Linearized Mathematical Programming approach - ULg

RECONCILIATION OF MATHEMATICAL
PROGRAMMING AND OPTIMALITY CRITERIA
APPROACHES IN STRUCTURAL OPTIMIZATION
Part II: Linearized MP approach
Pierre DUYSINX
Patricia TOSSINGS
LTAS – Automotive Engineering
Academic year 2015-2016
1
LAY-OUT
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Generalized optimality criteria

Linearized mathematical programming methods

Unified approach to structural optimization
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Examples
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INTRODUCTION
3
INTRODUCTION

Optimality criteria techniques (OC)
– Highly specific
– Intuitive techniques, simple
– Convergence to a design that is not necessarily optimal (KKT
conditions)
– Difficulties in identifying the set of active constraints
– Convergence instabilities
– Small number of reanalyses, independent of the number of
design variables
Résumé
– Low cost
– But uncertainty convergence
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INTRODUCTION

Pure Mathematical Programming methods
– Very general
– Rigorous methods, quite elaborated
– Convergence to a local minimum
– Stable and monotonic convergence
– Large number of reanalyses, growing with the number of
design variables
Résumé
– Rigorous framework & guaranteed convergence
– High cost (Growing with the size of the problem)
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INTRODUCTION

Generalized optimality criteria
– Dual scheme to solve the set of active constraints
– Inherent mechanism of selection of the set of active
constraints

Mixed methods
– Modification of pure MP methods to reduce the number of
function evaluations
– Projection scheme
– Linearization of constraints
– Restoring algorithm
– Controlling the convergence of the optimization process
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INTRODUCTION

Relation between optimality criteria (OC) and Mathematical
Programming (MP) approaches
– Generalized optimality criteria = Math Programming
linearization methods
– First order explicit approximation
– Constraint gradients
– Stress ratioïng
– Continuous transition between strict primal MP methods and
pure OC methods
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INTRODUCTION

UNIFIED OPTIMIZATION APPROACH
– First order approximation concept approach
– Sequence of explicit subproblems with linear or convex
approximations of constraints
– Efficient solution of the subproblems using primal and dual
mathematical programming algorithms
 Dual schemes  generalized optimality criteria
 Primal scheme  mixed method
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MATHEMATICAL PROGRAMMING
METHODS WITH LINEARIZATION

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Three actions:
– Use of reciprocal variables
– Restoration phase using scaling of variables
– Linearization = high quality approximations
Linearization approach
– Primal and mixed linearization methods
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Mathematical programming approach

Primal projection method: gradient projection method with NL
constraints
– Steadily improving (decreasing) objective function
– Always feasible designs

Several structural (FE) reanalyses / iterations
Large number of iterations : Growing with the number of design
variables
 PROHIBITIVE COST FOR LARGE SCALE PROBLEMS
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
Mathematical programming approach
1.
Feasible boundary point x(0).
2.
Minimization phase:
1. Search direction s= gradient projected onto the restriction tangent
planes
2. Step length is determined by line search
1.
Requires serveral reanalyses!
2.
x(1) = minimum along s
3.
4.
Restoration phase
1. Non linear constraints: x(1) is generally non feasible
2. Find x(2) back to the boundary of the feasible domain
3. Iterative process: it also requires several reanalyses
Feasible boundary point
x(2):
new minimization phase go to step 2
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Mathematical programming approach

Three actions to be able to apply MP to structural and
multidisciplinary optimization
1/ REDUCE THE NON LINEAR CHARACTER OF THE CONSTRAINTS
 Change of variables: switch to the reciprocal design space
New constraints very shallow
(assumption: weak structural redundancy)

Much larger steps can be taken
without seriously violating the constraints

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Mathematical programming approach
2/ RESTORATION PHASE :
 Use scaling
(No internal force redistribution during scaling)

