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.

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:

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).

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.

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.

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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. |

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.

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.

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.