Computational Design Strategies and Software for Lattice Structures and Functionally Graded Materials

Abstract

This study presents a comparative analysis of software platforms and computational methods used in the design of three-dimensional lattice structures and functionally graded materials (FGMs). Through systematic evaluation of 31 computational platforms across seven critical criteria (lattice type support, parametric control, conformal generation, multi-material capabilities, ease of use, FEA integration, and AM compatibility), this review identifies that specialized platforms significantly outperform general-purpose CAD tools, with scores exceeding 30/35 points compared to 15–20/35 for conventional systems. The analysis reveals that implicit and voxel-based representations dominate high-performance applications, while traditional boundary-representation methods approach fundamental limitations for complex lattice generation. Emerging machine learning-driven frameworks demonstrate 82% reduction in optimization iterations through Bayesian optimization and achieve property prediction speedups of nearly 100× compared to computational homogenization, enabling rapid inverse design workflows previously computationally infeasible. These insights provide researchers with evidence-based guidance for selecting computational approaches aligned with specific manufacturing capabilities and design objectives.

1. Introduction

The advancement of additive manufacturing (AM) technologies has catalysed a paradigm shift in structural design and material engineering. Unlike traditional subtractive manufacturing, AM enables the creation of geometrically complex structures and spatially varying material compositions that were previously unachievable or economically unfeasible. Two domains have emerged at the forefront of this revolution: lattice structures and functionally graded materials (FGMs) [1].

Lattice structures, defined by their periodic or quasi-periodic cellular architectures, exhibit highly favorable mechanical and functional properties relative to their structural mass. Owing to their engineered porosity and topologically optimised geometries, these architected materials demonstrate elevated specific strength, superior energy dissipation and crash mitigation performance, and enhanced acoustic attenuation and thermal management characteristics [2,3,4,5]. Contemporary lattice design paradigms span canonical configurations such as simple cubic, body-centered cubic (BCC), and derivative unit cell topologies, extending further to biomimetic architected materials inspired by hierarchical natural systems, including trabecular bone microstructures and honeycomb-derived cellular networks [6,7].

Functionally graded materials (FGMs) represent an advanced class of architected materials in which mechanical, thermal, or compositional properties vary continuously or discretely throughout the material’s volume [8]. Through the spatially controlled distribution of constituent phases, microstructures, or material compositions, FGMs enable engineered property gradients that can be optimised for application-specific loading conditions, extreme thermal environments, or broader multifunctional performance objectives.

When integrated with lattice-based architectures, the application of FGM principles, yielding graded lattice systems, introduces significant additional design complexity, including nonlinear spatial heterogeneity, multi-scale coupling, and increased sensitivity to local topological transitions. These complexities impose substantial computational demands, necessitating the use of advanced numerical optimization frameworks, multi-physics simulation tools, and high-fidelity geometric modeling techniques to accurately capture and exploit the behavior of such graded architected materials.

The computational design of these advanced structures demands sophisticated software platforms and methodologies capable of handling complex geometric representations, parametric modeling, multi-scale analysis, and integration with manufacturing constraints. Traditional CAD tools, while proficient in conventional solid modeling, often lack specialised features required for efficient lattice tessellation, functionally graded property definition, homogenization-based analysis, and topology optimization of cellular architectures [1,9,10,11]. Recent comparative studies demonstrate that boundary-representation (B-rep) CAD systems encounter fundamental scalability limitations when handling lattice structures with millions of geometric elements, with failure rates exceeding 90% for Boolean operations on TPMS geometries in conventional platforms such as Siemens NX and CATIA [1]. These limitations stem from fundamental challenges: defining continuous material gradients across irregular geometries, tessellating complex unit cells within conformal design spaces, computing effective properties through multi-scale homogenization, and optimizing structures with thousands of interdependent design variables. Consequently, specialised platforms integrating implicit modeling, density-based optimization, and automated homogenization workflows have emerged to address these computational challenges while maintaining compatibility with additive manufacturing process constraints.

This review synthesizes current computational approaches for the design and optimization of lattice structures and functionally graded materials through systematic literature analysis and software platform evaluation, as illustrated in Figure 1. Literature was identified through structured searches across major academic databases (Scopus, Web of Science, IEEE Xplore) covering publications from 2015 to 2025, using search terms combining lattice structures, functionally graded materials, computational design methods, software, machine learning and additive manufacturing. Articles were screened based on relevance to computational design tools, methodologies, and their integration with AM workflows. Software platforms were identified through cited tools in reviewed literature, manufacturer documentation, and consultation with AM practitioners. Platforms were included if they offered dedicated lattice generation, FGM design features, or topology optimisation with cellular structure export. Each platform was assessed across seven criteria: lattice type support, parametric control, conformal generation capability, multi-material support, ease of use, FEA integration, and AM compatibility. The scope encompasses parametric and non-parametric design strategies, geometric representation formats (NURBS, mesh-based, voxel, and implicit), simulation and optimisation frameworks including finite element analysis and topology optimisation, and emerging machine learning-driven approaches that are reshaping the design landscape. By providing this comprehensive comparative analysis, this review aims to assist researchers and practitioners in identifying suitable computational methodologies aligned with their specific application requirements, manufacturing capabilities, and design objectives.

2. Fundamental Concepts and Design Parameters

The design of lattice structures and functionally graded materials involve controlling multiple interdependent parameters that govern mechanical, thermal, and manufacturing performance. Understanding these fundamental design variables and their interactions is essential for effective computational modeling and optimization.

2.1. Lattice Structure Characteristics

Lattice structures are characterised by four fundamental factors that determine their macroscopic properties: unit cell morphology, tessellation strategy, relative density, and material composition [12]. The unit cell represents the smallest repeating element whose geometric and topological characteristics propagate throughout the entire structure when tessellated.

2.1.1. Unit Cell Morphology

Unit cell topology encompasses various configurations including cubic, body-centered cubic (BCC), face-centered cubic (FCC), diamond, truncated octahedron, gyroid, and other complex forms [12]. Each topology exhibits distinct mechanical characteristics, including different ratios of stiffness to weight and diverse deformation modes under loading. The strut diameter, length, and angular orientation within unit cells represent critical design parameters that significantly influence mechanical performance, particularly stress distribution at node junctions and overall structural stiffness.

2.1.2. Relative Density

Relative density, defined as the ratio of lattice material density to solid material density, is a primary design variable controlling macroscopic mechanical properties. By modulating relative density through variation of strut dimensions or unit cell size, engineers can achieve targeted stiffness, strength, and energy absorption properties [12]. This parameter is particularly important for gradient designs, where relative density varies spatially to create functionally graded lattices that optimize stress distribution and minimize weight.

2.1.3. Tessellation Strategies

Tessellation strategies describe how unit cells are arranged and connected within a design domain. Regular tessellations employ identical unit cells throughout the structure, simplifying computational analysis through periodic boundary conditions. Conformal tessellations adapt unit cell geometry to fit complex boundary conditions and design spaces, enabling integration of lattice structures into arbitrary component geometries while maintaining structural connectivity.

2.2. Types of Lattice Structures

• Strut-Based Lattice Structures: Strut-based lattices consist of rod-like elements (beams or struts) connected at nodes. Classification by coordination and topology includes simple cubic (6-connected, bending-dominated), body-centered cubic/BCC (14-connected, stretch-dominated), face-centered cubic/FCC (12-connected), octet truss (12-connected, Maxwell stable, highly efficient), Kelvin (14-sided tetrakaidecahedron cell), and diamond (4-connected tetrahedral network). By mechanical behavior [5], lattices are either bending-dominated (lower stiffness, higher energy absorption) or stretch-dominated (higher stiffness, more efficient load transfer).
• Surface-Based Lattice Structures (TPMS): Triply Periodic Minimal Surfaces (TPMS) are mathematical surfaces with zero mean curvature that repeat periodically in three dimensions. Major TPMS families include Schwarz surfaces (Primitive/P with simple cubic symmetry, Diamond/D with body-centered cubic symmetry, and IWP/I-WP wrapped package with complex cubic symmetry), Schoen surfaces (Gyroid/G with cubic symmetry and no straight lines, and I-Graph with body-centered symmetry), and Fischer-Koch surfaces (Neovius and F-RD/Fischer-Koch S). TPMS lattices can be realised as sheet networks (material follows the minimal surface) or solid networks (material fills regions on one or both sides of surface) [13].
• Hybrid and Custom Lattice Structures: These include functionally graded lattices (spatially varying cell size, strut thickness, or topology), multi-scale hierarchical lattices (lattices of lattices), interpenetrating lattices (multiple independent lattice networks occupying the same volume), and stochastic/random lattices (non-periodic arrangements such as Voronoi and random beam networks).

