Research highlight · Additive manufacturing

A better shape is only half the design.

When a printed material is stronger in one direction than another, choosing where to put it is not enough. We also need to choose which way it points.

Research by Bruno Denadai, Aldemar Siqueira, Xiaodong Huang and Josué Labaki

Published in Finite Elements in Analysis & Design (FEAD) · 2026

Read the open-access paper

Our optimized local material directions follow the branches of the bracket. Figure 4c, cropped from the paper.

The motivation

The manufacturing process is part of the design.

Topology optimization finds where material is most useful, often producing intricate shapes with branches and openings that are difficult to manufacture conventionally. Additive manufacturing, or 3D printing, is a major route for turning these optimized designs into real parts.

Yet the printing process also changes how those parts behave. Depositing material in layers and along particular paths can make a part stiffer or stronger in one direction than another, even when the starting material has no preferred direction. This effect is especially pronounced in extrusion-based printing and with fibre-reinforced filaments. Engineers call this directional behaviour anisotropy. Think of wood: loading along its grain is not the same as loading across it.

Many standard topology-optimization models assume that material behaves identically in every direction. If we use that assumption to design a printed part, we can overlook the directional stiffness and strength introduced by manufacturing. A shape that looks efficient in the model may behave differently once printed.

We bring that manufacturing reality into the design problem. In this work, we optimize the shape and local printing directions together, while enforcing a strength condition throughout the optimization. We ask not only where material should go, but which way it should point and whether it can carry the load.

Who led the work

Bruno Denadai

Lead author

Bruno led this research at SSI Lab, working with Aldemar Siqueira, Xiaodong Huang and Josué Labaki to bring shape, printing direction and anisotropic strength into a single optimization problem.

He is now pursuing a PhD at York University in Toronto, Canada.

Bruno on LinkedIn

Three questions, one design problem

Where should material remain?

We redistribute a limited amount of material using Floating Projection Topology Optimization, or FPTO, which extracts a smooth boundary from the numerical design.

Which way should it point?

We update local material directions alongside the shape, so the directional material can work with the routes through which the structure carries load.

Can it carry the load?

We use the Tsai–Wu criterion to account for directional strengths and the interaction of stresses. A smooth aggregate lets us handle the many local failure checks together.

Inside the paper · L-bracket benchmark

Same material budget. Different design freedom.

The L-shaped bracket is a demanding test because stress tends to concentrate near its inside corner. We compare two designs under the same loads and supports, using the same directional material and a target volume fraction of 50%. We include the failure constraint in both designs.

Optimized L-shaped bracket obtained while keeping the printing direction fixed at zero degrees. — Figure 7a. Material placement can change; its direction cannot.

Optimized L-shaped bracket obtained by changing both material placement and local printing direction. — Figure 4a. Both material placement and direction can change.

What does “compliance” mean?

Compliance is the load-weighted displacement of a structure, written C = Fᵀu. With the load F held fixed, a smaller value means less deformation in the directions in which the forces act. It measures stiffness, not the margin against every possible form of failure.

Why the strength check matters

A stiff design is not automatically an acceptable design.

In a third calculation, we removed the failure constraint while keeping the loading and material budget. We obtained a design with a sharp inner corner: a concentrated load path that looks efficient for stiffness but creates a critical stress concentration.

When we include the failure constraint, our method reshapes that region to reduce the concentration. We can therefore balance stiffness and material direction with the strength requirement, instead of accepting a shape on stiffness alone.

Numerical design without a failure constraint, showing a sharp inner corner in the L-shaped bracket. — Figure 8a. Without the failure constraint, the stress-concentrating inner corner remains.

What the evidence establishes

A computational method, tested on benchmark problems.

We tested our formulation on an L-bracket, a cantilever and an MBB beam. We used plane-stress models and in-plane material directions, relevant to planar deposition of reinforcing fibres.

Our results are numerical demonstrations. To assess fatigue life, interlayer performance and printer-specific manufacturing constraints, we would need additional models and tests of printed components.

When manufacturing gives a material direction-dependent properties, we need to design with those properties from the start. Our work brings that direction into the optimization, alongside shape and strength.

The paper

Smooth topology optimization of anisotropic structures for additive manufacturing applications

Bruno Benegra Denadai, Aldemar Siqueira, Xiaodong Huang and Josué Labaki.
Finite Elements in Analysis & Design 258 (2026), 104573.

Open paper · DOI 10.1016/j.finel.2026.104573

Figures 4a, 4c, 7a and 8a reproduced from Denadai et al. (2026) under CC BY 4.0. Panels cropped from the original paper; numerical geometry and directions are unchanged. Explanatory text by SSI Lab.