Fei HE , Hongqing LIAO , Jihong ZHU , Zhongze GUO

a Institute of Systems Engineering, China Academy of Engineering Physics, Mianyang 621900, China

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

KEYWORDS Dynamic response;Large mass method;Mode acceleration method;Pseudo excitation method;Random base acceleration excitations;Topology optimization

Abstract Structural topology optimization subjected to stationary random base acceleration excitations is investigated in this paper. In the random response analysis, the Large Mass Method(LMM) which attributes artif icial large mass values at each driven nodal Degree Of Freedom(DOF) to transforming the base acceleration excitations into force excitations is proposed. The Complete Quadratic Combination (CQC) which is commonly used to calculate the random responses in previous optimization has been proven to be computationally expensive especially for large-scale problems. In order to conquer this diff iculty, the Pseudo Excitation Method(PEM) and the combined method of PEM and Mode Acceleration Method (MAM) are adopted into the dynamic topology optimization, and random responses are calculated using these two methods to ascertain a high eff iciency over the CQC. A density-based topology optimization method minimizing dynamic responses is then formulated based on the integration of LMM and PEM or the combined method of PEM and MAM. Numerical examples are presented to verify the accuracy of the proposed schemes in dynamic response analysis and the quality of the optimized design in improving dynamic performance.

1. Introduction

Topology optimization method has been rapidly developed and widely applied in engineering design since the homogenization-based method for continuum structures was proposed by Bends?e and Kikuchi.1Till now, the idea has been successfully applied in both theoretical and practical engineering.2-4

As the inherent complexity of dynamic problems, topology optimization about dynamic problems is always a challenging topic. Relevant researches mainly focused on the topology optimization related to eigenfrequencies,such as maximization of fundamental eigenfrequency,high-order eigenfrequencies or the gap between two consecutive eigenfrequencies of given orders, etc.5-9

Topology optimization of dynamic response has also attracted broad attention in recent years. Ma et al.10studied the minimization of dynamic compliance under harmonic force excitation using the homogenization method. Tcherniak11maximized the magnitude of steady state vibrations of the resonating structures at a given excitation frequency. Shu et al.12minimized the frequency response based on the level set method.Vicente et al.13presented a concurrent topology optimization for minimizing frequency responses of a multiscale system composed of macro and micro phases using the BESO method. Du and Olhoff14dealt with topology optimization problems of minimizing the sound power radiated from the structural surfaces into a surrounding acoustic medium.Zhang and Kang15investigated topology optimization of the piezoelectric actuator/sensor coverage attached to a thin-shell structure to improve the active control performance for reducing the dynamic response under transient excitations. Beheron and Guest16presented a topology optimization framework for structural problems subjected to transient loading.

Due to the complex relationship between the dynamic responses and the design variables,topology optimization minimizing random responses is a more diff icult case, and only a few works have been found so far to deal with the topology optimization of structure under random excitations. Meanwhile,the existing results have mainly focused on the topology optimization under random force excitations. For example,Rong17and Rong18et al.presented the topology optimization of continuous structures using ESO method where stationary random responses were regarded as design constraints. Yang et al.19studied the topology optimization under static loads and narrow-band random excitations, where the static and dynamic response analyses were processed independently without any superposition. Zhang et al.20optimized the layout of multi-component under both static and stationary random excitations with density method.As a matter of fact,the above works were implemented by means of Complete Quadratic Combination (CQC). The CQC was the classical and basic method for random vibration. However, it was costineffective in random response analysis especially for largescale problems and it was the bottleneck problem to its application in engineering structures. The Pseudo Excitation Method (PEM)21-24which could transfer the solving of random responses into the solving of pseudo harmonic responses was thus introduced. Although both methods can completely achieve the same solution with the same number of structural modes, the eff iciency of the PEM is much higher than the CQC. With this advantage, Zhang et al.25studied topology optimization of large-scale structures subjected to stationary random force excitation with PEM. Fang et al.26dealt with an optimal layout design of the Constrained Layer Damping(CLD) treatment of vibrating structures subjected to stationary random excitation with PEM.

