REDUCTION OF FINITE ELEMENT MODELS BY PARAMETER IDENTIFICATION

Transcription

1 ISSN X INFORMATION TECHNOLOGY AND CONTROL, 008, Vol.37, No.3 REDUCTION OF FINITE ELEMENT MODELS BY PARAMETER IDENTIFICATION Riantas Barauskas, Vidantas Riavičius Departent of Syste Analysis, Kaunas University of Technology Studentų St , LT Kaunas Abstract. The study deals with the proble aong to forally reduce a coplex finite eleent structural odel to a sipler one. As a saple task, the reduction of a girder odel to the sipler equivalent ebrane odel has been investigated. The coincidence of odel displaceents at given loading conditions is eployed as a criterion of utual adequacy of the two odels. Both static and dynaic displaceents at selected reference points have been used in the expression of the penalty-type target function, the iniu of which indicates the best fit between the original and reduced odels. The target function has been iniized by using the geoetrical and physical paraeters of a typical ebrane eleent as optiization variables. The calculations have been perfored in MATLAB environent. The validity of obtained paraeters of the ebrane odel has been tested by investigating the original and reduced structures of different geoetrical shapes at coplex loadings. Keywords: Finite eleent odels, reduction, paraeter identification. 1. Introduction Finite eleent techniques in principle enable to analyse structures of any level of coplexity, including their essentially non-linear behaviour, peculiarities of internal texture, etc. [11]. The odels are often generated by the expense of very coplex structures, huge diensionalities and internal interactions, which require very large and often prohibitive aounts of coputational resources. Building siplified (reduced) coputational odels is a coon practice enabling to obtain solutions with practically acceptable costs. Modest size coputational odels of bodies the internal structure of which is coplex and heterogeneous are generally obtained by using appropriate constitutive equations describing the aterial behavior. In structural analysis, constitutive relations connect applied stresses or forces to strains or deforations. More generally, in physics, a constitutive equation is a relation between two physical quantities that is specific to a aterial or substance, and does not follow directly fro physical law. It is cobined with other equations that do represent physical laws to solve physical probles [10]. Soe constitutive equations are siply phenoenological; others are derived fro first principles. Most coercial finite eleent systes possess large libraries of constitutive odels. For exaple, LS-DYNA progra accepts a wide range of aterial and equation of state odels, each with unique nuber of history variables. Approxiately 100 aterial odels are ipleented in LS- DYNA. Soe aterials include strain-rate sensitivity, failure, equations of state and theral effects [7]. The concept of ulti-scale finite eleents has been introduced to describe the generalization of the traditional finite eleent by prescribing it to ore coplex behavior laws than could be possible by using traditional constitutive equation odels [4]. Multiscale finite eleent can be regarded as scaleable, atheatical acro-odel, which captures the response of a sub-region of the coputational doain, in a anner sealessly copatible with the finite eleent odeling infrastructure. Such generalized finite eleents are ipleented by using doain decoposition ethodology. Doain decoposition ethods solve a boundary value proble by splitting it into saller boundary value probles on subdoains and iterating to coordinate the solution between the subdoains. The probles on the subdoains are independent, which ake doain decoposition ethods suitable for parallel coputing. Doain decoposition is an active, interdisciplinary research area concerned with the developent, analysis, and ipleentation of coupling and decoupling strategies in atheatical and coputational odels of natural and engineering systes. The siplest applications of the idea of doain decoposition are super eleents in linear static probles, which are ipleented in ost finite eleent progras such as ANSYS, NASTRAN, etc. However, there are nuerous studies of application of doain decoposition in dynaics, 11

