MATLAB Solutions for Lubrication Theory with Code Implementation

Lubrication theory is a critical branch studying lubricant film behavior between two relatively moving surfaces, widely applied in mechanical engineering. MATLAB serves as a powerful numerical computation tool particularly suitable for solving such problems, especially for two典型典型 categories: elastohydrodynamic lubrication (considering surface elastic deformation) and rigid hydrodynamic lubrication (assuming rigid surfaces).

The core challenge in elastohydrodynamic lubrication solution lies in coupling the Reynolds equation with elastic deformation equations. The implementation involves discretizing Reynolds equation using finite difference method, while employing influence coefficient method for surface deformation calculation. This requires building pressure-deformation iteration loops until convergence. MATLAB's matrix operation capabilities efficiently handle such large-scale linear systems, and FFT can accelerate convolution computations through functions like fftn/ifftn.

Rigid hydrodynamic lubrication problems are relatively simpler, requiring only Reynolds equation solution when ignoring elastic deformation. After central difference discretization, Newton's iteration method handles nonlinear terms, combined with SOR (Successive Over-Relaxation) or direct matrix solvers like backslash operator () to obtain pressure distribution. MATLAB's sparse matrix storage (using sparse()) significantly reduces memory consumption for large-scale problems.

Key implementation techniques include: introducing dimensionless normalization to simplify calculations; applying multigrid methods (multigrid solvers) to accelerate convergence; for transient problems, combining Runge-Kutta methods (ode45/ode15s) for time integration. Visualization modules can plot pressure distributions and film thickness curves using contourf/mesh plots, facilitating lubrication performance analysis.

Advanced extensions can incorporate thermal EHL (coupling energy equations) or rough surface effects, which can be efficiently implemented using MATLAB's PDE toolbox (pdepe) or custom algorithms with adaptive mesh refinement.