Spot-Weld Fatigue Optimization

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Spot-Weld Fatigue

Optimization

FILIP ANDERSSON

RHODEL BENGTSSON

KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ENGINEERING SCIENCES

DEGREE PROJECT IN MECHANICAL ENGINEERING AND MATERIALS DESIGN, SECOND CYCLE,

30 CREDITS

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HESIS

Spot-Weld Fatigue Optimization

Authors:

Filip Andersson Rhodel Bengtsson

Supervisor: Ann-Britt Ryberg

A thesis submitted in fulfillment of the requirements for the degree of Master of Science in Engineering

in the

Solid Mechanics Department

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“If we knew what we were doing, it wouldn’t be called research”

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Abstract

The purpose of this thesis project is to develop a methodology that can be used to minimize the number of spot-welds in a mechanical structure, this is done in a re-liable manner via optimization methods. The optimization considers fatigue life in spot-welds and also stiffness and eigenfrequency values. The first chapter of this thesis presents a spot-weld fatigue model proposed by Rupp (1995), common FE-models of spot-welds and also important aspects about structural optimization in general. The second chapter further describes how topology optimization and size (parameter) optimization are applied on a simple multi-weld model with respect to the aforementioned structural constraints. The topology optimization is later used on a full-size car model, while the size optimization is used to optimize the multi-weld model by adding an non-linear structural constraint - a crash indentation con-straint. The spot-weld fatigue model proposed by Rupp (1995), is also verified by comparing FE results using different FE-models of spot-welds compared to fatigue data by Long and Khanna (2007). Both optimization methods successfully mini-mize the total amount of spot-welds on the multi-weld model. The topology opti-mization, accompanied with the gradient based MFD algorithm, minimizes the total spot-welds with around 15% and 3% on the multi-weld model and car body respec-tively. The size optimization, using design of experiments and response surfaces, manages to reduce the number of welds in the multi-weld model by 25%. However, with the size optimization the computational time is several orders of magnitude longer - even without the formulation of the crash constraint. The fatigue spot-weld model fares reasonably well compared to the experimental fatigue data, regardless of the FE model of the spot-weld. It is concluded that the ACM model would be recommended based on its compatibility with fatigue and optimization methods, mesh-independence and also other studies have shown its ability to represent stiff-ness and eigenfrequency correctly.

Keywords: Spot-weld, Fatigue, ACM, CWELD, Topology Optimization, Size Opti-mization, Response Surface, DOE, Genetic Algorithm, Most Feasible Direction, MFD

(1995), is also verified by comparing FE results using different FE-models of spot-welds compared to fatigue data by Long and Khanna (2007). Both optimization methods successfully minimized the total amount of spot-welds on the multi-weld model. The topology optimization, accompanied with the gradient based MFD algo-rithm, minimized the total spot-welds with around 15% and 3% on the multi-weld model and car body respectively. The size optimization, using design of experiments and response surfaces, managed to reduce the number of welds in the multi-weld model by 25%. However, with the size optimization it took orders of magnitude longer time to compute the results - even without the formulation of the crash con-straint. The fatigue spot-weld model fared reasonably well compared to the ex-perimentally.a, oberoende av FE-modell på punktsvetsarna. ACM-modellen rekom-menderas på grund av dess kompatibilitet med utmattning och optimeringsmetoder samt att andra studier har visat dess förmåga att representera styvhet och egen-frekvens.llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll

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ngsrikt den totala mängden punktsvetsar hos multisvetsmodellen. Topologi-optimeringen, tillsammans med den gradientbaserade MFD-algoritmen

minimer-ade mängden med cirka 15% och 3% på multisvetsmodellen respektive bilkarossen.llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll

Sammanfattning

Målet med detta examensprojekt är att ta fram en metodik som kan användas då antalet punktsvetsar minimeras i en mekanisk struktur, detta ska göras på ett tillför-litligt sätt via optimeringsmetoder. Optimeringen ska ta hänsyn till utmattning i punktsvetsarna men också styvhet och egenfrekvensvärde. Första kapitlet i denna avhandling presenterar en utmattningsmodell för punktsvetsar som föreslås av Rupp (1995), vanligt förkomna FE-modeller av punktsvetsar samt viktiga aspekter av struk-turell optimering i allmänhet. Det andra kapitlet beskriver vidare hur topologi-optimering och parameter-topologi-optimering tillämpas på en enkel multisvets modell med hänsyn till de ovannämnda strukturella bivillkoren. Topologi-optimeringen används sedan på en fullskalig bilmodell, parameter-optimeringen används istället för att op-timera multisvetsmodellen med hänsyn till ett krock-lastfall. Utmattningsmodellen som har föreslagits av Rupp (1995), verifieras också genom att jämföra FE-resultat med olika FE-modeller av punktsvetsar jämfört med utmattningsdata av Long and Khanna (2007). Båda optimeringsmetoderna minskade framgångsrikt den totala mängden punktsvetsar hos multisvetsmodellen. Topologi-optimeringen, tillsam-mans med den gradientbaserade MFD-algoritmen minimerade mängden med cirka 15% och 3% på multisvetsmodellen respektive bilkarossen. Parameter-optimeringen, tillsammans med användandet av DOE och responsytor, lyckades minska antalet med 25 % på multisvetsmodellen men det tog flera storleksordningar längre tid för att beräkna resultatet - oavsett om man inkluderade krockproblemet eller inte. Utmattningsmodellen uppträdde relativt bra jämfört med data, oberoende av FE-modell på punktsvetsarna. ACM-FE-modellen rekommenderas på grund av dess kom-patibilitet med utmattning och optimeringsmetoder, mesh oberoende samt att andra studier har visat dess goda förmåga att representera styvhet och egenfrekvens.

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Acknowledgements

We would like to express our appreciation to Dr. Ann-Britt Ryberg for her guidance during the term of our thesis. Without her help and counsel, always generously and unstintingly given, the completion of this work would have been immeasur-ably more difficult.

We are also indebted to the company of Combitech and their staff for their support and granting us access to their resources. The constant association with the mem-bers of Combitech CAE groups has been most pleasurable.

Finally, we would like to thank the staff of Altair, especially Ms. Britta Käck for imparting her knowledge and expertise during our work.

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List of Figures

1.1 Schematic description of RSW process (Pan, 2010) . . . 1 1.2 Rainflow counting example (Matsuishi and Endo, 1968). . . 3 1.3 Circular plate model for sheet metals (Kang, 2000). . . 5 1.4 Schematics of how t, θ and d relates to problem in question. In this

figure, only the mid-plane of the sheets are shown (NCode Theory Manual). . . 5 1.5 Beam model for nugget subjected to tension, bending and shear (Kang,

2000). . . 6 1.6 Spot weld analysis process summary with time, T. . . 7 1.7 Schematic description of the P2P model (Andersson and Deleskog,

2014). . . 8 1.8 Schematic description of the ACM model (Palmonella, 2005). . . 9 1.9 Schematic description of the CWELD model (Fang, 2011). . . 10 1.10 Example of a convex and non-convex (objective) function (Svanberg,

