A Hybrid Level Set Optimization Design Method of Functionally Graded Cellular Structures Considering Connectivity

With the continuous advancement in topology optimization and additive manufacturing(AM)technology,the capability to fabricate functionally graded materials and intricate cellular structures with spatially varying microstructures has grown significantly.However,a critical challenge is encountered in the design of these structures–the absence of robust interface connections between adjacent microstructures,potentially resulting in diminished efficiency or macroscopic failure.A Hybrid Level Set Method(HLSM)is proposed,specifically designed to enhance connectivity among non-uniform microstructures,contributing to the design of functionally graded cellular structures.The HLSM introduces a pioneering algorithm for effectively blending heterogeneous microstructure interfaces.Initially,an interpolation algorithm is presented to construct transition microstructures seamlessly connected on both sides.Subsequently,the algorithm enables the morphing of non-uniform unit cells to seamlessly adapt to interconnected adjacent microstructures.The method,seamlessly integrated into a multi-scale topology optimization framework using the level set method,exhibits its efficacy through numerical examples,showcasing its prowess in optimizing 2D and 3D functionally graded materials(FGM)and multi-scale topology optimization.In essence,the pressing issue of interface connections in complex structure design is not only addressed but also a robust methodology is introduced,substantiated by numerical evidence,advancing optimization capabilities in the realm of functionally graded materials and cellular structures.

Full-Scale Isogeometric Topology Optimization of Cellular Structures Based on Kirchhoff-Love Shells

Cellular thin-shell structures are widely applied in ultralightweight designs due to their high bearing capacity and strength-to-weight ratio.In this paper,a full-scale isogeometric topology optimization(ITO)method based on Kirchhoff-Love shells for designing cellular tshin-shell structures with excellent damage tolerance ability is proposed.This method utilizes high-order continuous nonuniform rational B-splines(NURBS)as basis functions for Kirchhoff-Love shell elements.The geometric and analysis models of thin shells are unified by isogeometric analysis(IGA)to avoid geometric approximation error and improve computational accuracy.The topological configurations of thin-shell structures are described by constructing the effective density field on the controlmesh.Local volume constraints are imposed in the proximity of each control point to obtain bone-like cellular structures.To facilitate numerical implementation,the p-norm function is used to aggregate local volume constraints into an equivalent global constraint.Several numerical examples are provided to demonstrate the effectiveness of the proposed method.After simulation and comparative analysis,the results indicate that the cellular thin-shell structures optimized by the proposed method exhibit great load-carrying behavior and high damage robustness.

Design Optimization of Irregular Cellular Structure for Additive Manufacturing

Irregularcellular structurehas great potential to be considered in light-weight design field. However, the research on optimizing irregular cellular structures has not yet been reporteddue to the difficulties in their modeling technology. Based on the variable density topology opti- mization theory, an efficient method for optimizing the topology of irregular cellular structures fabricated through additive manufacturing processes is proposed. The pro- posed method utilizes tangent circles to automatically generate the main outline of irregular cellular structure. The topological layoutof each cellstructure is optimized using the relative density informationobtained from the proposed modified SIMP method. A mapping relationship between cell structure and relative densityelement is builtto determine the diameter of each cell structure. The results show that the irregular cellular structure can be optimized with the proposed method. The results of simulation and experimental test are similar for irregular cellular structure, which indicate that the maximum deformation value obtained using the modified Solid Isotropic Microstructures with Penalization （SIMP） approach is lower 5.4× 10-5 mm than that using the SIMP approach under the same under the same external load. The proposed research provides the instruction to design the other irregular cellular structure.

Symplectic analysis for elastic wave propagation in two-dimensional cellular structures

On the basis of the finite element analysis, the elastic wave propagation in cellular structures is investigated using the symplectic algorithm. The variation principle is first applied to obtain the dual variables and the wave propagation problem is then transformed into two-dimensional （2D） symplectic eigenvalue problems, where the extended Wittrick-Williams algorithm is used to ensure that no phase propagation eigenvalues are missed during computation. Three typical cellular structures, square, triangle and hexagon, are introduced to illustrate the unique feature of the symplectic algorithm in higher-frequency calculation, which is due to the conserved properties of the structure-preserving symplectic algorithm. On the basis of the dispersion relations and phase constant surface analysis, the band structure is shown to be insensitive to the material type at lower frequencies, however, much more related at higher frequencies. This paper also demonstrates how the boundary conditions adopted in the finite element modeling process and the structures＇ configurations affect the band structures. The hexagonal cells are demonstrated to be more efficient for sound insulation at higher frequencies, while the triangular cells are preferred at lower frequencies. No complete band gaps are observed for the square cells with fixed-end boundary conditions. The analysis of phase constant surfaces guides the design of 2D cellular structures where waves at certain frequencies do not propagate in specified directions. The findings from the present study will provide invaluable guidelines for the future application of cellular structures in sound insulation.

