Derivatives za Nafasi za Shape Functions¶

Shape functions \(N_\alpha^e(\boldsymbol{r})\) zilizoanzishwa katika Shape Functions na Ukadiriaji wa Elementi Finiti hutolewa kama functions za natural coordinates \(\boldsymbol{r}\), ambazo ni coordinates za eneo la elementi. Kwa upande mwingine, kupitia uhusiano wa strain-displacement, integrands za weak form huwa na partial derivatives kwa physical coordinates (reference configuration \(\boldsymbol{X}\) au current configuration \(\boldsymbol{x}\)), yaani \(\partial N_\alpha^e/\partial \boldsymbol{X}\) au \(\partial N_\alpha^e/\partial \boldsymbol{x}\). Sura hii inaweka kwa mpangilio utaratibu wa kukokotoa derivatives za nafasi za shape functions kutoka coordinates za nodi za elementi na derivatives kwa natural coordinates.

Uwakilishi wa Derivatives za Nafasi kwa Chain Rule¶

Ukitumia chain rule chini ya interpolation formula \(\boldsymbol{X} = \sum_\alpha N_\alpha^e(\boldsymbol{r})\,\boldsymbol{X}^e_\alpha\) (sura iliyotangulia), na kutumia Jacobian matrix \(J_{ij} = \partial X_i/\partial r_j\) ya mapping kutoka natural coordinates kwenda physical coordinates, tunapata

Hapa, \(\partial \boldsymbol{N}^e/\partial \boldsymbol{X}\) na \(\partial \boldsymbol{N}^e/\partial \boldsymbol{r}\) ni matrices za \(n_e \times 3\) ambapo mstari \(\alpha\) unawakilisha vekta ya partial derivative ya nodi \(\alpha\).

Uundaji wa Jacobian Matrix¶

Ukidifferentiate interpolation formula kwa \(\boldsymbol{r}\),

hupatikana, hivyo Jacobian matrix inaweza kuundwa moja kwa moja kutoka element nodal coordinates \(\boldsymbol{X}^e_\alpha\) na natural-coordinate derivatives \(\partial \boldsymbol{N}^e/\partial \boldsymbol{r}\). Natural-coordinate derivatives huamuliwa na umbo la function la shape functions na hutekelezwa mapema kwa kila aina ya elementi.

Determinant ya Jacobian matrix \(\det(\partial \boldsymbol{X}/\partial \boldsymbol{r})\) hutumika katika ujumuishaji wa nambari kubadilisha volume element kama \(dV = \det(\partial \boldsymbol{X}/\partial \boldsymbol{r})\,d\boldsymbol{r}\). Ikiwa determinant ni \(0\), elementi imeharibika kijiometri na hesabu hushindwa.

Mtiririko wa Hesabu¶

Kutokana na uderivation hapo juu, hesabu ya spatial derivatives \(\partial \boldsymbol{N}^e/\partial \boldsymbol{X}\) inaweza kugawanywa katika hatua nne zifuatazo.

1. Kukokotoa derivatives za natural coordinates: Kwa kutumia aina ya elementi na natural coordinates \(\boldsymbol{r}\) za evaluation point kama ingizo, kokotoa natural-coordinate derivative matrix \(\partial \boldsymbol{N}^e/\partial \boldsymbol{r}\). Kwa sababu umbo la function la shape functions hutofautiana kwa aina ya elementi, uchakataji hugawanyika kulingana na aina ya elementi.
2. Kukokotoa Jacobian matrix: Kutoka element nodal coordinates \(\boldsymbol{X}^e\) na \(\partial \boldsymbol{N}^e/\partial \boldsymbol{r}\), unda Jacobian matrix \(\partial \boldsymbol{X}/\partial \boldsymbol{r}\) kwa equation ya sehemu iliyotangulia.
3. Kukokotoa inverse matrix na determinant: Kokotoa inverse \((\partial \boldsymbol{X}/\partial \boldsymbol{r})^{-1}\) na determinant ya Jacobian matrix. Determinant hutumika katika weight ya ujumuishaji wa nambari.
4. Kukokotoa derivatives za nafasi: Pata spatial derivatives \(\partial \boldsymbol{N}^e/\partial \boldsymbol{X}\) kama product ya \(\partial \boldsymbol{N}^e/\partial \boldsymbol{r}\) na \((\partial \boldsymbol{X}/\partial \boldsymbol{r})^{-1}\).

Hatua hizi nne hugawanyika kulingana na aina ya elementi na spatial dimension (2D au 3D), lakini mtiririko wa jumla ni mmoja.

Ushughulikiaji wa Pamoja wa Reference Configuration na Current Configuration¶

Utaratibu hapo juu unaweza kutumika pia kwa kubadilisha tu \(\boldsymbol{X}^e\) na current nodal coordinates \(\boldsymbol{x}^e_\alpha = \boldsymbol{X}^e_\alpha + \boldsymbol{u}^e_\alpha\), ili kupata spatial derivatives \(\partial \boldsymbol{N}^e/\partial \boldsymbol{x}\) katika current configuration. Hivyo, utaratibu huo huo unaweza kushirikiwa kati ya mbinu ya Total Lagrange na Updated Lagrange kwa kubadilisha tu nodal coordinates za ingizo.

Mada Zinazohusiana¶

• Shape Functions na Ukadiriaji wa Elementi Finiti — Ufafanuzi wa shape functions na element nodal coordinates
• Discretization ya Internal Virtual Work — Kuunda B matrix kutoka spatial derivatives
• Ujumuishaji wa Nambari — Matumizi ya Jacobian determinant katika Gauss quadrature