automatic design of marine structures by using successive response surface method

Struct Multidisc Optim (2014) 49:863–871DOI 10.1007/s00158-013-1013-7

INDUSTRIAL APPLICATION

Automatic design of marine structures by using successiveresponse surface method

Sami Pajunen · Ossi Heinonen

Received: 17 December 2012 / Revised: 9 September 2013 / Accepted: 8 October 2013 / Published online: 26 October 2013© Springer-Verlag Berlin Heidelberg 2013

Abstract An automated optimization procedure based onsuccessive response surface method is presented. Themethod is applied to weight optimization of a stiffened plateused in marine structures. In the design space the surrogatemodel is spanned sequentially into an optimally restrictedsubspace that is converging towards at least a local opti-mum. Both objective function and all constraint functionsare modeled using linear response surface method enablingthe use of a robust and efficient simplex algorithm forthe optimizations. Special attention is paid to CAD andFEM-model linking that plays a central role in practicalindustrial applications. In this project SOLIDWORKS andANSYS software are adopted for structural modeling andanalysis, respectively, and the optimization is carried outin a MatLab environment. The reported results achieved inthis project prove the robustness and effectiveness of theproposed approach.

Keywords Surrogate model · Response surface method ·Optimal design · Simplex-algorithm

1 Introduction

The main goal of automated product design is to set upa robust modeling, simulation and optimization procedurethat produces an optimized product with a minimum amount

S. Pajunen (�) · O. HeinonenDepartment of Mechanics and Design,Tampere University of Technology, Tampere, Finlande-mail: [email protected]

O. Heinonene-mail: [email protected]

of user effort. Automated product design has many advan-tages, routine work such as finite element model generationand re-meshing can be done automatically, structural opti-mization can be carried out systematically and the designercan handle multiple design cases simultaneously. The auto-mated design process becomes extremely efficient in caseswhere the product design is driven by customer require-ments. Typical examples of such individually designedproducts are beam cranes for which the customer defines themaximum lift capacity and the beam span length, as well asstiffened plate structures in shipbuilding and offshore equip-ment for which the customer defines the main dimensionswhile the loads are defined according to technical rules setby classification societies.

In this paper, an automated design procedure for a stiff-ened plate structure used in various marine constructions isreported. The procedure is developed to replace the manualdesign process formerly used in the tendering and designphases of the products at issue. The procedure is initi-ated with a generic CAD model containing the essentialunchanged data and the variable design parameters. Themodel is then linked to an FE-modeler and FE-solver thatare augmented with an iterative optimization scheme. Dur-ing the optimization, the original FE-problem is replaced bya surrogate model that carries out the numerous constraintfunction evaluations in a fast and computationally inexpen-sive manner. In this project, the standard response surfacemethod is adopted for the linear regression model. Moresophisticated models for the problem in hand are studiedin (Heinonen and Pajunen 2011), but the linear model isadopted for this project because of its efficiency to estimategradients.

In general, surrogate models have been widely appliedto various structural engineering problems like turbine discs(Huang et al. 2011), crash absorbers (Acar et al. 2011) and

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mailto:[email protected]

864 S. Pajunen, O. Heinonen

ship structures (Arai and Shimizu T 2001), just to namea few. There are several different types of surrogate mod-els, of which the response surface methodology, or in otherwords the polynomial regression model, is based on a prioridefined polynomial basis functions and has been success-fully applied to various cases (Abu-Odeh and Jones 1998;Ren and Chen 2010; Roux et al. 1998; Yoo et al. 2011).The kriging model includes a polynomial but also a stochas-tic basis functions and becomes a more attractive alternativeif the problem is more complex and non-linear (Sakataet al. 2003; Simpson et al. 2001). There are other methods,such as radial basis functions (Acar et al. 2011) or neu-ral networks (Gupta and Li 2000) but the response surfacemethod and the kriging method are the two most widelyused approaches in structural mechanics. A comprehensiveliterature survey on various surrogate models used in struc-tural optimization is given generally in (Queipo et al. 2005)and especially for the response surface methodology in(Kleijnen 2008). An encompassing study about the perfor-mance of various surrogate-based optimization techniquesapplied to a large number of test cases can be found in(Eldred and Dunlavy 2006).

