Model:GFlex: Difference between revisions

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|Source code availability=Through CSDMS repository
|Source code availability=Through CSDMS repository
|Source web address=
|Source web address=
|ViewVC web address=http://csdms.colorado.edu/viewvc/flexure/
|Program license type=GPL v3
|Program license type=GPL v3
|Program license type other=
|Program license type other=

Revision as of 13:52, 26 September 2012



GFlex


Metadata

Also known as
Model type Single
Model part of larger framework
Note on status model
Date note status model
Incorporated models or components:
Spatial dimensions 2D
Spatial extent Continental, Regional-Scale, Landscape-Scale
Model domain
One-line model description Direct 2D finite difference solution of lithospheric flexure
Extended model description Finite difference solution allows for calculations of flexural response in regions of variable elastic thickness / flexural rigidity. The direct solution technique means that it takes time to populate a cofactor matrix, but that once this has been done, flexural solutions may be obtained rapidly via a Thomas algorithm. This makes it less good for an individual solution where an iterative approach may be more computationally efficient, but better for modeling where elastic thickness does not change (meaning that you do not need to create a new cofactor matrix) but loads do.
Keywords:

lithospheric flexure,

Name Andy Wickert
Type of contact Model developer
Institute / Organization University of Colorado
Postal address 1 Department of Geological Sciences, UCB 399
Postal address 2 2200 Colorado Ave
Town / City Boulder
Postal code 80309
State Colorado
Country US"US" is not in the list (Afghanistan, Albania, Algeria, Andorra, Angola, Antigua and Barbuda, Argentina, Armenia, Australia, Austria, ...) of allowed values for the "Country" property.
Email address wickert@colorado.edu
Phone
Fax


Supported platforms
Unix, Linux
Other platform
Programming language

Python

Other program language
Code optimized Single Processor
Multiple processors implemented
Nr of distributed processors
Nr of shared processors
Start year development 2010
Does model development still take place? Yes
If above answer is no, provide end year model development
Code development status
When did you indicate the 'code development status'?
Model availability As code
Source code availability
(Or provide future intension)
Through CSDMS repository
Source web address
Source csdms web address
Program license type GPL v3
Program license type other
Memory requirements
Typical run time Minutes to populate cofactor matrix, ~1 second for solution


Describe input parameters
  • Elastic thickness map (ASCII)
  • Load map (ASCII)
  • dx, dy
  • Material properties
    • Young's modulus
    • Poisson's ratio
    • density of load
    • density of infilling material (optional; this can also be done via iteration for more complicated situations

Only the elastic thickness and load need to be actual input files. The rest (scalars) can be specified at the command line interface.

Input format ASCII
Other input format
Describe output parameters Cofactor matrix (*.mtx sparse matrix file; ASCII)

Flexural response map (ASCII)

Output format ASCII
Other output format
Pre-processing software needed? No
Describe pre-processing software
Post-processing software needed? No
Describe post-processing software
Visualization software needed? No
If above answer is yes
Other visualization software


Describe processes represented by the model Response of a lithospheric plate, potentially with nonuniform elastic thickness, to an applied surface load
Describe key physical parameters and equations [[Describe key physical parameters::See:#Key physical parameters and equations]]
Describe length scale and resolution constraints Insufficiently tested to know.
Describe time scale and resolution constraints Currently does not time-evolve. I would like to couple this to a 3D viscoelastic mantle at some point, but this hasn't happened yet.
Describe any numerical limitations and issues Insufficiently tested to know.


