Linköping University | Department of Management and Engineering

Master’s thesis, 30 credits| Master’s Programme

Autumn 2020| LIU-IEI-TEK-A--20/03904—SE

Design Parameter Identification and Verification for Thermoplastic Inserts

Malhar Shrikrishna Ozarkar

Supervisor: Lars Johansson

Examiner: Jonas Stålhand

Linköping University

SE-581 83 Linköping, Sweden

+46 013 28 10 00, www.liu.se

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Abstract

Inserts are a crucial part of household and industrial furniture. These small plastic parts which

often go unnoticed to the naked eye perform crucial functions like providing a base for the

furniture, leveling the furniture, safeguarding the user from edges of the tubes used and providing

an aesthetic finish. The inserts have a wing like structure on the exterior which enables them to be

inserted and securely held in the tubes. The inserts are assembled into the pipes manually or

through machines. The force required to install these inserts in the tube is called a push-in force

whereas a pull-out force is the force required for removal of the is called a pull-out force. These

forces are experienced by someone who assembles the furniture together. Thus, these forces

directly define the ease with which the furniture can be assembled. In the first part of the present

thesis, these push-in and pull-out forces are predicted using finite element simulations. These finite

element simulations were validated by performing physical assembly and disassembly

experiments on these inserts. It was found that the finite element simulations of the insert are useful

tool in predicting the push-in forces with a high accuracy.

These push-in and pull-out forces for a single insert vary by 2-5 times when the dimensional

variations in the tube are considered. The dimensional variations can be a result of the

manufacturing processes from which these tubes are produced. The maximum and minimum

dimensions that the tube can have are defined by the maximum material condition (MMC) and the

least material condition (LMC). To reduce the variation in push-in and pull out forces, a stricter

tolerance control can be applied to the manufacturing process. To avoid this cost while having a

lower variation in the push-in and pull out forces, the design of the insert was modified. To achieve

this enhanced design of the insert, a metamodel based optimization technique was used in the

second part of the thesis. For this optimization, the geometrical parameters - wing height, wing

diameter and stem thickness the of the insert were identified as the crucial factors which govern

the assembly/disassembly forces. The identification of these parameters was done through a design

of experiments. These parameters were then varied simultaneously in a metamodel based

optimization which had an objective to minimize the variation in forces observed for an insert

when the maximum material condition and the least material conditions are considered. The result

for the enhanced design of the insert was then stated in terms of the ratio of these identified

parameters. The modified design of the insert not only enables the manufacturer to have better

performance, but also reduces the amount of plastic material required for manufacturing of the

insert.

Keywords: Contact modelling, experimental validation, thermoplastic material, metamodel based

design optimization.

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Acknowledgement

Firstly, I would like to sincerely thank Gustav Holstein at Ikea Components AB who has been a

motivation and was extremely supportive throughout the duration of the project. His guidance and

mentorship in the project have been crucial in achieving the results for the thesis. I would like to

thank him for being directly involved in major discussions decisions regarding the project and

giving me the opportunity to apply my learnings to the simulations done in the industry.

In addition, I would like to thank Björn Stoltz for leading many fruitful and interesting discussions

regarding the project. His inputs and experience regarding contact simulations and material

behaviour were valuable and crucial while carrying out the experimentation. I would also like to

express sincere gratitude to Jenny Gurell and Marko Kokkonen at Ikea Test Lab for their insights

and support in the experimentation phase. I would also like to thank Carolina Kroon and Carina

Skov Pedersen for their prompt arrangement of the resources required for the project. I would also

like to extend my gratitude towards the entire design team for the valuable knowledge on the design

and the materials of the insert.

I would also like to thank my supervisor, Lars Johansson and my examiner, Jonas Stålhand at

Linköping University for their advice and guidance. Their constructive feedback on the theoretical

framework have been valuable contributions to the project. I would also like to thank my opponent

Puneeth Ballakkuraya for his thorough opposition and valuable insights on the project.

Last but not the least, I would like to thank my family and friends for their continuous motivation

during the project.

Table of Contents

1. Introduction ..................................................................................................................... 1

2. Methdology ....................................................................................................................... 4

3. Literature Review ............................................................................................................ 6

4. Contact Mechanics .......................................................................................................... 9

5. Metamodel Based Optmization .................................................................................... 14

6. Model Building ............................................................................................................... 17

7. Testing of Inserts ........................................................................................................... 21

8. Results ............................................................................................................................. 25

9. Discussion of Results ..................................................................................................... 29

10. Design of Experiments................................................................................................... 33

11. Optimization of Inserts for Enhanced Performance .................................................. 37

12. Conclusions..................................................................................................................... 45

13. Future Work .................................................................................................................. 46

14. Bibliography ................................................................................................................... 47

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List of Figures

Figure 1: Examples of applications of inserts in household furniture ............................................ 1

Figure 2: Parameters needed to be investigated for their effect on the push-in and pull-out Forces

......................................................................................................................................................... 3

Figure 3: Flowchart of the methodology used for the thesis ........................................................... 4

Figure 4: Conventional design optimization process ...................................................................... 8

Figure 5: Illustration of contact detection zone .............................................................................. 9

Figure 6: Simplified illustration of a contact detection algorithm ................................................ 10

Figure 7: Simplified version of a penalty-based contact ............................................................... 12

Figure 8: Geometries of the inserts used for verifying push-in and pull-out forces ..................... 17

Figure 9: Classification of models based on mesh properties (* indicates applicable to Insert-1

only) .............................................................................................................................................. 18

Figure 10: Overview of boundary conditions used in the simulations ......................................... 19

Figure 11: Example of displacement applied to the pipe in the simulations ................................ 20

Figure 12: Example of difference between stress-strain curves for dry and conditioned PA6

material.......................................................................................................................................... 21

Figure 13: Comparison in geometries of the designed and manufactured insert .......................... 22

Figure 14: Experimental setup for mapping the push-in forces .................................................... 23

Figure 15: Example of set of curves obtained from the experiments for push-in force. .............. 23

Figure 16: Experimental setup for mapping the pull-out forces ................................................... 24

Figure 17: Example of set of curves obtained from the experiments for pull-out force ............... 24

Figure 18: Comparison of simulated and experimental push-in forces for Insert-1 ..................... 25

Figure 19: Comparison of simulated and experimental push-in forces for Insert-2 ..................... 26

Figure 20: Comparison of simulated and experimental push-in forces for Insert-3 ..................... 26

Figure 21: Comparison of simulated and experimental pull-out forces for Insert-1 .................... 27

Figure 22: Comparison of simulated and experimental pull-out forces for Insert-2 .................... 28

Figure 23: Comparison of simulated and experimental pull-out forces for Insert-3 .................... 28

Figure 24: Effect of coefficient of static friction on contact force ................................................ 29

Figure 25: Effect of change in coefficient of kinetic friction on contact force ............................. 30

Figure 26: An example of weld-line present on the inner surface of the pipe .............................. 31

Figure 27: Effect of variation in pipe diameter on contact force .................................................. 32

Figure 28: Example of a forces acting on a single wing of the insert ........................................... 32

Figure 29: Dimensions of the insert used as design variables ...................................................... 33

Figure 30: Effect of variation in wing diameter (DVAR1) on contact force ................................ 35

Figure 31: Effect of variation in stem thickness (DVAR2) on contact force ............................... 36

Figure 32: Effect of variation in wing height (DVAR3) on contact force .................................... 36

Figure 33: Example of difference in forces for MMC and LMC for pipe .................................... 37

Figure 34: Flowchart of the approach used for optimization of the insert .................................... 38

Figure 35: Illustration of peaks and differences in peaks (Diff) for MMC and LMC conditions.

...................................................................................................................................................... .40

Figure 36: Comparison of forces observed in MMC and LMC conditions for baseline and

optimized designs by difference method ...................................................................................... 41

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Figure 37: Comparison of cross-sections of original and optimized design by difference method

.................................................................................................................................................. 42

Figure 38: Comparison of cross-sections of original and optimized design by division method

.................................................................................................................................................. 43

Figure 39: Comparison of forces observed in MMC and LMC conditions for baseline and

optimized designs by division method ..................................................................................... 44

List of Tables

Table 1: Summary of number of nodes and elements used in different types of models ........ 18

Table 2: Tolerances specified by the manufacturer for different components ......................... 31

Table 3: Summary of Design Variables and their limits .......................................................... 34

Table 4: Comparison of the optimization methods used .......................................................... 43

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1 Introduction

1.1 Background

Tube inserts have a wide range of application in furniture, especially as bottoms of legs for chairs

as shown in Figure 1. During assembly of furniture components, these inserts are press-fitted in a

hollow tube. The inserts are designed to be assembled using an interference-fit method and require

a certain amount of force for their insertion and removal, depending upon the design and

application of the insert. The forces during the assembly and disassembly of these products are

crucial as they directly address the effectiveness of the design and their ability to be customer

friendly. Traditionally, these inserts are physically tested to map the forces involved. The

simulation of this assembly-disassembly process is necessary to reduce the costs incurred during

repetitive physical testing of the components. This also enables the manufacturer to provide a

better performance to the customers. Due to the wide range of designs of these inserts, it is desired

to identify the effect of the design parameters on the forces.

Figure 1: Examples of applications of inserts in household furniture (Picture Courtesy: www.ikea.com)

The inserts commonly used in industry are usually manufactured by injection molding and are

made up of thermoplastic materials namely Poly Amide 6 (or PA6, for short) or Polyethylene

(PE, for short) or Acrylonitrile Butadiene Styrene (ABS, for short). The composition of the

insert’s material depends on the application and the intended use of the insert. In cases where a

stiffer insert is needed, fibers are added to the matrix of the above-mentioned materials. These

fibers are made of glass, nylon, cotton, or other materials. By altering the amount of fibers added,

the stiffness of the insert can be altered. In furniture applications, usually glass fibers are added

which typically constitute up to 30% of the volume.

