COMPOSITION OF ISOGEOMETRIC ANALYSIS WITH LEVEL SET METHOD FOR STRUCTURAL TOPOLOGY OPTIMIZATION

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1 INTERNATIONAL JOURNAL OF OPTIMIZATION IN CIVIL ENGINEERING Int. J. Optm. Cvl Eng., 202; 2():47-70 COMPOSITION OF ISOGEOMETRIC ANALYSIS WITH LEVEL SET METHOD FOR STRUCTURAL TOPOLOGY OPTIMIZATION S. Shojaee *, a,, M. Mohamadan b and N. Valzadeh a a Department of Cvl Engneerng, Shahd Bahonar Unversty, Kerman, Iran b Cvl Engneerng Department, Ferdows Branch, Islamc Azad Unversty, Ferdows, Iran ABSTRACT In the present paper, an approach s proposed for structural topology optmzaton based on combnaton of Radal Bass Functon (RBF) Level Set Method (LSM) wth Isogeometrc Analyss (IGA). The correspondng combned algorthm s detaled. Frst, n ths approach, the dscrete problem s formulated n Isogeometrc Analyss framework. The objectve functon based on complance of partcular locatons of materals n the structure s used and fnd the optmal dstrbuton of materal n the doman to mnmze the complance of the system under a volume constrant. The refnement s employed for constructon of the physcal mesh to be consstent wth the mesh s used for level set functon. Then a parameterzed level set method wth radal bass functons (RBFs) s used for structural topology optmzaton. Fnally, several numercal examples are provded to confrm the valdty of the method. Receved: 5 January 202; Accepted: 30 March 202 KEY WORDS: sogeometrc analyss; topology optmzaton; shape optmzaton; level set method; radal bass functons. INTRODUCTION An extensve development n structural shape and topology optmzaton has been experenced n recent three decades. Consderable researches and varous topology optmzaton methods such as materal dstrbuton method [-4], Sold Isotropc Materal wth Penalzaton (SIMP) methods [5-8], Bubble method [9] and Evolutonary Structural Optmzaton (ESO) method * Correspondng author: S. Shojaee, Department of Cvl Engneerng, Shahd Bahonar Unversty, Kerman, Iran E-mal address: saeed.shojaee@uk.ac.r

2 48 S. Shojaee, M. Mohamadan and N. Valzadeh [0] have been proposed. In recent years, the level set methods [, 2] have been ncorporated nto structural shape and topology optmzaton feld effectvely. The major strength of level set method s that, t s an mplct method for movng nteror and exteror boundares and durng the process; boundares may jon to each other. Among the frst researchers, Sethan and Wegmann [3] extended the level set method n shape optmzaton. In ther work, equvalent stress crtera update level set functons nstead of solvng equatons. Osher and Santosa [4] also apply the level set method to the topology desgn problem of a two-densty nhomogeneous drum membrane. In addton Allare et al [5] and Wang et al [6] studed the structural topology optmzaton by combnng the shape dervatve [7-9] or senstvty analyss wth the level set model. Recently Wang et al [20] and Luo et al [2] have proposed Radal Bass Functons as a means to parametrze level set method. Shojaee and Mohamadan [22-24] have proposed bnary and pecewse constant level set method and combned wth a Merrman-Bence-Osher scheme to solve a structural shape and topology optmzaton problem. There are generally two phases n the structural optmzaton process, the analyss phase and the boundary evoluton phase. The frst phase s usually performed wth the fnte element method (FEM). One drawback of FEM s that some approxmaton s nvolved n the geometrcal defnton of the boundares of the problem doman. Furthermore, the mposton of the essental boundary condtons on the boundares cannot be exactly accomplshed. Also the adaptablty and refnement of the soluton n the FEM requres several communcatons between the dscretzed geometry and the analyss tool whch s qute costly [25, 26]. Isogeometrc Analyss (IGA) has been developed wth the am of ntegratng Non-Unform Ratonal B-Splnes (NURBS) based fnte element analyss nto the CAD. One man dea n developng sogeometrc analyss has been to prevent tme-consumng data converson between CAD systems and the fnte element analyss (FEA) n engneerng problems. IGA s based on Non-Unform Ratonal B-Splnes (NURBS) bass functon appled for both the soluton feld approxmaton and the geometry descrpton. Therefore, CAD and FEA can be unfed effcently n one package. Ths leads to the ablty of modelng complex geometres accurately. Moreover, smple and systematc refnement strateges, exact representaton of common engneerng shapes, robustness and superor accuracy can be acheved n comparson wth the conventonal fnte element method. Accordng to these features, sogeometrc analyss s expected to become a powerful computatonal approach n the analyss of varous engneerng problems; as t has already been appled to structural problems, flud mechancs, flud-structure nteracton, structural optmzaton [27-30]. Isogeometrc shape optmzaton has been dscussed by Cho et al. [3], Wall et al. [32], Nagy et al. [33], Qan [34], Seoa et al [35] and Shojaee et al [36] recently. The valdty and effcency of the approaches have been verfed n a good manner. The present paper proposes an ntegrated optmzaton approach based on the concept of sogeometrc analyss and Radal Bass Functon (RBF) level set method. The presented sogeometrc analyss framework employs NURBS n the analyss stage. The optmzaton stage performs a RBF s based level set method to optmze topology. The sogeometrc method s naturally assocated wth RBF level set method provdes a very effcent treatment. In the followng sectons, a topology optmzaton problem s formulated based on the RBF level set method, and the method of sogeometrc analyss s explaned. The proposed topology

