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Bachelor in Aerospace Technology Engineering

Final Bachelors Thesis, Spring 2023

STUDY OF INTERIOR POINT METHODS AND MULTIPLE SOLUTIONS IN TOPOLOGY OPTIMIZATION

Director: Alex Ferrer Ferre

Co-director: Jose Antonio Torres Lerma

ESCOLA SUPERIOR DE ENGINYERIES INDUSTRIAL, AEROESPACIAL I AUDIOVISUAL DE TERRASSA

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CONTENTS

Background

01

Interior Point Methods

02

Code Validation

03

Results

04

Conclusions

05

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01. BACKGROUND

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FIXATIONS

HANGING POSITIONS

MOTIVATION

AIRCRAFT PYLON

LEADING EDGE RIB

SOURCES

[1][2]

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FIXATIONS

HANGING POSITIONS

MOTIVATION

AIRCRAFT PYLON

LEADING EDGE RIB

OPTIMIZED

STRUCTURE

OPTIMIZED

STRUCTURE

PRELIMINARY

DESIGN

SOURCES

[1][2]

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TOPOLOGY OPTIMIZATION

MATHEMATICAL FORMULATION

BOX CONSTRAINTS

DOMAIN DEFINITION

STRUCTURAL TYPICAL COST

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REFERENCE AND MATERIAL DOMAINS

HOW DOES IT WORK?

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DENSITY APPROACH

STRUCTURAL DOMAIN

PROBLEM FORMULATION

PROBLEM RELAXATION

WEAK FORMULATION

SIMP METHOD

INTERPRETATION OF MID DENSITY VALUES

LET

CONSTITUTIVE TENSOR

MECHANICAL PROPERTIES PENALIZATION

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CONSTRAINED OPTIMIZATION PROBLEM

EQUALITY CONSTRAINED

OPTIMIZATION PROBLEM

DUAL PROBLEM

DUAL FUNCTION

DUAL VARIABLE:

MAXIMIZING THE DUAL FUNCTION

KKT CONDITIONS OF THE

CONSTRAINED OPTIMIZATION PROBLEM

CONSTRAINED OPTIMIZERS

INSIDE SWAN REPOSITORY

    • DUAL NESTED IN PRIMAL
    • AUGMENTED LAGRANGIAN
    • NULL SPACE
    • FMINCON
    • MMA
    • IPOPT

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AIM

INTERIOR POINT METHODS

    • COMPUTATION OF A MATHEMATICAL OPTIMIZER INSIDE AN ENTIRELY NEW ENVIRONMENT SUCH AS SWAN REPOSITORY
    • THE LINEAR ALGEBRA REQUIRED FOR THE ALGORITHM IS FAST, IDEAL FOR LARGE, SPARSE PROBLEMS
    • CAN FIND MULTIPLE LOCAL MINIMUMS DEPENDING ALSO ON THE INITIAL VALUE OF THE VARIABLE
    • KEEPS THE DESIGN VARIABLE EVOLUTION INSIDE THE FEASIBLE REGION OF THE PROBLEM

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02. INTERIOR POINT METHODS

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PROBLEM DEFINITION

INEQUALITY CONSTRAINED

OPTIMIZATION PROBLEM

PENALIZATION

OBJECTIVE FUNCTION

INDICATOR FUNCTION

BARRIER FUNCTION

INTRODUCTION OF THE

BARRIER FUNCTION

BARRIER PROBLEM

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    • THE OPTIMIZER WILL FOLLOW A CENTRAL PATH INSIDE THE FEASIBLE REGION

​

    • DECREASING THE BARRIER TERM MULTIPLYER WILL APPROXIMATE MORE THE DESING VARIABLE TO THE OPTIMAL SOLUTION

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KKT CONDITIONS

OF THE BARRIER PROBLEM

EQUALITY AND INEQUALITY

CONSTRAINED OPTIMIZATION PROBLEM

INTRODUCTION OF THE BARRIER FUNCTION

DUAL FUNCTION

LINEAR SYSTEM TRANSFORMATION

WHERE

AND

,

SOLVING THE BARRIER PROBLEM

WHERE

WITH

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    • ALGORITHM EXTRACTED AND ADAPTED FROM [3]
    • ALGORITHM REFACTORY:
    • TRANSLATION TO OBJECT ORIENTED PROGRAMMING
    • ADAPTATION TO SWAN'S ENVIRONMENT

ALGORITHM TO FOLLOW

1. INITIALIZE

2. WHILE NOT CONVERGED

DO

3. COMPUTE SEARCH DIRECTION

AND

4. WHILE NOT ACCEPTABLE STEP DO

5. UPDATE

AND

WITH

6. IF

THEN

8. ELSE

7. ACCEPTABLE STEP = TRUE

9.