The pipes in which the inserts are inserted are usually made from sheet metal like steel and then

bent and welded to form a hollow pipe. The surface properties of these pipes are often altered by

applying a coating or by painting. This is for aesthetic purposes as well as to provide protection

from corrosive environments.

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1.2 Problem Description

During the assembly of furniture components, the inserts are assembled with the hollow pipes.

This is done manually by the customer or by assembly technicians at the factory. The two parts –

insert and the pipe – are designed using interference fit where the dimensions of the parts have

some overlap between them. This is necessary to secure the insert inside the pipe while in use and

in idle conditions. Due to the interference fit, these parts require a force to be applied for proper

assembly. These forces for installation of the insert and for its removal are called push-in and pull-

out forces respectively, and they vary for different designs of the insert and the application. The

forces during the assembly and disassembly of these products are crucial as they directly determine

the effectiveness of the design and its ability to be customer friendly.

Traditionally, the inserts are physically tested to map the forces involved. The design of the insert

allows a small of amount of deviation in the dimensions. This deviation, also called the tolerance,

affects the dimensions of the insert as well as the tube, which leads to variation in the push-in and

pull-out forces. To avoid repetitive physical testing, finite element (FEM) simulations are

explored. These simulations not only help reduce the costs incurred during testing, but they also

help the designers of these inserts to achieve better performance by simulating the behavior of the

insert under various loading conditions. As the parts consists of plastic materials, these simulations

have a high degree of non-linearity in terms of deformation behavior and the contact conditions.

Thus, to replicate the behavior of the insert during the assembly-disassembly, it is important that

the contact between the insert and the tube is simulated correctly.

1.3 Contact Between the Insert and the Pipe

The interaction between the insert and the pipe during the assembly/disassembly operations results

in a contact between the interacting surfaces. This contact or interaction of the parts results in a

force which is required to be overcome for continuous sliding of the insert inside the pipe. The

contact forces are made of normal pressure exerted by the insert and tangential components known

as frictional components. As the material of the insert (plastic) is significantly softer than that of

the pipe (steel), the contact results in deformation of the inserts and a change the area of the contact.

Thus, to simulate the assembly/disassembly operation, it is important that these forces are

determined accurately to replicate this process through finite element methods.

1.4 Project Objectives

The objectives of the present thesis are:

i. Verification of push-in and pull-out forces: Corelating the forces from the finite

element simulations to the forces observed during physical experiments or testing.

ii. Geometrical parameter identification and optimization: The effect of the geometrical

dimension (parameters) of the insert will be investigated and the design of the insert

will be optimized to reduce the difference in recorded forces for the maximum material

condition and least material conditions of the pipe.

Through these stages, the thesis will address the following questions:

• What parameters and contact properties of the FE-model are needed to obtain pull-out and

push-in forces which agrees with the experimentally obtained values?

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The parameters in this phase of simulation consist of different contact conditions, mesh

densities and method of load applications. These factors are major influencers on results of

the simulation. The conditions needed to replicate the physical experiments through the

FE-model will be determined largely on these parameters. The contact properties as stated

in the problem statement depend on the static and dynamic friction coefficients of the

interacting materials. The thesis intends to find the correct parameters needed for such an

optimization.

• Which geometrical parameters of the design of the insert have a profound effect the forces

experienced during the installation and removal of the insert? What causes the variation

of these forces and how can this variation be reduced?

Figure 2: Parameters to be investigated for their effect on the push-in and pull-out forces

The geometry of the inserts varies on the type of application where it is used. The important

features of the geometry of the insert in contact with the pipe are firstly, the wing tip

diameter(∅𝐷𝑤), the rib height (h) as shown in Figure 2 and secondly the assembly

tolerance. These parameters alter the contact area of the insert with the pipe and in-turn the

force required for assembly/dis-assembly. Also, the stem diameter (∅𝐷𝑠) and the wing

base diameter (∅𝐷𝑏) which contribute to the wall thickness of the insert stem which

provides the radial stiffness to the insert. Thus, it is crucial to examine the effect of these

parameters on the push-in and pull-out forces to optimize the insert.

1.5 Environmental and Sustainability Considerations

As mentioned earlier, the thesis deals with finding out the correct way of simulating the actual

tests carried out on the inserts and finding the correct geometrical dimensions in order to enhance

the performance of the insert. Thus, no direct impact on the environment are expected. In turn,

having an optimum design of the insert may result in a reduced rejection of these plastic parts as

the performance is improved.

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2 Methdology To achieve the desired outcomes of the thesis, the work is divided into several phases. These phases

are defined in such a way that through each of them, a desired outcome is stated and the outcomes

from each phase are needed as in input for the subsequent phases. An overview of this approach

is shown in Figure 3

Figure 3: Flowchart of the methodology used for the thesis

2.1 Literature Review

Due to the complex nature of the problem and due to company policies regarding privacy of

research, the material available is limited. Nevertheless, the material available for similar problems

in the different phases of this thesis will be reviewed to reach an understanding of theory and the

approach behind the problem. This phase helps in laying a foundation for the thesis.

2.2 Model Building and Preliminary Simulations

This phase mainly consists of discretizing the geometry of the insert (meshing) to setup the

constraints and boundary conditions required to simulate the load cases. For the simulation results

to be accurate, the mesh of the insert needs to be sufficiently converged, i.e. further refinement of

the mesh should not significantly affect the results. Also, during this phase, various forms of

elements will be tested to observe their response to the contact conditions required for the problem.

As the contact between the insert and the pipe is dependent on the mesh, a robust method that

handles the variation in the number of elements and the element formulation is needed. This phase

aims at finding a way to develop a robust model which meets these requirements. In these

simulations, a perfect geometry was considered which did not consider the differences in

dimensions due to tolerances. These conditions are simulated for preliminary results prior to

physical experiments.

2.3 Experimentation

To map the forces that occur during the actual assembly/disassembly of the insert in the pipe, an

experimental phase is included. The process of recording the forces will be done using a universal

testing machine on which a force transducer is set up. From the experiments, a force-displacement

curve will be obtained for each insert design. Moreover, to reduce the error in measurements and

to obtain an average curve for each of the insert designs, a sample set consisting of multiple

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samples of each design was tested. This phase provided real life force values which could be

compared to the preliminary simulation results. In this phase, the deviation in dimensions from a

perfect geometry are also measured. In case of a water absorbing material like PolyAmide-6, the

samples were tested for dry and conditioned states of the inserts. Due to machine limitations, it

was not possible to measure the coefficients of friction (static or dynamic).

2.4 Simulations to Obtain Push-In and Pull-Out Forces

After the experimental phase, some of the sources of errors that were observed during the testing,

like the actual dimensions of the components and the rate of loading were included into the

simulation models. As the friction properties of the interacting surfaces is not known, the exact

contact conditions could not be simulated through the preliminary simulations and due to the

limitations in measuring the actual friction coefficients, the coefficients were obtained from a

literature review of similar interacting materials. Through these simulations, force vs displacement

curves were obtained for different mesh densities and element types which were compared with

the experimental curves

2.5 Geometrical Parameter Identification and Enhancement

In this phase, the geometrical parameters that affect the push-in and pull-out forces in the

simulation and experimentation phase are identified. The effect of these identified parameters on

the contact forces is then investigated. These geometrical parameters are later modified in such a

way that for Maximum Material Conditions (MMC) and Least Material Conditions (LMC) of the

tube, the difference in the assembly forcers observed is minimized. Thus, the outcome of this stage

will be a modified design of the insert, the assembly forces for which have a reduced dependency

on the tube dimensions.

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3 Literature Review This section discusses previous research carried out for similar problems as well as their limitations

and the things that can be learned from previous research.

3.1 Finite Element Modelling

In the area of finite element modelling, there are several different elements which differ in their

configuration and the mathematical model from which the stresses and strains are calculated. The

elements can be divided into 1-D, 2-D and 3-D elements based on the number of dimensions the

element has. As the insert geometries have solid sections, they are commonly modelled using

hexahedrons and tetrahedrons. The elements can be further classified based on the order of the

shape function they have as linear, quadratic, or cubic. Linear solid elements are comparatively

simple elements and are often an extension of linear plane elements like a linear quadrilateral or

linear triangular elements. Tetrahedron elements are commonly used for meshing of complex

geometries like foam parts, cast components etc. In case of large bending, the linear tetrahedral

elements have poor performance and often exhibit shear locking. In such cases, quadratic elements

which have a higher degree of the shape function and can be used to simulate such problems with

better accuracy (Cook et al. 2001).

However in cases where the geometry is modelled using smaller elements, the use of quadratic

elements leads to increased computational costs as the number of nodes doubles and because of

the computation of second derivatives of element in order to simulate their behavior (Jansen 1999).

In case of modelling, contradictory research exists. Some research suggests that the use of

hexahedral elements provides more accuracy compared to the use of tetrahedral elements if the

order of elements used is the same (Bussler and Ramesh 1993), whereas some suggest that the use

of bi-linear hexahedral elements provide similar results when compared to quadratic tetrahedral

elements in terms of accuracy and cost (Cifuentes and Kalbag 1992). When small deformations

are expected, the use of both linear hexahedral and linear tetrahedral elements are sufficient to

capture the behavior (Benzley et al. 1995). In some insert designs, the cross section of the insert

has cyclic symmetry. In such cases, the geometry can be modelled using axi-symmetric elements,

which reduces the computational costs significantly (Hüyük et al. 2014).

Apart from their configurations, elements can be classified based on the numerical integration

methods used to calculate the element stiffness matrix as fully integrated or under integrated

elements. When fully integrated elements are used, the computation costs can be high (Livermore

Software Technology Corporation 2006). In order to save computational costs, under-integrated

elements are used. This also helps in reducing the volumetric locking which is observed in fully

integrated elements. However, this reduced integration technique can induce zero-energy modes

in the elements or ‘hour-glassing’ and to overcome this modes, hourglass control is needed

(Schultz 1985).

