Computer-Aided Design 45 (2013) 1524–1537

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Computer-Aided Design

journal homepage: www.elsevier.com/locate/cad

Engineering feature design for level set based structural optimization

Mingdong Zhou, Michael Yu Wang ∗

Department of Mechanical and Automation Engineering, The Chinese University of Hong Kong, Shatin, NT, Hong Kong

h i g h l i g h t s

• A method to design engineering features in structural optimization is proposed.• It combines CSG modeling and level set based shape and topology optimization.• Feature design and structural optimization are unified under the level set framework.• A truly optimal structure with features can be designed conveniently.

a r t i c l e i n f o

Article history:Received 22 September 2012Accepted 28 June 2013

Keywords:Engineering feature designStructural optimizationLevel set methodConstructive solid geometry

a b s t r a c t

Engineering features are regular and simple shape units containing specific engineering significance. Itis useful to combine feature design with structural optimization. This paper presents a generic methodto design engineering features for level set based structural optimization. A Constructive Solid Geometrybased Level Sets (CSGLS) description is proposed to represent a structure based on two types of basicentities: a level set model containing either a feature shape or a freeform boundary. By treating bothentities implicitly and homogeneously, the optimal design of engineering features and freeform boundaryare unified under the level set framework. For featuremodels, constrained affine transformations coupledwith an accurate particle level set updating scheme are utilized to preserve feature characteristics, wherethe design velocity approximates continuous shape variation via a least squares fitting. Meanwhile,freeform models undergo a standard shape and topology optimization using a semi-Lagrangian level setscheme. With this method, various feature requirements can be translated into a CSGLS model, and theconstrained motion provides flexible mechanisms to design features at different stages of the model tree.As a result, a truly optimal structurewith engineering features can be created in a convenientway. Severalnumerical examples are provided to demonstrate the applicability and potential of this method.

© 2013 Elsevier Ltd. All rights reserved.

1. Introduction

In structural design, engineering features refer to regular andsimple shape units containing specific engineering significance [1].They generally serve as a bridge between computer-aided design(CAD) and computer-aided manufacture (CAM), and also have agreat impact on assembly [2]. Recently, as structural optimizationtechniques have been widely utilized to design innovative andlightweight products, it is practically meaningful to generate anoptimal structural layout containing engineering features at anearly stage of product lifecycle. However, this has been a challengefor standard structural shape and topology optimization.

Current structural optimization techniques for continuumstructure can be categorized into three mainstreams based ondifferent model representations. Firstly, the B-rep model based

This paper has been recommended for acceptance by Horea T. Ilies.∗ Corresponding author. Tel.: +852 39438487.

E-mail addresses: mdzhou@mae.cuhk.edu.hk (M. Zhou),yuwang@mae.cuhk.edu.hk (M.Y. Wang).

0010-4485/$ – see front matter© 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.cad.2013.06.016

approaches [3,4] are the most suitable for feature design, as engi-neering significance is captured directly through the dimensionalparameters or constraints in a CAD modeler. But it rarely supportstopology optimization over an explicit model by modifying its pa-rameters. In comparison, the density based approaches, such as theHomogenization Method [5], the method of Solid Isotropic Mate-rial with Penalization [6,7] and the Evolutionary Structural Opti-mization [8], are able to optimize structural topology conveniently.However, it is difficult to employ geometric constraints into opti-mization, because neither an explicit geometry nor feature con-cept is readily available from a finite element (FE) mesh model.Implicit model based approaches, such as level set based optimiza-tion [9,10], have the advantage of maintaining a clear structuralboundary during a shape and topology optimization process. Butdue to its infinite dimensional nature, to track geometric consis-tency between consecutive updatedmodels is nontrivial, such thatfeature constraints can hardly be imposed during optimization.

To enhance the applicability of structural optimization, severaltechniques have beendeveloped for a simultaneously optimizationof engineering features and structural layout. The key to address

M. Zhou, M.Y. Wang / Computer-Aided Design 45 (2013) 1524–1537 1525

Fig. 1. (a) A right-angle feature; ((b)–(c)) feature in fixed geometry; (d) feature with a freeform boundary.

this problem is that the model representation must support bothfeature definition and topological change. In [11,12], a finite circlemethod was proposed to approximate the exact geometry ofpredefined feature components with circumcircles. The locationand orientation of these circles can be determined together witha density based structural topology optimization. Besides, Kangand Wang [13] presented a novel topology description modelto design movable hole-features by combining material densityand level set models. On the other hand, for dynamic featuredesign problems, that the feature shape is not fixed a priori, thestate-of-the-art solutions are mainly geometric primitives based.In [14], the Bubble method [15] was utilized to insert basic hole-features, such as circles and triangles, into a structure accordingto topological derivative analysis. The optimal design eventuallycomprises several simple feature shapes, which approximate themerged holes. Another parametric solution can be found in [16,17],in which they represented feature primitives by R-function andcombined them with a B-spline model.

The integration of feature and freeform boundary design hasprofound meaning in structural optimization. For example, if aright-angle feature shown in Fig. 1(a) is expected in final design,all the shape of Fig. 1(b)–(d) are potential candidates to be mod-eled into an initial structure for optimization. Obviously, the onewith a freeform boundary, as shown in Fig. 1(d), can lead to a bet-ter structural performance than the other two fixed geometry be-cause of the extra design freedom over the non-critical boundary.This example reveals that under certain circumstances, it is un-necessary to use fixed high-level primitives to capture the criticalengineering significance. Instead, if one can separate out featurecharacteristics and freely optimize non-critical regions, a truly op-timal design of maximum structural performance can be eventu-ally realized. In the literature, the method proposed in [16,17] canaccomplish this task by carefully conceiving parametric constraintsover the primitive’s boundary.

