1 Introduction

The field of structural optimization has been recently thoroughly studied by the scientific community with many promising practical industrial applications. Among the types of structural optimization problems, the present work deals with shape and topology optimization.

Considering shape optimization problems, the subject of optimization is the boundary of the structure itself. The boundary is varied according to the sensitivity analysis of the structure domain, however, the initial topology of the domain does not change. The numerical procedure related to shape optimization has been presented by Zienkiewicz and Campbell (1973), additional comprehensive reviews and studies can be found, among others, in the works of Haftka and Grandhi (1986) and Sokolowski and Zolesio (1992).

Topology optimization could be interpreted as an upgrade of shape optimization problems, since it allows topological changes, and aims to determine the optimal shape of the structure boundary, as also the optimal number of voids and connectivity of the domain. Comprehensive surveys regarding topology optimization approaches can be found in the literature by Rozvany (2001), Sigmund and Maute (2013) and Deaton and Grandhi (2013).

The most commonly used topology optimization methods are based on a density approach. One of the density approach methods is the solid isotropic material with penalization (SIMP) method. The idea behind the SIMP method is to use a fictitious isotropic material whose elastic modulus tensor is assumed to be a function of penalized material density. The SIMP method has been widely accepted and applied to many industrial problems, additional references and examples can be found in the monograph by Bendsøe and Sigmund (2003). Regarding the use of the SIMP method for structural optimization problems subjected to stress constraints, a comprehensive study has been presented by Le et al. (2010).

The other major topology optimization procedure is based on the advection of implicit interfaces and has been developed in recent years by the authors Sethian and Wiegmann (2000), Osher and Santosa (2001), Wang et al. (2003) and Allaire et al. (2004). The framework is based on the level-set method (LSM) which has been originally proposed by Osher and Sethian (1988) for numerically tracking fronts and free boundaries. A survey of the level-set method for different applications has been given by Burger and Osher (2005). The level-set method in structural topology optimization defines the interfaces between material phases implicitly by means of the level-set scalar function contours. This definition allows us to accurately capture the structure boundaries and avoids intermediate densities, which are common for the density-based methods. A comprehensive review of the level-set method for structural topology optimization problems has been made by Dijk et al. (2013).

Studies considering stress-related objective functions in the combination with the level-set method, which is also the subject of the present paper, have been published in more recent years. In the seminal paper by Allaire and Jouve (2008) the shape and topological derivatives of the stress-related objective functions have been considered, where the ersatz material approach has been used for the extension of the elasticity equations to the whole working domain. The topological derivatives of the penalty functional mimicking point-wise constraint of the von Mises stress field have been presented by Amstutz and Novotny (2010). The authors Guo et al. (2011) have considered different formulations of stress-related topology optimization problems, where the numerical solutions have been obtained by the extended finite element method (XFEM). A novel global stress measure of the optimization problem subjected to the volume constraint has been proposed by Xia et al. (2012). In the paper by Wang and Li (2012) the approach of shape equilibrium constraint strategy with the level-set/XFEM framework has been presented. A solution to the problem of multiple stress constraints in different regions has been given by Li and Wang (2014). The authors Zhang et al. (2013) have developed effective numerical techniques for designing stiff structures with less stress concentrations. Stress-related topology optimization of continuum structures involving multi-phase heterogeneous materials has been considered by Guo et al. (2014). Mass minimization in the combination with local stress constraints has been presented by Emmendoerfer and Fancello (2014).

The XFEM method, in contrast to ersatz material approach, represents the geometry of the structure with clear boundaries. Comprehensive studies of the XFEM method in the combination with structural optimization via the level-set method can be found in several papers. The authors Van Miegroet and Duysinx (2007) have used the XFEM method in the combination with the level-set method for the minimization of stress concentration in a two-dimensional fillet loaded in tension. In the paper by Wei et al. (2010) the FEM and XFEM have been compared for the classical compliance maximization problems. Computational accuracy and efficiency of the XFEM method have been considered by Li et al. (2012). The authors Makhija and Maute (2013) have studied numerical instabilities arising from the use of the XFEM method in structural topology optimization via the level-set method.

In the present paper, the main focus is on the shape sensitivity analysis and optimization efficiency of the global stress-deviation measure used as a novel stress-related objective function in minimum stress design via the level-set method. Considering the commonly used global stress measure (Allaire and Jouve 2008; Xia et al. 2012), even moderate values of the penalty parameter cause significant numerical and convergence problems (as noted by many authors, see for instance Xia et al. 2012), which mean that large penalty parameter values in the objective function are not feasible. Therefore, the main idea of the presented work is that the use of stress-deviation in the objective function would increase an average stress, which would indirectly help to push the stress-deviation based solution closer to the actual minimum stress problem solution at smaller values of the penalty parameter in the stress-deviation based objective function. The global stress-deviation measure is introduced through the definition and shape derivative of a general form of the global stress-related objective function. The partial differential equations related to the optimization problem are solved by the finite element based methods that use the same numerical framework. Finally, the optimization results for the global stress-deviation objective function are compared to the results for the commonly used global stress measure.

2 Generalized stress-related optimization problem

The optimization problem considered in this paper is concerned with determining the shape \( {\Omega } \subset \mathbb {R}^{d} \; (d = 2,3) \) which gives a minimum value to the global stress-related objective function. A structure of interest, shape Ω, is an open bounded set with a sufficiently smooth boundary ∂Ω occupied by an isotropic linear elastic material. A boundary ∂Ω is composed of three disjoint parts

$$ \partial {\Omega} = {\Gamma} = {\Gamma}_{D} \cup {\Gamma}_{N} \cup {\Gamma}_{0} , $$
(1)

where Dirichlet boundary conditions are imposed on Γ D , Neumann boundary conditions on Γ N and homogeneous Neumann boundary conditions (traction free) on Γ0. Boundaries Γ D and Γ N are fixed, while Γ0 is subjected to optimization and is allowed to move freely.

A general form of the global stress-related objective function J is written as

$$ J \left( {\Omega} \right) = {\int}_{\Omega}{ j\left( x, \sigma, J_{1}, J_{2}, \ldots, J_{m} \right) }{\mathrm{d}x}, $$
(2)

where j is a smooth integrand dependent on the domain Ω, σ = σ(Ω,x) is the stress tensor, and the subordinate functionals J i are expressed as

$$ J_{i} = J_{i} \left( {\Omega} \right) = {\int}_{\Omega}{j_{i}\left( x,\sigma \right) }{\mathrm{d}x}; \quad i=1, 2, \ldots, m , $$
(3)

where j i are smooth integrands also dependent on the domain Ω. Stress tensor is calculated by the Hooke’s law for linear elastic material as σ = A ε(u), where A represents the elastic modulus tensor, and ε is the infinitesimal strain tensor dependent on the displacement vector field u according to the relation \( \varepsilon (u) = \frac {1}{2} \big (\nabla u + {(\nabla u)}^{T} \big ) \). The displacement vector field u is a unique solution of the state equation and the corresponding boundary conditions, represented by the system

$$\begin{array}{@{}rcl@{}} - \text{div} \left( A \varepsilon(u) \right) & =& 0 \quad \text{in} \;\, {\Omega}, \\ u &=& 0 \quad \text{on} \;\, {\Gamma}_{D}, \\ \left( A \varepsilon(u)\right) n &=& g \quad\text{on} \;\, {\Gamma}_{N}, \\ \left( A \varepsilon(u)\right) n &=& 0 \quad \text{on} \;\, {\Gamma}_{0}, \end{array} $$
(4)

where g represents the boundary traction force, and n is the outward normal to the boundary ∂Ω.

