Differences

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--- doc:user:integration:scheme:dynimpl [2016/01/18 12:31] – [Generalized Midpoint Rule] papeleux
+++ doc:user:integration:scheme:dynimpl [2017/06/16 10:27] – 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 :
+$$\boldsymbol{C} =  a_m \boldsymbol{M} + a_k \boldsymbol{K}$$
+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 ====
@@ Line 195: / Line 229: @@
 <code>
-ti = AlphaGeneralizedIntegration(metafor)
+ti = AlphaGeneralizedTimeIntegration(metafor)
 ti.setAlphaM(_AlphaM)
 ti.setAlphaF(_AlphaF)