A level set based optimization method has good potentials ofdesigning freeformboundarymeanwhile supporting flexible shapeand topology optimization. However, there are two fundamentalchallenges to design features with the level set method. Firstly,as a structural boundary conventionally embeds inside an implicitmodel, a continuous shape evolutionwillmake the underlying geo-metric consistency unpredictable. Hence, it is complicated to trackany predetermined feature shape during optimization. Secondly,engineering features usually contain sharp characteristics, such ascorners in 2D model or corners and edges in 3D model. Due to theinherent dissipation nature of numerical calculation, all of thesehigh curvature regions will be gradually smoothed out during op-timization.

In this paper, the above modeling difficulties are resolved byleveraging a CSG representation and an accurate constrained mo-tion scheme. Inspired from the multiphase level set descriptionin [18], a structural model here is built upon two types of entities:a level set model containing either a feature shape or a freeformboundary. These entities are the operands in a CSG model tree andserve as either feature models containing necessary engineeringsignificance or freeformmodels (non-featuremodel) otherwise. Aninherited advantage from the CSG modeling is that a structure canbe flexibly constructed with different levels of entities according

to design requirements. In this way, feature characteristics can beidentified either through the shape of a particular feature modelor the relation between lower-level feature entities (e.g. the right-angle determined by two line feature models in Fig. 1), rather thanby simply resorting to a fixed high-level geometry.

Besides modeling features, it is important to devise a workablemechanism to preserve and optimize them. The idea of impos-ing motion constraints has been proven a viable way in designingstructures for practical requirements. In [19], the design velocityof level set equation was regularized to ensure the optimal struc-ture can be formed by casting. Moreover, the rigid body motionwas justified in [20], which spurs polygon-shaped components tobe positioned and oriented optimally inside design region. In thiswork, a constrained affine transformation coupled with an accu-rate particle level set updating scheme [21] is adopted to designfeature characteristics. Specifically, it consists of translation, rota-tion and scaling, which can simulate most of the effects in deform-ing a geometric primitive by modifying its parameters [17]. Thetransformation velocity is determined from a least square fittingto the continuous shape variation. Meanwhile, non-featuremodelsjust undergo a conventional level set updating for freeform shapeand topology optimization.

To demonstrate the proposed approach, this paper is organizedas follows. Section 2 introduces the CSG based level sets repre-sentation. Section 3 presents a sensitivity analysis for linear elas-tic structural optimization problems. The constrained motion andimplementation details are described in Sections 4 and 5 respec-tively. Section 6 shows several numerical examples. Finally, dis-cussion and conclusion are stated in Section 7.

2. CSG based level sets

Constructive Solid Geometry is a ubiquitous solid model rep-resentation, which facilitates both set operations and boundaryevaluation. In CSG modeling, because a structure can be assem-bled flexibly with different solid entities, it becomes possible to in-terpret practical machining or assembly requirements in terms ofthe geometry of and relation between feature entities (e.g. a low-level linear entity for a flat edge or surface, an angle intersectedby two entities, or a high-level predefined mating geometry). Thelevel set method, on the other hand, provides an effective way tofreely optimize the boundary of a solid model. Intuitively, by com-bining the strengths of CSGmodeling in feature definition and levelset method in shape and topology optimization, a CSG based LevelSets (CSGLS) model description is adopted to address the problemof feature design in structural optimization.

The CSGLS represents a solid structure in terms of m individ-ual sub-level set models Φ = [φ1, φ2, . . . , φm−1, φm]. Each φi (i =

1, 2, . . . ,m) is a well defined half-space model, denoting a geo-metric entity in the model tree. Among these models, a ‘‘featuremodel’’ contains feature geometry according to the engineering re-quirement, as opposed to a ‘‘freeformmodel’’ (non-feature model)with freely designable boundary embedded. The design domain Dwith a underlying structureΩ is thus formulated asΦD =

mi=1 φi,

and the followings are hold by convention:

φi(x) > 0 ∀x ∈ Ωi/∂Ωi (inside)φi(x) = 0 ∀x ∈ ∂Ωi (on the boundary)φi(x) < 0 ∀x ∈ D/Ωi (outside),

(1)

1526 M. Zhou, M.Y. Wang / Computer-Aided Design 45 (2013) 1524–1537

Fig. 2. CSGbased level sets: (a) a structurewith two slots; ((b)–(c)) sub-level setmodel containing a rectangular hole; (d) reconstruction of a sharp corner by local refinement.(For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

H(φi) =

1 if φi ≥ 00 if φi < 0 (2)

where H is a Heaviside function. A simple combination rule im-plies that there are at most n = 2m non-overlapping regionsωk (k = 1, 2, . . . , n) inside the domain D, such that:

D =

nk=1

ωk and ωp ∩ ωq = ∅, p = q. (3)

Each region is uniquely tagged with a constant Heaviside vectorH(Φ) = [H(φ1),H(φ2), . . . ,H(φm)]:

ωk = x ∈ D : H(Φ(x)) = constant vector. (4)

Note that, since there is only one material region assumed in thispaper, any point locating inside it should belong to the interior ofall sub-level set models:

ωk =

material if H(Φ(x)) = [1, . . . , 1] x ∈ ωkvoid otherwise. (5)

