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Kazi halisi ya nguvu za nje na assembly ya mlinganyo wa jumla

Katika Udiscretishaji wa kazi halisi ya nguvu ya ndani, upande wa kushoto wa weak form ulikusanywa kuwa element internal-force vector \(\boldsymbol{q}^e\) (UL) au \(\boldsymbol{Q}^e\) (TL). Sura hii inaanzisha element nodal external-force vector \(\boldsymbol{F}^e\) kutoka kazi halisi ya nguvu za nje, kisha kupitia operesheni ya assembly inayopanga na kujumlisha quantities za nodi za kipengele kwa global node numbers, tunafikia mfumo wa milinganyo isiyo ya mstari katika nodal displacements unaotatuliwa na uchanganuzi wa miundo usio wa mstari wa FrontISTR.

Mgawanyo wa kazi halisi ya nguvu za nje kwa vipengele

Upande wa kulia wa kanuni ya kazi halisi unaweza kugawanywa kwa kila elementi kama kazi halisi ya nguvu za nje inayojumuisha body force na prescribed surface traction kwenye mechanical boundary. Ili kuandika displacement interpolation iliyoanzishwa katika Shape functions na makadirio ya finite element kwa matrix form, kwa node \(\alpha\) tunatumia shape function \(N_\alpha^e\), block ya diagonal ya \(d \times d\) \(\boldsymbol{N}_\alpha\), na kisha \(\boldsymbol{N} = [\boldsymbol{N}_1, \ldots, \boldsymbol{N}_{n_e}]\) inayopanga blocks hizo kwa mlalo, hivyo \(\delta\boldsymbol{u} = \boldsymbol{N}\, \delta\boldsymbol{u}^e\). Tukibadilisha hii katika kazi halisi ya nguvu za nje iliyoandikwa kwenye reference configuration, tunapata

\[ \delta W^{\mathrm{ext}} = \sum_e \delta\boldsymbol{u}^{eT} \boldsymbol{F}^e, \qquad \boldsymbol{F}^e_\alpha = \int_{\Omega^e_0} \boldsymbol{N}_\alpha^T \rho_0 \boldsymbol{g}\, dV + \int_{\Gamma^e_{0t}} \boldsymbol{N}_\alpha^T \bar{\boldsymbol{t}}_0\, d\Gamma_0 \]

Hapa element nodal external-force vector imepangwa kama \(\boldsymbol{F}^e = (\boldsymbol{F}^{eT}_1, \ldots, \boldsymbol{F}^{eT}_{n_e})^T\). Kwa hiyo kazi halisi ya nguvu za nje inaweza kukusanywa katika umbo lilelile la "element nodal vector × test function" kama upande wa nguvu za ndani (hata ikiandikwa kwenye current configuration, umbo ni lilelile kwa kubadilisha \(dV \to dv\), \(\rho_0 \to \rho\), \(\bar{\boldsymbol{t}}_0 \to \bar{\boldsymbol{t}}\)).

Assembly ya quantities za nodi za kipengele

Nodal quantities \(\boldsymbol{Q}^e_\alpha, \boldsymbol{F}^e_\alpha\) zilizopatikana kwa kila elementi hukusanywa katika global vectors zilizopangwa kwa global node numbers. Kwa elementi \(\Omega^e\), local node number \(\alpha\) inayolingana na global node number iandikwe

\[ \mathrm{gdx}(e, \alpha) = i_g \]

basi quantity ya element node inalingana na component husika ya global nodal quantity (kama \(\boldsymbol{u}^e_\alpha = \boldsymbol{u}_{i_g}\)). Kwa kawaida node \(i_g\) hushirikiwa na elementi nyingi; kwa hiyo, kwa global node number \(i_g\), tunafafanua set ya jozi \((e, \alpha)\) kama

\[ \mathcal{E}(i_g) = \{ (e, \alpha) \mid \mathrm{gdx}(e, \alpha) = i_g \} \]

