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OPTIMIZATION OF SUB COMPONENTS WITHIN
A LARGE SYSTEM

presented by

PRAVEEN HALEPATALI

has been accepted towards fulfillment
of the requirements for the

 

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OPTIMIZATION OF SUB COMPONENTS
WITHIN
A LARGE SYSTEM

By

Praveen Halepatali

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE
Department of Mechanical Engineering

2004

ABSTRACT

OPTIMIZATION OF SUB COMPONENTS WITHIN

A LARGE SYSTEM
By

Praveen Halepatali
Advancements in mathematical tools are increasingly encouraging optimization of
problems with a large number of design variables. These approaches are still constrained
in terms of practical computational times, especially in the case of a nonlinear problem. A
computationally efficient and effective method has been developed to solve large and
complex optimization problems using a global-local method. This method tries to exploit
the advantage of performing local optimization of regions using submodeling,
simultaneously retaining its global nature by providing periodic updates to the local
optimization conditions. Local optimization performed under perturbed conditions has
shown increased robustness of the optimal design, consequently improving the
convergence characteristics of the process. The approach is evaluated with a shape
optimization problem of a hole in a square plate to study the effects of local model size
and magnitude of variations. To understand the problem better, a beam problem is
optimized using the global local approach. Crashworthiness shape optimization of an
automobile energy absorber was performed with the proposed global local approach.
Decomposition of this problem is achieved by using different levels of design variable
discretization and problem specific methods for increased performance. The method
pr0posed is generic in nature and has been successfully implemented in ABAQUS and

LSDYNA with capabilities of being used in future studies.

To my parents and my brother

iii

ACKNOWLEDGMENT

I would like to thank everyone who has directly and indirectly been involved with my
effort. I am grateful for the support provided by the Mechanical Engineering department

at Michigan State University during the program.

I would like to thank my advisor Dr. Ronald C. Averill for his all his advice,

understanding and for having provided me with the academic freedom to grow.

I would also like to thank Dr. Alejandro R. Diaz and Dr. Erik D. Goodman for taking out

time from their busy schedule to be in my thesis committee and guiding me.

Praveen Halepatali

iv

 

 

CI

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TABLE OF CONTENTS

TABLE OF FIGURES ................................................................................ viii
LIST OF TABLES ......................................................................................... xi
CHAPTER 1 .................................................................................................... 1
INTRODUCTION ................................................................................... 1

1.1 Preliminary Information ................................................................ 1

1.2 Literature Review .......................................................................... 3

1.2.1 Shape Optimization ............................................................ 4

1.2.2 Coupled Analysis ............................................................... 4

1.2.3 Global Local Approach and Boundary Effects .................. 5

1.2.4 Optimization with Evolutionary Algorithms ..................... 7

1.2.5 Crashworthiness Optimization ........................................... 7

1.3 HEEDS Overview ....................................................................... 10

1.3.1 Introduction to HEEDS .................................................... 10

1.3.2 HEEDS Terminology ....................................................... 12

1.4 Objective of Present Study .......................................................... 14

1.5 Organization of the Thesis .......................................................... 14
CHAPTER 2 .................................................................................................. 16
GLOBAL-LOCAL OPTIMIZATION .................................................. 16

2.1 Introduction to Global-Local Optimization ................................ 16

2.1.1 Application of Submodeling Optimization ...................... 16

2.1.2 Performance Analysis ...................................................... 21

CHAPTER 3 .................................................................................................. 24
OPTIMIZATION WITH PERTURBATIONS ..................................... 24

3.1 Global-Local Optimization with Perturbations ........................... 24

3.2 Study of Variations ............................................................. 30

3.2.1 Variation in 1- Elements .................................................. 30

3.2.2 Variation in 2-D Elements ............................................... 32

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3.3 ABAQUS Implementation .......................................................... 34

3.3.1 Two Element — ABAQUS Verification ........................... 34

3.3.2 Four Element — ABAQUS Verification ........................... 35

3.4 Effect of Number of PBIE .......................................................... 36
3.5 Factors Influencing the Optimization Process ............................ 38
3.6 Magnitude of Variations-Issues .................................................. 38
3.7 Plate With a Hole Optimization with Variation .......................... 40
3.7.1 Injection Island Optimization .......................................... 45

3.8. Beam Deflection Control Using Global-Local Approach ......... 48
3.8.1 Comparison of Beam Optimization Processes ................ 49

3.8.2 Multi Agent - Injection Island Optimization ................... 52
CHAPTER 4 .................................................................................................. 54
CRASHWORTHINESS OPTIMIZATION .......................................... 54
4.1 Introduction to crashworthiness design ...................................... 54
4.2 Introduction to robust design ...................................................... 54
5.3 Optimization of Energy Absorbers ............................................. 55
4.3.1 Problem Setup .................................................................. 56

4.4 Global-Local Setup ..................................................................... 59
4.4.1 Beam Model Test ............................................................. 59

4.4.2 Truck Model Test ............................................................. 61

4.5 Variations in Rail Optimization .................................................. 63
4.5.1 Linear Variation with Time ............................................. 64

4.5.2 Spatial Variation Factor ................................................... 65

4.5.3 Total Variation Factor ...................................................... 66

4.6 Rail Optimization Results ........................................................... 67
4.6.1 Global Local Optimization — Without Variations ........... 67

4.6.2 Global-Local Optimization — 5% variation ..................... 69
CHAPTER 5 .................................................................................................. 72
PERFORMANCE ANALYSIS AND CONCLUSIONS ...................... 72

vi

5.1 Performance Analysis ................................................................. 72

5.1.1 Global-Local Time Savings ............................................. 72

5.1.2. Effectiveness Ratio ......................................................... 74

5.2. Conclusion .................................................................................. 76

5.3. Scope for Future Research ......................................................... 76
BIBLIOGRAPHY ......................................................................................... 78

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TABLE OF FIGURES

Structure of a simple Genetic Algorithm ........................................................... 11
Quarter model of the plate with a hole problem ................................................ 18
Finite element mesh with PBIE ......................................................................... 18
Flow diagram of the Global-Local Optimization .............................................. 20
Iteration history for 4.0” sub model .................................................................. 21
Optimal design for 4.0” sub model .................................................................... 22
Iteration history for different sub model sizes ................................................... 23
Flow diagram of global-local Optimization with variations .............................. 25
Subdomains connected using the interface element .......................................... 26
. 1D cantilever beam with PBIE under axial load ............................................. 3O
. Displacement response with variation in Kl ................................................... 31
2D cantilever beam under axial load ............................................................... 32
Displacement response with variation in A, and A: ....................................... 34
Two-element Abaqus displacement profile ..................................................... 35
Four-element Abaqus displacement profile ..................................................... 36
Comparative study of accuracy with different number of PBIE ..................... 36
Optimal design after 50 cycles for sub model 2.8” with no variation ............. 40
Comparison of 2.8” without variation and 2.2”, 1.9” with 20% variation ...... 42
Iteration history of the1.9" sub model with 10%, 20% and 30% variation ..... 43
Injection Island Technology applied for Plate with a Hole problem ............... 46
Best design after 10 cycles using the Injection Island scheme ........................ 47

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Figure 23. Convergence of Injection Island topology analysis ......................................... 47
Figure. 24. Beam Global-Local setup. .............................................................................. 48
Figure 25. Comparison of the optimization runs .............................................................. 50
Figure 26. Local and global agent performances for 10% and 20% variation ................. 51
Figure 27. Multi Agent —Injection Island optimization of beam ...................................... 52
Figure 28. Multi Agent — Injection Island Optimization Flow Diagram .......................... 53
Figure 29. Comparison of Optimal and Robust Design .................................................... 55
Figure 30. Global Model of the Truck .............................................................................. 56
Figure 31. Local model for the Rain analysis ................................................................... 57
Figure 32. Rail profile showing the features spline and the cross-section ........................ 57
Figure 33. Cross-section definition of the rails ................................................................. 58
Figure 34. Cantilever beam global model ......................................................................... 60
Figure 35. Resultant velocity of node 5 for local model ................................................... 60
Figure 36. Resultant velocity of node 5 for global model ................................................ 61
Figure 37. Global model analysis ..................................................................................... 62
Figure 38. Local model with written boundary input file ................................................. 63
Figure 39. Local model with LSDYNA Interface file option ........................................... 63
Figure 40. Time varying factor with time ......................................................................... 64
Figure 41. Shape varying factor function of node number ............................................... 65
Figure 42. Visualization of Shape Varying Factor on a cross-section .............................. 66
Figure 43. Energy absorbed by local and global models for analysis without variation .. 68
Figure 44. Total energy absorbed by the truck system ..................................................... 68
Figure 45. Percentage energy absorbed by designed rails to system energy absorbed ..... 69

ix

Figure 46. Energy absorbed by local and global models for analysis with 5% variation. 70
Figure 47. Total energy absorbed by the truck system ..................................................... 71

Figure 48. Percentage energy absorbed by designed rails to system energy absorbed ..... 71

LIST OF TABLES

Table 1. Variations in Al and A2 ...................................................................................... 33
Table 2. Optimization results comparison for plate with a hole ....................................... 44
Table 3. Effectiveness ratio with TUIG and NG ............................................................. 75

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v-v

CHAPTER 1
INTRODUCTION

1.1 Preliminary Information

Competitive markets coupled with demanding standards are continuously driving today’s
engineers to develop products with abilities to perform better at reduced costs. New and
improved designs are evaluated by finite element methods even before prototyping and
testing. Over the last decade or so analytical design optimization approaches have
received considerable attention principally because of the exponential growth in the
computational capabilities. The effort to arrive at the best possible design at the earliest
and by the most economical means has lead to a stage where these optimization

procedures need to be conducted in an ‘optimized’ manner.

Computational time is a practical constraint especially in applications like optimization
with genetic algorithms, where multiple evaluations are performed to obtain the best
design. The principle of the technique is to simulate evolution roughly on the basis of the
principle ‘Survival of the Fittest’. The genetic algorithm generates a pool of solutions,
with some better than the others. Then crossover and mutation of these solutions is

employed to yield a new pool of designs with the goal of identifying a better design.

In a number of design optimization problems it is not difficult to identify the local

regions where the right modifications could mean an improved design with a better

performance. In other cases inefficiency of repetitive evaluation of the whole model leads
to a much more practical option of analyzing the sub region independently, being

subjected to environments predicted by the whole model.

Further, any analysis involving complex geometries with regions of specific interest
would demand a refined mesh to capture accurate solutions. Aspect ratio and continuity
constraints force a gradual change in the mesh density from refined to coarse, resulting in
a higher mesh density in the regions where the solution and its gradient are relatively
smooth, again making the analysis computationally expensive in terms of the time
consumed. One approach would be to divide the problem into subdomains on the basis of
the gradient of the solution, construct a refined'mesh in regions of interest, mesh other

regions in a coarse manner and connect theses subdomains with ‘Interface Elements’.

