Structured Mesh
Introduction
A structured mesh is a type of grid used in numerical simulations and computational modeling, characterized by a regular, repeating pattern of elements. Structured meshes are commonly employed in computational fluid dynamics (CFD), finite element analysis (FEA), and other areas of computational science and engineering. The regularity of the mesh simplifies the implementation of numerical algorithms and often leads to more efficient computations.
Characteristics of Structured Meshes
Structured meshes are defined by their orderly arrangement of elements, which can be quadrilaterals in 2D or hexahedra in 3D. Each element in a structured mesh can be indexed by a set of integers, making it straightforward to navigate and manipulate the mesh. This regular structure allows for efficient storage and retrieval of data, as well as simplified implementation of numerical methods.
Grid Generation
The process of generating a structured mesh involves defining a grid over the computational domain. This can be done using various techniques, such as:
- **Cartesian Grids**: The simplest form of structured mesh, where the elements are aligned with the coordinate axes. Cartesian grids are easy to generate and use but may not conform well to complex geometries.
- **Curvilinear Grids**: These grids allow for elements to follow the contours of the domain, providing better conformity to complex shapes. Curvilinear grids are more challenging to generate but offer improved accuracy in simulations involving irregular geometries.
- **Body-Fitted Grids**: These grids are specifically designed to align with the boundaries of the domain, ensuring that the mesh accurately represents the physical boundaries of the problem.
Mesh Quality
The quality of a structured mesh is crucial for the accuracy and efficiency of numerical simulations. Key factors influencing mesh quality include:
- **Element Shape**: Ideally, elements should be as close to regular shapes (e.g., squares or cubes) as possible to minimize numerical errors.
- **Element Size**: Uniform element size is preferred, but in practice, mesh refinement techniques are often used to increase resolution in regions of interest.
- **Orthogonality**: The angles between adjacent elements should be close to 90 degrees to reduce numerical diffusion and improve solution accuracy.
Applications of Structured Meshes
Structured meshes are widely used in various fields of science and engineering due to their simplicity and efficiency. Some notable applications include:
Computational Fluid Dynamics (CFD)
In CFD, structured meshes are used to discretize the fluid domain, allowing for the numerical solution of the Navier-Stokes equations. The regularity of the mesh facilitates the implementation of finite difference and finite volume methods, which are commonly used in CFD simulations.
Finite Element Analysis (FEA)
Structured meshes are also employed in FEA to model the behavior of solid structures under various loading conditions. The regular grid structure simplifies the assembly of the stiffness matrix and the application of boundary conditions.
Electromagnetic Simulations
Structured meshes are used in the simulation of electromagnetic fields, particularly in the finite difference time domain (FDTD) method. The regular grid allows for efficient computation of field values and the implementation of absorbing boundary conditions.
Advantages and Disadvantages
Advantages
- **Simplicity**: The regular structure of the mesh simplifies the implementation of numerical algorithms.
- **Efficiency**: Structured meshes often lead to more efficient computations due to their regularity.
- **Data Storage**: The orderly arrangement of elements allows for efficient storage and retrieval of data.
Disadvantages
- **Geometric Flexibility**: Structured meshes may not conform well to complex geometries, leading to inaccuracies in the representation of the domain.
- **Mesh Generation**: Generating a high-quality structured mesh for complex domains can be challenging and time-consuming.
- **Adaptivity**: Structured meshes are less flexible in terms of local refinement compared to unstructured meshes.