A key factor in the use of surrogate models is han-dling of the approximation error. The approximation errordepends crucially on the size of the sub-space in which theapproximation is set, number and location of the design-of-experiment points, and on the basis functions used toapproximate the model response. In the case of the responsesurface method, it means that larger sub-space requireshigher order basis functions. It is, however, noticed byHuang et al. (2011), that higher than second-order poly-nomial basis functions should not be used because theyproduce oscillations. It is also noticed by Heinonen andPajunen (2011) that the simplest, linear basis functions maylead to the most efficient method due to their low number ofdesign-of-experiment points. However, it is case-dependentwhether it is more efficient to use small sub-spaces with lin-ear basis or somewhat larger sub-spaces with second orderbasis functions. The selection between linear and higherorder models must be carried out separately for differenttypes of design problems. In this paper we consider only onedesign problem type, for which the linear model is foundto be very competitive (Heinonen and Pajunen 2011), andthus, the selection between linear and higher order modelsis outside of the scope of this paper.

In this work, linear basis functions are used, while theapproximation sub-space bounds and the number of design-of-experiments points are defined in a way that the approx-imation error remains below an acceptable, a priori definedlevel. The sub-space is re-defined at each optimization iter-ation step based on the latest iteration configuration asexplained in chapter 4. The proposed method can be seenas a variant of the Successive Response Surface Method

(SRSM) described in (Roux et al. 1998; Stander and Craig2002). The other sections of the paper are arranged asfollows. In chapter 2, topics concerning geometry and sim-ulation model construction and linking are discussed indetail. In this work, SOLIDWORKS and ANSYS softwareare adopted, but the developed method can be set up in everymodern CAD and FE software environment that enablesthe software linking as described in (Park and Dang 2010).Chapter 3 outlines the basic formulas for the adopted sim-ple surrogate modeling. In chapter 5 a benchmark problemis solved after which a design process of a typical marinestructure is carried out using the proposed method.

2 Geometry and simulation models

In this project, our main goal is to create an automatic designtool, which for given input data, results in an optimizedstructure that meets the order and strength requirements ofthe customer. Additionally, the tool should be robust, fastand accurate. The structure considered is a stiffened platestructure commonly used in marine industries. The mainmembers of the structure are the top plate, the longitudi-nal and transversal support beams, the top plate stiffenersand the bottom plate as depicted in Fig. 1. The devel-oped tool should be as general as possible so that e.g. themain dimension of the structure and the number of stiff-ener beams must be variables. Similarly, all possible loadingand supporting conditions, as well as the material type ofthe parts are parameterized in the model. For this reason,a generic surface model was first constructed in CAD soft-ware SOLIDWORKS. In the tendering phase, the customergives the main dimension data and loading conditions, afterwhich these variables can be fixed in the generic model.All the other CAD-model parameters can then be used asdesign variables in optimization. In order to fulfill the strictrequirements set for the developed tool, special attentionmust be paid to CAD and FE model construction and linkingas explained in what follows.

The main support beams are modeled by creating a planefor each longitudinal and each transversal stiffener beam.The surfaces for each beam web are then modeled on theseplanes. Locations of the planes and thereby of the beamsare controlled by simple sketch lines. Support beams canbe removed from the model by suppressing the associatedplane. Figure 2 shows two different beam configurations.

