Variable offsets and processing of implicit forms toward the adaptive synthesis and analysis of heterogeneous conforming microstructure

TL;DR

This work addresses the challenge of designing and analyzing heterogeneous lattices built from implicit TPMS tiles with controllable wall thickness. It introduces distance-field based, B-spline representations to produce constant and variable offsets for implicit tiles, and shows how to map tiles through a macro-shape to form continuous lattices. An unfitted finite element framework (cutFEM) is employed to simulate PDEs directly on these implicit geometries, enabling a closed design–analysis loop. The experimental results demonstrate variable-thickness lattices and mechanical behavior consistent with tile geometry, highlighting practical potential for graded porosity and tailored stiffness in additive manufacturing.

Abstract

The synthesis of porous, lattice, or microstructure geometries has captured the attention of many researchers in recent years. Implicit forms, such as triply periodic minimal surfaces (TPMS) has captured a significant attention, recently, as tiles in lattices, partially because implicit forms have the potential for synthesizing with ease more complex topologies of tiles, compared to parametric forms. In this work, we show how variable offsets of implicit forms could be used in lattice design as well as lattice analysis, while graded wall and edge thicknesses could be fully controlled in the lattice and even vary within a single tile. As a result, (geometrically) heterogeneous lattices could be created and adapted to follow analysis results while maintaining continuity between adjacent tiles. We demonstrate this ability on several 3D models, including TPMS.

Figures (15)

• Figure 1: Variations of implicit trivariate B-spline tiles, created using a distance field from curves ((a) and (b)) and a curve and a surface (c). Considering the complexity of the central joint (with ten arms!) in (a) and (b), the recreation of a similar tile using parametric forms will be painstakingly difficult. Note the corners are forming one octant of a joint with seven other neighboring tiles. (b) and (c) shows a closed implicit tile whereas in (a) the tile is created open (so it can be connected to some neighbors).
• Figure 2: A counter example showing why level sets cannot be employed to compute precise offsets. The distance field to the four curves in (b) is used to derive the two different implicit trivariates in (a) and (c), that are identical at level set $c_0$ and shown inred as ${\mathcal{I}}(u, v, w) = c_0$ (interior in magenta). Adjusting the level sets, in (a) and (c), from $c_0$ to $c_1$, yields the result in green as ${\mathcal{I}}(u, v, w) = c_1$ (interior in yellow). The green level set in (c) is different from (a) due to graded scaled up gradients, in (c) as we move away from the origin, along $x$. Note the pairs of green arms in $+x$(in the dotted ovals) that are much closer to the original red arms in (c), compared to (a), while other arms are similar (e.g., in the dashed ovals).
• Figure 3: The relationship between an implicit trivariate tile ${\mathbf I}(u, v, w)$ and the parametric trivariate macro-shape ${\mathcal{T}}(t_x, t_y, t_z)$. Note that, for every tile ${\mathbf I}$ in ${\mathcal{T}}$, the tiles' $(u, v, w)$ parametric values affinely equates (denoted by $\approx$) with the parameters of a box sub-domain of the domain $D$ of ${\mathcal{T}}$, or $(u, v, w) \approx (t_x, t_y, t_z)$, for every tile in $D$.
• Figure 6: An implicit trivariate TPMS (Schwarz-P) approximation tile (created using an interpolatory fitting scheme similar to Hu2021) in (a) and an implicit trivariate 3D cross tile (created using a distance field to curves) in (d) are employed to demonstrate (the zero sets of the) variable distance offset computations, as are shown in (b) and (e). (c) and (f) show (the zero sets of the) implicits of the base as well as the variable distance offset implicits, in the same images.
• Figure 7: Creation of controlled offsets for a Schwarz-P tile in lattice space. In (a), the tile $\mathcal{I}(u,v,w)=0$ in $D$ is represented (through a non-conforming high-order reparatemerization). In (b), the tile is mapped by a trivariate $\mathcal{T}$. A cloud of points lying on the mapped tile is represented in (c): these points are used for accelerating the search of the closest point $\bar{\mathbf{x}}$ to a given point $\mathbf{x}$. Finally, the material between $\mathbf{I}_{-e}$ and $\mathbf{I}_{+e}$ is represented in the domain $D$ (in (d)) and in the lattice space in (in (e)).

Theorems & Definitions (2)

• Remark A.1
• Remark A.2