Effect of Cellular Structure on Mechanical Properties of Polyurethane Foam Curing Materials

Based on the mechanical properties and microstructure of polyurethane foam solidified material, a two-dimensional model of polyurethane foam solidified material was constructed. Polyurethane foam was obtained by fully and uniformly mixing the two components. The research was carried out through the combination of experimental test and finite element simulation. The experimental results show that when the pore density is constant, the size of the bubble hole is an important factor affecting the mechanical properties of the model. The smaller the size of the bubble hole, the less likely it is to produce stress concentration inside the model, and the stronger the resistance to material deformation. Under the random distribution, the lower the density of the polyurethane cured material, the higher the probability of damage between the adjacent bubbles, which is not conducive to the stability of the material. The density of the cured material should not be lower than 199 kg/m^3.

Functionally Graded Cellular Structure Design Using the Subdomain Level Set Method with Local Volume Constraints

Functional graded cellular structure(FGCS)usually shows superiormechanical behaviorwith lowdensity and high stiffness.With the development of additivemanufacturing,functional graded cellular structure gains its popularity in industries.In this paper,a novel approach for designing functionally graded cellular structure is proposed based on a subdomain parameterized level set method(PLSM)under local volume constraints(LVC).In this method,a subdomain level set function is defined,parameterized and updated on each subdomain independently making the proposed approach much faster and more cost-effective.Additionally,the microstructures on arbitrary two adjacent subdomains can be connected perfectly without any additional constraint.Furthermore,the local volume constraint for each subdomain is applied by virtue of the augmented Lagrange multiplier method.Finally,several numerical examples are given to verify the correctness and effectiveness of the proposed approach in designing the functionally graded cellular structure.From the optimized results,it is also found that the number of local volume constraints has little influence on the convergence speed of the developed approach.

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.

Mechanical property and cellular structure of an additive manufactured FeCoNiCrMo_(0.2) high-entropy alloy at high-velocity deformation

The microstructural modification for cellular structures can achieve the improvement of dynamic me-chanical properties of a selective laser melted FeCoNiCrMo_(0.2)high-entropy alloy(SLM-FeCoNiCrMo_(0.2)HEA)and can expand its promising applications in the field of high-velocity deformation.In this work,FeCoNiCrMo_(0.2)HEAs with cellular structures in different sizes were produced by selective laser melt-ing(SLM)with different process parameters.The dynamic mechanical properties and microstructure of the SLM-FeCoNiCrMo_(0.2)HEA were studied.The dynamic mechanical properties of the SLM-FeCoNiCrMo_(0.2)HEA increased with decrease of average size of cellular structures,and the values of them were sensitive to strain rates.The energy absorption,compressive strength and yield strength of the SLM-FeCoNiCrMo_(0.2)HEAs reached 315.6 MJ/m^(3),2.2 GPa and 775.6 MPa,respectively at a strain rate of 2,420 s^(−1),under the process parameters of laser power and scanning speed of 330 W and 800 mm/s,respectively,where the corresponding average size of cellular structures in the HEAs was 483.6 nm.The value of strain-hardening rate of the SLM-FeCoNiCrMo_(0.2)HEA was about 5.1 GPa at a strain level of 0.1,which was much higher than that of the powder-metallurgy FeCoNiCrMo_(0.2)HEA.The cellular structure was formed inside the molten pool with segregation of Mo on the boundary.Deformation localization appeared in the cellular structures,forming several deformation bands after high strain-rate deformation.The elemental segre-gation strengthening and dislocation strengthening are considered to be the main strengthening mecha-nisms in SLM-FeCoNiCrMo_(0.2)HEA.

