Geometry and size optimization of stiffener layout for three-dimensional box structures with maximization of natural frequencies

Tinnn HU, Xiohong DING,*, Heng ZHANG, Lei SHEN, Ho LI

a School of Mechanical Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China

b Department of Mechanical Engineering and Science, Kyoto University, Kyoto 615-8540, Japan

KEYWORDS Box structure;Geometry optimization;Improved adaptive growth method;Maximum natural frequency design;Stiffener layout

Abstract Based on the growth mechanism of natural biological branching systems and inspiration from the morphology of plant root tips,a bionic design method called Improved Adaptive Growth Method (IAGM) has been proposed in the authors’ previous research and successfully applied to the reinforcement optimization of three-dimensional box structures with respect to natural frequencies. However, as a kind of ground structure methods, the final layout patterns of stiffeners obtained by using the IAGM are highly subjected to their ground structures, which restricts the optimization effect and freedom to further improve the dynamic performance of structures. To solve this problem, a novel post-processing geometry and size optimization approach is proposed in this article.This method takes the former layout optimization result as start,and iteratively finds the optimal layout angles, locations, and lengths of stiffeners with a few design variables by optimizing the positions of some specific node lines called active node lines. At the same time, thicknesses of stiffeners are also optimized to further improve natural frequencies of threedimensional box structures. Using this method, stiffeners can be successfully separated from their ground structures and further effectively improve natural frequencies of three-dimensional box structures with less material consumption. Typical numerical examples are illustrated to validate the effectiveness and advantages of the suggested method.

1. Introduction

As a promising design method to distribute material within the design domain for optimal structural performance, structural topology optimization has been widely developed since the first research of Michell1on the optimal topology of a truss structure. Many topology optimization methods, such as the Homogenization Design (HD) method,2-3the Solid Isotropic Material with Penalization (SIMP) method,4-5and the Evolutionary Structural Optimization (ESO)/Bi-dimensional ESO(BESO) method,6-8have been proposed to optimize the topological configurations with a wide range of application (automobile, aerospace, architecture, biodegradable implant,etc.,9-13considering different design objectives and constraints.

Among them, due to the significance in engineering fields,topology optimization with respect to natural frequencies has drawn researchers’ great attention after the first research of optimizing a single frequency of plate disks by Diaz and Kikuchi.14Pedersen15studied the efficiency of some modified SIMP models and optimized plate structures with maximization of the fundamental eigenfrequency.Du and Olhoff16considered the optimal design of freely vibrating structures using different approaches for multiple eigenfrequencies.Zuo et al.17proposed an improved BESO method for dynamic problems and successfully solved multiple eigenvalues and stiffnessfrequency optimization problems. Although these methods can offer rational material distribution for reference, their optimal configurations often have vague boundaries or bitmap like patterns, and need post-processing to further deal with. Besides, some important features might be eliminated during the post-processing procedure, and extra time will be consumed.18-19

Different from those vague boundaries obtained by using popular density-based methods, results of ground structure methods have unmatched advantages in manufacturability because those optimal topologies are directly presented by bar, beam, or shell elements. Since it was firstly proposed by Herbert et al.,20the ground structure approach has been widely utilized, especially in the field of civil engineering and mechanical engineering. With the consideration of gravity loading, Fairclough et al.21studied the theoretically optimal bridge form to span a long distance with the help of ground structures. Bolbotowski et al.22proposed a ground-structurebased method to identify the optimal layout of a grillage. On the basis of ground structures, Lu et al.23used plastic design layout optimization formulation to design the frame of buildings with consideration of carrying gravity and lateral loads.He et al.24introduced an adaptive ‘‘member adding” solution scheme in the numerical ground-structure-based layout optimization to efficiently optimize trusses under single or multiple load cases.

Stiffeners are common but effective components to reinforce structures from vibration or buckling25-26and stiffener layout patterns greatly determine the structural performance.27Another great application of ground structure methods is to reinforce structures with optimal stiffener layout patterns.Considering the design of stiffener layout as an evolutionary procedure,Ding and Yamazaki28-29and Ji et al.30got inspiration from the growth mechanism of natural branching systems in the plant kingdom and proposed a novel bionic design approach, named Adaptive Growth Method (AGM), for the reinforcement optimization of plate/shell structures. On the basis of Ding’s method,Li et al.31-33studied a venation growth algorithm for the static and dynamic reinforcement optimization of shell structures, and a grinding machine tool bed was optimized by Yan et al.34using Li’s method. It is wellknown that many structures in engineering can be simplified as three-dimensional box structures, such as the pedestal of machine tools and the wing of aircraft.35Therefore, Dong36and Zhang37et al. further developed the adaptive growth method for the static reinforcement optimization of threedimensional box structures. Recently, an Improved Adaptive Growth Method (IAGM) has been proposed by the authors and successfully applied to the reinforcement optimization of three-dimensional box structures with maximization of natural frequencies.38

Although they have great success in truss layout optimization and reinforcement optimization, the results obtained by ground structure methods have certain limitations. Taking our previously proposed IAGM as an example, the optimized layout of stiffeners can greatly reinforce structures but is highly subjected to their ground structures. To be specific,those stiffeners that we optimized have certain layout angles,locations, and lengths due to the discrete geometry of ground structures, which restricts the improvement of structural mechanical performance.

