FERUM 4.0 User's Guide
Contents
1. 4 Getting started 5 Organization of FERUM 4 0 m files 1 Introduction FERUM Finite Element Reliability Using Matlab is a general purpose structural reliability code whose first developments started in 1999 at the University of Cafifornia at Berkeley UCB DKHF06 This code consists of an open source Matlab toolbox featuring various structural reliability methods As opposed to commer cial structural reliability codes see e g PS06 for a review in 2006 the main objective of FERUM is to provide students with a tool immediately comprehensible and easy to use and researchers with a tool very accessible which they may develop for research purposes The scripting language of Matlab is perfect for such objectives as it allows users to give commands in a very flexible way either in an interactive mode or in a batch mode through input files FERUM was created under Prof Armen Der Kiureghian s leadership and was managed by Terje Haukaas at UGB until 2003 It benefited from a prior experience with CalRel structural reliability code which features all the methods implemented in the last version of FERUM It also benefited from the works of many researchers at UCB who made valuable contributions in the latest available version version 3 1 which can be down loaded at the following address http www ce berkeley edu FERUM Since 2003 this code is no longer officially maintained This document is a draft user s manual of a n2. 0 no assessment analysisopt total_indices 1 total_indices 1 assessment of total indices 0 no assessment analysisopt all_indices f all_indices 1 assessment of first order second order h all order indices h analysisopt first_indices must be set to 1 0 no assessment analysisopt NbCal 30 Number of replications of Sobols indices assessments based on the same SVR surrogate if analysisopt SVR yes A NbCal 1 one single assessment h NbCal gt 1 assessment of mean and variance for each Sobols indice Global Sensitivity Analysis GSA SVR options analysisopt SVRbasis CVT_unif SVRbasis h Sobol_norm Points are deterministically generated vf in the standard space normal distribution h use of Sobol sequences in 0 1 hypercube h CVT_unif Points are deterministically generated in an hypersphere with a specific radius A use of Voronoi cells analysisopt SVR_Nbasis 200 Number of points in the design of experiments 14 analysisopt gridsel_option 1 SVR parameter for cross validation search do not modify Parameters C epsilon and sigma RBF kernel is chosen are determined in the train_SVR m function However the user may have some knowledge of the problem in order to define a grid search for the hyperparameters C epsilon sigma analysisopt n_reg_radius 200000 Number of samples for assessing the radius of the learning hypers
3. IS and distribution analysis options analysisopt sim_point origin dspt design point origin origin in standard normal space analysisopt stdv_sim 1 Standard deviation of sampling distribution in the simulation analysis 10 Simulation analysis MC IS analysisopt target_cov analysisopt lowRAM 0 05 Target coef of variation for failure probability 0 1 memory savings allowed 0 no memory savings allowed 3 5 Directional Simulation The n dimensional normal vector U is expressed as U RA R gt 0 where R is a chi square distributed random variable with n degrees of freedom d o f independent of the random unit vector A which is uni formly distributed on the n dimensional unit sphere Q The failure probability p can be written as follows conditioning on A a Bje88 n P G RA lt 0 A al fa a da 11 acn where f a is the uniform density of A on the unit sphere Practically a sequence of N random direction vectors a u u j 1 N is generated first thenr r G raY 0 are found iteratively and p is finally estimated from the following expression rR yb a a where 7 is the chi square cdf with n d o f Figure 6 Directional Simulation DS Uniform distribution on the 2D unit sphere Uy Figure 7 Directional Simulation DS Determination of rj radius In FERUM 4 0 the user must select option 22 for a Directional Simulation D
4. Both CMC and QMC sampling procedure are implemented in FERUM 4 0 Another available option consists in building a Support Vector surrogate function by regression on a sam ple set of well chosen sampling points analysisopt SVR set to yes The size of the learning database number of these well choosen sampling points is specified by the field analysisopt SVR_Nbasis This option based on statistical learning theory proves to be a rather cost efficient technique for evaluating sensi tivities of models of moderate complexity For a Global Sensitivity Analysis select option 50 in FERUM 4 0 Parameters specific to this analysis are listed hereafter Global Sensitivity Analysis GSA analysis options analysisopt sampling i sampling 1 Sobol indices are assessed from h Crude Monte Carlo CMC simulations 2 Sobol indices are assessed from Quasi Monte Carlo QMC simulations th analysisopt Sobolsetopt 0 Sobolsetopt 0 Sobol sequences from Matlab Statistic toolbox 1 Sobol sequences from Sobol Broda http www broda co uk analysisopt SVR Ino s SVR yes a SVR is built as a surrogate to the physical model Sobol indices are assessed on the SVR surrogate no no SVR surrogate is built Sobol indices are assessed on the physical model by means of gfundata data structure analysisopt first_indices 1 first_indices 1 assessment of first order indices
5. for small failure probabilities and a number of variance reduction techniques have been proposed in the past to lower this computational effort One of these variance reduction techniques is known as Importance Sampling IS Equation 9 is rewritten now in the following form ae 10 A X Py l I x ne h x dx Ep x where h is called a sampling density For an IS analysis it is usual to take h x h x u u u where Yn is the n dimensional standard normal pdf and u is the vector of MPP coordinates coming from a previous FORM analysis Note that p is now obtained from the sample mean of q I x f x h x 10 Figure 5 Crude Monte Carlo CMC vs Importance Sampling IS FERUM 4 0 features both methods and calls to the limit state function are sent in a distributed manner the maximum number of jobs sent being adjusted by the variable analysisopt block_size For a crude MCS analysis in FERUM 4 0 set analysisopt sim_point to origin and select option 21 For an IS analysis set analysisopt sim_point to dspt and select option 21 Parameters specific to a crude MCS IS analysis are listed hereafter Simulation analysis MC IS DS SS and distribution analysis options analysisopt num_sim 1e4 Number of samples analysisopt rand_generator 1 0 default rand matlab function 1 Mersenne Twister to be preferred Simulation analysis MC
