A novel displacement constrained optimization approach for black and white structural topology designs under multiple load cases

Abstract

The gray problem of displacement constrained topology volume minimization under multiple load cases still is an opening topic of research. A series of topologies with clear profiles generated from an optimization process are very beneficial to method engineering applications. In this paper, a novel displacement constrained optimization approach for black and white structural topology designs under multiple load cases, is proposed to obtain a series of topologies with clear profiles. Firstly, a distribution feature of constraint displacement derivatives is investigated. Secondly, an adaptive adjusting approach of design variable bounds is proposed, and an improved approximate model with varied constraint limits and a volume penalty objective function are constructed. Thirdly, an improved density-based optimization method is proposed for the displacement constrained topology volume minimization under multiple load cases. Finally, several examples are given to demonstrate that the results obtained by the proposed method provide a series of topologies with clear profiles during an optimization process. It is concluded from examples that the proposed method is effective and robust for generating an optimal topology.

1 Introduction

Now, the SIMP (Solid Isotropic Material with Penalty) method is a well-known major topology optimization method (Sigmund and Maute 2013, Deaton and Grandhi 2014). One of its major difficulties is the presence of intermediate densities or gray-scale material in density-based structural topology optimization. Black-and-white solutions are often sought, that is, one (such as Sigmund 2007) expects that the final design includes either elements with full material or no material, excluding gray areas. In past works, researchers usually ensured continuous design variables to be forced toward a black and white solution by applying the SIMP model or the Rational Approximation for Material Properties (RAMP) model and their variants (Sigmund and Maute 2013, Deaton and Grandhi 2014). These models are prone to lead to the checkerboard, the mesh dependence and one-point hinge connection problems (Fujii and Kikuchi 2000, Deaton and Grandhi 2014). Some restriction methods, such as the perimeter method (see Haber 1996), the local gradient method (see, e.g. Petersson and Sigmund 1998) and mesh in-dependency filters (see, e.g. Sigmund and Peterson 1998 and etc.) were proposed to deal with these problems.

But gray bands could thus appear around the topological boundary again. Then, Heaviside projections (Guest et al. 2004; Xia et al. 2012 and etc.) were proposed to deal with the aforementioned problems. Another method is to employ a specified threshold value to obtain elements with normalized densities equivalent to either 1.0 or 0.0 (see, e.g. Kikuchi et al. 1998). However, it is not evident how to choose the threshold value appropriately in these works.

In the SIMP method, Olhoff et al. (1991) proposed a two-stage approach in which initially the intermediate densities are largely suppressed, with a second-stage design procedure generating the desired solid/empty solution. To achieve the solid/empty target in density-based optimization, Fuchs et al. (2005) introduced a new constraint that is labeled the sum of the reciprocal variables (SRV). Based on the work of Zhou and Rozvany (1991), Guedes and Taylor (1997), and Rietz (2001), Bruns (2005) introduced a penalty into the volume constraint, which does result in more predominantly black-and-white solutions. Groenwold and Etman (2009) developed a simple heuristic approach for the gray-scale suppression. Wood and Groenwold (2010) showed that the non-convex topology problem resulting from the application of the SIMP-like volumetric penalization can be solved by adopting the Falk dual. A method combining a separable constraint and a post-processor based on the sequential linear integer programming was proposed by Werme (2008). In addition to the above methods, other methods include the methods with filtering techniques based on image processing (Sigmund 2007, Wang and Wang 2005), a method based on a complex-shaped beam element and graph-based optimization (Sauter et al. 2008), and hybrid methods combining simulated annealing and SIMP (Garcia-Lopez et al. 2011), or standardized elements (Jang et al. 2009).

To the best of our knowledge, the gray problem of displacement constrained topology volume minimization under multiple load cases still is an opening topic of research. Moreover, a series of topologies with clear profiles can not be obtained by the aforementioned density-based methods during an optimization process. The goal of this paper is to develop a new density-based optimization method in an approximate binary fashion of density variables, and provide a different procedure obtaining an optimal topology, so that it can be adopted to easily obtain a predominantly black-and-white optimal topology and a series of topologies with clear profiles during an optimization process.

Recently, by introducing a series of varied constraint limits, a new structural topological optimization method with displacement constraints, was proposed by Rong and Yi (2010), Rong et al. (2011). However, the scheme of the varied constraint limits can not guarantee that the feasible domain of an approximate optimization model at each later iteration step of an optimization process, was completely coincided with the feasible domain of the original model. And because reciprocal topology variables, which may vary between 1.0 and 10,000.0, were selected as design variables in the method, there still existed a lot of elements with topology variables between 0.15 and 0.001 in each topology obtained during an optimization process. Then, a strategy was adopted to heuristically remove these elements with small topology variable values.

Although the displacement constrained optimization problem is related with stiffness optimization problems to a large extent, the stiffness optimal design is not the same as the optimal displacement design. The stiffness optimization increases the structural overall stiffness and thus the displacements are roughly reduced as an indirect effect. Despite vast researches on stiffness optimization, the displacement constrained continuum topology optimization is found in relatively less literature (see, Liang et al. 2001, Yin and Yang 2001, Huang and Xie 2010, Liu et al. 2011, Zuo et al. 2012, Zuo and Xie 2014, Deng and Suresh 2015). A basic feature during the process of obtaining a series of topologies with clear profiles, is that the topology variables of only a few of elements are allowed to change within a large range at each iteration step. However, a lot of simulations demonstrate that Lagrange multipliers of the conventional approximate SIMP model (Yin and Yang 2001) all are zeros at the many beginning steps in this kind change fashion of design variables (i.e., relative density variables), if the maximum design domain filled with solid material is selected as its initial design. At this case, design variable changes of the conventional approximate model are not affected by the static equations at the many beginning iteration steps because of the Karush-Kuhn-Tucker (KKT) condition action (Nocedal and Wright 1999). Therefore, in view of the fact, the displacement constrained optimization problem basically is different from the maximization problem of a structural overall stiffness subject to the volume constraint. Moreover, for some structures such as an aircraft wing, the displacements of the exterior surface should remain within a certain limit in order to maintain the aerodynamic performance (Sobieski and Haftka 1997).

To obtain a predominantly black-and-white optimal topology along the approximate binary path of the structural topology optimization, an improved displacement constrained topology optimization method of continuum structures is proposed. The structure of this paper is organized as follows. In Section 2, a feature of constraint displacement derivatives is introduced and analyzed. In Section 3.1 and 3.2, an improved approximate model with constraint limit adjustments and a volume penalty objective function, is proposed for the displacement constraint problem under multiple load cases. In Section 3.3, an adaptive approach of adjusting design variable bounds is proposed and integrated into the approximate model with varied constraint limits. In Section 4, a smooth dual algorithm of the approximate problem is developed. And a filtering scheme, a stabilization operation, and a stopping iteration criterion are given in Sections 5 and 6. The numerical results, analyses and a conclusion of the proposed method are given in Sections 7 and 8.

2 A feature of displacement derivatives

In this paper, the binary topological variable ρ _i (i.e., relative density) is replaced by the continuous topological variable ρ _i that varies between ${\rho}_i^{\min }=0.00001$ and 1. The volume and stiffness matrix expressions of a structural element are given as follows:

$$ {V}_i={f}_v\left({\rho}_i\right){V}_i^0,\kern0.5em {\mathbf{K}}^i={f}_k\left({\rho}_i\right){\mathbf{K}}_0^i $$

(1)

where V _i and K ⁱ are the volume and stiffness matrices of the ith element, respectively. And ${V}_i^0$and ${\mathbf{K}}_0^i$ are the original volume and stiffness matrices of the ith element, respectively. ρ _i is the topology variable of the ith element. And f _v (ρ _i ) and f _k (ρ _i ) denote the interpolation (identification or penalty) functions of the volume and stiffness matrices of the ith element, respectively.

Being referred to the RAMP model (Stolpe and Svanberg 2001a), interpolation functions f _k (ρ _i ) = ρ _i /(1 + υ(1 − ρ _i )) and${f}_v\left({\rho}_i\right)={\rho}_i^{\alpha_v}$ are selected. Being referred to (18) of Wood and Groenwold (2010), α _v  ≤ 1.0 denotes the penalty parameter, and a penalty is introduced into the volume objective function. Some references(Stolpe and Svanberg 2001a, Wood and Groenwold 2010 and etc.)showed that υ = 6.0 and α _v  = 0.8 are appropriate for obtaining an optimal topology. And υ = 6.0 and α _v  = 0.8 are also used in this paper examples.

Figure 1 gives a cantilever beam under a plane stress condition. The left hand side of the beam is fixed. A vertical load of 3kN is applied at the middle point of the free end. The dimension of the beam is 0.32 m × 0.16m × 0.001m. The Young’s modulus E = 200 GPaand Poisson’s ratio ν = 0.33 are specified. The maximum design domain is divided into a mesh of 64×32 with 2048 equal-size four-node square plane stress elements. Displacement constraints are imposed on the vertical displacement u _a y (ρ) at the middle point of the free end and the vertical displacement u _q y (ρ) at the point of interest ‘q’ located in the middle of the top edge of the beam in Fig. 1, respectively. Where ρ is a topological variable vector.

Fig.1

A single-load problem with displacement constraints for a cantilever beam

A displacement derivative with respect to the topological variable ρ _i can be expressed as

$$ \begin{array}{c}\hfill \partial \left|{u}_{a\; y}\left({\boldsymbol{\uprho}}^{(0)}\right)\right|/\partial {\rho}_i=\underset{\boldsymbol{\uprho} \to {\boldsymbol{\uprho}}^{(0)}}{ \lim}\left(\left|{u}_{a\; y}\left(\boldsymbol{\uprho} \right)\right|-\left|{u}_{a\; y}\left({\boldsymbol{\uprho}}^{(0)}\right)\right|\right)/\left({\rho}_i-{\rho}_i^{(0)}\right)\hfill \\ {}\hfill =\left[\underset{\boldsymbol{\uprho} \to {\boldsymbol{\uprho}}^{(0)}}{ \lim}\left({u}_{a\; y}^2\left(\boldsymbol{\uprho} \right)-{u}_{a\; y}^2\left({\boldsymbol{\uprho}}^{(0)}\right)\right)/\left({\rho}_i-{\rho}_i^{(0)}\right)\right]/\underset{\boldsymbol{\uprho} \to {\boldsymbol{\uprho}}^{(0)}}{ \lim}\left(\left|{u}_{a\; y}\left(\boldsymbol{\uprho} \right)\right|+\left|{u}_{a\; y}\left({\boldsymbol{\uprho}}^{(0)}\right)\right|\right)\hfill \\ {}\hfill \begin{array}{l}=\left(2{u}_{\; a\; y}\left({\boldsymbol{\uprho}}^{(0)}\right)\partial {u}_{a\; y}\left({\boldsymbol{\uprho}}^{(0)}\right)/\partial {\rho}_i\right)/\left(2\left|{u}_{a\; y}\left({\boldsymbol{\uprho}}^{(0)}\right)\right|\right)\\ {}=\mathit{\operatorname{sign}}\left({u}_{a\; y}\left({\boldsymbol{\uprho}}^{(0)}\right)\right)\partial {u}_{a\; y}\left({\boldsymbol{\uprho}}^{(0)}\right)/\partial {\rho}_i\end{array}\hfill \end{array} $$

(2)

where u _a y (ρ ⁽⁰⁾) ≠ 0.0.

