Tarkibga o'tish

Shakl funksiyalarining fazoviy hosilalari

Shakl funksiyalari va chekli element yaqinlashuvi da kiritilgan \(N_\alpha^e(\boldsymbol{r})\) shakl funksiyalari elementning mahalliy koordinatalari bo‘lgan tabiiy koordinatalar \(\boldsymbol{r}\) funksiyasi sifatida beriladi. Boshqa tomondan, kuchsiz formadagi integral osti ifodalarida deformatsiya–siljish bog‘lanishi orqali fizik koordinatalar (tayanch konfiguratsiya \(\boldsymbol{X}\) yoki joriy konfiguratsiya \(\boldsymbol{x}\)) bo‘yicha \(\partial N_\alpha^e/\partial \boldsymbol{X}\) yoki \(\partial N_\alpha^e/\partial \boldsymbol{x}\) xususiy hosilalari paydo bo‘ladi. Ushbu bobda bu shakl funksiyalarining fazoviy hosilalarini element tugun koordinatalari va tabiiy koordinatalar bo‘yicha hosilalardan hisoblash tartibi bayon qilinadi.

Zanjir qoidasiga ko‘ra fazoviy hosilalarni ifodalash

\(\boldsymbol{X} = \sum_\alpha N_\alpha^e(\boldsymbol{r})\,\boldsymbol{X}^e_\alpha\) interpolatsiya formulasiga (oldingi bob) zanjir qoidasini qo‘llab, tabiiy koordinatalardan fizik koordinatalarga akslantirishning Yakobian matritsasi \(J_{ij} = \partial X_i/\partial r_j\) dan foydalansak,

\[ \frac{\partial \boldsymbol{N}^e}{\partial \boldsymbol{X}} = \frac{\partial \boldsymbol{N}^e}{\partial \boldsymbol{r}}\,\left(\frac{\partial \boldsymbol{X}}{\partial \boldsymbol{r}}\right)^{-1} \]

hosil bo‘ladi. Bu yerda \(\partial \boldsymbol{N}^e/\partial \boldsymbol{X}\) va \(\partial \boldsymbol{N}^e/\partial \boldsymbol{r}\) matritsalarida \(\alpha\)-qator \(\alpha\)-tugunning xususiy hosilalar vektorini ifodalaydi; ular \(n_e \times 3\) o‘lchamli matritsalardir.

Yakobian matritsasini tuzish

Interpolatsiya formulasini \(\boldsymbol{r}\) bo‘yicha differensiallasak,

\[ \frac{\partial \boldsymbol{X}}{\partial \boldsymbol{r}} = \left[\boldsymbol{X}^e_1, \ldots, \boldsymbol{X}^e_{n_e}\right]\frac{\partial \boldsymbol{N}^e}{\partial \boldsymbol{r}} \]

bo‘ladi va Yakobian matritsasini element tugun koordinatalari \(\boldsymbol{X}^e_\alpha\) hamda tabiiy koordinatalar bo‘yicha hosilalar \(\partial \boldsymbol{N}^e/\partial \boldsymbol{r}\) dan bevosita tuzish mumkin. Tabiiy koordinatalar bo‘yicha hosilalar shakl funksiyalarining funksional ko‘rinishi bilan aniqlanadi va har bir element turi uchun oldindan amalga oshiriladi.

Yakobian matritsasining determinanti \(\det(\partial \boldsymbol{X}/\partial \boldsymbol{r})\) sonli integrallash da hajm elementini \(dV = \det(\partial \boldsymbol{X}/\partial \boldsymbol{r})\,d\boldsymbol{r}\) ko‘rinishida almashtirish uchun ishlatiladi. Determinant \(0\) bo‘lsa, element degeneratsiyalangan bo‘ladi va hisoblash buziladi.

Hisoblash oqimi

Yuqoridagi hosiladan fazoviy hosilalar \(\partial \boldsymbol{N}^e/\partial \boldsymbol{X}\) ni hisoblash quyidagi to‘rt bosqichga ajratiladi.

  1. Tabiiy koordinatalar bo‘yicha hosilalarni hisoblash: element turi va baholash nuqtasining tabiiy koordinatalari \(\boldsymbol{r}\) ni kirish sifatida olib, tabiiy koordinatalar bo‘yicha hosilalar matritsasi \(\partial \boldsymbol{N}^e/\partial \boldsymbol{r}\) hisoblanadi. Shakl funksiyalarining funksional ko‘rinishi element turiga qarab farq qilgani uchun ishlov element turiga qarab tarmoqlanadi.
  2. Yakobian matritsasini hisoblash: element tugun koordinatalari \(\boldsymbol{X}^e\) va \(\partial \boldsymbol{N}^e/\partial \boldsymbol{r}\) dan oldingi bo‘limdagi formula orqali Yakobian matritsasi \(\partial \boldsymbol{X}/\partial \boldsymbol{r}\) tuziladi.
  3. Teskari matritsa va determinantni hisoblash: Yakobian matritsasining teskari matritsasi \((\partial \boldsymbol{X}/\partial \boldsymbol{r})^{-1}\) va determinanti hisoblanadi. Determinant sonli integrallash og‘irligida ishlatiladi.
  4. Fazoviy hosilalarni hisoblash: \(\partial \boldsymbol{N}^e/\partial \boldsymbol{r}\) va \((\partial \boldsymbol{X}/\partial \boldsymbol{r})^{-1}\) ko‘paytmasi sifatida fazoviy hosilalar \(\partial \boldsymbol{N}^e/\partial \boldsymbol{X}\) olinadi.

Ushbu to‘rt bosqich element turi va fazo o‘lchamiga (ikki yoki uch o‘lcham) qarab tarmoqlanadi, biroq umumiy oqim bir xil.

Tayanch va joriy konfiguratsiyalar uchun umumiy ishlov

Yuqoridagi tartibda \(\boldsymbol{X}^e\) ni joriy konfiguratsiyadagi tugun koordinatalari \(\boldsymbol{x}^e_\alpha = \boldsymbol{X}^e_\alpha + \boldsymbol{u}^e_\alpha\) bilan almashtirishning o‘zi joriy konfiguratsiyadagi fazoviy hosilalar \(\partial \boldsymbol{N}^e/\partial \boldsymbol{x}\) ni olish uchun yetarli. Shuning uchun Total Lagrange va Updated Lagrange usullarida faqat kirish tugun koordinatalarini almashtirish orqali ayni tartibdan birgalikda foydalanish mumkin.

Tegishli mavzular

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