SENSITIVITY ANALYSIS WITH UNSTRUCTURED FREE MESH GENERATORS IN 2-D AND 3-D SHAPE OPTIMIZATION.
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1 SENSITIVITY ANALYSIS WITH UNSTRUCTURED FREE MESH GENERATORS IN 2-D AND 3-D SHAPE OPTIMIZATION. P. Duysnx, W.H. Zhang, C. Fleury. Aerospace Laboratory, LTAS, Unversty of Lège B-4000 LIEGE, BELGIUM. ABSTRACT. Shape optmzaton has to consder new progress n mesh generators to take full benefts from the use of free unstructured meshes n realstc ndustral optmzaton. The man dffculty reles n the mesh flow determnaton durng senstvty analyss. The authors present ther experence n applyng smoothng technques to transmt boundary perturbatons to the nner mesh. It s explaned how smoothng can be used to evaluate velocty felds n a smple and nexpensve way whle guarantyng relablty, effcency and precson. A wde range of smoothng procedures are compared and confronted wth other velocty feld approaches to nvestgate ther advantages and drawback. Fnally, applcatons of smoothng procedures n the commercal optmzaton package OPTI-SAMCEF are descrbed. 1. INTRODUCTION. As a result of prolfc research efforts, structural shape optmzaton has reached suffcent maturty to handle real-lfe problems n ndustral envronment. Now, t s stated as a mathematcal programmng problem n terms of an objectve functon f(z), desgn restrctons c (z) and desgn varables z. j (1.1) Due to the functon evaluaton cost, ths non-explct and nonlnear optmzaton problem s replaced by a sequence of conservatve convex approxmated explct sub-problems whch are generally solved by mathematcal programmng algorthms. These frst order approxmatons requre frst to perform an accurate effcent and relable senstvty analyss to get the necessary gradent nformaton. In shape optmzaton software, great care s devoted to senstvty analyss snce t requres the major part of computatonal effort. 3 As reported by Brabant, ntegratng new free mesh generators and computer aded desgn facltes nto a modern and powerful shape optmzaton tool s a problem whch s not completely solved. Frst shape 1-2 optmzaton tools used only regular F.E. meshes generated by a transfnte technque. In ths case, senstvty analyss s performed n an easy and natural way by usng the ntrnsc surface coordnates of transfnte mesh for grd deformaton. Even though ths natural method proved to be relable and robust n many problems, t revealed several lmtatons such as severe mesh dstortons for large geometry varatons
2 and somewhat tedous and nflexble when appled to complcated shapes. Snce few years, the emergence of new relable and effcent free meshers s very attractve, but ther applcaton s much more complcated for senstvty analyss. Unlke transfnte mesh, one cannot trust n free mesh generator for grd perturbaton. Mesh topology has to be frozen and a repostonng crteron has to be used to determne mesh flows relatve to boundary changes. Snce the end of 80's, several research works proposed to lnk mesh flows to physcal laws. The velocty feld s determned based on the mechancal behavor of the structure (In Ref.5 fcttous loads are used whereas prescrbed dsplacements are preferred n the last ones). Recently, local approaches have also appeared; they consst n relocatng only the outer layer of F.E. whle nteror nodes reman unchanged. Ref.9 has compared ths method transfnte approach. In the present research, the authors focussed on applyng Laplacan smoothng technques to the velocty feld problem wth unstructured meshes. Ths method can ether be related to the physcal approach as a soluton to the Posson's equaton or be regarded as a geometrc approach to the mesh flow problem. The effcency and relablty of these nexpensve methods were successfully tested and then ntegrated n the commercal F.E. and optmzaton 4 package OPTI-SAMCEF. 2. DISCRETE SEMI-ANALYTICAL APPROACH IN SENSITIVITY ANALYSIS WITH UNSTRUCTURED FREE MESHES. As the F.E. analyss s used, the sem-analytcal method s appled to the dscrete model. The F.E. dscretzaton of the equlbrum equaton of the structure s wrtten n terms of the generalzed stffness matrx K, dsplacements q and forces g : Dfferentatng equaton (2.1) gves the dsplacements senstvty wth respect to desgn varable z : (2.1) (2.2) The sem-analytcal approxmaton conssts n evaluatng the dervatves of the stffness matrx and generalzed forces by fnte dfference: (2.3) In order to generate the stffness matrx K(z +äz ) and load vectors g(z +äz ), the new poston of any nodal pont P after perturbaton of varable z s frst requred: (2.4) Usng the materal dervatve concept of contnuum mechancs, the velocty feld vector V s defned as the