The solution of the topology optimization problem is an admissible shape Ω contained in the working domain D. The set U a d of all admissible shapes is therefore defined as

$$ U_{ad} = \left\{ {\Omega} \subset D \; \text{such that} \; V({\Omega}) = V_{0} \right\}, $$
(5)

where the volume of the domain Ω is calculated as

$$ V({\Omega}) = {\int}_{\Omega}\mathrm{d}{x} , $$
(6)

and V 0<V(D) is a chosen value.

In the discussed minimum stress design problem the optimal solution may not exist due to insufficiently constrained topology (Allaire et al. 2004), which can lead to pathological behavior of the problem. In order to obtain a well-posed optimization problem, which admits at least one optimal solution, one possible method is to introduce the perimeter term in the original optimization problem as

$$ \underset{\Omega \in U_{ad}}{\inf} \; \left( J({\Omega}) + \ell P({\Omega}) \right), $$
(7)

where P is the perimeter of the shape, calculated as

$$ P({\Omega}) = {\int}_{\Gamma}\mathrm{d}{s}, $$
(8)

and ℓ>0 is the weighting parameter. The perimeter term is used as an effective way of regularizing the optimization problem, avoiding extreme boundary oscillations, and preventing the emergence of very small geometric features.

3 Shape derivative of stress-related objective function

In order to solve the minimization problem stated in the previous section, a gradient-based optimization algorithm is used, and therefore a shape sensitivity analysis of a global stress-related objective function given by the functional (2) should be applied.

In terms of a shape sensitivity analysis, so called shape derivative of a functional \( J:{\Omega } \rightarrow \mathbb {R} \) is defined. Starting from an initial shape Ω0, suppose that every admissible shape Ω can be obtained by applying a smooth vector field 𝜃, which represents the direction of a deformation of Ω0, as

$$ {\Omega} = (Id + \theta )({\Omega}_{0}), $$
(9)

where I d is the identity mapping. The shape derivative J ′(Ω)(𝜃) is then defined as the directional derivative of a functional J(Ω) in the direction of a vector field 𝜃, and is calculated as

$$ J^{\prime}({\Omega})(\theta) = \lim\limits_{\delta \to 0} \frac{ J((Id+\delta \theta)({\Omega}) )-J({\Omega})}{ \delta }. $$
(10)

The shape derivatives for the two types of functionals, where in the first case an integrand f Ω(x) is defined over the domain Ω, while in the second case an integrand f Γ(s) is defined on the boundary Γ, and integrands in both cases do not depend on the domain Ω, have the following forms

$$\begin{array}{@{}rcl@{}} J({\Omega}) &=& {\int}_{\Omega}{f^{\Omega}(x)}{\mathrm{d}x} \Rightarrow J^{\prime}({\Omega})(\theta) \\&=& {\int}_{\Gamma}{\theta \cdot n \; f^{\Omega}(x) }{\mathrm{d}s}, \\ J({\Omega}) &=& {\int}_{\Gamma}{f^{\Gamma}(s)}{\mathrm{d}s} \Rightarrow J^{\prime}({\Omega})(\theta) \\&=& {\int}_{\Gamma}{\theta \cdot n \left( \frac{ \partial f^{\Gamma}(s) }{\partial n}+ \kappa f^{\Gamma}(s) \right) }{\mathrm{d}s}, \end{array} $$
(11)

where κ is the mean curvature of Γ and is calculated as

$$ \kappa = \text{div} \, n. $$
(12)

Regarding the proof of the relations (11) see for instance the work of Sokolowski and Zolesio (1992).

For the calculation of the shape derivative of the functional J(Ω) written in the form (2) the method of Céa (Allaire et al. 2004; Allaire and Jouve 2008; Michailidis 2014) is followed. An integrand j in the discussed functional (2) depends through the solution u = u(Ω,x) of the state equation with the corresponding boundary conditions (4) also on the domain Ω. The main advantage of using the method of Céa is the avoidance of direct calculation of the shape derivative of u(Ω,x), meaning that the shape derivative can be calculated using the relations (11). In this method the state equation (4) is considered as a constraint of the optimization problem that the variable u needs to satisfy.

First, let us formulate the Lagrangian function \( \mathcal {L} \) as

$$\begin{array}{@{}rcl@{}} \mathcal{L}({\Omega},v,q)& = & {\int}_{\Omega}{j \left( x, A\varepsilon(v), J_{1},J_{2}, \ldots, J_{m} \right) }{\mathrm{d}x} \\ &&+ {\int}_{\Omega}{A\varepsilon(v):\varepsilon(q)}{\mathrm{d}x} - {\int}_{{\Gamma}_{N}}{g \cdot q}{\mathrm{d}s} , \end{array} $$
(13)

where v and q are the vector fields which do not depend on the domain Ω, and the condition v = q=0 holds on Γ D . The subordinate functionals J 1,J 2,…,J m in the expression (13) are given as

$$ J_{i}=J_{i} ({\Omega} ) = {\int}_{\Omega}{j_{i}\left( x, A\varepsilon(v) \right) }{\mathrm{d}x}; \quad i=1,2,\ldots, m. $$
(14)

The dependence indicators of the integrand j in the objective function, as also of the integrands j i in the subordinate functionals, are dropped in further equations to simplify the notations.

Now, let us discuss the conditions for a stationary point (Ω,u,p) of the Lagrangian \( \mathcal {L} \), defined by the relation (13). The partial derivative of \( \mathcal {L} \) with respect to q in the direction of virtual displacement field ϕ gives

$$\begin{array}{@{}rcl@{}} \left\langle \frac{\partial \mathcal{L}}{\partial q }({\Omega},u,p), \phi \right\rangle &= & {\int}_{\Omega}{A\varepsilon(u):\varepsilon(\phi)}{\mathrm{d}x} \\ && - {\int}_{{\Gamma}_{N}}{g \cdot \phi }{\mathrm{d}s} = 0. \end{array} $$
(15)

The expression (15) represents the weak form of the state equation (4). After the integration by parts, considering the condition u=0, ϕ=0 on Γ D , the following form is obtained

$$\begin{array}{@{}rcl@{}} \left\langle \frac{\partial \mathcal{L}}{\partial q }({\Omega},u,p), \phi \right\rangle &= & - {\int}_{\Omega}{ \text{div} (A\varepsilon(u) ) \cdot \phi}{\mathrm{d}x} \\ && + {\int}_{{\Gamma}_{N}}{ (A \varepsilon(u) n - g ) \cdot \phi }{\mathrm{d}s} \\ && + {\int}_{{\Gamma}_{0}}{A \varepsilon(u) n \cdot \phi}{\mathrm{d}s} = 0. \end{array} $$
(16)

Taking ϕ with compact support on Ω gives the state equation (4), and varying the trace of ϕ on Γ N ∪Γ0 gives the non-homogeneous and homogeneous Neumann boundary conditions.

