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--- doc:user:integration:scheme:dynimpl [2015/10/09 16:54] – [New Metafor Version > 2422] wautelet
+++ doc:user:integration:scheme:dynimpl [2017/06/16 10:28] (current) – papeleux
@@ Line 71: / Line 71: @@
+==== 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.
+<code>
+ref 1 : "The analysis of the Generalized- $\alpha$  method for non linear dynamic problems" -
+        S.Erlicher, L.Bonaventura, O.S.Bursi -  Computanional Mechanics 28 (2002) 83-104
+ref 2 : "Théorie des vibrations - Application à la dynamique des structures" -
+         M.Géradin, D.Rixen - Editions Masson
+</code>
+=== Parameters to the scheme ===
+  * Name of the scheme : ''DampedAlphaGeneralizedTimeIntegration''
+  * Parameters :
+    * AlphaGeneralizedTimeIntegration parameters has to be defined as usual
+    * update of the damping matrix managed through ''ti.setDampingMatrixUpdate(DMUxxx)''
+      * 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:user:elements:volumes:volumeelement?&#parameters|element properties]] (''DAMPSTIFF'',''DAMPMASS''). Parameters can depend of time.
 ==== Generalized Midpoint Rule ====
-The equilibrium i computed for a time defined by the parameter  $\theta$  (''MDR_THEM'') (see [[doc:user:integration:general:parameters]]):
+The equilibrium is computed for a time defined by the parameter  $\theta$  (''MDR_THEM'') (see [[doc:user:integration:general:parameters]]):
  $\boldsymbol{F}^{\text{inert}}(t^{n+\theta}) + \boldsymbol{F}^{\text{int}}(t^{n+\theta})= \boldsymbol{F}^{\text{ext}}(t^{n+\theta})$
@@ Line 173: / Line 207: @@
 (see [[doc:user:integration:general:parameters]])
-==== New Metafor Version > 2422 ====
+=== New Metafor Version > 2422 ===
-=== Consistent Time Integration (EDMA or EDMC) ===
+== Consistent Time Integration (EMCA or EDMC) ==
 <code>
@@ Line 182: / Line 216: @@
 </code>
-=== Chung Hulbert ===
+== Generalized Midpoint Rule ==
 <code>
-ti = ChExplicitTimeIntegration(metafor)
+ti = MpgTimeIntegration(metafor)
-ti.setRhoB(_rhoB)
+ti.setTheta(_theta)
 metafor.setTimeIntegration(ti)
 </code>
-The parameter "_rhoB" is the spectral radius at bifurcation point ([0, 1]). The default value is 0.8182.
+The parameter "_theta" is defined above. The default value is 1.1.
-=== Tchamwa ===
+== Alpha Generalized Family ==
 <code>
-ti = TchamwaTimeIntegration(metafor)
+ti = AlphaGeneralizedTimeIntegration(metafor)
-ti.setRhoB(_rhoB)
+ti.setAlphaM(_AlphaM)
+ti.setAlphaF(_AlphaF)
+ti.setBeta0(_Beta0)
+ti.setGamma0(_Gamma0)
 metafor.setTimeIntegration(ti)
 </code>
-The parameter "_rhoB" is the spectral radius at bifurcation point ([0, 1]). The default value is 0.8182.
+The parameters "_AlphaM", "_AlphaF",  "_Beta0" and "_Gamma0" are defined above. The default values are -0.97, 0.01, 0.25 and 0.5 respectively.
 Useful parameters : see [[quasistatique|quasi-statique]].