The local subdomain can be independently optimized under the conditions predicted by
the global model, which is constructed by using the interface elements. In shape
optimization problems the conditions that the local model is subjected to are the
displacements of the boundary nodes connecting it to the global model. This technique of
Global-Local optimization will not converge in cases where the boundary is not
sufficiently far away, as the boundary is not completely inert to the changes made during
different evaluations. The other situations where this technique would not converge are
situations with large deformations or a non-linear analysis, where the boundary
conditions predicted by the global model would be extremely sensitive to the design

changes made during the process, thus rendering the approach ineffective in practical

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problems like a selective region optimization or crash simulation, to name a few.

A possible approach to solve the above mentioned problem is first try to and obtain the
optimal design for the local model, subjecting it to the boundary conditions that vary
randomly by a small amount for each of the design evaluations. Thus not only solving a
problem which might have previously not converged, but also ensuring that the design
obtained is robust to changes in loading condition, material quality fluctuations or other
real-life uncertainties. Thus the design has to survive different environments, which in
turn depend on the present nature of the designs. So philosophically the approach is in
coherence with the natural idea of evolution with environments becoming increasingly
demanding. This study will primarily focus on determining the magnitude and the
manner in which the perturbation has to be employed at the boundary to obtain the best

results.

1.2 Literature Review

Analysts and researchers have worked over the past decade to develop reliable, effective
and robust optimization algorithms. All of them have special advantages and associated
drawbacks when applied to complex real time problems. Further, the objective is not only
to optimize the problem in the mathematical sense, but also to be robust to the
uncertainties not visualized in the problem statement. The following discussion gives an

overview of avenues explored by each of these methods.

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1.2.1 Shape Optimization

Structural optimization has branched into two clearly different approaches i.e., gradient
based techniques and evolutionary algorithms. Shape optimization has received a
considerable amount of attention with structural, computational and practical
perspectives. Topology optimization is a popular shape optimization technique; the
bibliography by Mackerle [1] provides a good overview of the present state of research.
Evolutionary Genetic Algorithms (EGA), though entirely different in their approach, has
its own rewards. The principal advantage of the latter approach is capability of material
addition and to have stress as an objective / constraint in shape optimization. GA has

been shown to have a multi—objective optimization capability in diverse applications [2].

1.2.2 Coupled Analysis

The finite element interface technology developed by Aminopour and colleagues [3,4,5]
allows the coupled analysis of incompatible meshes. Recently an alternative approach
using penalty parameters has been developed by Pantano et al [6] to efficiently solve and
counter the incompatibility in meshes. The principal advantage with the above-mentioned
technology is that refined regions of interest can be modeled and analyzed at minimum
computational expense. Flexibility offered by this Penalty Based Interface Element

(PBIE) technology has encouraged its use in this work.

 

 

 

 

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1.2.3 Global Local Approach and Boundary Effects

Substructuring is a popular method to decouple a complex system by representing it with
an equivalent reduced system. Lesser number of degrees of freedom in the substructure
directly leads to higher computational efficiency. Bathe et. al [7] discusses the ability to
decompose problems into linear and non-linear substructures in dynamics problems,
which could greatly reduce computational time. A detailed report on the present state of
substructuring and use of superelements can be found in Noor [8], highlighting its

advantages while analyzing complicated vibration problems.

As the name suggests submodeling is a simple procedure in which a large model is
segregated into smaller regions, with the boundary conditions supplied by the coarse
global model evaluation. Mesh refinement during the local evaluation infuses
considerable interface error. Techniques such as the exact zooming method [9] and
Specified Boundary Stiffness/Force (SBSF) [10] are methods which to try and reduce the
error incorporated because of change in the interface stiffness, changing the response of

the subdomain.

A multilevel approach to design optimization of large structures was explored by
Ramanathan et. el [11] for minimal-weight design of beam structures. Further Bloebaum
and colleagues [12] extended the study by focusing on the development of a general
procedure for handling a decomposed problem optimization problem. They show the

economical gain in time and ability to run each subproblem separately by applying it to a

. beam structure design optimization.

The technique of iterative global local analysis was investigated by Whitcomb [13] to
study the debond growth in adhesive bonded tubular structures. Refined global local
analysis and adaptive mesh generation techniques [14] have been studied to reduce the
interface error. Corimier [15] used aggressive techniques of repeated submodeling to
analyze stress concentrations in regions around cracks, where one needs to provide an
extremely fine mesh to get reliable results. In their work they proposed a scheme of
stage-wise gradual grid refinement within each stage, the interface displacements are
compared to that of the sub modeled region for convergence, thus ensuring that the
solution obtained is within an acceptable range. The Global Local procedure has been
used to analyze composite panels [16] and solder joint design for reliability [17] to name

a few.

Lee et. el [18] have used a global-local optimization scheme, in which the global model
provides a feasible direction for the process to evolve in. After the final global optimal
design is reached local optimization is carried out to improve the designs. The approach
has shown positive results on mathematical as well as ship hull parametric optimization

studies.

The stiffness of the local mesh is considerably different compared to one with the global
mesh, which leads to a significant violation of equilibrium during optimization. This in

effect places a constraint in the degree of miniaturization of the local model. The study of

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the interface error in the global-local approach [19] and the scheme of adaptive
refinement strategy based on the interior error estimates provide a good understanding of

the limitations.

Walker [20], in an effort to optimize a plate with a hole through static condensation of the
mesh, suggests that the modifications made during the search be distributed over the
elements; this motivates the present approach adopted. Eldred et a1 [21, 22] have
attempted to obtain solutions to problems with uncertainties through surrogate-based
optimization techniques. Uncertainty of boundary conditions and the response of the

structural members were treated and solved using Fuzzy logic by Cherki [23].

1.2.4 Optimization with Evolutionary Algorithms

Genetic algorithms and evolutionary search related to structural shape optimization
problems have been increasingly investigated in the last decade. A wealth of literature
can be found on the application of the two; the papers referred [24, 25] give a good

overview of the approach and its potential.

The papers by Goodman [26] provide a good introduction, understanding of the

application of GA and other related techniques in structural shape optimization problems.

1.2.5 Crashworthiness Optimization

Crashworthiness of automobile members is a measure of the amount of energy the

 

 

 

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support structure can absorb, transmitting minimal force to the passenger cabin.
Extensive research work has gone into efficient ways of modeling this highly nonlinear
event. Optimization of energy absorbing rails is complex in terms computation and
nonlinear responses. Thus a number of researchers have attempted to simplify the system
and optimize it using response surface responses [27].

Optimal shape of the rails is complex and must consider the desired response in terms of
energy absorbed, stability and minimal acceleration, to name a few. Averill et. al [28, 29]
have efficiently approached the crash shape optimization of the lower rail segment of a
automobile by using a combination genetic algorithms and other gradient and non
gradient based methods. They achieved improved performance of the optimization and
increased computational efficiency when independent modules are used to look for
designs for different crash times, with the information being exchanged between these

modules.

A simplified crash model generation scheme by stochastically fitting it with experimental
data has been developed by Jasbir [30]. Similarly, Sergio et. al [31] updates the model
and uses experimentally updated response surfaces to obtain the optimize the design of
the front rails. The search for the best genetic algorithm scheme to achieve reliability and
reduced computational time is still elusive. Kurutaran et. el [32] propose a scheme of
crash optimization using successive response surface approximations with genetic
algorithms. They suggest that the response surface provides a better representation when

constructed around a subregion, thus helping in the convergence to an optimum design.

Rzesnitzek [33] successfully proposes a two—stage optimization process, initially
stochastic and later deterministic. The first aids in identifying a feasible starting point by
scanning the design space and helping to recognize the high sensitivity design variables.

The second stage is gradient based and leads to an optimal solution.

1.3 HEEDS Overview

1.3.1 Introduction to HEEDS

HEEDS (Hierarchical Evolutionary Engineering Design System) is multi faceted
optimization software capable of handling complex problems with a large number of
design variables. With Genetic Algorithm (GA) as its backbone it uses a combination of
search methods: gradient search principles, automated design of experiments, and
simulated annealing to optimize the design problem. A GA is a search procedure based
on the mechanics of natural selection. A GA creates and destroys designs based on their
performance, with the better survivors influencing future designs. Figure 1 shows the

basic structure of a simple GA.

A GA creates and destroys designs during a process that involves decoding each
chromosome (or design vector), evaluating its satisfaction of constraints and its
performance relative to the objectives, and then allowing a simulated "natural selection"
to determine which designs are eliminated and which survive to generate other, derivative
designs. Designs that perform well (relative to constraints and objectives) have a higher

probability of surviving to influence future designs.

During reproduction, the two genetic operators commonly modeled that produce new
chromosomes (or design vectors) are called crossover and mutation. HEEDS sets
problem-specific defaults for the amount and type of crossover and mutation performed

at each stage of evolution ("cycle"), but each can instead be controlled via keyword input

10

in the search parameters input file. The crossover operation forms a new solution by
combining parts of two existing solutions. Mutation is a reproduction operation that

produces a new solution from a single existing solution, through any of several ways.

 

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Figure 1. Structure of a simple Genetic Algorithm
The design space characteristics bring strong possibilities that many local suboptimal
solutions exist, which limit the applicability of gradient-based approaches. The main
problem with using a simple GA is the potentially large number of design evaluations
required to obtain a set of satisfactory solutions. HEEDS reduces the number of
evaluations required to obtain a set of satisfactory solutions by hierarchically
decomposing a problem with multiple agents that represent the problem in various ways,
while also using efficient local search methods (automated design of experiments,

nonlinear sequential quadratic programming, and simulated annealing).

11

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1.3.2 HEEDS Terminology

A principal feature of HEEDS is the flexibility offered to organize and solve an
optimization problem by using its file structure and generic commands. Optimization of a
problem in HEEDS requires a clear definition and understanding of the following process
parameters:

1. Baseline Design

Baseline design is the design on which improvement is sought. The response of this
design is used to set targets, normalizing coefficients and to identify the process design
variables.

2. Design Variables and Criteria

The design variables are chosen from the regions and parameters of interest and will be
varied intelligently. HEEDS provides a high degree of freedom in choosing the design
variables. They could be shape, thickness, material properties, etc. Further, the design
space from which samples will be evaluated has to be clearly defined. This is done
through specifying the maximum and minimum design variable change permitted, with
all the values in between forming a part of the design space. The values the design
variable can take depend on the number of divisions of the design space, called the
resolution.

3. Design Objectives

The design objective of the problem must be defined carefully to ensure that the problem
being mathematically solved is the desired goal. For example, in a crash analysis

problem, any of internal energy absorbed, acceleration, reaction force, stress. etc could be

12

chosen as design objectives

4. Design Constraints

Design constraints are practical design limits an engineer would like to conform to in his
product. Again HEEDS permits a number of independent or dependent combinations of
design constraints.

5. Population Size

This represents the number of designs competing within one Agent. A small population
size could cause difficulty in convergence of the analysis, whereas a large number would
result in increase in computational time. The population size should be set carefully
considering both factors.

