fem structural optimisation

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng (2010)Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.2874

A structural optimization method based on the level set methodusing a new geometry-based re-initialization scheme

Shintaro Yamasaki,, Shinji Nishiwaki, Takayuki Yamada, Kazuhiro Izuiand Masataka Yoshimura

Department of Aeronautics and Astronautics, Kyoto University, Yoshida Honmachi,Sakyo-ku, Kyoto City 606-8501, Japan

SUMMARY

Structural optimization methods based on the level set method are a new type of structural optimizationmethod where the outlines of target structures can be implicitly represented using the level set func-tion, and updated by solving the so-called HamiltonJacobi equation based on a Eulerian coordinatesystem. These new methods can allow topological alterations, such as the number of holes, during theoptimization process whereas the boundaries of the target structure are clearly defined. However, there-initialization scheme used when updating the level set function is a critical problem when seekingto obtain appropriately updated outlines of target structures. In this paper, we propose a new structuraloptimization method based on the level set method using a new geometry-based re-initialization schemewhere both the numerical analysis used when solving the equilibrium equations and the updating processof the level set function are performed using the Finite Element Method. The stiffness maximization,eigenfrequency maximization, and eigenfrequency matching problems are considered as optimizationproblems. Several design examples are presented to confirm the usefulness of the proposed method.Copyright q 2010 John Wiley & Sons, Ltd.

Received 17 June 2008; Revised 2 June 2009; Accepted 19 January 2010

KEY WORDS: structural optimization; level set method; re-initialization; finite element method

1. INTRODUCTION

This paper discusses a structural optimization method based on the level set method that uses anew geometry-based re-initialization scheme. Structural optimization has been successfully appliedto a wide range of design problems in many industries, such as mechanical, automotive, andcivil industries. Structural optimization can be generally classified into three categories: sizing

Correspondence to: Shintaro Yamasaki, Department of Aeronautics and Astronautics, Kyoto University, YoshidaHonmachi, Sakyo-ku, Kyoto City 606-8501, Japan.

E-mail: [email protected]

Contract/grant sponsor: Mizuho Foundation for the Promotion of Sciences

Copyright q 2010 John Wiley & Sons, Ltd.

S. YAMASAKI ET AL.

optimization, shape optimization, and topology optimization. Shape and topology optimizationsare more likely to yield high-performance designs, due to the large number of degrees of freedomavailable for design variable settings.

Shape optimization originated from the work of Zienkiewicz and Campbell [1]. The basic ideais to update the boundaries of the structure that are represented by the shapes of finite elementsusing sensitivity analysis and an optimization method. Later, this approach was found to encounternumerical convergence problems [2], and several methods have been proposed to mitigate them.Imam [2] proposed three techniques: the design element technique, the super-curves technique,and the shape superposition technique. Braibant and Fleury [3, 4] proposed a structural shapeoptimization method where the shape boundary is represented using spline curves, such as Bezierand B-spline curves, with design variables as spline curve control points. Bennett and Botkin [5, 6]developed a structural shape optimization method with automated mesh generation. Kikuchi et al.[7] applied an adaptive mesh method in the Finite Element Method (FEM) to obtain sufficientboundary smoothness during the optimization process. Belegundu and Rajan [8] developed theNatural Design Variable method where fictitious loads applied on the shape boundary are used asdesign variables. Azegami et al. [9, 10] proposed the Traction Method where the shape sensitivityis replaced with the displacement field obtained by applying a fictitious traction based on the shapesensitivity. A comprehensive review of shape optimization research can be found in the literature[11, 12].

Topology optimization allows changes not only in shape but also in the topology of target struc-tures, and can provide high-performance structural configurations, whereas topological changesare hard to perform in shape optimization. Topology optimization has been successfully applied toa variety of structural optimization problems, such as the stiffness maximization problem [13, 14]and eigenfrequency problems [15, 16], since Bendse and Kikuchi [13] first proposed the so-calledHomogenization Design Method (HDM). The basic ideas consist of (1) the extension of a designdomain to a fixed design domain and (2) replacement of the optimization problem with a materialdistribution problem, using the characteristic function expressed in the papers of Murat and Tartar[17], Belytschko et al. [18], and Norato et al. [19]. A homogenization method is utilized to dealwith the extreme discontinuity of the material distribution and to provide material properties viewedin a global sense as homogenized properties. This method, called the HDM, has been appliedto a variety of design problems, and the density approach [20, 21], also called the SIMP (SolidIsotropic Material with Penalization) method [22] in structural mechanics problems, is anothercurrently used method. The basic idea of SIMP is the use of a fictitious isotropic material whoseelasticity tensor is assumed to be a function of penalized material density, expressed by an exponentparameter. Bendse and Sigmund [23] asserted the validity of the SIMP method in view of themechanics of composite materials and it has been widely used and applied to a variety of designproblems because of its simple formulation and facile implementation.

Unfortunately, topology optimization methods tend to suffer from numerical problems [24, 25],such as mesh dependency, checkerboard patterns, and gray scales. Several methods have beenproposed to mitigate these problems, such as the use of high-order finite elements [24], filteringschemes [25], and the perimeter control method [26]. Although the filtering schemes and theperimeter control method are now popular means of avoiding these numerical problems, thesemethods crucially depend on artificial parameters for which there is no rational guideline fordetermining appropriate a priori parameter values.

On the other hand, Xie and Steven [27] developed a different type of topology optimizationmethod, the so-called Evolutionary Structural Optimization (ESO) method. In this method, the

Copyright q 2010 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng (2010)DOI: 10.1002/nme

A STRUCTURAL OPTIMIZATION METHOD BASED ON THE LEVEL SET METHOD

design domain is represented using a fixed finite element mesh and the shape and topology of thestructure is represented as a subset of finite elements. The contribution to the objective functionof each finite element is evaluated and finite elements whose contribution is relatively low aregradually removed from the subset representing the structure. As a result, the optimal configurationis obtained as an optimal subset of finite elements. The ESO method has been applied to variousstructural design problems, such as stiffness problems [27], eigenfrequency problems [28], andheat conduction problems [29].

Recently, a new type of structural optimization method based on the implicit moving interfaceusing the level set method has been proposed [3032]. The level set method was originally proposedby Osher and Sethian [33] as a versatile method to implicitly represent evolutional interfaces in aEulerian coordinate system. Complex shape and topological changes can be handled using the levelset function that is an implicit function of a higher dimension. The evolution of the interfaces withrespect to time is tracked by solving the so-called HamiltonJacobi partial differential equation(PDE), which is also called the level set equation, with an appropriate normal velocity that is themoving interface velocity normal to the interface. The level set method is potentially useful in avariety of applications, including fluid mechanics [3436], phase transitions [37], image processing[3840], and solid modeling in CAD [41].

Sethian and Wiegmann [30] first proposed a structural optimization method based on the levelset method. In their method, the shape boundaries of the structure are represented using the levelset function, and the level set function is updated using an ad hoc method based on the VonMises stress. Osher and Santosa [42] proposed a structural optimization method where the shapesensitivity is used as the normal velocity, and the structural optimization is performed by solving thelevel set equation using the upwind scheme. This proposed method was applied to eigenfrequencyproblems for an inhomogeneous drum using a two-phase optimization of the membrane where themass density assumes two different values, whereas the elasticity tensor is constant over the entiredomain.

Wang et al. [31] proposed a structural shape optimization method based on the level set methodfor structures composed of a linear elastic material. They formulated the normal velocity based onthe shape sensitivity and solved the level set equation using the upwind scheme. They demonstratedthat appropriate optimal configurations can be obtained using their proposed method for theminimum compliance problem. Wang and Wang [43] extended their method to obtain optimalstructures composed of multi-materials, where m level set functions are used to represent 2m

different material phases, a representation model called the color level set model. Wang and Wang[44] later proposed a new, improved optimization method where the superimposed FEM is usedto compute the displacement field, to obtain optimal configurations for the minimum complianceproblem. The displacement field computed using the superimposed FEM is comparatively moreaccurate than the simple ersatz material approach that Allaire et al. [32] proposed.

Allaire et al. [32] independently proposed a level set method based structural optimizationmethod that also employs the upwind scheme, and discussed cases concerning structures composedof a linear elastic material as well as non-linear hyperelastic material. They obtained optimal config-urations for the minimum compliance problem in both cases with structures composed of linearelastic and non-linear hyperelastic material, and obtained optimal configurations for compliantmechanism design problems for structures composed of a linear elastic material. A simple ersatzmaterial approach was employed to compute the displacement field of the structure. Allaire andJouve [45] extended a level set method based structural optimization method for lowest eigenfre-quency maximization problems and minimum compliance problems having multiple loads.


