Grid Generation Chaiwoot Boonyasiriwat November 5, 2020 Sources of - PowerPoint PPT Presentation

Grid Generation Chaiwoot Boonyasiriwat November 5, 2020

Sources of Error at Grid Points ▪ “Mathematical models do not represent physical phenomena with absolute accuracy.” ▪ Numerical approximation of the mathematical model gives rise to an error. ▪ “The error is influenced by the size and shape of the grid cells .” ▪ “The error is contributed by the computation of the discrete physical quantities satisfying the equations of the numerical approximation.” ▪ “The error is caused by the inaccuracy of the process of interpolation of the discrete solution.” Liseikin (2010, p. 1) 2

Two Definitions of Grid 1. Points in the domain and on the boundary are called grid nodes. 2. n -dimensional volumes covering the entire area of the domain are called grid cells. ▪ “The cells are bounded by curvilinear volumes whose boundaries are divided into a few segments which are ( n -1)- dimensional cells.” Liseikin (2010, p. 2) 3

Grid Cells ▪ “In 1D, the cell is a closed line or segment whose boundary is composed of two cell vertices .” ▪ “A 2D cell is a 2D simply connected domain whose boundary is divided into a finite number of 1D cells called the edges of the cell.” ▪ 2D grid cells are in the form of triangles or quadrilaterals .” ▪ The boundaries of triangular and quadrilateral cells are composed of 3 and 4 segments, respectively. quadrilateral line/segment triangle Liseikin (2010, p. 3) 4

Grid Cells ▪ “A 3D grid cell is a simply connected 3D polyhedron whose boundary is composed of 2D cells called faces .” ▪ “Typical 3D cells are in the form of tetrahedrons, hexahedrons, or a prism .” ▪ A tetrahedron has 4 triangular faces, 6 edges, 4 vertices. ▪ A hexahedron has 6 quadrilateral faces, 12 edges, and 8 vertices. ▪ A prism has 3 triangular and 3 quadrilateral faces, 9 edges, and 6 vertices. tetrahedron hexahedron prism Liseikin (2010, p. 3) 5

Simplex ▪ “The edges and faces of the cells are typically linear.” ▪ “Linear triangles and tetrahedrons are 2D simplexes and 3D simplexes, respectively.” ▪ An n -dimensional simplex is the space defined by the barycentric sum where x i are the vertices of the simplex. ▪ Barycentric sum is required for blending of points so that the result is independent of coordinate system (Salomon, 2006). Liseikin (2010, p. 3) 6

Point vs Vector ▪ “A point has no dimensions; it represents a location in space.” ▪ “A vector has no well -defined location; its only attributes are direction and magnitude.” ▪ “Both points and vectors are represented by pairs or triplets of real numbers.” ▪ Addition of point and vector is a point. ▪ Addition of points are not well defined as the result depends on the coordinate system. ▪ Addition of points is well defined only when it is a barycentric sum. Salomon (2006, p. 1) 7

Point and Vector Addition Salomon (2006, p. 3) 8

Grid Size and Cell Size ▪ “Grid size is indicated by the number of grid points.” ▪ “Cell size is the maximum length of the cell edges.” ▪ A grid generation technique should be able to increase the number of grid nodes. “At the same time the edge lengths of the resulting cells should approach zero as the number of nodes tends to infinity.” ▪ “Small cells are necessary to obtain more accurate solutions and to investigate phenomena associated with the physical quantities on small scales, such as boundary layers and turbulence.” ▪ “Reducing cell size also enables the study of the convergence rate of a numerical code.” Liseikin (2010, p. 5) 9

Grid Class 2 Fundamental Classes: ▪ Structured grid : regular grid-point topology ▪ Unstructured grid: irregular topology 3 Subclasses: ▪ Block-structured grid ▪ Overset grid ▪ Hybrid grid

Coordinate Grids ▪ Coordinate grids are structured grids in which the nodes and cell faces are defined by the intersection of lines and surfaces of a coordinate system in X n . ▪ The Cartesian and cylindrical grids are examples of coordinate grids. ▪ Nodes of a coordinate grid do not necessarily coincide with the curvilinear boundary of a complex domain. Liseikin (2017, p. 14)

Boundary-Conforming Grids ▪ Boundary-fitted or boundary-conforming grids are structured grids that are obtained from one-to-one transformations x (  ) which map the boundary of the computation domain  n on to the boundary of the physical domain X n . Liseikin (2017, p. 14)

