CSE351/ IT351Modeling And Simulation

Choosing a

Mesh Model

Dr. Jim Holten

Choose a Model Object Representation

Finite Elements? (Pick from a zoo of predefined cell element types)

Regular Polyhedral Mesh? (Homogeneous mesh cells)

General Polyhedral Mesh? (Hierarchy of generalized cells)

Any combination of the above?

Model Cell Criteria

Cell shapes?Cell properties (attributes)?Cell associations with neighbor cells?Cell associations with cells in other cell sets?Complexities of developing supporting code?

Finite ElementCell Types

1-D: Line, interpolated line, spline2-D: Triangle, rectangle, trapezoid, circle, ellipse, interpolated shape variations3-D: Tetrahedral, hexahedral, spherical, ellipsoidalSpecial types: Springs, shock absorbers, circuit components, other custom variations

Finite ElementAttributes

Vertex coordinates (how many, what ordering)Edges of a face?Faces of a 3-D “zone”?Adjacent element shared faces, edges, vertices?Element volume, surface, face shapes?Attributes over each of these?

Why UseFinite Elements?

Known common toolbox of elements (cell types).Only a limited number of element (cell) types are needed.Element interconnects well-defined and easily applied.Problem at hand makes other approaches difficult.

Why NOT UseFinite Elements?

The needed element set (cell type) implementation is not already available.The element set may need to be expanded.May need to interface with models with other element types.A generalized approach would reduce coding needs. -- N finite element types gives N^2 interfaces that may need implemented.Parallel partitioning is complex and usually done by hand.

Regular PolyhedralCell Types

Limited to “regular” shapes that will cover a “region”.1-D: No problem.2-D: Triangles, quadrangles, and hexagons only.3-D: Hexahedrals only.Does not cover irregularly shaped model objects/parts well.

Regular Polyhedral CellAttributes

Neighbor interfaces are usually indicated through shared vertices, edges, and/or faces.Index functions relate neighbors as in A(i,j) next to A(i, j-1).Index functions relate cell hierarchies (points, edges, faces, and zones)

Why UseRegular Polyhedral Cells?

All cell attributes may be stored as simple vectors and arrays, using the corresponding indices of their parent cell.Neighbors are easy to access (index functions).Cell component hierarchies are easy to access (index functions).The common cell forms used limits the algorithm variations needed.

Why NOT UseRegular Polyhedral Cells?

A regular “grid” doesn't fit the modeled part.The model may be in several “parts” (subsets) of interest, such as interiors, surfaces, and boundary layers.The model may need deformed or refined, and becomes incompatible with a regular grid representation.

General PolyhedralCell Types

A hierarchy of cells (nodes, edges, faces, and zones).0-D: node (point) has a location (usually)1-D: edge (line) connects two end points (nodes).2-D: face is surrounded by edges.3-D: zone is surrounded by faces.Fully generalized polyhedral shapes, allowing extreme shape representation.

General PolyhedralCell Attributes

Nodes may have associated coordinates.Other attribute values may be associated with any cell type and cell type element subsets, as needed.Neighbor cells may be found through common (shared) faces, edges, or vertices.

Why UseGeneral Polyhedral Cells

May approximate most shapes.May be refined by substituting other cells for existing cells.Relationships (subsetting, cell associations, etc) may be generalized to fit most needs.

Why NOT UseGeneral Polyhedral Cells

No “special shapes” such as spheres, ellipsoids, cones, circles, etc.Less rigidly organized, so representations are less standardized than the others.More generalized algorithms can be slower or require more “control flags”.

Sets, Relations, and Fields(A metadata wrapper)

Set – any collection of membersRelation – a mapping as a collection of tuples relating members of its domain set to members of its range set.Field – a collection of values (of a homogeneous type) associated with each member of its domain set.

Sets, Relations, and Fields

Set -- Indexed sets of elements N elements Reference members by their index in the set

Relation -- Indexed mappings between sets are similar to sparse adjacency matrices

Field – Homogeneous collection of attribute values (properties), one for each set member

Sets, Relations, and Fields(A metadata wrapper)

May represent any combination of cells from any of the other three approaches.Can easily encapsulate or replace the representations of each of them.Uses well-defined algebraic operations for relation, set, and fields to describe association derivations and time-step behaviors.

Sets, Relations, and FieldsCell Attributes

A “field” is a collection of attribute values, one for each member of the parent set and can contain property metadata such as data type, units, and representation forms.A field “value” may be a simple real (float or double), integer, enumerated variable value, string, vector, matrix, tensor, or specialized structure.A set may have one or many fields over it. Multiple fields can define all the attributes for a homogeneous cell set.

Why UseSets, Relations, and Fields

Allows mixing mesh model types (and other elements types) in a single model.Included metadata can prevent data misinterpretations. Objects, their relationships, and their behaviors can all be described using the SRF algebra.SRF Algebra can easily define and be implemented to automatically project a parallel partitioning onto complex mesh combinations, including generating all communications maps.

Why NOT UseSets, Relations, and Fields

In early stages, so there aren't many developers familiar with it.Inherits some drawbacks from the mesh forms you choose to include in your model.

Wide Application

Particle cloud, flock, and herd models

Generalized object collections

Mesh models, including cell hierarchies

Commonality of representation allows generalization of operators and I/O

Standardized higher level structure representation allows easier understanding

“Field algebras” are possible.

Sets, Relations, and FieldsParallel Partitioning

Partitioning is just relational operations on sets

Arbitrary numbers of partitions are possible

Simplifies load balance calculations

Enables dynamic partitioning (on-the-fly, as needed load migration)

Partitioning Matrix Mesh Models

Divide each matrix “index axis”: (the i and j are the axes in A(i, j)Each axis gets an integer “divisor”, xd or ydThe number of partitions is (xd * yd)The matrix is subdivided along the “index axes”All vectors need subdivided into

“local” data (used only in this partition) “shared” data (must be made available to other

partitions)