Struct Multidisc Optim (2014) 49:761–769DOI 10.1007/s00158-013-1004-8

RESEARCH PAPER

Stress isolation through topology optimization

Li Li · Michael Yu Wang

Received: 1 June 2012 / Revised: 1 July 2013 / Accepted: 21 September 2013 / Published online: 8 November 2013© Springer-Verlag Berlin Heidelberg 2013

Abstract This paper introduces a problem of stress iso-lation in structural design and presents an approach tothe problem through topology optimization. We model thestress isolation problem as a topology optimization prob-lem with multiple stress constraints in different regions.The shape equilibrium constraint approach is employed toeffectively control the local stress constraints. The levelset based structural optimization is implemented with theextended finite element method (X-FEM) for providing anadequately accurate stress analysis. Numerical examples ofstress isolation design in two dimensions are investigatedas a benchmark test of the proposed method. The results,from the force transmittance point of view, suggest thatthe guard “grooves” obtained can change the force path tosuccessfully realize the stress isolation in the structure.

Keywords Stress isolation · Topology optimization ·Shape equilibrium constraint · Level set method · Extendedfinite element method (X-FEM)

1 Introduction

Stress isolation is an important practical problem in struc-tural design. Generally speaking, in a stress-loaded structure

L. Li (�)School of Mechatronics Engineering and Automation, ShanghaiUniversity, Shanghai 200072, Chinae-mail: lili.shu@shu.edu.cn

M. Y. WangDepartment of Mechanical and Automation Engineering,The Chinese University of Hong Kong,Hong Kong, Chinae-mail: yuwang@mae.cuhk.edu.hk

(called base-structure as shown in Fig. 1), there is a spec-ified region (or area) of particular interest (called sub-structure as shown in Fig. 1). Normally, stresses wouldpropagate between the base structure and the sub-structure.The concept of stress isolation is to attenuate propagationof stresses by means to relieve and/or redirect the stressesbetween the sub-structure and the base structure.

There are typically two scenarios of stress isolation.In one case, the sub-structure represents the source ofunwanted stresses. The stress isolation is to design somestructure elements on the base structure to significantlyreduce undesirable propagation of the stresses into thebase structure. A good example of this scenario is thewelding stress isolation structure for hard-disk head sus-pension assemblies (Gorard et al. 1998), where the sub-structure is a welding area and the stresses induced bywelding process must be substantially relieved and con-tained from the surrounding areas of the structure to preventundesirable thermo-stress induced structural distortions anddeformations.

In the other case, the base structure is the source ofunwanted stresses and the stress isolation is to isolate thesub-structure from undesirable propagation of the stressesinto it. This is a typical situation in sensor designs. Gen-erally, sensors are sensitive to environmental disturbancessuch as force transmittance. For instance, for a piezoelectricacceleration sensor, any strain on the surface of specimenwill transmit to the piezoelectric unit through the sensorfoundation as a noise. The strain may change the sensingcapability or even destroy the senor. Especially, in microme-chanical sensors with a large sensitivity to mechanicalstress, the output signal can be substantially disturbed dueto the unwanted stress (Tsuchiya and Funabashi 2004, Tsaiet al. 2007, Hsieh et al. 2011). Since the sensor unit is fixedon the base structure, the unwanted stresses induced by the

762 L. Li, M. Y. Wang

Fig. 1 A schematic of stress isolation structure

environmental disturbances must be contained and/or redi-rected from the sub-structure where the sensor is bounded,so that the stress level is below an allowable level.

There are two types of approaches to the stress isolationproblem. The first approach is to consider using a differentmaterial in the sub-structure where the sensor is bounded(Rogers 1992; Rogers and Kowal 1995). The other approachis from the structure design point of view, by designingstructural elements on the base structure, for instance, theguard-ring structure, to reduce the stresses (Okada 1992;Takao et al. 1995). In this study, we focus on the method ofstructure design to achieve stress isolation. In the literature,specific stress isolation structures are designed for specificapplications, such as the welding stress isolation structuresfor hard-disk head suspension assemblies (Gorard et al.1998) and stress isolation guard-rings and pedestals for sil-icon micro-machined accelerometers (Ronald and Graeme2001; Hsieh et al. 2011) and pressure sensors (Jamie 2010).

