7.2 Convergence behavior of the SEM
7.2.1 h-Refinement in x 1 -direction
The model set-up and the strategy to evaluate the results have already been explained in Section 7.1. Therefore, only the convergence graphs are shown at this point, cf. Figs. 7.5 and 7.6. The results of the convergence studies are obtained using a uniform h-refinement.
Figs. 7.5 and 7.6 display the relative error in the time-of-flight or the area below the envelope versus the number nodes per anti-symmetric wavelength χA0. Each sub-figure represents the convergence results for a fixed polynomial degree in the in-plane direction (x1). It should be noted that the results of the proposed lumping techniques are virtually coincidental. There are no noticeable differences in the convergence behavior concerning the individual diagonalized mass matrix SEM approaches.
In reference [152], Dauksher and Emery already pointed out that a row-summing proce-dure can result in accurate solutions with SEs using a CGL nodal distribution. They were able to show that the solution characteristics of SEs based on a Chebyshev grid are only negligibly affected by the diagonalization [151, 152]. Additional insights are that an in-creased polynomial order moderates the effects of lumping and that no negative diagonal terms are generated. These properties are confirmed for SEs based on a GLL grid, cf.
Figs. 7.5 and 7.6. Moreover, Dauksher and Emery claim that for a given accuracy -their results indicate that explicit spectral solutions require fewer nodes per wavelength than comparable consistent mass matrix solutions [151]. Our results confirm these find-ings of Dauksher and Emery [151, 152] and highlight the fact that the different lumping techniques essentially produce very similar results. Generally speaking, we found that the relative error converges faster to zero with an increasing FE shape function order - and a finer grid resolution also serves to reduce the error in the time-of-flight.
The main conclusion that should be drawn considering the presented results is that none of the lumping techniques applied to the SEM causes deteriorated results in this particular example. The advantages of a diagonal mass matrix can consequently be exploited for wave propagation analysis using the SEM.
The peaks that are observed in the convergence graphs are explained in detail in Sec-tion 7.8. There, we found that this behavior can be attributed to eigenfrequencies on the element level.
Convergence Studies 107
1E-03 1E-02 1E-01 1E+00 1E+01 1E+02 1E+03
0 5 10 15 20 25 30 35 Erel[%]
χA0 [−] SEM: GLL-quad.SEM
SEM: Row-sum SEM: HRZ
(a)px1=2
1E-03 1E-02 1E-01 1E+00 1E+01 1E+02 1E+03
0 5 10 15 20 25 30 35 Erel[%]
χA0 [−] SEM: GLL-quad.SEM
SEM: Row-sum SEM: HRZ
(b)px1=3
1E-03 1E-02 1E-01 1E+00 1E+01 1E+02 1E+03
0 5 10 15 20 25 30 35 Erel[%]
χA0 [−] SEM: GLL-quad.SEM
SEM: Row-sum SEM: HRZ
(c) px1=4
1E-03 1E-02 1E-01 1E+00 1E+01 1E+02 1E+03
0 5 10 15 20 25 30 35 Erel[%]
χA0 [−] SEM: GLL-quad.SEM
SEM: Row-sum SEM: HRZ
(d)px1=5
1E-03 1E-02 1E-01 1E+00 1E+01 1E+02 1E+03
0 5 10 15 20 25 30 35 Erel[%]
χA0 [−] SEM: GLL-quad.SEM
SEM: Row-sum SEM: HRZ
(e) px1=6
Annotations: The relative error in time-of-flight tc
is plotted against the number of nodes per anti-symmetric wavelength χA0. The numerical re-sults have been obtained using the SEM. For in-plane polynomial degrees of px1 = 2,3, . . . ,6 an h-refinement in x1-direction has been conducted.
The out-of-plane polynomial degree px2, however, is fixed to 6for all convergence studies. Note that one element has been employed to discretize the thickness of the plate. The reference solution is based on analytical formulae derived in [87].
Figure 7.5: Convergence curves for the wave propagation analysis for the numerical model of a two-dimensional aluminum plate.
To judge the numerical performance of the different SE-approaches, the number of non-zero components (nnz) within the global stiffness matrix nnz(Kuu) and the global mass matrix nnz(Muu)are considered - as a means to describe the memory requirement (RAM) of the personal computer being used to run the simulations. Furthermore, we monitor the computational time tcpu, needed for the solution of the equations of motion. As the
Convergence Studies 108 software implementation of the different sets of shape functions is not equally optimized in our in-houseHO-FE program, this seems to be an appropriate value. From an engineering point of view, an error threshold of1 %is sufficient, at least according to our opinion. Thus, we record the memory requirements and the computational time for those simulations that reach the threshold first.