2017). . . 12 1.11 Simple sine function with infinitely many local optima. . . 12 1.12 Impact on stiffness using different penalization factors p. . . 13 1.13 Example I and II shows how Monte Carlo and LHS method samples

respectively. . . 17 1.14 Flowchart of the GRSM optimization algorithm. . . 18 1.15 Flowchart of the GA method. . . 19 1.16 Topology optimization model used by Puchner (2006) to redistribute

welds. . . 20 1.17 a ) distribution using stiffness b ) distribution using fatigue c ) damage

in individual welds along the weld line. . . 20 1.18 Spot-welding lines identified for the driver and passenger seats (d’Ippolito

et al., 2008). . . 21 1.19 Parameterization of spot-weld locations in CAD-CAE Virtual.lab (d’Ippolito et al., 2008). . . 21 1.20 Simulated annealing algorithm for minimization problem (Abasi, 2012). 22 2.1 Flowchart of thesis project . . . 24 2.2 Schematic description of single weld tensile shear (TS) specimen,

di-mensions in mm . . . 25 2.3 Schematic description of single weld coach peel (CP) specimen,

di-mensions in mm . . . 25 2.4 Uniaxially loaded tensile shear (TS) specimen, meshed with element

size 1mm. . . 26 2.5 Uniaxially loaded coach peel (CP) specimen, meshed with element

size 1mm. . . 26 2.6 Fatigue setup for benchmark structures . . . 27 2.7 Geometric description of multi-weld model. . . 28

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2.8 Bending load-case, forces and constraints (Forces in Newtons). . . 28

2.9 Torsion load-case, forces and constraints (Forces in Newtons). . . 29

2.10 Topology Optimization work-flow. . . 31

2.11 The Multi-Weld model with 12 spot-weld lines. . . 32

2.12 A 100kg mass is applied with a speed of 15m/s on one end of the multi-weld model, while constrained at the far end. d is the average beam length after the "crash", and of which the nominal value is 391.1 mm . . . 33

2.13 Nominal weld distribution. . . 34

2.14 a) global torsion b ) global bending c ) transverse bending front d ) longitudinal bending rear e ) modal analysis f ) rear belt pull. . . 35

3.1 The equivalent radial stresses on a single spot-weld subjected to sinu-soidal bending. . . 37

3.2 Benchmark test with CP specimen and element size 1mm. . . 38

3.5 Benchmark test with TS specimen and element size 1mm. . . 40

3.8 Mesh density dependency CP specimen . . . 41

3.9 Mesh density dependency TS specimen . . . 42

3.10 Optimized Spot-Weld configuration using minimization of volume and fatigue life constraint. . . 44

3.11 Optimized Spot-Weld configuration using minimization of weighted compliance and fatigue life constraint. . . 45

3.12 Optimized Spot-Weld configuration using minimization of volume fraction and all linear constraints. . . 46

3.13 Optimized Spot-Weld configuration using minimization of weighted compliance and all linear constraints. . . 47

3.14 Optimized Spot-Weld configurations with GRSM and GA method and only using the fatigue constraint . . . 48

3.15 Optimized Spot-Weld configuration with GRSM and GA and all linear constraints. . . 48

3.16 Optimized Spot-Weld configurations with GRSM and GA method and using all linear constraints and crash constraint . . . 49

3.17 a ) welds removed using only fatigue constraint b ) welds removed using all linear constraints. . . 52

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List of Tables

2.1 Material data for Weld nugget and Plates . . . 27

2.2 Linear Constraints for BIW topology optimization. . . 35

3.1 Optimization objective and constraints . . . 43

3.2 Optimized Spot-Weld configuration using minimization of volume and fatigue life constraint. . . 44

3.3 Optimized Spot-Weld configuration using minimization of weighted compliance and fatigue life constraint. . . 45

3.4 Optimized Spot-Weld configuration using minimization of volume fraction and all linear constraints. . . 46

3.5 Optimized Spot-Weld configuration using minimization of weighted compliance and all linear constraints. . . 47

3.6 Size Optimization results with only fatigue constraint. . . 47

3.7 The optimal spot-weld configuration (only fatigue constraint) . . . 48

3.8 Size optimization results with all linear constraints. . . 48

3.9 The optimal spot-weld configuration (all linear constraints) for both algorithms . . . 49

3.10 Size optimization results with all linear constraints + crash constraint . 49 3.11 Optimal spot-weld configuration (all linear + crash) for both algorithms 50 3.12 Responses of the suggested optimal design validated in the simula-tion model . . . 50

3.13 Relative difference between the responses given from the response surface (Kriging) and the actual simulated design point . . . 50

3.14 Diagnostics of different response surfaces in terms of R-Square. Based on the fatigue response. . . 51

3.15 Comparison of optimization results . . . 51

3.16 BIW topology optimization responses. . . 52

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Chapter 1

Introduction

Resistance spot-welding (RSW) is a commonly used technique when joining two or more sheets of metal. Electrodes are applied to both side of a sheet metal envelope, a high electric current is applied and the resistance in the sheet metal results in the development of heat causing the parts to join in a local melt-point, a schematic de-scription of the process is shown in figure 1.1.

FIGURE1.1: Schematic description of RSW process (Pan, 2010)

This is a relatively cheap and effective way of joining sheets in for example the auto-motive industry. However the introduction of spot-weld joint creates disturbances in the stress field, local stress-concentrations known as "hot spots" appear in the vicinity of joints and may lead to static or fatigue failure (Hobbacher, 2016). Thus, accurate stress and fatigue life estimations at these locations are needed to verify the per-formance of the structure. To relieve spot-welds subjected to high stresses, more welds may be added. A regular car body contains thousands of spot-welds. Dif-ferent methods for determining how many welds the structure need are available. However, empirical and practical approaches often lead to a higher number of spot-welds and thus a sub-optimal design. Some methods have been deployed to opti-mize the number of spot-welds with respect to noise and vibration, crash-worthiness and structural stiffness, examples are shown in section 1.6. When reviewing the cur-rent literature, the majority of reports handle weld fatigue and optimization as sep-arate matters, however methods from both fields may in fact be valuable in future structural analysis.

The fatigue mechanism is said to be responsible for more than 80% of structural failures and operational accidents, this is true for all structures subjected to cyclic loading (Lundh, 2000). Since the 1800s, originating from Wöhler’s work, engineers

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have developed methods on how to estimate the fatigue life of mechanical systems to minimize the risk of said failures.

Reducing the number of welds, and thus the number of weld processes, may be a contributing factor when developing a time and cost-effective production. By the use of reliable fatigue analysis combined with an optimized design, the performance of the structure can be maintained.

1.1 Objective

Different methods have been developed to model mechanical structures containing welds in order to analyze their complex geometry, load-distribution, material be-havior and failure. By applying and evaluating some of the current weld modeling methods, the objective of this thesis is to identify a methodology which can be used to accurately model spot-welds. This developed methodology will also combine the fields of fatigue analysis and structural optimization to reduce the number of spot-welds in a general mechanical structure, while maintaining key properties.

1.2 Delimiter

The methodology developed in this thesis work will focus on modeling technique and the combination and application of the fatigue and optimization theory. The fatigue analysis will be limited to stress based damage computation and methods including nominal stress approach and structural stress approach. The structural optimization is based on the topology and size optimization methods. Benchmark structures will be used to test, evaluate and compare both fatigue and optimization results. For the parallel but equally important aspects regarding welding techniques and micro-mechanical behavior in and around the weld, we will refer to the works of others. Models presented in this thesis will not fully capture the complexity of the spot-weld geometry nor the residual stresses and material behavior that are in-duced during the weld process. With that in mind, it will be shown that valuable conclusions can still be made using fatigue analysis and optimization methods in combination with welded structures.

