Precise output loads control of load-diffusion components with topology optimization

Yinfeng CAO, Xiojun GU, Jihong ZHU,b,c, Weihong ZHANG

a State IJR Center of Aerospace Design and Additive Manufacturing, Northwestern Polytechnical University, Xi’an 710072, China

b Institute of Intelligence Material and Structure, Unmanned System Technologies, Northwestern Polytechnical University,Xi’an 710072, China

c MIIT Lab of Metal Additive Manufacturing and Innovative Design, Northwestern Polytechnical University, Xi’an 710072, China

KEYWORDS Artificial bar elements;Dynamic response;Kreisselmeier-Steinhauser(KS) function;Precise control of output loads;Topology optimization

Abstract The purpose of this paper is to present a novel topology optimization approach to control precisely the output loads under static loads and harmonic excitations. We introduce the Artificial Bar Element (ABE) at the designated output positions, where the output loads are equivalently measured and constrained with the nodal displacements of ABE. Optimization model is then formulated considering the output load constraints as well as the minimization of strain energy and dynamic displacement responses respectively under the static and dynamic conditions.The influences of the ABEs stiffness, different material usages of the design domain, widths of the output loads constraint intervals and variation ratios of output loads are discussed in detail. The proposed method is verified with several numerical examples with clear and reasonable load transfer paths.

1. Introduction

Load-transmission components in aircraft or aerospace vehicles are designed to transfer external loads or loads from one component to another.1,2Generally, the components must serve below their load limits and the loads must be transferred in a certain way.A typical load-diffusion component is shown in Fig. 1. It is used to diffuse the concentrated load produced by the engine to the main structure as uniform as possible. A reasonable design of load-diffusion component not only helps the aircraft/aerospace vehicles reduce weight,but also transfers the load reasonably to the secondary structures according to their bearing capacities. Topology optimization method is a good tool to find the optimal load transfer path of these kind of structures.In the recent decades,topology optimization has been developed as one of the most effective methods for concept design of structure configuration. Until now, topology optimization has made remarkable achievements in both theoretical studies and engineering applications especially in aircraft and aerospace structure systems design.3-9

Fig. 1 Illustration of a typical loads-diffusion component:Rocket engine support.

The topology optimization of load-diffusion components has been widely studied in the literature. Zhang et al.10proposed a design optimization method of rocket connection section for concentrated force diffusion. In the formulation of topology optimization, minimum compliance objective is subject to constraints on material volume and interfacial nodal force variance, which is introduced as the evaluation criterion of concentrated force diffusion efficiency. Similar work was carried out by Mei et al.11Gao et al.12investigated a topology optimization method of attaining uniform reaction forces at the specific fixed boundary.The variance of the reaction forces at the boundary between the elastic solid and its foundation is introduced to attain the uniform reaction forces.In addition to designing the load diffusion structure, Buhl13; Zhu and Zhang14; Zhu et al.15; Xia et al.16demonstrated methods and benefits of simultaneously designing structure and support distribution using topology optimization. New design variables were introduced to represent the support location. Zhang et al.17also discussed the method of simultaneous topology optimization of structure and applied external loads.However,all studies above can’t handle the problem of output loads precise control.

For some load-diffusion components,such as rocket engine support, the great thrust force generated by the engine is diffused from the support to the rocket precisely according to the structural style of the primary structure. It is necessary to design the structure such that the external loads can be assigned to the supports/interfaces in a controlled manner.For this reason, we proposed a novel topology optimization approach to control precisely the output loads. Both static load and harmonic excitations are considered. The output loads are measured by the proposed Artificial Bar Element(ABE). Meanwhile, artificial bar element can be used to simulate the structural support by selecting its stiffness properly.

The manuscript is organized as follows: after the definition of the optimization problem with output loads control in Section 2, the influence of the ABEs stiffness is discussed in Section 3. The sensitivity analysis of the problem is given by Section 4, including analysis of the Kreisselmeier-Steinhauser(KS) function based precise-controlled output displacement constraint.Several numerical examples,including static as well as dynamic load cases,are presented and discussed in Section 5.The general conclusion and acknowledgements are finally provided in Section 6.