Kwa kutumia set hii na kuandika upya jumla kama \(\sum_e \sum_\alpha = \sum_{i_g} \sum_{(e,\alpha) \in \mathcal{E}(i_g)}\), tunapata nodal internal force na global internal-force vector juu ya nodi zote \(n_g\)

\[ \boldsymbol{Q}_{i_g} = \sum_{(e,\alpha) \in \mathcal{E}(i_g)} \boldsymbol{Q}^e_\alpha, \qquad \boldsymbol{Q} = (\boldsymbol{Q}^T_1, \ldots, \boldsymbol{Q}^T_{n_g})^T \]

\(\boldsymbol{Q}_{i_g}\) ni resultant ya element nodal internal forces zinazotenda kwenye nodi \(i_g\), na huwa \(\boldsymbol{0}\) ikiwa hakuna external force na nodi iko katika equilibrium. Kwa mbinu ya UL, \(\boldsymbol{q}_{i_g}, \boldsymbol{q}\) hupatikana kwa utaratibu huo huo, na kwa thamani \(\boldsymbol{q} = \boldsymbol{Q}\); kwa hiyo kuanzia hapa tunatumia \(\boldsymbol{Q}\) isipokuwa pale tofauti inapohitajika. Global external-force vector \(\boldsymbol{F}\) pia hupatikana kwa aggregation hiyo hiyo.

Katika utekelezaji, set \(\mathcal{E}(i_g)\) haiundwi wazi; badala yake components husika huongezewa ndani ya element loop.

Anzisha global internal-force vector Q kuwa 0: Q_{i_g} = 0  (i_g = 1, ..., n_g)
for e = 1 to (idadi ya vipengele)
    for α = 1 to n_e
        i_g = gdx(e, α)
        Q_{i_g} += Q^e_α
    end for
end for

Global external-force vector \(\boldsymbol{F}\) huundwa kwa utaratibu huo huo. Operesheni ya kuongeza na kuhifadhi element nodal quantities katika vectors/matrices zilizonambariwa kwa global node numbers huitwa assembly. Kwa second-order tensor inayohusisha node numbers mbili (kama stiffness matrix), assembly ya aina hiyo hiyo hupatikana kwa set \(\mathcal{E}^2(i_g, i_h) = \{ (e, \alpha, \beta) \mid \mathrm{gdx}(e, \alpha) = i_g\ \mathrm{and}\ \mathrm{gdx}(e, \beta) = i_h \}\) (kwa uundaji mahususi, rejelea Matriki ya ugumu wa tanjenti).

Mlinganyo usio wa mstari unaotatuliwa

Tukibadilisha matokeo ya assembly ya internal na external forces katika kanuni ya kazi halisi, na kwa kuwa lazima ishikilie kwa test function yoyote \(\delta\boldsymbol{u}^n\) inayotimiza geometric boundary conditions, tunapata

\[ \boldsymbol{Q}(\boldsymbol{u}^n) - \boldsymbol{F}(\boldsymbol{u}^n) = \boldsymbol{0} \]

Katika muktadha wa incremental analysis (Mfumo wa uchanganuzi wa nyongeza), tukirudisha time subscript \(_{n+1}\) na kuacha superscript \(^n\) inayoonyesha global nodal vector, mlinganyo unaotatuliwa ni

\[ \boldsymbol{Q}(\boldsymbol{u}_{n+1}) - \boldsymbol{F}(\boldsymbol{u}_{n+1}) = \boldsymbol{0} \]

Tatizo la boundary-value lililodiskretishwa la kupata, katika muda \(t_{n+1}\), nodal displacement \(\boldsymbol{u}_{n+1}\) linakuwa kutatua nonlinear equation hii ya displacement pamoja na geometric boundary conditions. Linearization ya equation na uundaji wa tangent stiffness matrix unaelezwa katika Matriki ya ugumu wa tanjenti, na iterative solution katika Mbinu ya Newton-Raphson.

Vipengee vinavyohusiana

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