Received: 7 April 2017 Revised: 1 February 2018 Accepted: 10 March 2018
Applying functional principal components to structural
Department of Civil Engineering and
Architecture, University of Pavia,
Department of Mathematics, Polytechnic
University of Milan, Milan, Italy
Consiglio Nazionale delle Ricerche
(CNR), Istituto di Matematica Applicata e
Tecnologie Informatiche (IMATI),
Ettore Lanzarone, Consiglio Nazionale
delle Ricerche (CNR), Istituto di
Matematica Applicata e Tecnologie
Informatiche (IMATI), Milan, Italy.
Structural topology optimization aims to enhance the mechanical performance
of a structure while satisfying some functional constraints. Nearly all approaches
proposed in the literature are iterative, and the optimal solution is found by
repeatedly solving a finite element analysis (FEA). It is thus clear that the bot-
tleneck is the high computational effort, as these approaches require solving
the FEA a large number of times. In this work, we address the need for reduc-
ing the computational time by proposing a reduced basis method that relies on
functional principal component analysis (FPCA). The methodology has been
validated considering a simulated annealing approach for compliance mini-
mization in 2 classical variable thickness problems. Results show the capability
of FPCA to provide good results while reducing the computational times, ie, the
computational time for an FEA is about one order of magnitude lower in the
reduced FPCA space.
finite element analysis, functional principal component analysis, structural topology optimization
Structural optimization techniques aim to obtain optimized performance from a structure while satisfying several func-
tional constraints, eg, the total mass to employ or stress limits. The need for optimized solutions in structural applications
has increased over the years and has become nowadays fundamental, due to the limited availability of commodities, the
environmental impact, the market competition, and the new manufacturing processes, eg, 3-dimensional (3D) printing.
Three main categories can be distinguished in the wide class of structural optimization methodologies, ie, size optimiza-
tion, shape optimization, and topology optimization. We focus on the structural topology optimization (STO) of continuum
structures, whose aim is to produce an optimized structural component by determining its best mass or volume distribu-
tion in a given design domain. Differently from other alternatives, which deal with predefined configurations, STO design
can attain any shape within the domain.
STO is essentially treated as a constrained minimization or maximization problem. Several approaches and algorithms
have been proposed for STO, as documented by the huge literature available on the topic.
The main distinction is
between gradient-based and non–gradient-based approaches; however, nearly all of the algorithms are iterative, and the
optimal solution is found by repeatedly performing a structural finite element analysis (FEA) that involves the solution
of the equilibrium equations of the problem under study.
On the one hand, such iterative approaches are powerful, as they allow determining optimized solutions in a vari-
ety of situations without additional assumptions. On the other hand, the bottleneck of such approaches is the high
Int J Numer Methods Eng. 2018;115:189–208. wileyonlinelibrary.com/journal/nme Copyright © 2018 John Wiley & Sons, Ltd. 189