In order to tackle this problem, several different optimization schemes have been proposed. Based on the venation growth algorithm, Li et al. introduced a special treatment called Enhanced Stiffness Transformation Approach (ESTA)into the optimization, which enables stiffeners to grow along with arbitrary directions. Their new method has been successfully applied to static39-40and dynamic41-42reinforcement optimization of plate/shell structures, achieving better structural performance than the previous one. Wang et al.43proposed a streamline stiffener path optimization method for curved stiffener layout design. In their method, stiffeners are parameterized by streamline functions, and streamline function values are set as design variables to generate smooth and optimal stiffener layout, which is similar as the level set method with specific constraints.Using the starting and ending stiffener angles and one shift direction, Wang44-45and Cui46et al. designed grid-stiffened composite panels by mirroring and shifting parameterized curved stiffener paths from a reference path with linearly varying stiffener layout angles.Another option is to further optimize the results obtained from the initial layout optimization as a post-processing step.He and Gilbert47performed rationalization on optimized truss structures using geometry optimization. Some practical issues, such as node move limits and merging nodes, are considered during the optimization of nodal positions and cross-sectional areas of truss bars to offer better results. Fairclough and Gilbert48used mixed-integer linear programming to enforce buildability constraints and sequentially obtained simplified truss forms,which can also alleviate the influence of ground structures on optimal truss layout patterns. In the research of Ohsaki and Hayashi,49a conventional method of nonlinear programming was also used to further find optimal nodal locations of trusses, as well as their optimal cross-sectional areas. Besides,those issues like melting nodes that appeared in the research of He and Gilbert47were successfully avoided with the help of force densities.

Theoretically,the results obtained from the direct treatment of arbitrary stiffener layout angles, locations, and lengths within the layout optimization are of better structural performance,just like the research of Li et al.39-41However,optimal solutions that Li’s team obtained are not interconnected42in certain cases and a necessary post-processing for fabrication needs to be performed.40From the view of authors,such treatment will increase the burden on designers as well as computational cost despite the better results it can generate. On the other hand, stiffener layout patterns derived from the initial layout optimization procedure generally comprise far fewer stiffeners than those that need to be treated during Li’s layout optimization with ESTA. As a result, the computational expense of a post-processing optimization would be more acceptable. Furthermore, due to the maintenance of interconnection within the geometry optimization, the further optimized stiffener layout would be totally manufacturable.Thus, directly performing geometry optimization as a postprocessing step might be a better choice to solve the problem concerned and is selected in this paper to further improve the results that we optimized by using the IAGM.

In this paper, based on the stiffener layout patterns generated by the IAGM,and by mainly getting inspiration from the work of He and Gilbert,47geometry optimization is extended to three-dimensional reinforcement optimization problems to help stiffener layout patterns free from their ground structures.An optimization procedure composed of geometry and size optimization is proposed and utilized as a post-processing step to further optimize the results obtained by using the IAGM with further maximization of natural frequencies, which is the main innovation of this work and the great difference from our previously published one.38Positions of active node lines(which will be introduced in detail in Sections 2 and 3) and thicknesses of stiffeners are selected as design variables during the optimization. With the proposed optimization method,stiffeners own the ability to ‘‘move” freely within the given design domain and present different properties (e.g., length,angle, location, and thickness), further improving the structural performance at the same time.The remainder of this article is organized as follows. In Section 2, the optimization problem is mathematically defined and the design flowchart is constructed. Section 3 discusses some practical implementation issues for the geometry optimization of stiffener layout for three-dimensional box structures. Sensitivity calculation for geometry and size optimization are respectively provided in Section 4,with consideration of unimodal and multiple natural frequencies.In Section 5,typical examples are taken to numerically verify the effectiveness of the proposed optimization method. Conclusions are drawn in Section 6.

2. Post-processing geometry and size optimization of stiffener layout

2.1.Initial stiffener layout optimization using improved adaptive growth method Taking a four-corner-fixed supported box structure as an example, Fig. 130generally illustrates the main procedure of the initial stiffener layout optimization using the IAGM,38in which panels on the top surface of the box structure are made to be translucent to clearly show the stiffener layout.

