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Newton-Raphson usuli

Chiziqlashtirish va iterativ rekurrent munosabat

Tashqi kuchlarning virtual ishi va global tenglamalarni yig‘ish bo‘limida \(t_{n+1}\) vaqtdagi tugun ko‘chishi \(\boldsymbol{u}_{n+1}\) uchun olinadigan nochiziqli tenglama Newton-Raphson usuli bilan yechiladi. \(t_n\) vaqtgacha bo‘lgan tugun ko‘chishi \(\boldsymbol{u}_n\) ma’lum deb olinadi va aniqlanishi kerak bo‘lgan noma’lum sifatida ko‘chish inkrementi \(\Delta\boldsymbol{u}\) tanlanadi:

\[ \boldsymbol{u}_{n+1} = \boldsymbol{u}_n + \Delta\boldsymbol{u} \]

Quyida tashqi kuch vektorining tugun ko‘chishiga bog‘liqligi e’tiborsiz qoldiriladi va \(\boldsymbol{F}(\boldsymbol{u}_{n+1}) = \boldsymbol{F}_{n+1}\) deb,

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

tenglama yechiladi.

Joriy yechim \(\Delta\boldsymbol{u}\) da tangensial qattiqlik quyidagicha aniqlanadi:

\[ \boldsymbol{K} = \left. \frac{\partial \boldsymbol{Q}}{\partial \boldsymbol{u}} \right|_{\boldsymbol{u}_n + \Delta\boldsymbol{u}} \]

Bundan foydalanib nochiziqli tenglamani chiziqlashtirish quyidagini beradi:

\[ \boldsymbol{K}\, d\boldsymbol{u} + \boldsymbol{Q}(\boldsymbol{u}_n + \Delta\boldsymbol{u}) - \boldsymbol{F}_{n+1} = \boldsymbol{0} \]

\(i\)-iteratsiyadagi tuzatish \(d\boldsymbol{u}_i\) bo‘lsin, iteratsiya boshidagi qoldiq vektor esa

\[ \boldsymbol{R}_{i-1} = \boldsymbol{F}_{n+1} - \boldsymbol{Q}(\boldsymbol{u}_n + \Delta\boldsymbol{u}) \]

bilan aniqlansin. U holda iterativ rekurrent munosabat

\[ \boldsymbol{K}_i\, d\boldsymbol{u}_i = \boldsymbol{R}_{i-1}, \qquad \Delta\boldsymbol{u} \leftarrow \Delta\boldsymbol{u} + d\boldsymbol{u}_i \]

ko‘rinishida bo‘ladi. Shunday qilib, \(\boldsymbol{R}_i\) qoldiq muvozanat holatidan kuch nomuvozanatini ifodalovchi kattalikdir.

Tangensial qattiqlik matritsasini tuzish

Tangensial qattiqlik \(\boldsymbol{K} = \partial\boldsymbol{Q}/\partial\boldsymbol{u}\) Ichki kuchlar virtual ishini diskretlashtirishda olingan element ichki kuch vektorini tugun ko‘chishi bo‘yicha qisman differensiallash, hosil bo‘lgan element darajasidagi integrandlarni har bir element sohasi bo‘yicha integrallash va yig‘ish orqali tuziladi. Element darajasidagi integrand \(\boldsymbol{K}^e_X\) (boshlang‘ich konfiguratsiya, TL formulasi) yoki \(\boldsymbol{K}^e_x\) (joriy konfiguratsiya, UL formulasi) bilan belgilansa, element tangensial qattiqligi

\[ \boldsymbol{K}^e = \int_{\Omega^e_0} \boldsymbol{K}^e_X\, dV \quad (\text{TL}), \qquad \boldsymbol{K}^e = \int_{\Omega^e} \boldsymbol{K}^e_x\, dv \quad (\text{UL}) \]

bo‘ladi. Quyida TL/UL integrandlarining yakuniy ko‘rinishlari beriladi. Har ikki holatda ular material qattiqligi hadi (boshlang‘ich ko‘chish hadi) va geometrik qattiqlik hadi (boshlang‘ich kuchlanish hadi) yig‘indisiga ajraladi.

