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浙江大学学报(工学版)
浙江大学学报(工学版)  2026, Vol. 60 Issue (11): 2329-2337    DOI: 10.3785/j.issn.1008-973X.2026.11.002
机械工程     
三周期极小曲面多孔结构全参数化设计与关联模型
刘海宇1(),万琴1,许鑫1(),文兆隆1,杨海峰1,李洋2,*(),梅德庆2
1. 中国矿业大学 机电工程学院，江苏 徐州 221116
2. 浙江大学 机械工程学院，浙江 杭州 310058
Fully parametric design and parameter correlation models of triply periodic minimal surface porous structures
Haiyu LIU1(),Qin WAN1,Xin XU1(),Zhaolong WEN1,Haifeng YANG1,Yang LI2,*(),Deqing MEI2
1. College of Mechanical and Electrical Engineering, China University of Mining and Technology, Xuzhou 221116, China
2. College of Mechanical Engineering, Zhejiang University, Hangzhou 310058, China
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摘要：

构建Sheet构型三周期极小曲面（TPMS）多孔结构全参数调控方法，并建立设计参数与结构参数（孔隙率、比表面积）的数学关联模型. 采用隐式等值面函数构建Sheet构型TPMS多孔结构统一数学表达式，通过周期参数与曲率参数实现对结构孔隙率、几何形貌的定量精准调控；利用Network构型TPMS多孔结构与Sheet构型TPMS多孔结构的布尔差集几何转换关系，建立设计参数与结构参数的数学关联模型；确定5种典型曲面曲率参数的有效取值区间，规避结构断开、闭孔之类不可制造缺陷；采用三因素六水平正交试验完成模型精度验证. 研究结果表明，所建立的数学模型能够精准控制TPMS多孔结构的几何形貌，且孔隙率与比表面积的平均计算误差均小于0.3%.
关键词： 三周期极小曲面;  参数化设计;  多孔结构;  数学关联模型;  几何形貌    
Abstract:

A full parametric control method for sheet-type triply periodic minimal surface (TPMS) porous structures was developed, and mathematical correlation models relating design parameters to porosity and specific surface area were established. Implicit isosurface functions were used to unify the mathematical expression of sheet-type TPMS porous structures, achieving precise quantitative regulation of porosity and structural morphology via periodic and curvature parameters. Based on the geometric transformation of Boolean difference operations between network-type and sheet-type TPMS structures, mathematical correlation models were constructed. The valid curvature parameter ranges of five typical TPMS surfaces were determined to avoid unmanufacturable defects such as discontinuous frameworks and enclosed pores. Three-factor six-level orthogonal tests were conducted for model accuracy verification. The results show that the proposed models can accurately control the geometric morphology of TPMS porous structures, with the average calculation errors of porosity and specific surface area both below 0.3%.
Key words: triply periodic minimal surface    parametric design    porous structure    mathematical correlation model    geometric morphology
收稿日期: 2026-04-30 出版日期: 2026-10-16
CLC:  TH 123  
基金资助: 江苏省国家青年科学基金资助项目（BK20251641）；中央高校基础科研基金资助项目（2025QN1051）.
通讯作者: 李洋     E-mail: lhy@cumt.edu.cn;ts25050152p31ty@cumt.edu.cn;meyangli@zju.edu.cn
作者简介: 刘海宇（1997—），男，硕导，博士，从事超结构设计与增材制造研究. orcid.org/0000-0002-4797-9290. E-mail：lhy@cumt.edu.cn
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刘海宇
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梅德庆

引用本文:

刘海宇,万琴,许鑫,文兆隆,杨海峰,李洋,梅德庆. 三周期极小曲面多孔结构全参数化设计与关联模型[J]. 浙江大学学报(工学版), 2026, 60(11): 2329-2337.

Haiyu LIU,Qin WAN,Xin XU,Zhaolong WEN,Haifeng YANG,Yang LI,Deqing MEI. Fully parametric design and parameter correlation models of triply periodic minimal surface porous structures. Journal of ZheJiang University (Engineering Science), 2026, 60(11): 2329-2337.