6. Number of Design Cycles

The number of design cycles (generations) the analysis is permitted to evolve, after which
the analysis will automatically stop. It should be noted that the analysis could be restarted
to extend previous runs.

7. Agent

An agent is an entity, which deals with one specified set of objectives, targets, resolution
and constraints for its search. A problem can have multiple agents working independently
or together by sharing information intelligently after each cycle to arrive at the optimal
solution. The overall structure of the agents and their relationships are referred to as the
topology of the problem.

8. Agent Topology

Hierarchical decomposition of a problem is a powerful feature of HEEDS, which permits

the frequent exchange of best and/or random designs from one agent to another. A good

13

topology design will help in better global searches and convergence of the process.

1.4 Objective of Present Study

The present study is focused on the development of an efficient scheme for the
optimization of problems by performed with submodeling using a global-local approach.
The goal is to gain an understanding of the complications involved in local optimization
of the submodel resulting from the mismatch of the boundary conditions provided by the

global model.

Analyze the effects of optimization performed with perturbed local boundary conditions.
Further establish the relationships and restrictions on the order of variations that can be
safely used during the optimization process to achieve maximum convergence. Apply the
concept of global-local optimization for the shape optimization of front rail members of
an automobile to make it robust to variations in the boundary conditions and to maximize

its crashworthiness.

1.5 Organization of the Thesis

Chapter 2 will provide an introduction to the global-local method of optimization applied
to a plate with a hole problem. The effect of the proximity of the local model boundary to

the center of the hole on optimization is investigated.

Chapter 3 provides details of the implementation boundary perturbation by making

appropriate changes to the user element subroutine in ABAQUS. Study optimization

l4

performance with parameters such as magnitude of variation and distance of the local
model boundary from the center of the hole. The approach is also evaluated and studied
using a simple beam problem in LS—Dyna. The results this problem would provide

understanding of the issues when applied to bigger problems.

In Chapter 4 the application of the global-local shape optimization of front rail members
with boundary condition variations is elaborated. The optimization process and results

obtained are discussed.

Chapter 5 discusses the conclusion of and plans for future work and research in global-

local optimization with boundary variations. Further, improvements in the application of

the developed method to crashworthiness design are explored.

15

CHAPTER 2

GLOBAL-LOC AL OPTIMIZATION

2.1 Introduction to Global—Local Optimization

Infeasibility in optimizing computationally large problems points toward an alternative
option of performing the analysis at the local level using the submodeling option
available in a number of commercial codes. In this chapter, a new approach for global-

local optimization is proposed, outlined and demonstrated.

The proposed scheme of global-local optimization has the local model constituting the
region of interest receiving the boundary conditions from a global analysis. Optimization
is performed at the local level and the best design is evaluated at the global level for its
performance. If the indicated performance is better when compared to the previous best,
the local model boundary conditions are updated with the results from the global model
evaluation. The role of global evaluations is to check the best local model design in every
cycle and providing the direction with the supplied boundary conditions. During this
process one has to ensure that the boundary condition provided by the global model is

reasonably insensitive to the design changes in the various local model runs.

2.1.1 Application of Submodeling Optimization

Optimization of the profile of a hole in a plate (Figure 2) has been chosen to evaluate the

approach mentioned in above. The square plate has edge dimension of 20” with a central

l6

circular hole of diameter 2”. The plate is loaded in the x and y directions with 2400 psi
and 7200 psi respectively. Taking advantage of the symmetry only Mt of the entire plate is
modeled. The global model ‘BASELINE’ design is constructed by having a refined mesh
in the region around the circular hole (Figure 3). The rest of the plate is modeled in a
coarse manner with 7 nodes on the boundary with the interface. The two domains are
connected by Penalty Based Interface Element (PBIE), which is described in Chapter 3.
The local model consists of the refined mesh region with 13 nodes lying on the circular
sector and the number of nodes on the edges with symmetric boundary conditions
depending on the proximity of the interface. The goal of the problem is to minimize the
maximum Von Mises stress in the plate by finding the optimal profile of the hole. The
shape is generated using a spline with 7 equally spaced control points. Each of these
control points is a design variable permitted to take any of the equally spaced 20
positions along a normal to the circle at each of these points within the set offset limits.
The offset in the normal direction is restricted between —0.5” and 1”. Using the profile
generated by the spline a structured mesh is created with 2 elements between each of the
control points. Here the distance from the center of the circular hole is a parameter of

interest.

The optimization software HEEDS [34] was used to search for the optimal local design.
The routine performing the local model optimization will be referred to as Agent 0
(local), and the boundary conditions for the same are retrieved from an Agent 1 (global)
analysis (Figure 4). The initial boundary conditions for Agent 1 and the Von Mises stress

that is to be minimized are obtained from the outputs of the ‘BASELINE’ design.

17

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= 10" ———>

 

Ti

 

Figure 2. Quarter model of the plate with a hole problem

 

 

\

//
//
/

ILIT/ / /
/

 

 

 

\

 

/

 

L1\\IL

 

 

 

 

 

 

 

 

 

 

 

Figure 3. Finite element mesh with PBIE

The boundary conditions passed are for the nodes belonging to the global model, which
are connected to the local model with the PBIE. The best design found by Agent 0 after

every cycle of 25 evaluations is then evaluated by Agent 1 to determine if it is better than

18

the one from the previous best (initially the Baseline). If the design has a lower maximum
Von Mises stress value, then the new boundary conditions for Agent 0 are obtained by
performing the Agent 1 analysis. If the optimization analysis converges, this process is
repeated for a smaller local model with the interface region closer to the center of the
hole and the convergence of the new process is examined. There is a certain critical
proximity of the interface region at which the optimization performed by Agent 0 does
not converge. Optimization with submodels of this and lower sizes of interest are

considered in a future section.

19

 

Baseline Evaluations to get Local
Boundary conditions

]< A
Local Model Design
Evaluations

 

 

 

 

 

Agent 0

 

 

 

l Best design

 

Global Model Design
Agent 1 Evaluation using
Best Local Design

 

 

 

v

 

   
  
 

Update
Local
Boundary
Conditions

I

Max
Cycle no
exceeded

 

 

 

 
   
 

No

  

 

 
   
 

 

Stop

 

 

 

Figure 4. Flow diagram of the Global-Local Optimization

20

2.1.2 Performance Analysis

The optimization process was performed with the submodel interface at a distance of 4.0”
from the center of the hole. It was observed that ‘Agent 0’ converged smoothly (Figure 5)
and monotonically around the 30‘11 cycle to the optimal shape of an ellipse (Figure 6) with
a Von Mises stress of 9.45E3 lb/inz. Since the submodel interface is sufficiently far from
the hole, it ensures that the design changes in each cycle are performed for the boundary
conditions the global model would have provided. In other words, the Boundary

condition does not depend strongly on the local region

 

ta———»»)AwnA—H , , ._LM_L_L r

 

1.7» ~

1.6~

1.5L 3, a

1.4»

1.3L I. z

1.2 ~ ,

Design Metric (Von Misses Stress x 1OE4)

k
1'1b \ i l i i I x 1, '7 ' '1

"V ‘ ““ W— e.— ran—k,

 

 

1%” -IL ._-_L__L_LL..LL LL_L r

o 5 10 15 ‘ “go“ 25 30
Cycle Number

 

Figure 5. Iteration history for 4.0” sub model

21

 

Figure 6. Optimal design for 4.0” sub model

From the iteration history of the 3.7” model (Figure 7), it can be observed that the
convergence is close to the 4.0” analysis, but after the 30‘h cycle the boundary conditions
supplied by Agent 1 do not help in smoothing out the ellipse contour, resulting in a
higher stress of 9.67E3 lb/inz. In comparison, the 3.4” model converges smoothly only
until the 15‘11 cycle after which the progress of optimization is at a very sluggish rate.

This can be associated with two principle reasons; firstly the problem itself is optimized
for the wrong boundary conditions in each of the cycles and secondly ‘use of the wrong
boundary conditions in ‘Agent 0’ results in the process getting stuck in a wrong local
minimum. Effectively, the local solution closer to an ellipse, which found its way into
existence in the previous cases, has not been allowed to survive in the 3.4” submodel
analysis because this solution is not optimal for the applied (incorrect) boundary
conditions. Thus the 3.4” model is the minimum permitted size the current optimization
procedure can handle, beyond which the best design found by the local model (i.e. Agent
0) after each cycle does not perform well in the global problem, indicating that its

Performance is of a local nature, thus not helping the convergence of the process.

22

 

 

 

 

Vonmisses stress

0.9

 

 

 

 

 

 

 

 

 

5 10 15 20 25 30 35 40 45
Cycle No

Figure 7. Iteration history for different sub model sizes

23

CHAPTER 3
OPTIMIZATION WITH PERTURBATIONS

3.1 Global-Local Optimization with Perturbations

As observed in Chapter 2, the optimization process does not converge because of the fact
that the iterative analysis does not converge to the right BC’s. But the BC’s are
determined from the best design found by Agent 0 in the previous cycle, making the
problem coupled. The suggested approach is to subject the designs to different BC’s,
making the identification process more exhaustive. Further evaluating the design with

different BC’s is equivalent to using different loading conditions, making the design

more robust.

It should be noted that the local boundary nodes are perturbed within defined limits about
the conditions predicted by Agent 1 in the previous run, only if the design is better than
the best found earlier (Figure 8). It is desired that the process of updating be at a high

frequency, which requires conforming to the previously mentioned conditions.

The extraction of boundary conditions from the global agent could be done through the
submodeling option available in a number of commercial finite element software’s.
However, the inability for the user to change them leads to a method to do so using

flexibility offered by the PBIE, which is explained in detail in the next chapter.

24

 

Baseline Design Evaluation to
get Local Boundary conditions

 

 

 

if ‘
Local Model Design Evaluations
(Perturbed Boundary Conditions)

 

 

Agent 0

 

 

 

l Best design

Global Model Design

Agent 1 USmg Best Local Desrgn
Evaluatron

 

 

 

 

    
     
   
       

 

Update
Better Yes L003]
Design? Boundary
Condition

 

 

 

 

Max Cycle
No.
Exceeded

 

 

Stop

Figure 8. Flow diagram of global—local optimization with variations

 

 

3.1.1 Perturbations Through PBIE

In Penalty Based Interface Elements the displacement continuity is imposed between the
two regions in a least square sense in the Total Potential Energy function (Equation 1),
forcing the integral term to approach zero by choosing relatively large penalty parameter

(Yr and Y2) values. The integral term represents the error between the two interfaces to be

25

attached, thus continuity is achieved across the interface (Figure 9).

where

not

Yr

U1

U2

H=nQ1+HQZ+—I(V—ul)2+ds+—2l

I(V— U2)2dS
2521 (21

Total potential energy

Potential energy of the Qi. sub domain

Penalty parameter for ith sub domain
displacement field of the interface nodes
displacement field of the local sub domain nodes

displacement field of the global nodes

 

 

Domain 1 ‘1, Domain 2

 

 

 

 

 

 

 

 

¥

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Ul ’VUZ

 

 

Figure 9. Subdomains connected using the interface element

(1)

Thus for higher value of the Penalty Parameters the interface continuity is expected to

been forced with lower interface error. Continuity across the interface is enforced in the

following way:

26

U1=V

U2=V

Random error in enforcing the continuity at the interface would act like the random
variations that we desire to effect at the boundary. Thus by deliberately not equating
(Equation 2) the displacement field of the global nodal displacement field (U2) and the
interface displacement field (V), we can bring about random perturbations on the
boundary. While doing so it has to be noted that the inequality in u1(s) with V(s) will
make the nodes u;(s) also vary to meet the penalty condition. To avoid it and gain control
over the variation, the change is incorporated at the positions in the matrix involving only

u1(s) and regions connecting it to V(s).

