MORPACK Interface Functionality for importing FE Structures into MBS-Code via alternative Model Order Reduction Methods

Transcription

1 MORPACK Interface Functonalty for mportng FE Structures nto MBS-Code va alternatve Model Order Reducton Methods MSc.-Math. P. Koutsovasls, Dr.-Ing. V. Quarz, Prof. Dr.-Ing. M. Betelschmdt, Dresden Abstract The Matlab-based MORPACK (Model Order Reducton PACKage) nterface s presented and ts functonalty s outlned for mportng Reduced Order Models (ROMs) nto Mult Body System (MBS) codes. MOR methods, such as the Guyan reducton and the Component Mode Synthess (CMS), have been used for the past decades as standard approaches mplemented n commercal Fnte Element (FE) software, e.g. ANSYS and MSC Nastran. When mportng the ROMs nto MBS-codes, e.g. from ANSYS nto SIMPACK, the necessary FE-MBS data converson takes place based on the FEMBS nterface. MORPACK, consstng of two nner nterfaces, combnes all necessary procedures requred for the automatc mport of ANSYS FE models nto SIMPACK. The FE dscretzed model s system nformaton s converted nto the Matrx Market fle format, the ROM s created based on a user selected varety of physcal, non-physcal, and hybrd MOR methods, the ROM s dynamc propertes and structure preservaton are valdated by certan Model Correlaton Crtera (MCC), and fnally the necessary Standard Input Data (SID) fle s generated for further usage of the ROM n SIMPACK. In all the above mentoned levels varous numerc and modellng schemes have been conceved, e.g. the dagonal perturbaton method or the backtransformaton approach, whch consttute feasbly the applcaton of the selected MOR method overcomng dffcultes such as: the ll-condtonalty of system matrces when teratvely producng the ROM, non-stable matrx polynomals by certan non-physcal and hybrd MOR methods and the mport of non-physcal space ROMs nto SIMPACK. All the above s appled for free or fxed models orgnatng from the feld of structural mechancs. 1. Introducton Varous parts or n certan cases the whole set of complex systems orgnatng from felds such asengne dynamcs, automoble smulaton, Mcro Electro Mechancal Systems (MEMS), etc., possess a dynamc behavour, whch cannot be neglected durng any knd of modellng procedure. In structural mechancs elastcty s a dynamc term, whch contrbutes

2 to a more realstc approach durng the smulaton process. The Fnte Element Method (FEM) s the method used, when modellng such elastc MBS or parts of MBS. There are cases, though, where the necessty of modellng the elastcty n structural mechancs s mnmzed and the structure s modelled as a rgd body accordng to the rgd-mbs theory. Whle the geometrc complexty of mechanc structures grows, the requrement of combng both the FE and the rgd MBS methods grows along. The am s to adjust the model to the best possble realstc terms and thus, mnmze the nformaton loss that occurs durng any of the above mentoned modellng technques. The modellng workflow of the FEM-MBS couplng conssts of four major approxmaton levels, whch contrbute to the model s nformaton loss, as shown n Table 1. User s nterventon s restrcted to the frst level, whle tryng to construct the CAD fle. For the rest of the levels, the algorthms are sem or fully black-boxed n the source codes of the software or nterfaces, respectvely,.e. the user should only defne certan parameter sets for the algorthmc ntalzaton. Table 1: Modellng workflow n structural mechancs: approxmaton procedures Approxmaton levels Modellng tools Informaton loss due to 1. Geometry CAD Software Small areas, angled curves 2. PDE ODE FE Software Dscretzaton, element choce 3. FE DoF FE Software DoF condensaton 4. Elastc DoF FEM/MBS nterface Rtz Ansatz A major aspect when couplng a FE structure nto a MBS code s the Model Order Reducton (MOR) approach needed to reduce the FE-model s dmenson. Ths consttutes feasbly the further usage of the model n a MBS code, snce memory capacty problems and vast computaton tmes are avoded. Thus, t s essental that the Reduced Order Model (ROM) captures well the dynamcs of the orgnal FE model. Whle Guyan reducton and Component Mode Synthess (CMS) are the standard MOR methods mplemented n commercal FE software, other condensaton schemes orgnatng from the feld of control theory could offer ROMs, whch capture much better the dynamcs [1] of the orgnal model wth less effort (smaller system matrces),.e. leadng to better and faster MBS-models n MBS codes. Therefore, the Matlab-based MORPACK (Model Order Reducton PACKage) nterface has been created.

