5.3 Finite strain problems
5.3.2 Complex cube connector under pressure
As the second example, we consider a cube connector with a more complex structure.
Fig. 5.24a shows the model of the problem under investigation. As it can be seen from the figure, the structure of the complex cube connector is composed of eight individual single cube connectors. Thereby, each individual cube connector is defined by the level set
x z
y
¯ uz
(a) (b)
Figure 5.24:Complex cube connector. Geometry, boundary conditions, and discretization.
function given in Eq. (5.8) and a bounding box of dimensions 30×30×30mm3where the parameters defining the surface of every cube connector are given in Tab. 5.4. As depicted in Fig. 5.24a, the complex cube connector is fixed at the bottom face. Further, a prescribed displacement ¯uzis applied at the top surface acting in negativez-direction. Furthermore, the top surface is fixed inx- andy-direction. Moreover, the spatial discretization used for the analysis is given in Fig. 5.24b. Here, we employ a FCM mesh using 2,912 finite cells.
Table 5.4:Complex cube connector. Geometry parameters.
cube id xc yc zc R r d
mm mm mm mm mm mm4
1 15.0 15.0 15.0 15.0 11.25 53000.0 2 45.0 15.0 15.0 15.0 11.25 49000.0 3 15.0 45.0 15.0 15.0 11.25 51000.0 4 45.0 45.0 15.0 15.0 11.25 47000.0 5 15.0 15.0 45.0 15.0 11.25 52000.0 6 45.0 15.0 45.0 15.0 11.25 48000.0 7 15.0 45.0 45.0 15.0 11.25 50000.0 8 45.0 45.0 45.0 15.0 11.25 46000.0
5 Basis function removal for the FCM
Finite strain hyperelasticity
As the first test case of the complex cube connector, we assume an isotropic hyperelastic material behavior, see Sec. 2.3.2. The material parameters are listed in 5.2. To study the robustness of the FCM analyses the prescribed displacement ¯uuis increased incrementally until the Newton-Raphson method (and, thus, the entire analysis) fails. Thereby, a load increment of 0.2 mm is applied for each load step. Further, for the computation of the integrals over the physical domain, we employ the adaptive moment fitting using an octree of refinement levelk = 3 for the resolution of the geometry. For the volume fraction tolerances of the octree utilized by the adaptive moment fitting, we thereby choose a value of 0.85 at tree depth levelka = 0, 0.7 at levelka = 1, and 0.9 at levelka = 2.
Furthermore, for the computation of the integrals over the fictitious domain, we use the fictitious integration points depicted in Fig. 5.15b and 5.15c where the first point set is utilized without and the second one with the basis function removal. Moreover, for α= 10−qwe utilize a parameter ofq= 5.
In Fig. 5.25a, 5.25a, 5.25a, and 5.25a, the energy-displacement curves are plotted for a fixed value of the basis function criterionµtand different orders of the ansatzp. As it can be deduced from the figures, the basis function removal improves the robustness of the FCM significantly. Applyingµt= 0.3 enables to increase the prescribed displacement ¯uz
by a factor bigger than 2 for all orders of the ansatzpas compared to the values of the analyses without basis function removalµt= 0.0.
To study the effect of the basis function removal on the solution in Fig. 5.26a, 5.26b, and 5.26c, the energy-displacement curves are plotted for a fixed order of the ansatzp and different values of the basis function removal criterionµt. As it can be seen from the figures, the results applying the basis function removal show a good agreement with the results usingµt= 0.0. Even the results utilizing a higher removal criterionµt= 0.5 do not show a significant influence on the solution.
Finally, Fig. 5.27 shows the evolution of the von Mises stressσvM during the loading.
Here, the contour plots of the von Mises stress are plotted for different load steps applying a basis function removal criterion ofµt= 0.3 and an ansatz of orderp= 2.
5.3 Finite strain problems
displacement uz (mm) p=2, q=5, µt=0.0
displacement uz (mm) p=2, q=5, µt=0.1
displacement uz (mm) p=2, q=5, µt=0.3
displacement uz (mm) p=2, q=5, µt=0.5
p=3, q=5, µt=0.5 p=4, q=5, µt=0.5
(d)µt= 0.5 Figure 5.25:Complex cube connector. Energy-displacement curves.
