Multivariate Rational Response Surface Approximation of Nodal Displacements of Truss Structures

The response surface methodology (RSM) explores the relationships between several explanatory variables and one or more response variables. The method was introduced by Box and Wilson in 1951. The main idea of RSM is to use a sequence of designed experiments to obtain an optimal response. Box and Wilson suggest using a second-degree polynomial model to do this [1].

Because response surface methodology can establish the unknown function between design variables and response variables, it has been widely applied to optimization. Choon-Man Jang and Ka-Ram Choi [2] carried out the optimization of the blower impeller by using the response surface method (RSM). Heng Jiang et al. [3] carried out a dynamic and static multi-objective optimization of a vertical machining center based on response surface method. Quadratic polynomials are employed to construct response surface (RS) model, which reflects the relationship between design inputs and structural response outputs, according to the response outputs of these samples obtained by analyzing the dynamic and static characteristics of the machining centre at these samples with the software ANSYS. In the field of vehicle optimization, RSM also get a widely application [4‐6]. Chun-Lin Wang et al. [7] carried out a variable curvature blade of fire pump based on experimental design theory and response surface approximation. Dong-Sheng Jia et al. [8] established the mass function of impeller of water valve-controlled hydrodynamic coupling with RSM and optimized the impeller key structural parameters. Response surface method is applied in the field of robust optimization design [9, 10]. Reliability analysis has an improvement based on response surface method [11, 12].

Generally speaking, in the mathematical models of structural optimization, the objective function is the explicit function of the design variables and the constraint function is the implicit function of the design variables. This characteristic is also the important difference between the structure optimization problem and the mathematical program problem and it greatly increases the complexity of the structure optimization problem [13]. In order to solve this problem, the implicit function must be made explicit and the response surface methodology, is the most commonly used method [14]. Hong-Wu et al. [15] used response surface methodology to establish the function expressions of deflection and structural weight,and optimized the truss structure. Roux et al. [16] carried out RS-based optimization of truss structures and concluded that RS accuracy depends on the range of the design area and the form of the explicit function. Zhang et al. [17] establish a fitted regression model for the dynamic transmission error(DTE) fluctuations to quantify the relationship between modification amounts and DTE fluctuations by using response surface method.

In order to increase the accuracy and efficiency, Fan, et al. [18] derived a criterion for judging the existence of cross terms and proposed an adaptive response surface method. Houten et al. [19] considered fifth-order polynomials to check if higher-order polynomials improved RSM accuracy. Venter et al. [20] solved plate optimization problems using higher order RS approximations.

Sui and his collaborators [21‐25] developed the central point accurate response surface method where the response value at the center point of the approximate model is equal to the true response value. This method was successfully applied to optimization of membrane structures, 2D-continuum structures, plate and shell structures subject to frequency constraints, and multidisciplinary optimization. Using the property of the Kreisselmerier-Steinhauser function, RS approximation was built with function derivation, Taylor expansion near experimental points and numerical iteration, trying to minimize the largest difference between response function values and true response values.

In the finite element displacement method, the nodal displacement is the basic unknown quantity. The stress, dynamic response and so on can be derived from the displacement, so this paper is devoted to discuss the response surface of truss node displacement, its research method and results can be easily extended to other types of structures. It should be noted that the analytical form of polynomial-basis response surface approximation is often very different from the functions describing the actual displacement field. As this may result in low quality approximation, it is important to assess the real analytical form of the displacement field. This study will show that the nodal displacement field determined by finite element method for truss structures can be expressed as multivariate rational functions. In view of this, a response surface methodology based on multivariate rational functions will be developed. It will be illustrated that the determinant of the stiffness matrix and the corresponding elements of adjoint matrix involved in displacement determination are polynomials depending on cross-sectional areas, with the same order as their respective matrices, each term of which is the product of cross-sectional areas. The validity of the proposed approach is verified for many design examples of statically determinate and statically indeterminate truss structures.

According to the finite element method theory, the balance equation is as

The displacement equation is as Where the inverse matrix can be obtained by the formula as

According to the theory of determinants, the n-order determinant is a number firmed by n² elements of a _ij .where \(\tau (j_{1} j_{2} \cdots j_{n} )\) is the inverted sequence number of natural numbers 1,2,…,n, and \(\sum\nolimits_{{j_{1} \cdots j_{n} }} {}\) is the sum of all permutations of natural numbers 1,2,…,n.