2. Definition of the output loads precise control using ABE

Considering a structure as shown in Fig. 2, the structure is fixed through its bottom supports marked as A, B and C.The external static force F or a time-varying load F(t) is applied at the top of the structure. The topology optimization design domain is Ω, which is designed properly to make the external load diffuse to the boundary supports. In practical engineering, the boundary support is not always rigid. For the problem of output loads precise control, it is of vital importance to consider it under both rigid and elastic support.To solve this problem,we replace the current system as shown in Fig. 2 with an equivalent one as shown in Fig. 3. In the equivalent model, the idea of Artificial Bar Element (ABE) is introduced and placed in the direction of output load with one end connected to the support point and the other end fixed as shown in Fig. 3. The ABEs can simulate the real support condition by the proper selection of their stiffness. Now,instead of controlling the magnitude of the output loads directly, the problem becomes to control the displacements on the ABEs.

2.1.Basic definition of the output loads precise control in statics

As for a structure subjected to static load, the magnitudes of the output loads on the boundary support need to be controlled within specific ranges. In the equivalent load output model,the output loads on the boundary support are the axial forces of corresponding ABEs. The output load in the ith ABEs direction is assumed to be Fi. The relationship between the output load Fiand the displacement uiof the corresponding ABE can be easily expressed as

where Eiand Aiare the elastic modulus and the cross-sectional area of the ith bar element, respectively. diis the length of the ith bar element. Therefore, the nodal displacement of the ABEs can be used as the indicator of the magnitude of output loads.

Fig. 2 Illustration of output loads precise control design problem.

Fig. 3 Equivalent load output model using ABE.

2.2. Basic definition of the output loads precise control under harmonic force excitation

For the dynamic problems, suppose the input load is a harmonic force excitation.The equilibrium equation of a structure under harmonic excitation can generally be written as

where the input load F(t)is the harmonic force vector of form F(t)=F ejωt(j2= -1).F and ω denote the magnitude vector of harmonic force and excitation frequency, respectively. K,C and M are the global stiffness matrix, damping matrix and mass matrix of the system, respectively. u(t), ˙u（t） and ¨u（t）are the displacement, velocity and acceleration vectors of the system, respectively. Here, C is defined by using the Rayleigh damping model that

where α and β are Rayleigh damping coefficients.

The Mode Acceleration Method (MAM) is applied here to solve this dynamic problem. The eigenfrequencies and eigenvector can be solved from the equation of free vibration given below

The ith circular eigenfrequency ωiand eigenvector φican be obtained by solving the following equation

The mode shape matrix φ=[φ1,φ2,···,φn] is normalized by mass matrix. Classical damping is supposed with ξibeing the ith damping ratio. The relations between them can be expressed as

By introducing the following notation,u(t)can be expressed as

where y(t) is the vector of the generalized coordinates. The number of r uncoupled equations of motion can be obtained by substituting Eq. (6) and Eq. (7) into Eq. (2) through premultiplying φT.

yi(t) can be expressed by changing the form of Eq. (8) as

Thus, the displacement response related to Eq. (7) under harmonic force can be expressed as

where l is the number of the modes used in the above equation.

The inverse of the stiffness matrix can be represented as18

It is worth noting that in MAM, the first term on the right side of Eq. (10) can be exactly solved by means of Eq. (11) to include all r modes. As for the second and third terms of Eq.(10), by combining Eq. (9), we can have

Therefore, combining Eqs. (10)-(12), u(t) can be written as

and the further explanation of Eq. (13) can be simplified as

Because ABEs have no mass and damping,the relationship between the output load and the displacement of the corresponding ABE also can be expressed as the form in Eq. (1).To control the output loads precisely, the displacement amplitudes of the concerned output positions should be controlled by topology optimization in the design domain.