1.3 Fatigue Theory

The fatigue phenomenon allows failure to happen even well below the ultimate ten-sile strength (Lundh, 2000). It has its basis from micro-cracks nucleating due to high stress concentrations from extremely small material and/or geometric defects. These cracks slowly propagates when the load varies cyclically in time. When the crack fi-nally reaches a critical size, a relatively low load is enough for failure to occur. Fatigue can be classified into two types (Socie, 2017). High Cycle Fatigue (HCF) im-plies that the structure remains predominantly (macroscopically) elastic, resulting in a fatigue life above 104 cycles. Low Cycle Fatigue (LCF) implies that the structure remains predominantly (macroscopically) plastic, resulting in a fatigue life below 104cycles.

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1.3. Fatigue Theory 3

In a controlled environment, fatigue testings can be done. Test specimens of the same material and surface polish are uni-axially loaded cyclically in a purely alter-nating or pulsating manner. During testing, a number of specimens are loaded in various magnitudes, and the number of cycles until failure are registered for each magnitude. The results are presented in an S-N diagram (Stress-Number diagram), where the stress amplitude, σA, of the specimens is plotted against its corresponding

logarithmic life, logN.

1.3.1 Variable and Constant Load Amplitude

The majority of fatigue testing is carried out at constant amplitude (CA), however mechanical systems are often subjected to variable-amplitude (VA) loading in real applications. There are a number of fatigue life prediction methods of structures in-volving VA load-histories. These differ from the methods used for CA load-histories and the most widely used one is the Palmgren-Miner rule (Alfredsson, Larsson, and Öberg, 2017). The aforementioned method involves applying a load with stress am-plitude∆Sionto the structure with nicycles and calculate the total damage, D, as,

D= k

∑

i=1 ni Ni (1.1)

where k is the number of amplitudes during the load history and Ni is the

corre-sponding fatigue life at∆Sifrom a CA fatigue test. If damage is truly a linear process

the damage at failure will be equal to unity, however experiments have shown this is not always the case. In simple two step block loading, a sequence of high ampli-tude cycles followed by low ampliampli-tude cycles will have a damage sum D < 1.0 at failure. Similarly, a sequence of low amplitude cycles followed by high amplitude cycles may have a damage sum D > 1.0. Random loading histories have D ≈ 1.0. CA load histories obviously are D = 1.0, due to their inherent correlation with the Palmgren-Miner Rule.

When observing a VA load-history, it can be difficult to distinguish the cycles and their corresponding amplitudes from each other. In 1968, the so called "Rainflow cy-cle counting" method was developed and it is a process to obtain equivalent constant amplitude cycles (Alfredsson, Larsson, and Öberg, 2017). Its name comes from the original description from the Japanese researchers Matsuiski and Endo where they describe the process in terms of rain falling off a pagoda roof (Matsuishi and Endo, 1968). By the use of rainflow counting the VA load history is translated to a series of CA load cycles that can be used in equation 1.1. An example of rainflow counting is shown in figure 1.2.

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1.3.2 Approach to Spot-Weld Fatigue

Considering HCF fatigue in welds there are three main approaches when determin-ing fatigue life, nominal stress approach, fracture mechanic approach and structural stress approach (Khanna, 2010). The nominal stress approach simply compute dam-age based on the stress calculated in the FE-elements, this approach will be applied in this study. When applying the fracture mechanical approach initial cracks is as-sumed, different methods are then used to estimate the growth of those cracks. This approach is not investigated in this thesis due to FE-software limitations, for more information we refer to the works of Swellam (1991). The structural stress approach uses a equivalent structural stress for damage calculations, this is the main approach for this thesis and is described below.

According to Rupp (1995), detailed stress FE-analyses at spot-welds are generally not practical for automotive applications with respect to the corresponding com-puting expense. Instead, he reasoned that a nominal local structural stress directly related to the loads carried by the spot-welded joint would be of more use in engi-neering. These analytical structural stresses are calculated based on beam and an-nular plate theory using the forces and moments transferred through the spot-weld seen as a beam element. For Rupp’s structural stress approach, two types of failure modes of spot-welded joints are considered: cracking in the sheet metal and crack-ing through the weld nugget.

Failure mode: Sheets

In the case of thin plates with relatively large spot-weld diameters, cracks often oc-cur next to the spot-weld in one of the two sheet metal parts. For this reason, the calculation of the stresses in the sheets should start with the cross-sectional forces and moments in the centre-plane of the sheets. Deriving the structural stresses for the sheet, the spot-welded joint is treated as a circular plate with a central, rigid circular kernel while the outer edges of the plate is treated as fixed. This is schemati-cally shown in figure 1.3. Using Roark’s formulae for stress and strain (Young, 1989), the radial plate stress around the spot-weld, σr, are assumed to vary as a function of

sine and cosine,

σr(θ) = −σ(Fx)cos θ−σ(Fy)sin θ+σ(Fz) +σ(Mx)sin θ−σ(My)cos θ (1.2) where, σ(Fx,y) = Fx,y πdt (1.3)

are the stress contributions from the (crossectional) shear forces Fxand Fy

σ(Fz) =κ(1.744Fz

t2 ) for Fz >0

σ(Fz) =0 for Fz ≤0

(1.4)

are the stress from the axial force Fz and,

σ(Mx,y) =κ(1.872Mx,y dt2 ) κ =0.6 √ t (1.5)

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1.3. Fatigue Theory 5

are the stress contributions from the moments Mxand My.

Parameters t, d and θ are the sheet thickness, spot-weld diameter and angle around the spot-weld respectively, they are illustrated in figure 1.4.

FIGURE1.3: Circular plate model for sheet metals (Kang, 2000).

FIGURE 1.4: Schematics of how t, θ and d relates to problem in question. In this figure, only the mid-plane of the sheets are shown

(NCode Theory Manual).

Failure mode: Nugget

As a rule of thumb, cracks in nuggets often occur in thick sheets with relatively small spot-weld diameters. The nugget can then be modeled as a circular cross-section beam subjected to tension, bending and shear loading.

σn= 4Fz πd2 σ_b= 32Mx,y πd3 τmax= 16Fx,y 3πd2 (1.6)

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Forces and moments are shown in Figure 1.5.

By super-positioning the above formulae, nominal nugget stresses can be deter-mined which in return allows one to take a stress-based critical plane approach for fatigue analysis. The resulting stresses are shown below:

τ(θ) =τmax(Fx)cos θ+τmax(Fy)cos θ

σ(θ) =σ(Fz) +σ(Mx)sin θ−σ(My)cos θ (1.7) where, τmax(Fx,y) = 16Fx,y 3πd2 σ(Fz) = ( 4Fz πd2) for Fz >0 σ(Fz) =0 for Fz ≤0 σ(Mx,y) = ( 32Mx,y πd3 ) (1.8)

FIGURE 1.5: Beam model for nugget subjected to tension, bending and shear (Kang, 2000).

According to Rupp, most spot-welds in a well-designed structure will not experi-ence significant torsional loads under normal conditions, thus the torsional fatigue is considered negligible and the torsional load Mz is therefore not needed.

Based on the information provided by the load provider, the analysis engine creates time histories of force and moment at the three calculation points (Sheet 1, Sheet 2 and Nugget). From the force and moment histories, stresses are calculated as de-scribed above, before rainflow counting and damage accumulation using a standard S-N approach, see figure 1.6.

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1.4. Finite Element Spot-Welds 7

FIGURE1.6: Spot weld analysis process summary with time, T.

To account for mean stress, Rupp also proposed his own correction based on the Goodman diagram. The equivalent stress amplitude at R= −1 is calculated by,

S0= S

+MsSm

Ms+1 (1.9)

where S is the stress amplitude for the cycle, Msis the down-ward slope of a

Good-man diagram, and Sm is the mean stress for the cycle.

