Difference between revisions of "Adaptivity"

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f_i = \begin{cases}
 
f_i = \begin{cases}
 
1 + \frac{\eta - \varepsilon_i}{\eta - m} (\frac{1}{\beta} - 1), & \varepsilon_i \leq \eta, \quad \text{i.e.\ decrease the density} \\
 
1 + \frac{\eta - \varepsilon_i}{\eta - m} (\frac{1}{\beta} - 1), & \varepsilon_i \leq \eta, \quad \text{i.e.\ decrease the density} \\
1, & \eta < \varepsilon_i < \eps,  \quad \text{i.e.\ no change in density}\\
+
1, & \eta < \varepsilon_i < \varepsilon,  \quad \text{i.e.\ no change in density}\\
1 + \frac{\varepsilon_i - \eps}{M - \eps} (\alpha - 1), & \varepsilon_i \geq \eps, \quad \text{i.e.\ increase the density}
+
1 + \frac{\varepsilon_i - \varepsilon}{M - \varepsilon} (\alpha - 1), & \varepsilon_i \geq \eps, \quad \text{i.e.\ increase the density}
 
\end{cases}
 
\end{cases}
 
$$
 
$$

Revision as of 13:15, 11 June 2019

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Basic concept

The adaptive methodology in this paper behaves similarly to "remeshing" used commonly in FEM. Some initial (possibly variable) nodal spacing h^0 is chosen, as well as its lower and upper bounds h_L and h_U, respectively. 3 Domain \Omega is filled with nodes, conforming to h^0 and the solution u^0 is obtained. An error indicator is employed to determine which nodes should be (de)refined and the nodal density h^0 is altered appropriately. This adaptive cycle below is repeated until the convergence criterion is met. The procedure on j-th iteration is written in more detail below:

  1. Fill \Omega with nodes conforming to h^j.
  2. Solve the problem to obtain u^j.
  3. Compute the error indicator values \varepsilon_i^j for each node p_i.
  4. If the mean of \varepsilon_i^j is below some tolerance \varepsilon return u^j as the solution and stop.
  5. Adapt h^j to obtain h^{j+1}.

More details can be found in our paper: https://arxiv.org/abs/1811.10368

Node density adaptation

f_i = \begin{cases} 1 + \frac{\eta - \varepsilon_i}{\eta - m} (\frac{1}{\beta} - 1), & \varepsilon_i \leq \eta, \quad \text{i.e.\ decrease the density} \\ 1, & \eta < \varepsilon_i < \varepsilon, \quad \text{i.e.\ no change in density}\\ 1 + \frac{\varepsilon_i - \varepsilon}{M - \varepsilon} (\alpha - 1), & \varepsilon_i \geq \eps, \quad \text{i.e.\ increase the density} \end{cases}



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