S. YAMASAKI ET AL.

Haber [46] proposed a structural optimization method for eigenfrequency minimization andmaximization problems, where the shape boundaries of the structure are represented using thelevel set function and the level set function is updated using Sequential-Quadratic-Programming.Wang et al. [47] proposed a structural optimization method based on the level set method wherean image processing based approach is employed to preserve the connectivity of the structure, anddemonstrated that the appearance of de facto hinges can be effectively suppressed in compliantmechanism design problems.

In the above level set method based structural optimization methods, the shape boundaries areevolved from an initial configuration using certain procedures (typically, solving the level setequation), and an optimal configuration is finally obtained. The obtained optimal configuration,therefore, strongly depends on the given initial configuration. As Allaire et al. [48] discussed,such dependency is problematic, especially in two-dimensional cases. Several methods have beenproposed in an attempt to mitigate this problem. Allaire et al. [48] proposed a level set methodbased structural optimization method incorporating a method for introducing holes, based ontopological derivatives [49], and optimal configurations were obtained for the minimum complianceproblem and compliant mechanism design problems. The shape boundary evolution based on theshape sensitivity, and topological changes using a method for introducing holes, were sequentiallyperformed, and appropriate optimal configurations were obtained using several different initialconfigurations, despite the possibility of local minima still existing, as was pointed out. Zhuang et al.[50] proposed a structural shape and topology optimization method for heat conduction problems,in which the shape boundary evolution and topological changes were sequentially performed, suchas with the method proposed by Allaire et al. [48].

Burger et al. [51] proposed a level set method based optimization method where the levelset equation is modified by adding a source term that depends on the topological derivative.They applied their proposed method to the minimization of the least squares functional anddemonstrated that the shape and topology optimization were successfully performed. Based ontheir contribution, He et al. [52] proposed a level set method based structural optimizationmethod incorporating the topological derivative and solved structural optimization problems withrespect to photonic crystals and acoustic drums. Amstutz and Andra [53] proposed a structuraloptimization method where the level set function is updated based on the topological deriva-tive instead of the shape sensitivity, and applied their method to the minimum complianceproblem, compliant mechanism design problems and a multidisciplinary optimization problemfor a ceramic filter. Belytschko et al. [18] proposed a level set method based topology opti-mization method where the nodal values of the implicit function are updated using a heuristicscheme.

Some proposed structural optimization methods have been based on an approach where thelevel set function is represented using a superposition of Radial Basis Functions (RBFs). Wangand Wang [54] and Wang et al. [55] represented the level set function using a superposition ofMultiquadratic RBFs. Such representation converts the original shape and topology optimizationperformed by solving the level set equation to a sizing optimization with respect to the expansioncoefficients. In their structural optimization method, a re-initialization process is not required, andthe shape and topology of the structure are concurrently changed using the normal velocity thatis extended to the entire design domain, and optimal configurations for the minimum complianceproblem were successfully obtained. Wang and Wang [56] and Wang et al. [57] also applied inversemultiquadratic RBFs in their structural optimization method and obtained optimal configurationsfor the minimum compliance problem. Luo et al. [58] employed compact support RBFs to represent



the level set function, and performed structural optimization for a compliant mechanism designproblem.

Chen et al. [59] constructed the level set function using B-splines and parameterized primi-tives combined with R-functions, and shape and topology optimization with parametric controlwas performed for the minimum compliance problem. Wei and Wang [60] proposed a struc-tural optimization method based on the piecewise constant level set method [61, 62] wheremulti-phases are represented using a level set function. Luo et al. [63] proposed a structuraloptimization method based on the level set method where the level set equation is solvedusing a semi-implicit additive operator splitting (AOS) scheme. In their method, topologicalchanges are automatically carried out during the optimization without the need for artificialschemes.

One of the challenging tasks, and a critical issue, in level set method based structural optimizationis to develop level set function updating schemes that can be easily implemented, since solvingthe level set equation to update the level set function requires sophisticated numerical techniqueslike the upwind scheme. In this paper, to mitigate the above problem, we propose a new structuraloptimization method that updates the level set function based on a simplified level set equation thatis obtained using the explicit method and ensuring that the level set function maintains the signeddistance characteristic with respect to the shape boundaries at every update iteration. Unlike theordinary level set equation formulated as a HamiltonJacobi PDE, the simplified level set equationused here does not include terms pertaining to a spatial differential. The FEM is employed for thespatial discretization of the simplified level set equation, and sophisticated numerical techniqueslike the upwind scheme are not needed. Furthermore, the Neumann boundary condition is notrequired when solving the simplified level set equation using the FEM, unlike the upwind schemewhere it is required, and this further simplifies the numerical implementation of the proposedmethod. This is of great benefit when performing structural optimizations based on the level setmethod, especially for three-dimensional cases.

The level set function is re-initialized at every iteration during the optimization procedure andthe proposed method provides that the level set function maintains the signed distance characteristicwith respect to the shape boundaries at every iteration, which enables the level set equation tobe simplified. As we observe that the numerical accuracy of the re-initialization using certainpreviously proposed re-initialization schemes [34, 41] depends upon the distribution of the level setfunction, we developed a new geometry-based re-initialization scheme where the shape boundaryis discretized to a set of points on the shape boundary and the value of the level set function ateach node is computed as the signed distance from the nearest point. The proposed re-initializationscheme strictly reconstructs the level set function based on the definition of the signed distancefunction, so that the signed distance characteristic is preserved, while the shape boundaries arealmost perfectly preserved.

Re-initializing the level set function at every iteration prevents topological changes like theintroduction of holes, especially for two-dimensional cases, hence the obtained optimal configu-rations strongly depend on the initial configurations. To avoid this problem, we employ a methodfor introducing holes based on the topological derivatives, such as that used by Allaire et al. [48].

The proposed method is applied to the minimum compliance problem, the lowest eigenfrequencymaximization problem and the eigenfrequency matching problem, and several numerical examplesare provided for both two- and three-dimensional cases to confirm the usefulness of the proposedmethod. The FEM is employed to solve both the equilibrium equations and the simplified levelset equation.


S. YAMASAKI ET AL.

The remainder of this paper is organized as follows. In Section 2, the minimum compli-ance problem, the lowest eigenfrequency maximization problem and the eigenfrequency matchingproblem are formulated based on the proposed structural optimization method. In Section 3, thenew geometry-based re-initialization scheme is introduced, and how it differs from previouslyproposed re-initialization schemes is also explained. Section 4 explains the optimization algorithmand the numerical techniques required to successfully obtain appropriate optimal solutions, andseveral numerical examples are provided in Section 5 to confirm the validity and utility of theproposed optimization method. Finally, conclusions are provided in Section 6.

2. FORMULATIONS

2.1. Concept of structural optimization based on the level set method

Here, we briefly discuss the basic concepts of the structural optimization method based on thelevel set method that was originally developed by Osher and Sethian [33] as a method capable ofimplicit geometrical representation. Let us define a reference and bounded domain D, a domainfilled with a solid material, i.e. a material domain that denotes the domain to be designed suchthat D, and another complementary domain representing a void that exists, i.e. a void domainD\. The level set function (x) is introduced to represent a boundary between the materialdomain and the void domain, where x stands for a position in D. That is, the boundary is expressedusing the level set function (x) as follows:

(x) > 0 for x\(\D)(x) = 0 for x\D(x) < 0 for xD\

(1)

where and D are, respectively, the boundaries of and D. Using the above level set function,an arbitrary shape and topology of the material domain in the reference domain D can beimplicitly represented. Now we introduce a fictitious time t , and assume that the level set functionis also implicitly a function of t , to represent shape and topology changes in the material domain over time. That is, as time t is advanced, the boundary (x)=0 is updated as an evolving boundaryprocess that reaches an optimal configuration. Furthermore, an arbitrary iso-contour S(, t) isembedded in the level set function, and is described as

S(, t)={x(t)|(x(t), t)=} (2)where is an arbitrary value, and x(t) is a point on the iso-contour. By differentiating the aboveequation with respect to time t , we obtain the following equation, the so-called HamiltonJacobiequation, which is also called the level set equation:

(x, t)t

+(x, t)dxdt

=0 (3)Let the normal velocity function be VN (x, t), which represents the normal velocity of the evolvingiso-contour expressed by S(, t), formulated as

VN (x, t)=V(x, t) (x, t)|(x, t)| (4)



where V(x, t) is the velocity vector of the evolving iso-contour which is defined as

V(x, t)= dxdt

(5)

Substituting Equation (4) into Equation (3), we have

(x, t)t

+VN (x, t)|(x, t)|=0 (6)

The above equation represents the evolution of the boundary (x)=0. Thus, the structural opti-mization problem can be solved by providing appropriate velocity function values of VN (x, t) foruse in the above equation. The optimal configuration is finally obtained when the KarushKuhnTucker (KKT) conditions are satisfied. However, obtaining numerical solutions to Equation (6)usually requires the use of sophisticated numerical techniques. Wang et al. [31] and Allaire et al.[32] solved this equation using the upwind scheme, a numerical technique of the Finite DifferenceMethod (FDM) that is used to stably solve hyperbolic type differential equations. Sethian [41] andPark et al. [64], respectively, introduced the PetrovGalerkin method and the least-squares FEMto solve Equation (6) using the FEM.

