Triply periodic minimal surface(TPMS)porous structures:from multi-scale design,precise additive manufacturing to multidisciplinary applications

Inspired by natural porous architectures,numerous attempts have been made to generate porous structures.Owing to the smooth surfaces,highly interconnected porous architectures,and mathematical controllable geometry features,triply periodic minimal surface(TPMS)is emerging as an outstanding solution to constructing porous structures in recent years.However,many advantages of TPMS are not fully utilized in current research.Critical problems of the process from design,manufacturing to applications need further systematic and integrated discussions.In this work,a comprehensive overview of TPMS porous structures is provided.In order to generate the digital models of TPMS,the geometry design algorithms and performance control strategies are introduced according to diverse requirements.Based on that,precise additive manufacturing methods are summarized for fabricating physical TPMS products.Furthermore,actual multidisciplinary applications are presented to clarify the advantages and further potential of TPMS porous structures.Eventually,the existing problems and further research outlooks are discussed.

Innovative Design and Additive Manufacturing of Regenerative Cooling Thermal Protection System Based on the Triply Periodic Minimal Surface Porous Structure

The new regenerative cooling thermal protection system exhibits the multifunctional characteristics of load-carrying and heat exchange cooling,which are fundamental for the lightweight design and thermal protection of hypersonic vehicles.Triply periodic minimal surface(TPMS)is especially suitable for the structural design of the internal cavity of regenerative cooling structures owing to its excellent structural characteristics.In this study,test pieces were manufactured using Ti6Al4V lightweight material.We designed three types of porous test pieces,and the interior was filled with a TPMS lattice(Gyroid,Primitive,I-WP)with a porosity of 30%.All porous test pieces were manufactured via selective laser melting technology.A combination of experiments and finite element simulations were performed to study the selection of the internal cavity structure of the regenerative cooling thermal protection system.Hence,the relationship between the geometry and mechanical properties of a unit cell is established,and the deformation mechanism of the porous unit cell is clarified.Among the three types of porous test pieces,the weight of the test piece filled with the Gyroid unit cell was reduced by 8.21%,the average tensile strength was reduced by 17.7%compared to the solid test piece,while the average tensile strength of the Primitive and I-WP porous test pieces were decreased by 30.5%and 33.3%,respectively.Compared with the other two types of unit cells,Gyroid exhibited better mechanical conductivity characteristics.Its deformation process was characterised by stretching,shearing,and twisting,while the Primitive and I-WP unit cells underwent tensile deformation and tensile and shear deformation,respectively.The finite element predictions in the study agree well with the experimental results.The results can provide a basis for the design of regenerative cooling thermal protection system.

The effect of porosity on the mechanical properties of 3D-printed triplyperiodic minimal surface (TPMS) bioscaffold

Prevailing tissue degeneration caused by musculoskeletal maladies poses a great demand on bioscaffolds,which are artificial,biocompatible structures implanted into human bodies with appropriate mechanical properties.Recent advances in additive manufacturing,i.e.,3D printing,facilitated the fabrication of bioscaffolds with unprecedented geometrical complexity and size flexibility and allowed for the fabrication of topologies that would not have been achieved otherwise.In our work,we explored the effect of porosity on themechanical properties of a periodic cellular structure.The structure was derived from the mathematically created triply periodic minimal surface(TPMS),namely the Sheet-Diamond topology.First,we employed a series of software including MathMod,Meshmixer,Netfabb and Cura to design the model.Then,we utilized additive manufacturing technology to fabricate the cellular structures with designated scale.Finally,we performed compressive testing to deduce the mechanical properties of each cellular structure.Results showed that,in comparison with the highporosity group,the yield strength of the low-porosity group was 3 times higher,and the modulus was 2.5 times larger.Our experiments revealed a specific relationship between porosity and Young’s modulus of PLA-made Sheet-Diamond TPMS structure.Moreover,it was observed that the high-and low-porosity structures failed through distinctive mechanisms,with the former breaking down via buckling and the latter via micro-fracturing.