5 Basis function removal for the FCM
0 5 10 15 20 25 30
0 1 2 3 4 5 6 7 8 9
strain energy (J)
displacement uz (mm) p=2, q=5, µt=0.0
p=2, q=5, µt=0.1 p=2, q=5, µt=0.3 p=2, q=5, µt=0.5
(a)p= 2
0 5 10 15 20 25 30
0 1 2 3 4 5 6 7 8 9
strain energy (J)
displacement uz (mm) p=3, q=5, µt=0.0
p=3, q=5, µt=0.1 p=3, q=5, µt=0.3 p=3, q=5, µt=0.5
(b)p= 3
0 5 10 15 20 25 30
0 1 2 3 4 5 6 7 8 9
strain energy (J)
displacement uz (mm) p=4, q=5, µt=0.0
p=4, q=5, µt=0.1 p=4, q=5, µt=0.3 p=4, q=5, µt=0.5
(c)p= 4
Figure 5.26:Complex cube connector. Energy-displacement curves.
5.3 Finite strain problems
(a)u¯z= 1.4 mm (b)u¯z= 3.8 mm
(c)u¯z= 6.2 mm (d)u¯z= 8.6 mm
Figure 5.27:Complex cube connector. Contour plots of the von Mises stressσvMfor different load steps.
5 Basis function removal for the FCM
Finite strain elastoplasticity
As the second test case of the complex cube connector, we study the influence of the basis function removal assuming a material model based on theJ2flow theory of plasticity, see Sec. 2.3.4. The material parameters are listed in Tab. 4.3. To improve the robustness of the FCM analysis, we employ the same material model for the fictitious domain usingα= 10−q withq= 5 in order to scale the material parameters of the fictitious material model appro-priately. Moreover, we assume an infinite yield stress. Thus, the material behavior of the fictitious domain is defined by the hyperelastic strain energy density function of the con-stitutive model. For the numerical integration, we further employ the same quadratures as in the previous test case. Furthermore, analogous to the previous test case, the prescribed displacement ¯uz is increased incrementally until the analysis fails. To this end, for the first 5 displacement increments, we apply values of 0.01 mm, 0.02 mm, 0.02 mm, 0.05 mm, and 0.1 mm. For each load step bigger than 5, we then use a displacement increment of 0.2 mm.
To show the influence of the basis function removal, Fig. 5.28a, 5.28b, 5.28c, and 5.28d show the load-displacement curves for a fixed value of the basis function removal criterion µtand different orders of the ansatzp. As it can be seen from the figures, the basis function removal increases the robustness of the FCM analyses significantly. By usingµt= 0.3, the value of the prescribed displacement ¯uz could thus be increased by a factor of about 2 employing an ansatz ofp= 2 and p= 3. Moreover, a high removal criterionµt = 0.5 allows to increase the robustness of the analysis by applying an ansatz of orderp= 4.
Here, the prescribed displacement could be increased by a factor of about 5.
Next, to study the effect of the basis function removal on the solution, Fig. 5.29a, 5.29b, and 5.29c show the load-displacement curves for a fixed order of the ansatzpand different values ofµt. As it can be seen from the figures, higher values of the basis function removal criterionµt= 0.3 andµt= 0.5 increase the deviation in the load values as compared to the results obtained without basis function removal (µt= 0.0). When employingµt= 0.1, however, the deviations in the load values are negligible.
Finally, the evolution of the equivalent plastic strain during the loading is given in Fig. 5.30. Here, the contour plots of ¯αare depicted for different load steps applying an ansatz ofp= 2 and a basis function removal criterion ofµt= 0.3.
5.3 Finite strain problems
displacement uz (mm) p=2, q=5, µt=0.0
displacement uz (mm) p=2, q=5, µt=0.1
displacement uz (mm) p=2, q=5, µt=0.3
displacement uz (mm) p=2, q=5, µt=0.5 p=3, q=5, µt=0.5 p=4, q=5, µt=0.5
(d)µt= 0.5 Figure 5.28:Complex cube connector. Load-displacement curves.
5 Basis function removal for the FCM
displacement uz (mm) p=2, q=5, µt=0.0
displacement uz (mm) p=3, q=5, µt=0.0
displacement uz (mm) p=4, q=5, µt=0.0 p=4, q=5, µt=0.1 p=4, q=5, µt=0.3 p=4, q=5, µt=0.5
(c)p= 4
Figure 5.29:Complex cube connector. Load-displacement curves.
5.3 Finite strain problems
(a)u¯z= 1.2 mm (b)u¯z= 3.6 mm
(c)u¯z= 6.0 mm (d)u¯z= 8.4 mm
Figure 5.30:Complex cube connector. Contour plots of the equivalent plastic strain α¯ for different load steps.
5 Basis function removal for the FCM