The level set function can maintain a characteristic of a specific function. When the level setfunction is a signed distance function with respect to the shape boundary (x)=0, the level setfunction is defined as

(x)={d(x,\D) if xd(x,\D) if x / (7)

where d(x,\D) is defined byd(x,\D)= inf

y\Dd(x,y) (8)

and d(x,y) is the distance between x and y. The signed distance function defined in Equation (7)also yields the following relation.

|(x)|=1 (9)Mulder et al. [65] demonstrated that initializing the level set function as a signed distance functionprovides more accurate numerical solutions than initializing the level set function as a colorfunction that is similar to the Heaviside function. In terms of reducing numerical errors like thetentpole phenomenon, Chopp [66] suggested that the level set function be periodically re-initializedas a signed distance function. Furthermore, Allaire and Jouve [45] periodically re-initialized thelevel set function to prevent it from becoming too flat or too steep, since these conditions causelarge errors in the location of the shape boundaries and also yield large errors in the evaluationof the level set function gradient when using finite differences. Thus, the need to periodicallyre-initialize the level set function as a signed distance function with respect to the shape boundariesis commonly accepted and several re-initialization schemes have been proposed to this end, ofwhich the Fast Marching Method based scheme [41] and the scheme based on PDE solutions [34]are most often used. In most previous research [31, 32, 45], since the re-initialization of the levelset function is performed once every few iterations during the optimization procedure, the level set


S. YAMASAKI ET AL.

function does not always maintain the signed distance characteristic represented by |(x)|=1,therefore, Equation (6) is solved both temporally and spatially.

Here, we propose a new scheme to solve Equation (6). In this new scheme, instead of solvingEquation (6) both temporally and spatially, the equation is solved only temporally with the explicitmethod and a re-initialization scheme. Equation (6) is discretized with respect to time using theexplicit method as follows:

(x, t+t)(x, t)t

+VN (x, t)|(x, t)|=0 (10)

where t is the time increment. As the first step in solving the above equation, (x, t) is re-initialized as a signed distance function. As |(x, t)|=1 when (x, t) is a signed distancefunction, Equation (10) can be simplified as follows by re-initializing (x, t) at every updateiteration.

(x, t+t)(x, t)t

+VN (x, t)=0 (11)

The weak form of the above equation can be easily derived asD

(x, t+t)(x, t)t

d+DVN (x, t)d=0 for (12)

where is the space of admissible level set functions. Then, as the second step in solvingEquation (10), the updated level set function (x, t+t) is calculated using the above-simplifiedequation. This updating procedure is iterated until the optimal solution is obtained. Note that(x, t+t) is not required to be a signed distance function just after the update, since Equation (10)contains no spatial differential term concerning (x, t+t) due to the discretization using theexplicit method.

Formerly, updating the level set function based on Equation (6) required the use of the sophisti-cated techniques mentioned above, but updating the level set function using Equation (12) can stablyprovide numerical solutions without the use of such techniques, in exchange for the computationalcosts of the re-initialization.

In this research, we always re-initialize the level set function before the update, as a signeddistance function, and then update the level set function by numerically solving Equation (12)using the FEM. We iterate this updating procedure until the optimal solution is obtained. Theeffectiveness of the updating scheme depends on the numerical accuracy of the re-initialization,hence we developed a new geometry-based re-initialization scheme that is better at guaranteeing thesigned distance characteristic represented by |(x)|=1 than previous methods. Further discussionappears in Section 5, along with the numerical results.

Concerning boundary conditions, appropriate boundary conditions on D must be set in somecases while solving Equation (6). Wang et al. [31] and Allaire and Jouve [45] imposed the followingNeumann boundary condition:

n

=0 on D (13)

where n is the normal of D. This boundary condition is required when the level set equation issolved using the upwind scheme based on the FDM, since the upwind side of always exists in



reference domain D that maintains the above condition. Furthermore, as Allaire and Jouve [45]discussed, topological changes due to the introduction of holes from outside the reference domainD as a consequence of spurious values created by the boundary conditions do not occur. However,in the proposed method, the above boundary condition is not required, since the level set functionis updated based on Equation (12), which has no spatial differential operator. That is, the level setfunction is updated based on Equation (12) alone, with no additional boundary conditions. As willbe demonstrated in Section 5 using several numerical examples, appropriate optimal configurationscan be obtained without applying the condition represented in Equation (13).

2.2. Formulation of structural optimization problems

Consider the minimization problem of an objective functional F() under a constraint functionalG(). The structural optimization problems are formulated as follows based on the level setmethod:

Minimize F() (14)

Subject to G()Gmax (15)

for where Gmax is the upper limit value of G(). The shape derivatives of F() and G() are thendefined as the Frechet derivatives with respect to . Here we consider the case where the Frechetderivative representing the shape derivative of F() can be described as

dF()

d,

=Df ((x))d (16)

where is the variation of the level set function such that , and f ((x)) is used to expressthe integrand with in the shape derivative of F(). We also consider the case where the shapederivative of G() can be described as

dG()

d,

=Dg((x))d (17)

where g((x)) is used to express the integrand with in the shape derivative of G(). For thestructural optimization problems discussed in this paper, the Frechet derivatives of F() and G()can be derived using the above expressions.

Here, we adopt the method presented in [31] to solve the above optimization problem. Thatis, we introduce a Lagrangian F() and minimize it by solving the level set equation. F() isformulated as

F()= F()+(G()Gmax) (18)where is the Lagrange multiplier. Based on the discussion in [31], we express VN (x, t) inEquation (12) as follows:

VN (x, t)= f ((x, t))+g((x, t)) (19)


S. YAMASAKI ET AL.

where is obtained as

=

0 if G()


The bilinear form a(u,v,) and the load linear form L(v,) are here defined by

a(u,v,) =D

(v) :D :(u)H((x))d (25)

L(v,) =Db vH((x))d+

Dt

tvd (26)

where is the linearized strain tensor, D is the elasticity tensor, and

U ={v=viei :vi H1(D) with v=0 on Du} (27)Next, the shape derivatives of F() and G() with respect to are derived. The shape derivative

of F() is obtained bydF()

d,

=D{2b u(u) :D :(u)}((x))d (28)

where ((x)) is the Dirac delta function. For details concerning the derivation of the aboveequation, see Appendix A. On the other hand, the shape derivative of G() is simply obtained by

dG()

d,

=G()

,

=D

((x))d (29)

Thus, we have

f ((x)) = {2b u(u) :D :(u)}((x)) (30)g((x)) = ((x)) (31)

2.4. The eigenfrequency optimization problems

Next, the eigenfrequency optimization problems are considered. Suppose that a reference domainD is fixed at boundary Du , and a linear elastic structure occupying the domain is freelyvibrating. Let k be the k-th eigenfrequency and uk be the corresponding k-th eigenmode. Thek-th eigenvalue k is expressed as k =2k . Body forces applied to the elastic body and dampingeffects are assumed to be ignored for simplicity in the formulation.

Here, we consider two optimization problems, i.e. the lowest (first) eigenfrequency maximizationand the matching of the eigenfrequency with a target value. For the lowest (first) eigenfrequencymaximization problem, the objective functional is formulated as follows:

F()=21=1 (32)Note that by prefixing a minus sign to the eigenvalue, the maximization problem is transformedinto a minimization problem. However, if the objective functional only takes the lowest eigen-frequency into account, the sequence of the eigenmodes may change during the optimization


S. YAMASAKI ET AL.

procedure, preventing the objective functional from converging. To avoid this problem, we adoptthe formulation proposed by Ma et al. [16]. The objective functional is then formulated as:

F()=(

qk=1

1

2k

)1=

( qk=1

1

k

)1(33)

where q is an appropriate number of eigenfrequencies from the lowest eigenmode. The optimizationproblem under the volume constraint of the entire reference domain is now formulated as:

Minimize F()=( qk=1

1

k

)1(34)

Subject to G()=DH((x))dGmax (35)

a(uk,v,)=kb(uk,v,) (36)for , uk U, vU, k=1, . . .,q

where the bilinear form b(uk,v,) is here defined by

b(uk,v,)=D

v ukH((x))d (37)

and is the mass density.The shape derivative of F() is described as follows:

dF()

d,

=( qk=1

1

k

)2{ qk=1

D{(uk) :D :(uk)+kuk uk}((x))d

2kD uk ukH((x))d

}(38)

For details concerning the derivation of the above equation, see Appendix B. As the eigenmodecan be normalized with respect to the mass density such that

Duk uk H((x))d=1 (39)

we finally obtain

f ((x)) =( qk=1

1

k

)2{ qk=1

{(uk) :D :(uk)+kuk uk}((x))2k

}(40)

g((x)) = ((x)) (41)

Next, the matching of the eigenfrequency with a target value is considered. Let obj,s be a targetvalue of the s-th eigenfrequency. The objective functional is formulated as follows:

F()= (2s 2obj,s)2= (sobj,s)2 (42)



However, in this formulation as well, the sequence between the target eigenmode and the adjacenteigenmodes may change during the optimization procedure, and to avoid this, the followingobjective functional is proposed:

F()=s+1

k=s1

wk

(2k2obj,k

2obj,k

)2=

s+1k=s1

{wk

(k obj,k

obj,k

)2}(43)

where wk is a weighting coefficient for the k-th eigenfrequency matching and obj,k =2obj,k . Bysetting obj,s1 to a sufficiently smaller value than obj,s and obj,s+1 to a sufficiently largervalue than obj,s , we can avoid sequence changes between the target eigenmode and adjacenteigenmodes. When the target eigenfrequency is the lowest eigenfrequency, only the lowest andsecond eigenfrequencies are taken into account, by setting w0=0.