3 COMPOSITION OF ISOGEOMETRIC ANALYSIS WITH LEVEL SET METHOD optmzaton method s appled to the mnmum mean complance problem. Fnally, to confrm the valdty and utlty of the proposed topology optmzaton method, several numercal examples are provded. 2. LEVEL SET METHOD The level set method s an mplct method for descrbng the evoluton of an nterface between two domans. It makes use of a functonφ, referred to as the level set functon, whch represents the boundary as the zero level set and nonzero n the doman [, 2]. Accordng to the value of the level set functon: φ( xt ()) > 0 : xt () D\ Ω φ( xt ()) = 0 : xt () Ω φ( xt ()) < 0 : xt () Ω\ Ω d where D R s a fxed doman n whch all admssble form of Ω are ncluded (.e. Ω D ). The level set functon s used to defne the nsde and outsde regons of nterface. The boundary or nterface s embedded as the zero level of the level set functon. Durng the optmzaton process, the level set surface may move up and down, and ths causes the embedded boundary to undergo drastc shape or topologcal changes. From begnnng to end, the value of the level set functon on the boundary s constantly kept to be zero Dfferentatng n t yelds, φ( x) = 0.0, x Ω (2) φ + φ.( vx) = 0.0 t () (3) dx where ν (x)= s the velocty vector feld, provded based on senstvty analyss. dt φ Consderng n = and v. φ = ( v n) φ, leads equaton (3) takes the form, φ φ + v n φ =0.0 t The solvng procedure of equaton (4) requres approprate choce of the upwnd schemes, rentalzaton algorthm and extenson velocty method, whch may requre excessve amount of computatonal efforts and thus lmt the utlty of the level set methods []. Therefore, a parameterzed level set method wth radal bass functons (RBFs) s used n the present study for structural topology optmzaton. (4)

4 50 S. Shojaee, M. Mohamadan and N. Valzadeh 3. SHAPE AND TOPOLOGY OPTIMIZATION PROBLEM 3.. Statement of optmzaton problem The optmzaton goal s to mnmze the complance (global stran energy) over the structural doman wth a constrant on total materal volume resource. There exst numerous equvalent formulatons of the mnmum complance problem that we use whch was gven n the work of Allare et al. [5]. Let Ω be a smooth bounded open set, and s occuped by a lnear sotropc elastc materal wth Hook s law A n desgn doman. A general objectve functon (complance) can be formulated as, J( Ω ) = fudv. + guds. = Aeu ( ). eudv ( ) (5) Ω ΓN Ω where ΓN s Neumann boundary condton, f, g are body force and surface load respectvely, and u s the dsplacement feld based on the followng lnear elastcty equatons dv( Aeu ( )) = f n Ω u = 0 on ΓD (6) ( Aeu ( )) n = g on ΓN where Γ D s Drchlet boundary condton. The standard noton for mnmum complance desgn problems can be mathematcally defned as follows Mnmze J( Ω ) = fudv. + guds. = Aeu ( ). eudv ( ) Ω ΓN Ω Subjectto: max 0 dv V Ω (7) 3.2. Shape dervatve In order to apply a gradent method to the mnmzaton of (7), we recall a classcal noton of shape dervatve. Murat and Smon [37] ntroduced a technque for constructng shape dervatve by parameterzaton of domans. We use ther approach as follows: Ω θ = ( I+ θ ) Ω (8) Where Ω s a smooth open set doman, I s dentty mappng nr N and, ( N N N θ W R, R ). The shape dervatve of objectve functon J ( Ω) : R R s defned as, the Frechet dervatve n ( N N W R, R ), J(( I+ θ) Ω ) = J( Ω ) + J ( Ω ) θ + O ( θ) (9)