2.3. Functionally Graded Materials

Functionally graded materials represent a compositional and microstructural engineering approach wherein material properties vary continuously or discretely from one region to another. The gradient can be defined through volume fraction laws that describe the spatial distribution of constituent phases, enabling tailored performance characteristics impossible with homogeneous materials.

2.3.1. Gradient Functions

The design of FGMs typically involves defining a gradient function that describes property variation along selected spatial coordinates. Common gradient definitions include linear, exponential, polynomial, and sigmoidal functions that control material composition distribution according to loading conditions, thermal environments, or functional requirements. Mathematically, a typical gradient function takes the form [14,15]:

$$\mathrm{P}\left(\mathrm{x}\right)={\mathrm{P}}_1+\left({\mathrm{P}}_2-{\mathrm{P}}_1\right)\cdot \mathrm{f}\left(\mathrm{x}\right)$$

(1)

where $\mathrm{P}\left(\mathrm{x}\right)$ is the material property at position $\mathrm{x}$, ${\mathrm{P}}_1$ and ${\mathrm{P}}_2$ are properties of constituent materials, and $\mathrm{f}\left(\mathrm{x}\right)$ is a normalised gradient function ranging from 0 to 1.

2.3.2. Volume Fraction Laws

The volume fraction law quantitatively specifies how the proportion of each material constituent changes spatially, typically expressed as a power-law relationship:

$$V_f\left(\mathrm{x}\right)= V_f^{max}·{\left({\displaystyle \frac{z-z_{min}}{z_{\min }-z_{\min }}}\right)}^n$$

(2)

where ${\mathrm{V}}_f\left(\mathrm{x}\right)$ is the volume fraction at position $\mathrm{x}$, $z$ represents the spatial coordinate along the grading direction, and $n$ is the power-law exponent controlling gradient steepness [16]. This formulation enables precise control over property transitions while satisfying physical constraints on volume fractions.

2.3.3. Multi-Material Complexity

Multi-material FGMs introduce additional complexity by incorporating three or more material phases, each with distinct mechanical, thermal, or functional properties, and distributing them spatially according to design objectives. The computational challenge intensifies when gradient definitions must simultaneously satisfy structural performance requirements, manufacturing constraints (such as material compatibility and bonding), and physical constraints (such as avoiding phase combinations with excessive thermal expansion mismatch).

2.4. Homogenization and Multiscale Approaches

Homogenization theory provides the theoretical foundation for multiscale lattice design, enabling computational linking of microscale unit cell properties to macroscale structural response.

2.4.1. Homogenization Framework

The homogenization approach involves three fundamental steps: (1) defining a representative volume element (RVE) corresponding to the unit cell with appropriate boundary conditions, (2) conducting microscale finite element analysis to determine effective elastic properties, yield stresses, and failure criteria through computational micromechanics, and (3) upscaling these homogenised properties for use in macroscale structural analysis and optimization.

The effective elastic tensor $C^{\mathrm{eff}}$ is computed by solving microscale boundary value problems [17]:

$$C_{ijkl}^{\mathrm{eff}}={\displaystyle \frac{1}{\left|V\right|}}{\int }_V{C}_{ijkl}^{\mathrm{micro}}\left(\mathit{x}\right)\hspace{0.17em}d\mathrm{V}$$

(3)

where $C^{\mathrm{micro}}$ represents the local constitutive tensor within the RVE and $V$ is the RVE volume.

2.4.2. Concurrent Multiscale Design

Concurrent multiscale design simultaneously optimizes lattice topology at microscale and macroscale structure topology, enabling more complex gradient distributions and superior overall performance compared to sequential optimization approaches. This methodology addresses the limitation of highly restricted unit cell designs, which may fail to achieve optimal desired properties despite lower computational cost. The concurrent formulation couples microscale unit cell variables with macroscale topology variables through homogenization relationships, solving a unified optimization problem that balances local microstructural efficiency with global structural performance [18].

3. Geometric Representations in Lattice and FGM Design

Efficient computational design of lattice architectures and functionally graded materials (FGMs) fundamentally relies on the selection of a suitable geometric representation. Each representational framework, whether discrete, implicit, parametric, or hybrid, entails specific advantages and limitations with respect to geometric fidelity, numerical tractability, manufacturability, and integration within optimization or multiscale simulation pipelines. Selection of representation profoundly influences available modeling operations, as detailed in Section 9. Figure 2 presents common lattice and FGM model representations: NURBS surfaces (four-sided patches that form boundary representation models), implicit surfaces defined by signed distance functions, voxel-based discretizations, and mesh representations.

3.1. NURBS-Based Representations

Non-Uniform Rational B-Splines (NURBS) provide a mathematically rigorous framework for representing smooth surfaces and complex geometries through control points and basis functions. NURBS are extensively utilised in conventional CAD systems and offer exceptional precision for defining continuous material property gradients.

Advantages: NURBS representations enable smooth, mathematically continuous descriptions of FGM compositions and lattice node positions. They integrate seamlessly with industry-standard CAD software—such as SolidWorks 2024 (Dassault Systèmes, Vélizy-Villacoublay, France), Autodesk Fusion 360 (Autodesk Inc., San Francisco, CA, USA), and Siemens NX 2312 (Siemens Digital Industries Software, Plano, TX, USA)—and facilitate parametric design workflows. The continuity properties of NURBS make them advantageous for FEM preprocessing and analysis.

Limitations: NURBS-based approaches become computationally expensive for highly complex lattice topologies with numerous nodes and struts. Representing highly irregular or biomimetic lattice patterns with NURBS is cumbersome. Additionally, NURBS surfaces may require complex trimming and patching operations for irregular geometries, introducing computational overhead and potential mesh generation challenges.

3.2. Mesh-Based Representations

Triangular and tetrahedral mesh representations discretize geometry into polygonal elements, serving as the primary interface between geometric definition and FEM analysis.

Advantages: Mesh representations are the standard format for FEM software—such as ANSYS Mechanical 2024 R1 (ANSYS Inc., Canonsburg, PA, USA) and Abaqus 2023 (Dassault Systèmes Simulia Corp., Johnston, RI, USA)—, eliminating conversion steps between design and analysis. They accommodate highly irregular geometries and complex topologies naturally. Mesh density can be locally refined in regions requiring higher geometric accuracy or analysis resolution.

Limitations: Mesh-based approaches are inherently discrete, necessitating careful refinement strategies to capture geometric details accurately. Material property gradients must be approximated through element-wise or nodal property assignments, potentially introducing discretization artifacts. Mesh generation for complex lattice structures remains computationally intensive and may require specialised algorithms to ensure element quality and manifold consistency.

3.3. Voxel-Based Representations

Voxel models discretize the design domain into regular three-dimensional grid cells, each assigned material or geometric properties.

Advantages: Voxel representations simplify material property assignment and gradation, as each voxel can directly encode local compositional or microstructural information. This approach is particularly suited for lattice structures with discrete reinforcement phases or multi-material configurations. Voxel grids interface naturally with volumetric image data from computed tomography (CT) scans, facilitating reverse engineering and biomimetic design workflows. Certain topology optimization algorithms operate natively on voxel grids, eliminating intermediate conversion steps. Figure 3 shows a shoe sole modeled by the authors using Crystallon [19], Pufferfish [20], Dendro [21], and Grasshopper [22], where it was possible to use custom plantar pressure data to control the geometry of the struts, as well as the unit cells inside the lattice structure.

Limitations: Voxel representations suffer from inherent staircase artifacts along curved surfaces and boundaries, requiring substantial refinement to achieve smooth geometries comparable to NURBS or high-quality meshes. The regular grid structure may be inefficient for representing elongated or thin-walled features. Memory requirements scale cubically with grid resolution, limiting practical voxel dimensions for large-scale structures.

3.4. Implicit Surface Representations

Implicit surfaces, defined as the zero-level set of a scalar function $\mathrm{f}\left(x\right)=0$, offer a mathematically elegant framework for describing complex geometries and topologies.

Advantages: Implicit representations naturally accommodate arbitrary topological changes and complex internal structures without explicit mesh connectivity constraints. They excel at representing smooth gradations and complex property fields through continuous scalar functions. Implicit methods integrate well with topology optimization and level-set-based design approaches. Boolean operations (union, intersection, difference) are computationally efficient in implicit space [23].