The mid-surface geometry model is linked to an FE-modeler–ANSYS Workbench in this case—in which thestructure is meshed using the program’s mesh control set-tings to ensure a good quality mesh. The element size isfixed between 50 mm and 125 mm and the advanced sizefunction on curvature is used. One half of the structure is

Automatic design of marine structures by using successive response surface method 865

Fig. 1 Half of the structurewith part of the top plate hiddento reveal the main structuralmembers

modeled using symmetric boundary conditions. The struc-ture is loaded with multiple uniform pressures at certainareas of the top plate and it is simply supported at designatedpoints. The loads cause mainly compressive stress in the topplate, tensile stress in the bottom plate and remarkable shearstresses in the webs of the support beams.

To enable automated creation of the simulation model,a template file is written in ANSYS. This template filecontains APDL (Ansys Parametric Design Language) com-mands that are run in the preprocessor phase which handlesthe load, support, and material model definitions needed.For example, the supports for the model are defined withAPDL command D that defines the displacement and rota-tions of certain nodes. To select certain nodes and elementsfor setting the supports and loads, a feature called NamedSelections is utilized in the geometry model.

Named Selections are selection groups that can con-tain nodes or elements, and they can be linked so that the

selection groups are common for both the geometry andsimulation models. This allows us to create a selectiongroup containing e.g. all the vertices in the geometry modelthat are simply supported in the simulation model. Likewise,as we utilize symmetry and use only half of the structure,all the edges in the symmetry plane can be defined as aNamed Selection for which the displacements and rotationsare defined using the command D. Loads are also defined inthe same manner but using the command SF for the surfaceload.

Since a surface model is used, the surface thicknessesmust be defined separately. To do this automatically, theNamed Selections are utilized so that all the surfaces withthe same thickness are collected to the same selectiongroups. The thickness of each element in each selectiongroup is defined with the commands SECTYPE and SEC-DATA.

Fig. 2 Two differentconfigurations for the structureconsidered having differentdimensions and beamarrangements


Besides automating the FE-model creation and FE-analysis, the solution of the optimization problem is alsoautomated. The actual optimization procedure will bedescribed in chapter 4. For the optimization automating, theNamed Selection groups and related APDL commands areutilized in the post-processing phase after running the FEanalysis.

In the optimization the mass of the structure is mini-mized with respect to constraint equations related to allowedstresses and stability. Design rules related to marine indus-tries are taken into account (IACS 2009). The stresses in thebottom plate are constrained so that the material will notyield and stresses in the support beam webs and top plateare constrained so that the structural members will not losetheir stability with respect to the design rules.

Critical areas from each structural member group canbe pointed out and the stresses are analyzed only at thesespecific locations. From the top plate e.g. the most criticalplate fields with respect to buckling are pointed out. Theseplate fields are defined as Named Selections in the geom-etry model and this way the stresses, dimensions etc. ofthe specific plate fields can be accessed with APDL post-processing commands. The same procedure is utilized withthe most critical parts of other structural members as well tocover all the constraint equations. The mass of the structurecan also be easily accessed through APDL commands.

In the optimization problem, the design variables areassociated to plate thicknesses of various areas of the struc-ture defined by separate Named Selection groups. In orderto parameterize the surface thicknesses the input argumentsrelated to the APDL commands were used allowing thearguments being used as design variables.

To complete the ANSYS template file, the input and out-put variables still need to be chosen as parameters in theproject. For the input parameters this means simply tickingthe boxes related to the input arguments. To extract outputparameters from the post-processing commands, a prefix isused with the output variables that the user wants to useas parameters. The default prefix is my and it can be usede.g. with the *GET command. For example, the command*GET, my sx, NODE, N, S, X would give the x-componentof stress in node number N . By using the prefix, ANSYSidentifies the parameters and lists them in the details of theAPDL command.

By defining the Named Selections in the geometry modeland adding related commands in both the preprocessing andthe post-processing phase in ANSYS Workbench, a fullyautomated parametric model is developed. The end usercan now modify the master geometry, loads and materi-als according to the customer’s order and a parametric FEmodel is then created fully automatically.

