# Robust Design Optimization of Car-Door Structures with Spatially Varied Material Uncertainties

Authors: Yuee Zhao; Hai Dong; Haibin Liang
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

This paper presents an effective approach for robust design optimization of car-door structures with spatially varied material properties. This spatially varied material property causes structural response quantities; for example, the natural frequency and the lateral stiffness coefficient become random variables. In this regard, the Karhunen-Loève expansion is first used to represent the elastic modulus and the mass density random fields as a series of random variables. Then, a stochastic finite-element model is formulated for uncertainty quantification of the car-door structure. Combined with a polynomial-based response surface model to mimic the true performance indicator, this allows one to efficiently evaluate probability constraints for the robust design optimization of the uncertain car-door structure. In numerical simulations, design variables of the uncertain car-door structure are defined as thickness values of the tailor rolled blank structure at various regions, whereas multiple design objectives are formulated via the structural weight, the first-order natural frequency, and the lateral stiffness coefficient. Results have shown that the mean value of performance indicators can be generally improved, whereas the response variance is further minimized to archive the robust design objective. The probability-based constraint is significant to relate the Pareto optimum set to the targeted structural safety level. The proposed approach is simple, suggesting an attractive tool for the robust design optimization of car-door structures with spatially varied material uncertainties.


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## Body

## 1. Introduction

The design optimization of car-door structures has received considerable attention due to its significant role in the daily safe commuting of students [1, 2]. In this regard, Kunakornong et al. [3] realized the design optimization of bus structures by considering Sandwich composite via the finite-element approach. Lan et al. [4] presented the comparative design optimization of bus side structures with and without supporting members between longitudinal beams, whereas the multiobjective design optimization of bus structures was considered in Zhong et al. [5] to simultaneously minimize the structural total weight and maximize the torsional stiffness. Together with the lightweight design objective, performance indicators are formulated as the structural lateral stiffness and the first-order natural frequency in this paper. Specifically, probability constraints to evaluate the exceeding probability of the two performance indicators over their design limits are further considered to account for structural safety concerns.Numerical iterations for design optimization of the car-door structure are usually realized based on a finite-element (FE) model that depicts the structural implicit function, for example, the first-order natural frequency and the lateral stiffness coefficient. The gradient-based optimization algorithm, however, needs to recursively run the FE model for optimum design variables [6, 7]. This is computationally demanding for the design optimization involving high-dimensional input parameters. Many surrogate modelling algorithms, for example, the Gaussian process regression method [8], the polynomial chaos expansion, and the artificial neural networks, were considered in the literature [9, 10]. Once a response surface model that mimics the true performance function is analytically or numerically available, the Pareto optimal set can be obtained for the lightweight and the dynamic/strength design optimization of the car-door structure.Another issue is about various sources of input uncertainties associated with the structural simulation model, for example, the geometry, the material property, and the loading parameters [11]. Take the elastic modulus and the mass density car-door structure as examples, which are actually realized as the high-strength alloy steel. This material property might show spatially varied uncertainties due to fluctuated manufacturing factors, for example, the pressure, the temperature, and the milling force. This motivates us to employ the random field theory for structural uncertainty simulation [12]. This is because the optimum obtained via the deterministic model sometimes becomes unrealistic and less meaningful in reality [13]. Along with various contributions of the deterministic optimization in the literature [14, 15], this paper considers optimizing the structural performance and minimizing the response variance, that is, the robust-based design optimization of car-door structures with uncertain input factors.In general, a realization of the spatially varied material property consists of an infinite number of random variables, which occur explicitly in the structural governing equation [16]. After considering the safety constraint for highly reliable structures, the spatially varied input uncertainty cannot be directly handled by ordinary simulation algorithms [17, 18]. The uncertainty quantification of the uncertain car-door structure, hence, becomes a challenging but highly demanding task [19]. To address the spatially varied input uncertainty, the Karhunen-Loève (K-L) expansion is used to represent the material random field as a small number of deterministic functions and independent random variables [20]. This is useful to develop a stochastic FE model for uncertainty simulation of the car-door structure.Once a simulation model is set up to depict the implicit design performance function of the car-door structure, the multiple design objectives can be numerically optimized via a global iteration algorithm [21], for example, the genetic algorithm in this paper. Therefore, the dynamic characteristics of the car-door structure are first modelled by the FE formulation and the Mindlin plate theory [22–24]. Herein, the stochastic stiffness and mass formulations are determined based on the numerical results of the material random field. Note that the material property has been subjectively assumed as deterministic constants, which implies that the Gauss-Legendre scheme with three to six points can be used to evaluate the stiffness and mass integrals. The introduction of the spatially varied random field, however, results in high-order eigenfunctions to define the stochastic FE matrix, and high-order Gauss-quadrature schemes are required to address the multivariate integration problem. Specifically, the FE model becomes stochastic due to input random variables associated with the material random field. This motivates the robust design optimization to minimize the structural response variance. Note that other advanced plate models can be alternatively used to perform a numerical simulation of the car-door structure. The presented stochastic FE model and the robust design method are general to determine the corresponding design optimization results.The objective of this manuscript is to present an effective approach for the robust design optimization of car-door structures with spatially varied material uncertainties. To begin with, the elastic modulus and the mass density of the car-door structure are modelled as the Gaussian random field, which is numerically represented through the K-L expansion method with a small number of deterministic spatial functions and Gaussian random variables. Then, a stochastic FE model for structural uncertainty analysis is set up based on the simulated material random field and the Mindlin plate theory. This allows one to numerically determine stochastic samples for uncertain analysis, for example, the natural frequency and the lateral stiffness coefficient of the car-door structure. To reduce the computational cost for recursively running the stochastic FE model, a polynomial-based response surface is used to mimic the true design performance function. The robust design objective is formulated via the structural lightweight indicator, the first-order natural frequency, and the lateral stiffness coefficient. Note that the minimization of the response variance has been taken into account to search for robust design solutions. Together with structural safety constraints in terms of the nonexceeding probability with respect to the natural frequency and the lateral stiffness, the Pareto optimum set is obtained for robust design optimization car-door structures with spatially varied material uncertainties.The rest of this manuscript is organized as follows. Section2.1 briefly summarizes the K-L expansion method for numerical representation of material random fields, whereas simulation results for the elastic modulus and the mass density random fields are demonstrated in Section 2.2. In Section 3, the stochastic finite-element model for structural uncertainty simulation is further developed. This is used to define multiple design objectives and safety constraints in Section 4.1. To reduce computational cost caused by recursively running the stochastic FE model, a polynomial-based response surface model is presented in Section 4.2 to mimic the true design performance indicator. Combined with the genetic algorithm to search for the global optima, the Pareto sets are determined in Section 4.3 for various structural safety levels. Conclusions are summarized in Section 5.


## 2. Uncertain Modelling of Spatially Varied Material Properties

To model spatially varied material properties of the car-door structure, this section first briefly summarizes the K-L expansion method. Numerical simulation results for the mass density and Young’s modulus random fields are followed to demonstrate the engineering applications. Due to the spatially varied material uncertainty, the structural response quantity, for example, the first-order natural frequency or the lateral stiffness coefficient, becomes a random variable. This motivates the robust design optimization to minimize the response variance of the car-door structure. To begin with, the numerical representation of the material random field is presented based on the K-L expansion method as follows.