6. from BRODA BRODA subdirectory e SVR training and evaluation based on the spider Toolbox spider subdirectory e Training samples possibly re arranged by means of Centroidal Voronoi Tessellation CVT subdirectory N2LA directory e Nested bi level RBDO analysis based on FORM and gradient of p index w rt design variables N2LA m e Polak He optimization algorithm ph_quadprog m e fun m auxiliary file ferum_in_loop directory e Example script files showing how to call FERUM in a silent mode aster_files directory e Files required for external calls to Code_aster Finite Element code in a sequential way inputfile gfun aster m function called by gfun m Windows specific files Code_aster template files gawk binary gawk file for post processing of FE results 17 References ABO1 Bje88 BLO8 BMDO09 DKD98 DKHF06 DKLH87 DM07 DNBF09 LDK86 Lem09 NBGLO6 Pol97 PS06 RP04 RPO7 Sob07 ZDK94 S K Au and J L Beck Estimation of small failure probabilities in high dimensions by subset simulation Probabilistic Engineering Mechanics 16 263 277 2001 P Bjerager Probability integration by directional simulation Journal of Engineering Mechanics 114 8 1285 1302 1988 J M Bourinet and M Lemaire FORM sensitivities to correlation Application to fatigue crack propagation based on Virkler data In Proc of the 4th International ASRANet Colloquium At
7. in reference BLO8 Parameters specific to FORM reliability sensitivities are listed hereafter Thet are set up in the probdata structure variable for sensitivities w rt distributions parameters 0 and in the gfundata structure variable for sensitivities w r t deterministic limit state function parameters 0 Flag for computation of sensitivities w r t means standard deviations parameters and correlation coefficients 1 all sensitivities assessed 0 no sensitivities assessment probdata flag_sens 1 Flag for computation of sensitivities w r t thetag parameters of the limit state function 1 all sensitivities assessed 0 no sensitivities assessment gfundata 1 flag_sens 1 analysisopt ffdpara_thetag 1000 Parameter for computation of FFD estimates of dbeta_dthetag perturbation thetag analysisopt ffdpara_thetag if thetag 0 h or 1 analysisopt ffdpara_thetag if thetag y Recommended values 1000 for basic limit state functions i 100 for FE based limit state functions 3 1 3 FORM with search for multiple design points Search for multiple MPPs such as described in DKD98 is also implemented in FERUM 4 0 option 11 Figure 2 illustrates the use of this method applied to a 2D example with a parabolic limit state function DKD98 g x g x x2 b x3 K x e 8 where b 5 x 0 5 and e 0 1 Both variables x and x are independent and identically distributed i i d standard normal ra
8. options analysisopt grad_flag ddm ddm direct differentiation ffd forward finite difference analysisopt ffdpara 1000 Parameter for computation of FFD estimates of gradients Perturbation stdv analysisopt ffdpara J Recommended values 1000 for basic limit state functions 50 for FE based limit state functions 3 1 2 Reliability sensitivities importance measures In addition to the reliability index 6 and the MPP coordinates coming from a FORM analysis the user may use FERUM 4 0 to calculate the sensitivities of 6 or of the failure probability pp to distribution parameters 8 or to deterministic limit state function parameters 8 For instance the sensitivity of w rt 0 reads T Vo B Juro 2 0 a 6 where Jusa x 0 au a JAE The Jacobian of the transformation is obtained by differentiating Equation 4 w r t 0 parameters ou aga Oz ts 7 3O 00 30 In FERUM 4 0 sensitivities w rt distributions parameters 0 are evaluated based on both terms of Equa tion 7 as opposed to FERUM 3 1 which only uses the first term Sensitivities to correlation are based on the second term of this expression only as the first one vanishes BLO8 Sensitivities are evaluated numerically with the same integration scheme as the one used for obtaining R matrix and it is required to differentiate the Cholesky decomposition algorithm in a step by step manner Examples of application are given
9. FERUM 4 0 User s Guide J M Bourinet September 22 2009 Abstract The development of FERUM Finite Element Reliability Using Matlab as an open source Matlab toolbox was initiated in 1999 under Armen Der Kiureghian s leadership at the University of California at Berkeley UCB This general purpose structural reliability code was developed and maintained by Terje Haukaas with the contributions of many researchers at UCB The present document aims at presenting the main features and capabilities of a new version of this open source code FERUM 4 0 based on a work carried out at the Institut Francais de M canique Avanc e IFMA in Clermont Ferrand France This new version offers improved capabilities such as simulation based technique Subset Simulation Global Sensitivity Analysis based on Sobol s indices Reliability Based De sign Optimization RBDO algorithm etc Beyond the new methods implemented in this code an emphasis is put on the new architecture of the code which now allows distributed computing either virtually through vectorized calculations within Matlab or for real with multi processor computers An important note about this User s Guide is that it does not contain a detailed description of FERUM usage and FERUM available methods The user must have some prior knowledge about probability concepts stochastic methods structural reliability sensitivity analysis etc More details about the FERUM inputs may be f