3 Response surface approximations

In the polynomial regression model, the unknown functionis expressed using usually either linear polynomial as

y(x) = β0 +n∑

i=1

βixi + ε (1)

or in the case of second-order polynomial as

y(x) = β0 +n∑

i=1

βixi +n∑

i=1

βiix2i +

n∑

i=1

n∑

j≥i

βij xixj + ε (2)

in which y is the unknown estimated function of factorsxi collected in vector x, and ε is a random error, whichin deterministic cases equals to zero. If the number ofdesign-of-experiment points is larger than the number ofcoefficients β0, βi , βii and βij , a regression analysis can beutilized in order to fine-tune the surface so that certain errormeasures are minimized. The quadratic polynomial is full ifall the cross-terms xixj are included in the surface expres-sion and if the cross-terms are omitted, the model is calleda pure quadratic model. The coefficients β0, βi , βii and βijcan be listed in vector β that can be solved using e.g. methodof least squares leading to the equation (Montgomery 2001)

β = [XTX]−1XTy (3)

in which X is a matrix containing the design points and yis a vector containing the estimated function values at thedesign-of-experiment points.

The number of design-of-experiment points and theirlocations can be defined in several ways. In this work, a D-optimal method (Khuri and Cornell 1987) with a minimumnumber of design-of-experiment points is adopted. For a lin-ear model, the minimum number of design points is n+1and the use of a minimum number of points results in zeroestimation error at these points. Other methods like the LatinHypercube Sampling (LHS) method (Queipo et al. 2005),the Central Composite Design (CCD) and orthogonal arrays(OA) are reviewed in e.g. (Montgomery 2001; Khuri andCornell 1987).

4 Optimization procedure

The optimization problem is written as

min m(x)g(x) ≤ 0xL ≤ x ≤ xU

(4)

in which m is the mass of the structure, x is a design variablevector containing continuous size variables and g is a con-straint function vector containing displacement, stress andbuckling functions, whereas subscripts L and U denote thelower and upper limits, respectively.


Fig. 3 Feasible design spaceand sequential optimization

2nd iteration optimum = 3rd ROI origin

initial ROI

initial configuration

1st iteration optimum = 2nd ROI origin

xi

xj

xi,L xi,U

xj,L

xj,U

feasible design space

…

final optimum

gk=0

In this work, optimization is applied to an industrialproduct design that has been designed manually during thelast decades. Hence, a vast amount of data from previousprojects makes it rather straightforward to select a feasibleinitial design point rather close to the final optimal solution.However, when the proposed method is applied to otherstructures, the initial design point selection does not play acentral role due to the fact that initial point feasibility doesnot affect the robustness of the method, but only the numberof required iterations.

After the initial point selection, the linear response sur-face method with D-optimal design-of-experiment pointselection is used to span responses of each output variableappearing in the optimization problem. The responses arespanned into the Region of Interest (ROI) centrally aroundthe initial point x0 as depicted in Fig. 3 so that

(1 − α1)x0i ≤ xi ≤ (1 + α1)x0

i (5)

in which α1 is the initial value of radius of the ROI, thatis chosen based on experience of the problem in hand andx0

i is the initial value of the design variable xi . In order tocalculate the approximation error, the true responses are cal-culated additionally in the centre of the ROI and the valuesare compared to the estimated responses.

Once the surrogate model is constructed using only lin-ear response surfaces, the original optimization problem canbe solved within the ROI very efficiently using the simplex-algorithm. Optimization is very robust and rapid due toanalytical and very simple estimation functions for m and g.