Next, to find the adjoint state equation, the Lagrangian \( \mathcal {L} \) is differentiated with respect to v in the direction of ϕ, which at a stationary point gives

$$\begin{array}{@{}rcl@{}} \left\langle \frac{\partial \mathcal{L}}{\partial v }({\Omega},u,p), \phi \right\rangle &= & {\int}_{\Omega}{ A \frac{\mathrm{d}{j} }{\mathrm{d}{\sigma} } : \varepsilon(\phi) }{\mathrm{d}x} \\ && + {\int}_{\Omega}{ A\varepsilon(p) : \varepsilon(\phi) }{\mathrm{d}x} = 0, \end{array} $$
(17)

where the total derivative \( \frac {\mathrm {d}{j} }{\mathrm {d}{\sigma } } \) is equal to

$$ \frac{\mathrm{d}{j} }{\mathrm{d}{\sigma} } = \frac{\partial j}{\partial \sigma} + \sum\limits_{i=1}^{m} \frac{\partial j_{i} }{ \partial \sigma } {\int}_{\Omega}{ \frac{\partial j}{\partial J_{i}} }{\mathrm{d}x} $$
(18)

The expression (17) represents the weak form of the adjoint state equation. After the integration by parts, considering the condition p=0, ϕ=0 on Γ D , the following form is obtained

$$\begin{array}{@{}rcl@{}} \left\langle \frac{\partial \mathcal{L}}{\partial v }({\Omega},u,p), \phi \right\rangle &= & - {\int}_{\Omega}{ \text{div} \left( A \frac{\mathrm{d} j }{\mathrm{d} \sigma } + A \varepsilon(p) \right) \cdot \phi }{\mathrm{d}x} \\ & &+ {\int}_{{\Gamma}_{N} \cup {\Gamma}_{0}}{ \left( A \frac{\mathrm{d} j }{\mathrm{d} \sigma } + A \varepsilon (p) \right) n \cdot \phi }{\mathrm{d}s} \\ &= & \; 0. \end{array} $$
(19)

Again, taking ϕ with compact support on Ω gives the adjoint state equation, and varying the trace of ϕ on Γ N ∪Γ0 gives the non-homogeneous Neumann boundary conditions. Therefore, the adjoint state equation with the corresponding boundary conditions is obtained as

$$\begin{array}{@{}rcl@{}} - \text{div} (A \varepsilon (p) ) &=& \text{div} \left( A \frac{\mathrm{d} j }{\mathrm{d} \sigma } \right) \quad \text{in} \;\, {\Omega}, \\ p &=& 0 \qquad\quad\qquad\;\; \text{on} \;\, {\Gamma}_{D}, \\ (A \varepsilon (p) ) n &=& - \left( A \frac{\mathrm{d} j }{\mathrm{d} \sigma } \right) n \quad \text{on} \;\, {\Gamma}_{N} \cup {\Gamma}_{0}, \end{array} $$
(20)

where the adjoint state vector field p represents the unique solution of the adjoint state equation.

Finally, the shape derivative of the Lagrangian at a stationary point is equal to the shape derivative of the objective function, which reads as

$$ \frac{\partial \mathcal{L}}{\partial {\Omega} }({\Omega},u,p)(\theta) = J^{\prime}({\Omega})(\theta). $$
(21)

Since v and q do not depend on Ω, the shape derivative of the Lagrangian can be calculated using the relations (11), and the shape derivative of the objective function can therefore be written in the general form as

$$ J^{\prime}({\Omega})(\theta)={\int}_{\Gamma}{\theta \cdot n \; f }{\mathrm{d}s}, $$
(22)

where the integrand f in the discussed optimization problem is equal to

$$ f = j + \sum\limits_{i=1}^{m} j_{i} {\int}_{\Omega}{ \frac{\partial j}{\partial J_{i}} }{x} + A \varepsilon(u): \varepsilon(p). $$
(23)

In order to assure a decrease of the objective function J(Ω), the directional vector 𝜃 on the boundary Γ is chosen as

$$ \theta = - f \; n. $$
(24)

Therefore, the obtained shape derivative of the objective function has the following form

$$ J^{\prime}({\Omega})(\theta) = - {\int}_{\Gamma}{ f^{2} }\mathrm{d}{s}. $$
(25)

3.1 Global stress measure

A common type of the stress-related objective function is defined as

$$ J_{s,\alpha}({\Omega}) = {\int}_{\Omega}{ k(x) \; \sigma_{v}^{\alpha}(\sigma)}{\mathrm{d}x}, $$
(26)

where \( \alpha \in \mathbb {N} \) is the penalty parameter, k(x) represents the localization function which does not depend on the domain Ω, and σ v is the von Mises stress given as

$$ \sigma_{v} = \sqrt{\frac{3}{2} \sigma : \sigma - \frac{1}{2}(\sigma:I)^{2} }, $$
(27)

where I represents the identity tensor.

Analysis of the objective function J s,α (Ω) given by the expression (26) in terms of the general form (2) shows that the number of the subordinate functionals is equal to m=0. The integrand in the expression (26) is recognized as \( j = k(x) \; \sigma _{v}^{\alpha } \), and the total derivative of j is equal to

$$ \frac{\mathrm{d} j }{\mathrm{d} \sigma } = \alpha k(x) \sigma_{v}^{\alpha-1} \frac{\mathrm{d}\sigma_{v}}{\mathrm{d}\sigma}, $$
(28)

where the derivative of σ v is

$$ \frac{\mathrm{d}{\sigma_{v}} }{ \mathrm{d}{\sigma} } = \frac{1}{2 \sigma_{v}}(3 \sigma - (\sigma : I ) I ). $$
(29)

According to the expressions (22) and (23), the shape derivative of J s,α (Ω) results in

$$ J_{s,\alpha}^{\prime}({\Omega})(\theta) = {\int}_{\Gamma}{\theta \cdot n \left( k(x) \;\sigma_{v}^{\alpha} + A \varepsilon(u) : \varepsilon(p) \right)}{\mathrm{d}s}. $$
(30)

3.2 Global stress-deviation measure

The main idea of the present work is to use the global stress-deviation measure as an objective function in the stress-related optimization problem. The stress-deviation based objective function is defined as

$$ J_{sd,\mu}({\Omega}) = {\int}_{\Omega}{ k(x) \left( \sigma_{v}(\sigma) - \overline{\sigma}_{v} \right)^{2\mu} }{\mathrm{d}x}, $$
(31)

where \( \mu \in \mathbb {N} \) is the penalty parameter, and the average von Mises stress \( \overline {\sigma }_{v} \) is given as

$$ \overline{\sigma}_{v} = \overline{\sigma}_{v} ({\Omega} ) = \frac{{\int}_{\Omega}{\sigma_{v}(\sigma)}{\mathrm{d}x}}{{\int}_{\Omega}\mathrm{d}{x}}. $$
(32)