Although the final stiffener layout patterns obtained by using the IAGM are pretty clear and can effectively improve the dynamic performance of box structures, stiffeners are subjected to their ground structures and can only grow along certain discrete angles as shown in Fig.1(e), which makes it hard to form a real optimal stiffener layout pattern. Theoretically,constructing a dense ground structure can partly eliminate the restriction of ground structures on the final optimal stiffener layout. However, such treatment leads to a significant growth in problem size and consumes lots of computational time and resources.

On the other hand,the stiffener number in the final stiffener layout pattern is much smaller than that in the initial layout optimization, which means that addressing arbitrary layout of stiffeners via a post-processing step results in comparatively smaller computational time. Therefore, the results obtained from the initial layout optimization are regarded as the start for the post-processing geometry and size optimization, and such treatment is also utilized in the field of truss layout optimization.47

Fig. 1 Illustration for the main procedure of stiffener layout optimization using IAGM.30

2.2. Basic formulation for post-processing geometry and size optimization

To further optimize the layout angle, location and length of stiffeners, positions of node lines of corresponding stiffeners are considered. According to the boundary conditions and structural characteristics,the node lines of stiffener layout patterns,typically shown in Fig.2,can be classified as active node lines,passive node lines,and fixed node lines.Active node lines are the direct geometry design objects which drive corresponding stiffeners to move,and the selection criteria of them will be discussed later in Section 3.1. Fixed node lines are those who have nodes fixed according to the boundary constraints of the structure, which are unchangeable during the geometry optimization. Obviously, those seed lines selected in the initial layout optimization are fixed node lines in the post-processing geometry and size optimization. Other node lines of stiffeners that are neither active nor fixed are passive node lines.

In this article, the fundamental natural frequency of threedimensional box structures is selected as the design object to maximize, which is the same as the initial layout optimization in Ref.38.The x,y positions of active node lines and the thickness T of stiffeners are set to be design variables. Once the positions of active node lines are optimized to new values,the length, angle, and location of corresponding stiffeners will be automatically updated and thus the limitation of ground structures can be broken through. It should be noted that the z position of nodes on a node line in the coordinate system is unchangeable in consideration of the draft direction, which refers to the parting direction of molds in the casting process.In this way,geometry optimization of stiffeners can be considered as a two-dimensional problem to some extent. However,corresponding sensitivity derivation and finite element analysis are still performed in three dimensions and thus it cannot be regarded as a ‘‘pure” two-dimensional optimization problem.

The basic mathematical formulation for the geometry and size optimization of stiffener layout patterns for threedimensional box structures can be given as

Fig.2 Definition of node lines and division of stiffener intervals(black solid line denotes fixed node line, purple dashed line denotes passive node line, and red solid line denotes active node line).

where M and N are respectively the total number of active node lines and stiffeners; Ω is the set of active node lines,and D is the set of stiffener plates; F(X,Y,T) is the objective function;K and M are global stiffness matrix and mass matrix,respectively; u and f are the nodal displacement vector and load vector, respectively; l and h are the length and height of a stiffener,respectively;V is the given upper volume of stiffeners; xminand yminare the minimum values of design variables xiand yi, respectively, while xmaxand ymaxare the maximum values of them,respectively,following the idea of‘‘node move limit” in the work of He and Gilbert47Tminand Tmaxare the minimum and maximum values of the design variable Tjin the final structure, respectively.

2.3. Optimization strategy

There are lots of publications corresponding to the optimization of two (or more) types of design variables (e.g., geometry and size, topology and location) that could be cited here.47-54Among them, many different simultaneous optimization strategies have been proposed to solve the problem. From the view of authors, they could be roughly divided into two groups:

(1) ‘‘True”simultaneous optimization.In this group,different types of design variables are directly and jointly dealt with,usually with a solver of nonlinear programming.47-49,53-54Within the framework of mathematical programming,this simultaneous treatment is on the basis of standard theory of nonlinear optimization. However, this direct treatment towards Eq.(1)is not efficient and only relatively small problems can be dealt with. Furthermore, most standard solvers of nonlinear optimization may fail when tackling the problem.55

(2) Simultaneous optimization with the alternating scheme.In this group, different kinds of optimization design variables are optimized within an iteration for the whole problem but with an alternating optimization procedure.56It means that for the optimization of one type of design variables, the other type(s) of design variables are fixed, and this treatment repeats until all kinds of design variables have been optimized in one iteration step. Based on this optimization rule, many different optimization ideas have been proposed, such as‘‘multi-level optimization”57or‘‘hybrid optimization”.58Despite its heuristic characteristic and no guarantee to the global optimality, this strategy can generate better and better feasible results of the problem. Besides, each optimization interaction with this strategy can be rigorously treated because sensitivity calculation needs to consider only one type of design variables at each time,which is a big advantage.

