Adaptive Meshing

AdaptiveMesh(grid; maxlevel)

Adaptively refinable mesh. Use refine_mesh! / coarsen_mesh! and build a new FerriteDiscretization on current_grid after each change.

• Quadrilateral or hexahedral grid: forest of quadtrees/octrees from Ferrite's AMR, at most maxlevel refinement levels (default 10; one level quarters a cell). Refined grids have hanging nodes.
• Linear triangle grid: newest vertex bisection, at most maxlevel bisections per cell (default 20; two bisections quarter a cell). Refined grids are conforming; facet and cell sets are carried over. Coarsening undoes bisections exactly.

Other cell types (tetrahedra, quadratic cells) are accepted with a warning and never refined.

current_grid(am::AdaptiveMesh)

The current (possibly non-conforming) grid.

refine_mesh!(am, cells)
refine_mesh!(am)

Refine the cells cells of the current grid (or all cells) once and restore the 2:1 balance between neighbouring cells. Cell numbers refer to current_grid(am).

coarsen_mesh!(am, cells)

Coarsen: every complete family of sibling cells among cells is merged into its parent. On triangle meshes a family is the two or four cells around a vertex created by bisection (merging them undoes that bisection exactly).

cell_levels(am::AdaptiveMesh)

Refinement level of every cell of current_grid(am) (0 = cell of the initial grid).

max_level(am::AdaptiveMesh)

Maximum refinement level; cells at this level are not refined further. Exclude them from marking (e.g. set their indicator to zero) so that the adaptive loop keeps making progress.

is_nonconforming(grid)

Whether grid has hanging nodes (a non-conforming grid from adaptive refinement).

residual_indicator(disc, fm, σ, X, currents = nothing; mode = :neumann, normalize = false)

Residual-based error indicator of the forward problem, one value η_K² per cell, summed over the patterns (columns of X, the states from forward_neumann or forward_dirichlet):

η_K² = Σₛ h_K² ‖∇⋅(σ∇uₛ)‖²_K + ½ Σ_F h_F ‖[σ∂ₙuₛ]‖²_F + Σ_{F ⊂ ∂Ω} h_F ‖gₛ - σ∂ₙuₛ‖²_F.

The boundary current density gₛ comes from the electrode model: the current density of the continuum model and Iₗ/|eₗ| of the gap model (both need currents), (Uₗ - u)/zₗ of the complete electrode model, zero in the gaps. Point loads are not estimated, and Dirichlet boundaries (mode = :dirichlet) have no boundary term. normalize = true scales every pattern's indicator to unit sum. √Σ_K η_K² bounds the energy error up to a constant; it works on conforming and hanging-node meshes.

flux_recovery_indicator(disc, σ, X; normalize = false)

Zienkiewicz–Zhu error indicator of the current density, one value per cell:

η_K² = Σₛ ∫_K |J*ₛ - σ ∇uₛ|² dx,

where J*ₛ is the L² projection of the discrete current density σ∇uₛ onto continuous linear vector fields. X holds the states of all patterns (n × s, e.g. from forward_neumann; only the first ndofs_u(disc) rows, the potential, are used). Large values mark electrode edges and conductivity interfaces, where the discrete current density jumps. With normalize = true every pattern's indicator is scaled to unit sum before summing, so that low-energy patterns (high spatial frequency) weigh as much as the dominant low-frequency ones.

goal_oriented_indicator(disc, fm, σ, X; estimator = :recovery, currents = nothing,
                        solver = DirectSolver(), normalize = false)

Goal-oriented indicator for the measured voltages, one value per cell:

η_K = η_K(u) η_K(z),   η_K(u)² = Σₛ ‖J*ₛ - σ∇uₛ‖²_K,   η_K(z)² = Σₘ ‖J*(zₘ) - σ∇zₘ‖²_K,

where zₘ are the dual solutions of the measurements (A zₘ = Qᵀ Πᵀ eₘ, the adjoint fields of the Jacobian rows). Both factors are flux_recovery_indicators (estimator = :recovery) or residual_indicators (estimator = :residual, needs the injected currents). The voltage error is bounded by products of primal and dual errors, so cells are refined where errors are made and influence the measurements. X holds the current-driven states (n × s). normalize weighs every pattern and every measurement equally.

In benchmark/adaptive_meshing.jl the residual version is the cheapest effective choice (indicator cost about that of one forward solve with all patterns).

jump_indicator(disc, σ)

Feature indicator of a piecewise constant conductivity, one value per cell: Σ_F |F| |σ_K - σ_K'| / 2 over the facets F of the cell. Refining where it is large resolves inclusion boundaries; cells with zero indicator are candidates for coarsening.

dorfler_marking(η, θ)

Dörfler (bulk) marking: the smallest set of cells, taken in decreasing order of η, whose indicators sum to at least θ times the total.

transfer_conductivity(disc_old, σ_old, disc_new)

Conductivity coefficients on disc_new from σ_old on disc_old (e.g. after refinement or coarsening): the L² projection of the old conductivity onto the new σ space, with the old conductivity evaluated at the quadrature points of the new mesh. For piecewise constants this is exact in both directions: refined cells inherit the value of their parent, coarsened cells get the mean of their children.