How Graded-Density Lattices Put Stiffness Where You Need It

A uniform lattice is already useful; a graded density lattice is more useful still, because most real loads are not uniform. A midsole sees higher stress under the heel than under the arch. A structural panel carries bending loads at the edges and almost none at the centre. A biomedical scaffold needs dense, stiff walls at the load-bearing cortical surface and open, porous geometry inside for vascular ingrowth. A single graded density lattice can match all of these requirements in one part, without joints, assembly steps, or abrupt material transitions.

What "functionally graded" means in practice

The term functionally graded material (FGM) comes from materials science, where it originally described parts whose chemical composition transitions gradually from one material to another across a cross-section, giving properties that vary spatially rather than jumping at a sharp interface. Applied to lattices, the idea is the same but achieved through geometry rather than chemistry: instead of changing the material, the density of the lattice structure varies continuously. The word "functionally" matters here; the gradient is not random variation but a deliberate spatial distribution engineered to match a specific load distribution, a thermal gradient, or a stiffness target.

In a printed lattice, density is controlled in several ways. The most direct is varying the relative density, the fraction of solid to total volume, by changing strut diameter or wall thickness while keeping cell size constant. Another approach changes cell size while keeping relative density constant, with smaller cells in high-stress zones and larger cells elsewhere. A third approach combines both. In practice, implicit modelling tools allow all of these to be expressed as a continuous field that smoothly transitions across the part, avoiding the step changes that can act as stress concentrators at the boundary between zones.

How density maps to stiffness and cushioning

For most lattice cell types, stiffness scales roughly with relative density raised to a power between 1 and 2, depending on whether deformation is dominated by bending or stretching of the cell walls. What this means practically is that doubling the relative density of a cell more than doubles its stiffness in the bending-dominated regime. The relationship is non-linear and cell-type dependent, which is why graded lattice design benefits from FEA rather than relying on intuition about proportionality.

The cushioning response is the complement of stiffness. A sparse zone, where relative density is low and cell walls are thin, deforms more easily under load, storing and then releasing elastic energy or absorbing it plastically. Where the load is compressive and the goal is energy absorption rather than elastic return, a gradient from stiff to compliant through the thickness of the part allows the dense layer to remain dimensionally stable while the compliant layer progressively collapses. This is a useful configuration for impact protection, where the protective element must remain in place structurally while absorbing the impact energy.

The adidas 4DFWD midsole as a density-tuning example

The adidas 4DFWD midsole, produced by Carbon using Digital Light Synthesis, demonstrates density tuning at commercial scale. The midsole uses a lattice of over 10,000 struts whose geometry varies across the footbed: the zone under the heel, which takes the highest impact loads, has a different cell density and geometry from the zones under the midfoot and forefoot, which need to flex and direct energy forward. The result is a single printed component that handles the distinct mechanical requirements of each foot zone without seams or bonded interfaces. The density tuning allows the lattice to be engineered to direct compression forward rather than simply dissipating it, which is how the reported forward propulsion performance is achieved. This is a working illustration of what graded density design makes possible at retail scale.

Designing the gradient deliberately

A graded density lattice is only as good as the load map it was designed against. The starting point is a finite element analysis of the part under its real load conditions, which maps stress and strain across the volume. High-stress regions are assigned higher density; low-stress regions are assigned lower density. The FEA can be run on a simplified solid model first, then the density field extracted from the stress map and applied to the lattice geometry. This is called topology-informed lattice grading, and it is the systematic way to ensure the gradient serves the structural function rather than being applied arbitrarily.

The gradient must also be smooth enough to print reliably. An abrupt step from high density to low density creates a plane of weakness at the transition, because the cells at each side of the boundary have different stiffnesses and the mismatch concentrates stress at that interface. Implicit modelling tools handle this naturally when the density field is expressed as a continuous function; the transition is smooth by construction. When designing transitions manually, a rule of thumb is to spread the density change over at least three to five cell lengths to avoid a stress concentration at the boundary.

Where isolating compression and tension fits in

Graded density controls how stiff or compliant different regions of a lattice are, but it does not directly address which regions carry compression and which carry tension. In many practical load cases, both forces exist simultaneously in different parts of the same volume. A midsole compresses under load while its outer skin is in tension; a structural panel bends so its top face compresses and its bottom face stretches. Density grading optimises the stiffness distribution, but it is a separate design decision from routing the compression and tension paths through deliberately chosen members or surfaces.

This is where the geometry-level expertise that Sam Lanahan at C6XTY works on becomes relevant. Separating compression and tension within a lattice, so that each force flows through members designed for it, allows a part to be simultaneously stiff where it must resist crushing and flexible where it must stretch, within one continuous geometry. Combined with density grading, which sets the local stiffness level, this gives a level of structural control that neither tool provides alone.

File size and tooling for graded lattices

Graded lattices are among the most file-size-intensive geometries in mesh-based CAD, because each zone has different cell dimensions and so the geometry cannot be represented by tiling a single repeated unit. In traditional tools, this means every cell must be stored individually, which pushes file sizes well beyond what uniform lattices of the same part volume would require. Implicit modelling tools, which express the gradient as a field equation, handle graded lattices without any additional file-size cost; the gradient is just another parameter in the field. For this class of geometry, the practical case for implicit modelling is stronger than for uniform lattices.

Key takeaways

• A graded density lattice varies cell density spatially so one printed part delivers different stiffness, strength, and cushioning in different zones.
• The gradient should be derived from an FEA load map and expressed as a smooth field to avoid stress concentrations at density transitions.
• Density grading sets local stiffness; separately isolating compression and tension paths extends that control to force routing within the geometry.
• Implicit modelling tools handle graded lattices without file-size penalties; mesh-based tools struggle because each cell in the gradient must be stored individually.

Related reading

• How Lattice Midsoles Turn Each Step Into Forward Motion
• Separating Compression and Tension Inside a Lattice
• How Geometric Lattices Outperform Solid Material on Strength-to-Weight

Frequently asked questions

What is the difference between a graded lattice and a uniform lattice?

A uniform lattice repeats the same cell geometry throughout the part; stiffness and strength are the same everywhere. A graded lattice varies cell density continuously across the part so that different regions have different mechanical properties, matching the local load requirements without adding mass where it is not needed.

How is a density gradient designed?

The standard workflow is to run FEA on a simplified solid model under the expected loads, extract a stress map across the volume, and use that map to define a density field: high density where stress is high, low density where stress is low. The density field is then applied to the lattice geometry using implicit modelling tools, which generate the graded geometry as a smooth continuous function.

Can any cell type be graded?

Yes. Both strut-based cells, such as BCC and octet trusses, and surface-based TPMS cells, such as gyroids, can be graded. The gradient is applied to the relative density or cell size parameter of whichever cell type is used. TPMS cells with density gradients need particular attention to powder removal design, since the channel widths narrow in high-density zones.

Does grading a lattice make it harder to print?

Grading complicates design and inspection more than printing. The print process does not distinguish between uniform and graded cells; it fuses material layer by layer regardless. The design challenge is ensuring smooth density transitions to avoid stress concentrations, and the inspection challenge is verifying that the gradient was printed as intended, typically requiring CT scanning on first articles.

Is graded density the same as topology optimisation?

They are related but distinct. Topology optimisation determines the overall shape and material distribution of a part to minimise mass for a given stiffness, often producing irregular solid regions. Density grading applies within a lattice that already fills a defined volume, varying local stiffness continuously rather than removing solid regions. The two approaches can be combined: topology optimisation defines the outer envelope and general load paths, and graded lattices fill the interior.