2.3. KS function based precise-controlled output displacement constraint aggregation

The above derivations indicate that, for both static and dynamic problems,one can achieve the precise control of output load through controlling the displacements of ABEs at the load-output locations. Here, ||um|| is used to represent the value of the concerned nodal displacement in statics or its amplitude in dynamic problems, and it can be defined as

where m represents the mth load-output direction.u is the global displacement vector.Tmis a coefficient matrix.Suppose the total number of output loads needed to be controlled precisely is M. The following formulations are constructed to control the corresponding output loads in optimization.

where Rm-1represents the target value of the ratio andstands for its tolerance. The total number of inequality constraints in the design problem is 2(M-1).

The KS function is applied as a common used aggregation method19to transform the multi-constraint problem into single constraint problem. The KS function has been proven to be effective in dealing with multi-constraint optimization problems and further detailed information about KS function can be seen in Martins and Poon20(2005); Gao et al.21(2015).The established KS function can be written as

where μ is the aggregation parameter.gmaxstands for the maximum value of all the constraints, and

In this paper, the following criterion is used to select the proper aggregation parameter.

where ε is a prescribed positive value. Here,the method based on the Steffensen iteration21is selected to calculate the μ.

Therefore, multiple constraint can be merged as one single KS function constraint

3. Problem formulation and influence of the ABE’s stiffness

3.1. Basic formulation of output loads precise control

In the static problems, the optimization problem is defined to find the maximum stiffness of the system satisfying the corresponding constraint conditions. The density-based topology optimization formulation is implemented with a prescribed upper bound of material volume fraction.

The formulation of the optimization problem can be defined as

where η is the design variables vector related to elements pseudo-density describing material distribution in the design domain.A lower bound ηminis assigned to avoid the singularity of the stiffness matrix,and it is set to be 0.001 in this paper.C is the global compliance.K is the global stiffness matrix.F is the force vector and u is the corresponding nodal displacement vector.V is the material volume and V*is its upper bound.neis the number of elements in the design domain.

The interpolation model used in the optimization is Polynomial Interpolation Scheme (PIS) proposed by Zhu et al.15.Compared to the Solid Isotropic Material with Penalization(SIMP) model,22,23PIS can avoid, for low density elements,localized deformation in static problems and localized modes in dynamic problems. The PIS is implemented and its formulation is given as

where Keand meare the stiffness matrix and mass of the eth element, respectively. Ke0and me0are the stiffness matrix and mass matrix with full solid material of the eth element,respectively. χ and p are the weighting factor and penalization factor of the scheme,respectively.In this paper,they are set to 16 and 5, respectively.

For the dynamic problem applied harmonic force excitation, the objective function in Eq. (21) is defined to minimize the amplitude of the concerned displacement

where||us(t)||is the displacement amplitude at the loaded point along the force excitation direction.

3.2. Analysis of the influence of ABE’s stiffness on the design problem

In this section,the influence of the stiffness of the ABE on the global compliance is discussed.The stiffness of the ABE is proportional to its elastic modulus. The derivative of the global compliance with respect to the ABEs elastic modulus Eican be written as

Considering the equations of equilibrium F=Ku, its derivative with respect to the elastic modulus can be expressed as

If F is design independent load, the Eq. (25) can be expressed as

Combining Eqs. (24) and (26) gives

Assume that the stiffness matrix of ABEs is K＊, which is dependent upon Ei. It should be noted that, the elastic modulus and sectional cross area of all the bar elements introduced by ABEs are set to the same values and Eq. (24) can be expressed as

5. Numerical examples of output loads precise control

In this section, several numerical examples are calculated to verify the proposed formulation,which is implemented to deal with the output loads precise control in both static and dynamic problems. In all the numerical cases, elastic modulus and Poisson’s ratio of the design domain are assigned to 210 GPa and 0.3, respectively. For ABE in all the numerical cases, its elastic modulus, length and radius are set to 3000 GPa, 1 mm and 0.5 mm to simulate the rigid support unless otherwise stated. ε is selected as 10-6to calculate the proper aggregation parameter. It should be noted that the Globally Convergent Method of Moving Asymptotes(GCMMA)25within the Boss-QuattroTM optimization platform26is used as the optimizer.