1E-03 1E-02 1E-01 1E+00 1E+01 1E+02 1E+03
0 5 10 15 20 25 30 35 Erel[%]
χA0 [−] SEM: GLL-quad.SEM
SEM: Row-sum SEM: HRZ
(a)px1=2
1E-03 1E-02 1E-01 1E+00 1E+01 1E+02 1E+03
0 5 10 15 20 25 30 35 Erel[%]
χA0 [−] SEM: GLL-quad.SEM
SEM: Row-sum SEM: HRZ
(b)px1=3
1E-03 1E-02 1E-01 1E+00 1E+01 1E+02 1E+03
0 5 10 15 20 25 30 35 Erel[%]
χA0 [−] SEM: GLL-quad.SEM
SEM: Row-sum SEM: HRZ
(c) px1=4
1E-03 1E-02 1E-01 1E+00 1E+01 1E+02 1E+03
0 5 10 15 20 25 30 35 Erel[%]
χA0 [−] SEM: GLL-quad.SEM
SEM: Row-sum SEM: HRZ
(d)px1=5
1E-03 1E-02 1E-01 1E+00 1E+01 1E+02 1E+03
0 5 10 15 20 25 30 35 Erel[%]
χA0 [−] SEM: GLL-quad.SEM
SEM: Row-sum SEM: HRZ
(e) px1=6
Annotations: The relative error in areaA(area be-low the envelope of the displacement signal) is plot-ted against the number of nodes per anti-symmetric wavelength χA0. The numerical results have been obtained using the SEM. For in-plane polynomial degrees ofpx1 = 2,3, . . . ,6anh-refinement inx1 -direction has been conducted. The out-of-plane polynomial degree px2, however, is fixed to 6 for all convergence studies. Note that one element has been employed to discretize the thickness of the plate. The reference solution is based on analytical formulae derived in [87].
Figure 7.6: Convergence curves for the wave propagation analysis for the numerical model of a two-dimensional aluminum plate.
Convergence Studies 109
0.7 0.8 0.9 1 1.1 1.2 1.3 1.4
2 3 4 5 6
nnz(Kuu)normalized
px1
(a)Number of non-zero components in the global stiffness matrix Kuu
0 0.2 0.4 0.6 0.8 1 1.2 1.4
2 3 4 5 6
nnz(Muu)normalized
px1
SEM: GLL quadratureSEM SEM: Row-sum SEM: HRZ
(b) Number of non-zero components in the global mass matrix Muu
0.6 0.81 1.2 1.4 1.6 1.82 2.2 2.4
2 3 4 5 6
tcpunormalized
px1
(c)Computational time for the solution of the equation of motion using the Newmark method Annotations: The Newmark method is used as an implicit time-integration algorithm. The results are assessed at a relative error in time-of-flight of Erel = 1 %. All values are normal-ized to the simulation results obtained using the in-plane polynomial degree px1 = 2. The out-of-plane polynomial degree px2 is fixed to6 for all conver-gence studies.
Figure 7.7: Comparison of the performance of different SE-schemes. The methods are eval-uated concerning their memory requirements (number of non-zero components in the system matrices - nnz) and the computational time needed for the time-integration of the equations of motion.
Convergence Studies 110 Initially, we note that all approaches reach an acceptable accuracy with considerably less than 20 nodes per wavelength, typically mentioned in current literature [76, 77]. The evaluation of the performance with respect to the memory requirements and the compu-tational time is shown in Fig. 7.7. All results are normalized with respect to the solution forpx1 = 2. From the bar graphs, it is apparent that for the SEM deploying a consistent mass matrix formulation px1 = 3 is the “optimal” choice, whereas px1 = 2 displays the best performance if a diagonalized mass matrix is used. With regard to the overall behavior, the SEM deploying a GLL nodal quadrature can be recommended. The most significant benefit is gained in terms of the computational-time, cf. Fig. 7.7c, while the memory requirements for the stiffness matrix are approximately equal. Naturally, a lumped mass matrix will serve to decrease the memory requirements for storing the global mass matrix significantly.