10. S analysis Beside a classical random generation of directions analysisopt dir_flag set to random a slightly modified version of this algorithm is proposed Instead of generating random directions on the unit sphere it is proposed to divide it into N evenly distributed points in a deterministic manner in order to gain an improved accuracy at a given computational cost This second option is selected by setting analysisopt dir_flag to det In the two methods intersections with the limit state function along each direction are obtained in a distribu tive manner based on a vectorized version of fzero m Matlab function The maximum number of jobs sent is adjusted by the variable analysisopt block_size It is worth noting that DS looses efficiency as the number of random variables n increases 11 Parameters specific to a DS analysis are listed hereafter Simulation analysis MC IS DS SS and distribution analysis options analysisopt num_sim 200 Number of directions DS analysisopt rand_generator 1 O default rand matlab function 1 Mersenne Twister to be preferred Directional Simulation DS analysis options analysisopt dir_flag det det deterministic points uniformly distributed h on the unit hypersphere using eq_point_set m h function random random points uniformly distributed h on the unit hypersphere analysisopt rho 8 Max search radius in standard normal space for Directi
11. analysis MC IS DS SS and distribution analysis options analysisopt num_sim 1e4 Number of samples analysisopt rand_generator I 0 default rand matlab function 1 Mersenne Twister to be preferred Simulation analysis MC IS and distribution analysis options analysisopt sim_point origin dspt design point origin origin in standard normal space analysisopt stdv_sim J Standard deviation of sampling distribution in the simulation analysis 3 4 Crude Monte Carlo Simulation Importance Sampling Equation 1 is rewritten as follows p 1 f x dx Ey 11000 9 Dix where D represents the integration domain of joint pdf f x I e is an indicator function which equals 1 if g x lt 0 and 0 otherwise and E e denotes the mathematical expectation w rt joint pdf fy x 0 and 0 parameters omitted for the sake of clarity The expectation in Equation 9 is estimated in a statistical sense for a crude Monte Carlo Simulation crude MCS The u space is randomly sampled with N independent samples u j 1 N These N sam ples are then transformed to the x space x x u and an unbiased estimate of p is finally obtained from the sample mean of q I x Note that a standard deviation is also obtained for this sample providing useful information regarding the accuracy of the estimated value of p It must be stressed out here that a crude MCS requires a high computational effort large N
12. des the random variable space in a safety domain g x 0 gt 0 and a failure domain g x 0 lt 0 The probability of failure therefore reads Pf fx x 0 dx 1 g x 8 lt 0 2 2 Probability distributions and transformation to standard normal space The joint density function fy x 0 is often unknown and replaced by its Nataf counterpart completely defined by specifying marginal distributions and the Gaussian correlation structure between random vari ables see LDK86 This Nataf joint distribution is completely specified by variables probdata marg and probdata correlation in FERUM input files FERUM has a rich library of probability distribution mod els including extreme value distributions and a truncated normal distribution These distributions can be specified through either their statistical moments or parameters See hereafter the corresponding part of the inputfile_template m file Marginal distributions for each random variable probdata marg type mean stdv startpoint p1 p2 p3 p4 input_type l type 1 Parameter in reliability analysis thetag Deterministic parameter cg x Ii h 1 Normal distribution h 2 Lognormal distribution vA 3 Gamma distribution h 4 Shifted exponential distribution h 5 Shifted Rayleigh distribution 6 Uniform distribution 4 7 Beta distribution vA 8 Chi square distribution hh vA 11 Type I largest value distribution same as G
13. distributed computing see e g examples of applications in NBGLO6 DNBFO9 The number of jobs sent simultaneously is tuned through the variable analysisopt block_size Such an option is available in FERUM assuming that the user has a suitable computer platform and all the necessary tools to create send and post process multiple jobs scripting language such as Perl queuing systems such as OpenPBS on Linux job schedulers The function in charge of the job allocation is obviously application specific and is called by gfun m analysisopt multi_proc 1 1 block_size g calls sent simultaneously vA gfunbasic m is used and a vectorized version of h gfundata expression is available h The number of g calls sent simultaneously block_size depends h on the memory available on the computer running FERUM h gfunxxx m user specific g function is used and able to handle h block_size computations sent simultaneously on a cluster yA of PCs or any other multiprocessor computer platform 0 g calls sent sequentially analysisopt block_size 50 Number of g calls to be sent simultaneously Based on the same developments of FERUM algorithms it is also possible to send multiple calls to a user defined Matlab limit state function written in a vectorized manner Vectorized calculations in the Matlab sense eliminate the need to cycle through nested loops and thus run much faster because of the way Matlab handles vectors internally The princip
14. e function gfun m calls gfunxxx m which evaluates gfundata lsf expression where gext variable is h a result of the external code Case of a limit state function which is defined by means of an analytical expression a Matlab m function gfundata 1 evaluator basic gfundata 1 type expression Do no change this field Expression of the limit state function gfundata 1 expression c1 X2 1000 X3 X1 200 X3 72 X5 1000 X6 X4 200 X6 72 Expression of the limit state function gfundata 1 expression gfun_nl_oscillator mp ms kp ks zetap zetas Fs S0 Case of a limit state function which requires a call to Code_Aster FE code gfundata 1 evaluator aster gfundata 1 type expression Do no change this field Expression of the limit state function gfundata 1 expression gext u0 2 4 Vectorized distributed computing A major change brought to FERUM 4 0 is that calls to the limit state function g can be evaluated in a distribu tive manner as opposed to the sequential manner of the previous version Every algorithm implemented in FERUM was revisited so as to send multiple calls to g whenever possible If one thinks of FE based Monte Carlo Simulation MCS on a multiprocessor computer the strategy consists in sending calls to the FE code in batches the number of jobs in each batch being equal to the number of available CPUs This strategy is known as