Thus, the optimization problem is formulated under the volume constraint of the entire referencedomain as follows:

Minimize F()=s+1

k=s1

{wk

(kobj,k

obj,k

)2}(44)

Subject to G()=DH((x))dGmax (45)

a(uk,v,)=kb(uk,v,) (46)for , uk U, vU, k= s1,s,s+1

The shape derivative of F() is described asdF()

d,

=

s+1k=s1

{2wk

(kobj,k

2obj,k

) D{(uk) :D :(uk)kuk uk}((x))d

D uk uk H((x))d

}(47)

For details concerning the derivation of the above equation, see Appendix C. As the eigenmodecan be normalized in the same way as the eigenmode in Equation (39), we have

f ((x)) =s+1

k=s1

{2wk

(kobj,k

2obj,k

)((uk) :D :(uk)kuk uk)((x))

}(48)

g((x)) = ((x)) (49)

3. RE-INITIALIZATION SCHEME

As briefly explained in Section 2.1, the level set function can maintain the characteristic of aspecific function. In discussions concerning the level set theory, Osher and Fedkiw [67] pointedout that one can obtain more accurate numerical results by preserving the characteristic of the


S. YAMASAKI ET AL.

level set function as a signed distance function. Mulder et al. [65] compared the numerical resultsobtained by either initializing the level set function as a signed distance function or initializing thelevel set function as a color function that is a non-smooth function used to track interfaces, andis similar to the Heaviside function, and demonstrated that more accurate numerical results can beobtained by initializing the level set function as a signed distance function. Several re-initializationschemes have been proposed to reconstruct the level set function as a signed distance functionwith respect to the shape boundaries while preserving the shape boundaries. Here, we first brieflydiscuss the previously proposed schemes and their drawbacks. Next, we explain the details of anewly developed re-initialization scheme that can maintain higher accuracy of the signed distancefunction than the previous schemes.

3.1. Previously proposed schemes

Chopp [66] proposed a re-initialization scheme where the iso-contour representing the shapeboundaries is discretized and then distances from the discretized iso-contour are computed. Theidea of this scheme is simple and straightforward, but the author does not provide details of themethod used to discretize the iso-contour.

Sussman et al. [34] proposed a different re-initialization scheme, where the level set function isre-initialized by solving a PDE called the re-initialization equation, instead of explicitly computingthe distance from the shape boundaries. The PDE includes an approximated sign function where anartificial parameter is used for numerical purposes. We denote this parameter as in the followingdiscussion. Sethian [41] proposed another re-initialization scheme using the Fast MarchingMethod.

The PDE-based scheme proposed by Sussman et al. [34] and the Fast Marching Method basedscheme [41] are often used to re-initialize the level set function in the level set method basedstructural optimization methods mentioned above, but these re-initialization schemes have certaindrawbacks, such as the PDE-based schemes requiring an artificial parameter . As demonstratedin Section 5.1, when the PDE-based scheme is implemented using the first-order upwind schemewith second-order Essentially Non Oscillatory (ENO) scheme [67], suitable values for the numberof iterations, the Courant number and all depend upon the distribution of the level set function.Since the distribution of the level set function varies during the optimization procedure, it is difficultto re-initialize the level set function successfully during the optimization procedure. In the FastMarching Method based scheme, although there is no artificial parameter like the , the level setfunction values at nodes close to the shape boundaries cannot be re-initialized. Therefore, anotherre-initialization scheme must be employed to re-initialize the level set function values at nodesclose to the shape boundaries when the level set function is re-initialized using the Fast MarchingMethod based scheme. And, as demonstrated in Section 5.1, when we employ the conventionalscheme introduced by Osher and Fedkiw [67], adherence to the signed distance characteristicrepresented by |(x)|=1 is not guaranteed. Therefore, we developed a new geometry-basedre-initialization scheme as described below.

3.2. A new geometry-based re-initialization scheme

As briefly discussed in Section 2.1, appropriate optimal solutions can be obtained by solvingEquation (12) while rigorously maintaining the signed distance characteristic represented by|(x)|=1 using a suitable re-initialization scheme during the optimization procedure. However,this maintenance cannot always be rigorously accomplished using the previously proposed schemes,as explained in the above section and Section 5. Here, a new re-initialization scheme is developed



to maintain the signed distance characteristic with sufficient rigor. The basic idea is to discretizethe iso-contour representing the shape boundaries and then compute distances from the discretizediso-contour, which is similar to the scheme proposed by Chopp [66]. The details for discretizingthe iso-contour representing the shape boundaries, and the re-initialization procedure, are explainedbelow. As the re-initialization scheme is performed after the numerical analysis for solving theequilibrium equations using the FEM, as shown in Figure 5 in Section 4.4, the re-initializationoccurs after discretization has been performed in the reference domain D using finite elements.

First, all finite elements that cross the shape boundary (x)=0 are selected. For each of theselected elements, an appropriate number of points are set on (x)=0 by interpolating the curverepresenting (x)=0 using the shape function of the FEM as shown in Figure 1. That is, thelevel set function is mapped on the parametric coordinates (,,) as shown in Figure 2, and isinterpolated as

(,,)=NTUe (50)where N is the shape function vector and Ue is the nodal value vector of the level set function inan element. For example, in the case where the reference domain D is discretized using four-nodequadrilateral elements in the two-dimensional problem, the level set function is interpolated as

(,) = 14(1)(1)e(1,1)+ 14(1+)(1)e(1,1)+ 14(1)(1+)e(1,1)+ 14(1+)(1+)e(1,1) (51)

\D( ) 0=x

Figure 1. Points on the shape boundary in an element.

\D

node1

node2

node4node3

x

y

node3( 1 , 1)

node4(1 , 1)

node2(1 , 1)

node1( 1 , 1)

\D

(a) (b)

Figure 2. Element coordinates: (a) original coordinate and (b) parametric coordinate.


S. YAMASAKI ET AL.

and, in the case where the reference domain D is discretized using eight-node hexahedral elementsin the three-dimensional problem, the level set function is interpolated as

(,,) = 18 (1)(1)(1)e(1,1,1)+ 18(1+)(1)(1)e(1,1,1)+ 18(1)(1+)(1)e(1,1,1)+ 18(1+)(1+)(1)e(1,1,1)+ 18(1)(1)(1+)e(1,1,1)+ 18(1+)(1)(1+)e(1,1,1)+ 18(1)(1+)(1+)e(1,1,1)+ 18(1+)(1+)(1+)e(1,1,1) (52)

An appropriate number of points on (,,)=0 are obtained as intersectional points betweenthe shape boundaries representing equations obtained using the above interpolation functions andthe following equations, respectively, as shown in Figure 3,

(,) =(

1+ 2np

k,

)for k=0,1, . . .,np

(,) =(

,1+ 2np

k

)for k=0,1, . . .,np

(53)

in the two-dimensional case, and

(,,) =(

1+ 2np

k,1+ 2np

l,

)for k, l=0,1, . . .,np

(,,) =(

1+ 2np

k,,1+ 2np

l

)for k, l=0,1, . . .,np

(,,) =(

,1+ 2np

k,1+ 2np

l

)for k, l=0,1, . . .,np

(54)

in the three-dimensional case, where np is a parameter that determines the number of intersectionalpoints. This parameter also affects the accuracy of the re-initialization. By increasing the valueof np , the level set function can be re-initialized more accurately, at the expense of increasedcomputational cost. The above operations are performed for each of the selected elements thatcross the shape boundary. Finally, a set of points representing the shape boundaries (x)=0 isobtained in the reference domain D, as shown in Figure 4, with the obtained set of points herecalled P .