Details of the advantages and drawbacks of these optimization strategies can be found in the research work of Achtziger55and the references therein. To avoid potential numerical and programming difficulties, we select simultaneous optimization with the alternating scheme as the optimization strategy to solve the problem. Following Dobbs and Felton,56the basic mathematical model (1) can be separated into two following sub-formulations:

Here, Eq. (2a) is the mathematical model for the geometry optimization and Eq. (2b) is for the size optimization of stiffener layout. It should be noted that the volume constraint is only considered in the size optimization part for simplicity.Besides,a full iteration step of the further post-processing optimization of stiffener layout for three-dimensional box structures is composed of one iteration step of geometry optimization and one iteration step of size optimization.

2.4. Design flowchart

Fig.3 illustrates the architecture of the design procedure of the post-processing geometry and size optimization, which mainly consists of four steps and the details of each step are shown as follows:

Step 1. Extract the geometrical model

As the starts for geometry and size optimization,the initial stiffener layout pattern and panels of a box structure are firstly extracted from those results obtained by using the IAGM.Here, those concentrated masses used in the initial stiffener layout optimization to maintain the modal shape of box structures are also removed.

Step 2. Select active node lines and set design parameters

Active node lines within the stiffener layout pattern are selected as design objects for the geometry optimization. Once the selection is done,stiffener plates are automatically divided into several stiffener intervals according to the positions of active node lines and fixed node lines. A typical division is shown in Fig. 2 to offer a clearer explanation. Those sixteen stiffener plates in the figure are grouped into four stiffener intervals including (A) Stiffener plates Nos.1-4; (B) Stiffener plates Nos.5-8; (C) Stiffener plates Nos.9-12; (D) Stiffener plates Nos.13-16. Then, the corresponding optimization design parameters are given by designers.

Fig. 3 Design flowchart of geometry and size optimization of stiffener layout for three-dimensional box structures.

Step 3. Calculate design sensitivity

During the optimization, sensitivities of the objective function with respect to the x, y positions of active node lines and the thickness T of stiffeners are calculated respectively. Considering three-dimensional box structures with symmetry,stiffener layout patterns ought to be designed symmetrically.However, numerical errors are pretty common in the calculation process of natural frequency optimization, and the final distribution of stiffeners often appears to be asymmetric. To solve this problem, a symmetry control must be performed in the geometry and size optimization procedure when dealing with symmetric box structures. The x1, y1and x2, y2positions of two symmetric node lines with regard to the plane P:y=Ax+B, where A and B are the slope and intercept of the plane P respectively, can be calculated as

with Eq. (3), the symmetric relationship can be effectively defined. The layout of symmetry planes for a threedimensional box structure of equal length and width, which has symmetric constraint conditions,can be applied,as shown in Fig. 4. As a result, sensitivity values of those symmetric active node lines are equal to ensure symmetric characteristics of the stiffener layout pattern during the geometry optimization. Also, those symmetric stiffeners would have the same thickness with the help of Eq. (3) when performing size optimization. Besides, the computational cost of sensitivity analysis can be greatly reduced.

Step 4. Update design variables and reconstruct the structure

Based on sensitivity information,the x,y positions of active node lines are optimized using the Sequential Quadratic Programming(SQP)algorithm,due to the nonlinear and nonconvex characteristics of Eq. (2a). With new values of positions,stiffeners can move with active node lines to optimal locations and angles step by step, and thus the limitation of the ground structure is partially eliminated. After an iteration step of geometry optimization, the structure is reconstructed and the finite element analysis is performed to offer information for the following size optimization step. Next, thicknesses of stiffeners are optimized using the Method of Moving Asymptotes(MMA) algorithm to further improve the structural performance.Then,thicknesses of stiffeners in the model are updated and the finite element analysis is re-performed for the next potential iteration step.

Steps 3 and 4 are repeated during the optimization. The whole optimization procedure stops and obtains the optimal stiffener layout pattern if either of the two following conditions is met: the difference of the objective function value between two consecutive integral iterations is smaller than the convergence tolerance ε;the number of optimization interactions reaches the given upper limit I, which is introduced to prevent the possible poor convergence of optimization caused by repeated nature frequencies.

3. Practical issues

During the post-processing geometry optimization, several practical issues need to be considered to ensure the effectiveness and feasibility of the proposed method.

Fig. 4 Layout of symmetry planes for a three-dimensional box structure which has symmetric constraint conditions and equal length and width.

3.1. Selection of active node lines

In the geometry optimization of stiffener layout for threedimensional box structures, the positions of active node lines are design variables. With the movement of active node lines,corresponding stiffeners will have different layout angles from those discrete ones in the ground structure.As a result,stiffeners can be separated from the ground structure and own optimal layout directions and locations for the concerned performance of three-dimensional box structures.

Before the implementation of geometry optimization,active node lines need to be firstly selected as design objects.The criteria for the selection of active node lines can be listed as follows:

(A) Node lines shared by three or more stiffeners, as shown in Fig. 5(a).