### 2.1. The Karhunen-Loève Expansion

The section considers the numerical representation of a two-dimensional material random fieldHx,y;ξ. A sample of the spatially varied material property contains infinity numbers of random variables. This motivates the utility of the K-L expansion method to represent the random field by means of random variables.Following the random field theory, the expectation of the two-dimensional material random field is defined as(1)μHx,y=∫DHx,y;ξdξ,and H′x,y;ξ=Hx,y;ξ−μHx,y denotes the fluctuation of Hx,y;ξ with respect to the mean-value function. Therefore, the covariance function of the material random field is defined as(2)Covx,x′;y,y′=EH′x,y;ξH′x,y;ξ,which is a nonnegative quantity for any function ϕx,y on a subregion Ωe⊆Ω:(3)∑x,x′,y,y′∈ΩeCovx,x′;y,y′ϕx,yϕx′,y′>0.Through the K-L expansion method, the two-dimensional random fieldHx,y;ξ can be represented as [25](4)HHx,y;ξ=μHx,y+∑i=1∞λiϕix,yξi,where μH(x, y) represents the mean value of the random field and the eigenvalues λi and eigenfunctions ϕi(x, y) are the solution of the two-dimensional homogeneous Fredholm integral equation of the second kind [26]:(5)∫ΩCovx,x′;y,y′ϕx,ydxdy=λϕx′,y′.Specifically, arbitrary two orders of the eigenfunction are orthogonally defined with respect to each other; that is,∫Ωϕix,yϕjx,ydxdy=δij, where δij is the Kronecker symbol. This determines uncorrelated random variables ξi as(6)ξi=1λi∫ΩH′x,y;ξϕix,ydxdy,and the eigendecomposition of the covariance model is given as(7)Covx,x′;y,y′=∑i=1∞λiϕix,yϕix′,y′,which contains infinity orders of the eigensolution λi,ϕix,y.After truncating the K-L expansion afterM terms, an approximation of the two-dimensional material random field Hx,y;ξ is given as(8)H^x,y;ξ=μHx,y+∑i=1Mλiϕix,yξi,which is approximated by M-order eigenfunctions and the corresponding Gaussian random variables ξii=1,…,M.Numerical simulation of the random field needs to solve the integral eigenvalue problem (IEVP) in equation (6), and the analytic solution is seldom available except for a few covariance models defined on canonical regions, for example, the rectangular and circles in the two-dimensional case. One has to resort to a numerical algorithm, for example, the collocation or the expansion optimal linear estimation (EOLE) method in the literature [27].Once numerical results for the eigensolutionλi,ϕix,y are derived, a realization of the two-dimensional material random field can be given as(9)H^x,y;ξ=μHx,y+∑i=1Mλ^iϕ^ix,yξi,where λ^i,ϕ^ix,y denote numerical solutions of the true but unknown eigenpairs, which are numerically determined via the high-order polynomial-based Ritz-Galerkin approximation approach [28]. The corresponding simulation results of the two-dimensional material random field are further presented as follows.


### 2.2. Numerical Realizations of the Material Random Field

To demonstrate the utility of the K-L expansion method for numerical representation of spatially varied material uncertainties, this section presents simulation results for the elastic modulus random field of the car-door structure. To begin with, Figure1 depicts the structural simulation domain, which has been discretized via 1028 Q4 elements. Note that this random field simulation algorithm can deal with other structural geometries once the corresponding FE mesh result is available.Figure 1
An illustrative model for random field simulation of the car-door structure. (a) The physical model. (b) Simulation domain and mesh result.
(a)(b)For the sake of illustration, the elasticity modulus of the car-door structure is assumed as the Gaussian random field with the exponential covariance model as follows:(10)Covx,y;x′,y′=σE2x,yexp−x−x′δx−y−y′δy,as∀x,x′,y,y′∈Ω.Herein, the correlation length parameters in thex- and y-dimension are given as δx=δy=1.0m. Note that the mean-value function is assumed as homogeneous; that is, μEx,y=2.068×105 MPa. Given the coefficient of variation (COV) 20%, the standard deviation function is defined as σEx,y=0.2×μEx,y. Based on the FE mesh result of the car-door structure in Figure 1(b), the high-order polynomial-based Rize-Galerkin approach is used to determine numerical results for the eigenvalues and eigenfunctions of the car-door structure.Figure2 first presents numerical results for eigenvalues of the exponential covariance model, whereas the corresponding results for the first-eight orders of the two-dimensional eigenfunction are depicted in Figure 3. Note that the eigenpair result λi,ϕix,y is no long analytically available for the car-door structure due to its noncanonical simulation region. The tensor-based approximation approach, therefore, is not applicable to simulate the multivariate material random field in this paper.Figure 2
Numerical results for eigenvaluesλii=1,…,30 of the elastic modulus random field.Figure 3
Numerical results for eigenfunctionsϕ^ix,yi=1,…,8 of the car-door structure based on the exponential covariance mode: the first-eight orders (a∼h) of the eigenfunction results.
(a)(b)(c)(d)(e)(f)(g)(h)Once numerical results for the eigenpair are available, numerical realizations of the elastic modulus random field can be generated as(11)H^Ex,y;ξ≈μEx,y+∑i=1Mλiξiϕix,y,where the truncation order parameter is given as M=10 in subsequent simulations.Four realizations of the elastic modulus random field are presented in Figure4. For the sake of illustration, they are randomly selected from totally 104 stochastic realizations of the material random field. To implement the Monte-Carlo simulation (MCS) of the material random field, 104 samples of the standard Gaussian variables ξii=1,…,M in equation (11) are first digitally generated. Substituting for the corresponding results of the eigenvalues the eigenfunctions in Figures 2 and 3, equation (11) is able to determine the corresponding sample of the elastic modulus random field in conjunction with the truncation order parameter M=10.Figure 4
Four realizations (a∼d) of the elasticity random field based on the truncation parameterM=10 (unit: MPa).
(a)(b)(c)(d)It is observed that numerical realizations of the elastic modulus random field are continuously varied around the mean value2.068×105MPa within the whole simulation domain x,y∈Ω, rather than being deterministically fixed as a constant value in conventional formations. With this spatially varied material uncertainty, structural response quantities, for example, the first-order natural frequency and the lateral stiffness coefficient, become random variables. This motivates the robust design of the car-door structure to simultaneously optimize the performance indicator and minimize the response variance as follows.Figure5 further presents relative errors for the mean and the standard deviation functions of the simulated random field, which are generally less than 0.5% and 6.0%, respectively, compared to the benchmark result provided by the MCS method with 104 samples. This implies the utility of the truncation parameter M=10 is able to include the majority of input uncertainties for uncertainty simulation of the car-door structure. Similar results for the mass density random field are determined, yet they are omitted here for the sake of brevity. This has verified the applicability of the K-L expansion method for digital simulations of the structural material random fields, and the corresponding stochastic FE model is further derived as follows.Figure 5
The mean value and the standard deviation errors for the simulation results of the elasticity random field. (a) Relative error for the mean-value function. (b) Relative error for the standard deviation.
(a)(b)


## 2.1. The Karhunen-Loève Expansion

The section considers the numerical representation of a two-dimensional material random fieldHx,y;ξ. A sample of the spatially varied material property contains infinity numbers of random variables. This motivates the utility of the K-L expansion method to represent the random field by means of random variables.Following the random field theory, the expectation of the two-dimensional material random field is defined as(1)μHx,y=∫DHx,y;ξdξ,and H′x,y;ξ=Hx,y;ξ−μHx,y denotes the fluctuation of Hx,y;ξ with respect to the mean-value function. Therefore, the covariance function of the material random field is defined as(2)Covx,x′;y,y′=EH′x,y;ξH′x,y;ξ,which is a nonnegative quantity for any function ϕx,y on a subregion Ωe⊆Ω:(3)∑x,x′,y,y′∈ΩeCovx,x′;y,y′ϕx,yϕx′,y′>0.Through the K-L expansion method, the two-dimensional random fieldHx,y;ξ can be represented as [25](4)HHx,y;ξ=μHx,y+∑i=1∞λiϕix,yξi,where μH(x, y) represents the mean value of the random field and the eigenvalues λi and eigenfunctions ϕi(x, y) are the solution of the two-dimensional homogeneous Fredholm integral equation of the second kind [26]:(5)∫ΩCovx,x′;y,y′ϕx,ydxdy=λϕx′,y′.Specifically, arbitrary two orders of the eigenfunction are orthogonally defined with respect to each other; that is,∫Ωϕix,yϕjx,ydxdy=δij, where δij is the Kronecker symbol. This determines uncorrelated random variables ξi as(6)ξi=1λi∫ΩH′x,y;ξϕix,ydxdy,and the eigendecomposition of the covariance model is given as(7)Covx,x′;y,y′=∑i=1∞λiϕix,yϕix′,y′,which contains infinity orders of the eigensolution λi,ϕix,y.After truncating the K-L expansion afterM terms, an approximation of the two-dimensional material random field Hx,y;ξ is given as(8)H^x,y;ξ=μHx,y+∑i=1Mλiϕix,yξi,which is approximated by M-order eigenfunctions and the corresponding Gaussian random variables ξii=1,…,M.Numerical simulation of the random field needs to solve the integral eigenvalue problem (IEVP) in equation (6), and the analytic solution is seldom available except for a few covariance models defined on canonical regions, for example, the rectangular and circles in the two-dimensional case. One has to resort to a numerical algorithm, for example, the collocation or the expansion optimal linear estimation (EOLE) method in the literature [27].Once numerical results for the eigensolutionλi,ϕix,y are derived, a realization of the two-dimensional material random field can be given as(9)H^x,y;ξ=μHx,y+∑i=1Mλ^iϕ^ix,yξi,where λ^i,ϕ^ix,y denote numerical solutions of the true but unknown eigenpairs, which are numerically determined via the high-order polynomial-based Ritz-Galerkin approximation approach [28]. The corresponding simulation results of the two-dimensional material random field are further presented as follows.