Boundary-Fitted Coordinates Consider the flow through the divergent duct. Let We can use the transformation 13

Boundary-Fitted Coordinates Consider now an airfoil shape with a curvilinear grid wrapped around the shape. C-type and O-type grids 14

Boundary-Fitted Coordinates ▪ “What transformation will cast this curvilinear grid into a uniform grid in the computational plane?” ▪ We know the coordinates ( x , y ) of the inner boundary  1 and outer boundary  2 ▪ This gives us a hint that we can form a boundary- value problem with boundary values specified everywhere. ▪ This BVP can be an elliptic problem! 15

Grid Examples Structured Grid Cartesian Grid Regular Grid Rectilinear Grid Curvilinear Grid Unstructured Grid Image sources: http://en.wikipedia.org/wiki/Regular_grid http://en.wikipedia.org/wiki/Unstructured_grid 16

Block-Structured Grid Grid blocks do not overlap but interfaces can be either continuous or discontinuous. Liseikin (2010, p. 17) 17

Block-Structured Grid Topology Liseikin (2010, p. 18) 18

Overset Grid Grid blocks can overlap. Liseikin (2010, p. 20) 19

Hybrid Grid Various grid types are combined. Liseikin (2010, p. 21) 20

Grid Generation Methods ▪ Mapping methods for structured grids • Algebraic methods: transfinite interpolation • Differential methods: PDE based • Variational methods: optimization based ▪ Unstructured grid • Quadtree/octree methods • Delaunay procedures • Advancing-front techniques Liseikin (2010, p. 22-25)

Mapping Methods ▪ Structured grids are typically generated through a mapping approach in which a computational domain  n of a simple shape is mapped into a physical domain X n . ▪ The most efficient structured grids are boundary- conforming grids typically generated for the finite difference method. ▪ 3 main mapping methods for structured grids are • Algebraic methods use interpolation. • Differential methods are based on elliptic, parabolic, or hyperbolic PDEs. • Variational methods are based on optimization of grid quality properties. Liseikin (2017, p. 29)

Coordinate Transformations ▪ “PDEs in the physical domain X n can be solved on a grid obtained by mapping a reference grid in the logical domain  n into X n with a coordinate transformation” ▪ “The mapping approach provides an alternative way to obtain a numerical solution to a PDE, by solving the transformed equation with respect to the new independent variables  ,  ,  ,  on the reference grid in the logical domain  n .” 23 Liseikin (2017, p. 47)

Coordinate Transformation ▪ “Consider a 2D unsteady flow with independent variables x , y , and t .” ▪ “We will transform the independent variables in physical space ( x , y , t ) to a new set of independent variables in transformed space (  ,  ,  ), where 24

Derivative Terms ▪ In the governing equations, there are many derivative terms with respect to physical space variables that must be transformed due to the coordinate transformation. ▪ Use chain rule to derive these operators: 25

Derivative Terms ▪ Let ▪ Then, ▪ Using the chain rule, the mixed derivative terms can be turned into derivative with respect to only  and  . 26 Anderson (1995, p. 174)

Derivative Terms ▪ The first mixed derivative operator can be rewritten as ▪ The second mixed derivative operator can be rewritten as ▪ We then have 27 Anderson (1995, p. 174)

Derivative Terms ▪ Similarly, we have 28 Anderson (1995, p. 175)

Metrics and Jacobians ▪ Metrics are used to call the derivatives ▪ It may be convenient to use the inverse transformation ▪ We then need to replace metrics by inverse metrics 29 Anderson (1995)

Metrics and Jacobians ▪ Let . Total differential of u is given by ▪ We then have the derivatives 30 Anderson (1995)

Metrics and Jacobians Solving for and yields where J is Jacobian determinant defined as 31 Anderson (1995)

Differential Operators From previous results we have where 32 Anderson (1995)

A Formal Approach Consider a direct transformation We have the total differentials 33 Anderson (1995)

A Formal Approach Now consider an inverse transformation We have the total differentials 34 Anderson (1995)

A Formal Approach We then have 35 Anderson (1995)

Stretch Transformations 36

Stretching Transformations ▪ “Algebraic grid generation may be used in combination with univariate stretching transformations to control grid density.” ▪ “Stretching transformations involves positive monotonic univariate functions, here given by x = x (  ) and y = y (  ), with inverse  =  ( x ) and  =  ( y ) .” 37 Farrashkhalvat and Miles (2003, p. 99)