Link¨oping Studies in Science and Technology
Thesis No. 1325
Optimization of Castings by using
Surrogate Models
Erik Gustafsson
LIU-TEK-LIC-2007:34 Department of Mechanical Engineering Link¨oping university, SE-581 85, Link¨oping, Sweden
Cover:
Residual stresses in a stress lattice. The colors indicating different stress levels along the direction of the legs.
Printed by:
LiU-Tryck, Link¨oping, Sweden ISBN 978-91-85831-25-8 ISSN 0280-7971
Distributed by: Link¨oping university
Department of Mechanical Engineering SE-581 85, Sweden
c
2007 Erik Gustafsson
This document was prepared with LATEX, October, 2007
No part of this publication may be reproduced, stored in a retrieval system, or be transmitted, in any form or by any means, electronic, mechanic, photocopying, recordning, or otherwise, without prior permission of the author.
Preface
There are many people I would like to thank for supporting and helping me to get to this stage but first I would like to thank my employer SweCast AB for giving me the opportunity to become an industry graduate student.
Among the persons I would like to thank, the first one to mention is my su-pervisor, Associate Professor Niclas Str¨omberg at JTH. Without his support and encouragement this work would not have been where it is today.
Thanks to my graduate student colleague at JTH, Magnus Hofwing, for great discussions regarding residual stress analysis and also all very nice discussions about soccer and sports in general.
Thanks to my colleagues at SweCast for great input regarding the casting pro-cess. Magnus Wihed for performing tensile testing, Ulf Gotthardsson and Ulla Ledell for help with questions regarding and testing of mold material, J¨orgen Blom for help with forming of molds, Martin Risberg for introduction to Hyper-Mesh, Ola Bj¨ork for help with some CAD and finally Stefan Gustafsson Ledell and Lars-Erik Bj¨orkegren for their support in general.
Thanks to Volvo Powertrain, Tomas Bandh, Kent Eriksson, Mari Larsson, P˚al Schmidt and Gunnar ˚Akerstr¨om.
Thanks to graduate student Niclas Wiker at LiTH for providing me with an excellent LATEX template for this thesis.
Thanks to the financiers Swedish Knowledge Foundation (KK-stiftelsen), VIN-NOVA and Nordisk InnovationsCenter.
Finally, thanks to my fiance Jennie and our lovely daughter Tindra for your endless support and thanks to my parents for helping out with various things when there just isn’t time enough to deal with everything.
Erik Gustafsson Aneby, September 2007
Abstract
In this thesis structural optimization of castings and thermomechanical analysis of castings are studied.
In paper I an optimization algorithm is created by using Matlab. The algorithm is linked to the commercial FE solver Abaqus by using Python script. The opti-mization algorithm uses the successive response surfaces methodology (SRSM) to create global response surfaces. It is shown that including residual stresses in structural optimization of castings yields an optimal shape that differs significantly from the one obtained when residual stresses are excluded.
In paper II the optimization algorithm is expanded by using neural networks. It is tested on some typical bench mark problems and shows very promising results. Combining paper I and II the response surfaces can be either analytical func-tions, both linear and non-linear, or neural networks. The optimization is then performed by using either sequential linear programming or by using a zero-order method called Complex. This is all gathered in a package called StuG-OPT.
In paper III and IV focus is on the thermomechanical problem when residual stresses are calculated. In paper III a literature review is performed and some nu-merical simulations are performed to see where nunu-merical simulations can be used in the industry today. In paper IV simulations are compared to real tests. Several stress lattices are casted and the residual stresses are measured. Simulations are performed by using Magmasoft and Abaqus. In Magmasoft aJ2-plasticity model is used and in Abaqus two simulations are performed using eitherJ2-plasticity or the ”Cast Iron Plasticity” available in Abaqus that takes into account the different behavior in tension and compression for grey cast iron.
List of Papers
This thesis is based on the following four papers, which will be referred to by their Roman numerals:
I. E. Gustafsson, N. Str¨omberg, Optimization of Castings by using Successive Response Surface Methodology, Accepted for publication in Structural and Multidisciplinary Optimization, 2006.
