Structural and Multidisciplinary Optimization

, Volume 56, Issue 6, pp 1589–1595 | Cite as

To avoid unpractical optimal design without support



In some cases, topology optimization of continuum structures subjected to applied loads having a zero resultant force may result in an unpractical design without support. This phenomenon occurs because the original optimization problem neglects the possible change of the direction of applied load. This brief note sheds the light on avoiding such an unpractical design from the engineering viewpoint. In our work, this usually neglected phenomenon is systematically illustrated by employing a series of two-dimensional (2D) cantilever design problems using a simple and efficient Bi-directional Evolutionary Structural Optimization (BESO) method. An alternative scheme is further recommended to tackle the concerned conundrum. The proposed scheme not only can avoid unpractical designs without any support, but also takes into account the inherent uncertainty property in designing actual engineering structures.


Topology optimization Zero resultant force Unpractical design BESO Uncertain applied load 

1 Introduction

This brief note investigates one phenomenon which is usually ignored by the researchers when a continuum structure is optimally designed by using the topology optimization methods (Suzuki and Kikuchi 1991; Sigmund 2001; Xie and Steven 1993; Wang et al. 2003; Allaire et al. 2004; Junker and Hackl 2015, 2016; Bendsøe and Kikuchi 1988; Jantos et al. 2016). Specifically, topology optimization may lead to an unpractical design without support when the applied loads are self-balanced with the zero resultant force. This phenomenon is abnormal as an optimal structure should provide the best transmission path between applied loads and supports. In other words, materials should be continuously distributed to connect from the loading points to the constraint points.

In this brief note, the concerning design problem is classified into two categories according to the applied loads being symmetrical or asymmetrical. The design of a structure subjected to symmetrical loads is one of the typical numerical examples by using the topology optimization methods (Sigmund 2001; Liu and Tovar 2014). The famous educational article in the field of continuum topology optimization written by Sigmund (2001) presented a cantilever optimization example with symmetrical applied loads (Fig. 1a) to demonstrate the validity of the algorithm. It can be apparently seen from the optimal structure (Fig. 1b) that there is no connection between the applied load and the left support, indicating that the obtained structure is not a cantilever anymore. Such an unpractical design is only correct when all loads are applied in the exact directions without any derivation. Once the direction of the applied load is slightly changed, the resulting design fails to undertake any such a load since there is no support. To further illustrate this phenomenon, an engineering example, namely, the design of a part of the pylon is presented in Fig. 2. Normally, the part highlighted by dotted lines (Fig. 2a) can be regarded as a cantilever. This part could be obtained by optimizing an initial cuboid design domain subjected to two symmetrically applied loads (Fig. 2b). The loads are generated by the tension of the wire. Unfortunately, we cannot produce the expected structure (Fig. 2c) by using the traditional topology optimization methods (Liu and Tovar 2014).
Fig. 1

Topology optimization of a cantilever beam with two symmetrical loads: (a) initial design domain; (b) final design

Fig. 2

Design of a part of the pylon: (a) the real part of the pylon; (b) an initial cuboid design domain; (c) final design

However, few researches can be found to focus on designing structures subjected to asymmetrical loads with a zero resultant force. Comparing with symmetric loads with a zero resultant force such as Fig. 1a, the asymmetrical loads, e.g., as shown in Fig. 3a, can also produce a zero resultant force. Generally, the applied loads can be expressed with
$$ \sum_{\mathrm{i}=1}^{\mathrm{m}}{F}_{\mathrm{Ui}}-\sum_{\mathrm{j}=1}^{\mathrm{n}}{F}_{\mathrm{Dj}}=0 $$
Fig. 3

Topology optimization of a cantilever beam with asymmetrical loads with a zero resultant force: (a) a general case; (b) a specific case with three loads; (c) final design for the specific case

where FUi and FDj represent the ith upward load and the jth downward load. m and n are the total numbers of the upward and downward loads, respectively.

For the convenience of investigating this phenomenon, one simple case is studied here as shown in Fig. 3b. The structure is subjected to two upward loads: one is 1 N at the middle of the right side and another is 1 N at top right corner, respectively. Meanwhile, one downward load at its bottom right corner, with a magnitude of 2 N is also applied. Once again, the unpractical design without support as shown in Fig. 3c is obtained by using traditional topology optimization technique.

