DIN 2093 Submodeling Method: Local Refined Stress Analysis

1. Definition and Purpose of the Submodeling Method

The submodeling method is a finite element analysis technique based on Saint-Venant's principle. It is used to perform a refined mesh re-solution of local critical areas (such as the OM point, tooth root, contact edge, and other stress concentration zones of disc springs) based on the results of a global coarse mesh model. This achieves high-precision local stress distribution at a lower computational cost.

The core idea is: if the cut boundary is sufficiently far from the stress concentration zone, the displacement field on the cut boundary is primarily determined by the global response, and local details (fillets, notches, etc.) have minimal influence on the far-field displacement. Therefore, a coarse mesh can first be used to calculate the overall structure, and then the displacements on the cut boundary are interpolated and applied as boundary conditions to the refined submodel.

2. Basic Procedure and Formulas

The steps for submodel analysis are as follows:

2.1 Global Coarse Mesh Analysis

• Establish a global model containing the entire disc spring (or assembly), using a relatively sparse mesh (which must still meet basic convergence requirements).
• Apply service loads (e.g., axial compression $s$) and boundary conditions.
• Perform a linear or nonlinear static analysis to obtain the nodal displacement field $\mathbf{U}^{coarse}$.

2.2 Submodel Cutting and Refined Mesh Generation

• Extract the local area of interest (e.g., a sector containing the OM point) from the global model. The cut boundary should be at least 2–3 times the characteristic length away from the stress concentration zone (the characteristic length can be the thickness $t$ or a local geometric dimension).
• Generate a refined mesh for this local geometry. There should be ≥ 4 layers in the thickness direction, and the mesh should be refined at stress concentration points to capture high stress gradients.

2.3 Cut Boundary Displacement Interpolation (Core Calculation Formula)

The boundary conditions for the refined submodel are obtained from the global coarse mesh solution via shape function interpolation. Let the displacement of a point on the cut boundary in the global model be $\mathbf{u}_G$, and the displacement of the corresponding boundary point in the submodel be $\mathbf{u}_{sub}$. Then:

where: - $\mathbf{x}_i$ — coordinates of node $i$ on the submodel boundary; - $\mathbf{X}_j$ — coordinates of node $j$ in the global model; - $N_j$ — shape functions of the global element, used for spatial interpolation on the cut boundary; - $m$ — number of nodes in the global element containing point $\mathbf{x}_i$.

Commercial software (e.g., Abaqus, ANSYS) performs this interpolation automatically: the user only needs to specify the submodel boundary, and the program reads the global results file (e.g., .odb, .rst) and maps the displacements from the corresponding locations to the submodel.

2.4 Submodel Solution

The submodel is treated as an independent analysis. It applies the displacement boundary conditions interpolated from the global model and re-solves for the stress and strain fields on the local refined mesh, yielding high-precision local stresses $\boldsymbol{\sigma}^{fine}$.

The submodel analysis can be linear or can inherit the material nonlinearity of the global model. However, it is crucial that the displacements on the cut boundary accurately reflect the global response; therefore, the global analysis must be sufficiently accurate.

3. Application in Disc Springs

The fatigue life of a disc spring is typically governed by the stress at the OM point (inner edge of the upper surface). Since this point is on a free edge and experiences bending stress, a global coarse mesh may underestimate the peak stress due to mesh discretization. Using the submodeling method:

1. Global Model: A relatively coarse mesh (e.g., 2–3 layers in the thickness direction) is used to analyze the entire disc spring and obtain the displacement field.
2. Submodel: A sector (e.g., 15°–30°) containing the OM point and the inner diameter edge is cut out. This sector is modeled with a very fine mesh (6–8 layers in thickness, refined at the edge). The cut boundary is placed sufficiently far from the inner and outer edges (≥ 2 t from the edge).
3. The global displacements are mapped to the cut surfaces of the submodel, symmetry boundary conditions are applied, and the solution yields the accurate compressive stress at the OM point.

4. Accuracy and Efficiency of the Submodeling Method

Accuracy Conditions: - Sufficiently Distant Cut Boundary: Generally, the cut boundary should be 2–3 times the local characteristic length away from the area of interest (for disc springs, 3–5 times the thickness $t$ can be used). If the cut boundary is too close to the stress concentration zone, errors in the global solution at that location will be transferred to the submodel, reducing accuracy. - Global Mesh Not Excessively Coarse: While the global mesh can be relatively coarse, it must still reasonably represent the overall stiffness and displacement trends. The mesh near the cut boundary should not be overly distorted. - Sufficiently Converged Submodel Mesh: The mesh within the submodel must satisfy convergence requirements, ensuring that the stress no longer changes significantly with further mesh refinement.

Efficiency Gains: - The global coarse mesh can save 60%–80% of computation time compared to a fully refined mesh model. - The submodel solves only a very small region, resulting in very low additional computational cost. - Multiple submodel analyses for different local details (e.g., simultaneously evaluating the OM point, I point, and contact edge) can be performed based on the same global model.

5. Example Analysis

Assume a precise stress analysis is required for the OM point of an M10 disc spring. The global model uses 36 elements around the circumference and 2 layers in thickness (fast computation). The submodel cuts out a 30° sector around the inner diameter, with 6 layers in thickness and 24 elements circumferentially (local only). The global solution gives a nominal stress of approximately 1 800 MPa at the OM point, while the refined submodel solution yields 2 050 MPa, a difference of about 12%. This indicates that the coarse mesh significantly underestimates the stress. The submodel result is ultimately used as the basis for fatigue verification.

6. Important Considerations

• Nonlinear Problems: If the global model is geometrically nonlinear, the submodel should inherit the same nonlinear settings, as the displacements on the cut boundary may correspond to a large deformation state.
• Contact: If the global model involves contact, the submodel cut boundary should not pass through the contact region; otherwise, the interpolated displacements will be discontinuous.
• Thermomechanical Coupling: The submodel can also inherit the temperature field for refined thermal stress analysis.
• Saint-Venant's Principle Verification: The adequacy of the boundary distance can be verified by gradually expanding the submodel region and observing whether the stress at the point of interest converges.

Summary: The submodeling method uses the displacement field from a global coarse mesh to drive a refined local model via shape function interpolation. It achieves an optimal balance between computational cost and accuracy in the stress concentration analysis of disc springs, making it an effective tool for extracting high-precision OM point stresses and optimizing fatigue design in engineering practice.