(B) Node lines shared by two non-collinear stiffeners, as shown in Fig. 5(b).

(C) Node lines that are not fixed, shared by panels of the structure with a stiffener or stiffeners, as shown in Fig. 5(c).

3.2. Collinear constraint

After the selection of active node lines, stiffeners will be automatically divided into several intervals with the help of fixed node lines. With this treatment, the layout angle, position,and length of stiffeners are directly controlled by certain active node lines. Those stiffeners within the same interval are obviously collinear, due to the selection of active node lines on the basis of the results of initial layout optimization. However,once the positions of active node lines are optimized,the structural characteristic of collinear among stiffeners within an interval might be destroyed if the positions of passive node lines are not updated simultaneously, which is definitely not an appropriate result in consideration of the manufacturability and optimality. Thus, a collinear constraint must be added to avoid this kind of situation. For a certain stiffener interval starting with node line A and ending with node line C, as shown in Fig. 6, the collinear constraint can be simply formulated as.

Fig. 5 Typical cases of active node lines for geometry optimization of three-dimensional box structures.

Fig. 6 A stiffener interval starting with node line A and ending with node line C.

With the proposed collinear constraint Eq. (4), the x, y positions of passive node lines can be automatically adjusted according to the updated values of those active node lines within a stiffener interval. As a result, the collinear structural characteristic of stiffeners within a stiffener interval can be ensured, as shown in Fig. 7, by firstly updating the positions of active node lines and secondly updating the positions of corresponding passive node lines.

3.3. Merging active node lines

In the research of He47and Achtziger55et al. on the geometry optimization of truss structures, nodes of truss structures may migrate towards others. Similarly, some active node lines may be very close to each other during the geometry optimization process and they should be regarded as the same one. Therefore, an approach to merge active node lines into a concentrated active node line is introduced in this paper and it involves two major steps with the help of a given active node line merging radius rM.

In the first step, the distance among all active node lines is calculated to judge whether there are some active node lines that need to be merged. For a certain active node line, those active node lines lying in its merging room(a cylindrical room with the radius of rM, the height of an active node line) are added in the same merging group and will be merged together,as shown in Fig.8.When the distance among active node lines A, B,and C is smaller than rM,a single merging group will be created and all three active node lines are contained in it(Fig.8(a)). When the distance between active node lines A and C is smaller than rM, and the distance between node lines A and B, B and C, are larger than the merging radius rM(Fig. 8(b)), a single merging group only containing active node lines A and C will be created.

In the second step, all active node lines in a merging group are merged to the centroid of them as a single active node line,shown as solid blue lines in Fig. 8. Obviously, such a merging process will affect the volume of stiffeners and may violate the given volume constraint. However, the volume constraint will be satisfied by optimizing the thickness of stiffeners in the following size optimization step.Stiffener intervals are iteratively updated according to the situation of merging active node lines. Besides, with the merging of active node lines, the positions of corresponding passive node lines should be updated at the same time. Thus, the collinear constraint is also used in this step.

Fig. 7 Effects of collinear constraint.

Fig. 8 Merging a group of active node lines.

3.4. Model reconstruction

In the results of the IAGM,geometric models are composed of panels and stiffeners, and they are all constructed with rectangular shell elements according to the structural characteristics of initial ground structures.When the positions of active node lines are updated, stiffeners have new lengths, locations, and angles, and they can be constructed with rectangular shell elements too. However, those shell elements representing panels of box structures would become irregular with the movement of stiffeners, especially those for the top and bottom panels.To avoid such phenomenon of elemental imperfection and ensure the accuracy of finite element analysis, triangular shell elements are used to reconstruct stiffeners and panels in the geometry and size optimization.

During the geometry optimization, a small move limit gap r* (r*＜rM) is also given to all active node lines when calculating their optimal values. For two random active node lines A and B, the restriction of r* can be described as With the help of Eq. (7), stiffeners cannot cross over each other and thus potential troubles in the geometrical reconstruction process can be successfully avoided, as well as corresponding numerical difficulties.

4. Sensitivity analysis

4.1. Sensitivity analysis of a unimodal natural frequency

The well-known description for the fundamental free vibration of a structure without damping can be written as

where K is the global stiffness matrix of the structure;Φ1is the global eigenvector corresponding to the fundamental natural frequency; M is the global mass matrix.

In the geometry and size optimization,if Eq.(8)has distinct natural frequencies, the sensitivity of the fundamental natural frequency ω1with respect to a specific design variable δ can be derived by differentiating the free vibration equation with respect to the x, y positions of a single active node line or the thickness T of a stiffener element, and then we obtain

where δ could be the positions of an active node line xior yi,or the thickness of a stiffener Tj; Kδand Mδare the stiffness matrix and mass matrix related to the design variable δ,respectively. In the geometry optimization,Kxi，yiand Mxi，yiare assembled by corresponding matrixes of all the stiffeners within those intervals having a relationship with the i-th active node line, while in the size optimization, KTjand MTjrefer to the stiffness matrix and mass matrix of the j-th stiffener element, respectively.