## 2.2. Numerical Realizations of the Material Random Field

To demonstrate the utility of the K-L expansion method for numerical representation of spatially varied material uncertainties, this section presents simulation results for the elastic modulus random field of the car-door structure. To begin with, Figure1 depicts the structural simulation domain, which has been discretized via 1028 Q4 elements. Note that this random field simulation algorithm can deal with other structural geometries once the corresponding FE mesh result is available.Figure 1
An illustrative model for random field simulation of the car-door structure. (a) The physical model. (b) Simulation domain and mesh result.
(a)(b)For the sake of illustration, the elasticity modulus of the car-door structure is assumed as the Gaussian random field with the exponential covariance model as follows:(10)Covx,y;x′,y′=σE2x,yexp−x−x′δx−y−y′δy,as∀x,x′,y,y′∈Ω.Herein, the correlation length parameters in thex- and y-dimension are given as δx=δy=1.0m. Note that the mean-value function is assumed as homogeneous; that is, μEx,y=2.068×105 MPa. Given the coefficient of variation (COV) 20%, the standard deviation function is defined as σEx,y=0.2×μEx,y. Based on the FE mesh result of the car-door structure in Figure 1(b), the high-order polynomial-based Rize-Galerkin approach is used to determine numerical results for the eigenvalues and eigenfunctions of the car-door structure.Figure2 first presents numerical results for eigenvalues of the exponential covariance model, whereas the corresponding results for the first-eight orders of the two-dimensional eigenfunction are depicted in Figure 3. Note that the eigenpair result λi,ϕix,y is no long analytically available for the car-door structure due to its noncanonical simulation region. The tensor-based approximation approach, therefore, is not applicable to simulate the multivariate material random field in this paper.Figure 2
Numerical results for eigenvaluesλii=1,…,30 of the elastic modulus random field.Figure 3
Numerical results for eigenfunctionsϕ^ix,yi=1,…,8 of the car-door structure based on the exponential covariance mode: the first-eight orders (a∼h) of the eigenfunction results.
(a)(b)(c)(d)(e)(f)(g)(h)Once numerical results for the eigenpair are available, numerical realizations of the elastic modulus random field can be generated as(11)H^Ex,y;ξ≈μEx,y+∑i=1Mλiξiϕix,y,where the truncation order parameter is given as M=10 in subsequent simulations.Four realizations of the elastic modulus random field are presented in Figure4. For the sake of illustration, they are randomly selected from totally 104 stochastic realizations of the material random field. To implement the Monte-Carlo simulation (MCS) of the material random field, 104 samples of the standard Gaussian variables ξii=1,…,M in equation (11) are first digitally generated. Substituting for the corresponding results of the eigenvalues the eigenfunctions in Figures 2 and 3, equation (11) is able to determine the corresponding sample of the elastic modulus random field in conjunction with the truncation order parameter M=10.Figure 4
Four realizations (a∼d) of the elasticity random field based on the truncation parameterM=10 (unit: MPa).
(a)(b)(c)(d)It is observed that numerical realizations of the elastic modulus random field are continuously varied around the mean value2.068×105MPa within the whole simulation domain x,y∈Ω, rather than being deterministically fixed as a constant value in conventional formations. With this spatially varied material uncertainty, structural response quantities, for example, the first-order natural frequency and the lateral stiffness coefficient, become random variables. This motivates the robust design of the car-door structure to simultaneously optimize the performance indicator and minimize the response variance as follows.Figure5 further presents relative errors for the mean and the standard deviation functions of the simulated random field, which are generally less than 0.5% and 6.0%, respectively, compared to the benchmark result provided by the MCS method with 104 samples. This implies the utility of the truncation parameter M=10 is able to include the majority of input uncertainties for uncertainty simulation of the car-door structure. Similar results for the mass density random field are determined, yet they are omitted here for the sake of brevity. This has verified the applicability of the K-L expansion method for digital simulations of the structural material random fields, and the corresponding stochastic FE model is further derived as follows.Figure 5
The mean value and the standard deviation errors for the simulation results of the elasticity random field. (a) Relative error for the mean-value function. (b) Relative error for the standard deviation.
(a)(b)


## 3. Stochastic Finite-Element Model of the Car-Door Structure

Rather than being assumed as deterministic constants, the elastic modulus and the mass density of the car-door structure are further modelled via the random field theory. The utility of the K-L expansion method can represent the material random field by means of a limited number of random variables. Due to this spatially varied material property, uncertain simulation of the car-door structure requires stochastic finite-element models. In this regard, the Mindlin plate theory is first presented for deterministic FE simulation of the car-door structure, whereas stochastic stiffness and mass formulations are further derived to account for the spatially varied material uncertainties. This determines a stochastic FE model for uncertainty analysis of the car-door structure.


### 3.1. The Mindlin Plate Theory

Following the Mindlin plate theory, the general displacement for the car-door structure can be represented as [29](12)ux,y=zx,yθxx,y=zx,y∂wx,y∂x,vx,y=zx,yθyx,y=zx,y∂wx,y∂y,wx,y=wx,y,where u, v, and w denote displacements in the x-, y-, and the z-direction, respectively, whereas θx and θy represent the corresponding rotation angles as a reference to the neutral surface of the x-z and y-z planes, respectively.Given the shape functionNx,y for the Q4 element, the structural displacement vector can be further interpolated as(13)wx,y=∑iNix,ywix,y,θxx,y=∑iNix,yθxx,y,θyx,y=∑iNix,yθyx,y.Herein,wix,y, θxix,y, and θyix,y denote the displacement components at an ith node of the FE mesh result.Following the linear stress-strain relation, the stress vector is expressed as(14)σ=Dε,where σ=σx,σyγxy,γxz,γyz,T and ε=εx,εyεxy,εxz,εyz,T denote the stress and strain tensors, respectively, whereas the matrix D is given in a diagonal form as(15)D=Db00Ds.Herein, the bending and shearing matrices are(16)Db=E1−ν21ν0ν10001−ν2,Ds=E21+ν1001.This determines the stiffness matrix for an element of the FE model as(17)Ke=h312∬ΩeBbTDbBbdxdy+αh∬ΩeBsTDsBsdxdy.Note thatΩe defines the element of representing the car-door structure, whereas α=5/6 represents the correction factor associated with the thickness parameter h. Note that the strain matrices Bb and Bs are generally defined by partial derivatives of ∂Nix,y/∂x and ∂Nix,y/∂y of the shape function [30].Similarly, the mass matrix is formulated as(18)Me=∬ΩeρNTdiagh,h312,h312Ndxdy,where the two-dimensional integration can be exactly evaluated via the three-order Gauss-Legendre quadrature [30].Once spatially varied mass and density random fields are considered, the K-L approximation in equation (9) determines the eigenfunction ϕ^ix,y as high-order polynomials. Therefore, a high-order Gauss-quadrature scheme is required for numerical evaluation of the mass and the stiffness matrix integrals. Besides, the finite-element formulation is derived via the standard Mindlin plate theory [31], and other advanced plate theories can be alternatively used. Once an ordinary FE model is available, a similar stochastic simulation model can be developed as follows for uncertainty simulation of the car-door structure with spatially varied material uncertainties.