Therefore, ∂|u _a y (ρ ⁽⁰⁾)|/∂ρ _i easily is obtained by using the second expression of (1), seeing the paper provided by Huang and Xie (2010). In this paper, a sensitivity filtering approach similar to it proposed by Sigmund (2001), is adopted to deal with numerical instabilities (Sigmund and Peterson 1998) in topology optimization. Fig. 2a gives a topology variable distribution of the beam. And Fig. 2b and Fig. 2c depict the logarithmic distributions of the ratios between the absolute values of filtered constraint displacement derivatives and the absolute values of corresponding displacements at the constraint directions of the structure in Fig. 2a, respectively. Here, filtered constraint displacement derivatives mean that the constraint displacement derivatives are modified by use of the filtering approach.

Fig. 2

A topology variable distribution, and the logarithmic distributions of the ratios between the absolute values of filtered constraint displacement derivatives and the absolute values of corresponding vertical displacements, respectively, at the middle point “a” of the free end and the “q” point of the top side of the beam in Fig.1, corresponding to Fig.2a: (a) topology variable distribution; (b) derivatives logarithmic ln(|∂|u _a y (ρ)|/∂ρ _i |/|u _a y (ρ)|)distributions with respect to topology variables; (c) derivatives logarithmic ln(|∂|u _q y (ρ)|/∂ρ _i |/|u _q y (ρ)|) distributions with respect to topology variables. Note: In Fig.2a and color topology variable distributions of the consequent sections, the greenest elements denote elements whose topology variable values all are between 0.95 and 1.0; the dark green elements denote elements whose topology variable values all are between 0.9 and 0.95;the yellow elements denote elements whose topology variable values all are between 0.7 and 0.9; the blueish elements denote elements whose topology variable values all are between 0.5 and 0.7; the red elements denote elements whose topology variable values all are between 0.3 and 0.5; the dark red elements denote elements whose topology variable values all are between 0.1 and 0.3; the darkish elements denote elements whose topology variable values all are between 0.001 and 0.1; the silver white elements denote elements whose topology variable values all are below 0.001. |• | denotes the absolute value of •

It is found from Fig. 2b, c that there are huge differences between the maximums and corresponding minimums of the absolute values of these displacement derivatives within all the structural elements in the structures of Fig. 2b and Fig. 2c, respectively. For example, the ratios of these maximums and corresponding minimums, obtained by numerical computations, are 3.80 × 10⁶ and 4.54 × 10⁷ in the structures of Fig. 2b and Fig. 2c, respectively. And for some constraint structural displacement under a load case, there also exist some structural regions including elements that possess relative small absolute values of displacement derivatives with respect to their maximum. Here, the aforementioned huge difference and some structural regions with relative small absolute values of displacement derivatives in a structure, are thought as a feature of structural displacement derivatives.

The load case in Fig. 1 is replaced by another load case, in which a uniform distributed load of 600kN/m is applied on the free end of the beam along the vertical direction. The logarithmic ln(|∂|u _a y (ρ)|/∂ρ _i |/|u _a y (ρ)|) distribution of the ratios between the absolute values of filtered constraint displacement derivatives and the absolute value |u _a y (ρ)| of the vertical displacement at the middle point of the free end of the beam under the uniform distributed load case, is shown in Fig.3. The physical feature in Fig. 2 also occurs in Fig. 3. Generally, the constraint displacement function |u _a y (ρ)| in Fig. 2 or Fig. 3 can approximately be expressed as an one-order Taylor series expansion, namely

$$ \left|{u}_{a\; y}\left(\boldsymbol{\uprho} \right)\right|\approx \left|{u}_{a\; y}\left({\boldsymbol{\uprho}}^{\left( k-1\right)}\right)\right|+\sum_{q=1}^Q\frac{\partial \left|{u}_{a\; y}\left({\boldsymbol{\uprho}}^{\left( k-1\right)}\right)\right|}{\partial {\rho}_{n_q}}\left({\rho}_{n_q}-{\rho}_{n_q}^{\left( k-1\right)}\right) $$

(3)

Fig. 3

The logarithmic ln(|∂|u _a y (ρ)|/∂ρ _i |/|u _a y (ρ)|) distribution of the ratios between the absolute values of filtered constraint displacement derivatives and the absolute value |u _a y (ρ)| of the vertical displacement at the middle point “a” of the free end of the beam, where topology variables all are equal to 1.0

where,Q is the number of the designable elements, and the designable element numbering may be represented by n _q  , q = 1 , 2 ,  ⋯  , Q. It is assumed that a few (such as $\overline{n}=0.015 Q$, i.e. a small value with respect to Q) elements with the minimum absolute values of one-order displacement derivatives within the elements whose topology variables are bigger than 0.1 in Fig. 2 or Fig. 3, are selected. And their topology variables are denoted by ${\rho}_{n_{ql}}, l=1,2,\cdots, \overline{n}$. Where, $\left\{{n}_{ql}, l=1,2,\cdots, \overline{n}\right\}$ is a subset of {n q, q = 1, 2,  ⋯ , Q}. It is assumed that there exist big changes (such as ${\rho}_{n_{ql}}$from $\kern0.5em {\rho}_{n_{ql}}^{\left( k-1\right)}$ to ${\rho}_{n_{ql}}^{\min }=0.00001$) of the topology variables ${\rho}_{n_{ql}}, l=1,2,\cdots \overline{n}$, and topology variables of other elements in the structure are kept as constants. From (3), one total change quantity of the displacement function can approximately be estimated as follows.

$$ \left|\varDelta \left|{u}_{a\; y}\left(\boldsymbol{\uprho} \right)\right|\right|\approx \left|\sum_{l=1}^{\overline{n}}\left({\rho}_{n_{ql}}^{\min }-{\rho}_{n_{ql}}^{\left( k-1\right)}\right)\frac{\partial \left|{u}_{a\; y}\left({\boldsymbol{\uprho}}^{\left( k-1\right)}\right)\right|}{\partial {p}_{n_{ql}}}\right|\le 2.0\sum_{l=1}^{\overline{n}}\left|\frac{\partial \left|{u}_{a\; y}\left({\boldsymbol{\uprho}}^{\left( k-1\right)}\right)\right|}{\partial {p}_{n_{ql}}}\right| $$

(4)

From Fig. 2 and Fig. 3, it can easily be known that the term at the right-hand side of (4) is a very small quantity with respect to |u _a y (ρ)|. Therefore, big changes of topology variables of these elements have a little effect on the structural constraint displacement |u _a y (ρ)|. In view of volume minimizing, if a small quantity addition (such as 0.01|u _a y (ρ)|) on the constraint point displacement value |u _a y (ρ)| of the current design, is allowed, these $\overline{n}$ elements will be removed, and topology variables of other elements will not be changed. Namely, if a small quantity addition on the constraint point displacement value |u _a y (ρ)|of the current design is imposed on the optimization model, it may explanatively be justified that this element change way approximately follows the binary path of the structural topology optimization at the current iteration. It also may empirically be justified from Fig. 2 and Fig. 3 that the aforementioned feature of displacement derivatives just is the essential foundation of the success of the ESO and BESO methods.

3 A structural optimization model and its novel approximate model

3.1 Problem statement

In this study, a fixed maximum finite element mesh is adopted. Structural elements can be categorized into two kinds: (a) designable elements and (b) non-designable elements. The non-designable element numbering may be expressed as i _p  , p = 1 , 2 ,  ⋯  , P, its topological variables ${\rho}_{i_p}, p=1,2,\cdots, P$ do not change during the optimization process, and P is the number of the non-designable elements. The designable element numbering may be represented by n _q  , q = 1 , 2 ,  ⋯  , Q, and Q is the number of the designable elements. The displacement constrained optimization problem is formulated as follows.

$$ \left\{\begin{array}{c}\hfill \min \kern1em V=\sum_{q=1}^Q{\rho}_{n_q}^{\alpha_v}{V}_{n_q}^0+\sum_{p=1}^P{\rho}_{i_p}^{\alpha_v}{V}_{i_p}^0\hfill \\ {}\hfill s. t.\kern2em \left|{u}_{NC_j}^l\left(\boldsymbol{\uprho} \right)\right|\le {U}_j,\kern1em j=1,2,\cdots, J;\hfill \\ {}\hfill \mathbf{K}\left(\boldsymbol{\uprho} \right)\kern0.24em {\mathbf{u}}^l\left(\boldsymbol{\uprho} \right)={\mathbf{F}}^l\hfill \\ {}\hfill {\rho}_{n_q}^{\min}\le {\rho}_{n_q}\le 1,\kern0.5em q=1,2,\cdots, Q\hfill \end{array}\right.\kern0.5em l=1,2,\cdots, L $$

(5)

where V is the structural volume, and ${V}_{n_q}^0$ and ${V}_{i_p}^0$ are the original volumes of the n _q -th element and the i _p -th element, respectively. K(ρ) is the structural total stiffness matrix, F ^l is the lth load acting on the structure, and u ^l(ρ) is the displacement vector under the lth load case. And L is the number of the load cases. ${u}_{NC_j}^l\left(\boldsymbol{\uprho} \right)$ is the displacement at the NC _j -th degree of freedom of the structure under the lth load case. And U _j is the constraint limit of the jth displacement constraint.J is the number of the displacement constraints for each load case. ${\rho}_{n_q}$ is the qth topological variable, and ${\rho}_{n_q}^{\min }$ is its lower limit, and ${\rho}_{n_q}^{\min }=0.00001$ is used in the examples of this paper. In order to deal with the checkerboard problem of topology optimization, a displacement sensitivity filtering approach is adopted to modify displacement sensitivities.