5.1.Output loads precise control of a plate on the elastic base in statics

Considering a plate on the elastic base as shown in Fig. 4, the design domain is discretized into 5545 shell elements with thickness of 1 mm. elastic modulus and Poisson’s ratio of the elastic base are assigned to 100 MPa and 0.3, respectively.An input force of 200 N is applied in the middle of the top edge.The bottom edge of elastic base is fixed.Before the equivalent load output model as shown in Fig.5 is used,the stiffness of ABEs should be selected properly to simulate the elastic base.

Taking the selection of stiffness of ABE A1B1for example,am unit force is applied at the top left corner of elastic base as shown in Fig.6.The displacement of loading point in the loading direction can be calculated easily. Then the stiffness of ABE A1B1equals one divided by the displacement. The stiffness of ABEs is obtained as shown in Table 1 through repeating the above procedure.In this section,the length and radius of ABEs are assigned to 5 mm and 0.1 mm, respectively.

In order to verify the validity of equivalent model using ABEs,the stiffness topology optimization for the whole structure, the plate with rigid support and the plate with ABEs are calculated to make a comparison. The optimized results are listed in Table 2. In the remainder of this paper, F, its superscript and its subscript stand for the magnitude, position and direction of output load, respectively.

Fig. 4 Schematic diagram of a plate on elastic base.

Fig.5 Schematic diagram of equivalent load output model using ABEs.

Fig. 6 Schematic diagram of calculating equivalent stiffness for elastic base.

As shown in Table 2, it is obvious that the plate with rigid support can’t simulate the plate on the elastic base correctly.The optimized result for plate with ABEs is consistent with the real condition. Therefore, the proposed method can simulate the elastic support reasonably by the proper selection of stiffness of ABE according to the stiffness of elastic support.

For the equivalent load output model using ABEs as shown in Fig. 5, twelve output loads are distributed at the bottom of the structure,where the corresponding ABEs are marked form A1B1to E4F4.The output loads in the vertical direction for A1and E4are constrained to be 15 N and 20 N,the ratio of which is 3:4.The bottom of ABEs are fixed.The nodal displacements in the vertical direction for A1and E4are the optimization constraints.

Under a prescribed 30 % material volume constraint and R*1being 10%, the optimization result is compared with the standard optimization result as shown in Table 3.

As shown in Table 3, the optimization considering output loads control can control the output loads according to design requirements effectively by sacrificing the structural stiffness.It is worth mentioning that the proposed method can reduce the computation cost to some degree if the elastic base is complex.

5.2.Output loads precise control of a plate on the elastic base in statics

In order to save the computation time, a quarter of inverted cone is used instead of the complete one. As illustrated inFigs. 7 and 8, a quarter of inverted cone is assigned as the design domain which is discretized into 7354 shell elements.The base and top radii of the inverted cone are 30 mm and 50 mm, respectively. And its height and thickness are 70 mm and 1 mm, respectively. An input uniform force of 100 N is applied in the middle of the bottom edge.Six output load positions are distributed uniformly at the top of the structure as shown in Fig. 8, where the corresponding ABEs are marked from A1A2to F1F2. The output loads in the vertical direction should be distributed symmetrically. The ratio of the output loads for A&F, B&E and C&D is constrained to be 1:1:3.The degrees of freedom of nodes A2, B2, C2, D2, E2and F2are constrained completely and the degrees of freedom of nodes A1,B1,C1,D1,E1and F1are constrained in the horizontal direction. Symmetric boundary condition is applied at the two sides of design domain. The nodal displacements in the vertical direction for A1, B1, C1, D1, E1and F1nodes are the optimization constraints.

Table 1 Stiffness of ABEs used to simulate elastic base.