3 2. MORPACK nterface MORPACK s a Matlab-based nterface adapted both to the modellng requrements and compatbltes of couplng FE models orgnatng from ANSYS [2] nto SIMPACK [3]. It conssts of two nner nterfaces and four applcaton levels, as depcted n Fg. 1. Fg. 1: MORPACK general workflow The four applcaton levels operate ether as data transfer and converson tools or as functons, whch valdate the model s dynamcs or transform ther state-space defnton n order to mport and export compatble models for the nner nterfaces. 2.1 Model Order Reducton (MOR) nterface n The general concept of MOR s to fnd a low dmenson subspace T m, m<< n n order n to approxmate the unknown state (or dsplacement) vector x,.e. x Tx of the 2 nd - order Ordnary Dfferental Equaton (ODE), whch descrbes the dynamcs of a FE dscretzed model: R Mx () t + Dx () t + Kx() t = Bu() t where {,, } Bu B n n q M D K n n are respectvely the mass, dampng, and stffness matrces and the vector of external forces actng on the structure consstng of the scatter matrx q responsble for the doman allocaton of the nput vector u. (1)

4 By projectng (1) on the subspace spanned by T, a lower dmenson lnear 2 nd -order ODE s obtaned: Mx () t + Dx () t + Kx() t = Bu() t R R R R R R R wth m m [ ] : = T [ ], [ ] = {,, } R T T M D K beng the reduced system matrces and the reduced load vector m B : = T T B. R (2) The MOR nner-nterface of MORPACK ntegrates varous MOR technques [1, 4, and 5] orgnatng from three general categores: (C1) modal truncaton, sub-structurng, and statc condensaton, (C2) Padé and Padé-type approxmatons, and (C3) balancng related truncaton technques. Dependng on whether or not the Degrees of Freedom set (DoF) of the reduced vector x R les completely (S1- physcal MOR), partally (S2- sem-physcal MOR) or not at all (S3- non-physcal MOR) on the physcal subspace a,.e.: x R a 1 S1: = 1, : 1D a a a a 2 S2: = { ℵ : ℵ = }, a= {1,2,3}, ndof = 3, : 2D (3) 3 a a a S3: = { ℵ : ℵ = } 6, : 3D each of the (C1)-(C3) categores s further categorzed [6] and MOR methods are apponted as summarzed n Table 2. Table 2: MOR methods mplemented n MORPACK (S1): S1 1. Guyan reducton S1 2. Dynamc Reducton S1 3. Improved Reducton System Method (IRS) S1 4. System Equvalent Reducton Expanson Process (SEREP) (S2): S2 1. Component Mode Synthess (CMS) (S3): S3 1. Second Order Krylov Subspace Method (KSM) S3 2. Second Order Balanced Truncaton (SOBT) S3 3. Second Order Modal Truncaton (SOMT) (S4): Hybrd MOR (combnaton of S1, S2, S3)

5 The applcaton of solely S1, S2 or S3 consttute the 1-step MOR, whle the S4 category the 2-step MOR, snce a combnaton of the prevous MOR takes place (e.g. SEREP-KSM, KSM- SOBT, SOMT-KSM, etc.), as shown n Fg. 2. Fg. 2: MOR methods ntegrated n MORPACK Computaton and numerc advances The MOR nterface ntegrates and utlzes sparse solvers for the drect soluton [7] of the 5 selected MOR. Stll, for systems (1) wth dmenson n 10 and under normal hardware confguraton the drect soluton fals due to memory capacty problems. Therefore, teratve approaches [8] are ntegrated, e.g. the Precondtoned Conjugate Gradent method (PCG), wth whch the ROMs are generated. The drawback when usng teratve methods s the possblty of slow convergence rate and thus, vast computaton tmes, manly due to ll-condtoned system matrces (such matrces occur often n structural mechancs). The MOR nner nterface ntegrates the dagonal perturbaton method [1], whch accelerates the teratve scheme used. The dynamc propertes of the model are not damaged f the perturbaton scheme proceeds by selectng