1.4 Finite Element Spot-Welds

The development of finite element (FE) theory and computational power has changed the way structural analysis is conducted. Empirical approaches based on physical testing are often complemented or substituted by FE models. This is also true for joints in structures. Joint connectors include e.g. bolts, glue and welds. Modern FE softwares offer a large number of weld models, the majority of them are different setups of bar, beam and solid finite elements. Originally, a spot-welded joint was modeled with a rigid beam element, however improved models including flexible beams and solid elements are now available.

To ensure that our developed methodology is based on valid modeling assumptions, some well established spot-weld models are investigated in this thesis. These can be ordered in three main categories,

• Point to point (P2P)

• Area Contact Method (ACM) • CWELD

Several models of these three types are available in the software HyperMesh 2017.2, which was used as the pre-processor in this thesis project.

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1.4.1 P2P

The oldest and simplest model of a spot-weld is the point to point connector. It consists of a single element that connects one FE node to another, illustrated in figure 1.7.

FIGURE1.7: Schematic description of the P2P model (Andersson and Deleskog, 2014).

Different formulations for such elements are available in todays FE-softwares. In HyperMesh 2017.2 we identified four different P2P candidate-elements for spot-weld modeling. These are called CROD, CBAR, CBEAM and rigid connector. The "C" points to the fact that it is a connector element type. Rod connectors can only carry axial loading and is limited to one degree of freedom. The bar and the beam both carry axial, bending and torsional loading and have six degrees of freedom per node, the difference between the bar and the beam is that the bar has a constant cross sectional area. A rigid connector simply constrain one of the nodes (slave node) to match the displacement of the other (master node). The forces and moments are transmitted through the rigid connector, but no stress is calculated in this type of element (Castelo, 2004).

1.4.2 ACM

One of the most commonly used spot-weld models was proposed by Heiserer et.al (1999) and is called the area contact method or ACM. The weld is modeled with a solid hexahedral element which represents the weld nugget, in Hypermesh 2017.2 this element is by default a first order element with 8 nodes called HEX8 (HyperWorks Help). A special type of interpolation elements known as rigid bar element 3 (RBE3) is used to connect the nodes in this hexahedral element with adjacent nodes in the corresponding shell elements. A schematic description of the ACM model is shown in figure 1.8.

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1.4. Finite Element Spot-Welds 9

FIGURE1.8: Schematic description of the ACM model (Palmonella, 2005).

The set of shell elements that is connected to the solid makes up the patch surface. The reference node in a RBE3 element (e.g. the corner node of the solid element shown in figure 1.8 ) will move as a weighted average of one or more nodes nected to the reference node (HyperWorks Help). This model does not require con-gruent mesh on the opposing shell faces, however the RBE3 connections and patch surface are affected by the size of the elements. The RBE3 elements transmit forces between the solid and the sheet, the forces is distributed using the position of the RBE3 connection and the individual weight functions of the shell elements. This weld type element can be modeled with a single solid or several solids, more inter-polation elements is needed when modeling ACM with multiple solids.

1.4.3 CWELD

The CWELD element was introduced by Fang (2011) and implemented in the pre-processor of the FE-solver NASTRAN. The element was designed specifically for weld modeling of both congruent and non-congruent meshes (Palmonella, 2005). The CWELD is formulated as a shear flexible Timoshenko beam with two nodes and 12 degrees of freedom. When connecting two faces A and B, the nodes of the beam element grid point A (GA) and grid point B (GB) are connected to adjacent elements in any of the three ways listed below,

• Node to node connection • Node to patch connection • Patch to patch connection

The set of elements that is connected to the CWELD is called the patch area, the most general connection is the patch to patch connection in which neither of the grid-points are located in a node, this is illustrated in figure 1.9.

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FIGURE 1.9: Schematic description of the CWELD model (Fang, 2011).

The connections between the point GA and GA1,GA2,GA3 and GA4 is formulated with weight functions and translational and rotational constraint equations based on Kirchhoff shell theory. For the full mathematical formulation we refer to the original report of Fang (2011).

1.5 Structural Optimization

Structural optimization is a general term involving techniques used to optimize the design of a load carrying structure. The design variables x influence the design of the structure and these are updated in such a way that an objective function f0(x)is

minimized and the constraints gi are satisfied.

A general (structural) optimization problem reads: minimize

x∈Rm f0(x)

subject to gi(x) ≤ gi, i=1, . . . , c.

ji(x) =ji, i=1, . . . , h.

xe ≤xe≤xe, e=1, . . . , m.

where the c+h-many constraints state that the (state) functions gi need to return a

value not greater than the upper limits gi and the (state) functions ji(x)need to

re-turn exactly ji, while the m-many design variables lie between the lower and upper

bounds xeand xe, respectively.

The mathematical problem shown above is solved iteratively using optimization al-gorithms. For each evaluation of the objective function a new solution to the state problem is needed. In this case, the new solutions are often via FE-analysis, and thus one need to keep in mind how computationally expensive the problem is to solve. However, there are some methods involving regression-models of the objective- and state functions, reducing the computation significantly.

In this thesis, it is of interest to minimize the total amount of spot-welds, S(x), in a given structure. Furthermore, the problem have to comply in such a way that the fatigue acceptance criteria, A(x)have to be strictly larger than a value Aconstraint

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1.5. Structural Optimization 11

when returning the optimal amount of spot-welds.

The optimization problem of this thesis is formulated in a simple manner: minimize

x∈Rm S(x)

subject to −A(x) ≤ −Aconstraint (∗)

xe ≤xe≤ xe, e=1, . . . , m.

* Minus signs were added for the sake of not switching the inequality sign.

The design variable x of the above formulation involves the spot-welds themselves, but are perceived somewhat differently depending of what kind of optimization ap-proach is used, this is thoroughly explained in section 1.5.2 and 1.5.3. The location of the spot-weld is also of major importance to a constraint such as fatigue life, and will be handled and described further in other sections. Additional constraints will also be introduced later, e.g stiffness and eigenfrequency, for the sake of benchmark-ing different optimization methods compatibility and performance usbenchmark-ing multiple constraints.

1.5.1 Pitfalls and Possibilities of Optimization Problems

During the design process of a product, characteristics such as efficiency, reliability, economy and structural capabilities are important to consider and makes the prod-uct more desirable to customers. Embodying all these parameters into one single optimization when designing is called multidisciplinary design optimization, MDO. The optimum from an MDO problem is a superior design compared to one that has been iteratively found by optimizing each discipline sequentially. MDO has the ben-efit of exploiting the interactions between the disciplines. However, optimizing for several constraints simultaneously will significantly increase the complexity of the problem definition.

Convexity

When an optimization problem increases in size it is crucial to identify whether it is convex or not. An identified optimum for a non-convex problem may be a local one, leaving the global optimum still undiscovered. Contrary, a convex optimization problem can only have one optimum which is consequently globally optimal. Considering a case in one variable, a function is convex if a line segment that is drawn from any arbitrary point(x, f(x))to another (x0, f(x0))is strictly above the function graph (Svanberg, 2017). This is shown in figure 1.10.

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FIGURE1.10: Example of a convex and non-convex (objective) func-tion (Svanberg, 2017).

Although an objective function may be non-convex, the optimization problem itself does not necessarily have to be non-convex as the constraints and bounds to the problem may constrict the function into a convex region. A simple sine function is overall a non-convex function but if restricted to be evaluated between−πto 0 it is convex with only one possible optimum, see figure 1.11.