Therefore, it is of great importance to find a way to tackle such a critical engineering design problem. In this brief note, this problem is systematically investigated on designing a series of cantilever structures by using the BESO approach (Huang and Xie 2007, 2009). Finally, we suggest one alternative scheme to handle the concerned design problem.

2 Investigation of the concerned problem

In this section, the BESO method is used to investigate the concerned problem. It should be noted that this method can certainly be replaced by any other traditional continuum topology optimization methods. The concept of the BESO approach is very simple by removing inefficient elements from and adding efficient materials to the design domain.

Two design problems are presented to illustrate the concerned issue. The design domain for both problems has the same dimensions, with a length of 60 mm and a width of 60 mm, as shown in Fig. 1a . For the first design problem, the loading condition is the same as that shown in Fig. 1a and F1 = F2 = 1 N. While for the second design problem, the loading condition is the same with that shown in Fig. 3b. Design problems with various desired volumes are investigated. Optimal material layouts for the first and second design problems are shown in Fig. 4 and Fig. 5, respectively. Note that Vol means the volume factor of the final layout. From Fig. 4 and Fig. 5, it can be clearly found that there is no connection between the applied loads and the support except for the optimal topology with Vol = 0.9.
Fig. 4

Final optimized designs with various desired volumes for symmetrical loads with a zero resultant force

Fig. 5

Schematic diagram for describing the uncertain loads

In the above examples, symmetric or asymmetric applied loads with a zero resultant force may result in optimal designs without supports, which are totally different from expected cantilever structures. The reason for this phenomenon is that the BESO method like other traditional topology optimization methods only provides the optimal solution under the given conditions. Once the assumed conditions are slightly changed, e.g., the direction of applied loads in the above examples, the resulting designs may be extremely non-optimal. Therefore, it is necessary to find an alternative scheme to deal with the possible change of the direction of applied loads with a zero resultant force.

3 An alternative scheme

In this section, we present an alternative scheme to figure out the conundrum based on our previous work (Liu et al. 2015; Liu et al. 2017). We employ interval mathematics to equivalently transform the topology optimization problem with the uncertainties of the direction of applied loads into a deterministic one with multiple load cases. The scheme is simply presented here. For more details, the readers are referred to references (Liu et al. 2015, 2017).

We define the uncertainty of load directions in 2D space in terms of their angles of the applied loads,θ = (θ1, θ2,  ⋯ , θi,  ⋯ , θs), where θi is regarded as an interval and s denotes the total number of external loading. Let θi be positive when an uncertain load goes counterclockwise around the deterministic load; otherwise, θi is negative. Let fi be in the range from \( {\boldsymbol{f}}_i^L \) to\( {\boldsymbol{f}}_i^R \), which can be described by the interval \( \left[{\theta}_i^L,{\theta}_i^R\right] \) (Fig. 6). Thus, θiL and \( {\theta}_i^R \)are negative and positive, respectively. The interval angle can be expressed as
$$ {\boldsymbol{\theta}}^I=\left[{\boldsymbol{\theta}}^L,{\boldsymbol{\theta}}^R\right] $$
Fig. 6

Final optimal designs of various desired volume with asymmetrical loads with a zero resultant force