Thus, from the above research, it could be concluded that in cases of large deformation, where the

behavior of the element is relatively unknown, the type of element to be used along with the

formulation and integration technique needs to be carefully reviewed in order to tailor it to the

specific model requirements.

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3.2 Contact Modelling

As the deformation of the insert and the force observed during its installation is dependent on its

interaction with the pipe, the contact area for undeformed and deformed shape needs to be

modelled correctly in FEM to obtain accurate results. The field of contact modelling has gained

increased focus over the last few years as more and more industries demand robust contact

algorithms which are stable and capable of handling material and geometrical non-linearity

(Rodriguez-Tembleque and Aliabadi 2018). Contact friction models based on adhesion theory of

friction have been developed and are supported by the experimental measurement of static and

dynamic friction coefficients (Manninen et al. 2007). In FEM, there are two major types of contact

algorithms - constraint-based contact and penalty-based contact. In constraint-based contact, a

kinematic set of constraints is created between the two interacting surfaces and these constraints

are checked at every time step in the simulation. If the constraints are being violated, a reaction

force is applied at the nodes to the elements to satisfy the constraint requirement. The more widely

used method, the penalty method, allows for minute penetrations observed in the contacting

elements. This makes the penalty method economical and useful for solving problems pertaining

to contact friction (Stefancu et al. 2011). The main difficulties in modelling a penalty based

contacts are that parameter tuning is required and it does not treat simultaneous contact well

(Rengifo 2009). If a large penetration is observed, the contact area will be altered and will also

have an effect on the contact forces observed (Bay and Wanheim 1976). Moreover, due to the way

the penalty method works, there will always be small penetrations between the pipe and the insert

which will minutely alter the real dimensions of the tube and the contact area between them. Thus,

it is necessary to limit the penetrations in the simulation.

In finite element methods the implementation of the contact algorithms can be classified be broadly

classified into two stages:

i. Contact detection: Here the finite element program establishes if the interacting

surfaces are considered as ‘in contact’ or ‘not in contact’

ii. Contact treatment: Here, if the interacting surfaces are in contact, a method or

algorithm is specified by the program which defines how the contacting surfaces will

be handled in the simulation.

The area of contact mechanics will be discussed in detail in Section 4

3.3 Optimization

A traditional approach for optimizing a design shown in Figure 4. The design of a

component is modified based on the results of the simulation by the designers and the new

designs are simulated again to learn more about their behaviour. This iterative process

incurs recurring costs to the manufacturer. However, this process can be reduced by

carrying out continuous simulation through optimization techniques. In recent times, due

to advancements in computational capacities, the finite element simulations are coupled

with mathematical optimization algorithms. Through these optimization algorithms, it is

possible to tailor the design of the insert to the required response. In such optimization

calculations, one or more sets of design parameters are varied to either minimize or

maximize an objective function. The objective function is most often expressed in terms of

a response which is obtained from the finite element simulations. When several parameters

in the design are varied, it is essential to identify the effect of each parameter varied and

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then achieve optimal design for given conditions. To explore multiple design parameters

simultaneously, a multi-parameter optimization is used. One technique for such multi-

parameter optimization is called a metamodel based design optimization (MBDO). This

technique allows numerical optimization to be applied even to compute complex problems

and thus reduce the computationally intensive conventional process of design re-iteration

(Wang and Shan 2007). The metamodels or surrogate models are created using the

response obtained from a robust simulation and input datasets for the varied parameters.

The process of placing the design points in space is called design of experiments. The

surrogate models created compute the response for variation in datasets (Ryberg et al.

2012). A detailed description of the working of such an optimization is given in Section 5.

Figure 4: Conventional design optimization process

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4 Contact Mechanics This section describes the typical implementation of contact mechanics in a finite element

program. The section is divided into two sub-sections - Contact Detection and Contact Treatment.

Under the treatment of contacts, the various methods that are used in handling contact phenomena

are described along with different discretization techniques for the contact.

4.1 Contact Detection

To understand how contacts are treated in a finite element code, it is essential to understand the

conditions that define a contact phenomenon. These conditions are needed to invoke the part of

the program which specifies the treatment of the contact. Consider two arbitrary bodies Ω1 and Ω2

in space as shown in Figure 5. The position of the bodies in space is known from their co-ordinates

with respect to a global co-ordinate system. This position of the body in a simulation is generally

defined using a design tool. In continuum mechanics, it is assumed that two distinct points cannot

occupy the same position in space simultaneously (Gonzalez and Stuart 2008). In case of distinct

bodies in space, no bodies can penetrate each other (Heinstein et al. 1993). If there are two or more

elements from two different bodies are intersecting each other at the start of a finite element

simulation, an initial penetration is said to occur. In such conditions, the contact algorithm may

not work.

Figure 5: Illustration of contact detection zone

The bodies Ω1 and Ω2 obtained from a design software (CAD) are discretized using a pre-processor

into elements which are connected by nodes. To detect the contact, each of these points or nodes

on body Ω1 is projected on to body Ω2 and the normal distance to the body form the point is

calculated. If this normal distance from the body, nd is less than a specified value, dmax a contact is

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said to occur. The conditions for maximum distance and the normal traction force acting on a body

are written as inequality constraints (Kloosterman 2002):

𝑛𝑑 ≤ ⅆmax (1)

𝑡𝑛 ≤ 0 (2)

𝑡𝑛 𝑛𝑑 = 0; (3)

where 𝑡𝑛 is the traction component in the outward normal direction.

Along with the constraints in Equations 1, 2 and 3, frictional constraints are also applicable in

simulations where friction is considered. Different finite element solvers use different friction laws

which are governed by the displacements or velocities of the contacting bodies. In this thesis, the

friction law used in the LS-DYNA solver is:

𝜇 = 𝜇𝑠 + (𝜇𝑠 − 𝜇𝑘)𝑒−𝐷𝐶 ∣𝑣𝑟𝑒𝑙∣; (4)

where, 𝜇𝑠 is the coefficient of static friction, 𝜇𝑘is the coefficient of kinetic friction, DC is the

damping coefficient and 𝑣𝑟𝑒𝑙 is the relative tangential velocity between the contacting surfaces.

Equation 1 creates a constraint for the contact to be initiated. This specified distance dmax, creates

an envelope of a detection area in the vicinity of the body. Equation 2 states that the normal traction

acting on the body should be compressive whereas Equation 3 states a complementary condition

for impenetrability. A simplified version of such an envelope and example of contact detection is

shown in Figure 6. It should be noted that the distance of this envelope is negligible compared to

the size of the contacting bodies and its discretized parts. If the nodes of one such body are in this

envelope, a contact is established between these two bodies. In this example, the detection of the

nodes of is considered through one arbitrarily chosen body. However, in contact algorithms, the

detection of these nodes is based on the contact discretization selected.

Figure 6: Simplified illustration of a contact detection algorithm

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4.2 Contact Discretization and Types

When contact is detected between two bodies, the contact algorithm kicks in. In the LS-DYNA,

contact algorithm, the contacting surfaces are termed as a master surface and a slave surface. In

most common contact algorithms, for the detection of contact, the penetration of the nodes on the

slave side are checked at the master side. That is, only the master body will have an envelope

around it to detect this contact, and if the slave body is inside this envelope, contact is said to occur.

The contact algorithms can be classified based on how the master and slave sides are defined. This

discretization of the contact is based on the elementary units which transmit the stresses from one

contacting surface to the other. Typically, these elementary units can be:

a. Nodes,

b. Edges

c. Segments (Face of the element)

Based on these units, the contacts between the master and slave sides are classified as:

i. Nodes to segment

ii. Nodes to node

iii. Edge to segment

iv. Edge to edge

v. Segment to segment or surface to surface

The node to node discretization passes Taylor’s patch test, which is a used to assess the robustness

of the contact condition (Taylor and Papadopoulos). Due to this robustness, the node to node

discretization is comparatively simple to implement. However, this method requires the mesh on

both the contacting bodies to be approximately of the same size and the simulation is limited to

small slip and deformation conditions (Kikuchi and Oden 1988). To handle large deformations

and slip, the node to segment contact was developed which is independent of the mesh density

used in the contacting bodies (Bathe and Chaudhary 1985). These contact algorithms work well

with simple geometries, but a drawback is that they do not pass Taylor’s patch test. A segment to

segment based contact is typically used for modelling contact behavior of soft materials (Mayer

and Gaul 2007). Any of the above contact algorithms can be optimized by inclusion of treatment

of segments in the contacting bodies, for example, a node to surface contact algorithm can be

improved by including an algorithm for the contacting segments. The segment-to-segment contact

provides a robust solution for most of the contacting conditions. In this type of contact, the

contacting surfaces are discretized using the faces of the elements modelling these bodies. As a

result, this method has a better ability to handle the mesh differences in the bodies.

As mentioned earlier, the contact formulations can also be divided into kinematic constraint-based

contact and penalty-based contacts. In the penalty-based formulation, the penetration between the

two interacting surfaces is detected and then this penetration is minimized by applying a force to

remove the penetration between the interacting elements. This force is applied by placing virtual

springs which are normal to the interacting surfaces. A simplified diagram of this type of contact

algorithm is shown in Figure 7, where the force required to minimize the penetration is provided

by an imaginary spring with stiffness 𝑘𝑐𝑜𝑛𝑡𝑎𝑐𝑡. The contact forces are then calculated using:

𝐹𝑐𝑜𝑛𝑡𝑎𝑐𝑡 = 𝑘𝑐𝑜𝑛𝑡𝑎𝑐𝑡𝑥, (5)

12

where, 𝑥 is the penetration depth. If solid elements are used for modelling the contacting interfaces,

the contact stiffness used in LS-DYNA is given by (Livermore Software Technology Corporation

2005):

𝑘𝑐𝑜𝑛𝑡𝑎𝑐𝑡𝑠𝑜𝑙𝑖𝑑 =

(Sf𝐾𝐴2)𝑉

⁄ , (6)

where, Sf is the stiffness scale factor, K is the bulk viscosity of the solid element, A is the contact

segment area and V is the element volume.