6 the ntal perturbaton parameter k accordng to the algorthm s crtera. Fgure 3 llustrates the computaton tme reducton for a rather medum sze model ( n = ) and varous choces of the perturbaton parameter. The dynamc preservaton of the ROM vares accordng to the choce of k. DoF Fg. 3: Elastc rod teratvely reduced to the dmenson of m = 30, dagonal perturbaton 5 When dealng wth very large models,.e. n 10, MORPACK offers the selecton of 2-step MOR approaches (Fg. 2). Thus, hybrd methods are nvoked and combned wth dagonal perturbaton, computaton tme reducton s acheved. Table 3 gathers the CPU tme nformaton when applyng the SEREP-KSM hybrd MOR for reducng the dmenson of an elastc crankshaft ( n DoF = ). The orgnal model s frstly reduced to a medum sze dmenson n SEREP = 450 by the use of SEREP MOR, such that perfect correlaton s assured. Then, the reduced model s used for further reducton by

7 applyng KSM and generatng the fnal ROM of dmenson m = 25. The qualty of the MOR s depcted n Fg. 4 by comparng the frst 25 non-rgd egenvectors of both the orgnal and the fnal ROM. Table 3: Hybrd SEREP-KSM MOR: SEREP CPU tmes CPU Tme CPU Tme MOR: SEREP (secs) [%] Modal Analyss Algorthm - Lumped mass - PCG Lanczos TSEREP Formulaton Sparse SVD SEREP { } [ ], [ ] = M, D, K Sparse matrx algebra SEREP Fg. 4: Elastc crankshaft reduced by hybrd SEREP-KSM MOR: correlaton of dynamcs MOR modellng advances The MOR modellng aspects, whch are ntegrated nto MORPACK, concern the qualtatve mprovement and feasblty when applyng two non-physcal MOR methods, namely the KSM and SOBT. Concernng KSM, although beng a master DoF free method, the qualty of the ROM depends on the approprate selecton of a startng vector for the ntalzaton of the Arnold procedure. Two optmum startng vector algorthms OSV 1 and OSV 2 are ntegrated n the KSM MOR, whch assure qualtatvely good KSM ROMs n comparson to others generated

8 usng a random startng vector choce (Fg. 5). Research s stll open n ths area, snce a proof should be gven for ther applcablty (.e. to prove the fact that the result s due to the algorthms and not model dependent). Fg. 5: OSV 1 and OSV 2 appled to the UIC60-ral durng KSM MOR (b 1 -b 5 random vectors) The SOBT method can be used for reducng 2 nd -order systems (1) wth the prerequste that the matrx polynomal of (1) 2 P( λ)= λ M + λd+ K or certan transformatons of (3) must be stable,.e. all the egenvalues must have negatve real parts. Ths s to assure the unqueness of postve defnte 2 nd -order gramans or ther Cholesky factors when solvng the equvalent Lyapunov equatons. (3) The problem arses wth the fact that the structure s zero egenfrequences ( 0 Hz) contrbute to egenvalues of the matrx polynomal wth postve real parts and thus, damage the condton for applyng SOBT. Ths s the case of structures wth none or few fxed DoFs. Ths problem s overcome by the drect or teratve applcaton of the egenfreequency-shftng propertes of dagonal perturbaton. The unwshed egenvalues are shfted (Table 4), such that the polynomal matrx s stablzed wthout vastly damagng the model s dynamcs (Fg.7). Table 4: Free UIC60-ral shftng of f for SOBT usng dagonal perturbaton f rgd k = 4 k = 1 k = e e e e e e e e e e e e+003

