# A Hybrid Level Set Method for the Topology Optimization of Functionally Graded Structures

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

^{0}continuity of neighboring unit cells. A HLSM-based topology optimization model for the FGSs is established to maximize structural stiffness. The optimization model is solved by the optimality criteria (OC) algorithm. Two typical FGSs design problems are investigated, including thin-walled stiffened structures (TWSSs) and functionally graded cellular structures (FGCSs). In addition, additively manufactured FGCSs with different core layers are tested for bending performance. Numerical examples show that the HLSM is effective for designing FGSs like TWSSs and FGCSs. The bending tests prove that FGSs designed using HLSM are have a high performance.

## 1. Introduction

## 2. Geometric Modeling Based on HLSM

#### 2.1. Level Set Modeling

**X**). The dynamic structural interfaces are obtained by tracking the movement of the higher dimensional function. Its mathematical formulation is defined as follows [35,36]:

**X**is the physical spatial coordinate vector of the higher-dimensional space. D is the design domain that contains the complete structure. ∂Ω =

**Γ**

_{D}∪

**Γ**

_{N}∪

**Γ**

_{f}contains Γ

_{D}as the Dirichlet boundary, Γ

_{N}as the Neumann boundary and Γ

_{f}as the traction free boundary.

**X**).

_{p}(

**X**) of the geometric structures. The SDF is given as:

**X**).

**X**

_{b}is any point on the curve (or surface). ||•|| is the Euclidean norm. In addition, explicit modeling techniques have certain limitations in the parametric modeling of the microstructure with complex geometric features and rich variations. Theoretically, any type of microstructures can be digitally constructed based on the SDF. For example, using the above implicit modeling strategy, some microstructure unit cells with different topological forms are built for various FGSs, as shown in Figure 2.

#### 2.2. Hybrid Level Set Function

**w**(t) is an m × 1 time-dependent vector:

**w**(t):

**S**

_{C}is a selection matrix used to select the weight coefficient ${w}_{C}^{G}$(t) of the unit cell from the global weight coefficient

**w**(t). The dimension of the selection matrix

**S**

_{C}is 8 × m, which is defined as follows:

**N**

_{C}(

**x**) is written as:

_{0}= ξ

_{i}ξ, η

_{0}= η

_{i}η, ζ

_{0}= ζ

_{i}ζ, ξ

_{i }= ±1, η

_{i}= ±1, and ζ

_{i}= ±1. ξ, η and ζ are the internal node coordinates of the unit cell in local coordinate system.

**w**

_{C}(t) of the unit cell is given as:

_{1}, x

_{2}∈ Q and s∈R, we have sx

_{1}+ (1 − s)x

_{2}∈ Q. In a similar concept, the HLSF of two pre-defined unit cells can generate a new unit cell with a different geometrical configuration and volume fraction. The hybrid process of unit cells is defined as follows:

_{H}(

**X**) is the LSF of the unit cell with new structural features. Φ

_{1}(

**X**) and Φ

_{2}(

**X**) are the LSFs of two pre-defined master unit cells, respectively.

**s**is the design variable, which is the weight coefficient vector of the two pre-defined unit cells.

**w**(t) is a variable vector with respect to the pseudo-time t, and the value range is 0 ≤

**w**(t) ≤ 1. φ

_{1}(

**x**) and φ

_{2}(

**x**) are the LSFs of two pre-defined unit cells, respectively. The pseudo-time t adds dynamics to the weight coefficient

**w**(t), indirectly changing the implicit interface ∂Ω.

**w**(t) are all the same, the geometric configuration of the new unit cell changes uniformly in space. (2) Conversely, when the values of the

**w**(t) are different, the geometry of the unit cell takes on an asymmetric evolution. Therefore, when the weight coefficients correspondingly change with the external loading and boundary conditions during the optimization process, a series of microstructure unit cells with diverse geometries are distributed in the macrostructure. The obtained unit cells can exhibit directional stiffness, and thus provide enormous design freedom for the optimal design of the FGSs.

**w**

_{C}(t), the continuous geometry transition of neighboring unit cells can be expected. For example, I-WP TPMSs are taken as the representative volume elements (RVEs) to illustrate the interpolation process, as shown in Figure 5. C

_{1}–C

_{4}are RVEs with different volume fraction. This diagram also demonstrates the basic idea of the strategy of geometrical continuity. The geometry continuity of neighboring unit cells depends on the continuity of the weight coefficient vector

**w**

_{C}(t). During the optimization process, the local design variables are updated by the interpolation of global design variables. According to the definition in Figure 3, neighboring unit cells share the same weight coefficients on common boundaries. Thus, the interpolation function the global weight coefficients have at least C

^{0}continuity. With this updating strategy of design variables, the proposed HLSM can naturally guarantee geometry continuity without imposing any extra constraints.

