Received: 1 June 2017 Revised: 10 November 2017 Accepted: 20 November 2017
A geometry projection method for the topology optimization
of curved plate structures with placement bounds
Arun L. Gain
Julián A. Norato
Department of Mechanical Engineering,
School of Engineering, University of
Connecticut, Connecticut, USA
Champaign Simulation Center,
Caterpillar Inc., Illinois, USA
Julián A. Norato, Department of
Mechanical Engineering, School of
Engineering, University of Connecticut,
191 Auditorium Road, U-3139, Storrs,
Connecticut 06269, USA
Single-curvature plates are commonly encountered in mechanical and civil
structures. In this paper, we introduce a topology optimization method for the
stiffness-based design of structures made of curved plates with fixed thickness.
The geometry of each curved plate is analytically and explicitly represented by
its location, orientation, dimension, and curvature radius, and therefore, our
method renders designs that are distinctly made of curved plates. To perform the
primal and sensitivity analyses, we use the geometry projection method, which
smoothly maps the analytical geometry of the curved plates onto a continuous
density field defined over a fixed uniform finite element grid. A size variable
is ascribed to each plate and penalized in the spirit of solid isotropic material
with penalization, which allows the optimizer to remove a plate from the design.
We also introduce in our method a constraint that ensures that no portion of
a plate lies outside the design envelope. This prevents designs that would oth-
erwise require cuts to the plates that may be very difficult to manufacture. We
present numerical examples to demonstrate the validity and applicability of the
curved plate structures, design for manufacturing, placement constraint, topology optimization
Topology optimization techniques are powerful tools to explore light-weight structural designs by providing insight on
where to place or remove material within a design envelope. These insights help design engineers translate the optimal
topology into a computer-aided design (CAD) model. During this translation, however, it is common to introduce depar-
tures from the optimal topology due to the fact that prevalent free-form topology optimization methods render organic
structures that cannot be easily manufactured. This shortcoming becomes more severe if one seeks a design that uses only
commercially available stock material such as plates. In this case, additional design iterations are required in producing
a structurally feasible design made of plates, during which additional weight is inevitably introduced.
Recently, several works have proposed topology optimization techniques for structures made of geometric components.
These works integrate an analytical description of the geometric components with a fixed finite element grid for primal
and sensitivity analysis, thus, they circumvent remeshing upon design changes. The geometry projection method maps
the analytical descriptions of discrete geometric primitives such as 2-dimensional (2D) bars and 3-dimensional (3D) plates
onto a continuous density field defined over the finite element grid.
A hallmark of these methods is that a size variable
128 Copyright © 2017 John Wiley & Sons, Ltd. wileyonlinelibrary.com/journal/nme Int J Numer Methods Eng. 2018;114:128–146.