Fabrication Sequence Optimization for Minimizing Distortion in Multi-Axis Additive Manufacturing

1. Introduction

Multi-axis additive manufacturing (AM), exemplified by robotic Wire Arc Additive Manufacturing (WAAM), introduces manufacturing flexibility by allowing reorientation of the print head or component. This breaks the constraint of planar layer deposition inherent in conventional AM. However, metal AM involves significant thermal gradients and phase transformations, leading to uneven thermal expansion/contraction and consequential distortion, which critically impacts dimensional accuracy and structural performance for assembly.

Optimizing the fabrication sequence—the order in which material is deposited—presents a novel avenue to mitigate this distortion. The challenge lies in representing the sequence as differentiable optimization variables suitable for gradient-based methods. This work addresses this by proposing a computational framework for fabrication sequence optimization to minimize distortion.

2. Methodology

2.1 Pseudo-Time Field Encoding

The core of the framework is the representation of the fabrication sequence. Each material point x in the component domain Ω is assigned a scalar pseudo-time $T(x)$. The fabrication process is modeled as the sequential materialization of points according to this field: a point with a smaller $T$ is deposited before a point with a larger $T$. This transforms the discrete sequence optimization into a continuous field optimization problem.

2.2 Distortion Modeling

A simplified yet physically representative model is used to predict distortion. It mimics the inherent strain method, where each newly deposited material element experiences a prescribed shrinkage strain (e.g., thermal contraction) upon cooling. The accumulated distortion $\mathbf{u}$ is computed by solving the linear elasticity equilibrium equations over the entire domain, considering the history-dependent strain fields.

2.3 Gradient-Based Optimization

The objective is to minimize a measure of final distortion, e.g., the compliance of the distortion field or its maximum displacement. The design variable is the pseudo-time field $T(x)$. The gradient of the objective with respect to $T(x)$ is computed using the adjoint method, allowing for efficient large-scale optimization. Constraints ensure the time field is monotonic to represent a valid, non-reversing deposition sequence.

3. Numerical Studies & Results

3.1 Benchmark Case: Cantilever Beam

The framework was tested on a 3D cantilever beam geometry. The baseline case used conventional vertical planar layers. The optimization algorithm was then tasked with finding a pseudo-time field that minimizes the vertical deflection at the beam's free end due to deposition-induced shrinkage.

3.2 Comparison: Planar vs. Curved Layers

The study visually and quantitatively compared the distortion fields. The planar layer sequence led to a predictable, cumulative bending effect. In contrast, the optimized curved-layer sequence strategically "balanced" the shrinkage strains throughout the volume, often by depositing material in a way that induces counter-acting distortions, leading to a near-net-shape final part.

4. Technical Analysis & Framework

4.1 Mathematical Formulation

The optimization problem can be summarized as: $$ \begin{aligned} \min_{T} \quad & J(\mathbf{u}) = \int_{\Omega} \mathbf{u} \cdot \mathbf{u} \, d\Omega \\ \text{s.t.} \quad & \nabla \cdot \boldsymbol{\sigma} + \mathbf{b} = \mathbf{0} \quad \text{in } \Omega \\ & \boldsymbol{\sigma} = \mathbf{C} : (\boldsymbol{\epsilon} - \boldsymbol{\epsilon}^{sh}(T)) \\ & \boldsymbol{\epsilon} = \frac{1}{2}(\nabla \mathbf{u} + \nabla \mathbf{u}^T) \\ & T_{\min} \leq T(x) \leq T_{\max}, \quad \nabla T \cdot \mathbf{n} \geq 0 \, (\text{monotonicity}) \end{aligned} $$ Where $J$ is the distortion objective, $\boldsymbol{\epsilon}^{sh}(T)$ is the shrinkage strain dependent on the pseudo-time, and the monotonicity constraint ensures a feasible deposition order.

4.2 Analysis Framework Example

Scenario: Optimizing the print sequence for a WAAM-produced bracket to minimize warpage for subsequent assembly.

1. Input: 3D CAD model of the bracket, material shrinkage parameters (from calibration).
2. Discretization: Mesh the domain. Initialize a pseudo-time field (e.g., corresponding to planar layers).
3. Simulation Loop: For the current $T$ field, simulate the sequential deposition and compute the final distortion field $\mathbf{u}$ and objective $J$.
4. Adjoint & Gradient: Solve the adjoint equation to compute $\partial J / \partial T$ efficiently.
5. Update: Use a gradient-based optimizer (e.g., MMA, SNOPT) to update the $T$ field, respecting constraints.
6. Output: The optimized $T$ field, which is then interpreted into a robot toolpath for curved-layer WAAM deposition.

5. Application Outlook & Future Directions

The framework opens several impactful avenues:

• Integration with Full Thermo-Mechanical Models: The current shrinkage model is a simplification. Future work must integrate high-fidelity, transient thermo-mechanical simulations, akin to the multi-physics challenges tackled in models for laser powder bed fusion. This increases accuracy but also computational cost, necessitating model order reduction.
• Path Planning for Robotic WAAM: The optimized pseudo-time field must be translated into collision-free, kinematically feasible robot trajectories. This bridges computational design with robotic execution.
• Multi-Objective Optimization: Simultaneously optimize for distortion, residual stress, build time, and support structure volume. This aligns with the holistic process optimization seen in advanced manufacturing research from institutions like Oak Ridge National Laboratory.
• Machine Learning Surrogates: To achieve real-time or near-real-time sequence planning, neural networks can be trained as surrogates for the expensive physics simulation, following trends set by works like CycleGAN for image-to-image translation, but applied to mapping geometry to optimal deposition sequences.
• In-Situ Distortion Correction: Combine the optimized plan with in-process monitoring (e.g., laser scanning) to create a closed-loop system that adjusts the sequence in real-time based on measured distortion.

6. References

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3. Wang, W., van Keulen, F., & Wu, J. (2023). Fabrication Sequence Optimization for Minimizing Distortion in Multi-Axis Additive Manufacturing. arXiv preprint arXiv:2212.13307.
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5. Oak Ridge National Laboratory. (2017). 3D Printed Excavator Project. Retrieved from https://www.ornl.gov/news/3d-printed-excavator-project.
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