Enhanced energy-absorbing and sound-absorbing capability of functionally graded and helicoidal lattice structures with triply periodic minimal surfaces

Lattice structures have drawn much attention in engineering applications due to their lightweight and multi-functional properties.In this work,a mathematical design approach for functionally graded(FG)and helicoidal lattice structures with triply periodic minimal surfaces is proposed.Four types of lattice structures including uniform,helicoidal,FG,and combined FG and helicoidal are fabricated by the additive manufacturing technology.The deformation behaviors,mechanical properties,energy absorption,and acoustic properties of lattice samples are thoroughly investigated.The load-bearing capability of helicoidal lattice samples is gradually improved in the plateau stage,leading to the plateau stress and total energy absorption improved by over 26.9%and 21.2%compared to the uniform sample,respectively.This phenomenon was attributed to the helicoidal design reduces the gap in unit cells and enhances fracture resistance.For acoustic properties,the design of helicoidal reduces the resonance frequency and improves the peak of absorption coefficient,while the FG design mainly influences the peak of absorption coefficient.Across broad range of frequency from 1000 to 6300 Hz,the maximum value of absorption coefficient is improved by18.6%-30%,and the number of points higher than 0.6 increased by 55.2%-61.7%by combining the FG and helicoidal designs.This study provides a novel strategy to simultaneously improve energy absorption and sound absorption properties by controlling the internal architecture of lattice structures.

Adaptive enhancement design of triply periodic minimal surface lattice structure based on non-uniform stress distribution

The Schwarz primitive triply periodic minimal surface(P-type TPMS)lattice structures are widely used.However,these lattice structures have weak load-bearing capacity compared with other cellular structures.In this paper,an adaptive enhancement design method based on the non-uniform stress distribution in structures with uniform thickness is proposed to design the P-type TPMS lattice structures with higher mechanical properties.Two types of structures are designed by adjusting the adaptive thickness distribution in the TPMS.One keeps the same relative density,and the other keeps the same of non-enhanced region thickness.Compared with the uniform lattice structure,the elastic modulus for the structure with the same relative density increases by more than 17%,and the yield strength increases by more than 10.2%.Three kinds of TPMS lattice structures are fabricated by laser powder bed fusion(L-PBF)with 316L stainless steel to verify the proposed enhanced design.The manufacture-induced geometric deviation between the as-design and as-printed models is measured by micro X-ray computed tomography(μ-CT)scans.The quasi-static compression experimental results of P-type TPMS lattice structures show that the reinforced structures have stronger elastic moduli,ultimate strengths,and energy absorption capabilities than the homogeneous P-TPMS lattice structure.

Computing Quantum Bound States on Triply Punctured Two-Sphere Surface

We are interested in a quantum mechanical system on a triply punctured two-sphere surface with hyperbolic metric. The bound states on this system are described by the Maass cusp forms （MCFs） which are smooth square integrable eigenfunctions of the hyperbolic Laplacian. Their discrete eigenvalues and the MCF are not known analytically. We solve numerically using a modified Hejhal and Then algorithm, which is suitable to compute eigenvalues for a surface with more than one cusp. We report on the computational results of some lower-lying eigenvalues for the triply punctured surface as well as providing plots of the MCF using GridMathematica.

TRIPLY BRIDGED BINUCLEAR COMPLEX OF OXOMOLYBDENUM(Ⅴ)WITH MERCAPTOPHENOLATE LIGANDS

Reaction of(n-Bu_4N)_xNa_(4-x)[a-Mo_8O_(26)](x=1-4)and mpH_2(mpH_2=o-mercaptophenol) leads to the formation of compound(n-Bu_4N)[Mo_2O_3(mp)_3Na(EtOH)_2(MeOH)],It crystallizes in the space group P2_1/n with four molecules per unit cell.The cell dimension is a=12.776(3),b=14.154(5),c=26.281(7)(?)=95.50(2)°,V=4730.5 (?),d_c=1.407 g/cm^3.Full-matrix refinement of 496 parameters on 3348 reflection data gave final R(R_w) value of 0.072(0.072).The two oxomolybdenum-mercaptophenolate,units are triply bridged by an inorganic oxygen atom and a doubly bridging mp^(2-)ligand.

Doubly and Triply Periodic Waves Solutions for the KdV Equation

Based on the arbitrary constant solution, a series of explicit doubly periodic solutions and triply periodic solutions for the Korteweg-de Vries (KdV) equation are first constructed with the aid of the Darboux transformation method.