Mechanical properties and shape memory behavior of 4D printed functionally graded cellular structures

The properties of functionally graded(FG) cellular structures vary spatially, and the varying properties can meet the requirements of different working environments. In this study, we fabricated FG cellular structures with shape memory effect by 4D printing and evaluated the compressive performance and shape memory behavior of these structures with temperature through experimental analysis and finite element simulations. The results show that the maximum energy absorption gradually decreases but the compressive modulus gradually increases with increasing gradient parameters. Moreover, the finite element simulations also show that the compressive deformation mode of the structure shifts from uniform to non-uniform deformation with increasing gradient parameters. The compressive modulus and compressive strength of 4D printed FG structures decrease with increasing temperature due to the influence of the shape memory polymer, and they exhibit outstanding shape recovery capability under high-temperature stimulus. The proposed 4D printed FG structures with such responsiveness to stimulus shed light on the design of intelligent energy-absorbing devices that meet specific functional requirements.

Review of Crashworthiness Studies on Cellular Structures

The application of lightweight structures with excellent energy absorption performance is crucial for enhancing vehicle safety and energy efficiency.Cellular structures,inspired by the characteristics observed in natural organisms,have exhibited exceptional structural utilization in terms of energy absorption compared with traditional structures.In recent years,various innovative cellular structures have been proposed to meet different engineering needs,resulting in significant performance improvements.This paper provides a comprehensive overview of novel cellular structures for energy absorption applications.In particular,it outlines the application forms and design concepts of cellular structures under typical loading conditions in vehicle collisions,including axial loading,oblique loading,bending loading,and blast loading.Cellular structures have evolved to meet the demands of complex loading conditions and diverse research methods,focusing on achieving high-performance characteristics across multiple load cases.Moreover,this review discusses manufacturing techniques and strate-gies for enhancing the manufacturing performance of cellular structures.Finally,current key challenges and future research directions for cellular structures are discussed.The aim of this study is to provide valuable guidelines for researchers and engineers in the development of next-generation lightweight cellular structures.

New insight into fabrication of shaped Mg-X alloy foams with cellular structure via a gas release reaction powder metallurgy route

Shaped Mg alloy foams with closed-cell structure are highly interested for a great potential to be utilized in the fields where weight reduction is urgently required.A powder metallurgical method,namely gas release reaction powder metallurgy route to fabricate Mg-X(X=Al,Zn or Cu)alloy foams,was summarized.The principles on shaped Mg-X foams fabrication via the route were proposed.In addition,the effects of alloying elements,sintering treatment and foaming temperatures on fabrication of shaped Mg-X alloy foams were investigated experimentally.The results show that the key to ensure a successful foaming of Mg-X alloy foams is to add alloying metals alloyed with Mg to form lower melting(<600℃)intermetallic compounds by the initial sintering treatment.The foaming mechanism of Mg-X alloy foams also has been clarified,that is,the low-melting-point Mg-based intermetallic compounds melt first,and then reactions between the melt and CaCO_(3),a foaming agent,release CO gas to make the precursor foamed and finally shaped Mg-X alloy foam with a promising cellular structure is prepared.This route has been verified by successful fabrication on shaped Mg-Al,Mg-Zn and Mg-Cu foams with cellular structure.

Effect of film types on thermal response,cellular structure,forming defects and mechanical properties of combined in-mold decoration and microcellular injection molding parts

Different types of polymer films were used in the combined in-mold decoration and microcellular injection molding(IMD/MIM)process.The multiphase fluid-solid coupled heat transfer model was established to study the thermal response at the melt filling stage in the IMD/MIM process.It was found that the temperature distributed asymmetrically along the thickness direction due to the changed heat transfer coefficient of the melt on the film side.When polyethylene terephthalate(PET)films were applied,the temperature of the melt-film interface increased faster and to be higher at the end of melt filling stage in comparison with the application of polycarbonate(PC)and thermoplastic polyurethane(TPU)films.And the effects of film types on the cellular structure,forming defects and mechanical properties of IMD/MIM parts were also studied experimentally.The results showed that the film types had no obvious effect on the cells size in the transition layer and the mechanical properties of the parts.Under certain film thickness,the offset distance of core layer was the largest with PET film used,while the offset distance was the smallest with TPU film used.And similar results were found for the warpage of the parts.However,an exactly opposite change occurred for the thickness of film-side transition layer and the bubble marks on the surface of the parts.