Analysis of the objective function J s d,μ (Ω) given by the expression (31) in terms of the general form (2) shows that the number of the subordinate functionals is equal to m=2. The integrand in the expression (31) is recognised as \( j= k(x)\big (\sigma _{v}- \frac {J_{1}}{J_{2} } \big )^{2 \mu } \), the integrands in the subordinate functionals are j 1 = σ v , j 2=1, and the total derivative of j is equal to

$$\begin{array}{@{}rcl@{}} \frac{\mathrm{d} j }{\mathrm{d} \sigma } &= & 2 \mu \left( k(x) (\sigma_{v} - \overline{\sigma}_{v} )^{2 \mu - 1} - \frac{{\int}_{\Omega}{ k(x) (\sigma_{v} - \overline{\sigma}_{v} )^{2 \mu - 1} }{\mathrm{d}x}}{ {\int}_{\Omega}{\mathrm{d}x}} \right) \frac{\mathrm{d}{\sigma_{v}} }{ \mathrm{d}{\sigma}}.\\ \end{array} $$
(33)

According to the expressions (22) and (23), the shape derivative of J s d,μ (Ω) results in

$$\begin{array}{@{}rcl@{}} J_{sd,\mu}^{\prime}({\Omega})(\theta) &= & {\int}_{\Gamma}\theta \cdot n \big(k(x) (\sigma_{v} - \overline{\sigma}_{v})^{2\mu} - 2 \mu (\sigma_{v} - \overline{\sigma}_{v})\\&&\times \frac{{\int}_{\Omega} {k(x) (\sigma_{v} - \overline{\sigma}_{v})^{2 \mu - 1}}{\mathrm{d}x}}{{\int}_{\Omega}\mathrm{d}{x}} \\ && + A \varepsilon(u) : \varepsilon(p)\big){\mathrm{d}s} . \end{array} $$
(34)

4 Level-set method

The basis for the level-set method as proposed by the authors Osher and Sethian (1988), and the usage in structural optimization (Allaire et al. 2004; Wang et al. 2003) are briefly described in this section. The level-set method is particularly adequate for the domain boundary control which is implicitly defined via the zero level-set of an auxiliary function. Besides, its main advantage is the ease of handling the topological changes.

A bounded domain D which includes all admissible shapes Ω⊂D is called the working domain. A structure of interest Ω contained in the working domain D and an example of boundary conditions are shown in Fig. 1. The working domain boundary is also decomposed into three parts ∂ D = ∂ D D ∪∂ D N ∪∂ D 0. The Dirichlet boundary conditions are constrained as Γ D ⊆∂ D D and the Neumann boundary conditions are fixed as Γ N = ∂ D N during the optimization procedure.

Fig. 1
figure 1

Working domain with structure of interest and boundary conditions

Full size image

The boundary of the domain Ω is parameterized by the level-set function ψ as

$$ \psi(x) \left\{\begin{array}{l} > 0, \: \forall \: x \in {\Omega} \\ = 0, \: \forall \: x \in {\Gamma} \\ < 0, \: \forall \: x \in D \backslash \overline{\Omega} , \end{array}\right. $$
(35)

where \( \overline {\Omega } = {\Omega } \cup {\Gamma } \). The parameterization of the boundary Γ is presented in Fig. 2.

Fig. 2
figure 2

Parameterization of boundary Γ using level-set function

Full size image

The domain Ω=Ω(t) which evolves in time \( t \in \mathbb {R}^{+} \) under an influence of the advection velocity field is therefore described by the time dependent level-set function ψ = ψ(t,x). The governing partial differential equation which describes an evolution of the level-set function is called the Hamilton-Jacobi transport equation. It is derived by differentiating the zero level-set ψ(t,x)=0 with respect to t, which results in

$$ \frac{\partial \psi }{\partial t} + \theta \cdot \nabla \psi = 0 \quad \text{in} \;\, D, $$
(36)

where the directional vector field 𝜃 = 𝜃(t,x) is written using the normal velocity scalar field v n = v n (t,x) as

$$ \theta = v_{n} \;n . $$
(37)

The unit outward normal vector n in the working domain D is given as

$$ n = - \frac{\nabla \psi }{ | \nabla \psi | } \quad \text{a.e. in} \; D. $$
(38)

Combining the (36) and (37) gives the classical form of the Hamilton-Jacobi equation

$$\begin{array}{@{}rcl@{}} \frac{\partial \psi }{\partial t} - v_{n} | \nabla \psi | &=& 0 \quad \text{in} \;\, D , \\ \psi(0,x) &=& \psi_{0}(x), \end{array} $$
(39)

where ψ 0(x) is the initial level-set function.

During an evolution the level-set function can become too flat or too steep. To obtain accurate numerical results, the level-set function should be reinitialized to the signed distance function d Ω = d Ω(t,x) which satisfies the Ekional equation

$$ | \nabla d_{\Omega} | = 1 \quad \text{a.e. in} \;\, D. $$
(40)

In this paper, the method of solving a partial differential equation is used to compute d Ω (Osher and Fedkiw 2003; Xing et al. 2010). For another possible method based on the geometry reinitialization scheme, see for instance Yamasaki et al. (2010). Starting form the level-set function ψ(x) obtained by (39), the corresponding signed distance function is obtained as the stationary solution of the following partial differential equation

$$\begin{array}{@{}rcl@{}} \frac{\partial d_{\Omega} }{\partial t} + \text{sgn}(\psi)(|\nabla d_{\Omega}|-1) &=& 0 \quad \text{in} \;\, D , \\ d_{\Omega}(0,x) &= &\psi(x). \end{array} $$
(41)

The relation (41) can also be written in the following form

$$ \frac{\partial d_{\Omega} }{\partial t} + w \cdot \nabla d_{\Omega} = \text{sgn}(\psi) \quad \text{in} \;\, D , $$
(42)

where w=sgn(ψ)(∇d Ω/|∇d Ω|) .

5 Numerical implementation

In this section, the numerical implementation of the elasticity equations and the Hamilton-Jacobi equation is presented. The main feature of the used numerical solution techniques is that all numerical solutions are obtained by the finite element based methods, which use the same finite element mesh. Finally, the iterative algorithm for the optimization problem solution is established.

5.1 Solution of the elasticity equations

Instead of the usual ersatz material approach (Allaire et al. 2004; Wang et al. 2003), the extended finite element method (Wei et al. 2010; Li et al. 2012) is used for the solution of the elasticity equations. To prevent a singular static equilibrium of free floating material pieces, soft springs are added between every material point and a fictitious support (Makhija and Maute 2013). This is usually not the problem with ersatz material approach, since the void phase is approximated by soft material.