3.2 An improved approximate model with varied constraint limits

In engineering structural optimization, an objective function or/and constraint functions (including structural mechanics behaviors) are nonlinear functions of design variables, and also are implicit functions of design variables. Generally speaking, at each iteration of an optimization process, an approximate optimization model replaces its original structural optimization model, in which approximate explicit functions replace their original implicit objective and constraint functions (Nocedal and Wright 1999, Sigmund and Maute 2013, and Deaton and Grandhi 2014). An approximate optimization model should approximately be equivalent to the original structural optimization model in a certain domain of the current design point. The original structural optimization model possesses at least two performances: (a) the change requirements of its objective function and constraint functions can strongly control the change domain of design variables; (b) design variable changes within a small domain, can make that the objective function and constraint functions change in a small quantity way. Moreover, if a series of topologies with clear profiles must be obtained during an optimization process, the proposed optimization method should possess a feature. This feature is that the topology variables of only a few of elements are allowed to change within a large range at each iteration step. Namely, this feature requirement must be imposed on its approximate model or its algorithm at most iteration steps. A lot of simulations demonstrate that varied displacement limits are beneficial to big changes of topology variables of only a few of elements at each iteration step. At same time, these limits also can make that Lagrange multipliers of the approximate model possess at least one non-zero value in the case of approximate binary changes of only a few of design variables to a certain extent, even if at many beginning iteration steps when the maximum design domain filled with solid material is selected as its initial design.

An approximate optimization model with varied displacement constraint limits, is constructed as follows:

$$ \left\{\begin{array}{c}\hfill \min \kern2em \sum_{q=1}^Q{\rho}_{n_q}^{\alpha_v}{V}_{n_q}^0/{V}^{(0)}\kern20em \hfill \\ {}\hfill s. t.\kern2em \left|{\tilde{u}}_{NC_j}^l\left(\boldsymbol{\uprho} \right)\right|/\left|{u}_{NC_j}^{l,(0)}\right|\le {\tilde{U}}_j^{(k)}/\left|{u}_{NC_j}^{l,(0)}\right|,\kern1em j=1,2,\cdots, J; l=1,2,\cdots, L; k=1,2,\cdots \hfill \\ {}\hfill {\rho}_{n_q}^{\min}\le {\rho}_{n_q}\le 1,\kern0.5em q=1,2,\cdots, Q\kern11em \hfill \end{array}\right. $$

(6)

where $\left|{\tilde{u}}_{NC_j}^l\left(\boldsymbol{\uprho} \right)\right|$ denotes a one-order Taylor series expansion of $\left|{u}_{NC_j}^l\left(\boldsymbol{\uprho} \right)\right|$ at ${\rho}_{n_q}^{\left( k-1\right)}, q=1,2,\cdots, Q$, and $\left|{u}_{NC_j}^{l,(0)}\right|$is its initial value, and V ⁽⁰⁾ is the initial design structural volume. The varied displacement limit ${\tilde{U}}_j^{(k)}$ at the kth iteration is introduced and expressed as

$$ {\tilde{U}}_j^{(k)}={\overline{U}}_j^{(k)},\kern1em j=1,\cdots, J;\kern1em k=1 $$

(7)

$$ {\tilde{U}}_j^{(k)}=\left\{\begin{array}{l} \max \left({\overline{U}}_j^{(k)},{\widehat{U}}_j^{(k)}\right),\kern2em if\underset{l=1,2,\cdots, L}{ \max}\;\left(\left|{u}_{NC_j}^{l,(0)}\right|\right)\le {U}_j\\ {} \min \kern0.62em \left(\kern0.5em {\overline{U}}_j^{(k)},{\widehat{U}}_j^{(k)}\right),\kern2.12em if\underset{l=1,2,\cdots, L}{ \max}\;\left(\left|{u}_{NC_j}^{l,(0)}\right|\right)>{U}_j\end{array}\right.,\kern0.5em j=1,\cdots, J; k=2,3,\cdots $$

(8)

where, ${\overline{U}}_j^{(k)}$and ${\widehat{U}}_j^{(k)}$ are given by use of following formulas.

$$ {\overline{U}}_j^{(k)}=\left\{\begin{array}{l} \min \kern0.5em \left(\kern0.5em \left(\underset{l=1,2,\cdots, L}{ \max}\kern0.5em \left(\left|{u}_{NC_j}^{l,\left( k-1\right)}\right|\right)+{\beta}_1{\eta}_j{U}_j\right),{U}_j\right),\kern1.5em if\underset{l=1,2,\cdots, L}{ \max}\kern0.5em \left(\left|{u}_{NC_j}^{l,(0)}\right|\right)\le {U}_j\\ {} \max \kern0.5em \left(\left(\underset{l=1,2,\cdots, L}{ \max}\kern0.5em \left(\left|{u}_{NC_j}^{l,\left( k-1\right)}\right|\right)-{\beta}_1{\eta}_j{U}_j\right),{U}_j\right),\kern1.5em if\underset{l=1,2,\cdots, L}{ \max}\kern0.5em \left(\left|{u}_{NC_j}^{l,(0)}\right|\right)>{U}_j\end{array}\right.; j=1,\cdots, J; k=1,2,\cdots $$

(9)

$$ {\widehat{U}}_j^{(k)}=\left\{\begin{array}{l} \min \kern0.5em \left(\;\left(\;{\tilde{U}}_j^{\left( k-1\right)}+{\beta}_1{\eta}_j{U}_j\right),\kern0.62em {U}_j\right),\kern2em if\underset{l=1,2,\cdots, L}{ \max}\kern0.5em \left(\left|{u}_{NC_j}^{l,(0)}\right|\right)\le {U}_j\\ {} \max \kern0.5em \left(\kern0.5em \left(\kern0.5em {\tilde{U}}_j^{\left( k-1\right)}-{\beta}_1{\eta}_j{U}_j\right),\kern0.5em {U}_j\right),\kern2.12em if\underset{l=1,2,\cdots, L}{ \max}\kern0.5em \left(\left|{u}_{NC_j}^{l,(0)}\right|\right)>{U}_j\end{array}\right.,\kern0.5em j=1,\cdots, J; k=2,3,\cdots $$

(10)

$$ {\eta}_j= \min \left\{\frac{\underset{l=1,\cdots, L}{ \min}\left(\left|{U}_j-\right|\left.{u}_{NC_j}^{l,(0)}\right\Vert \right)}{\underset{l=1,\cdots, L; j=1,2,\cdots, J}{ \min}\left(\left|{U}_j-\right|\left.{u}_{NC_j}^{l,(0)}\right\Vert \right)},1.5\right\}, j=1,\cdots, J $$

(11)

where ${u}_{NC_j}^{l,\left( k-1\right)}$ is the displacement at the NC _j -th degree of freedom of the structural topology obtained at the (k-1)th outer loop iteration step under the lth load case. η _j is a displacement limit ratio factor, and η _j  ≥ 1.0 for j = 1 , 2 ,  ⋯  , J. β ₁ is a displacement limit empirical parameter, and one value of [0.002,0.012] is selected as β ₁ in the examples of this paper. Here, the varied displacement limit scheme given by (7‑11) is different from the varied limit scheme proposed by Rong and Yi (2010), Rong et al. (2011). For example, for approximate optimization model (6) with $\underset{l=1,2,\cdots, L}{ \max}\left(\left|{u}_{NC_j}^{l,(0)}\right|\right)\le {U}_j$, j = 1 , 2 ,  ⋯  , J, there always exists a limited integer number k ₀ (k ₀ ∈ {1, 2,  ⋯ ,  k , ⋯}, and k ₀ is a constant for the prescribed β ₁ and displacement limits U _j  , j = 1 , 2 ,  ⋯  , J) so that the following (12, 13) can be guaranteed by adopting the varied displacement limit ${\tilde{U}}_j^{(k)}$ in (7, 8) and (9, 10).

$$ {U}_j=\cdots ={\tilde{U}}_j^{\left({k}_0+1\right)}={\tilde{U}}_j^{\left({k}_0\right)}\ge \cdots >{\tilde{U}}_j^{\left( k+1\right)}>{\tilde{U}}_j^{(k)}>{\tilde{U}}_j^{\left( k-1\right)}>\cdots >{\tilde{U}}_j^{(2)}>{\tilde{U}}_j^{(1)},\kern1em j=1,\cdots, J $$

(12)

$$ {\Omega}_{f\; d}=\cdots ={\Omega}_{\left({k}_0+1\right)}={\Omega}_{\left({k}_0\right)}\supset \cdots \supset {\Omega}_{\left( k+1\right)}\supset {\Omega}_{(k)}\supset {\Omega}_{\left( k-1\right)}\supset \cdots \supset {\Omega}_{(2)}\supset {\Omega}_{(1)} $$

(13)

where Ω_(k) (k ≥ 1) denotes the feasible domain of the approximate optimization model (6) at the kth iterative step in Fig. 4.

Fig. 4

A simple illustration of optimization iteration solutions for the model (6)

Now, the feature of the model (6) is introduced. Fig. 4 gives a simple interpretation of optimization iteration solutions for the model (6) when $\underset{l=1,2,\cdots, L}{ \max}\left(\left|{u}_{NC_j}^{l,(0)}\right|\right)\le {U}_j$, j = 1 , 2 ,  ⋯  , J. ρ ^{(k − 1)}, ρ ^(k) andρ ^(k + 1) , k ≥ 2 in Fig. 4, respectively, represent the solutions obtained by adopting the proposed method at the (k-1)th, kth and (k+1)th iteration steps. And Ω _fd and ρ ^optrepresent the feasible domain and an optimal solution of the original optimization model (5) in Fig. 4, respectively.

The feasible domain Ω _f d of the original optimization model (5) may be divided into a feasible domain Ω_(k) and a non-feasible sub-domain (Ω _f d  − Ω_(k))of the approximate model (6) at the kth iteration step because (12, 13) is satisfied. Namely, a feasible sub-domain (Ω _f d  − Ω_(k))of the original model (5) may become a non-feasible sub-domain of the approximate model (6) at the kth iteration step. And a non-feasible sub-domain (Ω_(k + 1) − Ω_(k)) of the approximate model (6) at the kth iteration step may become a feasible sub-domain of the approximate model (6) at the (k+1)-th iteration step.

Therefore, the varied displacement limit parameter β ₁ of the (7‑11) may be manipulated to make that the model (6) possesses a smaller feasible domain Ω _(k) shown in Fig. 4 than the model (5). The optimal solution within Ω _(k) and the next iteration solution ρ ^(k)of the model (6) will fall in a smaller local feasible domain (Ω _(k) − Ω _{(k − 1)}) shown in Fig. 4 at the kth iteration step because of the volume minimizing requirement, the varied displacement limits in (7‑11), and the features of Section 2. It is found from simulations that the approximate model (6) with more tight displacement constraints at all iteration steps, the smaller local feasible domain Ω _(k), and the features of Section 2, can provide very strong effect to enhance the robustness of an optimization process and black/white distributions of a series of topologies obtained.

Moreover, for $\underset{l=1,2,\cdots, L}{ \max}\left(\left|{u}_{NC_j}^{l,(0)}\right|\right)\le {U}_j$ or $\underset{l=1,2,\cdots, L}{ \max}\left(\left|{u}_{NC_j}^{l,(0)}\right|\right)>{U}_j$, it can easily be found from (6–13) that the feasible domain of an approximate optimization model after a limited iteration step (such as  k ₀) of an optimization process, is completely coincided with the feasible domain of the original model. If a mathematical programming algorithm is adopted to solve the models (6), the convergence of the algorithm for the approximate model (6) may completely be determined by the algorithm performance. The scheme of the (7, 8) and (9, 10), is a modification of the varied limit scheme proposed by Rong and Yi (2010), Rong et al. (2011).