### 3.2. The Stochastic Finite-Element Model

Following the random field simulation procedure presented inSection 2.2, the elastic modulus and the mass density of the car-door structure are realized as spatially varied input uncertainties. The corresponding stiffness and mass matrices of the FE model become spatially dependent within the whole simulation domain. This derives the stochastic FE model as follows.Following the K-L approximation result, numerical realizations of the elastic modulus and the mass density are given as(19)H^Ex,y;ξ≈μEx,y+∑i=1Mλ^Eiϕ^Eix,yξEi,H^ρx,y;ξ≈μρx,y+∑i=1Mλ^ρiϕ^ρix,yξρi.Substituting for the uncertain material property, the matrixD in equation (14) is further realized as(20)Dbx,y;ξ=H^Ex,y;ξ1−v21v0v10001−v2,Dsx,y;ξ=H^Ex,y;ξ21+v00H^Ex,y;ξ21+v.This determines the corresponding stiffness matrix similar to the deterministic result in equation (17) as(21)Kex,y;ξ=h312∬ΩeBbTDbx,y;ξBbdxdy+αh∬ΩeBsTDsx,y;ξBsdxdy,which becomes spastically varied due to random factors ξi and eigenfunctions ϕ^Eix,yi=1,…,M.Similarly, the mass matrix that accounts for the material random field is given as(22)Μex,y;ξ=∬ΩeH^ρx,y;ξNTx,ydiagh,h312,h312Nx,ydxdy.Substituting for the mass densityH^ρx,y;ξ result, the stochastic mass matrix is finally realized as(23)Μex,y;ξ=∬Ωeμρx,yNTx,ydiagh,h312,h312Nx,ydxdy+∑i=1Mλ^ρiξρi×∬Ωeϕ^ρix,yNTx,ydiagh,h312,h312Nx,ydxdy.Compared to the deterministic result in equation (18), an introduction of the spatially varied material uncertainties results in high-order integrals with respect to the mean-value function μx,y and eigenfunctions ϕ^ρix,y. Note that μx,y is assumed to be constant for a homogeneous random field in this paper, whereas approximation results for ϕ^ρix,y in Figure 3 require high-order polynomials. Therefore, a high-order Gaussian quadrature scheme is necessary for a reliable result of Kex,y;ξ and Mex,y;ξ in reality.Given the isotropic transform of the shape functionNx,y, the Gauss-Legendre quadrature is used to evaluate the stochastic stiffness and mass matrices. This further determined the global FE model for uncertainty analysis of the car-door structure:(24)Kx,y;ξ=∑eTeKex,y;ξTeMx,y;ξ=∑eTeMex,y;ξTe.Herein,Te represents the coordinate transformation matrix, whereas ∑e denotes the procedure to assembly all element results together towards the global matrix.In summary, the stochastic simulation of the car-door structure with spatially varied material uncertainties includes the representation of material random fields via the K-L expansion inSection 2.1, the simulation of the elastic modulus and the mass random fields in Section 2.2, the derivation of stochastic stiffness and mass matrices in equation (21) and (23), and the formulation of stochastic FE models via equation (24).To demonstrate numerical applications of the presented stochastic FE model for uncertainty simulation of the car-door structure, average models for the elastic modulus and the mass density are constantly assumed as(25)μEx,y=2.068×105MPa,μρx,y=7.9×103kg/m3.Given the coefficient of variation (COV) as 20%, the standard deviation function becomesσx,y=0.2×μx,y.The K-L expansion with the truncation parameterM=10 is first used to simulate the stochastic material property as shown in Figure 4. Then, a sample of the standard Gaussian random variables ξEi and ξρii=1,…,10 is digitally generated to realize the stiffness and mass matrices, and a solution of the general eigenvalue problem determines the corresponding sample of the structural natural frequency.Repeating this simulation procedure 104 times, a collection of all samples allows one to determine an estimation of the mean-value result for the mode shape as shown in Figure 6. Note that the outer boundary of the car-door structure is fully fixed to mimic the closed-door scenario for the natural frequency analysis. In addition, the empirical probability distribution for the first-order natural frequency of the car-door structure is depicted as shown in Figure 7, which has been closely fitted by the Gaussian distribution.Figure 6
Mean-value results for the first-six orders of the structural natural frequency: from (a) to (g).
(a)(b)(c)(d)(e)(f)Figure 7
The empirical probability distribution for the first-order nature frequency of the car-door structure with spatially varying material properties. (a) The probability density function. (b) The cumulative distribution function.
(a)(b)To account for the stiffness characteristic of the car-door structure, the lateral stiffness coefficient is defined as the ratio between the applied force at the door-handle position and the maximal structural deflection. Herein, the applied force is deterministically assumed as 500 N. Therefore, a solution of the equationKx,y;ξUx,y;ξ=Q determines the deflection Ux,y;ξ and the corresponding sample of the lateral stiffness coefficient.Figure8 demonstrates the empirical distribution result of the lateral stiffness coefficient based on 10−4 samples of the elastic modulus random field. It is observed that the natural frequency and the lateral stiffness coefficient become random variables after taking into account the spatially varied material uncertainties. This motivates the robust design optimization of the uncertain car-door structure to optimize the performance indicator and simultaneously minimize the response variance as follows.Figure 8
The empirical probability distribution for the lateral stiffness coefficient of the car-door structure with spatially varying material properties. (a) The probability density function. (b) The cumulative distribution function.
(a)(b)


## 3.1. The Mindlin Plate Theory

Following the Mindlin plate theory, the general displacement for the car-door structure can be represented as [29](12)ux,y=zx,yθxx,y=zx,y∂wx,y∂x,vx,y=zx,yθyx,y=zx,y∂wx,y∂y,wx,y=wx,y,where u, v, and w denote displacements in the x-, y-, and the z-direction, respectively, whereas θx and θy represent the corresponding rotation angles as a reference to the neutral surface of the x-z and y-z planes, respectively.Given the shape functionNx,y for the Q4 element, the structural displacement vector can be further interpolated as(13)wx,y=∑iNix,ywix,y,θxx,y=∑iNix,yθxx,y,θyx,y=∑iNix,yθyx,y.Herein,wix,y, θxix,y, and θyix,y denote the displacement components at an ith node of the FE mesh result.Following the linear stress-strain relation, the stress vector is expressed as(14)σ=Dε,where σ=σx,σyγxy,γxz,γyz,T and ε=εx,εyεxy,εxz,εyz,T denote the stress and strain tensors, respectively, whereas the matrix D is given in a diagonal form as(15)D=Db00Ds.Herein, the bending and shearing matrices are(16)Db=E1−ν21ν0ν10001−ν2,Ds=E21+ν1001.This determines the stiffness matrix for an element of the FE model as(17)Ke=h312∬ΩeBbTDbBbdxdy+αh∬ΩeBsTDsBsdxdy.Note thatΩe defines the element of representing the car-door structure, whereas α=5/6 represents the correction factor associated with the thickness parameter h. Note that the strain matrices Bb and Bs are generally defined by partial derivatives of ∂Nix,y/∂x and ∂Nix,y/∂y of the shape function [30].Similarly, the mass matrix is formulated as(18)Me=∬ΩeρNTdiagh,h312,h312Ndxdy,where the two-dimensional integration can be exactly evaluated via the three-order Gauss-Legendre quadrature [30].Once spatially varied mass and density random fields are considered, the K-L approximation in equation (9) determines the eigenfunction ϕ^ix,y as high-order polynomials. Therefore, a high-order Gauss-quadrature scheme is required for numerical evaluation of the mass and the stiffness matrix integrals. Besides, the finite-element formulation is derived via the standard Mindlin plate theory [31], and other advanced plate theories can be alternatively used. Once an ordinary FE model is available, a similar stochastic simulation model can be developed as follows for uncertainty simulation of the car-door structure with spatially varied material uncertainties.


