Multigrid barrier and penalty methods for large scale topology optimization of solid structures

Brune, Alexander (2022). Multigrid barrier and penalty methods for large scale topology optimization of solid structures. University of Birmingham. Ph.D.

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Abstract

We propose two algorithms for solving minimum compliance topology optimization
problems defined on finite element meshes with several million elements, where the
design geometry is parameterized on the discretized problem domain by an elementwise constant density field and we use the variable thickness sheet formulation to
map the density to the material stiffness. The first method is an interior point (IP)
method and the second follows the penalty-barrier multiplier (PBM) or nonlinear
rescaling framework. To solve the linear systems arising in each optimization iteration, we use a multigrid-preconditioned Krylov solver. We employ a reformulation
of the linear system to obtain a symmetric positive definite matrix that is amenable
to standard multigrid transfer operators. We test the performance of both our algorithms on a wide range of numerical examples, comparing their performance to
each other and to that of the well-established optimality criteria (OC) method. Our
PBM algorithm proves to be more robust and efficient than both the IP and OC
method.
We then extend our approach to problems defined on unstructured meshes, which
necessitates switching to an algebraic multigrid preconditioner. Using the (adaptive)
smoothed aggregation method of Vaněk, Mandel, and Brezina, we propose and test
different non-standard setup strategies for the multigrid transfer operators in order
to identify the most efficient one for our type of problem.
The PBM method is applied to the dual of the compliance minimization problem,
which permits an easy integration of unilateral contact constraints. We include
examples featuring such constraints in our numerical experiments, both for problems
on uniform structured meshes and on unstructured meshes.