FIGURE1.11: Simple sine function with infinitely many local optima.

1.5.2 Topology Optimization

The goal of a topology optimization is to find the optimal distribution of material in a pre-defined design-domain. To describe this mathematically the element densities are parametrized with the design variable ρe. Each design variable may take on any

value between 0 and 1, where 0 results in zero density and 1 results in elements with full density. When conducting linear analysis the local stiffness tensor Ke of

each element is scaled with the design variable ρein the same manner as the element

density, this can be formulated as,

Ke =ρeK0,e

(1.10) Where K0,e is the initial local stiffness matrix. The continuous characteristic of the

design variables results in non trivial solutions when searching for the optimal dis-tribution. To make the results more discrete a penalization factor p is commonly used. Many FE softwares, including Optistruct 2017.2, use the "Solid Isotropic Mate-rial with Penalization method" called SIMP (HyperWorks Help). This is a power law formulation which adjust equation 1.10 accordingly,

Ke =ρepK0,e

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This formulation results in a penalization of intermediate values and a more discrete design. The final design will include more elements with zero or full stiffness when using a higher penalization factor. The impact of the penalization factor is illustrated in the graph shown in figure 1.12.

FIGURE1.12: Impact on stiffness using different penalization factors p.

One consequence of using ρ as design variables is that the optimal solution becomes mesh dependent, by increasing the mesh density thinner structural members are made possible during the topology optimization process. This may result in unreal-istic designs such as extremely thin structural members or a phenomenon known as checkerboard (Sadek et.al 2017). To avoid too thin structures and the checkerboard phenomena each element density is set to affect the adjacent elements, mathemati-cally this is formulated as a sensitivity filter which is controlled with the minimum member size parameter rmin, the full mathematical formulation reads,

d∂ f ∂ρk = 1 ρk∑eN=1Hˆe N

∑

e=1 ˆ Heρe∂ f ∂ρe (1.12) where, ˆ He=rmin−dist(k, e) (1.13)

f is the objective function and dist(k, e)is the distance between a specific element k and adjacent element e (Bendsoe, 2004). In a topology optimization the design vari-ables ρ is used to formulate the objective function f(ρ), a general formulation can be written,            minimize f(ρ) subject to      0≤ρ≤1

State function constraint Manufacturing constraints

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commonly used, namely minimize volume or minimize structural flexibility. Using the volume the objective function reads,

minimize V=

N

∑

e=1

ρepV0,e (1.14)

Where e denotes the individual element number and V0,e is the initial element

vol-ume. Using the flexibility, (also known as compliance) the objective function reads,

minimize C=

N

∑

e=1

ρepuTeK0,eue (1.15)

Where ueis the local displacement vector and K0,eis the initial local stiffness matrix.

Method of Feasible Directions

The topology optimization setup results in a non-linear optimization problem. To solve such a problem, a non-linear solver algorithm is used. HyperMesh 2017.2 uses a gradient based solver algorithm called method of feasible directions, MFD, considering the non-linear optimization problem,

minimize

x∈Rm f(xi)

subject to gj(xi) ≤0 ∀j

(1.16)

where i denotes the iteration number. The method of feasible direction is based on a initial starting point xi that is within the feasible space. This point is iteratively

updated in the following manner,

xi+1 =xi+λidi (1.17)

Where λi is the step size and di is the directional vector. By moving in a direction

such that the objective function decreases and the constraints is met, the optimum is reached (Zoutendijk, 1959). To find such a direction among the infinite number of directions a set of conditions is used,

d_i∇f(x_i) <0 d_i∇gj(xi) ≤0 ∀j

(1.18)

where the first condition renders the descent direction and the second condition renders the feasible direction. These conditions are closely related to the conditions known as the Karush-Kuhn-Tucker or KKT-conditions (Zoutendijk, 1959). ∇f(xi)

and∇gj(xi)are the gradients of the objective function and constraints respectively.

Directional vectors di that satisfies both conditions is sought, these are called

us-able directions. One way to find a usus-able direction is to solve the following linear optimization, minimize x∈Rm −β subject to β=- max{d_iT∇f(xi), dTi ∇gj(xi)} ∀j d_iT∇f(xi) +β≤0 d_iT∇g_j(x_i) +β≤0 ∀j 0< β d_iTd=1 (1.19)

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If βopt < e1then a optimal solution has been found (xi = xopt) and the solver stops,

e1 is a pre-defined small number. If βopt ≥ e1 then di,opt is combined with equation

1.17 , the step size is determined by,

λi : (_{∂ f}₍_λ i) ∂λi =0 gj(λi) ≤0 (1.20) The solver stops at this stage (xi = xopt) if the following conditions is met,

f(x_i) − f(x_i+1) f(x_i) <e2 ||x_i−x_i+1|| <e3 (1.21)

Where e2and e3are small numbers. If the conditions in equation 1.21 is not met xi+1

is put back into equation 1.19 for a new iteration.

1.5.3 Size Optimization with Response Surfaces and Genetic Algorithms

Size optimization is a broad term that involves parameterizing anything that is of interest to be changed on a structure into a design variable, e.g thickness, with a minimum expense of certain factors such as weight and economy. Unlike topology optimization, the design variables are not penalized and forced to an upper or lower limit.

Most size optimization problems require experiments and/or simulations to evalu-ate objective and constraints as functions of the desired design variables. However, a single experiment or simulation can take hours or even days to complete. As a result, design optimization become unreasonable since they may require thousands or even millions of simulation evaluations.

Response Surfaces and Design of Experiments

In consideration of the aforementioned problem, generating models called response surfaces (also referred to as metamodels or surrogate models) would be more feasible. Response surfaces mimic the behavior of the simulation model as closely as possi-ble while using as few as possipossi-ble evaluation points for regression. When only one single design variable is involved, the process is simply known as curve fitting. A general (linear) regression model is of the form

y(x) =c1ψ1(x) +c2ψ2(x) +. . .+cnψn(x) (1.22)

and can be used to create response surfaces. ψ are the arbitrary linearly independent basis functions and ciare the coefficients that has to be determined by a least square

fit by minimizing the sum of the square of the errors y(xi) −gi, where gi are the

sample data shown in equation 1.23.            c1ψ1(x1) +c2ψ2(x1) +. . .+cnψn(x1) =g1 c1ψ1(x2) +c2ψ2(x2) +. . .+cnψn(x2) =g2 .. . c1ψ1(xm) +c2ψ2(xm) +. . .+cnψn(xm) =gm (1.23)

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The regression model for a second-order response surface can be formulated as, y= β0+ k

∑

j=1 βjxj+

∑ ∑

i<j βijxixj+ k

∑

j=1 βjjx2j (1.24)

This is equivalent to a first-order regression model, with exception for βjjx2_j which is

the quadratic terms and βijxixjwhich are the interactions.

Referring back to equations 1.22 and 1.23, instead of using simple polynomials such as in the equation above, the functions ψi can be described as radial basis functions,

ψi(x) =ψi(kx−xik) =ψi(r) (1.25)

where r is the distance between the points x and xi, hence why this model is called

the radial basis model (RBF). The functions ψi(e) have many forms and are always

radially symmetric. The most common is the Gaussian function: ψ(r) =e−(

r

k)2 (1.26)

where k is a shape parameter.

An additional response surface model is the Kriging model, this model has the bene-fit of interpolating between data points. Kriging is a combination of the polynomial regression model and a random function Z(x). In a polynomial regression model, the coefficients are chosen so that the error between the given data and model are minimized. However the random function Z(x)reduces this error to zero so that the model interpolates the given data points.

