Experiment no. 01 (1)
- 1. Experiment No. 01 TITLE: Study of different mesh generation schemes OBJECTIVE: To study the different types of mesh generation schemes THEORY: 1. Introduction The partial differential equations that govern fluid flow and heat transfer are not usually amenable to analytical solutions, except for very simple cases. Therefore, in order to analyze fluid flows, flow domains are split into smaller subdomains (made up of geometric primitives like hexahedra and tetrahedra in 3D and quadrilaterals and triangles in 2D). The governing equations are then discretized and solved inside each of these subdomains. Typically, one of three methods is used to solve the approximate version of the system of equations: finite volumes, finite elements, or finite differences. Care must be taken to ensure proper continuity of solution across the common interfaces between two subdomains, so that the approximate solutions inside various portions can be put together to give a complete picture of fluid flow in the entire domain. The subdomains are often called elements or cells, and the collection of all elements or cells is called a mesh or grid. The origin of the term mesh (or grid) goes back to early days of CFD when most analyses were 2D in nature. For 2D analyses, a domain split into elements resembles a wire mesh, hence the name. An example of a 2D analysis domain (flow over a backward facing step) and its mesh are shown in pictures below. Fig. 1: 2D analysis domain and its mesh
- 2. The process of obtaining an appropriate mesh (or grid) is termed mesh generation (or grid generation), and has long been considered a bottleneck in the analysis process due to the lack of a fully automatic mesh generation procedure. Specialized software progams have been developed for the purpose of mesh and grid generation, and access to a good software package and expertise in using this software are vital to the success of a modeling effort. 2. Mesh classification As CFD has developed, better algorithms and more computational power has become available to CFD analysts, resulting in diverse solver techniques. One of the direct results of this development has been the expansion of available mesh elements and mesh connectivity (how cells are connected to one another). The easiest classifications of meshes are based upon the connectivity of a mesh or on the type of elements present. 1. Connectivity-Based Classification a. Structured Meshes b. Unstructured Meshes c. Hybrid Meshes 2. Element-Based Classification 2.1 Connectivity-Based Classification Structured Meshes A structured mesh is characterized by regular connectivity that can be expressed as a two or three dimensional array. This restricts the element choices to quadrilaterals in 2D or hexahedra in 3D. The above example mesh is a structured mesh, as we could store the mesh connectivity in a 40 by 12 array. The regularity of the connectivity allows us to conserve space since neighborhood relationships are defined by the storage arrangement. Additional classification can be made upon whether the mesh is conformal or not. Unstructured Meshes An unstructured mesh is characterized by irregular connectivity is not readily expressed as a two or three dimensional array in computer memory. This allows for any possible element that a solver might be able to use. Compared to structured meshes, the storage requirements for an unstructured mesh can be substantially larger since the neighborhood connectivity must be explicitly stored. Hybrid Meshes A hybrid mesh is a mesh that contains structured portions and unstructured portions. Note that this definition requires knowledge of how the mesh is stored (and used). There is disagreement as to the correct application of the terms "hybrid" and "mixed." The term "mixed" is usually applied to meshes that contain elements associated with structured meshes and elements associated with unstructured meshes (presumably stored in an unstructured fashion).
- 3. 2.2 Element-Based Classification Meshes can also be classified based upon the dimension and type of elements present. Depending upon the analysis type and solver requirements, meshes generated could be 2-dimensional (2D) or 3-dimensional (3D). Common elements in 2D are triangles or rectangles, and common elements in 3D are tetrahedra or bricks. As noted above, some connectivity choices limit the types of element present, so there is some overlap between connectivity-based and element-based classification. For a 2D mesh, all mesh nodes lie in a given plane. In most cases, 2D mesh nodes lie in the XY plane, but can also be confined to another Cartesian or user defined plane. Most popular 2D mesh elements are quadrilaterals (also known as quads) and triangles (tris), shown below. Fig. 2: 2D mesh shapes 3D mesh nodes are not constrained to lie in a single plane. Most popular 3D mesh elements are hexahedra (also known as hexes or hex elements), tetrahedra (tets), square pyramids (pyramids) and extruded triangles (wedges or triangular prisms), shown below. It is worth noting that all these elements are bounded by faces belonging to the above mentioned 2D elements. Some of the current solvers also support polyhedral elements, which can be bounded by any number and types of faces. Fig. 3: 3D mesh shapes
- 4. Since all 3D elements are bounded by 2D elements, it is obvious that 3D meshes have exposed 2D elements at boundaries. Most of the meshing packages and solvers prefer to club exposed elements together in what is known as a surface mesh (for the purposes of applying boundary conditions, rendering meshed domains and visualizing results). A surface mesh does not have to be 2D, since volume meshes may conform to domains with non-planar boundaries. Many meshing algorithms start by meshing bounding surfaces of a domain before filling the interior with mesh nodes (such algorithms are also known as boundary to interior algorithms). For such algorithms, generation of good quality surface meshes is of prime importance, and much research has been done in the field of efficient and good quality surface mesh generation. Since surface meshes are geometrically somewhere between 2D and 3D meshes, they are also sometimes known as 2.5D meshes. CONCLUSION ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ QUESTION 1. Give examples of an elliptic, hyperbolic & parabolic equation. Explain how they are classified? 2. Using Taylor’s series, derive the first order forword difference, backward difference and central difference approximation for the term 𝜕𝑢 𝜕𝑦 .