Four benchmark examples of structural topology optimization are used to test the per- formance of the proposed GPU-based matrix-free PCG solver discussed in Algorithm 1, using both ebe and nbn strategies given in Algorithm 2, and 3 respectively. For each benchmark example, five different mesh sizes are used to evaluate the scalability of the proposed strategies.
The simulations are performed on a CPU with Intel Xeon E5-1650 processor equipped with 6 cores and 12 threads. It has clock speed of 3.2 GHz and offers maximum memory bandwidth of 51.2 GB/s. The GPU instances are run on NVIDIA Tesla K40c card that has 2880 cores and 12 GB of memory with bandwidth of 288 GB/s. The computational
steps performed on the CPU were written in C, whilst GPU kernels were developed using CUDA 10.0. For topology optimization an artificial material is considered and its properties are given in Table 3.1. It is noted that the implementation is made unit- independent, and changes in material properties do not affect the topology. The other common parameters are also listed in Table3.1. The optimization loop terminates when the difference between compliance value of two subsequent iterations falls below 10−2 or the number of optimization iterations exceeds 150.
Table 3.1: Common parameters used for topology optimization of three examples.
Young’s Poisson’s SIMP Min. PCG Change in
Parameter modulus ratio penalty parameter density tolerence compliance
E ν p ρmin ε ∆ =Ci−Ci−1
Value 1.0 0.3 3.0 10−2 10−5 10−2
The four benchmark examples are discussed in the following subsections.
3.3.1 3D Cantilever Beam
The first example is a 3D cantilever beam [50] and its domain and boundary conditions are shown in Figure 3.7(a). The L:B :H ratio of the cantilever beam is 2 : 1 : 1. Each node of the left face is subjected to the Dirichlet boundary condition, while each node at the lower edge of the right face is subjected to the Neumann boundary condition. Figure 3.7(b) shows the discretized domain of the beam. It can be observed that the mesh is structured and composed of identical 8-noded hexahedral elements.
The volume fraction (Vf) for this example is taken as 30% of the initial domain volume. The total number of elements, nodes, DoFs and the sensitivity filter radius (R) in the five mesh sizes used for this example are given in Table 3.2.
3.3.2 3D L-Beam
The second benchmark example is a 3D L-beam that is shown in Figure3.8(a). The figure shows the design domain, loading, and boundary conditions of the example. The face ‘C’
is subjected to the Dirichlet boundary condition, while the edge ‘AB’ is subjected to the Neumann boundary condition. The beam thickness is taken as 0.25×(2L/5). During
L H
B
F
(a) Initial domain, loading, and boundary condi- tions
(b) Discretized domain
Figure 3.7: 3D Cantilever beam example.
Table 3.2: Number of elements, nodes, DoFs and corresponding filter radius for mesh sizes of 3D cantilever beam example.
Mesh # Elements # Nodes # DoFs Filter radius (R)
CB1 31,250 34,476 103,428 3.0
CB2 85,750 92,016 276,318 3.5
CB3 182,250 192,556 577,668 4.0
CB4 432,000 450,241 1,350,723 4.5 CB5 1,024,000 1,056,321 3,168,963 5.0
optimization, a unit load is applied at each node of the edge ‘AB’. The discretized design domain of 3D L-beam example is shown in Figure3.8(b). As can be seen, this example is also meshed using a structured mesh of 8-noded hexahedral elements.
For optimization, Vf = 45% is considered. The details of the mesh sizes used for 3D L-beam example are given in Table 3.3.
3.3.3 Michell Cantilever
In topology optimization, the Michell cantilever is a well-known benchmark example [97]. This example has a semi-circular boundary that is supported as shown in Figure 3.9(a). The supported semi-circular boundary is subjected to the Dirichlet boundary condition, whereas the loaded points in Figure 3.9(a) are subjected to the Neumann
A F
B C
L
2L/53L/5
(a) Initial domain, loading, and boundary condi- tions
(b) Discretized domain
Figure 3.8: 3D L-beam example.
Table 3.3: Number of elements, nodes, DoFs and corresponding filter radius for mesh sizes of 3D cantilever beam example.
Mesh # Elements # Nodes # DoFs Filter radius (R)
LB1 32,768 38,313 114,939 3.0
LB2 85,184 95,580 286,740 3.5
LB3 188,384 206,205 618,615 4.0
LB4 438,976 469,700 1,409,100 4.5 LB5 1,000,000 1,053,026 3,159,078 5.0
boundary condition. The L:H :Rratio is taken as 5 : 4 : 1. Vf is limited to 45% of the design domain volume. Since this example has a semi-circular boundary, unstructured mesh can be seen in Figure 3.9(b). The five mesh sizes used for this example are listed in Table 3.4.
H
L R
F
(a) Initial domain, loading, and boundary condi- tions
(b) Discretized domain
Figure 3.9: Michell cantilever example.
Table 3.4: Number of elements, nodes, DoFs and corresponding filter radius for mesh sizes of Michell cantilever example.
Mesh # Elements # Nodes # DoFs Filter radius (R)
M C1 38,148 52,008 156,024 3.0
M C2 85,674 116,012 348,036 3.5
M C3 183,432 232,125 696,375 4.0 M C4 453,175 526,998 1,580,994 4.5 M C5 1,170,894 1,374,415 4,123,245 5.0
3.3.4 Connecting Rod of an Auto-mobile Engine
The fourth benchmark example is a connecting rod of an auto-mobile engine taken from Nana et al. [98]. Figure 3.10(a) shows the design domain and its loading and boundary conditions. The following dimensions are considered; height = 350 units, length = 150 units, inner and outer diameters of the upper bore = 60 units and 80 units, and inner and outer diameters of the lower bore = 35 units and 50 units. Both bores are considered non-design material, and during optimization, no material is removed from these bores.
As shown in Figure3.10(a), the lower half of the bottom bore is subjected to the Dirichlet boundary condition, while the upper half of the top bore is subjected to the Neumann boundary condition. Vf = 30% is considered for this example.
(a) Initial domain, loading, and boundary condi- tions
(b) Discretized domain
Figure 3.10: Connecting rod of an auto-mobile engine example.
Since the design domain has two bores, unstructured meshes can be seen in Figure 3.10(b). The mesh sizes used for connecting rod are listed in Table 3.5.
Table 3.5: Number of elements, nodes, DoFs and corresponding filter radius for mesh sizes of connecting rod example.
Mesh # Elements # Nodes # DoFs Filter radius (R)
CR1 33,178 48,248 144,744 3.0
CR2 85,376 124,305 372,915 3.5
CR3 182,637 238,578 715,734 4.0
CR4 465,708 572,978 1,718,934 4.5 CR5 1,033,820 1,223,392 3,670,176 5.0