V(S)= U2(S)
V(s): ul (s)+ f(s) (2)
Where
nozzrew‘
u1(s) = 2 u] ”“11i
fi : U|i*gi
-RS gi S R

g i : random number

‘1’i : interpolation function of the element at node i

R : maximum random variation (input by the user)

Here g, is a random number obtained from a random number generating subroutine. The
random number is normalized to a value between —R and R, where R (user defined) is the
maximum percentage variation of the nodal displacement.

Let the random matrix H be defined as

27

Hi=(1+gi)

Where H (n x n) is an array of random numbers defined from the values of g.

 

 

_H1 o 0 0 0
o H 2 o 0 0
0 0 H' 0 0
o 0 0 H"_1 0
L 0 0 0 0 H"_
Thus
V(S)=H*UI(S) -- (3)

Using the above equations formulating the varying nodal displacements of edge u1(s), we
obtain

n
lll(.S')=ZI‘1l *1/11 *ulj
i=1

Collecting the terms and substituting in Equation (1), we get the Modified Potential

Energy equation (MPEE).

71 i2 72 i 2
IT=ITQI+I192+7J qu—Hqul (15+—2'—Jr (quwNZqz) d5 " (4)
S S

Where,

T : Matrix of cubic spline interpolation functions.

N, : Linear interpolation functions.

qli and qzi : Nodal displacements on the two subdomains £21 and 822 respectively.

The variations can be applied on both sides of the interface or applied on one side, the
second case will provide more control to monitor the same. Thus taking the variation

with respect to q1. q2 and q5 and incorporating the changes we get the following

28

 

 

   

   

an an
-—+a]mMImasajmautaa—afhw
1
r1
fl+y2ISN2TT(1Y(15‘—Y7J [N2TN2]([7dS-a Q2 :0
dq ([2 L .— S H 8612
an T T T
JTT Tq,ds—y2jT Tq,ds+y1j [HNI] quds+y2j N2 Tq2d5=0
qu s ‘ s j s 5

Collecting the terms and representing them as
ii T
0] =Y1LIHNII [N1] d5
ii T
02 =y2j IN2I INzl ds
5

. T
Gis ZYILIHNII [TI (15
Gr=kil
0? =72jiN2TTJ ‘13
S

G)” = ijjTTTJ ds

Organizing them in matrix form

 

 

 

 

 

 

KP“ Ki” 0 0 0 "qu 'fr‘)‘
Ki” <KI"+GI"> —Gt’ 0 0 all fit
0 —GI“ (Gt5+02‘“> ~02“ 0 iq. i=< 0 i
0 0 -01; (K? +0?) Ki” qt f2‘

l. 0 0 0 K50 Kgogflg, tfzo

Here q? is the boundary condition imposed on the local model, obtained from the global

analysis. Thus the displacements q: computed would be stochastically varying.

29

3.2 Study of Variations

The change induced through the change in interpolation functions in the PBIE has to be
studied to determine the factors to be considered. The parameters of interest are the order,
magnitude of variation to be applied and its relationship with the penalty parameter and

the material constants.
3.2.1 Variation in 1- Elements

A two-element beam (Figure 10) with PBIE in the middle is chosen to investigate the
effect on the displacements and stresses arising from the induced randomly varying

interpolation functions.

1’1 EAL EAL
'Q :0: ¢——bP

:4 U1 U2 v U3 U4
,1

 

h.. \\ l \\ \K
| I V ‘. ‘s
1‘ ’- L .A_L_\_J_

u
I

Figure 10. 1D cantilever beam with PBIE under axial load
The variation of Total Potential Energy (Equation 1) with respect to all the DOF’s gives

the following system of equations (Equation 5).

 

c —c 0 0 0"U1‘ 7,]
—c (c+kly) 0 0 -y U2 0
0 O (c+y) —c ~y<U3j= 0 (5)
0 0 —c c 0 U4 P
_0 —k1}/ —y 0 2y_[VJ LO-

 

 

 

 

 

. . EA .
In the above system of equations the material constant CZT’ y 18 the penalty

parameter with U,, V indicating the nodal displacements and f,, P, respectively,

indicating the reaction force and the applied load. Further kl is a randomly varying

30

coefficient, taking any value between —R and R. Using the boundary condition U1: 0 the

general solution is obtained (Equation 6).

Pill-—
\__/
"U

kn
*—

Q
DJ

I
+
”U

a I

Q
N
I II
/——-—\ /——-\

(6)

+

 

VIN VIN

‘3

Q
a.
II

A
pa
+
P‘s"
y.—
W
\.________,./
'1:

<
II
/_—_\
Vli—
+
all"
\_____../
”0

It can be observed that the response to the variation is linear in nature with U3 and U4
following each other to enforce the perturbed constraints (Figurel).

1.15

 

 

 

 

 

 

 

 

-8- U2
-0-U3
V
1.1 . "T“ .
H
C
031.05 b u
E
(D
Q
9
Q.
.9 ' i
C)
1195 - a
09 1 L 1 1
[1.9 [1.95 1 1.05 1.1 1.15

Variation in K1 ‘

Figure 11. Displacement response with variation in Kl
The values of material constant ‘c’ and load ‘P’ are taken as unity; the value of the

penalty parameter y is set as 1E3 for obtaining the response.

31

3.2.2 Variation in 2-D Elements

Though the study of variation in 1D elements provided a lot of insight into varying PBIE,
the response of the same in 2D has to be checked and evaluated. Two square elements
modeling a cantilever beam with PBIE in between under axial load is chosen to study the
effect of the variation (Figure 12). The variations of the interface nodes Vi are assumed to

be linear in nature to study the same.

 

 

 

 

 

 

 

 

 

aw U3 is .U8 U; . a2
/
/
/
/
/
/
/
/
%
am U2. #2 d'US U? pprz

Figure 12. 2D cantilever beam under axial load

The variation of it with respect to the nodal displacements U , and V gives the following

stiffness matrix K, shown below. Where k, and k2 are the factors inducing the variation
at node U2 and U3 respectively and c is the material constant. P/2 is the load'applied and

f,the reaction forces in the load vector {f, 0 0 f4 0 0 0 P/2 P/2 0}T.

Here k, and k2 may be defined as
, 2 1+ A, (7)
k2 = 1+ A2 (8)
where A, and A2 are independently varying random numbers between prescribed limits

of variation.

The assembled stiffness matrix of the system is presented below.

32

 

 

' 'l
3c —l( --l-c —l(‘ O 0 0 0 0 O
3 6 3 6
k »,
—lc (:C+,_}’) (-lC+L-_y) _1 -Z -Z 0 O 0 0
6 3 3 6 6 3 3 6
' 2 k1
——c kiwi—"1) (—c+ J) --l-c J: J— 0 o 0 0
3 6 6 3 3 6 6 3
l l l 2
--—c —-c ——c —c 0 0 0 0 0 0
6 3 6 3 7
-]‘I_7 -I‘°y 0 21 :1 -2’_ 0 0 -1
K: 3 6 3 3 3 6
0 -£1_y. _k3y 0 El .21 -2’__ 0 0 -2.
6 3 3 3 6 3
o 0 0 0 —Z -1 3H1 ——c —lc —1 +1
3 6 3 3 6 3 6 6
l 2 l
0 0 0 0 ———c —c -—c -—c
6 3 6
0 0 0 0 ‘1C —1C 35' —_l-c
6 3 6
o 0 o 0 -2 -2 “HI _1(. _1C 4+1
L 6 3 6 6 3 6 3,

 

 

It is observed that the coefficients common to the varying nodes and the PBIE interface
nodes have the varying coefficients in them and the rest of the matrix remains unchanged.

The values of A, and A2 are varied as shown in Table 1.

 

 

 

 

 

 

 

 

 

 

Al A2
Case 1 -0.1 -0.1
Case 2 +0.1 -0.1
Case 3 0.0 0.0
Case 4 -0.1 +0.1
Case 5 +0.1 +0.1

 

Table 1. Variations in A, and A2
With values of c =1E7, ,8 =1E3 and P: 100 the displacements of nodes U2,U3,U5,U8
and V1, V2 are plotted in Figure 13. The nodal displacements in cases 1 and 5 are in
agreement with the 1D model (Figure 11). Case 3 with no variation in A, and A2 gives

the exact solution with only the PBIE. In the Cases 2 and 4 where the variation applied on

U 2 ,U3 are of opposite signs, the displacements follow the applied variations. The

33

average displacement is the same as in Case 3 but the stress would change fractionally,
which is desired. It is observed that the mean variation matches the no-variation case,
thus when randomly perturbed, the mean response magnitude will be the same as the
response with no variation.

RESPONSE TO VARIATION

 

 

 

 

 

 

 

 

 

 

OHS ' ' T ' . ' -+CASE I
—e CASE 2
_ _ ..__ —CASE3
”____*_'_‘—*_———‘_’“ ’1” -+~CASE4
0'“ " ‘ —EICASE5
‘T
h.)
a
t; 0.105 .
E t + a + r -
Lu \ ‘ " .\
:1 0.1 __ a
Q q + (j s t)
0095 ~ ..
—--——Ei——---EI—————El——-—-—~ —B———t3
009C}—— 1 1 L 1 L l r r 1
U2 U3 V1 v2 U5 U8
NODES

Figure 13. Displacement response with variation in A, and A2

3.3 ABAQUS Implementation

The variation is implemented by making suitable changes to the user PBIE code and is

tested using simpler applications to ensure reliability of the application.
3.3.1 Two Element — ABAQUS Verification

A two-element job is constructed with similar properties as in Section 3.2.2 with
variations given through modified PBIE. The results for randomly varying perturbations

were found to be in agreement with the numerical study. Figure 14 shows one such run

34

 

with A, and A, allowed a maximum of 10% variation.

 

 

 

 

 

Figure 14. Two-element Abaqus displacement profile

3.3.2 Four Element — ABAQUS Verification

The Abaqus user element PBIE is called every time it has to construct an interface
element, which means a different random matrix H gets multiplied to the coefficients of
the nodes being varied. At the node where two interface elements join each of the PBIE
evaluations, the common node coefficients have a combination of two random numbers.