15. ew version of this code FERUM 4 0 which results from a work carried out at the Institut Francais de M canique Avanc e IFMA in Clermont Ferrand As previously achieved in the past the main intention is to provide students and researchers with a developer friendly com putational platform which facilitates learning methods and serves as a basis for collaborative research works FERUM should still be viewed as a development platform for testing new methods and applying them to various challenging engineering problems either represented by basic analytical models or more elaborated numerical models through proper user defined interfaces The main architecture of FERUM was preserved in general see Section 2 for more details In order to improve its efficiency in terms of computational time all algorithms have been revisited to extend FERUM capabilities to distributed computing For example in its new version FERUM makes Monte Carlo Simulation MCS much faster thanks to limit state functions defined in a vectorized form or real distributed computing according that a proper interface is defined for sending multiple jobs to a multi processor computer platform 2 Problem definition and structure of FERUM This section briefly presents the general formulation of time invariant structural reliability problems In ad dition to some very brief details about theoretical concepts this section highlights how these concepts are translated to FERUM structu
16. g is a deterministic model and Y is a scalar random output In order to determine the importance of each input variate we consider how the variance of the output Y decreases when variate X is fixed to a given x value V Y X x 15 where V e denotes the variance function Since x value is unknown we take the expectation of Equation 15 and by virtue of the law of to tal variance we can write V E Y X V Y E V V1X 16 13 The global sensitivity index of the first order is defined as follows for i 1 n _v E yIx _ i 1 i v Y V a Indices of higher orders are defined in a similar manner e g for the second order _V E Y IAL V i _ Yy a ij v Y V First order indices inform about the influence of each variate taken alone whereas higher order indices ac count for possible influences between various parameters Total sensitivity indices are also usually introduced They express the total sensitivity of Y variance to X input including all interactions that involve X Sr Sk 19 ick where i C k denotes the set of indices containing i From a computational viewpoint Sobol indices can be assessed using a Crude Monte Carlo CMC or a Quasi Monte Carlo QMC sampling procedure analysisopt sampling respectively set to 1 or 2 This latter technique is based on low discrepancy sequences which usually outperform CMC simulations in terms of accuracy at a given computational cost
17. gation problems In Proc of the 3rd International ASRANet Colloquium Glasgow UK 2006 E Polak Optimization Algorithms and consistent approximations Springer Verlag 1997 M E Pellissetti and G I Schu ller On general purpose software in structural reliability an overview Structural Safety 28 3 16 2006 J O Royset and E Polak Reliability based optimal design using sample average approximations Probabilistic Engineering Mechanics 19 4 331 343 2004 J O Royset and E Polak Extensions of stochastic optimization results to problems with system failure probability functions Journal of Optimization Theory and its Application 132 2 1 18 2007 I M Sobol Sensitivity estimates for nonlinear mathematical models Mathematical Modelling and Computational Experiments 1 407 414 2007 Y Zhang and A Der Kiureghian Two improved algorithms for reliability analysis In R Rackwitz G Augusti and A Borri editors 6th IFIP WG 7 5 Working Conference on Reliability and Optimi sation of Structural Systems Reliability and Optimization of Structural Systems Chapman amp Hall 1994 18
18. hens Greece 2008 J M Bourinet C Mattrand and V Dubourg A review of recent features and improvements added to FERUM software In Proc of the 10th International Conference on Structural Safety and Reliability ICOSSAR 09 Osaka Japan 2009 A Der Kiureghian and T Dakessian Multiple design points in first and second order reliability Structural Safety 20 1 37 49 1998 A Der Kiureghian T Haukaas and K Fujimura Structural reliability software at the University of California Berkeley Structural Safety 28 1 2 44 67 2006 A Der Kiureghian H Z Lin and S J Hwang Second order reliability approximations Journal of Engineering Mechanics 113 8 1208 1225 1987 O Ditlevsen and H O Madsen Structural reliability methods Internet Edition 2 3 7 2007 V Dubourg C Noirfalise J M Bourinet and M Fogli FE based reliability analysis of the buckling of shells with random shape material and thickness imperfections In Proc of the 10th Interna tional Conference on Structural Safety and Reliability ICOSSAR 09 Osaka Japan 2009 P L Liu and A Der Kiureghian Multivariate distribution models with prescribed marginals and covariance Probabilistic Engineering Mechanics 1 2 105 112 1986 M Lemaire Structural reliability John Wiley 2009 L Nespurek J M Bourinet A Gravouil and M Lemaire Some approaches to improve the computational efficiency of the reliability analysis of complex crack propa