9 e e e e e e e e+003 Fg. 7: Free UIC60-ral shftng of f for SOBT usng dagonal perturbaton damage control 2.2 Standard Input Data (SID) nterface The purpose of the SID nterface s to obtan the elastc body nformaton avalable after the MOR and prepare t, such that t can be used n a MBS code. Ths s accomplshed by the applcaton of the Rtz approxmaton. The flexble body s nerta as well as the generalzed elastc (nternal) and external forces gve the necessary nformaton for the full knetc descrpton of a deformed structure [9, 10]. Takng nto account the knematc constrants that descrbe mechancal jonts as well as defned trajectores, the generalzed DAE, whch gves the moton of constraned flexble MBS, s derved: T My + Ky + G λ = h + h, = 1,2,..., n y p gy (, t) = 0, ω b (4) wth n b beng the total number of bodes n the MBS, y = y t yr q the state vector T consstng of the translatonal, rotatonal and the elastc q coordnates, G the y t y r y constrant Jacoban matrx, λ the vector of Lagrange multplers, h p the vector of

10 generalzed forces, and h ω the quadratc velocty vector contanng the gyroscopc and Corols force components. The vector of algebrac equatons constrants for the system s dynamcs. gy (, t) descrbes the set of The system matrces (4) are parttoned accordng to the coordnates contrbuton (ndext for translatonal and - r for rotatonal) for both the rgd and the elastc (ndex- e ) parts of the MBS: T M sym. tt G yt T T M = rt rr,, M M K = G = G, y r y T Met Mer Mee 0 0 Kee G q ( h ) ( ) ( hp ) ( hω ) ( ω) ( hω ) p t t hp = h p, h = r ω h r q q The SID nterface generates the ASCII object-orented structured SID fle [11], based on the evaluaton of the mass, stffness and external forces ntegrals (5). The calculaton of the MBS s pure elastc parts,.e. the evaluaton of the ROM s (free and/or fxed) FE data s accomplshed by Rtz approxmaton usng smple vector-matrx multplcatons (analogue to FEMBS [3]). (5) Back-projecton approach The structure s dynamc nformaton (5) n the SID fle s obtaned by usng certan ROM s FE data, e.g. the system and deformaton matrces as well as the translatonal and rotatonal rgd body modes, etc. When a non-physcal or a certan hybrd MOR s used, the DoF set of a a a the reduced vector x R (2) les on a non-physcal space { ℵ : ℵ = }. Thus, the drect generaton of the knetc nformaton n MBS codes (SIMPACK) s not possble, snce SIMPACK requres that all the system s nformaton s defned n the Eucldean space. An extra subspace s requred [6] n order to project the ROM (2) back onto the physcal confguraton space (Fg. 9):

11 Mx + Dx + Kx = b, m m m T T = ( T a a) ( T a a) b = ( T a a) b = { M D K} [ ] [ ],, [ ],,. ℵ R ℵ ℵ R (6) The SID nner nterface of MORPACK ntegrates the back-projecton approach enablng thus, the mport of non-physcal or hybrd MOR nto SIMPACK. Fg. 9: Back-projecton scheme for reducton-expanson methods 2.3 Applcaton levels The two nner MOR and SID nterfaces of MORPACK operate wth data provded by four applcaton functons, whch operate accordngly: frstly the Import FE Data -applcaton s actvated amng at convertng the FE data nto the matrx market fle format. Ths data s used for the selected MOR and then the second Export ROM s Data -applcaton s actvated, whch gathers all ROM s nformaton necessary for the SID fle generaton (Table 4). Table 4: Applcaton levels 1-2 Levels 1-2 Import FE Data Export ROM s Data 1. System matrces 2. DoF nformaton (free and fxed structure) 3. DoF allocaton 3. Mode dsplacement matrces (modal and/or statc matrx) 4 Egenfreqency nformaton (value, rgd or non-rgd) 5. Stress matrces and unt load vectors at nodes