Thus the response to the variation is not linear in nature.

A possible way to counter this is not to have variations on the first and last nodes
connecting two PBIE’s. In order to maximize the number of points at which variation is

obtained a minimum number of interface elements has to be used.

In the four-element job with one varying PBIE, only the mid node is allowed to vary

(Figure 15). The results are satisfactory in comparison to the scheme in which every node

was allowed to vary.

35

 

 

—————~A—.—d ___.'—_7 2_.-

 

 

___ ___-__________-._..L. ._- .-_ -._.-_i

Figure 15. Four-element Abaqus displacement profile

3.4 Effect of Number of PBIE

The idea of using a minimal number of interface elements goes against the prescribed
practice of joining the incompatible meshes with as many PBIE’s as possible to increase
the accuracy. The effect of decreasing the number of PBIE’s is evaluated on the plate

with a hole problem.

 

3.5 ._._______2___ - -- ----—----——~——»—-—--——— ——-_-—v ______.__

34
3:3-

 

. .....:::-—:::::?~_

 

 

32

FT:::::::T‘-A ' -—+— noIE

 

~‘—' 315

 

31

-—w— 215
TIE

 

29

 

 

 

 

ISPLACEMENT

D
DJ
rm

 

 

27

 

 

157

 

r I

158 159 160
NODE N0

181

152

183

 

 

Figure 16. Comparative study of accuracy with different number of PBIE

Figure 16 shows the displacement magnitudes of the local model nodes for different
number of PBIE’s used. The number of interfaces constructed does not have a major

effect on the accuracy. The profile of the nodal displacements in all cases matches the

36

_hnv_§. 1;... n

 

 

 

case with no interface elements. Thus for the optimization runs only two PBIE’s are used

to join the meshes to obtain a maximum number of points being subjected to variation.

37

 

3.5 Factors Influencing the Optimization Process

The relationship of the percentage variation limit (P) with the rest of the factors has to be
understood. The following are the features of the problem:

1) The baseline design.

2) Extent of design change permitted.

3) Loading conditions.

4) Magnitude of variations (MV).

5) Extent of change in the Interface Boundary Conditions.

The base line design indicates the degree of global-local nature of the problem, this may
be observed by studying the stress gradients or in the case of a nonlinear analysis the
response with time at the interface. The extent of design changes would have a direct
influence on the behavior of the BC’s. Loading conditions determine the magnitude of
the displacements and have to be taken into account. The magnitude of variation, which

has to be determined is directly influenced by these factors and influence them indirectly.

3.6 Magnitude of Variations-Issues

To understand the relationships and to non-dimensionalize the factors, the response of the
baseline design subjected to variations has to be established. It has to be ensured that the
average amount of variation induced does not exceed maximum change in the BC
brought about by Agent 1. It is not possible to determine the maximum BC change a
priori, as it would mean knowledge of the optimal design a priori. The rate at which the

design changes initially is rapid and large in magnitude, thus the limit can be set on the

38

 

 

 

basis of the first few cycles of optimization performed without the variations.

The following example will illustrate the method adopted:

The array of nodal displacements of the Baseline Design at the interface: [Db]

The array of nodal displacements of the local-global interface after N cycles: [Dn]

The absolute maximum difference in displacements between the baseline and the Nth

cycle design is F
Max ([Dbr- (Dar) = ID... - Dnbl

The maximum absolute change in displacements is at the nth node is: ID", - Dnbl

The percentage variation (Equation 8) is a fraction of the change and can be represented

 

by using a multiplying factor F.

D -—D
R=F Lfl—J—b—l x100 (9)
Dub

Thus for every local model size there is a unique factor to determine the recommended

amount of variation (R) in the displacement.

39

 

 

3.7 Plate With a Hole Optimization with Variation

It was previously observed that the original global-local approach of optimization
converged to an ellipse in 25 cycles for the 3.4” submodel. The 2.8” submodel took
longer to converge (50 cycles) because of the conflicting local model boundary
conditions borrowed from Agent 1. Figure 17 shows the optimal profile of an ellipse

obtained after 50 cycles

 

Vlewport: 1 008: Rielnemmepataltempr’gd28_c50_mvar.odb

I 008: gdZQ_c50_novar.odb ABAQUS/Standard 6.3-1 Wed Aug 06 22:43:26 EDT 2003
1

Step: Step-'1
Increment 1: Stcp Tim. - 2.2200E-16

 

 

 

Figure 17. Optimal design after 50 cycles for sub model 2.8” with no variation
In spite of an extended run of 80 cycles for the 2.2” submodel, the lowest final stress
found was 1.18134 1b/in2 which is 20% more than the known optimal (Table 2). Further
the design did not show signs of convergence to an ellipse. It was observed that after 10
cycles the frequency of boundary condition being updated reduced, implying that Agent 1
did not provide the correct boundary condition to Agent 0, resulting in the optimization

process being carried out in incorrect conditions. The design search seemed to be stuck in

40

 

 

a local optimum.

Using the modified PBIE the unconverged 2.2” sub model was subjected to
maximumlO%, 20% and 30% displacement variations at the local global interface. Here
the percentage denoted is the amount of maximum change permitted as a ratio to the
maximum change observed in the first few cycles. With 10% variation Agent 0 was able
to find better designs to be checked by Agent l.Which were scrutinized and filtered by
other boundary conditions in the next cycle, providing a better identification and
checking process. It showed evidence of higher rate of updating of the boundary
condition by Agent 1. Though the final stress was higher and the profile was not an exact
ellipse it performed better than the analysis with no variation and helped in moving out of
the local minima into a flattened profile. The final state of the analysis was used as a
starting point for the extended analysis using the EXTEND option in HEEDS with no

variation for another 25 cycles leading to the desired optimal profile of an ellipse.

The process of optimization for sub models smaller than 2.8” shows marked
improvement in convergence rates compared to the analysis with only global local
analysis. Figure 18 shows the iteration history of the 2.8” without variation and sub
models 1.9”, 2.2” with 20% variation where the better performance of the latter two can
be clearly observed. In the 1.9” sub model analysis in the 11'h cycle Agent 0 had
identified profiles whose inferior performance was spotted and the design was filtered out
in the next cycle. Further a clear saving in the computational time for optimization was

observed as the sub model size was reduced.

41

 

 

 

 

 

 

 

f Tg ‘ 1
l - - 2.8" ,
1 6i L 2 2" H
v 1.5} ..
“J l
x l f
5’» 1.4 - a
9 g,
(75 1
8 .
ca 1.3} _
.22
2 r .
O .11 . .12: l j
> 1.2 ~ . , . . . . , z
1.1 r ,k..,‘*,—(\‘- 1 ,‘ . . ' /,‘\\ )7 i ' .2
4‘ w: 4 ‘1 ‘- ”I 1» J h at it ~. “,sfir—k :4
1 __ I L ___ __.' _ I I ___E
O 5 10 15 20 25 30
cycle no

Figure 18. Comparison of 2.8” without variation and 2.2”, 1.9” with 20% variation
From the local-global optimization canied out for the 1.9” sub model with maximum
variation limits 10%, 20% and 30% it was found that the higher variations are
objectionable. Random variations with maximum of 20% gave the best results for both
2.2” and 1.9” models. The 1.9” model subjected to 10% and 30% variations showed poor
convergence characteristics in the first 15 cycles (Figurel9). Agent 1 in the 10% case
never updated the boundary condition since the variation given was not enough to
explore and find better solutions. In the case of 30% variation it is observed that from the
2"d to 6th cycles the best designs found by Agent 0 had high metrics of 1.6E4 lb/inz,
attributed to the fact that because of the high variations a different problem was being

solved each time. It is observed that between 15 and 25 cycles, 30% variation performs

42

 

better than the 10% case providing a better starting point for the extended run which

converged to an optimal structure with metric of 1.09 E4 lb/inz.

 

 

'4‘ 10%
“B— 20%
*4" 30%

 

 

 

 

Von Misses Stress x 1E4

 

 

 

 

cycle

Figure 19. Iteration history of thel.9” sub model with 10%, 20% and 30% variation
The optimal design obtained of the ellipse was not stretched to maximum permitted
limits. The perturbed local model boundary causes the loading in the Y direction to be
greater, requiring the optimal profile not to be the extreme ellipse. The profile obtained

(Figure 20) augments the theory of robustness expected through the applied variations.

43

 

 

2

l 008: msrl-Ouadb ABAQUS/Standard 6.3-1 Tue Sep 02 08:17:59 EDT 2003 l.
1

Step: Step-1
Incrmmt

Vlewporr: 1 0011: :hoMalepatalemp’mrtOodb

 

1: 811.9 Time - 2.22002-16 L 1 _ . E

 

 

Figure 20. Optimal design after 25 cycles for sub model 1.9” with 20% variation

Table 2 provides the comparison of convergence comparison with sub model size,

variation, minimal stress reached and the number of cycles taken to reach the optimal

solution. These results clearly show that too much variation has a negetive effect on the

process. The reason is the effect of best design found and the variations get cancelled out

and the design oscillates, showing poor convergence.

 

 

 

 

 

 

 

 

Optimal
Size Variation Stress (lb/inz) Convermze
3.4" 0% 9.94E+03 Good ellipse found and good convergence rate 425 cycles)
2.8” 0% 9.99E+03 Good ellipse found and good convergence rate Q50 cycles)
2.2” 0% 1 .18E+04 Bad design (80 cycles)
2.2” 10% 1.35E+04 Shows signs of moving out of the local minima (25 cycles)
2.2" 20% 1.03E+04 Good ellipse found and good convergence rate (25 cycleg)
1.9” 0% 1.40E+04 Boundary too close --stuck at bad distorted design
1.9” 10% 1.42E+04 Boundary too close «stuck at bad distorted design
1.9” 20% 1.09E+O4 Good ellipse found and good convergence rate (25 cycles)
1.9” 30% 1.10E+O4 Converged after extension with no variation (25 cycles)

 

 

 

Table 2. Optimization results comparison for plate with a hole

44

3.7.1 Injection Island Optimization

As discussed earlier a major problem with the proposed linear structure of the local-
global optimization with variation is that if the percentage variation is lesser or greater
than the critical limits then the analysis might not converge. Further correct estimation of
the order of variation requires knowledge of the solution a priori or performing multiple
runs, requiring monitoring and analysis of each of them.

A practical solution is to take advantage of the Injection Island Topology with Variations
(IITV) to cover a broader range of design space. In IITV a number of local agents
perform optimization with different orders of variations. They information among
themselves about the best designs. One local model performs the local search with
boundary conditions without variations. This local agent receives the best designs from
the agents evaluating with variations. The loop is completed with a global analysis to
determine if the conditions need to be updated.

The advantages with this structure are that

1) More searches are performed at the local level, which is computationally cheap.

