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


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.


Gradient method Non-parameter Principal compliance Robust design Shape optimization 

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

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



Shape derivative

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

Material derivative


Iteration history of domain variation


Partial differential notation


Virtual work of rigidity


Mid-surface of shell structures


Cross-section of beams in frame structures


Mid-surface of shell structures after domain variation


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


Diameter of members in frame structures


Young’s modulus

f = {fi}i = 1, 2, 3

External loadings

F = {Fi}i = 1, 2, 3

Admissible loading space


Shape gradient function of frame structures


Shape gradient function of shell structures


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


Sobolev space of square integrable and differentiable




Principal compliance


Lagrange functional




Initial volume

\( \widehat{M} \)

Constraint value of M


Direction vector


Unit vector in x1 direction


Unit vector in x2 direction


Normal vector at the bottom surface of shell structures


Normal vector at the mid-surface of shell structures


Normal vector at the top surface of shell structures


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


Time of domain variation


Centroidal axis of frame structures

Sj (j = 1, 2, 3, ..., N)

Centroidal axis of member j in frame structures


Centroidal axis of frame structures after domain variation


Thickness of shell structures



u = {ui}

Displacement vector of frame structures


Admissible function space satisfying Dirichlet boundary condition


Design velocity field


Design velocity field in n1 direction


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


Circumference of the cross-section of members in frame structures

One-dimensional space


Two-dimensional space


Three-dimensional space


A small positive value


Spring constant


Delta function


Tolerance of repeated eigenvalues




Lagrange multiplier of principal compliance


Maximum value of η


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

λ(r) (r = 1, 2, 3, ...)

rth eigenvalue


Shear modulus of frame structures


Poisson’s ratio


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



{ξij}i,j = 1,2,3,{ξαβ}α,β = 1,2

Weighting coefficient matrix in terms of F


Decrease of volume


Boundary of domain Ω

Γj (j = 1, 2, 3, ..., N)

Circumference surface of member j in frame structures


Boundary of domain Ωs


Lagrange multipliers for volume constraint


Initial domain


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


Updated domain after variation



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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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