This paper presents a robust topology optimization design approach for multi-material functional graded structures under periodic constraint with load uncertainties. To characterize the random-field uncertainties with a reduced set of random variables, the Karhunen-Loève (K-L) expansion is adopted. The sparse grid numerical integration method is employed to transform the robust topology optimization into a weighted summation of series of deterministic topology optimization. Under dividing the design domain, the volume fraction of each preset gradient layer is extracted. Based on the ordered solid isotropic microstructure with penalization (Ordered-SIMP), a functionally graded multi-material interpolation model is formulated by individually optimizing each preset gradient layer. The periodic constraint setting of the gradient layer is achieved by redistributing the average element compliance in sub-regions. Then, the method of moving asymptotes (MMA) is introduced to iteratively update the design variables. Several numerical examples are presented to verify the validity and applicability of the proposed method. The results demonstrate that the periodic functionally graded multi-material topology can be obtained under different numbers of sub-regions, and robust design structures are more stable than that indicated by the deterministic results.

Functionally graded structure, known as non-uniform structure, has extensive application prospects in the fields of aerospace, aircraft design, communication and electronics, etc., due to excellent performances such as vibration absorption, heat insulation and noise reduction.

In recent years, there is a trend to optimize the design of functionally graded structures using topology optimization methods [

Currently, topology optimization under loading uncertainties is generally classified into two categories, namely reliability-based topology optimization (RBTO) [

This paper optimizes the topology of the gradient layer with the mean and standard deviation weights of the structural compliance as the objective function and the volume fraction as the constraint to identify the optimal multi-material layout scheme. Meanwhile, considering the application in practical engineering, periodic constraints are applied in the macrostructure topology optimization model for easy manufacturing and assembly. A robust topology optimization design method for multi-material functionally graded structures with periodic constraints is proposed, which applies the Ordered-SIMP method to characterize the multi-material interpolation model. In terms of load random field distribution, the K-L expansion is employed to transform the random field into a finite number of unrelated load random variables, and the sparse grid numerical integration method is adopted to transform the RTO into a set of multi-weighted deterministic topology optimization (DTO).

The rest of the paper is organized as follows. Section 2 introduces the periodic multi-material functionally graded structure. Section 3 gives a brief introduction of multi-material structure interpolation model. The uncertainties in the characterization of the loading uncertainty with random fields are described in Section 4. Section 5 presents the establishment development and solution procedure of the proposed RTO formulation. Several numerical examples are given in Section 6 to verify the proposed method. Section 7 summarizes the remarkable conclusions.

The macro design domain of a periodic multi-material functionally graded structure is divided into multiple gradient layers (

The SIMP method introduces cell design variables that vary continuously in the interval of 0–1 [

In terms of the design of multiple materials structure, the elastic modulus and density of each material are firstly normalized, and the material properties are transformed into dimensionless relative values

where

The multiple material interpolation model based on the ordered-SIMP method [

where _{e}_{m}_{m}_{m}

where

The load uncertainties can be characterized by random field or random variable. For distributed loads with spatial correlation, the commonly used discrete and simulation methods include the extended optimal linear estimation method [

where

where

where

When the K-L expansion is applied to the discretization of random field

where

where the correlation matrix

In practical problems, the main probability characteristics of a stochastic process are usually approximated by the random variables corresponding to the first few eigenvalues descending from the maximum eigenvalue. Taking the first

when

The purpose of robust topology optimization design is to reduce the sensitivity of performance function to uncertain random variables while optimizing the objective function, so as to minimize the variance of performance index fluctuation [

According to the mathematical description of uncertain variables, robust topology optimization design can be divided into non-probabilistic and probabilistic models [

where _{e}_{e}_{0} and _{e}

where

In order to calculate the overall layout of a functionally graded structure [

Then, each gradient layer is individually designed for optimization. The topology optimization model of the functionally graded structure design can be defined as

where _{z}_{z}

The optimization goal of this paper is the weighted sum of the mean and standard deviation of structural compliance. Among them, the sensitivity of the mean and standard deviation relative to the design variable

In order to solve the objective function of the robust topology optimization design, a sparse grid method based on the Smolyak criterion [

where

where

where

The corresponding weights are determined as

Thus, the configuration points are given as

By adjusting the level accuracy

The sequence of figuration points is

Then, the corresponding weights are expressed as follows:

The sparse grid numerical integration method is employed to solve the mean and the standard deviation of the robust topology optimization design. The expression of mean and standard deviation can be modified as

According to

The commonly used optimization solution algorithms include mathematical programming method and optimization criterion method. In the iterative process, MMA [

In order to obtain the periodic functionally graded structure, the elements of the different substructures of each gradient layer at the same position have the same material properties. The periodic layer-wise design is shown in _{xj}_{yj}

By reallocating the average element compliance in sub-regions, the periodic constraint setting of the gradient layer is achieved.

At this point, the sensitivity of elements at the same position is equal to that of different substructures to achieve periodic geometric constraints.

