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Numerical integration

Element internal-force vectors \(\boldsymbol{q}^e, \boldsymbol{Q}^e\) na element stiffness matrix \(\boldsymbol{K}^e\) zilizopatikana katika Discretization ya virtual work ya internal force na Virtual work ya external force na assembly ya global equation huwa katika umbo la integrals juu ya element domain \(\Omega^e\) au \(\Omega^e_0\). FrontISTR hutathmini integrals hizi kwa nambari kwa kutumia Gauss quadrature.

Gauss quadrature na variable transformation

Gauss quadrature hukadiria integral juu ya standard domain \(\Xi\) kwa linear sum ya thamani za integrand na weights \(\boldsymbol{\xi}_i \in \Xi\) kwenye integration points \(w_i\). Matumizi yake kwenye element domain \(\Omega^e\) yanahusisha variable transformation kupitia mapping \(\boldsymbol{x}: \Xi \to \Omega^e\),

\[ \int_{\Omega^e} f(\boldsymbol{x})\, dv \approx \sum_{i=1}^{n_q} w_i\, f(\boldsymbol{x}(\boldsymbol{\xi}_i))\, J_{\xi_i}, \qquad J_{\xi_i} = \left.\det\!\left(\frac{\partial \boldsymbol{x}}{\partial \boldsymbol{\xi}}\right)\right|_{\boldsymbol{\xi}_i} \]

ambapo \(n_q\) ni idadi ya integration points na \(J_{\xi_i}\) ni determinant ya transformation Jacobian. Standard domain \(\Xi\) huamuliwa kwa kila aina ya elementi (hexahedron ni \([-1,1]^3\), huku triangle, tetrahedron na wedge zikitumia reference shapes zao), na integration points \(\boldsymbol{\xi}_i\) pamoja na weights \(w_i\) hutolewa kama jedwali la nambari. Surface integrals hushughulikiwa kwa umbo lilelile kwa ku-map element face kutoka standard domain ya dimensionality mbili.

Idadi ya kawaida ya integration points inayotumiwa na FrontISTR imeonyeshwa hapa chini (kwa uhusiano na element types, rejelea Mfumo wa namba za elementi na shape-function library).

Aina ya elementi Integration formula Idadi ya integration points
Tetrahedron ya nodi 4 (tet4n) Formula ya point 1 1
Tetrahedron ya nodi 10 (tet10n) Formula ya point 4 4
Triangular prism ya nodi 6 (prism6n) Formula ya point 2 2
Triangular prism ya nodi 15 (prism15n) Formula ya point 9 9
Hexahedron ya nodi 8 (hex8n) 2×2×2 Gauss-Legendre 8
Hexahedron ya nodi 20 (hex20n) 3×3×3 Gauss-Legendre 27
Quadrilateral ya nodi 4 (quad4n) 2×2 Gauss-Legendre 4
Quadrilateral ya nodi 8 (quad8n) 3×3 Gauss-Legendre 9
Triangle ya nodi 3 (tri3n) Formula ya point 1 1
Triangle ya nodi 6 (tri6n) Formula ya point 3 3

Kwa hexahedron, quadrilateral na line elements, tensor product ya Gauss-Legendre formula katika kila axis hutumiwa. Triangle, tetrahedron na triangular prism hutumia formula maalum zinazofaa simplex shapes, yaani point arrangements zinazointegrate polynomials kwa usahihi moja kwa moja juu ya triangle au reference shape husika.

Matumizi katika element integration

Katika element integration, natural coordinates \(\boldsymbol{r}\) hutumiwa kama coordinates za standard domain (\(\boldsymbol{r} = \boldsymbol{\xi}\)), na mapping kwenda physical coordinates hutolewa na interpolation ya nodal coordinates kupitia shape functions. Kulingana na uchaguzi wa reference configuration (Mfumo wa incremental analysis), hutumiwa kama ifuatavyo.

Total Lagrange method (integration juu ya reference configuration \(\Omega^e_0\)): mapping na Jacobian ni

\[ \boldsymbol{X}^e(\boldsymbol{r}) = \sum_{\alpha} N_\alpha^e(\boldsymbol{r})\, \boldsymbol{X}^e_\alpha, \qquad J_{r_i} = \left.\det\!\left(\frac{\partial \boldsymbol{X}}{\partial \boldsymbol{r}}\right)\right|_{\boldsymbol{r}_i} \]

na element internal force na stiffness matrix hukadiriwa kwa

\[ \boldsymbol{Q}^e \approx \sum_{i=1}^{n_q} w_i\, (\boldsymbol{B}_L + \boldsymbol{B}_{NL})^T\, \boldsymbol{S}\, J_{r_i}, \qquad \boldsymbol{K}^e \approx \sum_{i=1}^{n_q} w_i\, \boldsymbol{K}^e_{x}\, J_{r_i} \]

\(\boldsymbol{B}_L, \boldsymbol{B}_{NL}, \boldsymbol{S}, \boldsymbol{K}^e_{x}\) zote hutathminiwa kwenye integration point \(\boldsymbol{r}_i\).

Updated Lagrange method (integration juu ya current configuration \(\Omega^e\)): mapping na Jacobian ni

\[ \boldsymbol{x}^e(\boldsymbol{r}) = \sum_{\alpha} N_\alpha^e(\boldsymbol{r})\, \boldsymbol{x}^e_\alpha = \sum_{\alpha} N_\alpha^e(\boldsymbol{r})\, (\boldsymbol{X}^e_\alpha + \boldsymbol{u}^e_\alpha), \qquad J_{r_i} = \left.\det\!\left(\frac{\partial \boldsymbol{x}}{\partial \boldsymbol{r}}\right)\right|_{\boldsymbol{r}_i} \]

na

\[ \boldsymbol{q}^e \approx \sum_{i=1}^{n_q} w_i\, \boldsymbol{B}_L^T\, \boldsymbol{\sigma}\, J_{r_i}, \qquad \boldsymbol{K}^e \approx \sum_{i=1}^{n_q} w_i\, \boldsymbol{K}^e_{x}\, J_{r_i} \]

hutumiwa.

Tofauti kati ya formulations hizi mbili ni point moja tu: nodal coordinates zinazotumika kama input ya mapping ni \(\boldsymbol{X}^e_\alpha\) au \(\boldsymbol{x}^e_\alpha\). Muundo wa integration points, weights na integration-point loop ni wa pamoja.

Full integration na reduced-order integration

Integration inayotumia idadi ya integration points inayoweza ku-integrate kwa usahihi polynomial order ya integrand huitwa full integration; integration inayotumia kiwango kimoja cha points chache kuliko hicho huitwa reduced-order integration. Reduced-order integration hutumika kupunguza shear na volumetric locking, lakini huhitaji kushughulikia spurious deformation modes kama hourglass modes. Idadi ya integration points kwa kila element type na uchaguzi wa full/reduced integration vinaelezwa kuanzia Mfumo wa namba za elementi na shape-function library na katika High-performance element formulations.

Vipengee vinavyohusiana

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