Yng LI, Tong GAO,b,*, Qinying ZHOU, Ping CHEN, Dezheng YIN,Weihong ZHANG
aState IJR Center of Aerospace Design and Additive Manufacturing, Northwestern Polytechnical University, Xi’an 710072, China
bInstitute of Intelligence Material and Structure, Unmanned System Technologies, Northwestern Polytechnical University, Xi’an 710072, China
cBeijing Aerospace Technology Institute, Beijing 100074, China
dBeijing Institute of Long March Space Vehicle, Beijing 100076, China
KEYWORDSLayout design;Thin walled structures;Topology optimization;Lattice;Stiffener
AbstractIn this paper, the thin-walled structures with lattices and stiffeners manufactured by additive manufacturing are investigated.A design method based on the multi-material topology optimization is proposed for the simultaneous layout optimization of the lattices and stiffeners in thin-walled structures.First, the representative lattice units of the selected lattices are equivalent to the virtual homogeneous materials whose effective elastic matrixes are achieved by the energybased homogenization method.Meanwhile, the stiffeners are modelled using the solid material.Subsequently,the multi-material topology optimization formulation is established for both the virtual homogeneous materials and solid material to minimize the structural compliance under mass constraint.Thus, the optimal layout of both the lattices and stiffeners could be simultaneously attained by the optimization procedure.Two applications, the aircraft panel structure and the equipment mounting plate, are dealt with to demonstrate the detailed design procedure and reveal the effect of the proposed method.According to numerical comparisons and experimental results,the thin-walled structures with lattices and stiffeners have significant advantages over the traditional stiffened thin-walled structures and lattice sandwich structures in terms of static,dynamic and antiinstability performance.
Demand is increasing for thin-walled structures with high mechanical performance and light-weight design in an array of engineering fields including aerospace, transportation,nuclear reactors, and civil engineering.Two representative thin-walled structures, stiffened thin-walled structures and sandwich structures are widely used in aerospace vehicles to meet the light-weight requirement, as illustrated in Fig.1.Here, the sandwich structures include several light-weight cores, such as the honeycombs, foams and lattices.
Nowadays, stiffened structures can be found in a large number of mechanical structures.The research on stiffened thin-walled structures has been carried out for a long time and obtained abundant research results1–4, the research on the performance of stiffened structures has been quite mature.Due to the requirements of lightweight design,how to find the layout and control the size of the stiffened structures under the requirements of ensuring the overall structural performance is a new research hotspot, more and more researchers solve this problem by using the topology optimization method.
The recent research shows that the lattice structures have great potential in achieving lightweight properties and desirable structural performances, such as high specific stiffness5,energy absorbability6, shock resistance7, and heat insulation8.Gibson and Ashby9took the lead in systematically studying the bearing performance of lattice structure through theoretical calculation and mechanical experiment,and gave the fitting function relationship between modulus, strength and relative density of lattice structure in detail.According to the deformation form under working load, Deshpande et al.10divided lattice structures into two types: Stretching-dominated architecture and Bending-dominated architecture.Ushijima et al.11proposed a theoretical derivation method for the initial stiffness and plastic yield limit of lattice structures.Gu¨mru¨k and Mines12researched the static mechanical response of different types of lattice structures under tension, compression,shear and combined loads.Ostos et al.13found that the compressive stress–strain curve of lattice structures generally includes three stages: linear elastic stage, plateau stage and densification stage, among which the plateau stage determines the energy absorption capacity of the lattice structure.Tian et al.14systematically studied the influence of cell configuration, material properties, porosity and other factors on the heat transfer performance of lattice structures through numerical calculation and experiment.In recent years, due to the development of additive manufacturing technology,the manufacturing capacity with complex geometric features has been improved, which makes the actual manufacturing of multiscale lattice structures possible, and greatly promotes the development and application of multi-scale lattice research in aerospace, power energy, medical equipment and other engineering fields15.In 2002, Rodrigues et al.16first proposed a multi-scale structure design method considering both micromaterial distribution and macro-structure layout based on topology optimization technology.Xia and Bieitkopf17realized the coupling calculation of mechanical mechanics on the macroscale and microscale through a nonlinear solution strategy.Gao et al.18proposed a multi-scale collaborative optimization design method considering multi-domain microstructures.In order to reduce the computational cost and improve the efficiency of structural design,Liu et al.19proposed a periodic hierarchical topology optimization method.Yan et al.20proposed a multi-scale structure design method considering uniform microstructure materials based on the Bidirectional Evolutionary Structural Optimization (BESO)method.In recent research,the level set method(LSM)21,multiple variable cutting (M-VCUT) level set method22, isogeometric analysis method23, data-driven methods24and other topology optimization methods were employed in the multiscale structure design.Wu et al.25reviewed the related research progress of multi-scale structure topology optimization design and prospected the potential applications of multi-scale structures in various engineering fields.
