dual System with low memory requirements we:
Employ the Limited-Memory Quasi-Newton Method (storing
few vector to represent quasi-Newton matrix).
Employ the Gradient Conjugate Method (avoiding system
matrix storage).
Compute the product of the constraint gradient matrix times
a vector (avoiding the storage of constraint gradient matrix).
José Herskovits
Introduction
FAIPA
FAIPA-SAND
FAIPA-MDO
FAIPA-Low Memory
To solve the Dual System at each iteration of the PCG method, we
must compute:
where:
h
∇t g(x) B −1 ∇g(x) − Λ−1 G(x)
i
z
∈ Rm
v = ∇g(x) z s the gradient of an auxiliary constraint g t (x) z.
w = B −1 v is obtained with limited memory formulation.
∇g t (x) w is a directional derivative of the constraints.
instead of storing the whole derivative matrix, we just compute
and store the products ∇g(x) z and ∇g(x)t w .
José Herskovits
Introduction
FAIPA
FAIPA-SAND
FAIPA-MDO
FAIPA-Low Memory
∇g(x) z can be computed with the adjoint variables method.
For linear elastic structures, one system with the stiffness
matrix must be solved.
∇g(x)t w Directional derivatives of displacements in linear
elastic structures follows from directional derivation of the
equilibrium equation.
THUS: two linear systems with the stiffness matrix are solved
at each iteration of the Conjugate Gradient Method.
José Herskovits
Introduction
FAIPA
FAIPA-SAND
FAIPA-MDO
Numerical example
16875, 67500 and 270000
elements.
Initial thickness 0.95 cm.
Lower bound of thickness
tmin = 1mm.
Upper bound of thickness
tmax = 1cm.
for all elements: isotropic material
and Young module: 210 GPa Poisson: 0.3.
Stress constraint Mises stress in
center of element less than 2.5 104
Pa.
José Herskovits
Introduction
FAIPA
FAIPA-SAND
FAIPA-MDO
Numerical examples 1
José Herskovits
Introduction
FAIPA
FAIPA-SAND
FAIPA-MDO
Numerical example 1
José Herskovits
Introduction
FAIPA
FAIPA-SAND
FAIPA-MDO
Outline
1
Introduction
2
SAND Optimization
3
FAIPA
FAIPA-SAND
FAIPA-MDO
4
José Herskovits
Introduction
FAIPA
FAIPA-SAND
FAIPA-MDO
FDIPA-SAND
Is a quasi-Newton Algorithm for SAND Optimization
Makes iterations simultaneously in the design variables and in
the state variables
It does not require, at each iteration, the restoration of the
reduced equality constraints. That is, the state is not solved
at each iteration
The state equation is satisfied only at the final convergence of
the algorithm.
José Herskovits
Introduction
FAIPA
FAIPA-SAND
FAIPA-MDO
FDIPA-SAND
The Nonlinear Program for SAND Optimization:


Minimize x,u
f (x, u)


 Submitted to:

g(x, u) ≤ 0



e(x, u) = 0
The first linear system solved by FAIPA is:

B

 Λ ∇g(x, u)t
∇e(x, u)t
 



∇f (x, u)
d0
∇g(x, u) ∇e(x, u)



 
0
G(x, u)
0

  λ0  = − 
h(x, u)
µ0
0
0
José Herskovits
Introduction
FAIPA
FAIPA-SAND
FAIPA-MDO
FDIPA-SAND
The first linear system can be written as follows:


Bxx
Bux
Λ ∇x g(x, u)t
∇x e(x, u)t
Bxu
Buu
Λ ∇u g(x, u)t
∇u e(x, u)t
∇x g(x, u)
∇u g(x, u)
G(x, u)
0
where:
B=
"
Bxx
Bux
∇x e(x, u)
∇u e(x, u)
0
0
Bxu
Buu
 
 
d0 x
d0 u
λ0
µ0

 = −
#
we eliminate d0 u and µ0 from these equations.
José Herskovits

∇x f (x, u)
∇u f (x, u)
0
h(x, u)