Since elements a _ij of stiffness matrix include products of linear terms corresponding to cross-sectional areas, the determinant can be simplified aswhere f(A ⁿ ) is an n-order polynomial of cross-sectional area variables A, and n is the order of the stiffness matrix, equal to the DOF of the structure system.

Elements of the adjoint matrix adjK are polynomials g _ij (Aⁿ⁻¹) which are one order lower than the determinant of the stiffness matrix. Thus, the form of the adjoint matrix is

Combining all of the above formulas, it can be obtained the equation to analytically determine nodal displacements of truss structures:

Consequently, the displacement of a certain node in a given direction can be expressed as

The previous derivation indicate that the determinant of the stiffness matrix of the truss structure and the elements of adjoint matrix involved in the determination of nodal displacements are respectively n-order and (n–1)-order polynomials. Stiffness matrix determinant is an algebraic sum including n! terms each of which is the product of n factors belonging to different rows and columns. The sign of each term depends on the corresponding inverted sequence number. Therefore, there are at least n! coefficients. Similarly, determinants of adjoint matrix elements are algebraic sums of (n–1)! terms each of which corresponds to the product of (n–1) factors belonging to different rows and different columns. The adjoint matrix is an n-order square matrix. Equation (26) implies that 2n! function evaluations must be done. However, if matrix order changes, the number of coefficients also will change in a factorial way. Hence, if the equation order gets large, too many combinations should be considered. Variable linking and relationships between cross-sectional area variables may greatly reduce computational complexity of combinations and multivariate rational function response surfaces can be built based on these relationships. The above mentioned relationships will be illustrated by the following examples.

where

The following conjecture can be made from the previous relationships. The determinant of stiffness matrix and the adjoint matrix term entailed in the determination of a certain nodal displacement are polynomials of the same order as the corresponding matrices. Each term of these polynomials is the product of first power terms corresponding to cross-sectional areas.where \(i\,1 \ne i2 \ne i3 \ne \cdots \ne i(n - 1) \ne in\).

The number of DOF n is the same as the order of stiffness matrix; m is the number of design variables. A ⁿ are the possible combinations of cross-sectional areas A _i selected from the set of m design variables; m₁ is the number of these combinations, namely, \(m_{1} = C_{m}^{n} = {{m!} \mathord{\left/ {\vphantom {{m!} {[(m - n)!n!]}}} \right. \kern-0pt} {[(m - n)!n!]}}\). Similarly, Aⁿ⁻¹ are other possible combinations of cross-sectional areas A _i ; m₂ is the number of these combinations, namely, \(m_{2} = C_{m}^{n - 1} = {{m!} \mathord{\left/ {\vphantom {{m!} {[(m - n + 1)! \cdot (n - 1)!]}}} \right. \kern-0pt} {[(m - n + 1)! \cdot (n - 1)!]}}\); α and β are unknown coefficients to be determined.

The number of terms included in matrix stiffness determinant decreases from n! to \({{m!} \mathord{\left/ {\vphantom {{m!} {[(m - n)!n!]}}} \right. \kern-0pt} {[(m - n)!n!]}}\); while the number of terms in the corresponding adjoint matrix decreases from (n-1)! to \({{m!} \mathord{\left/ {\vphantom {{m!} {[(m - n + 1)! \cdot (n - 1)!]}}} \right. \kern-0pt} {[(m - n + 1)! \cdot (n - 1)!]}}\), thus reducing the computational cost of the fitting process.

Let′s now consider a statically determinate structure with m bars and n DOFs. Since there are no redundant bars, it holds m = n. Hence, in Eq. (29), it holds \(m_{1} = C_{m}^{n} = C_{n}^{n} = 1,\) \(m_{2} = C_{m}^{n - 1} = C_{n}^{n - 1} = n\) and the equation simplifies to

By substituting Eq. (30) into Eq. (26), it followswhere \(C_{j} = a_{n - j + 1} /b_{0}\).