Fig. 2 illustrates the CSGLS description, in which a simplemodelis constructed with two sub-level set models Φ = [φ1, φ2]

representing a rectangular hole feature as shown in Fig. 2(b) and(c) respectively. There are in total four separating regions in thedesign domain, where H(Φ) = [1, 1] denotes the material regionand the others refer to void. In practice, the two feature modelsφi can be further decomposed into lower level entities φi =

[φi1, φi2, φi3, φi4], i = 1, 2, where each φij, j = 1, 2, 3, 4 denotesa half space model defined by one dash line shown in Fig. 2(b)and (c). Such decomposition is important to preserve and retrievesharp characteristics during optimization. As shown in Fig. 2(d),a 2D sharp corner is rebuilt by firstly cutting the voxel accordingto one zero level set (red dash line), and then the other (purpledash line). If such a sharp feature is expected in final design,the exact shape and relation of φ1 and φ2 in Fig. 2(d) must bewell maintained throughout optimization. This is the motivationto apply constrained affine transformation with a highly accurateupdating scheme. Note that, to reconstruct complex geometrylocally (e.g. curves defined by one or multiple level sets inside anelement), several refinement schemes are available in [22].

3. Structural optimization with CSGLS

In level set based structural optimization, a solid structure isrepresentedby adynamic signed-distance functionΦ(x, t) andop-timized via solving the following Hamilton–Jacobi (H–J) equation:

∂Φ

∂t+ Vn · |∇Φ| = 0, Φ(x, 0) = Φ0(x), (6)

where Φ0(x) denotes an initial structural configuration and Vnrefers to a normal boundary velocity derived from the sensitivityanalysis of continuum shape variation [9,10]. However, for the pro-posed CSGLS model, because each sub-level set model φi requires

an independent design velocity V in for shape updating, the conven-

tional sensitivity analysis for a single level set modelmust bemod-ified accordingly.

By leveraging the idea in [23], a characteristic function χk(x) isfirstly introduced for each region ωk as:

χk(x) =

1 if x ∈ ωk0 otherwise. (7)

Let the Heaviside function H(φi) be expressed as following:

H1i = H(φi), H0

i = 1 − H(φi). (8)

Eq. (7) can be re-written in terms of Heaviside function as:

χk(Φ) =

mi=1

HIkii , (9)

where the superscript Iki indicates the relationship between eachωk and φi as follows:

Iki =

1 if ωk ∈ φi0 otherwise. (10)

In this paper, the optimization problem of a linear elastic struc-ture under a volume constraint is studied. The mathematical for-mulation with CSGLS representation is expressed as:

MinimizeΦ

J(u, Φ) =

nk=1

DF k(u)χk(Φ)dΩ, (11)

Subject to G(Φ) =

nk=1

Dχk(Φ)dΩ ≤ V0, (12)

a(u, v, Φ) = l(v, Φ) for all v ∈ U, (13)

a(u, v, Φ) =

nk=1

DEkε(u) : ε(v)χk(Φ)dΩ, (14)

l(v, Φ) =

nk=1

D(fk · v)χk(Φ)dΩ +

Γk

hk· vdΓ

, (15)

where F(u) denoting the objective function, V0 the upper allow-able volume,U the space of kinematically admissible displacementfields with predefined displacement at Dirichlet boundary, u theelastic displacement under external loads, E the elasticity tensor,ε the strain tensor, f the body force, h the boundary traction force,andΓk the boundary ofωk. Eq. (13) corresponds to the linear elasticequilibrium, where a(u, v, Φ) and l(v, Φ) are the energy bilinearand load linear variational form respectively.

By taking thematerial derivative [6,24] of Eq. (11), the followingexpressions can be obtained as in [23]:

dJ(u, Φ)

dt=

mi=1

DPi(u,w, Φ)δ(φi)∇φiV i

ndΩ

=

mi=1

Γi

Pi(u,w, Φ)V indΓ , (16)

M. Zhou, M.Y. Wang / Computer-Aided Design 45 (2013) 1524–1537 1527

Pi(u,w, Φ) =

nk=1

(2Iki − 1)(F k(u) + fk · w

− Ekε(u) : ε(w))

me=1,e=i

H Ikee

+ (∇(hi · w) · ni + (hi · w)κ(φi)), (17)

where Γi = x : φi(x) = 0 denoting the full boundary of level setfunction φii = (1, 2, . . . ,m),n the normal, κ the curvature, andwan adjoint variable for the following adjoint equation:

nk=1

DEkε(w) : ε(v)χk(Φ)dΩ

=

nk=1

D

∂F k(u)

∂uχk(Φ)dΩ, ∀v ∈ U andw = 0 on ΓD. (18)

Considering that the physical properties and corresponding mea-sures are defined only at the material region, Eq. (17) reduces tothe following expression by canceling out zero terms that relatedto the void regions:

Pi(u,w, Φ) = (F ks(u) + fks · w − Eksε(u) : ε(w))

×

me=1,e=i

H(φe) + (∇(hi · w) · ni

+ (hi · w)κ(φi)), (19)

where ks is the index for solid region. Meanwhile, the materialderivative of the volume constraint in Eq. (12) can be determinedas:

dG(Φ)

dt=

mi=1

Γi

m

e=1,e=i

H(φe)

· V i

ndΓ . (20)

For simplicity, a traction-free boundary and body-force free con-ditions are assumed in this paper. (Readers may refer to [23] andreferences therein for a complete discussion about the problemsubject to a boundary dependent loading.) Consequently, by con-structing an augmented objective functional as follows:

J = J + λ(G − V0), (21)

where λ is a Lagrange multiplier, the steepest descent direction ofthis composite objective can be readily found by setting the normalvelocity V i

n for each level set model φi as:

V in = −(F ks(u) − Eksε(u) : ε(w) + λ)

me=1,e=i

H(φe), (22)

where λi+1 = max0, λi +µ(

ΩdΩ −V0) and µ > 0 is a penalty

parameter.Eq. (22) implies that non-zero design velocity exists only at a

boundary portion Γ ′

i of φi, where Γ ′

i = x ∈ D : φi(x) = 0and Φ(x) = 0. The other part Γ ∗

i = x ∈ D : φi(x) = 0and Φ(x) = 0 submerged in solid region has zero velocity,which indicates no contribution in improving the overall structuralperformance. This result coincides with the single level set modelproblem in [24], as the final design velocity are the same over thestructural boundary Γ = x ∈ D : Φ(x) = 0. However, it revealsan interesting fact that the evolution process of a single level setmodel can be decomposed into a parallel behavior of lower-levelentities with a coherent but zero-extended velocity. Therefore, itis practically viable to separate out feature level sets from a CSGLSmodel and control their evolution individually.

4. Constrained motion with affine transformation

In feature based design,manufacture or assembly requirementsare interpreted through carefully conceived feature geometry. Tomaintain such geometric information throughout optimization, aconstrained motion strategy is pertinent for the proposed CSGLSscheme to preserve the shape and relation of well initiated featuremodels. Three basic affine transformations, namely translation,rotation, and scaling, are adopted here to reproduce the same neteffect to that by changing a primitive’s parameters as in [14,17].They can be applied to entities at different levels of a CSG treefor various purposes, such as to change the shape of a compositemodel in a rigid body manner, to maintain the relation betweenentities, or to preserve the shape of a low-level primitive. Thecorresponding applications are demonstrated in Section 6.

For simplicity and conciseness, only a 2D algorithm is intro-duced below. Extension to 3D problems is straightforward. Givena feature level set model φf

i with boundary Γi, the constrained ve-locity is defined as the composition of two linear translation veloci-ties Vx and Vy for X and Y axis respectively, a counter-clock angularvelocity θ about the centroid ci of Γi, and a homogeneous scalingcoefficient α about ci for both axes. For any point p ∈ Γi, its normalvelocity Vnp can be thus expressed in homogeneous coordinate as:

Vnp = n · W · (p − ci)T, (23)

where (p− ci) = [px − cix, py − ciy, 1],n = [nx, ny, 0] the normalvector at point p, and W the velocity matrix:

W =

α −θ Vx

θ α Vy0 0 1

. (24)

The unknown Vx, Vy, θ , α are determined through the follow-ing least square fitting problem, which approximates the uncon-strained normal velocity V i

n inf in Eq. (22):

Find Vx, Vy, θ , α,

Min Z =12

Γi

(Vnp − V in inf)

2dΓ . (25)

This quadratic problem can be readily solved as:

X =Vx Vy θ α

T= A−1B, (26)

where A =

O P Q RP M S TQ S N UR T U L

,

B =

Γi

V in inf · nxdΓ

Γi

V in inf · nydΓ

Γi

V in inf · [ny(px − cix) − nx(py − ciy)]dΓ

Γi

V in inf · [nx(px − cix) + ny(py − ciy)]dΓ

,

(27)

O =

Γi

nxnxdΓ

Q =

Γi

nx[ny(px − cix) − nx(py − ciy)]dΓ

P =

Γi

nxnydΓ

1528 M. Zhou, M.Y. Wang / Computer-Aided Design 45 (2013) 1524–1537

Fig. 3. Cut elements with one zero level set.

Fig. 4. Cut elements with two zero level sets.

S =

Γi

ny[ny(px − cix) − nx(py − ciy)]dΓ

R =

Γi

nx[nx(px − cix) + ny(py − ciy)]dΓ

T =

Γi

ny[nx(px − cix) + ny(py − ciy)]dΓ(28)

N =

Γi

[ny(px − cix) − nx(py − ciy)][ny(px − cix)

− nx(py − ciy)]dΓ

L =

Γi

[nx(px − cix) + ny(py − ciy)][nx(px

− cix) + ny(py − ciy)]dΓ

M =

Γi

nynydΓ

U =

Γi

[ny(px − cix) − nx(py − ciy)][nx(px − cix)

+ ny(py − ciy)]dΓ .

Note that the matrix A in Eq. (27) may become singular undercertain circumstances. For example, if a feature model containsonly a linear boundary with normal component nx = 0 or ny = 0,many items in matrix A will vanish. In such case, the translationmotion alongX orY axis should be suppressed andhence Eqs. (25)–(28) must be reformulated accordingly. In the Appendix, all thepossible combinations of 2D transformations and correspondingdesign velocities are listed for reference. Properly combining thethree basic transformations can produce flexible feature designcapabilities.

5. Implementation details

To make this paper self-contained, several implementation de-tails are described in this section. Firstly, an extended finite ele-mentmethod (XFEM) [25] is adopted for structural analysis, whichcomputes a more accurate result than density based FEM. Sec-ondly, to alleviate the side effect due to unavoidable numerical

dissipation in solving the H–J equation (6), an accurate particlelevel setmethod [21] is employed to update featuremodels. Mean-while, a numerically stable semi-Lagrange method [26,27] is im-plemented for freeformboundary optimization. Lastly, a divide andconquer strategy is proposed to design sharp characteristics.