3.3 An adaptive approach of adjusting design variable bounds

3.3.1 The case of at least one non-zero Lagrange multiplier

Gradient-based optimization algorithms that rely on nonlinear but convex separable quadratic approximation functions, have been proven to be very effective for large-scale structural optimization (see, e.g. Groenwold and Etman 2009; Groenwold et al. 2009). Fleury (1989) proposed a SQP(Sequential quadratic programming)method that uses diagonal Hessian information only. The conservative convex separable approximation (CCSA) framework proposed by Svanberg (1987) may be used to ensure a global convergence and termination. Wood and Groenwold (2010) investigated the topology optimization problem with SIMP-like volumetric penalization, in which minimum compliance is sought subject to a single concave constraint on the volume. They showed that it is sometimes possible to solve non-convex problems directly using a dual method. They also concluded that the presence of the concave constraint may increase the difficulty of the problem dramatically if one employs a method based on strictly convex approximations. Their numerical results exhibited large-scale oscillatory behaviors of physical quantities during an optimization process unless an additional external move limit was applied.

Moreover, ${\partial}^2\left|{u}_{NC_j}^l\left(\boldsymbol{\uprho} \right)\right|/\partial {\left({\rho}_{n_q}\right)}^2$ at some design points with partial small topology variable values, may be negative. Namely, the quadratic Taylor series expansion of $\left|{u}_{NC_j}^l\left(\boldsymbol{\uprho} \right)\right|$ with respect to topology variables may be a non-convex function in a small region of some design point with partial small topology variable values. If separable strict convex quadratic approximations of the constraints $\kern0.5em \left|{u}_{NC_j}^l\left(\boldsymbol{\uprho} \right)\right|/\left|{u}_{NC_j}^{l,(0)}\right|\le {\tilde{U}}_j^{(k)}/\left|{u}_{NC_j}^{l,(0)}\right|\kern0.5em$replace displacement constraints of the model (6), (13) can not be guaranteed by these approximation constraints, when a lot of elements of the optimized structure possess small topology variable values.

Here, topology variables are selected as design variables, and the varied constraint limit scheme of the (7‑11), and one-order Taylor series expansions of displacement constraints are adopted in the approximate model (6). Namely, the approximate model (6) replaces the model (5) at each iteration step. However, the approximate displacement constraint functions and the objective function of the model (6) do not possess strong controlling ability of the topology variable changes. Some simulations also demonstrate that the approximate model (6) still do not completely possess the two performances aforementioned (at least one non-zero Lagrange multiplier in (14)), and also can not satisfy the requirement aforementioned in Section 3.2 at the most iteration steps.

$$ L\left(\boldsymbol{\uprho}, \boldsymbol{\uplambda} \right)=\sum_{q=1}^Q{\rho}_{n_q}^{\alpha_v}{V}_{n_q}^0/{V}^{(0)}+\sum_{l=1}^L\sum_{j=1}^J{\lambda}_{\left( l-1\right)\times J+ j}\left(\;\left|{\tilde{u}}_{NC_j}^l\left(\boldsymbol{\uprho} \right)\right|-{\tilde{U}}_j^{(k)}\right)/\left|{u}_{NC_j}^{l,(0)}\right| $$

(14)

Therefore, in order to simply obtain a series of topologies with clear profiles during an optimization process, except that the approximate model (6) is adopted, change restrictions of the design variables are given so that the approximate model (6) possesses the two performances and the requirement aforementioned in Section 3.2 at the most iteration steps. Generally speaking, the change region $\left[{x}_i^L,{x}_i^U\right]$ of any design variable x _i in trust region optimization methods, may be determined by using the objective and constraint function values and their derivative values at continuous several iteration steps (Tang 2014). Here, in order to effectively and simply construct trust regions of a lot of design variables, a novel trust region scheme being different from that of the paper published by Rong et al. (2016),is proposed.

• (a) The basic characteristics of an optimal solution

It is assumed that {n ₁, n ₂,  ⋯ , n _q ,  ⋯ , n _Q } is denoted by a set Q. When a structural optimization solution has evolved its optimal state, (15) should approximately be satisfied according to the KKT condition, and the approximate model (6) should possess at least one non-zero Lagrange multiplier (because of a non-zero volume derivative for any element with a topology variable value being close to 1.0). Therefore, increasing the topology variables of some design elements with the most minimum values of $\left|\partial L\left(\boldsymbol{\uprho}, \boldsymbol{\uplambda} \right)/\partial {\rho}_{n_q}\right|$ in (15) within Q, and reducing the topology variables of some design elements with the most minimum values of $\left|\partial {L}_{ca}\left(\boldsymbol{\uprho}, \boldsymbol{\uplambda} \right)/\partial {\rho}_{n_q}\right|$ in (16) within Q, will be beneficial to obtaining the optimal topology .

$$ \begin{array}{cc}\hfill \partial L\left(\boldsymbol{\rho}, \lambda \right)/\partial {\rho}_{n_q}\approx 0.0,\hfill & \hfill {n}_q\in \hfill \end{array}\boldsymbol{Q} $$

(15)

$$ \partial {L}_{ca}\left(\boldsymbol{\uprho}, \boldsymbol{\uplambda} \right)/\partial {\rho}_{n_q}\approx \sum_{l=1}^L\sum_{j=1}^J{\lambda}_{\left( l-1\right)\times J+ j}\left(\partial \left|{u}_{NC_j}^l\left({\boldsymbol{\uprho}}^{\left( k-1\right)}\right)\right|/\partial {\rho}_{n_q}\right)/\left|{u}_{NC_j}^{l,(0)}\right| $$

(16)

where, L(ρ, λ) is the Lagrange function of the model (6), and L _ca (ρ, λ) is called as the Lagrange penalty function of the model (6).

• (b) A novel trust region scheme

When there exists at least one non-zero Lagrange multiplier at the previous iteration step, i.e. the (k-1) iteration step, λ _{(l − 1) × J + j}  , j = 1 , 2 ,  ⋯  , J ; l = 1 , 2 ,  ⋯  ,  L in the (15, 16) are replaced by the Lagrange multipliers $\lambda \left( l-1\right)\times J+ j\left( k-1\right)$(j = 1, 2,  ⋯ , J; l = 1, 2,  ⋯ , L) obtained at the (k-1)th iteration step.

At first, a small empirical parameter δ ₁ is prescribed, and a value of [0.003,0.012] is adopted as δ ₁. At each iteration step, the set Q with Q design elements, is divided into following four sub-sets:

• (1) Constructing a design element subset Q ₁ with int(δ ₁ × Q) elements

Topology variables of elements in the subset Q ₁ all are bigger than 0.01, and will be reduced, and these int(δ ₁ × Q) elements possess the most minimum values of $\left|\partial {L}_{ca}\left(\boldsymbol{\uprho}, \boldsymbol{\uplambda} \right)/\partial {\rho}_{n_q}\right|$ in (16) within Q.

• (2) Forming a design element subset Q ₂ with int(δ ₁ × Q) elements

Topology variables of elements in the subset Q ₂ all are between 0.01 and 0.999, and will be increased, and these int(δ ₁ × Q) elements possess the most minimum values of $\left|\partial L\left(\boldsymbol{\uprho}, \boldsymbol{\uplambda} \right)/\partial {\rho}_{n_q}\right|$ in (15) within (Q − Q ₁).

• (3) Constructing another design element subset Q ₃

All elements in the subset Q ₃ belong to the subset (Q − Q ₁ − Q ₂), their topology variables all are below 0.999, and will be changed.

• (4) Forming the fourth element subset Q ₄

${\mathbf{Q}}_4=\mathbf{Q}-\underset{i=1}{\overset{3}{\cup }}{\mathbf{Q}}_i$ is set, and topology variables of elements in the subset Q ₄ all are held as constants at the current iteration step.

At least one of Q ₁,Q ₂ and Q ₃ is a non-empty set, and anyone of these three subsets may be an empty set at each iteration step. And following relations are satisfied.

$$ \mathbf{Q}=\underset{i=1}{\overset{4}{\cup }}{\mathbf{Q}}_i;\kern1.62em {\mathbf{Q}}_i\cap {\mathbf{Q}}_j=\mathbf{0},\kern1em i, j=1,2,3,4; i\ne j $$

(17)

where ∩ and ∪ represent the intersect set and sum set operators of several sets, respectively.

It is assumed that ${\overline{n}}_q, q=1,2,\cdots, {Q}_{red}$ is the element numbering of the subset Q ₁, and Q _red is equal to int(δ ₁ × Q); ${\tilde{n}}_q, q=1,2,\cdots, {Q}_{inc}$ is the element numbering of the subset Q ₂, and Q _inc is less than and equal to int(δ ₁ × Q); ${\tilde{\overline{n}}}_q, q=1,2,\cdots, {Q}_{cha}$ is the element numbering of the subset Q ₃, and Q _cha is its element number. In this paper, the topology variables ${\rho}_{{\overline{n}}_q}$ for any ${\overline{n}}_q\in {\mathbf{Q}}_1$, ${\rho}_{{\tilde{n}}_q}$ for any ${\tilde{n}}_q\in {\mathbf{Q}}_2$ and ${\rho}_{{\tilde{\overline{n}}}_q}$ for any ${\tilde{\overline{n}}}_q\in {\mathbf{Q}}_3$ are temporarily selected as design variables, and the topology variable ${\rho}_{{\widehat{n}}_q}$ for any ${\widehat{n}}_q\in {\mathbf{Q}}_4$ is set as a constant at each iteration step.