Next, for each node, we calculate the distance between it and each of the points in P , and obtainthe shortest distance. We then examine the sign of the level set function at each node, according tothe rule that if a node is located in the material domain, its sign must be positive, whereas nodeslocated in the void domain are negatively signed, with all values being calculated distances. Wedetermine the updated value of the level set function at the i-th node reinit,i using the followingequation derived from Equations (7) and (8)

reinit,i = sgn(prereinit,i) infpP d(xi ,p) (55)



\D

Intersection points of

and += ,1 2 kn

\D

Intersection points of

and += kn21,)( )(

Figure 3. Intersection points on the shape boundary in an element.

\D

( ) 0=x

Figure 4. A set of points P .

where prereinit,i is the level set function value before the re-initialization at the i-th node, xi isthe coordinates of the i-th node, and d(xi ,p) is the distance between xi and p.

As the above scheme sets the points inside the finite elements when computing distances, moreaccurate computation of the level set function values can be expected when compared with previousschemes based on the PDE [34], since these evaluate the level set function values only at nodes. InSection 5.1, we compare the proposed scheme with previous schemes and show that the proposedscheme provides more accurate computation results.

Furthermore, since the proposed scheme is a FEM-based scheme that allows discretization usingan unstructured mesh, the re-initialization here can also be performed based on a discretizationusing an unstructured mesh, whereas discretization using a structured mesh is required when thelevel set function is re-initialized using a FDM-based scheme such as the PDE-based schemeproposed by Sussman et al. [34], and the Fast Marching Method based scheme [41]. Also, inour proposed method, any type of finite element can be used for the re-initialization, in additionto four-node quadrilateral elements and eight-node hexahedral elements. In Sections 5.2.1 and5.3.1, we demonstrate that essentially identical optimal configurations can be obtained using theproposed scheme regardless of whether the reference domain is discretized using a structured orunstructured mesh.


S. YAMASAKI ET AL.

4. NUMERICAL IMPLEMENTATIONS

4.1. Relaxed heaviside function

In structural optimization methods based on the level set method, a numerical technique to solvethe equilibrium equations in a Eulerian coordinate system is introduced. For this purpose, we adoptthe idea proposed by Wang and Wang [43]. That is, we relax the Heaviside function H((x))in Equation (24) and the Dirac Delta function ((x)), and introduce a scheme to deactivateelements whose element stiffness and mass matrices are the zero matrix. Here, we denote therelaxed Heaviside function as H ((x)) and employ the approximated Heaviside function used in[68] for H((x)). We also denote the relaxed Dirac Delta function as ((x)) and employ theapproximated Dirac Delta function used in [68] for ((x)). In these two approximated functions,parameter h is used to represent the bandwidth between the complete material domain (whereh


Similarly, f ((x)) for the lowest eigenfrequency maximization problem and the eigenfrequencymatching problem are, respectively, reformulated as,

f ((x))=( qk=1

1

k

)2{ qk=1

{(uk) :D :(uk)+kuk uk}((x))H((x))2k

}(58)

and

f ((x))=s+1

k=s1

{2wk

(kobj,k

2obj,k

)((uk) :D :(uk)kuk uk) ((x))H((x))

}(59)

On the other hand, for the three optimization problems, g((x)) is reformulated as,

g((x))= ((x)) (60)The normal velocity VN is computed based on the above reformulations.

4.3. Level set function updating scheme

Once the value of the level set function (x) and the normal velocity VN in the entire referencedomain are computed, the level set function is evolved forward in time. The following equationwith respect to the evolution of the level set function is derived by discretizing Equation (12) usingthe FEM.

E{Ut+t Ut }t

+VtN =0 (61)

where Ut is the nodal value vector of the level set function at time t . E and VtN are described asfollows:

E =nej=1

V e

NNT d

VtN =nej=1

V e

VN (x, t)Nd

(62)

where ne is the number of elements andne

j=1 represents the union set of the elements. Based onEquation (61), the following equation is derived:

Ut+t Ut =tE1VtN (63)Next, a procedure for computing appropriate values of t in the above equation is discussed.

By introducing dlim, a parameter representing the maximum variation of the level set functionvalues in the reference domain when the level set function is updated, the following relation withrespect to t is derived:

t= dlimE1VtN(64)


S. YAMASAKI ET AL.

where dlim must satisfy the following CourantFriedrichsLewy (CFL)-condition given by

dlimx

1 (65)

and x is the distance between adjacent nodes. Thus, t is computed using Equation (64), andthe level set function is then updated using Equation (63).

4.4. Optimization algorithm

We now construct the optimization algorithm using the numerical techniques explained inSections 4.14.3. Figure 5 shows a flowchart of the optimization process. As shown in thisfigure, the initial configuration is first set and then the level set function is initialized as a signeddistance function. Next, the equilibrium equations are solved using the FEM. That is, in theminimum mean compliance problem, the displacement field is computed by solving Equation (23),and in the eigenfrequency optimization problems, the eigenmodes are computed by solvingEquations (36) or (46). The objective functional and f ((x)) are then computed. If the objectivefunctional converges, the optimization process terminates, otherwise the constraint functionaland g((x)) are computed. Next, the level set function is updated based on Equation (63) usingthe FEM, and re-initialized. As the level set function is numerically updated using temporaland spatial discretization, the inequality constraint in Equation (15) may be violated when thelevel set function is updated. Therefore, if the constraint is violated, the level set function ismodified to satisfy the constraint. Finally, we examine whether a topological change, i.e. theintroduction of one or more holes based on the topological derivatives, is needed. If a topologicalchange is required, an appropriate number of holes are introduced. The optimization process thenreturns to the first step. Concerning the level set function modification scheme to satisfy theconstraint, we adopt a level set function modification scheme similar to the modification schemeproposed by Osher and Santosa [42] where the Lagrange multiplier is modified using Newtonsmethod. In our scheme, we modify the level set function value at each node by adding a constantvalue to it.

5. NUMERICAL EXAMPLES

In this section, we first compare the effectiveness of the proposed re-initialization scheme withthe above-mentioned Fast Marching Method- and PDE-based re-initialization schemes. We thenexamine the results for both two- and three-dimensional cases for several numerical examples toconfirm the utility of the proposed method for the minimum compliance and the eigenfrequencyoptimization problems.

5.1. Performance of re-initialization schemes

We now compare the effectiveness of the proposed re-initialization scheme, the Fast MarchingMethod based re-initialization scheme, and the PDE-based re-initialization scheme proposed bySussman et al. [34]. Four level set functions are prepared for this examination. In Case 1, the levelCopyright q 2010 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng (2010)

DOI: 10.1002/nme


Initialize the level set function

( )( )xf

( )( )xgCompute the constraint functional and

Convergence?

Update the level set function using the FEM

Modify the level set functionto satisfy the constraint

EndYesNo

Re-initialize the level set function

Topologicalchange needed?

Introduce holes usingthe topological derivatives

Yes

No

The constraint satisfied?

Yes

No

Solve the equilibrium equations using the FEM

Compute the objective functional and

Figure 5. Flowchart of the optimization process.

set function (x, y) is given by

(x, y) = 2(x2+ y2+1.5)

for (x, y) D, D={(x, y)|5x5,5y5} (66)In Case 2, the level set function (x, y) is given by

(x, y) =min(2x+2y+3,2x+2y+3,2x2y+3,2x2y+3)for (x, y) D, D={(x, y)|5x5,5y5} (67)

In Case 3, the level set function (x, y) is given by

(x, y) = (x2+ y2+1.5)+0.1cos(4x)+0.1cos(4y)

for (x, y) D, D={(x, y)|5x5,5y5} (68)


S. YAMASAKI ET AL.

-5

0

5

5

0

-5

-15

-10

-5

0

5

x y

-5

0

5

5

0

-5x y

-5

0

5

5

0

-5x y

-5

0

5

5

0

-5x y

(a) (b)

-7.5

-5

-2.5

0

2.5

(c) (d)

-37.5

-25

-12.5

0

12.5

-15

-10

-5

0

5

Figure 6. Distributions of (x) for Cases 14: (a) Case 1; (b) Case 2; (c) Case 3; and (d) Case 4.

and in Case 4, the level set function (x, y) is given by

(x, y)= (x2+ y2+1.5){0.9cos(4x)+1}{0.9cos(4y)+1}

for (x, y)D, D={(x, y)|5x5,5y5} (69)In these four cases, the reference domain D is discretized using a structured mesh with 100100elements. The distributions of (x) for Cases 14 are shown in Figure 6.

The numerical implementation and parameter settings with respect to the three re-initializationschemes are explained below. For the proposed re-initialization scheme, Parameter n p explainedin Section 3.2 is set to 8. For the Fast Marching Method based re-initialization scheme, theconventional scheme introduced by Osher and Fedkiw [67] is employed to re-initialize the levelset function values at nodes close to the shape boundaries. That is, if the level set function (x)changes sign in a coordinate direction, linear interpolation is used to locate the shape boundaryand determine a candidate value for (x), and then (x) is re-initialized using the candidate thathas the smallest magnitude. For the PDE-based re-initialization scheme, the first-order upwindscheme with second-order ENO scheme [67] is employed, and the number of iterations, theCourant number, and parameter are, respectively, set to 600, 0.01, and 0.1 in Cases 1 and 2.These parameters are, respectively, set to 400, 0.01, and 0.2 in Case 3, and 200, 0.01, and 0.2 inCase 4.