Total Lagrange formulasi

Total Lagrange formulasida ikkinchi Piola-Kirchhoff kuchlanishi tezligi \(\dot{\boldsymbol{S}}\) bilan Green-Lagrange deformatsiyasi tezligi \(\dot{\boldsymbol{E}}\) orasida \(\dot{\boldsymbol{S}} = \boldsymbol{\mathsf{C}}:\dot{\boldsymbol{E}}\) chiziqli munosabat faraz qilinadi. Bu chiziqli elastik materiallar (St. Venant-Kirchhoff materiallari) va giperelastik materiallar konstitutiv qonunlariga mos keladi; FrontISTR ushbu materiallar uchun Total Lagrange formulasidan foydalanadi. Element tangensial qattiqlik integrandi tenzor ko‘rinishida

\[ \delta\boldsymbol{u}^{eT}\, \boldsymbol{K}^e_X\, \dot{\boldsymbol{u}}^e = \dot{\boldsymbol{S}}:\delta\boldsymbol{E} + \boldsymbol{S}:(\delta\boldsymbol{F}^T \dot{\boldsymbol{F}}) \]

deb yoziladi. O‘ng tomondagi birinchi had material qattiqligi hadi (boshlang‘ich ko‘chish hadi), ikkinchi had esa geometrik qattiqlik hadi (boshlang‘ich kuchlanish hadi).

FrontISTR amalga oshirishida ushbu integrand Voigt belgilashida matritsa ko‘rinishida baholanadi:

\[ \boldsymbol{K}^e_X = (\boldsymbol{B}_L + \boldsymbol{B}_{NL})^T\, \tilde{\boldsymbol{C}}\, (\boldsymbol{B}_L + \boldsymbol{B}_{NL}) + \boldsymbol{F}_9^T\, \boldsymbol{S}_9\, \boldsymbol{F}_9 \]

Har bir matritsa quyidagicha. \(\boldsymbol{B}_L, \boldsymbol{B}_{NL}\) — Ichki kuchlar virtual ishini diskretlashtirishda kiritilgan B matritsalar, \(\tilde{\boldsymbol{C}}\) esa \(\boldsymbol{\mathsf{C}}\) konstitutiv tenzorining Voigt ifodasi, ya’ni \(6\times 6\) material qattiqlik matritsasi (Tenzor belgilanishi va matematik asoslar). \(\boldsymbol{S}_9, \boldsymbol{F}_9\) geometrik qattiqlik hadini matritsa ko‘paytmasi ko‘rinishida ifodalash uchun ishlatiladigan quyidagi qayta tartiblash matritsalaridir. Avval \(3\times 3\) ikkinchi tartibli tenzor \(\boldsymbol{A}\) uchun uni 9 komponentli vektorga qayta tartiblaydigan \([\,\cdot\,]\) belgilashni aniqlaymiz:

\[ [\boldsymbol{A}] = (A_{11}, A_{21}, A_{31}, A_{12}, A_{22}, A_{32}, A_{13}, A_{23}, A_{33})^T \]

Shu ta’rif bilan \(\boldsymbol{F}_9\) deformatsiya gradienti variatsiyasini \([\delta\boldsymbol{F}] = \boldsymbol{F}_9\, \delta\boldsymbol{u}^e\) ko‘rinishida ifodalaydi va \(9\times d n_e\) matritsadir. \(\alpha = 1, \ldots, n_e\) element tuguni uchun mos \(9\times d\) blok