Describe available calibration data sets
Upload calibration data sets if available:
Describe available test data sets
Upload test data sets if available:
Describe ideal data for testing


Do you have current or future plans for collaborating with other researchers?
Is there a manual available? No
Upload manual if available:
Model website if any
Model forum / discussion board
Comments


This part will be filled out by CSDMS staff

OpenMI compliant No but possible
BMI compliant No but planned
WMT component In progress
PyMT component
Is this a data component
DOI model 10.1594/IEDA/100123
For model version 0.6
Year version submitted 2012
Link to file
Can be coupled with:
Model info

  • Download GFlex [ version: 0.6]
    Doi: 10.1594/IEDA/100123
Nr. of publications: 2
Total citations: 34
h-index: 1
m-quotient: 0.08
Qrcode GFlex.png
Link to this page



Introduction

The model flexure implements multiple (user-selectable) solution methods to solve for flexure and isostasy due to surface loading in both one (line loads) and two (point loads) dimensions. It works for elastic lithospheric plates of both constant and spatially variable elastic thickness and allows the user to select the solution method.

Solution methods

Analytical

An analytical approach to solving the flexure equations is computed by the superimposition of analytical solutions for flexure with a constant elastic thickness. Current implementations perform this superposition in the spatial domain. This works both on uniform grids and arbitrary meshes.

The good:

  • The analytical solution method for an arbitrary mesh is useful for coupling Flexure with finite element models such as CHILD without requiring any regridding.

The bad:

  • Analytical solutions by superposition are an N2 problem, making this method become increasingly problematic for larger grids and numbers of nodes.
  • Analytical solutions are computed based on sets of point loads at nodes or centers of cells, so they will fail and show too much isostatic response if the cells become larger than a modest fraction of a flexural wavelength; this is because at too large of a grid size, the approximation of summing immediately adjacent loads breaks down.
  • Analytical solutions work only with approximations of constant elastic thickness.

Analytical solutions using spectral techniques are not yet implemented.

Numerical

For the numerical implementation, Flexure computes a direct finite difference (lower/upper decomposition) solution to the flexure equations for a lithospheric plate of nonuniform (or uniform, if one so desires) elastic thickness via a thin plate assumption. It uses the UMFPACK direct solvers to compute the solutions via a lower-upper decomposition of a coefficient matrix. The coefficient matrix in the 1D case is a pentadiagonal sparse matrix that is trivial to generate. In the 2D case, we reorder the 2D grid into a 1D vector, which allows us to use sparse matrix construction operators to fairly rapidly build this N2 by N2 matrix. UMFPACK solution routines are then able to copmute solutions to flexure in around a second or less.

The good:

  • This method permits spatial variability in lithospheric elastic thickness. This allows the use of real maps of elastic thickness in models or synthetic maps of elastic thickness variability to test hypotheses.
  • This rapid solution once the coefficient matrix is built makes this method a good choice for numerical models that require frequent updating of flexural deformations of the lithosphere.
  • The rapid solution technique likewise allows efficient calculations of mixed sediment and/or water loading, or water loading with onlap and offlap, such that a constant fill density cannot be assumed and solutions must be produced iteratively.
  • This model will not "blow up" when grid sizes are increased, as the superposition of analytical solutions will.

The bad:

  • This model will provide unrealistic solutions when grid spacing is too fine when compared to a flexural wavelength - this requires wariness, but the fact that the lithosphere acts as a low-pass filter on surface loads means that an appropriately-spaced grid can be interpolated with a large degree of accuracy.
  • This model cannot handle very well sudden (adjacent cell) large discontinuities in elastic thickness of the lithosphere

Key physical parameters and equations

We solve the PDE for lithospheric flexure in 2 dimensions:

[math]\displaystyle{ \nabla^2\left(D(x,y)\nabla^2 w \right) + \Delta \rho g w = q(x,y) }[/math]

Here, D is the flexural rigidity, w is the vertical displacement at each (x,y), Δρ is the mantle density minus the density of infilling material, g is gravitational acceleration, and q is the applied load. We follow Wees and Cloetingh (1994) in acknowledging that flexural rigidity is a tensor property:

[math]\displaystyle{ D = \frac{E T_e^3}{12\left(1-\nu^2\right)} \left[ \begin{array}{ccc} 1 & \nu & 0 \\ \nu & 1 & 0 \\ 0 & 0 & \frac{1-\nu}{2} \end{array} \right] }[/math]

History

Flexure was developed first in MATLAB (Spring / early Summer 2010) and then in Python with Numpy, Scipy, and Matplotlib (translated October 2010).