First, the shape boundaries after the re-initialization are examined. Figures 710, respectively,show the shape boundaries before and after the re-initialization for Cases 14. As shown in these



(a) (b) (c) (d)

Figure 7. Shape boundaries before and after re-initialization for Case 1: (a) shape boundary beforere-initialization; (b) shape boundary after re-initialization using the proposed re-initialization scheme; (c)shape boundary after re-initialization using the Fast Marching Method based re-initialization scheme; and

(d) shape boundary after re-initialization using the PDE-based re-initialization scheme.

(a) (b) (c) (d)



(a) (b) (c) (d)



figures, each of the re-initialization schemes discussed here almost perfectly preserves the shapeboundaries. That is, the given shape boundaries after the re-initialization are almost identical withthe shape boundaries before the re-initialization.


S. YAMASAKI ET AL.

(a) (b) (c) (d)



Table I. ||AVG in Cases 14.||AVG

Level set function Case 1 Case 2 Case 3 Case 4

Before re-initialization 2.000 2.000 1.828 1.222After re-initialization using

the proposed scheme 1.001 1.002 0.9745 0.9754the Fast Marching Method based scheme 1.081 1.210 1.085 1.072the PDE-based scheme 1.001 1.018 1.065 1.150

Next, adherence to the signed distance characteristic represented by |(x)|=1 is evaluated.As a measure of this adherence, we introduce ||AVG which is formulated as

||AVG=nbjb=1

V e |(x)|d

Ve d(70)

where nb is the number of elements that cross the shape boundary andnb

jb=1 represents the unionset of such elements. Thus, the adherence to the signed distance characteristic is evaluated withrespect to the elements that cross the shape boundary, since the normal velocity VN formulated inSection 4.2 has a non-zero value only near the shape boundaries. When ||AVG is sufficientlyclose to 1, we deem that the adherence near the shape boundaries is sufficient. Table I showsthe values of ||AVG for Cases 14, and the numerical error for the worst case is 15.0% whenthe PDE-based re-initialization scheme is used and 21.0% when the Fast Marching Method basedscheme is used. In contrast, when the proposed re-initialization scheme is used, the numericalerror in the worst case is only 2.6%.

Although we searched for suitable values for the number of iterations, the Courant numberand parameter for use with the PDE-based re-initialization scheme, adherence to the signeddistance characteristic represented by |(x)|=1 was not always preserved. Furthermore, suitablevalues for these parameters depend upon the distribution of the level set function, hence weconjecture that the PDE-based scheme implemented using the first-order upwind scheme withsecond-order ENO scheme is unsuitable for the proposed structural optimization method where



Table II. CPU time in Cases 14.

Computation time (s)

Number of elements Case 1 Case 2 Case 3 Case 4

5050 0.037 0.051 0.046 0.038100100 0.280 0.397 0.368 0.283200200 2.220 3.076 3.200 2.219400400 17.41 24.23 25.20 17.48800800 136.6 192.6 198.2 135.4

the level set function is re-initialized at every iteration. For the Fast Marching Method basedre-initialization scheme, we employed the conventional scheme introduced by Osher and Fedkiw[67] to re-initialize the level set function values at nodes close to the shape boundaries. Asthis is a conventional scheme based on linear interpolation, the re-initialization using the FastMarching Method based re-initialization scheme yielded large numerical errors when the shapeboundaries crossed the finite elements diagonally. On the other hand, the proposed scheme strictlyreconfigures the level set function based on the definition of the signed distance function, and allowsreasonable values to be set for parameter np as demonstrated in the above examples, where suitablevalues for this parameter do not depend upon the distribution of the level set function. Therefore,the re-initialization using the proposed re-initialization scheme was successfully performed inCases 14.

Finally, the time complexity for a single re-initialization process using the proposed re-initialization scheme is evaluated. To evaluate the time complexity, we measure the CPU timerequired to carry out calculations for Cases 14, using different mesh sizes. That is, for eachof the cases, the reference domain D is discretized using 5050 elements, 100100 elements,200200 elements, 400400 elements, and 800800 elements. A UNIX operating system isused with a 3.00GHz Intel Pentium D processor. Parameter n p is set to 8. Table II shows theaverage CPU time for 10 re-initializations performed in Cases 14. As this table show, for eachof the cases, when the number of elements quadruples, the CPU time increases roughly by a

factor of 8. Therefore, the time complexity for two-dimensional cases is O(N32 ), where N is

the number of elements. We conjecture that these results are due to the following reasons. For

two-dimensional cases, the number of points in P shown in Figure 4 is of order O(N12 ), since the

shape boundary is a one-dimensional line and the reference domain is a two-dimensional area.With the number of nodes in the entire reference domain of order O(N), the time complexity is

O(N32 ), since the CPU time is proportional to the product of the number of points P and the

number of nodes in the entire reference domain. For three-dimensional cases, the time complexity

is O(N53 ), since the number of points in P shown in Figure 4 is O(N

23 ). Note that since the time

complexity of the Fast Marching Method based scheme is O(N logN) [67], the time complexityof the proposed re-initialization scheme could be improved by incorporating the Fast MarchingMethod. That is, using the Fast Marching Method to re-initialize the level set function values atnodes beyond the neighborhood of the shape boundaries and using the proposed method for nodesclose to the shape boundaries, the time complexity issue could be improved. Unfortunately, thishybrid scheme cannot be used for unstructured meshes.


S. YAMASAKI ET AL.

(a)

10

5

x

y

(b)

Material domain

Figure 11. Reference domain and initial configuration for the two-dimensional case: (a) reference domainand (b) initial configuration.

0.2

Forc

e : 1

000

x

y

Figure 12. Boundary conditions for the two-dimensional minimum compliance problem.

We conclude that the proposed re-initialization scheme provides sufficient adherence to thesigned distance characteristic in the neighborhood of the shape boundaries, while almost perfectlypreserving the shape boundaries. Note that the adherence to the signed distance characteristic wasevaluated only near the shape boundaries in these examples, but the proposed re-initializationscheme provides sufficient adherence to the signed distance characteristic represented by|(x)|=1 almost everywhere in the reference domain, except at singular points, since the levelset function is simply reconstructed based on the distance from the shape boundaries.

5.2. Minimum compliance problem

In this section, numerical examples are presented to confirm the utility of the proposed optimizationmethod for the two- and three-dimensional minimum compliance problems. The isotropic linearelastic material has Youngs modulus =2.1108 and Poissons ratio=0.3. Parameter h explainedin Section 4.1 is set to 0.1, and parameter dlim in Equation (64) is set to 0.1.

5.2.1. Two-dimensional problem. Figure 11(a) shows the reference domain. The reference domainis discretized using a structured or unstructured mesh and four-node quadrilateral plane stresselements of length 0.1. Figure 11(b) shows the initial configuration. The maximum allowablearea of the material domain Gmax is set to 15, i.e. 30% of the area of the reference domain D.Parameter np explained in Section 3.2 is set to 8. Figure 12 shows the boundary conditions for thetwo-dimensional minimum compliance problem. As shown, the left side of the reference domainis fixed, and a vertical traction is applied at the center of the right side.

First, we examine the relation between the optimal configurations and the reference domaindiscretizations where a structured or unstructured mesh is used. Figure 13(a) shows the optimal



(a) (b)

Figure 13. Optimal configurations for the two-dimensional minimum compliance problem: (a)the reference domain is discretized using a structured mesh and (b) the reference domain is

discretized using an unstructured mesh.

configuration where the reference domain is discretized using a structured mesh and Figure 13(b)shows the optimal configuration where the reference domain is discretized using an unstructuredmesh. In both cases, the level set function is re-initialized using the proposed re-initializationscheme. The mean compliance of the optimal configurations shown in Figure 13(a),(b) are,respectively, 0.4443 and 0.4446. As these figures show, essentially identical optimal configu-rations are obtained using the proposed method regardless of whether the reference domain isdiscretized using a structured or unstructured mesh. These obtained optimal configurations, withthe volume constraint active, are nearly identical to the optimal configurations that other researchersobtained using previously proposed structural optimization methods based on the level set method[32, 48, 53, 54]. Note that the optimal configuration in Figure 13(b) shows that an appropriateoptimal configuration can be obtained when the reference domain is discretized using an unstruc-tured mesh, but this does not guarantee that appropriate optimal configurations will always beobtained regardless of the mesh shape and size.

