Table 3 Influence of the order-approximation of \(1/\mu \) with respect to \(\phi \)—one layer \([0^{\circ } ]\)—\(S=4\)—mesh 26 \(\times \) 10—\(N_z = NC\)
\({\varvec{\phi }}\) | Expansion | Model | \(\bar{{{\varvec{u}}}}{{\varvec{(0,e/2)}}}\) max | \(\bar{{{\varvec{w}}}}{{\varvec{(L,0)}}}\) | \(\bar{\varvec{\sigma }}_{{{\varvec{11}}}}{{\varvec{(-e/2)}}}\) | \(\bar{\varvec{\sigma }}_{{{\varvec{13}}}}{{\varvec{(0)}}}\) | \(\bar{\varvec{\sigma }}_{{{\varvec{33}}}}\) max |
|---|---|---|---|---|---|---|---|
\(\phi =\pi /8\) | Zero-order | VS-LM4 | 0.1758 | 0.0206 | \(-\) 0.1721 | 0.1465 | 0.9699 |
Error | 20.37% | 14.13% | 35.61% | 8.30% | 3.01% | ||
One-order | VS-LM4 | 0.1943 | 0.0237 | \(-\) 0.2138 | 0.1676 | 0.9782 | |
Error | 11.98% | 1.10% | 20.01% | 4.94% | 2.18% | ||
Two-order | VS-LM4 | 0.1925 | 0.0235 | \(-\) 0.2117 | 0.1670 | 0.9698 | |
Error | 12.79% | 1.97% | 20.78% | 4.57% | 3.02% | ||
Exact | 0.2207 (\(-e/2\)) | 0.0240 | \(-\) 0.2673 | 0.1597 | 1.0000 | ||
\(\phi =\pi /3\) | Zero-order | VS-LM4 | 2.3665 | 0.2792 | \(-\) 1.0406 | 0.5287 | 0.9225 |
Error | 10.39% | 10.50% | 21.81% | 7.87% | 7.75% | ||
One-order | VS-LM4 | 2.6745 | 0.3153 | \(-\) 1.3452 | 0.5958 | 0.9163 | |
Error | 1.28% | 1.08% | 1.08% | 3.82% | 8.37% | ||
Two-order | VS-LM4 | 2.6386 | 0.3112 | \(-\) 1.3261 | 0.5909 | 0.9062 | |
Error | 0.09% | 0.25% | 0.36% | 2.97% | 9.38% | ||
Exact | 2.6408 | 0.3120 | \(-\) 1.3309 | 0.5739 | 1.0000 | ||
\(\phi =\pi /2\) | Zero-order | VS-LM4 | 19.7700 | 1.2434 | \(-\) 2.4271 | 0.9709 | \(-\) 1.4613 |
Error | 9.90% | 9.85% | 21.19% | 8.83% | 15.41% | ||
One-order | VS-LM4 | 22.3131 | 1.4019 | \(-\) 3.1382 | 1.0922 | \(-\) 1.6628 | |
Error | 1.69% | 1.63% | 1.89% | 2.57% | 3.75% | ||
Two-order | VS-LM4 | 21.9662 | 1.3801 | \(-\) 3.0886 | 1.0822 | \(-\) 1.6379 | |
Error | 0.10% | 0.06% | 0.28% | 1.62% | 5.19% | ||
Exact | 21.9433 | 1.3793 | \(-\) 3.0799 | 1.0649 | \(-\) 1.7275 | ||
\(\phi =5\pi /6\) | Zero-order | VS-LM4 | 1268.7500 | 39.7516 | \(-\) 15.1031 | 4.0278 | \(-\) 13.0670 |
Error | 9.01% | 8.99% | 20.62% | 10.01% | 10.39% | ||
One-order | VS-LM4 | 1436.1875 | 44.9875 | \(-\) 19.6075 | 4.5312 | \(-\) 14.8620 | |
Error | 3.00% | 3.00% | 3.05% | 1.24% | 1.92% | ||
Two-order | VS-LM4 | 1409.4375 | 44.1469 | \(-\) 19.2512 | 4.4970 | \(-\) 14.5900 | |
Error | 1.08% | 1.08% | 1.18% | 0.48% | 0.05% | ||
Exact | 1394.3688 | 43.6759 | \(-\) 19.0264 | 4.4756 | \(-\) 14.5826 |