II. E. Gustafsson, N. Str¨omberg, Successive Response Surface Methodology by using Neural Networks, Presented at 7thWorld Congress on Structural and Multidisciplinary Optimization, 2007.
III. E. Gustafsson, Residual Stresses in Castings - ’a literature review’, to be published as SweCast report.
IV. E. Gustafsson, M. Hofwing, N. Str¨omberg, Simulation and Measurement of Residual Stresses in a Stress Lattice, Presented at 2ndInternational Confer-ence on Simulation and Modelling of Metallurgical Processes in Steelmak-ing, 2007
Contents
Preface v Abstract vii List of Papers ix Contents xi 1 Introduction 1 2 Structural Optimization 3 2.1 Surrogate Models . . . 32.1.1 Analytical Response Surface . . . 5
2.1.2 Neural Networks . . . 6
2.1.3 Radial Basis Function Network . . . 7
2.1.4 Kriging . . . 8
3 Residual Stresses in Castings 11 4 Results 13 5 Summary of Appended Papers 17 5.1 Paper I . . . 17 5.2 Paper II . . . 17 5.3 Paper III . . . 17 5.4 Paper IV . . . 18 Bibliography 19 Paper I 29 Paper II 63
CONTENTS
Paper III 87
Paper IV 115
1
Introduction
There are higher and higher demands on casted components in modern products. They should withstand higher loads and at the same time be as light as possible. In the automotive industry there is also emission regulations and this also put high demands on casted components in engines since emission for example can be reduced by increasing the pressure or temperature in the engine. These regulations were originally introduced in [1] followed by a number of amendments. In 2005, the regulations were re-cast and consolidated by [2]. The emission regulations are often referred to as Euro I . . . V. In short these regulations drastically reduce the allowed NOxand particles from the engine. This is summarized in Table 1. It should be noted that the actual date for Euro IV and Euro V are 2006.09 and 2009.09 respectively, i.e. the introduction of these regulations are a little behind schedule.
Table 1: EU Emission standards for heavy duty diesel engines, [g/kWh]. Gate Date NOx Particles
Euro I 1992 8.0 0.36 Euro II 1998.10 7.0 0.15 Euro III 2000.10 5.0 0.10 Euro IV 2005.10 3.5 0.02 Euro V 2008.10 2.0 0.02
In order to be able to solve the problem of higher loads on casted components there is a need to include properties from the casting process into the structural stress calculations and further on perform structural optimization on these com-ponents while including casting constraints. An easy to use and correct method-ology where casting simulation and structural stress analysis are combined should therefore be very useful. Casting simulation is though a wide name of several simulation disciplines within the casting process. There are fluid flow simulation to determine the filling of the mould, solidification analysis from a metallurgical point of view where the location and size of pores and local material properties are
CHAPTER 1. INTRODUCTION
determined and finally thermo-mechanical solidification analysis where residual stresses, shape distortions and hot tear are determined. Since the casting simula-tion covers all these disciplines it is hard (impossible??) to cover all this in one project and the focus in this thesis therefore lies in combining thermo-mechanical solidification analysis with structural stress analysis. There are though ongoing projects, for example VIKTOR, in Sweden today where research groups consist-ing of people representconsist-ing several disciplines are workconsist-ing together to combine more disciplines from casting simulation with structural stress analysis.
Today there are a couple of leading softwares on the market for casting simu-lation like Magmasoft, NovaFlow & Solid and ProCast. The two first, Magmasoft and NovaFlow & Solid uses the Finite Difference Method (FDM) and ProCast uses Finite Element Method (FEM). All of these softwares are capable of simu-lating the complete casting process from filling to solidified material.
The casting constraint from the casting process that we would like to include is residual stresses and in order to do this the FE software Abaqus was used because it is easily handled by scripts and since the FE calculations are done in Abaqus it is convenient to have the residual stress calculation done in the same software. The scripts for Abaqus are written in the program language Python and the residual stresses from the casting process are then easily included in the structural stress analysis.
By doing structural optimization on casted components in trucks a significant weight saving can be achieved since casted components approximately represents a 10-15 % [3] of the total weight of the vehicle.