5 COMPOSITION OF ISOGEOMETRIC ANALYSIS WITH LEVEL SET METHOD... 5, where J ( Ω ) s a contnuous lnear form on ( N N W R, R ) gven as the unque soluton to equaton. 7. The senstvty of the mean complance (equaton 7) s gven as follows [5, 7], (. ) ( Ω ) θ gu J = (2[ + Hg. u+ fu. ] Aeu ( ). eu ( )). + ( ). ( ). Γ θ nds Aeu eu θ nds N n (0) ΓD where H = dvn s mean curvature, Ω s the boundary of materal doman. Ω can be decomposed as three parts, Γ,, Γ. Γ s admssble Drchlet boundary condtons D D N O D such that ΓD DD. Γ N = DN ΓOs Neumann boundary condtons where DN supports a non-homogeneous one and ΓO supports homogeneous one. Let us suppose that there s no body force then n (5) f = 0.0, thus the objectve functon s defned as: J( ) guds. () Ω = D N Therefore, the Frechet dervatve of the mean complance and the volume constrant can be reformulated as, J ( Ω ) θ = ( Aeu ( ). eu ( )) θ. nds (2) ΓO Ω = Ω V ( ) θ θ( xnxds ) ( ) (3) In order to solve the optmzaton problem, we use the augmented Lagrangan method. For K a gven penalty parameter ( Λ ) and Lagrange multpler ( λ K ) the augmented problem can be stated as follows, 2 J( Ω ) = J( Ω ) + λ K max + max dv V Ω K 2Λ dv V (4) Ω The Lagrange multpler and penalty parameters are updated as follows at each teraton of the optmzaton process, λ K+ K = λ + Λ dv V K max (5) Ω K+ K Λ = αλ (6) where α (,) 0 s a constant parameter. Usng the shape dervatve of equaton (4) where there s no body force, gves, Ω = Γ J ( ) θ vθ. nds (7) O

6 52 S. Shojaee, M. Mohamadan and N. Valzadeh 2 v= λ + max ( ). ( ) 2Λ dv V Aeu eu Ω (8) To ensure the decrease of the objectve functon n level set method, the normal velocty feld must be chosen approprately. The fast descent or the steepest descent method s used such as proposed n Allare et. al. [5] and Wang et. al.[6], whchθ = vn. The normal velocty feld n equaton (4) s substtuted wth normal component of ths drectonθ. n = v. Φ v Φ =0 (9) t 3.3. The RBFs based level set method The use of explct schemes for solvng level set equaton requres approprate choce of the upwnd schemes, rentalzaton algorthms and extenson velocty methods, whch lmt the applcaton of the standard level set method to shape and topology optmzaton problems. For the sake of overcomng these drawbacks whle retan the topologcal benefts of the mplct representaton, we apply the Radal Bass Functons (RBFs) to represent the mplct level set modelng to reconstruct the shape and topology of an admssble desgn n a parametrc way [6,20]. By usng of ths scheme, the orgnal Partal Dfferental Equaton (PDE) based level set method converts nto a set of much easer Ordnary Dfferental Equaton (ODE) system and makes the level set method more effcent to mplement RBFs mplct modelng Radal bass functons are used to approxmate an admssble desgn wth a sngle functon whch s globally contnuous and dfferentable. RBFs are radally-symmetrc functons centered at partcular ponts whch can be expressed as, ϕ( x) = ϕ( x x ), x D (20) Where ϕ : R + R wth ϕ(0) 0 and x s the poston of the pont or the knot. There are a large number of dfferent radal bass functons whch can be roughly dvded nto two classes. One of these s the globally supported radal bass functons (GSRBFs) such as: thn-plate splne, Sobolev splnes, polyharmonc splnes, multquadrcs (MQ), nverse multquadrcs (IMQ) [39] and so on. The other s the compactly supported radal bass functons (CSRBFs) whch s presented by Wendland [40]. In ths paper the CSRBF wth C2 contnuty s chosen due to ts strctly postve defnteness and the sparsty of collecton matrces. The CSRBF C2 kernel (Fgure -a) s defned as, 4 ( ) ϕ () r = max(0,( r)).(4r + ) (2) where the radus of support r s gven n two dmenson Eucldean spaces as

7 COMPOSITION OF ISOGEOMETRIC ANALYSIS WITH LEVEL SET METHOD r = 2 2 ( x x) + ( y y) d (22) sp where d sp usually set large enough to guarantee there are enough knots located n the neghborhood of the current knot [39,40]. where Fgure. C2-CSRBF and ts partal dervatve n x drecton The dervatves of these CSRBFs can be easly obtaned by the chan rule, ϕ( xy, ) 3 r = (max(0, r)).( 20 r). x x ϕ( xy, ) 3 r = (max(0, r)).( 20 r). y y r x x = x d. ( x x) + ( y y ) 2 2 sp r y y = y d. ( x x) + ( y y) 2 2 sp (23) (24) The mplct level set functon φ( x) can be approxmated wth radal bass functon N (25) = T φ( x) = αϕ ( x) = ϕ ( x) α where α s the weght, whch s usually called expanson coeffcent RBF-level set optmzaton method As aforementoned, we use RBF-level set method to transform the level set PDE equaton nto