The next optimization iteration is taken as depicted inFig. 3. The subspace size is changed adaptively so thatapproximation error, measured in the centre of the ROI sat-isfies the accuracy tolerance. Since the D-optimal methodused tends to locate the design points at ROI corners and theuse of a minimum number of design points results in zeroestimation error at design point locations, the centre of theROI is a natural choice for the location of estimation error

definition. The radius α of the ROI is modified to match thea priori defined target error e as

αj+1 =max

(yi−yiyi

)

eαj (6)

in which the superscript j denotes the iteration cycle andsubscript i denotes the computed response, y is the truevalue and y is the estimated value of the associated responsecomputed in the origin of the ROI. However, in order toavoid oscillation, maximum change of the radius of the ROI

Define model dimensions and beam configuration

SOLIDWORKS

Modify ROI according to Fig. 3

Define design points within ROI

Create response surface

Run optimization on response surface

MATLAB

Run analyses on defineddesign points

ANSYS

Optimum reached?

Yes

No

Run binary integer optimization

MATLAB

Fig. 4 Flowchart illustrating the optimization procedure


-70

-60

-50

-40

-30

-20

-10

0

1 3 5 7 9 11 13 15 17 19

ob

ject

ive

fun

ctio

n

Iteration cycle

init (1,1), ROI fixed

init (1,1), ROI adaptive

init (5,5), ROI fixed

init (5,5), ROI adaptive

0

1

2

3

4

5

6

1 3 5 7 9 11 13 15

des

ign

var

iab

les

Iteration cycle

x1, init (1,1), ROI adaptive

x2, init (1,1), ROI adaptive

x1, init(5,5), ROI adaptive

x2, init(5,5), ROI adaptive

Fig. 5 Convergence of the objective function and the design variablesfrom different initial points

is limited to 50 %. In the jth optimization iteration the sur-rogate model is spanned so that the ROI origin is set toan optimum point xj−1 found at previous iteration j -1 andusing the ROI limits

(1 − αj)xj-1i ≤ x

ji ≤ (1 + αj)x

j-1i (7)

In a case where the sequentially constructed ROI crosses theglobal design variable limits as given in (4), the subspaceis shifted in a normal direction of the feasible design spaceso that the ROI face is coincident with the feasible designspace face.

The iteration is terminated once the change in the objec-tive function is below required tolerance as

mj −mj−1

mj≤ tol (8)

in which the superscript denotes the optimization iterationcycle. Since the estimation error is handled adaptively dur-ing the iteration, there is no need for a separate check ofconstraint function convergence.

Being a practical optimization problem, continuousdesign variables are not sufficient in general. In order togenerate a discrete variable optimum solution from thecontinuous problem solution, a simple binary integer opti-mization is adopted as follows. Let x* be the continuousoptimum and y an integer design variable vector. The integerdesign problem can be written as

min m(y)g(y) ≤ 0yL ≤ y ≤ yU

(9)

The integer design problem can be modified to a binaryinteger optimization problem as

min m(z)g(z) ≤ 00 ≤ z ≤ 1

(10)

in which the binary variable vector z components are

zi = y − yi,L

yi,U − yi,L(11)

The binary integer optimization problem is solved usingthe branch-and-bound method (Nemhauser and Laurence1988). It should be noted that the binary integer opti-mization method is very suitable for problems where thecontinuous design variable can be rounded to two distinctdiscrete values. The proposed optimization procedure isoutlined in Fig. 4.

Fig. 6 Main dimensions of theoptimized structure inmillimeters


5 Validation of the method

The proposed method is first applied to a benchmark prob-lem no 12 reported in Hock and Schittkowski (1981) afterwhich the method is adopted for a stiffened plate designproblem frequently appearing in marine constructions.

The benchmark problems reads as

min 0.5x21 + x2

2 − x1x2 − 7x1 − 7x2

25 − 4x21 − x2

2 ≥ 0(12)

in which the continuous design variables x1 and x2 areunlimited. In the problem, both the objective and con-straint functions are quadratic and the optimum point issought from various initial points as depicted in Fig. 5.The proposed optimization method finds the optimum in arobust way starting for feasible and infeasible initial pointsx0 = (1,1) and x0 = (5,5), respectively. This simple testalso highlights, that the use of adaptive iteration step sizeimproves the convergence rate of the method remarkably. Inthe reported computations, α = 0.1 (fixed ROI) or α1 = 0.1& e = 0.001 (adaptive ROI), and tol = 0.001. From boththe initial points, the optimum (x1, x2)∗ = (2, 3) is foundexactly.