19. ic method are listed hereafter Simulation analysis MC IS DS SS and distribution analysis options analysisopt num_sim 10000 Number of samples per subset step SS analysisopt rand_generator 1 0 default rand matlab function 1 Mersenne Twister to be preferred Subset Simulation SS analysis options analysisopt width 2 Width of the proposal uniform pdfs analysisopt pf_target O21 5 Target probability for each subset step analysisopt flag_cov_pf_bounds 1 1 calculate upper and lower bounds of the is coefficient of variation of pf 0 no calculation analysisopt ss_restart_from_step inf i gt 0 restart from step i inf all steps no record default 1 all steps record all analysisopt flag_plot 0 1 plots at each step 2 r v examples only 0 no plots analysisopt flag_plot_gen 0 1 intermediate plots for each MCMC chain h 2 r v examples only 0 no plots 3 7 Global Sensitivity Analysis Global sensitivity analysis aims at quantifying the impact of the variability in each or group of input variates on the variability of the output of a model in apportioning the output model variance to the variance in the input variates Sobol indices Sob07 are the most usual global sensitivity measures They can be evaluated in FERUM 4 0 We consider here a model given by Y g X g X1 X Xn 14 where X X1 X X is a vector of n independent random input variates
20. igin to the MPP in the standard space The first order approximation of the failure probability is then given by pr amp f where e is the standard normal cdf As in FERUM 3 1 the new version is based on the iHLRF algorithm see ZDK94 for further details In order to take advantage of distributed computing g calls required for gradient evaluations by finite differences at a specific point of the standard space are sent in a single batch The same technique is applied to step size evaluation with Armijo rule where all corresponding g calls are sent simultaneously p u Jcs a O st u Figure 1 First Order Reliability Method FORM Parameters specific to FORM in the analysisopt block_size structure variable are listed hereafter FORM analysis options analysisopt i_max 100 Maximum number of iterations allowed in the search algorithm analysisopt el 0001 Tolerance on how close design point is to limit state surface analysisopt e2 0 001 Tolerance on how accurately the gradient points towards the origin 0 step size by Armijo rule otherwise given value is the step size 0 u vector not recorded at all iterations 1 u vector recorded at all iterations 0 x vector not recorded at all iterations 1 x vector recorded at all iterations Il oO we x analysisopt step_code Ul p x analysisopt Recorded_u Ul Rp x analysisopt Recorded_x FORM SORM analysis
21. l 3 rbdo_parameters alpha 0 5 ph_quadprog parameter rbdo_parameters beta 0 6 ph_quadprog parameter rbdo_parameters gamma 8 ph_quadprog parameter rbdo_parameters delta a ph_quadprog parameter rbdo_parameters steplim 50 Max number of steps in stepsize calculation see ph_quadprog m rbdo_parameters max_iter Max number of iterations of N2LA algorithm rbdo_parameters target_beta 53 Target beta reliability index rbdo_parameters method FORM 3 9 Random fields nothing available yet 4 Getting started You might want to check out how FERUM works by running one of the examples provided with the program package To do so proceed as follows 1 2 Go to the FERUM 4 0 homepage http www ifma fr FERUM Download the FERUM4 0 zip archive file and extract all compressed files to a directory of your choice on your harddisk where you wish to save all FERUM files extraction with pathnames Before you can start using the toolbox you must to add all FERUM subdirectories aster_files dir_simu lations etc to your Search Path so that Matlab knows where to look for them This is done by using one of the three following options e select Set Path in the File menu in Matlab e select subdirectories in the Current Directory Browser then right click Add to path Selected folder and Subfolders e use the addpath m function Read one of the example inputfiles in
22. le is similar to distributed computing the difference being that the multiprocessor computer is virtually replaced by a single computer which can handle a number of runs simul taneously this maximum number being directly dependent on the memory available on the computer Here again the maximum number of runs sent simultaneously is controlled through analysisopt block_size variable For illustration purpose on an Intel T7800 2 6GHz dual core CPU with 4Gb RAM a crude MCS takes 31 min with 1 5 10 samples for a basic g r s problem where R and S are normal random variables in a vectorized manner FERUM 4 0 as opposed to 6 days 15 hours in a sequential manner 3 Overview of available methods 3 1 FORM and reliability sensitivities importance measures 3 1 1 Basic FORM First Order Reliability Method FORM option 10 of FERUM 4 0 aims at using a first order approximation of the limit state function in the standard space at the so called Most Probable Point MPP of failure P or design point which is the limit state surface closest point to the origin Finding the coordinates u of the MPP consists in solving the following constrained optimization problem u argmin llull g x u 0 G 4 0 0 gt Once the MPP P is obtained the Hasofer and Lind reliability index 6 is computed as 8 a u where a V G u VuG u is the negative normalized gradient vector at the MPP P It represents the distance from the or
23. lity equals a sufficiently large value a in order to be efficiently estimated with a rather small number of samples in practice a 0 1 0 2 set by analysisopt pf_target parameter In essence there is a trade off between minimizing the number m of subsets by choosing rather small intermediate conditional probabilities and maximizing the same probabilities so that they can be estimated efficiently by simulations The first threshold y is obtained by a crude MCS so that P F a see Figure 8 subplot a For further thresholds new sampling points corresponding to F F _ conditional events are obtained from Markov Chains Monte Carlo MCMC based on a modified Metropolis Hastings algorithm see green star points in fig ure 8 subplot b corresponding to i 2 case The process is repeated until a negative threshold y is found This is therefore the last step i m and y is set to zero The corresponding probability P F Fm 1 is then evaluated See Figure 8 subplot d The last step is reached for m i 3 in the present case The coeffi cient of variation of the failure probability estimated from SS can be evaluated by intermediate coefficients of variation weighted by the correlation that exists between the samples used for the estimation of intermediate conditional probabilities please refer to reference AB01 for more details For a Subset Simulation analysis select option 23 in FERUM 4 0 Parameters specific to this specif
