Basic data types, constants, structures
FEAT2 provides a set of data types, constants and basic structures. The following list contains an overview about these data types, constants and the most important high-level structures which are necessary for the discretisation.
Data types (fsystem.f90)
Fortran 90 defines a set of standard data types:
| Name | data type |
|---|---|
| integer | Standard integer type |
| real | Standard float type |
| complex | Standard complex type |
| character | A character value |
| logical | A boolean value, only .true. or .false. |
The module fsystem.f90 declares a set of additional data types which are frequently used throughout the whole FEAT2 library and in the applications. The following table contains a brief overview about these types:
| Name | data type |
|---|---|
| integer(I8) | 8-bit integer |
| integer(I16) | 16-bit integer |
| integer(I32) | 32-bit integer |
| integer(I64) | 64-bit integer |
| real(SP) | Single precision, corresponds to float in C/C++ |
| real(DP) | Double precision, corresponds to double in C/C++ |
| real(QP) | Quad precision, corresponds to long double in C/C++ |
| complex(SP) | Single precision complex, corresponds to float complex in C/C++ |
| complex(DP) | Double precision complex, corresponds to double complex in C/C++ |
| complex(QP) | Quad precision complex, corresponds to long double complex in C/C++ |
Constants (fsystem.f90)
The module fsystem.f90 defines a set of constants for all relevant data types that describe the set of possible values:
| Name | Description |
|---|---|
| SYS_EPSREAL_SP | Minimal difference to unity for real values |
| SYS_EPSREAL_DP | |
| SYS_EPSREAL_QP | |
| SYS_MINREAL_SP | Minimal positive values for real variables |
| SYS_MINREAL_DP | |
| SYS_MINREAL_QP | |
| SYS_MAXREAL_SP | Maximal values for real variables |
| SYS_MAXREAL_DP | |
| SYS_MAXREAL_QP | |
| SYS_MAXINT | Maximal values for integer variables |
| SYS_MAXI8 | |
| SYS_MAXI16 | |
| SYS_MAXI32 | |
| SYS_MAXI64 |
Of mathematical importance are furthermore the following constants:
| Name | Description |
|---|---|
| SYS_PI | Mathematical constant Pi; double precision |
| SYS_INFINITY_SP | Internal constant for infinity |
| SYS_INFINITY_DP | |
| SYS_INFINITY_QP |
For string handling, fsystem.f90 defines a set of default string length constants:
| Declaration | Description |
|---|---|
| character(LEN=SYS_STRLEN) | Declares a string with standard string length |
| character(LEN=SYS_NAMELEN) | Declares a string used for identifiers, names, etc. |
Mesh and domain
t_triangulation (
triangulation.f90)- Encapsules a mesh in 1D, 2D and/or 3D
- Simple meshes can be created using the routines in
meshgeneration.f90, e.g.meshgen_rectangular2DQuadMesh: Creates a regular 2D mesh with quadrilaterals.
- Extended meshes can be imported from
.TRIfiles.
t_boundary (
boundary.f90)- Encapsules analytical descriptions of complex boundaries
- Currently used only in 2D
- Used to adapt a mesh to the boundary during refinement
- Cal be imported from
.PRMfiles
t_meshregion (
meshregion.f90)- Encapsules parts of a mesh, e.g., a set of cells or vertices
- Used to collect boundary vertices during the generation of boundary conditions
t_boudaryRegion (
boundary.f90)- Encapsules parts of a domain, e.g., boundary segments
- Used to in the definition of boundary conditions to mark parts of the boundary
Discretisation
t_scalarDiscretisation (
spatialdiscretisation.f90)- Encapsules a finite element space $V_h$ over a mesh, e.g., $Q_1$
- Contains links to a mesh, transformation formula and (possibly) a description of the boundary
t_blockDiscretisation (
spatialdiscretisation.f90)- Encapsules tensor products of finite element spaces, e.g., $X_h = V_h \times V_h$ with $V_h$ a finite element space like $Q_1$
- Contains links to an underlying mesh and (possibly) a description of the boundary
- Realised as a list of
t_scalarDiscretisationstructures, one for each FE space. - Access to the subspaces is possible via
(t_blockDiscretisation)%RspatialDiscr(:)
All equations are realised based on t_blockDiscretisation. A Poisson equation for example is described by a t_blockDiscretisation structure with only one block, which results in a 1x1 block matrix in the linear system of equations.
Matrices/Vectors
Corresponding to t_scalarDiscretisation and t_blockDiscretisation, there exist "scalar" and "block" versions of matrices and vectors:
t_vectorScalar (
linearsystemscalar.f90)- Encapsules the degrees of freedom (DOFs) of a finite element space $V_h$
- Contains links to a
t_scalarDiscretisationstructure describing the finite element space $V_h$ - Linear algebra on such vectors is provided by the routines in
linearsystemscalar.f90
t_matrixScalar (
linearsystemscalar.f90)- Describes a discrete operator $A_h : V_h \to W_h$ for $V_h$ and $W_h$ finite element spaces
- Contains links to the finite element trial space V_h and to the finite element test space $W_h$
t_vectorBlock (
linearsystemblock.f90)- Encapsules the degrees of freedom (DOFs) of a tensor product space X_h, e.g., $X_h = V_h \times V_h$
- Realised as a list of
t_vectorScalarstructures, i.e., as a list of "scalar" vectors. Every scalar subvector is a "block" in the block vector. - Access to the blocks is possible via
(t_vectorBlock)%RvectorBlock(:) - Linear algebra on such vectors is provided by the routines in
linearsystemblock.f90
t_matrixBlock (
linearsystemblock.f90)- Describes an operator $A_h : X_h \to Y_h$ between finite element tensor product spaces $X_h$ and $Y_h$
- Contains links to the underlying trial space $X_h$ and test space $Y_h$
- The block matrix is realised as 2D array of "scalar matrices"
t_matrixScalar, one submatrix for every combination of the finite element spaces in $X_h$ and $Y_h$. - Access to the submatrices is possible via
(t_matrixBlock)%RmatrixBlock(:,:)
Cubature
Cubature formulas are defined as constants CUB_xxxx declared in cubature.f90. For every type of cell (triangle, quad, hex,...), there exist a set of cubature formulas. To simplify working with them, a special cubature structure is provided to the assembly routines:
t_scalarCubatureInfo (
spatialdiscretisation.f90)- Encapsules all cubature formulas to be used during an assembly. (-> Triangle-Formulas for triangles, Quad-formulas for quads,...)
- Provides the use of "automatic" cubature formulas which automatically adapt themself depending on the cell type.
Example: At_scalarCubatureInfois typically set up with the cubature formulaCUB_GEN_AUTO_G3(seecubature.f90) which means:- Use 3x3 Gauss formula for quads in 2D
- Use 3-point Gauss formula for triangles in 2D
- Use 3x3x3 gauss formula for hexas in 3D
- ...
- Currently, there is no "block" variant available. In "block assembly" routines (which assemble a complete matrix), the same cubature formula is used for all blocks.
Boundary conditions
- t_discreteBC (
discretebc.f90)- Encapsules a discrete version of the boundary conditions of the system
- Typically used for Dirichlet boundary conditions
Linear solver
t_linsolNode (
linearsolver.f90)- Encapsules a linear solver
- The solver can be simple (Gauss elimination) or very complex (multigrid with smoothers, etc.)
Postprocessing
t_ucdExport (
ucd.f90)- Provides an export structure for writing
.VTKfiles which can be postprocessed with "Paraview" - Created based on a mesh. Finite element solution vectors can be attached. At the end, a postprocessing file can be written.
- Provides an export structure for writing