Composite Finite Element Representations

Table of Contents

• Defining a composite representation
  • Construction from existing ML representations
• Matrix construction
• Storage Layout

The Open FUSION Toolkit supports the construction of composite vectors and matrices from sets of finite element fields. This is primarily used to support Multi-Physics solution where multiple fields are advanced simultaneously. It can also be used to represent vector valued fields with scalar finite elements for each component. A composite FE representation is defined in terms of the sub-field representations through the oft_fem_comp_type class.

Defining a composite representation

For most use cases composite representations can be assmbled by simply linking to the sub-field representations as shown in the example below. The oft_fem_comp_type class requires the number of sub-fields along with a reference to the FE representation and a character tag for each sub-field. The character tags are used when for restart files I/O using the vec_save and vec_load methods.

USE fem_composite, ONLY: oft_fem_comp_type
USE lagrange_basis, ONLY: oft_lagrange
TYPE(oft_fem_comp_type) :: fem_3vec
!---Specify field count
fem_3vec%nfields = 3
ALLOCATE(fem_3vec%fields(fem_3vec%nfields))
ALLOCATE(fem_3vec%field_tags(fem_3vec%nfields))
!---Define field 1 (X-direction)
fem_3vec%fields(1)%fe => oft_lagrange
fem_3vec%field_tags(1) = "x"
!---Define field 2 (Y-direction)
fem_3vec%fields(2)%fe => oft_lagrange
fem_3vec%field_tags(2) = "y"
!---Define field 3 (Z-direction)
fem_3vec%fields(3)%fe => oft_lagrange
fem_3vec%field_tags(3) = "z"

Construction from existing ML representations

If you intend to define and use a multi-level hierarchy for multi-grid this can be done automatically from existing ML FE definitions. The oft_ml_fem_comp_type class can be constructed in a similar way to the oft_fem_comp_type class, but using references to ML FE objects instead of single level definitions. Once defined the setup method is called and the sub-levels are created automatically. An example of this for the 3-field Lagrange vector is shown below.

USE fem_composite, ONLY: oft_ml_fem_comp_type
USE lagrange_basis, ONLY: ml_oft_lagrange
TYPE(oft_ml_fem_comp_type) :: ML_fem_3vec
!---Specify field count
ml_fem_3vec%nlevels = ml_oft_lagrange%nlevels
ml_fem_3vec%nfields = 3
ALLOCATE(ml_fem_3vec%fields(ml_fem_3vec%nfields))
ALLOCATE(ml_fem_3vec%field_tags(ml_fem_3vec%nfields))
!---Define field 1 (X-direction)
ml_fem_3vec%fields(1)%fe => ml_oft_lagrange
ml_fem_3vec%field_tags(1) = "x"
!---Define field 2 (Y-direction)
ml_fem_3vec%fields(2)%fe => ml_oft_lagrange
ml_fem_3vec%field_tags(2) = "y"
!---Define field 3 (Z-direction)
ml_fem_3vec%fields(3)%fe => ml_oft_lagrange
ml_fem_3vec%field_tags(3) = "z"
!---Setup sub-levels
CALL ml_fem_3vec%setup

Matrix construction

Most matrices for composite representations can be defined using the mat_create method as with single field FE representations. For composite representations the mat_create method supports simple masking of sub-matrices to eliminated unnecessary storage when only certain sub-fields have non-zero interaction. The example below illustrates creating a matrix for the 3-field Lagrange vector representation shown above, where sub-fields "y" and "z" do not interact with each other.

USE oft_matrices, ONLY: oft_matrix
INTEGER(i4) :: mask(3,3)
CLASS(oft_matrix), POINTER :: mat
!---Setup graph mask
mask=1      ! Include all blocks by default
mask(2,3)=0 ! Skip block (2,3)
mask(3,2)=0 ! Skip block (3,2)
!---Create block matrix
CALL fem_3vec%mat_create(mat,mask)

Storage Layout

Currently the code creates composite fields by stacking the sub-fields in memory. Although there are instances where an interleaved layout would be desired this method is not amenable to cases where the finite element representations are not homogeneous, ie when Lagrange and Conforming/Nedelec representations are used. This is due to the fact that the elements do share a common node ordering so that there is no way to consistently interleave elements.