8 54 S. Shojaee, M. Mohamadan and N. Valzadeh a system of frst-order ordnary dfferental equaton (ODE). In ths paper the RBF mplct modelng s used to approxmate φ ( x) by usng CSRBF centered at knots, T φ = φ( xt,) = ϕ ( x) α() t (26) Wth ths parameterzaton the space and tme become separate. Snce the present RBF based nterpolaton scheme s performed under assumpton that all the knots are fxed n the desgn doman thus, unlke the conventonal dscrete level set approach n the RBFs based parametrc way, the desgn varables are the expanson coeffcentsα. Usng equaton (26) and equaton (4) obtan, T dα T ϕ + Vn ϕα= 0 (27) dt To determne n unknown coeffcent, one can use the collocaton method and obtan a system of ODEs as follow dα H + B( α ) = 0 dt (28) Equaton (28) can be solved by several dfferent ODE solvers such as the frst-order forward Euler s method and hgher-order Runge-Kutta, Rung-Kutta-Fehlberg, Adames- Bashforth or Adams-Moulton method. In the present study the forward Euler method s used and an approxmate soluton to equaton (28) can be gven by α t α t th B α t (29) ( n+ ) = ( n) + ( ( n)) where t s the tme step. After obtanng the approxmate soluton n equaton (29) at each tme step, the level set functon can be updated usng equaton (25). 4. ISOGEOMETRIC ANALYSIS The tradtonal Fnte Element formulatons are based on nterpolaton schemes wth Lagrange, Legendre or Hermte polynomals to approxmate geometry, physcal feld and ts dervatves. Ths approach often requres a substantal smplfcaton of the geometry, partcularly for curved boundares of the analyss doman. Generally, adaptve refnement of the dscretzed doman s appled to better approxmate the boundary and to acheve suffcent convergence. The concept of sogeometrc analyss has been proposed by Hughes and coworkers recently [25]. The man dea of the sogeometrc analyss s to apply the same nterpolaton scheme that s used accurately to descrbe the geometry for the approxmaton of the physcal varables. Snce NURBS bass functons have become standard bass for descrbng and modelng the geometry n CAD and computer graphcs, they are used for descrbng both geometry and soluton spaces.

9 COMPOSITION OF ISOGEOMETRIC ANALYSIS WITH LEVEL SET METHOD B-splne and NURBS bass functon NURBS are a generalzaton of pecewse polynomal B-splnes curves. The B-splne bass functons are defned n parametrc space on a knot vector Ξ. A knot vector n one dmenson s a non-decreasng sequence of real numbers: Whereξ s the { ξ ξ ξ 2 } + + Ξ=,,..., n p (30) th knot, s the knot ndex,,2,..., n p = + +, p s the order of the B- splne, and n s the number of bass functons. The half open nterval [, ) ξ ξ + s called the th knot span and t can have zero length snce knots may be repeated more than one, and the nterval ξ, ξ + + n p s called a patch. In the sogeometrc analyss, always open knot vectors are employed. A knot vector s sad to be open f t has p + repeatng knots at the two ends. Wth a certan knot span, the B-splne bass functons are defned recursvely as, and N,0 ( ξ ) + = (3) 0 f ξ ξ ξ otherwse ξ ξ ξ ξ N ( ξ) = N ( ξ) + N ( ξ), p =, 2, 3,. + p+ p,, p +, p ξ+ p ξ ξ+ p+ ξ+ A B-splne curve of order p s defned by n =, p (32) C( ξ) = N ( ξ) P. (33) where N, ( ξ ) s the th B-splne bass functon of order p and P are control ponts, gven n p d-dmensonal space d. -D B-splnes bass functons, that are bult from open knot vectors, are nterpolatory at the ends of parametrc space. Fgure 2 shows the quadratc B-splne bass functons. In two dmensons, B-splne bass functons are nterpolatory at the corners of the patches. The non-unform ratonal B-splne (NURBS) curve of order p s defned as, n C( ξ) = R ( ξ) P (34) = p,