Our goal is to develop a general method to designstress isolation structures through the topology optimizationmethod. In principle, we model a general stress isolationproblem into a structural topology optimization problemwith multiple stress constraints in different regions of thestructure. Through topology optimization, we achieve struc-tural elements on the base structure with required func-tions of stress isolation. The shape equilibrium constraintapproach and X-FEM schemes are employed to solve thestress isolation problem.

The structure of this paper is as follows. We first describethe problem formulation of the stress constrained topol-ogy optimization in Section 2 as a model of stress isolationdesign. Then the shape equilibrium constraint method isapplied to the topology optimization problem to achievestress control in Section 3. The shape sensitivity analy-sis of the optimization problem with the adjoint methodis described in Section 4. Key details of the numerical

implementation are provided in Section 5. Two numericalexamples are presented in Section 6 to illustrate the pro-posed stress isolation approach. Finally, conclusions aremade in Section 7.

2 Problem formulation for stress isolation constrainedtopology optimization

We use a linear elastic structure to describe the problem ofstress constrained topology optimization. In this study, weemploy the level set model to represent the continuum struc-ture (Wang et al. 2003; Allaire et al. 2004). The structuralboundary is represented as the zero level set of an implicitfunction φ(x), i.e.,

φ(x) = 0, f or x ∈ Γ

φ(x) < 0, f or x ∈ Ω(1)

On the structural boundary Γ , the unit outward normal isdefined as n = ∇φ/|∇φ| and the curvature as κ = ∇ · n.

Using the level set method, the structural topology opti-mization with an objective function and stress isolationconstraints can be specified as follows:

⎧⎪⎪⎨

⎪⎪⎩

min J = ∫

ΩF(u)dV

s.t. a(u, v) = l(v), ∀v ∈ U, u|∂Ω = u0

G1 = σrs − Ss ≤ 0G2 = σrb − Sb ≤ 0

(2)

where J is the objective function with a specific physi-cal or geometric function F, such as unit weight or strainenergy. The elastic equilibrium equation is written in itsweak variation form, where u denotes the displacement fieldin the space U of kinematically admissible displacementfields, v presents the test function, and u0 is the pre-scribed displacement on Dirichlet boundary ΓD . The bilin-ear terms of the state equation of the linear elastic system aredescribed as

a(u, v) =∫

Ω

Eε(u)ε(v)dV (3)

l(v) =∫

Ω

f vdV +∫

ΓN

tvdS (4)

where E is the elasticity tensor, ε represents the strain ten-sor, f is the body force, t is the boundary tractions appliedon Neumann boundary ΓN .

There are two different constraints on the parts of thestructure as shown in Fig. 1. σrs denotes the stress level at apoint in the sub-structure region, and G1 defines the stress

Stress isolation through topology optimization 763

Fig. 2 Model of the square plane with a circular sub-structure forstress isolation

constraint in the sub-structure such that the stress level is nolarger than an allowable level of Ss . For the rest of the basestructure, the constraint G2 requires that its stress level σrbis no larger than its allowable level of Sb. These allowablelevels of stress in the sub-structure and the base-structurerespectively are usually defined with an appropriate materialstrength and a design factor of safety. Without losing gener-ality, we assume in the paper that Ss is far below Sb. In otherwords, we consider the scenario that a sensor is mountedon the sub-structure region and the stress isolation for thesensor is achieved with the constraint G1, while a safe basestructure is also maintained with constraint satisfaction ofG2. We use the Von Mises stress as the stress measure.

3 Shape equilibrium constraint approach

To apply the stress constraints, the shape equilibrium con-straint approach is used. The reader can find a detailedexplanation of the shape equilibrium constraint approach

using level set method in (Wang and Li 2013). In the gen-eral approach, a stress constraint can be considered in anyregion of the structure under optimization. We may considera domain Ω as an union of two distinct sub-domains sepa-rated by an interface. The first sub-domain denoted by Ω+is considered to be an active zone, in which a stress con-straint is violated, meaning the stress level σr within Ω+ isover the allowable value σa , i.e.,

σr (x) > σa, ∀x ∈ Ω+ (5)