## 3.2. The Stochastic Finite-Element Model

Following the random field simulation procedure presented inSection 2.2, the elastic modulus and the mass density of the car-door structure are realized as spatially varied input uncertainties. The corresponding stiffness and mass matrices of the FE model become spatially dependent within the whole simulation domain. This derives the stochastic FE model as follows.Following the K-L approximation result, numerical realizations of the elastic modulus and the mass density are given as(19)H^Ex,y;ξ≈μEx,y+∑i=1Mλ^Eiϕ^Eix,yξEi,H^ρx,y;ξ≈μρx,y+∑i=1Mλ^ρiϕ^ρix,yξρi.Substituting for the uncertain material property, the matrixD in equation (14) is further realized as(20)Dbx,y;ξ=H^Ex,y;ξ1−v21v0v10001−v2,Dsx,y;ξ=H^Ex,y;ξ21+v00H^Ex,y;ξ21+v.This determines the corresponding stiffness matrix similar to the deterministic result in equation (17) as(21)Kex,y;ξ=h312∬ΩeBbTDbx,y;ξBbdxdy+αh∬ΩeBsTDsx,y;ξBsdxdy,which becomes spastically varied due to random factors ξi and eigenfunctions ϕ^Eix,yi=1,…,M.Similarly, the mass matrix that accounts for the material random field is given as(22)Μex,y;ξ=∬ΩeH^ρx,y;ξNTx,ydiagh,h312,h312Nx,ydxdy.Substituting for the mass densityH^ρx,y;ξ result, the stochastic mass matrix is finally realized as(23)Μex,y;ξ=∬Ωeμρx,yNTx,ydiagh,h312,h312Nx,ydxdy+∑i=1Mλ^ρiξρi×∬Ωeϕ^ρix,yNTx,ydiagh,h312,h312Nx,ydxdy.Compared to the deterministic result in equation (18), an introduction of the spatially varied material uncertainties results in high-order integrals with respect to the mean-value function μx,y and eigenfunctions ϕ^ρix,y. Note that μx,y is assumed to be constant for a homogeneous random field in this paper, whereas approximation results for ϕ^ρix,y in Figure 3 require high-order polynomials. Therefore, a high-order Gaussian quadrature scheme is necessary for a reliable result of Kex,y;ξ and Mex,y;ξ in reality.Given the isotropic transform of the shape functionNx,y, the Gauss-Legendre quadrature is used to evaluate the stochastic stiffness and mass matrices. This further determined the global FE model for uncertainty analysis of the car-door structure:(24)Kx,y;ξ=∑eTeKex,y;ξTeMx,y;ξ=∑eTeMex,y;ξTe.Herein,Te represents the coordinate transformation matrix, whereas ∑e denotes the procedure to assembly all element results together towards the global matrix.In summary, the stochastic simulation of the car-door structure with spatially varied material uncertainties includes the representation of material random fields via the K-L expansion inSection 2.1, the simulation of the elastic modulus and the mass random fields in Section 2.2, the derivation of stochastic stiffness and mass matrices in equation (21) and (23), and the formulation of stochastic FE models via equation (24).To demonstrate numerical applications of the presented stochastic FE model for uncertainty simulation of the car-door structure, average models for the elastic modulus and the mass density are constantly assumed as(25)μEx,y=2.068×105MPa,μρx,y=7.9×103kg/m3.Given the coefficient of variation (COV) as 20%, the standard deviation function becomesσx,y=0.2×μx,y.The K-L expansion with the truncation parameterM=10 is first used to simulate the stochastic material property as shown in Figure 4. Then, a sample of the standard Gaussian random variables ξEi and ξρii=1,…,10 is digitally generated to realize the stiffness and mass matrices, and a solution of the general eigenvalue problem determines the corresponding sample of the structural natural frequency.Repeating this simulation procedure 104 times, a collection of all samples allows one to determine an estimation of the mean-value result for the mode shape as shown in Figure 6. Note that the outer boundary of the car-door structure is fully fixed to mimic the closed-door scenario for the natural frequency analysis. In addition, the empirical probability distribution for the first-order natural frequency of the car-door structure is depicted as shown in Figure 7, which has been closely fitted by the Gaussian distribution.Figure 6
Mean-value results for the first-six orders of the structural natural frequency: from (a) to (g).
(a)(b)(c)(d)(e)(f)Figure 7
The empirical probability distribution for the first-order nature frequency of the car-door structure with spatially varying material properties. (a) The probability density function. (b) The cumulative distribution function.
(a)(b)To account for the stiffness characteristic of the car-door structure, the lateral stiffness coefficient is defined as the ratio between the applied force at the door-handle position and the maximal structural deflection. Herein, the applied force is deterministically assumed as 500 N. Therefore, a solution of the equationKx,y;ξUx,y;ξ=Q determines the deflection Ux,y;ξ and the corresponding sample of the lateral stiffness coefficient.Figure8 demonstrates the empirical distribution result of the lateral stiffness coefficient based on 10−4 samples of the elastic modulus random field. It is observed that the natural frequency and the lateral stiffness coefficient become random variables after taking into account the spatially varied material uncertainties. This motivates the robust design optimization of the uncertain car-door structure to optimize the performance indicator and simultaneously minimize the response variance as follows.Figure 8
The empirical probability distribution for the lateral stiffness coefficient of the car-door structure with spatially varying material properties. (a) The probability density function. (b) The cumulative distribution function.
(a)(b)


## 4. Robust Design Optimization of the Car-Door Structure

This section considers the robust design optimization (RDO) of the car-door structure with spatially varied material uncertainties. To begin with, the multiobjective design functions in terms of the lightweight indicator, the first-order natural frequency, and the lateral stiffness coefficient are formulated. To overcome the computational demanding cost associated with the uncertainty analysis, a polynomial-based regression model is proposed to mimic the true performance function. Together with the probability-based constraint on the structural safety level, this finalizes an effective procedure for the robust design optimization of car-door structures with spatially varied material properties.


### 4.1. The Robust Performance Function

The performance indicators, for example, the first-order natural frequency, the total weight, and the lateral stiffness coefficient, are directly linked with spatially varied material uncertainties of the car-door structure. This is modelled via a stochastic FE formulation based on the Mindlin plate theory. In this respect, the proposed approach can be extended to other plate theories for uncertainty analysis and robust design optimization of the car-door structure.Due to the input material uncertainties, performance indicators of the car-door structure become random variables. To achieve the robust design optimization, the objective function is usually defined via the average performanceμYt and the variance σY2t. Therefore, three types of the robust design model have been considered in the literature [32]:(1)
Nominal-the-best:(26)FμY,σY2;ξ,t=δ1μY−ytargetμ0−y02+δ2σYσ02.(i)
Herein,ytarget and y0 are the target and the initial values of the performance indicator Yt,ξ, respectively, and δ1∈0,1 and δ2∈0,1 are weights with the relation δ1+δ2=1. To reduce the dimensionality of the mean and the variance objectives, each term has to be normalized by the initial design results μ0 and σ0.(2)
Smaller-the-best:(27)FμY,σY2;ξ,t=δ1μYμ02+δ2σYσ02.(3)
Largest-the-best:(28)FμY,σY2;ξ,t=δ1μ0μY2+δ2σYσ02.In this paper, the main objective of the design optimization is to maximize the first-order natural frequency and the structural lateral stiffness coefficient and yet minimize the structural total weight and the response variance. This determines the design objective functions as(29)Fmξ,t=δm1μmξ,tμm0ξ,t+δm2σmξ,tσm0ξ,t,Ffξ,t=δf1μf0ξ,tμfξ,t+δf2σfξ,tσf0ξ,t,Fsξ,t=δs1μs0ξ,tμsξ,t+δs2σsξ,tσs0ξ,t.Herein, symbols with the subscript “0” represent the nominal value, whereasμ(⋅) and σ(⋅) define the mean value and the standard variation of the performance indicator. Together with weighting coefficients δ(⋅), the mass, the natural frequency, and the stiffness functions are defined for the robust-based design optimization.Specifically, the door structure is manufactured as the tailor rolled blank with different thickness values, that is, the hinge reinforcement panelt1, the window frame t2, the lock reinforcement panel t3, the window reinforcement panel t4, and the interior and outer panels t5 and t6 [33]. Note that the initial values of the design variable are assumed as t0=1.5,2.4,0.8,1.2,1.0,1.7T (unit: mm). Together with the lower and upper design limits tL=1.0,1.8,0.5,0.7,0.5,1.0T and tU=2.5,3.0,1.5,2.0,1.5,2.5T, the parameters will be used for design optimizations of the car-door structure.Following the stochastic simulation procedure of the door structure, the mean value of the total weight, the first-order natural frequency, and the lateral stiffness coefficient are estimated as 6.66 kg, 23.88 Hz, and 136.01 N/mm, respectively, whereas the corresponding standard deviation results are given as 1.03 kg, 2.38 Hz, and 22.57 N/mm. To efficiently realize the robust design optimization, a polynomial-based response surface method is determined as follows to reduce the embedded computational cost for probabilistic simulations.