2) The design space is searched more thoroughly because of the different agents looking
with different variation conditions.

3) Sharing of the designs improves the robustness for higher order variations.

4) The local agent performing the analysis without variation acts like a damping factor
bringing the gradual change effect.

For this problem, four local agents were used, with 10%, 20%, 30% and no variation,

working with an agent managing the global evaluation (Figure 21).

45

 

 

A

Local 213% LocallU% Local 500
variation variation variation

Local N o
variation

 

 

 

 

 

 
   
  
   

Global Agent

 

 

 

 

 

 

 

  

 

 

 

 

 

N
0 Yes
Design
No change performance #
check Update Boundary conditions

 

 

 

Figure 21. Injection Island Technology applied for Plate with a Hole problem
Two best designs of the previous cycle from Agents 0,1and 2 are received by Agent 3,
which performs the local model run with no variation. The global model evaluates the
best design found in the cycle and depending on its performance updates the Boundary
Conditions. The local designs shared between Agents 0,1 and 2 are evaluated for
different orders of variation. It was observed that the procedure is better than the linear
model in terms of the convergence rate. The large number of local runs are justified in
case of a complicated problem, where performing a global analysis is computationally
expensive. Figure 22 shows the best design found after the 10‘h cycle and Fig 23 shows

the convergence performance of the IITV.

46

 

 

 

‘Vlewporr: 1

0 DB: mo memalepatanem p’MSlli - l .odb

2

I one: MSR1-1.odb ABAQUS/Standard 6.3-1 Mon Sep 25 04:34:40 :0? 2003
1

Step: Step-1
Islet-nan: 1: Step Tim. - 2.220012-16

 

Figure 22. Best design after 10 cycles using the Injection Island scheme

Von Misses Stress x 1E4

 

 

 

 

 

 

2.5 . 4 41 W _fifl , 2.2.2.“. .1
T
i
2— l i
l
l
l
1
l
l
1.5‘1 i q
‘ l
i i f--#-— §\
\‘1/
l t .
i 4 . y: ,4 y 1“ P i—~ .y__-.._..__" I, ) \\
I V: , "at: ‘1 1
1L 1 ___- r r 1
5 10 15 20 25

Cycle no

Figure 23. Convergence of Injection Island topology analysis

47

 

 

3.8. Beam Deflection Control Using Global—Local Approach

Two cases of an indeterminate beam have been considered for analysis through the global
local approach. Because of symmetry only half the beam is modeled with a uniformly
distributed load. The following objective was chosen to be achieved through the

optimization process:
. . * . *
Objective: (5 <6 and mass constrained at m

Where 6 is the midpoint deflection, 6* is the limiting displacement and m* the mass
constraint value. The beam is broken into 5 regions (see Figure 24) with the height and
width of regions 1 through 4 being designed to achieve the above-mentioned objectives.
By varying both the height and width the second moment of inertia of the beam changes
to control the deflection of the midpoint.

TG W0 Global Model Ti Wf

l l l I
If f 4 o 4 g? i ; § 53,411;

I
If

r/

 

1
Boundary Condition Transfer Local MUUBI

\. . l l L 1
A"--.\‘ 1 Y ’1" i:
O k 1)::
l L L"

1::1 Global Region

it LOAD L—J Local Region
Ti W1 Design Variables
rG we Global Constants

 

—.— 1D BEAM

Figure. 24. Beam Global—Local setup.
For the analysis the design variables Ti and Wi could take values between 60 and 80mm,

while the value of TG and WG were kept constant at 80mm for each of the analyses.

48

 

3.8.1 Comparison of Beam Optimization Processes

The beam optimization was carried out in three ways. In global beam optimization the
entire problem setup was analyzed to achieve the objectives. The local beam optimization
involved studying just the local region for the same objectives, where the local region got
the boundary condition from the baseline global model. The third scheme was a global-
local set up done by transferring the results from the global model after every cycle being
evaluated by the global agent. The objective of these optimization runs is not simply to
try to determine the global maxima or minima but also to understand and accelerate the
progress towards it. Therefore all the analyses were started from the same random seed to

ensure the process technique is the major governing factor in the study.

Though the problem seems really straightforward it must be noted that there are a number
of combinations of the design variables capable of satisfying any mass constraint, but the
individual section properties dictate the response of the beam. Comparing the local model
and global model runs the importance of direction for the local agent can be realized
(Figure 25). In fact as the boundary conditions do not change in the local model the value
of the performance predicted was found to have a considerable amount of error. Thus
again requiring a different scheme for good prediction. The optimal design found by the
local agent after 10 cycles was not even remotely close in nature to the one determined by

the global agent.

The objective of the problem is to determine the most effective method to accelerate the

49

 

initial stages of the optimization process, through quickly identifying good designs being
fed to the global agent, which in turn provides the right direction for the process to
evolve. Boundary condition getting updated after every global cycle showed a lot of
promise but still lagged behind the global best. Further, the best designs found were also

different from the global best design.

 

 

 

 

 

 

 

 

 

0) "fl 2--” - '
Q .
s: h V
w __ _
g ,2 _
a) K
o.
-15 ~ / ———4——-— Local only 5
. . -— Global
.1 6 ( ——&—— Global-Local -
- —- - --D -— - ‘ GL-10% variation
------ 9- - - - " GL- 20% variation
.1.7 L l 1 L i l l l
o 1 2 3 4 5 e 7 s 9

Global Cycle No

Figure 25. Comparison of the optimization runs
Analysis of the local model with variation was again studied by varying the local model
boundary conditions by 5%, 10% and 20%. The idea behind the analysis is to explore and
ensure the probability of this design being a good one at the global level. In order to
achieve it, the local agent is looped 4 times within itself resulting in the best design found

in a generation being checked for robustness approximately 3 times.

50

 

 

 

 

 

 

 

 

 

 

.12
8 1.25
C
(U
E -1.3 y
‘1:
CD
0. 1.35
.1 .4 If” + Local 10% ‘
t —‘9— Global10%
-145 _ _53_. Loca|20°16 -
' J 6 Globalmifi
-15 L 1 l 1 1 1 l
o 5 10 15 20 25 so as 40

Global Agent Cycle No

Figure 26. Local and global agent performances for 10% and 20% variation
The 5% analysis performance was similar to the no-variation analysis, which meant that
the amount of variation provided was too small to have an impact on the response. The
10% variation analysis was able to fight its way away from the no-variation run after the
4'h global cycle. The 20% variation run ensured that the design found could stand the test
of slight variation in boundary conditions, so the right direction was established and the
analysis moved faster than any of the previous analysis. Thus the 20% analysis performed
could not only ensure good direction but also resulted in accelerating the process (Figure
26). It is interesting to observe that the local and global performances for thelO% and
20% lines get closer in the latter cycles of the analysis, indicating that the global agent is

providing better boundary conditions there.

51

 

I'm".-- a.

3.8.2 Multi Agent - Injection Island Optimization

We observe that there are two principal issues with the variation case performed in the
earlier segment:

1) There were 4 generations of evaluation inside each local agent cycle. This provided us
with the robustness but could mean that the same design, which was determined as good
in a generation could get knocked out of the process by the given variations.

2) There exists a problem of being unable determine exactly what is the best order of

variation to be given to a local model. Further it was observed that there was a need to

 

somehow decrease the variations in the latter stages of the analysis.

 

-1 .05

 

 

-1.2 d
(D
3 125
(U ' . ‘1
E
E -1.3 -
31’
-1 .35 ' + Locd20% i

   

-e— Global Agent
-A- Local 10%

{1- Local 20%

-0- Local no yan'dion
—~ Global Agent optimizaticn ‘

 

 

 

 

 

l I l l l l 1 l

0 2 4 8 8 10 12 14 18 18 2]
Global Agent Cycle No

Figure 27. Multi Agent —Injection Island optimization of beam
An injection island topology (Figure 28) was proposed to ensure that the designs found

by different agents evaluating the local model at different orders of variation and no

52

variation agents exchange designs, thus helping to ensure robustness. The design space is
also searched more exhaustively. The other advantage is that the no-variation local agent
is an inexpensive local check before sending the design to be globally examined. Finally
this also ensures that the local agent, which performs the search without variation can

independently look for good designs, helping the process solve with more efficiency.

 

: A

 

 
    

   
  

 

Local 10%
variation

Local 5%
variation

  
    

   

Local 20%
vafiafion

  
  

Local No
vanafion

  

 

 

 

 

 
 

Global Agent

 
  
 

 

 

 

 

 

 

 

 

 

 

    

 

 

 

      
 

No
Yes
Design
No change performance H
check Update Boundary
Conditions

 

 

 

Figure 28. Multi Agent — Injection Island Optimization Flow Diagram

53

 

CHAPTER 4
CRASHWORTHINESS OPTIMIZATION

4.1 Introduction to crashworthiness design

Crashworthiness shape optimization is a challenging problem made so by the
complications in design, analysis and computational requirements. The role of the front
rail members of an automobile is to absorb the crash impact and to transmit the least
amount of force to the passenger cabin. The design targets that can be investigated are
energy absorbed, force transmitted to the cabin, maximum acceleration, weight of the rail

member and vibration minimization to name a few.

The shape of the front part of the rail is expected to be complex, in order to absorb energy
by progressively failing, and with minimal buckling. Thus the design of this member is a

complex task that isvery difficult with conventional design engineering principles.

Principal concerns of passenger safety and robustness of rail to multiple crash scenarios
encouraged the selection of energy absorbed by the rails as an objective and the

maximum reaction wall force as a constraint for optimization.

4.2 Introduction to robust design

In most optimization processes the optimal design is determined for a specific set of
conditions and constraints, which might not survive the expected variations in reality. In

a practical problem it not possible to eliminate all deviations during manufacture of the

54

 

product or ascertain and ensure that the environment it is subjected to remains the same.

 

OPTIMAL DESIGN
ROBUST DESIGN

A,

DESIGN / OPERATING VARIABLE

4

 

 

PERFORMANCE

 

 

Figure 29. Comparison of Optimal and Robust Design
The obtained optimal design may fare extremely well for certain design considerations or
specific environments but might not respond with the same performance with small
variations (Figure 29).
This is especially valid in the case of crash optimization problems where the best design
should perform well in non-frontal crashes as well. Further designing a product to satisfy
every worst-case scenario may be an expensive option. Thus justifying the need for

stochastic tools to be used while engineering the optimized design of such products.

5.3 Optimization of Energy Absorbers

Shape optimization of front rail members of an automobile is a complex problem, which
ideally suits the use of Global- Local approach for efficient handling of the system. The

model Chevrolet C1500 Pickup available over the intemet truck from the National Crash

55

 

I.

Analysis Center (George Washington University) has been used in this investigation
(Figure 30).

4.3.1 Problem Setup

The front rails of the truck model are critical members, which absorb maximum energy
during a crash process (Figure 31). Further, their position and nature make the Global—

Local technique of analysis ideal for efficiently optimizing them.