2
Structural Optimization
Structural optimization is today a well known method for designing components that are optimized according to some objective function such as for example min-imization of weight or stresses. Within the industry today this is a useful tool. Structural optimization of castings when taking into account the residual stresses from the casting process or other information from the casting process such as location or size of pores has not reach an industry standard yet. Suggested in this thesis is to use surrogate models when performing structural optimization on castings since it is a non-linear problem with perhaps many local minima.
The optimization problem that we would like to solve is to minimize some function g0(x) where x = (x1; x2; : : : ; xn) is a vector containing the design variables.g0(x) can be any non-linear function subjected to constraint functions gj(x). The optimization problem can be written as
SO 8 > < > : min g0(x) s.t. ( gj(x) 0 j = 1; 2; : : : ; m xlower i xi xupperi ;
wherexloweri andxupperi are lower and upper bounds respectively for the variable xiandm are the number of constraint equations.
2.1
Surrogate Models
Surrogate models or response surface methodology is an established method for optimization of various non-liner optimization problems. Let us consider a re-sponsey dependent of a set of variables x. The exact functional relationship then reads
y = g0(x): (1)
We would then like to approximate the unknown functiong0 with a surrogate function (response surface)f that we know
CHAPTER 2. STRUCTURAL OPTIMIZATION
over some region of interest (RoI)0.
The response is then evaluated atN points within the RoI so we at our experi-mental points havey = (y1; y2; : : : ; yN)Tandf(x) = f(x1); f(x2); : : : ; f(xN)T. The model can now be written as
y = f(x) + (x); (3)
where (x) are the modeling errors. If physical experiments are performed a random error is also present in (3).
The method then works such that the response surface is approximated to num-ber of simulations that are made and the optimization is then performed on these surfaces. These responses can e.g. be analytical functions, Kriging or neural net-works. Roux, Stander and Haftka were among the first to use RSM for structural optimization in 1998 [4]. Since then the method have been used for optimization of crashworthiness [5], optimization of multibody-systems [6] and optimization of sheet metal forming [7]. Other useful and interesting references are [8, 9, 10] and references therein. For more basic theory regarding RSM read [11] that gives a good introduction to RSM. In [12] there is an extensive list of various RSM activities from 1989.
An extension of RSM is the successive response surface methodology (SRSM) as proposed by Stander and Craig [13] where the region of interest (RoI) is grad-ually decreased around an optimal point. SRSM has been used successfully in for example [14, 15, 16] and [17].
In this thesis SRSM using both analytical response surfaces and neural net-works have been performed. Two other ways of representing response surfaces, radial basis function network and Kriging, are also described in coming subchap-ters.
2.1. SURROGATE MODELS
2.1.1
Analytical Response Surface
The response surfaces can as mentioned earlier be approximated by different ana-lytical functions. The most common form of these anaana-lytical functions are linear, parabolic and quadratic. The general analytical function for linear, parabolic and quadratic surface approximation will be as follows
yi= 0+ X j jxij+ "i; i = 1; 2; : : : ; Nj = 1; 2; : : : ; M; (4) yi= 0+ X j jxij+ X j jjxijxij+ "i; i = 1; 2; : : : ; Nj = 1; 2; : : : ; M ; (5) yi= 0+ X j jxij+ X j X k jkxijxik+ "i; i = 1; 2; : : : ; N j = 1; 2; : : : ; M k = 1; 2; : : : ; M ; (6) wherexiare the design points in the region of interest (RoI),"iis the error in-cluding both modeling (bias) and random errors,N is the number of evaluations andM are the number of parameters. The bias error is the difference between the response approximation function and the exact evaluated response.
In order to determine the unknown variables , the analytical function is writ-ten in matrix notation as
y = X(x) + ": (7)
is then found by minimizing the error " in a least square sense where this spe-cific value for denoted is given by
= XT(x)X(x) 1XT(x)y: (8)
To determine the value of the parameters a minimum number of simulations, Nmin, is needed. For the different analytical approximations the value ofNmin
are determined by the following equations gathered in Table 2. It has though been shown in [4] and [9] that1:5 Nminsimulations in every iteration might be used to get an efficient coverage of the design domain.