Therefore, the state equation is solved using the weak form of the state equation (15), which is extended with an additional stiffness term as

$$ {\int}_{\Omega}{A \varepsilon(u) :\varepsilon(\phi)}{\mathrm{d}x} - {\int}_{{\Gamma}_{N}}{g \cdot \phi }{\mathrm{d}s} + c {\int}_{\Omega}{ \; u \cdot \phi }{\mathrm{d}x} = 0, $$
(43)

where c represents the stiffness of the distributed system of springs and is chosen as a small value to prevent possible singularities. The displacement field u(x) is interpolated as

$$ u(x) = \sum\limits_{a=1}^{m_{n}} \varphi_{a}(x) \widetilde{u}_{a}, $$
(44)

where m n represents the number of nodes in the finite element mesh, \( \widetilde {u}_{a} \) correspond to the displacement nodal values and φ a (x) are the usual finite element shape functions. In a similar way, the virtual displacement field ϕ(x) is interpolated as

$$ \phi(x) = \sum\limits_{a=1}^{m_{n}} \varphi_{a}(x) \widetilde{\phi}_{a}, $$
(45)

where \( \widetilde {\phi }_{a} \) represent the virtual displacement nodal values. The interpolation fields for u(x) and ϕ(x) are substituted in equation (43), and the following linear system for the displacement nodal values is obtained

$$ \sum\limits_{b=1}^{m_{n}}(K_{ab}^{\Omega} + c \; M_{ab}^{\Omega}\; I) \widetilde{u}_{b} = F_{a}^{{\Gamma}_{N}}; \quad a=1,2,\ldots, m_{n}, $$
(46)

where the stiffness matrix element \( K_{ab}^{\Omega } \) is

$$ K_{ab}^{\Omega} = {\int}_{\Omega}{ A \nabla \varphi_{a} \nabla \varphi_{b} }{\mathrm{d}x} , $$
(47)

the matrix element \( M_{ab}^{\Omega } \) is

$$ M_{ab}^{\Omega} = {\int}_{\Omega}{ \varphi_{a} \varphi_{b} }{\mathrm{d}x} , $$
(48)

and the load vector element \( F_{a}^{{\Gamma }_{N}} \) is

$$ F_{a}^{{\Gamma}_{N}} = {\int}_{{\Gamma}_{N}}{ g \varphi_{a} }{\mathrm{d}s}. $$
(49)

The determination of the adjoint field p(x) is also necessary for the calculation of the shape derivative given by the expressions (22) and (23). Adding an additional stiffness term to the weak form of the adjoint state equation (17) results in

$$\begin{array}{@{}rcl@{}} &&{\int}_{\Omega}{A \frac{\mathrm{d}{j} }{\mathrm{d}{\sigma} } : \varepsilon(\phi) }{\mathrm{d}x} + {\int}_{\Omega}{A \varepsilon(p) :\varepsilon(\phi)}{\mathrm{d}x} \\ && \quad + c {\int}_{\Omega}{ \; p \cdot \phi }{\mathrm{d}x} = 0. \end{array} $$
(50)

Again, using the same interpolation form, the adjoint field p(x) is interpolated as

$$ p(x) = \sum\limits_{a=1}^{m_{n}} \varphi_{a}(x) \widetilde{p}_{a}, $$
(51)

where \( \widetilde {p}_{a} \) represent the adjoint nodal values. Substituting the interpolation fields for p(x) and ϕ(x) results in the linear system for the adjoint nodal values

$$ \sum\limits_{b=1}^{m_{n}}(K_{ab}^{\Omega} + c \; M_{ab}^{\Omega}\; I) \widetilde{p}_{b} = P_{a}^{\Omega}; \quad a=1,2,\ldots, m_{n}, $$
(52)

where the adjoint load vector element \( P_{a}^{\Omega } \) is

$$ P_{a}^{\Omega} = - {\int}_{\Omega}{ A \frac{\mathrm{d}{j} }{\mathrm{d}{\sigma} } \nabla \varphi_{a}}{\mathrm{d}x}. $$
(53)

The coefficient matrices of the linear systems (46) and (52) are the same, so only one numerical factorization and two back-substitutions are necessary to obtain the solutions for the nodal values of the displacement and the adjoint field.

In (47), (48) and (53) the integration is carried over Ω in the working domain D. This means that the integrals over the finite elements of the domain \( D \backslash \overline {\Omega } \) can simply be omitted (Daux et al. 2000), which also reduces time to perform numerical factorization of the linear system given by (46). Finite elements with the ratio r e = V(Ω e )/V(D e )<10−2, where the working domain and the structure of interest corresponding to a given finite element e are denoted by D e and Ωe respectively, are also omitted. Removing the finite elements with small r e circumvents the problem of overestimating the stress and possible singularities in the stiffness matrix (see for instance the work of the authors Van Miegroet and Duysinx 2007). On the contrary, finite elements that lie entirely inside the structure of interest Ω are integrated using the usual four point Gauss quadrature rule.

However, there are also some elements intersected by the domain boundary Γ which should be partitioned in a way that integration is possible. Different partitioning procedures are described by the authors Sukumar et al. (2001), Min and Gibou (2007) and Wei et al. (2010). A finite element is intersected by the domain boundary if the relation \( \min (\widetilde {\psi }_{i}^{e})\; \max (\widetilde {\psi }_{i}^{e}) < 0 \) holds for that element. The notation \(\widetilde {\psi }_{i}^{e} \) , where i=1,2,3,4 is a node index, represents a nodal value of the level-set function for a given finite element e. The partitioning procedure, based on the work of Min and Gibou (2007), is described for the first order quadrilateral finite element (Q4) shown in Fig. 3 left, and is comprised of the following steps:

  • Every intersected Q4 finite element of the corresponding nodal coordinates \( \widetilde {x}_{i} \) is middle cut into two triangular finite elements (Fig. 3).

  • For each triangular finite element the number of nodes with \( \widetilde {\psi }_{i}^{e} > 0 \) is determined and denoted by η +. Possible values of η + are 0,1,2,3. The value of η +=0 represents an empty triangular finite element where integration is omitted. The value of η +=3 represents a full triangular finite element, which is further partitioned into three sub-triangles and the usual Gauss quadrature integration with one integration point per sub-triangle is used. The values of η +=1 and η +=2 represent the two possible cases where a triangular finite element is intersected by the domain boundary, as shown in Fig. 3 right and center, respectively. Based on the condition \( \widetilde {\psi }_{i}^{e} \widetilde {\psi }_{j}^{e} < 0 \) for an intersected edge of a triangular finite element, the corresponding intersection point x i j is calculated as

    $$ \widetilde{x}_{ij}= \widetilde{x}_{j} \frac{\widetilde{\psi}_{i}^{e}}{\widetilde{\psi}_{i}^{e}-\widetilde{\psi}_{j}^{e}} - \widetilde{x}_{i} \frac{\widetilde{\psi}_{j}^{e}}{\widetilde{\psi}_{i}^{e}-\widetilde{\psi}_{j}^{e}}. $$
    (54)

  • The two possible shapes of Ωe that correspond to a triangular finite element intersected by the domain boundary (η +=1,2) are, based on their respective center point \( \widetilde {x}_{c} \), further partitioned into sub-triangles. Finally, all the contributions of integration on the obtained sub-triangles are summed.

