NAME

KEYWORDS

SYNOPSIS

DESCRIPTION

pip install "gflex>=2.0.0"

NOTES

FD — Finite Difference

Typically faster for large grids and supports spatially variable elastic thickness via the van Wees and Cloetingh (1994) stencil. Memory requirements scale with grid size; choose a grid spacing appropriate for the flexural wavelength of the problem. Supports all boundary conditions and in-plane stresses.

FFT — Fast Fourier Transform (spectral)

Requires a scalar (uniform) elastic thickness. Supports in-plane stresses. Fastest option for large uniform-T_e grids. Each opposite edge pair (west/east, north/south) is handled independently: setting both sides of a pair to periodic makes that axis exactly periodic; any other value (including the default infinite) zero-pads that axis to approximate no_outside_loads. Mixed-axis configurations are valid.

SAS — Superposition of Analytical Solutions

Each grid cell is treated as a point load and its deflection computed analytically. Best for smaller grids or when accuracy near individual loads matters. Requires scalar elastic thickness. Boundary conditions are not applicable; the solver inherently assumes an infinite plate with no outside loads.

The flexural solution is generated for the current computational region, so be sure to check g.region before running the module.

input is a 2-D raster of loads in units of stress [Pa], equal to the load material density times gravitational acceleration times column thickness. This is computed by the user prior to running the module, and is not affected by the g parameter below (which applies only to the restoring isostatic force).

te, written in standard notation as T_e, is the lithospheric elastic thickness. It may be either a scalar value or (for FD only) a raster of spatially variable elastic thickness.

Boundary conditions apply to the FD and FFT solvers. SAS assumes the plate extends to infinity with no outside loads and ignores boundary condition settings.

The boundary conditions are:

infinite — gFlex canonical name: no_outside_loads

The domain is automatically extended on that side by one flexural wavelength with zero loads, the solve is performed on the enlarged domain, and the deflection is trimmed back to the original region before returning. For variable-T_e FD runs, the elastic thickness is also smoothly tapered across the padding ring to suppress spurious deflections from abrupt rigidity gradients at the domain edge (via the D-derivative terms in the van Wees & Cloetingh 1994 stencil). For FFT, the load array is zero-padded by fft_pad_n_alpha × α per side (default 4α; α = (4D/Δρg)^1/4) and trimmed internally. This is the most physically realistic choice for isolated loads far from any genuine plate edge, and is the default.

clamped — gFlex canonical name: zero_displacement_zero_slope

Displacement w = 0 and slope dw/dx = 0 (or dw/dy = 0) at the boundary. The plate is rigidly attached to a fixed wall: it can neither deflect nor rotate at the edge. Use when the domain boundary coincides with a stable craton or rigid block that prevents deformation, or as a far-field approximation when the load is many flexural wavelengths from the edge. Note: if any load is within one flexural wavelength of a clamped boundary, the forebulge may be artificially suppressed; gFlex will issue a warning in that case.

pinned — gFlex canonical name: zero_displacement_zero_moment

Displacement w = 0 and bending moment M = D d²w/dx² = 0 at the boundary. The plate cannot deflect at the edge but is free to rotate; no bending moment is transmitted across the boundary. Corresponds to a simply-supported (pinned) end in structural mechanics. Use when the plate rests on a rigid foundation at its edge without being clamped to it.

free — gFlex canonical name: zero_moment_zero_shear

Bending moment M = 0 and shear force V = D d³w/dx³ = 0 at the boundary. The plate end is entirely free: no moment and no transverse force are transmitted. Equivalent to the free end of a cantilever (diving board). Use for rifted or passive continental margins where the lithosphere has a broken, unsupported edge, for subduction trenches with an applied edge load, and for broken-plate flexure problems.