For statically determinate structures, if the external force P is known, internal forces N _j developed in each bar are constant. If the generalized displacement δ _i in the direction i is computed using Mohr’s theorem, a unit generalized force N _i ⁰ is applied in the direction i and the corresponding internal forces N _j in each bar also are constant. Element lengths l _j and cross-sectional areas A _j are constant values as well. Therefore, nodal displacements of a truss structure can be expressed aswhere the constant C is equal to \(C_{j} = {{N_{j} N_{j}^{0} l_{j} } \mathord{\left/ {\vphantom {{N_{j} N_{j}^{0} l_{j} } E}} \right. \kern-0pt} E}\).

The consistency of Eqs. (31) and (32) proves that it may be reasonable to approximate nodal displacements of statically determinate truss structures with multivariate rational response surfaces. The following examples will prove the validity of this conclusion also for statically indeterminate trusses.

The corresponding adjoint matrix element involved in the determination of the displacement of node C in the direction of applied load F is as follows:where {a, b, c, d, e, f, g, h} ∈ {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, α and β are the unknown coefficients of the products of design variable A _i , and a ≠ b ≠ c ≠ d ≠ e ≠ f ≠ g ≠ h.

The corresponding adjoint matrix element involved in the determination of the displacement of node A in the direction of applied load F is as follows:where {a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r} ∈ {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18}, α and β are the unknown coefficients of the products of design variable A _i , and a ≠ b ≠ c ≠ d ≠ e ≠ f ≠ g ≠ h ≠ i ≠ j ≠ k ≠ l ≠ m ≠ n ≠ o ≠ p ≠ q ≠ r.

Other examples not documented in this article for the sake of brevity confirm the validity of the conjecture made on the mutual relationship between the stiffness matrix of the truss structure and adjoint matrix elements.

Response surface approximation of truss nodal displacements can be built as follows:where α _i and \(\beta_{ij}^{'}\) are the RS fitting coefficients to be determined with \(\beta_{ij}^{'}\) = β _ij F _j ; m₁ and m₂, respectively, are the number of terms included in polynomials f(A ⁿ ) and g _ij (Aⁿ⁻¹).

Suitable design points are defined including the cross-sectional areas of the M bars forming the truss. For the generic design “C”, it holds A _k  = A _k ^C with k = 1, 2,…, M.

After the displacement δ _i ^C corresponding to design points is determined from finite element analysis, Eq. (37) is rewritten as follows:or

The above expression is a homogeneous equation. Since the ratio between numerator and denominator in Eq. (37) stays the same if both terms are multiplied by the same quantity, RS model coefficients can be normalized with respect to any fitting coefficient. In order to keep generality, let us set α₁ = 1 and rewrite Eq. (39) as follows:

Selecting M design points (M ≥ m₁ + Km₂, where K is the number of loads F _j that are not equal to zero), a system of M linear equations in the unknown coefficients α _i and \(\beta_{ij}^{'}\) is obtained:

The determination of these coefficients by solving this system can be done in two ways.

According to best uniform approximation theory, quality of RS approximation improves as the fitting model resembles the real structural response. Since the multivariate rational RS model developed in this study reproduces the actual analytical model behind determination of nodal displacements of truss structures, the present model can give results very close to the real structural response of a truss structure.

The accuracy of the multivariate rational response surface approximation model developed in this study is verified in some design examples. The cross-sectional area of truss elements is selected as the design variable which can range between 0.01 and 0.05 m²; cross sections of all elements are supposed to be circular. The Young’s modulus is 2.1 × 10¹¹ Pa while Poisson’s ratio is 0.3. Node displacement is taken as the structural response to be approximated. Value of design variables selected for design points P_D is \(A_{{{\text{P}}_{\text{D}} ,i}} \in \{ 0.01,0.02,0.03,0.04,0.05\}\) while value of design variables selected for check points P_E is \(A_{{{\text{P}}_{\text{E}} ,i}} \in \{ 0.015,0.025,0.035,0.045\}\).

It can be seen that relative error between response results obtained by multivariate rational RS approximation and ANSYS is very small for all design examples. Evaluation coefficients are always very high, thus satisfying engineering requirements. The same conclusion holds true for both statically determinate and indeterminate truss structures.