Fig. 3
figure 3

Partitioning of quadrilateral finite element intersected by domain boundary

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Additionally, the patch recovery method is used to calculate stress values at the nodal points (Zienkiewicz and Zhu 1992). The method works by defining a local patch of elements connected to a node. At those elements, sampling of stress values at the super-convergent points (Gauss integration points for Q4 finite element) is carried out. Using the least squares fitting to those sampling points smoothed stress values are obtained at the nodal points.

5.2 Solution of the Hamilton-Jacobi equation

The Hamilton-Jacobi equation (36) is of hyperbolic type, and various methods can be used to obtain its unique solution. Among finite difference based methods, the most common is upwind finite difference scheme, however, more efficient are higher-order accurate essentially non-oscillatory and weighted methods (Osher and Fedkiw 2003). In this work, instead of solving the Hamilton-Jacobi equation with one of the mentioned finite difference methods, the streamline diffusion finite element method, described by Tornberg and Engquist (2000) and Xing et al. (2010), is used.

Details regarding the derivation of the weak form of the Hamilton-Jacobi equation using the stream line diffusion finite element method are omitted, see for instance the work of Xing et al. (2010). For the time increment number k, the weak form of the Hamilton-Jacobi equation (36), defined on the whole working domain D, is given as

$$\begin{array}{@{}rcl@{}} {\int}_{D}{}{ \psi^{(k+1)} \; \phi }{\mathrm{d}x} &=& - {\Delta} t {\int}_{D}{ v_{n}^{(k)} n_{h}^{(k)} \cdot \nabla \psi^{(k)} \; \phi_{t}^{(k)} }{\mathrm{d}x} \\ && + {\int}_{D}{ \psi^{(k)} \; \phi }{\mathrm{d}x}, \end{array} $$
(55)

where

$$ \phi_{t}^{(k)} = \phi + \beta_{t} \; v_{n}^{(k)} n_{h}^{(k)} \cdot \nabla \phi. $$
(56)

The normal is calculated as \( n_{h}^{(k)} = - \nabla \psi ^{(k)} / | \nabla \psi ^{(k)} |_{h} \), where the notation |.| h =(|.|2 + h 2)1/2 means the regularized norm used to avoid numerical instabilities due to small gradients. The characteristic element length is determined as h=(V(D)/m e )1/d, where m e is the number of finite elements. The parameter β t is given as

$$ \beta_{t} = \left( 2 \sqrt{{\Delta t}^{-2}+| v_{n}^{(k)} n_{h}^{(k)} \; J_{e}^{-1} | }\right)^{-1}, $$
(57)

where J e is the Jacobian matrix for mapping from the reference to the physical finite element. The time step Δt is determined by the Courant-Friedrichs-Lewy (CFL) condition as

$$ {\Delta} t = \alpha_{t} \frac{h}{v_{n,\max}}; \quad \alpha_{t} \in (0,1], $$
(58)

where v n,max is a maximum value of the normal velocity, and α t is a chosen value that can be adjusted throughout the optimization procedure based on a decrease or increase of the objective function.

The solution procedure of the Hamilton-Jacobi equation starts with ψ 0(x), and the final solution ψ(x) is obtained after a chosen time t, which in turn is also based on a decrease or increase of the objective function. During the solution procedure, the level-set function ψ (k)(x) for the time increment number k is interpolated as

$$ \psi^{(k)}(x) = \sum\limits_{a=1}^{m_{n}} \varphi_{a}(x) \widetilde{\psi}_{a}^{(k)}, $$
(59)

where \( \widetilde {\psi }_{a}^{(k)} \) represent the level-set function nodal values at the time increment number k. Substituting the interpolation fields (59) and (45) in equation (55) results in the linear system for the level-set function nodal values at the time increment number k+1

$$ \sum\limits_{b=1}^{m_{n}} M_{ab}^{D} \widetilde{\psi}_{b}^{(k+1)} = Q_{a}^{D \:(k)}; \quad a=1,2,\ldots,m_{n} , $$
(60)

where the matrix element \( M_{ab}^{D} \) is

$$ M_{ab}^{D} = {\int}_{D}{ \varphi_{a} \varphi_{b} }{\mathrm{d}x} , $$
(61)

and the vector element \( Q_{a}^{D \:(k)} \) is calculated as

$$\begin{array}{@{}rcl@{}} Q_{a}^{D \:(k)} &= & - {\Delta} t \; {\int}_{D} v_{n}^{(k)} n_{h}^{(k)} \cdot \nabla \psi^{(k)}\\&&\times\left( \varphi_{a} +\beta_{t} \; v_{n}^{(k)} n_{h}^{(k)} \cdot \nabla \varphi_{a} \right) {\mathrm{d}x} \\&&+ {\int}_{D}{ \psi^{(k)} \varphi_{a} }{\mathrm{d}x}. \end{array} $$
(62)

Solution of the linear system (60) is straightforward, only one factorization of the matrix \( M_{ab}^{D} \) is needed, since it does not change through the iterations. Alternatively, the matrix \( M_{ab}^{D} \) can be lumped into a diagonal matrix, and no factorization is needed. In the present work, matrix lumping by the row sum method is used, since it provides an additional dissipation and helps preventing numerical oscillations (Xing et al. 2010).

Next, the weak form of the reinitialization (42) for the time increment k is given as

$$\begin{array}{@{}rcl@{}} &&{\int}_{D}{d_{\Omega}^{(k+1)} \; \phi }{\mathrm{d}x} + {\Delta} \tau \gamma {\int}_{D}{ \nabla d_{\Omega}^{(k+1)} \cdot \nabla \phi }{\mathrm{d}x} = {\Delta} \tau\\&&\times{\int}_{D}{\left( s_{h} - w_{h}^{(k)} \cdot \nabla d_{\Omega}^{(k)} \right) \phi_{\tau}^{(k)} }{\mathrm{d}x} + {\int}_{D}{d_{\Omega}^{(k)} \; \phi}{\mathrm{d}x} , \end{array} $$
(63)

where

$$ \phi_{\tau}^{(k)} = \phi + \beta_{\tau} \; w_{h}^{(k)} \cdot \nabla \phi, $$
(64)

the vector field \( w_{h}^{(k)} = s_{h} \nabla d_{\Omega }^{(k)} / | \nabla d_{\Omega }^{(k)} |_{h} \), and the smooth sgn function of ψ is s h = ψ/(ψ 2 + h 2)1/2 . The parameter β τ is given as

$$ \beta_{\tau} = \left( 2 \sqrt{{\Delta \tau}^{-2}+| w_{h}^{(k)} J_{e}^{-1} | }\right)^{-1} . $$
(65)

The time step Δτ is determined by the CFL condition as

$$ {\Delta} \tau = \alpha_{\tau} \; h ; \quad \alpha_{\tau} = 1/2 . $$
(66)

The factor of diffusion γ is given as

$$ \gamma = \frac{\alpha_{\gamma} \; h^{2}}{ {\Delta} \tau}; \quad \alpha_{\gamma} \in [0.1,1] . $$
(67)