2 R. Barauskas, V. Riavičius such as coponent ode synthesis ethods. Generally, coputational odel reduction ethodologies are iportant in research area leading to efficient and reasonably adequate odels of various engineering systes [6]. Nuerous engineering approaches to construction of ultiscale finite eleent approaches have been reported in the scientific literature. As an exaple, the very high velocity contact interaction probles are aong the ost coplicated in coputational echanics. The failure processes that follow the interaction are initiated in icro-volues considerably saller than the easureents of the interacting bodies. It is practically ipossible to odel the behavior of the aterial at the icro-level the nuber of degrees of freedo of such a odel would be too large and unrealistic for coputer resources nowadays and probably in the nearest future as well. In practice, usually the coputations are perfored by using acroscopic aterial odels that approxiately describe real processes taking place in the aterial. In [5] real and nuerical shooting-through experients are presented for the Nextel fabrics and the Kevlarepoxy shield. The icro- and ezzo-echanical odels have been used to siulate the behavior of sall speciens, and in coparison of the nuerical results with experiental data the aterial odel characteristics have been found. The stiffness coefficients were used for deterining the deviatoric stresses and by eans of state equation of the relationship between the pressure and volue change was established. Further coputations have been perfored by using the acro-echanical odel where a layered structure has been presented as the porous continuu. It enabled to disregard the real geoetry of the weave and to present the averaged strength paraeters of fabrics. The resulting odel was axisyetric, of reasonable diension, and the obtained results were satisfactorily close to the experiental ones. As an exaple, a woven textile structure can be represented by using odels of different levels of detalization. A woven structure coposed of shell eleents [], sipler and ore efficient cobined particles odel [3], orthotropic ebrane odels [1], have been eployed in order to represent the dynaic behavior of textile cloths under conditions of echanical ipact and penetration. A special attention and prospective deserve odels, in which central and distant zones of the sae structure are presented by different odels. As in [1], the zone of ballistic interaction of textile structure has been odeled by using the coplex contact odel of a woven structure, eanwhile the distant zones have been presented by ebrane eleents. The coupling between the zones has been ipleented by eans of tie constraint. The ain purpose of this cobination was to ipleent the alost infinite surrounding. As a rule, such cobined odels are obtained by using a lot of engineering intuition and basing on profound knowledge of physical properties of the investigated phenoena. More regular approaches are necessary, which enable to synthesize siplified or reduced odels of internally coplex structures. The paraeters of the reduced odel can be adjusted by perforing the iniization of error functions, quantitatively indicating the non-coincidence of the response between the siplified and reference odels. An alternative approach can be based on neural network techniques in order to synthesize the odels exhibiting the required structural response [9]. Generally, siplified (reduced) odels of coplex structures can be obtained on the base of coparison of their response to appropriate set of excitations against the response of a ore detailed odel exposed to the sae excitations. In this work, we deonstrate probably the siplest approach to the construction of reduced odels based on iniization of penaltytype target function representing the residual between the two sets of responses. The work presents the procedure and results of synthesis of the continuous ebrane odel, which iitates the behavior of girder structure under static, as well as, dynaic loadings.. Proble Forulation The analyzed source structure is a D girder, which is coposed of tiny bea eleents, and the approxiating reduced odel is a planar ebrane. The girder consists of rods of unifor width and thickness. It is necessary to find the paraeters of the equivalent orthotropic ebrane. Assue that the ebrane odel presents a satisfactory approxiation of the girder if the displaceents of nodes at the sae loading are obtained nearly the sae by using both odels. Consider rectangular plate and rectangular girder, which have identical diensions (Figure 1). The girder geoetry is described by rod width h, rod thickness b and N nuber of cells along the side of the girder. Physical paraeters used in the sall displaceent elasticity odel are Young odule E g and ass density ρ g. The ebrane odel is characterized by thickness s and orthotropic aterial paraeters: Young odule E, Poisson ratio ν, shear odule G and ass density ρ. Mebrane paraeters ν, E, G and s have to be established, which enable the ebrane to exhibit the sae or siilar behaviour in ters of displaceents of respective eleent nodes at a given loading. As a easure of quality of the approxiation of the girder odel by equivalent ebrane odel we eploy the iniu of a penalty-type target function expressed as a su of squares of differences between the displaceents of corresponding nodes of each odel. The static as well as dynaic behaviour of the 1