Following formulas are directly adopted by use of the aforementioned scheme.

$$ 1.0\times {10}^{-5}={\rho}_{{\overline{n}}_q}^L\le {\rho}_{{\overline{n}}_q}\le {\rho}_{{\overline{n}}_q}^U=1.0,\kern1em {\overline{n}}_q\in {\mathbf{Q}}_1 $$

(18)

$$ 1.0\times {10}^{-5}={\rho}_{{\tilde{n}}_q}^L\le {\rho}_{{\tilde{n}}_q}\le {\rho}_{{\tilde{n}}_q}^U=1.0\kern0.5em ,\kern1em {\tilde{n}}_q\in {\mathbf{Q}}_2 $$

(19)

$$ \max \left\{\;\left({\rho}_{{\tilde{\overline{n}}}_q}^{\left( k-1\right)}-\Delta \right),1.0\times {10}^{-5}\right\}={\rho}_{{\tilde{\overline{n}}}_q}^L\le {\rho}_{{\tilde{\overline{n}}}_q}\le {\rho}_{{\tilde{\overline{n}}}_q}^U= \min \left\{\;\left({\rho}_{{\tilde{\overline{n}}}_q}^{\left( k-1\right)}+\Delta \right),\kern0.5em 1.0\right\}\kern0.5em ,\kern1em {\tilde{\overline{n}}}_q\in {\mathbf{Q}}_3 $$

(20)

where, Δ may be treated as an empirical parameter, and a value of [0.01,0.02] can be selected as Δ according to small variable region requirements in general trust region optimization methods (Nocedal and Wright 1999). Δ=0.015 is adopted, and is not a sensitive parameter in all examples of this paper. Based on (18‑20) and the aforementioned change scheme of design variables, the approximate model (6) may be transferred into the approximate model (21):

$$ \left\{\begin{array}{c}\hfill \mathit{\min}\kern1.5em \left(\sum_{q=1}^{Q_{red}}{\rho}_{{\overline{n}}_q}^{\alpha_v}{V}_{{\overline{n}}_q}^0+\sum_{q=1}^{Q_{inc}}{\rho}_{{\tilde{n}}_q}^{\alpha_v}{V}_{{\tilde{n}}_q}^0+\sum_{q=1}^{Q_{cha}}{\rho}_{{\tilde{\overline{n}}}_q}^{\alpha_v}{V}_{{\tilde{\overline{n}}}_q}^0\right)/{V}^{(0)}\kern2em \hfill \\ {}\hfill s. t.\kern2em \left(\;\left|{\tilde{u}}_{NC_j}^l\left(\boldsymbol{\uprho} \right)\right|/\left|{u}_{NC_j}^{l,(0)}\right|-{\tilde{U}}_j^{(k)}/\left|{u}_{NC_j}^{l,(0)}\right|\right)\kern0.5em \le 0.0\kern0.5em \hfill \\ {}\hfill {\rho}_{{\overline{n}}_q}\in \left[{\rho}_{{\overline{n}}_q}^L,{\rho}_{{\overline{n}}_q}^U\right],\kern1em {\overline{n}}_q\in {\mathbf{Q}}_1\kern1em \hfill \\ {}\hfill {\rho}_{{\tilde{n}}_q}\in \left[{\rho}_{{\tilde{n}}_q}^L,{\rho}_{{\tilde{n}}_q}^U\right],\kern1.5em {\tilde{n}}_q\in {\mathbf{Q}}_2\kern1em \hfill \\ {}\hfill {\rho}_{{\tilde{\overline{n}}}_q}\in \left[{\rho}_{{\tilde{\overline{n}}}_q}^L,{\rho}_{{\tilde{\overline{n}}}_q}^U\right],\kern1.5em {\tilde{\overline{n}}}_q\in {\mathbf{Q}}_3\kern1em \hfill \end{array},\kern0.5em \begin{array}{c}\hfill j=1,\cdots, J; l=1,2,\cdots, L;\hfill \\ {}\hfill k=1,2,\cdots \hfill \end{array}\right. $$

(21)

3.3.2 The case of zero Lagrange multipliers

Some simulations show that the case of zero Lagrange multipliers of the model (21), will occur at a few of outer iteration steps during the early process of an optimization iteration solving, when a big value of β ₁ in (9‑11) is adopted. For the case of that Lagrange multipliers all are zero at the end of the (k-1)th iteration step (for example, the first iteration step), firstly setting

$$ {g}_1^{\ast}\left({\boldsymbol{\uprho}}^{\left( k-1\right)}\right)=\left(\;\left|{\tilde{u}}_{NC_{j_s}}^{l_s}\left({\boldsymbol{\uprho}}^{\left( k-1\right)}\right)\right|/\left|{u}_{NC_{j_s}}^{l_s,(0)}\right|-{\tilde{U}}_{j_s}^{(k)}/\left|{u}_{NC_{j_s}}^{l_s,(0)}\right|\;\right) $$

(22)

where, l _s and ${NC}_{j_s}$ satisfy following equation.

$$ \left|\kern0.5em \left(\;\left|{\tilde{u}}_{NC_{j_s}}^{l_s}\left({\boldsymbol{\uprho}}^{\left( k-1\right)}\right)\right|/\left|{u}_{NC_{j_s}}^{l_s,(0)}\right|-{\tilde{U}}_{j_s}^{(k)}/\left|{u}_{NC_{j_s}}^{l_s,(0)}\right|\;\right)\kern0.5em \right|=\underset{\begin{array}{c}\hfill l=1,2,\cdots, L\hfill \\ {}\hfill j=1,2,\cdots, J\hfill \end{array}}{ \min}\left|\kern0.5em \left(\;\left|{\tilde{u}}_{NC_j}^l\left({\boldsymbol{\uprho}}^{\left( k-1\right)}\right)\right|/\left|{u}_{NC_j}^{l,(0)}\right|-{\tilde{U}}_j^{(k)}/\left|{u}_{NC_j}^{l,(0)}\right|\;\right)\kern0.5em \right| $$

(23)

For another prescribed small parameter δ ₂, if (24) is satisfied, the displacement constraint function at the NC _j -th degree of freedom of the structure under the lth load case will be considered as a potential tight constraint.

$$ \left|\left|\left(\left|{\tilde{u}}_{NC_j}^l\left({\boldsymbol{\uprho}}^{\left( k-1\right)}\right)\right|/\left|{u}_{NC_j}^{l,(0)}\right|-{\tilde{U}}_j^{(k)}/\left|{u}_{NC_j}^{l,(0)}\right|\right)\right|-\left|{g}_1^{\ast}\left({\boldsymbol{\uprho}}^{\left( k-1\right)}\right)\right|\right|\le {\delta}_2 $$

(24)

where, any value of an empirical parameter δ ₂ within [0.03,0.06] does not affect the final optimization solution and solving speed of the proposed method, because the proposed method makes that the objective function and constraint functions change in a small quantity way at each iteration step, and the case of zero Lagrange multipliers of the model (21) scarcely occurs at outer iteration steps.

It is assumed that, except the (l _s  − 1) × J + j _s -th displacement constraint of (22, 23), there exist (m-1) potential tight constraints in the model (6), which satisfy (24), and they are represented by ${g}_r^{\ast}\left({\boldsymbol{\uprho}}^{\left( k-1\right)}\right), r=2,\cdots, m$. Therefore, when Lagrange multipliers of the optimization model (21) all are zeros at the end of the (k-1)-th iteration step, the design element subsets Q ₁, Q ₂ and Q ₃ are easily determined by using the $\left|\partial {g}^{\ast}\left({\boldsymbol{\uprho}}^{\left( k-1\right)}\right)/\partial {\rho}_{n_q}\right|$ of (25). Namely, all int(δ ₁ × Q) elements in Q ₁ possess the most minimum values of $\left|\partial {g}^{\ast}\left({\boldsymbol{\uprho}}^{\left( k-1\right)}\right)/\partial {\rho}_{n_q}\right|$ within Q, and their topology variables all are bigger than 0.01. And all int(δ ₁ × Q) elements in Q ₂ possess the most maximum values of $\left|\partial {g}^{\ast}\left({\boldsymbol{\uprho}}^{\left( k-1\right)}\right)/\partial {\rho}_{n_q}\right|$ in (25) within (Q − Q ₁), and their topology variables all are between 0.01 and 0.999 . Q ₃ and Q ₄ are similarly determined by the approach of Section 3.3.1.

$$ \frac{\partial {g}^{\ast}\left({\boldsymbol{\uprho}}^{\left( k-1\right)}\right)}{\partial {\rho}_{n_q}}=\left(\prod_{r=1}^m{g}_r^{\ast}\left({\boldsymbol{\uprho}}^{\left( k-1\right)}\right)\right)\left(\sum_{r=1}^m\frac{\partial {g}_r^{\ast}\left({\boldsymbol{\uprho}}^{\left( k-1\right)}\right)/\partial {\rho}_{n_q}}{g_r^{\ast}\left({\boldsymbol{\uprho}}^{\left( k-1\right)}\right)}\right) $$

(25)

Therefore, the optimization model (6) at the kth iteration step can approximately be transferred into the model (21) by considering the subsets Q ₁, Q ₂, Q ₃and Q ₄, and (18‑20), when Lagrange multipliers of the optimization model (21) all are zeros at the (k-1)-th iteration step.

4 A smooth dual solving method of the approximate problem

Although the optimization model (21) is a linear optimization problem, it is an approximate model of the nonlinear optimization problem (5), and possesses a lot of design variables. If there exist a lot of design variables and only several constraints in an optimization problem, the dual solving method can dramatically reduce the computational quantity (Nocedal and Wright 1999). In order to robustly obtain the solutions of the model (21), following the approach proposed by Svanberg (1987), an artificial variable vector $\mathbf{y}={\left({y}_1,{y}_2,\cdots, {y}_{m_{\lambda}}\right)}^T$ is introduced into the problem (21), and ${x}_q={\rho}_{{\overline{n}}_q},{\overline{n}}_q\in {\mathbf{Q}}_1$, ${x}_{Q_{red}+ q}={\rho}_{{\tilde{n}}_q},{\tilde{n}}_q\in {\mathbf{Q}}_2$ and ${x}_{Q_{red}+{Q}_{inc}+ q}={\rho}_{{\tilde{\overline{n}}}_q},{\tilde{\overline{n}}}_q\in {\mathbf{Q}}_3$ are set at each iteration step. The following enlarged approximate problem at the initial point ${x}_q^{(0)}={\rho}_{{\overline{n}}_q}^{\left( k-1\right)},{\overline{n}}_q\in {\mathbf{Q}}_1$, ${x}_{Q_{red}+ q}^{(0)}={\rho}_{{\tilde{n}}_q}^{\left( k-1\right)},{\tilde{n}}_q\in {\mathbf{Q}}_2$ and ${x}_{Q_{red}+{Q}_{inc}+ q}^{(0)}={\rho}_{{\tilde{\overline{n}}}_q}^{\left( k-1\right)},{\tilde{\overline{n}}}_q\in {\mathbf{Q}}_3$, is given for the displacement constrained optimization problem under multiple load cases.

$$ \left\{\begin{array}{l}\mathit{\min}\kern1em {f}_0\left(\mathbf{x}\right)+\sum_{v=1}^{m_{\lambda}}\left({c}_v{y}_v+\frac{1}{2}{d}_v{y}_v^2\right)\\ {} s. t.\kern1em {g}_v\left(\mathbf{x}\right)-{y}_v\le 0,\kern0.5em v=1,2,\cdots, {m}_{\lambda}\kern0.5em \\ {}\kern3.5em \mathbf{x}\in {\mathbf{x}}^{(k)},\kern1em \mathbf{y}\ge 0\end{array}\right. $$