CHAPTER 2. STRUCTURAL OPTIMIZATION
Table 2: Minimum number of simulations needed for different D-optimal selec-tions. Surface Nmin Linear N + 1 Parabolic 2N + 1 Quadratic (N + 1)(N + 2) 2
2.1.2
Neural Networks
In addition to the more traditional way to represent the response surfaces by using analytical functions neural networks can be used. An artificial neural network might be seen as a mathematical model of a human brain. A human brain is a complex biological system of 1011 neurons. A neuron consists of a cell body, an axon and dendrites. Incoming signals are first interpreted by the cell body, which in turn sends the interpretation as a new signal by the axon to a network of dendrites. Here, the signal is captured by new neurons through synapses. In the artificial network the first step is modeled by a summation and a transfer function. The output is weighted and linked to layers of new transfer functions. This step represents the axon, dendrites and synapses. A brain is learned by adjusting the synapses. In a similar manner, the artificial neural network is trained by finding optimal weights. Neural networks were introduced in 1943 by McCulloch and Pitts [18] where a single neuron was modeled. An example of a single neuron is shown in Figure 1 wherexiare the inputs,wiare the weights andb is the bias. The expression forv is then calculated as
v = b +Xwixi; (9)
which is feed forward to the transfer functionf. The transfer function might have several formats, e.g. see [19, 20]. The final output from the neuron is then
y = f(v): (10)
Further information regarding neural networks can for example be found in [19, 20, 21]. In this work we use a feed-forward multi-layer network with back-propagation developed in [22]. The network consists of two hidden layers and one output layer. The transfer functions in the hidden layers are log-sigmoid functions and a linear function is used in the output layer. A principle sketch of the network is shown in Figure 2.
2.1. SURROGATE MODELS
Figure 1: A single neuron withn input variables.
Figure 2: The neural network used in this work.
2.1.3
Radial Basis Function Network
Another similar type of network is the Radial Basis Function Network. In [23] an introduction to this methodology is given. The responsef from such a network is given by
f(x) =Xm
j=1
wjhj(x) (11)
wherex are the design variables, wjare the weights to be determined,hjare basis functions andm is the number of basis function one would like to use. Any set of function can be used as basis function but it is preferred if it is a differentiable equation. A simple example is the straight line
f(x) = ax + b; (12)
which is a linear model whose two basis functions are
h1(x) = 1; (13)
h2(x) = x; (14)
and whose weights arew1 = b and w2 = a. To make it more general the radial basis functions are used, that have the special feature that their response is affected
CHAPTER 2. STRUCTURAL OPTIMIZATION
by the distance from a central point. An example is the Gaussian function which for the case of a inputx looks like
h(x) = e (x c)T (r2x c) (15)
Its parameters are the centerc and the radius r. These functions can then be gathered in a network. An example of a network is shown in figure 3.
Figure 3: Example of a radial basis function network. [23]
Further and recent information regarding this method is found in e.g. [24] and [25].
2.1.4
Kriging
Kriging is when the responsey(x) is a combination of a global linear regression modelfT(x) and a random process Z(x)
y(x) = fT(x) + Z(x): (16)
By doing this a local correction of the global model is obtained and an exact representation of the responses at all training points can be achieved. The values of and Z(x) in (16) are then obtained by the following procedure as explained in [26]. After performingN simulations we have the following system of equations
y = X + Z: (17)
This is similar to (7) but now the residualsZ are correlated according to a corre-lation functionR(x; w) where x and w are different training points.
cov(Z) = 2 0 B @ R(x1; x1) : : : R(x1; xN) .. . . .. ... R(xN; x1) : : : R(xN; xN) 1 C A = 2R D (18) 8
2.1. SURROGATE MODELS
whereR(xk; xl) = e (xk xl)2, andk; l 2 [1; 2; : : : ; N], is the correlation func-tion and2 is the product of the process variance. Other types of correlation functions can also be used. is known as the correlation function parameter.