Scaling factor determined by the most violated constraint

Only one reanalysis per analysis / restoration
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Mathematical programming approach
3/ APPROXIMATE THE LINE SEARCH DURING THE MINIMIZATION
PHASE
 Instead of using an exact line search that costs several FE
analyses, consider that linearized behavior constraint in the
reciprocal constraints are very good explicit approximations
Skip reanalyses: accept solutions if feasible according to
approximations
CONCLUSION:
 Only one single reanalysis per iteration

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Projection and restoration with mixed MP
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Mathematical programming approach
3 actions  One single FE reanalysis per iteration
BUT
 Number of iteration is still growing with the number of design
variables


Proposed approach
– Sequence of several search directions evaluated without
reanalyses
– Periodic update of the linearized constraints (after k search
directions)
– Linearized approximation in the reciprocal design variables
are high quality explicit approximations of the responses
constraints
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Mathematical programming approach

Linearized problem
– New minimization phase = solving partly this problem by gradient
projected method for linear constraints
– Perform k iterations then reanitialization (update)
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Mathematical programming approach

Number of solution steps of the subproblem solution is a convergence
control parameter:
– PRIMAL APPROACH : k small:
 Sequence of steadily improved feasible design
 Pure projection method k=1
 Monotonic convergence
– LINEARIZATION APPROACH : k large:
 Sometimes increase of the weight after rescaling because of
the constraint violations. More or less complete solution of the
subproblem
 Pure linearization method (k ∞)
 Fast but risk of unstable convergence
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Mathematical programming approach
Ten-bar truss – Mixed methods
Stress and
displacement
constraints
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RELATIONS BETWEEN OC AND MP

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Generalized OC = MP linearization method
Approximation concept approach
Constraint gradients
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Generalized OC = MP linearization method

GOC : sequence of explicit subproblems where real constraints
are approximated by
– cij = virtual energy densities

Mixed method: sequence of linearized problems with
– cij = derivatives of the responses functions with respect to
the reciprocal variables
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Generalized OC = MP linearization method

Reminder: Equality of the derivatives wrt to reciprocal variables
and of the virtual strain energy
For OC: the strain energy density

For the mixed method
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Generalized OC = MP linearization method

For the mixed method
– The change of variable to the reciprocal variable

So it comes
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Generalized OC = MP linearization method

Unified approach:
– Sequence of explicit subproblems obtained by linearizing the
behavior constraints with respect to the reciprocal variables
Later: linearizing any
behavior constraints with
respect to the reciprocal
variables!

– Independence wrt the number of design variables!
Solution of the explicit subproblems
– Dual solution scheme: generalization of conventional OC
techniques (GOC)
– Primal solution scheme: mixed method: gradual transition
between pure MP and OC approaches
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Generalized OC = MP linearization method


Moderate structural redundancy:
– shallow constraints
– Non linearity in reciprocal space is weak  OC OK!
Strong structural redundancy:
– highly non linear constraint  mixed approach can help!
Convergence of the OC
and mixed methods
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Approximation concepts (MP methods)

Brief history of MP approaches
– Primal methods:
 Feasible direction: Schmidt, Gellaty, Tocher,
Vanderplats…
 Gradient projection: Brown and Ang
– Barrier functions: Moe, Kavlie
– Linearization methods: Moses, Pope Pedersen
Prohibitive cost
Convergence
to a vertex
Instabilities
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Approximation concepts (MP methods)

Use of reciprocal variables
– Pure linearization techniques: Reinschmidt et al.
– Pure linearization using inscribed hyper spheres: Schmidt
and Farshi
– Objective function not linearized: Schmidt and Miura

Approximation concept approach
– Basically the same approach as mixed methods
– Primal philosophy: partial solution of the linearized problem
 Feasible directions (CONMIN)
 Extended barrier function (NEWSUMT)
– Dual methods (DUAL2, DUAL1, CONLIN, MMA…)

MP have evolved towards OC techniques
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Approximation concepts (MP methods)



Symmetry  2 design variables
Stress constraints only
Optimum: only one active constraint (not FSD)
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Approximation concepts (MP methods)