Fig.1 Illustration of representative thin-walled structures.
In contrast, the lattice structures and stiffened structures have their own advantages and disadvantages in mechanical performance and engineering application.Compared with the stiffened structures, the lattice structures have important advantages such as effective absorption of impact energy and greatly reduced vibration response while satisfying the loadbearing performance,and having better dynamic performance.However, the lattice structures also have their shortcomings.For example, compared with the stiffened structures, the lattice structures can’t provide sufficient structure stiffness at the specific location where has concentrated loads.In order to give full play to the advantages of the two structures, the structure with lattices and stiffeners can be employed.In the structure with lattices and stiffeners, the stiffened structures play the role of main support to ensure the overall stiffness of the structure; the lattice structures mainly assume the functions of buffering, energy absorption and shock absorption.Through the reasonable layout design of the two structures,it can achieve an overall structure with good static and dynamic properties.Some researchers have studied the layout design method of the structure with lattices and stiffeners,Dong et al.26proposed a general design method based on BESO to design solid-lattice hybrid structures.Boccini and Furferi27proposed an optimization method named lattice structural optimization(LSO).Wang et al.28proposed a novel multi-scale design method to create solid-lattice hybrid structures and lattice-based optimization is performed to obtain the optimal cross-section area of the lattice structures.
The current research on the layout design method of structure with lattices and stiffeners mainly follows the steps that first perform a conventional topology optimization to achieve the layout result of the stiffened structures, then fill the void prat with lattice structures.This method only considers the influence of solid materials on the optimization result but does not consider the influence of the layout of different structures during the optimization process on the overall structural mechanical performance, thus, the optimization result achieved by this method is not necessarily the optimal structure optimization result.Inspired by the research on multimaterials and multi-types lattice layout optimization21,29–31based on multi-scale structure optimization in recent years,the layout design problem of structure with lattices and stiffeners could be transformed into a topology optimization problem with multi-materials.Continuum topology optimization with multi-materials was first investigated by Thomsen32.Within the scope of the density method of topology optimization, several material interpolation models, such as the DMO(Discrete Material Optimization)scheme33,SFP(Shape Functions with Penalization) scheme34, and BCP scheme (Bi-value Coding Parameterization)35are proposed and varieties of topology optimization problems with multi-materials are handled, for example, the design of micro-structures36, the thermo-elastic problem37and multi-physics actuator design38.Meanwhile, the ESO was also applied to address multimaterials39.Alternatively, the level set method and the phase-field method were applied to address multi-material topology optimization problems due to the ability to describe the implicit interfaces between distinct solid materials40.Additionally, some other multi-material interpolation models were proposed for topology optimization41–43.
This work tries to combine the stiffened structures with the lattice structures in the thin-walled structures, and focuses on the simultaneous layout optimization of the lattices and stiffeners in thin-walled structures.This paper is organized as follows.In Section 2, the thin-walled structures with lattices and stiffeners are introduced and the general design process is presented.The equivalent method of the elastic matrixes of the lattice structure is discussed in Section 3.And then,the layout design scheme of the thin-walled structures with lattices and stiffeners is proposed in Section 4 within the framework of the multi-material topology optimization.In Section 5, two applications illustrate the validity and the effectiveness of the proposed optimization method and the advantages of the thin-walled structures with lattices and stiffeners.In the last section, the conclusions and contributions are highlighted.
A typical thin-walled structure with lattices and stiffeners is illustrated in Fig.2.The structure could be considered as a sandwich structure and consists of three parts: the skins, lattices and stiffeners.The last two pates compose the core.In this work, the thickness of both skins is supposed constant,while the layout of the stiffeners and lattices will be simultaneously designed.The main thought of the design method can be summarized as follows:First,the representative lattice units of the selected lattices are equivalent to the virtual homogeneous materials whose effective elastic matrixes are achieved by the energy-based homogenization method.Meanwhile,the stiffeners are modelled using the solid material.Thus, the optimal layout of both the lattices and stiffeners could be simultaneously attained by the multi-material topology optimization.