If shell elements are used for modelling the contact interface, the stiffness used is related to the

characteristic length (L) of the element as (Livermore Software Technology Corporation 2005):

𝑘𝑐𝑜𝑛𝑡𝑎𝑐𝑡𝑠ℎ𝑒𝑙𝑙 =

(𝑆𝑓𝐾𝐴)𝐿

⁄ , (7)

Figure 7: Simplified version of a penalty-based contact

As mentioned earlier, the contacting bodies are termed as master and slave bodies. Usually, the

selection of the slave body is done in such a way that its stiffness is less than or equal to that of the

master side. Traditionally, the contact is detected by the master body by searching for the nearest

point on the slave body. This method is called the one-way treatment of contact. However, in case

of large deformations, this may lead to an unwanted penetration of the slave body into the master

body as this condition is not checked for. To check this condition, two-way treatment of contacts

is needed. In such a case, the selection of the master and slave sides becomes redundant as both

the bodies are checked for penetration into each other.

As the penalty-based contact work with nodes contacting the master and slave nodes or segments,

there is a high dependency of the element type used in the contacting surfaces. Due to this, this

method has drawbacks like poor contact in deformable bodies, failure in general patch tests, and

13

reduced convergence rate and low robustness (Yang et al. 2004). To improve these conditions, two

numerical approaches can be added to the traditional penalty-based contact. Firstly, a soft

constraint method, where the excess penetration is removed by the calculation of an additional

stiffness based on the Courant-Freidrichs-Lewy stability conditions in such a way that the contact

stiffness is given by (Baranowski et al. 2006):

𝑘𝑐ontact =1

2𝑆𝐹𝑠𝑜𝑓𝑡𝑚∗ (

1

Δ𝑡𝑖) (8)

Where, 𝑚 ∗ is function dependent on the master and slave side nodes and ∆𝑡𝑖 is the initial timestep

for the simulation.

A second method for improving the standard penalty-based contact is by adding an algorithm for

the detection of segments in the contacting bodies. However, neither of these methods guarantee

a solution for any contact conditions. To have a robust contact algorithm which can handle a wide

range of materials and geometries a so-called mortar formulation for the contact was developed.

This formulation involves the decomposition of the domains of the contacting parts (Belgacem et

al. 1998). It can also be classified as an advanced segment-to-segment penalty-based contact. This

method couples the different variational approximations for interacting bodies and is a relatively

new method for numerical approximations. In the domain decomposition, large computational

problems are broken down into several smaller size problems. In contact problems, this method

divides the element boundaries into different sub-domains (Nikishkov 2007).The decomposition

of the domain is implemented with the help of a direct method or a iterative solver. As a result of

advances in computational capabilities, an active strategy was derived by applying a semi-smooth

Newton method. This strategy is known as primal-dual active set strategy or PDASS (Hintermuller

et al. 2003). This strategy was further developed by carrying out a linearization of the normal and

frictional contact forces (Gitterle et al. 2010). This linearization results in so-called ‘mortar

surfaces’ at the contacting element boundaries. Due to the discretized nature of these surfaces, the

contact kinematics are relatively easily evaluated as compared to traditional segment-based

contacts. Through this discretization, it is possible to apply the continuity conditions of the contact

in a more efficient way. This provides an accurate modelling of the contact conditions (McDevitt

and Laursen 2000).

Through these mortar surfaces, the transfer of contact stress is facilitated. The contact stresses

observed are given by (Borrvall 2012):

𝜎𝑐 = 𝑆𝑓ε𝐾slave𝑓 (x

𝜀𝑑𝑐) (9)

where 𝑆𝑓 is the scale factor for the contact stiffness, 𝐾𝑠𝑙𝑎𝑣𝑒 is the stiffness modulus of the slave

side segment, x is the penetration depth, 𝑑𝑐 is the characteristic length of the element used for

modelling the body and 𝜀 is a constant (= 0.03).

Previously, this method was widely used in implicit calculations as it is an accurate and robust

contact algorithm, capable of handling geometrical and material non-linearities, which led to

higher likelihood for convergence. But due to advances in computational capacities, this method

is now being used for explicit simulations as well. This procedure is computationally expensive

for implementation in explicit simulations involving large problems (Livermore Software

Technology Corporation 2006).

14

5 Metamodel Based Optimization This chapter discusses the approach of metamodel-based optimization used in this thesis. When a

component or a product is in the development stage, it is often needed to find out the correct

geometrical parameters that achieve an optimal performance for its design. If the finite element

simulations are performed for these models, then it is possible to achieve this optimal design by

manually changing the required geometrical parameters and then simulating the design again based

on the behaviour of the insert. Such a process can be called as an iterative optimization. However,

this process can be repetitive and expensive as it increases the computational cost significantly and

affects the efficiency of the development process.

Moreover, when complex models and multiple geometrical parameters are involved, this iterative

analysis can be difficult. In such cases, a metamodel based optimization can be performed. This

technique can significantly reduce the computational time and cost required to find the optimal

design parameters. In this process, the repeated analyses are replaced by simpler lower order

models. These simpler models or metamodels can be very useful in optimization problems which

involve multiple objective and constraints. Through this technique, not only an optimization is

achieved but it can also help in understanding the problem formulation for multiple variables and

also predict the value of outputs of design variables (Bonte et al. 2005).

This chapter is divided into 4 stages essential stages in an optimization algorithm: sampling of

variables, building of metamodels and domain reduction.

5.1 Sampling of Design Points

For an optimization, a set of design variables need to be tested to find their optimum values. The

design variables can be a set of continuous points between two given limits or can be discrete value

based on the design values. For example, if the upper and lower limits of a certain design space

are 1 and 4 respectively, the discrete values can be (1.0, 1.2, 2.5, 2.8…) and a continuous set of

points can be (1.00, 1.01, 1.02, 1.03…).A continuous set of points is simulated to explore the entire

design space. While using such an input, the random design points are used to simulate the load

case and record the response. The selection of design points from a design space is called sampling.

Various algorithms are available in commercial FE-programs which select pseudo-random points

from the design space provided. Some of the widely used sampling methods are:

i. Monte Carlo Method: This method is an adaptation of the direct Monte Carlo method

to a metamodel based optimization. In the direct Monte Carlo method, the sampling is

based on random variables an no meta model is built. In case of the adapted method,

the sampling is done based on metamodels and it is based on the probability distribution

of the variables. Through this technique, the different possibilities of the values of the

design variables occurring are established and then, a random set of values for these

design variables is selected from the design space. The results from the optimization

are then saved to the corresponding design points so that the results or responses can

be used in later processes for statistical modelling (Paltani 2010).

ii. Latin Hypercube Method: The Latin Hypercube sampling method is used for

improving the sampling efficiency of the sampling problem. This method is superior to

the Monte Carlo methods (Olsson et al. 2003). The increased efficiency of this method

is due the joint probability density function that are constructed for the problem

(Huntington and Lyrintzis 1998). The method reduces the design space of each of the

15

variables into smaller spaces based on the probability distribution. Then a single value

is chosen from each of these intervals and then paired with random values of the other

design variables to form randomly unique design points.

iii. Space Filling: Space-filling algorithms are modifications of the traditional latin

hypercube method. For a good optimization problem, it is essential that the design

points are random but cover the entire range of the variables being tested. Space-filling

algorithms are useful tools for replicating such conditions. They do not impose strong

constraints on the model and thus allow different number of subsets for each variable

(Stander et al. 2019). Space filling sampling schemes work by maximizing the

minimum distance between two design points in the design space. This kind of

algorithm is being widely used as a basis to set up different metamodeling techniques

like neural networks and radial basis functions.

iv. D-optimal: This method is used to generate model-specific design points. This is done

by selecting the best set of points for the response surface from a given set of points.

This method is useful when having several variables with a large design space. The

algorithm for the d-optimal design is based on iterative search algorithm and have the

objective to maximize the determinant 𝐷 = |𝑋𝑇𝑋| , Where X is the design matrix of

model terms evaluated in specific design spaces (Tu and Choi 1999).

5.2 Metamodeling Techniques

Once the sampling of the design variables is completed, the simulations are run to find the

responses for each of the set of design variables. Using these design points and the respective

responses, metamodels are constructed. By using metamodel based methods, the simulation time

is significantly reduced which makes such methods favorable to use. A few ways these metamodels

can be built are explained in short below.

i. Polynomial: In this method, the metamodels are based on first order or second order

polynomials which can be linear, elliptical, or quadratic. When the order of the polynomial is

increased, the number of unknown coefficients to be determined increases as the number of

terms in the polynomial are increased. Increasing the order results in better accuracy but also

requires additional computational time. The polynomials are used to determine the significance

of the design variables on the responses from the simulation.

ii. Neural Networks: This method is an extension of the linear regression methods. Neural

networks are series of algorithms of these linear regressions with the objective to recognize the

relationships between different design variables and responses. These algorithms adapt to

changes in input and thus the algorithm can optimize to find the best solution without a need to

re-simulate based on the progress in iterations. The algorithms learn more and more about the

nature of the responses by reducing the mean square error between the predicted and computed

responses. However, the results from such an algorithm are dependent on the input of the design

variables chosen and thus can lead to unexpected errors. Neural networks can be of two types

– feed forward network or the radial basis function network. While feed forward networks use

non-linear regression, they tend to cost more compared to radial basis functions which use linear

regression.

iii. Kriging: Kriging is a gaussian process regression which uses weights functions on design

points to predict the joint probability distribution. This method is based on empirical

observations from the response surface, which makes it more useful in generating estimated

response surfaces. The main advantage of this technique is that it generates the uncertainty of

each design point in the model (Remy et al.). This method is highly dependent on the correlation

16

of the spatial design points and is used mostly as a prediction tool to find a response to a given

set of design points.