Table 2 Comparison of optimization results among whole structure, plate with rigid support and plate with ABEs.

Table 3 Comparison of optimization results under different

Table 3 Comparison of optimization results under different

Strategy Output load(N) C(J) FA1y /FC4 y Optimized design FA1 y FC4 y Optimization considering output loads control 16.07 20.55 1.25 0.782images/BZ_195_1830_1612_2230_1851.pngStandard optimization 16.91 16.37 1.08 1.033

Fig. 7 Schematic diagram of a quarter of inverted cone.

Fig. 8 Schematic diagram of output loads precise control of a quarter of inverted cone.

Under a prescribed 50 % material volume constraint and all the values of R*mfor different ABEs being 1 %, the optimization is carried out to find the maximum global stiffness of the system.Material layouts of different iterations are illustrated in Fig.9.The optimization converges after 64 iterations with the optimized design shown in Fig.9(f).In the optimized result, the magnitude of output loads on A1, B1and C1achieves their feasible design with 10.13 N, 10.04 N and 29.84 N, respectively. This ratio is with good agreement with the optimization constraints.

In order to discuss the influence ofon the optimization results,which equals 1 %, 5 %, 15 %, 30 %, 50 %, 70 %and 90%under the same material volume fraction being 50%and ratio of output loads being 1:1:3 is calculated in the topology optimization,respectively.The optimized results are listed in Table 4. And also the optimized result obtained using standard method is listed here to make a comparison.

As shown in Table 4, the compliance of the optimized design decreases with the increase ofConsidering the optimized material layouts shown in Table 4,most of the materials in the optimization result without output loads precise control are distributed in the both sides of the structure to obtain a better stiffness. It can be seen that larger output loads will be obtained at the positions near the both sides of the structures (i.e., A and F) with the increase of the parameterAnd also a better stiffness will be achieved if the parameterincreases. When the output load precise control constraint is introduced,i.e.becomes smaller,materials tend to be distributed in the middle of the structure to ensure a relatively small compliance of the structure. Therefore, the relatively weaker branches will undertake smaller loads while the stronger ones will bear larger loads correspondingly. In this problem, the dilemma of trade-off between the output loads precise control constraint and the whole stiffness is faced and the optimized design is the result of the compromise between the global stiffness and the local load precise control.Then the effect of the different ratios of output loads on the material layout and the global compliance under the same input load is studied. Whenequals 1 %, the ratios of out-put loads among A1, B1and C1are constrained to be 1:1:4,1:1:3, 1:1:2, 1:1:1, 3:1:1, 4:2:1 and 1:1:0, respectively.

Fig. 9 Material layouts under different iterations.

Table 4 Comparison of optimization results under different

Table 4 Comparison of optimization results under different

R*m Output load (N) C (J) Optimized design A&F B&E C&D 1% 10.13 10.04 29.84 4.13×10-1 5% 10.69 10.19 29.12 3.92×10-1images/BZ_197_1897_822_2215_1175.png15% 12.08 10.51 27.42 3.87×10-1images/BZ_197_1897_1183_2219_1538.png30% 14.10 10.85 25.05 3.81×10-1images/BZ_197_1897_1547_2217_1901.png50% 16.67 11.11 22.23 3.75×10-1images/BZ_197_1897_1910_2215_2263.png70% 19.03 11.20 19.78 3.72×10-1images/BZ_197_1897_2272_2217_2625.png90% 20.56 10.82 18.63 3.71×10-1images/BZ_197_1897_2634_2216_2987.png

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Table 4 (continued)

The global compliance obtained without the output loads precise control constraint is shown in Fig. 10 as a dotted line with its optimized design above the line,where the ratio of output loads is 2.08:1:0. Considering different ratio of output loads, the output loads precise control constraint is introduced. Meanwhile, the curve of the global compliance and optimized design varied with the ratio of output loads are illustrated in Fig. 10. It can be seen that the global compliance tends to approach to its lower bound when the ratio of output loads is near 2.08:1:0.

