(a) FP (b) P1 (c) P2
(d) P3 (e) S1 (f) S2
Figure 4.1: The configurations evaluated in this thesis, including both new config-urations and the ones studied by Granum and Løken [29]. Configconfig-urations already tested at NTNU are the FP and P2.
4.2 Models
The models presented in this chapter include only the blast-exposed area of the plates and utilize symmetry about two axes. Since the models are geometrically simple and exclude the clamping frames that hold the blast-exposed plate in place, the outer boundaries are fixed. Only models that utilize symmetry about two axes will be run.
It is shown in [29], [28], and [30], that symmetric models differ negligibly from full models for this specific problem. The perforations are modeled as perfect squares and the slits as stars with a center gap of 0.01 mm.
Different configurations, mesh sizes, and element types will be evaluated. To keep track of all parametric changes, a naming convention is introduced in Table 4.1.
The naming convention states the applied element type, the approximate size of the elements, the geometrical configuration of the plate tested and the applied loading.
This convention is applied throughout both this chapter and Chapter 5.
Chapter 4.
Table 4.1: Naming convention used to label different models in the preliminary numerical study. The labels will be used in plots and in tables.
General label name: XX EE ZZ phh tt Part of name Possible configurations Explanation
FP Full Plate
XX PX Perforated plate, geometry X
SX Slit plate, geometry X
EE S4R Shell element, reduced integration
C3D8R Brick element, reduced integration
ZZ 0xx Mesh size of 0.xx mm
xx Mesh size of x.x mm
phh tt p77 tt hh = driver length in [cm]
tt = nominal firing pressure [bar]
4.2.1 Description of the shell element models
The shell element models apply quadratic S4R or triangular S3R shell elements with an approximate element size on the range 3.2 mm to 0.8 mm. These elements utilize reduced integration, built-in hourglass control, drilling stiffness, and are linear general purpose shell elements meant to handle both thin and thick shell problems [43].
Meshing of the shell models
When modeling fracture, the geometry of the mesh and the element size may be of great importance [44] [30]. To investigate this effect, a mesh study was conducted.
Several different discretizations of the S2 configuration has been evaluated and are given in Figure 4.2(e) - 4.3(f), referred to as M1 to M4. Due to the angular difference of 45◦ between the slits and the outer boundary, the mesh for the S2 configuration requires a transition zone to adjust. M1 and M2 apply only squared S4R elements across the entire plate. The two meshes are similar but differ in the mesh transition zone in front of the tip of the slit. The M3 mesh is constructed by using triangular S3R elements in a squared zone around the slit, and S4R elements in the rest of the plate. This mesh is regular but introduces a mesh with mixed element types. For M4, S4R elements are used for the entire plate and the medial axis meshing technique is applied [45]. All four meshes use an approximate size of 1.0 mm but some elements are smaller in the transition zones, which leads to higher computational costs.
For the S1, P1, P2, and P3 configurations, only regular meshes was applied. The different element sizes for the P1 configuration in presented Figures 4.2(a) to 4.2(d).
4.2. Models
(a) 3.2 mm (b) 2.0 mm
(c) 1.0 mm (d) 0.8 mm
(e) Mesh M1 (f) Mesh M1, zoomed
Figure 4.2: Meshes applied to the different configurations in the simulations in Abaqus. (a)-(d) Show different mesh sizes and geometries used in the simulations for the P1, P2, and the P3 configuration. (e) and (f) shows the M1 mesh for the S2 configuration.
Chapter 4.
(a) Mesh M2 (b) Mesh M2, zoomed
(c) Mesh M3 (d) Mesh M3, zoomed
(e) Mesh M4 (f) Mesh M4, zoomed
Figure 4.3: Meshes applied to the S2 configuration. (a),(c) and (e) shows the entire mesh, used for the meshing methods M2, M3 and M4 respectively. (b), (d) and (f) are showing zoomed images of the mesh around the slit for M2, M3 and M4.
4.2. Models
4.2.2 Description of the solid element models
All models apply the C3D8R element. This is a linear 8-noded element with reduced integration and built-in hourglass control [46]. Simulations have been run with two, three, and four elements across the thickness of the plate. Three elements result in an approximate element size of 0.26 mm, and approximately 1 million elements. When modeled with two and four, the numbers are approximate 250 000 and 2 million elements respectively. The number differs slightly between the slit and perforated configuration as the perforations remove 16% of the total meshed area.
For P1, P2, P3, and S1, the models use regular mesh geometries similar to what is shown in figure 4.2(a). The S2 configuration is modeled slightly different for the solid models than for the shell models. Figure 4.4(a) shows the partitions made to construct the mesh for the S2 model. The green area is meshed using a structured mesh, while the yellow area applies the meshing algorithm sweep advancing front [47]. Figure 4.4(b) shows the corresponding mesh in the area close to the slit. The mesh is mostly uniform but there is a transition zone between the areas where the two different meshing techniques are used. Due to much higher computational costs when running the solid element model, no study on the mesh geometry was included.
(a) (b)
Figure 4.4: Partitioning and meshing of the S2 configuration. (a) shows the mesh-ing algorithms applied in the simulations. Green represents the structured meshmesh-ing algorithm and sweep advanced front meshing is represented with yellow [47]. (b) shows the result of sweep advanced meshing for the yellow area.
Chapter 4.