 

Figure 30. Global Model of the Truck
The local model is generated by HEEDS using a combination of spline and closed surface
generation capabilities (Figure 32). The rails are created using an automatic mesh
generator within HEEDS. The spline provides the backbone for the closed surfaces to be
generated on it. The rails are modeled by dividing them into 9 appropriately positioned
cross-sections with 20 nodes on each of them and are used to create 20 cross-sections,
generating an efficient representation of the rails. Thus the model is a structured mesh

with 1000 nodes and 980 elements.

56

 

   
    

Local Moch Reginn a a;

  
 

Figure 31. Local model for the Rain analysis
The spline geometry is modeled with 12 points, with these points acting as design
variables that could be permitted to vary spatially. This would however result in a

number of designs displaying entirely different behavior (instabilities). Hence this

representation was not used in this study.

  
   

  
   

1‘.
\
ii
‘1
(‘1‘
l"

.a
”it:
i)

“‘15,.

       
 

 
 
 

.
-—:
-
a.
Q
.

   

     

.1
y
a

Figure 32. Rail profile showing the features spline and the cross-section

The main design variables are the control points on the cross-sections along the length of

57

the rail. All cross-sections initially have a predefined rectangular shape with the nodes 1
to 8, l9 and 20 acting as the master nodes with nodes 9 through 18 as slaves governed
respectively by their master nodes positioned symmetrically across the Y’ axis (Figure
33). Since a new mesh is created for every new analysis the above relationships reduce
the possible complications of mesh distortion and facilitate ease in manufacturing the rail.

Non-symmetry of the automobile requires the rails to be independently designed.

 

 

 

A u
.L Y
6 T A 3 9 10 ll
0 e:
l
5 12
D
3 14
2 15
© mmnom
. SLAVENODE ,

20 19 18 I7 16

Figure 33. Cross-section definition of the rails
The continuity of the meshes between the generated model and the extension of the rail in
the global model force the last cross-section not to be varied. This does not have a
pronounced effect as present interest is primarily in the front parts of the rails, which

account for maximum initial energy absorption.

58

mill

 

 

Each of the shape control points is given a resolution of 7 and the spline points are fixed
for the present problem. This optimization process involves finding the right combination

of the 160 design variables.

The objective of the rail optimization problem is to maximize the design performance,
which is measured by the total energy absorbed (EA) by the first five segments of the
designed rails. The maximum reaction force in the rail cross-section is treated as a

constraint set to a standard value of 170 KN.

4.4 Global—Local Setup

The global-local setup is achieved by providing the local model with the required input
from the previous global model run. The nodal translation and rotational displacements of
the nodes forming the interface between the global- local models are written at a desired
frequency during global analysis. This data is read by a C++ program and rewritten as a
LS-DYNA input file, where the nodal displacements rewritten as function of time drive

the local analysis in the form of a Boundary Prescribed Motion problem.
4.4.1 Beam Model Test

To ensure that the logic and approach work smoothly, they were tested for a beam-
bending problem. The cantilever beam is modeled with shell elements with a nodal force
in the negative Y direction loading on node 18 (Figure 34) with the local model as the
region composed of the region to the left of nodes 7 and 17. The displacement histories of

nodes 7 and 17 run the local analysis.

 

11 12

 

19 20

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1 2
9 10
Figure 34. Cantilever beam global model
25 2D BEAM MODEL
Node No
/ 5
20 j _A_
2:
815
76
> /
§1o
3
3 /
[I 5 /
0 I l l l l l l

 

 

 

 

 

 

 

 

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014
Time
Figure 35. Resultant velocity of node 5 for local model
The resultant velocities of the local and global analyses of node 5 are compared to
evaluate the effectiveness of the process. Comparison of Figures 35 and 36 reveals that
the velocity responses of the local analysis are in complete agreement with the global run.

The analysis was performed considering the region to the right of nodes 7 and 17 as a

local model and found to be in total agreement with the global analysis.

60

20 GLOBAL BEAM MODEL

 

 

 

 

 

25 / Node No
l _A_5

20 /

3‘ /

§15 A

(D

> /

§1o /

3 /

3

tr 5 //

 

 

 

 

 

 

 

 

 

 

0 1 1 l 1 1
0 0.002 0-004 0.006 0.008 0-01 0.012 0.014
Time
Figure 36. Resultant velocity of node 5 for global model

4.4.2 Truck Model Test

In spite of satisfactory results for the Beam Model test it has to be ensured that the
approach works well in highly nonlinear crash problems. The local analysis was
performed with analysis results obtained from the global run of the entire truck. On
comparing the response of the two systems it was found that the local model encapsulates
the global model run for small values of time increments at which the results are written
and fed to the local model. This is performed to capture features such as stress waves and
complex deformation histories that should be properly replicated in the local run.
However for frequencies at which the data is read and written the input file size rapidly
increases, placing a practical computational constraint on the frequency of the nodal input

to the local model.

61

 

 

20 . CZ 500.P1CKUP TRLJQKMQQELINQZLQLQL -___--

 

 

 

 

,.__- Node No
A A 43100
15.1. A
A
a r
+
E
z... 10 *- .l
'8
3
>0 A
>34 A
5-.. a
can
0 L g r g 1 g 1 g 1 4‘ g
0.005 0.01 0.015 0.02 0.025
TIME
Figure 37. Global model analysis
g.

It was found that when the input frequency to the local rail model is at lE—4 it sufficiently
simulates the global model run and ensures comfortable handling of the files. Figures 37,
38 and 39 show the X velocity results of a representative node 43100, in the global
analysis, the local model run with the C++ program and the local model run with the
LSDYNA Interface option, respectively. It may be observed the local model and the
LSDYNA Interface option performed at a frequency of lE—4 sufficiently capture the

crash process.

62

20 (32500 PICKUP TRUCK MODEL merge vs;

 

 

 

 

 

 

 

 

 

Node No
A ' A 43100
__ . . A . . _
15 A
8
+
E
2:10“: ‘
'6 A
0
To A
’t
X 5 .» ...
1.
0 1 g r g 1 g n g r 41 4L
0.005 0.01 0.015 0.02 0.025
TIME
Figure 38. Local model with written boundary input file
C2500 PICKUP TRUCK MODEL (NCAC V8)
20 , r i .
. : Node No
l
' A 43100
A
15 “r A A
3
+
B
2:10 “P , ~
'6
0 P A
3
>.
X 5‘1" 1A
1 . .
o l 1 l . t . i . 4
0.005 0.01 0.015 0.02 0.025

Time
Figure 39. Local model with LSDYNA Interface file option

4.5 Variations in Rail Optimization

The local model boundary conditions have the nature of prescribed boundary conditions

and thus are time dependent. This requires a scheme for variation as a function of the

63

analysis time. This difference would in effect capture and simulate variation in crash
velocity and the nature of resistance it would meet. Further, the spatial variation when
incorporated would account for variations from frontal impact. Thus when the two
variations are used together they would simulate a slightly different crash scenario every

evaluation.

4.5.1 Linear Variation with Time

The variation in input to the local analysis is varied linearly with analysis time. The
maximum magnitude of variation (MMV) is a user specified percentage, which may be
obtained from prior knowledge or an analysis without variation. Two random number,
Time Factor Start (TFS) and Time Factor End (TFE), are generated inside the MMV
limits; the Time Varying Factor (TVF) is determined as a function of the analysis time
(equivalently time step number) and these quantities. Figure 40 shows a representative

diagram of how the TVF is varied within each analysis.

 

 

 

 

 

l l
———-— ——————+MMV
TFS‘ TVF
z
9 TIME
E \ ’-
3 TFE
3’
__________ ”-va
' 1

Figure 40. Time varying factor with time

64

 

The total analysis time is the product of the number of time steps (NTS) and time step
increment value. Thus the relationship equations as a function of the present time step
(PTS) are given by:

-MMVS TFS _<_+MMV

-MMV S TFE S +MMV

TVF:

[__TFE ‘TFS )PTS + TFS

Thus each local analysis performed has variations different in magnitude and in the rate

at which they change with time.
4.5.2 Spatial Variation Factor

Since the generated profile is closed in nature, the variation must be closed as well. Thus
for spatial variation a parabolic function is chosen as it fits the requirements. This Option
can be used with code involved here only if the profile is structured and the node
numbers on the local global region increases uniformly. The representative Figure 41
shows how the Spatial Variation Factor (SFV) is determined as a function of the node

number. Figure 42 provides a visualization of how SVF changes around a cross-section.

sve
Mid Node /

\/ NODE no

SVM

 

 

SHAPE VARIATION FACTOR

 

 

Figure 41. Shape varying factor function of node number

65

 

 

 

 

\x SHAPE VARYIN G FACTOR
“H MAGNITUDE OF SHAPE VARYING FACTOR

Figure 42. Visualization of Shape Varying Factor on a cross-section

Two random numbers Spatial Variation End (SVE) and Spatial Variation Middle (SVM)
are generated between 1 and —l. A parabolic equation is constructed such that the SVF at
the mid node (MN) is SVM and the first, last nodes (FN, LN) have SVE amounts of
variations, respectively. Thus the spatial variation factor can be randomly generated for

every node point on the interface.
4.5.3 Total Variation Factor
The total Variation Factor (VP) at any time and at any nodal point of the local analysis is
the product of the Time Variation Factor and Spatial Variation Factor.
VF = TVF * SVF
A randomness of the VF generated for every local analysis is assured by making the seed

of each random number generated a function of the machine time.

66

 

4.6 Rail Optimization Results

4.6.1 Global Local Optimization — Without Variations

The optimization scheme was set up as per the scheme discussed in the second and third
chapters. It was observed that the optimization performed at the local level by Agent 0
would yield better results if performed with four generations within each cycle. This
provides opportunity for the design to evolve for a certain boundary condition before
being passed to Agent 1 for performing a global evaluation. The above scheme provides

the computational advantage by avoiding expensive global runs.

Figure 43 shows the local and global energy absorbed by the first five sections of the rail
for the 29ms analysis performed. The global model absorbs more energy than the local
model because of some external contact. But as observed, the difference is almost
constant, supporting the use of the local model results in the analysis. Further, the mode

of global and local crush is very similar in nature.

One can observe monotonic increase in the performance of the rails both in the global
and local agents with 20% increase in the measured sections. It may be observed that the
local model experiences difficulty in identifying better designs after the 12‘h cycle, which
could have prevented evolution of the design. In effect the analysis was stuck in a local
optimum. It must however be noted that a major amount of effort was spent in trying to

determine designs which do not buckle and have better energy absorbing attributes.

67

 

3.2

 

 

 

 

 

 

 

 

3 _, HB—G—O—Q-G-O-O-H‘O _
{MJ + Local
23 _ 940‘“ a Global
/ '1
uz-Gzee-o—é *
(D
.Q
I—
8 2.4 ~ .
_D
<
E22 _ a
(D
C
L“ 2 . .
.4
1.8 l l 1 L
n s 10 is m 25

Global Cycle no
Figure 43. Energy absorbed by local and global models for analysis without variation

 

The system level energy absorbed (Figure 44) shows an increase of 3.22%. The plot
shows that the best design passed to the global agent afier every four cycles of local

evaluation is better every time.