During the calculation of the reinitialization equation, the boundary Γ tends to drift because of discretization errors and the diffusion term in (63). To prevent a drift of the boundary Γ, the Dirichlet boundary condition should be enforced for \( d_{\Omega }^{(k+1)} \) on Γ, which is written in the weak form as

$$ {\int}_{\Gamma}{ d_{\Omega}^{(k+1)}\; \phi }{\mathrm{d}s} = 0. $$
(68)

During the solution procedure, the signed distance function \( d_{\Omega }^{(k)}(x) \) for the time increment number k is interpolated as

$$ d_{\Omega}^{(k)}(x) = \sum\limits_{a=1}^{m_{n}} \varphi_{a}(x) \widetilde{d_{\Omega}}^{(k)}_{a}, $$
(69)

where \( \widetilde {d_{\Omega }}^{(k)}_{a} \) represent the signed distance function nodal values at the time increment number k. Substituting the interpolation fields (69) and (45) in (63) results in the linear system for the signed distance function nodal values at the time increment number k+1

$$\begin{array}{@{}rcl@{}} {\sum}_{b=1}^{m_{n}} (M_{ab}^{D} + {\Delta} \tau \: \gamma \: L_{ab}^{D} + \rho \; M_{ab}^{\Gamma} ) \widetilde{d_{\Omega}}_{b}^{(k+1)} = R_{a}^{D \: (k)} ; && \\ a=1,2,\ldots,m_{n}\!\!\!\!\! &&, \end{array} $$
(70)

where the matrix element of the diffusion term \( L_{ab}^{D} \) is

$$ L_{ab}^{D} = {\int}_{D}{ \nabla \varphi_{a} \cdot \nabla \varphi_{b} }{\mathrm{d}x}, $$
(71)

the parameter ρ represents the penalization factor related to preventing a drift of the boundary Γ, the matrix element \( M_{ab}^{\Gamma } \) is

$$ M_{ab}^{\Gamma} = {\int}_{\Gamma}{ \varphi_{a} \varphi_{b} }{\mathrm{d}s} , $$
(72)

and the vector element \( R_{a}^{D \: (k) } \) is calculated as

$$\begin{array}{@{}rcl@{}} R_{a}^{D\:(k)} &= & {\Delta} \tau {\int}_{D}\left( s_{h} - w_{h}^{(k)} \cdot \nabla d_{\Omega}^{(k)} \right)\\&&\times \left( \varphi_{a} + \beta_{\tau} \; w_{h}^{(k)} \cdot \nabla \varphi_{a} \right) {\mathrm{d}x} \\ && + {\int}_{D}{ d_{\Omega}^{(k)} \varphi_{a} }{\mathrm{d}x}. \end{array} $$
(73)

Since the left hand side of (70) does not change through the iterations, only one matrix factorization is needed. The solution procedure of the reinitialization equation starts with ψ(x), and the stationary solution d Ω(x) is obtained when the convergence criterion related to the Ekional (40) is satisfied, which reads as

$$ \left( {\int}_{D}{(|\nabla {d_{\Omega}}|-1)^{2}}{\mathrm{d}x}\right)^{1/2} \leq \varepsilon_{d}, $$
(74)

where ε d >0 is a small number.

5.3 Optimization procedure

The equality constrained optimization problem, given by the expression (7), is solved using the agumented Lagrangian method (Nocedal and Wright 2006). The augmented Lagrangian function L including the equality volume constraint is defined as

$$\begin{array}{@{}rcl@{}} L({\Omega}) &= & J({\Omega}) + \ell P({\Omega}) + \lambda (V({\Omega})-V_{0}) \\ && + \frac{\eta}{2}(V({\Omega})-V_{0})^{2}, \end{array} $$
(75)

where λ is the Lagrange multiplier and η is the penalty parameter.

The shape derivative of the augmented Lagrangian function L ′(Ω)(𝜃) is calculated according to the general rules (11) as

$$\begin{array}{@{}rcl@{}} L^{\prime}({\Omega})(\theta) &=& J^{\prime}({\Omega})(\theta) + {\int}_{\Gamma}\theta \cdot n (\ell \kappa + \lambda \\ && + \eta (V({\Omega})-V_{0})) {\mathrm{d}s}, \end{array} $$
(76)

where κ is the mean curvature of Γ defined by (12), which is written in the weak form, after the integration by parts, as

$$ {\int}_{D}{ \kappa \; \phi }{x} = - {\int}_{D}{ n \cdot \nabla \phi }{\mathrm{d}x}. $$
(77)

The mean curvature κ(x) is interpolated in the usual way as

$$ \kappa(x) = \sum\limits_{a=1}^{m_{n}} \varphi_{a}(x) \widetilde{\kappa}_{a}, $$
(78)

where \( \widetilde {\kappa }_{a} \) represent the mean curvature nodal values. Substituting the interpolation fields (78) and (45) in (77) results in the linear system for the mean curvature nodal values

$$ \sum\limits_{b=1}^{m_{n}} M_{ab}^{D} \widetilde{\kappa}_{b} = {C_{a}^{D}}; \quad a=1,2,\ldots,m_{n}, $$
(79)

where the vector element \( {C_{a}^{D}} \) is

$$ {C_{a}^{D}} = {\int}_{D}{ n \cdot \nabla \varphi_{a}}{\mathrm{d}x}. $$
(80)

Finally, using the shape derivative form given by relations (22) and (23), and the directional field (37) with its selection according to (24) in the expression (76), the normal velocity field is identified as

$$\begin{array}{@{}rcl@{}} v_{n} &= & - \left( j + \sum\limits_{i=1}^{m} j_{i} {\int}_{\Omega}{ \frac{\partial j}{\partial J_{i}} }{\mathrm{d}x} + A \varepsilon(u): \varepsilon(p) + \ell \; \kappa \right.\\ &&\qquad\left. + \lambda + \eta (V({\Omega})-V_{0}) {\vphantom{\sum\limits_{i=1}^{m}}}\right). \end{array} $$
(81)

Other choices for the normal velocity v n are also possible, see for instance the articles by the authors Burger (2003) and Gournay (2006).

Putting it all together, the optimization procedure is comprised of the following steps:

  1. 1.

    Define initial shape Ω(0) and corresponding level-set function ψ (0).

  2. 2.

    Start optimization iteration l.

  3. 3.

    Calculate state u (l) and adjoint p (l) field, from the linear system (46) and (52), respectively.

  4. 4.

    Calculate objective function J (l) from the relation (2) and check the convergence criterion

    $$|J^{(l)}-J^{(l-1)}| < \varepsilon_{J}, $$

    where ε J >0 is a chosen small number representing the convergence tolerance. If the convergence criterion is satisfied then finish, else continue.

  5. 5.

    Update Lagrange multiplier as

    $$\lambda^{(l)} = \lambda^{(l-1)} + \eta^{(l-1)} (V({\Omega}^{(l-1)}) - V_{0}), $$

    and penalty parameter as

    $$\eta^{(l)} \geq \eta^{(l-1)}. $$
  6. 6.

