Properly embedded minimal planar domains

Additive manufacturing is an advanced technology used to fabricate complex geometries with unique properties like cellular structures which accommodate repeated unit cells located in the x, y and z direction. These structures can be used as infill patterns due to their self-supporting structure. Among the cellular structures, Triply Periodic Minimal Surface (TPMS) structures such as Gyroid, Diamond and Schwarz Primitive (SchwarzP) structures can be tailored to produce complex structures for various applications like tissue engineering scaffolds and replace the conventional polymeric foams. TPMS structures are designed and manufactured by using the Fused Deposition Modelling (FDM) technique using Poly-Lactic Acid (PLA) as material. Among TPMS structures, Gyroid is having a unique property like structurally symmetric which design was modified to enhance the mechanical properties. The modified Gyroid or deformed Gyroid undergone a quasi-static compression test and compare the results with Diamond and SchwarzP structures. Porosity and permeability coefficients are evaluated and an optical microscope is used to verify the fabricated components. As well as, Failure patterns of the structures were evaluated and energy absorption capabilities determined. The main objective of this paper is to evaluate the impact of design and porosity on the mechanical and morphological properties of TPMS structures. In conclusion, the deformed Gyroid has more energy absorption capability up to the 11.6% strain than other TPMS structures. After 11.6% of strain, SchwarzP structure dominates.

On the effect of porosity and functional grading of 3D printable triply periodic minimal surface (TPMS) based architected lattices embedded with a phase change material

Heat transfer and pressure drop characteristics of heat exchangers based on triply periodic minimal and periodic nodal surfaces

Design and performance assessment of a triply-periodic-minimal-surface structures-enhanced gallium heat sink for high heat flux dissipation: A numerical study

Microstructures with minimal surfaces can be often found in natural porous architectures, where the surface tension minimizes the area. The triply periodic minimal surfaces (TPMS) [1] are an example of such microstructures. Compared with other porous structures, TPMS have three significant features: firstly, their geometries can be completely expressed via analytical functions; secondly, TPMS are periodic in three independent directions and thirdly, the mean curvature of TPMS is zero [2]. Transforming the TPMS-based unit cell into a lattice structure has particular usage in aerospace, nuclear energy, and biomedical applications where light weight, high stiffness, and temperature resistance are of critical importance. In the presented studies, the failure behavior of four typical TPMS structures (Primitive, Gyroid, Neovius, and IWP) under compression was studied using finite element analysis. Numerical modeling of the damage propagation and strength prediction was performed by removing the finite elements in which the appropriate damage criterion is reached. Utilizing the equations of the generated TPMS structures, the wall thickness of unit cell was considered the main parameter that defined the ceramics volume fraction and should be taken into consideration. Therefore, various unit cell models for different wall thicknesses were generated and used to investigate the impact of the cell geometry on the damage initiation, propagation, and overall compression strength. The results of compression strength and damage development were compared with those of other TPMS structures for the same wall thickness and volume fraction. Finally, the grade TPMS porous structure was provided to verify the effect of wall thickness variation on damage evolution on the macroscale.

Triply periodic minimal surfaces (TPMS) inspired by nature serve as a foundation for developing novel nanomaterials, such as templated silicas, graphene sponges, and schwarzites, with customizable optical, poroelastic, adsorptive, catalytic, and other properties. Computer simulations of reactions on TPMS using reactive intermolecular potentials hold great promise for constructing and screening potential TPMS with the desired properties. Here, we developed an off-lattice, surface-constrained Metropolis Monte Carlo (SC-MMC) algorithm that utilized a temperature quench process. The presented SC-MMC algorithm was used to investigate the process of graphitization reactions on the Schwarz primitive, Schwarz diamond, and Schoen gyroid TPMS, all with a cubic lattice parameter of 8 nm. We show that the optimized carbon TPMS exhibits a low energy, approximately -7.1 eV/atom, comparable to that of graphite and diamond crystals, along with a variety of topological defects. Furthermore, these structures showcase extensive and smooth surfaces characterized by a negative discrete Gaussian curvature, a distinctive feature indicative of an interconnected morphology. They possess specific surface areas of ∼2700 m2/g, comparable to graphene, and exhibit a significant porosity of around 90%. The theoretical X-ray correlation functions and nitrogen adsorption isotherms confirm that the constructed TPMS exhibit remarkably similar surface properties, although the pore space topology varies significantly.

