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--- team:gdeliege:nnm [2015/08/13 10:29] – geoffrey
+++ team:gdeliege:nnm [2016/03/30 15:23] (current) – external edit 127.0.0.1
@@ Line 6: / Line 6: @@
 This is one of the research topics of the
 [[http://www.ltas-s3l.ulg.ac.be/cmsms/|Space Structures & Systems Lab.]] (Prof. Kerschen).
-Actually, I knew nothing on the subject until Ludovic Renson
+Actually, I knew nothing of the subject until Ludovic Renson
 told me about it
 and about the nonlinear equations he wanted to solve with finite elements.
@@ Line 26: / Line 26: @@
   \\
   v\, \partial_u Y_i +f_k\, \partial_v Y_i &=& f_i
+  \\
+  i\in\{1,...,N\},\; i&\neq& k
 \end{eqnarray*}
 $$
- $i\in\{1,...,N\}$ ,  $i\neq k$ .
+The unknowns are now space variables  $X_i(u,v)$  and velocities  $Y_i(u,v)$ .
-The unknowns are now space variables  $X_i(u,v)$  and velocities  $Y_i(u,v)$
+The $f_i(u,v,X_j,Y_j),\; j\neq k$ are elastic and dissipative forces in the equations of motion.
-and  $f_i$  are elastic and dissipative forces in the equations of motion.
 These equations are solved in a two-dimensional invariant manifold in phase space.
- $\vec{v}=[v, f_k]$
+It turns out that the best way to solve these equations with finite elements is to define a pseudo-velocity  $\vec{v}=[v, f_k]$  and to use a classical upwinding technique (Streamline Upwind Petrov-Galerkin formulation) to deal with the oscillations induced by the convective terms. The formulation reads : \\
-Find  $X_i\in V$  et  $Y_i\in V$ , $i=\{1, \ldots, 5\}$, such that
+//Find  $X_i\in V$  et  $Y_i\in V$ , $i=\{1, \ldots, N\} $,$ i\neq k$, such that//
 $$
 \begin{eqnarray*}
@@ Line 48: / Line 49: @@
   \;{\rm d}\Omega = 0\;,
   && \forall X'_i \in V \;,
+  \\
+  V = \{ f(u,v)\in H^1(\Omega):\, f(0,0)=0\}
+  \;. &&
 \end{eqnarray*}
 $$
-with
+The domain boundary must be tangent to the velocity field to avoid problems with the definition of boundary conditions. Ludovic came up with clever and effective solutions to deal with this issue [1].
-$$
+Fig. 1 shows results I obtained with my code for a simple 2-DOF system, also described in [1]
-\begin{equation*}
-  V = \{ f(u,v)\in H^1(\Omega):\, f(0,0)=0\}
-  \;.
-\end{equation*}
-$$
+{{ :team:gdeliege:nnm02.png |}}
+//Figure 1. Finite element solution of a 2-DOF conservative system, calculated with my own code (mesh and visualization by [[http://www.geuz.org/gmsh|Gmsh]]): (1) mesh of the elliptical domain, (2) pseudo-velocity field, (3) displacement  $X2$ , (4) velocity  $Y2$ .//
+I also attempted to solve a 6-DOF system corresponding to a cantilever beam. This requires the solution of a system of 10 equations with 10 unknown fields.
+Although my code has been designed to allow the definition of an arbitrary number of unknown fields, I had never tried more than 2 or 3 fields coupled in one single formulation. The Jacobian matrix for the Newton-Raphson algorithm is written below; I wrote a Python script to define this formulation in my code avoiding copy-paste errors.
+To my own surprise, it worked, although efficiency issues allowed me to solve the system on a small domain only.
 {{ :team:gdeliege:nnm01.png |}}
+=== References ===
 [1] L. Renson, G. Deliége, G. Kerschen. //An effective finite-element-based method for the computation of nonlinear normal modes of nonconservative systems.// Meccanica, vol. 49(8), pp. 1901-1916, 2014. \\
 [2] L. Renson. //Nonlinear Modal Analysis of Conservative and Nonconservative Aerospace Structures.// PhD Thesis, Université de Liège, 2014. \\
+\\
+[[team:gdeliege|Back to main page]]