The accuracy of the response surface depends on the number and locations of the data points that have been evaluated in the design space. It is important that the data points are chosen so that the response surface has a proper input-output behav-ior. There are various so called design of experiments (DOE) that cater to different sources of errors, in particular errors due to noise in the data or due to an improper response surface. The following are three common design of experiments:

• Randomized sampling, often known as Monte Carlo simulations, samples from stochastic variables based upon a chosen probability density function. The most common probability density function used is the Gaussian function. Al-though simple to apply, the drawback is that many simulations are needed to get good accuracy.

• The Latin Hypercube method inherits the stochastic properties of the Monte Carlo method, but work in a different manner as the points chosen are more computationally efficient. A square grid containing sample positions is a Latin square if, and only if, there is only one sample in each row and each column. A Latin Hypercube DOE, categorized as a space filling DOE, is the generaliza-tion of this concept to an arbitrary number of dimensions. The applicageneraliza-tion is relatively more difficult than the normal Monte Carlo method since the DOE has to remember all the previous sample points when creating a new one. See figure 1.13.

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FIGURE 1.13: Example I and II shows how Monte Carlo and LHS

method samples respectively.

• Factorial DOE often involves having k many input variables that have n many levels to be considered, and thus creating all possible combinations of input variables and levels, e.g. two input variables with only two levels would yield four possible combinations to be evaluated. While k many input variables with n many levels each would yield nk many possible combinations to be evalu-ated.

Global Response Surface Method

Global Response Search Method (GRSM), which is an exclusive optimization al-gorithm by HyperStudy, generates a response surface based approach for solving parameterized structural optimization problems. Each iteration creates a response surface (a radial basis function regression model) that is adaptively updated with a space filling DOE. The iteration suggests an optima as well as verifying the previous suggested optimum from the last iteration, see figure 1.14.

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FIGURE1.14: Flowchart of the GRSM optimization algorithm.

Unlike gradient based schemes, there is no numerical convergence in GRSM that indicates when to terminate the algorithm. The same optimum may be obtained in sequential iterations of GRSM. The terminating condition of GRSM is instead con-stricted by the maximum number of evaluations set by the user.

Genetic Algorithm

Genetic algorithms (GA) are a form of optimization algorithms which are modeled after the evolutionary process theory. GA starts with the creation of a population of designs (a generation). These designs are then ranked with respect to their fitness. Fitness is calculated as a function of constraint violation and objective function val-ues, describing how good the design is. New designs are created with inheriting characteristics from design points with high fitness (cross-over) as well as allow-ing to mutate. These newly produced design points become members of the next generation. This process is repeated for many generations until the evolution of a population converges to the optimal solution, see figure 1.15.

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1.6. Previous Works on Optimization and Spot-Welds 19

FIGURE1.15: Flowchart of the GA method.

The most common genetic operators that generate a new population are cross-over and mutation:

• Cross-over combines two design points to produce a new design point. The idea behind cross-over is that the new design may be better than the previous ones if it inherits the characteristics from the so called "parent" design points. As to what characteristics from which parents the "offspring" design point inherits, is according to a user-definable cross-over probability.

• Mutation alters one or more characteristic in a design point from its initial state. Rather than fully resembling like its parents, certain characteristics are chosen to be slightly mutated (altered). Mutation is an important part of the genetic search, as it helps to "explore" the design domain for possible minimums and preventing the population from stagnating at any local optima. As to how much the characteristics are mutated is by a user-definable mutation probabil-ity.

1.6 Previous Works on Optimization and Spot-Welds

While conducting a literature review, projects related to optimization and fatigue was scrutinized and some guided the research in this project. This section presents some important reports of earlier studies related to topic of this thesis.

1.6.1 Topology Optimization and Spot-Welds

Besides the extensive available material concerning fatigue estimation in spot-welds some studies also involve the fatigue analysis in topology optimization. An Aus-trian study written by Puchner (2006) combines topology optimization and fatigue analysis. The goal of their optimization is to improve structural characteristics by

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redistributing the initial welds. To maintain the same number of welds while dis-tributing them differently weld density is initially increased, the number of welds along certain weld lines is doubled and an optimization constraint is used to reduce them to the initial number. Compliance and fatigue life are used separately as ob-jective functions and the method is applied to a simple sheet-metal beam shown in figure 1.16. The results from their study is shown in figure 1.17.

FIGURE1.16: Topology optimization model used by Puchner (2006) to redistribute welds.

FIGURE 1.17: a ) distribution using stiffness b ) distribution using fatigue c ) damage in individual welds along the weld line.

The welds are modeled with solid hexahedral elements and the fatigue is calcu-lated using a local stress approach developed by Nakahara (2000) which is similar to Rupp’s method. The study show that both optimizations render similar results, that is the minimization of compliance results in similar results as minimization of damage or maximization of life. According to this study, zero-density spot-welds are removable and does not influence the fatigue and compliance results. The great benefit of this method is that the final result is obtained with only 30-40 iterations that have a reduction rate up to 30 % in each iteration.

1.6.2 Size Optimization and Spot-Welds

d’Ippolito et al. (2008) adopted a multi-disciplinary design optimization (MDO) pro-cedure, in order to identify the optimal spot weld layout distribution (along chosen

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1.6. Previous Works on Optimization and Spot-Welds 21

connection lines) that minimizes the fatigue damage of a car floor component while satisfying NVH constraints. A spot-weld redistribution tool was used in MSC Nas-tran to parameterize the spot-weld density independently along chosen connection lines, yielding a flexible parametrization with a limited number of design variables, see figure 1.18 and 1.19.

FIGURE1.18: Spot-welding lines identified for the driver and passen-ger seats (d’Ippolito et al., 2008).

FIGURE1.19: Parameterization of spot-weld locations in CAD-CAE Virtual.lab (d’Ippolito et al., 2008).

d’Ippolito approached their optimization problem based on a preliminary design space exploration using a Latin Hypercube design of experiments combined with an interpolating radial basis function (RBF) response surface. The actual optimiza-tion algorithm itself used was the aforemenoptimiza-tioned Genetic Algorithm. Results of the optimization process obtained from the response surface had to be verified with a single final direct simulation, in order to assess the error between the response model and simulation analysis at the optimal point. In conclusion, they managed to reduce the amount of spot-welds by 12% and recommended to apply the same methodology to other vehicle components as well. Additionally, they concluded that the approximate model allowed a fast optimization process to locate an optimal configuration. However, a subsequent local optimization had to be done to refine

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and check the validity of the results.

In addition to the works of Siemens, a similar approach by Ryberg and Nilsson (2016) was demonstrated. However, in this case, the Genetic Algorithm only relied upon simulated design points (no response surface), thus no subsequent local opti-mization was needed. A response surface-based approach was indeed still used by Ryberg, similarly to the GRSM algorithm, a feed-forward neural network response surface is iteratively built with 16 simulations added in each iteration, combined with an adaptive simulated annealing (ASA) optimization algorithm. In conclusion, Ryberg managed to remove far more spot-welds with the response surface based approach compared to the direct approach with GA although half the amount of evaluations were used.

How the ASA optimization algorithm works is described by Abasi (2012). A basic structure of the simulated annealing algorithm is presented in figure 1.20.

FIGURE1.20: Simulated annealing algorithm for minimization prob-lem (Abasi, 2012).