### 4.2. The Response Surface Model

The performance indicators defined in equation (29) are formulated by standard Gaussian random variables ξ and design parameters t=t1,…,t6T. The corresponding design iterations need to recursively run the stochastic FE model for the mean value and the variance results of the design function. To reduce the total amount of the computational cost, a polynomial-based response surface model is used to mimic the true performance function of the car-door structure as follows.The polynomial-based response surface model (p-RSM) has been widely used to represent complex input-output relations in engineering reality. However, the total number of samples for a reliable predictor is positively proportional to the highest-order of the polynomial function, which motivates us to use quadratic p-RSM including mixed terms in this paper [34–36]:(30)y=a+∑i=1mbiui+∑i=1mciui2+∑i=1m−1∑j>imdijuiuj+∈err.Herein, the variabley represents a performance indicator, that is, the structural weight, the first-order natural frequency, and the lateral stiffness coefficient, whereas the vector u contains all design variables ti and the standard Gaussian variables ξi due to the discretization result of the material random field. Totally, l=2m+6m=10 variables are finally involved to develop the response surface model.Or in a matrix form, the second-order p-RSM model can be expressed as(31)y=uTβ+∈err.To determine the regression vectorβ, totally p=3×2m+6 low-discrepancy realizations of the structural performance indicator are gathered as the training dataset y=y1,…,ypT:(32)Ξ=1u11⋯u1l2⋯u1l−1u1l1u21⋯u2l2⋯u2l−1u2l⋮⋱⋱⋮⋱⋮1up1⋯upl2⋯upl−1uplp×q,where the parameter q is defined as l+1l+2/2.This determines the regression coefficients in equation (31) as(33)β^=ΞΤΞ−1ΞΤy,as well as the covariance matrix(34)Covβ^=σerr2ΞΤΞ−1.Here, the symbolσerr2 denotes the global variance of the residual error ∈err, and its unbiased estimator is given as(35)σ^err2=zTIp−Hyq−p,where Ip denotes a p×p identity matrix and H=ΞΞTΞ−1ΞT.Once the training matrixΞ defined in equation (32) and the corresponding performance samples y are available, the minimization of the residual errors allows one to derive the p-RSM as(36)y^ξ,t=uTβ^,whereu=tT,ξTT,which is used to mimic the true but computationally intensive stochastic finite-element response of the car-door structure.Figure9 presents the polynomial-based surrogate model for the structural weight ℳξ,t, the first-order natural frequency ℱξ,t, and the lateral stiffness coefficient Sξ,t. As a reference to the benchmark result provided by the stochastic finite-element simulation, the numerical accuracy of the surrogate model y^ can be evaluated by the root-of-mean-square error (RMSE):(37)RMSE≔1my¯∑i=1myi−y^i2,where the mean value of the prediction result is defined as y¯=1/m∑i=1myi.Figure 9
Numerical verification of the response surface model for the performance indicator of the car-door structure. (a) The total weight. (b) The first-order natural frequency. (c) The lateral stiffness coefficient.
(a)(b)(c)Results for the RMSE in Table1 have shown the high accuracy of the p-RSM for reliable prediction results. The small statistical error RMSE≤1.956×10−3 has validated the applicability of the quadratic surrogate model for design optimization of the uncertain car-door structure.Table 1
Results for the RMSE of the polynomial-based response surface model.
Total weightℳξ,tNatural frequencyℱξ,tLateral stiffnessSξ,tRMSE2.139 × 10−48.365 × 10−41.956 × 10−3Therefore, a mathematical model for the probability-based robust design optimization of the car-door structure is finally formulated as(38)Find:t∗=t1∗,…,t6∗TminFmξ,t∩maxFfξ,t∩maxFsξ,ts.t.tiL≤ti≤tiUPrℱξ,t≥f0≥P1PrSξ,t≥s0≥P2.Herein,F⋅ξ,t denotes {design} objectives listed in equation (29), whereas the weighting coefficients are given as 0.5. This implies that the mean and variance of the performance function have been equivalently treated for the robust design optimization. Specifically, the operator Pr⋅ evaluates the probability of a performance indicator over its design limit. Together with the targeted probability Pii=1,2, this allows one to determine the corresponding design result associated with the targeted safety level, for example, 90% or 95% in numerical examples.


### 4.3. Numerical Results for the Design Optimization

The robust design optimization of the car-door structure with spatially varied material properties is numerically realized based on the flowchart depicted in Figure10. This includes the reproduction of material random field simulation with the truncated K-L expansion in equation (11), the development of the stochastic finite-element model for uncertain mass, stiffness coefficient, and natural frequency analysis in Section 3.2, and the utility of the polynomial-based response surface model to mimic the structural performance indicators in equation (30). Once the robust design optimization model in equation (38) is set up, numerical iteration to optimize design variables is realized with the genetic algorithm in the literature [21].Figure 10
The flowchart for robust design optimization of the car-door structure with spatially varied material properties.To begin with, the structural design optimization is considered based on various objectives in equation (29). In this regard, the lightweight design objective has been considered as the priority. Combined with the natural frequency and the lateral stiffness indicators, this determines two design cases for the biobjective design optimization.Table2 summarizes the robust design optimization result for the single, bi, and triobjective scenarios. In general, the design optimization is able to simultaneously improve the performance indicator and minimize the response variance. However, the uniobjective design can determine the most optimum result among all investigated scenarios. For instance, the mean value of the lateral stiffness coefficient Sξ,t has been improved from 136.01 N/mm to 145.25 N/mm with an increasing rate 6.79%, whereas the response variance has been significantly minimized over 61.4% (from 508.95 N2/mm2 to 196.28 N2/mm2). A small response variance implies the improved robustness to accommodate potential variabilities caused by input material uncertainties. As a comparison, results based on the bi- and triobjective functions are not improved as much as that of the single-objective case. Therefore, the biobjective scenarios, that is, minFmξ,t∩maxFfξ,t and minFmξ,t∩maxFsξ,t, will be further investigated in conjunction with various probability-based constraint levels.Table 2
Results for the robust design optimization based on various combinations of the objective function.
Initial resultUniobjectiveBiobjectiveTriobjectiveℳ0ℱ0S0ℳ∗ℱ∗S∗ℳ∗∩ℱ∗ℳ∗∩S∗ℳ∗∩ℱ∗∩S∗Mean6.6623.88136.014.3726.04145.255.01, 24.734.87, 142.55.06, 24.28, 140.25Std.D1.032.4122.560.511.5814.100.61, 1.840.57, 15.220.63, 1.96, 16.17COV15.5%10.09%16.58%10.98%10.98%9.70%12.17%, 7.44%11.49%, 10.68%12.45%, 8.07%, 11.53%Std.D: standard deviation; COV: coefficient of variation.To account for the effect of a probability-based constraint on the lightweight design of the uncertain car-door structure, threshold values for the first-order natural frequency and the lateral stiffness coefficient are given as 22 Hz and 123 N/mm, respectively. Following the procedure in equation (38), Table 3 summarizes the corresponding lightweight design result associated with the targeted safety levels 90% and 95%. Note that initial values and design boundaries of the design variable t are similar to those in the foregoing section.Table 3
Results for the lightweight design optimization associated with various probability-based constraints.
Probability-based constraintt1∗t2∗t3∗t4∗t5∗t6∗ℳξ,t∗MeanCOV (%)Prℱξ,t≥f0≥90%1.42841.94270.64060.77910.59851.26174.98899.78Prℱξ,t≥f0≥95%1.45311.98190.67800.81210.63691.32915.171210.66PrSξ,t≥f0≥90%1.47752.13640.73850.89760.76431.49175.66798.91PrSξ,t≥f0≥95%1.47492.12450.73460.91150.75851.47315.857810.52f0 = 22 Hz and s0 = 123 N/mm.Compared to the initial design result, the mean value of the total massℳξ,t has been generally minimized for all investigated cases, yet they are all slightly heavier than those provided in Table 2. This is mainly due to additional safety constraints on the first-order natural frequency and the lateral stiffness. To better understand the effect of this safety-related constraint, the Pareto optimum sets for the weight-frequency and the weight-stiffness objectives are determined as follows.Figure11 depicts the Pareto result for the weight-frequency and the weight-stiffness biobjective functions in conjunction with the 90% and 95% safety levels. The third performance function, for example, the lateral stiffness coefficient in the weight-frequency case, has been used to define the probability constraint. It is observed that the 95% probability level determines a more conservative result than that of the 90% nonexceeding probability. For instance, a realization of the Pareto optimum set can be determined as (24 Hz, 5.81 kg) for the 90% safety level, whereas this total weight result will be increased to 7.0 kg for the 95% probability level and the identical natural frequency result. As a comparison, the Pareto curve for the weight-stiffness case is almost identical for the two investigated probability scenarios. In this regard, one can conclude that the first-order natural frequency is more sensitive to the variation of thickness variables than that of the lateral stiffness coefficient.Figure 11
The Pareto results for the weight-frequency and the weight-stiffness biobjective functions of the uncertain car-door structure associated with 90% and 95% probability-based constraints. (a) The weight-frequency biobjective function. (b) The weight-frequency biobjective function.
(a)(b)To investigate the effect of the input variability on the design optimization result, Figure12 presents the corresponding Pareto optimum set based on various COV values of the material random field. In this regard, the COV value of the Exponential covariance model has been assumed to vary from 10% to 30%. Combined with the 95%-based probability constraint, significantly divergent Pareto optimum sets were obtained for the weight-frequency objective function. This has confirmed the large variation of the first-order natural frequency with respect to the COV value of the material random field. This is obvious for the case of COV = 30%, in which a small increase of the structural weight can result in a dramatic increase in the first-order natural frequency. The lateral stiffness coefficient, however, is almost insensitive to the COV value of the material random field. Therefore, the Pareto optimum sets for the weight-stiffness case are almost identical for various COV values. In summary, the effectiveness of the proposed procedure has been further justified by numerical robust design optimization of the uncertain car-door structure with various COV values of the material random field.Figure 12
The Pareto result for the lightweight design optimization of the car-door structure (the legends indicate the COV values of the material random field). (a) The weight-frequency objective function. (b) The weight-frequency objective function.
(a)(b)