Further, in order to discuss the influence of the stiffness of ABE on the optimization result, different tests with different elastic moduli under the ratio of output loads being 1:1:3 are implemented. i.e., the elastic moduli of ABEs which equal 8 MPa, 10 MPa, 20 MPa, 30 MPa, 80 MPa, 200 MPa,400 MPa, 600 MPa, 800 MPa, 1000 MPa, 2000 MPa,3000 MPa,4000 MPa and 5000 MPa are adopted in the calculation. The curve of the optimization results varied with the elastic modulus of ABEs is illustrated in Fig. 11.

Fig. 10 Comparison of global compliance and material layout under different ratios of output loads.

Fig. 11 Comparison of global compliance and material layout under different elastic modulus of ABEs.

As shown in Fig. 11, when the elastic modulus is small enough, the top of the optimized structure which is similar to the compliant mechanisms undergoes large deformation together with ABEs. Its deformation is shown in Fig. 12. As shown in Fig. 12, the ratio of deformations for A1A2, B1B2and C1C2is controlled to be 1:1:3 through the deformation of compliant mechanisms. With the increase of elastic modulus, the top of optimized structure tends to be rigid due to the small deformation occurring in ABEs. The global compliance reduces constantly and almost stays constant finally with the increase of the elastic modulus of ABEs,which agrees with the theory obtained in the Section 3.2. This indicates that the elastic modulus of ABEs should be set appropriately on the basis of the supporting forms in the practical engineering problems.

5.3. Output loads precise control in both directions for the rectangular plate

Considering a rectangular plate(100 mm×30 mm×1 mm)as shown in Fig. 13, the design domain is discretized into 12000 shell elements.An input force of 200 N is applied at the top left corner of the structure. The four output load positions distribute at the bottom of the structure, where A is fixed and the corresponding artificial elements are referred to as B1B2,B1B3, C1C2, C1C3and D1D2. In these ABEs, B1B2and C1C2which are used to constrain output shear loads are set in the horizontal direction. The other ABEs are set in the vertical direction to constrain output tension loads. Position B or C should only export one kind of load, such as only shear load or tension load. In the optimization, the ratio of the output loads in the vertical and horizontal direction for B and the ratio of the output loads in the horizontal and vertical direction for C should be zero. Meanwhile, the ratio of the output loads in the vertical direction for C1and D1should be 1:1.The degrees of freedom for A, B2, B3, C2, C3and D2are constrained completely and the degrees of freedom for D1are constrained in the horizontal direction.The degrees of freedom for B1and C1are free.

Fig. 12 Deformation of top of optimum layout.

In this section, the influence of the material usage on the optimization result is studied. The material volume fraction equals 15 %, 20 %, 30 %, 40 % and 50 % with the value of R*1,2=10 % and R*3=10 %, respectively. The optimized design, the global compliance, the magnitude of the output loads and the error value of output loads are shown in Table 5.

Further the larger volume fraction constraints(60%,70%,80 %, 90 %) are applied. The curve of the global compliance varying with the volume fraction is shown in Fig. 14, where the global compliance remains nearly unchanged when the material volume fraction tends to be 1.

When the amount of materials increases, the increased materials can improve the global stiffness. In Table 5, the increased materials distribute into the left part of the structure with the increase of volume fraction to improve the stiffness.However, the material layout in the right part of the structure varies little with the increase of volume fraction, because the right part of structure is used to maintain the unchanged ratio of output loads.Although a not that clear configuration is generated when the volume fraction is more than 60%,the output loads ratios almost maintains unchanged with the redundant materials distributing into the positions which are no use for load transmission and the improvement of stiffness. When the volume fraction is up to 90 %, the topology optimization is difficult to obtain a clear load-transferred path. At this moment, a similar breach configuration is obtained to control the output loads precisely.