链接本文:

https://www.zjujournals.com/eng/CN/10.3785/j.issn.1008-973X.2026.11.002    或    https://www.zjujournals.com/eng/CN/Y2026/V60/I11/2329

图 1  5种典型的TPMS曲面
TPMS函数公式
Primitive (P)$ \phi (x,y,z)=\cos\; ({\omega }_{x}x)+\cos\; ({\omega }_{y}y)+\cos\; ({\omega }_{z}z)=C $
Gyroid (G)$ \phi (x,y,z)=\sin\; ({\omega }_{x}x)\cos\; ({\omega }_{y}y)+\sin\; ({\omega }_{y}y)\cos\; ({\omega }_{z}z)+\sin\; ({\omega }_{z}z)\cos\; ({\omega }_{x}x)=C $
Diamond (D)$ \phi (x,y,z)=\cos\; ({\omega }_{x}x)\cos\; ({\omega }_{y}y)\cos\; ({\omega }_{z}z)+\sin\; ({\omega }_{x}x)\sin\; ({\omega }_{y}y)\sin\; ({\omega }_{z}z)=C $
I-Wrapped Package (IWP)$\phi(x, y, z)= 2[\cos\; (\omega_x x) \cos\; (\omega_y y)+\cos\; (\omega_y y) \cos\; (\omega_z z)+\cos\; (\omega_z z) \cos\; (\omega_x x)]-[\cos\; (2 \omega_x x)+\cos\; (2 \omega_y y)+\cos\; (2 \omega_z z)]=C $
Frenet-Rodrigues (FRD)$\phi(x, y, z)= 4 \cos\; (\omega_x x) \cos\; (\omega_y y) \cos\; (\omega_z z)-[\cos\; (2 \omega_x x) \cos\; (2 \omega_y y)+\cos\; (2 \omega_x x) \cos\; (2 \omega_z z)+\cos\; (2 \omega_y y) \cos\; (2 \omega_z z)]=C$
表 1  TPMS函数表达式
图 2  Network构型TPMS和Sheet构型TPMS多孔结构生成示意图
图 3  全参数Sheet-TPMS多孔结构生成示意图
图 4  Sheet-P多孔结构随着参数C、ω的变化情况
图 5  Network-TPMS和Sheet-TPMS多孔结构
图 6  不同曲率参数下的Network-P多孔结构
图 7  Network-TPMS多孔结构的孔隙率和比表面积与参数C或ω的关系
图 8  Network-TPMS多孔结构的结构参数计算
TPMS数学关联模型R2
P$ {\varphi }_{\rm{P}}=-0.286\;27C+0.5 $$ {R}^{2}=1.000 $
$ {S }_{{\rm{v}}{\text{-P}}}=(-0.062\;24{C}^{4}-0.177\;64{C}^{2}+1.175\;89)\cdot \omega $$ {R}^{2}=1.000 $
G$ {\varphi }_{{\mathrm{G}}}=-0.010\;45{C}^{3}-0.319\;85C+0.5 $$ {R}^{2}=1.000 $
$ {S }_{{\rm{v}}{\text{-G}}}=(-0.122\;99{C}^{6}+0.158\;89{C}^{4}-0.378\;96{C}^{2}+1.552\;62)\cdot \omega $$ {R}^{2}＞0.999 $
D$ {\varphi }_{{\mathrm{D}}}=-0.029\;3{C}^{3}-0.578\;26C+0.5 $$ {R}^{2}=1.000 $
$ {S }_{{\rm{v}}{\text{-D}}}=(-3.225\;11{C}^{6}+0.605\;17{C}^{4}-1.309\;35{C}^{2}+1.921\;71)\cdot \omega $$ {R}^{2}=1.000 $
IWP$ {\varphi }_{{\mathrm{IWP}}}=-4.728\;57\times 10^{-4} \times {C}^{4}-0.001\;77{C}^{3}-0.003\;49{C}^{2}-0.129\;32C+0.530\;04 $$ {R}^{2}＞0.999 $