This paper proposes three numerical examples to illustrate the difference between the optimal topologies under loading uncertainties and deterministic loading conditions. For comparison, the settings in the DTO are the same as those in the RTO examples. In the calculation examples, the design domain is discretized by square elements, and the weight coefficients of mean value and standard deviation are set to 0.5, respectively. In order to prove the effectiveness of the proposed method, gradient layers of 2, 3, 4 and 6 are preset for different design domains in this paper, and for more intuitive comparison of DTO and RTO,

The double-sided fixed beam is shown in

Number | Material | Density normalized value |
Elastic modulus normalized value ^{N} |
Color |
---|---|---|---|---|

1 | Void | 0 | 0 | White |

2 | Material A | 0.4 | 0.2 | Blue |

3 | Material B | 0.6 | 0.5 | Red |

4 | Material C | 1.0 | 1.0 | Black |

The design domain is discretized into 12800 (

Deterministic design | Robust design | ||||||||
---|---|---|---|---|---|---|---|---|---|

Layer 1 | Layer 2 | Layer 3 | Layer 4 | Layer 1 | Layer 2 | Layer 3 | Layer 4 | ||

Schemes I–IV | 0.2000 | 0.2000 | 0.2000 | 0.2000 | |||||

Scheme I | 0.1870 | 0.2130 | 0.2130 | 0.1870 | 0.1910 | 0.2090 | 0.2090 | 0.1910 | |

Scheme II | 0.1991 | 0.2009 | 0.2009 | 0.1991 | 0.2012 | 0.1988 | 0.1988 | 0.2012 | |

Scheme III | 0.1877 | 0.2123 | 0.2123 | 0.1877 | 0.1950 | 0.2050 | 0.2050 | 0.1950 | |

Scheme IV | 0.1900 | 0.2100 | 0.2100 | 0.1900 | 0.1948 | 0.2052 | 0.2052 | 0.1948 |

The optimization results show that the proposed method can effectively obtain the periodic functionally gradient structure under different gradient layers, periodic division and material combination schemes, and determine the reasonable material distribution, which shows the effectiveness of the proposed method. Compared with the DTO, the materials of the robust optimization structure are more concentrated on the horizontal force transmission route, thus improving the horizontal load capacity of the structure. It can be seen from

Case | Deterministic design | Robust design | ||||||
---|---|---|---|---|---|---|---|---|

Periodic | Mean | Standard deviation | Objective function | Time(/s) | Mean | Standard deviation | Objective function | Time(/s) |

193 | 292 | 242.5 | 4905 | 121 | 183 | 152 | 2830 | |

83 | 125 | 104 | 4914 | 82 | 124 | 103 | 2839 | |

56 | 84 | 70 | 4850 | 54 | 82 | 68 | 2770 |

For the periodic functionally graded structure under the layered setting, the optimal periodic structure of each gradient layer can be obtained. According to the

The design domain of cantilever beam is a _{1}, and the lower right corner acted as a vertical downward uncertain load _{2} that are assumed to be 1.0. The load condition and the parameters of material A and C are the same as those of Section 6.1. Let the normalized value of density ^{N} = 0.8. The structural volume fraction is limited within 30%.

The design domain is discretized into 14400 (

The optimization results show that the optimized structures are symmetrical because the design domain, boundary conditions, and load effects of the two working conditions are all symmetrical. Compared with the deterministic design, the optimized results of the robust design have thicker upper and lower edges and better horizontal load bearing capacity due to the uncertain loads that generates horizontal partitioning. Although such a structure has a reduced vertical load bearing capacity, the horizontal load bearing capacity is enhanced and the overall structure becomes more stable.

Meanwhile, considering the influence of periodic functional gradient constraints, we observed that the material distribution of the fixed edge of the DTO was not obvious, and the load bearing capacity was weak. On the contrary, the overall material distribution of the RTO results is reasonable, which further proves that it exhibits better stability.

The convergence process is rapid and stable, indicating the stability of the algorithm. According to the

Case | Deterministic design | Robust design | ||||||
---|---|---|---|---|---|---|---|---|

Periodic | Mean | Standard deviation | Objective function | Time(/s) | Mean | Standard deviation | Objective function | Time(/s) |

2594 | 3926 | 3260 | 7730 | 2229 | 3375 | 2802 | 3450 | |

1278 | 1930 | 1604 | 7524 | 1085 | 1642 | 1364 | 3460 | |

848 | 1281 | 1065 | 7650 | 606 | 916 | 762 | 3470 |

The simply supported beam structure is shown in

where

The design domain is discretized into 10800 (

where

It can be seen from the

According to the

Case | Deterministic design ( |
Robust design ( |
|||||
---|---|---|---|---|---|---|---|

Layer | Periodic | Mean | Standard deviation | Objective function | Mean | Standard deviation | Objective function |

3 | 1.23 | 3.07 | 2.15 | 1.17 | 2.89 | 2.03 | |

1.05 | 2.61 | 1.83 | 1.01 | 2.53 | 1.77 | ||

0.65 | 1.59 | 1.12 | 0.63 | 1.53 | 1.08 | ||

6 | 1.27 | 3.15 | 2.21 | 1.25 | 3.09 | 2.17 | |

1.05 | 2.59 | 1.82 | 1.05 | 2.59 | 1.82 | ||

0.64 | 1.60 | 1.12 | 0.63 | 1.57 | 1.10 |

This paper presents a robust topology optimization design method for multi-material functional gradients considering periodic constraints. In order to optimize the topology of the multi-material functionally graded structure and minimize the structural compliance under the volume constraint, an ordered-SIMP interpolation is proposed. Meanwhile, the structure is set periodically considering the practical engineering applications. Then, the sparse grid numerical integration method is introduced to calculate the objective function, the mean and standard deviation solutions for the robust topological optimization design. The design is explored in terms of both deterministic and robust design for different load conditions. By three arithmetic examples, the effectiveness of the design method is demonstrated. From the results, it can be seen that the robust design provides a more reliable and effective design solution compared with the corresponding deterministic topology optimization design. In addition, the proposed approach yields better design results for different functional gradient settings and material combination schemes, further demonstrating the practicality of the design solution.

The authors are thankful for Professor Krister Svanberg for MMA program made freely available for research purposes and the anonymous reviewers for their helpful and constructive comments.