 

 

 

 

6.15
6.‘ '- and
0'0 HO—O-O—O—O-O—O'O

63$ - .
'0 04-0-4
°’ 1
.Q
L
O
m 6.3 . I -
.Q
<[
" 1
0'1
33
C 625 . .
LU

Ibo-6'
6.2 AA A l l l 1
0 5 10 15 Z] 25

Global Cycle no

Figure 44. Total energy absorbed by the truck system

68

The percentage of energy absorbed by the rails to the system energy showed 6%

improvement in performance (Figure 45).

i
{Tb—*3 (\P—J"
.. “ «,1 \V-

 

 

Percentage energy absobed by rail to system

 

 

44 i
43:» l
I l
42f ,
l
!
41L-.. __..__A-___l_ l L ...- l
0 5 10 15 20 25

Cycle no

Figure 45. Percentage energy absorbed by designed rails to system energy absorbed

4.6.2 Global-Local Optimization — 5% variation

Global local optimization with 5% variation in the boundary conditions was studied to
understand and observe its effects. It was observed that the region connecting the local
model to the global model experiences almost uniform deformation characteristics.
Therefore the Shape Variation Factor (SVF) was not used to ensure the local model still
performs analysis with reasonably similar characteristics to the global model. Thus the

total Variation Factor (VF) is equal to the Time Variation Factor, which is set to 5%.

69

 

 

 

 

 

l ___._________
' l # Local EA
sl l_53_'E"_E§
N |
LU
35 l
1: l
(D
E
8 I
.0 l 3' F
(U , _ .
> ; 9 Pi» é»r+—§--—b——¢‘—‘+E
E" l
0C3 2.57 " ‘l
UJ i ‘ 71
l ,/I
I 1' I a r I l
24 -_- l __ .l _- » AM" 1 i
0 5 10 15 2O 25
Cycle no

Figure 46. Energy absorbed by local and global models for analysis with 5% variation

Figure 46 shows the local and global energy absorbed for the 5% variation optimization
run, where the best design determined is better than the no-variation case by 6%. Though
there was an increase in the local model performance between cycles 8 and 15, these
designs did not show similar performance in the global agent evaluation. Continual
search with variations around the same boundary conditions helped the designs eliminate
a probable local minimum and find designs eventually that were better than those found
in the run without variations. Both the runs were started with the same random seed,

demonstrating the increase in efficiency of the 5% variation process.

70

 

 

9’

4:.

U1
r.

6.4l

8

__L-___ _.__.L.___.. _J.___._ __ 1.

System energy absobed 1xE7

9’
(40
E -7" "'T’ “M 4—Y"

. .

.07
M
01

 

 

6.2—4) 1;. +1» 4.5%. “___—J --—--—- I - l

5 1o 15 20 25
Cycle no

 

Figure 47. Total energy absorbed by the truck system

 

 

48 l" “__ — Y _Tf' '_ "___ T" '_ ‘—"T‘ " f

.1
J
(\
l D
C)
r l
t D

a
\o
2‘

b
\J

 

31 3
T a-_——"‘TT ‘m'v’f

fi

 

8

_J___z_.___._-. “ha—”___Lwfi.___l_.__~__i___ ___J

percentage energy absorbed by rail to system

A
N

l /\
"y:
‘

 

l.. l l

5 V 10 15 20 25
Cycle no

5
_5
f

Figure 48. Percentage energy absorbed by designed rails to system energy absorbed

Total system level energy absorbed by the truck showed an increase of 4.5%, almost
1.5% better than the no-variations analysis. The percentage energy absorbed by the rails

to the system level energy absorbed again shows an increase of 6% (Figure 47 and 48).

71

CHAPTER 5
PERFORMANCE ANALYSIS AND CONCLUSIONS

5.1 Performance Analysis

Total computational time is the most important feature the present study tried to reduce
through the incorporated schemes. Thus the total computational time was chosen as the
basis to compare efficiency of the global-local approach to the normal global
optimization method. A number of process parameters and known quantities have been
used to determine the relationships, measuring the efficiency of the developed approach.
Further, these equations would provide an understanding of when decisions have to be
made on the process parameters for any other analysis. It has to be observed that this
approach is for problems whose computational times are very large. Thus more often than
not the measures from the global analysis may not be available a priori. The following
analysis provides the framework to work backward from the time available to perform an

optimization run and choose the ideal parameters in the best possible manner.

5.1.1 Global-Local Time Savings

In order to determine the total global-local advantage we must define the following

optimization parameters:
1V0: Number of global evaluations required to do the global optimization in the

traditional manner (depends on the number of design variables and nature of the design

space)

72

 

NG : Number of global evaluations required for the global—local process.
NL : Number of local evaluations required for the global-local process.
NL/ C3 Number of local evaluations required per global-local cycle.

NC : Number of global-local cycles

TG : Total time for the global optimization in the traditional way.

Fifi

TGL : Total time for global-local optimization.

 

TC : Analysis time for a single global evaluation.

TL : Analysis time for a single local evaluation.

The number of local evaluations per global-local cycle would always be less than the

number of global evaluations (i.e. NU C S 1V0) because the local cycles need not totally

. . N
converge. Thus, as a very conservative assumption, we let NL,C=—ZG—-. Further the

number of global-local cycles in the global-local set up is assumed to be equal to the

number of global evaluations (i.e. NC 2 NC ). This again is conservative since it would
always be one less than the number of global evaluations required (i.e. NC = N6 - l ).

Thus the following equations could be deduced using the defined quantities:

NOW
2

G
NL=NL/C*NC=

The analysis times for each of the optimization schemes can be represented as the

following:

73

fG:NG*TG

 

_ iv" T
TGLzNL*TL+NG*TG=NG[ (32L+TG]

5.1.2. Effectiveness Ratio

Effectiveness ratio (77) may be defined as the fraction of time taken by the global-local

analysis setup to the global analysis, providing a direct measure of the computational

efficiency of the proposed process.

 

 

 

Grouping and rearranging the parameters:

UZNG fl— +—_:l—
ZTG NG

It may be observed that the efficiency of the process increases when the local model is
smaller and can be performed quickly. The efficiency is also high when the number of
evaluations required to do it the traditional way is large indicating the problem must be

complex and demand computational time efficiency. More importantly, the effectiveness

ratio is linearly dependent on N G .

Table 3 indicates how the effectiveness ratio gets better, for IV C set as 1000 evaluations.

To illustrate the advantage of the global-local approach the table has been divided into

three zones based on the effectiveness ratio Le, less than 15%, 15-30% and greater than

74

 

30%. The top left hand comer is the ideal zone where the process displays highest
efficiency, but would mean a low number of global-local cycles, which may have not
identified the right boundary conditions. It is better to choose values towards the right of
the table, which would mean more design evaluations thus more robustness. The values
when multiplied with the global analysis time would show how the approach could make
a problem conventionally optimized in a month possible to accomplish in a few

days,doing more design evaluations and with variations making it more robust.

 

 

 

 

 

 

NC 5 6 7 8 9 10 11 12 13 14
Tl/Tg
0.01 3 3.6 4.2 4.8 5.4 6 6.6 7.2 7.8 8.4
0.02 5.5 6.6 7.7 8.8 9.9 11 12.1 13.2 14.3 15.4 L
0.03 8 9.6 11.2 12.8 14.4 16 17.6 19.2 20.8 22.4
0.04 10.5 12.6 14.7 16.8 18.9 21 23.1 25.2 27.3 29.4
0.05 13 15.6 18.2 20.8 23.4 26 28.6 31.2 33.8 36.4
0.06 15.5 18.6 21.7 24.8 27.9 31 34.1 37.2 40.3 43.4
0.07 18 21.6 25.2 28.8 32.4 36 39.6 43.2 46.8 50.4
0.08 20.5 24.6 28.7 32.8 36.9 41 45.1 49.2 53.3 57.4
0.09 23 27.6 32.2 36.8 41.4 46 50.6 55.2 59.8 64.4
0.1 25.5 30.6 35.7 40.8 45.9 51 56.1 61.2 66.3 71.4
0.11 28 33.6 39.2 44.8 50.4 56 61.6 67.2 72.8 78.4
0.12 30.5 36.6 42.7 48.8 54.9 61 67.1 73.2 79.3 85.4
0.13 33 39.6 46.2 52.8 59.4 66 72.6 79.2 85.8 92.4
0.14 35.5 42.6 49.7 56.8 63.9 71 78.1 85.2 92.3 99.4
0.15 38 45.6 53.2 60.8 68.4 76 83.6 91.2 98.8 106.4
0.16 40.5 48.6 56.7 64.8 72.9 81 89.1 97.2 105.3 113.4
0.17 43 51.6 60.2 68.8 77.4 86 94.6 103.2 111.8 120.4
0.18 45.5 54.6 63.7 72.8 81.9 91 100.1 109.2 118.3 127.4
0.19 48 57.6 67.2 76.8 86.4 96 105.6 115.2 124.8 134.4
0.2 50.5 60.6 70.7 80.8 90.9 101 111.1 121.2 131.3 141.4

 

 

 

 

 

 

 

 

 

 

 

 

Table 3. Effectiveness ratio with TUTG and NO

75

 

5.2. Conclusion

Through the proposed global-local approach, practical design optimization problems
previously almost impossible to solve could be handled efficiently. The optimization
scheme consistently showed computational efficiency compared to the traditional global
approach. The scheme to perform the local analysis with variation showed clearly better
ability in finding a more robust solution, thus leading to a good local boundary condition
and improving the efficiency of the process in the important initial cycles. The approach
was used with the previously developed interface element, which would show
dramatically increased efficiency if used to couple the local and global regions. The
method was studied on different problems using LSDYNA and ABAQUS as solvers and
HEEDS as the optimizing tool. Further, it was run with a number of different Optimizing
schemes available within HEEDS, clearly showing its applicability using any of the other
tools. The developed set of equations help the user to choose the optimal configuration to
run the setup considering practical constraints, guided by knowledge of the problem

being solved.

5.3. Scope for Future Research

As the method is in its infant stage, there is a lot of scope and promise for improvement
and understanding in almost every direction conceivable. Firstly the powerful concept of
injection island optimization could be used inside the local search agent with different

resolution levels to get to better designs and arrive at good boundary conditions faster.

76

Secondly the amount of variation supplied can be intelligently managed depending on the
amount of change in boundary conditions and status of the process. Thirdly the scheme
can be tested for larger problems using multiple processors performing the search at
different levels and with different orders of variation. A study of design robustness could
complement the approach in a number of ways. Finally a fundamental understanding of
the statistical nature of the approach and the concept of design information feedback

mathematically will provide the basis for future improvements in the proposed method.

77

 

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