As of October 2011, Flexure became IRF- and CMT-compliant, and it was coupled to the landscape evolution model CHILD for the Fall 2011 CSDMS meeting. (Abstract and presentation are here, though I haven't dared to watch myself.)

Current work is being done to improve boundary condition handling and the speed of finite difference solutions to constant elastic thickness problems (Early 2012).

Papers

Nothing published yet, so please email Andy Wickert (see contact info box at top) if you are going to use this model for a publication, and we'll figure out how you can cite it. But there is now manuscript in the works:

Wickert, A. D. (in prep. for 2012 submission), Lithospheric Flexure and Earth-Surface Processes: 1. Rapid Solutions with Nonuniform Elastic Thickness.

Issues

Instructions

[ Spot for modern version instructions, 29 OCT 11]

Version 0.1

(Modified from instructional emails)

This is the old version of Flexure, though it is downloadable via the tags/ directory; go to Download models and select "flexure".

Prerequisites

You need to have python, numpy, scipy, and matplotlib installed to use flexure.

Structure

The version 0.1 structure (prone to change) consists of two main files.

  • flexcalc.py is a python module which contains all of the functions needed to execute flexure.
  • flexit.py is the frontend that, via optparse, gives you the ability to specify inputs and outputs and run flexural solutions via an interactive command-line interface.

In addition to these files, version 0.1 comes with some basic test loads and elastic thickness maps.

Trial run

For the basic functionality on the first runthrough, you navigate to the directory with the flexit.py and type something in like:

python flexit.py -vcrp --dx=20000 --Te=Te_sample/bigrange_Te_test.txt --q0=q0_sample/top_bottom_bars.txt

You select which of the sample Te and q0 files you want. Andy_output.png is the flexural response (variable=w) output from:

python flexit.py -vcrp --dx=20000 --Te=Te_sample/bigrange_Te_test.txt --q0=q0_EW_bar.txt

The flags are explained in the help file (python flexit.py -h), as are other options for running the code. Basically you will want to run "-c" the first time, but not again unless you are going to redo the coefficient matrix (i.e., use a different pattern of elastic thickness). This calculation can take a long time for large grids, so you will want to store these files. When running without making a coefficient matrix, unless you're using the default coefficient matrix name (as we do above), you will have to specify its location with "--coeff-file=NAME". For example:

python flexit.py -vrp --Te=Te_sample/bigrange_Te_test.txt --q0=q0_sample/top_bottom_bars.txt --coeff_file=coeffs.txt

If all else fails

Have you typed:

python flexit.py -h

? This gives you all of the in-program help information.

Feel free to email Andy Wickert for anything related to this model (see contact info in box at top).

Input Files

The coefficient matrix for the 2D Thomas algorithm solution requires a map of elastic thicknesses in *.txt / ASCII format. This elastic thickness map must be two cells wider on each side than the map of loads; this is because the finite difference solution must "look" two cells in every direction. It also requires the specification of several parameters, including:

  • Young's modulus (defaults to 1011 Pa)
  • Poisson's ratio (defaults to 0.25)
  • Mantle density (defaults to 3300 kg/m3)
  • Density of infilling material (defaults to 0 kg/m3)

This outputs an ASCII sparse matrix file (Matrix Market *.mtx format).

The flexural solution requires the ASCII file for the sparse coefficient matrix generated above and an imposed array of loads (also ASCII), along with the specification of input and output file names.

Output Files

The coefficient matrix creator writes a *.mtx sparse matrix ASCII file that is used in the direct solution. This matrix is characteristic to a given pattern of elastic thickness, and therefore can be reused if elastic thickness does not change.

The real solver outputs an ASCII grid of deflections due to the load. This is the output that is of scientific interest and/or useful to plug into other modules (e.g., for flexural subsidence).