S = the current solution S∗= the best solution Sn= neighboring solution

f(S)= the value of objective function at solution S n = repetition counter

T0= initial temperature

L = number of repetition allowed at each temperature level p = probability of accepting Snwhen it is not better than S

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23

Chapter 2

Method

In this thesis project, the research is conducted in accordance with the work-flow shown in figure 2.1. The study starts with an investigation of current weld meth-ods. By reviewing the literature and by consultation with FE software developers, different approaches are identified. Ways of evaluating fatigue in and around the candidate welds are then considered. By exploring current methods and their ap-plicability and compatibility to spot-welded joints, a number of viable methods are identified. The candidate weld models and fatigue calculation methods are then evaluated using single-weld benchmark structures, for more details, see section 2.1. These steps mark the first stage in the work-flow described in figure 2.1.

By evaluating the results from the first stage the weld model and fatigue calculation that renders best results is then considered for optimization. The second stage of this study is dedicated to investigating different optimization approaches. Strengths and limitations are identified by a literature review and, at this stage, the compatibility with weld optimization is considered. The single weld benchmark structures is then replaced by a multi-weld model which can be used to implement two different op-timization approaches, topology and size opop-timization. This concludes the second stage of the work-flow shown in figure 2.1.

The multi-weld model originates from a paper written by Ryberg and Nilsson (2016), stage three to five involves the model and optimization problem setup for topology and size optimization, this is described in greater detail in section 2.3 and 2.4 re-spectively. As indicated in figure 2.1, the results from the two approaches are then compared in stage six and more about the comparison is found in section 2.5. In a final stage the topology optimization with fatigue constraints is applied to a full size car model supplied by Combitech, to evaluate the method on a large-scale applica-tion. The size optimization is combined with a non-linear load-case to evaluate its ability to handle more complex analysis. The last stage of the work-flow is described in section 2.6.

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FIGURE2.1: Flowchart of thesis project

2.1 Verification of Rupp’s approach and comparison of weld

model types

Model setup

To evaluate different weld models and fatigue model proposed by Rupp (1995), two simple benchmark structures is used. These structures are recurring in several pa-pers with included load-cycle data. The benchmark structures is modeled using specified geometric data, see Long (2007), and is shown in figures 2.2 and 2.3 respec-tively. The benchmark structures are referred to as the tensile shear (TS) specimen and the coach peel (CP) specimen. A uniaxial pulsating load is applied to the ends of each specimen resulting in a shear loading in the TS-specimen weld and a bending load in the CP-specimen weld.

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2.1. Verification of Rupp’s approach and comparison of weld model types 25

FIGURE2.2: Schematic description of single weld tensile shear (TS) specimen, dimensions in mm

FIGURE 2.3: Schematic description of single weld coach peel (CP) specimen, dimensions in mm

The sheet metal in each specimen is modeled with first order quadrilateral shell ele-ments called QUAD4. The single weld is modeled with the weld element candidates described in the previous chapter. The models are prepared in the pre-processor Hy-permesh 2017.2. The meshed TS and CP specimens are shown in figures 2.4 and 2.5 respectively. An element size of 1mm is initially used. The spot-weld diameter is set to match the physical diameter of the weld, the material of the sheet metal are defined in the fatigue module and is presented in table 2.1.

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FIGURE2.4: Uniaxially loaded tensile shear (TS) specimen, meshed with element size 1mm.

FIGURE 2.5: Uniaxially loaded coach peel (CP) specimen, meshed with element size 1mm.

Fatigue setup

The fatigue evaluation can be summarized as a separate workflow shown in figure 2.6. Fyshown in figures 2.4 and 2.5 is applied as line loads, which are used to solve

the linear static sub-cases. These loads are then scaled in the fatigue tool to match the experimental fatigue loads. In the fatigue sub-case, these loads are applied in a sinusoidal cycle using the load ratio R=0.1 which is defined as

R= σmin

σmax

= Fmin

Fmax

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2.2. Multi-Weld Model 27

FIGURE2.6: Fatigue setup for benchmark structures

The material data is set to match a high strength low alloy (HSLA) steel, the material data from (Long and Khanna, 2007) was cross-checked and complemented with data from nCode material library. The weld nugget is modeled with HSLA spot weld material and the sheets with a generic steel sheet material, the material data is shown in table 2.1.

TABLE2.1: Material data for Weld nugget and Plates

Material Parameter Weld Nugget Sheet metal

Yield Strength [MPa] 320 355

Ultimate Tensile Strength [MPa] 484 500

Elastic Modulus [GPa] 200 210

Elastic Poisson’s Ratio 0.3 0.3

Stress Range Intercept [MPa] 3496 2900 First Fatigue Strength Exponent -0.1818 -0.1667 Second Fatigue Strength Exponent -0.1 -0.09091

Standard Error of Log(N) 0.33 0.33

Given this material data the fatigue are calculated using the theory presented in previous chapter, fatigue results based on the nominal stress-field and fatigue results based on Rupp’s local structural stresses is extracted. This process is repeated for both the TS and the CP specimens with different element sizes and weld element types.

2.2 Multi-Weld Model

The multi-weld model described in the first section in this chapter is used to set up the topology and size optimization. The structure is a symmetric square tube with a longitudinal center plate of sheet metal. The sheets are welded together with 120 spot-welds distributed over four straight weld lines. The geometric description of the structure is shown in figure 2.7.

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FIGURE2.7: Geometric description of multi-weld model.

Note that the center plate is connected through 3-layer spot-welds, that is three sheets and two weld elements shown in 2.7 (B). Starting from either side of the struc-ture, the first, fifth and sixth weld in each weld line is considered a fixed weld. The sheets are modeled with first order shell elements called QUAD4 and have a thick-ness of 1mm. The welds are modeled with any of the candidate spot-weld elements and the weld diameter is set to 5mm. The structure is meshed in HyperMesh 2017.2 and the element size is set to 5mm. Two static load-cases are considered. Forces and constraints together with meshed structure are shown in figure 2.8 and 2.9, respec-tively.

FIGURE 2.8: Bending load-case, forces and constraints (Forces in

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2.3. Topology Optimization Setup 29

FIGURE2.9: Torsion load-case, forces and constraints (Forces in New-tons).

For both optimization methods, a fatigue constraint on each individual spot-weld is implemented. The fatigue is calculated via Rupp’s local structural stresses approach. Both static load cases are superposed and applied as a sinusoidal cycle with the load ratio R = −1. In addition to optimizing with respect to spot-weld fatigue, the multi-weld model will also be optimized with respect to several stiffness constraints and an modal analysis constraint. One torsional stiffness constraint, Ct[N/deg], is

introduced via the torsional load case shown in figure 2.9, two stiffness constraints, Cbr (rocking motion) [N/mm] and Cbt (tunnel motion) [N/mm] are introduced via

the bending load case shown in figure 2.8. The modal analysis constraint is the second eigenfrequency which coincides with the first torsional mode. All constraints are defined as 90% of the nominal run of the multi-weld model. This setup was used for both optimization methods described in section 2.3 and 2.4.

2.3 Topology Optimization Setup

The topology optimization is set up in the software Optistruct (OS), the design do-main is limited to the set of non-fixed weld elements. Nominally, there are 84 weld elements which may be removed. However these are distributed on 56 weld point since some welds are 3 layer welds (see previous section). The density and stiffness of the elements are scaled by the design variables ρe, resulting in 84 design variables

in OS. One fatigue load-case is considered and the spot-weld fatigue is computed using Rupp’s local structural stress approach. The static load-cases are applied as a sinusoidal cycle with the load ratio R = −1. Since only one fatigue load-case is considered, the local stresses are computed using superpositioning of stress from the bending and the torsion. Material parameters for welds and sheets are the same as for the single weld simulations, which were shown in table 2.1. The optimization problem is solved using two different setups in OS,

• Minimization of volume fraction • Minimization of weighted compliance.

