show details of the fine meshes near the fault. Thesmallest element dimension of thefinest meshwas 2.75% of the fault radius; 3 × 3 ×3 Gaussintegrationwas used when forming the finest mesh stiffness matrix. Figure 7.6 shows the displacement boundary conditions that were applied to the finite element model to describe the loading and symmetry conditions.
7.2.2 Performance of the Multigrid Method
As in the case of the crack problem discussed in chapter6, the MGι-GS multigrid method was used to solve thecircularfault problem. Table 7.2 shows thenumber of multigrid cycles and thesolutiontimes required for the solution of this problem for various values of7, vi, and z∕2∙ Figure 7.7 shows the convergence behavior of these MGι-GS methods.
The optimal method was given by 7 = 3 and ι∕1 = v2 = 10. The reason for this particular choice of the parameter 7 was discussed in section 6.2.1. It is interesting to note that the MGι-GS method is muchless sensitive to the choice of7 for the circular fault problem than for the crack problem; this is due to the absense ofany significantbendingdeformationin the former problem. The times required to form the finest mesh stiffness matrix and the coarse mesh stiffness and interpolation matrices were 43% and 7% of the solution time of the fastest MG1-GSmethod. Thisonce again shows that the computational effort associated with the coarse meshes was fairly small.
The Jacobi preconditonedconjugategradient method (JCG) was also used tosolve the circular fault problem usingthe finest mesh shown in figure 7.5. The convergence behavior of the method is shown in figure 7.8. 310 cyclesof iteration
were required, which took 4.00 × 104 CPU secs. Therefore the solution time of the JCG method was 1.6 that of the fastest MGι-GS scheme. However, the total time required by the JCG method was 5.06X 104 CPU secs, which was 1.4 times aslong as the fastest MG1-GS method. The JCG method needed3.25× 107bytes of storage, which is 56% of the multigrid storage requirement (5.77 × 107 bytes).
The datapresented in this section once again demonstrates the effective
ness of the multigrid method when used to solve a large linear elastic solid me chanics problem. Although the optimal MGj-GS method appears to be given by 7 = 3, ιq = i'2 = 10, the parameters that were used in the production runs for the solution of the fault problems discussed in sections 7.3 and 7.4 were γ = 1, i∕1 = i√2 = 5. Thiswas done because it was considered too expensiveto determine the optimumparameters for each of the different geometries considered.
7.2.3 Numerical Results
One of the benefits of using the multigrid method is that the coarse meshes are available to solve the problem under consideration. This allows the convergence propertyof the finite element meshes to be observed. Improved numerical results can then be obtained by extrapolation using the measured convergence rate. Fig ures 7.9 and 7.10 show plots of u1 and u3 versusx1 along the aq—axis, respectively, computed for the circular fault using the three finest meshes. Also shown is the analytical solution given in equation (7.3). The meshes are identified by means ofa typical mesh size; the finest mesh is referred to as mesh 1, the next coarsest as mesh 2, etc. The plots of u1 and u3 clearlyindicate that the numerical results are converging to the analytical solution as the meshes are refined. Figure 7.10
demonstratesthe rapid decay in the u3-displacements away from the fault. Fig ure 7. Il· shows the valuesof C computed fromthenumerical results obtained from the three meshes and the analytical result given in equation (7.4). Thenumerical values of ul were integrated using3 × 3 Gauss integration. The errorin the value ofC computedon the finest mesh was 2.8% ofthe exact solution. This computed valueofC was improved bylinearlyextrapolating the results computedusingthe two finest meshes. This results in a new value ofC of 1.37, which gives an error of only 0.3% of the exact solution. Thus, a simple linear extrapolation between the values ofC computed on the two finest meshesreduced the error to less than 1%.
The near linear convergence of C can be explained in the following way.
There are two sources of errorthat need tobeconsidered: the discretizationerror present in the computed displacements, and the error that is associated with the numerical integration of these displacements to obtain C. The theoretical convergence rate of the displacements computed with the linear brick elements used in the finite element model of the circular fault is quadratic (which means that the error will be reduced by a factor of four for a single mesh refinement).
However, the singularitypresent at the edge of the fault violates the smoothness of the displacements required to yield this rate, and therefore lowers the rate of convergence. This reduced rate applieseveninplaces where the solutionis smooth [55]. Figure 7.12 shows the values ofu1 computed at the center of thecircular fault (where the solution is smooth) using the three finest meshes and the analytical solution given in equation (7.3). This indicates that the errorin the displacements is decreasing at a rate slower than quadratic. In fact, the error in ui computed on the finest mesh is 42% of the error on the next finest mesh, suggesting that
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.