Tangent to the contours of level sets determines the local fiber orientations
Fiber orientations are decoupled from the macroscale parametrization
Manufacturability
To avoid gaps or overlaps, we require the level set function 𝜙 to be a distance field, i.e., \(\phi\) satisfies the eikonal equation: \[ \| \nabla \phi \| =1 \]
Eikonal equation is highly nonlinear; instead minimize the residual: \[ \mathscr{F}_\| := \rho(\| \phi \| −1)^2 \]
\(\mathscr{F}_‖\) is referred to as parallel misalignment penalty
Singularities appear when \(\| \nabla \tilde{s}_\theta \| \to 0\)
Reinforces the no gaps/overlaps requirement
Singularity penalty function \(\mathscr{P}(\cdot)\)
Influence of Initial Design
Minimum compliance design of a short beam with different initial level set functions
Our parametrization is quite sensitive to initial design
Three-step Heuristic
Step 1: run the optimization problem with direct parametrization \[
\begin{align}
\underset{s_\rho,s_\theta}{min} \quad& z(s_\rho,s_\theta,u) \\
s.t. \quad& g_l \le g(u;s_\rho,s_\theta) \le g_u \\
\quad& h(s_\rho, s_\theta, u)=0 \\
\quad& R(u; s_\rho, s_\theta)=0
\end{align}
\]
\(z = \frac{c}{c_0}\)
\(g = V - V_0 / 2\)
\(R \equiv [K]\{u\} - \{f\} = 0\)
Step 2: dehomogenize using the projection method to obtain a scalar field \(\alpha\) that best aligns with optimal \(\theta\) from step 1
Step 3: run the optimization problem with the proposed parametrization with initial \(s_\theta=\alpha\) and \(s_\rho\) from step 1
The parallel misalignment penalty is added to the objective such that \(z = \frac{1}{2} \frac{c}{c_0} + \frac{1}{2} \frac{\mathscr{F_\|}}{\mathscr{F_\|}_0}\)
The material stiffness is penalized for singularities
The second step now requires computing two scalar fields \(s_{lam}\) and \(s_{fib}\) that satisfy the equation \(\nabla \tilde{s}_{lam} \times \nabla \tilde{s}_{fib} = \hat{t}\)
\(\hat{t}\) is the optimized fiber orientation field obtained from step 1
The optimization statement for step 3:
Objective
Minimize compliance (i.e., maximize stiffness)
Reduce gaps/overlaps among fiber paths and mitigate singularities
Minimally compliant design of a beam subject to 50% volume constraint
\(| \vec{F} |\) = 1 MPa, \(a_1\) = 120 mm, \(a_2\) = 40 mm, \(a_3\) = 20 mm, \(a_4\) = 10 mm
Design of a Beam: Results
Constant stiffness laminate
Uniform lamination and fibration
Optimized macroscale structure
Realizable via material extrusion and laminated object manufacturing
Design of a Beam: Results
Uniform fibration and variable lamination
Optimized macroscale structure
Realizable via non-planar material extrusion and curved-layer laminated object manufacturing
Design of a Beam: Results
Uniform lamination and variable fibration
Optimized macroscale structure
Manufacturable via material extrusion or AFP
Design of a Beam: Results
The general case
Variable lamination and variable fibration
Optimized macroscale structure
Requires multi-axis material extrusion
Design of a Beam: Summary
Problem Setup
Strain Energy (kJ)
Lamination Misalignment Penalty
Fibration Misalignment Penalty
Lamination
Fibration
Uniform
Uniform
197.16
0
0
Variable
Uniform
130.62
71.83
8.95
Uniform
Variable
114.76
0
8.02
Variable
Variable
92.1
360.96
394.38
Results
MTO for Manufacturable 2.5D AFRCs
Many aerospace and automotive composites are thin-walled composite shells
Characterized by curved geometry with thickness much smaller than the other dimensions
How can the framework be extended to such structures?
Composite skin of an aircraft fuselage Source: erau.edu
Manufacturability for 2.5D AFRCs
To avoid gaps or overlaps, the eikonal equation can still be used: \[ \| \nabla \phi \| =1 \]
However, gradient operator on a manifold has to be used instead of the Cartesian gradient operator
For a smooth scalar function \(f:\mathscr{M}\rightarrow\mathbb{R}\) defined on an arbitrary manifold, (\(\mathscr{M}, g\)) (\(\mathscr{M}\) and \(g\) denote the manifold space and its metric, respectively)
Gradient of \(f\)\[ \nabla f := \frac{df}{d\mathbf{X}}= \frac{\partial f}{\partial \xi^k} \mathbf{G}^k \]
\(\xi^k\): k-th curvilinear coordinate
\(\mathbf{G}^k\): contravariant basis vector
Design of a Spherical Shell: Problem Setup
\(\vec{F}\)=1 MPa, \(b_1\)=200, thickness = 1 mm
Design of a Spherical Shell: Results
With a single layer
Optimized topology with fiber paths
Misalignment penalty \(\phi\)
Design of a Spherical Shell: Results
With two layers
Bottom layer
Optimized topology with fiber paths
Misalignment penalty \(\phi\)
Top layer
Conclusion
A robust framework for design of AFRCs that are readily manufacturable
Future Work
Including curvature constraints to make the framework industry-ready
Extend the framework to design soft composites with potential applications in biomedical and robotics
Leverage singularities to induce complex shape changes in liquid crystal elastomers
Other Ongoing Research
Mechanics of soft multiphase architected materials
Homogenization theory for soft material spheroidal liquid inclusions