4.1.1. Sensitivity analysis for geometry optimization

In the geometry optimization, δ in Eq. (11)is x or y.The partial derivative of a stiffener element’s stiffness matrix and mass matrix with respect to design variables can be shown as

Fig. 9 A stiffener interval starting with node line A and ending with node line C.

With the above Eq.(17),the partial derivative of the transformation matrix Tewith respect to the length of a stiffener l can be obtained.

In three-dimensional box structures, the stiffness matrix and mass matrix of a stiffener element with 12 DOFs are assembled by those of a rectangular membrane element and a rectangular thin plate element, having the same geometrical parameters and material properties. Fig. 10 shows the freedoms of these two kinds of elements,in which u,v,w are translational freedoms in the x, y, z direction, respectively; α is the in-plane rigid rotational freedom of the membrane element; ψ is the out-plane rotational freedom for nodes of the thin plate element. Following the description of Long et al. [59], the length and the height of a stiffener element are expressed using a and b in this section.For a stiffener element, l is equal to 2a and h equals 2b.square brackets in Eq. (19) all have a relationship with the length of a stiffener, and thus they all need to be taken the derivative in the sensitivity calculation for the geometry optimization. The details of the matrix Eq. (19) can be found in Appendix A. Then, its corresponding derivation can be calculated.

Fig. 10 Illustration of elemental freedoms.

where ρ is the material density; M1is a 3×3 identity matrix;M0is a 3×3 null matrix.

With the above-mentioned matrixes, the derivations of Eq.(14)can be successfully obtained.According to the criteria for the selection of active node lines discussed in Section 3.1, an active node line must be the starting node line or the ending node line of a stiffener interval. Thus, based on Eq. (6), we have.

Corresponding details of the matrix Eq. (20) are given in Appendix B and its corresponding derivation can be thus obtained.

where γ is a variable parameter. When x in Eq. (22) is xA, γ equals 1; when x is xC, γ equals -1.

Finally, by substituting Eqs. (12)-(22) into the dominant sensitivity Eq.(11),the sensitivity calculation for the geometry optimization of stiffener layout can be performed.

4.1.2. Sensitivity analysis for size optimization

Fig. 11 Four -corner-fixed supported square box structure.

Fig. 12 Geometry and size optimization of stiffener layout for four-corner-fixed supported square box structure with maximization of fundamental natural frequency.

4.2. Sensitivity analysis of multiple natural frequencies

Multiple frequencies may manifest themselves in different ways in structural natural frequency optimization problems.For example, geometrically symmetric structures usually have coincident natural frequencies showing different modal shapes because of the symmetry. Besides, as the fundamental natural frequency increases during the optimization, the subsequent originally unimodal natural frequency may gradually reduce and become very close to each other. Sometimes they even coincide with the fundamental natural frequency and cause serious interference among themselves.In these cases,sensitivities of multiple natural frequencies cannot be calculated straightforwardly by Eq. (11) due to the nondifferentiable properties of subspaces associated with multiple natural frequencies.To tackle with the occurrence of multiple natural frequencies in the geometry and size optimization of stiffener layout for three-dimensional box structures, a multiplicity judgment parameter β(e.g.,β=5%)is introduced to identify the number of natural frequencies within the β-neighborhood of the fundamental natural frequency, following the research of Manickarajah et al.60.The modification of the fundamental natural frequency towards a case of Q-fold(Q ≥1)natural frequencies can be given as.

With Eq. (26), frequencies of different orders are regarded as the same order of the fundamental natural frequency if their separation is smaller than β,and are different orders if the separation is larger than β. After the modification, the sensitivity calculation of multiple natural frequencies can be solved as the case of a unimodal natural frequency.

Furthermore, in order to guarantee that the updated thicknesses of stiffener elements are identical along the same draft direction, a formula is introduced to correct sensitivities of stiffener elements by Eq. (27). The mean value of sensitivities is taken if the corresponding stiffener elements are in the same draft direction.38

Here, Seis the sensitivity of a stiffener element; Zeis the total number of stiffeners in the same draft direction; Stis the sensitivity of the t-th stiffener element.

Fig. 13 Four-midpoint-fixed supported unenclosed square box structure.