Elastic mechanical property hybridization of configuration-varying TPMS with geometric continuity

Triply periodic minimal surface (TPMS) structures have unique geometries and excellent mechanical properties, which have attracted much attention in many fields. However, the relationship between different filling forms and different directions of functionally graded TPMS structures on energy absorption has not been fully studied. In this study, a functionally graded strategy was proposed to investigate the effect of filling form and direction gradient on the energy absorption of TPMS structures. The design of functionally graded Gyroid and Diamond TPMS cellular structures with multiple forms was characterized, and the structures were fabricated using additive manufacturing technology. The effects of uniformity and different directional gradients on the deformation and energy absorption properties of the structures were studied experimentally and numerically. According to the compression test results, it was found that different filling forms of the TPMS structure behave differently in terms of yield plateau and deformation pattern, and the sheet structures can develop a better deformation pattern to enhance energy absorption capacity. Functionally graded sheet Diamond TPMS cellular structures along the compression direction exhibit a 32% reduction in initial peak force, providing more advantages in structural deformation and energy absorption. More closely, it is possible to further reduce the initial peak force, delay the densification point, and thus increase the energy absorption capacity by designing functionally graded sheet Diamond TPMS based cellular structures. The results of this study provide valuable guidance for the design of high-performance impact-protection components.

In recent years, lightweight and material-efficient cellular structures, such as triply periodic minimal surfaces (TPMS), have garnered increasing attention for advanced engineering applications. This study introduces a novel homogenization approach for TPMS structures, including Gyroid, Primitive, IWP, Diamond, and Neovius, employing Non-Uniform Rational B-Splines (NURBS) for precise geometric representation. Unlike traditional finite element method (FEM) models requiring fine meshes to represent TPMS curvature, NURBS ensures accurate geometric fidelity independent of mesh refinement, eliminating volume fraction inconsistencies. Homogenization is conducted without periodic boundary conditions, utilizing standard boundary conditions across representative volume elements (RVEs) from 1 × 1 × 1 to 4 × 4 × 4 unit cells. Effective material properties are derived and validated in Abaqus by comparing the mechanical response of homogenized TPMS structures to equivalent solid structures, verifying consistent stiffness predictions. The results are compared with previous TPMS homogenization studies, helping to clarify the change in material properties when changing the geometric description method. The NURBS-based approach facilitates efficient computations and enables isogeometric analysis (IGA), offering a versatile framework for analyzing TPMS-based cellular structures in additive manufacturing applications.

Enhanced convective heat transfer in new triply periodic minimal surface structures: Numerical and experimental investigation

Topological interface modes in 3D-printed triply periodic minimal surface phononic crystals

In this note, we study an overdetermined system of partial differential equations whose solutions determine the minimal surfaces in of constant Gaussian curvature. If the Gaussian curvature is positive, the solution to the global problem was found by [Calabi], while the solution to the local problem was found by [Wallach]. The case of nonpositive Gaussian curvature is more subtle and has remained open. We prove that there are no minimal surfaces in Sn of constant negative Gaussian curvature (even locally). We also find all of the flat minimal surfaces in and give necessary and sufficient conditions that a given two-torus may be immersed minimally, conformally, and flatly into S. 0. Introduction. In this paper, we classify the connected minimal surfaces of constant Gaussian curvature in the unit n-sphere Sn C Enll for all n. It is a classical fact (and follows easily from the structure equations) that the only examples up to rigid motion in S3 are the open subsets of the geodesic spheres (with K 1) and the Clifford torus (with K 0). In an early paper Boruvka constructed a linearly full immersion S2 C S2m for each m > 0, where the induced metric had K 2/m(m + 1). Later [Calabi] showed that, up to rigid motion, these Boruvka spheres were the only compact minimal surfaces with K Ko > 0 in S' for any n. [Wallach] proved that any connected piece of minimal surface with K a positive constant in Sn was a subset of a Boruvka sphere. [Kenmotsu, 1976] found all of the flat minimal surfaces (K 0) in S' and quite recently, [Kenmotsu, 1983] showed that K Ko < 0 is impossible for minimal surfaces in S4. The techniques used in the above proofs range from harmonic analysis to rather involved calculations with the moving frame. On the other hand, in this paper, we take a somewhat different point of view. If (M2, ds2) is a surface of constant Gaussian curvature K, then the minimal surfaces in S n C E n ? 1 of constant Gaussian curvature K are given locally by smooth maps f: M En' + lwhich satisfy three conditions: first, Kf, f) 1; second, Af =2f (A is the Laplace-Beltrami operator of ds2); and third, that f should be an isometry, Kdf, df) = ds2. We study the more general class of maps f: M En + 1 which only satisfy the first two conditions. Note that this set of equations is already overdetermined (by one equation). Received by the editors August 14, 1984. 1980 Mathematics Subject Classification. Primary 53A10; Secondary 53B25 53C40.