mirror — gFlex canonical name: zero_slope_zero_shear

Slope dw/dx = 0 and shear force V = 0 at the boundary. Mathematically equivalent to reflecting the load field (and, for variable-T_e runs, the elastic thickness field) across the boundary, so the full domain is treated as the symmetric half of a larger problem. Use when the loading geometry and lithospheric structure are symmetric about the domain edge, for example along the axis of a mid-ocean ridge or the crest of a symmetric mountain belt.

periodic (FD: matched pairs only; FFT: per opposite-edge pair)

The opposite edges are treated as adjacent: the east edge connects to the west edge and/or the north edge connects to the south edge. For FD, periodic must be applied in matched pairs (both east–west or both north–south); a one-sided periodic BC will produce a warning. For FFT, each axis pair is handled independently: setting both sides of a pair to periodic makes that axis exactly periodic, while leaving the other pair at infinite (the default) zero-pads that axis.

Boundary conditions may be mixed freely across the four sides, except that periodic must be applied in matched pairs. To minimize boundary effects with explicit BCs, place domain edges at least one flexural wavelength from the nearest load, or use infinite to let gFlex extend the domain automatically.

In-plane stresses (sigma_xx, sigma_yy, sigma_xy) add membrane (tectonic) stress terms to the governing equation, following the full variable-D formulation. These are supported by the FD and FFT solvers and default to 0 (no in-plane stress). Positive values are tensional.

r.flexure may be run in latitude/longitude coordinates with the -l flag. Because the module requires a single dx and dy, it computes dx at the midpoint latitude of the domain. This approximation degrades near the poles.

EXAMPLES

Asymmetric volcanic load on a laterally heterogeneous lithosphere

This example is based on the scenario in the gFlex documentation: a circular volcanic edifice on a lithosphere whose elastic thickness rises from 15 km in the west to 35 km in the east across a sigmoid transition. The thinner western lithosphere deflects more steeply and over a shorter wavelength; the stiffer eastern side produces a broader depression with a more prominent forebulge.

# Domain: 750 × 750 km at 5 km resolution (projected CRS required)
g.region n=750000 s=0 e=750000 w=0 res=5000

# Sigmoid elastic thickness: 15 km (west) to 35 km (east)
# tanh is unavailable in r.mapcalc; use the equivalent logistic form:
#   0.5*(1 + tanh(u)) = 1/(1 + exp(-2u)), u = (x - cx)/(8*dx)
r.mapcalc expression="te_sigmoid = 15000 + 20000 / (1 + exp(-(x()-375000) / 20000))"

# Circular volcanic load (30 MPa) within 61.3 km of the domain centre
# 30 MPa ≈ 1055 m of mantle-density rock; radius is 1.5 × the flexural parameter α
r.mapcalc expression="load_volcano = if(sqrt((x()-375000)^2+(y()-375000)^2) <= 61300, 30000000.0, 0)"

# FD flexural deflection; default infinite BCs approximate an unbounded plate
r.flexure method=fd \
    input=load_volcano \
    te=te_sigmoid te_units=m \
    output=w_volcano

d.rast w_volcano
d.legend w_volcano

The result shows roughly 54 m of central subsidence and about 1 m of forebulge uplift. The depression is deeper and narrower on the soft (thin T_e) western side and broader with a clearer forebulge on the stiff eastern side.

For a broken-plate scenario (e.g. a passive margin or oceanic trench) set the relevant edge to free:

r.flexure method=fd \
    input=load_volcano \
    te=te_sigmoid te_units=m \
    output=w_volcano_free_south \
    southbc=free

SEE ALSO

REFERENCES

van Wees, J. D., and S. Cloetingh (1994), A Finite-Difference Technique to Incorporate Spatial Variations In Rigidity and Planar Faults Into 3-D Models For Lithospheric Flexure, Geophysical Journal International, 117(1), 179–195, doi:10.1111/j.1365-246X.1994.tb03311.x.

AUTHOR

SOURCE CODE

Available at: r.flexure source code (history)

Latest change: Wednesday Jun 17 14:05:16 2026 in commit: 2b69c1e5403d2a3377c287af027fcbad020a088c

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