Update (2.0) to MiniSurf —A minimal surface generator for finite element modeling and additive manufacturing

This is an update to PII: S2665963820300178. We present an updated version v2.0 of our minimal surface generator MiniSurf v1.0 , that creates triply periodic minimal surface (TPMS) computer-aided design (CAD) files for both finite element modeling and additive manufacturing. Besides making the GUI more user-friendly, in this new version we significantly improve the mesh quality of the generated CAD files by incorporating a mesh smoothing feature. With this new smoothing feature, MiniSurf v2.0 can now produce high quality CAD files for more accurate finite element modeling and more precise additive manufacturing.


Keywords:
Triply periodic minimal surfaces (TPMSs) Finite element modeling (FEM) Additive manufacturing (AM) Computer-aided design (CAD) files 3D printing Architected materials A B S T R A C T This is an update to PII: S2665963820300178.
We present an updated version v2.0 of our minimal surface generator MiniSurf v1.0, that creates triply periodic minimal surface (TPMS) computer-aided design (CAD) files for both finite element modeling and additive manufacturing. Besides making the GUI more user-friendly, in this new version we significantly improve the mesh quality of the generated CAD files by incorporating a mesh smoothing feature. With this new smoothing feature, MiniSurf v2.0 can now produce high quality CAD files for more accurate finite element modeling and more precise additive manufacturing. The code (and data) in this article has been certified as Reproducible by Code Ocean: (https://codeocean.com/). More information on the Reproducibility Badge Initiative is available at https://www.elsevier.com/physical-sciences-and-engineering/computer-science/journals.

Introduction
One of the major limitations we discussed in MiniSurf v1.0 is the generation of CAD files with suboptimal meshes (long and narrow triangular facets resulting from the meshing of very complex topologies based on the 3D uniform grid points) [1]. The low-quality mesh often results in the inaccurate prediction of the local stress field (undesirable especially when the peak stress in an implant is estimated [2] or when a local failure criterion is used such as in the case of the fracture toughness modeling of cellular materials [3]) in a finite element simulation and the poor manufacturing of 3D printed parts (missing faces and geometric imprecision). To address this issue, we added the mesh smoothing feature that was previously used to obtain the smooth spinodal shell-based structures [4][5][6][7].

Smoothing algorithm and feature
The mesh smoothing algorithm is based on the Laplacian smoothing detailed in [8] and briefly described below: where N is the total number of the neighboring nodes i, each of which connects to a mesh node j by a single mesh edge; ′ , ′ , and ′ are the new x, y, and z coordinates of the mesh node j found by taking the average of , , and , the current x, y, and z coordinates of nodes i respectively.
To preserve the periodicity of the generated TPMS mesh, the Laplacian smoothing is only applied to the interior mesh nodes, so the boundary nodes remain unchanged. The smoothing algorithm is incorporated in the smoothing push-button in the left control panel of the GUI as shown in Fig. 1. Note that every time the button is pushed, it is equivalent to applying equation (2.1), (2.2) and (2.3) six times (this has been found in general to produce an optimal balance between the smoothness and shrinkage of a given mesh) as shown in Fig. 2. The mesh smoothing can be executed iteratively to the generated triply periodic minimal surface (TPMS) mesh at user's discretion. The smoothing algorithm is also parallelized (currently set to six processors or less depending on its availability) for the efficient computation of the smoothed mesh. Furthermore, we decouple the visualization and the CAD file generation in the GUI with the addition of the ''Export CAD'' button as shown in Fig. 1 providing a more convenient, intuitive, and user-friendly experience.

New advantages and impact overview
We previously showed that MiniSurf v1.0 can efficiently create CAD (a mesh made of many triangular facets) files of 19 common TMPSs for finite element modeling and additive manufacturing. These TPMSs have been studied extensively in multidisciplinary fields [9][10][11][12][13][14][15][16][17] due to their large surface area with very uniform double curvatures (directly related to the quality of underlying mesh [18][19][20]); hence it is extremely essential to ensure that the generated CAD files have high quality meshes consisting of triangular facets with aspect ratio close to one (not realizable in MiniSurf v1.0).
MiniSurf v2.0 is built to address this challenge by incorporating the Laplacian smoothing [18,21] feature detailed in Section 2. We emphasize the advantages of MiniSurf v2.0 over v1.0 as follows: (1) Users can control the smoothness of the generated mesh and hence the underlying mesh curvatures. (2) The mesh smoothing is highly parallelized (up to six processors by default). A finely meshed surface (300 × 300 × 300 triangular facets) can be smoothed under three seconds. (3) Most importantly, the generated TPMS CAD files now have high quality mesh for more accurate finite element modeling and more precise additive manufacturing. MiniSurf v2.0 is currently being used in the following ongoing projects: (a) Whipple shield design -SPH modeling of hypervelocity impact on sandwiched TPMSs. We replace the traditional whipple shield design (a front bumper and a back plate) with the sandwiched primitive TPMS (smoothed and generated by MiniSurf v2.0). We then model the hypervelocity impact of a space debris (represented by a sphere projectile traveling at the speed of 6 km/s) using smoothed particle hydrodynamics (following similar procedures in [22]). (b) Fracture toughness of 3D cellular materials -minimal surface-based versus strut-based topologies. We investigate the mode-I fracture toughness of primitive TPMS (smoothed and created by MiniSurf v2.0), octet, and cubic lattices by performing finite element calculations of the J-integral on a single-edge-notch-bend (SENB) specimen (the numerical approach is detailed in [3]).
We expect MiniSurf v2.0 to be impactful in multidisciplinary fields just as the MiniSurf v1.0 given the recent growing interest in building new multifunctional materials with TPMS topologies. Furthermore, with high quality CAD files (smoothed and generated by MiniSurf v2.0) researchers can now make more accurate finite element predictions and more precise prototyping to better design the materials.

Conclusion
In this work, we added the mesh smoothing functionality based on the Laplacian smoothing to our previously developed software MiniSurf. TPMS CAD files can now be efficiently created with a user-controlled smooth mesh for more accurate finite element modeling and sound additive manufacturing. In the future, we will continue to tackle the remaining limitations mentioned in MiniSurf v1.0 [1]. Furthermore, we will regularly add new TPMS to our existing library.