Next Generation Domain-Services in PL-Grid Infrastructure for Polish Science Daniel Bachniak 1, Jakub Liput 2, Łukasz Rauch 1, Renata Słota 2,3, Jacek Kitowski 2,3 1 AGH University, Department of Applied Computer Science and Modelling, 2 AGH University, ACC Cyfronet AGH 3 AGH University, Department of Computer Science, Krakow, Poland PPAM’15 Krakow, Poland, September 6-9, 2015 Massively Parallel Approach to Sensitivity Analysis on HPC Architectures by using Scalarm Platform

Agenda Example results Sensitivity Analysis and Scalarm Scalarm architecture Integration Sensitivity Analysis methods SA results for crankshaft deformation Scalability tests Local methods Global methods Conclusions Introduction

Design of production processes Rolling Forging Stamping Welding Flow forming Dimensions: L > 10m, W > 30t Precision: ϕ < 50µ

How to design the production process? Build the model p p – N p input process properties Apply material properties p m – N m input material properties Apply boundary conditions Perform numerical simulation OBJECTIVE – to obtain a set of optimal p p and p m !!! X Solve ill-posed inverse problem (optimization procedure) SENSITIVITY ANALYSIS Allows to determine influence of input parameters of the model on its output parameters. Select and configure SA method Sample of (N p +N m ) hypercube to receive N samples Perform N simulations of the process Analyse results

Sensitivity Analysis Supervisor Separated approach Integrated approach

Scalarm architecture

Local methods 1 Local sensitivity analysis investigates of model behavior around random point x of parameter space. Influence of parameter x i on the model output f(x) can be estimated by calculation of partial derivatives: x0x0 x0x0 x1x1 x2x2 >? S 1 estimation: S 2 estimation: x1x1 x2x2 2D parameter space: x0x0

Local methods 2 Advantages: easy low computational cost: 2 parameters -> 3 points 3 parameters -> 4 points Application: often used problems with many parameters: 10 parameters -> 11 points as a preliminary sensitivity analysis x1x1 x2x2 Disadvantages: only local measurement x0x0 ? ? ? ? ? ? ?

One step before global methods x1x1 x2x2 x 01 x 02 Add new random points strategy: 2 measurements-> 6 points Structured distribution strategy: x1x1 x2x2 x 01 x 02 2 measurements-> 4 points x 01 x 02 vs Structured distribution strategy problem: periodical functions. S 1 = 0, S 2 = 0

Global methods – Morris Design Screening designs Qualitative estimation of the parameters importance One-At-a-Time (OAT) approach – Methods based on the OAT technique investigate the impact of the variation of each factor in turn Morris Design – Estimation of the main effect of the factor Local measures at different points in the input space are computed Points selection: each factor covers the whole interval in which it was defined Sensitivity measures: estimation of the mean value  i and standard deviation  i 0 1/3 2/ /3 1/3

Morris Design example Steps of algorithm: normalization of parameters to range choose the smallest step: dx = 1/3 random starting points: (for example 3 points) create trajectories (one change for each direction): calculate the means (u 1 and u 2 ) of derivatives for directions x 1 and x 2. compare the means (u 1 >? u 2 ). dx1 1 dx1 2 dx1 3

Global methods – Factorial Design Factorial Design, a) two factors, Two-level b) two factors, Three-level c) three factors, Two-level. reduce the number of runs of the model by studying multiple factors simultaneously commonly used for computationally intensive models x2x2 Uniform distribution strategy: x1x1 A high A low B high B low number of HIGH points number of LOW points „level” factor as discretization parameter:

Global methods – Sobol’ Variance- Based Method total variance of the output partial variances main effects two parameters interactions higher order influences divide by: first order indices second order indices higher order indices 512 points of a two-dimensional a) standard random sequence and b) Sobol’ quasi-random sequence. Calculate first-order indices: Calculate total-effect: Variances estimation by using Monte-Carlo approach, and Sobol’ sequence. Total Sensitivity Indices = main effect + interactions Variance decomposition

Case study Investigation of parameters influence on the crankshaft deformation Deformation of crankshaft in a furnace at different times: Supporting points of crankshaft: Importance of parameters on deflection angle?Sensitivity Analysis

Results (1) Cooling and heating sequence after forging operations during crankshaft manufacturing: Results of sensitivity analysis:

Results (2) Results of sensitivity analysis: Investigated parameters: initial temperature (t_start) of normalization, material parameters (Young modulus – E20, yield strength – Sp20)

Weak Scaling Efficiency Measurements of the weak scaling efficiency for Sensitivity Analysis conducted with use of Scalarm platform. where: d 1 is the referential problem to calculate t 1 (d 1 ) is the time of referential measurements for d 1 t N (d N ) is the time measured for N cores and problem size d N = N * d1 efficiency drop (due to Zeus’ queue-waiting time)

References 1.T. Meyer and G. Horne, “NATO Data Farming Report Published in March 2014 Launches New Possibilities”, in: Proceedings and Bulletin of the International Data Farming Community, Issue 15, Workshop 27, May A. Saltelli, M. Ratto, T. Andres, F. Campolongo, J. Cariboni, D. Gatelli, M. Saisana, and S. Tarantola, “Global Sensitivity Analysis”, The Primer. John Wiley & Sons Ltd, (2008). 3.Szeliga D., Kusiak J., Rauch L., Sensitivity Analysis as Support for Design of Hot Rolling Technology of Dual Phase Steel Strips, Steel Res. Int, Special Issue, (2012), M.D. Morris: Factorial sampling plans for preliminary computational experiments, Technometrics, 33 (1991), F. Yates and K. Mather, “Ronald Aylmer Fisher”, Biographical Memoirs of Fellows of the Royal Society of London, 9 (1963), 91– I.M. Sobol’: Sensitivity analysis for non linear mathematical models, Math. Model. Comput. Exp., 1 (1993), D. Król, M. Wrzeszcz, B. Kryza, L. Dutka and J. Kitowski. “Massively Scalable Platform for Data Farming Supporting Heterogeneous Infrastructure”. In: CLOUD COMPUTING 2013, The Fourth International Conference on Cloud Computing, GRIDs, and Virtualization. 2013, pp. 144– M. Kvassay, L. Hluchy, S. Dlugolinsky, M. Laclavik, B. Schneider, H. Bracker, A. Tavcar, M. Gams, D. Krol, M. Wrzeszcz, and J. Kitowski. An integrated approach to mission analysis and mission rehearsal. In Proceedings of the Winter Simulation Conference, page 362. Winter Simulation Conference, Accessed: Accessed:

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