The method is then applied to the design process of astiffened plate structure depicted in Fig. 6. The structure issimply supported at support beam ends and the boxes areconnected with bar elements modeling the hinges that con-nect the panels and their vertical displacements at specificpoints.

The structure is loaded uniformly on top panels with apressure of 45 kPa. The load is based on the design rules(IACS 2009) and consists of a static load and a dynamiccoefficient.

The generic CAD-model is used with the initial inputconcerning the main dimensions, beam configuration, platestiffener locations and their thickness. The optimizationproblem contains 17 thickness design variables, eight of

Fig. 7 The most critical areas (shaded) of the top plate are checkedagainst buckling

18000

18500

19000

19500

20000

20500

21000

21500

22000

0 2 4 6 8 10 12

Mas

s (k

g)

iteration cycle

initially infeasible

initially feasible

Fig. 8 Convergence of the mass. First cycle stands for the initialdesign

which belong to various top plate areas, four to bottom plateareas, three to transversal beam webs and two to longitudi-nal beam webs. As side constraints, the minimum thicknessis 7 mm for the bottom plate and 8 mm for the other parts.The maximum thickness for all parts is 14 mm, but theseconstraints never become active.

Constraints of the optimization problem are derived fromboth the design rules and the empirical knowledge gath-ered from previous projects. The top plate is mostly under

0,4

0,5

0,6

0,7

0,8

0,9

1

1,1

0 2 4 6 8

con

stra

int f

un

ctio

n v

alu

e

iteration cycle

initially feasible

constraint 1

constraint 4

constraint 10

constraint 19

0,4

0,5

0,6

0,7

0,8

0,9

1

1,1

1,2

1,3

1,4

0 2 4 6 8 10 12

con

stra

int f

un

ctio

n v

alu

e

iteration cycle


constraint 1

constraint 4

constraint 10

constraint 13

Fig. 9 Convergence of a selected set of critical constraint functions


0

0,02

0,04

0,06

0,08

0,1

0,12

0 2 4 6 8 10

RO

I rad

ius

size

iteration cycle


initially feasible

Fig. 10 Radius of the Region of Interest during optimization iteration

compressive and shear stress and is constrained againstbuckling. Figure 7 highlights the nine critical plate fieldsthat are checked against buckling according to the designrules. The locations of areas prone to buckling are chosenwith the aid of empirical knowledge. Constraints 1–9 are topplate buckling constraints.

The beam webs are mostly under shear stress and are con-strained against buckling. Constraints 10–18 are transversaland 19–25 longitudinal beam web buckling constraints. Thebottom plate is mostly under tension and shear stresses, sothat the most critical areas of the bottom plate are con-strained against material yielding according to the designrules (IACS 2009). Constraints 26–34 are bottom plate

Table 1 Design variable values at the optimum

Infeasible initial design Feasible initial design

Initial Continuous Integer Initial Continuous Integer

8 8,44 9 9 8,44 9

8 8,00 8 9 8,00 8

8 8,80 9 9 8,81 9

8 8,38 9 9 8,38 9

8 9,15 10 9 9,15 10

8 8,80 9 9 8,79 9

8 9,10 10 9 9,08 9

8 8,00 8 9 8,00 8

7 7,44 8 9 7,44 8

7 8,27 9 9 8,27 9

7 9,97 10 9 9,97 10

7 7,00 7 9 7,00 7

8 8,00 8 9 8,00 8

8 8,00 8 9 8,00 8

8 8,25 9 9 8,24 9

8 9,91 10 9 9,90 10

8 8,00 8 9 8,00 8

material yielding constraints. There are altogether 34 con-straint inequalities in the optimization problem, all of whichare scaled so that their allowed value is below one.