\[ [\boldsymbol{F}_9]_\alpha = \begin{bmatrix} (\partial N_\alpha^e/\partial X_1)\, \boldsymbol{I} \\ (\partial N_\alpha^e/\partial X_2)\, \boldsymbol{I} \\ (\partial N_\alpha^e/\partial X_3)\, \boldsymbol{I} \end{bmatrix} \qquad (\boldsymbol{I} \text{ — } 3\times 3 \text{ birlik matritsa}) \]

ko‘rinishida va bloklar element tugunlari tartibida gorizontal joylashtirilib \(\boldsymbol{F}_9 = [[\boldsymbol{F}_9]_1, \ldots, [\boldsymbol{F}_9]_{n_e}]\) hosil qilinadi. \(\boldsymbol{S}_9\) shu matritsa bilan birga geometrik qattiqlik hadini \(\delta\boldsymbol{u}^{eT}\, \boldsymbol{F}_9^T \boldsymbol{S}_9 \boldsymbol{F}_9\, \dot{\boldsymbol{u}}^e\) ko‘rinishida ifodalash uchun tanlanadi; u quyidagi \(9\times 9\) matritsadir:

\[ \boldsymbol{S}_9 = \begin{bmatrix} S_{11} \boldsymbol{I} & S_{12} \boldsymbol{I} & S_{13} \boldsymbol{I} \\ S_{21} \boldsymbol{I} & S_{22} \boldsymbol{I} & S_{23} \boldsymbol{I} \\ S_{31} \boldsymbol{I} & S_{32} \boldsymbol{I} & S_{33} \boldsymbol{I} \end{bmatrix} \]

Natijaviy matritsa shu ko‘rinishda olinadi.

Updated Lagrange formulasi

Updated Lagrange formulasida nisbiy Kirchhoff kuchlanish tenzorining Jaumann tezligi \(\hat{\boldsymbol{\sigma}}^{\nabla J}\) bilan deformatsiya tezligi tenzori \(\boldsymbol{D}\) orasida \(\hat{\boldsymbol{\sigma}}^{\nabla J} = \boldsymbol{\mathsf{C}}:\boldsymbol{D}\) chiziqli munosabat faraz qilinadi. Bu chiziqli elastik, elastoplastik va krip materiallar uchun umumiy gipoelastik konstitutiv qonun ko‘rinishidir; FrontISTR ushbu materiallar uchun Updated Lagrange formulasidan foydalanadi. Joriy konfiguratsiyada ifodalangan element tangensial qattiqlik integrandi tenzor ko‘rinishida

\[ \delta\boldsymbol{u}^{eT}\, \boldsymbol{K}^e_x\, \dot{\boldsymbol{u}}^e = \boldsymbol{\sigma}^{\nabla T}:\delta\boldsymbol{A}_{(L)} + \boldsymbol{\sigma}:(\delta\boldsymbol{F}_t^T\, \boldsymbol{L}) \]

deb yoziladi. Bu yerda \(\boldsymbol{\sigma}^{\nabla T}\) — Truesdell tezligi, \(\boldsymbol{A}_{(L)}\) — Almansi deformatsiyasining chiziqli qismi, \(\boldsymbol{F}_t = \partial\boldsymbol{u}/\partial\boldsymbol{x}\) — joriy konfiguratsiyaga nisbatan ko‘chish gradienti, \(\boldsymbol{L}\) esa tezlik gradienti tenzori. O‘ng tomondagi birinchi had material qattiqligi hadi, ikkinchi had geometrik qattiqlik hadi.

FrontISTR amalga oshirishida ushbu integrand Voigt belgilashida matritsa ko‘rinishida baholanadi:

\[ \boldsymbol{K}^e_x = \boldsymbol{b}^T\, (\tilde{\boldsymbol{C}} - \boldsymbol{G})\, \boldsymbol{b} + \boldsymbol{f}_9^T\, \boldsymbol{\sigma}_9\, \boldsymbol{f}_9 \]

Bu yerda \(\boldsymbol{b}\) — joriy konfiguratsiyada tuzilgan B matritsa (Ichki kuchlar virtual ishini diskretlashtirish). \(\boldsymbol{\sigma}_9, \boldsymbol{f}_9\) TL formulasidagi \(\boldsymbol{S}_9, \boldsymbol{F}_9\) dan ikkinchi PK kuchlanishi \(\boldsymbol{S}\) ni Cauchy kuchlanishi \(\boldsymbol{\sigma}\) ga va boshlang‘ich konfiguratsiya gradienti \(\partial N_\alpha^e/\partial X_i\) ni joriy konfiguratsiya gradienti \(\partial N_\alpha^e/\partial x_i\) ga almashtirish orqali olinadi.