The second sub-domain denoted by Ω− consists of allother regions of Ω where the stress constraint is satisfied.At the interface ΓS of the two sub-domains,

σr (x) = σa, ∀x ∈ ΓS (6)

For a topology optimization problem with the stressconstraint, the goal is to find an optimal shape Ω so asthe objective function is minimized and the active stressconstraint sub-domain Ω+ reduces to of zero volume atthe same time. The optimization process is carried out byevolving the boundary Γ of the design domain Ω with atransformation velocity in pseudo time τ ,

V (x) = dx/dτ = Vn n (7)

Thus, the active stress constraint problem changes to ashape equilibrium problem. We construct a shape equilib-rium constraint function, giving rise to an exact single globalconstraint as

G(x) =∫

Ω+(σr (x)− σa)dV ≤ 0 (8)

For the shape equilibrium constraint function G(σ), itsshape derivative is given with respect to a shape deforma-tion on Ω+ with respect to pseudo time τ as defined in (7),

Fig. 3 Example 1: a two-waystretch plane

764 L. Li, M. Y. Wang

Fig. 4 The optimization results and Von Mises stress distributions at different stages of the first case of the two-way stretch plane

as a consequence of the transformation velocity Vn on thefree-change boundary Γ . The shape derivative of G(σ) isfound to be (Wang and Li 2013)

G′ ={ ∫

Ω+ σ ′rdV + ∫

∂Ω+ (σr − σa) VndS (9)

4 Shape sensitivity analysis with adjoint equation

Now, the original topology optimization problem (2) isconverted to an equivalent form in terms of the shapeequilibrium constraints

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

min J = ∫

ΩF(u)dV

s.t. a(u, v) = l(v), ∀v ∈ U, u|∂Ω = u0

G1 = ∫

Ω+s(σrs − Ss)dV ≤ 0

G2 = ∫

Ω+b(σrb − Sb)dV ≤ 0

(10)

where Ω+s is the active zone in the sub-structure, and Ω+

b isthe active zone in the base-structure.

We can derive the shape sensitivity for the stress isolationoptimization problem, using material derivative, Lagrangemultiplier, and the adjoint method. The total Lagrangian isdefined as

L = J + a(u, v)− l(w) +Λ1

(∫

Ω+s

(σrs − Ss

)dV

)

+Λ2

(∫

Ω+b

(σrb − Sb

)dV

) (11)

where w is a Lagrange multiplier for the state equation oflinear elasticity system (also known as the adjoint variable),Λ1 > 0 and Λ2 > 0 are the Lagrange multipliers for theglobal stress constraint function G1(σ ) and G2(σ ) respec-tively. Thus, the shape derivative of the total Lagrangian isgiven by

L′ = J ′ + a′(u, v)− l′(w)+Λ1G′1(σ )+Λ2G

′2(σ ) (12)

Stress isolation through topology optimization 765

Fig. 5 The iteration history ofstain energy and stressconstraints of the first case ofthe two-way stretch plane

where

J ′ =∫

Ω

F ′(u)dV +∫

Γ

F (u)VndS (13)

a′(u, w) = ∫ΩEε(u′)ε(w)dV + ∫

ΩEε(u)ε(w′)dV+ ∫

Γ Eε(u)ε(w)VndS

(14)

l′(w) = ∫

Ωfw′dV + ∫

Γ fwVndS + ∫

ΓNtw′dS

+ ∫

ΓN

(∇(tw)T n+ κtw)VndS

(15)

G′1 =

∫

Ω+s

σ ′rsdV (16)

G′2 =

∫

Ω+b

σ ′rbdV +

∫

∂Ω+b

(σrb − Sb

)VndS (17)

Collecting all the terms that contain w’ in the derivativeof the Lagrangian and letting the sum of the terms to bezero, we recover the weak form of the state equation. Col-lecting all the terms that contain u’ in the derivative of theLagrangian and letting the sum of the terms to be zero, weobtain the adjoint equation for the adjoint variable w,∫

ΩEε(u′) ε(w)dV = − ∫

ΩF ′(u)dV

−Λ1

(∫

Ω+sσ ′rsdV

)−Λ2

(∫

Ω+bσ ′rbdV

) (18)