The specific process of the present design method is summarized in Fig.3 and described as follows.
Firstly, select the suitable lattice unit cell according to the characteristics of the design structure and the manufacturing process constraints.Some commonly used lattice unit cells are illustrated in Fig.4.The main factors affecting the selection of lattice unit cells include printing direction,printing process, the mechanical properties of the material, functional requirements, etc.
Subsequently,the selected lattices unit cell is treated as virtual homogeneous materials and their macroscopic equivalent elastic property is calculated by the energy-based homogenization method under periodic boundary conditions.Meanwhile,the stiffeners are modelled with the given structure material.
Finally, the layout optimization design of the thin-walled structure with lattices and stiffeners is carried out.In the finite element model of the design domain of the thin-walled structure, each discrete element can be composed of two materials in proportion.The proportion coefficient of each material is described by a given interpolation model.According to the design objectives and constraints, the overall layout topology optimization formulation is constructed, the sensitivity of the topology optimization problem is analyzed, and the optimization problem is solved by the gradient-driven algorithm to obtain the optimal layout of the lattices and stiffeners.
Fig.2 Illustration of thin-walled structure with lattices and stiffeners.
Fig.3 Flowchart of the design process.
The asymptotic homogenization method44and the energybased homogenization method45are the most commonly used methods to achieve the equivalent properties of the lattice structure.The asymptotic homogenization method is a mathematical method based on the perturbation expansion theory and the calculation process is complicated and timeconsuming.In this paper, the energy-based homogenization method under periodic boundary is employed to calculate the equivalent properties of the lattice structure.The configuration and the geometric parameters of the BCC-wf are shown in Fig.5.The representative lattice unit cell is equivalent to a virtual homogeneous medium.The equivalent stress and the equivalent strain should follow the Hooke’s law:
Fig.4 Several popular kinds of lattice unit cells.
Fig.5 Schematic diagram of homogenization for lattice unit cell.
where DHis the equivalent elastic matrix of the virtual homogeneous medium, σ-is the equivalent stress and ε- is the equivalent strain of the virtual homogeneous medium.Considering the structural characteristic of the lattice unit cell, the virtual material can be regarded as a three-dimensional orthotropic material, and its equivalent elastic matrix DHcan be further expressed as.be found in the reference45.According to the homogenization definition and continuum mechanics theory, the periodic and continuous conditions must be satisfied when the strain fields are applied to the lattice unit cell model.Therefore, the periodic boundary constraints should be applied to couple the displacement of the corresponding nodes on the vertex,edge, and surface of the unit cell during the solving process of strain energy in the finite element analysis.
In this section,the formulation of the multi-material topology optimization problem under mass constraint is introduced.And then, the sensitivity analysis of the involved structural response and mass constraint are both derived based on the material interpolation model.
4.1.Topology optimization with multi-materials
Generally, upon using the finite element method, the discretized finite element formulation is written as
where K is the global stiffness matrix.u and F are the nodal displacement vector and the nodal force vector, respectively.In this paper, the compliance of the whole structure Ω,hereinafter referred to as ‘‘overall compliance”, is taken as the objective function and is computed by
Herein,n and m are the number of designable elements and candidate materials, respectively.For example, m = 2 in the case of one considered lattice structure and the solid material.In this formulation, x denotes the set of design variables and xijrepresents the presence (1) or absence (0) of the jth candidate material in the ith finite element.A lower bound for the design variables of xmin=10-3is introduced in order to avoid the singularity of the structural stiffness matrix in the finite element analysis.Notice that Fdrefers to the inertial load vector dependent upon the design variables while the applied mechanical force Fais supposed to be design-independent.For the design-dependent inertial forces, we have
where a is the acceleration vector and Miis the mass matrix of the ith designable element.The structural mass M is considered as the constraint and should be less than its upper bound M.
4.2.Material interpolation model
In static finite element analysis, the element stiffness matrix could be calculated with.
where Biis the strain–displacement matrix consisting of derivatives of element shape functions.Considering that the lattice structures are usually equivalent to the anisotropic materials,the element elasticity matrix Dibut not the Young’s Modulus is used for the material interpolation model.In a previous work46, the UMMI scheme, which is known as DMO and firstly proposed by Stegmann and Lund33, was found to be superior to the RMMI because the former made it possible to formulate the mass constraint in a linear form with separable design variables.One such formulation benefits the problem resolution by means of mathematical programming approaches, specifically convex programming methods.Thus,Dican be expressed as the weighted summation of all of the candidate material phases
in which the subscripts i and j indicate the ith designable element and the jth candidate material, respectively.D （j）is the elasticity matrix of the jth candidate material.