10 56 S. Shojaee, M. Mohamadan and N. Valzadeh Here R, p Fgure 2. Quadratc bass functons for an open knot vector Ξ= { 0,0,0,0.2,0.4,0.6,0.8,,,} R p, ( ξ ) = N ( ξ ) w p, n Np, ( ξ ) w (35) = s the NURBS bass functons, P s the control pont and w s the th weght that must be non-negatve. In the two dmensonal parametrc space[ ] 2 0,, NURBS surfaces are constructed by tensor product through knot vectors { ξ, ξ2,..., ξ n + p + } Ψ= { ψ, ψ2,..., ψ m + q + }. It yelds to, n m = j= pq,, j, j Ξ= and S( ξψ, ) = R ( ξψ, ) P (36) where, j P s the (, ) j -th of n mcontrol ponts, also called the control mesh. The nterval ξ, ξ + + n p ψ, ψ + + ξ, ξ + ψ j, ψ j + s a knot span. NURBS bass functon n two dmensonal space, m q s a patch and [ ) ) R ξη s the pq,, j(, ) and R pq,, j N ( ξ) M ( ψ) w ( ξψ, ) = W ( ξψ, ) p, jq,, j, j n m, j = p, jq,, j = j= W ( ξψ, ) N ( ξ) M ( ψ) w (37) (38) p, q The dervatve of R ( ξ, ) and W, ( ξψ, ) wth respect to ξ s derved by smply, j η j

11 COMPOSITION OF ISOGEOMETRIC ANALYSIS WITH LEVEL SET METHOD applyng the quotent rule to (37) and (38), and R W ( ξψ, ) N ( ξ) M ( ψ) ww ( ξψ, ) N ( ξ) M ( ψ) w, j pq, p, jq,, j, j p, jq,, j, j( ξψ, ) ξ = 2 ξ ( W, j( ξψ, )) W ( ξψ, ) (39) n m, j = N p, ( ξ ) M jq, w (40), j ξ = j= The doman of problem s dvded nto patches and each patch s dvded nto knot spans or elements. Patches play the role of sub-domans wthn whch element types and materal models are assumed to be unform [25]. Nevertheless, many complcated domans can be represented by a sngle patch NURBS based sogeometrc analyss formulaton Consderng a 2-D lnear elastcty problem wth the presence of body force b and tracton force t. Implementng the vrtual dsplacement method, the followng weak form equaton s obtaned, T T T δεσdω δ d δ d 0 Ω ub Ω Γ= Ω u t Γt (4) where σ s the stress tensor and ε s the stran tensor. In sogeometrc approach, the dscretzaton s based on NURBS. Hence, the geometry and soluton feld are approxmated as, x( ξη, ) = RP ξη, Ωpatch (42) h u ( ξη, ) = Rd ξη, Ω where Ω = {( ξη, ) ξ ξ, ξ + +, η η, η + + } patch (43) patch n p m q. The matrx-form of R, jand P, jcan be changed nto vector-form by mappng from, j subscrpts to k by: k = + ( j ) n, wth k =,2,..., nm. (44) So, the control ponts are defned as, P= ( P, P, P, P,..., P ) x y x y y T,, 2, 2, nm, (45) The values of soluton feld at the control ponts, also called control varables, n the present IGA formulaton are dsplacements and can be arranged smlar to the control ponts n a

12 58 vector-form, S. Shojaee, M. Mohamadan and N. Valzadeh d = ( d, d, d, d,..., d ) (46) x y x y y T,, 2, 2, nm, The matrx R s obtaned from NURBS bass functons, R, 0 R2, 0... Rnm, 0 R = 0 R, 0 R2,... 0 Rnm, Next, the stffness matrx for a sngle patch s computed as, (47) T K = t B ( ξη, ) DB( ξη, ) J d Ω% patch Ω% (48) where t s the thckness, Ω % s the parametrc space, B ( ξη, ) s the stran-dsplacement matrx, and Js the jacoban matrx whch maps the parametrc space to the physcal space. D s the elastc materal property matrx for plane stress. The force vector on a sngle patch n the presence of body forces b and tracton forces t s obtaned as, f RbJ % R tj % T T = d Ω+ dγ b b Ω% Γ% where Γ % s the tracton boundary n the parametrc space, R s the NURBS bass functon b evaluated on the tracton boundary and J s the Jacoban that maps the tracton boundary nto b a part of physcal space boundary. The control varables can then be solved by the followng dscretzed equlbrum equaton, (49) Kd =f (50) The soluton feld at each pont of the physcal space can be approxmated by equaton (50). For numercal ntegraton, the standard Gauss-quadrature s used over each element (knot span). The number of quadrature ponts depends on the NURBS order. The detals of spaces, mappng and ntegraton n sogeometrc analyss are shown n Fgure 3. Note that the physcal mesh s only an mage of knot spans on the physcal space H-refnement or knot nserton There are three types of refnement n sogeometrc analyss: h-refnement or knot nserton, p- refnement or order elevaton and k-refnement [25]. In ths paper, we employed only the h- refnement. Knot nserton s a procedure that arbtrary new knots are added to a knot vector wthout any change n the shape of the B-splne curve. If there are m=n+p+ knots n the knot vector of the B-splne curve, where n s the number of control ponts and p s the order of B-splne