3 frst dervatve of the poston of any pont P(x) wth respect to a change of desgn varable z. One can also nterpret the velocty feld V as the law gvng frst order perturbatons äx of nodal poston after a small varaton of boundares assocated to a change of desgn varable z. To understand the problem of senstvty analyss wth free mesh generators, the dervaton of stffness matrx has to be nspected. Lke matrx K tself, ts dervatves are bult by assemblng each elementary contrbuton (where NEL s the number of F.E. n the mesh): (2.5) where L s an ncdent matrx for each element. As shown by prevous equaton, accurate fnte dfferences of stffness matrx and load vector need to be performed wth the same mesh topology (connectvty and F.E. number) durng the perturbaton process. Unfortunately, free mesh generators do not gve these guarantees to remesh and a crteron has to be ntroduced to reposton the F.E. nodes after boundary varaton. Lke matrx K, the velocty feld V s computed n a fnte dfference form; ths means that the crteron s used to determne the perturbaton of poston äx=v äz relatve to a varaton of desgn varable äz nstead of the feld V tself. Lke the velocty feld, node perturbatons have to ft boundary shape modfcatons due to desgn varable varatons. In other words, the velocty feld s explctly determned on the boundares and s accordngly used as non homogeneous boundary condtons to determne the nner mesh moton wth the velocty feld law. In the present research, the mesh flow s actually realzed n computng the perturbaton of postons äx=väx wth a Laplacan smoothng.
4 3. SMOOTHING PROCEDURES TO SOLVE THE VELOCITY FIELD PROBLEM. Relocaton and smoothng procedures are well-known procedures and have already been successfully appled n several domans from aero-elastcty to sold mechancs wth relablty and effcency. Because of ther smplcty and easness of mplementaton, they seem very (3.1) attractve to mesh flow determnaton wth unstructured free meshes. The basc scheme of Laplacan smoothng (3.1) acts as the applcaton of the Laplacan fnte dfference operator to unknown postons. If a regular rectangular mesh was used, t would gve the rgorous soluton to a bharmonc equaton. Unknown postons can also be related to the centrods Q of neghborng elements wth weghted factors w : e e (3.2) The basc scheme can be also modfed to take nto account addtonal nformaton about grd geometry: (3.3) where stffness terms k j are ntroduced between any neghborng nodes and j (connected by F.E. edge or a dagonal). These lnear equatons can be expressed n a matrx formulaton n separatng the boundary nodes of ndces b from the nner ones of ndces : (3.4) As explaned, mesh flow computaton s realzed through a smoothng procedure appled to the ncrements of nodal postons. On the outlne, nodal perturbatons are ruled by the boundary parametrc equatons whle the nner mesh moton s computed based on the relocaton crteron wth those non homogeneous boundary condtons. So, the velocty feld can be wrtten advantageously as a lnear matrx relaton lnkng the nner perturbaton to boundary node dsplacements: (3.5)
5 Practcally, smoothng s used n ths lnear matrx formulaton and planar or sold problems are replaced by a sequence of scalar uncoupled problems for each desgn varable perturbaton, resultng n a drastc reducton of the computer cost. For more effcency, ths relaton s treated n a subdoman strategy. Ths procedure exhbts numerous advantages. At frst, the velocty feld s ndependent of the smoothng state of the ntal mesh, otherwse smoothng should be appled one more tmes to extract the relocaton moton of the ntal mesh from the mesh flow. Due to the smoothng property of Laplace operator, the procedure generates velocty felds free of local mnma or maxma. In addton, pseudo-laplace problems generate mesh flows that dffuse deeply the perturbatons nto the grd, whch s hghly desrable for accuracy. Fnally, the smoothng procedure s hghly relable and robust as long as small ncrements of node postons are manpulated. The nherent drawbacks of Laplacan technques for mesh relocaton (such as mesh degeneraton, nverted F.E., or potental movements of nodes out of the doman due to non convex local geometry) are allevated snce only small ncrements of node postons of a basc F.E. grd of farly good qualty (ths condton s generally fulflled by good mesh generators) are consdered. Table 1: Summary of Laplacan Smoothng Schemes. Pure Laplacan Smoothng k = 1. f and j connected k j = 0. f and j unconnected Modfed Laplacan Smoothng w e = 1. f element e contans node 10 Isoparametrc Laplacan Smoothng k = 1. f and j connected Power Laplacan Smoothng F.E. Error Weghted Scheme 11 j p 12 j e e e e e j j k = -0.5 f and j on a dagonal k = 1./L L nterface (, j) length w = E /A E /A element error densty 4.COMPARISON OF VARIANT SMOOTHING PROCEDURES NUMERICAL TESTS AND APPLICATIONS. The frst concern s to select a good smoothng crteron.e. adequate stffness terms between the nodes. Table 1 summarzes several tested smoothng schemes. As stated before, the easest technques are the pure Laplacan smoothng and the modfed