12 Before actvatng the SID nterface the thrd applcaton level s nvoked. It conssts of varous Modal Correlaton Crtera (MCC) [1], wth whch the dynamc propertes of the ROM s are valdated (Table 5). Thus, the user valdates the ROM pror to the SID generaton and only f successfully the SID nterface s actvated. Otherwse the reducton scheme should be repeated. Table 5: Applcaton level 3 Level 3 Model Correlaton Crtera Dynamc propertes checked 1. Normalzed Relatve Frequency Dfference 2. Modal Assurance Crteron 3. Modfed Modal Assurance Crteron 4. Mass Normalzed Vector Dfference 5. Stffness Normalzed Vector Dfference 1. Matrces defnteness 2. Structure preservaton The fourth applcaton level concerns ROMs, whch are created by the applcaton of the back-projecton approach. Thus, t concerns the non-physcal or certan hybrd MOR schemes both of whch generate ROMs lyng on a non-physcal subspace. Although the back-projecton approach transforms the ROM s data n order to be defned on the Eucldean space, the ROM s dynamc propertes (6) mght be damaged due to ll-condtonalty of the back-projecton matrx T a a ℵ. The Back-Projected ROM Control -applcaton verfes whether or not the ROM s structure s preserved (Table 6.). If not, the ROM wth ths specfc confguraton cannot be mported nto SIMPACK. Table 6: Applcaton level 4 Level 4 Back-Projected ROM Control Dynamc propertes checked Normalzed Relatve Frequency Dfference Structure preservaton 3 Conclusons and outlook The MORPACK nterface consttutes a necessary tool for the FEM-MBS couplng when alternatve MOR methods are appled,.e. MOR schemes not supported by commercal FE

13 and MBS software. MORPACK ntegrates n a user-frendly bass a varety of physcal, semphyscal, and non-physcal MORs, whch are supported by new numerc and modellng schemes: the dagonal perturbaton method, two optmum startng vector algorthms, and the back-projecton approach. Herewth, the effcency of the MOR algorthm s computatonally mproved and the mport of qualtatvely better ROMs nto SIMPACK s realzed. MORPACK s the lnk-nterface between ANSYS and SIMPACK. Further development of ths package could also nclude other commercal packages both from the FE feld, e.g. MSC Nastran and the MBS codes, e.g. ADAMS. Ths corresponds to mplementng the adequate applcaton levels for the data transfer from and nto the new software packages, whle the rest of the source code would reman as t s. Lterature [1] Koutsovasls, P., Betelschmdt, M.: Comparson of Model Reducton Technques for Large Mechancal Systems; A Study on an Elastc Rod. Multbody System Dynamcs, 20 (2008), p [2] ANSYS. ANSYS Advanced Analyss Technques Gude, [3] SIMPACK User s Gude; SIMPACK Release 8.6, [4] Antoulas, C. A.: Approxmaton of Large-Scale Dynamcal Systems. Seres Advances n Desgn and Control. Phladelpha: SIAM, 2005 [5] Benner, P., Mehrmann, V., Sorensen, D. C.: Dmenson Reducton of Large-Scale Systems. Seres Lecture Notes n Computatonal Scence and Engneerng. Sprnger Berln/Hedelberg: Sprnger-Verlag, 2005 [6] Koutsovasls, P., Quarz, V., Betelschmdt, M.: Standard Input Data for FEM-MBS Couplng; Importng Alternatve Model Reducton Methods nto SIMPACK. Mathematcal and Computer Modellng of Dynamcal Systems, 2008 (accepted) [7] Golub, G. H., Van Loan, C. F.: Matrx Computatons. Thrd Edton. Baltmore, Maryland: The John Hopkns Unversty Press, 1996 [8] Saad, Y.: Iteratve Methods for Sparse Lnear Systems. Second Edton. Mnneapols, Mnnesota: SIAM, 2003 [9] Shabana, A. A.: Dynamcs of Multbody Systems. Thrd Edton. Cambrdge Unversty Press, 2005 [10] Schwertassek, R., Wallrapp, O.: Dynamk flexbler Mehrkörpersysteme. Rehe Grundlagen und Fortschrtte der Ingeneurwssenschaften. Braunschweg/Wesbaden: VIEWEG, 1999

14 [11] Wallrapp, O.: Standardzaton of flexble modellng n multbody system codes; Part I defnton of standard nput data. Mech. Struct. Mach. 22 (1994), p