5.3 Domain Reduction

This stage of the optimization is used to reduce the design space in successive iterations of the

simulations. The method used to reduce the domain is known as ‘Sequential Response Surface

Methodology (SRSM). This method uses information gained from previous iteration to design the

subsequent iteration. This is done by identifying the region of interest from the given design space

to form a sub-space based on the approximate optimal solution of each iteration. This makes the

optimization more efficient as the area of design space is reduced. If the distance between the

starting design point and the predicted design point is smaller, the region of interest is reduced

rapidly (Stander et al. 2019). If the optimum point lies outside the region of interest, then the region

of interest remains unchanged for the next iteration in the simulation. The movement of the

reduction of the design space is based on two factors - contraction and zooming parameters- which

are designed to prevent premature convergence. The design spaces created strictly use a d-optimal

algorithm to create new design points (Stander and Craig 2002). When using coarse convergence

criteria, this method provides good accuracy when compared with direct-optimization techniques

and it is also more stable and has robust convergence characteristics.

17

6 Model Building This section discusses the model building phase of the thesis. For pre-processing (meshing and

applying boundary conditions), Hypermesh from Altair was used while for model building, the

solver template for LS-DYNA was used.

6.1 Model Geometry and Properties

The area of concentration of this thesis is the wings of the insert and its surrounding areas.

Therefore, the part of the insert which is distant from these wings was not considered in the mesh

models in order to reduce the computational costs by reducing the number of FE-entities (nodes

and elements). It is noted that the geometries of the insert are symmetrical in at least 2 planes and

Insert-1 has cyclic symmetry. Hence, only a quarter portion of the insert was modelled, and

symmetry conditions were used as described below. The geometries of the insert that were used in

the verification of the pull-in and push-out forces are shown in Figure 8 along with the modelled

part.

Figure 8: Geometries of the inserts used for verifying push-in and pull-out forces

The inserts were modelled primarily using hexahedral elements while pentahedral elements were

used where sharp corner was observed. The mesh models were divided into three categories - fine,

medium, and coarse to check the effect of mesh densities on the contact forces of each insert as

shown in Figure 9. In case of Insert-1, an additional model using axisymmetric elements was

18

created. Table 1 summarizes the number of FE-entities present in each model for the 3 designs of

the insert.

Figure 9: Classification of models based on mesh properties (* indicates applicable to Insert-1

only)

For hexahedral and pentahedral elements, different formulations were tested for the boundary and

contact conditions. Through trial-and-error, it was found that the element formulation ‘Fully

integrated selectively reduced (S/R) elements (Livermore Software Technology Corporation 2017)

intended for poor aspect ratios worked best for all conditions of mesh and contact for hexahedron

elements. These elements result in improved behavior over conventional fully integrated elements

by reducing the transverse shear locking observed in the later formulation of elements. For

pentahedron elements, automatic sorting was used which sorted hexahedrons to the above element

formulation and pentahedrons to 2-point integrated formulation.

In case of the model for the pipe, the hexahedral elements were used. As the pipe was considered

rigid, and deformation of the pipe and the stresses observed on the pipe were not calculated in the

FE program. Hence, it was possible to reduce the calculation times and costs. As the geometry of

the pipe was fairly simple, tetrahedral elements were not used.

Table 1: Summary of number of nodes and elements used in different types of models

Nodes Elements Nodes Elements Nodes Elements

Fine 150944 119390 106358 88743 130538 110762

Medium 82685 56262 68103 56900 60074 51632

Coarse 10952 7172 29442 19662 20364 16580

Tetrahedrons 9076 22168 25168 75432 18820 54898

Axisymmetric

Elements1324 1029

Mesh

Density/Type

Insert-1 Insert-2 Insert-3

Not Applicable

19

Figure 10: Overview of Boundary Conditions used in the simulations

6.2 Initial and Boundary Conditions

6.2.1 Constraints and Loading:

As only a quarter portion of the insert was modelled, appropriate constraints were used at faces of

the cross section of the insert. Single point constraints (SPCs) were used for this purpose to restrict

the movement of the nodes in these respective symmetry planes. The insert position was fixed in

space with the help of SPC-constraints at the bottom face of the insert which constrained the nodes

in all directions. An overview of the boundary conditions is shown in Figure 10. To simulate the

motion during assembly and dis-assembly, displacement was applied to the pipe using a

displacement-time curve as shown in Figure 11.

6.2.2 Contact Conditions:

Initially, the well-established penalty-based surface to surface contact was tested using nominal

values of friction coefficients. It was observed that for coarse meshes, large amounts of

penetrations were observed. As it was important to have minimal penetration in the contacting

surfaces, the stiffness scale factor for the master side elements was increased. Even though the

penetration observed by changing this scale factor was less, it was observed that the scale factor

was dependent on the area of the contact and the mesh size. Thus, different scale factors were

needed for different insert designs and mesh densities. It was noted that by using a mortar contact

formulation, this parameter tuning for scale factors was reduced, thereby reducing the number of

variable parameters in the contact algorithm. With the use of such a contact algorithm, it was

possible to obtain a reasonable behavior of the insert under heavy deformation. For the preliminary

simulations, the values of static friction and dynamic friction were both set to 0.2 as per standard

practices for preliminary simulations for the manufacturer. After the inserts were tested, the

20

diameter of the tube was modified as per the observations during the testing phase. Also, the

friction parameters were set to 0.4 and 0.1 for static and kinetic friction coefficients, respectively

as observed from the literature (Lancaster 1968) and design handbooks (American Institute of

Physics 1972).

Figure 11: Example of displacement applied to the pipe in the simulations

6.2.1 Material Parameters:

As discussed above, the pipe was considered as rigid, the material assigned to the pipe was steel

using the MAT020 material card in LS-DYNA. Nominal values of Young’s Modulus (207 GPa),

density (7890 km/m3) and Poisson’s ration (0.3) were used for the purpose of mass calculations.

For the insert, a linear elastic-plastic material was assigned using the MAT024 material card in

LS-DYNA. The stress-strain response for this material was obtained from the material testing and

material database from the manufacturer. For confidentiality purposes, the data for material data

for the thermoplastic insert is not provided in the current thesis. However, a representative data for

the thermoplastic of a similar type is shown in the subsequent sections.

21

7 Testing of Inserts In this phase, the inserts were physically tested to measure the push-in and the pull-out forces. The

experimental phase can be divided into 3 stages, which are explained in the subsections below.

7.1 Preparation of Samples

This procedure is applicable to inserts made of PA-6 material i.e. Insert-2 and Insert-3, because of

their inherent capacity of PA-6 to absorb moisture. The inserts produced are transported by various

means for a period of 2-3 weeks. Hence, environmental factors affect the moisture content in these

inserts. The insert is classified as a dry sample if there is no water absorption whereas it is classified

as a conditioned sample if the water absorbed by the material of the insert is maximum. A

comparison of the of the stress-strain curves is seen in Figure 12, where a significant difference in

the Young’s modulus and the yield limit is observed for dry and conditioned states of the material.

Thus, while testing, it is important to determine the state of the samples to know the correct

material parameters for the simulation. Before placing the samples in the climate chamber, the

baseline mass of the samples was noted, and the procedure stated in ISO 62:2008 (International

Organization for Standardization 2008) was followed. To measure and remove the absorbed water,

the insert samples were place in a climate chamber at 50° C for a period of one week and weighed

every 2 days. This standard states that the samples can be considered as completely dry if the

change in mass between two successive weight measurements is less than ±0.001 grams. At the

same time another set of samples was kept in a controlled climatic condition of temperature 70⁰

Celsius and humidity of 62% relative humidity for a period of 4 weeks. Like the dry samples, the

baseline mass of the samples was noted and the procedure per ISO1110:2019 (International

Organization for Standardization 2019) was followed. This standard states that the samples can be

considered as completely conditioned when the change in the mass of the samples for 2 successive

measurements during the climatization is less than ±0.1 grams. Both the conditioned and the dry

samples were then scanned as described in Section 7.2 and tested as described in Section 7.3 and

Section 7.4 below.

Figure 12: Example of difference between stress-strain curves for dry and conditioned PA6

material

22

7.2 Measuring the Samples

As the samples are made from injection molding, there are possible deviations in the manufactured

components as the mold cannot have sharp edges unlike in CAD models, i.e., they will always

have a small radius in place of sharp corners due to manufacturing constraints. In addition to this,

the contraction of the manufactured part depends on the mold temperature and the material liquid

temperature and the moisture content in the material. These parameters are difficult to control

during the manufacturing process which result in the component deviating from the ideal

geometry. To measure these deviations, the insert samples were scanned in a 3D scanner. As shown

in Figure 13, the actual geometry of the insert varies from the ideal design (CAD design) by a

substantial amount in the region of interest. The color contour plots indicate deviation in

geometries of the designed and manufactured part. This deviation can lead to a change in the

contact area of the insert resulting in a higher push-in and pull-out forces. In addition to this, the

diameters for the pipes in which the inserts are assembles were measured. As the pipes were not

perfectly circular, the average diameters were calculated based on 5 readings of each of the pipes.

4 such pipes were used to simultaneously test multiple inserts.

Figure 13: Comparison in geometries of the designed and manufactured insert

7.3 Testing for Push-in Force

To test the push-in forces, an Instron 5944 Universal Testing machine was used along with a load

cell having an upper limit of 2000 Newtons. The resolution of the load cell was 0.02 Newtons and

an accuracy of ±0.5% of the reading. The forces were reported at time intervals of 60 milliseconds

during the test duration. The pipe was placed in a vice grip and the insert was placed lightly on the

top opening of the tube as shown in Figure 14. The UTM machine was then started and the insert

was pushed inside the tube with the help of an extension arm. A set of force vs displacement graphs

were obtained from the test as shown in Figure 15. From this set of graphs, an average graph was

obtained for each insert which was then compared to the corresponding curves from the

simulations.