5.1. An extended finite element method

For a CSGLS based structural model, its explicit boundary (evensharp characteristics) can be reconstructed [22] for an XFEMbased structural analysis. Similar to previous works [25,28,29], thedisplacement field u is approximated as:

u(x) =

i∈I

Ni(x)H(Φ(x))ui, (29)

where a point x locates inside the ith element of the entire elementset I,Ni the standard finite element shape function, ui the corre-sponding nodal values, and H a Heaviside function as in Eq. (2).In practice, the global stiffness matrix reads in the following dis-cretized form as:

K =

i∈I

Ke =

i∈I

Ω

BTDsBdΩ, (30)

where Ke the element stiffness matrix, B the displacement differ-entiation matrix, Ω the solid domain inside a element, Ds the elas-ticity matrix of solid material.

Eq. (30) is evaluated numerically with a standard Gaussianquadrature rule for solid elements. But the cut elements, contain-ing both material and void inside, require a special treatment.Figs. 3 and 4 illustrate different types of 2D cut elements. The solidregion inside each element is firstly identified according to zerolevel set intersections and then divided into several sub-triangles(tetrahedrons in 3D case) connecting to its barycenter. As a result,the element stiffness equals to the sumof that of each sub-triangle.In current implementation, weak material is assumed in void ele-ment to avoid numerical singularity, which provides a natural ve-locity extension for level set update [10].

M. Zhou, M.Y. Wang / Computer-Aided Design 45 (2013) 1524–1537 1529

Fig. 5. Convergence history of MOI example: (a) initial configuration; ((b)–(c)) intermediate steps; (d) optimal result.

5.2. Model update

Generally, shape evolution with the level set method suffersfrom numerical dissipation, which causes difficulty in maintainingparticular feature geometry. To mitigate such side effect in updat-ing feature entities, a particle level set method [21] of acceptablecomputational accuracy is leveraged for the proposed affine trans-formations. This scheme utilizes marker particles to correct errorsdue to numerical dissipation and is able to preserve feature char-acteristics during optimization. Table 1 briefly summarizes thealgorithm according to a publicly available C++ library [21]. The al-gorithm is re-implemented using MATLAB for the 2D examples inthis paper.

On the other hand, freeform models undergo a standard semi-Lagrangian updating scheme [26,27] for simultaneously shape andtopology design. This scheme follows the fact that theH–J equationpropagates Φ values along the characteristic curves x = c(t), thatdefined by the following equations:

c(t) = v(c(t), t), c(0) = x0. (31)

By approximating the curve with a first order Courant–Isaacson–Rees formula [30]:

c(t) ≈ x − (ti+1 − t) · v(x, ti), ti+1 = ti + 1t, (32)

where v(x, ti) is the velocity of any spatial point x at previous timeti, the current Φ value at ti+1 can be explicitly determined througha backward path tracing as follows:

Φ(x, ti+1) = Φ(x − 1t · v(x, ti), ti). (33)

5.3. Design sharp characteristics

With the proposed CSGLS description and constrained motion,sharp feature can also be designed by firstly identifying the lower-level feature entities, which contribute to sharp characteristics,and then maintaining their geometry and relation throughoutoptimization.

For example, all the sharp corners in Fig. 2(a) can be recon-structed during optimization only if φ1 and φ2 in Fig. 2(b) and (c)keep their shape exactly as the origin under the constrained mo-tion. However, simply updating φ1 or φ2 will still smooth out thecorner regions after cycles of rotation and scaling, because the fun-damental dissipation nature cannot be eliminated completely evenwith the accurate particle level set method [21]. As a remedy, oncethe constrained velocity ofφ1 orφ2 is obtained, it is applied to drivethe lower-level entities φi = [φi1, φi2, φi3, φi4] i = 1, 2 individu-ally, as shown in Fig. 2(b) and (c), tomaintain the linear hole edges.In this way, all the sharp corners are implicitly preserved and canbe reconstructed on the fly for visualization or structural analysis.Experiments show that this ‘‘divide and conquer’’ strategy workswell in designing sharp features, which makes full use of the pro-posed approach.

6. Numerical examples

6.1. Moment of inertia (MOI) maximization

The following MOI maximization problem under a volumeconstraint is firstly studied:

MinimizeΦ

J(Φ) =

nk=1

D−|x − c|2χk(Φ)dΩ, (34)

Subject to G(Φ) =

nk=1

Dχk(Φ)dΩ ≤ V0, (35)

where c denoting the domain center and V0 the upper volumeconstraint. According to the sensitivity analysis in Section 3, designvelocity for each level set φi is obtained as:

V in = (|x − c|2 − λ)

me=1,e=i

H(φe),

where λi+1 = max0, λi + µ

Ω

dΩ − V0

. (36)

Fig. 5(a) shows the initial configuration, where three level setmodels are used for computation: a rectangular hole, a circularhole, and a fixed rectangular outer boundary. The dimension ofdesign domain is 1.0 × 1.0 and the volume constraint is set asV0 = 0.58. During the optimization process shown in Fig. 5, bothholes move gradually towards the center of the domain throughtranslation, rotation, expansion and shrinking. The final result con-taining one circular hole at the center indicates the MOI has beenmaximized. Meanwhile, the volume constraint is also satisfied.