verson wth a unt stffness or a unt weght. To the authors' experence, these smple methods are relable and robust n numerous practcal applcatons provded that the 10 ntal mesh s suffcently smooth. The soparametrc Laplacan scheme of Herrmann s desgned to reflect n a better way the curvature and the mesh dstrbuton nformaton of the boundary. When the element szes are very dfferent, t s preferable to take nto account ther nfluence and to adopt a scheme proposed by 11 Robnson. In ths one, stffness are weghted by the edge lengths L through an nverse power functon k j = p 1./L (exponent p s generally selected between 1 and 2). Snce the stffness effect s ncreased on the smallest edges, the mesh deformaton s the largest n the bggest elements far from the crtcal zones. Ths method exhbts very good results when appled to F.E. meshes presentng refned zones. Other smoothng procedures 12 usng the element area or F.E. estmated error nformaton can also be mplemented. Lke Daz who suggested to use the F.E. error for grd adaptaton, the authors attempts to extend the use of F.E. error estmators as weghts for smoothng. Such a mesh flow localzes the most mportant mesh deformatons n the F.E. havng small error estmators.
6 Fgure 2: Comparason of Four Dfferent Laplacan Smoothng Schemes. Applcatons of several relocaton schemes are presented n fgures 2a-b-c-d. The velocty felds are -4 relatve to a modfcaton of the crcular notch curvature (wth a relatve perturbaton of 10 ). The free mesh 4 has been generated by an offset generator and the drawn node dsplacements are amplfed by a 10 factor. The perturbated meshes (n dot lnes) of fgures 2 are yelded when applyng respectvely Laplacan p smoothng, modfed Laplacan smoothng, soparametrc Laplacan and nverse power (1/L wth p=2) schemes. Ths example shows hat the deformed mesh s very smlar to the orgnal one, preservng the mesh qualty. The boundary modfcaton yelds a deep mesh flow. In the present case, the smallest depth of perturbaton s gven by the soparametrc Laplacan smoothng whle the power scheme dffuses t far away because of the ncreased rgdty of the small trangular elements. Nevertheless, senstvty analyss of dsplacements and stress reveals closed results. Other numercal tests have been realzed to compare relocaton schemes wth other velocty feld methods. In fact, there s no contradcton between smoothng and transfnte velocty feld snce transfnte remeshng can be regarded as a perfect Laplacan smoothng of the regular mesh n the ntrnsc coordnates system. On the other hand, the local approach (where only the outer layer of nodes are relocated whle the nner nodes reman unchanged) can be consdered as a lmt case of Laplacan smoothng technque when the depth of mesh flow s reduced to the outer elements. Despte ts smplcty, studes n Ref.7 demonstrated that ths technque exhbted a degradaton of senstvty analyss qualty resultng n a slower convergence when appled to complcated problems. Fnally, lke natural methods, Laplacan smoothng s related to a physcal phenomenon as steady state thermal dstrbuton; t could also be retreved from lnear elastcty equaton f uncouplng dsplacements are assumed. Ref.7 showed that assumpton gves smlar results but smoothng s much more nexpensve than addtonal F.E. analyss.
7 Now ths new tool s fully ntegrated n the OPTI-SAMCEF package to handle free unstructured meshes. Its ablty to handle real-lfe problems s llustrated by the well known applcaton of a torque arm shape optmzaton (see Ref.13, 2, 7). The weght of ths rear automotve torque arm s to be mnmzed whle 2 the maxmum stress s lmted to N/cm. Eght desgn varables descrbe the outlne of the arm as an assembly of straght lnes and crcle arcs wth tangent condtons. An offset free mesh generator was selected and appled at each teraton to mantan mesh qualty wth the same mesh densty. Fgure 3 shows the ntal 2 and fnal shapes (after 8 teratons wth CONLIN optmzer ).. Fgure 3: Optmzaton of a Torque Arm 5. CONCLUSIONS. In order to take nto account the new free mesh generators n a modern shape optmzaton tool, the authors have turned ther attenton towards smoothng procedures to solve the mesh flow problem. The research led to create a procedure based on applyng smoothng to the ncrements of node poston after perturbatng the outlne defnton. Provded the velocty feld exhbts a suffcent depth of perturbaton propagaton n the mesh, the tested relocaton schemes have revealed robust and relable. In addton of ts effcency and relablty, senstvty analyss s also nexpensve snce smoothng s formulated as the repeated soluton of a scalar problem. These procedures are now ntegrated n the commercally supported software OPTI-SAMCEF. After applcaton of smoothng to 2-D problems, the method s now beng ntegrated and tested n 3-D problems. Even though applcatons seem to be heaver, encouragng results are expected.