Besides the force excitations, base acceleration excitations are actually more common in engineering.Unfortunately,only a few works have been found dealing with topology optimization under base excitations. Rong et al.27optimized the structural topology using the ESO method with stationary random responses constrained in design. Hao et al.28designed the strap-down inertial navigation support under random loads.Lin et al.29adopted the PEM as an eff icient optimization procedure in the maximization of the energy harvesting performance under stationary random excitation. Zhang30optimized the structure with CQC under random base excitation. Allahdadian et al.31optimized the layout of the bracing system of multi-story structures under harmonic base excitation.Allahdadian and Boroomand32investigated the topology optimization of planner frames under base excitations using the SIMP method. Zhu et al.33studied topology optimization minimizing dynamic responses under harmonic base accelerations based on the integration of Large Mass Method(LMM) and Mode Displacement Method (MDM) or Mode Acceleration Method (MAM).

In recent decades,the LMM34has been widely used in engineering structure analyses.35-38LMM attributes artif icial large mass at each driven nodal Degree Of Freedom(DOF)to transforming the base acceleration excitations into force excitations.Then high eff icient methods such as PEM or the combination of MAM and PEM can thus be applied to the dynamic response calculation.

Therefore, we propose combining LMM with PEM or the combined method of PEM and MAM to solve topology optimization problems of linear dynamic system with classical damping and steady-state responses under random base acceleration excitations. This method may be applied to some potential real-world applications where random acceleration excitations widely exist,such as wing,missile,rocket and some other aeronautic and aerospace structure designs, automobile skeleton design,ship structure design and so on.In this paper,the design sensitivities of the dynamic responses are also derived based on LMM.Several numerical examples including those with complicated design domains and large numbers of DOFs are tested to verify the effect of the proposed optimization scheme.

2. Basic formulations of LMM and stationary random analysis methods

In general, the equation of a discretized n-DOF dynamic system subjected to stationary random force excitation can be written as

where M,C and K represent the mass matrix,damping matrix and stiffness matrix;p(t)is a d-dimensional stationary random force vector of non-zero values,whose Power Spectral Density(PSD) matrix is of d-dimension and denoted by Sp(ω). Notice that b is a n×d transformation matrix representing the force distribution.

If the structure suffers base acceleration excitations instead of force excitations, Eq. (1) can be written as

2.1. Large mass method

The dynamic equilibrium equation in Eq.(2)can be expressed in partitioned form where the subscript index f represents free DOFs and s represents supported DOFs;and Uf(t) are absolute acceleration, velocity and displacement vectors corresponding to free DOFs respectively;and Us(t) are absolute acceleration, velocity and displacement vectors of supported DOFs respectively. In LMM, in order to apply acceleration excitations, large mass values MLare attached at the supported DOFs. The inertia forces will then dominate the response. To reproduce the specif ied accelerogram, the inertia forces generated at the supported DOFs should also be considered as external driving forces. Eq. (3) thus can be written as

To explain the method clearly, we assume that DOF I is a driven support DOF. The Ithequation is written as

where MLIis the large mass added at the Ithdriven support DOF.MLIis assigned as 107times of the total structural mass in this paper.According to our test,stable response values can be obtained when MLIis between 102and 1014times of the total structural mass. But when it is larger than 1016times of the total structure,the responses will be signif icantly inf luenced by the numerical errors.¨Ub(t)means the excitation acceleration corresponding to the large mass.

As MII+MLIis much larger than the coeff icients of other items,small items can be omitted.Then Eq.(5)can be approximately written as

So we can obtain

Accordingly, equivalent force excitation vector can be obtained as

In this way, the dynamic response analysis under base acceleration excitations is transformed into the one under force excitations. Moreover, relative responses can be obtained directly if rigid body modes are excluded in the response analysis procedure. To be clarif ied, by introducing a local coordinate system def ined on the base, we can denote the motion of the structure relative to the base.In this case,the governing equation can be reduced to the original equation of motion with an equivalent force dependent on the excitation frequency.

2.2. Structural random response analysis with CQC

To solve Eq. (1), the following notation is introduced:

where Z(t) is the vector of generalized coordinates.Φ= [Φ1Φ2...Φn] is the mode shape matrix and normalized by mass matrix. According to the theory of classical damping with ξibeing the ithdamping ratio,the following relations then hold

where ωirepresents the ithcircular eigenfrequency. The Rayleigh damping corresponds to

Notice that constants α and β are Rayleigh damping coeff icients of the structure.From the third relation of Eq.(10),the modal damping ratio for the ithnormal modecan be expressed as

Then a number of n uncoupled equations of motion can be derived by substituting Eq.(9)into Eq.(1)and premultiplying ΦT.

