FEA Mesh Guidelines: Element Size and Density Criteria for Disc Spring Analysis

1. Purpose of Mesh Guidelines

The accuracy of Finite Element Analysis (FEA) is highly dependent on mesh quality. Mesh guidelines refer to the minimum element size, number of element layers, and distribution density that must be adhered to in order to ensure that calculation results (displacement, stress, frequency, etc.) achieve engineering-acceptable accuracy. For thin-walled bending structures like disc springs, a coarse mesh can lead to issues such as "shear locking," excessively high bending stiffness, and significant underestimation of stress.

Mesh guidelines typically originate from three sources: - Theoretical Requirements: Based on the interpolation capability of element shape functions, the minimum number of element layers required through the thickness for bending problems. - Empirical Rules: Recommended mesh densities derived from previous analyses and experimental comparisons. - Convergence Verification: By refining the mesh and comparing result changes, the mesh is deemed sufficient when the change is less than the target tolerance.

2. Through-Thickness Mesh Layer Guidelines (Bending-Dominated Problems)

Disc springs primarily undergo bending deformation, with cross-sectional stress distributed linearly through the thickness (purely elastic). To accurately capture the extreme stresses on the upper and lower surfaces, the number of nodes through the thickness must be sufficient to describe the linearly varying bending normal stress.

• Linear Fully Integrated Elements (e.g., C3D8): Prone to "shear locking" during bending, requiring a sufficient number of layers to mitigate this. Recommended minimum: 4 layers.
• Linear Reduced Integration Elements (e.g., C3D8R): Exhibit better bending performance with proper hourglass control. No fewer than 3 layers.
• Quadratic Elements (e.g., C3D20, C3D15): Each node has higher-order interpolation capability. At least 2 layers through the thickness are sufficient for reasonably accurate bending stress description.
• Quadratic Tetrahedral Elements (C3D10): Also require no fewer than 2 layers through the thickness, though the total element count is typically higher.

Recommendation for Disc Spring Thickness: For thickness $t$, when using linear hexahedral elements, divide the thickness into 4 to 6 layers; for quadratic hexahedral elements, use 2 to 3 layers. This ensures stress errors at the OM point and I point are less than 3%.

3. Circumferential Mesh Element Guidelines

Disc springs are axisymmetric bodies, but under uniform axial loading, the deformation remains axisymmetric. Insufficient circumferential discretization leads to a coarse geometric description of the inner and outer edges, causing stress distortion.

• Full Model: Recommended minimum of 48 elements in the circumferential direction (i.e., one element every 7.5°).
• Symmetric Model (e.g., using a 1/4 or 1/8 sector with symmetry boundary conditions): The number of circumferential elements within the sector should be no less than 12 to 16, equivalent to a minimum of 48 elements for the full circumference.
• If Only Concerned with Force-Displacement Curve: The number of circumferential elements can be slightly reduced, e.g., 36, but stress evaluation still requires 48 or more.

4. Mesh Size Estimation Based on Stress Gradient

In regions with high stress gradients (e.g., inner and outer edges), the element size should be smaller than the characteristic length of stress variation. A common empirical relationship for estimating local element size is:

where $L_{grad} = \left| \frac{\sigma}{\partial \sigma/\partial x} \right|$ is the characteristic length of stress variation. A preliminary analysis with a coarser mesh can identify regions with the highest stress gradients. Local refinement should then be applied so that the element size in the refined zone satisfies the above relationship.

For disc springs, stress gradients are extremely high near inner and outer edge chamfers or fillets. Local mesh refinement is necessary, ensuring that the fillet radius is covered by at least 4 to 6 elements.

5. Contact Region Mesh Requirements

If the analysis considers contact between disc springs (parallel/series stacking), the mesh on the contact surfaces must be sufficiently fine to accurately calculate contact pressure.

• The element size in the contact region should be less than 1/3 of the expected contact half-width.
• For surface-to-surface contact, the mesh densities of the master and slave surfaces should be similar to avoid penetration caused by significant stiffness disparity.
• The through-thickness direction should still maintain ≥ 4 layers. While contact occurs only on the surface, accurate transmission of bending stress requires sufficient through-thickness layers.

6. Mesh Convergence Criteria

To determine if the mesh meets accuracy requirements, calculations should be performed with at least three different mesh densities (e.g., coarse, medium, fine), comparing key result quantities:

• Force Value (Reaction Force): Error requirement < 1%;
• Maximum Compressive Stress at OM Point: Error requirement < 3%;
• Tensile Stress at I Point: Error requirement < 5%.

The mesh is considered converged only when the change between two consecutive densities falls below the above thresholds. Typically, a medium mesh (4 layers through thickness, 48 elements circumferentially) satisfies most requirements.

7. Explicit Dynamics Mesh Guidelines

For explicit impact analysis, element size not only affects accuracy but also directly influences the critical time step $\Delta t_{crit}$. Smaller elements lead to smaller $\Delta t_{crit}$, drastically increasing computational cost. Therefore, a balance between accuracy and efficiency is necessary:

• The minimum element size $L_{min}$ should be no smaller than required for reliable contact and stress distribution, but should not be excessively fine to avoid an overly small time step.
• Empirical rule: The through-thickness element size for a disc spring should not be less than $t/5$; otherwise, the time step may drop below $10^{-8}$ seconds, making the calculation infeasible within a reasonable time.
• Mass scaling can be used to appropriately increase the time step, but the change in kinetic energy must be kept < 2% to avoid significantly affecting the dynamic response.

8. Submodeling Mesh Guidelines

When using the submodeling technique, the global coarse mesh can have relaxed requirements (though it must still reasonably transmit displacements), while the mesh within the submodel must be fine. The cut boundary of the submodel should be located at least 2 to 3 times the thickness away from stress concentration zones. The mesh within this region can transition gradually. Inside the submodel, the critical areas should satisfy the bending mesh requirements mentioned above.

Summary: The core FEA mesh guidelines for disc springs consist of 4 layers through the thickness (for linear elements) and 48 elements circumferentially. This is combined with local refinement based on stress gradients, matching contact mesh densities, and convergence verification to ensure analysis results fall within acceptable engineering error tolerances. A rational mesh guideline is a prerequisite for obtaining reliable load, stress, and fatigue life predictions.