Structural insights into the unusual core photocomplex from a triply extremophilic purple bacterium,Halorhodospira halochloris

Halorhodospira(Hlr.)halochloris is a triply extremophilic phototrophic purple sulfur bacterium,as it is thermophilic,alkaliphilic,and extremely halophilic.The light-harvesting-reaction center(LH1–RC)core complex of this bacterium displays an LH1-Q_(y)transition at 1,016 nm,which is the lowest-energy wavelength absorption among all known phototrophs.Here we report the cryo-EM structure of the LH1–RC at 2.42?resolution.The LH1 complex forms a tricyclic ring structure composed of 16αβγ-polypeptides and oneαβ-heterodimer around the RC.From the cryo-EM density map,two previously unrecognized integral membrane proteins,referred to as protein G and protein Q,were identified.Both of these proteins are single transmembrane-spanning helices located between the LH1 ring and the RC Lsubunit and are absent from the LH1–RC complexes of all other purple bacteria of which the structures have been determined so far.Besides bacteriochlorophyll b molecules(B1020)located on the periplasmic side of the Hlr.halochloris membrane,there are also two arrays of bacteriochlorophyll b molecules(B800 and B820)located on the cytoplasmic side.Only a single copy of a carotenoid(lycopene)was resolved in the Hlr.halochloris LH1–α3β3 and this was positioned within the complex.The potential quinone channel should be the space between the LH1–α3β3 that accommodates the single lycopene but does not contain aγ-polypeptide,B800 and B820.Our results provide a structural explanation for the unusual Q_(y)red shift and carotenoid absorption in the Hlr.halochloris spectrum and reveal new insights into photosynthetic mechanisms employed by a species that thrives under the harshest conditions of any phototrophic microorganism known.

Gyroid Triply Periodic Minimal Surface Lattice Structure Enables Improved Superelasticity of CuAlMn Shape Memory Alloy

Improving the shape memory effect and superelasticity of Cu-based shape memory alloys(SMAs)has always been a research hotspot in many countries.This work systematically investigates the effects of Gyroid triply periodic minimal surface(TPMS)lattice structures with different unit sizes and volume fractions on the manufacturing viability,compressive mechanical response,superelasticity and heating recovery properties of CuAlMn SMAs.The results show that the increased specific surface area of the lattice structure leads to increased powder adhesion,making the manufacturability proportional to the unit size and volume fraction.The compressive response of the CuAlMn SMAs Gyroid TPMS lattice structure is negatively correlated with the unit size and positively correlated with the volume fraction.The superelastic recovery of all CuAlMn SMAs with Gyroid TPMS lattice structures is within 5%when the cyclic cumulative strain is set to be 10%.The lattice structure shows the maximum superelasticity when the unit size is 3.00 mm and the volume fraction is 12%,and after heating recovery,the total recovery strain increases as the volume fraction increases.This study introduces a new strategy to enhance the superelastic properties and expand the applications of CuAlMn SMAs in soft robotics,medical equipment,aerospace and other fields.

Bioceramic scaffolds with triply periodic minimal surface architectures guide early-stage bone regeneration

The pore architecture of porous scaffolds is a critical factor in osteogenesis,but it is a challenge to precisely configure strut-based scaffolds because of the inevitable filament corner and pore geometry deformation.This study provides a pore architecture tailoring strategy in which a series of Mg-doped wollastonite scaffolds with fully interconnected pore networks and curved pore architectures called triply periodic minimal surfaces(TPMS),which are similar to cancellous bone,are fabricated by a digital light processing technique.The sheet-TPMS pore geometries(s-Diamond,s-Gyroid)contribute to a 3‒4-fold higher initial compressive strength and 20%-40%faster Mg-ion-release rate compared to the other-TPMS scaffolds,including Diamond,Gyroid,and the Schoen’s I-graph-Wrapped Package(IWP)in vitro.However,we found that Gyroid and Diamond pore scaffolds can significantly induce osteogenic differentiation of bone marrow mesenchymal stem cells(BMSCs).Analyses of rabbit experiments in vivo show that the regeneration of bone tissue in the sheet-TPMS pore geometry is delayed;on the other hand,Diamond and Gyroid pore scaffolds show notable neo-bone tissue in the center pore regions during the early stages(3-5 weeks)and the bone tissue uniformly fills the whole porous network after 7 weeks.Collectively,the design methods in this study provide an important perspective for optimizing the pore architecture design of bioceramic scaffolds to accelerate the rate of osteogenesis and promote the clinical translation of bioceramic scaffolds in the repair of bone defects.