In this example, although the feature models (rectangular andcircular holes) are updated implicitly with level set method, allthe required feature shapes and sharp characteristics are well pre-served during optimization. It validates that the proposed velocityapproximation and affine transformations can trigger a reasonablefeature based optimization without jeopardizing the effectivenessof gradient based search.

6.2. Feature design in structural topology optimization

One interesting application of the proposed method is struc-tural optimization for assembly. In Fig. 6, a static circular hole is ex-pected inside a cantilever beam, while the structural compliance isto beminimized under a volume constraint. Previous implementa-tions, like in [31–33], simply treat the circular region as non-designdomain while optimizing the shape and topology of the rest re-gions. This is straightforward if such a determinate requirement isknown as a priori. However, it may become difficult for traditionalSIMP or level set method to dynamicallymodify the configurationsof feature (e.g. position and radius of the circle), unless its geome-try is explicitly parameterized as in [16].

1530 M. Zhou, M.Y. Wang / Computer-Aided Design 45 (2013) 1524–1537

Table 1Algorithm: Particle level set method.

1. Structure initialization and particles seeding2. Level set update: first order semi-Lagrangian advection3. Particle update: second order Runge–Kutta advection4. Correction of the level set function according to the value of particles5. Level set reinitialization6. Correction of the level set function according to the value of particles7. Particle reinitialization (optional)8. Go to 2

Fig. 6. A cantilever beam with a circular hole.

To further demonstrate the generality and applicability of theproposed approach in structural shape and topology optimization,this benchmark example is re-studied and compared with a dy-namic feature design. As shown in Fig. 6, the structure is subject toa fixed boundary condition at the left edge and a downward load atthe right bottom corner. The geometric dimensions and force aredefined as follows: H = 120, L = 180, R = 30,W = 60,D =

60, F = 10. The Young’s modulus of weak and solid material areset as 1e− 3 and 1 respectively. The mathematical formulations ofthis linear elastic problem are written as:

MinimizeΦ

J(Φ) =

nk=1

D

12Ekε(u) : ε(u)χk(Φ)dΩ, (37)

Subject to G(Φ) =

nk=1

Dχk(Φ)dΩ ≤ V0, (38)

a(u, v, Φ) = l(v, Φ) for all v ∈ U . (39)

Table 2Comparison of different optimal designs.

Case Compliance Volume Radius of hole

Fixed hole 5780 8208 30Variable-size hole 5334 8208 24.6

For simplicity, both the boundary tracking force and body force areneglected. Thus, a sensitivity analysis gives the following designvelocity for each φi for steepest descent search:

V in =

12Eksε(u) : ε(u) − λ

me=1,e=i

H(φe),

where λi+1 = max0, λi + µ

Ω

dΩ − V0

. (40)

In this example, the initial structure shown in Fig. 7 is builtupon one feature model for circular hole and another freeformmodel. Figs. 8 and 9 record two different optimization processesrespectively. In Fig. 8, a fixed feature hole is realized by simplyassigning the feature model as non-design domain. In comparison,a scaling transformation is enabled in Fig. 9 to optimize the circularradius simultaneously with the structural shape and topology.Although the two optimal designs (Fig. 10) share a similar topologyto the benchmark results in [31–33], the one with a variable-sizehole has a lower compliance value (Table 2), thus a better structuralperformance. In fact, this example demonstrates that the proposedapproach generalizes and enhances the feature design capability oftraditional structural optimization techniques.

6.3. Generative feature design

Another promising capability of this method is to design trulyoptimal structures with generative features. In this paper, threemotion constraints (design rules), namely group, relation andprimitive constraints, are proposed to accommodate differenttypes of feature. As illustrated in Fig. 11, the constraints at differentlevels of a CSG model tree have different purposes to (i) preservehigh-level feature shape, (ii) keep the relation between low-levelentities and (iii) maintain geometry of basic entities, respectively.Their practical usage is demonstrated by a cantilever beam exam-ple as shown in Fig. 12, which initially contains two rectangularfeature holes inside. The structure is subject to a fixed Dirichletboundary condition at the left edge and a downward force loadat the middle of the right edge. Parameters are consistently set

Fig. 7. Initial configuration represented by two level set models.

Fig. 8. Structural optimization with a static hole: ((a)–(c)) intermediate designs; (d) optimal result.

M. Zhou, M.Y. Wang / Computer-Aided Design 45 (2013) 1524–1537 1531

Fig. 9. Structural optimization with a variable-size hole: ((a)–(c)) intermediate designs; (d) optimal result.

Fig. 10. Comparison of two optimal designs: (a) fixed circle, (b) optimized circle.

Fig. 11. Group, relation, and primitive constraints at different levels of a CSG model tree.

Fig. 12. A cantilever beam with two rectangular holes.

for all the following experiments as: L1 = 200,H1 = 100, L2 =

20,H2 = 10, L3 = 35, L4 = 155, F = 10, the upper volumeV0 = 0.5Vdomain, where Vdomain denotes the volume of designdomain, and the Young’s modulus 1e − 3 and 1 for weak and solidmaterial respectively. The optimization problem is to minimizestructural compliance under a volume constraint.

The beam is firstly optimized using the standard level setmethod with one single model. As shown in Fig. 13, both of the

rectangular features have disappeared in the final design and nogeometric consistency can be explicitly tracked during the opti-mization process.