5. Numerical verification

5.1. Four-corner-fixed supported square box structure

To verify the effectiveness of the proposed post-processing geometry and size optimization method, the four-cornerfixed supported three-dimensional square box structure, as shown in Fig. 11(a), is selected as the first example.38Both the length and width of the box are 0.8 m, and the height of it is 0.2 m.Panels of the box structure are of 0.015 m in thickness. The material properties are 200 GPa, 0.3, and 7800 kg˙sm-3for elastic modulus, Poisson’s ratio, and density,respectively. Fig. 11(b) shows the geometric model extracted from the result obtained by using the IAGM,and the elements on the top panel are made to be translucent to clearly show the stiffener layout pattern. Obviously, stiffeners in Fig. 11(c) are of certain layout angles and locations because of the geometrical characteristics of the ground structure. The minimum thickness Tminand maximum thickness Tmaxof them are 0.01 m and 0.02 m, respectively. The prescribed material volume of stiffeners is equal to that of the initial optimized stiffener layout pattern of the box structure, which is 1.37×10-3m3in this example. The convergence tolerance for the optimization procedure is 0.01%. The symmetry control shown in Fig. 4 is used in this example and the next one.

Fig. 14 Geometry and size optimization of stiffener layout for four-midpoint-fixed supported unenclosed square box structure with maximization of fundamental natural frequency.

The main corresponding geometry and size optimization procedure is presented in Fig. 12(a)-(f). Here, stiffener layout patterns are reconstructed with rectangular shell elements,viewed from the top to offer a clearer expression,and such representation is also utilized in the following examples. These layout patterns are the results after a full optimization interaction, which means that both of geometry and size design variables have been optimized. It can be clearly seen that the positions of active node lines are updated during the optimization and the thicknesses of stiffeners are also varied.At Step 5,the scheme of merging active node lines is activated.As shown in Fig. 12(e), several active node lines are merged and some stiffeners in the former layout disappear.Stiffeners in the final optimal layout pattern, as shown in Fig. 12(f), are mainly distributed in the diagonals and some stiffeners with small thickness form an appropriate circle to reinforce the deformation at the central areas of the top and bottom panels. Fig. 12(a) and(f) clearly show the difference of stiffener layout before and after the geometry and size optimization, and it can be generally seen that the final layout pattern is much more concise and has fewer stiffeners than the initial layout optimization result.The thicknesses of some critical stiffeners reach the maximum thickness Tmax,which reveals that T ≤Tmaxis the most critical constraint in this example.

The iteration history of the objective function is shown in Fig. 12(g). It can be seen that the fundamental natural frequency gradually increases in the first three iteration steps and it gradually stabilizes after a dramatic increment at Step 4. Finally, the fundamental natural frequency of the threedimensional square box structure reaches 563.54 Hz at Step 6, which is higher than its initial value by more than 5% with a satisfaction of the termination condition. The total volume of stiffeners in the final optimal stiffener layout pattern reduces by more than 30%compared to the initial volume of stiffeners.This is because the initial stiffener volume is obtained with the help of concentrated masses in the initial stiffener layout optimization and the proposed geometry and size optimization algorithm can find the optimal stiffener volume considering the design objective after the removement of those concentrated masses. Another important information that we can get from the iteration history is that both geometry and size optimization within an iteration of the post-processing optimization design can effectively increase the fundamental natural frequency of the box structure.

It is worth noting that coarse element mesh in all numerical examples is only used to offer structural illustration of optimization results in a clearer way.During the geometry and size optimization, structural models, including stiffeners and panels,are constructed using triangular shell elements with the size of one-fifth of the initial length of stiffeners, which can ensure the accuracy of finite element analysis.

5.2. Four-midpoint-fixed supported unenclosed square box structure

In engineering fields, there are some three-dimensional box structures without the bottom panel, named unenclosed box structures in this paper. To test the efficiency of the proposed method, a three-dimensional unenclosed square box structure is selected as the second example with its four midpoints of bottom edges fixed supported,as shown in Fig.13(a).The geometric model for the post-processing geometry and size optimization is presented in Fig. 13(b).38The initial stiffener layout pattern obtained by using the IAGM is shown in Fig. 13(c). The geometric parameters of the box structure are the same as those of example 1, as well as material properties and other design parameters.The initial volume of stiffeners in this example is 1.05×10-3m3.

The main optimization procedure of the stiffener layout pattern is also given, as shown in Fig. 14(a)-(f). The variation of the positions of active node lines is pretty clear and it shows a tendency to form a nearly square pattern within the threedimensional box structure. The stiffener layouts obtained by using the IAGM and the proposed method are respectively shown in Fig. 14(a) and (f), and there are similarities and differences between them. In both of them, several stiffeners are concentrated around the fixed midpoints of the structure.However,there are some stiffeners that grow towards the central area of the top panel in the result of the IAGM, and this does not happen during the geometry and size optimization.The reasons for this phenomenon are the restriction of the ground structure and the growth process of stiffeners in the IAGM.

