I am working on implementing periodic boundary conditions in a finite element problem and I want to pair the nodes on boundary A with nodes on boundary B given a vector trans
that lines boundary A up with boundary B. The nodes in boundary A are given in a list g1
; in B they are g2
. The node coordinates are looked up in mesh%nodes(:,nodenum)
.
I have decided to do this by creating a distance matrix for each node, which I realise is not the most efficient way, and to be honest I don't expect to save significant time by optimising this algorithm. The question is more academic.
I know that Fortran stores in column-major order, on the other hand, the array will be symmetric and when the array is completed I want to take column slices of it to find the nearest node. So the question is how should one populate this?
Here is my naive attempt.
subroutine autopair_nodes_in_groups(mesh, g1, g2, pairs, trans)
type(meshdata) :: mesh
integer(kind=sp) :: i,j
integer(kind=sp),dimension(:) :: g1,g2
integer(kind=sp),dimension(:,:) :: pairs
real(kind=dp) :: trans(3) !xyz translate
real(kind=dp) :: dist_mat(size(g1),size(g2))
real(kind=dp) :: p1(3), p2(3)
dist_mat = -1.0_wp
! make a distance matrix
do j=1,size(g2)
p2 = mesh%nodes(1:3,g2(j))-trans
do i=1,j
p1 = mesh%nodes(1:3,g1(i))
dist_mat(i,j) = norm2(p1-p2) !equivalent to norm2(n1pos trans-n2pos)
if (i.ne.j) dist_mat(j,i) = dist_mat(i,j) !fill symmetry
end do
end do
! Remainder of routine to find nearest nodes
end subroutine autopair_nodes_in_groups
The problem as far as I can tell is that this is efficient in terms of memory access until one symmetrises the array.
CodePudding user response:
To do a fast nearest-neighbor search, you should implement a tree structure that has search complexity O(log(N)) instead of looking at all point-to-point distances which are O(N^2).
Anyways, regarding symmetric matrix handling, you'll have:
! Storage size of a symmetric matrix
elemental integer function sym_size(N)
integer, intent(in) :: N
sym_size = (N*(N 1))/2
end function sym_size
! Compute pointer to an element in the symmetric matrix array
elemental integer function sym_ptr(N,i,j)
integer, intent(in) :: N,i,j
integer :: irow,jcol
! Column-major storage
irow = merge(i,j,i>=j)
jcol = merge(j,i,i>=j)
! Skip previous columns
sym_ptr = N*(jcol-1)-((jcol-1)*(jcol-2))/2
! Locate in current column
sym_ptr = sym_ptr (irow-jcol 1)
end function sym_ptr
then do your job:
N = size(g2)
allocate(sym_dist_mat(sym_size(N)))
do j=1,size(g2)
p2 = mesh%nodes(1:3,g2(j))-trans
do i=j,size(g2)
p1 = mesh%nodes(1:3,g1(i))
sym_dist_mat(sym_ptr(N,i,j)) = norm2(p1-p2)
end do
end do
The minloc function should then look something like this (untested):
! minloc-style
pure function symmetric_minloc(N,matrix,dim) result(loc_min)
integer, intent(in) :: N
real(8), intent(in) :: matrix(N*(N 1)/2)
integer, intent(in) :: dim
real(8) :: dim_min(N),min_column
integer :: loc_min(N)
select case (dim)
! Row/column does not matter: it's symmetric!
case (1,2)
dim_min = huge(dim_row)
loc_min = -1
ptr = 0
do j=1,N
! Diagonal element m(j,j)
ptr=ptr 1
min_column = matrix(ptr)
if (min_column<=dim_min(j)) then
loc_min(j) = j
dim_min(j) = min_column
end if
! Lower-diagonal part at column j
do i=j 1,N
ptr=ptr 1
! Applies to both this column,
if (matrix(ptr)<=dim_min(j)) then
loc_min(j) = i
dim_min(j) = matrix(ptr)
end if
! And the i-th column
if (matrix(ptr)<=dim_min(i)) then
loc_min(i) = j
dim_min(i) = matrix(ptr)
end if
end do
end do
case default
! Invalid dimension
loc_min = -1
end select
end function symmetric_minloc