where θL and θR denote the lower bound and upper bound of the interval vector, respectively; the superscript I represents the interval number or vector.
The interval vector [θL, θR] is first divided into t small interval vectors, as
$$ {\boldsymbol{\theta}}^I=\left[{\boldsymbol{\theta}}^L,{\boldsymbol{\theta}}^R\right]=\left({\boldsymbol{\theta}}_1^I,{\boldsymbol{\theta}}_2^I,\dots, {\boldsymbol{\theta}}_j^I,\dots, {\boldsymbol{\theta}}_t^I\right),\kern1.5em {\boldsymbol{\theta}}_j^I=\left[{\boldsymbol{\theta}}_j^L,{\boldsymbol{\theta}}_j^R\right],\kern1.5em j=1,2,\dots, t $$
where \( {\boldsymbol{\theta}}_j^I \) denotes the jth small interval vector.
The deviation and midpoint of an arbitrary interval vector can be defined based on interval mathematics as
$$ \kern0.5em {\boldsymbol{\theta}}_j^r=\frac{{\boldsymbol{\theta}}_j^R-{\boldsymbol{\theta}}_j^L}{2} $$
$$ \kern0.5em {\boldsymbol{\theta}}_j^c=\frac{{\boldsymbol{\theta}}_j^R+{\boldsymbol{\theta}}_j^L}{2} $$
Then, the jth small interval vector and its midpoint can be expressed by
$$ {\boldsymbol{\theta}}_j^I=\left[{\boldsymbol{\theta}}_j^L,{\boldsymbol{\theta}}_j^R\right]=\left[{\boldsymbol{\theta}}^L+\left( j-1\right)\frac{2{\boldsymbol{\theta}}^r}{t},\kern0.5em {\boldsymbol{\theta}}^L+ j\frac{2{\boldsymbol{\theta}}^r}{t}\right] $$
$$ \kern0.5em {\boldsymbol{\theta}}_j^c={\boldsymbol{\theta}}^L+\frac{\left(2 j-1\right){\boldsymbol{\theta}}^r}{t} $$
Thus, the following expression can be approximately obtained given t is sufficiently large
$$ \kern0.5em {\boldsymbol{\theta}}_j^c\approx {\boldsymbol{\theta}}_j^L\approx {\boldsymbol{\theta}}_j^R $$
The design problem subjected to loading f with direction uncertainty θI as to minimize the mean compliance with a volume constraint can be approximated as multiple load cases, which can be stated as
$$ \begin{array}{l}\mathrm{Minimize}:\kern0.5em C=\sum_{j=1}^t{\omega}_j{C}_j=\frac{1}{2}\sum_{j=1}^t{\omega}_j{\boldsymbol{f}}_j{\boldsymbol{u}}_j\\ {}\mathrm{Subject}\kern0.5em \mathrm{to}:\kern0.5em {V}^{\ast }-\sum_{e=1}^N{V}_e{x}_e=0,\\ {}\kern8em {\boldsymbol{K}}_j{\boldsymbol{u}}_j={\boldsymbol{f}}_j,\kern0.5em j=1,\kern0.5em 2,\kern0.5em \dots, \kern0.5em t\\ {}\kern8em {x}_e={x}_{\min}\kern0.5em \mathrm{or}\kern0.5em 1\end{array} $$
where ωj is the prescribed weighting factor for the jth load with a direction represented by \( \kern0.5em {\theta}_j^c \), and \( \sum_{j=1}^t{\omega}_j=1 \).
Considering that t is sufficiently large, it would be time consuming to solve the design problem under multiple load cases directly. Normally, a large deviation of the interval angle will correspond to a large t and vice versa. Thus, in order to improve the computational efficiency, two unit forces are defined such that it applied to the force bearing point of f in x-direction and y-direction, respectively, as shown in Fig. 6. The displacement uje of the eth element subjected to the loading fj can be expressed by
$$ {\boldsymbol{u}}_{j e}\kern0.5em ={\boldsymbol{f}}_j \sin \left({\boldsymbol{\theta}}_j^c\right){\boldsymbol{u}}_e^x+{\boldsymbol{f}}_j \cos \left({\boldsymbol{\theta}}_j^c\right){\boldsymbol{u}}_e^y $$
where \( {\boldsymbol{u}}_e^x \) and \( {\boldsymbol{u}}_e^y \) are the displacement fields of the eth element subjected to the unit forces in x-direction and y-direction, respectively.
Thus, the sensitivity number can be formulated by (Bendsøe and Sigmund 1999)
$$ {\alpha}_e=\left\{\begin{array}{l}\frac{1}{2}\sum_{j=1}^t{\omega}_j{\left({\left({\boldsymbol{f}}_j \sin \left({\boldsymbol{\theta}}_j^c\right){\boldsymbol{u}}_e^x+{\boldsymbol{f}}_j \cos \left({\boldsymbol{\theta}}_j^c\right){\boldsymbol{u}}_e^y\right)}^T{\boldsymbol{k}}_e^0\left({\boldsymbol{f}}_j \sin \left({\boldsymbol{\theta}}_j^c\right){\boldsymbol{u}}_e^x+{\boldsymbol{f}}_j \cos \left({\boldsymbol{\theta}}_j^c\right){\boldsymbol{u}}_e^y\right)\right)}_j\kern3.5em \boldsymbol{when}\kern0.5em {x}_e=1\\ {}\frac{x_{\min}^{p-1}}{2}\sum_{j=1}^t{\omega}_j{\left({\left({\boldsymbol{f}}_j \sin \left({\boldsymbol{\theta}}_j^c\right){\boldsymbol{u}}_e^x+{\boldsymbol{f}}_j \cos \left({\boldsymbol{\theta}}_j^c\right){\boldsymbol{u}}_e^y\right)}^T{\boldsymbol{k}}_e^0\left({\boldsymbol{f}}_j \sin \left({\boldsymbol{\theta}}_j^c\right){\boldsymbol{u}}_e^x+{\boldsymbol{f}}_j \cos \left({\boldsymbol{\theta}}_j^c\right){\boldsymbol{u}}_e^y\right)\right)}_j\kern2em \boldsymbol{when}\kern0.5em {x}_e={x}_{\min}\end{array}\right.\kern0.5em $$

