Deformation gradient averaging regularization for third medium contact

1. Problem Statement

The Third Medium Contact (TMC) method is an alternative to traditional contact mechanics (e.g., penalty or Lagrange multiplier methods) that avoids the need for explicit contact search algorithms. It achieves this by filling the void space between bodies with a fictitious, highly compliant material that stiffens significantly upon compression (near-zero volume), thereby enforcing contact constraints.

While TMC is highly effective for topology optimization and complex geometries where contact detection is difficult, it faces specific challenges in finite strain settings:

• Excessive Deformation: In the pre-contact phase, the highly compliant third medium tends to deform unrealistically (excessive distortion).
• Regularization Complexity: Existing solutions to stabilize the third medium rely on penalizing second gradients of displacement (e.g., $\nabla^2 u$).
  • This typically requires second-order finite elements (quadratic shape functions).
  • Alternatively, it requires additional degrees of freedom (DOFs) when using first-order elements.
  • These requirements increase computational cost and implementation complexity.

The authors aim to develop a regularization method that stabilizes the third medium using first-order finite elements without introducing additional degrees of freedom, thereby simplifying implementation and maintaining computational efficiency.

2. Methodology

The proposed approach introduces a new material model for the third medium consisting of three distinct energy components:

A. Combined Hyperelastic and Linear Elastic Terms

The total strain energy density ($W_{3M}$) is composed of:

1. Volumetric Contact Term: Based on the logarithmic volumetric part of the Neo-Hookean law ($\frac{1}{2}\kappa_{vol} \ln^2 J$). This term reacts asymptotically to extreme compression ($J \to 0$), preventing penetration. The isochoric (distortional) part of the Neo-Hookean law is omitted to improve stability.
2. Constant Stiffness Term: A linear elastic term ($\frac{1}{2}(F-I):D_{3M}:(F-I)$) applied in the pre-contact phase. Unlike traditional models where stiffness scales with deformation, this term enforces a constant stiffness (scaled by a small parameter $\gamma_e$) to prevent excessive distortion of low-stiffness elements before contact occurs.
3. Deformation Averaging Regularization: The core innovation of the paper.

B. Deformation Gradient Averaging Regularization

Instead of penalizing the second gradient of displacement ($\nabla \nabla u$), the authors propose penalizing the deviation of the deformation gradient ($F$) from its element-wise average value ($\bar{F}$).

The regularization energy term is defined as:
$$W_{\bar{F}} = \frac{1}{2} \kappa_{\bar{F}} \| F - \bar{F} \|^2$$
Where:

• $F$: Deformation gradient evaluated at Gauss integration points.
• $\bar{F}$: Deformation gradient evaluated at the element centroid.
• $\kappa_{\bar{F}}$: A penalty parameter controlling the uniformity of deformation across the element.
• $\| \cdot \|$: Frobenius norm.

Key Implementation Details:

• No Second Gradients: The method does not require computing second derivatives of the displacement field.
• First-Order Elements: It works with standard bilinear (first-order) quadrilateral elements.
• Integration Points: The method relies on elements having multiple integration points (e.g., 4 or 9 Gauss points) where $F$ can vary. The penalty forces $F$ at these points to remain close to the centroid value $\bar{F}$, effectively suppressing spurious hourglassing and excessive bending modes without needing extra DOFs.
• Implementation: The centroid value $\bar{F}$ is computed by evaluating shape function gradients at the element center (natural coordinates $\xi=0, \eta=0$).

3. Key Contributions

1. Novel Regularization Scheme: Introduction of a deformation gradient averaging term that penalizes spatial variations of $F$ within an element, replacing the need for second-gradient formulations.
2. Simplified Formulation: The method enables the use of standard first-order finite elements without adding extra degrees of freedom, significantly reducing implementation complexity compared to previous TMC models.
3. Constant Stiffness Pre-Contact: The adoption of a linear elastic term with constant stiffness in the pre-contact phase improves the robustness of the simulation by preventing unrealistic distortion of the void mesh.
4. Energy Potential: The entire formulation is derived from an energy potential, ensuring the symmetry of the global stiffness tangent matrix, which is crucial for Newton-Raphson solvers and stability analysis.

4. Results and Validation

The authors validated the method using the open-source software OOFEM across four numerical benchmarks:

• C-Shape Benchmark (Mesh Sensitivity):

  • Simulated two cantilevers closing a gap.
  • Demonstrated that the method produces consistent results across different mesh refinements (M3 to M21) when the penalty parameter $\kappa_{\bar{F}}$ is scaled with mesh size.
  • Results matched traditional node-to-segment contact methods closely, with only minor "rounding" at the contact point characteristic of TMC.

• Closed Box Benchmark (Self-Contact):

  • Simulated a box undergoing large deformation and self-contact (inner and outer surfaces).
  • Showed excellent agreement with traditional contact methods in force-displacement curves.
  • Successfully handled complex self-contact scenarios without contact search algorithms.

• Constrained Buckling and Instability:

  • Simulated a beam clamped at both ends, compressed by a piston, leading to buckling, wall contact, and self-contact.
  • The method correctly captured the Eulerian critical load (within 0.6% error) and the subsequent stiffening/softening behavior during complex contact events.
  • Demonstrated robustness in tracking unstable equilibrium paths.

• Configurational Forces:

  • Validated against an analytical solution for the J-integral and configurational force in a large deformation contact problem.
  • The third medium approach accurately recovered the analytical horizontal reaction forces, proving its ability to handle combined normal and tangential interactions and curved contact geometries.

5. Significance and Conclusion

This paper presents a robust, simple, and efficient regularization technique for the Third Medium Contact method. By shifting from second-gradient penalization to element-wise deformation gradient averaging, the authors have removed the primary barrier to using TMC in standard finite element workflows: the need for high-order elements or complex DOF management.

Significance:

• Topology Optimization: The method is ideally suited for topology optimization where the contact interface evolves, and mesh quality varies, as it avoids the computational overhead of contact search algorithms.
• Multiphysics: The approach can be extended to other multiphysical simulations where "air-like" spaces need to be meshed and regularized.
• Accessibility: The ease of implementation (using standard first-order elements) makes advanced contact mechanics more accessible to researchers and engineers using standard FEM codes.

The authors conclude that this approach offers a universal, stable, and easy-to-implement alternative for simulating contact in finite strain problems, particularly in contexts involving instability and complex topology changes.