Limitations: Implicit representations require dense sampling or surface reconstruction algorithms to generate exportable geometry (mesh or point clouds) for visualization and AM. Interpreting implicit functions for design intent communication with fabrication partners can be challenging. Integration with conventional CAD workflows remains limited, as most industry-standard software operates on explicit representations.

4. Design Strategies: Parametric and Non-Parametric Approaches

Design methodologies for lattice structures and FGMs categorize into parametric and non-parametric strategies. Parametric approaches employ mathematical rules and functional relationships to define geometric features, offering precise control through explicit design variables. Non-parametric methods leverage optimization algorithms, implicit representations, or data-driven techniques to generate structures without predefined geometric rules. This section examines both paradigms and their capabilities for managing geometric complexity and achieving targeted performance outcomes.

4.1. Parametric Design Strategies

Parametric design defines geometry through explicit mathematical relationships and controllable parameters. These approaches enable systematic exploration of design spaces while maintaining clear relationships between input parameters and resulting structures. Two primary categories characterize parametric strategies: rule-based systems that apply geometric transformations according to predefined logic, and function-based systems that use continuous mathematical functions to describe material distribution and structural features.

4.1.1. Rule-Based Parametric Systems

Rule-based approaches encode design intent through explicit mathematical rules or algorithms that govern lattice topology, strut dimensions, and material composition as functions of input parameters. Common implementations include:

Cellular Solid Architectures: Regular or quasi-regular unit cell patterns (cubic, BCC, face-centered cubic, auxetic configurations) are defined parametrically through lattice constant, strut diameter, and spatial frequency parameters. These architectures enable efficient FEM analysis and fabrication, as their periodic nature facilitates mesh generation and material specification. Tools like nTop [24], parametric CAD, and specialised lattice design software implement these approaches.

Graded Lattice Definitions: Parametric functions modulate unit cell dimensions, strut thickness, or material composition as functions of spatial coordinates. For example, a graded density lattice might employ:

$$\mathrm{d}\left(x\right)=d_0+\mathrm{k}\cdot g\left(x\right)$$

(4)

where $\mathrm{d}\left(x\right)$ represents strut diameter at position $x$, $d_0$ is a baseline diameter, $\mathrm{k}$ is a scaling factor, and $\mathrm{g}\left(x\right)$ is a spatial grading function. This approach maintains computational tractability while achieving performance-optimised gradients [25].

Advantages: Parametric systems offer explicit control over design intent and computational efficiency. Parameter sensitivity studies are straightforward to execute. Designs remain flexible and easily modified through parameter adjustment. Generated geometries integrate well with conventional CAD and FEM workflows.

Limitations: Complex irregular features or highly optimised topologies are difficult to express through explicit parametric rules. Parametric systems may impose artificial constraints on achievable designs. Design space exploration requires manual iteration or coupled optimization loops.

4.1.2. Topology Optimization Approaches

Topology optimization mathematically solves for optimal material distributions within a defined design domain subject to performance objectives (e.g., minimize compliance, maximize buckling load) and constraints (e.g., material volume fraction, stress limits).

Formulation: The fundamental topology optimization problem is typically formulated as:

$$\underset{\mathrm{\rho }\left(\mathit{x}\right)}{\min } u^{T}Ku$$

(5)

$$\mathrm{subject} \mathrm{to} Ku=f, 0<{\mathrm{\rho }}_{min}\le \mathrm{\rho }\left(x\right)\le 1, {\int }_{\mathrm{\Omega }}\mathrm{\rho }\left(x\right)d\mathrm{\Omega }=V_{\mathrm{target}}$$

(6)

where $\mathrm{\rho }\left(x\right)$ is the material density field, $K$ is the global stiffness matrix, $u$ and $f$ are displacement and force vectors respectively, and $V_{\mathrm{target}}$ is the target volume fraction [26].

Density-based methods treat each voxel or finite element as a design variable with material density varying between zero (void) and one (solid). These methods, exemplified by SIMP (Solid Isotropic Material with Penalization), are computationally efficient and readily integrate with commercial FEM software. However, they often produce ambiguous intermediate-density solutions requiring post-processing interpretation and thresholding to obtain manufacturable geometries. For lattice structures, density-based approaches can directly map optimised densities to relative lattice densities, enabling automated generation of graded cellular architectures [1].

Level set methods employ implicit surface representations to track material-void boundaries during optimization. The level set function $\mathsf{\varphi }\left(x,t\right)$ evolves according to shape sensitivity information derived from adjoint analysis [27]:

$${\displaystyle \frac{\partial \mathsf{\varphi }}{\partial \mathrm{t}}}+V_n\left|\nabla \mathsf{\varphi }\right|=0$$

(7)

where $V_n$ is the normal velocity field determined by objective function sensitivities. These methods naturally incorporate geometric constraints such as minimum feature sizes and enable direct control of structural boundaries, producing crisper material-void interfaces than density-based approaches.

Stress-constrained topology optimization specifically targets lattice design by directly incorporating stress limitations, enabling designers to control stress concentrations and ensure structural safety while optimising lattice topology. This approach integrates homogenization theory to characterize effective elastic constants and yield stresses of lattice metamaterials, ensuring that microscale stresses within unit cell struts remain below material limits. The optimization formulation becomes:

$$\underset{\mathrm{\rho }\left(\mathit{x}\right)}{\min } u^{T}Ku$$

(8)

$$\mathrm{subject} \mathrm{to} \hspace{1em}{\mathrm{\sigma }}_{\mathrm{eff}}\left(x\right)\le {\mathrm{\sigma }}_{\mathrm{allow}},\hspace{1em}{\int }_{\mathrm{\Omega }}\mathrm{\rho }\left(x\right)d\mathrm{\Omega }\le V_{\mathrm{target}}$$

(9)

where ${\mathrm{\sigma }}_{\mathrm{eff}}$ is the effective stress computed through homogenization and ${\mathrm{\sigma }}_{\mathrm{allow}}$ is the allowable stress limit [28].

Capabilities: Topology optimization often reveals counterintuitive designs with performance characteristics superior to conventional engineer-designed solutions. The method naturally produces internal lattice structures and material gradients. Optimization-based approaches facilitate multi-objective optimization considering strength, weight, thermal properties, and manufacturability simultaneously.

Challenges: Optimised topologies often feature complex, irregular geometries poorly suited to conventional AM processes. Fabrication constraints (minimum feature size, overhang angles) must be incorporated into the optimization formulation, complicating problem setup. Computational cost scales significantly with design domain resolution. Post-processing to convert optimised density fields into manufacturable structures often involves ad hoc filtering and thresholding steps.

4.2. Non-Parametric Design Strategies

Non-parametric approaches generate designs without explicit mathematical or rule-based relationships, employing data-driven, generative, or heuristic methods to explore design possibilities.

4.2.1. Data-Driven Design and Reverse Engineering

Data-driven approaches leverage experimental data, historical designs, or digitised natural structures to inform new designs. Applications include:

Biomimetic Design: Natural structures (trabecular bone, hierarchical composites, cellular organisms) are digitized via computed tomography or 3D scanning, analysed computationally, and adapted for engineered applications. Voxel or implicit representations naturally accommodate irregular biomimetic structures. Density gradients observed in natural systems inform FGM design strategies.

Design Space Databases: Curated databases of lattice configurations, material combinations, and performance characteristics enable rapid prototyping and design reference. Machine learning models trained on these datasets can predict performance or suggest design modifications for specific objectives [29].

Advantages: Data-driven approaches ground designs in real-world performance or natural precedent. Biomimetic structures often exhibit emergent properties not anticipated by conventional design thinking. Reverse engineering enables rapid design iteration when baseline structures are available.

Limitations: Biomimetic data digitization is labor-intensive and subject to interpretive decisions. Scaling or adapting natural structures to new applications requires careful consideration of similarity and dimensional analysis. Database-driven approaches are limited by database scope and completeness.

4.2.2. Generative and AI-Driven Design

Generative adversarial networks (GANs), variational autoencoders (VAEs), and reinforcement learning (RL) algorithms represent emerging approaches for non-parametric design generation [30].

Generative Models: GANs and VAEs learn latent representations of geometric design space from training datasets. These learned representations enable generation of novel designs by sampling latent space or interpolating between known designs. Conditional variants allow control over design generation through performance objectives, constraints, or material specifications [30].

Reinforcement Learning: RL frameworks formulate design generation as sequential decision-making problems. An agent iteratively modifies a design to maximize cumulative reward (e.g., structural performance, manufacturability score). RL is particularly suited to multi-step design refinement where global optimization is intractable.