Next, the effect that the three different re-initialization schemes have upon the optimal config-urations is examined. Figure 14(c) shows the optimal configuration where the level set functionis re-initialized using the Fast Marching Method based re-initialization scheme, and Figure 14(d)is an enlarged view of Figure 14(c). The Fast Marching Method based re-initialization scheme isimplemented using the conventional scheme introduced by Osher and Fedkiw [67]. Figure 14(a)shows the optimal configuration where the level set function is re-initialized using the proposedre-initialization scheme, that is Figure 14(a) is the same as Figure 13(a). Figure 14(b) is anenlarged view of Figure 14(a). Comparing Figures 14(b) and (d), the shape boundaries showobvious corrugation when the Fast Marching Method based scheme is used. As explained inSection 5.1, re-initialization using the Fast Marching Method based re-initialization scheme yieldslarge numerical errors when the shape boundaries cross the finite elements diagonally, and weconjecture that these errors are the cause of the corrugations when the Fast Marching Method basedre-initialization scheme is used. On the other hand, when re-initialization is performed using thePDE-based re-initialization scheme where the first-order upwind scheme with second-order ENOscheme is employed, suitable values for the number of iterations, the Courant number, and param-eter must be given a priori. However, finding suitable values that remain constant during theoptimization procedure is problematic, hence the structural optimization could not be performed.

The effect of the initial configurations upon the resulting optimal configurations are now exam-ined. The initial configurations are shown in Figure 15(a),(c),(e), and (g). When Figure 15(a) is the


S. YAMASAKI ET AL.

(a) (b)

(c) (d)

Figure 14. Optimal configurations for the two-dimensional minimum compliance problem: (a) obtainedusing the proposed re-initialization scheme; (b) enlarged view of (a); (c) obtained using the Fast Marching

Method based re-initialization scheme; and (d) enlarged view of (c).

initial configuration, lacking a hole, Figure 15(b) is the obtained optimal configuration. Figure 15(c)shows the initial configuration with one hole and Figure 15(d) is the obtained optimal configuration.Figure 15(e) is the initial configuration with four holes and Figure 15(f) is the obtained optimalconfiguration. Finally, Figure 15(g) is the initial configuration with a large number of holes andFigure 15(h) is the obtained optimal configuration. For all these cases, the level set function is re-initialized using the proposed re-initialization scheme. These optimal configurations are essentiallyidentical and appropriate. That is, in cases where the boundary conditions are imposed as shown inFigure 12, an appropriate optimal configuration can be obtained for all initial configurations here.When the initial configuration is as shown in Figure 15(a), the area of the material domain obvi-ously violates the volume constraint, hence the maximum allowable area of the material domainGmax was gradually decreased during the optimization procedure. The ultimately obtained optimalconfiguration shown in Figure 15(b) satisfies the volume constraint, and this was true in the othercases as well.

5.2.2. Three-dimensional problem. Figure 16(a) shows the reference domain. The referencedomain is discretized using a structured mesh and eight-node hexahedral elements whose lengthis 0.1. Figure 16(b) shows the initial configuration. The maximum allowable volume of thematerial domain Gmax is set to 4.8, i.e. 30% of the volume of the reference domain D. Parameter



(g) (h)

(a) (b)

(c) (d)

(e) (f)

Figure 15. Initial configurations and resulting optimal configurations: (a) initial configuration lacking ahole; (b) optimal configuration when initial configuration is (a); (c) initial configuration with a hole; (d)optimal configuration when initial configuration is (c); (e) initial configuration with four holes; (f) optimalconfiguration when initial configuration is (e); (g) initial configuration with a large number of holes; and

(h) optimal configuration when initial configuration is (g).

np explained in Section 3.2 is set to 4. The re-initialization is performed using the proposedre-initialization scheme. Figure 17 shows the boundary conditions, with the left side of thereference domain fixed, and a vertical traction applied at the center of the right side.

Figure 18 shows the optimal configuration. In this case, the volume constraint is active. Themean compliance of the optimal configuration is 0.2157 whereas the mean compliance of the initial


S. YAMASAKI ET AL.

2

2 4

x

yz

Material domain

(a) (b)

Figure 16. Reference domain and initial configuration for the three-dimensional case: (a) reference domainand (b) initial configuration.

x

yz

0.2

0.2

Forc

e : 1

000

Figure 17. Boundary conditions for the three-dimensional minimum compliance problem.

configuration is 0.7376. Interestingly, the obtained optimal configuration is similar to the optimalconfiguration for the two-dimensional case obtained using the HDM [32]. In the non-penalizedoptimal configuration obtained using the HDM, the edges of the cantilever are dense, althoughgrayscales appear inside the cantilever boundaries. Similarly, in the three-dimensional optimalconfiguration obtained using the proposed method, the outer portions of the cantilever are ratherwide, but thin suddenly toward the center.

5.3. Lowest eigenfrequency maximization problem

The utility of the proposed method for the two- and three-dimensional lowest eigenfrequencymaximization problems is now examined. In the numerical examples here, the lowest, second andthird eigenfrequencies are the target eigenfrequencies, that is parameter q in Equation (33) is setto 3. The isotropic linear elastic material has Youngs modulus =2.1108, Poissons ratio =0.3,and mass density =7.8106. Parameter h explained in Section 4.1 is set to 0.1, and parameterdlim in Equation (64) is set to 0.1.



x

yz

Figure 18. Optimal configuration for the three-dimensional minimum compliance problem.

0.20.2

Concentrated mass : 0.16

x

y

Figure 19. Boundary conditions for the two-dimensional eigenfrequency optimization problems.

5.3.1. Two-dimensional problem. Figure 11(a) shows the reference domain. The referencedomain is discretized using a structured or unstructured mesh where four-node quadrilateralplane stress elements of length 0.1 are used. Figure 11(b) shows the initial configuration. Theupper limit for the area of the material domain Gmax is set to 15, i.e. 30% of the area ofthe reference domain D. Parameter np explained in Section 3.2 is set to 8. Figure 19 showsthe boundary conditions, where the left and right sides of the reference domain are fixed,and a concentrated mass is set at the center of the reference domain. The lowest, second,and third eigenfrequencies of the initial configuration, respectively, reach 1189, 4077, and14050Hz.

First, we examine the effect that using reference domain discretizations with a structured orunstructured mesh has upon the optimal configurations. The level set function is again re-initializedusing the proposed re-initialization scheme. Figure 20(a) shows the optimal configuration wherethe reference domain is discretized using a structured mesh. Figure 20(b) shows the optimalconfiguration where the reference domain is discretized using an unstructured mesh. In bothcases, the volume constraint is active. The lowest, second, and third eigenfrequencies of the twooptimal configurations are shown in Table III. These results show that essentially identical optimalconfigurations are obtained using the proposed method regardless of whether the reference domainis discretized using a structured or unstructured mesh, similar to the results for the two-dimensionalminimum compliance problem. The lowest eigenmode of the two optimal configurations is they-directional first bending mode where the vibration amplitude at the center of the structure islargest, and the stiffness of the optimal configurations with respect to this first bending mode


S. YAMASAKI ET AL.

(a) (b)

Figure 20. Optimal configurations for the two-dimensional lowest eigenfrequency maximizationproblem: (a) the reference domain is discretized using a structured mesh and (b) the reference domain

is discretized using an unstructured mesh.

Table III. Lowest, second, and third eigenfrequencies for the optimal configurations in Section 5.3.1.

Lowest Second ThirdFigures eigenfrequency (Hz) eigenfrequency (Hz) eigenfrequency (Hz)

Figure 20(a) 2019 3370 13 720Figure 20(b) 2019 3372 13 730Figure 21 1430 3713 14 070Figure 22 2056 2289 12 540

is effectively improved. Thus, the proposed method provides a good maximization of the lowesteigenfrequency.

Next, we examine the effect that introducing holes has on the optimal configurations. Figure 20(a)shows a case where topological changes using the scheme for introducing holes are allowed,whereas Figure 21 shows the optimal configuration where topological changes using the schemefor introducing holes are not used, and whose lowest, second, and third eigenfrequencies are shownin Table III. Comparing Figures 20(a) and 21, higher stiffness in the y-direction at the location ofthe non-structural mass is obtained when topological changes due to the introduction of holes areallowed. Such stiffness increases the frequency of the lowest eigenmode, the first bending modewhere the anti-node is located at center of the reference domain. Therefore, the topological changesthat occur as a consequence of introducing holes offer the possibility of optimal configurationsthat have higher eigenfrequencies.

