Overview
A novel generative design system I’m developing that creates flexure geometries optimized to match specific and arbitrary force-displacement profiles. By inputting a target force profile, the system generates compliant mechanism geometries such that the reaction force at any displacement closely matches the desired response curve.
This enables the creation of mechanisms with precisely tailored mechanical behaviors such as exponential stiffnening, softening responses, bi-stable regions with negative reaction force, or complex multi-phase force profiles for specialized applications in robotics, precision instruments, and energy absorption systems.
Technical Approach
- Profile Matching: I’m developing an advanced optimization formulation to minimize the difference between target and achieved force-displacement curves across the entire actuation range, automatically weighting unstable/imfortant regions of the curve.
- MMA Sensititvities/Gradients: Using my custom adjoint calculations to tell optimizer each elements effect on the REST of the curve at later equilibrium points, not just that current equilibrium point (for EVERY discritized point on the curve!).
- Multi-Objective Optimization: Simultaneously optimizes for profile matching, manufacturability, stress distribution, and material efficiency.
- Nonlinear Analysis: Custom arc-length solver handles snap-through, snap-back, and negative stiffness behaviors critical for complex force profiles.
- Density-Based TO: SIMP method with custom sensitivities for gradient-based optimization of the material distribution.
Custom Arc-Length Nonlinear FEA
This animation was created with my custom arc-length solver that is able to simulate snap-through/snap-back behavior and integrates with MMA optimization through custom adjoint-derived sensitivities that I developed.
Before designing these force profile flexures, I need to model the most nonlinear behaviors including elemental negative stiffness, near-singular stiffness matrices, and buckling phenomena. This requires a highly specialized Arc-length solver capable of handling extreme geometric nonlinearity and material instabilities that occur in compliant mechanisms by dynamically tracing the equalibrium path of the actuated object.
While open-source nonlinear FEA solvers exist, they cannot be used for this application as I need direct control of the elemental values for SIMP density-based topology optimization, as well as the ability to create custom adjoint sensitivities and gradients required by my MMA optimizer. This necessitated developing a fully custom arc-length solver based on PETSc and UFL (to expose and create custom FEniCS forms/expressions for discritization).
Force-displacement curve demonstrating negative stiffness behavior
Early Resutls With My Formulation
Exponential stiffening flexure: Force increases exponentially with displacement, ideal for impact protection systems and progressive-rate suspensions.
Softening response flexure: Force plateaus with increasing displacement, perfect for constant-force mechanisms and precision assembly tools.