(26)

where m _λ  = J × L $\begin{array}{c}\hfill {f}_0\left(\mathbf{x}\right)=\left(\sum_{q=1}^{Q_{red}}{x}_{{\overline{n}}_q}^{\alpha_v}\;{V}_{{\overline{n}}_q}^0+\sum_{q=1}^{Q_{i nc}}{x}_{\left( q+{Q}_{red}\right)}^{\alpha_v}{V}_{{\tilde{n}}_q}^0+\sum_{q=1}^{Q_{cha}}{x}_{\left( q+{Q}_{red}+{Q}_{i nc}\right)}^{\alpha_v}{V}_{{\tilde{\overline{n}}}_q}^0\right)/{V}^{(0)}\hfill \\ {}\hfill \kern0.5em {g}_v\left(\mathbf{x}\right)=\left({u}_{NC_j}^l\left({\boldsymbol{\uprho}}^{\left( k-1\right)}\right)-{\tilde{U}}_j^{(k)}\right)/\left|{u}_{NC_j}^{l,(0)}\right|+\sum_{i=1}^{\left({Q}_{red}+{Q}_{i nc}+{Q}_{cha}\right)}\;\left(\partial \left|{u}_{NC_j}^l\left({\boldsymbol{\uprho}}^{\left( k-1\right)}\right)\right|/\partial {x}_i/\left|{u}_{NC_j}^{l,(0)}\right|\right)\kern0.5em \left({x}_i-{x}_i^{(0)}\right)\kern2em ,\hfill \end{array}$

$$ v=\left( l-1\right)\times J+ j,\kern1em j=1,2,\cdots, J; l=1,2,\cdots, L\kern8em $$

(27)

$$ {\mathbf{x}}^{(k)}=\left\{\left.{x}_i\right|{x}_i\in \left[{x}_i^L,{x}_i^U\right],\kern0.5em i=1,2,\cdots, \left({Q}_{red}+{Q}_{i nc}+{Q}_{cha}\right)\right\} $$

(28)

Being referred to the paper published by Svanberg (1987), c ₁ = 1000 and d ₁ = 1.0 are selected in the examples of this paper. At the beginning of the sub-optimization iterations, it is necessary to choose x ⁽⁰⁾, and then to compute y ⁽⁰⁾, and obtain an initial feasible estimate (x ⁽⁰⁾, y ⁽⁰⁾)^T of the problem (26). Considering only the main constraints, since the simple box and the non-negativity constraints of the artificial variables $\mathbf{y}={\left({y}_1,{y}_2,\cdots, {y}_{m_{\lambda}}\right)}^T$ will be incorporated in the minimization process, the Lagrange function corresponding to the problem (19) may be given by

$$ \tilde{L}\left(\mathbf{x},\boldsymbol{\uplambda} \right)={f}_0\left(\mathbf{x}\right)+\sum_{v=1}^{m_{\lambda}}{\lambda}_v\left({g}_v\left(\mathbf{x}\right)-{y}_v\right)+\sum_{v=1}^{m_{\lambda}}\left({c}_v{y}_v+\frac{1}{2}{d}_v{y}_v^2\right) $$

(29)

The programming problem (26) can be transferred into following dual programming problem by the dual theory.

$$ \begin{array}{l} \max :\kern0.5em \phi \left(\boldsymbol{\uplambda} \right)\\ {} s. t.\kern1em \boldsymbol{\uplambda} \ge 0.0\end{array} $$

(30)

where, $\phi \left(\boldsymbol{\uplambda} \right)=\underset{\mathbf{x},\mathbf{y}}{ \min}\left\{\tilde{L}\left(\mathbf{x},\mathbf{y},\boldsymbol{\uplambda} \right),{x}_i\in \left[{x}_i^L,{x}_i^U\right],\mathbf{y}\ge 0\right\}$.

To obtain an iteration solution of (26) by adopting an iteration solution of Lagrange multipliers in the model (30), the function relations of the design variables with respect to the Lagrange multiplier vector must approximately be given. Based on the KKT condition (Nocedal and Wright 1999), following equation may be gotten.

$$ \frac{\partial \tilde{L}\left(\mathbf{x},\boldsymbol{\uplambda} \right)}{\partial {x}_i}=\frac{\partial {f}_0\left(\mathbf{x}\right)}{\partial {x}_i}+\sum_{v=1}^{m_{\lambda}}{\lambda}_v\frac{\partial {g}_v\left(\mathbf{x}\right)}{\partial {x}_i} = 0.0,\kern1em i=1,2,\cdots, \left({Q}_{red}+{Q}_{i nc}+{Q}_{cha}\right) $$

(31)

Multiplying (31) by ${x}_i^{(r)}\left(\boldsymbol{\uplambda} \right)$, following equation can be obtained.

$$ \frac{-\sum_{v=1}^{m_{\lambda}}{\lambda}_v\frac{\partial {g}_v\left(\mathbf{x}\right)}{\partial {x}_i\;}}{\frac{\partial {f}_0\left(\mathbf{x}\right)}{\partial {x}_i}}=\frac{-\sum_{v=1}^{m_{\lambda}}{\lambda}_v\frac{\partial {g}_v\left(\mathbf{x}\right)}{\partial {x}_i\;}{x}_i^{(r)}\left(\boldsymbol{\uplambda} \right)}{\frac{\partial {f}_0\left(\mathbf{x}\right)}{\partial {x}_i}{x}_i^{(r)}\left(\boldsymbol{\uplambda} \right)}=1\kern0.5em ,\kern1em i=1,2,\cdots, \left({Q}_{red}+{Q}_{i nc}+{Q}_{cha}\right) $$

(32)

where ${x}_i^{(r)}\left(\boldsymbol{\uplambda} \right)$ denotes the ith design variable at the kth outer iteration and the rth sub-iteration.

From (32), an iteration formula can be built as follows:

$$ {x}_i^{\ast \left( r+1\right)}\left(\boldsymbol{\uplambda} \right)=\frac{-\sum_{v=1}^{m_{\lambda}}{\lambda}_v\frac{\partial {g}_v\left({\mathbf{x}}^{(0)}\right)}{\partial {x}_i\;}}{\frac{\partial {f}_0\left({\mathbf{x}}^{(0)}\right)}{\partial {x}_i}}{x}_i^{(r)}\left(\boldsymbol{\uplambda} \right) $$

(33)

where ${x}_i^{\ast \left( r+1\right)}\left(\boldsymbol{\uplambda} \right)$ denotes the ith design variable obtained by using the KKT condition at the kth outer iteration and the (r+1)-th sub-iteration, and x ⁽⁰⁾ denotes the design variable vector at the beginning sub-iteration of the kth outer iteration.

A so-called fixed point iteration method has widely been used in solving the roots of a nonlinear equation. In order to ensure iteration convergence, a scalar relaxation parameter γ(Burden and Faires 1985; Du and Taylor 2002) is introduced into (33), and the (33) becomes

$$ {x}_i^{\ast \left( r+1\right)}\left(\boldsymbol{\uplambda} \right)=\gamma \frac{-\sum_{v=1}^{m_{\lambda}}{\lambda}_v\frac{\partial {g}_v\left({\mathbf{x}}^{(0)}\right)}{\partial {x}_i\;}}{\frac{\partial {f}_0\left({\mathbf{x}}^{(0)}\right)}{\partial {x}_i}}{x}_i^{(r)}\left(\boldsymbol{\uplambda} \right)+\left(1-\gamma \right)\;{x}_i^{(r)}\left(\boldsymbol{\uplambda} \right) $$

(34)

Really, the change quantities of design variables can be appropriately adjusted by a change of γ, and γ will affect all design variables and ensure iteration convergence. And a value of [0.02, 0.12] is selected as γ in the examples of Burden and Faires (1985)and Du and Taylor (2002) and this paper. From (34), following equation may be derived

$$ \frac{\partial {x}_i^{\ast \left( r+1\right)}\left(\boldsymbol{\uplambda} \right)}{\partial {\lambda}_v} = -\gamma \left(\frac{\partial {g}_v\left({\mathbf{x}}^{(0)}\right)/\partial {x}_i}{\partial {f}_0\left({\mathbf{x}}^{(0)}\right)/\partial {x}_i}\right){x}_i^{(r)}\left(\boldsymbol{\uplambda} \right) $$

(35)

Considering the upper and lower limits of the design variables, the design variable ${x}_i^{\left( r+1\right)}\left(\boldsymbol{\uplambda} \right)$ and the artificial variable ${y}_v^{\left( r+1\right)}\left(\boldsymbol{\uplambda} \right)$ can be calculated by

$$ {x}_i^{\left( r+1\right)}\left(\boldsymbol{\uplambda} \right)= \max \left\{{x}_i^L, \min \left\{{x}_i^U,{x}_i^{\ast \left( r+1\right)}\left(\boldsymbol{\uplambda} \right)\right\}\kern0.5em \right\},\kern1em i=1,2,\cdots, \left({Q}_{red}+{Q}_{i nc}+{Q}_{cha}\right) $$

(36)

$$ {y}_v^{\left( r+1\right)}\left({\lambda}_v\right)= \max \left(0.0,\kern0.5em \frac{\lambda_v-{c}_v}{d_v}\right),\kern0.5em v=1,2,\cdots, {m}_{\lambda} $$

(37)

Note that ${x}_i^{\left( r+1\right)}\left(\boldsymbol{\uplambda} \right):\kern0.5em {R}^{m_{\lambda}}\to R$ and ${y}_v^{\left( r+1\right)}\left({\lambda}_v\right):\kern0.5em R\to R$ are continuous functions of the Lagrange multiplier vector λ, but not differentiable at the point λ that ${x}_i^{\left( r+1\right)}\left(\boldsymbol{\lambda} \right)={x}_i^L$ or ${x}_i^{\left( r+1\right)}\left(\boldsymbol{\lambda} \right)={x}_i^U$ for i = 1 , 2 ,  ⋯  , (Q _red  + Q _inc  + Q _cha ) and λ _v  = c _v for v = 1 , 2 ,  ⋯  , m _λ .

