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Design and Optimization of Conforming Lattice Structures
Jun Wu, Weiming Wang, Xifeng Gao
TL;DR
The paper addresses the challenge of designing lightweight lattices that conform to principal stress directions and optimized boundaries. It combines simultaneous shape-and-lattice optimization with field-aligned compilation of locally defined material configurations into globally consistent structures. The resulting method works in 2D and 3D, produces highly detailed printable lattices, and is supported by numerical and physical evaluations.
Problem
Existing grid-aligned microstructures restrict the solution space and do not conform to curved boundaries, despite anisotropic materials performing best when oriented with principal stress directions.
Method
The method simultaneously optimizes shape and spatially varying lattice orientation, porosity, and anisotropy, then compiles the local configuration into a globally consistent lattice using field-aligned parameterization.
Results
The method generates conforming lattices that align with principal stress directions and optimized boundaries, with numerical analyses and physical verifications demonstrating their structural properties.
Takeaways & Limitations
The workflow supports scalable, highly detailed lattice structures for both 2D and 3D domains that can be fabricated by 3D printing.
Takeaways & Limitations
Graph extraction can produce T-junctions with near-flat angles near parameterization singularities, which are suboptimal for lattice stiffness.
Abstract
from arXiv · showhide
Inspired by natural cellular materials such as trabecular bone, lattice structures have been developed as a new type of lightweight material. In this paper we present a novel method to design lattice structures that conform with both the principal stress directions and the boundary of the optimized shape. Our method consists of two major steps: the first optimizes concurrently the shape (including its topology) and the distribution of orthotropic lattice materials inside the shape to maximize stiffness under application-specific external loads; the second takes the optimized configuration (i.e. locally-defined orientation, porosity, and anisotropy) of lattice materials from the previous step, and extracts a globally consistent lattice structure by field-aligned parameterization. Our approach is robust and works for both 2D planar and 3D volumetric domains. Numerical results and physical verifications demonstrate remarkable structural properties of conforming lattice structures generated by our method.
1 INTRODUCTION
The paper introduces a robust workflow for conforming lattice structures that jointly optimizes shape and lattice distribution, then compiles the result into a globally connected structure. It targets anisotropic materials whose orientations follow principal stress directions while conforming to optimized boundaries.
- The method jointly evolves the optimized shape and spatially varying lattice orientation through stress analysis and numerical optimization.
- Homogenization-based topology optimization efficiently simulates and optimizes lattice material distribution at a relatively coarse resolution.
- A parameterized unit cell maintains uniform strut thickness while permitting substantial lattice anisotropy.
- Lattice compilation extracts a globally consistent structure from locally defined orientation, porosity, and anisotropy using extended field-aligned meshing techniques.
- The workflow supports conforming lattice design in both 2D and 3D through homogenization-based topology optimization and field-aligned parameterization.
- The optimized lattice chair contains 178,291 struts on a 140 × 100 × 200 simulation resolution.
2 RELATED WORK
Prior work often aligns lattice or microstructures to regular grids or prescribed shapes, while this paper combines stress alignment, concurrent shape optimization, and field-aligned compilation for conforming 2D and 3D lattices.
- Prior approaches: Earlier microstructure methods commonly assume alignment with a prescribed regular grid, limiting the solution space and achievable structural performance.The paper contrasts this assumption with its stress-aligned approach.
- Prior approaches: Existing approaches conform lattices to prescribed 3D boundaries, whereas this method concurrently optimizes the shape and aligns lattices with stress directions.Alignment with principal stress directions is associated with improved structural performance.
- Prior approaches: Stress-tracing and ground-structure methods produce appealing 2D results but do not directly extend to 3D volumetric lattices because of initialization challenges.The proposed method instead constructs stress-aligned 3D volumetric lattices using homogenization-based topology optimization.
- Homogenization and compilation: Homogenization-based methods must compile differently oriented hollow cells into continuous structures, with prior projection methods producing high-resolution pixel or voxel models.The paper positions its field-aligned compilation method as an alternative representation.
- Paper’s distinctions: The proposed approach uses constant-thickness anisotropic cells, simultaneous shape and lattice optimization, and field-aligned meshing that compactly represents the result as a graph.These three differences distinguish it from earlier homogenization-based conforming-lattice approaches.
- Field-aligned compilation: Field-aligned parameterization is repurposed from mesh generation to lattice generation, avoiding mesh-extraction stability and geometric-topological complications because lattices are encoded as graphs.The paper extends local parameterizations to incorporate anisotropy and heterogeneity.
- Field-aligned compilation: Compared with stress-line tracing, the method directly generates struts through graph operations and uses optimized stress fields without smoothing them.The authors describe this as fast and efficient.
3 OVERVIEW
The method takes application-specific loads and a design domain, then jointly optimizes a stress-aligned lattice structure and boundary-conforming shape for stiffness.
- Method goals: Given a design domain and application-specific loads, the method generates a lattice structure that maximizes structural stiffness.The optimized struts conform with principal stress directions.
- Method goals: Boundary struts span a smooth surface that faithfully approximates the optimized shape.This couples the internal stress-aligned lattice with the optimized external boundary.
- Two-step workflow: The workflow first optimizes shape and lattice-material distribution, then compiles the optimized fields into a globally connected lattice structure.The first stage includes topology changes, while the second uses the resulting field data for compilation.
4 LATTICE OPTIMIZATION
The optimization represents each element with adaptable lattice cells whose occupancy, scaling, and orientation are optimized under stiffness and material constraints, then iteratively solved using homogenized finite-element analysis.
- Discretization and parameterization: The design domain is discretized into bilinear quadrilateral elements in 2D or trilinear cubic elements in 3D, each filled by a scaled and rotated periodic cell.Scaling factors and rotation matrices are optimized per element.