Electrochemical behavior of open-cellular structured Ti-6Al-4V alloy fabricated by electron beam melting in simulated physiological fluid:the significance of pore characteristics

The cellular structured titanium alloys have attracted significant attention for implants because of their lower Young’s modulus,which is comparable to human bone and has the capability of providing space for bone tissue in-growth.However,there is a gap in the knowledge in regard to the relationship between the pore characteristics and the electrochemical performance of open-cellular structured titanium alloys.In this study,we elucidate the influence of pore characteristics on the electrochemical performance of open-cellular structured Ti-6Al-4V alloys produced by electron beam melting(EBM).Intriguingly,the passive film formed on cellular structured Ti-6Al-4V alloy with a larger pore size was more stable and protective,and the corrosion performance was superior compared to the samples with a smaller pore size in phosphate buffered saline(PBS),mainly because of relatively smaller exposed surface area and unlimited flow of electrolyte.However,in acidic PBS containing fluoride ions,the pore characteristics did not play an important role in the corrosion resistance.It was considered that the protective film breaks down such that the corrosion performance of cellular structured alloys was comparable to each other in this harsh environment.

Development of Composite Cellular Cores for Sandwich Panels Based on Folded Polar Quadra-Structures

An idea to develop a family of cellular cores for sandwich panels using a technology of prepreg folding is presented.Polar folded quadra-structures are regarded as a geometric basis for these cores whose standard fragment has the fourth degree of axial symmetry.The classification of the polar structures are described and a method of various quadra-structure synthesis is developed.A possibility to provide high strength of the structure due to preservation of faces reinforcement pattern is presented.Arrangement of the plane core on a bi-curvature surface is also introduced.Besides,provision of isotropy of the core in two or three directions are described.Finally,examples of cellular folded cores manufactured from basalt reinforced plastic are demonstrated.

Direct design to stress mapping for cellular structures

This paper aims to instantly predict within any accuracy the stress distribution of cellular structures under parametric design,including the shapes or distributions of the cell geometries,or the magnitudes of external loadings.A classical model reduction technique has to balance the simulation accuracy and interaction speed,and has difficulty achieving this goal.We achieve this by computing offline a design-to-stress mapping that ultimately expresses the stress distribution as an explicit function in terms of its design parameters.The mapping is determined as a solution to an extended finite element analysis problem in a high-dimension space,including both the spatial coordinates and the design parameters.The well-known curse of dimensionality intrinsic to the high-dimension problem is(partly)resolved through a spatial separation using two main techniques.First,the target mapping takes a reduced form as a sum of the products of separated one-variable functions,extending the proper generalized decomposition technique.Second,the simulation problem in a varied computation domain is reformulated as that in a fixed-domain,taking an integration function as the sum of the products of separated one-variable functions,in combination with high-order singular value decomposition.Extensive 2D and 3D examples are shown to demonstrate the approach’s performance.

Design and mechanical properties analysis of a cellular Waterbomb origamistructure

Cellular structures are commonly used to design energy-absorbing structures,and origami structures are becominga prevalent method of cellular structure design.This paper proposes a foldable cellular structure based on theWaterbomb origami pattern.The geometrical configuration of this structure is described.Quasi-static compressiontests of the origami tube cell of this cellular structure are conducted,and load-displacement relationship curvesare obtained.Numerical simulations are carried out to analyze the effects of aspect ratio,folding angle,thicknessand number of layers of origami tubes on initial peak force and specific energy absorption(SEA).Calculationformulas for initial peak force and SEA are obtained by the multiple linear regression method.The degree ofinfluence of each parameter on the mechanical properties of the single-layer tube cell is compared.The resultsshow that the cellular structure exhibits negative stiffness and periodic load-bearing capacity,as well as foldingangle has the most significant effect on the load-bearing and energy-absorbing capacity.By adjusting the designparameters,the stiffness,load-bearing capacity and energy absorption capacity of this cellular structure can beadjusted,which shows the programmable mechanical properties of this cellular structure.The foldability andthe smooth periodic load-bearing capacity give the structure potential for application as an energy-absorbingstructure.

