Shape functions za vipengele vya solid vya 3D¶

Kwa shape functions \(N_\alpha^e(\boldsymbol{r})\) zilizoanzishwa katika Shape functions na makadirio ya vipengele vya mwisho, sura hii inakusanya natural-coordinate systems na maumbo maalum ya 3D solid elements zinazotumiwa na FrontISTR katika continuum analysis. Aina sita zinazohusika ni linear/quadratic tetrahedra (341/342), linear/quadratic triangular prisms (351/352), na linear/quadratic hexahedra (361/362). Kwa node ordering na face numbers katika HEC-MW mesh format, rejelea Maktaba ya vipengele katika function reference.

Utaratibu wa kuunda element integrals kutoka shape functions hizi na Jacobian determinant unafuata Numerical integration. B-bar, F-bar na incompatible formulations za 361, pamoja na shape functions za beam na shell elements, zinaelezwa katika Advanced element formulations.

Natural-coordinate systems na internal node ordering¶

Shape functions hufafanuliwa kwenye natural coordinates \(\boldsymbol{r}\) za kila kipengele. Kwa 3D solid elements, natural-coordinate systems tatu zifuatazo hutumika kulingana na umbo la kipengele.

Hexahedra (361/362): \(\boldsymbol{r} = (\xi, \eta, \zeta)\), \(-1 \le \xi, \eta, \zeta \le 1\).
Tetrahedra (341/342): volume coordinates \((L_1, L_2, L_3, L_4)\), \(L_1 + L_2 + L_3 + L_4 = 1\), \(L_i \ge 0\). Independent variables ni \(\boldsymbol{r} = (L_2, L_3, L_4)\) na \(L_1 = 1 - L_2 - L_3 - L_4\).
Triangular prisms (351/352): volume coordinates \((\xi, \eta)\) za triangular base (\(0 \le \xi, \eta\), \(\xi + \eta \le 1\)) huunganishwa na axial coordinate \(\zeta \in [-1, 1]\) kuwa \(\boldsymbol{r} = (\xi, \eta, \zeta)\). Auxiliary quantity \(a = 1 - \xi - \eta\) hutumika kama volume coordinate ya tatu ya base.

Kwa linear elements, element node numbers katika HEC-MW input format na internal format zinafanana. Kwa quadratic elements, kanuni ya ordering ya midside nodes inatofautiana kati ya formats hizo. FrontISTR inapokea HEC-MW format kama ingizo (corner nodes kwanza kuzunguka, kisha midside nodes kwa edges 1-2, 2-3, ...), wakati shape functions katika element library zimetekelezwa kwa node ordering tofauti.

Tofauti kati ya formats hizo huondolewa na hecmw2fstr_mesh_conv, inayoitwa mara tu baada ya kusoma mesh na kupanga upya nodi kwa kila kipengele. Kati ya 3D solid elements, aina mbili zifuatazo zinahitaji conversion.

Aina ya kipengele	Jedwali la conversion	Maelezo
342 (quadratic tetrahedron)	`Table342`	Panga upya nodi 10 kuwa \(\{1,2,3,4,7,5,6,8,9,10\}\)
352 (quadratic triangular prism)	`Table352`	Panga upya nodi 15 kuwa \(\{1,2,3,4,5,6,9,7,8,12,10,11,13,14,15\}\)

Kwa kuwa reordering hufanywa moja kwa moja kwenye array inayohifadhi element connectivity, simu zote zinazofuata za element library hutumia internal node ordering ya FrontISTR. Kwa quadratic hexahedron 362, ordering ya midside nodes katika HEC-MW input format na internal format ya FrontISTR inafanana, kwa hiyo hakuna conversion table. Wakati wa kutoa matokeo, reverse conversion fstr2hecmw_mesh_conv huitwa kurudisha HEC-MW node ordering kabla ya kuandika, hivyo mtumiaji anahitaji kuzingatia HEC-MW format pekee kwa ingizo na toleo. Node numbers za shape functions zilizo hapa chini ni za internal format ya FrontISTR baada ya conversion.

Shape functions za linear elements¶

Shape functions za linear tetrahedron (341) ni volume coordinates zenyewe:

\[ N_1 = L_1, \qquad N_2 = L_2, \qquad N_3 = L_3, \qquad N_4 = L_4. \]

Shape functions za linear triangular prism (351) ni bidhaa ya linear shape functions za triangular base (\(a, \xi, \eta\)) na axial linear shape functions (\((1-\zeta)/2, (1+\zeta)/2\)):

\[ \begin{aligned} N_1 &= \tfrac{1}{2}\, a\,(1-\zeta), & N_2 &= \tfrac{1}{2}\, \xi\,(1-\zeta), & N_3 &= \tfrac{1}{2}\, \eta\,(1-\zeta), \\ N_4 &= \tfrac{1}{2}\, a\,(1+\zeta), & N_5 &= \tfrac{1}{2}\, \xi\,(1+\zeta), & N_6 &= \tfrac{1}{2}\, \eta\,(1+\zeta). \end{aligned} \]

Shape functions za linear hexahedron (361) ni tensor product ya linear shape functions katika axes zote tatu:

\[ N_\alpha = \tfrac{1}{8}\,(1+\xi_\alpha \xi)(1+\eta_\alpha \eta)(1+\zeta_\alpha \zeta), \qquad \alpha = 1, \ldots, 8. \]

Hapa \((\xi_\alpha, \eta_\alpha, \zeta_\alpha)\) ni natural-coordinate values za nodi \(\alpha\) (kila component ni \(\pm 1\)).