Mechanical Response of Triply Periodic Minimal Surface Structures Manufactured by Selective Laser Melting with Composite Materials

It is of significance but remains a pivotal challenge to simultaneously enhance the strength and lightweight levels of porous structures.We provide an innovative strategy to improve the strength of porous structures with unchanged lightweight levels by applied composite materials.Selective laser melting(SLM)is convenient for integral forming of materials and structures.Hence,in this study,the research about the mechanical response of triply periodic minimal surfaces(TPMS)porous structures with 316 L and composites fabricated by SLM was conducted.The compression test and finite element method(FEM)were used to characterize mechanical properties.The composite structures exhibit enhanced elastic modulus,yield strength,unvaried lightweight level and refined grain microstructure,which are difficult to realize for porous structures made by pure 316 L materials.The elastic modulus,yield strength,plateau stress and energy absorption of composites were 3187.50,67.73,15.24 and 17.09 MJ/m^(3),respectively.

3D-printed strontium-incorporatedβ-TCP bioceramic triply periodic minimal surface scaffolds with simultaneous high porosity,enhanced strength,and excellent bioactivity

In bone tissue engineering,scaffolds with excellent mechanical and bioactive properties play prominent roles in space maintaining and bone regeneration,attracting increasingly interests in clinical practice.In this study,strontium-incorporatedβ-tricalcium phosphate(β-TCP),named Sr-TCP,bioceramic triply periodic minimal surface(TPMS)structured scaffolds were successfully fabricated by digital light processing(DLP)-based 3D printing technique,achieving high porosity,enhanced strength,and excellent bioactivity.The Sr-TCP scaffolds were first characterized by element distribution,macrostructure and microstructure,and mechanical properties.Notably,the compressive strength of the scaffolds reached 1.44 MPa with porosity of 80%,bringing a great mechanical breakthrough to porous scaffolds.Furthermore,the Sr-TCP scaffolds also facilitated osteogenic differentiation of mouse osteoblastic cell line(MC3T3-E1)cells in both gene and protein aspects,verified by alkaline phosphatase(ALP)activity and polymerase chain reaction(PCR)assays.Overall,the 3D-printed Sr-TCP bioceramic TPMS structured scaffolds obtained high porosity,boosted strength,and superior bioactivity at the same time,serving as a promising approach for bone regeneration.

Fabrication of Ceramic-Polymer Piezo-Composites with Triply Periodic Minimal Interfaces via Digital Light Processing

The geometry of the phase interface in co-continuous piezoelectric composites is critical in improving their piezo-electric properties.However,conventional co-continuous piezoelectric composites are mostly simple structures such as wood stacks or honeycombs,which are prone to stress concentrations at the joints,thus reducing the fatigue service performance and force-electric conversion efficiency of piezoelectric composites.Such simple structures limit further improvements in the overall performance of co-continuous piezoelectric composites.In this study,based on the digital light processing 3D printing method,we investigated the influence of three dif-ferent structures-the gyroid,diamond,and woodpile interfaces-on the piezoelectric and mechanical properties of co-continuous ceramic/polymer piezoelectric composites.These findings demonstrate that the gyroid and di-amond interfaces outperformed the ceramic skeleton of the woodpile interface in terms of both mechanical and electrical properties.When the ceramic volume percentage was 50%,the piezo-composite of the gyroid surface exhibited the greatest hydrostatic figure of merit(HFOM),reaching 4.23×10^(−12) Pa^(−1),and its piezoelectric coeffi-cient(d_(33))and relative dielectric constant(εr)reached 115 pC/N and 748,respectively.The research results lay the foundation for the application of co-continuous piezoelectric composites in underwater communication and detection.