In this work is presented a family that generalizes Bour’s, Enneper’s, Richmond’s and Henneberg’s minimal surfaces. First of all is introduced a generalization for the well-known Bour’s family $$B_m$$ by including a new index r, determining a new bigger family $$B_{m r}$$ , which includes Bour’s, Enneper’s and Richmond’s families of minimal surfaces. After that, realizing some similarities between the Enneper–Weierstrass data of the Henneberg’s minimal surface and the $$B_{m r}$$ family, it is established a new family of minimal surfaces $$H_{m r}$$ that includes the classical Henneberg minimal surface. Finally, it is presented a bigger family $$C_ {m r}^{\kappa h}, \ m,r \in {\mathbb {R}}, \ \kappa \in {\mathbb {C}}$$ which includes as particular cases $$B_{m r}$$ and $$H_{m r}$$ families. Consequently, catenoid, helicoid, Enneper’s minimal surfaces, Bour’s minimal surfaces, Richmond’s minimal surfaces and Henneberg’s minimal surfaces are just some elements of the set of $$C_ {m r}^{\kappa h}$$ surfaces.

We develop Teichmuller theoretical methods to construct new minimal surfaces in $\BE^3$ by adding handles and planar ends to existing minimal surfaces in $\BE^3$. We exhibit this method on an interesting class of minimal surfaces which are likely to be embedded, and have a low degree Gau\ss map for their genus; the (Weierstrass data) period problem for these surfaces is of arbitrary dimension. In particular, we exhibit a two-parameter family of complete minimal surfaces in the Euclidean three-space $\BE^3$ which generalize the breakthrough minimal surface of C. Costa; these new surfaces are embedded (at least) outside a compact set, and are indexed (roughly) by the number of ends they have and their genus. They have at most eight self-symmetries despite being of arbitrarily large genus, and are interesting for a number of reasons. Moreover, our methods also extend to prove that some natural candidate classes of surfaces cannot be realized as minimal surfaces in $\BE^3$. As a result of both aspects of this work, we obtain a classification of a family of surfaces as either realizable or unrealizable as minimal surfaces.

PurposeThe purpose of this paper is to develop a general mathematic approach to model the microstructures of porous structures produced by additive manufacturing (AM), which will result in fractal surface topography and higher roughness that have greater influence on the performance of porous structures.Design/methodology/approachThe overall shapes of pores were modeled by triply periodic minimal surface (TPMS), and the micro-roughness details attached to the overall pore shapes were represented by Weierstrass–Mandelbrot (W-M) fractal representation, which was integrated with TPMS along its normal vectors. An index roughly reflecting the irregularity of fractal TPMS was proposed, based on which the influence of the fractal parameters on the fractal TPMS was qualitatively analyzed. Two complex samples of real porous structures were given to demonstrate the feasibility of the model.FindingsThe fractal surface topography should not be neglected at a micro-scale level. In addition, a decrease in the fractal dimension Ds may exponentially make the topography rougher; an increase in the height-scaling parameter G may linearly increase the roughness; and the number of the superposed ridges has no distinct influence on the topography. Furthermore, the synthesis method is general for all implicit surfaces.Practical implicationsThe method provides an alternative way to shift the posteriori design paradigm of porous media to priori design mode through numeric simulation. Therefore, the optimization of AM process parameters, as well as the porous structure, can be potentially realized according to specific functional requirement.Originality/valueThe synthesis of TPMS and W-M fractal geometry was accomplished efficiently and was general for all implicit freeform surfaces, and the influence of the fractal parameters on the fractal TPMS was analyzed more systematically.