Shape functions za quadratic elements¶

Shape functions za quadratic tetrahedron (342) hugawanywa katika corner nodes 4 na midside nodes 6:

\[ \begin{aligned} N_1 &= L_1(2L_1 - 1), & N_2 &= L_2(2L_2 - 1), \\ N_3 &= L_3(2L_3 - 1), & N_4 &= L_4(2L_4 - 1), \\ N_5 &= 4\,L_1 L_2, & N_6 &= 4\,L_2 L_3, \\ N_7 &= 4\,L_1 L_3, & N_8 &= 4\,L_1 L_4, \\ N_9 &= 4\,L_2 L_4, & N_{10} &= 4\,L_3 L_4. \end{aligned} \]

Midside nodes 5–10 zinalingana na midpoints za edges 1-2, 2-3, 1-3, 1-4, 2-4, na 3-4, mtawalia.

Shape functions za quadratic triangular prism (352) zinajumuisha corner nodes 6, midside nodes 6 kwenye triangular faces, na midside nodes 3 za axial edges:

\[ \begin{aligned} N_1 &= \tfrac{1}{2}\, a\,(1-\zeta)(2a - 2 - \zeta), & N_2 &= \tfrac{1}{2}\, \xi\,(1-\zeta)(2\xi - 2 - \zeta), \\ N_3 &= \tfrac{1}{2}\, \eta\,(1-\zeta)(2\eta - 2 - \zeta), & N_4 &= \tfrac{1}{2}\, a\,(1+\zeta)(2a - 2 + \zeta), \\ N_5 &= \tfrac{1}{2}\, \xi\,(1+\zeta)(2\xi - 2 + \zeta), & N_6 &= \tfrac{1}{2}\, \eta\,(1+\zeta)(2\eta - 2 + \zeta), \\ N_7 &= 2\,\xi\, a\,(1-\zeta), & N_8 &= 2\,\xi\, \eta\,(1-\zeta), \\ N_9 &= 2\,\eta\, a\,(1-\zeta), & N_{10} &= 2\,\xi\, a\,(1+\zeta), \\ N_{11} &= 2\,\xi\, \eta\,(1+\zeta), & N_{12} &= 2\,\eta\, a\,(1+\zeta), \\ N_{13} &= a\,(1 - \zeta^2), & N_{14} &= \xi\,(1 - \zeta^2), \\ N_{15} &= \eta\,(1 - \zeta^2). \end{aligned} \]

Midside nodes 7–9 ni midpoints za edges za bottom triangle (\(\zeta = -1\)), 10–12 ni midpoints za edges za top triangle (\(\zeta = +1\)), na 13–15 ni midpoints za axial edges.

Quadratic hexahedron (362) ni element ya nodi 20 ya familia ya Serendipity. Natural-coordinate values za nodi \(\alpha\) ziwe \((\xi_\alpha, \eta_\alpha, \zeta_\alpha)\). Kwa corner nodes (\(\alpha = 1, \ldots, 8\), kila component ni \(\pm 1\)),

\[ N_\alpha = -\tfrac{1}{8}\,(1+\xi_\alpha \xi)(1+\eta_\alpha \eta)(1+\zeta_\alpha \zeta)\bigl(2 - \xi_\alpha \xi - \eta_\alpha \eta - \zeta_\alpha \zeta\bigr). \]

Kwa midside nodes (\(\alpha = 9, \ldots, 20\), moja ya natural coordinates tatu ni \(0\) na nyingine mbili ni \(\pm 1\)), kwa nodi zenye \(\xi_\alpha = 0\) (edge direction ni \(\xi\) axis),

\[ N_\alpha = \tfrac{1}{4}\,(1 - \xi^2)(1 + \eta_\alpha \eta)(1 + \zeta_\alpha \zeta). \]

Midside nodes kwenye \(\eta\) na \(\zeta\) axes hutolewa vivyo hivyo, kwa kuondoa natural coordinate husika kupitia squared term.

Vipengee vinavyohusiana¶

Shape functions na makadirio ya vipengele vya mwisho — Ufafanuzi wa kihisabati wa shape function \(N_\alpha^e\)
Spatial derivatives za shape functions — Global-coordinate derivatives na Jacobian
Numerical integration — Integration points, weights na approximation ya element integrals
Advanced element formulations — B-bar, F-bar, incompatible elements, shell na beam
Maktaba ya vipengele (kipengele) — Node ordering na face numbers katika HEC-MW format
Orodha ya alama za quantities za fizikia