Since the residuals are correlated (17) is multiplied by a weighting matrixW and the following is obtained
W y = W X + W Z ) y= X + Z: (19)
W is defined as
W = (cov(Z)) 1= 2R 1
D : (20)
The coefficients can now be found in a least square sense. This specific value of denoted is given by
= (XTR 1
D X) 1XTRD1y; (21)
and we can also determine by maximizing the log-likelihood function L() = (Nln(s2) + ln(det(R
D))); (22)
where
s2= 1
N(y X ∗)TRD1(y X ∗): (23)
Our actual expression for the response at a pointx0can then be written as y(x0) = fT(x) + rT(x0)RD1ZD; (24) where r(x0) = 0 B @ R(x0; x1) .. . R(x0; xN) 1 C A (25)
is the vector of residuals andZDrepresents the residuals for the training designs
ZD= y X : (26)
In (24) the second term is an interpolation of the residuals of the regression model fT(x
0) . This results in a correct prediction if the response surface at the
train-ing points.
More information regarding Kriging can be found in [27]. How Kriging is being used in structural optimization are shown in for example [28] and [29].
3
Residual Stresses in Castings
Residual stresses in castings has been studied for approximately 45 years. One of the first to study this was Weiner and Boley that in 1963 published a paper where they analytically determined the stresses in a solidifying slab [30]. Tien and Koump [31] also solved this analytically but introduced some temperature dependent material properties. Around this time Perzyna presented his theory regarding visco-plasticity [32] that could be very useful for thermomechanical solidification analysis. This was implemented by Zienkiewicz and Cormeau, [33]. In [34] viscoplasticity was used for thermal stress calculations in castings. In [35] there is good overview of research field of residual stresses in castings until 1995. Among the work done in recent years there are for example [36, 37, 38, 39] and references therein.
Regarding residual stress analysis on casted components for the automotive industry there are a few interesting papers like [40] but here they only study stresses caused by quenching since a stress relieving process has removed all stresses caused by the casting process. The same questions are studied in [41]. In [42] the software MagmaSoft is used to determine the thermal history in a gray iron casting and the following thermo-mechanical analysis is done in Abaqus. In [42] the fact that gray iron is behaving different in tension and compression is considered by using the material model ”Cast Iron Plasticity” that is available in Abaqus. A material model for gray iron similar to the one available in Abaqus is the one presented by Hjelm [43, 44, 45]. Wiese and Dantzig [46] has proposed another constitutive model for cast iron plasticity and showed how it can be used in solidification analysis.
The use of Abaqus for residual stress calculations is well known. Some ex-amples are [47] where Abaqus is used to determine residual stresses in cast iron calender rolls and [48] where the coupled thermal-stress model in Abaqus is used to calculate temperature, strain, and stress fields in an aluminium alloy. In [49] and [50] the work in Abaqus has been focused on developing new sand surface elements to reduce calculation time.
In this thesis residual stresses has been calculated by using either Abaqus or Magmasoft. TheJ2-plasticity model in Abaqus has been used when structural
CHAPTER 3. RESIDUAL STRESSES IN CASTINGS
optimization has been performed on a simple casted component. The residual stress calculations in Abaqus and Magmasoft have also been compared to real tests. In Magmasoft aJ2-plasticity model has been used and in Abaqus bothJ2 -plasticity and ”Cast Iron Plasticity” have been used for comparisons.
4
Results
The developments in this thesis is that an optimization algorithm is written in Matlab and an investigation of the capabilities of simulating residual stresses in casting has been performed. The optimization algorithm uses the SRSM which has proven to work very good for non-linear structural optimization problems. The response surfaces can be either analytical functions, linear and non-linear, as well as neural networks. The optimization code has today been used together with Abaqus and an in-house FEM-software called TriLab and can by very easy manipulations be used with any FEM-software that can be run via scripts.
Regarding the residual stress simulations, several stress lattices have been casted and the residual stresses have been measured and compared to simulation results from the two softwares Magmasoft and Abaqus. Some numerical exam-ples of how residual stress simulations can be used in the product development process is also shown. Next follows some of the most important results obtained in this thesis.
The developed optimization routines has been gathered in a package called StuG-OPT and the optimization code has been tested on several problems. In paper I the SRSM algorithm with analytical non-linear response surfaces has been tested on a beam where the objective is to minimize the weight of the beam under a constraint on the maximum von Mises stress in the beam. The optimization is performed while both including and excluding residual stresses from the casting process. In Figure 4 the optimized geometries can be seen.
CHAPTER 4. RESULTS
(a) Optimal shape obtained for pure me-chanical problem, SRSM optimization.