Finally, collecting all the remaining terms that containVn assuming there exists no body force, i.e., t = 0, andfurther assuming that ΓN is fixed and ΓH is the only partthat subjects to optimization, we obtain the shape derivativeof the total Lagrangian as

L′ = ∫

ΓF (u)VndS + ∫

ΓHEε(u)ε(w)VndS

+Λ2

(∫

ΓH+(σrb − Sb)VndS) (19)

According to (19), we readily obtain a descent direc-tion for the optimization solution. In the simplest form, itamounts to choosing the steepest descent direction (Wanget al. 2003; Allaire et al. 2004) with

Vn = −F(u)− Eε(u)ε(w) −Λ2

⎧⎨

⎩

(σrb − Sb),

∀x ∈ ΓH+0, ∀x ∈ ΓH−

(20)

5 Numerical implementation

The numerical steps for solving the stress isolation problemare as follows:

Step 1: Define the stress isolation problem with stress con-straints in two regions using shape equilibriumconstraint approach with level set method.

Fig. 6 Force lines of the initialdesign and the final design ofthe first case of the two-waystretch plane

766 L. Li, M. Y. Wang

Fig. 7 The final results of thesecond case of the two-waystretch plane

Step 2: Perform stress analysis using X-FEM (Li et al.2012; Wei et al. 2010; Duysinx et al. 2006;Miegroet and Duysinx 2007).

Step 3: Construct the augmented Lagrange function usingthe augmented Lagrange multipliers method.

Step 4: Evolve the level set function according to theupwind scheme for Hamilton-Jacobi equation.

If the constraints are not satisfied or the process is notconvergent, the optimization process will return to Step2. Key aspects of the numerical implementation are fullydescribed in (Wang and Li 2013) and we shall not repeatthem here.

6 Numerical examples

In this section, the proposed stress isolation approach isapplied to several examples in two dimensions, althoughthere exist no conceptual difficulties to extend it into threedimensions. The main purpose is to demonstrate the effec-tiveness of the present approach.

Fig. 8 Example 2: A four-way stretch plane

In the numerical examples, the representative stress is theVon Mises stress, which in planar case is given as

σr =√

σ 2xx + σ 2

yy − σxxσyy + 3σ 2xy (21)

σ ′r = ∂σr

∂σ· ∂σ∂u

= 12σr

[2σxx − σyy, 2σyy − σxx, 6σxy ] · σ(u′)(22)

We consider two examples of a square plate with a spec-ified sub-structure shown in Fig. 2. The sub-structure is acircular region, and the stress in this region is required lowso as to not exceed the allowable stress constraint. Out-side this region, the stress constraint is set at a prescribedvalue, such as practical yield strength. The dimensions ofthe square plate and the circular sub-structure are given inFig. 2 with a unit thickness. In these plane stress exam-ples, it is assumed that the material has a Young’s modulusE = 1N/m2 and Poisson’s ratio ν = 0.3. For all examples,the background mesh has 6400 four-node bilinear quadrilat-eral elements for X-FEM analysis, and a 81× 81 rectilineargrid of the square is used for level set computations. Thesecond order Gauss quadrature is used for solving the stateand the adjoint equations.

6.1 Two-way stretch plane

The first example is of a uniform load force of 1N/m intension in the horizontal direction as shown in Fig. 3a. Inthis case, the plate has a uniform Von Mises stress equal to1. The force lines (Kokcharov and Burov 2001; Rathkjen1997) in the plane are shown in Fig. 3b. It is noted that theforce transmits in a straight line and the stress is uniformin the whole structure. Now, we consider the inner circularregion as the sub-structure and we require the stress insideit to be lower than 1 but allows the stress in other area of thebase-structure to be relatively larger. It is assume that theallowable stress Ss in the sub-structure is equal to 0.85 whilethe stress constraint Sb in the base-structure is set to be 2.5.