3 Reduction of Finite Eleent Models by Paraeter Identification two structures has been analyzed. The schees of two static loading cases are presented in Figures and, where both structures have been exposed to static load F and the differences of displaceents of 4 selected nodes have been included into the penalty function expression. The obtained paraeters of the equivalent ebrane shall be tested by using several freely selected test odels (loading cases), two of which are presented in Figures c) and d). Figure 1. The girder ( and equivalent steel ebrane ( c) d) Figure. The finite eleent odels: 1 st analysis odel; nd analysis odel; c) 1 st test odel; d) nd test odel In the case of dynaic analysis, the differences between displaceents of nodes are iniized at selected tie oents. The analysis has been perfored by using ANSYS and MATLAB software. The displaceents obtained in ANSYS have been used when foring the target function, which subsequently has been iniized by eploying MATLAB function FMINCON(). 13

4 R. Barauskas, V. Riavičius 3. Analysis of Results 3.1. Static analysis Further, the analysis of the statics of selected girder as well as paraeters of ebrane resulted in the optiization, are presented. In order to facilitate the optiization proble assue E g = E, ν = 0, ρg = ρ and h= b. Assue the girder rods being thin enough to aintain the echanical features of the girder: 1 b 1 L L L, L = b, 0 L 8 N 0 N 8 N (1) where L is the side length of the rectangular eleent and N nuber of divisions of the side, which is equal to the nuber of cells along the side of the girder. In this exaple, nubers of divisions of the ebrane and the girder are selected the sae, however, generally the grid spacing ay be uch saller than side length of the ebrane eleent. Consider the odels in Figure 3. In the first load case (LC1) (Figures 3, ), all nodes of the botto side are fixed eanwhile all nodes of the right hand side are exposed to forces iitating distributed loading along the Ox direction. In the second load case (LC) (Figures 3 c), d)) the top side is exposed to distributed loading along the Oy direction. c) d) Figure 3. The finite eleent esh and loads of the : 1 st analysis odel of the ebrane ( and the girder (; nd analysis odel of the ebrane (c) and the girder(d) The target function is read as follows: () T( p, p, q, q ), ( ) ( ) ( ) ( ) n n i i i i ( p1 q1) ( p q) i i i i i= 1 i= = + n n n n i i i i p1 + q1 p + q i= 1 i= 1 i= 1 i= 1 i i where, p are the vectors of i-node displaceents p1 of the 1 st and nd i i odels of ebrane, q1 and q are the vectors of i-node displaceents of the 1 st and nd odels of girder, n= ( N + 1) total nuber of the nodes of each odel. After the iniization of () we obtained the relationship of optiu thickness s of the equivalent ebrane against the girder rod thickness b and against the shear odule G. By applying the least squares approxiation (LSA), the quadratic and linear relationships between the geoetric paraeters of the 14

5 Reduction of Finite Eleent Models by Paraeter Identification girder and the equivalent ebrane have been established as in Figure 4. where n total nuber of nodes of each odel, i i p, p are x and y displaceents of node i of the x y ebrane, of the girder. i i q, q x y are x and y displaceents of node i Figure 4. The pairs of optial paraeters of the girder and the equivalent ebrane (points arked by crosses) and the regression curves,( square fit; ( linear fit The analytical expressions of the regression curves presented in Figure 4 are read as s = s(n, = b ( 1175, , N), 3 (3) G = G(N, = N 10 ( 93, 09 3,660 N ) b 10 ( 16818,, 0013 N) + N b 10 ( 9, 450 ) The estiation of derived forulas (3) against calculated pairs of optiu paraeters at different values of girder paraeter b and side division N is presented in Figure 5. It follows fro Figure 5 that the increase of the esh division paraeter N, akes each rod of the girder thinner, see forula (1). The corresponding values of the thickness of the equivalent ebrane and its shear odule tend to decrease. The deviations of calculated optiu values of the ebrane paraeters with respect to the obtained regression (3) have been evaluated as n ( p q ) + ( p q ) Δ = i i i i i x x y y n i i ( qx + qy ) i= 1, (4) Figure 5. D regressions of depending paraeters Further, we present the evaluation of the derived dependencies (3). The nuber of divisions along the side of tested odels was N = 38, and the values b of the girder was chosen in accordance with the forula L b =. The paraeters of equivalent ebrane 8 N have been calculated according to forula (3). The sae loading of the odel has been used in all investigated cases as in Figures and. The estiation of the deviations of ebrane displaceents fro the reference displaceents of the girder is presented in Figure 6. The largest deviations of the first odel (Figure 6 ) are at the nodes in the vicinity of the constrained side of the ebrane. On the contrary, in the second odel (Figure 6 ) the deviations at the nodes in the vicinity of the constrained side of the ebrane are the sallest. The next nuerical experient is perfored by loading the sae girder and equivalent ebrane by eans of the force applied in the plane xoy at the corner at angle 45 0, (odel Figure c), Figure 7 ). 15