(b) Optimal shape obtained for thermome-chanical problem, SRSM optimization.
Figure 4: Comparison of obtained optimal shapes.
In paper II neural networks is added to the SRSM and this will increase the capabilities of the response surface to capture more non-linearities. The optimiza-tion is tested on some typical benchmark problems like an elastic plate with a hole in the center and a rivet through a plate [51]. For the plate with the hole the objective is to minimize the von Mises stress in the plate under a constraint on volume when it is subjected to a tensile load. The von Mises stress in the plate with initial shape and the plate with optimal shape can be seen in Figure 5. The decrease in maximum von Mises stress is approximately 37%. For the rivet, the objective is to minimize maximum contact pressure between the rivet and plate under a constraint on the volume of the rivet when a tensile stress is applied to the bottom of the rivet. In Figure 6(a) the optimal shape of the rivet is seen and in Figure 6(b) the contact pressure for initial and optimal design is shown.
(a) Initial geometry, = 414MPa. (b) Optimal geometry, = 262MPa.
Figure 5: Von Mises stress in plate with hole.
(a) Optimal shape of rivet. (b) Contact pressure between rivet and plate.
Figure 6: To the left optimal shape of rivet is shown and on the right the contact pressure for initial design (solid line) and optimized design (dashed line) is shown. In paper IV the two softwares Magmasoft and Abaqus are compared to real tests on stress lattices. In Magmasoft, the residual stresses are simulated by using aJ2-plasticity model and in Abaqus two different constitutive models are used, J2-plasticity and the ”Cast Iron Plasticity” (CI-plasticity) mentioned in previous
chapter. After casting the lattices are cut and the area of the remaining fracture surface is measured. Tensile test bars are then prepared with a cut similar to the cut that causes the lattices to fracture. The prepared test bar are then tested and the force that is needed for fracture is recorded and compared to the forces seen in the simulations. The results are summarized in Table 3. It is also possible to see how the residual stresses develop through the solidification/cooling and this is seen in Figure 7. Notice that there is a difference in the obtained results from the two softwares.
Table 3: Simulated and the average of measured reaction forces after cutting. Material model of plasticity Software Reaction force [kN]
J2 Abaqus 55.6
J2 Magmasoft 39.9
CI-plasticity Abaqus 75.0
(a) Positions of where the stresses are taken in the stress as a function of time re-sults.
(b) Abaqus -J2-plasticity.
(c) Abaqus - CI- plasticity. (d) Magmasoft -J2-plasticity.
5
Summary of Appended Papers
5.1
Paper I
In this paper an optimization algorithm is developed and used for structural op-timization of a casted beam. The algorithm is written in Abaqus and linked to the FE-solver Abaqus through python scripts. The optimization algorithm uses successive response surface methodology (SRSM) and the optimization is per-formed using sequential linear programming. It is shown that by including resid-ual stresses in structural optimization of casting this yields an optimal shape that might differ significantly from the optimal shape obtained when residual stresses are excluded.
5.2
Paper II
The SRSM algorithm is in this paper extended to include neural networks. By doing this the response surface is capturing non-linearities in the response more efficiently. The network used is a so called feed-forward backpropagation net-work with two hidden layers and one output layer. The optimization algorithm is benched towards one classical linear elastic problem and two non-linear problems where one is including contact and the other is the thermomechanical problem studied in paper I
5.3
Paper III
This is a survey of the research in the area of residual stress calculations. A short introduction to plasticity is given and some constitutive models that can be used for residual stress calculations are presented. Different techniques available for measuring residual stresses are also presented. Finally, two numerical examples are given to show how casting simulation can be used.
CHAPTER 5. SUMMARY OF APPENDED PAPERS
5.4
Paper IV
In this paper two commercial softwares, Magmasoft and Abaqus, are compared to real tests in order to evaluate their residual stress calculation capabilities. Sev-eral stress lattices are casted and compared to simulations. In Magmasoft aJ2 -plasticity model is used and in Abaqus two different -plasticity models are used,J2 -plasticity and cast iron -plasticity which is a -plasticity model available in Abaqus that takes into account the different behavior in tension and compression for grey cast iron.
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