Three bar truss problem: trajectories in the reciprocal design
space
Solution by ACCESS 3
with NEWSUMT and
DUAL2
Strictly feasible approach
Follow the constraint
Newsumt (0.5 x 1):
-0.5 response factor
decrease ratio
-1: number of unconstrained
minimization
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Approximation concepts (MP methods)

Three bar truss problem: convergence of the weight (objective
function)
Convergence acceleration
as the explicit
subproblems are solved
more deeply
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NUMERICAL APPLICATIONS
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Comparison of 1st and 2nd order algorithms
72 bar truss
25 bar truss
200 bar truss
63 bar truss
Composite box beam
I beam
Delta wing
Aircraft spoiler
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Optimization algorithms

Sequence of linearly constrained subproblems

General purpose optimizers
– CONMIN: feasible directions
– NEWSUMT: barrier function
– PRIMAL1: gradient projection
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Optimization algorithms


Because of the simple algebraic structure:
– Explicit
– Strictly convex
– Separable
Second order primal and dual algorithms
– Primal second order algo: partial solution of the optimization
subproblem is possible (mixed method)
– Dual algo: complete solution of the subproblem
 Equivalent to Generalized OC techniques
– Based on Newton’s method
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Test problem Primal vs Dual solutions
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72 bar truss problem (four level tower)



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Stress and minimum size
constraints
Displacement limits at nodes
1, 2, 3, 4 (X and Y directions)
Symmetry  16 design
variables (using dv linking)
Two load cases
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72 bar truss problem (four level tower)

Stress constraints not
very critical  treated
by stress ratioing (not
expensive)
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63 bar truss: Wing Carry through box

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
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Two load cases
Upper limit on the relative displacement in the X direction
(torsion rotation)
Stress and minimum size constraint
No linking : 63 independent design variables
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63 bar truss: Wing Carry through box
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63 bar truss: Wing Carry through box
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63 bar truss: Wing Carry through box
Much lower
computation
al cost
Faster convergence
but CPU increase
Slow and
unstable
convergence
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I- beam
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I- beam
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Delta Wing problem

Delta wing structure

Fiber reinforced skins
(Carbone epoxy)
– 0°/45°/90°/-45°
– Symmetric and
balanced laminates
Metallic webs

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
Single load case
(pressure)
Temperature change
conditions
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Delta Wing problem
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Thickness of each
composite ply i.e. 252
orthotropic membranes
for skins
Thickness of aluminum
webs: 70 symmetric
shear panels
Design variable linking
 60 design variables
Constraints: deflection of
wing tip nodes
Max strain criteria in
composite panels
Lower bound on
eigenfrequency with
fixed mass (fuel)
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Delta Wing problem
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Delta Wing problem
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Weight minimization of an aircraft spoiler
Light aluminum alloy
Front spar and secondary spar joined by 12 ribs
And covered by 2 skins
2 load cases (pressure)
Stress constraints
(Fleury, 1976)
Landing: trailing edge must remain straight
within a tolerance e< 0.5 mm
Difficult displacement constraint for any
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nodes on trailing edge
Weight minimization of an aircraft spoiler
627 design variables
OC techniques failed
Mixed methods and
Sequential programming
have to be used
(Fleury, 1976)
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Weight minimization of an aircraft spoiler
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Two simplified FE models
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Weight minimization of an aircraft spoiler
(Fleury, 1976)
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Weight minimization of an aircraft spoiler
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Weight minimization of an aircraft spoiler
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Weight minimization of an aircraft spoiler
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Weight minimization of an aircraft spoiler
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(Fleury, 1976)
Weight minimization of an aircraft spoiler
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(Fleury, 1976)
Weight minimization of an aircraft spoiler
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(Fleury, 1976)
CONCLUSION
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CONCLUSION

Sequence of explicit subproblems
– First Order Approximations

Primal / dual solution schemes
DUAL
 Generalized of OC
 Computationally economical but
convergence instability
 Discrete design variable possible
 Reliable computer
implementation
 Dual bound = monitoring
PRIMAL
 Mixed method (OC/PM)
 Control over convergence at a
higher cost
 Other objective functions non
separable explicit functions
 Sophisticated algorithms
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