Advantages: Generative models explore design space more broadly than parametric or optimization-driven approaches. These methods naturally accommodate complex, irregular geometries. Integration of domain knowledge through network architecture or reward function design enables flexible objective formulation.

Limitations: Generative models require substantial training data. Model interpretability remains challenging, as understanding why a particular design was generated aids neither design communication nor optimization insights. Computational cost of training is substantial. Fabrication feasibility constraints are difficult to enforce strictly within current generative frameworks.

5. Computational Methods: Simulation and Optimization

Rigorous computational evaluation of lattice structures and functionally graded materials (FGMs) requires integrated simulation and optimization frameworks that predict multiscale performance, reveal critical failure mechanisms, and iteratively refine design configurations.

5.1. FEA for Lattice Structures

Finite Element Analysis is the primarynumerical method for predicting mechanical behavior, thermal properties, and multiphysics responses in lattice structures and FGMs [31]. Meshing for these systems is challenging due to their complex topology and variable feature sizes. Thin struts require high mesh aspect ratios; beam elements (1D) offer efficient analysis for global behavior, while solid tetrahedral or hexahedral elements deliver higher fidelity for regions of complex geometry or high stress concentration. Hybrid approaches mix both element types as needed. Junction regions, where multiple struts intersect, require fine mesh resolution for accurate stress prediction, with geometric preparation like node rounding (fillets) impacting stress results and manufacturability [31]. For graded lattices, spatially varying material properties are assigned at element integration points using appropriate averaging schemes to ensure physics is captured accurately.

5.2. Constitutive Modeling for FGMs

Standard isotropic models are not suitable for FGMs that have spatially varying and anisotropic properties. Constitutive parameters (elastic modulus, Poisson’s ratio, density, thermal properties) are defined as functions of spatial coordinates. Element integration points interpolate from these fields, and sharp property gradients (e.g., at phase boundaries) require careful numerical handling to avoid artificial stress concentrations.

Lattice structures with oriented struts display directional elastic anisotropy, best represented by suitable constitutive models (transversely isotropic or orthotropic). FGMs often serve multifunctional roles, prompting coupled analyses, that are supported by the FEM with built-in multi-physics coupling [8].

5.3. Computational Efficiency Strategies

Large-scale lattice analysis demands computational efficiency. For periodic lattices, periodic boundary conditions allow analyzing single unit cells to predict global behavior, leveraging homogenization theory [32]. Adaptive mesh refinement uses error estimators or stress gradients to concentrate resolution where it matters most. Parallel computing in commercial FEM codes utilizes multi-core and GPU architectures for large systems.

5.4. Isogeometric Analysis (IGA)

IGA couples FEA directly with CAD geometric representations using NURBS basis functions [33]. This eliminates geometry conversion errors, allows higher-order continuity ($C^{p-1}$ where $p$ is polynomial degree), and reduces mesh density requirements for equivalent accuracy, leading to seamless CAD-FEA integration.

5.5. Optimization Frameworks

Optimization for lattices and FGMs uses gradient-based, gradient-free, or hybrid approaches, typically within iterative FEM analyses. Gradient-based methods harness sensitivity analysis for rapid convergence and scalability when parametric representations are used. However, they can get trapped in local minima and are less applicable for objectives that are discontinuous or non-differentiable [12].

Gradient-free methods like genetic algorithms and Bayesian optimization are suitable for discrete, non-differentiable, or black-box objectives, though they require more function evaluations [12].

Lattice and FGM designs often involve competing objectives—minimizing weight, maximizing stiffness or energy absorption—requiring multi-objective optimization. Pareto-optimal solutions represent trade-offs, created using multi-objective genetic algorithms (NSGA-II, SPEA2) or Bayesian optimization, enabling informed selection of designs based on application-specific priorities [12].

Multi-Disciplinary Optimization Workflow Platforms

Complex lattice and FGM design often requires orchestrating multiple software tools—CAD for geometry, FEA for analysis, and specialised optimizers—into integrated workflows. Multi-disciplinary optimization (MDO) platforms automate these workflows, managing design variable propagation, simulation execution, and optimization algorithm coordination across heterogeneous software ecosystems.

modeFRONTIER [34] provides a comprehensive workflow automation environment that connects diverse CAD, FEA, and CFD tools into unified optimization loops. The platform supports numerous optimization algorithms including NSGA-II, MOGAII (Multi-Objective Genetic Algorithm II), simulated annealing, and particle swarm optimization, with particular strength in multi-objective problems producing Pareto-optimal solutions. For lattice structure applications, modeFRONTIER has been successfully coupled with ANSYS Mechanical 2024 R1 (ANSYS Inc., Canonsburg, PA, USA) for parametric lattice topology optimization, nTop for automated lattice parameter studies, and various mesh morphing tools for design space exploration. The platform’s grid computing capabilities enable distributed simulation execution across networked machines, significantly accelerating computationally expensive lattice optimization campaigns involving hundreds or thousands of design evaluations.

Simcenter HEEDS [35] offers similar workflow automation with emphasis on efficient design space exploration through advanced search algorithms. The platform interfaces with all major commercial CAD and CAE tools, automating repetitive tasks while systematically exploring design variations. HEEDS employs proprietary search strategies optimized for reducing the number of required simulations—a critical advantage when each lattice structure evaluation involves computationally expensive FEA or homogenization analysis. The software supports multidisciplinary optimization considering coupled mechanical, thermal, and manufacturing objectives simultaneously, making it particularly suitable for functionally graded lattice structures where performance depends on complex interactions between geometry, material distribution, and loading conditions.

Practical Application: These MDO platforms are typically employed when lattice design involves coupling specialised geometry generators (nTop, Grasshopper) with external FEA solvers (ANSYS, Abaqus) and manufacturing constraint validation. They excel at parameter sensitivity studies, multi-objective trade-off analysis, and systematic design space exploration where manual iteration would be prohibitively time-consuming. However, they require upfront investment in workflow setup and inter-software integration.

6. Software Platforms, Tools, and Additive Manufacturing Integration

The landscape of software platforms for lattice structure and FGM design spans general-purpose CAD and FEA environments to highly specialised tools for lattice generation, grading control, and multiscale analysis. These platforms differ in their geometry representations, computational abilities, and integration with manufacturing workflows. As geometric modeling, simulation, and optimization converge, hybrid environments now support parametric design, implicit modeling, automated meshing, performance prediction, and process-aware validation in unified frameworks. Understanding the strengths, limits, and intended use-cases of these ecosystems is essential for rigorous computational design of architected materials.

6.1. Specialised Lattice Design Platforms

For complex, conformal, and field-driven lattice generation, specialised platforms often outperform traditional CAD by utilizing implicit representations or voxel-based approaches to handle high geometric complexity.

nTop (formerly nTopology) [24] uses implicit geometry and parametric rule-based systems to excel at conformal lattice generation. It supports TPMS, strut, and custom topologies, alongside multi-material workflows and topology optimization. While it is highly capable across all key metrics, it presents a steeper learning curve and higher cost. Similarly, Carbon Design Engine [36] offers rapid, distributed cloud-based lattice design focusing on accessibility. It utilizes inverse design to generate lattices for desired properties and integrates seamlessly with Carbon DLS printers, although it relies on cloud connectivity.

Altair Sulis [37] combines custom geometry kernels with drag-and-drop interfaces for Design for Additive Manufacturing (DfAM), supporting TPMS/strut lattices and performance zones for multi-region optimization. Synera “Fabulous Freya” (Synera GmbH, Bremen, Germany) (formerly Elise) [38] automates these complex workflows in a visual node-based style, integrating partner lattice modules and connecting to external CAD/FEA solvers.

Hyperganic HyDesign [39] is a cloud-native, browser-based platform launched in 2024 for rapid lattice structure creation. It leverages voxel-based implicit modeling to generate conformal TPMS and custom lattices, with export capabilities. The platform integrates Forward AM’s Ultrasim 3D Lattice Library, providing validated material and printer pairings for footwear, seating, and protection applications. A beta meshless simulation feature achieves results within 5% deviation from industry-standard FEA packages, reducing trial-and-error iterations.

General Lattice [40] provides intuitive lattice modeling tools and functions as a specialised environment for creating complex geometries that might be cumbersome in traditional CAD, often offering commercial design services alongside their tools.

6.2. Industrial AM Preparation & Optimization (AM Software)

This category includes software specifically designed to bridge design and manufacturing. These tools often focus on lightweighting via lattice infills, ensuring manufacturability, and validating print strategies.