In comparison, two group constraints are employed to preservethe two rectangular holes. Accordingly, the model in Fig. 14 isconstructed with nine entities: eight feature models (four line en-tities for each hole) and one freeform model for the outer bound-ary. During optimization, the constrained velocity of rectangularholes is firstly obtained and used to update the lower-level lineentities. As illustrated in Fig. 15(a), the size and location of fea-ture holes are eventually optimized with all the sharp corners wellpreserved. Besides, the outer freeform boundary transforms ide-ally towards a maximal structural performance. Comparing to thesmooth convergence history in Fig. 13(b) of the previous infinitedimensional optimization, the wiggles shown in Fig. 15(b) are dueto the extra motion constraint over feature models, which causessmall-scale geometric modulation around to the optimum. How-ever, such phenomenon only occurs after a proper optimal designhas been found, and thus has little impact to overall optimization.

In the third example, two 90° angles intersected by the yellowedges in Fig. 16will be preserved under a relation constraint, whilethe left hole is subject to a group constraint. Similar to the previous

1532 M. Zhou, M.Y. Wang / Computer-Aided Design 45 (2013) 1524–1537

Fig. 13. Infinite dimensional design: (a) optimal configuration; (b) convergence curves.

Fig. 14. Constraints and model tree group constraint: to maintain the shape of rectangular holes (black color).

Fig. 15. Group constraint: (a) optimal configuration; (b) convergence curves.

M. Zhou, M.Y. Wang / Computer-Aided Design 45 (2013) 1524–1537 1533

Fig. 16. Constraints and model tree: (1) relation constraint: to preserve two 90° angles (yellow color); (2) group constraint: to maintain the rectangular shape (black color).(For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 17. Relation constraint: (a) optimal configuration; (b) convergence curves.

example, each feature hole is further decomposed into low-levelentities as in Fig. 16(b). But the one containing the blue edge isregarded as a freeform entity. In the final structural configurationshown in Fig. 17, the right rectangular hole eventually transformsinto a truly optimal shape together with the outer boundary andall the required features are properly satisfied.

Finally, two primitive constraints are exerted to the right holein Fig. 18 to keep the linearity of two edges (in red color). Theleft hole is still subject to a group constraint and the otherboundary (including blue edges and outer boundary) can be freelydesigned. Fig. 19 shows the optimal result and convergence curves.

During optimization, the two feature edges maintain their lineargeometry while the adjacent edges are freely optimized, althoughthey together form a rectangular primitive initially. Note that,numerical singularity of velocity approximation occurs in thisexample, because the two feature line entities initially have a zeronormal component nx = 0 along its boundary. Disabling horizontaltranslation at the beginning can avoid such problem.

To summarize, Table 3 records the structural compliance andvolume of each optimal design. A stricter and more conservativeconstraint (group constraint) essentially leads to a larger compli-ance value compared with the infinite dimensional optimization

1534 M. Zhou, M.Y. Wang / Computer-Aided Design 45 (2013) 1524–1537

Fig. 18. Constraints and model tree: (1) primitive constraint: to preserve the linear edges (red color); (2) group constraint: to maintain the rectangular shape (black color).(For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 19. Primitive constraint: (a) optimal configuration; (b) convergence curves.

(no constraint), which gives the most effective structural layoutwith the same amount of material.

This example clearly demonstrates that the CSGLS representa-tion and proposed motion constraints constitute an elegant so-lution to the problem of generative feature design in structuraloptimization. By imposing motion constraints to different levels ofa CSG tree, various feature geometry can be flexibly designed tomeet practical engineering requirements. Compared with the tra-ditional methods using fixed geometric primitives, this approachfocuses on preserving the necessary engineering significance andmeanwhile allows for a truly optimized shape to be designed in aneffective and reasonable way.

Table 3Comparison of different optimal designs.

Motion constraints Compliance Volume

Group constraint 8000 10000Relation constraint 7790 10000Primitive constraint 7359 10000No constraint 6811 10000

6.4. 3D structural design with features

The proposed approach is further justified by a 3D designexample. As shown in Figs. 20(a) and Fig. 21(a), an initial beam

M. Zhou, M.Y. Wang / Computer-Aided Design 45 (2013) 1524–1537 1535

Fig. 20. Feature based optimization of a 3D structure: (a) initial configuration withboundary condition; ((b)–(c)) the optimal design.

containing a cylindrical feature is to be optimized for maximumstiffness under a volume constraint. Two level set models are usedto model the initial beam: one for the cylinder hole and the otherfor the outer freeform boundary. In both examples, the followingparameters are set: L = 40,H = 20,W = 15, d = 8, an uppervolume constraint V0 = 0.4Vdomain, and Young’s modulus 1e − 3and 1 forweak and solidmaterial respectively. Besides, a structuralmesh of 32 × 16 × 12 elements is utilized for structural analysis.

In Fig. 20(a), a vertical force load F = 20 is firstly appliedat the middle of front face while the back side is subject to afixed boundary condition. Fig. 20(b) and (c) show the optimizeddesign from different views. The location and size of the cylindricalhole are properly determined through affine transformationsand the outer freeform boundary evolves simultaneously for amaximum structural performance. To demonstrate the robustnessof proposed approach, another horizontal force load F2 = F1 = 10is added to the beam in Fig. 21(a) and the optimal designs arepictured in Fig. 21(b) and (c). In this example, not only the size andlocation of the cylindrical feature are optimized but also the mosteffective orientation is found accordingly.