The iteration history of the objective function of the box structure is shown in Fig. 14(g). The fundamental natural frequency generally increases as the number of iteration steps increases, and it begins to slowly stabilize with slight increments from Step 6. The final value of the fundamental natural frequency of the three-dimensional box structure is 588.25 Hz, which is higher than the initial value by more than 13%. Although the volume of stiffeners is reduced after the geometry optimization, the thicknesses of stiffeners grow during size optimization in this example. The final volume of stiffeners is 1.04×10-3 m3, which satisfies the corresponding constraint condition and is almost the same as the initial one, which means that the maximum volume constraint is the most critical constraint for this optimization.

Fig. 15 Four-corner-fixed supported rectangular box structure.

Fig.16 Geometry and size optimization of stiffener layout for four-corner-fixed supported rectangular box structure with maximization of fundamental natural frequency.

5.3. Four-corner-fixed supported rectangular box structure

In Example 3, a rectangular box structure with four corners fixed supported, as shown in Fig. 15(a), is selected to further verify the validity of the proposed method. The geometric dimensions of this box structure are 0.6 m in length, 0.3 m in width,and 0.1 m in height,with panels of 0.010 m in thickness.The minimum and maximum thicknesses for stiffeners are 0.007 m and 0.012 m, respectively. Material properties of the structure and other design parameters are the same as those in the above examples. Fig. 15(b) shows the geometric model for the post-processing geometry and size optimization, and the initial stiffener layout obtained by using the IAGM is presented in Fig. 15(c). The initial volume of stiffeners for this rectangular box structure is 1.94×10-3m3. Considering the rectangular box shape,the symmetry control without the symmetry plane of y=x shown in Fig. 4 is added to ensure the symmetric layout of stiffeners.

Fig. 16(a)-(f) show the main optimization procedure of geometry and size of the internal stiffener layout. From Step 1 to Step 9, the active node lines of stiffeners gradually tend to move closer to each other and the distance among those(near) parallel stiffeners in the middle area of the structure keeps decreasing. Then, the scheme of merging active node lines is activated at Step 10 and a‘‘＞—＜”shape stiffener layout is thus obtained with a great reduction in the total number of stiffeners.In the following steps,the pattern of stiffener layout generally remains the same but has some variations in thickness.At Step 15,merging active node lines happens again and the number of internal stiffeners is further reduced. Obviously, the final stiffener layout shown in Fig. 16(f) is much more concise than the initial stiffener layout. Same as the first example, some critical stiffeners own the maximum thickness Tmax.

The iterative history of the objective function is shown in Fig. 16(g). Similarly, the fundamental natural frequency of the box structure gradually increases as the number of optimization interactions increases and finally reaches 859.48 Hz,which is more than 5% higher than its initial value. It can be clearly seen from the iterative history that the fundamental natural frequency of the box structure has a dramatic increment when some active node lines merge at Step 10. From the view of the authors, this increment proves the necessity of the proposed merging scheme. The volume of stiffeners in the final result is 1.40×10-3m3,which is significantly reduced by more than 27%.

6. Conclusions

In the present work,a post-processing geometry and size optimization approach is proposed to optimize the initial layout patterns of stiffeners for three-dimensional box structures obtained by using the IAGM, with further maximization of the fundamental natural frequency. Using the IAGM, the initial layout patterns of stiffeners are pretty clear and they are the starts for geometry and size optimization. With the proposed method, stiffeners in the initial layout patterns own the ability to randomly move to their optimal lengths, angles,and locations with the help of the optimization towards the positions of active node lines. Then, the thickness of stiffeners is optimized to further improve the dynamic performance of structures in consideration of the volume constraint. Therefore, the potential of three-dimensional box structures with respect to natural frequencies can be further developed and typical numerical examples demonstrate the following advantages of the proposed method: (A) As a post-processing approach,the optimization strategy can be easily implemented and only a few design variables need to be treated by optimizing the positions of active node lines, which can effectively reduce the cost and complexity of numerical calculation. (B)The final stiffener layout patterns are well simplified with the maintenance of great clarity and manufacturability. (C) Even if being optimized in a post-processing way, stiffeners can get rid of the geometrical characteristics of their initial ground structures to some extent and own optimal lengths,angles,and locations to further effectively improve the fundamental natural frequency of three-dimensional box structures with less material consumption. Although the strategy of simultaneous optimization with the alternating scheme in the proposed post-processing optimization method only guarantees nearoptimal layout patterns of stiffeners, the suggested method is a good choice for engineers to further develop the potential of structures and may offer some inspirations to other researchers to further tackle the issue.

Acknowledgements

This work was financially supported by National Natural Science Foundation of China (Nos. 51975380 and 52005377),China Postdoctoral Science Foundation (No. 2020M681346)and Japan Society for the Promotion of Science (No.JP21J13418).

Appendix A.Items for the stiffness matrix Eq. (19) are shown as follows: