A novel triple periodic minimal surface-like plate lattice and its data-driven optimization method for superior mechanical properties

Lattice structures can be designed to achieve unique mechanical properties and have attracted increasing attention for applications in high-end industrial equipment,along with the advances in additive manufacturing(AM)technologies.In this work,a novel design of plate lattice structures described by a parametric model is proposed to enrich the design space of plate lattice structures with high connectivity suitable for AM processes.The parametric model takes the basic unit of the triple periodic minimal surface(TPMS)lattice as a skeleton and adopts a set of generation parameters to determine the plate lattice structure with different topologies,which takes the advantages of both plate lattices for superior specific mechanical properties and TPMS lattices for high connectivity,and therefore is referred to as a TPMS-like plate lattice(TLPL).Furthermore,a data-driven shape optimization method is proposed to optimize the TLPL structure for maximum mechanical properties with or without the isotropic constraints.In this method,the genetic algorithm for the optimization is utilized for global search capability,and an artificial neural network(ANN)model for individual fitness estimation is integrated for high efficiency.A set of optimized TLPLs at different relative densities are experimentally validated by the selective laser melting(SLM)fabricated samples.It is confirmed that the optimized TLPLs could achieve elastic isotropy and have superior stiffness over other isotropic lattice structures.

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.

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.

Control mesh representation of a class of minimal surfaces

Minimal surface is extensively employed in many areas. In this paper, we propose a control mesh representation of a class of minimal surfaces, called generalized helicoid minimal surfaces, which contain the right helicoid and catenoid as special examples. We firstly construct the Bézier-like basis called AHT Bézier basis in the space spanned by {1, t, sint, cost, sinht, cosht}, t∈[0,α], α∈[0,5π/2]. Then we propose the control mesh representation of the generalized helicoid using the AHT Bézier basis. This kind of representation enables generating the minimal surfaces using the de Casteljau-like algorithm in CAD/CAGD mod- elling systems.

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.

DEGREE 3 ALGEBRAIC MINIMAL SURFACES IN THE 3-SPHERE

We give a local analytic characterization that a minimal surface in the 3-sphere S3 C R4 defined by an irreducible cubic polynomial is one of the Lawson＇s minimal tori. This provides an alternative proof of the result by Perdomo （Characterization of order 3 algebraic immersed minimal surfaces of S3, Geom. Dedicata 129 （2007）, 23 34）.

Robust construction of minimal surface from general initial mesh

We analyze three commonly used energy functions in solving Plateau-Mesh Prob- lem, that is, Dirichlet, area, and the discrete mean curvature（DMC）. They all possess unique advantages compared to others, but their drawbacks restrict their usages individually. Our algo- rithm combines the three steps together to make full use of their features. At first the Dirichlet energy is optimized for faster approximation with better topology. Then the area energy is used to come close to the constrained domain. Finally the DMC energy is engaged to achieve a better converging step. Results show that our method can work under a rather noisy initial mesh, which is even topologically different from the final result.

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.

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.

Minimal Surfaces in S 6 with Constant Khler Angles and Constant Curvature *

本文给出了近Ｋａｈｌｅｒ球面Ｓ６中具有常数Ｋａｈｌｅｒ角和常数曲率的极小曲面的例子，同时证明了两个唯一性定理．

Heat Flow for the Minimal Surface with Plateau Boundary Condition

The heat flow for the minimal surface under Plateau boundary condition is defined to be a parabolic variational inequality, and then the existence, uniqueness, regularity, continuous dependence on the initial data and the asymptotics are studied. It is applied as a deformation of the level sets in the critical point theory.

Some Remarks on the Symmetry of Self-Conjugate Minimal Surfaces in R^3

Symmetry properties of self-conjugate minimal surfaces,i.e.minimal surfaces which are congruent with their conjugate ones in R^3 are studied.

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.

Laguerre Minimal Surfaces in ■~3

Laguerre geometry of surfaces in R^3 is given in the book of Blaschke, and has been studied by Musso and Nicolodi, Palmer, Li and Wang and other authors. In this paper we study Laguerre minimal surface in 3-dimensional Euclidean space R^3. We show that any Laguerre minimal surface in R^3 can be constructed by using at most two holomorphic functions. We show also that any Laguerre minimal surface in R^3 with constant Laguerre curvature is Laguerre equivalent to a surface with vanishing mean curvature in the 3-dimensional degenerate space R0^3.

Topologically minimal surfaces versus self-amalgamated Heegaard surfaces

Let V ∪SW be a Heegaard splitting of M,such that αM = α-W = F1 ∪ F2 and g(S) = 2g(F1)= 2g(F2). Let V * ∪S*W * be the self-amalgamation of V ∪SW. We show if d(S) 3 then S* is not a topologically minimal surface.

Computing Open-Loop Optimal Control of the q-Profile in Ramp-Up Tokamak Plasmas Using the Minimal-Surface Theory

The q-profile control problem in the ramp-up phase of plasma discharges is consid- ered in this work. The magnetic diffusion partial differential equation （PDE） models the dynamics of the poloidal magnetic flux profile, which is used in this work to formulate a PDE-constrained op-timization problem under a quasi-static assumption. The minimum surface theory and constrained numeric optimization are then applied to achieve suboptimal solutions. Since the transient dy- namics is pre-given by the minimum surface theory, then this method can dramatically accelerate the solution process. In order to be robust under external uncertainties in real implementations, PID （proportional-integral-derivative） controllers are used to force the actuators to follow the computational input trajectories. It has the potential to implement in real-time for long time discharges by combining this method with the magnetic equilibrium update.

New minimal surfaces in the hyperbolic space

We obtain new complete minimal surfaces in the hyperbolic space H3, by using Ribaucour transformations. Starting with the family of spherical catenoids in H^3 found by Mori(1981), we obtain 2-and 3-parameter families of new minimal surfaces in the hyperbolic space, by solving a non trivial integro-differential system. Special choices of the parameters provide minimal surfaces whose parametrizations are defined on connected regions of R^2 minus a disjoint union of Jordan curves. Any connected region bounded by such a Jordan curve, generates a complete minimal surface, whose boundary at infinity of H^3 is a closed curve. The geometric properties of the surfaces regarding the ends, completeness and symmetries are discussed.