the computed displacements are converging at a rate which is nearly linear. A similar observation has been reported in a solution of a two dimensional crack problem where linear quadrilateral elements were used [55], further supporting the nearly linearrate ofconvergence seenfor the circular fault.
The numerical integration of the displacements necessary to compute C also leads to an error. It can be shown for the meshes used in this studythat the error present in the numerical integration of Uι is reduced by a factor of about 0.3 as the mesh size is halved. This was done by assuming that the computed nodal displacements were exact and by making use of the analytical solution for u1 given in equation (7.3). This gives an upper bound on the convergencerate of C, which is faster than linear.
This discussion shows that it is reasonable to use a linear convergence rate when computing an improved value of C by extrapolation. The accuracy of the extrapolated value obtained for the circular fault shows that the artificial boundary was placed sufficiently far (ten fault radii) from the center of thefault.
ψ = Ο, 0.02, 0.165, and ∞. The meshes used to model these geometries had the same basic features as those shown in figures 7.2, 7.3, 7.4, and 7.5; i.e. the elements near the fault had similar dimensions, and the artificial boundary was placed ten times half the width of the fault away from the center of the fault.
The applied boundary conditionswere similar to those shown in figure 7.6. The degrees-of-freedom in the meshes for the four depths of burial numbered 90,891, 121,827, 105,099, and 58,419, respectively. The larger numbers for the surface breaking and finitely buried faults were due to the need to model one quarter of the domain, rather than the one eighth possible for the infinitely buried fault.
The detail of the fault region for the meshes with — 1 and ⅛ = 0,0.02,0.165, and ∞ are shown in figures 7.13, 7.14, 7.15, and 7.16. The meshes used to model the faults with = 2 were generated by modifying thesemeshes.
For example, figure 7.17 shows the detail of the finest mesh usedfor the = 2,
= 0.165 fault. This meshwas formed by moving all of thenodes in the coarsest mesh used for the = 1, = 0.165 fault one unit in the xi-direction (except those on the x2 — X3 plane). This coarsest mesh was then uniformly subdivided three times to produce the mesh shown in figure 7.17.
7.3.2 Performance ofthe Multigrid Method
The solution ofthe rectangular strike-slip fault problems discussed above enables a great deal of data concerning the performance of the multigrid method to be collected. Thisinformation is summarized in tables 7.3 and 7.4. Table 7.3 shows the number ofMG1-GS (7 = 1, ι∕1 = v2 = 5) cycles required to solve the = 1 fault problems, together with the solution and total times, and the storage
requirements of the method. The corresponding data for the = 2 fault is shownin table 7.4. These tablesnot only show the performanceof the multigrid method when it was used to solve the fault problem on the finest meshes, but also on the next two coarsest meshes. Thisdata demonstrates the effectiveness of the method when it was used to solve large solidmechanics problems on a VAX 11/750(for example, the ψ = 1, ψ = 0.165 fault with 105,099 degrees-of-freedom was solved in about 27 CPU hours).
Tables 7.3 and 7.4 show that the number of cycles requiredfor convergence of each fault problem was not independent of the problem size, n. Figures 7.18and 7.19 show plots of the number of multigrid cycles versus log10n for the = 1 and 2 faults, respectively. These graphs show that the number of cycles were proportional to thelogarithm of the problem size. However, this can be considered a weak dependence on n, especially given the large range of n. Tables 7.3 and 7.4 also demonstrate that the storage requirements ofthe multigrid method are linearly proportionalto n.
Another interesting feature of the data presentedinthis sectionis that the fault problems with ψ = 2 required significantly more multigrid cycles (approxi mately twice as many) for convergence thanthosewith = 1. Since the elements in the meshes used to model the = 2 faults had a larger aspect ratio than the elements in the ∙jy = 1 meshes,they would have been subjected to agreaterbend
ing deformation. As was noted in chapters 5 and 6, the bending behavior of the coarse meshes can significantly effect the convergence of the multigrid method.
7.3.3 Numerical Results
As in the caseof the circular fault, values ofCwere computed from thefinest mesh and the nextfinest mesh for each fault geometry. A linearextrapolationwas then performed to obtain an estimate of C,, which is expected to have an error of less than 1% of the exact value. Table 7.1 and figure 7.20 show these estimated values for fault lengths of = 1 and 2. These results exceed those values given in [10];
for example, for thefaults that breakthe surface, the values of Cpresentedinthis chapter are higher by 20% for = 1 and by58% for ψ = 2. The present results are thought to be correct since verification of the computational technique was performed on the circular fault. The square fault stiffness is now 1.05 that of the circular fault, amorereasonablevalue (see section 7.1). Note also that thevalues for C computed in the thesis decreaseby a factor of less than 2 as is increased from 1 to 2 for all values of which is physically realistic. It is possible that the previous formulation omitted the natureof the stress singularities present at the cornersof the rectangular fault. This would explain the agreement in the results for → ∞, since the effect of the singularities would be reduced as the length of the fault is increased.
Figure 7.21 shows the distribution of slip (the ιz1-displacement compo nent) on the rectangular faults at various depths of burial. The main feature of the distributions shown infigure 7.21 is the insensitivity of the ιz1-displacement component to the depth of burial. The faults at depths = 0.02, 0.165, and
∞ all have similar distributions of slip; the distribution changes only when the faults break the surface (i.e. = 0). Thisis reflected in the sharp drops in C as the faults break the surface shown in figure 7.20.