## 4.1. The Robust Performance Function

The performance indicators, for example, the first-order natural frequency, the total weight, and the lateral stiffness coefficient, are directly linked with spatially varied material uncertainties of the car-door structure. This is modelled via a stochastic FE formulation based on the Mindlin plate theory. In this respect, the proposed approach can be extended to other plate theories for uncertainty analysis and robust design optimization of the car-door structure.Due to the input material uncertainties, performance indicators of the car-door structure become random variables. To achieve the robust design optimization, the objective function is usually defined via the average performanceμYt and the variance σY2t. Therefore, three types of the robust design model have been considered in the literature [32]:(1)
Nominal-the-best:(26)FμY,σY2;ξ,t=δ1μY−ytargetμ0−y02+δ2σYσ02.(i)
Herein,ytarget and y0 are the target and the initial values of the performance indicator Yt,ξ, respectively, and δ1∈0,1 and δ2∈0,1 are weights with the relation δ1+δ2=1. To reduce the dimensionality of the mean and the variance objectives, each term has to be normalized by the initial design results μ0 and σ0.(2)
Smaller-the-best:(27)FμY,σY2;ξ,t=δ1μYμ02+δ2σYσ02.(3)
Largest-the-best:(28)FμY,σY2;ξ,t=δ1μ0μY2+δ2σYσ02.In this paper, the main objective of the design optimization is to maximize the first-order natural frequency and the structural lateral stiffness coefficient and yet minimize the structural total weight and the response variance. This determines the design objective functions as(29)Fmξ,t=δm1μmξ,tμm0ξ,t+δm2σmξ,tσm0ξ,t,Ffξ,t=δf1μf0ξ,tμfξ,t+δf2σfξ,tσf0ξ,t,Fsξ,t=δs1μs0ξ,tμsξ,t+δs2σsξ,tσs0ξ,t.Herein, symbols with the subscript “0” represent the nominal value, whereasμ(⋅) and σ(⋅) define the mean value and the standard variation of the performance indicator. Together with weighting coefficients δ(⋅), the mass, the natural frequency, and the stiffness functions are defined for the robust-based design optimization.Specifically, the door structure is manufactured as the tailor rolled blank with different thickness values, that is, the hinge reinforcement panelt1, the window frame t2, the lock reinforcement panel t3, the window reinforcement panel t4, and the interior and outer panels t5 and t6 [33]. Note that the initial values of the design variable are assumed as t0=1.5,2.4,0.8,1.2,1.0,1.7T (unit: mm). Together with the lower and upper design limits tL=1.0,1.8,0.5,0.7,0.5,1.0T and tU=2.5,3.0,1.5,2.0,1.5,2.5T, the parameters will be used for design optimizations of the car-door structure.Following the stochastic simulation procedure of the door structure, the mean value of the total weight, the first-order natural frequency, and the lateral stiffness coefficient are estimated as 6.66 kg, 23.88 Hz, and 136.01 N/mm, respectively, whereas the corresponding standard deviation results are given as 1.03 kg, 2.38 Hz, and 22.57 N/mm. To efficiently realize the robust design optimization, a polynomial-based response surface method is determined as follows to reduce the embedded computational cost for probabilistic simulations.


## 4.2. The Response Surface Model

The performance indicators defined in equation (29) are formulated by standard Gaussian random variables ξ and design parameters t=t1,…,t6T. The corresponding design iterations need to recursively run the stochastic FE model for the mean value and the variance results of the design function. To reduce the total amount of the computational cost, a polynomial-based response surface model is used to mimic the true performance function of the car-door structure as follows.The polynomial-based response surface model (p-RSM) has been widely used to represent complex input-output relations in engineering reality. However, the total number of samples for a reliable predictor is positively proportional to the highest-order of the polynomial function, which motivates us to use quadratic p-RSM including mixed terms in this paper [34–36]:(30)y=a+∑i=1mbiui+∑i=1mciui2+∑i=1m−1∑j>imdijuiuj+∈err.Herein, the variabley represents a performance indicator, that is, the structural weight, the first-order natural frequency, and the lateral stiffness coefficient, whereas the vector u contains all design variables ti and the standard Gaussian variables ξi due to the discretization result of the material random field. Totally, l=2m+6m=10 variables are finally involved to develop the response surface model.Or in a matrix form, the second-order p-RSM model can be expressed as(31)y=uTβ+∈err.To determine the regression vectorβ, totally p=3×2m+6 low-discrepancy realizations of the structural performance indicator are gathered as the training dataset y=y1,…,ypT:(32)Ξ=1u11⋯u1l2⋯u1l−1u1l1u21⋯u2l2⋯u2l−1u2l⋮⋱⋱⋮⋱⋮1up1⋯upl2⋯upl−1uplp×q,where the parameter q is defined as l+1l+2/2.This determines the regression coefficients in equation (31) as(33)β^=ΞΤΞ−1ΞΤy,as well as the covariance matrix(34)Covβ^=σerr2ΞΤΞ−1.Here, the symbolσerr2 denotes the global variance of the residual error ∈err, and its unbiased estimator is given as(35)σ^err2=zTIp−Hyq−p,where Ip denotes a p×p identity matrix and H=ΞΞTΞ−1ΞT.Once the training matrixΞ defined in equation (32) and the corresponding performance samples y are available, the minimization of the residual errors allows one to derive the p-RSM as(36)y^ξ,t=uTβ^,whereu=tT,ξTT,which is used to mimic the true but computationally intensive stochastic finite-element response of the car-door structure.Figure9 presents the polynomial-based surrogate model for the structural weight ℳξ,t, the first-order natural frequency ℱξ,t, and the lateral stiffness coefficient Sξ,t. As a reference to the benchmark result provided by the stochastic finite-element simulation, the numerical accuracy of the surrogate model y^ can be evaluated by the root-of-mean-square error (RMSE):(37)RMSE≔1my¯∑i=1myi−y^i2,where the mean value of the prediction result is defined as y¯=1/m∑i=1myi.Figure 9
Numerical verification of the response surface model for the performance indicator of the car-door structure. (a) The total weight. (b) The first-order natural frequency. (c) The lateral stiffness coefficient.
(a)(b)(c)Results for the RMSE in Table1 have shown the high accuracy of the p-RSM for reliable prediction results. The small statistical error RMSE≤1.956×10−3 has validated the applicability of the quadratic surrogate model for design optimization of the uncertain car-door structure.Table 1
Results for the RMSE of the polynomial-based response surface model.
Total weightℳξ,tNatural frequencyℱξ,tLateral stiffnessSξ,tRMSE2.139 × 10−48.365 × 10−41.956 × 10−3Therefore, a mathematical model for the probability-based robust design optimization of the car-door structure is finally formulated as(38)Find:t∗=t1∗,…,t6∗TminFmξ,t∩maxFfξ,t∩maxFsξ,ts.t.tiL≤ti≤tiUPrℱξ,t≥f0≥P1PrSξ,t≥s0≥P2.Herein,F⋅ξ,t denotes {design} objectives listed in equation (29), whereas the weighting coefficients are given as 0.5. This implies that the mean and variance of the performance function have been equivalently treated for the robust design optimization. Specifically, the operator Pr⋅ evaluates the probability of a performance indicator over its design limit. Together with the targeted probability Pii=1,2, this allows one to determine the corresponding design result associated with the targeted safety level, for example, 90% or 95% in numerical examples.