6 R. Barauskas, V. Riavičius Figure 7 is obtained using odel (odel Figure d)) by applying the force at the id-side node. Figure 6. The estiated values of differences of displaceents of the corresponding nodes of ebrane and girder by using 1 st ( and nd ( odels (Figure ) We have found the equivalent ebrane for the selected girder by considering the extra loading cases (odels Figure c) and d)) and deterined that the deviations of relevant nodes have increased up to 6 ties. The axiu value of relative displaceent deviation between the two odels was equal to approxiately 16%. The biggest deviations of displaceents appear at the nodes affected by the force. In order to reduce the deviations we should include the displaceents of nodes of the two odels into the target function to be optiized. Forula () is to be odified by adding the suation ters representing the reference nodes of all investigated odels. As a consequence, the regression forulas (3) could have been altered. The approxiation of the reference girder odel by the equivalent ebrane has practical iportance only if linear diensions of ebrane eleents are uch bigger than the distance between the adjacent rods of the girder. The paraeter optiization procedure reains essentially the sae. The only difference is that the reference points on the girder ay correspond to the points between the nodes of the ebrane and therefore the interpolation of ebrane displaceents is necessary. Figure 7. The estiated values of differences of displaceents of the corresponding nodes of ebrane and girder with free selected odels The estiations of displaceent differences between the two odels in the first loading case is presented in Figures 8, for the coincident (N=38) and non-coincident (N=4) esh, respectively. It can be observed that the axiu estiation values did not change, however, the estiations at individual nodes ay change significantly (up to 6 ties in this case). 3.. Dynaic analysis Here we extend the regression forulas deterined in section 3.1 for the static analysis to the dynaic analysis. Tie is introduced as continuous variable t [0; T] and optiization is perfored by using a new target function, obtained by integrating expression () over tie. Now P and Q are threediensional atrices, in which nodal displaceents are stored at all tie steps t k. The integration over tie is perfored nuerically by using the 5 th order Newton Cottes quadrature forula [8]. The tie law of the loading is read as 1 E g T L. We chose the tie interval of the ρ g transient dynaic analysis equal to the tie necessary for the longitudinal elastic wave to travel distance L equal to the side length of the odel. The sine-pulse 16

7 Reduction of Finite Eleent Models by Paraeter Identification shaped by tie law of force F we assued to have period T TF, Figure 9. Forces at individual nodes have been applied Fax sin( π t) F =. In order to N 1 iitate the distributed loading of the side of the ebrane, the nodes at vertices of the rectangular ebrane are affected by only half of the force, which is applied to other nodes of the ebrane boundary. Figure 9. The reference nodes of the odel, The tie law of the loading force Figure 8. The estiated values of differences of displaceents of respective nodes of equivalent ebrane and reference girder at coincident eshing N=38 ( and non-coincident eshing N=4 ( We select 8 reference nodes (Figure 9 ), at which displaceent tie laws of both structures are copared against each other. In order to perfor the iniization of the penalty type target function, the paraeters of the ebrane are calculated by using forula (3) derived for the static analysis case. In the siplest case we are using only one optiization variable as equivalent ass density ρ of the ebrane. The esh in both structures is eployed. The nuerical values of the paraeters are presented in Table1. First colun presents the paraeter naes, second colun forulas, by which they are calculated, and the third the values of the paraeters. We use FMINCON() function in order to deterine ρ value. The iniization process is shown in Figure 10. Figure 10. The variation of the value optiization paraeter during iterations We get the equivalent ebrane by having the * * kg ρ density ρ = 1615,, The obtained density value of the equivalent ebrane does not neces- ρg sarily correspond to the density of a real aterial. It is rather a atheatical approxiation of the density of equivalent ebrane, which provides good coincidence of displaceents of both structures. On the other hand, when estiating the reality of the equivalent density we should take into account the surface density ( ρ s ) rather than voluetric density ρ, because the planar behavior of the structure is investigated. 17