24. ndom variables This problem is characterized by two MPPs at similar distances from the origin and basic FORM algo rithm results are therefore not valid 1 83 10 instead of 3 02 107 reference value for p y Results in Figure 2 are obtained with parameter values recommended in DKD98 i e y 1 1 6 0 75 and e 0 5 These parameters are set in the form_multiple_dspts m function All valid MPP results obtained with FERUM 4 0 are stored in an ALLformresults array which is added as an extra field to the analysisopt structure variable U space U space U space CO bseostes ne a 3 Q QA 5 5 5 x1 Figure 2 FORM with search for multiple design points 3 2 SORM curvature fitting and point fitting As in the previous version FERUM 4 0 offers two ways for computing a second order approximation of the failure probability In both methods the SORM approximation of the failure probability p is computed with Breitung or Tvedt formulae as in FERUM 3 1 The first method consists in determining the principal curvatures and directions by solving an eigen problem involving the Hessian of the limit state function option 12 of FERUM 4 0 The Hessian is computed by finite differences the perturbations being set in the standard normal space All calls to the limit state func tion corresponding to perturbated points are potentially sent simultaneously as being all independent from each other The sec
25. o rithm Pol97 and requires the gradients of both cost and constraints functions which themselves require the gradient of the reliability index 6 w rt design variables 0 Previous RBDO applications of the Polak He algorithm showed that its rate of convergence was highly dependent on the order of magnitude of the design parameters cost and constraints functions In FERUM 4 0 all these values are normalized at each Polak He iteration thus improving and ensuring convergence what ever the initial scaling of the problem in Equation 20 Convergence to an optimum is assumed to be obtained when the cost function has reached a stable value and all the constraints are satisfied i e f 0 f 0 lt 0 In order to carry out a N2LA FORM based RBDO analysis select option 40 in FERUM 4 0 The current implementation is so far restricted to deterministic parameters as design parameters i e 9 0 The cost function rbdo_fundata cost field is expressed in the form c 0 co cpps where py represents the failure probability Deterministic constraints are defined by means of the rbdo_fundata constraint field Parameters specific to the RBDO analysis are listed hereafter Cost function in the form hk C0 tef pst rbdo_fundata cost 4 3 pi r1 73 10 73 c_O term 0e2 4 3 pi r1 73 r0 73 c_f term rbdo_fundata constraint rO 40 Deterministic constraints ri 150 hfi lt 0 i ty 2 0q 1 rO r
26. oint of the FORM analysis in the physical space hh vA Refer to ferum_pdf m ferum_cdf m and distribution_parameter m functions for more information h on a specific distribution Correlation matrix This matrix is a square matrix with dimension equal to size probdata marg 1 Lines columns corresponding to parameters in reliability analysis thetag or deterministic parameters cg are removed in a pre processing stage of FERUM by means of the update_data m function The structural reliability problem of Equation 1 expressed in the original space of physical random vari ables X is transformed to a standard normal space where U becomes an independent standard normal vector This mapping is carried out in FERUM using the Nataf model LDK86 Physical random variables X are transformed to correlated standard normal variables Z whose correlation structure obeys integral relation of Equation 3 and Z is then transformed to uncorrelated standard normal variables U fx gt i 1 n X with R et acm Z N 0 R gt U N 0 1 2 For each ij pair of variables with known correlation p Equation 3 should be solved to determine correlation poi between mapped z variables TRER Xi Hi xj Bj Pij P2lZis Zj Poij dz dz 3 ee SAN Oi dj where u and o respectively stand for the mean and standard deviation of the ith component of X and 2 e 0 p is the 2D standard normal probability density function pdf with co
27. onal Simulation analysis analysisopt tolx 1e 5 Tolerance for searching zeros of g function analysisopt keep_a 0 Flag for storage of a values which gives axes along which simulations are carried out analysisopt keep_r 0 Flag for storage of r values for which g r 0 3 6 Subset Simulation Starting from the premise that the failure event F g x 0 lt 0 is a rare event S K Au and J L Beck pro posed to estimate P F by means of more frequent intermediate conditional failure events F i 1 m called subsets so that F gt Fy D D Fm F ABO1 The m sequence of intermediate conditional failure events is selected so that F g x 0 lt yi where y s are decreasing values of the limit state function and Ym 0 As a result the failure probability py P F is expressed as a product of the following m conditional probabilities Dp PCF P Fn P E Frnt PF PED PG Fea 13 i 2 Unknown Unknown limit state gt 0 gt 0 x x First threshold n First threshold y al 5 5 5 0 5 5 0 5 u u a b Unknown Unknown limit state limit state s 0 a 0 Second threshold y a Last threshold y_ y 0 al 5 5 5 0 5 5 0 5 u u c d Figure 8 Main steps in Subset Simulation 12 Each subset event F and the related threshold value y is determined so that its corresponding condi tional probabi