    Calculate normal velocity field \(\hspace *{-.2pt}v^{(l)\hspace *{-.2pt}}_{n}\) from equation (81).

  7. 7.

    Update level-set function ψ (l) by solving the linear system (60) for k time increments. Here, the usual procedure of adjusting the parameter α t and the number of time increments k based on a decrease or increase of the objective function J (l), defined by Allaire et al. (2004), is followed.

  8. 8.

    Reinitialize level-set function ψ (l) by solving equation (70) for signed distance function \( d_{\Omega }^{(l)} \) and set

    $$\psi^{(l+1)} = d_{\Omega}^{(l)}. $$

6 Numerical examples

In the numerical examples, the two stress-related objective functions are considered: the commonly used global stress measure (26) and the novel global stress-deviation measure (31), respectively. The optimization efficiency is presented in terms of maximum von Mises stress, which tends to occur at the sharp re-entrant corners that are deliberately set up in the working domains. To speed up the optimization procedure, the optimal geometries for values α=2 and μ=1 of the respective objective function are taken, and used as the initial shapes in the optimization procedure for higher values of α and μ.

The working domains in the numerical examples are uniformly meshed with Q4 finite elements. The Young modulus is equal to E=1 MPa and the Poisson ratio is equal to ν=0.3. The volume constraint is given in terms of the volume ratio V(Ω)/V(D), where the ratio of 1/3 is chosen in the discussed examples.

In all cases, the localization function k(x) is used and set to zero in the small areas around the vertical load application points (colored white in Figs. 4 and 9), everywhere else is set to one. This localization function is needed to exclude the stress peaks at the load application points, and allow the stress peaks at the sharp re-entrant corners. The value of the level-set function in the areas where the localization function is set to zero is imposed as ψ>0.

Fig. 4
figure 4

Initial shape of L-beam with boundary conditions

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6.1 L-beam benchmark

As a numerical benchmark, the two-dimensional L-beam is used to present the performance of the novel stress-deviation based objective function in minimum stress design. The use of an L-shaped beam is adequate to test the efficiency of the discussed global stress-related objective functions, since a significant stress peak tends to occur at the re-entrant corner of the L-beam. The initial shape of L-beam included in the working domain with the corresponding boundary conditions is shown in Fig. 4. The L-beam working domain is uniformly meshed with finite elements of the characteristic element length h=1.2⋅10−2 mm.

The optimization results in terms of the objective function values for all used objective functions are given in Table 1. The values that correspond to the respective optimization problem are written in bold font. Of course, in an individual row the bold value is always minimal. This confirms that the corresponding shape is indeed the solution of the optimization problem for the respective objective function. Furthermore, for the objective functions J s,4 and J s d,2 the convergence history with the corresponding volume ratio is shown in Figs. 5 and 6, respectively.

Fig. 5
figure 5

Convergence history of objective function J s,4 and volume ratio for L-beam

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Fig. 6
figure 6

Convergence history of objective function J s d,2 and volume ratio for L-beam

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Table 1 Optimization results for L-beam
Full size table

The optimal shapes of L-beam with the corresponding von Mises stress distributions for the used values of the penalty parameters α and μ are shown in Fig. 7. In general, as expected, higher values of α and μ in the corresponding objective functions give lower maximum values of stresses, which can be observed in the magnifications of the re-entrant corner in Fig. 7. On the other hand, high values of the penalty parameters also give rise to numerical problems.

Fig. 7
figure 7

Optimal shapes of L-beam with corresponding stress distributions

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In the perspective of the efficiency of the global stress-related objective functions, the most interesting is the comparison of the maximum and average stress values corresponding to the optimal solutions for the used two types of objective functions, presented in Fig. 8. Note that the results for a given value of μ should be compared to the results for the value of α=2μ in order to have consistent comparisons. It can be found that in the case of global stress-deviation measure J s d,μ the maximum stress value is significantly reduced, even at lower values of the penalty parameter. This is an important observation, since an efficient stress-related optimization problem solution also allows low values of the penalty parameter, which considerably reduce numerical issues. Therefore, the main assumption of the present work that, at comparable values of the penalty parameters, the use of global stress-deviation measure would result in lower maximum stress than the use of common global stress measure is confirmed. Besides, the average stress values are higher in the case of using J s d,μ as the objective function, which implies better material utilization.

Fig. 8
figure 8

Maximum and average stress corresponding to optimal solutions for L-beam

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6.2 Notched beam example

As an additional numerical example, the two-dimensional notched beam with four-point bending load is chosen. The triangular notch with a sharp re-entrant corner is placed in the area of maximum bending moment. The initial shape of notched beam included in the working domain with the corresponding boundary conditions is shown in Fig. 9. In the case of notched beam, working domain is uniformly meshed with finite elements of the characteristic element length h=1⋅10−2 mm.

Fig. 9
figure 9

Initial shape of notched beam with boundary conditions

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The optimal shapes of notched beam with the corresponding von Mises stress distributions for the used values of the penalty parameters α and μ are shown in Fig. 10. Again, higher values of α and μ in the corresponding objective functions give lower maximum stress values, which can be observed in the magnifications of the notch in Fig. 10. Interestingly, the re-entrant corner stress reduction in the case of global stress measure J s,α converges to a rigid alternative of the notch, while on the other hand it converges to a flexible alternative of the notch in the case of global stress-deviation measure J s d,μ .

Fig. 10
figure 10

Optimal shapes of notched beam with corresponding stress distributions

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The comparison of the maximum stress values corresponding to the optimal solutions, presented in Fig. 11, again shows that in the case of global stress-deviation measure, lower maximum stress is achieved than in the case of common global stress measure, particularly at low values of the penalty parameters.

Fig. 11
figure 11

Maximum and average stress corresponding to optimal solutions for notched beam

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7 Conclusions

In this paper, a novel global approach to the solution of the stress-related structural optimization problem via the level-set method is presented. Based on the given optimization procedure and the obtained numerical results, the following concluding remarks are offered:

  • The global stress-deviation measure, which penalizes deviations from an average stress, is used as a novel objective function in the minimum stress design problem. Considering calculated maximum stress values in the presented numerical examples, the stress-deviation based objective function has proven to be significantly more effective than the classical stress-related objective function, particularly at low values of the penalty parameter. In practice, low maximum values of stress peaks are important to improve fatigue life of a structure, while the use of low penalty parameter values in the objective function considerably reduce numerical issues.

  • The generalized form of the global stress-related objective function which includes subordinate functionals is defined and analyzed in terms of shape sensitivity, meaning that its shape derivative is calculated.

  • The finite element based methods, which use the same finite element mesh, are employed to solve all partial differential equations related to the optimization problem. Namely, the elasticity equations are solved by the extended finite element method (XFEM), while the Hamilton-Jacobi equation is solved by the streamline diffusion finite element method (SDFEM). The unified numerical approach has a significant practical potential, since it uses only one numerical framework, and can easily handle more complex shapes of the working domains.