Finite Element Method - 2D Mesh Generator - Metfem2D

Generator Optimization via the Metropolis method

Boundary mesh elements

The Metropolis algorithm[1, 2] applied to boundary nodes slightly differs from the above– described case and could be summarize in the following steps:

  1. Find all the boundary or edge nodes i. e. nodes for which pk, edge ∈ Γ.
  2. Find triangles in the closest neighbourhood of the considered pk,edge node. Then calculate an area of each triangle Al,edge.
  3. Calculate the force acting on each boundary node except the constant nodes and coming from only two boundary nodes connected to it. This imposes the following constrain on the motion of the k-th node in order to keep it in the boundary Γ
  4. Fi = - j=1 Fj δ |rji|
    rji

    ,
    |rji|
    where δ|rjk| is defined as previously. The force is tangential to the boundary Γ.
  5. Similarly, find an area of each triangle Anewl,edge after shifting pk,edge → pnewk,edge according to the force Fk.
  6. Adopt the Metropolis energetic condition to the boundary case i.e.
  7. δE = l ( (Anewl, edge - A)2 - (Al, edge - A)2 ) l = 1, 2, …, L.

    If

    e-δE⁄T > r
    the new configuration is accepted otherwise is rejected. T denotes temperature and a random number r ∈ U(0; 1) as previously.
  8. The main point of this part is to ensure that the boundary nodes are moved just along the boundary Γ.

finite element method, fem, numerical integration, differential equations, lde, 2D mesh generator, Metfem2D, taketechease

References

  1. ^ O. C. Zienkiewicz, R. L. Taylor and J. Z. Zhu, The Finite Element Method: Its Basis and Fundamentals, Sixth edition., Elsevier 2005
  2. ^ N. Metropolis, S. Ulam, The Monte Carlo Method, J. Amer. Stat. Assoc., 44, No. 247., pp. 335-341, 1949