PETSc & Nonlinear Solvers

Notes on using PETSc for matrix-free nonlinear conjugate gradient methods, preconditioning, and multi-level strategies in large-deformation FEA.

1. Matrix-Free NCG

For large elastomer problems, forming the full tangent matrix can be expensive. Matrix-free NCG uses only residual evaluations and approximate directional derivatives while still capturing the structure of the PDE.

• Concept of a model problem with fixed contact and controlled stiffness
• Residual-only Krylov / CG-like methods
• Trade-offs between robustness and memory footprint

2. Preconditioning as Structural Shaping

In the Rᵀ–L–S language, preconditioning can be seen as reshaping how the solver interrogates the operator. Practical PETSc topics:

• Left vs right preconditioning and their structural implications
• Using block preconditioners for coupled thermo-mechanical systems
• Embedding multi-grid as a structured preconditioner

3. Multi-Level ALM-Based Framework

Multi-level solvers with Augmented Lagrange control of material, contact, or failure constraints can be implemented in PETSc with:

• Level-0: NCG on a model problem (fixed contact, scaled stiffness)
• Higher levels: update multipliers and penalties, refresh constraints
• Monitoring constraint violation and convergence structurally

4. Example Snippets (coming)

Future content here will include:

• PETSc NCG driver skeleton (code / PDF)
• Matrix-free residual callback examples
• Prototype ALM multi-level solver outline

Implementation details may live in separate code repositories; this page will remain the conceptual and documentation index.