Meanwhile, only the applied loads are involved in this paper.Hence,the SIMP scheme is utilized in combination with the UMMI scheme46,to formulate models of material properties.Supposing p as the penalty factor in the SIMP scheme,the weighting functions in the above parameterization models then correspond to
To formulate the structural mass, the linear formulation47of the density in element i is interpolated as
where qiis the density of the ith element and q （j）is the density of the jth candidate material.
The corresponding mass constraint of multi-materials is then formulated as a linear expression.
Herein Videnotes the volume of the ith element.It has been proven that the linear form and the separability of the design variables are favorable to the mathematical programming approaches, especially the convex programming methods37.
To calculate the design-dependent nodal force vector, the mass matrix of the ith designable element Micould be expressed as
where Mi0is the mass matrix of the ith designable element filled with solid material of unit density.
4.3.Sensitivity analysis
Based on the Eq.(4), the sensitivity of the structural overall compliance C then corresponds to
To calculate the partial derivative of weightwi, the interpolation scheme given in Eq.(9) is used to produce.
And then, ∂Fd/∂xijcan be expressed as
Evidently, ∂C/∂xijcan be easily calculated at the element level.
The sensitivity of this formulation of mass constraint can easily be derived as.
In this section, two typical thin-walled structures are studied and the layout design of the lattices and stiffeners is achieved by the proposed optimization method.The gradient-based GCMMA algorithm48is utilized here to solve the TO problem and the convergence criterion is set to be that the relative variations of both the objective function and the constraints between two consecutive iterations are less than 0.1%.The sensitivity filtering technique49is introduced to achieve the checkerboard-free configurations.In both tests, AlSi10Mg is selected as the structural material and the specific performance is shown in Table 1.
5.1.Example 1: Aircraft panel structure
The geometric model and dimensions of the aircraft panel structure are illustrated in Fig.6.The panel structure is composed of the designable domain, upper skin, lower skin and connecting strips.The outer and inner skins are mainly used to keep the feature of the aircraft.Furthermore,the outer skin also plays the role of carrying the aerodynamic loads, and theinner skin is usually used to install some equipment.The thickness of both skins is 1 mm.The designable domain should be filled with the stiffeners and lattices,and its thickness is 18 mm.The finite element model of the panel structure is established in which the skins are discretized into 47750 shell elements and the designable domain is discretized into 48,000 hexahedron solid elements.A uniform aerodynamic pressure of 11.5 kPa is applied to the upper skin and fixed constraints are imposed on the connecting strips.According to the design requirements,the mass of the design domain must be less than 20 kg.In order to verify the mechanical properties of the panel structure with lattices and stiffeners, the panel structure designed by the proposed method will be compared to the stiffened panel structure and the lattice sandwich panel structure.The three types of structures are designed and reconstructed under the same mass constraints.
Table 1 Properties of AlSi10Mg.
Fig.6 Aircraft panel structure.
5.1.1.Panel structure with lattice and stiffeners
(1) Panel structure with BCC lattice and stiffeners.
In Section 2, we have proposed the design process of the thin-walled structures with lattices and stiffeners.In this example, the BCC lattice unit cell is selected and the main reasons are as follows: (a)The load applied to the structure is the uniform aerodynamic pressure and the BCC lattice unit cell has an excellent performance in bearing unidirectional pressure11.(b) The panel structure is fabricated by SLM (selective laser melting)3D printing method,and the panel structure is printed along the length direction.Using BCC lattice unit cells can avoid the usage of the supporting structures to im prove manufacturing efficiency and equality.(c)The BCC lattice unit cell is symmetrical along the arrangement direction, so it guarantees the connectivity and continuity at the boundary between the unit cells.
One layer of BCC lattice unit cells is used to fill the panel structure in the thickness direction.The dimension of the lattice unit cell is 18 mm × 18 mm × 18 mm, the diameter of the lattice rod is 1.8 mm.The parameters of the selected lattice unit cell are illustrated in Table.2.
The optimization iteration curves of the overall compliance and the mass of the designable domain are shown in Fig.7(a).The optimization result is obtained after 32 iterations.The stiffeners grow from the connecting strips and finally connect together.The final topology configuration of the thin-walled panel structure with lattices and stiffeners is shown in Fig.7(b).
Current research work shows that the mass constraint plays an important role in the multi-material topology optimization50.The effects of the mass constraint upon the optimization configurations and the overall compliance are illustrated in Fig.8.It is found that more stiffeners appear and their thicknesses increase with the upper bound of the mass constraint.
According to the obtained optimization configuration, the reconstructed panel structure is illustrated in Fig.9 and its mass is 18.594 kg.Half of the outer skin is eliminated to show the lattices and stiffeners.It is noticed that there exist slight differences between the reconstructed structure and the optimized configuration.The reason is that this example is derived from the actual engineering project and the influences of the manufacturability, manufacturing cost, requirements of assembly and trade standards should be considered in the reconstruction.
(2) Panel structure with BCC & Carbon lattices and stiffeners.
Both BCC and Carbon lattices are selected in the optimization of the aircraft panel structure to illustrate the ability of the proposed optimization method for multiple lattices.
The equivalent elastic matrix of the selected Carbon lattice unit cell is
The optimization iterations and results curves are illustrated in Fig.10.The optimization results verify that the design method proposed in this paper can optimal the structure with multiple kinds of lattice unit cells.It is noted that the distribution of the stiffeners in Fig.10(a) is basically the same as the result in Fig.7(a).Meanwhile, the BCC latticesare filled in the area among the stiffeners and a few Carbon lattices mainly locate between the stiffeners and the BCC lattices.Considering the connectivity and manufacturability,the design of the panel structure with BCC lattice and stiffeners is adopted in the following comparisons.
Table 2 Parameters of BCC lattice unit cell.
Fig.7 Optimization results of panel structure with BCC lattices and stiffeners.
Fig.8 Effects of mass constraint on optimization results.
5.1.2.Stiffened panel structure
Fig.9 Model of panel structure with lattices and stiffeners.
The design of stiffened panel structure is obtained by the conventional topology optimization to minimize the overall compliance.The iterations curves of the overall compliance and the mass of the design domain are shown in Fig.11(b).According to the optimization configuration in Fig.11(a), the reconstructed stiffened panel structure is obtained and shown in Fig.12 and its mass is 19.045 kg.
5.1.3.Lattice sandwich panel structure
The panel structure with the BCC lattice core is also constructed for comparisons.The rod diameter of the lattice unit cell is set to be 3.7 mm.And then,the model of the lattice sandwich panel structure is shown in Fig.13 and its mass is 18.812 kg.
Fig.10 Optimization results of panel structure with BCC & Carbon lattices and stiffeners.
Fig.11 Optimization results of stiffened panel structure.
Fig.12 Model of stiffened panel structure.
5.1.4.Comparisons
The remodeled structures of the panel structure are compared.Their global deformation and von-Mises stress distribution under the uniform aerodynamic pressure,the first three natural frequencies and buckling factor are illustrated in Table 3,respectively.It notes that it is hard and unnecessary to accurately control the mass of the reconstruction result from the engineering viewpoint.Thus, the masses of the remodeled structures are similar and less than the upper bound.
Fig.13 Model of lattice sandwich panel structure.
The panel structure with lattices and stiffeners exhibits the highest stiffness.Its maximum deformation is 39.4% and 33.6% less than that of the stiffened panel structure and the lattice sandwich panel structure, respectively.Benefitting from the lattices, no obvious local deformation occurs in the panel structure with lattices and stiffeners and the lattice sandwich panel structure.In addition, their Von-Mises stress distribute more uniformly than the stiffened panel structure.The latter’s stress distribution is obviously affected by the stiffeners andthe stress concentration exists on the connecting area of the skin and the stiffeners.
Table 3 Numerical analysis results of three designs of panel structure.
The panel structure with lattices and stiffeners has higher natural frequencies.It is found that the first three mode shapes of the panel structure with lattices and stiffeners and the lattice sandwich panel structure are quite similar, while the former has higher natural frequencies.The resonances of the stiffened panel structure occur on the skin without stiffeners.
The buckling analysis results under the given uniform aerodynamic pressure indicate that the critical buckling factor of the panel structure with lattices and stiffeners is much higher than the other two designs.Especially, the stiffened panel structure has weak anti-buckling capacity and the skin without support is prone to local-buckling.The results prove that the infilled lattice structure can increase the critical buckling load and prevent the thin-walled structure from buckling.