NO_3^-/NH_4^+ ratios affect plant growth, chlorophyll content,respiration rate, and morphological structure in Malus hupehensis seedlings

Malus hupehensis （Pamp.） Rehd. is a widely cultivated rootstock in China. We studied the effect of three NO3-/NH4＋ ratios （100/0, 50/50, and 0/100, molar basis） at total nitrogen （N） concentration of 8 mmoL L-1 in a nutrient solution on M. hupehensis seedlings. Plant biomass, NO3- and NH4＋concentrafion, chlorophyll con- tent, respiratory rate, and cellular structure were investi- gated. M. hupehensis seedlings at the NO3-/NH4＋ ratio of 50/50 had the highest level of fresh weight, dry weight, shoot length, and chlorophyll （a, b, and a ＋ b） content, but the lowest respiration rate in the leavesand roots. In addition, thickness and numbers of palisade and spongy tissue cells of the leaves were greater with this treatment than with other treatments. At the NO3-/NH4＋ ratio of 100/0, the leaves and roots had higher NO3- concentration and lower NH4＋ concentration. However, the opposite trend occurred at the NO3-/NH4＋ ratio of 0/100. Chloro- phyll （a, b, and a ＋ b） content was lowest at the NO3-/NH4＋ ratio of 100/0 than at the other ratios. At the NO3-/ NH4＋ ratio of 0/100, oxygen （02） consumption increased in the leaves and roots, and irregular epidermis and cortex cells were observed in the root apical meristematic and mature region. Our results indicated that the NO3-INH4＋ ratio at 50/50 was suitable for growth of M. hupehensis seedling to achieve the highest biomass production and efficiency.

Multifunctional cellular carbon foams derived from chitosan toward self-cleaning, thermal insulation, and highly efficient microwave absorption properties

To adapt the practical demand,designing and constructing the multifunctional microwave absorbers(MAs)is the key future direction of research and development.However,effective integrating the multiple functions into a single material remains a huge challenge.Herein,cellular carbon foams(CCFs)with different porous structures were elaborately designed and fabricated in high efficiency through a facile continuous freeze-drying and carbonization processes using a sustainable biomass chitosan as the precursor.The obtained results revealed that the thermal treated temperature and g-C_(3)N_(4) amount played a great impact on the carbonization degrees,pore sizes,and morphologies of CCFs,which led to their tunable electromagnetic(EM)parameters,improved conduction loss,and polarization loss abilities.Owing to the special cellular structure,the designed CCFs samples simultaneously displayed the strong absorption capabilities,broad absorption bandwidths,and thin matching thicknesses.Meanwhile,the as-prepared CCFs exhibited the strong hydrophobicity and good thermal insulation,endowing its attractive functions of self-cleaning and thermal insulation.Therefore,our findings not only presented a facile approach to produce different porous structures of CCFs,but also provided an effective strategy to develop multifunctional high-performance MAs on basis of three-dimensional CCFs.

Fabrication and Characterization of Nano-CaCO_3/Polypropylene Foam Sheets

By applying the reinforcing and toughening effect of calcium carbonate （CaCO3） nanoparticles on polypropylene, foam sheets of good performance were successfully fabricated by extrusion. The equipment and conditions of the extrusion were explored. The mechanical properties of the produced foam sheets were tested. The effect of CaCO3 nano-particles on the mechanical properties and the cellular structure of the sheets was comprehensively studied. The experimental results show that the optimum content of CaCO3 nano-particles in the composite material was -4wt%. At this content, the nano-particles were well dispersed in the substrate, and the composite material had maximum tensile strength and impact strength. Surface treatment of the nano-particles only affected the impact strength of the composite material. CaCO3 micro-particles, on the other hand, showed little effect on the properties of the composite material when the micro-particles content was less than 5 wt%. At a content higher than 5wt%, the properties of the composite material significantly worsened.

Preparation of Controllable Cross-Linking Polyethylene Foaming Materials and Their Properties

In order to prepare the polyethylene materials with controlling properties,we developed two kinds of controllable cross-linking polyethylene foaming system.2,5-dimethyl-2,5-bis (tert-butyl peroxy) hexane was used as cross-linking agent and TEMPO as cross-linking inhibitor,azodicarbonamide (AC) was used as foaming agent and citric acid as foaming promoter.The density,expansion ratio,cellular structure and mechanical property of these two kinds of controllable materials were studied.Experimental results show that,properties of these two kinds of materials appear similar trend:cellular size and expansion ratio are enlarged with the amount of cross-linking inhibitor or foaming promoter increasing,while density and mechanical strength appear decreasing trend.Through comparing those two material systems’ properties,cross-linking polyethelene foaming system with citric acid as foaming promoter has better properties.