23

Figure 14: Experimental setup for mapping the push-in forces

Figure 15: Example of set of curves obtained from the experiments for push-in force.

7.4 Testing for Pull-out Force

After testing the samples for the push-in force, the pipe along with the inserted sample was turned

180° so that the end which did not have the insert faced the arm of the testing machine. The arm

was then replaced with a solid pipe of a diameter smaller than the inner diameter of the insert as

24

shown in Figure 16. Displacement was applied to the solid pipe through the machine, which pulled

the insert out from the pipe. The force required for this displacement was measured through the

force transducer. The set of graphs for pull-out forces against the displacement that was obtained

from the testing for each insert. An example of this set of graphs is shown in Figure 17.

Figure 16: Experimental setup for mapping the pull-out forces

Figure 17: Example of set of curves obtained from the experiments for pull-out force

25

8 Results After the experimental phase, the results from each set of samples were processed and an average

curve was obtained for the both push-in and pull-out cases for all the inserts. This average curve

was then compared to the contact forces observed in the simulation.

8.1 Push-in Forces

It was observed that from Figure 18 for Insert-1, that the forces from the simulation are about 80%

of the actual forces observed from the experiments. For Insert-2 and Insert-3 it can been from

Figure 19 and Figure 20 respectively that the results from the simulation are significantly lower

than the forces observed in the experiments for samples in the dry state. However, for the

conditioned samples, the experimental forces were close to contact forces obtained from the

simulations. It can also be observed from the graph that the simulations were fairly converged as

the mesh density parameter did not affect the contact forces to a large extent.

Figure 18: Comparison of simulated and experimental push-in forces for Insert-1

In case of Insert-2, as seen in Figure 19, breakage of the first two wings was observed, which

resulted in a higher displacement of the wings than the ideal. Due to this breakage, the peaks in

the force-displacement curve appear to be spread out over a larger displacement. This results in a

deviation of the experimental curve from the simulated curves as the damage and failure was not

captured through the material model used in the simulation. Also, it was found that a higher push-

in force is required for the dry samples. This force is approximately 1.4 to 1.7 times the push-in

forces observed for the conditioned samples. For conditioned samples, the simulations predict the

experimentally observed forces accurately. These predicted forces are 85-90% of the experimental

forces. The peculiar nature of this graph results from successive deformation of the wings. Thus,

each peak indicates the forces required to deform the wing.

26

Figure 19: Comparison of simulated and experimental push-in forces for Insert-2

Unlike Insert-2, no breakage was observed for Insert-3. As shown in Figure 20, this results in better

correlation in terms of the peaks for the experimental and simulation curves. The simulations for

this insert resulted in a lower convergence. In this case, the model with a fine mesh, predicts the

experimental forces to about 80-85% of the actual push-in forces. Like Insert-2, the experimental

forces observed for the dry samples are 1.45 to 1.6 times the forces observed for conditioned

samples.

Figure 20: Comparison of simulated and experimental push-in forces for Insert-3

27

It can be concluded from the above section that finite element simulations can be used to predict

the actual behavior of the insert for the push-in case with a reasonably accuracy.

8.2 Pull-out Forces

For pull-out forces, the simulations predict a considerably higher force compared to the

experimental forces for all insert designs. From Figure 21, Figure 22 and Figure 23, for the

simulations, as the insert is pulled out of the pipe, due to the successive disengagement of the

wings, the forces required keeps decreasing with the displacement. However, in the experimental

curve for Insert-1, a local peak in the forces is observed just before the disengagement of a wing.

This peak maybe due to the cyclic geometry of the insert due to which may cause the air to be

trapped in between the spaces of two wings. These peaks are less distinct in case of Insert-2 and

Insert-3 which have cut-outs in the wings.

Figure 21: Comparison of simulated and experimental pull-out forces for Insert-1

In case of dry samples of the insert, the forces pull-out forces are considerably lower compared to

the conditioned inserts. This large difference can be due to the breakage of the wings of the insert

during the push-in tests. The breakage may result in a lower contact area between the insert and

the tube which contributes to a decreased stiffness of the insert during the pull-out condition. In

case of a conditioned sample, no such breakage was observed, which show higher pull out forces.

In general, the forces obtained through simulations were about 70-80% of the experimental forces

for conditioned samples, whereas the simulated forces were 1.5 to 2 times the experimental forces

observed for dry inserts.

28

Figure 22: Comparison of simulated and experimental pull-out forces for Insert-2

Figure 23: Comparison of simulated and experimental pull-out forces for Insert-3

29

9 Discussion of Results In this section, the results from the simulations and experiments stage are critically assessed. The

sources of errors that may have occurred in theses stages are listed and their impacts on the results

are evaluated. This section is divided into three sections based on major sources of error.

9.1 Friction Coefficients in Simulations

The friction coefficients used in the simulations were obtained from published values found in the

literature review. These studies do not take the surface roughness and the surface finish into

consideration. The manufactured pipes for the furniture have a surface coating made of a fine

powder or paint. These surface finishes will alter the coefficient of friction, both static and

dynamic, and it will rarely be close to the values found in the literature. A study of these friction

coefficients was conducted to observe the effect of change in friction coefficients on the contact

forces. The effect of change in static coefficient on the contact forces can be seen in Figure 24. It

is observed that the coefficient of static friction significantly alters the peaks observed in the

contact forces. In contrast, there is no effect of the altering of the kinetic/dynamic friction

coefficient as seen in Figure 25. Only the amount of noise observed changes when varying this

parameter

Figure 24: Effect of coefficient of static friction on contact force

30

9.2 Limitations of the Material Model

For the simulations, the material model used was a MAT24 LS-Dyna material model, which uses

a stress vs plastic strain curve (or the plasticity curve) to estimate the strains in the simulation.

However, in this material model, the elastic region is independent of the strain rate up to the yield

point of the material. Another drawback of this material model is that variation in the failure strains

because of the strain rate is not considered. Although the MAT24 material model is useful in most

applications while modelling the behavior of metals, it can seldom exhibit high accuracy for

plastics. The damage accumulation in the material is not considered, which limits the modelling

of the failure at large strains. This incapability of modelling the failure of the material will have a

larger impact in the pull-out forces than the push in forces.

Figure 25: Effect of change in coefficient of kinetic friction on contact force

9.3 Geometrical Deviations in the Manufactured Parts

As seen from the discussion in the Section 7.2, as the manufactured part deviates from the nominal

dimensions. These deviations are enough to alter the dimensions to the extent that they affect the

result. This is because these deviations change the contact area between the insert and the pipe and

thus in turn will affect the push-in and pull-out forces. The following geometrical deviations were

observed during the testing phases.

9.3.1 Circularity of the Pipe:

The manufactured part is not perfectly circular. The dimensions of the pipe are altered in

the production of the pipe due to improper cooling and minute warping of the pipe due to

cooling. Moreover, clamping of the pipe in the vice during the experimental phases results

in the pipe attaining a slightly elliptical shape. In the simulations, a perfectly circular pipe

was used.

31

9.3.2 Manufacturing Defects in Pipe and Insert:

The manufactured tube is produced by bending sheet metal rolls around a mandrel. The

free edges of the sheet are then welded together by the electric resistance welding process

to form a hollow pipe. This welding results in a weld line on the inner surface of the pipe

as shown in Figure 26. This results in a local reduction of the diameter of the pipe. From

measurements it was found that the wall thickness of this pipe is increased locally by 0.13

to 0.26 millimeters around this weld line. This increase in thickness should result in a

higher contact forces as the inner diameter of the tube is reduced. Also, there are deviation

in the dimensions of the insert due to various effects like initial mold temperature, cooling

methods, ejection temperature, material flow rate of the molten plastic material for

injection molding and its temperature.

Figure 26: An example of weld-line present on the inner surface of the pipe

9.3.3 Tolerance of Pipe and Insert:

The above-mentioned effects could be controlled by implementing more stringent quality

control while manufacturing these parts. However, this leads to higher production costs

and may result in more rejection of manufactured parts. Hence, these effects are considered

during the design process by setting limits on the maximum and minimum dimensions for

both the insert and the tube. This helps the manufacturer to allow for some deviations in

the production of these components while controlling the costs incurred in production. For

the pipes and insert used in the present thesis, the tolerances are summarized in the Table

2 below. Due to confidentiality reasons imposed by the manufacturer, dummy values are

specified.

Table 2: Tolerances specified by the manufacturer for different components

1 Outer Diameter ±a

2 Wall Thcikness ±a or ± b

3 Outer Diameter ±b

4 Height ±c

Number Component Dimension NameSpecified

Tolerance

Pipe

Insert

32

The effect of change in contact forces due to variation in the wall thickness of the tube is

shown in Figure 27. As seen, a tighter fit between the tube and the insert can result in a

contact force which is three times that of a looser fit. This difference affecting the contact

forces can be directly translated into the performance since because of this difference in

fit, the force required for assembly/disassembly of the components is altered. Thus, it can

be concluded that the normal contact force is a major influencer of the total contact force.

The major components of the force acting on a one of the wings of the insert are shown in

Figure 28. The effect of changes in dimension of the insert is not studied in this thesis, but

it is assumed that if the diameter of the insert is increased, the contact forces observed will

also increase due to a tighter fit.

Figure 27: Effect of variation in pipe diameter on contact force

Figure 28: Example of a forces acting on a single wing of the insert

33

10 Design of Experiments In this section, the parameters which affect the performance of the insert are identified and

experiments designed to investigate the effect of these parameters on the push-in and pull out

forces are described. This section is classified into 3 topics describing the selection of parameters,

their ranges and the results showing their effect on the contact forces.