- Pipeline: The pipeline optimizes lattice-material distribution first and extracts a continuous lattice structure from the optimized configuration second.The 2D example shows the progression from design specification to optimized distribution to continuous structure.
- Cell parameterization: Keeping strut thickness t constant while elongating cell side lengths creates cells with graded solid fractions across the domain.The unit-cell side length l and thickness t are user-specified.
- Cell parameterization: Per-axis elongation increases cell anisotropy, which is beneficial because stress tensors are typically anisotropic.The resulting material distribution can place lower-solid-fraction cells in relatively low-stress regions.
- Design variables: The relaxed occupancy variable ϕe indicates whether each element contains lattice material and thereby determines the optimized structure’s overall shape.Intermediate values support gradient-based optimization, while the variable is interpreted as lattice-material fraction.
- Optimization objective: The optimization minimizes external-force work, equivalent to compliance minimization and stiffness maximization, subject to a solid-material budget and bounds on occupancy and scaling.The force is fixed, displacements come from static equilibrium, and scaling bounds are user-defined.
- Optimization formulation: Optimizing scaling factors instead of strut thickness preserves uniform strut thickness, while optimizing occupancy allows shape and lattice distribution to evolve together.Setting ϕ = 1 fills the prescribed design domain; allowing ϕ to vary enables simultaneous shape evolution.
- Iterative solution: Finite-element equilibrium supplies displacements, strains, and stresses; gradient-based updates then modify occupancy and scaling variables with filtering and Heaviside projection.The iteration continues until design changes fall below a threshold or the iteration limit is reached.
5 LATTICE COMPILATION
Lattice compilation converts optimized orientation, scaling, and occupancy fields into a graph that preserves anisotropic directions and target resolution while conforming to the extracted shape. Local parameterizations are optimized over the input graph, then integer translations guide edge collapsing, retention, and diagonal removal.
- Overview: The compilation input is a graph whose vertices encode position, orientation, and axis scaling; the output graph aims to preserve directions, anisotropy, and resolution.The method allows relaxed input geometry and topology, while the lattice output need not contain solid face elements.
- 5.1 Parameterization: A local parameterization assigns each vertex an origin, orientation matrix, and fixed axis-dependent unit lengths based on a global target edge length.Unlike rotationally symmetric coordinate systems, the mutable coordinates may only flip axis signs while preserving the right-hand rule; origins are optimized.
- 5.1 Parameterization: The parameterization energy sums squared differences between neighboring local parameterizations after interpolating directions and scales and applying integer translations.The translation term moves a neighboring parameterization to the nearest integer-shifted position, so the energy measures fractional-part differences without integer jumps.
- 5.1 Parameterization: Local parameterizations are optimized by iterative neighbor smoothing, with a hierarchical graph structure used to accelerate computation and retain scalability.The local formulation is easily parallelizable, and the implementation uses 50 iterations for 2D levels and 200 for 3D levels.
- 5.2 Graph Extraction: Integer translations classify edges by the number of nonzero ±1 components: zero collapses vertices, one retains stress-direction edges, and two or three removes diagonal edges.This extraction produces mostly right angles, while relabeling nearby diagonal edges can preserve missing rotational directions before diagonal removal.
6 RESULTS
The method produces optimized 2D and 3D lattices with spatially varying orientation, porosity, and anisotropy, including lattices conforming to curved shapes. Greater design flexibility improves stiffness, while compiled structures closely match homogenization predictions and support detailed physical fabrication.
- Examples: Optimized 2D and 3D lattices vary in orientation, porosity, and anisotropy across prescribed or optimized domains.
- Design options: 110.84 →96.03 →85.85 compliance for the cantilever and 230.52 →177.86 →149.96 for the bridge as design flexibility increases.
- Design options: 44.39% lower compliance is achieved by the most flexible design than by the fixed baseline in the 2D design-option study.The study attributes the improvement to adaptive porosity and anisotropic features.
- Validation: 2.89%–6.46% compliance differences show that compiled lattices closely match homogenization-based predictions.The comparison uses full-resolution finite element analysis of the compiled structures.
- Performance: The optimization and compilation together take less than 1 hour even for complex models such as the chair and femur.
- Physical comparison: Physical tests found that the lattice structure supported a maximum force twice as large as the topology-optimized solid before buckling, with slightly lower stiffness.
- Physical comparison: The conforming lattice had compliance 177.29 versus 184.64 for bone-like infill, while optimization took 1 minute 7 seconds versus 40 minutes.
- Scalability: Highly detailed compiled structures contained 462k struts in the bridge and 351k in the cantilever.
7 CONCLUSIONS
The paper presents a scalable workflow for conforming lattice structures that aligns lattices with principal stress directions and optimized boundaries. Numerical and physical evaluations support its performance, fabrication, and buckling-stability claims.
- The workflow combines homogenization-based topology optimization with field-aligned parameterization for 2D and 3D conforming lattices.
- The method generates lattices over optimized subregions and prescribed curved shapes while conforming to principal stress directions and shape boundaries.
- Numerical studies support aligning anisotropic lattices with internal stresses and grading porosity and anisotropy.
- Physical tests show optimized lattices can support twice the buckling load of topology-optimized solids at a slight stiffness cost.
APPENDIX
The appendix presents the engineering-notation form of a 6×6 rotation matrix for an elasticity tensor and lists its component expressions.
- The appendix introduces a 6×6 rotation matrix for the elasticity tensor in engineering notation.
- The supplied appendix passages do not state a further computational or optimization role for the matrix.
- The displayed expressions contain products of direction components such as m, n, and l.