8 R. Barauskas, V. Riavičius Table 1. Paraeters of the girder and equivalent ebrane obtained as a result of target function iniization Paraeter nae Forula, by which paraeter is calculated Paraeter value E g,pa b,. L 8 N g h,. b kg ρ g, N g 48 ν 0 E, Pa. E g s. s ( N, , G Pa. G ( N, ,, kg ρ. 3 ρ 1615 N 48 Figure 11. The agnitudes of displaceents at several selected tie oents in the girder ( and equivalent ebrane structure ( Figure 11 presents the agnitudes of displaceents caused by the transient vibration process (the wave traveling along the Ox direction, Figure ) at several selected tie oents in the girder ( and equivalent ebrane structure (. The two graphs are practically equivalent to each other. Figure 1 presents errors of displaceents of reference nodes. The obtained errors are quite sall indicating that the regression forulas (3) are suitable for eploying the for the dynaic analysis only with the equivalent density value of the ebrane being adjusted properly. 4. Conclusions A foral approach to the reduction of a coplex finite eleent structural odel to the sipler one has been proposed. The procedure is based on the penalty type target function iniization in the space of paraeters of the reduced odel. As a saple task, the synthesis of the reduced equivalent continuous ebrane odel, which iitates the behavior of the girder structure, has been solved. For the static analysis case the equivalent ebrane paraeter set has been deterined at which the 18

9 Reduction of Finite Eleent Models by Paraeter Identification odels worked satisfactorily in the case of coincident, as well as, non-coincident eshes of the reference and reduced equivalent structure and different loading configurations. Regression forulas for obtaining the equivalent paraeters have been derived. The odel based on the equivalent paraeters obtained for static analysis has been deonstrated to work also in the dynaic analysis case, provided that a proper equivalent ass density value of the equivalent reduced structure is selected. Figure 1. Differences between corresponding displaceents of reference nodes of the girder and ebrane odel References [1] R. Barauskas. Modeling of the bullet penetration into textile targets by using cobined woven structure ebrane approach. WSEAS Transactions on Inforation Science and Applications, Issue 11, Vol., Noveber 005, ISSN , [] R. Barauskas, A. Abraitiene, A. Vilkauskas. Siulation of a ballistic ipact of a deforable bullet upon a ultilayer fabric package. nd International Conference on Coputational Ballistics, WIT Press, Southpton, Boston, 005, [3] R. Barauskas, M. Kuprys. Collision detection and response of yarns in coputational odels of woven structures, Proceedings of 10th International Conference Matheatical Modeling and Analysis and nd International Conference Coputational Methods in Applied Matheatics, June 1-5, 005, Trakai, Lithuania, 1-6. [4] A.C. Cangellaris, Wu Hong. Doain decoposition and ulti-scale finite eleents for electroagnetic analysis of integrated electronic systes. Electroagnetic Copatibility, 005, EMC 005, 005 International Syposiu, Vol.3, Issue, 8-1 Aug. 005, [5] R. Clegg, et al. Application of a coupled anisotropic aterial odel to high velocity ipact response of coposite textile aror. 18th International Syposiu and Exhibition on Ballistics, San Antonio, Texas USA, Noveber 15-19,1999, TP05. [6] A.Y. Lee, et al. Model reduction etodologies for articulated ulti-flexible body systes. Structural Dynaic Systes Coputational Techniques and Optiization:Nonlinear Techniques, Gordon and Breach Science Publishers, 1999, [7] LSDYNA. Theoretical Manual Liverore Software Technology Corporation, 006. [8] K. Plukas. Skaitiniai etodai ir algoritai. Kaunas: Naujasis lankas, 001, ISBN , 48. [9] A. Verikas, A. Gelžinis. Neuroniniai tinklai ir neuroniniai skaičiaviai. Kaunas: Technologija, 003, 175. [10] Wikipedia, [11] O.C. Zienkiewitcz. The Finite Eleent Method. Butterworth-Heineann, 000. Received January