10.1 Determining the Variables

To optimize the geometry of the insert, the parameters that can be varied for a typical insert design

need to be identified. The design of base of the insert varies as per the area of the application and

the functionality of the insert. Hence, it was necessary to find common parameters that these inserts

have, for an optimal design for all the insert applications. During the simulation phase, it was

observed that the base is relatively unaffected by the deformation near the area of the wings. The

parameters of the wing Dw and h were chosen as variables. When varying these parameters, the

angle of the wing (α) is varied. The base angle (β) is restricted to be 90⁰ due to the manufacturing

processes. Similarly, the distance between the two wings is also constrained due to manufacturing

considerations. The inner and outer radius of the stem were converted into a single parameter

called as stem thickness t. While varying the stem thickness, the outer radius of the stem is kept

constant. This is done to maintain a fixed ratio between the wing height and the stem thickness. If

this ratio is higher, manufacturing defects like voids would be observed in the component, whereas

if the ratio is smaller, it will result in the breakage of the insert during the operation.

After careful considerations of the manufacturing and design constrains, the variables were

selected for the shape optimization. These variables are henceforth called as design variables and

named as shown in the Table 3 along with a summary of the starting, minimum, and maximum

values for all the design variables.

Figure 29: Dimensions of the insert used as design variables

34

10.2 Determining the Range of Selected Design Variables

After the variables for the optimization are selected, it is necessary to set minimum and maximum

values of these variables. These values determine the design limits for the design of the insert. The

starting value of the variables were based on the dimensions of the current design.

Wing Diameter (Dw = DVAR1): For wing diameter, the limits for the values are based on the

minimum and maximum overlap that can be present with the pipe. If this diameter is smaller, there

will be no interference with the pipe. In such a case, the forces required for the insertion will be

negligible. This condition is undesirable since then the pull-out force required will be small. Due

to this small force, the insert may easily come out of the pipe and become loose. This is a safety

issue and needs to be avoided. If the wing diameter is too large, the wings will interfere for the

entire dimension of the pipe thickness. This may result in breakage of the wings rendering the

insert useless.

Stem Thickness (t = DVAR2): For stem thickness, the minimum value is limited by the

manufacturability as the insert cannot be manufactured if the stem thickness is too small. For the

maximum thickness, there is no manufacturability limit. Hence, the largest thickness is only

limited by the inner stem radius as in that case, the insert may become completely solid.

Wing Height (H = DVAR3): The maximum wing height is limited by the distance between two

successive wings. When the rib height is increased, this distance is reduced. This reduced distance

between the wings causes issues in manufacturing of the insert. Hence it is limited to a maximum

value. On the minimum side, the wing height is not limited by manufacturability, but for the wing

to be present in the model for simulation, a nominal minimum value is set as 0.5 mm.

A summary of the starting, minimum, and maximum values for all the design variables is shown

in Table 3. To maintain confidentiality, the dimensions are scaled by an arbitrary number.

Table 3: Summary of Design Variables and their limits

10.3 Effect of Individual Parameters

After the range and the starting values for the design variables were determined, a study was

carried out to observe the effect of these design variables individually on the contact forces. In this

stage, for each of the design variable, design points are selected from the prescribed range of

variables and then simulations are carried out for each of these design points. For each of these

simulations, the desired result is plotted in terms of the range of variables. In case of this part of

the thesis, the variation of contact forces was obtained against the simulation time. For the selection

of design points in the design space, a space-filling algorithm was used.

Geometrical

DimensionSymbol

Name of Design

Variable

Starting

Value

Minimum

value

Maximum

value

Wing Diamater Dw DVAR1 1.92 1.80 2.04

Stem Thickness t DVAR2 0.17 0.09 0.40

Wing Height H DVAR3 0.25 0.05 0.37

35

i. Wing Diameter (DVAR1)

Increase in the wing diameter results in an increase in the contact forces observed as seen from

Figure 30. When the radius of the wing increases, the total volume of the wing increases and as a

result, a higher force is required for the deformation of the wing. Also, because of the increase in

the wing diameter, the tube contacts the earlier resulting in a deviation at the base of the of the 1st

crest in the curve. It can also be observed that by changing the wing diameter from 18.2 mm to

20.4 mm, the push-in forces doubles. In case of pull-out forces, no significant effect is observed.

Out of the three design variables used, the wing diameter is the parameter that has the greatest

effect on the contact force.

ii. Stem Thickness (DVAR2)

Decreasing the stem thickness results in a reduction in the contact forces observed. When the stem

thickness of an insert is reduced, the push-in and the pull-out forces are also reduced. This

reduction in the contact forces is observed due to a weaker section modulus of the insert wall. As

the section modulus of the wall is reduced, it deforms more easily due to which the normal force

exerted by the insert on the pipe is reduced. It can be observed from Figure 31 that when the stem

thickness is reduced from 2.45 mm to 0.5 mm, the contact forces observed are reduced by 20-25%.

Figure 30: Effect of variation in wing diameter (DVAR1) on contact force

36

Figure 31: Effect of variation in stem thickness (DVAR2) on contact force

iii. Wing Height (DVAR3)

It was observed that contact force is directly proportional to the wing height. An increase in the

wing height leads to an increase in the contact force as shown in Figure 32. The wing height does

not affect the contact force as much as the wing diameter or the stem thickness and is the least

important variable. The variation in wing height from 1 mm to 4 mm increases the observed

contact forces by 10-12%.

37

Figure 32: Effect of variation in wing height (DVAR3) on contact force

11 Optimization of Inserts for Enhanced Performance In this section the process of optimizing the geometry of the insert to achieve a better performance

while assembling the insert and the pipe is described. This performance is related to the force

required for assembly of the inserts. One possible solution to achieve this could be by reducing the

tolerance limits to more stringent tolerances on the pipe or the inserts. However, this incurs

additional quality control costs for the manufacturer. Hence, in this section, two possible ways to

improve the inserts to achieve better performance are explored. In this part of the thesis, Insert-3

is selected for optimization.

11.1 Need for Optimization

As seen in previous sections, the push-in force required for a given pipe varies for different

dimensions of the pipe. As the inner diameter of the tube is controlled by the outer diameter and

the wall thickness, the dimension of the inner radius can be written as:

For Max Material condition (MMC),

∅MMCinner = ∅outer + ∅tol − 2(t + ttol) (10)

And for Least Material Condition (LMC),

∅LMCinner = ∅outer − ∅tol − 2(t − ttol) (11)

Thus, for MMC and LMC for a given pipe, there will be a variation in the observed push-in force

due to the change in inner diameter of the pipe. This difference in forces is shown in Figure 33.

Figure 33: Example of difference in forces for MMC and LMC for pipe

As seen, the force required for LMC is approximately half of the force observed for MMC. For

the insert to be more customer-friendly, this difference in forces needs to be reduced. Thus, the

38

insert should be modified to fit in pipes of both these conditions without a significant difference

in the force required. In this thesis, 2 ways of reducing this difference are explored.

11.2 Approach

To improve the insert, an insert design should be simulated for both the least material condition

and the maximum material condition. Then, the results for both these cases should be analyzed

and the difference between the two conditions should be analyzed. Figure 34 shows an overview

of the approach used for the optimization of the insert and the steps are elaborated below:

i. Setup: In this stage, the design variables that are identified in Section 10.1 are provided

as an input to the optimization program. The nature of the variables i.e. if the variable

is discrete or continuous in a specified range or the dependency of a variable on another

is specified. In this thesis, all the variables were chosen as continuous. The starting,

minimum, and the maximum values are specified for each of the variables and one such

value of in this specified range is chosen by a sampling algorithm.

ii. Sampling: In this stage, the value of the variable from the specified range is chosen

through a specified sampling algorithm or the point selection technique. The number

of points selected in the design space for each iteration was specified as 10 i.e. for each

iteration 10 values of each variable will be chosen from the design space specified in

the setup stage. A D-optimal algorithm as discussed in Section 5.1 was used to choose

these 10 points in the design space.

Figure 34: Flowchart of the approach used for optimization of the insert

iii. Generation of models: Based on the design points created in the sampling stage, the

FE-model of the insert is modified using a preprocessor. This stage generates different

for each of the 10 design points. The starting value of the selected range is set as a 0

value for all the variables. This indicates that for the baseline run, no modification is

39

needed. From this baseline design, the other designs are generated as per the design

points. Hypermorph was used as preprocessor to build these models.

iv. Checking of the models: In this stage, the FE- models built in the model generation

stage are checked. This phase is essential to check that the FE-model does not create

undesirable shapes based on the design points. An undesirable model can be one where

the elements used are heavily distorted or where the elements are not formed at all due

to improper inputs from the preprocessor. After the FE-model is checked, it is sent to

the solver to run simulations for 2 different criterions – maximum material condition

and least material condition.

v. Simulations: In this stage the generated model are simulated using the LS-DYNA

solver. This stage is divided into 2 categories where the ‘SimulateMMC’ stage

simulates the behavior and records the forces observed during the assembly and

removal of the insert in a pipe of maximum material condition. Similarly, the

‘SimulateLMC’ simulates the insert in the least material condition of the pipe. The

responses that need to be recorded for optimizing the insert are specified in this stage.

These responses are then analyzed in the further stages and compared with the

responses from the previous iterations in the metamodel and optimization stages.

vi. Metamodel building: In this stage, the metamodels are constructed by the

optimization program. The metamodels are constructed using the values of the design

variables from the sampling stages and the responses from the simulation stages. From

the finite number of design points and responses, approximations are made for

responses for the entire design space and response surfaces are constructed. These

response surfaces are then optimized by reducing the mean square errors for the

approximations made in successive iterations. The number of iterations that are needed

for the model to accurately predict the response varies for different optimization

criterions used.

vii. Optimization: In this stage, the constraints for the responses and the objectives for the

optimization simulation are specified. The constraints are used to determine if the

results obtained from a simulation are feasible or infeasible. If the repose from the

simulations violate the constraints specified, the design point is classified as infeasible,

else it is deemed as feasible. The objective of the optimization can be to minimize or

maximize a response depending on the optimization setup created. The objectives and

constraints used in the thesis are described in the respective methods in Section 11.3

and Section 11.4 below.

viii. Termination criteria: In this stage a termination criterion and the maximum number

of iterations are specified. The simulation is completed if this termination criteria is

met. The simulation is completed if the design change factor and the accuracy of the

metamodel are below a set tolerance level. If the termination criteria are met, the

simulation proceeds to verify the design points obtained from the meta model by

running a simulation using these design points. If the termination criteria are not met,

the optimization proceed to simulate another iteration of the design space.

ix. Domain reduction: As discussed in Section 5.3, the design space is narrowed down

for successive iterations in the optimization. This helps in achieving a faster

convergence of the optimization towards an optimal solution within the design space.