8 REFERENCES: 1. Brabant, V. and Fleury, C.: "Shape optmal desgn usng B-splnes", Computer Methods n Appled Mechancs and Engneerng, vol. 44, 1984, pp Brabant, V. and Fleury, C.: "An approxmaton-concepts approach to shape optmal desgn", Computer Methods n Appled Mechancs and Engneerng, vol. 53, 1985, pp Brabant, V. and Morelle, P.: "Shape optmal desgn and free mesh generaton", Structural Optmzaton, vol 2, 1990, pp Morelle, P., Duysnx, P., Fleury, C.: "CAD/FEM couplng n shape optmzaton", FEM'92 IKOSS Congress, Baden-Baden, Germany, November Belgundu, A. and Rajan, S.: "A shape optmzaton approach based on natural desgn varables and shape functons", Computer Methods n Appled Mechancs and Engneerng, 1988, pp Yao, T.S., Cho, K.K.: "3-D shape optmal desgn and automatc fnte element regrdng", Internatonal Journal of Numercal Methods n Engneerng, vol. 28, 1989, pp Zhang, W. H.: "Calcul des sensbltés et optmsaton de forme par la méthode des éléments fns", Ph. D. dssertaton, Unversty of Lège, Appled Scences Faculty, 1991, (In French). 8. Beckers, P.: "Recent developments n shape senstvty analyss: the physcal approach", Engneerng Optmzaton, vol 18, 1991, pp Kbsgaard, S.: "Senstvty analyss - The bass for optmzaton", Internatonal Journal of Numercal Methods n Engneerng, vol. 34, 1992, pp Herrmann, L.R.: "Laplacan-soparametrc grd generaton scheme", J. Engng Mech. Dv., ASCE, vol. 102, 1976, pp Robnson, B., Batna, J. and Yang, H.: "Aeroelastc analyss of wngs usng Euler equatons wth deformng mesh", Journal of Arcraft, vol 28, n 11, November Daz, A., Kkuch, N. and Taylor, J.: "A method of grd optmzaton for fnte element methods", Computer Methods n Appled Mechancs and Engneerng, vol.41, 1983, pp Bennet, J.A. and Botkn, M.E.: "Shape optmzaton of two-dmensonal structures wth geometrc problem descrpton an adaptatve mesh refnement", presented at AIAA/ASME/ASCE/AMS Structures, Structural Dynamcs and Materal Conference, Lake Tahoe, Nevada, 1983.
9 SENSITIVITY ANALYSIS WITH UNSTRUCTURED FREE MESH GENERATORS IN 2-D AND 3-D SHAPE OPTIMIZATION. P. Duysnx, W.H. Zhang, C. Fleury. Aerospace Laboratory, LTAS, Unversty of Lège B-4000 LIEGE, BELGIUM. ABSTRACT. Shape optmzaton has reached suffcent maturty to handle real-lfe problems n ndustral envronment. However, shape optmzaton packages have now to consder new progress n mesh generators and have to ntegrate these new capabltes to take full benefts from the use of free unstructured meshes n realstc ndustral optmzaton. To take nto account those nterestng new tools, the man dffculty reles n the velocty feld determnaton durng senstvty analyss. From all the possble velocty feld procedures, the authors' experence n applyng smoothng technques s presented. Frst t s shown that smoothng technques satsfy fundamental requrements to carry out successfully the senstvty analyss task for unstructured meshes wth success. Boundary perturbatons are transmtted to the nner mesh through a smoothng crteron. It s explaned how smoothng can be used to evaluate velocty felds n a cheap and smple way whle guarantyng relablty, effcency and precson. A wde range of smoothng procedures (pure Laplacan methods, length weghted smoothng or relocaton based on fnte element error crtera) are presented. Orgnal mprovements and modfcatons are presented to deal wth refned meshes, F.E. errors nformaton, etc.... Smoothng procedures are also compared wth other velocty feld approaches to provde an analyss of advantages and drawbacks of the method. Extensons from 2-D planar problems to 3-D shell and volume shape optmzaton are then dscussed. Fnally, applcatons of smoothng procedures n the commercal optmzaton package OPTI- SAMCEF are descrbed. And several practcal examples are provded to llustrate the proposed approach.
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