13 COMPOSITION OF ISOGEOMETRIC ANALYSIS WITH LEVEL SET METHOD curve, by addng a new knot, a new control pont must be added. Also, some current control ponts must be redefned. Consder a knot vector Ξ= { ξ, ξ2,..., ξ m = n + p + } wth n control ponts P, P 2,..., P n and the order of p. Let ξˆ [ ξk, ξ k + ] be a desred new knot. The knot nserton procedure has the followng 3 steps [3],. Fnd k such thatξˆ belongs to [, ] ξk ξ k Fnd p + control ponts P k p, P k p+,..., P k. 3. Compute p new control pontsq from the above p + control ponts usng equaton (27). Q = ( α) P + αp (5) where α s obtaned from, α ξˆ ξ = for k p + k ξ ξ + p (52) Fgure 3. Physcal space Ω s mapped nto the parametrc space ~ Ω usng NURBS bass functons. For numercal ntegraton n the parametrc space, each knot span s mapped to the parent element, where the ntegraton s performed on By performng the above procedure, the new knot vector and control ponts are obtaned as,

14 60 S. Shojaee, M. Mohamadan and N. Valzadeh { ξ, ξ ˆ 2,..., ξk, ξξ, k+,..., ξm } { P, P2,..., Pk p, Qk p+, Qk p+ 2,..., Qk, Pk, Pk+,..., Pn} Now, ths knot nserton algorthm s extended to a NURBS curve. For ths purpose, a gven NURBS curve n d-dmensonal space s converted nto a B-splne curve n (d+)- dmensonal space, then by applyng the knot nserton algorthm n ths B-splne curve, the new control ponts are obtaned. These new control ponts should then be projected to d- dmensonal space to obtan the new control ponts of the NURBS curve. Consder control ponts P = ( x, y) wth the weghts w. By convertng these control ponts to 3-dmensonal w space, P = ( wx, wy, w), the new control ponts are then computed from equaton (27), Q = ( α ) P + α P w w w The locaton of control ponts n 2D are obtaned by the followng projecton technque: and the weghts are: Q (53) (54) w w ( α) P + αp = ( α ) w + α w (55) w = ( α ) w + αw Q 5. ISOGEOMETRIC ANALYSIS WITH LEVEL SET METHOD FOR STRUCTURAL TOPOLOGY OPTIMIZATION In proposed approach, the dscrete problem s formulated n sogeometrc analyss framework. The general optmzaton problem stated n (5) s specfed for an sogeometrc dscretzaton. We apply objectve functons based on complance of partcular locatons of the structure. In geometry constructon, we construct the geometry by performng knot nserton algorthm on the ntal geometry model s shown n Fgure 4. The ntal geometry s modeled as a NURBS surface of b-quadratc order wth 6 4 control ponts. The open knot vectors n ξ drecton s {0,0,0,0.25,0.5,0.75,,,} and n ψ drecton s {0,0,0,0.5,,,}, whch yelds 4 2 knot spans. By dvdng each knot span n ξ and ψ drecton nto 0 equal parts, the physcal mesh wth equal knot spans and the control mesh wth control ponts s obtaned that s shown n fgure 5. Note that the locaton of the control ponts n the ntal geometry model plays an mportant role n reachng the desred analyss model. We used the local support property of NURBS bass functon,.e., there are only( p+ ) ( q+ ) number of nonzero bass functons wthn each knot span, where p and q are the orders of NURBS. We know that each NURBS bass functon has a correspondng control pont. So, we can assgn to each knot span, ( p+ ) ( q+ ) control ponts. In ths work, p= q= 2 and we have 3 3 control ponts for each knot span, as shown n Fgure 5. (56)

15 COMPOSITION OF ISOGEOMETRIC ANALYSIS WITH LEVEL SET METHOD... 6 (a) Fgure 4. Intal geometry model. (a) Control mesh. (b) Physcal mesh (a) (c) (b) (d) Fgure 5. Geometry representaton for the analyss model. (a) Physcal mesh that s an mage of knot spans. (b) Control mesh that s consstng of control ponts. (c) Local vew of the left top corner of physcal mesh, shows elements a, b and c. (d) Control ponts have supported on elements a, b and c (b) 6. NUMERICAL IMPLEMENTATION The objectve of ths secton s to descrbe the numercal mplementaton of the proposed method. These mplementatons are developed to mprove the performance of the proposed method. 6.. Flterng As mentoned before, shape dervatve causes the level set functon to have non- unform value at ponts n desgn doman and can lead to numercal errors. To avod quck changes and suppress the non smooth varaton a flterng technque orgnally developed n mage processng s used. One of the flterng approaches may be employed n structural topology optmzaton problems s convoluton technque. In convoluton based methods the densty of each pxel s changed accordng to nformaton from ts neghborhood. The convoluton

