Research Papers: Design Automation

Anisotropic Mesh Adaptation and Topology Optimization in Three Dimensions

[+] Author and Article Information
Kristian Ejlebjerg Jensen

Villum Foundation,
Department of Earth Science and Engineering,
Imperial College London,
London SW7 2AZ, UK
e-mail: kristianejlebjerg+asme@gmail.com

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received April 29, 2015; final manuscript received December 8, 2015; published online April 15, 2016. Assoc. Editor: James K. Guest.

J. Mech. Des 138(6), 061401 (Apr 15, 2016) (8 pages) Paper No: MD-15-1332; doi: 10.1115/1.4032266 History: Received April 29, 2015; Revised December 08, 2015

Anisotropic mesh adaptation has been used to accelerate computation in several engineering fields, and we show that it can also be used for topology optimization. We use a combination of filtered continuous sensitivities and filtered design variables to drive the mesh adaptation. The filtered design variables are computed for this purpose only, while the filtered sensitivities are used as input to the optimizer. We test mesh independence for a cantilever problem and also show results for two other test cases. Finally, speedup relative to isotropic adaptation is estimated at 50 using average element aspect ratios.

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Grahic Jump Location
Fig. 1

Two metrics can be combined using the inner ellipse method [16]. It is not uncommon to see one ellipse within the other, but the case of intersection is plotted, which always gives rise to loss of anisotropy. Note that metrics generally have units of inverse squared length, so it is in fact the inverse square rootof the metrics that have been plotted. In this work, one ellipse represents the metric coming from the filtered design, while the other comes from the sensitivity, which resembles the elastic strain energy, because we only consider compliance minimization.

Grahic Jump Location
Fig. 2

The smoothing, coarsening, and refinement operations are illustrated in two dimensions, while the swapping operation is illustrated in three dimensions. Coarsening serves to remove short edges, and it is the only operation that is allowed to decrease the local element quality. It works by preferentially collapsing the shortest edge. All operations rely on coloring, but the refinement is special in the sense that it only assigns the first color and prioritizes long edges (in metric space) in doing so. The swapping of edges in three dimensions iteratively creates edges so as to generate the best elements. Note that we do not perform face to edge operations, also known as 2–3 swapping.

Grahic Jump Location
Fig. 3

The cantilever is a classic benchmark for compliance minimization, but we have chosen to also include the stool and crank described in Ref. [10]. The gray areas to the left on the cantilever and top of the crank denote fixed boundaries, while the four corner patches for the stool only fix the normal component of the displacement vector. The load vector is applied in the y, x, and z directions on a L1 × L1 patch for the cantilever, stool, and crank, respectively. The second load cases for the cantilever problem are not shown, but the only difference is that the load is in the z direction.

Grahic Jump Location
Fig. 4

The two different types of symmetry boundary conditions are illustrated. Loading along the boundary gives rise to a symmetry boundary, while loading normal to the boundary gives rise to an antisymmetry boundary. For two dimensions, either case leads to a scalar Dirichlet boundary condition, but in three dimensions the antisymmetry boundary leads to two scalar Dirichlet boundary conditions.

Grahic Jump Location
Fig. 5

The construction of a nodal Hessian, H¯¯(ρ̃) from a nodal scalar, ρ̃, is illustrated in two dimensions. The derivative is taken twice, and therefore, two Galerkin projections are needed to recover continuous representations, see Eqs. (4) and (5). Numerically, the Hessian is not symmetric, but we make it so by taking the average for the off-diagonal components.

Grahic Jump Location
Fig. 6

A flowchart of the algorithm is sketched with the operations associated with the mesh adaptation to the right. These operations are positioned between the sensitivity calculation and the optimizer. Note that we use a crude convergence criteria based on the number of iterations.

Grahic Jump Location
Fig. 7

The best cantilever designs that satisfy the volume constraint to a relative tolerance of 0.2% are shown in terms of the ρ = 0.5 and ρ = 1.0 isosurfaces as well as slices at x/Lx equal to 0, 0.25, 0.5, 0.75, and 1. Lmin is fixed at 5 × 10−3, while η takes on the values 0.08, 0.04, and 0.02 (problem nos. 1, 2, and 3 in Table 1). They all give the same topology, but the slice through x = 0.25Lx shows significant variations.

Grahic Jump Location
Fig. 8

The compliance for the cantilever problem is plotted throughout three optimizations with different tolerances for the mesh adaptation (Nos. 1, 2, and 3 in Table 1). Note that the PE parameter is smaller than 4 before iteration 200, and that the oscillations decrease with mesh refinement.

Grahic Jump Location
Fig. 9

The result of a cantilever optimization with Lmin = 5 × 10−2 is shown for η = 0.02 (No. 4 in Table 1). The larger minimum length scales cause a change of topology compared to that of Fig. 7.

Grahic Jump Location
Fig. 10

The cantilever optimization with σ¯¯load·n=0.5z (No. 5) is shown. This results in an internal hole as visible on the slice where x/Lx = 0.5.

Grahic Jump Location
Fig. 11

An example of multiple load cases is illustrated. The loading is both in the y and z direction, which causes a shell-like design, No. 6 in Table 1.

Grahic Jump Location
Fig. 12

The best stool design (No. 7) is plotted in terms of its ρ = 0.5 and ρ = 1.0 isosurfaces. The ice support causes a connecting plate between the legs. This is the only flat section in this otherwise truss-dominated design, and therefore, the average AR is comparatively low.

Grahic Jump Location
Fig. 13

The best crank design (No. 8) is shown from the side (upper) with the support visible and from the bottom where the load is applied (lower). Note that the title statistics pertain to the actual computational domain as illustrated with the wireframe.




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