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Structural and Multidisciplinary Optimization

, Volume 59, Issue 5, pp 1543–1565 | Cite as

Non-parametric shape optimization method for robust design of solid, shell, and frame structures considering loading uncertainty

  • Masatoshi Shimoda
  • Tomohiro Nagano
  • Jin-Xing ShiEmail author
Research Paper
  • 157 Downloads

Abstract

The robust design of structures is essential to improve their stabilities in structural design optimization and has been studied based on a variety of optimization methods. In this study, we propose a non-parametric optimization method for the robust shape design of solid, shell, and frame structures subjected to uncertainty loadings. We adopt the concept of principal compliance to perform the robust shape design considering loading uncertainty and transform the principal compliance minimization problem into the fundamental eigenvalue maximization problem associated with the weighting coefficients of the unknown loadings. The proposed non-parametric shape optimization method for robust design consists of four main procedures: the eigenvalue analysis of structures, derivation of shape gradient functions considering repeated eigenvalues, velocity analysis based on the H1 gradient method, and shape updating. We perform several design examples to confirm the validity of the proposed non-parametric shape optimization method. The optimal results show that the proposed optimization method works efficiently to reduce the principal compliance and enhance the robust behavior of each design example. As a feature, by setting the weighting coefficients, we can enhance the robust of the structures subjected to the unknown loadings at different loading positions and with different magnitudes of the directions of the admissible loading space.

Keywords

Gradient method Non-parameter Principal compliance Robust design Shape optimization 

Nomenclature Note that vectors are indicated as BOLD font in this study.

\( \left(\overline{\cdotp}\right) \)

Variation

(·)

Shape derivative

\( \left(\overset{\cdotp }{\cdotp}\right)\kern0.24em \left(={\left(\cdotp \right)}^{\prime }+{\left(\cdotp \right)}_{,i}{V}_i\right) \)

Material derivative

(·)s

Iteration history of domain variation

(·),i(=(·)/∂xi)

Partial differential notation

a(·,·)

Virtual work of rigidity

A

Mid-surface of shell structures

Ab

Cross-section of beams in frame structures

As

Mid-surface of shell structures after domain variation

b(·,·)

Virtual work of pseudo-inertia

{Cijkl}i, j, k, l = 1, 2, 3

Stiffness tensor of solids structures

{Cαβγδ}α, β, γ, δ = 1, 2

Stiffness tensor of shell structures with respect to membrane stress

\( {\left\{{C}_{\alpha \beta}^S\right\}}_{\alpha, \beta =1,2} \)

Stiffness tensor of shell structures with respect to transverse shear stress

\( {C}_{\varTheta}^{\mathrm{solid}} \)

Kinematically admissive function space of solid structures

\( {C}_{\varTheta}^{\mathrm{shell}} \)

Kinematically admissive function space of shell structures

\( {C}_{\varTheta}^{\mathrm{frame}} \)

Kinematically admissive function space of frame structures

d

Diameter of members in frame structures

E

Young’s modulus

f = {fi}i = 1, 2, 3

External loadings

F = {Fi}i = 1, 2, 3

Admissible loading space

G|frame(=G|framen)

Shape gradient function of frame structures

G|shell(=G|shelln)

Shape gradient function of shell structures

G|solid(=G|solidn)

Shape gradient function of solid structures

G(1)|solid, \( {\left.{G}_f^{(1)}\right|}_{\mathrm{solid}} \)

Shape gradient density functions of solid structures

G(1)|shell, \( {\left.{G}_f^{(1)}\right|}_{\mathrm{shell}} \)

Shape gradient density functions of shell structures

\( {\left.{G}_1^{(1)}\right|}_{\mathrm{frame}} \), \( {\left.{G}_2^{(1)}\right|}_{\mathrm{frame}} \)

Shape gradient density functions of frame structures

H1

Sobolev space of square integrable and differentiable

l

Compliance

lp

Principal compliance

L

Lagrange functional

M

Volume

M0

Initial volume

\( \widehat{M} \)

Constraint value of M

n

Direction vector

n1

Unit vector in x1 direction

n2

Unit vector in x2 direction

nbtm

Normal vector at the bottom surface of shell structures

nmid

Normal vector at the mid-surface of shell structures

ntop

Normal vector at the top surface of shell structures

nφ

Unit vector according to φ

nφ +  π

Unit vector according to angle φ + π

P1, P2, P3, P4

Positions subjected to unknown loadings

r (≥2)

Multiplicity of repeated eigenvalues

s

Time of domain variation

S

Centroidal axis of frame structures

Sj (j = 1, 2, 3, ..., N)

Centroidal axis of member j in frame structures

Ss

Centroidal axis of frame structures after domain variation

t

Thickness of shell structures

Ts(X)

Mapping

u = {ui}

Displacement vector of frame structures

U

Admissible function space satisfying Dirichlet boundary condition

V

Design velocity field

V1

Design velocity field in n1 direction

V2

Design velocity field in n2 direction

w = {wi}i = 1, 2, 3

Displacement vector

w0 = {w0α}α = 1, 2

In-plane displacement vector of shell structures

x = {x1, x2, x3}

Position vector

\( \widehat{x}=\left\{{\widehat{x}}_1,{\widehat{x}}_2,{\widehat{x}}_3\right\} \)

Vector of loading position

X(={X1, X2, X3})

Position vector in Ω

Xs(={Xs1, Xs2, Xs3})

Position vector in Ωs

∂Ab

Circumference of the cross-section of members in frame structures

One-dimensional space

2

Two-dimensional space

3

Three-dimensional space

Δs

A small positive value

α

Spring constant

δ(·)

Delta function

ϕ

Tolerance of repeated eigenvalues

φ

Angle

η

Lagrange multiplier of principal compliance

ηmax

Maximum value of η

κ

Twice the mean curvature of shell structures or the curvature of frame structures

λ(r) (r = 1, 2, 3, ...)

rth eigenvalue

μ

Shear modulus of frame structures

ν

Poisson’s ratio

θ

Circumferential angle of compliance

θ = {θi}(i = 1, 2, 3)

Rotation angle vector of frame structures in the local coordinate system

θ = {θα}(α = 1, 2)

Rotation angle vector of shell structures in the local coordinate system

ρ

Density

{ξij}i,j = 1,2,3,{ξαβ}α,β = 1,2

Weighting coefficient matrix in terms of F

ΔM

Decrease of volume

Γ

Boundary of domain Ω

Γj (j = 1, 2, 3, ..., N)

Circumference surface of member j in frame structures

Γs

Boundary of domain Ωs

Λ

Lagrange multipliers for volume constraint

Ω

Initial domain

Ωf

Domain subjected to external loadings

Ωj (j = 1, 2, 3, ..., N)

Domain of member j in frame structures

\( {\varOmega}_f^j\;\left(j=1,2,3,...,{N}_f\right) \)

Domain subjected to external loadings in frame structures

Ωs

Updated domain after variation

Notes

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Advanced Science and TechnologyToyota Technological InstituteNagoyaJapan
  2. 2.Graduate School of Advanced Science and TechnologyToyota Technological InstituteNagoyaJapan
  3. 3.Department of Production Systems Engineering and SciencesKomatsu UniversityKomatsuJapan

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