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Hello Mohammad Alneari

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In COMSOL, the error you are encountering—related to a "nonsymmetric matrix" and a "singular matrix"—typically arises due to issues with the setup or numerical solution of the partial differential equations (PDEs) in 3D.

Here's a step-by-step guide to troubleshooting and fixing this issue in COMSOL for your magnetization dynamics simulation:

1. Check Mesh Quality Reason: The 3D model typically requires a finer and more structured mesh compared to 2D. Poor mesh quality can lead to instability in the matrix solver, resulting in nonsymmetric or singular matrices.

Solution: In COMSOL, use Mesh > Statistics to check for distorted or low-quality elements in your 3D mesh. Refine the mesh where needed, especially in regions where the magnetization or current density gradients are large. You may also switch to a finer mesh or adaptive mesh refinement.

2. Solver Settings Reason: The solver might be inappropriate for the complexity of your 3D model, which may include complex boundary conditions or anisotropy.

Solution: Direct Solver: Go to Study > Solver Configurations > Solver. Try switching to a direct solver (e.g., MUMPS) instead of iterative solvers like GMRES. Direct solvers are more robust for ill-conditioned systems.

Iterative Solver Settings: If using an iterative solver, enable preconditioning and increase the number of iterations to help with convergence. Tolerance: Lower the relative tolerance in the solver settings to improve accuracy. Symmetric Solver Option: In some cases, enabling or forcing the symmetry option in the Advanced Solver Settings might help, but this depends on whether your problem is fundamentally symmetric.

3. Check Boundary and Initial Conditions Reason: Boundary and initial conditions in 3D are more complex than in 2D. Incorrect or inconsistent boundary conditions can lead to singular matrices. Solution: Revisit the boundary conditions and ensure they are physically meaningful and properly defined for all the 3D boundaries. Ensure that initial conditions for the magnetization field are realistic, especially for 3D, where small inconsistencies can lead to large numerical issues.

4. Material Properties

Reason: In 3D, the material properties such as magnetic permeability or electrical conductivity may need to be defined more precisely. Any anisotropy in the material might be incorrectly represented, leading to matrix issues.

Solution: Double-check that the magnetic permeability and conductivity tensors (if anisotropic) are defined correctly for the 3D geometry. Also, check that material properties vary smoothly throughout the 3D domain.

5. Equation Formulation Reason: Extending from 2D to 3D can introduce additional terms or complexities (like demagnetizing fields) in the Landau-Lifshitz-Gilbert (LLG) equations or other governing equations.

Solution: Re-check the formulation of the magnetization dynamics equations in COMSOL. Ensure that all terms are correctly extended to 3D (e.g., including the demagnetizing fields and stray fields). If necessary, enable or explicitly define terms for anisotropy, exchange energy, and demagnetizing field in the 3D domain, which may not be as crucial in the 2D case but are essential in 3D simulations.

6. Magnetic Vector Potential vs. Scalar Potential Reason: In 3D, the choice between using a magnetic vector potential formulation (for transient magnetic fields) versus a magnetic scalar potential formulation (for static fields) can affect stability and convergence. Solution: Verify the potential formulation used in your simulation: Use magnetic vector potential for dynamic simulations involving currents and moving magnetization.

Use magnetic scalar potential for simpler static field calculations. In COMSOL, you can adjust this under Physics > Magnetic Fields.

7. Singular Matrix and Regularization

Reason: Singular matrices often indicate that the system of equations is not properly constrained or ill-posed.

Solution: In COMSOL, you can use regularization techniques to handle singular matrices. Check the options in Physics > Magnetic Fields to add regularization terms, particularly for ill-posed boundary conditions or problematic initial conditions.

8. Current Density Distribution Reason: In 3D, the current density distribution applied to the model may differ significantly from the 2D case, leading to unexpected behavior.

Solution: Ensure that the current density is applied correctly in 3D. If the current is distributed non-uniformly, increase the resolution of the applied current and check for regions with high gradients.

9. Plotting and Post-Processing Reason: The fact that your normal magnetization plot shows straight lines in 2D but not in 3D suggests that the magnetization is behaving differently in 3D, which is expected due to additional degrees of freedom.

Solution: Use Post-processing in COMSOL to visualize the magnetic fields, current density, and magnetization dynamics in the 3D model. Compare the 2D and 3D results to understand where the magnetization dynamics differ and adjust parameters accordingly.

By following these steps, you should be able to address the matrix singularity and nonsymmetric matrix warnings in COMSOL, improving the accuracy and stability of your 3D magnetization dynamic sitution.