A numerical methodology to assess the quality of the design velocity field computation methods in shape sensitivity analysis

Abstract

A numerical methodology to assess the quality of the design velocity field computation methods in shape sensitivity analysis J. J. Ródenas * , † , F. J. Fuenmayor and J. E. Tarancón Departamento de Ingeniería Mecánica y de Materiales, Universidad Politécnica deValencia, E-46022 Valencia, Spain SUMMARYThe velocity field in shape sensitivity analysis is not uniquely defined although it must meet numerous theoretical and practical criteria. These practical criteria can be used to compare the existing velocity field computation methods which meet the theoretical criteria, but only in qualitative terms. When the FEM is used in design sensitivity analysis (DSA), due to the FE discretization error, the DSA errors will depend on the design velocity field considered. This paper presents a numerical methodology for quality evaluation of design velocity field computation methods in quantitative terms based on the analysis of the DSA discretization error.The sensitivity of the squared energy norm ( m = * u 2 /*a m , a m being a design variable) has been taken as the magnitude to measure the error of the DSA. For h-refinements, the squared error in energy norm ( e(u) 2 ) and the error in m (e( m )) are theoretically related by a constant which is independent of the refinement degree of the FE model. The quality of the design velocity field computation methods can therefore be assessed in terms of the stability of e( m )/ e(u) 2 in sequences of meshes. An example of the use of this methodology, where six design velocity field computation methods are compared, is presented.

On the need for the use of error‐controlled finite element analyses in structural shape optimization processes

“…For nodes located on the boundary of the domain, these sensitivities can be easily obtained from the first‐order sensitivity analysis of the B ‐spline curves used to define the boundary. A description and comparison of methods commonly used to evaluate the sensitivities of the internal nodes can be found in 17. The classical Laplacian smoothing technique has been used in the numerical examples presented in this work.The evaluation of ∂(η)/∂ v i also requires the evaluation of the recovered stress field σ * .…”

A gradient-based shape optimization scheme using an interface-enriched generalized FEM

“…We use a standard isoparametric FEM wherein the integrals that define the element sti↵ness matrices and load vectors are computed via Gaussian quadratures. However, this requires special care over the enriched elements due to discontinuity of the interpolation, as the enriched element ⌦ e must be split into its two integration elements ⌦ (1) e and ⌦ (2) e . To this end and without loss of generality, refer to Fig.…”

“…The computation of the design sensitivities has been the topic of multiple studies [34, 35,36,37,1,38], which can be classified in four broad categories: (i) overall or global finite di↵erences, (ii) discrete derivatives or discretize then di↵erentiate, (iii) continuum derivatives or differentiate then discretize, and (iv) computational or automatic di↵erentiation 35 [38]. The latter three methods encompass either direct or adjoint variants.…”

“…In shape optimization studies, the position of each material point in the domain or minimally on the boundary and material interfaces is a function of the shape design variables [1]. Therefore, initial finite-element-based shape optimization studies used the nodal coordinates as the design variables [2].…”

An integration of a low cost adaptive remeshing strategy in the solution of structural shape optimization problems using evolutionary methods

“…A description and comparison of methods commonly used to evaluate mesh sensitivities can be found in [7]. These sensitivities are used to asses how the mesh evolves when the design variables change.…”