$ {S }_{{\rm{v}}\text{-IWP}}=(-0.006\;86{C}^{4}-0.010\;5{C}^{3}-0.036\;03{C}^{2}+0.042\;46C+1.766\;96)\cdot \omega $$ {R}^{2}＞0.999 $
FRD$ {\varphi }_{{\mathrm{FRD}}}=-0.004\;76{C}^{3}-0.018\;57{C}^{2}-0.213\;31C+0.567\;81.000 $$ {R}^{2}=1.000 $
$ {S }_{\text{v-FRD}}=(-0.083\;17{C}^{5}-0.137\;23{C}^{4}+0.016\;8{C}^{3}-0.225\;55{C}^{2}+0.059\;39C +2.425\;02)\cdot \omega $$ {R}^{2}=1.000 $
表 2  5种Network-TPMS多孔结构的孔隙率与比表面积的数学关联模型
TPMS数学关联模型(C2 > C1)
P$ {\varphi }_{\rm{P}}=1-0.286\;27({C}_{2}-{C}_{1}) $
$ {S }_{{\rm{v}}{\text{-P}}}=[-0.062\;24（{C}_{1}^{4}+{C}_{2}^{4}）-0.177\;64({C}_{1}^{2}+{C}_{2}^{2})+2.351\;78]\cdot \omega $
G$ {\varphi }_{{\mathrm{G}}}=1-0.319\;85({C}_{2}-{C}_{1})-0.010\;45({C}_{2}^{3}-{C}_{1}^{3}) $
$ {S }_{{\rm{v}}{\text{-G}}}=[-0.122\;99({C}_{1}^{6}+{C}_{2}^{6})+0.158\;89({C}_{1}^{4}+{C}_{2}^{4})-0.378\;96({C}_{1}^{2}+{C}_{2}^{2})+3.105\;24]\cdot \omega $
D$ {\varphi }_{{\mathrm{D}}}=1-0.578\;26({C}_{2}-{C}_{1})-0.029\;3({C}_{2}^{3}-{C}_{1}^{3}) $
$ {S }_{{\rm{v}}{\text{-D}}}=[-3.225\;11({C}_{1}^{6}+{C}_{2}^{6})+0.605\;17({C}_{1}^{4}+{C}_{2}^{4})-1.309\;35({C}_{1}^{2}+{C}_{2}^{2})+3.843\;42]\cdot \omega $
IWP$ {\varphi }_{{\mathrm{IWP}}}=1-0.129\;32({C}_{2}-{C}_{1})-0.003\;49({C}_{2}^{2}-{C}_{1}^{2})-0.001\;77({C}_{2}^{3}-{C}_{1}^{3})-4.728\;57\times 10^{-4} \times ({C}_{2}^{4}-{C}_{1}^{4}) $
$ {S }_{{\rm{v}}\text{-IWP}}=[-0.006\;86({C}_{1}^{4}+{C}_{2}^{4})-0.010\;5({C}_{1}^{3}+{C}_{2}^{3})-0.036\;03({C}_{1}^{2}+{C}_{2}^{2})+0.042\;46({C}_{1}+{C}_{2}) +3.533\;92]\cdot \omega $
FRD$ {\varphi }_{{\mathrm{FRD}}}=1-0.213\;31({C}_{2}-{C}_{1})-0.018\;57({C}_{2}^{2}-{C}_{1}^{2})-0.004\;76({C}_{2}^{3}-{C}_{1}^{3}) $
${S }_{{\rm{v}}\text{-FRD}}=[-0.083\;17({C}_{1}^{5}+{C}_{2}^{5})-0.137\;23({C}_{1}^{4}+{C}_{2}^{4})+0.016\;8({C}_{1}^{3}+{C}_{2}^{3})-0.225\;55({C}_{1}^{2}+{C}_{2}^{2})+0.059\;39({C}_{1}+{C}_{2})+4.850\;04]\cdot \omega $
表 3  Sheet-TPMS多孔结构孔隙率与比表面积的数学关联模型
图 9  Sheet-G多孔结构的参数化生成和参数反向求解
图 10  结构参数拟合函数的误差
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