Autodesk Netfabb [41] is a comprehensive AM preparation suite featuring the “Lattice Commander” and “Lattice Assistant” modules. It includes unit cell libraries, simulation capabilities for AM processes, and topology optimization tools to validate performance before printing. Three-dimensional Systems 3DXpert allows for the creation of conformal lattices that align with design domain boundaries, streamlining the transition from design to print for industrial users.

Materialise Magics [42] is widely used for print preparation and standard lattice infills. It features slice-based lattices that avoid heavy mesh file creation, supporting various unit cell libraries for weight reduction, heat exchange applications, and medical implants. The software focuses on functional lattice generation with control over beam filtering and perforations to aid powder removal. Separately, Materialise 3-matic [43] extends lattice capabilities to the mesh level, enabling conformal lattice design that aligns with complex part geometries. It allows for optimised lattice structures, complex textures, and Boolean operations directly on mesh data, making it suitable for design optimization workflows that reduce material usage and print time.

6.3. Mainstream CAD with Lattice Capabilities

Traditional CAD vendors are increasingly integrating lattice functionalities, functional grading, and topology optimization directly into parametric environments.

Fusion 360 [44] supports volumetric lattices (gyroid, Schwarz, Octet Truss) with real-time parametric editing and generative design capabilities for AM. PTC Creo integrates Generative Topology Optimization (GTO) and lattice creation directly into the engineering workflow, allowing functional requirements to drive lattice features. SolidWorks [45] and Siemens NX [46] have also evolved; SolidWorks focuses on parametric strut lattices with add-ins for specific generation, while Siemens NX includes a dedicated “Lattice command” that supports both homogenous and graded structures [47]. CATIA [48] (via Lattice Designer in 3DEXPERIENCE) supports sophisticated TPMS and wizard-driven workflows with associative CAD updates.

6.4. FEA and Simulation-Driven Optimization

These platforms focus on performance validation and optimization, often importing lattice data to define gradients based on stress or thermal loads.

Altair Inspire [49] & OptiStruct [50] deliver generative design and large-scale topology optimization (SIMP, level-set), capable of exporting density fields that translate into lattice structures. ANSYS [51] integrates topology optimization with SpaceClaim’s direct modeling; SpaceClaim [52] includes unit cell libraries and offers functionalities for conformal lattice generation aligned with boundary conditions.

COMSOL Multiphysics 6.2 (COMSOL AB, Stockholm, Sweden) [53] excels in multiphysics and optimization for graded lattices via scripting and direct CAD integration. Abaqus [54] remains a powerhouse for nonlinear analysis; its “Abaqus Additive Suite” and ecosystem of plugins allow for lattice creation, Representative Volume Element (RVE) analysis, and homogenization, enhancing its built-in finite element capabilities.

6.5. Grasshopper Ecosystem (Computational Design)

The Grasshopper [22] visual programming environment for Rhinoceros is a hub for research and complex lattice generation, offering high connectivity between different logic components.

Crystallon [19] generates TPMS and strut lattices via voxel tessellation, making it effective for conformal and hybrid designs. Intralattice [55] specializes in generating strut lattices with customizable cells and integrated property tools. Axolotl [56] allows users to create 3D sine isosurfaces (TPMS) and adjust them via a visual slider interface. Pufferfish [20] extends these capabilities with tweening and lattice transformations.

For volumetric operations, Dendro [21] uses OpenVDB to transform curve lattices into volume meshes with controlled thickening. Karamba enables FEA-driven optimization within the canvas, while Monolith uses voxel representations to thicken wireframes and handle complex multi-material distributions.

6.6. Open-Source & Research Tools

Academic tools often offer high control over specific algorithms, particularly for TPMS generation using mathematical approximations.

MSLattice [57] is a MATLAB R2024b (The MathWorks, Inc., Natick, MA, USA) -based tool for TPMS generation using level-set approximations, supporting effective property homogenization and density gradients. OpenVCAD [11] is a free, open-source volumetric geometry compiler for multi-material additive manufacturing. With its scripting interface, users can design complex lattices, functionally graded structures, and meta-materials, specifying spatially varying materials and exporting directly for AM or simulation. It supports medical image inputs and scales efficiently for large, high-resolution prints. FlattPack [58], also MATLAB-based, creates TPMS structures with thickness grading and supports STL/NASTRAN export. ASLI [59] provides a C++/Qt GUI for TPMS lattice generation with graded infill. STL Lattice Generator [60] is a minimalist tool for prototyping BCC/FCC lattices.

Fogleman’s sdf [61] is a lightweight Python 3.11 (Python Software Foundation, Wilmington, DE, USA) library enabling the generation of 3D meshes from signed distance functions (SDFs). It focuses on code-driven design using constructive solid geometry (CSG), allowing for precise mathematical definition of lattice structures without a graphical interface.

To systematically assess the current software ecosystem, 31 platforms spanning specialised lattice generators, industrial AM preparation suites, mainstream CAD environments, FEA-driven optimization tools, computational design plugins, and open-source research utilities were evaluated across seven critical criteria. The comparison considers: (1) Lattice Types—range of supported geometries including TPMS, strut-based, and custom topologies; (2) Parametric—degree of algorithmic control over structural parameters; (3) Conformal—ability to generate boundary-aligned lattices; (4) Multi-Material—support for spatially varying material compositions; (5) Ease of Use—learning curve and interface intuitiveness; (6) FEA Integration—coupling with simulation environments for performance validation; and (7) AM Compatible—readiness for direct manufacturing workflows. Each criterion is scored on a 5-point scale, with higher values indicating superior capabilities. This matrix enables objective platform selection based on specific project requirements, balancing geometric sophistication, computational rigor, and manufacturing readiness.

Comparative evaluation of 31 software platforms for lattice structure and FGM design across seven key criteria: lattice type support, parametric control, conformal generation, multi-material capabilities, ease of use, FEA integration, and AM compatibility. Scores range from 1 (limited) to 5 (excellent), as illustrated in

Table 1. Comparative evaluation of 31 software platforms.

Software	Lattice Types	Parametric	Conformal	Multi-Material	Ease of Use	FEA Integration	AM Compatible
nTop	TPMS, Strut, Custom	5	5	4	4	4	5
Altair Sulis	TPMS, Strut	4	4	3	4	5	5
Carbon Design Engine	Various	4	5	3	5	3	5
Synera	TPMS, Strut (via partner)	4	3	2	4	4	4
Hyperganic HyDesign	TPMS, Custom	5	5	2	5	4	5
Autodesk Netfabb	Strut, Unit Cell Lib.	3	3	2	4	4	5
3D Systems 3DXpert	Strut, Conformal	3	4	2	4	3	5
Materialise Magics	Strut, Unit Cell Lib.	3	2	1	4	2	5
Materialise 3-matic	Strut, Conformal	3	4	2	3	2	5
General Lattice	Strut, Custom	4	3	2	4	2	5
Fusion 360	TPMS, Volumetric	4	3	1	4	3	4
PTC Creo	Strut, GTO	4	3	2	3	4	4
SolidWorks	Strut, Limited TPMS	3	2	1	3	2	3
Siemens NX	TPMS, Strut	3	2	2	3	3	3
CATIA (Lattice Designer)	TPMS, Strut	4	3	2	4	3	4
Altair Inspire	TPMS, Strut	4	3	2	4	5	4
OptiStruct	Density-based	3	2	1	3	5	3
ANSYS (SpaceClaim)	Custom, Conformal	3	4	2	3	5	3
COMSOL	Custom (scripting)	3	2	2	3	5	2
Abaqus	Integrated Module	3	2	2	3	5	4
ASLI	TPMS	3	3	4	3	4	4
FlattPack	TPMS	4	2	1	3	3	4
OpenVCAD	Lattice, TPMS, Custom	5	5	5	3	3	4
MSLattice	TPMS	3	2	1	3	3	4
Crystallon (GH)	TPMS, Strut	4	3	2	3	2	3
Axolotl (GH)	TPMS (Isosurface)	4	2	1	3	1	3
Intralattice (GH)	TPMS, Strut	4	2	1	3	2	3
Dendro (GH)	Volumetric	3	3	2	3	2	2
Karamba (GH)	Custom/Structural	3	2	1	3	4	2
fogleman/sdf	Custom (SDF)	5	5	1	2	1	3
Blender	Voxel/Custom	2	2	1	2	1	3

.