## 3. Topology Optimization Model and Sensitivity Analysis

#### 3.1. Topology Optimization Model

**ε**is the strain field.

**u**is the displacement field.

**E**is the material elasticity tensor of solid material. a

_{Φ}(

**u**,

**v**) and l

_{Φ}(

**v**) are the bilinear and linear term of the state equation of the linear system.

**v**is the virtual displacement field belonging to the space U spanned by the kinematically admissible set of displacement. μ is the prescribed volume fraction. g(Φ) is volume of the whole design domain. w

_{k}(t) represents the global weight coefficient of the k’th node in the design domain, k is the number of discrete element nodes in the design domain.

_{Φ}(

**u**,

**v**) = l

_{Φ}(

**v**) is given in its weak form. The energy bilinear term and load liner term are defined as:

#### 3.2. Sensitivity Analysis

**n**is the normal vector.

_{n}. This paper takes the global weight coefficients as design variables. By substituting the HLSF into the Hamilton–Jacobi partial differential equation, the original space- and time-coupled Hamilton–Jacobi partial differential equation is decoupled into a system of ordinary differential equations. Additionally, the ordinary differential equation only for pseudo-time t is given as:

_{n}is obtained to drive the movement of ∂Ω in the cell:

## 4. Numerical Implementation

^{−3}or the maximum loop number is reached.

## 5. Numerical Examples

#### 5.1. Thin-Walled Stiffened Structures

_{i}(t) is the original sensitivity of the objective function J(Φ) with respect to the design variable

**w**(t), which is on the node of the parallel element in the Z-direction. ∂J(Φ)/w

_{i}(t) is the averaged sensitivities. NE is the number of nodes (global design variables) in the Z-direction of unit cell. N is the number of unit cell nodes (global design variables) on the same layer section. The sensitivity constraints can perfectly control the geometry of the stiffener unit cell, as shown in Figure 7c,d. It should be noted that the sensitivity averaging operation is included in the optimization process of the TWSSs, rather than the post-processing. Therefore, it does not affect the mechanical properties and simulation accuracy of the optimized results. On the contrary, the proposed method ensures the regular geometry shape of stiffening ribs and effectively improves the manufacturability of the optimized results.

_{1}= 2, and H

_{2}= 30. The upper thin wall is the non-design domain, and the lower thin wall is the design domain. The structure is fixed at both the left and right side. A concentrated load F = 10 is applied to the center of the upper surface of the structure. The design domain is discretized into 6 × 6 × 1 unit cells, and each unit cell is further discretized by 30 × 30 × 32 eight-node hexahedral elements. The upper thin wall is the non-design panel and contains two layers of elements. The volume fraction of the stiffened structure is set to μ = 0.3.

#### 5.2. Functionally Graded Cellular Structures

#### 5.2.1. Graded Cellular Structure

^{3}. The loading–deformation and loading–stress curves for the two structures are plotted in Figure 15. In the linear elastic region, the maximum deformation and stress of the uniform cellular structure are significantly larger than those of the optimized FGCSs. The stiffness of the structures is calculated according to the loading–deformation curve, which is 631.01 N/mm and 1323.94 N/mm, respectively. The FEA results show that the proposed method significantly improves the bearing capacity of cellular structure. Figure 16 shows the deformation and stress nephograms of the uniform cellular structure and the optimized FGCS under a loading condition of 800 N. Compared with the uniform cellular structure, more materials are distributed in the area of high stress for FGCS. This strategy will help us to improve the utilization ratio of the materials and load-bearing property of the macrostructure. It should be noted that the unit cells distributed in the cellular structure all evolve from the I-WP TPMS. The configurations of unit cells are similar in geometry, which may restrict the design space and its properties to some extent. However, they are still flexible enough for similar unit cells to achieve satisfactory properties and various functionalities [44,45].

#### 5.2.2. Lattice Sandwich Structure

_{1}= 96, H

_{2}= 3. The core layer is the design domain and the thin and hard panels are non-design domains. The support domain is L/10 from both sides of the bottom of the design domain. The distributed force F = 10 N is applied vertically downward in the middle of the upper face. The design domain is discretized into 10 × 1 × 3 unit cells, and each unit cell is further discretized by 24 × 24 × 4 eight-node hexahedral elements. The volume fraction of the LSS is set to μ = 0.35.

## 6. AM and Experimental Validation

#### 6.1. Geometric Modeling and AM Process

#### 6.2. Mechanical Test

## 7. Conclusions

**(1) High versatility and effectiveness:**This paper constructs a design strategy to drive the topology optimization of macro/microstructures. By presetting the level set functions of various structural unit cells, high-performance FGSs with different structural forms and functional properties could be obtained using level set evolution. Hence, HLSM is a convenient tool for designers.

**(2) Perfect geometric continuity:**According to the definition of weight coefficients, neighboring unit cells share the same global weight coefficients on common boundaries. Additionally, the local design variables are updated by the interpolation of global design variables. Thus, the interpolation function of the global weight coefficients has at least C

^{0}continuity. With this updating strategy of design variables, the proposed HLSM can naturally guarantee perfect geometry continuity without imposing any extra constraints.