The structure is optimized using two different initial con-figurations. First the optimization is carried out so thatall the design variables have their allowed minimum valuedefined in the design rules (IACS 2009). For the bottomplate the allowed minimum is 7 mm and for the other partsit is 8 mm. Another initial configuration is chosen from

Table 2 Objective (1st row) and constraint function (2nd–34th rows)values at the optimum

Estimation True response Estimation error (%)

Continuous Integer Integer

variables variables variables

19995 20631 20631 0,0

1,00 0,92 0,92 0,1

1,00 0,85 0,85 0,2

0,74 0,74 0,74 0,3

1,00 0,82 0,82 0,6

0,90 0,90 0,90 −0,1

1,00 0,87 0,86 1,1

0,99 0,73 0,73 0,3

1,00 1,00 1,00 −0,1

0,36 0,35 0,35 0,1

1,00 0,92 0,92 0,0

0,71 0,68 0,67 0,2

0,81 0,80 0,80 −0,3

1,00 0,94 0,94 −0,1

0,94 0,88 0,88 0,0

1,00 0,99 0,98 0,1

0,96 0,95 0,95 0,2

0,93 0,91 0,91 0,2

0,69 0,64 0,64 −0,1

1,00 0,97 0,97 0,4

1,00 0,86 0,85 0,7

0,77 0,79 0,79 −0,1

0,25 0,27 0,27 −0,6

0,00 0,00 0,00 −0,1

0,75 0,75 0,75 −0,3

0,52 0,53 0,53 −0,2

0,72 0,71 0,71 0,0

0,66 0,66 0,66 0,0

0,67 0,66 0,66 −0,2

0,46 0,46 0,46 −0,2

0,63 0,63 0,63 0,6

0,29 0,28 0,28 −0,2

0,33 0,33 0,33 −0,1

0,18 0,17 0,18 −0,8

0,72 0,71 0,71 0,0


the feasible design space so that all variables equal to 9mm. The convergence of objective function is depicted inFig. 8. Altogether 11 constraints become active for the con-tinuous optimization problem, and a representative set oftypically behaving active constraint functions are depictedin Fig. 9. The ROI radius size evolution is shown in Fig. 10and the design variable optimum values and final objectiveand constraint function values are listed in Tables 1 and 2,respectively. In the computations, the maximum estimationerror within each ROI is set to 1 % and the initial ROI radiusα1 is set to 0.1 meaning that each design variable value canchange ±10 %. The optimization is terminated when therelative change of the mass is less than 1 %.

Both initial design points result in similar objective andconstraint functions’ convergence as depicted in Figs. 8 and9, respectively. In the figures, the last values stand for theinteger optimization cycle. The values of objective and con-straint functions at the optimum, reported in Table 2, showthat the estimation error remains below the specified levelof 1 % in all cases expect one having a minor conflict withthe criterion. This small discrepancy follows from the factthat the error is measured only in the center of the ROI. Thesize of the ROI radius α decreases from the initial value of0.1 to ca. 0.025 and it tends to oscillate within the allowed50 % change between iteration cycles.

6 Conclusions

This paper presents a methodology for automatic productdesign including structural optimization. The methodologyis applied to a panel box commonly used in marine struc-tures. By using such an approach, a carefully designedgeneric parametric CAD-model can be adopted for a widevariety of structures, and an optimized structure can beobtained fast and robustly with minimum user input. Theoptimization is based on linear response surface method andthe classical simplex-algorithm that are used together itera-tively to find at least a local optimum. The paper presentsan industrial application project in which the methodologygives excellent results in terms of product design robustnessand efficiency. When compared to the results of a traditionaldesign process, the proposed automatic methodology resultsin a substantially faster design process and slightly lighterfinal configuration. The robustness of the optimization isensured by using linearization of the responses and relax-ation of integer variables to continuous ones during the opti-mization. In the last optimization cycle, the design variablesare optimized to integer values. Although such a roundingprocess may deteriorate the final objective function value, itis still chosen due to prioritization of trustworthiness insteadof exactness.