10. END IF

11. END WHILE

12. CHECK CONVERGENCE WITH

13. DECREASE

WITH

14. END WHILE

MERIT FUNCTION DEFINITION

BEING

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WHERE

HESSIAN FUNCTION

APPROXIMATION

QUASI NEWTON METHODS: BFGS METHOD

NEWTON'S METHOD

WHERE

GRADIENT

RESPECT TO

    • CONVERSION WITH ONLY FIRST ORDER INFORMATION

    • DOES NOT NEED ANY LINEAR SYSTEM SOLVING

NEWTON

STEP

INITIAL APPROXIMATION:

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03. CODE VALIDATION

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ACADEMIC TESTS

TEST 1

TEST 2

FORMULATION

DESIGN VARIABLE EVOLUTION

COST AND CONSTRAINTS EVOLUTION

FORMULATION

DESIGN VARIABLE EVOLUTION

COST AND CONSTRAINTS EVOLUTION

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ACADEMIC TESTS

TEST 3

TEST 4

FORMULATION

DESIGN VARIABLE EVOLUTION

COST AND CONSTRAINTS EVOLUTION

FORMULATION

DESIGN VARIABLE EVOLUTION

COST AND CONSTRAINTS EVOLUTION

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04. RESULTS

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2D CANTILEVER BEAM

GEOMETRIC LAYOUT

BOUNDARY DEFINITION

CASE 1

DUAL NESTED IN PRIMAL

INTERIOR POINT METHOD

MESH SIZE

80 X 40

NULL SPACE

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2D CANTILEVER BEAM

CASE 2

AUGMENTED LAGRANGIAN

GEOMETRIC LAYOUT

BOUNDARY DEFINITION

INTERIOR POINT METHOD

MMA

MESH SIZE

80 X 40

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2D ARCH

GEOMETRIC LAYOUT

BOUNDARY DEFINITION

CASE 1

DUAL NESTED IN PRIMAL

INTERIOR POINT METHOD

AUGMENTED LAGRANGIAN

MESH SIZE

80 X 40

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2D ARCH

GEOMETRIC LAYOUT

MMA

BOUNDARY DEFINITION

CASE 2

INTERIOR POINT METHOD

MESH SIZE

80 X 40

NULL SPACE

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2D BRIDGE

GEOMETRIC LAYOUT

CASE 1

CASE 2

CASE 3

BOUNDARY DEFINITION

MESH SIZE

200 X 10

INTERIOR POINT METHOD

NULL SPACE

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05. CONCLUSIONS

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CONCLUSIONS AND FURTHER RESEARCH

LEVEL SET IMPLEMENTATION

3D OPTIMIZATION PROBLEMS

    • SUCCESSFUL REFACTORY
    • INTEGRATION INTO SWAN REPOSITORY
    • VALIDATED WITH ACADEMIC TESTS COMPARED WITH OTHER OPTIMIZERS
    • ABLE TO SOLVE TOPOLOGY OPTIMIZATION PROBLEMS
    • TOPOLOGY OPTIMIZATION SOLUTIONS COMPARED WITH OTHER OPTIMIZERS

INTERIOR POINT METHODS ALGORITHM IMPLEMENTATION

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QUESTIONS?

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REFERENCES

[1]

ZHU, J., ZHANG,W. AND XIA, L., "TOPOLOGY OPTIMIZATION IN AIRCRAFT AND AEROSPACE STRUCTURES DESIGN" ARCHIVES OF COMPUTATIONAL ENGINEERING, 2015.

KROG, L., ET. AL., "APPLICATION 0F TOPOLOGY, SIZING AND SHAPE OPTIMIZATION METHODS TO OPTIMAL DESIGN OF AIRCRAFT COMPONENTS". IN PROCEEDINGS OF 3RD ALTAIR UK HYPERWORKS USERS CONFERENCE , 2002.

APMONITOR, "INTERIOR POINT METHODS", AVAILABLE AT: HTTP://APMONITOR.COM/ME575/INDEX.PHP/MAIN/INTERIORPOINTMETHOD, 2022

[2]

[3]

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