Stefano Serra-Capizzano and Eddie Wadbro
University of Insubria, Italy and Umeå University
The possibility of allowing a zero lower bound on the physical density in material distribution based topology optimization is studied. We limit our attention to the standard test problem in one dimension of minimizing the compliance of a linearly elastic structure subject to a constant forcing. First order tensor product Finite Elements discretize the problem. An elementwise constant material indicator function defines the discretized, elementwise constant, physical density by using filtering and penalization.
To alleviate the ill-conditioning of the stiffness matrix, due to the variation of the elementwise constant physical density, we precondition the system. We provide a specific spectral analysis for large matrix sizes for the one-dimensional problem with Dirichlet--Neumann conditions in detail, even if most of the mathematical tools apply also in a d-dimensional setting, d larger or equal to 2, with multigrid replacing preconditioning as a numerical technique.
Finally, we critically report and illustrate results from numerical experiments: as a conclusion, it is indeed possible to solve large-scale topology optimization problems, allowing a vanishing physical density, and the technique is flexible enough to handle the same problem in more dimensions.