The cumulative scoring analysis (Figure 4) reveals a clear stratification within the software ecosystem, with specialised platforms significantly outperforming general-purpose tools. nTop and OpenVCAD emerge as the highest-scoring solutions, achieving strong marks across all criteria through their implicit modeling foundations and multi-material workflows, though OpenVCAD trades ease of use for research-grade flexibility. Commercial AM preparation suites (Netfabb, 3DXpert, Materialise products) cluster in the mid-range, prioritizing manufacturing readiness over parametric sophistication, while mainstream CAD platforms (SolidWorks, Siemens NX) score lowest due to limited TPMS support and weaker conformal capabilities. A notable trend is the inverse relationship between advanced capabilities and accessibility: cloud-native platforms like Hyperganic HyDesign and Carbon Design Engine successfully bridge this gap through intuitive interfaces backed by powerful implicit engines.

The Grasshopper ecosystem demonstrates specialization, with individual plugins excelling in narrow domains but requiring assembly into custom workflows. Open-source tools consistently underperform in usability and integration metrics despite strong algorithmic foundations, highlighting the persistent barrier between academic innovation and industrial adoption. The dominance of implicit and voxel-based platforms in the top tier suggests that traditional boundary-representation CAD is approaching its limits for complex lattice generation, while the weak multi-material scores across most tools indicate this remains a frontier capability concentrated in cutting-edge platforms.

The software ecosystem for functionally graded lattice structure design is transitioning from fragmented, script-based workflows toward unified engineering platforms that integrate design, simulation, and manufacturing. While academic tools continue pioneering novel TPMS geometries and computational methods, commercial CAD and AM software increasingly adopt these capabilities through implicit modeling and cloud-native architectures, broadening access to previously specialised techniques. This convergence is accelerated by machine learning-driven optimization that reduces design iterations by up to 80%, the adoption of interpretable AI frameworks for revealing design variable influences on performance, and cloud-based platforms enabling distributed computation without high-end local hardware.

Future platforms will extend beyond geometry generation to offer predictive digital twins with integrated performance validation and automated functional grading driven by multi-objective optimization. As implicit modeling, generative algorithms, and data-driven approaches mature, computational design, performance prediction, and process planning will merge into seamless workflows capable of synthesizing application-specific, functionally graded architected materials for aerospace, biomedical, and structural applications.

6.7. Integration with Additive Manufacturing Workflows

The practical realization of computationally designed lattice structures and functionally graded materials (FGMs) relies on effective integration within additive manufacturing workflows. This integration demands geometric representations and data pipelines compatible with process-specific constraints, enabling complex digital architectures to transfer into manufacturable toolpaths. Achieving consistency between virtual design and fabricated artefact requires consideration of process-induced deviations, material anisotropy, and post-processing needs.

The complete AM workflow encompasses geometric translation to industry-standard formats (STL, AMF, 3MF), process planning to verify design feasibility, and slicing with toolpath generation. Slicing algorithms must accurately accommodate internal feature representation—lattices and graded property regions—while multi-material AM requires layer-by-layer material assignment to ensure correct phase distribution in FGMs. Modern design optimization frameworks increasingly incorporate manufacturability constraints directly: minimum strut diameter and spacing requirements ensure fabricability, while property gradients must remain within achievable bounds determined by allowable material phase combinations and volume fractions. Connectivity constraints prevent isolated features or regions unreachable during fabrication.

Implicit geometry representation has transformed AM workflows for lattice structures. Unlike mesh-based approaches that discretize geometry into triangular facets, implicit modeling represents structures as continuous mathematical functions, offering native support for complex lattice architectures and functionally graded materials.

nTop offers direct integration with leading additive manufacturing platforms including EOS and Materialise Magics, where the nTop Implicit File format dramatically reduces file sizes and export times for complex lattice structures. Through integration with Materialise Magics, users can robustly prepare both mesh-based and implicit files, complete build setup, and manage downstream workflows.

Implicit direct slicing provides significant benefits throughout the manufacturing chain by preserving design intent, reducing computational overhead for complex geometries, and offering native support for continuous gradients without discretization artifacts. This approach enables seamless integration between design optimization and process planning. Major AM slicers including EOS EOSPRINT and Materialise Magics now support implicit and hybrid geometry inputs, allowing direct processing without requiring intermediate tessellation steps. These capabilities prove particularly valuable in aerospace, biomedical, and high-performance applications where precise control over lattice architecture and material distribution is essential.

7. Machine Learning Integration for Lattice Structures and Functionally Graded Materials

Recent advances in machine learning (ML) have fundamentally transformed the design paradigm for lattice structures and functionally graded materials (FGMs), enabling rapid exploration of vast design spaces and accurate prediction of complex structure-property relationships. By leveraging deep neural networks as surrogate models, ML approaches achieve property evaluations orders of magnitude faster than traditional finite element analysis—enabling inverse design workflows where target properties directly guide geometry generation rather than requiring iterative trial-and-error [62]. Beyond lattice-specific studies, broader progress in ML-driven materials design has established general frameworks for data-efficient learning, inverse structure–property mapping, and closed-loop optimization across materials classes, highlighting ML as a unifying paradigm for accelerated architected material discovery [63]. Integration of ML with topology optimization, computational homogenization, and additive manufacturing is creating intelligent, data-driven design pipelines specifically suited to the multi-scale complexity of architected lattices and FGMs [30,62,64,65,66,67].

ML approaches for lattice design fall into three categories: (1) surrogate modeling: neural networks replacing expensive FEA to predict properties from geometry, (2) generative design: V data scarcity AEs and GANs creating novel geometries by learning design space distributions, and (3) inverse design: mapping target properties to geometries through optimization (indirect), learned parameter mappings (semi-direct), or conditional generation (direct). Each serves distinct workflow stages. Surrogates accelerate evaluation, generative models explore unconventional topologies, and inverse methods enable property-driven design.

7.1. Surrogate Modeling and Property Prediction

7.1.1. Neural Network Architectures for Lattice Design

Modern neural surrogates, including multilayer perceptrons (MLPs), convolutional neural networks (CNNs), and graph neural networks (GNNs), enable rapid prediction of mechanical properties from geometric representations of lattice unit cells. These models learn mappings from parametric descriptors (geometric parameters, relative density) or direct representations (voxels, point clouds) to key performance metrics including stiffness tensors, yield strength, Poisson’s ratio, and energy absorption capacity. Computational speedups are dramatic: deep neural networks can predict effective properties in seconds compared to 500–770 s required for computational homogenization methods [62], representing speed improvements of nearly 100× while maintaining prediction accuracies above R² = 0.95 [64]. For 3D lattice structures, CNN-based surrogate models achieved R² = 0.971 when predicting effective modulus from voxelised geometries [67], demonstrating that well-trained models can reliably replace expensive simulations during iterative design.

7.1.2. Dataset Considerations and Training Efficiency

The performance of ML surrogates critically depends on dataset quality and size. Studies have successfully trained models on datasets ranging from 1500 lightweight lattice structures to 250,000 topologically optimised microstructures [30,65], with practical applications often using 5000–10,000 samples for single-property prediction [62,67]. Dataset preparation typically combines topology optimization, phase-field methods, or parametric generation algorithms with homogenization-based property evaluation [65,66]. Training time ranges from minutes to hours on modern GPUs depending on architecture complexity [62,66], but once trained, surrogate models enable near-instantaneous property queries essential for optimization and inverse design [64,65].

7.2. Generative Design and Inverse Workflows

7.2.1. Inverse Design Strategies for Functionally Graded Lattices

ML enables three distinct inverse design approaches for lattice structures: indirect methods combine property prediction networks with evolutionary algorithms (genetic algorithms, Bayesian optimization) to search design space [64,65,68]; semi-direct methods map target properties to design parameters which then generate geometries via modeling algorithms [30,62,66]; and direct methods employ generative models (VAEs, GANs) to produce pixel/voxel representations satisfying target properties [65,67]. Each approach presents trade-offs: indirect methods offer flexibility but require many function evaluations, semi-direct methods are computationally efficient but need additional geometry generation steps, while direct methods immediately produce geometries but demand larger training datasets and more complex architectures [65].

7.2.2. Bayesian Optimization and Active Learning

Bayesian optimization (BO) dramatically reduces the number of simulations required to discover optimal lattice designs by intelligently balancing exploration and exploitation. For TPMS lattice families, BO with Gaussian process surrogate models achieved 82% reduction in simulations (50 evaluations vs. 270 for grid search) when targeting performance thresholds of p ≥ 0.90 [64]. The method proved especially effective for mixed search spaces combining discrete variables (lattice topology family) with continuous parameters (geometric dimensions), where one-hot encoding enables unified optimization across topologically distinct unit cells. These active learning approaches are particularly valuable when simulation or fabrication costs are high, enabling efficient discovery with limited computational budgets.

