[Paper Review] Topology Optimization of self-contacting structures

Andreas Henrik Frederiksen, Ole Sigmund+1 more|arXiv (Cornell University)|May 11, 2023

Topology Optimization in Engineering4 citations

TL;DR

This paper proposes a robust topology optimization framework that enables internal self-contact in compliant structures using the third medium contact method, where the void phase acts as a compliant contact medium. By integrating Helmholtz filtering, SIMP, threshold projection, and MMA, with volume constraint applied on a dilated design to penalize small features, the method achieves stable, contact-enabled designs under large deformations, demonstrated in lifting, coupling, and bending mechanisms with controlled force-displacement responses.

ABSTRACT

Inclusion of contact in mechanical designs opens a large range of design possibilities, this includes classical designs with contact, such as gears, couplings, switches, clamps etc. However, incorporation of contact in topology optimization is challenging, as classical contact models are not readily applicable when the boundaries are not defined. This paper aims to address the limitations of contact in topology optimization by extending the third medium contact method for topology optimization problems with internal contact. When the objective is to maximize a given contact load for a specified displacement, instabilities may arise as an optimum is approached. In order to alleviate stability problems as well as provide robustness of the optimized designs, a tangent stiffness requirement is introduced to the design objective. To avoid a non-physical exploitation of the third medium in optimized designs, small features are penalized by evaluating the volume constraint on a dilated design. The present work incorporates well-established methods in topology optimization including Helmholtz PDE filtering, threshold projection, Solid Isotropic Material Interpolation with Penalization, and the Method of Moving Asymptotes. Three examples are used to illustrate how the approach exploits internal contact in the topology optimization of structures subjected to large deformations.

Motivation & Objective

• To enable the inclusion of internal self-contact in topology optimization for compliant mechanisms subjected to large deformations.
• To address the instability and non-physical exploitation of void regions that arise when using the third medium contact method in density-based topology optimization.
• To develop a robust optimization framework that maintains contact integrity while preventing small-scale features from dominating the design.
• To extend the solution space of topology optimization by enabling programmable force-displacement responses through controlled contact formation.
• To validate the method on complex kinematic mechanisms such as lifting, coupling, and bending structures.

Proposed method

• The third medium contact method is adapted to density-based topology optimization, using the void phase as a compliant contact medium with finite, low stiffness under compression.
• Helmholtz PDE filtering is applied to enforce length-scale control and improve mesh convergence.
• Threshold projection is used to generate clear solid-void designs from the optimized density field.
• The Solid Isotropic Material with Penalization (SIMP) model is employed to interpolate material properties and penalize intermediate densities.
• The Method of Moving Asymptotes (MMA) is used as the optimizer to solve the constrained optimization problem.
• A dilated design is used to evaluate the volume constraint, penalizing small features and preventing non-physical exploitation of the void contact medium.

Experimental results

Research questions

• RQ1 Can the third medium contact method be effectively integrated into standard density-based topology optimization to enable internal self-contact in compliant mechanisms?
• RQ2 How can numerical instabilities arising near the optimum be mitigated when contact is included in the optimization process?
• RQ3 To what extent can void regions be autonomously formed during optimization to serve as contact media, without pre-specifying contact zones?
• RQ4 How does the inclusion of contact affect the force-displacement response in large deformation mechanisms?
• RQ5 Can the proposed method produce robust, manufacturable designs with controlled tangential stiffness and stable contact engagement?

Key findings

• The optimized designs successfully exploit both pre-defined and autonomously formed void regions as contact media, enabling stable self-contact under large deformations.
• Contact engagement is clearly observed in the bending mechanism at approximately 14° and 16° for the two designs, marked by a soft kink in the moment and force curves.
• The force remains low until contact is established, confirming the objective of maintaining low contact force prior to engagement.
• The bending stiffness increases significantly around the contact angle, indicating effective load transfer through the contact interface.
• The design with α₀ = 14° forms two distinct contact regions, while the α₀ = 18° design forms only one, demonstrating adaptive contact topology formation.
• The use of dilated volume constraint effectively suppresses small-scale features and improves numerical stability, especially in complex contact scenarios.