SCEC Award Number 13096 View PDF
Proposal Category Individual Proposal (Integration and Theory)
Proposal Title Improved Numerical Methods for Earthquake Cycle Simulations
Investigator(s)
Name Organization
Paul Segall Stanford University
Other Participants Andrew Bradley
SCEC Priorities 2, 1, 2 SCEC Groups CS, Simulators, SDOT
Report Due Date 03/15/2014 Date Report Submitted N/A
Project Abstract
 dc3dm is a software package that efficiently forms and applies the linear operator relating quasistatic dislocation and traction components on a nonuniformly discretized rectangular fault with rectangular elements in a homogeneous elastic half space. This linear operator implements what is called the displacement discontinuity method (DDM). The key properties of dc3dm and the algorithms it implements are: 1. The mesh can be nonuniform. 2. Work and memory scale roughly linearly in the number of elements (rather than quadratically). 3. The order of accuracy (OOA) on a nonuniform mesh is the same as that of the standard method on a uniform mesh.
Property 2 is achieved using our package hmmvp, which implements hierarchical-matrix (H-matrix) compression.
A nonuniform mesh (property 1) is natural for some problems. For example, in a rate-state friction simulation,
nucleation length, and so required element size, scales reciprocally with effective normal stress, and the factor
difference between smallest and largest required element sizes is frequently 16 to 100 by area.
On a uniform mesh, straightforward application of a constant-slip Green's function (GF) yields a DDM we refer
to as DDMu. On a nonuniform mesh, this same procedure leads to artifacts that degrade the OOA of DDMu. We
have developed a method we call IGA that implements the DDM using linear combinations of the same GF for a nonuniformly discretized mesh. dc3dm implements an approximate form of IGA. Importantly, IGA's OOA on a nonuniform mesh is the same as DDMu's on a uniform one (property 3).
dc3dm and hmmvp are available at pangea.stanford.edu/research/CDFM/software.
Intellectual Merit hmmvp has a rigorous and demonstratively effective error control framework with a clear interpretation of
error, thus allowing for approximations built using hmmvp also to have rigorous error control. IGA is a Displacement Discontinuity Method (DDM) that, for
one useful class of nonuniform discretizations, has the same order of accuracy as the standard method on a uniform
mesh. IGA and its implementation in dc3dm are supported by extensive order of accuracy analysis. The assessment
methodology reveals potential errors researchers may make when using nonuniform constant-slip elements. It also
provides a framework for our future work in developing DDMs for more complicated geometries.
Broader Impacts hmmvp enables researchers to run rate-state friction simulations on faults requiring about 100 more
elements than the straightforward method permits, allowing more realistic rheology. It permits arbitrary geometry
and (non-oscillatory) Green's function. hmmvp has been fully integrated into the simulators CFRAC (Mark Mclure,
U. of Texas, Austin) and Unicycle (Sylvain Barbot, Nanyang Technological U.) and is being evaluated for use in
another. It was used in Johnson et al. [2013]. dc3dm increases efficiency above that provided by hmmvp for a limited geometry, permitting even more efficient theoretical studies to be done. It also demonstrates that analyzing the convergence behavior of a DDM can lead to a more efficient method. Both hmmvp and dc3dm are free and open source software available at pangea.stanford.edu/research/CDFM/software.
Exemplary Figure Figure 4. Tractions and pointwise absolute errors (relative to maximum traction magnitude) for the naive method, left two columns) and AIGA (right) Both methods applied to the strike-on-strike Green's function for an example problem.
From top to bottom, the level of refinement increases successively by 2 in each dimension. The naive method causes errors where adjacent elements differ in size (outlined by black lines in the top-left image). Peak magnitude of the error
stays approximately constant with refinement, causing a drop in order of accuracy from 2 to 1/2. Color scales are
the same in respectively columns 1 and 3, and 2 and 4. Images are zoomed to a region of interest.