Stress isolation through topology optimization 767

Fig. 9 The initial design andfinal design with its Von Misesstress distribution of the firstcase of the four-way stretchplane

In the first case, we consider the mean compliance as theobjective function defined as

J = 1

2

∫

Ω

Eε(u)ε(u)dV (23)

The initial design of the base structure is shown inFig. 4a with multiple holes inserted as potential stress reliefelements and the corresponding Von Mises stress distribu-tion is displayed in Fig. 4b. The designs and contour maps ofthe Von Mises stress are illustrated for different stages of theoptimization process as shown in Fig. 4. The convergencehistory is shown in Fig. 5. At the convergence, the maximumVon Mises stress in the sub-structure is 0.82, smaller thanthe allowable value of 0.85 while the maximum Von Misesstress in the base-structure is 2.49. As expected, the stressisolation is successfully carried out. The force lines in theinitial and final structures are shown in Fig. 6. From theforce lines plots, the force transmittance is clearly shown tobe changed by the “grooves” that reduce the stress in thesub-structure. The mean compliance is also reduced fromthe original value of 144.71 to the final value of 116.61.

In the second case, the objective function is the totalmaterial volume defined as

J =∫

Ω

dV (24)

With the same initial design as in Fig. 4, the finalstructure is shown in Fig. 7a and the corresponding Von

Mises stress distribution is displayed in Fig. 7b. In theoptimized structure, the maximum Von Mises stress in thesub-structure is equal to 0.82 while the maximum Von Misesstress in the base-structure is equal to 2.42. The final design,its stress distribution, and its force lines are illustrated inFigs. 7a–c. The volume of the structure is reduced by 7.4 %from the original design.

6.2 Four-way stretch plane

In the second example, a four-way tension force of 1N/m

is applied in both horizontal and vertical directions simulta-neously as shown in Fig. 8. Similarly to the first example, itis assumed that the allowable stress Ss in the sub-structureis equal to 0.85 while the stress constraint Sb for thebase-structure that is set to be 2.5.

In the first case of this example, the mean compliance isset as the objective function as defined in (23). The initialdesign is shown in Fig. 9a, and the final structure is obtainedafter 300 iterations as shown in Fig. 9b with the correspond-ing Von Mises stress distribution displayed in Fig. 9c. Theconvergence history of this case is illustrated in Fig. 10.At the convergence, the maximum Von Mises stress in thesub-structure is equal to 0.85, satisfying the stress constraintin this region, and the maximum Von Mises stress in thebase-structure is equal to 2.15, which is smaller than theallowable value of 2.5. The goal of the stress isolation is

Fig. 10 The iteration history ofstain energy and stressconstraints of the first case ofthe four-way stretch plane

768 L. Li, M. Y. Wang

Fig. 11 The final design and itsVon Mises stress distribution ofthe second case of the four-waystretch plane

obtained, as shown in Fig. 9c. Finally, the strain energy isreduced from the original value of 197.08 to the final valueof 157.83.

In the second case of this four-way stretch plane exam-ple, we define the total material volume as the objectivefunction as described in (24). The initial design is same asthe first case shown in Fig. 9a and the final results are dis-played in Fig. 11. In the final design, the maximum VonMises stress in the sub-structure is equal to 0.83, satisfyingthe stress constraint in this region, while the maximum VonMises stress in the base-structure is equal to 2.48. Similarto the first case, the stress isolation is achieved by satisfyingall the stress constraints, as shown in Fig. 11b. The differ-ence is that the stress isolation slots in this case are widerand more, since we aim for a minimal weight base structurein the second case.

7 Conclusion

This paper introduces the problem of stress isolation and anapproach to it using the topology optimization method. Theproblem is modeled with multiple stress constraints in dif-ferent structure regions. The constraints can be controlledeffectively in the shape equilibrium constraint approach.In this optimization process, the level set model providesa geometric representation of the general shape of a con-tinuum solid with the flexibility of topological changes. Adirect analysis of the sensitivities of the shape equilibriumconstraints in a form of the adjoint method is also applied.

In the paper, the proposed method is demonstrated tobe able to deal with the stress isolation problem effec-tively. The stress level in a sub-region for stress isolationis reduced to a low level according to its stress constraint.The design process produces structural elements, similar toslots and grooves that are intuitively known, on the basestructure. These structural elements change or redirect theforce path to successfully realize stress isolation in the

structure. Potential applications of the proposed approachinclude MEMS sensor designs and thermo-stress isolationin welded structures.

Acknowledgments The financial support from the Research GrantsCouncil of Hong Kong S.A.R. (project No. CUHK417309) is grate-fully acknowledged.

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