## 4.3. Numerical Results for the Design Optimization

The robust design optimization of the car-door structure with spatially varied material properties is numerically realized based on the flowchart depicted in Figure10. This includes the reproduction of material random field simulation with the truncated K-L expansion in equation (11), the development of the stochastic finite-element model for uncertain mass, stiffness coefficient, and natural frequency analysis in Section 3.2, and the utility of the polynomial-based response surface model to mimic the structural performance indicators in equation (30). Once the robust design optimization model in equation (38) is set up, numerical iteration to optimize design variables is realized with the genetic algorithm in the literature [21].Figure 10
The flowchart for robust design optimization of the car-door structure with spatially varied material properties.To begin with, the structural design optimization is considered based on various objectives in equation (29). In this regard, the lightweight design objective has been considered as the priority. Combined with the natural frequency and the lateral stiffness indicators, this determines two design cases for the biobjective design optimization.Table2 summarizes the robust design optimization result for the single, bi, and triobjective scenarios. In general, the design optimization is able to simultaneously improve the performance indicator and minimize the response variance. However, the uniobjective design can determine the most optimum result among all investigated scenarios. For instance, the mean value of the lateral stiffness coefficient Sξ,t has been improved from 136.01 N/mm to 145.25 N/mm with an increasing rate 6.79%, whereas the response variance has been significantly minimized over 61.4% (from 508.95 N2/mm2 to 196.28 N2/mm2). A small response variance implies the improved robustness to accommodate potential variabilities caused by input material uncertainties. As a comparison, results based on the bi- and triobjective functions are not improved as much as that of the single-objective case. Therefore, the biobjective scenarios, that is, minFmξ,t∩maxFfξ,t and minFmξ,t∩maxFsξ,t, will be further investigated in conjunction with various probability-based constraint levels.Table 2
Results for the robust design optimization based on various combinations of the objective function.
Initial resultUniobjectiveBiobjectiveTriobjectiveℳ0ℱ0S0ℳ∗ℱ∗S∗ℳ∗∩ℱ∗ℳ∗∩S∗ℳ∗∩ℱ∗∩S∗Mean6.6623.88136.014.3726.04145.255.01, 24.734.87, 142.55.06, 24.28, 140.25Std.D1.032.4122.560.511.5814.100.61, 1.840.57, 15.220.63, 1.96, 16.17COV15.5%10.09%16.58%10.98%10.98%9.70%12.17%, 7.44%11.49%, 10.68%12.45%, 8.07%, 11.53%Std.D: standard deviation; COV: coefficient of variation.To account for the effect of a probability-based constraint on the lightweight design of the uncertain car-door structure, threshold values for the first-order natural frequency and the lateral stiffness coefficient are given as 22 Hz and 123 N/mm, respectively. Following the procedure in equation (38), Table 3 summarizes the corresponding lightweight design result associated with the targeted safety levels 90% and 95%. Note that initial values and design boundaries of the design variable t are similar to those in the foregoing section.Table 3
Results for the lightweight design optimization associated with various probability-based constraints.
Probability-based constraintt1∗t2∗t3∗t4∗t5∗t6∗ℳξ,t∗MeanCOV (%)Prℱξ,t≥f0≥90%1.42841.94270.64060.77910.59851.26174.98899.78Prℱξ,t≥f0≥95%1.45311.98190.67800.81210.63691.32915.171210.66PrSξ,t≥f0≥90%1.47752.13640.73850.89760.76431.49175.66798.91PrSξ,t≥f0≥95%1.47492.12450.73460.91150.75851.47315.857810.52f0 = 22 Hz and s0 = 123 N/mm.Compared to the initial design result, the mean value of the total massℳξ,t has been generally minimized for all investigated cases, yet they are all slightly heavier than those provided in Table 2. This is mainly due to additional safety constraints on the first-order natural frequency and the lateral stiffness. To better understand the effect of this safety-related constraint, the Pareto optimum sets for the weight-frequency and the weight-stiffness objectives are determined as follows.Figure11 depicts the Pareto result for the weight-frequency and the weight-stiffness biobjective functions in conjunction with the 90% and 95% safety levels. The third performance function, for example, the lateral stiffness coefficient in the weight-frequency case, has been used to define the probability constraint. It is observed that the 95% probability level determines a more conservative result than that of the 90% nonexceeding probability. For instance, a realization of the Pareto optimum set can be determined as (24 Hz, 5.81 kg) for the 90% safety level, whereas this total weight result will be increased to 7.0 kg for the 95% probability level and the identical natural frequency result. As a comparison, the Pareto curve for the weight-stiffness case is almost identical for the two investigated probability scenarios. In this regard, one can conclude that the first-order natural frequency is more sensitive to the variation of thickness variables than that of the lateral stiffness coefficient.Figure 11
The Pareto results for the weight-frequency and the weight-stiffness biobjective functions of the uncertain car-door structure associated with 90% and 95% probability-based constraints. (a) The weight-frequency biobjective function. (b) The weight-frequency biobjective function.
(a)(b)To investigate the effect of the input variability on the design optimization result, Figure12 presents the corresponding Pareto optimum set based on various COV values of the material random field. In this regard, the COV value of the Exponential covariance model has been assumed to vary from 10% to 30%. Combined with the 95%-based probability constraint, significantly divergent Pareto optimum sets were obtained for the weight-frequency objective function. This has confirmed the large variation of the first-order natural frequency with respect to the COV value of the material random field. This is obvious for the case of COV = 30%, in which a small increase of the structural weight can result in a dramatic increase in the first-order natural frequency. The lateral stiffness coefficient, however, is almost insensitive to the COV value of the material random field. Therefore, the Pareto optimum sets for the weight-stiffness case are almost identical for various COV values. In summary, the effectiveness of the proposed procedure has been further justified by numerical robust design optimization of the uncertain car-door structure with various COV values of the material random field.Figure 12
The Pareto result for the lightweight design optimization of the car-door structure (the legends indicate the COV values of the material random field). (a) The weight-frequency objective function. (b) The weight-frequency objective function.
(a)(b)


## 5. Conclusions

This paper presents an effective approach for the robust design optimization of car-door structures with spatially varied material uncertainties. To implement, the Karhunen-Loève expansion method is first used to represent the elastic modulus and the mass density random fields as a few deterministic functions and random variables. Then, a stochastic finite-element model is developed for uncertainty simulation of the car-door structure. To reduce the computational cost for iteratively running the mechanistic model, a polynomial-based surrogate model is developed to predict stochastic responses of the uncertain car-door structure. Combined with various probability-based constraints to address the safety issue, robust design optimization of the uncertain car-door structure is finally realized through the genetic algorithm to search for optimum mean-value performance results and simultaneously minimizing the response variance.Numerical results have shown that the polynomial-based surrogate model is able to provide reliable prediction results for uncertain response quantities of the uncertain car-door structure, whereas the mean and the variance of the performance function can be generally improved via the proposed robust design model. The 95%-based probability constraint function determines a more conservative result than that of the 90% safety level. The COV value of the covariance model has shown a significant effect on the Pareto optimum set for the first-order natural frequency performance function, whereas results for the weight-stiffness biobjective function are determined almost identical for various input COV levels. This has demonstrated potential applications of the proposed approach for the robust design optimization of uncertain car-door structures with spatially varied material properties.



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