28. ond method consists in approximating the limit state function by a piece wise paraboloid sur face DKLH87 option 13 of FERUM 4 0 This approximate surface must be tangent to the limit state at the design point and coincides with the limit state at two points on each axis selected on both sides of the origin see Figure 4 It is built iteratively with a limited number of iterations and all calls to the limit state function at each iteration are potentially sent simultaneously as well This second approach is advantageous for slightly noisy limit state functions e g when a FE code is involved for problem with a large number of random variables or when the computation of curvatures turns out to be problematic U Figure 4 Second Order Reliability Method Point fitting method SORM pf 3 3 Distribution Analysis FERUM 4 0 offers a way to assess qualitatively visual inspection of histograms and quantitatively to some extent only mean and variance in the current version the distribution of the random variable which corre sponds to the limit state function g option 20 of FERUM 4 0 The limit state function may either return a scalar value if gfundata lsf ng field is set to 1 in the input file or multiple values if gfundata 1sf ng is set to a value strictly greater than 1 please refer to the gfun m function for more details Parameters specific to a distribution analysis are listed hereafter Simulation
29. option offered in FERUM 4 0 is that the limit state function can be defined through a user provided Matlab function which calls a third party soft ware such as a Finite Element code Such merging of FERUM with problem specific external codes was made in various applications such as probabilistic buckling DNBF09 and crack propagation NBGLO6 For con trolling such external codes extra variables are provided to FERUM through the structure variable femodel and the user must create an application specific function One more option available in FERUM 4 0 is that it takes advantage of gradients w rt all or parts of the basic variables when available from the external code This proves to be very useful when limit state functions involve very computationally demanding numerical models as it avoids tedious estimations by finite differences FERUM 4 0 is compatible with Code Aster running on a Windows OS Code_Aster version STA9 1 compiled on Windows was developed by NECS and is available at http www necs fr gb telechargement php See hereafter the corresponding part of the inputfile_template m file Type of limit state function evaluator basic the limit state function is defined by means of an analytical expression or a Matlab h m function using gfundata lsf expression The function gfun m calls gfunbasic m which evaluates gfundata lsf expression A mes the limit state function evaluation requires a call to an external code Th
30. ound in the template input file template_inputfile m provided with the FERUM package This document has been mainly elaborated from FERUM 4 0 reference paper BMD09 The FERUM 4 0 package is available under the conditions of the GNU General Public License like FERUM 3 1 This implies that you may download the program for free However if you make changes or additions you must make them available for free under the same general public license Contents 1 Introduction 2 Problem definition and structure of FERUM 2 1 Time invariant structural reliability lt o sao eraa ee ee ee ee aw a 2 2 Probability distributions and transformation to standard normal space 2 3 Definition of limit state functions 2 4 Vectorized distributed computing 3 Overview of available methods 3 1 FORM and reliability sensitivities 3 1 1 Basi FORM as ics eas importance MEASUFES si Er a cee sa er See he Be GP tw RO Sa A 3 1 2 Reliability sensitivities importance measures 1 0 00 eee eee 3 1 3 FORM with search for multiple design points 2 3 2 SORM curvature fitting and point fitting i so eses ee 3 3 Distribution Analysis 3 4 Crude Monte Carlo Simulation Importance Sampling 0 0 0 00 3 5 Directional Simulation 3 6 Subset Simulation 3 7 Global Sensitivity Analysis 3 8 Reltability Bas d Design OpGMIZAHON wise ie Se ee ke ee RO o ee 3 9 Random fields
31. phere when analysisopt SVRbasis CVT_unif analysisopt svm_buf_size 350072 RAM size dependent parameter for eval_svm m function 3 8 Reliability Based Design Optimization FERUM 4 0 now offers Reliability Based Design Optimization RBDO capabilities The problem of interest reads in its most basic and general formulation fiO f 0 S 0 min c 0 s t f x0 B B x 0 lt 0 20 where e 6 stands for the design variables of the problem either purely deterministic variables 0 or distribution parameters 0 e c is the cost function to be minimized e f 8 f 1 8 is a vector of q 1 deterministic constraints over the design variables 0 e f x 9 is the reliability constraint enforcing the respect of the design rule referred to as the limit state function and considering the uncertainty to which some of the model parameters x are subjected to f is the targeted safety index One way to answer the problem in Equation 20 consists in a brute force outer optimization loop over the reliability evaluation here termed Nested bi level approach N2LA This might be computational expensive in the case of simulation based methods such as MCS and DS as addressed in RP04 and RP07 respec tively However if based on FORM this brute force method gives a solution within a reasonable amount of calls to the limit state function The outer optimization loop makes use of the Polak He optimization alg
32. points form_multiple_dspts m via calls to form m Auxiliary files 16 sensitivities directory e All files necessary to assess FORM sensitivities w r t distribution parameters 6 means standard devia tions correlation distribution parameters and deterministic parameters 0 in the limit state function SORM directory e SORM curvature fitting method sorm_cfh m e SORM point fitting method sorm_pf m e Auxiliary files distributions directory e Distribution of the limit state function based on a Monte Carlo simulation distribution_analysis m simulations directory e Crude Monte Carlo CMC simulation or Importance Sampling IS simulation_single_dspt m simula tion_single_dspt_lowRAM m e Mersenne Twister pseudo random number generator source file twister cpp Windows dll file Linux 32 bit mexglx file dir_simulations directory e Directional Simulation dir_simulation m with either random or deterministic directions through calls to eq_point_set m function and auxiliary files in eq_sphere_partitions directory subset_simulations directory e Subset Simulation subset_simulation m e Auxiliary files Sobol directory e Global Sensitivity Analysis based on Sobol indices Sobol SA m either through calls to the original limit state function or to a SVR surrogate Support Vector Regression e Sobol quasirandom sequences generated by means of the Matlab Statistical Toolbox or the sobolseq51 Windows dll file