7. Conclusions

This paper presents a general approach to designing engineer-ing features in structural shape and topology optimization. It com-bines CSG modeling in representing features and level set methodin optimizing freeform boundary. Several well-known techniques

Fig. 21. Another 3D optimal design under a different force load: (a) initial config-uration with boundary condition; ((b)–(c)) the optimal design .

are employed, such as XFEM, particle level set method and semi-Lagrangemethod, and constitutes a viable solution to the proposedproblem.

The CSGLS description distinguishes feature entities from non-feature ones in a CSG model tree, and allows engineering sig-nificance to be precisely interpreted by the geometry or relationbetween feature models. For practical applications, the rule ofthumb to build an effective CSGLSmodel is to use aminimumnum-ber of feature entities (such as line or plane primitives) to satisfyengineering requirements, while ensuring non-critical regions canbe freely optimized as much as possible.

The proposed constrained motion provides several basic butuseful design functions to preserve and design features (includingsharp characteristic) during optimization. It provides a practicalmeans tomanipulate features through an implicit design approach,especially for level set based structural optimization. However, thecurrent affine transformation is by no means a complete solution.It would be desirable to develop alternative constrained motionsto address more complex design requirements.

Remarkably, this method unifies feature design and structuralshape and topology optimization under the level set framework. Itshows an appealing design flexibility to create truly optimal struc-tural configurations with features considered. The whole schemecan be easily implemented into any level set based structural op-timization framework without model parameterization. Based on

1536 M. Zhou, M.Y. Wang / Computer-Aided Design 45 (2013) 1524–1537

Table A.1

Constrained motion Velocity matrix X = A−1B

X-translation X = [Vx] ; Vy = 0, θ = 0, α = 0 A = [O] B =

Γi

−GnxdΓ

Y -translation X =Vy

; Vx = 0, θ = 0, α = 0 A = [M] B =

Γi

−GnydΓ

Scaling X = [α] ; Vx = 0, Vy = 0, θ = 0 A = [L] B =

Γi

−G[nx(px − cix) + ny(py − ciy)]dΓ

Rotation X =θ; Vx = 0, Vy = 0, α = 0 A = [U] B =

Γi

−G[ny(px − cix) − nx(py − ciy)]dΓ

XY -translation X =Vx Vy

T; θ = 0, α = 0 A =

O PP M

B =

Γi

−GnxdΓΓi

−GnydΓ

X-translation & scaling X =Vx α

T; Vy = 0, θ = 0 A =

O RR L

B =

Γi

−GnxdΓΓi

−G[nx(px − cix) + ny(py − ciy)]dΓ

X-translation & rotation X =Vx θ

T; Vy = 0, α = 0 A =

O QQ N

B =

Γi

−GnxdΓΓi

−G[ny(px − cix) − nx(py − ciy)]dΓ

Y -translation & scaling X =Vy α

T; Vx = 0, θ = 0 A =

M TT L

B =

Γi

−GnydΓΓi

−G[nx(px − cix) + ny(py − ciy)]dΓ

Y -translation & rotation X =Vy θ

T; Vx = 0, α = 0 A =

M SS N

B =

Γi

−GnydΓΓi

−G[ny(px − cix) − nx(py − ciy)]dΓ

XY -translation & scaling X =Vx Vy α

T; θ = 0 A =

O P RP M TR T L

B =

Γi

−GnxdΓΓi

−GnydΓΓi

−G[nx(px − cix) + ny(py − ciy)]dΓ

XY -translation & rotation X =Vx Vy θ

T; α = 0 A =

O P QP M SQ S N

B =

Γi

−GnxdΓΓi

−GnydΓΓi

−G[ny(px − cix) − nx(py − ciy)]dΓ

X-translation & scaling & rotation X =Vx θ α

T; Vy = 0 A =

O Q RQ N UR U L

B =

Γi

−GnxdΓΓi

−G[ny(px − cix) − nx(py − ciy)]dΓΓi

−G[nx(px − cix) + ny(py − ciy)]dΓ

Y -translation & scaling & rotation X =Vy θ α

T; Vx = 0 A =

M S TS N UT U L

B =

Γi

−GnydΓΓi

−G[ny(px − cix) − nx(py − ciy)]dΓΓi

−G[nx(px − cix) + ny(py − ciy)]dΓ

the current implementation, future development could take 3D ge-ometric editing into account for an integrated CAD/CAE solution. Itwould also be interesting to apply the proposed methodology todesign sharp features for stress related design problems.

Appendix

For the constrained affine transformation, combination oftranslation, rotation and scaling can produce different motionpatterns for practical design requirements. The following table

summarizes all the possible constrained motions and the corre-sponding velocity matrix of 2D problem (see Table A.1).

O =

Γi

nxnxdΓ

Q =

Γi

nx[ny(px − cix) − nx(py − ciy)]dΓ

R =

Γi

nx[nx(px − cix) + ny(py − ciy)]dΓ

M. Zhou, M.Y. Wang / Computer-Aided Design 45 (2013) 1524–1537 1537

P =

Γi

nxnydΓ

S =

Γi

ny[ny(px − cix) − nx(py − ciy)]dΓ

T =

Γi

ny[nx(px − cix) + ny(py − ciy)]dΓ

N =

Γi

[ny(px − cix) − nx(py − ciy)]

× [ny(px − cix) − nx(py − ciy)]dΓ

L =

Γi

[nx(px − cix) + ny(py − ciy)]

× [nx(px − cix) + ny(py − ciy)]dΓ

M =

Γi

nynydΓ

U =

Γi

[ny(px − cix) − nx(py − ciy)]

× [nx(px − cix) + ny(py − ciy)]dΓ .

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