Coordinate Systems and Transformations

In this chapter, the coordinate systems and the relations between them, as well as the coordinate transformations are described.

Coordinate systems and related transformations are of primary importance to the understanding of the basic mechanisms of B2000++. The description of meshes in B2000++ is substructure (branch) oriented. A B2000++ mesh can consist of an arbitrary assemblage of smaller meshes collected in branches. However, the branches themselves may not contain other branches. In this sense, the branches can be seen as macro-elements composed of the individual finite elements. To provide generality, the properties of all components are defined upwards context-free, meaning that they are described without any reference to entities higher up in the hierarchy of structural elements. Hence, geometry and other properties of the finite elements can be described independently of the branch they are embedded in. Equally, branches are described irrespective of topology and connectivity of the structure they help to synthesize. It is only on the highest level that a user must specify how the total structure has been modelled and how the branches are connected with the join command.

1. Coordinate systems

B2000++ employs a variety of coordinate systems for the description of relevant quantities. These coordinate systems are independent of any system which might be used inside any element formulation. Their main purpose is to unify the flow and exchange of information between various modules, and to standardize the way in which data are accepted, interpreted and produced by participating functional units. In addition, information can be stored in any of a number of formats suitable for a specific application. Whereas the specification of a coordinate system provides a frame of reference for the description of mathematical quantities, the format describes the logical structure imprinted on the data, e.g. wether in a vector all declared degrees of freedom are mentioned or only those which appear in a particular equation system. The figure below illustrates the five coordinate systems:

1.1. Global coordinate system

The global coordinate system is a Cartesian coordinate system used for describing the geometry of the structure as a whole. Its main purpose is to define how the various branches in a structure are joined together (global geometry). In some instances, global- global coordinates are used as a reference system for the force and displacement vectors as well as for graphical representation. The global coordinate system should not to be confused with the global computational coordinate system (described below).

1.2. Branch coordinate system

The geometry of each branch is described in a branch related Cartesian coordinate system. The branch coordinate system is used to define the shape of the surface. In some cases, it also serves as a basis for describing the forces and displacements as well as for specifying boundary conditions.

1.3. Node local coordinate system

Node related properties such as forces (natural boundary conditions), displacements (DOF's), or boundary conditions (essential boundary conditions) at given node points are described in a node related local coordinate system whose orientation is determined by a local base relative to the branch coordinate system. By default the local coordinate system is identical to the branch coordinate system.

Note Although natural boundary conditions, such as forces and essential boundary conditions, such displacement constraints, are formulated with respect to the node-local coordinate system, B2000++ will compute relevant output quantities, such as force vectors or displacement vectors, in the branch coordinate system. Example: A force vector (1, 0, 0) at a specific node specified in the MDL input file will not translate to (1, 0, 0) in the FORC vector of the database if the node-local coordinate system is not identical to the branch coordinate system, because the MDL specification is with respect to the node-local system, and FORC with respect to the branch system.

1.4. Element coordinate system

The element (local) coordinate system defines a reference frame for formulating certain quantities, such as composite ply angles, strains, or stresses. B2000++ makes assumptions on how the element coordinate system is defined by means of the element node connectivity indices, see Element Naming and Numbering Conventions.

1.5. Material coordinate system

Certain materials, like orthotropic or anisotropic materials, need to be formulated with respect to an own coordinate system called the material coordinate system. The mbase element property attribute defines the material coordinate system.

1.6. Computational coordinate system

The computational coordinate system is the coordinate system in which the computational degrees of freedoms of a node are expressed. The term computational refers to the degrees of freedom which are used in the final equation system and which express the relation between displacements and applied forces, i.e. to the unknowns used in the computational process.

On the nodal level, the global computational coordinate system is frequently identical with the node local coordinate systems. This is certainly the most convenient alternative for shell structures (with the exception of points located on a juncture line between several substructures). Also for 3D structures the global computational and the node local systems usually coincide, or can be made to coincide with little difficulty.