5.4.Output loads precise control of a 2D plate under a harmonic force excitation

In this section, the proposed formulation is implemented to control the output loads precisely for a 2D plate under a harmonic force excitation. As shown in Fig. 15, the rectangular plate (100 mm×60 mm×1 mm) is assigned as the design domain with 12,334 DOFs which is discretized by 6000 plane elements. An input harmonic load with the amplitude 90 N is applied in the middle of the top edge. The objective is to minimize the displacement amplitude at the loaded point. Six uniform-distributed positions at the bottom of the structure are chosen as the output positions of the loads. Here, the corresponding ABEs are marked from A1A2to F1F2.The displacement amplitudes at the load-output points in the load-output directions should be distributed symmetrically about the vertical center line. The ratio of the displacement amplitudes for A1, B1and C1is set to be 4:3:2. The degrees of freedom for nodes A2,B2,C2,D2,E2and F2are constrained completely and the degrees of freedom for A1, B1, C1, D1, E1,F1nodes are constrained in the horizontal direction.

Fig. 13 Schematic diagram of output loads precise control in both directions for rectangular plate.

Table 5 Optimization results and magnitude of output loads under different volume fractions.

Fig. 14 Variation trend of global compliance under different volume fractions.

Fig. 15 Schematic diagram of output loads precise control design of rectangular plate.

Fig. 16 Variation trend of displacement amplitude at loaded point under different excitation frequencies.

The same Rayleigh damping is adopted in all cases in this paper with α=10-3and β=10-6. MAM is implemented with the first l=30 order modes for this structure. In this condition,the initial structure system as shown in Fig.15 with all pseudo-densities being 0.3 resonates at its 2-order eigenfrequency which is 3710 Hz and the curve is illustrated in Fig. 16.

Fig. 17 Optimized designs under different R*m.

Table 6 Comparison of optimization results under different

Table 6 Comparison of optimization results under different

R*m ||us(t)|| Resonant frequency (Hz)5% 3.116×10-3 14453 10% 3.054×10-3 13499 15% 3.018×10-3 14507 20% 2.972×10-3 16551 25% 3.097×10-3 14740 30% 2.855×10-3 15387

In order to discuss the influence ofon the optimization results,equals 5 %, 10 %, 15 %, 20 %, 25 % and 30 %under the volume fraction being 30 % and excitation frequency being 3225 Hz in the topology optimization. The optimized material layouts and the optimization results under differentare listed in Fig. 17 and Table 6, respectively.

As shown in Table 6, the displacement amplitude at the loaded point almost reduces with the increase ofexcept isolated optimization results. However, the compliance always reduces with the increase of the interval width of output displacement constraint in the static optimization problem. Different from the static optimization problem, both the interval width of constraint and the resonant frequency of optimized layout have effects on the objective function. As a result, just the biggerin the optimization may not obtain the smaller ||um|| which is different from the regularity in the static optimization problem.

6. Conclusions

In this paper,a novel formulation dealing with the precise control of output loads with topology optimization is proposed.The idea of Artificial Bar Element (ABE) is introduced at the designable output positions to replace the standard boundary conditions. By introducing the ABE, the traditional constraint on the precise control of output loads is transferred into an equivalent constraint on the ABE’s nodal displacement. Minor prescribed upper and lower limits are assigned to each nodal displacement constraint to obtain the effect of precise control. To constrain the output nodal displacements of ABEs effectively, a large amount of these constraints are aggregated into a single KS function. Numerical examples,both static problems and dynamic problems with harmonic load excitation, are calculated by using the proposed formulation. In all cases, the clear and reasonable load-transferred paths are obtained with the desired ratio of output loads.More examples are discussed to investigate the influence of the material usages, the interval widths and the ratios of output loads on the design result.The proposed formulation makes it easier to deal with the precise control of output loads under both rigid and elastic support during the conceptual design of structure systems with topology optimization.

Acknowledgements

This work is supported by National Key Research and Development Program(No.2017YFB1102800),NSFC for Excellent Young Scholars (No. 11722219), Key Project of NSFC (Nos.51790171, 5171101743).