7.2.3. Functionally Graded Design Acceleration

For functionally graded lattices where many spatially-varying unit cells must be individually tailored, ML provides critical efficiency gains. A DNN-based inverse design framework for graded auxetic lattices reduced design time from 773 s (computational homogenization) to under 8 s per unit cell—enabling rapid spatial property gradation that would be prohibitively expensive with traditional methods [62]. Similar approaches combining pixel-based material databases with neural network generators enable gradient modulus and Poisson’s ratio distributions for biomimetic prosthetic designs, with complete lattice structure generation completed in approximately 2 s [30].

7.3. Interpretability and Design Insight

Understanding which geometric features drive performance is essential for developing design principles and building designer intuition. Shapley Additive Explanations (SHAP) analysis quantifies the contribution of each design parameter to predicted properties, revealing that for TPMS lattices under compression, unit cell dimension in the loading direction has the largest influence on effective modulus while wall thickness exhibits comparatively minor effects [64]. Such interpretability methods help identify critical design variables, validate model physics, and guide dimensionality reduction when design spaces become intractably large. However, SHAP and similar techniques primarily capture individual parameter effects rather than coupled interactions, highlighting an ongoing need for methods that elucidate complex, nonlinear feature dependencies in lattice design [64].

7.4. Methodological Challenges, Reproducibility, and Outlook

7.4.1. Dataset Limitations and Standardization

Current ML approaches for lattice structures face significant dataset challenges. Most studies focus on narrow geometric families (specific TPMS types, parametric lattice variants) limiting model generalization [64,65,66]. Dataset sizes remain modest compared to other ML domains, ranging from 270 samples for TPMS lattices to 10,000 samples for voxelised structures [30,62,64,65,67], and lack of standardised benchmarks complicates comparison across methods [65]. Addressing these limitations requires: (1) development of larger, more diverse datasets spanning multiple lattice topologies, density ranges, and loading conditions [64,65]; (2) community-driven benchmark datasets enabling fair model comparison [65]; and (3) transfer learning strategies to leverage knowledge across related structure families [65].

7.4.2. Manufacturability and Experimental Validation

Bridging the gap between ML-designed lattices and fabricated structures requires explicit consideration of additive manufacturing constraints. Recent work has begun addressing connectivity between graded unit cells [65], minimum feature sizes, and overhang angles [62,68], but systematic integration of process-specific constraints (powder removal for LPBF, support requirements for stereolithography) into ML design workflows remains underdeveloped. Experimental validation of ML-designed lattices through mechanical testing provides essential ground truth, but remains limited in the literature—highlighting the need for tighter integration between computational design, fabrication, and experimental characterization loops.

7.4.3. Physics-Informed and Hybrid Approaches

Physics-informed neural networks (PINNs) that embed governing equations, symmetries, or physical constraints into ML architectures offer potential to improve generalization and reduce data requirements for lattice design [65,66]. These approaches remain largely unexplored for lattice structures despite promise in other mechanics domains. Similarly, hybrid frameworks combining traditional topology optimization with ML-accelerated subroutines (property evaluation, sensitivity analysis) can leverage strengths of both paradigms while mitigating individual weaknesses [30,65,66]. Future research integrating physical laws, symmetry operations, and homogenization theory into neural architectures could enable more robust and physically consistent lattice design tools.

Two additional challenges limit ML adoption: model generalization and reproducibility. ML models trained on specific lattice families often fail when applied to different topologies, loading conditions, or material systems, while transfer learning strategies to leverage knowledge across structure types remain underexplored despite success in other materials domains. Furthermore, inconsistent train-test splitting, inadequate hyperparameter documentation, and absence of code/data sharing impede reproducibility. Adopting standardised reporting protocols (analogous to PRISMA guidelines for reviews), including methodology pre-registration and mandatory code release, would substantially improve research rigor.

8. Computational Strategies for Modeling Lattice and Functionally Graded Materials

8.1. Computational Challenges in Lattice Modeling

Traditional CAD systems rely on boundary representation (B-rep) schemes that store explicit geometric boundaries, an approach well suited for conventional solid parts but fundamentally incompatible with lattice architectures. A cubic lattice structure measuring 0.1 m per side with 0.1 mm unit cells requires approximately 6.4 TB of memory to instantiate all geometric and topological information, far exceeding the capabilities of current commercial software [1]. This model data scaling problem becomes critical when structures grow from thousands of elements (manageable with existing tools) to millions or billions of elements (beyond current CAD capabilities). The challenge extends beyond mere storage: maintaining consistency across multiple geometric scales (from microscopic strut features to macroscopic component boundaries) during editing operations remains largely unsolved, as modifications at one scale fail to propagate automatically to others. Additionally, the geometric complexity of lattice structures leads to frequent degenerate cases including tangencies and overlaps during Boolean and blending operations, with failure rates exceeding 90% for TPMS structures in software like Siemens NX and CATIA [1].

8.2. Geometric Modeling Operations

8.2.1. Boundary Generation and Mesh Creation

Converting compact representations (implicit functions, procedural definitions, parametric descriptions) into explicit surface meshes enables visualization and manufacturing preparation, though this process confronts significant computational barriers for large scale structures. Several research groups have developed approximation strategies to manage this complexity. Verma et al. [69] simplified node connections using convex hull approximations that sacrifice geometric accuracy for improved robustness, while Wu et al. [70] created approximate nodal geometry evaluation methods specifically for lattice architectures. For structures approaching billion element counts, Zou et al. [71] demonstrated GPU accelerated tessellation using spatial decomposition strategies that process independent regions in parallel. Recent developments in feature-preserving meshing address persistent challenges in dual contouring implementations. Liu and Xia [72] proposed an enhanced framework that mitigates the partial volume effect through dichotomous sampling grid refinement and vertex position optimization, successfully generating manifold surface meshes that preserve sharp features inherent to implicit functions. Their approach generates body-fitted tetrahedral meshes compatible with commercial FEM software (ABAQUS, ANSYS) through standard .inp file formats, enabling complete design-to-simulation workflows for complex lattice architectures. Zhao et al. [73] developed specialised conversion methods transforming implicit TPMS equations into C² continuous B-spline surfaces suitable for finite element analysis.

8.2.2. Stress Relief Through Junction Blending

Sharp intersections between struts and nodes create stress concentrations that significantly reduce structural performance and fatigue resistance. Early approaches by Chen [74] applied point based blending to mesh format lattices, but function representation (F-rep) methods developed by Pasko et al. [75] provided more robust solutions by converting geometric blending operations into algebraic manipulations of implicit scalar fields. Liu et al. [76] advanced this framework using convolution surfaces where implicit fields emerge from skeletal geometry convolved with smoothing kernels, producing physically motivated material distributions aligned with stress patterns.

8.2.3. Boundary Conformity Strategies

Achieving conformity between lattice microstructures and complex boundary geometries prevents dangling elements and maintains structural integrity. Volumetric parameterization methods construct trivariate B-spline mappings from regular parametric domains to physical design spaces, enabling simultaneous deformation of all lattice elements through control point manipulation while preserving topological connectivity [1]. Hex-mesh based decomposition fills each hexahedral element with deformed unit cells conforming to element boundaries, naturally handling complex topologies though dependent on high quality hex-mesh generation. Distance field methods offer an alternative by distributing Voronoi seeds according to signed distance fields, generating irregular lattices that inherently conform to boundaries with spatially varying density control. Liu et al. [77] constructed conformal Voronoi diagrams within distance field boundaries. The computationally simplest approach by Aremu et al. [78] directly trims regular lattices against boundaries then thickens surface skins to cover exposed elements, producing suboptimal but manufacturable results.

8.2.4. Property Variation Through Offsetting

Spatially varying structural properties in functionally graded materials require controlled geometric scaling operations that expand or contract lattice features. Traditional explicit offsetting algorithms encounter robustness failures when applied to complex lattice cell geometries, making implicit distance field methods the preferred implementation approach. These operations directly enable FGM design by modulating strut radii, wall thicknesses, or cell dimensions according to prescribed spatial functions, creating continuous material property gradients optimised for local loading or thermal conditions.

8.3. Boolean and Topological Operations

Boolean operations (union, intersection, difference) enable complex assembly construction and boundary trimming operations essential for lattice design. Current implementations rely primarily on voxel discretization or implicit field representations. Software platforms like nTop employ implicit F-rep frameworks where Boolean operations map to algebraic operations on scalar fields (maximum for union, minimum for intersection, complement for difference), avoiding fragile explicit surface intersections [23]. However, these approaches scale poorly for very large structures, requiring development of lattice optimised algorithms. Hollowing operations generate controlled internal porosity by distributing micro holes throughout volumes using Voronoi diagram optimization or offsetting based methods that create self-supporting hole configurations.

