doc:user:integration:scheme:dynexpl
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doc:user:integration:scheme:dynexpl [2013/07/12 15:24] – joris | doc:user:integration:scheme:dynexpl [2022/12/21 11:35] (current) – [New Metafor Version > 2422] boman | ||
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- | ====== | + | ====== |
===== Description ===== | ===== Description ===== | ||
- | Il s'agit d' | + | The equilibrium equation between internal |
$$Ma+F^{int}=F^{ext}$$ | $$Ma+F^{int}=F^{ext}$$ | ||
- | ==== Le schéma de la différence centrée | + | ==== Central difference method |
- | Les relations entre les déplacements | + | Relations between displacements |
$$v(t^{n+1/ | $$v(t^{n+1/ | ||
$$x(t^{n+1}) = x(t^n) + (t^{n+1}-t^n) v(t^{n+1/ | $$x(t^{n+1}) = x(t^n) + (t^{n+1}-t^n) v(t^{n+1/ | ||
- | L' | + | The equilibrium equation becomes |
$$a(t^{n+1}) = (F^{ext}(t^{n+1}) - F^{int}(t^{n+1}))/ | $$a(t^{n+1}) = (F^{ext}(t^{n+1}) - F^{int}(t^{n+1}))/ | ||
- | Ce schéma est conditionnellement | + | This scheme is conditionally |
- | ==== Le schéma alpha-généralisé | + | ==== Alpha-generalized scheme |
- | Il s'agit des mêmes | + | Same relations |
- | $$(1-\alpha_M) a(t^{n+1}) + \alpha_M a(t^n) = \frac{F^{ext}(t^n) - Fint(t^n)}{M}$$ | + | $$(1-\alpha_M) a(t^{n+1}) + \alpha_M a(t^n) = \frac{F^{ext}(t^n) - F^{int}(t^n)}{M}$$ |
- | Les relations entre les déplacements | + | Relations between displacements |
$$x(t^{n+1}) = x(t^n) + (t^{n+1}-t^n) v(t^n) + (t^{n+1}-t^n)^2 \left( (0.5-\beta)a(t^n) + \beta a(t^{n+1})\right) $$ | $$x(t^{n+1}) = x(t^n) + (t^{n+1}-t^n) v(t^n) + (t^{n+1}-t^n)^2 \left( (0.5-\beta)a(t^n) + \beta a(t^{n+1})\right) $$ | ||
$$v(t^{n+1}) = v(t^n) + (t^{n+1}-t^n) {(1-\gamma)a(t^n) + \gamma a(t^{n+1})} $$ | $$v(t^{n+1}) = v(t^n) + (t^{n+1}-t^n) {(1-\gamma)a(t^n) + \gamma a(t^{n+1})} $$ | ||
- | Les valeurs particulières des paramètres de pondération qui conduisent à une dissipation | + | Specific values leading to an optimal numerical |
$$\alpha_M = (2\rho_\beta-1)/ | $$\alpha_M = (2\rho_\beta-1)/ | ||
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$$\beta = \frac{5-3\rho_\beta}{(1+\rho_\beta)^2 (2-\rho_\beta)}$$ | $$\beta = \frac{5-3\rho_\beta}{(1+\rho_\beta)^2 (2-\rho_\beta)}$$ | ||
- | Ce schéma est conditionnellement | + | Conditionally |
- | ==== Le schema Tchamwa ==== | + | |
- | Algorithme explicite avec dissipation numérique controlée par le paramètre $\phi$. | + | ==== Tchamwa Scheme ==== |
- | L' | ||
- | $$a(t^{n+1}) = \frac{Fext(t^{n+1}) - Fint(t^{n+1})}{M}$$ | ||
- | Les relations entre les déplacements | + | Explicit algorithm where numerical dissipation is monitored by the parameter $\phi$. |
+ | |||
+ | Equilibrium computed with | ||
+ | |||
+ | $$a(t^{n+1}) = \frac{F^{ext}(t^{n+1}) - F^{int}(t^{n+1})}{M}$$ | ||
+ | |||
+ | Relations between displacements $x$, velocities $v$ and accelerations $a$ are: | ||
$$x(t^{n+1}) = x(t^n) + (t^{n+1}-t^n) v(t^n) + \phi (t^{n+1}-t^n)^2 a(t^n) $$\\ | $$x(t^{n+1}) = x(t^n) + (t^{n+1}-t^n) v(t^n) + \phi (t^{n+1}-t^n)^2 a(t^n) $$\\ | ||
$$v(t^{n+1}) = v(t^n) + (t^{n+1}-t^n) a(t^n) $$ | $$v(t^{n+1}) = v(t^n) + (t^{n+1}-t^n) a(t^n) $$ | ||
- | La stabilité est assurée pour $\phi \geq 1 $ et les hautes fréquences sont annihilées en un pas de temps pour $\phi = 2$. Le schema est d' | + | Stability guaranteed for |
- | * 2 pour $\phi = 1$ (pas de dissipation | + | * second order for $\phi = 1$ (no numerical |
- | * 1 pour $\phi > 1$ (dissipation | + | * first order for $\phi > 1$ (numerical |
+ | |||
+ | Relation between $\phi$ and spectral radius for the bifurcation $\rho_\beta$ (user parameter '' | ||
+ | * $$\phi = \frac{2(1- \rho_\beta^{1/ | ||
+ | * $$\phi = 1 \mbox{ if } \rho_\beta = 1 $$ | ||
+ | |||
+ | ===== Input file ===== | ||
+ | |||
+ | See [[dynimpl|dynamic implicit]] scheme for definition of density and initial velocities. | ||
+ | |||
+ | ==== Old Metafor Version <= 2422 ==== | ||
+ | |||
+ | === Choosing the algorithm === | ||
+ | |||
+ | ^ | ||
+ | | Certered difference | ||
+ | | Chung Hulbert | ||
+ | | Tchamwa | ||
+ | |||
+ | (see [[doc: | ||
+ | |||
+ | ==== New Metafor Version > 2422 ==== | ||
+ | |||
+ | === Centered Difference === | ||
+ | |||
+ | < | ||
+ | ti = CentralDifferenceTimeIntegration(metafor) | ||
+ | metafor.setTimeIntegration(ti) | ||
+ | </ | ||
- | La relation entre $\phi$ et le rayon spectral à la bifurcation $\rho_\beta$ (paramètre utilisateur '' | + | === Chung Hulbert === |
- | * $$\phi | + | |
- | * $$\phi | + | |
- | ===== Jeu de données ===== | + | < |
+ | ti = ChExplicitTimeIntegration(metafor) | ||
+ | ti.setRhoB(_rhoB) | ||
+ | metafor.setTimeIntegration(ti) | ||
+ | </ | ||
- | Voir schéma | + | The parameter '' |
- | ==== Choisir l' | + | === Tchamwa |
- | ^ | + | < |
- | | Chung Hulbert | + | ti = TchamwaExplicitTimeIntegration(metafor) |
- | | Différence centrée | + | ti.setRhoB(_rhoB) |
- | | Tchamwa | + | metafor.setTimeIntegration(ti) |
+ | </ | ||
- | (voir [[doc: | + | The parameter '' |
- | Paramètres supplémentaires: voir [[quasistatique]] | + | Other parameters |
doc/user/integration/scheme/dynexpl.1373635459.txt.gz · Last modified: 2016/03/30 15:22 (external edit)