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

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## 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 H^{1} 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}(=*∂*(·)/*∂x*_{i}) Partial differential notation

*a*(·,·)Virtual work of rigidity

*A*Mid-surface of shell structures

*A*_{b}Cross-section of beams in frame structures

*A*_{s}Mid-surface of shell structures after domain variation

*b*(·,·)Virtual work of pseudo-inertia

- {
*C*_{ijkl}}_{i, j, k, l = 1, 2, 3} Stiffness tensor of solids structures

- {
*C*_{αβγδ}}_{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***f*_{i}}_{i = 1, 2, 3}External loadings

= {**F***F*_{i}}_{i = 1, 2, 3}Admissible loading space

|**G**_{frame}(=*G*|_{frame})**n**Shape gradient function of frame structures

|**G**_{shell}(=*G*|_{shell})**n**Shape gradient function of shell structures

|**G**_{solid}(=*G*|_{solid})**n**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

*H*^{1}Sobolev space of square integrable and differentiable

*l*Compliance

*l*_{p}Principal compliance

*L*Lagrange functional

*M*Volume

*M*_{0}Initial volume

- \( \widehat{M} \)
Constraint value of

*M***n**Direction vector

**n**_{1}Unit vector in

*x*_{1}direction**n**_{2}Unit vector in

*x*_{2}direction**n**^{btm}Normal vector at the bottom surface of shell structures

**n**^{mid}Normal vector at the mid-surface of shell structures

**n**^{top}Normal vector at the top surface of shell structures

**n**^{φ}Unit vector according to

*φ***n**^{φ + π}Unit vector according to angle

*φ*+*π**P*_{1},*P*_{2},*P*_{3},*P*_{4}Positions subjected to unknown loadings

*r*(≥2)Multiplicity of repeated eigenvalues

*s*Time of domain variation

*S*Centroidal axis of frame structures

*S*^{j}(*j*= 1, 2, 3, ...,*N*)Centroidal axis of member

*j*in frame structures*S*_{s}Centroidal axis of frame structures after domain variation

*t*Thickness of shell structures

*T*_{s}(*X*)Mapping

= {**u***u*_{i}}Displacement vector of frame structures

*U*Admissible function space satisfying Dirichlet boundary condition

**V**Design velocity field

**V**_{1}Design velocity field in

*n*_{1}direction**V**_{2}Design velocity field in

*n*_{2}direction= {**w***w*_{i}}_{i = 1, 2, 3}Displacement vector

**w**_{0}= {*w*_{0α}}_{α = 1, 2}In-plane displacement vector of shell structures

= {**x***x*_{1},*x*_{2},*x*_{3}}Position vector

- \( \widehat{x}=\left\{{\widehat{x}}_1,{\widehat{x}}_2,{\widehat{x}}_3\right\} \)
Vector of loading position

(={**X***X*_{1},*X*_{2},*X*_{3}})Position vector in

*Ω***X**_{s}(={*X*_{s1},*X*_{s2},*X*_{s3}})Position vector in

*Ω*_{s}*∂A*_{b}Circumference of the cross-section of members in frame structures

*ℝ*One-dimensional space

*ℝ*^{2}Two-dimensional space

*ℝ*^{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

*η*_{max}Maximum value of

*η**κ*Twice the mean curvature of shell structures or the curvature of frame structures

*λ*^{(r)}(*r*= 1, 2, 3, ...)*r*th eigenvalue*μ*Shear modulus of frame structures

*ν*Poisson’s ratio

*θ*Circumferential angle of compliance

= {**θ***θ*_{i}}(*i*= 1, 2, 3)Rotation angle vector of frame structures in the local coordinate system

= {**θ***θ*_{α}}(*α*= 1, 2)Rotation angle vector of shell structures in the local coordinate system

*ρ*Density

- {
*ξ*_{ij}}_{i,j = 1,2,3},{*ξ*_{αβ}}_{α,β = 1,2} Weighting coefficient matrix in terms of

*F**ΔM*Decrease of volume

*Γ*Boundary of domain

*Ω**Γ*^{j}(*j*= 1, 2, 3, ...,*N*)Circumference surface of member

*j*in frame structures*Γ*_{s}Boundary of domain

*Ω*_{s}*Λ*Lagrange multipliers for volume constraint

*Ω*Initial domain

*Ω*_{f}Domain subjected to external loadings

*Ω*^{j}(*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

*Ω*_{s}Updated domain after variation

## Notes

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