Finally, we examine a case where the objective functional is as formulated in Equation (32),where only the lowest eigenfrequency is regarded as an objective functional. In such cases, theobjective functional may fail to converge due to changes in the sequence of the eigenmodes.Figure 22 shows the optimal configuration for a case where the objective functional is formulatedas in Equation (32), and the lowest, second, and third eigenfrequencies are as shown in Table III. Asthe sequence of the eigenmodes did not change during the optimization procedure in this setting,the objective functional converged. As shown in Figure 22, two holes are observed near the centerof the reference domain, while such holes are not present in the optimal configuration shown inFigure 20(a). However, comparing the lowest eigenfrequencies for the cases shown in Figures20(a) and 22, the values only differ by 2%. Therefore, the formulation explained in Section 2.4



Figure 21. Optimal configuration obtained when a hole introducing method based on thetopological derivative is not used.

Figure 22. Optimal configuration obtained when only the lowest eigenfrequency is maximized.

that aims to protect the convergence of the objective functional does not significantly degrade theobtained optimal configuration.

5.3.2. Three-dimensional problem. Figure 16(a) shows the reference domain of the three-dimensional problem. The reference domain is discretized using a structured mesh and eight-nodehexahedral elements of length 0.1. Figure 16(b) shows the initial configuration. The maximumallowable volume of the material domain Gmax is set to 4.8, i.e. 30% of the volume of thereference domain D. The re-initialization is performed using the proposed re-initialization scheme.Parameter np explained in Section 3.2 is set to 4. Figure 23 shows the boundary conditions,where the left and right sides of the reference domain are fixed, and a concentrated mass is set atthe center of the reference domain. The lowest, second, and third eigenfrequencies of the initialconfiguration respectively reach 45 460, 49 290, and 85560Hz.

Figure 24 shows the optimal configuration of this three-dimensional case. The lowest, second,and third eigenfrequencies of the optimal configuration, respectively, reach 52 390, 52 590, and82680Hz. The optimal configuration of this three-dimensional case is an axially symmetricconfiguration of the two-dimensional case shown in Figure 20(a), which we consider veryreasonable.

5.4. Eigenfrequency matching problem

Numerical examples for the two- and three-dimensional eigenfrequency matching problem are nowpresented. In this section, parameter h explained in Section 4.1 is set to 0.1, and parameter dlim in


S. YAMASAKI ET AL.

x

yz

Concentrated mass : 4 10 4

( size : 0.2 0.2 0.2)

Figure 23. Boundary conditions for the three-dimensional eigenfrequency optimization problem.

x

yz

Figure 24. Optimal configuration for the three-dimensional lowest eigenfrequency maximization problem.

Equation (64) is set to 0.01. The material properties are the same as for the lowest eigenfrequencymaximization problem.

5.4.1. Two-dimensional problem. The reference domain shown in Figure 11(a) is discretized usinga structured mesh and four-node quadrilateral plane stress elements of length 0.1. The initialconfiguration is shown in Figure 11(b). The upper limit for the area of the material domain Gmaxis set to 15, i.e. 30% of the area of the reference domain D. The re-initialization is performedusing the proposed re-initialization scheme. Parameter n p explained in Section 3.2 is set to 8. Theboundary conditions are shown in Figure 19. The lowest, second, third, and fourth eigenfrequenciesof the initial configuration, respectively, reach 1189, 4077, 14 050, and 16290Hz.

First, the lowest eigenfrequency matching cases are examined. The weighting coefficientsw1 andw2 in Equation (43) are, respectively, set to 0.9 and 0.1. Figure 25 shows the optimal configurationswhose lowest eigenfrequency is matched to a specified target value. The lowest eigenfrequencytarget values for the cases shown in Figure 25(a)(e) are, respectively, set to 300, 600, 900,1500, and 1800Hz. In all of these cases, the target value of the second eigenfrequency is set to4000Hz to avoid changes in sequence between the lowest and second eigenmodes. The lowest andsecond eigenfrequencies of the optimal configurations are shown in Table IV. For the cases shownin Figure 25(a)(c), topological changes are not performed and the volume constraint remains



(a) (b)

(c) (d)

(e)

Figure 25. Optimal configurations whose lowest eigenfrequency is matched to specified target value: (a)lowest eigenfrequency target value=300Hz; (b) lowest eigenfrequency target value=600Hz; (c) lowesteigenfrequency target value=900Hz; (d) lowest eigenfrequency target value=1500Hz; and (e) lowest

eigenfrequency target value=1800Hz.

inactive. On the other hand, topological changes are carried out in the cases shown in Figure 25(d,e), and the volume constraint becomes active. As the lowest eigenfrequency reaches a maximumvalue of 1430Hz in the case shown in Figure 21 where topological changes using the scheme forintroducing holes are not allowed, it appears that topological changes must be allowed when theaim is to achieve target values above 1430Hz.

Next, the second and third eigenfrequencies matching cases are examined. Figure 26(a) showsthe optimal configuration whose target value for the second eigenfrequency is specified as 3400Hz.In this case, the target values of the lowest and third eigenfrequencies are, respectively, set to1200Hz and 14000Hz to avoid changes in the sequence among the three eigenmodes, and theweighting coefficients w1, w2, and w3 are, respectively, set to 0.1, 0.8, and 0.1. Figure 26(b) showsthe optimal configuration whose target value for the third eigenfrequency is 7000Hz. The targetvalues of the second and fourth eigenfrequencies are, respectively, set to 4000 and 16000Hz toavoid changes in the sequence among the three eigenmodes, and the weighting coefficients w2,w3, and w4 are, respectively, set to 0.1, 0.8, and 0.1. In both cases, the volume constraint is


S. YAMASAKI ET AL.

Table IV. Lowest, second, third, and fourth eigenfrequencies for the optimal configurations inSection 5.4.1.

Lowest Second Third Fourtheigenfrequency eigenfrequency eigenfrequency eigenfrequency

Figures (Hz) (Hz) (Hz) (Hz)

Figure 25(a) 299.6 3537 Figure 25(b) 601.4 3816 Figure 25(c) 900.5 3983 Figure 25(d) 1499 3932 Figure 25(e) 1786 3678 Figure 26(a) 1201 3399 13 870 Figure 26(b) 3432 6980 16 273Figure 27 1800 3357

(a) (b)

Figure 26. Optimal configurations whose 2nd or 3rd eigenfrequency is matched to specified target value:(a) 2nd eigenfrequency target value=3400Hz and (b) 3rd eigenfrequency target value=7000Hz.

Figure 27. Optimal configuration whose lowest eigenfrequency target value=1800Hz(2nd eigenfrequency is not matched).

inactive. The eigenfrequencies of these optimal configurations are listed in Table IV. As shown inthese figures, clear optimal configurations are obtained even for cases where the second or thirdeigenfrequencies are specified.

Finally, we examine a case where only the lowest eigenfrequency is matched to 1800Hz whereasthe second eigenfrequency is not matched to a specified target value. Figure 27 shows the optimalconfiguration, and the lowest and second eigenfrequencies, respectively, reach 1800 and 3357Hz. Inthis case, the volume constraint is active and as a result, the sequence between the lowest and secondeigenmodes is not changed during the optimization procedure. The obtained optimal configuration



is similar to Figure 25(e) where the target values for the lowest and second eigenfrequencies are,respectively, set to 1800 and 4000Hz. Comparing the cases shown in Figures 25(e) and 27, thesecond eigenfrequency in Figure 25(e) is closer to 4000Hz, since the target value for the secondeigenfrequency is given as 4000Hz. On the other hand, the lowest eigenfrequency in Figure 25(e)fails to reach 1800Hz, whereas the lowest eigenfrequency in Figure 27 matches the 1800Hz targetexactly. However, since the lowest eigenfrequency in Figure 25(e) reaches 1786Hz, and deviatesfrom the target value by only 1%, we consider that the lowest eigenfrequency is sufficientlymatched to the target value, because the target value for the second eigenfrequency is also given.

5.4.2. Three-dimensional problems. The reference domain shown in Figure 16(a) is discretizedusing a structured mesh where eight-node hexahedral elements of length 0.1 are used. The initialconfiguration is shown in Figure 16(b). The maximum allowable volume of the material domainGmax is set to 4.8, i.e. 30% of the volume of the reference domain D. The re-initialization isperformed using the proposed re-initialization scheme. Parameter n p explained in Section 3.2 isset to 4. The boundary conditions are shown in Figure 23. The lowest and second eigenfrequenciesof the initial configuration, respectively, reach 45 460 and 49290Hz.

Figure 28 shows the obtained optimal configurations whose lowest eigenfrequency is matchedto a specified target value. The target values of the lowest eigenfrequency for the cases shown inFigure 28(a)(c) are, respectively, set to 15 000, 25 000, and 35000Hz. In these cases, the targetvalue for the second eigenfrequency is set to 48000Hz to avoid changes in the sequence betweenthe lowest and second eigenmodes, and the weighting coefficients w1 and w2 in Equation (43) are,respectively, set to 0.9 and 0.1. The eigenfrequencies of these optimal configurations are shownin Table V. For these cases, topological changes are not performed and the volume constraintremains inactive. The lowest eigenmode for these cases is the y-directional first bending modewhere the ant

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