Automatic Mesh Generation Based on Boundary Equations

In engineering calculations and scientific simulations, mesh generation serves as a crucial preprocessing step. MATLAB enables efficient automatic mesh generation based on boundary equations, producing mesh numbering and element-node information. This approach is particularly suitable for scenarios where geometric shapes are defined by mathematical equations.

Core Methodology Boundary Equation Input: Users provide mathematical equations defining geometric boundaries, such as circles, rectangles, or more complex parametric curves. In MATLAB implementation, these equations can be programmed as anonymous functions (e.g., @(x,y) x.^2 + y.^2 - 1 for circular boundaries). Mesh Parameter Configuration: Specify grid dimensions like length (Δx) and width (Δy) to control mesh density. Code implementation typically involves creating dimension vectors using linspace or meshgrid functions. Discretization Process: The region is discretized into small elements (quadrilaterals or triangles) based on boundary equations and grid parameters. Algorithmically, this involves checking boundary conditions for each potential grid point using logical indexing. Node Numbering Generation: Unique identifiers are assigned to each element's vertices, establishing element-node topology relationships for subsequent finite element analysis. This can be implemented through systematic indexing schemes using reshape or sparse matrix operations.

Advanced Applications Adaptive Meshing: Dynamic mesh refinement based on physical field gradients, with increased density in regions of rapid variation. Implementation involves gradient calculation algorithms and conditional grid scaling. Complex Boundary Handling: Supports combination of multiple boundary equations and parametric variables (like polar coordinates) for irregular domains. MATLAB's symbolic math toolbox can facilitate coordinate transformations.

Through MATLAB's matrix operations and loop structures, these workflows can be efficiently implemented without external mesh generation tools. Users can rapidly generate discretized models for various scenarios simply by adjusting boundary equations and mesh parameters, leveraging functions like pdepe for partial differential equation setups or custom spatial partitioning algorithms.