33. re This includes for instance the stochastic model the transformation to stan dard normal variates limit state functions and more generally other points regarding computational aspects It is important here to recall that the main structure of input data in FERUM is preserved compared to version 3 1 same Matlab structure variables probdata analysisopt gfundata femodel Changes brought to FERUM are applied to core m functions and within the fields of the existing structure variables Similarly to version 3 1 results are stored in structure variables with the following syntax results keyword appended to the name of the method applied such as e g formresults sormcfhresults etc 2 1 Time invariant structural reliability We consider here only time invariant structural reliability problems see e g DM07 Lem09 The probability w r t an undesired or unsafe state is expressed in terms of a n dimensional vector X of random variables with continuous joint density function fx x 0s where 8 stands for a vector of distribution parameters Failure is defined in terms of a limit state function g x 8 where x is a realization of the random vector X and 0 denotes a vector of deterministic limit state function parameters We restrict here the analysis to component reliability with a single g function but this function may represent multiple failure modes in subset simulation in Section 3 6 without lack of generality This limit state function divi
34. rrelation coefficient p Independent standard normal variables U are then obtained from Z variables such as follows v 1 2 4 where Lg is the lower triangular Cholesky decomposition of R Poi jl matrix such that Lol Rp Solutions of Equation 3 can be found by formulae of reference LDK86 for most common statistical distributions These formulae are most often obtained by least square fitting and therefore approximate ex cept for a few pairs of distributions FERUM 4 0 is now based on accurate solutions obtained by 2D numerical Gauss integration of Equation 3 A particular attention is paid to strongly correlated random variables where the number of integration points along each dimension in z z space must be selected carefully for accurate Po values A practical rule adopted here consists in increasing these numbers of points with corre lation ranging from 32 points along each dimension for absolute values of correlation lower than 0 9 to 1024 points for absolute values larger than 0 9995 2 3 Definition of limit state functions As in FERUM 3 1 the limit state function is defined through the structure variable gfundata of the input file and called by the file named gfun m Various strategies are now offered in FERUM 4 0 The limit state func tion can either be a simple expression directly written in the input file or a Matlab function For both cases gfun m calls another function called gfunbasic m Another interesting
35. to your Matlab workspace This can be done by issuing the com mand gt gt inputfile_xxx in Matlab You now have the necessary input parameters for a reliability analysis in your current Matlab workspace Issue the command gt gt ferum in the Matlab workspace 6 Choose a specific option e g option 10 for a FORM analysis 7 The program now performs an analysis corresponding to the selected option and gives you intermediate 5 results information as it runs If the analysis is successful you will finally see an overview of the available result parameters on the screen These results are gathered in the fields of a specific structure in the Matlab workspace e g formresults if a FORM analysis is run Organization of FERUM 4 0 m files Main directory ferum m main file definition of distributions FERUM pre processing files update_data m distribution_parameter m Nataf model specific files including mod_corr_solve m main file files necessary for FORM sensitivities in the sensitivities directory Mapping from physical space to standard space and reverse Limit state function main file gfun m and gfunbasic m inputfile directory Examples of inputfiles Input file template inputfile_template m gfunction directory Examples of limit state functions defined through Matlab m files FORM directory Search for a single design point with the iHL RF algorithm form m Search for possible multiple design
36. umbel distribution h 12 Type I smallest value distribution h 13 Type II largest value distribution h 14 Type III smallest value distribution vi 15 Gumbel distribution same as type I largest value distribution h 16 Weibull distribution same as Type III smallest value distribution with epsilon 0 hh h 18 Reserved for Laplace distribution h 19 Reserved for Pareto distribution hh h 51 Truncated normal marginal distribution hh Notes hh vA Each field type mean stdv startpoint pl p2 p3 p4 input_type must be filled in If not used input a dummy nan value hh h input_type 0 when distributions are defined with mean and stdv only 1 when distributions are defined with distribution parameters pi i 1 2 3 or 4 depending on the total number of parameters required hh h For the Type III smallest value marginal distribution you must give the value of epsilon parameter as p3 when using the mean and stdv input input_type 0 hh For the Beta marginal distribution you must give the value of a parameter as p3 and hh b parameter as p4 when using the mean and stdv input input_type 0 For the Truncated normal marginal distribution you must set input_type 1 and give the mean and stdv of the untruncated marginal distribution as p1 and p2 respectively the lower bound xmin and the upper bound xmax as p3 and p4 respectively hh startpoint stands for the starting p
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