Introduction
FAIPA
FAIPA-SAND
FAIPA-MDO
FDIPA-SAND
Let us consider:
ũ = [∇u e t (x, u)]−1 e(x, u)
D ũ = ∇u e t (x, u)−1 ∇x e t (x, u)
∇u e t (x, u) is the derivative of the state equation
ũ is a linearization of the state variables
Dũ and ũ can be computed using the Analysis Code.
José Herskovits
Introduction
FAIPA
FAIPA-SAND
FAIPA-MDO
FDIPA-SAND
Then, we have
d0 u = −δ u − Dud0 x
and
µ0 = −[∇u e t ]1 (−∇u f (x, u) − Bxu d0 x − Bxu d0 x − Bxu d0 x )
José Herskovits
Introduction
FAIPA
FAIPA-SAND
FAIPA-MDO
FDIPA-SAND
By substitution, a Reduced Linear System is obtained:
h
Λ [∇x
gt
∇x g − Du t ∇u g
G
B̄
− ∇u g t Du]
i h
d0 x
λ0
i
=
h
b
−Λ∇tu g δu
where
B̄ = M B M t is the reduced quasi-Newton matrix,
M = [I Du t ]
Ix = [0 I], Iu = [I
0]
b = −∇x f (x, u) + Du t ∇u f (x , u) − {Ix + Du t Iu }BItu δ u
José Herskovits
i
Introduction
FAIPA
FAIPA-SAND
FAIPA-MDO
FDIPA-SAND
The linear systems and the quasi-Newton matrix have the
same sizes as in the classical implicit techniques.
Requires the solution of additional linear systems.
The matrix of these systems is the sensitivity of the state
equation and can be solved by the Analysis Code.
Employs a Limited Memory representation of quasi-Newton
matrices.
José Herskovits
Introduction
FAIPA
FAIPA-SAND
FAIPA-MDO
FDIPA-SAND
FAIPA-SAND can interact with Industrial Codes:
Employing existing calculus techniques and routines.
Using industrial solvers, that usually take advantage of the
structure of each problem.
José Herskovits
Introduction
FAIPA
FAIPA-SAND
FAIPA-MDO
Outline
1
Introduction
2
SAND Optimization
3
FAIPA
FAIPA-SAND
FAIPA-MDO
4
José Herskovits
Introduction
FAIPA
FAIPA-SAND
FAIPA-MDO
FDIPA-MDO
The mathematical program for MDO is:


Minimize x,z,u f (x, z, u)




Submitted
to:





g(x, z, u) ≤ 0




h(x, z, u) = 0
e1 (x, z, u1 ) = 0
e2 (x, z, u2 ) = 0
..
.














er (x, z, ur ) = 0
José Herskovits
Introduction
FAIPA
FAIPA-SAND
FAIPA-MDO
FDIPA-MDO
Is a generalization of FAIPA-SAND.
Makes iterations simultaneously in the design variables and in
all the state variables and in the auxiliary variables.
All the state equations and the compatibility conditions are
satisfied only at the final convergence of the algorithm.
We employ the same formulation as for FAIPA-SAND.
Since the State Equations of the disciplines are uncoupled
with respect to the Sate Variables, we have to compute.
ũ = [∇u e t (x, u)]−1 e(x, u)
D ũ = ∇u e t (x, u)−1 ∇x e t (x, u)
José Herskovits
Introduction
FAIPA
FAIPA-SAND
FAIPA-MDO
FDIPA-MDO
The linear systems and the quasi-Newton matrix have the
same sizes as in the classical implicit techniques.
Requires the solution of an additional linear systems per
discipline.
The matrices of these systems are the sensitivities of the state
equations and can be solved by the Analysis Codes of each
discipline.
The Analysis Codes solvers take advantage of the structure of
the state equation
José Herskovits
Introduction
José Herskovits
FAIPA
FAIPA-SAND
FAIPA-MDO
Introduction
Conclusions
1
Our approach can be employed with complete models.
2
Low fidelity models can be also employed
3
SAND formulation is very efficient for iterative analysis
MDO can be introduced in a smooth way:
4
First, each discipline optimization is developed
Then, all disciplines are integrated for MDO
José Herskovits
Introduction
Further developments
1
The challenge now is putting teams together for MDO
2
Appropriate MDO environments must be developped
José Herskovits
Introduction
THANKS !
José Herskovits

Numerical Models and Algorithms for Multidisciplinary Design

Transcription

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