16 62 S. Shojaee, M. Mohamadan and N. Valzadeh process can be formulated as, n n c (, j) = hkl (, ) A ( + m k, j+ m l ) (57) k= l= where hkl (,) s the densty of the pxel located n the k th th row and l column of the mage and c s the fltered densty of the pxel. The so called mpulse response matrx A (, j ) s a n n square matrx that has to be chosen accordng to the purpose of the flter. The varable m= ( n+ )/ 2, where n s the number of pxel n each sde of the flter wndow. It should be noted that by mplementng ths method, the orgnal optmzaton problem s changed and as result of seekng clear mages, suboptmal results for the value of the objectve functon are obtaned Ersatz materal approach A challenge to structural topology optmzaton s the fact that the sogeometrc mesh wll become dstorted after the shape and topology change. Under these crcumstances, the structure doman must be remeshed. However, remeshng s a complcatng and tme consumng task, and wll brng down the effcency of optmzaton. Instead, n ths paper, the so-called ersatz materal approach [5] whch has been wdely used n senstvty analyss of the complance optmzaton problem. In ths approach the vod doman s assumed to be replaced by a type of weak materal, whose Young s modulus s very low. More precsely, we defne a Young s modulus E 0 as, E = ce 0 (58) Where E s the Young s modulus of the sold materal of the structure and c s a coeffcent. The amount of ths coeffcent s selected as c = for sold materal and c = for vod doman. Note that c cannot be too small, otherwse the stffness matrx wll be sngular. Moreover, for the elements ntersected by the boundary, Young s modulus s calculated accordng to the fracton of sold materal. For example, n one element, f the volume of sold takes one half of the volume of the element, then the Young s modulus of ths element s set to E 0 = 0.5E. 7. NUMERICAL EXAMPLES In ths secton, two wdely studed examples n structural topology optmzaton are used to llustrate the potentals of the proposed method. The optmzaton problem s consdered as the complance mnmzaton subjected to volume constrant. The unts of all the parameters can be defned flexbly, but they should reman unchanged durng all dfferent stages. All numercal examples have the followng data, Young s modulus of real materal s assumed, ersatz materal 0-3, Posson rato for two materal s assumed 0.3 and the order of NURBS bass functons n each drecton s 2. In ths method, the level set functon s ntally

17 COMPOSITION OF ISOGEOMETRIC ANALYSIS WITH LEVEL SET METHOD embedded as a sgned dstance functon, but no further rentalzaton s appled n the rest of teratons. Also, the present algorthm s termnated when the relatve dfference between four 4 successve objectve functon values s less than 0 or when the gven maxmum number of teratons has been reached. 7.. Cantlever beam Fgure 6 shows the desgn doman of a cantlever beam wth a sze of L =. The boundary of the left sde s fxed, and a vertcal concentrated force F=N s appled at the center pont of the rght sde boundary. The specfed materal volume fracton s 50%. In ths example, we use Wendland C2-CSRBF and the knot ponts are dstrbuted unformly n the desgn doman. L 2L Desgn Doman Fgure 6. A cantlever beam For comparson purpose, frst, we apply fnte element method as an analyzer when the level set mesh concdes wth the fnte element mesh. The desgn doman s dscretzed wth fnte elements and the ntal desgn s completely sold. It s clearly observed that the results of IGA by 800 elements agree well wth FEA by 3200 elements. Ths IGA feature smplfes usng of relatvely coarse meshes n the topology optmzaton procedure. Other parameters that are used for solvng ths example are ds=5, λ =0, Λ = 20, α = Fgure 7 dsplays the evoluton process of the cantlever beam at dfferent stages. These fgures show that the proposed level set method can handle shape fdelty and topologcal changes smultaneously by retanng a smooth boundary. In ths method the ntal level set surface s embedded as a sgned dstance functon and we do not apply rentalzaton at the rest of the topology optmzaton procedure. Also, because the Hamlton-Jacob partal dfferental equaton s converted to an ordnary dfferental equaton, we do not need to meet the CFL condton for tme step, thus we choseτ = 0 as a tme step. These crcumstances caused the nucleaton of holes n desgn doman durng the optmzaton process. F (a) Intal desgn (b) Step 0 (c) Step 5