**(3) Easy-to-manufacture:**For TWSSs, the geometry constraint is proposed to ensure regular geometry shape of the stiffening ribs, so that TWSSs can be manufactured by welding or casting processes. Hence, excellent manufacturability is obtained. For FGCSs, the self-supporting lattice unit cell could be employed to design FGCSs, which are easy to fabricate through AM.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Implicit modeling based on LSF and 3D geometric model. (

**a**) The LSF. (

**b**) Contour of zero level set. (

**c**) 3D geometric model.

**Figure 2.**Unit cells constructed by LSF. (

**a**) Triply periodic minimal surface (I-WP). (

**b**) Cross-shaped stiffening rib unit cell. (

**c**) X-shaped stiffening rib unit cell. (

**d**) Hybrid stiffening rib unit cell. (

**e**) Lattice sandwich structure unit cell.

**Figure 7.**Geometry constraints on stiffening rib unit cell. (

**a**) Unit cell without geometry constraint. (

**b**) Discrete elements with different sensitivities. (

**c**) Discrete elements with average sensitivities. (

**d**) Unit cell with geometry constraint.

**Figure 8.**Pre-defined unit cells and design domain of TWSS. (

**a**) X-shaped stiffening rib unit cells with different volume fractions. (

**b**) Loading and boundary conditions for 3D TWSS.

**Figure 10.**Optimized TWSS using the HLSM. (

**a**) Optimized TWSS. (

**b**) Optimized layout of stiffening ribs. (

**c**) Top view of stiffening ribs.

**Figure 11.**FEA results of TWSS. (

**a**) Deformation nephogram of TWSS with uniform stiffening ribs layout. (

**b**) Deformation nephogram of TWSS with optimized graded stiffening ribs layout. (

**c**) Comparison of the FEA results.

**Figure 12.**Pre-defined unit cells and design domain of FGCS. (

**a**) I-WP TPMS with different volume fractions. (

**b**) Loading and boundary conditions for 3D Michell beam.

**Figure 15.**Comparison of maximum deformation and stress between uniform cellular structure and optimized FGCS.

**Figure 17.**Pre-defined unit cells and design domain of LSS. (

**a**) LSS unit cells with different volume fractions. (

**b**) Loading and boundary conditions for 3D LSS.

**Figure 19.**Optimized GLSS using the HLSM. (

**a**) Optimized GLSS. (

**b**) Front view of GLSS. (

**c**) First core layer of GLSS. (

**d**) Second core layer of GLSS. (

**e**) Third core layer of GLSS.

**Figure 20.**CAD models and printed structures of the test samples. (

**a**) CAD model of optimized GLSS. (

**b**) CAD model of LSS with uniform I-WP TPMS. (

**c**) CAD model of LSS with uniform BCC unit cells. (

**d**) Printed sample of (

**a**). (

**e**) Printed sample of (

**b**). (

**f**) Printed sample of (

**c**).

Model Size | Material Properties | Loading | ||||
---|---|---|---|---|---|---|

W | L | H_{1} | H_{2} | Elastic Modulus | Poisson Ratio | Concentrated Load |

180 mm | 180 mm | 2 mm | 5 mm | 2 × 10^{5} MPa | 0.3 | 1000 N |

Pre-Defined Unit Cells | Topological Configuration | Uniform Structure | Optimized TWSS |
---|---|---|---|

Volume fractions: 0.15, 0.4 | μ = 0.3, J = 229 | MD: 0.37 mm AD: 0.12 mm | MD: 0.31 mm AD: 0.09 mm |

Volume fractions: 0.4, 0.5 | μ = 0.45, J = 215 | MD: 0.32 mm AD: 0.11 mm | MD: 0.29 mm AD: 0.09 mm |

**Table 3.**The influence of pre-defined unit cells with different volume fractions and topological configurations on optimized results.

Pre-Defined Unit Cells | 3D Michell Beams | 3D Cantilever Beams |
---|---|---|

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## Share and Cite

**MDPI and ACS Style**

Fu, J.; Shu, Z.; Gao, L.; Zhou, X.
A Hybrid Level Set Method for the Topology Optimization of Functionally Graded Structures. *Materials* **2022**, *15*, 4483.
https://doi.org/10.3390/ma15134483

**AMA Style**

Fu J, Shu Z, Gao L, Zhou X.
A Hybrid Level Set Method for the Topology Optimization of Functionally Graded Structures. *Materials*. 2022; 15(13):4483.
https://doi.org/10.3390/ma15134483

**Chicago/Turabian Style**

Fu, Junjian, Zhengtao Shu, Liang Gao, and Xiangman Zhou.
2022. "A Hybrid Level Set Method for the Topology Optimization of Functionally Graded Structures" *Materials* 15, no. 13: 4483.
https://doi.org/10.3390/ma15134483