After the domain is reduced, the iteration loops through stages ii to vii until the

termination criteria is achieved or the maximum number iterations is completed.

40

Based on this approach, two methods were tested using different objectives and constraints to find

an improved a design of the insert.

11.3 Method 1: Minimizing the Absolute Difference in Peak Forces

In this method, the objective of the optimization is to reduce the difference in the peak forces

observed during the push-in case of the insert for MMC and LMC cases. The peak value is

determined by the maximum value occurring within a particular period. Three such peaks were

identified for the insert for each of the conditions. These peaks correspond to deflection of

successive wings of the insert. The absolute value of the peaks in the MMC cases are then

subtracted from the respective peaks in the LMC case. The differences defined are illustrated in

Figure 35. The differences in peaks were calculated by:

Diffa = 𝐹aMMC − 𝐹aLMC

(12)

Where a = 1,2,3 is the wing number in the order of insertion and F indicates the peak forces

observed for the wing

Figure 35: Illustration of peaks and differences in peaks (Diff) for MMC and LMC conditions

The objective of the optimization is then set to reducing these differences. As this optimization

may result in a design of the insert which has minimal interference with the pipe, constraints are

needed to maintain minimum forces required for insertion and removal of the insert. The

manufacturer sets the constraint on the pull-out force which must be greater than 25 Newtons for

safety purposes. In the simulation, a safety factor of 25% is applied to this value and the constraint

is set to 31.25 Newtons. Also, it was observed that to achieve a higher pull out force, a higher push

in force is needed. Hence, a minimum push-in force of 25 Newtons was specified as a constraint

for the optimization. Based on this value, similar constraints were applied to other peaks. The

minimum force requirements were applied to the responses from the LMC condition, as the lower

41

forces would be observed for this condition. Also, the ratio of stem thickness to wing height was

maintained as per the original design to maintain the manufacturability of the insert.

Mathematically, the optimization problem can be written as:

min (FaMMC− FaLMC

)

s.t.

FminLMC < −31.25

F1LMC > 25

(FaMMC− FaLMC

) ≥ 0

(13)

Where a=1,2,3 is the wing number in the order of insertion and F indicates the peak forces observed

for the wing. For better understanding, these terms are shown in the graph in Figure 35.

The results for contact forces observed from the optimization after 10 iterations are compared in

Figure 36 and the resulting cross-section of the insert is shown in Figure 37. It was observed that

the optimized insert had a ratio of 1:1.78:1.12 for stem thickness (t): wing height (h): interference.

The interference was calculated by the following formula:

𝐼𝑛𝑡𝑒𝑟𝑓𝑒𝑟𝑒𝑛𝑐𝑒 = ∅𝐷𝑤 − ∅𝑃𝑖𝑝𝑒𝑛𝑜𝑚; (14)

where ∅𝐷𝑤 is the wing diameter and ∅𝑃𝑖𝑝𝑒𝑛𝑜𝑚 is the nominal diameter of the pipe used to

assemble the insert.

Figure 36: Comparison of forces observed in MMC and LMC conditions for baseline and

optimized designs by difference method

42

Figure 37: Comparison of cross-sections of original and optimized design by difference method

11.4 Method 2: Minimizing the Ratio of Peak Forces

In Method 1, the tendency of the optimization is to weaken the overall insert to reduce the peak

forces for both the MMC and LMC cases. Hence, another method was tested to achieve an

enhanced performance of the insert.

In this method, the objective of the optimization was set to minimize the ratio between the peaks

observed in the LMC and MMC conditions. In this method of optimization, by minimizing the

ratio between the peak forces, the possibility of having higher push-in forces in the LMC

conditions to achieve forces closer to the MMC condition was explored. The ratio between the

peaks was defined by

Divx = PeakxMMC / PeakxLMC

(15)

Where x is the wing number in the order of insertion and Peak indicates the peak forces observed

for the wing

The objective of the optimization was set to minimize these ratios. The constraints for the

responses were applied in the same way as for Method 1. The mathematical formulation of the

objective can be written as:

min (FaMMC/ FaLMC

)

s.t.

FminLMC < −31.25

F1LMC > 25

(FaMMC/ FaLMC

) ≥ 1

(16)

Where a=1,2,3 is the wing number in the order of insertion and F indicates the peak forces observed

for the wing. For better understanding, these terms are shown in the graph in Figure 35.

43

The results for the optimized insert after 10 iterations contact forces are compared with the original

design as shown in Figure 39 and the comparison of shapes is shown in Figure 38. Through this

method, the ratio for stem thickness (t): wing height (h): interference was observed to be 1: 2.05:

1.22. Table 4 shows the comparison for the objective, constraints and results for both the

optimization methods tested in this thesis.

Table 4: Comparison of the optimization methods used

Figure 38: Comparison of cross-sections of original and optimized design by division method

Diff1 Div1

Diff2 Div2

Diff3 Div3

Diff1 64.94% Div1 60.00%

Diff2 54.85% Div257.20%

Diff3 55.68% Div3 61.6%

5

Optimized Ratio

(Stem thickness (t): Wing

height (h): Interference)

4Reduction in Objectives

Achieved

1: 1.78: 1.12 1: 2.05: 1.22

2 Definition of Objective Diffa = Fa(LMC) - Fa(MMC) Diva = F1(LMC) / F1(MMC)

3 ConstraintsPeak3(LMC) > 25 N

Peakmin(LMC) > -31.25 N

Number Description Method 1 Method 2

1 Objetives Minimize Minimize

44

Figure 39: Comparison of forces observed in MMC and LMC conditions for baseline and

optimized designs by division method

45

12 Conclusions This section lists the various conclusions and findings from this thesis.

I. The mortar contact formulation can be a useful contact algorithm in simulation of softer

materials under heavy deformation and sliding. It has an excellent capability to handle most

types of mesh and element formulation without a need for tweaking of the contact

parameters.

II. Even though the data on static and kinetic friction coefficients is limited, the data obtained

from literature for similar conditions predicts the push-in forces with a good accuracy.

Even though the variation in the values of friction coefficients will affect the contact forces

observed, the main component of force that alters the push-in and pull-out forces are the

normal forces exerted by the wings of the insert on the pipe.

III. The behavior of the inserts during the experiments can be simulated to a fair amount of

accuracy even by using approximate boundary conditions if these conditions do not

interfere with the behavior of the insert. In this thesis, the boundary conditions used in the

simulations were not exactly same as the conditions under which the insert was tested but

the applied boundary conditions in the simulation were similar to those during testing.

IV. To predict the contact forces and the behavior of the insert during the pull-out operation, a

better material model is needed which can simulate the ductile damage and the

thermoplastic material behaviour beyond the ultimate tensile strength. Through such a

model, it would be possible to accurately simulate the real-world response for a pull-out

load case.

V. Even though the simulations were able to predict the forces observed in the experiments,

it is possible that some of the sources of error cancel or dampen the effect of other sources

of errors. For example, the deviations in dimensions of the insert were not considered in

this thesis, which may counterbalance the effect of approximate coefficients of friction

VI. There will always be a difference in the recorded forces for an insert assembled with pipes

of different inner diameters. This is due to the change in the contact area which alters the

normal force exerted by the wings of the insert on the inner surfaces of the pipe.

The above conclusions (I-VI) answer the project objective 1 stated in Section 1.4, whereas the

statements below answer the project objective 2

VII. The most important geometrical factor which influences the push-in and pull out forces is

the wing diameter of the insert. The wing diameter also affects the contact area of the insert

with the tube there-by having a greater impact on the forces observed.

VIII. The effect of difference in recorded forces for different dimension of the pipe can be

reduced by optimizing the geometry of the insert. This optimized insert can eliminate this

effect by approximately 50% thereby enhancing the performance of the insert. Such an

optimized insert should have a stem thickness (t): wing height (h): interference in the ratio

of 1:1.78 :1.17 or 1: 2.05 :1.22.

IX. The optimized geometry of the insert not only enhances the performance of the insert, but

also reduced the use of plastic material by 29%, thus reducing manufacturing costs.

46

13 Future Work Even though some conclusive results were obtained in this thesis, there is still a wide scope of

activities that can be carried out as supplement or addition to this thesis. This chapter lists a few

of such activities that can be the next steps in the project.

I. Implementation of a material model capable of simulating damage accumulation and

failure beyond necking of the thermoplastic material. Such a model should be capable of

predicting the pull-out forces and the behavior of the insert during this operation. From this

simulation, the deformation of the insert can be compared to the actual deformation using

3D-scanners. This will make the current simulations replicate the real-world scenarios

closely

II. The statistical study of deviations in the insert dimensions can be performed and

implemented in these simulations. By considering the deviations in the insert, the variation

in the forces that occur in the reality can be simulated. This will help predict the amplitude

and the range of forces that would be observed during the assembly is such furniture

components.

III. In this thesis, the outer diameter of the stem was fixed when the stem thickness was varied.

An insert with varying the stem diameter can be studied for further optimization. In case

of such an insert, a model with adaptive mesh may be needed.

IV. Testing of the optimized geometry of the insert and comparison with the current design of

the insert to verify the findings from this thesis.

47

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