For a convergence analysis of the Euclidean norm of the gap between the contact partners with respect to the considered JIM a series of nonlinear analysis has been performed. The structure is mounted at the location of the reference frame, which is outlined green in Figure 2. The orange arrows in Figure 2 denote a force acting along the y axis and a torque acting around the x axis. The resulting deformation is outlined on the right hand side of Figure 2. Note, that a nonlinear penalty contact model has been implemented for the contact pressure. 6.1.2. Results Table 1 contains a visualization of the first seven JIM due to the different approaches. Note, that the JIM of the generalized eigenvalue bases approach (first column of Table 1) is sorted by increasing eigenvalues while the others (column 2 to 5) are sorted by decreasing eigenvalues (or Hankel singular values). Figure 3 contains on the right hand side a convergence analysis of the modelled energy with respect to the number of considered JIM according to equation (18). Using all available JIMs leads to a modelled energy of 100%. It can be seen, that already much less than the available 400 JIM covers almost all energy which can be modelled. On the left hand side the convergence of the Euclidian norm of the gap is depicted. Figure 3: Convergence of modeled energy and of Euclidian gap in the contact with respect to the number of JIM The latter 400 available JIM indicates, that the joint area is subdivided into 400 subareas in order to determine the mechanical joint characteristics, refer to [11]. In a final computation the JIM according to the POD based method of section 5.2 has been computed with 800, 400, 200, 100, 50, 25 and 13 subareas and the energy distribution has been evaluated in Figure 4. The according convergence analysis of the Euclidian norm of the gap and the joint pressure can be seen in Figure 5. 6.1.3. Conclusions The conclusions drawn from the Table 1, Figure 3, Figure 4 and Figure 5 are: x A visual evaluation of the different JIM in Table 1 reveals no significant qualitative difference. This ‘intuitive’ impression is underlined by the convergence analysis of the gap due to a certain static load. The five approaches lead to almost the same convergence rate. x Figure 3 shows, that in case of a POD based approach which uses the stiffness matrix K according to sections 5.2 and 5.4, the convergence of the energy is somehow similar to the convergence of the Euclidian norm of the gap. Consequently, the Hankel singular values can be used as kind of ‘a – priory’ estimation of the required number of JIM. Note that the POD based approaches without using the stiffness matrix K (sections 5.1 and 5.3) yields Hankel singular values which do not represent the gap convergence. This is because just the deformations itself are approximated, while in the other case the strain energy is approximated. x Figure 3 reveals that a nonlinear discretization of the joint area according to [11] is not necessary. There is no gain of accuracy while the nonlinear computation of X will take much more computational effort as the linear does. x The question whether a joint is discretized fine enough can be answered by the results depicted in Figure 4 and Figure 5. A necessary criterion for a satisfying discretization seems to be a distinct convergence to the 100% limit. For this particular example, 50 to 100 subareas lead to an energy convergence with distinct convergence characteristics to the 100% limit. 25
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