By doing so, we can only run finite element analysis 2s times per iteration to solve the problem in Eq. (9) by using the BESO method, which is much more efficient than that of conducting traditional topology optimization under multiple load cases directly (Liu et al. 2017).

Fig. 7 and 8 show the optimal solutions for the design problems with symmetrical and asymmetrical loads, respectively, when the proposed scheme is implemented. Each applied load in the design problems is assumed having an uncertain interval angle of [−π/4, π/4] and [−π/3, π/3] for symmetrical and asymmetrical load cases, respectively. From Fig. 7 and 8, it can be evidently found that all the optimal structures connect the applied loads and supports, indicating that the concerned design without any support is successfully avoid.
Fig. 7

Final optimized designs of various desired volume for symmetrical loads with a zero resultant force using the proposed scheme

Fig. 8

Final optimized design of various desired volume for asymmetrical loads with a zero resultant force using the proposed scheme

It should be emphasized that, the alternative scheme can also be integrated into other topology optimization algorithms by considering the uncertainty of directions of applied loads (Csébfalvi 2017a, 2017b; Csébfalvi and Lógó 2015; Lógó et al. 2009; Dunning et al. 2011; Dunning and Kim 2013; Liu et al. 2016). However, the proposed scheme used in this work can deal with design problems under large uncertainties without considering the accurate distribution of the uncertain variables.

4 Concluding remarks

One problem in designing structures subjected to applied loads with a zero resultant force is systematically studied by using BESO method in this brief note. It is revealed that traditional topology optimization methods may produce unpractical design without support, which is extremely non-optimal when the load directions are slightly changed. One alternative scheme is subsequently presented to tackle such conundrum. The proposed scheme considers the inherent uncertainty property in designing actual engineering structures, which is demonstrated by a series of 2D cantilever design problems. It is worth noting that we only suggest one possible solution, namely, considering the uncertainty of the loading direction, to figuring out the unpractical design problem. There are certainly some other possible solutions for the problem, for example, we can also define the non-design domains (Zuo and Xie 2015; Huang and Xie 2008a, 2008b; Zuo et al. 2009), which enforce the connection between the applied loads and supports. This brief note provides the useful methodology for designing structures subjected to applied loads with a zero resultant force.



The first author is partially supported by the scholarship No. 201506130053 of China Scholarship Council. This work was supported jointly by the project of “Chair Professor of Lotus Scholars Program” in Hunan province, the National Natural Science Foundation of China (No. 11672104) and the National Science Fund for Distinguished Young Scholars in China (No. 11225212). The authors also would like to thank the support from the Collaborative Innovation Center of Intelligent New Energy Vehicle, and the Hunan Collaborative Innovation Center for Green Car.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.State Key Laboratory of Advanced Design and Manufacturing for Vehicle BodyHunan UniversityChangshaChina
  2. 2.Centre for Innovative Structures and Materials, School of Civil, Environmental and Chemical EngineeringRMIT UniversityMelbourneAustralia
  3. 3.Faculty of Science, Engineering and TechnologySwinburne University of TechnologyMelbourneAustralia

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