18 64 S. Shojaee, M. Mohamadan and N. Valzadeh (d) Step 20 (e) Step 25 (f) Fnal desgn Fgure 7. The evoluton of optmal topology of the cantlever beam usng FEA Also, Fgure 8 shows the stran energy and the volume fracton n dfferent teratons. Fgure 8. The Hstory of objectve functon and volume In second step, we optmzed ths example when the sogeometrc analyss s appled. Here, the sogeometrc mesh does not correspond wth the level set mesh. More precously, the desgn doman s dvded wth level set mesh and sogeometrc mesh. The accuracy of the analyss when we apply sogeometrc method wth coarse mesh s partally the same as we use the fnte element mesh wth fne sze. Fgure 9 llustrate the desgns n dfferent stages of the optmzaton process, when we use the sogeometrc method for analyzng. As shown n ths fgure, one can fnd that the two dfferent schemes can lead to the smlar desgns. But, because n later scheme we use a coarse mesh for dvdng the desgn doman, the total tme of the optmzaton process when we apply the sogeometrc method s much better than we use fnte element method as an analyzer. The results obtaned wth the two dfferent schemes are lsted n Table. Fgure 0 shows that the convergent curves of the objectve functon and the volume rato. (a) Intal desgn (b) Step 0 (c) Step 5

19 COMPOSITION OF ISOGEOMETRIC ANALYSIS WITH LEVEL SET METHOD (d) Step 20 (e) Step 30 (f) Fnal desgn Fgure 9. The evoluton of optmal topology of the cantlever beam usng IGA Fgure 0. The Hstory of objectve functon and volume rato Table. Comparson of the FEA and IGA Analyzer J(Ω) (objectve) T(s) (total tme) N (teratons) FEA IGA (a) Wth 2 holes (b) Wth 33 holes

20 66 S. Shojaee, M. Mohamadan and N. Valzadeh (c) Wth 44 holes Fgure. The effect of the ntal desgn of the cantlever beam We also nvestgate the nfluence of dfferent number of ntal holes on the fnal desgn. Fgure a, b and c show that the dentcal optmal structures are obtaned regardless of number of ntal holes n the desgn doman and the complexty of the fnal topologes do not change obvously wth the dfferent number of ntal holes. In table 2 the results of these dfferent ntal desgns are lsted. Intal desgn Table 2. Results of dfferent desgns of the cantlever beam J(Ω) (objectve) T(s) (total tme) N (teratons) Fgure 3.a wth 2 holes Fgure 3.b wth 33 holes Fgure 3.c wth 44 holes 7.2. A MBB beam The desgn doman of a MBB beam s shown n Fgure 2. A load F=300 s appled at the center of the top edge. Its left corner at the bottom s fxed and the rght corner s supported as a roller. We also consder L= and the desgn doman s dscretzed wth 20 30elements. For ths example, The IGA mesh s correspondng wth level set mesh. The volume fracton s 40% and the other parameters, usng for solvng ths problem, are ds = 5, λ = 0, Λ= 30, α = 0.9. In Fgure 3, the evoluton of optmal topology s dsplayed by use of the present method L F 2L L Desgn Doman Fgure 2. A MBB beam

21 COMPOSITION OF ISOGEOMETRIC ANALYSIS WITH LEVEL SET METHOD (a) Intal desgn (b) Step 5 (c) Step 5 (d) Step 20 (e) Step 25 (f) Fnal desgn Fgure 3. The evoluton of optmal topology of the MBB beam 8. CONCLUSION In the present study, the composton of RBF level set method wth sogeometrc analyss has been successfully appled to the structural shape and topology optmzaton. In ths method, the dscrete problem s formulated n sogeometrc analyss framework. A parameterzed level set method wth RBFs s used for structural topology optmzaton. The proposed sogeometrc based topology optmzaton method has several advantages compared to the fnte element based method. Unlke the standard FEM-based desgn optmzaton, the computatonal tme can be reduced by usng ths analyzer, whle obtanng the same optmal topology. Due to the desrable characterstcs of NURBS n IGA, t does not have any destructve effect on the qualty of dscretzaton. Furthermore, the proposed method n ths paper has ts strength on the capacty of dealng wth the desgn dependent load problem or stress optmzaton problem. REFERENCES. Allare G, Kohn RV. Optmal bounds on the effectve behavor of a mxture of two wellordered elastc materals, Quat Appl Math, 993; 5: Allare G, Bonneter E, Francfort G, Jouve F. Shape optmzaton by the homogenzaton method, Numer Math, 997; 76: Suzuk K, Kkuch N. A homogenzaton method for shape and topology optmzaton, Comp Meth Appl Mech Eng, 99; 93: Bendsøe M.P, Kkuch N. Generatng optmal topology n structural desgn usng a homogenzaton method, Comput Meth Appl Mech, 988; 7(2): Bendsøe M.P, Optmal shape desgn as a materal dstrbuton problem, Struct Optm,

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