"... In PAGE 9: ...2 Mesh Untangling and Improvement To demonstrate the e ectiveness of optimization- based untangling, we used a randomized smoothing scheme on the original meshes given in Table 1 to create \tangled quot; meshes with valid connectivity but several hundred inverted elements. In Table3 , we... ..."

"... In PAGE 9: ...2 Mesh Untangling and Improvement To demonstrate the e ectiveness of optimization- based untangling, we used a randomized smoothing scheme on the original meshes given in Table 1 to create \tangled quot; meshes with valid connectivity but several hundred inverted elements. In Table3 , we... ..."

"... In PAGE 10: ...2 Mesh Untangling and Improvement To demonstrate the e ectiveness of optimization- based untangling, we used a randomized smoothing scheme on the original meshes given in Table 1 to create \tangled quot; meshes with valid connectivity but several hundred inverted elements. In Table3 , we give the number of inverted elements, NI, the number of sliver elements, NS, as well as the quality metrics avg, max, A avg, and A max for each of the four tangled meshes. The results of the untangling proce- dure described in Section 3 are reported for each ge- ometry in the rows labeled \Untangle quot;.... ..."

"... In PAGE 8: ... With the preselection process, we can considerably speed up the LP problem- solving process, as indicated in Table 4. Table5 lists the topological network lifetime achieved through random BS location (by exhaustive grid search), time routes, and (b) node energy and lifetime. # Iterations Max RN 55 8.... ..."

"... In PAGE 8: ... In each case, we iterate over the interior nodes in the mesh until the change in all node point positions is smaller than some tolerance. In Table2 we report the results of each technique... ..."

"... In PAGE 8: ... In each case, we iterate over the interior nodes in the mesh until the change in all node point positions is smaller than some tolerance. In Table2 we report the results of each technique... ..."

"... In PAGE 37: ... The m variables are branched on next, and the r variables follow. The results using this branching strategy, a maximum of one hundred BB iterations and no bound updates are shown in Table16 under strategy (a). Although the average CPU time per run is greatly improved, the standard deviation on every run is large.... In PAGE 37: ....3.2 Variable bound updates The bounds on all integer variables are now tightened through the optimization-based pro- cedure. The results, presented in Table16 under strategy (b), show that the mean number of nodes decreases by 53 to 93% and the mean CPU time decreases by 74 to 94%. 5.... In PAGE 37: ....3.3 Automatic BB run termination The test of lower bound improvement rate is imposed on the BB runs using values of m and r that were successful for the pump network synthesis example, namely m = 3 iterations and r = 30%. The results, shown in Table16 under strategy (c), indicate that the criterion used is too stringent for this problem as a few more nodes must be explored on average before the global optimum solution is found than when one hundred BB iterations are allowed. However, the computational requirements are su ciently close to the best values obtained so far to warrant adopting the automatic stopping criterion with m = 3 iterations and r = 0:3.... ..."

"... In PAGE 34: ...ound updates for the success of a GMIN- BB run. As discussed in Section 4.1, the ideal value of itermax, the maximum number of BB iterations, depends on the structure of the problem. The results shown in Table14 are obtained by imposing the adaptive test of Eq. (20), with di erent values of m and r.... ..."