Lattice transport elements

Lattice transport elements

These are two-node lattice transport elements intended for coupled and stand-alone mass- or heat-transport analyses on Delaunay/Voronoi lattice networks. Each node carries one pressure (or temperature) degree of freedom. Flow along the element follows a linear pressure interpolation; the cross-section polygon is the dual Voronoi facet and enters strength/stiffness-like quantities through polycoords.

The element assembles a capacity matrix that can optionally be switched from the default consistent form to a diagonal (row-sum-preserving) lumped form. Lumping makes the scheme equivalent to a two-point-flux-approximation (TPFA) finite-volume discretisation on the Voronoi dual and is recommended when the capacity \(c(p)\) is strongly nonlinear (e.g. drying, Richards-type problems), where the consistent form is known to produce pressure oscillations and an apparent loss of conductivity near sharp fronts. Lumping is opt-in via lumpedcapacity; the default preserves existing behaviour.

latticemt2d element

Two-node 2D lattice mass-transport element. The element features are summarised in Table Table 80.

Table 80 latticemt2d element summary

Keyword	latticemt2d
Description	2d lattice mass-transport element
Specific parameters	thick (rn) width (rn) gpcoords (ra) [dim (rn)] [crackwidth (rn)] [couplingflag (in)] [couplingnumber (in)] [lumpedcapacity (in)]
Parameters	thick: out-of-plane thickness. width: Voronoi-facet width (cross-section length). gpcoords: integration-point coordinates in the global system. dim: dimension factor (optional, default 2). crackwidth: crack width used in conductivity scaling (optional). couplingflag: flag (optional, default 0) activating coupling with a mechanical lattice element. couplingnumber: number of the coupled mechanical element. lumpedcapacity: flag (optional, default 0). If set to 1, the capacity matrix is replaced by its diagonal (row-sum-preserving) lumped form, yielding a TPFA-equivalent, monotone scheme. Recommended for strongly nonlinear \(c(p)\).
Unknowns	Single dof (P_f - pressure / moisture) per node.
Approximation	Linear pressure along the element.
Integration	One Gauss point at gpcoords.
CS properties	simplecs.
Status	Reliable
Tests/Examples	lm/lattice2dcrackinput.in, lm/latticetransmat.in

latticemt3d element

Two-node 3D lattice mass-transport element. The cross-section is a polygonal Voronoi facet supplied via polycoords. The element features are summarised in Table Table 81.

Table 81 latticemt3d element summary

Keyword	latticemt3d
Description	3d lattice mass-transport element
Specific parameters	polycoords (ra) [dim (rn)] [area (rn)] [mlength (rn)] [crackwidths (ra)] [couplingflag (in)] [couplingnumber (ia)] [lumpedcapacity (in)]
Parameters	polycoords: coordinates of the mid-cross-section (Voronoi facet) vertices in the global system. dim: dimension factor (optional, default 3). area: prescribed cross-section area (optional; otherwise computed from polycoords). mlength: minimum element length threshold (optional). crackwidths: per- vertex crack widths used in conductivity scaling (optional). couplingflag: flag (optional, default 0) activating coupling with a mechanical lattice element. couplingnumber: array of coupled mechanical element numbers. lumpedcapacity: flag (optional, default 0). If set to 1, the capacity matrix is replaced by its diagonal (row-sum-preserving) lumped form, yielding a TPFA-equivalent, monotone scheme. Recommended for strongly nonlinear \(c(p)\).
Unknowns	Single dof (P_f - pressure / moisture) per node.
Approximation	Linear pressure along the element.
Integration	One Gauss point at the element midpoint.
CS properties	simplecs.
Status	Reliable
Tests/Examples	lm/lattice3d_mt2.in, lm/lattice3d_mt1.in

latticemt3dboundary element

Three-noded 3D lattice mass-transport element for boundaries of 3D periodic cells. The first two nodes carry the single pressure / moisture dof as in latticemt3d; the third is a control node that supplies the macro gradients driving the periodic cell. The dof of the node that lies outside the periodic cell is reconstructed from its periodic image inside the cell plus the control-node dofs, using the same translation pattern as lattice3dboundary (Fig. 35). The element features are summarised in Table 82.

Table 82 latticemt3dboundary element summary

Keyword	latticemt3dboundary
Description	3d lattice mass-transport boundary element
Specific parameters	polycoords (ra) location (ia) [dim (rn)] [area (rn)] [mlength (rn)] [crackwidths (ra)] [couplingflag (in)] [couplingnumber (ia)] [lumpedcapacity (in)]
Parameters	polycoords: coordinates of the mid-cross-section (Voronoi facet) vertices in the global system. location: array of two integers between 1 and 26 specifying the location of the two transport nodes with respect to the 3D periodic cell — same encoding as lattice3dboundary. dim: dimension factor (optional, default 3). area: prescribed cross-section area (optional; otherwise computed from polycoords). mlength: minimum element length threshold (optional). crackwidths: per- vertex crack widths used in conductivity scaling (optional). couplingflag: flag (optional, default 0) activating coupling with a mechanical lattice element. couplingnumber: array of coupled mechanical element numbers. lumpedcapacity: flag (optional, default 0). Replaces the capacity matrix with its diagonal lumped form for monotone TPFA-equivalent behaviour.
Unknowns	Single dof (P_f) per transport node. The third (control) node carries the periodic-cell dofs as for lattice3dboundary but reduced to the transport-relevant components.
Approximation	Linear pressure along the element.
Integration	One Gauss point at the element midpoint.
CS properties	simplecs.
Status	Reliable
Tests/Examples	lm/lattice3d_mt2.in