\(\boldsymbol{G}\) — gipoelastik konstitutiv qonun \(\hat{\boldsymbol{\sigma}}^{\nabla J} = \boldsymbol{\mathsf{C}}:\boldsymbol{D}\) ni Truesdell tezligiga asoslangan konstitutiv qonun sifatidagi tangensial qattiqlik doirasiga moslashtirish uchun zarur Cauchy kuchlanishiga bog‘liq tuzatish matritsasi. U to‘rtinchi tartibli tenzor komponentlari \(G_{ijkl} = \delta_{il}\sigma_{kj} + \delta_{jl}\sigma_{ik}\) ni \(6\times 6\) Voigt ko‘rinishida joylashtirish orqali olinadi:

\[ \boldsymbol{G} = \begin{bmatrix} 2\sigma_{11} & 0 & 0 & \sigma_{12} & 0 & \sigma_{31} \\ 0 & 2\sigma_{22} & 0 & \sigma_{12} & \sigma_{23} & 0 \\ 0 & 0 & 2\sigma_{33} & 0 & \sigma_{23} & \sigma_{31} \\ \sigma_{12} & \sigma_{12} & 0 & \tfrac{\sigma_{11}+\sigma_{22}}{2} & \tfrac{\sigma_{12}}{2} & \tfrac{\sigma_{23}}{2} \\ 0 & \sigma_{23} & \sigma_{23} & \tfrac{\sigma_{12}}{2} & \tfrac{\sigma_{22}+\sigma_{33}}{2} & \tfrac{\sigma_{12}}{2} \\ \sigma_{31} & 0 & \sigma_{31} & \tfrac{\sigma_{23}}{2} & \tfrac{\sigma_{12}}{2} & \tfrac{\sigma_{33}+\sigma_{11}}{2} \end{bmatrix} \]

Natijaviy matritsa shu ko‘rinishda olinadi.

Global qattiqlik matritsasini yig‘ish

Global tangensial qattiqlik \(\boldsymbol{K}\) har bir element qattiqligi \(\boldsymbol{K}^e\) ni har bir tugun jufti uchun \(d\times d\) bloklar \(\boldsymbol{K}^e_{\alpha\beta}\) ga ajratib va Tashqi kuchlarning virtual ishi va global tenglamalarni yig‘ishda kiritilgan ikkinchi tartibli tenzor yig‘ish to‘plami \(\mathcal{E}^2(i_g, i_h)\) dan foydalanib olinadi:

\[ \boldsymbol{K}_{i_gi_h} = \sum_{(e,\alpha,\beta) \in \mathcal{E}^2(i_g, i_h)} \boldsymbol{K}^e_{\alpha\beta} \]

Olingan qiymatlar qatori \(i_g\), ustuni \(i_h\) bo‘lgan matritsa sifatida joylashtiriladi. Amalda \(\mathcal{E}^2\) to‘plami aniq tuzilmaydi; uning o‘rniga tegishli bloklar element sikli ichida bevosita qo‘shiladi. Matritsa o‘lchami tugundagi erkinlik darajalari soni \(\times\) umumiy tugunlar soni \(n_g\) ga teng kvadrat matritsa, biroq element orqali bog‘langan tugunlar orasidagidan boshqa komponentlar \(0\) bo‘lgani uchun siyrak matritsa ko‘rinishida saqlanadi.

