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Figure 11. Adapted meshes (1%, 3, 5", 7", 9" and 12") using 1:2 refinement The improvement in the convergence rate is due to the progressive reduction and confinement of the error throughout the adaptation process. In figure 14, the distribution of the error density in the eigenvector (the electric field) for the meshes of the adaptive process (figure 12) is shown. Comparing these distributions with those obtained from the uniform meshes (figure 15) it is evident that the adaptation reduces the area with the highest error.

Figure 11 Adapted meshes (1%, 3, 5", 7", 9" and 12") using 1:2 refinement The improvement in the convergence rate is due to the progressive reduction and confinement of the error throughout the adaptation process. In figure 14, the distribution of the error density in the eigenvector (the electric field) for the meshes of the adaptive process (figure 12) is shown. Comparing these distributions with those obtained from the uniform meshes (figure 15) it is evident that the adaptation reduces the area with the highest error.

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Figure 1 shows the flow diagram of a general adaptive procedure, where, at each iterative step, the problem is solved, an indication of the error for each element is obtained and the mesh is refined in those zones with a bigger error.
Figure 1 shows the flow diagram of a general adaptive procedure, where, at each iterative step, the problem is solved, an indication of the error for each element is obtained and the mesh is refined in those zones with a bigger error.
where 7° is the solution of the FEM in the element i and 7* is that solution at the edge k. The internal and singular residuals are: tensor a , and p is the degree of the interpolation functions employed in the FEM (in this work p=1). This measure of the error is qualitative, that is, allows to detect elements or zones in the mesh with a bigger error, but it does not provide a bound of the error in the problem.
where 7° is the solution of the FEM in the element i and 7* is that solution at the edge k. The internal and singular residuals are: tensor a , and p is the degree of the interpolation functions employed in the FEM (in this work p=1). This measure of the error is qualitative, that is, allows to detect elements or zones in the mesh with a bigger error, but it does not provide a bound of the error in the problem.
is the maximum error in the mesh and ck (0<xK <1) isa parameter which controls the number of elements to be refined and, therefore, the number of new elements at each step of the adaptive process. When x is close to 1, the refinement takes place only at the elements with an error measure close to the maximum. This behavior implies a lot of steps in the adaptive process for obtaining accurate results. On the other hand, when « is close to 0, a great quantity of elements are refined, even those with a little error. So, the computational cost of the FEM solver is highly increased and most of the new unknowns do not improve the accuracy of the solution. The results presented in this paper have been obtained with « =0.5. Figure 2 shows a scheme of this refinement.
is the maximum error in the mesh and ck (0<xK <1) isa parameter which controls the number of elements to be refined and, therefore, the number of new elements at each step of the adaptive process. When x is close to 1, the refinement takes place only at the elements with an error measure close to the maximum. This behavior implies a lot of steps in the adaptive process for obtaining accurate results. On the other hand, when « is close to 0, a great quantity of elements are refined, even those with a little error. So, the computational cost of the FEM solver is highly increased and most of the new unknowns do not improve the accuracy of the solution. The results presented in this paper have been obtained with « =0.5. Figure 2 shows a scheme of this refinement.
In an h-refinement approach, there are different ways for refining the mesh. For instance, by adding new nodes in the selected elements and running a Delaunay triangulation [9]. In this work, procedures based on the “longest edge” bisection [10] and the regular split of triangles have been employed. The first technique splits the triangle up into two new elements (figure 4), and the second one into four elements with the same aspect ratio (figure 5a). Moreover, a maximum angle criterion (figure 5b) has been applied in order to obtain an additional improvement in the aspect ratio of some triangles. It is very important to maintain or improve the regularity of the elements throughout the adaptive procedure because the exactness of the FEM solution is directly related to it.
In an h-refinement approach, there are different ways for refining the mesh. For instance, by adding new nodes in the selected elements and running a Delaunay triangulation [9]. In this work, procedures based on the “longest edge” bisection [10] and the regular split of triangles have been employed. The first technique splits the triangle up into two new elements (figure 4), and the second one into four elements with the same aspect ratio (figure 5a). Moreover, a maximum angle criterion (figure 5b) has been applied in order to obtain an additional improvement in the aspect ratio of some triangles. It is very important to maintain or improve the regularity of the elements throughout the adaptive procedure because the exactness of the FEM solution is directly related to it.
Figure 6. Non-conformal meshes: a) initial mesh, b) first refinement, c) second refinement (triangles). In order to obtain a conformal mesh, that is, to avoid “hanging nodes”, a new generation of elements is needed (figure 7).
Figure 6. Non-conformal meshes: a) initial mesh, b) first refinement, c) second refinement (triangles). In order to obtain a conformal mesh, that is, to avoid “hanging nodes”, a new generation of elements is needed (figure 7).
Figure 8. Solution of non-conformal cases
Figure 8. Solution of non-conformal cases
The quality of the adaptive procedure can be established by comparing the convergence of the solution (the eigenvalue) with that obtained from the classical FEM. The convergence for this one has been obtained using different uniform meshes (figure 10) for the same problem, which have been generated, as the graded meshes for the other examples, by means on an hybrid advancing front / interpolation method [11] [12]. The first of these meshes is, also, the initial mesh for the adaptive process.
The quality of the adaptive procedure can be established by comparing the convergence of the solution (the eigenvalue) with that obtained from the classical FEM. The convergence for this one has been obtained using different uniform meshes (figure 10) for the same problem, which have been generated, as the graded meshes for the other examples, by means on an hybrid advancing front / interpolation method [11] [12]. The first of these meshes is, also, the initial mesh for the adaptive process.
Figure 9. Transversal component of the first-mode electric field in an L-shaped homogeneous waveguide
Figure 9. Transversal component of the first-mode electric field in an L-shaped homogeneous waveguide
Figure 12. Adapted meshes (1°, 2", 3", 4", 5 ane 6") using 1:4 refinement
Figure 12. Adapted meshes (1°, 2", 3", 4", 5 ane 6") using 1:4 refinement
Figure 14. Distribution of the estimated error density in the adapted meshes
Figure 14. Distribution of the estimated error density in the adapted meshes
Figure 15. Distribution of the estimated error density in the uniform meshes
Figure 15. Distribution of the estimated error density in the uniform meshes
Figure 16. Cross-section of the microstrip line Figure 20 shows the convergence of the phase constant for the adaptation process compared with the convergence for consecutive graded meshes (figure 18). The convergence is clearly accelerated with the adaptive refinement.
Figure 16. Cross-section of the microstrip line Figure 20 shows the convergence of the phase constant for the adaptation process compared with the convergence for consecutive graded meshes (figure 18). The convergence is clearly accelerated with the adaptive refinement.
Figure 17. Transversal component of the first mode electric field in the microstrip line
Figure 17. Transversal component of the first mode electric field in the microstrip line
Figure 19. Adapted meshes (1%, 2"°, 3, 4°", 5" and 6") using 1:4 refinement Figure 18. Graded meshes in the microstrip line
Figure 19. Adapted meshes (1%, 2"°, 3, 4°", 5" and 6") using 1:4 refinement Figure 18. Graded meshes in the microstrip line
Figure 20. Convergence in the microstrip line 3.3. Shielded Unilateral Finline
Figure 20. Convergence in the microstrip line 3.3. Shielded Unilateral Finline
In the transversal component (the dominant one) of the first- mode electric field (figure 22) appears again a singularity, in this case near the edge of the strip.
In the transversal component (the dominant one) of the first- mode electric field (figure 22) appears again a singularity, in this case near the edge of the strip.
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Adaptive Mesh RefinementAdaptive Mesh
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