Rong et al. (2016) investigated continuum structural topological optimization with stress constraints based on an active constraint technique and reciprocal topology variables. In the method, a strategy with two optimization phases and a phase transferring step, and a smooth dual solving approach were proposed to heuristically remove a lot of elements with topology variables between 0.15 and 0.001 at some iterations. Here, the smooth dual solving method proposed by Rong et al. (2016) is adopted to solve the approximate sub-problem. ${x}_i^{\left( r+1\right)}\left(\boldsymbol{\uplambda} \right)$ and ${y}_v^{\left( r+1\right)}\left({\lambda}_v\right)$ in (36) and (37) may be approximately transformed into the following smoothing functions ${\tilde{x}}_i^{\left( r+1\right)}\left(\boldsymbol{\uplambda} \right)$ and ${\tilde{y}}_v^{\left( r+1\right)}\left({\lambda}_v\right)$.

$$ {\tilde{x}}_i^{\left( r+1\right)}\left(\boldsymbol{\uplambda} \right)={x}_i^L+{p}_1 \ln \left(1+{e}^{z_i^{\left( r+1\right)}\left(\boldsymbol{\uplambda} \right)/{p}_1}\right),\kern1em i=1,2,\cdots, \left({Q}_{red}+{Q}_{i nc}+{Q}_{cha}\right) $$

(38)

$$ {\tilde{y}}_v^{\left( r+1\right)}\left({\lambda}_v\right)={p}_1 \ln \left(1+{e}^{\left({\lambda}_v-{c}_v\right)/\left({p}_1{d}_v\right)}\right),\kern1.5em v=1,2,\cdots, {m}_{\lambda} $$

(39)

where ${z}_i^{\left( r+1\right)}\left(\boldsymbol{\uplambda} \right)=\left({x}_i^U-{x}_i^L\right)-{p}_2 \ln \left(1+{e}^{\left(-{x}_i^{\ast \left( r+1\right)}\left(\boldsymbol{\uplambda} \right)+{x}_i^U\right){p}_2}\right)$,p ₁ and p ₂ are two small positive parameters. At the rth sub-iteration step, the p ₁ and p ₂ of the (38, 39) are replaced by ${p}_1^{(r)}$ and ${p}_2^{(r)}$.

Therefore, an approximate quadratic programming model of the dual programming problem (30) can easily be obtained by adopting (29, 30, 34–39), combining the approach of Section 4.2 in Rong et al. (2016). And the detail approach of the approximate quadratic programming model and its simple solving approach are referred to Section 4.2 of the paper published by Rong et al. (2016).

5 A filtering scheme and a stabilization of the optimization process

To circumvent numerical instabilities, such as checkerboard patterns and mesh-dependency, a sensitivity filtering scheme being similar to that of Sigmund and Peterson (1998), is adopted. The defined filtering function is based on a filtering length scale r _min that does not change with its mesh refinement. A circle of a radius r _min, centered at the centroid of the eth element, and the circular sub-domain ${\overline{\Omega}}_e$ may be generated. Elements, whose centroids are located inside ${\overline{\Omega}}_e$, contribute to the computation of the improved sensitivity of the eth element as

$$ \partial \left|{\overline{u}}_{NC_j}^l\left({\boldsymbol{\uprho}}^{\left( k-1\right)}\right)\right|/\partial {\rho}_e=\frac{\sum_{q=1}^Q w\left({r}_{e\kern0.24em {n}_q}\right)\;{\left({\rho}_{n_q}^{\left( k-1\right)}\right)}^{\alpha}\partial \left|{u}_{NC_j}^l\left({\boldsymbol{\uprho}}^{\left( k-1\right)}\right)\right|/\partial {\rho}_{n_q}}{{\left({\rho}_e^{\left( k-1\right)}\right)}^{\alpha}\sum_{q=1}^Q w\left({r}_{e\kern0.24em {n}_q}\right)}, \kern1em e\in \mathbf{Q};\kern0.5em j=1,2,\cdots, J; l=1,2,\cdots, L $$

(40)

The convolution operator (weight factor) $w\left({r}_{e\kern0.24em {n}_q}\right)$ is written as

$$ w\left({r}_{e\kern0.24em {n}_q}\right)=\left\{\begin{array}{c}\hfill {r}_{\min }-{r}_{e\kern0.24em {n}_q},\kern0.5em {r}_{e\kern0.24em {n}_q}<{r}_{\min}\hfill \\ {}\hfill 0.0,\kern3.62em {r}_{e\kern0.24em {n}_q}\ge {r}_{\min}\hfill \end{array}\right. $$

(41)

where ${r}_{e\kern0.24em {n}_q}$ is the distance between the centre of the eth element and the centre of the n _q -th element, and the value selection of ${r}_{e\kern0.24em {n}_q}$ is discussed in many references (Sigmund 2001, Wood and Groenwold 2010 and etc.). α of (40) is an empirical parameter, and its value is between 0.15 and 0.25. To further improve the sensitivity accuracy, the simple way provided by Huang and Xie (2007) is adopted, and its formula is expressed as:

$$ \partial \left|{\widehat{u}}_{NC_j}^l\left({\boldsymbol{\uprho}}^{\left( k-1\right)}\right)\right|/\partial {\rho}_e=\left(1-{\chi}_s\right)\partial \left|{\overline{u}}_{NC_j}^l\left({\boldsymbol{\uprho}}^{\left( k-2\right)}\right)\right|/\partial {\rho}_e+{\chi}_s\partial \left|{\overline{u}}_{NC_j}^l\left({\boldsymbol{\uprho}}^{\left( k-1\right)}\right)\right|/\partial {\rho}_e,\kern1em e\in \mathbf{Q} $$

(42)

Then, $\partial \left|{u}_{NC_j}^l\left({\boldsymbol{\uprho}}^{\left( k-1\right)}\right)\right|/\partial {\rho}_e$in (15,16) and (27) is replaced by $\partial \left|{\widehat{u}}_{NC_j}^l\left({\boldsymbol{\uprho}}^{\left( k-1\right)}\right)\right|/\partial {\rho}_e$ in (42). χ _s is another empirical parameter, and 0.7 is selected as its value in examples of this paper according to the suggestion of Huang and Xie (2007).

6 A stopping iteration criterion and a black and white fraction

The outer loop finishes successfully whenever the following two conditions all are satisfied.

$$ \underset{j=1,2,\cdots, J;\kern0.62em l=1,2,\cdots, L}{ \min}\left\{\;\left({U}_j-\left|{u}_{NC_j}^l\left({\boldsymbol{\uprho}}^{(k)}\right)\right|\right)/{U}_j\;\right\}\le {\varepsilon}_1 $$

(43)

$$ \left|\left({V}^{(k)}-\frac{V^{\left( k-1\right)}+{V}^{\left( k-2\right)}}{2}\right)/{V}^{(k)}\right|\le {\varepsilon}_1 $$

(44)

where, ε ₁ is an allowable small convergence parameter. ε ₁ = 0.001 often was adopted in examples of papers published by many authors, such as Huang and Xie (2007), Xia et al. (2012) and Zuo and Xie (2014), and also is adopted in the examples of this paper.

Similarly, the elemental ‘black and white fraction φ _B & W ’ in Groenwold et al. (2009) and Wood and Groenwold (2010), is adopted. Its expression is given as follows:

$$ {\varphi}_{B\& W}=\left({N}_{\left[0\right]}+{N}_{\left[1\right]}\right)/ N $$

(45)

where N _[0] and N _[1], respectively, represent the element number of the subset { i |ρ _i  ≤ 0.001}, and the element number of the subset { i |ρ _i  ≥ 0.999}. And N denotes the total number of the structural elements in the fixed design domain.

7 Numerical examples

7.1 Optimization designs of the classic 2-D cantilever beam

Two displacement constraints in Fig. 1 (Deng and Suresh 2015) are imposed: one at the point of force application, and the other at a point of interest ‘q’ located in the middle of the top edge in Fig. 1:

$$ \begin{array}{c}\hfill \left|{u}_{a\; y}\left(\boldsymbol{\uprho} \right)\right|-{d}_a\kern0.5em \left|{u}_{a\; y}\left({\boldsymbol{\uprho}}^{(0)}\right)\right|\le 0\hfill \\ {}\hfill \left|{u}_{q\; y}\left(\boldsymbol{\uprho} \right)\right|-{d}_q\kern0.5em \left|{u}_{q\; y}\left({\boldsymbol{\uprho}}^{(0)}\right)\right|\le 0\hfill \end{array} $$

(46)

Here, the filtering length scale r _min = 2Δ _min = 0.01m (Δ _min is the minimum side length of all structural elements) being the same as that of Deng and Suresh (2015), is adopted. And the empirical parameters β ₁=0.004, δ ₁ = 0.1, and γ=0.1 are adopted. Other five parameters are selected according to the suggestion values in corresponding sections of this paper. Here, the empirical parameters δ ₁,γ and β ₁ are treated as important empirical parameters, and are determined according to a small change principle of structural characteristic quantities. Really, β ₁ is the most active and important empirical parameter. δ ₁ means that the topology variables of only certain number elements are allowed to change within a large range at each iteration step. γ and β ₁ mean that certain change quantities of design variables are allowed, and β ₁ is a total control quantity of design variable changes. When small values of γ and β ₁ are given, the quantity of δ ₁ will affect the gray distribution and its optimization iteration numbers. A small value of δ ₁ will lead to very good black/white distributions and many iteration numbers required. A big value of δ ₁ will lead to poor black/white distributions. The empirical parameter value δ ₁ is easily determined by a trial calculation and a black/white distribution requirement. A lot of simulations demonstrate that if a value of [0.003,0.012] is adopted as δ ₁, a series of topologies with good black/white distributions and convergence, can be guaranteed before about 120 iteration steps. The empirical parameter values γ and β ₁ are easily determined by a trial calculation and the requirement of at least one non-zero Lagrange multiplier of the approximate model at most iteration steps for any example.γ and β ₁ must be increased before structural optimization when the optimization solution needs too many iteration numbers. And γ and β ₁ must be reduced before structural optimization when there exist some bad topological configurations during an optimization process. A value of [0.03, 0.12] may be selected as γ, and a value of [0.002, 0.008] may be selected as β ₁.

Figure 5 gives the topology distribution optimization history of the cantilever beam, which is obtained by adopting the proposed method for the allowable constraint parameters d _a =1.5 and d _q =1.5, and the volume penalty parameter α _v =1.0, when topology variables of the initial structural model all are 1. Fig. 6 gives the volume fraction, the black and white fraction and displacement ratio optimization histories of the cantilever beam, corresponding to Fig. 5. The structural topology characteristic data in Fig. 5, are given in Table 1. Its computational time employed by adopting the computer with CPU 2.66G is 283s.

Fig. 5

The topology distribution optimization history of the cantilever beam, which is obtained by the proposed method for the volume penalty parameter α _v =1.0, and d _a =1.5 and d _q =1.5. Note: In Fig. 5 (l), the white elements denote elements whose topology variable values are equal to 0.00001

Fig. 6

The volume fraction, black and white fraction and displacement ratio optimization histories of the cantilever beam, which is obtained by the proposed method for the volume penalty parameter α _v =1.0, and d _a =1.5 and d _q =1.5

Table 1 The characteristic data corresponding to the topologies in Fig.5

It is seen from Fig. 5 that although the topologies obtained at all iterative steps include gray elements, the number of the elements whose topology variables are below 0.001, increases with an increase in outer iteration number; and there only are a few of the gray elements, whose topology variables are between 0.001 and 0.95. It is found from its optimization process that at least one displacement constraint is an active constraint at almost all iteration steps. Fig. 6 and the Table 1 also show that black and white fractions of all topologies are bigger than 0.85. Fig. 6 shows that constraint displacements and the structural volume stably and smoothly change, and the black and white fraction changes in a way of very small fluctuations during the optimization process.