TL va UL formulalari uchun element qattiqlik matritsalari boshlang‘ich konfiguratsiyani almashtirish (tugun koordinatalari va B matritsasini tuzish manbasi) hamda \(\boldsymbol{G}\) matritsasining bor-yo‘qligidan tashqari bir xil ko‘rinishga ega. Shu sababli FrontISTR har ikki formulani umumiy quyi dastur ichida amalga oshiradi.

Iteratsiya algoritmi

Yuqoridagilarni jamlab, iteratsiya boshida \(\Delta\boldsymbol{u} = \boldsymbol{0}\) deb o‘rnating va boshlang‘ich qoldiq \(\boldsymbol{R}_0 = \boldsymbol{F}_{n+1} - \boldsymbol{Q}(\boldsymbol{u}_n)\) ni hisoblang. So‘ng \(i\)-iteratsiyada quyidagi tartibni bajaring.

  1. Joriy ko‘chish \(\boldsymbol{u}_n + \Delta\boldsymbol{u}\) da Tangensial qattiqlik matritsasini tuzish tartibidan foydalanib tangensial qattiqlik \(\boldsymbol{K}_i = \partial\boldsymbol{Q}/\partial\boldsymbol{u}|_{\boldsymbol{u}_n + \Delta\boldsymbol{u}}\) ni hisoblang.
  2. Geometrik chegaraviy shartlarni qo‘llash uchun ko‘chish cheklovi ostidagi erkinlik darajalari uchun tangensial qattiqlik matritsasi va qoldiq vektorni o‘zgartirib, \(\tilde{\boldsymbol{K}}_i, \tilde{\boldsymbol{R}}_{i-1}\) ni oling (Geometrik chegaraviy shartlarni qayta ishlash).
  3. \(\tilde{\boldsymbol{K}}_i\, d\boldsymbol{u}_i = \tilde{\boldsymbol{R}}_{i-1}\) chiziqli tenglamani yechib, \(d\boldsymbol{u}_i\) tuzatishni oling. Bu jarayon ko‘pincha iterativ hisoblash xarajatining asosiy qismini tashkil etadi.
  4. Ko‘chish inkrementini \(\Delta\boldsymbol{u} \leftarrow \Delta\boldsymbol{u} + d\boldsymbol{u}_i\) bo‘yicha yangilang va mos ravishda ichki kuch vektori \(\boldsymbol{Q}(\boldsymbol{u}_n + \Delta\boldsymbol{u})\) hamda qoldiq \(\boldsymbol{R}_i = \boldsymbol{F}_{n+1} - \boldsymbol{Q}(\boldsymbol{u}_n + \Delta\boldsymbol{u})\) ni hisoblang.
  5. Yaqinlashishni tekshiring va erishilgan bo‘lsa iteratsiyani tugating. Geometrik chegaraviy shart qo‘yilgan erkinlik darajalarida cheklangan reaksiyalarga mos komponentlar qoldiq \(\boldsymbol{R}_i\) da paydo bo‘ladi, shu sababli yaqinlashish ko‘rsatkichi bu komponentlarni chiqarib tashlagan \(\tilde{\boldsymbol{R}}_i\) dan tuziladi. Aniq yaqinlashish ko‘rsatkichlari va chegaralari Yaqinlashuv mezonlarida bayon etiladi. Yaqinlashishga erishilmasa va iteratsiya chegarasiga yetilsa, iteratsiya muvaffaqiyatsiz deb hisoblanadi.

Iteratsiya yaqinlashganda, yaqinlashgan \(\Delta\boldsymbol{u}\) ni \(\boldsymbol{u}_n\) ga qo‘shib \(t_{n+1}\) vaqtdagi jamlangan ko‘chish \(\boldsymbol{u}_{n+1} = \boldsymbol{u}_n + \Delta\boldsymbol{u}\) olinadi va keyingi vaqt qadamiga o‘tiladi.

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