doc:user:integration:scheme:dynimpl
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doc:user:integration:scheme:dynimpl [2015/10/09 17:01] – [New Metafor Version > 2422] wautelet | doc:user:integration:scheme:dynimpl [2017/06/16 10:28] (current) – papeleux | ||
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+ | ==== Damped Alpha-Generalized family ==== | ||
+ | Adding damping forces to the Alpha-Generalized time integration scheme family allow user to dissipate energy in a regulated way. According to the reference below, damping forces can be balanced between previous and current time as others (internal or external) forces through $\alpha_F$ parameter. The global system is then written by : | ||
+ | |||
+ | $$(1-\alpha_M) \boldsymbol{F}^{\text{inert}}(t^{n+1}) + \alpha_M \boldsymbol{F}^{\text{inert}}(t^n) + (1-\alpha_F) \boldsymbol{F}^{\text{damp}}(t^{n+1}) + \alpha_F \boldsymbol{F}^{\text{damp}}(t^n)$$ | ||
+ | $$+ (1-\alpha_F) \boldsymbol{F}^{\text{int}}(t^{n+1}) + \alpha_F \boldsymbol{F}^{\text{int}}(t^n) | ||
+ | = (1-\alpha_F) \boldsymbol{F}^{\text{ext}}(t^{n+1}) + \alpha_F \boldsymbol{F}^{\text{ext}}(t^n)$$ | ||
+ | |||
+ | where the damping forces are computed proportional to velocities : | ||
+ | $$ \boldsymbol{F}^{\text{damp}} = \boldsymbol{C} * v $$ | ||
+ | |||
+ | According to ref2 (chapter 3), an easy way to build a diagonal damping matrix is to compute a ponderated sum of the mass and stiffness matrices : | ||
+ | $$\boldsymbol{C} = a_k \boldsymbol{K} + a_m \boldsymbol{M}$$ | ||
+ | |||
+ | The modal damping factor corresponding to each eigen pulsation $\omega_{0r} = 2 \pi \phi$: | ||
+ | $$\epsilon_r=\frac{1}{2}(a_k \omega_{0r} + \frac{a_m}{\omega_{0r}} )$$ | ||
+ | |||
+ | telling us that the mass damping factor $a_m$ will induice damping on lowest eigen frequencies and that the stiff damping factor $a_k$ will induce damping on the higher eigen frequencies. | ||
+ | |||
+ | < | ||
+ | ref 1 : "The analysis of the Generalized-$\alpha$ method for non linear dynamic problems" | ||
+ | S.Erlicher, L.Bonaventura, | ||
+ | ref 2 : " | ||
+ | | ||
+ | </ | ||
+ | === Parameters to the scheme === | ||
+ | * Name of the scheme : '' | ||
+ | * Parameters : | ||
+ | * AlphaGeneralizedTimeIntegration parameters has to be defined as usual | ||
+ | * update of the damping matrix managed through '' | ||
+ | * DMUINIT : Damping matrix computed on initial configuration only | ||
+ | * DMUPERSTAGE : Damping matrix computed at each stage change | ||
+ | * DMUPERSTEP : Damping matrix computed at the beginning of each time step (base on previous equilibrated solution to avoid need of stiffness computation) | ||
+ | * Mass Damping Factor and Stiffness Damping Factor are defined though the [[doc: | ||
+ | | ||
==== Generalized Midpoint Rule ==== | ==== Generalized Midpoint Rule ==== | ||
- | The equilibrium | + | The equilibrium |
$$\boldsymbol{F}^{\text{inert}}(t^{n+\theta}) + \boldsymbol{F}^{\text{int}}(t^{n+\theta})= \boldsymbol{F}^{\text{ext}}(t^{n+\theta})$$ | $$\boldsymbol{F}^{\text{inert}}(t^{n+\theta}) + \boldsymbol{F}^{\text{int}}(t^{n+\theta})= \boldsymbol{F}^{\text{ext}}(t^{n+\theta})$$ | ||
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(see [[doc: | (see [[doc: | ||
- | ==== New Metafor Version > 2422 ==== | + | === New Metafor Version > 2422 === |
- | === Consistent Time Integration (EMCA or EDMC) === | + | == Consistent Time Integration (EMCA or EDMC) == |
< | < | ||
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</ | </ | ||
- | === Generalized Midpoint Rule === | + | == Generalized Midpoint Rule == |
< | < | ||
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The parameter " | The parameter " | ||
- | === Alpha Generalized Family | + | == Alpha Generalized Family == |
< | < | ||
- | ti = AlphaGeneralizedIntegration(metafor) | + | ti = AlphaGeneralizedTimeIntegration(metafor) |
ti.setAlphaM(_AlphaM) | ti.setAlphaM(_AlphaM) | ||
ti.setAlphaF(_AlphaF) | ti.setAlphaF(_AlphaF) |
doc/user/integration/scheme/dynimpl.1444402900.txt.gz · Last modified: 2016/03/30 15:22 (external edit)