Acknowledgments Support from the Finnish Metals and Engineer-ing Competence Cluster (FIMECC) Innovations & Network- researchand the research project Computational methods in mechanical engi-neering product development - SIMPRO are gratefully acknowledged.

References

Abu-Odeh A-Y, Jones H-L (1998) Optimum design of compositeplates using response surface method. Compos Struct 43:233–242

Acar E, Guler MA, Gerceker B, Cerit ME, Bayram B (2011) Multi-objective crashworthiness optimization of tapered thin-walledtubes with axisymmetric indentations. Thin-Walled Struct 49:94–105

Arai M, Shimizu T (2001) Optimization of the design of ship structuresusing response surface methodology. In: Wu Y-S, Cui W-C, ZhouG-J (eds) Practical design of ships and other floating structures.pp 331–339

Eldred MS, Dunlavy DM (2006) Formulations for surrogate-basedoptimization with data fit, multifidelity, and reduced-order mod-els. In: Proceedings of the 11th AIAA/ISSMO MultidisciplinaryAnalysis and Optimization Conference, paper AIAA-2006-7117.Portsmouth, VA, 6-8 Sept

Gupta KC, Li J (2000) Robust design optimization with mathematicalprogramming neural networks. Comput Struct 76:507–516

Heinonen O, Pajunen S (2011) Optimal design of stiffened plate usingmetamodeling techniques. J Struct Mech 44(3):218–230

Hock W, Schittkowski K (1981). Test examples for nonlinear pro-gramming codes. Lecture Notes in Economics and MathematicalSystems, 187. Springer

Huang Z, Wang C, Chen J, Tian H (2011) Optimal design of aero-engine turbine disc based on kriging surrogate models. ComputStruct 89:27–37

IACS (2009) Common Structural Rules for Bulk CarriersKhuri AI, Cornell JA (1987) Response surfaces: design and analyses.

Marcel Dekker IncKleijnen JPC (2008) Response surface methodology for constrained

simulation optimization: an overview. Simul Model Pract Theory16:50–64

Montgomery DC (2001) Design and analysis of experiments. WileyNemhauser GL, Laurence AW (1988) Integer and Combinatorial

Optimization. WileyPark H-S, Dang X-P (2010) Structural optimization based on CAD-

CAE integration and metamodeling techniques. Comput AidedDes 42:889–902

Queipo KN, Haftka RT, Shyy W, Goel T, Vaidyanathan R, Tucker PK(2005) Surrogate-based analysis and optimization. Prog AerospSci 41:1–28

Ren W-X, Chen H-B (2010) Finite element model updating in struc-tural dynamics by using response surface method. Eng Struct32:2455–2465

Roux WJ, Stander N, Haftka RT (1998) Response surface approx-imations for structural optimization. Int J Numer Methods Eng42:517–534

Sakata S, Ashida F, Zako M (2003) Structural optimization usingKriging approximation. Comput Methods Appl Mech Eng192:923–939

Simpson TW, Mauery TM, Korte JJ, Mistree F (2001) Kriging mod-els for global approximation in simulation-based multidisciplinarydesign optimization. AIAA J 39(12):2233–2241

Stander N, Craig KJ (2002) On the robustness of a simple domainreduction scheme for simulation-based optimization. Eng Comput19(4):431–450

Yoo K-S, Eom Y-S, Park J-Y, Im M-G, Han S-Y (2011) Reliability-based topology optimization using successive standard responsesurface method. Finite Elem Anal Des 47:843–849

automatic design of marine structures by using successive response surface method

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