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--- doc:user:integration:scheme:dynexpl [2013/07/12 15:23] – created joris
+++ doc:user:integration:scheme:dynexpl [2022/12/21 11:35] (current) – [New Metafor Version > 2422] boman
@@ Line 1: / Line 1: @@
+====== Explicit dynamic integration schemes ======
+===== Description =====
+The equilibrium equation between internal forces $F^{int}$, inertial forces $Ma$ (where $M$ is the diagonalized mass matrix and $a$ the acceleration) and external forces $F^{ext}$ :
+$$Ma+F^{int}=F^{ext}$$
+==== Central difference method ====
+Relations between displacements $x$, velocities $v$ and accelerations $a$ are:
+$$v(t^{n+1/2}) = v(t^{n-1/2}) + (t^{n+1}-t^n) a(t^n) $$ \\
+$$x(t^{n+1}) = x(t^n) + (t^{n+1}-t^n) v(t^{n+1/2}) $$
+The equilibrium equation becomes :
+$$a(t^{n+1}) = (F^{ext}(t^{n+1}) - F^{int}(t^{n+1}))/M $$
+This scheme is conditionally stable (time step limited) and non dissipative.
+==== Alpha-generalized scheme ====
+Same relations as in the implicit [[dynimpl|alpha-generalized]] scheme, but with the parameter used to weight internal and external forces equal to 1, leading to :
+$$(1-\alpha_M) a(t^{n+1}) + \alpha_M a(t^n) = \frac{F^{ext}(t^n) - F^{int}(t^n)}{M}$$
+Relations between displacements $x$, velocities $v$ and accelerations $a$ are:
+$$x(t^{n+1}) = x(t^n) + (t^{n+1}-t^n) v(t^n) + (t^{n+1}-t^n)^2 \left( (0.5-\beta)a(t^n) + \beta a(t^{n+1})\right) $$
+$$v(t^{n+1}) = v(t^n) + (t^{n+1}-t^n) {(1-\gamma)a(t^n) + \gamma a(t^{n+1})} $$
+Specific values leading to an optimal numerical dissipation are given as function of the spectral radius $\rho_\beta$ (''MDR_ECHR'') for a bifurcation frequency (a spectral radius equal to 1 leads to a conservative algorithm when a spectral radius lower than 1 leads to a dissipative one ((see real parameters)):
+$$\alpha_M = (2\rho_\beta-1)/(1+\rho_\beta) $$\\
+$$\gamma = 3/2 - \alpha_M $$\\
+$$\beta = \frac{5-3\rho_\beta}{(1+\rho_\beta)^2 (2-\rho_\beta)}$$
+Conditionally stable.
+==== Tchamwa Scheme ====
+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:
+$$x(t^{n+1}) = x(t^n) + (t^{n+1}-t^n) v(t^n) + \phi (t^{n+1}-t^n)^2 a(t^n) $$\\
+$$v(t^{n+1}) = v(t^n) + (t^{n+1}-t^n) a(t^n) $$
+Stability guaranteed for  $\phi \geq 1 $ and high frequencies killed over a single time step for \phi = 2$. the scheme is of :
+  * second order for $\phi = 1$ (no numerical dissipation)
+  * first order for $\phi > 1$ (numerical dissipation)
+Relation between $\phi$ and spectral radius for the bifurcation $\rho_\beta$ (user parameter ''MDR_ECHR'') is:
+  * $$\phi = \frac{2(1- \rho_\beta^{1/2})}{(1-\rho_\beta)} \mbox{ if } \rho_\beta < 1 $$
+  * $$\phi = 1 \mbox{ if } \rho_\beta = 1 $$
+===== Input file =====
+See [[dynimpl|dynamic implicit]] scheme for definition of density and initial velocities.
+==== Old Metafor Version <= 2422 ====
+=== Choosing the algorithm ===
+^       Scheme         ^  ''MDE_NDYN''  ^  ''MDR_ECHR''  ^
+| Certered difference  |       1        |                |
+| Chung Hulbert        |       3        |       X        |
+| Tchamwa              |       6        |       X        |
+(see [[doc:user:integration:general:parameters]])
+==== New Metafor Version > 2422 ====
+=== Centered Difference ===
+<code>
+ti = CentralDifferenceTimeIntegration(metafor)
+metafor.setTimeIntegration(ti)
+</code>
+=== Chung Hulbert ===
+<code>
+ti = ChExplicitTimeIntegration(metafor)
+ti.setRhoB(_rhoB)
+metafor.setTimeIntegration(ti)
+</code>
+The parameter ''_rhoB'' is the spectral radius at bifurcation point ([0, 1]). The default value is 0.8182.
+=== Tchamwa ===
+<code>
+ti = TchamwaExplicitTimeIntegration(metafor)
+ti.setRhoB(_rhoB)
+metafor.setTimeIntegration(ti)
+</code>
+The parameter ''_rhoB'' is the spectral radius at bifurcation point ([0, 1]). The default value is 0.8182.
+Other parameters : see [[quasistatique]]