《船舶与海洋工程结构风险评估》课程教学课件（讲稿）Lecture 15 Response SurfaceMethods

ASRAnetStructural Reliability & Risk Assessment4-8 July 2016Wuhan, ChinaLecture 18: Response surface methodsProfessor Purnendu K. DasB.E., M.E., PhD, CEng, CMarEng, FRINA, FIStructE, FIMarEST

Structural Reliability & Risk Assessment 4 – 8 July 2016 Wuhan, China Lecture 18: Response surface methods Professor Purnendu K. Das B.E., M.E., PhD, CEng, CMarEng, FRINA, FIStructE, FIMarEST 1

ContentsBackgroundandbasicconceptsAnalysis of some simple examples. AccuracyApplication to buckling of a stiffened panel based on FEManalysis(iterativeresponsesurfacemethod)Application to analysis of riserfatigue due to Vortex-lnduced-Vibrations (VIV)Summaryandconclusions2

Contents • Background and basic concepts • Analysis of some simple examples. Accuracy • Application to buckling of a stiffened panel based on FEM analysis (iterative response surface method) • Application to analysis of riser fatigue due to Vortex-Induced￾Vibrations (VIV) • Summary and conclusions 2

Backgroundand basic concepts3

Background and basic concepts 3

BackgroundFreguently,limit statefunction is onlyknown implicitlythroughaprocedure such as a Finite Element Analysis.Safe domain can then be defined only through point-by-pointdiscovery (i.e. by repeated numerical analysis with different inputvalues)Twobasicschemesarerelevant:1.Thecomputerprogramtobeappliedforreliabilityanalysisisconnecteddirectlytothecomputerprogramfornumericalanalysisofthe limitstatefunction-ll(iThelimitstatefunctionisfirstevaluatedforanumberofvaluesoftheinputparameters(i.e.atanumberofpoints).(i)Asmoothfunction issubsequentlyfitted based onthesepoints.(ii)Thesmoothfunction is subsequently appliedbythe reliability analysisprogram

Background • Frequently, limit state function is only known implicitly through a procedure such as a Finite Element Analysis. • Safe domain can then be defined only through point-by-point discovery (i.e. by repeated numerical analysis with different input values) • Two basic schemes are relevant: – I. The computer program to be applied for reliability analysis is connected directly to the computer program for numerical analysis of the limit state function – II (i)The limit state function is first evaluated for a number of values of the input parameters (i.e. at a number of points). (ii)A smooth function is subsequently fitted based on these points. (iii) The smooth function is subsequently applied by the reliability analysis program

Background. Option l: Direct coupling of reliability analysisprogram and computer program fornumerical evaluation of limit state functionReliability analysis computer programEvaluation of limit state function forgiven parameter combinationFinite Element computer program

Background • Option I: Direct coupling of reliability analysis program and computer program for numerical evaluation of limit state function Reliability analysis computer program Evaluation of limit state function for given parameter combination Finite Element computer program

Background· Option Il: (i) The limit state function isevaluated at a number of points. (i) Asmooth analytical function is.fitted to theseReliabilityanalysisPasemter(m) Based on th eomanth rogrdiorfunctioncQmbinatiapiilyehalysis is performed.userFinite ElementEvaluation of limit statefunction for givencomputerparameter combinationprogram

Background • Option II: (i) The limit state function is evaluated at a number of points. (ii) A smooth analytical function is fitted to these points. (iii) Based on the smooth function, the reliability analysis is performed. Reliability analysis computer program Evaluation of limit state function for given parameter combination Finite Element computer program Parameter combinations given by user

Background A third option is also possible, where thepoints which define the response surface aredefined iteratively during the reliabilityReliabilityanalys'sPaas. This is referredomst eptivecombipatispsaiyanbymethod.TuserFinite ElementEvaluation of limit statefunction for givencomputerparameter combinationprogram

Background • A third option is also possible, where the points which define the response surface are defined iteratively during the reliability analysis. This is referred to as an iterative response surface method. Reliability analysis computer program Evaluation of limit state function for given parameter combination Finite Element computer program Parameter combinations given by user

Background· The failure function, G(x), is evaluated for a set of points inthe space of basic variables (i.e. X-space)We then seek a function G,(x) which best fits the discrete setof values of G(x). Typically, is taken to be an nth orderpolynomial.The difference between G(x) and Gz(x) can e.g.be quantifiedby squaring the difference at a number of control points andsummingthesesquares

Background • The failure function, G(x), is evaluated for a set of points in the space of basic variables (i.e. X-space) • We then seek a function G2(x) which best fits the discrete set of values of G(x). Typically, is taken to be an nth order polynomial. • The difference between G(x) and G2(x) can e.g. be quantified by squaring the difference at a number of control points and summing these squares

Second orderpolynomial responsesurfaces. A second order polynomial is very much applied fortheresponse surface (see e.g.Faravelli(1989),Bucher andBourgund (1990), Rajashekhar and Ellingwood (1993)·This hasthefollowing formG2(x) = A + XTB + XTCXwhere A is a constant, BT = [By, B2,...B,] is a vector ofconstants, and Cis an (n x n) matrix containing thecoefficientsforthe secondorderterms:11nC=SVn

Second order polynomial response surfaces • A second order polynomial is very much applied for the response surface (see e.g. Faravelli (1989), Bucher and Bourgund (1990), Rajashekhar and Ellingwood (1993)) • This has the following form: G2(x) = A + X TB + X TCX where A is a constant, B T = [B1 , B2 ,.Bn ] is a vector of constants, and C is an (n x n) matrix containing the coefficients for the second order terms:              n n 1 1 1n sym.C . . C .C C

Selection of points where failure functionisevaluated: Coefficients of second order popynomial areobtained by conducting a series of “numericalexperiments" One simple approach is to select poins aroundthe mean point, parallell to the axis of eachvariablé, see circles in figture belowMeanvalue Additid nar of-aiagonal points can also beadded,|see stars in figure below+

Selection of points where failure function is evaluated • Coefficients of second order popynomial are obtained by conducting a series of “numerical experiments” • One simple approach is to select poins around the mean point, parallell to the axis of each variable, see circles in figure below • Additional off-diagonal points can also be added, see stars in figure below Mean value point