This paper proposes a new element-based multi-material topology optimization algorithm using a single variable for minimizing compliance subject to a mass constraint. A single variable based on the normalized elemental density is used to overcome the occurrence of meaningless design variables and save computational cost. Different from the traditional material penalization scheme, the algorithm is established on the ordered ersatz material model, which linearly interpolates Young's modulus for relaxed design variables. To achieve a multi-material design, the multiple floating projection constraints are adopted to gradually push elemental design variables to multiple discrete values. For the convergent element-based solution, the multiple level-set functions are constructed to tentatively extract the smooth interface between two adjacent materials. Some 2D and 3D numerical examples are presented to demonstrate the effectiveness of the proposed algorithm and the possible advantage of the multi-material designs over the traditional solid-void designs.

Topology optimization aims at finding optimal material distribution within the prescribed design domain and achieving the best performance of the structure. Since the seminar paper of Bendsøe et al. [

Multi-material topology optimization traditionally seeks the best distributions of materials under multiple volume constraints so that the objective performance of the resulting multi-material structure is optimal. Sigmund et al. [

In practical engineering applications, the total weight of a structure may be more concerned. This is a lightweight design, where the total mass should be reasonably adopted as a constraint or objective function for a multi-material design. However, when a single mass function is implemented with multiple design variables, some meaningless combinations of multiple design variables may occur and further bring some numerical difficulties for multi-material topology optimization. Yin et al. [

The multi-material topology optimization using a single variable has an obvious advantage in saving computational cost [

Suppose that a multi-material structure is composed of M-phase materials within the design domain, where void is also as one material.

When the design domain is made up of multiple materials, as shown in

For an element with the normalized density

However, such an ordered ersatz material model becomes non-differentiable at points,

To consider the mass constraint defined in

The optimality criterion in

In the traditional solid/void topology optimization, the implicit floating projection constraint simulates 0/1 constraints of design variables and further modifies the design variables after filtering. In the multi-material design, all design variables should be constrained to the discrete values,

With the increase of

Once the algorithm is convergent for a given

The threshold,

Obviously, the above level-set functions are purely based on imaging processing and the resulting multi-material design may be far different from the element-based design expressed by the design variables,

In this section, several 2D and 3D numerical examples are presented to demonstrate the effectiveness of the proposed multi-material topology optimization algorithm. It is assumed that Poisson's ratio of all candidate materials is

As shown in

The optimized multi-material design is shown in

This example shows the multi-material design for a 2D beam as shown in

Material | ^{3}) |
Color | |
---|---|---|---|

m1 (solid) | 400 | 50 | Blue |

m2 (solid) | 700 | 80 | Red |

m3 (solid) | 1000 | 100 | Black |

Under the given mass constraint,

Different from the ordered SIMP method, the structural topology is formed by the floating projection constraint in the proposed multi-material topology optimization algorithm. For the comparison, the above example is re-calculated using 100

Material | ^{3}) |
Color | |
---|---|---|---|

m1 (solid) | 0.4 | 0.2 | Blue |

m2 (solid) | 0.7 | 0.6 | Red |

m3 (solid) | 1 | 1 | Black |

In practice, it preferably achieves a lightweight structure under single or multiple displacement constraints. The proposed multi-material topology optimization, similar to the FPTO method [

Taking the simply supported beam shown in

This example shows the multi-material designs for a 3D cantilever shown in

Material | ^{3}) |
Color phase | |
---|---|---|---|

m1 (solid) | 300 | 40 | Blue |

m2 (solid) | 600 | 70 | Red |

m3 (solid) | 1000 | 100 | Black |

When the compliance is minimized subject to the total mass constraint,

Next, the multi-material lightweight design is applied by specifying the displacement constraint at point A,

This paper proposed a new and simple multi-material topology optimization algorithm for minimizing compliance subject to a single mass constraint based on the FPTO method. Under the framework of the finite element analysis, the elemental normalized density is used as a single variable for designing structures composed of multiple materials without the increase of the computational burden. The ordered ersatz material model is proposed to interpolate the material property for the relaxed design variables linearly. Some 2D and 3D examples are presented to demonstrate the effectiveness of the proposed multi-material topology optimization algorithm and optimized multi-material designs are represented by the smooth interfaces between any two adjacent materials. Besides, the proposed algorithm can be extended to minimizing the total mass subject to single or multiple displacement constraints for a lightweight design of structures. Numerical results show that the multi-material designs could outperform the traditional solid/void designs, and this performance improvement increases when more candidate materials are involved in optimization.