If the volume penalty parameter α _v =0.8 and the previous other conditions are adopted, optimization results obtained by using the proposed method are shown in Fig. 7, Fig. 8 and Table 2. Fig. 7, Fig. 8 and Table 2 show that a series of topologies obtained, possess a more predominant black-and-white distribution than topologies in Fig. 5. Fig. 7 also depicts that the results obtained after several iterations provide a series of topologies with clear profiles.

Fig. 7

The topology distribution optimization history of the cantilever beam, which is obtained by the proposed method for the volume penalty parameter α _v =0.8, and d _a =1.5 and d _q =1.5

Fig. 8

The volume fraction, black and white fraction and displacement ratio optimization histories of the cantilever beam, obtained by the proposed method for the volume penalty parameter α _v =0.8, and d _a =1.5 and d _q =1.5

Table 2 The characteristic data corresponding to the topologies in Fig. 7

If the volume penalty parameter α _v =0.8 and the aforementioned other empirical parameter values are adopted, the results obtained by using the proposed method for the problem in Fig. 1 and corresponding different values for the allowable relative displacements at both constraint points of interest, are listed in Table 3. Table 4 gives constraints and relative results obtained by Deng and Suresh (2015) for the problem. Comparing Table 3 and Table 4, it is concluded that two more optimal topologies with different configurations and more 4.5% reductions of the objective function than the results of the Deng and Suresh method (2015), are obtained by using the proposed method for this example with the allowable relative parameters d _a =1.5 and d _q =10., and d _a =1.5, and d _q =1.5, respectively .

Table 3 Constraints and their results obtained by adopting the proposed method for the problem in Fig. 1

Table 4 Constraints and their results obtained by Deng and Suresh (2015) for the problem in Fig. 1

For the allowable relative parameters d _a =10. and d _q =1.5., although the convergent optimal solution of the problem in Fig. 1 is not obtained by adopting the proposed method, two topologies satisfying the allowable constraints (d _a =10. and d _q =1.5.) of the problem in Fig. 1, are obtained by use of the proposed method, and are shown in Table 5. Comparing Table 3, Table 4 and Table 5, it is concluded that the proposed method can obtain a more optimal topology than the Deng and Suresh method (2015) for a reasonable displacement constraint model of Fig. 1. Table 5 (b) also shows that although the topology with a bigger volume reduction, satisfies the allowable constraints (d _a =10. and d _q =1.5.) of the problem in Fig. 1, the constraint displacement point q is disconnected from the support. Therefore, there exists a reasonable displacement constraint modeling problem or a disconnected constraint point problem in the proposed method. In view of the fact that the topology of Table 5(a) possesses a smaller volume than that of Table 4 (c), a reasonable displacement constraint modeling problem or a disconnected constraint point problem (which is similar to an ill load optimization problem) will be a problem to be investigated.

Table 5 Two topologies satisfying the allowable constraints (d _a =10. and d _q =1.5.) of the problem in Fig. 1 and their characteristic data

7.2 A classic MBB beam design subjected to one concentrated load

To further estimate the proposed method, the SIMP method (Sigmund 2001) combined with the simple heuristic approach for the gray-scale suppression (Groenwold et al. 2009), is programmed in Fortran. And the compliance minimization problem of the well-known MBB beam in Fig. 9 with a volume ratio limit 0.34 is solved by the SIMP method Fortran code to obtain the vertical displacement value at the loading point and the optimum topological configuration. Here, the SIMP penalty parameter p=3 and the gray-scale suppression parameter q=2 (Groenwold et al. 2009), a move limit parameter 0.1 and a convergence error 0.001 are adopted. Then, the vertical displacement value at the loading point of the optimum topology is taken as a displacement constraint value of the proposed model, and the MBB beam design is solved by the proposed method.

Fig. 9

The load case and the support condition of a MBB beam, and an initial structural model for its analysis and optimization

Considering the symmetries of the structure and the load, only half design domain in Fig. 9b is to be optimized. And the problem settings are as follows: a finite element mesh model with 150 × 50 equal-size four-node plane stress elements and a linear sensitivity filter with radius r _min = 4Δ _min = 0.08 are adopted.

The topology variable distribution optimization history of the half MBB structure is presented in Fig. 10, which is obtained by adopting the SIMP method (Groenwold et al. 2009) for the initial structural model with topology variables all being 0.8. Fig. 11 gives the volume fraction, black and white fraction and total compliance optimization histories of the half MBB beam. Numerical characteristic results corresponding to the topologies in Fig. 10, are presented in the Table 6. Its computational time employed by adopting the computer with CPU 2.66G is 1455s. It is found from Fig. 10 and Fig. 11 that there exist some big jumps in the compliance at the beginning iteration steps.

Fig. 10

The topology distribution optimization history of the half MBB beam, which is obtained by using the SIMP method for the initial structural model with topology variables all being 0.8

Fig. 11

The volume fraction, black and white fraction and total compliance optimization histories of the half MBB beam, which are obtained by using the SIMP method for the initial structural model with topology variables all being 0.8

Table 6 The characteristic data corresponding to the topologies in Fig.10

The results of the half MBB beam optimization are presented in Fig. 12, Fig. 13 and Table 7, which are obtained by adopting the SIMP method when topology variables of the initial structural model all are 1. Here, all empirical parameter values are the same as that adopted in Fig. 10. Fig. 10h and Fig. 12h show that the SIMP method can obtain an optimal topology with a predominantly black-and-white distribution for each of two different initial topologies, but the optimal topology obtained by adopting the SIMP method for the initial topology with topology variables all being 1.0, possesses one-point hinge connection elements and a bigger compliance.

Fig. 12

The topology distribution optimization history of the half MBB beam, which is obtained by using the SIMP method for the initial structural model with topology variables all being 1.0

Fig. 13

The volume fraction, black and white fraction and total compliance optimization histories of the half MBB beam, obtained by using the SIMP method for the initial structural model with topology variables all being 1.0

Table 7 The characteristic data corresponding to the topologies in Fig.12

Here, the displacement value 131.30 abstracted from Fig. 10h and Table 6(h), is taken as a displacement constraint value of the proposed model for Fig. 9. Here,r _min = 4Δ _min = 0.08 being the same as that of the SIMP method, and empirical parameters β ₁=0.003, δ ₁ = 0.08 and γ=0.05 are adopted. The results of the half MBB beam optimization are presented in Fig. 14, Fig. 15 and Table 8, which are obtained by adopting the proposed method when topology variables of the initial structural model all are 1.0. It is seen from Fig. 15 that the black and white fraction, the volume fraction and the constraint displacement stably and smoothly change with an increase in outer iteration number. It is found from Fig. 10h, Table 6h, Fig. 14h and Table 8h that the proposed method can obtain an optimal topology with a smaller volume and a basic same black and white fraction than the SIMP method for the same vertical displacement of the applied load point.

Fig. 14

The topology distribution optimization history of the half MBB beam, which is obtained by using the proposed method for the initial structural model with topology variables all being 1.0

Fig. 15

The volume fraction, black and white fraction and total compliance optimization histories of the half MBB beam, obtained by using the proposed method for the initial structural model with topology variables all being 1.0

Table 8 The characteristic data corresponding to the topologies in Fig.16

7.3 An arch bridge example under multiple load cases

Figure 16 shows the two-dimensional design domain of an arch bridge, where the most left and most right sides in the range of 0.6 m at the bottom of the bridge, and the upper ends in the range of 0.4 m at its left hand and right hand sides, respectively, are fixed. A uniform distributed load of ρ₁ = 30kN/m is acting on the top side of a non-designable deck layer (its thickness is 0.4 m) of the bridge. And another load case with a uniform distributed load of ρ₂ = 460kN/m being applied on the partial top side with a 0.8 m interval centered at point A of the bridge top, is considered. Here, points A, B and C are uniformly distributed at the top side of the bridge. And the third load case with three same uniform distributed loads of ρ₃ = 270kN/m being applied on three partial top sides with three same 0.8 m intervals centered at points A, B and C of the bridge, respectively, is given. The gravity load is not considered in the structural analyses and optimization. The bridge is assumed to be constructed by reinforced concrete. The Young’s modules E ⁰ = 4.25MPa and Poisson’s ratio ν = 0.2 are specified in the analyses.

Fig. 16

The initial structural model of an arch structure

The structural maximum domain is modeled by adopting a finite element mesh model with 152×50 equal-size four-node plane strain elements, its two-dimension size is 30.4m × 10m, and its thickness is 10m. And a linear sensitivity filter with radius r _min = 4Δ _min = 0.8m is adopted. The thickness of the bridge desk is 0.4m and the bridge desk is defined as a non-design domain.

A prescribed displacement 14.9um is taken as the vertical displacement constraint values of A, B and C points under the three load cases for the model (5) of Fig. 16. Here, empirical parameters β ₁=0.003, δ₁ = 0.06 and γ=0.04 are adopted. The results of the bridge optimization are presented in Fig. 17 and Fig.18, which are obtained by the proposed method when topology variables of the initial structural model all are 1. The initial and final structural volumes are 3040m ³and 877.0m ³, respectively. Table 9 gives the constraint direction displacement data of the initial and final topologies. Fig. 18 shows that the black and white fraction changes in a way of fluctuations during the optimization process. When components disappear at some iteration steps (Fig. 17), there are several big jumps in the black and white fraction curve of Fig. 18. The aforementioned good feature can be found in Figs.17- 19.

Fig. 17

The topology distribution optimization history of an arch bridge structure, which is obtained by using the proposed method

Fig. 18

The volume fraction and black-white fraction histories of the bridge structure, obtained by using the proposed method

Table 9 The constraint direction displacement data of the initial and final topologies

Fig. 19

The vertical displacement optimization histories of the bridge constraint points, obtained by using the proposed method

8 Conclusions

A novel displacement constrained optimization approach for black and white structural topology designs under multiple load cases, is proposed. Some examples are given to demonstrate that the feasibility and effectiveness and features of the proposed method. The following conclusions can be drawn:

• (1) Two improved schemes dealing with trust regions and adjustments of constraint limits, respectively, are proposed. The proposed approximate optimization model possesses a tight constraint feature and can approximately adhere to the binary change nature requirement of structural topology optimization.

• (2) An improved optimization method for a displacement constrained topology optimization problem is proposed, which possesses convergence and the robustness of the optimization process.

• (3) The proposed method can obtain an optimal topology with a predominantly black-and-white distribution, and a series of topologies with clear profiles during an optimization process for the topology optimization problem under multiple load cases.

• (4) The disconnected constraint point problem and the solid/empty solving issue of topology optimization with more structural behavior constraints under multiple load cases, require further mathematical and scientific developments.