According to the above comparisons, the proposed panel structure with lattices and stiffeners offers comprehensive performance benefits over the conventional stiffened structure and the lattice sandwich structure.
5.2.Example 2: Equipment mounting plate
Fig.14 Equipment mounting plate.
An equipment mounting plate is designed to further verify the effectiveness and superiority of the proposed method.Its geometric model is illustrated in Fig.14 and its thickness is 10 mm.Three equipments are installed to the equipment mounting plate and their mass are 625 g, 625 g and 2500 g.The plate is connected with the aircraft cabin through six bolts and all degrees of freedom of the connecting holes are fixed in the optimization process.32 times the gravitational acceleration along the negative direction of the X-axis and 41 times along the positive direction of the Z-axis are applied.
5.2.1.Structural design
The BCC-cubic lattice unit cell is selected for the layout design.The dimension of the unit cell is 10 mm×10 mm×10 mm and the diameter of the lattice rod is 1 mm.The equivalent elastic matrix of the selected BCC-cubic lattice unit cell is.
The main reasons to select the BCC-cubic unit cell are as follows:(a)BCC-cubic lattice unit cell has an excellent performance in bearing and vibration suppression51.(b) BCC-cubic lattices are support-free for the Selective Laser Melting(SLM).By using the BLT-S400 platform, an advanced SLM platform from Xi’an Bright Laser Technologies Co., ltd(BLT), the final design of the equipment mounting plate was additively manufactured using the AlSi10Mg powder.As shown in Fig.15(a),the BCC-cubic lattice unit cell can be fabricated without support structures under the griven print angle.
The upper bound of the mass constraint is set to 600 g and the optimized configuration of the equipment mounting plate with lattices and stiffeners is illustrated in Fig.15(b).The stiffeners grow from the six connecting holes and go across the plate, and several fine stiffeners are distributed along the Xaxis direction.The lattice structures infill the rest part.The reconstructed model is illustrated in Fig.15(c) and the mass of the fabricated sample shown in Fig.15(d) is 520 g.
The stiffened equipment mounting plate is also obtained for comparisons and the optimized configuration is illustrated in Fig.16(a).Based on the reconstructed model shown in Fig.16(b), the sample fabricated by milling is illustrated in Fig.16(c) and its mass is 524 g.
Fig.16 Stiffened equipment mounting plate.
Fig.17 Experimental settings for static experiments.
Fig.18 Comparison of strain–time curves of two designs.
5.2.2.Static experiment
Static experiments were carried out to validate the mechanical performance of the two fabricated samples of the equipment mounting plate under the extreme loading condition.The loading and measuring system is shown in Fig.17.The equipment mounting plate is fixed on the fixtures by bolt connections;three metal boxes are installed on the equipment mounting plate to simulate the equipment.The load is applied by the indenter on the metal boxes through a metal plate and is loaded step by step with an increment of 100 N during the experiment,the initial load is 100 N and the final load is 700 N.Two measuring points are set on the equipment mounting plates and the strain values are extracted by the strain gauges.
The Strain-Time curves of two fabricated samples of the equipment mounting plate are illustrated in Fig.18.During the loading process, the strain of the equipment mounting plate with lattices and stiffeners is less than that of the stiffened equipment mounting plate at the corresponding measuring point.The experiment results show that the optimized design with lattices and stiffeners can effectively improve the static performance of the equipment mounting plate.
This paper proposes a design method based on the multimaterial topology optimization for the layout design of the thin-walled structures with lattices and stiffeners.By using the energy-based homogenization method, the lattice unit cell is equivalent to a virtual homogeneous material.The layout design optimization problem of thin-walled structures with lattices and stiffeners is successfully transformed into a multimaterial topology optimization problem.Two typical aircraft thin-walled structures are designed to demonstrate the practical validity and advantage of the proposed thin-walled structure design.
The simulation and experiment results show that the thinwalled structure with lattices and stiffeners has better mechanical performance than the conventional stiffened structure and the lattice sandwich structure under the same mass constraint.It has been found that the panel structure with lattices and stiffeners has higher natural frequencies.Therefore, topology optimization for dynamic responses is a valuable question worthy of further study.Besides, the rod diameter of the lattice unit cells employed in this work is constant.Based on recent achievements, a size optimization could achieve better nonuniform lattices and further improve the performance of the thin-walled structure.This is also a valuable question in the future work.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
This work was supported by the National Natural Science Foundation of China(No.12172294,51735005 and 12032018).
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