Triply Factorised Groups and the Structure of Skew Left Braces

The algebraic structure of skew left brace has proved to be useful as a source of settheoretic solutions of theYang-Baxter equation.We study in this paper the connections between left and right π-nilpotency and the structure of finite skew left braces.We also study factorisations of skew left braces and their impact on the skew left brace structure.As a consequence of our study,we define a Fitting-like ideal of a left brace.Our approach depends strongly on a description of a skew left brace in terms of a triply factorised group obtained from the action of the multiplicative group of the skew left brace on its additive group.

Analysis of the Triply Heavy Baryon States with QCD Sum Rules

In this article, we study the (1/2) ± and (3/2)± triply heavy baryon states in a systematic way by subtracting the contributions from the corresponding (1/2)■ and (3/2)■ triply heavy baryon states with the QCD sum rules, and make reasonable predictions for their masses.

基于点阵结构的混合结构的各向同性研究

点阵多孔结构在力学性能上通常表现为各向异性,在多孔骨植入物工程应用中,可能因弹性模量各向异性在部分方向发生应力屏蔽效应。针对上述问题,本文借助均匀化理论和有限元方法分析了典型点阵结构的力学性能及各向同性。研究表明,结构孔隙度对各向同性有较大影响,孔隙度越大,齐纳因子越大。其中,Gyroid、边心立方单元各向同性最优,I-WP、体心立方单元均匀性较差。对Primitive、体心立方单元进行单元混合,优化结构各向同性,P60BCC65,P65BCC70,P70BCC70和P75BCC80结构齐纳因子均接近于1,其中P75BCC80点阵结构的齐纳因子为1.02,孔隙度为62.4%,在适用的骨科植入体孔隙度范围之内。

Co-continuous Composite Material Design Using the Volumetric Distance Field Based on Analytic Shape Functions

An effective and simple design method for co-continuous composite material construction is proposed by using a hybrid methodology with triply periodic minimal surface( TPMS) cellular topology and the volumetric distance field( VDF). After generating a set of VDF-based features for the given exterior shape and desired internal core structure,a series of simple modifications in distance fields enabled us to obtain an arbitrarily-shaped complex co-continuous composite material computational model. Design results and manufactured prototypes through 3 D printing technology show that the proposed methodology has the potential to open a new paradigm for producing multifunctional next generation co-continuous composite materials which are impossible to design and manufacture using traditional CAD and CAM.

Design and Mechanical Characterization of an S-Based TPMS Hollow Isotropic Cellular Structure

Cellular structures are regarded as excellent candidates for lightweight-design,load-bearing,and energy-absorbing applications.In this paper,a novel S-based TPMS hollow isotropic cellular structure is proposed with both superior load-bearing and energy-absorbing performances.The hollow cellular structure is designed with Boolean operation based on the Fischer-Koch(S)implicit triply periodic minimal surfaces(TPMS)with different level parameters.The anisotropy and effective elasticity properties of cellular structures are evaluated with the numerical homogenization method.The finite element method is further conducted to analyze the static mechanical performance of hollow cellular structure considering the size effect.The compression experiments are finally carried out to reveal the compression properties and energy-absorption characteristics.Numerical results of the Zener ratio proved that the S-based hollow cellular structure tends to be isotropic,even better than the sheet-based Gyroid TPMS.Compared with the solid counterpart,the S-based hollow cellular structure has a higher elastic modulus,better load-bearing and energy absorption characteristics.

（e, 2e） Investigations of the Simultaneous Ionization and Excitation of Helium to He＋（n=2）

Three-Coulomb-wave method is employed to treat the process of （e, 2e） simultaneous ion- ization and excitation to the n=2 state of helium, with radial and angular correlated wave-function of He target. The triple differential cross sections are calculated and analyzed in very asymmetric coplanar geometry at incident energies of 5.50, 1.50 and 0.57 keV. Results are compared with the absolute measurements and the theoretical first and second Born approximation. The present triply differential cross section （TDCS） is found to be in good agreement with experimental data qualitatively. The distinguishing feature noted in TDCS structure is the presence of intense recoil peak that for certain parameters is even larger than the binary peak, an unusual feature for the single-ionization process at high and intermediate energies.