高效设计半规则多孔结构的变形行为控制

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"本文主要探讨了如何高效地设计具有所需变形行为的半规则多孔结构，利用双尺度设计方法，结合优化设计、模型简化和同质化理论，为增材制造提供理论支持。" 在工程和自然科学中，多孔材料和结构因其独特的性能和广泛的应用而受到广泛关注。它们在许多自然物体中都能找到，如骨骼、植物组织等，展现出非凡的物理和力学特性。随着增材制造技术的进步，我们现在能够精确控制地制造出这些复杂的多孔结构，但设计出具有特定变形行为的最优多孔结构仍然是一个计算复杂度极高的难题。这篇研究论文提出了一种新颖且高效的方法来解决这一问题。该方法专注于设计一种半规则的多孔结构，其中每个给定四边形或六边形网格元素内都包含一个孔洞。这种设计策略使得结构的变形行为可以根据需要进行定制。作者通过分析表达式和优化算法，将设计过程分为几个关键步骤： 1. **模型构建**：基于给定的四边形或六边形网格布局，创建结构的基础框架，其中每个单元内部包含一个孔洞。这样的半规则结构可以模拟不同的力学响应。 2. **设计优化**：采用优化算法来调整孔洞的形状、大小和位置，以达到期望的变形行为。这可能涉及到有限元分析和其他数值方法，以预测结构在不同载荷下的行为。 3. **模型简化**：为了降低计算复杂性，研究中采用了模型简化技术，如降阶模型，将复杂的多孔结构转化为简化的代数模型，同时保持对结构性能的准确预测。 4. **同质化理论**：利用同质化理论将多孔结构的宏观行为与微尺度特征联系起来。这种方法允许将复杂的多孔结构视为均匀的材料，简化了分析和设计过程。 5. **增材制造考虑**：考虑到实际制造过程，设计阶段就考虑了增材制造的限制，确保设计的可行性，并减少制造过程中的误差。这篇论文提供了一种创新的双尺度设计策略，旨在高效地创建具有特定变形特性的半规则多孔结构。这种方法对于优化材料利用率、提高结构性能以及在生物医学、航空航天、能源等领域实现定制化设计具有重要意义。通过结合先进的计算工具和理论，它为多孔结构的设计和制造开辟了新的可能性。

2.2. Topology optimization

Topology optimization aims to ﬁnd the optimal solid-void

material distribution within a design domain for structure perfor-

mance optimization. Various approaches have been proposed to

solve this problem including the homogenization-based approach,

SIMP, level set, ESO; see Ref. [38] for a recent review. Basically, the

approaches iteratively update the design’s geometry after simulat-

ing its physical behavior until convergence, and are computation-

ally very expensive.

These topology optimization approaches were initially studied

at the single macroscale [38]. Later on, approaches to design an

optimal RVE were proposed, without considering the macrostruc-

ture. The concurrent topology optimization approaches were also

proposed to simultaneously optimize the macroscale and micro-

scale topology [39] and are much more expensive to compute.

Zhang et al. proposed an approach to design the porous structure

of variable density for compliance minimization with a regular

hex-mesh conﬁguration and is very related to the work proposed

here [40]. In this work [40], the microstructure and material prop-

erties were ﬁrst built using the homogenization theory. Then, the

conventional topology optimization approach was used to gener-

ate the cell density distribution, which was then mapped to cell

structural parameters (circle/sphere radius here, for example).

However, the previous work only studied the problem of compli-

ance minimization, whereas the desired behavior is studied here.

Moreover, owing to the introduction of various numerical tech-

niques, the porous structure optimization is efﬁciently resolved.

2.3. Performance-oriented structure optimization

The design of the optimal topology of a structure, particularly

for additive manufacturing, has also attracted much interest in

geometric computing and computer graphics. Wang et al. opti-

mized skin-frame structures with the interior of model via updat-

ing the radii and numbers of struts of an initial frame [41].A

honeycomb-like Voronoi structures was introduced by Lu et al.

[42] by adaptively updating the centroidal Voronoi diagram and

its interior holes from an initially constructed Voronoi cells. These

approaches initiated the studies on interior design in the graphics

community, but do not fully address the optimization process, and

the aspect of computational cost. Recently, Panetta et al. [43] and

Schumacher et al. [44] also proposed excellent approaches for

microstructure design for elastic performance control. They basi-

cally constructed a library of microstructures by topology opti-

mization or geometric approaches, and then properly placed

these structures with each prescribed hexahedron cell according

to the design requirements or users’ prescriptions. However,

improper connection between the adjacent cells may appear, and

the computations are also expensive.

Model reduction is a commonly used technique in mechanical

engineering or geometric computing [28,45]. Xu et al. [28] demon-

strated an interactive method to design the Young’s modulus dis-

tribution of a material, and Li [45] proposed an approach for

elastic animation editing with space-time constraints based on

rotation-strain (RS) coordinates. In these approaches, model reduc-

tion was performed in the physical space, and the problem of por-

ous structure construction was not studied.

3. Problem statement and approach overview

A linear elasticity problem was studied here: Design a semireg-

ular porous structure so that the displacements of some of its

points are close to certain values, under prescribed external forces.

See also Fig. 2. In the following, the basics of the problem are ﬁrst

explained. Then, the problem of semiregular porous structure

design is mathematically deﬁned, followed by the approach

overview.

3.1. Semiregular structure and linear elasticity

3.1.1. Semiregular structure

A regular porous structure is a set of completely identical con-

nected porous cells arranged under a speciﬁed order, as illustrated

in Fig. 1(a).

A stochastic porous structure is a set of connected porous cells

arranged randomly without any order, as illustrated in Fig. 1(b).

A semiregular porous structure is a set of connected porous cells

of similar geometries arranged under a speciﬁed order, as illus-

trated in Fig. 1(c).

3.1.2. Notations about the design domain

is the solid domain under study (as in Fig. 4(a));

is the ﬁxed boundary of

;

is the boundary of

exerted by external forces

;

is the boundary of

when the displacements are set as the

design target u

;

is the quadrilateral mesh associated to

(as in Fig. 4(c));

e ¼fe

; 1 6 i 6 ng is the quad mesh set; e

2 e for the i-th mesh

element; e 2 e for a general mesh element;

r ¼fr

min

6 r

max

; 1 6 i 6 ng is the radius distribution over

model

;

M;r

is the semiregular porous structure obtained by digging a

hole within the i-th element e

of radius r

2 r (as in Fig. 4(e));

is a density model of prescribed density distributions

(as

in Fig. 4(g));

k¼0

are the density bases after the model reduction, for the

number of bases N.

3.1.3. Linear elasticity

Following the principle of linear elasticity studied here, the

deformation of

can then be stated as follows: Find the displace-

ment ﬁeld u satisfying the following equations including a PDE

(partial differentiable equation).

div

ðuÞ¼f; on X;

ðuÞn ¼

; on

;

u ¼ 0; on

;

u ¼ u

; on

;

ð1Þ

where divðÞ is the divergence of a vector, f is the body force vector

deﬁned on mesh vertices,

s is the exerted external force on the

boundary

of outer normal direction n; u

is the preset displace-

ment ﬁeld on boundary

and r is the stress tensor deﬁned follow-

ing the generalized Hooke’s Law:

Fig. 2. A simple illustration of the design problem input and output.

286 C. Xu et al. / Computers and Structures 182 (2017) 284–295

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