8.4. Current Capabilities and Future Directions

Contemporary modeling systems handle simple lattice structures containing thousands of elements using conventional CAD operations effectively. Complex structures with millions of elements require specialised approaches, while highly complex structures approaching billions remain at research frontiers. Critical limitations include slow boundary evaluation preventing interactive visualization, computationally expensive Boolean operations, and absence of unified frameworks for semi-regular topology representation. Recent algorithmic advances in octree-based adaptive sampling and feature-preserving mesh repair [72] show promise for handling structures with millions of elements while maintaining geometric fidelity. Advancing the field requires continued development of GPU parallel algorithms, compressive pattern recognition methods, generative frameworks, and hybrid representations capable of addressing large scale computation challenges in lattice and functionally graded material design.

9. Challenges and Future Directions

9.1. Computational Challenges

9.1.1. Scalability and Optimization Complexity

Computational cost remains a critical barrier to widespread industrial adoption. Analyzing large scale lattice structures with thousands of design variables imposes prohibitive computational costs for routine engineering workflows. While advances in high performance computing, surrogate modeling techniques, and hierarchical decomposition strategies show promise, their effectiveness varies substantially across problem classes and requires continued algorithmic development.

9.1.2. Multiscale Coupling

Bridging microscale behavior, which encompasses microstructural architecture and constituent material properties, to macroscale component performance represents a fundamental computational challenge. Current homogenization theories and hierarchical modeling frameworks provide theoretical foundations for this linkage, yet computational expense scales unfavorably with model fidelity. Developing efficient multiscale frameworks that maintain physical accuracy while achieving tractable solution times deserves prioritization, particularly for functionally graded lattices where properties vary spatially across multiple length scales.

9.1.3. Uncertainty Quantification and Robust Design

Incorporating manufacturing uncertainties, material property variability, and model approximation errors into design workflows remains underdeveloped. Fabricated lattice structures exhibit geometric deviations from nominal designs due to process variations in strut diameter, surface roughness, and nodal geometry. Similarly, powder based additive manufacturing introduces material property scatter through microstructural heterogeneity. Probabilistic design methodologies and robust optimization frameworks capable of systematically accounting for these uncertainties represent active research frontiers.

9.2. Manufacturing and Design Integration

9.2.1. Process Aware Design Optimization

Current design processes insufficiently capture manufacturing realities, often optimizing structures for idealised geometric and material assumptions that fail to materialize in practice. Tighter coupling between process simulation, which predicts thermal histories, residual stresses, and geometric distortions, and design optimization is essential to ensure fabricated structures achieve intended performance. This integration requires bidirectional workflows where manufacturing constraints actively inform design space exploration rather than serving merely as validation filters applied after optimization.

9.2.2. Machine Learning Validation Requirements

Emerging machine learning models for manufacturing outcome prediction demonstrate impressive correlations with simulation data, yet substantial experimental validation remains necessary before deployment in critical applications. Expanding experimental characterization of additive manufacturing processes, particularly for novel lattice architectures and multi-material systems, is resource intensive but essential for establishing validated workflows connecting design to manufacturing. Standardised testing protocols for lattice structures across different AM platforms would accelerate community knowledge accumulation.

9.3. Standardization and Interoperability

9.3.1. Data Format Maturation

While 3MF and AMF represent significant advances over legacy STL formats, standardised representations for functionally graded properties, multi-material specifications, and complex implicit geometries remain immature. Industry consensus on standardised metadata structures, property encoding schemes, and validation procedures is needed to enable seamless data exchange across the entire workflow from design through manufacturing. Particular attention should focus on representing continuous material gradients, anisotropic properties, and manufacturing annotations within unified file specifications.

9.3.2. Software Ecosystem Fragmentation

The computational design ecosystem remains fragmented, requiring manual data translation between specialised platforms for geometry creation, topology optimization, finite element analysis, and manufacturing preparation. Each translation step introduces opportunities for geometric degradation, property information loss, and human error. Development of standardised application programming interfaces and neutral data exchange protocols would reduce workflow friction and enable modular tool combinations tailored to specific application requirements. Open-source reference implementations of these standards could accelerate adoption across commercial and academic software platforms.

10. Conclusions

Computational design of lattice structures and functionally graded materials has evolved into a mature interdisciplinary field supported by established methodologies, specialised software ecosystems, and expanding industrial applications. However, this maturity reveals significant opportunities for advancement. The comparative analyses presented throughout this review demonstrate that effective design practice requires informed alignment of geometric representations, computational strategies, and optimization frameworks with specific performance requirements, manufacturing constraints, and available computational resources.

The choice of geometric representation fundamentally influences all subsequent design and analysis workflows. NURBS-based formalisms provide smooth, mathematically continuous geometries with natural integration into conventional CAD ecosystems, but struggle with topologically complex architectures. Mesh-centric approaches offer direct finite element analysis compatibility yet require careful refinement to capture geometric fidelity. Voxel representations enable explicit multi-material specification and align naturally with volumetric optimization methods, though at the cost of boundary discretization artifacts. Implicit representations accommodate extraordinary topological complexity with mathematical elegance and support advanced optimization strategies, yet their integration into traditional engineering workflows remains constrained by limited software interoperability.

No single computational approach universally optimizes all design objectives. Parametric methodologies provide explicit control over design variables, facilitating systematic sensitivity analysis and design space exploration within predefined geometric families. Topology optimization frameworks discover unconventional, high-performance configurations that often surpass engineer-intuited designs, though they necessitate careful regularization and manufacturability post-processing. Emerging machine learning paradigms demonstrate potential to expand accessible design spaces by orders of magnitude while accelerating optimization cycles, yet require extensive experimental validation and methodological standardization before deployment in safety-critical applications.

10.1. Path Forward: Recommendations for Researchers and Industry

The convergence of implicit modeling, generative algorithms, multi-physics simulation, and data-driven optimization is creating computational environments capable of synthesising application-specific architected materials with unprecedented geometric complexity and functional sophistication. Translating these advances from research demonstrations to industrial deployment requires addressing key technical and methodological challenges.

10.1.1. For Academic Research Communities

Priority should focus on developing publicly accessible benchmark datasets with experimental validation to enable reproducible model comparison and accelerate community progress. Current ML approaches suffer from limited generalization across lattice families, topologies, and loading conditions. Investigating physics-informed neural networks that embed conservation laws and material symmetries can reduce data requirements while improving model robustness. Multi-fidelity frameworks that leverage both high-accuracy experimental data and large approximate simulation datasets offer promising pathways to overcome data scarcity while maintaining prediction reliability. Establishing standardised reporting protocols for model training, validation, and performance metrics will improve reproducibility and facilitate fair comparison across methodologies.

10.1.2. For Industry Practitioners

Platform selection should prioritise capabilities aligned with specific design requirements and manufacturing constraints. For periodic lattices with established unit cell topologies, specialized platforms such as nTop and Altair Sulis offer integrated implicit modeling and direct AM slicing that eliminate file conversion bottlenecks. Functionally graded structures requiring conformal boundary adaptation benefit from voxel-based workflows available in OpenVCAD and Hyperganic HyDesign, which enable spatially varying property assignment without complex NURBS operations. Multi-disciplinary optimization platforms like modeFRONTIER and Simcenter HEEDS should be implemented when design workflows couple multiple specialized software tools. For single-platform workflows, native optimization modules typically provide sufficient capability with reduced setup complexity.

10.1.3. For Manufacturing Integration

Establishing bidirectional data exchange between process simulation and design optimisation is critical for robust manufacturing outcomes. Incorporating thermal distortion prediction and residual stress analysis directly into topology optimisation formulations, rather than relying on post hoc validation, improves as-built performance. Implicit geometry file formats such as nTop Implicit File reduce computational overhead for complex lattices, while STL and mesh formats remain appropriate for simple structures or legacy systems lacking implicit support. Standardised experimental protocols for characterising as-built lattice properties across AM platforms are essential. Systematically documenting relationships between nominal designs and fabricated geometries will inform robust design methodologies that account for manufacturing-induced variations.

Success in realising the full potential of lattice structures and functionally graded materials hinges on continued collaboration between computational researchers, materials scientists, and manufacturing engineers to address the data, standardisation, and process integration challenges outlined throughout this review.