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By scaling the formulation given in equation 1.14 with the initial volumes, the first setup reads, minimize ρ∈Rm Vf(ρ) subject to Vf(ρ) = ∑ N e=1ρ p eV0e V0, e K(ρ) = N

∑

e=1 ρp_eK0,e K(ρ)U=F Aconstraint ≤ A 0≤ρ≤1 (2.2)

where Aconstraintis a pre-defined life or damage acceptance criteria. The second setup

uses the formulation that is stated in equation 1.15 with the extra condition that sev-eral load-cases are considered. This is called a multi-objective optimization problem and uses weighted compliance. The formulation with constraints reads

minimize ρ∈Rm C(C1(ρ), C2(ρ)) subject to C(ρ) = 2

∑

j=1 wjCj(ρ) 2

∑

j=1 wj =1 Cj(ρ) = N

∑

e=1 ρepuTejK0,euej Kj(ρ)Uj =Fj Aconstraint ≤ A Vf(ρ) ≤Vfconstraint 0≤ρ≤1 (2.3)

where wj is the weight, j is the load case number and e is the element number. For

our optimization the weights for the torsional and bending load case are set equal, the penalization factor that is discussed in the previous chapter is set to 30 to get a discrete solution. A life acceptance criteria, Aconstraint is used to constrain the

fa-tigue solution, in OS this constraint is defined on the topology design variable solver card. 90% of the number of cycles the structure endured with all welds in place is used as constraint. Note that the volume fraction constraint, V_f_constraint is added in the second setup, this constraint may be based on a volume removal target value or it could be determined iteratively. The non-linear optimization problem is solved us-ing the method of feasible directions discussed in section 1.5.2. The work-flow for the topology optimization is illustrated in figure 2.10.

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2.4. Size Optimization Setup 31

FIGURE2.10: Topology Optimization work-flow.

The optimization is also conducted using multiple linear constraints, this is done in order to evaluate how well the optimization works when fatigue life is mixed with other constraints.

2.4 Size Optimization Setup

In the size optimization, a total of 12 lines are defined between the fixed spot-welds in the multi-weld model. The number of spot-welds along each individual line are used as design variables. Thus, e.g. l4 = 3 would result in having three equally

spaced spot-welds onto line 4 in the model. The problem is further reduced by applying a symmetry condition on the xz-plane, restricting lines 4 to 6 to have the same amount of spot-welds as lines 10 to 12 respectively. This results in total of nine design variables. The variables l2, l8and l11may vary in amount of spot-welds

between 0 and 8, while variables l1, l3, l7, l9, l10 and l12 may vary between 0 and 3.

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FIGURE2.11: The Multi-Weld model with 12 spot-weld lines.

The above formulation is summarized in to an optimization problem shown below,

minimize f0 = 12

∑

i=1 li subject to gi(x) ≤gi, i=1, . . . , c. l4=l10 l5=l11 l6=l12 l1,3,7,9,10,12∈ {0, 1, 2, 3} l2,8,11∈ {0, 1, 2, . . . , 8}

The state functions gi(x)are in this case the torsional stiffness Ct, the bending

stiff-nesses Cbrand Cbt, the first torsional eigenfrequency Fm2. The constraints gi are 90%

of the nominal values of the state functions.

To further show the capabilities of size optimization and what kind of state func-tions and constraints can be defined, an average beam length (after crash) constraint d is added in addition to the aforementioned linear constraints on the multi-weld optimization.

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2.4. Size Optimization Setup 33

FIGURE2.12: A 100kg mass is applied with a speed of 15m/s on one end of the multi-weld model, while constrained at the far end. d is the average beam length after the "crash", and of which the nominal

value is 391.1 mm

For the size optimization, two different search algorithm are evaluated - the Global Response Surface Method (GRSM) and the Genetic Algorithm (GA) as implemented in HyperStudy. Both approaches are based on DOEs and response surfaces.

• GRSM creates an initial DOE of 20 randomly chosen sample points (Latin Hy-percube), creating an initial response surface from a radial basis function (RBF) fit. The current response surface is evaluated and suggests an optimal design that is validated in the next iteration, as well as nine other points to "explore" and improve the response surface. For every iteration, the response surface is iteratively updated and an optima is suggested for the current response sur-face. This algorithm is restricted to only run for a maximum of 200 evaluations (the total amount of sample points may not exceed 200).

• GA is supplied with a Kriging response surface model fitted to a DOE of 200 randomly chosen design points. The algorithm proceeds as described in the introduction, with an initial population of 200 evaluations of the response sur-face. It iterates in such a way that for each iteration, the top 10 % of the popula-tion are used as basis for creating the next generapopula-tion. The iterapopula-tion continues until the overall fitness improvement is negligible (< 1%) or the amount of unique design points evaluated (via the response surface) exceeds 10 000. For the size-optimization, HyperStudy is used as the optimization software. How-ever, to evaluate the state functions of the design points in the DOE, the solutions are solved via Optistruct and Radioss. Where Optistruct is used to evaluate the fatigue life, stiffness and eigenfrequency values, while Radioss is for the crash problem. This is unlike the topology optimization where the algorithm and state function evalua-tions are all integrated within Optistruct.

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2.5 Comparison of Optimization Techniques

Extracted results from the topology and size optimizations using the multi-weld model are compared. To be able to compare the results the size optimization was limited to include only one fatigue load-case, in which the two static loads was su-perposed. This was done due to a limitation in the Optistruct topology module, only one fatigue load case may be included when adding a fatigue life constraint. The re-sults from optimization on the multi-weld model are compared using both single fatigue life constraint as well as multiple linear constraints.

2.6 Final Application

Weld reduction BIW

By applying the topology optimization method on a realistic structure the ability of combining optimization and fatigue estimation can be represented. A BIW car body model is used for this purpose, The model has been a target for spot-weld opti-mization in previous projects. However it has never been investigated using fatigue analysis. A schematic description of the model and the initial weld distribution is shown in figure 2.13.

FIGURE2.13: Nominal weld distribution.

The sheet metal are modeled with first order tetra and quad shell elements. The ACM weld model described in section 1.4 are used to model the welds. The ma-terial and fatigue data that was described in table 2.1 was used. Six pre-defined linear load-cases are considered, global bending, global torsion, transverse bending in front, longitudinal bending in rear, modal analysis and belt pull. The loadcases are shown in figure 2.14

Spot-Weld Fatigue Optimization

Spot-Weld Fatigue

Optimization

FILIP ANDERSSON

RHODEL BENGTSSON

R

OYAL

I

NSTITUTE OF

T

ECHNOLOGY

M

T

Spot-Weld Fatigue Optimization

Abstract

Sammanfattning

Acknowledgements

Contents

List of Figures

List of Tables

Chapter 1

Introduction

1.1

Objective

1.2

Delimiter

1.3

Fatigue Theory

∑

1.4

Finite Element Spot-Welds

1.5

Structural Optimization

∑

∑

∑

∑

∑ ∑

∑

1.6

Previous Works on Optimization and Spot-Welds

Chapter 2

Method

2.1

Verification of Rupp’s approach and comparison of weld

model types

2.2

Multi-Weld Model

2.3

Topology Optimization Setup

∑

∑

∑

∑

2.4

Size Optimization Setup

∑

2.5

Comparison of Optimization Techniques

2.6

Final Application