By means of Duhamel integral,the time-domain solution of Eq. (13) can be stated as39

where hi(τ)is the unit impulse response function related to the single DOF system of Eq. (13)

with

Then the displacement response U(t) can be derived as

The autocorrelation function of displacement response U(t)reads39

The PSD matrix of random displacement response SU(ω)can then be obtained by Fourier transformation of the above autocorrelation function

where Hidenotes the frequency domain transfer function between loading and response, andis the conjugated function of Hi

with j2=-1.In reality,only the f irst l modes are employed in the computing due to the computing eff iciency, and then Eq.(19) can be approximated as

This is well-known CQC formula.24,39The conventional CQC will compute Eq. (21) directly to get the PSD matrix of random displacement. Since it involves the cross-correlation terms between all l participant modes, the computing cost will be very expensive for large values of l.

2.3. Formulations for conventional PEM and combined method of PEM and MAM

Since the PSD matrix is Hermitian,it can be decomposed into

in which Q is the rank of SU(ω).Substituting Eq.(22)into Eq.(21) and expanding it give

Suppose

According to the MDM,39,40Xq(t) is the displacement response vector of Eq.(1)under the qthpseudo harmonic force vector bγqejωt. Eq. (23) then can be rewritten as

This is the conventional PEM proposed by Lin and his cooperators.21-24According to this method, the computing of PSD matrix of random displacement responses can be solved through the common harmonic displacement responses under pseudo harmonic excitations. The PEM is deduced equally from the CQC method. Therefore, both methods will obtain the same solution with the same number of structural modes, but the PEM is more eff icient than the CQC for random vibration analysis.Zhang et al.25compared the eff iciency of PEM and conventional CQC in detail.

According to the conventional PEM,the computing of random responses is transformed into the solving of pseudo harmonic responses with the famous MDM in Eq. (24). Only the lower l modes are employed in MDM. The MAM adds one more static analysis to take into account the contribution of the omitted higher modes. Namely the error of truncation mode with MDM in Eq. (24) is further remedied by the MAM through a pseudo-static solution. With MAM, Eq.(24) can be modif ied as

where (Xq)staticis the pseudo-static displacement under static force vector (bγq). As Eq. (26) just needs one more additional static analysis than Eq. (24), eff iciencies of MAM and MDM would be very close.33Eq. (26) can also be written as40-42

The second term of Eq.(27)related to MAM could be treated as the correction term to Eq. (24) of MDM. Detailed definitions and derivations of these two methods can be found in some previous literatures such as Ref. 43.

It should be mentioned that during the solution of MAM in Eq. (26), the additional static analysis should be performed with the equivalent static loads actually applied on the original supported DOFs.This will become static analysis without any f ixation and will lead to a rigid body motion. Therefore, the inertia relief analysis method is employed here to obtain the relative displacement responses in the structure as if the structure was freely accelerating due to the applied loads.Boundary conditions are applied only to restrain rigid body motion,while the reaction forces corresponding to these boundary conditions are zero because the external loads are balanced by these accelerations. In this way, the boundary conditions are applied at the originally supported DOFs to eliminate the rigid body motion, an additional acceleration is then applied to the structural body to obtain the relative displacement responses.

In order to check the computing accuracy of the conventional PEM and the combined method of PEM and MAM,the exact solution of PSD obtained by means of the Full Method (FM) is chosen as a benchmark. The exact displacement response vector can be obtained by the FM with directly solving Eq. (1) under the qthpseudo harmonic force vector bγqejωt.

Eq.(28)can be dealt with by sparse direct solver as used in static analysis but with complex arithmetic by default. By replacing Eq. (24) with Eq. (28), the PSD of random response would be the exact solution and will be used as the benchmark.It can be seen that each excitation frequency ω implies one independent complex analysis of Eq. (28) with the FM. This is the reason why the exact PEM is not suggested in topology optimization due to the fact that the computing cost is prohibitive in practice for the multiple excitation frequencies, as detailed in Ref. 43.

2.4. Numerical tests for response analysis

In order to investigate the computing accuracy of different methods under stationary random base acceleration excitations, two numerical examples are investigated in this section. In all the tests of this paper, Young's modulus, Poisson ratio and density of the solid material are 200 GPa, 0.3 and 7800 kg/m3, correspondingly. To simplify the discussion,the combined method of PEM and MAM is shortly called modif ied PEM.

2.4.1. 2D cantilever beam

The structure is a 2D rectangle of the size 1.0 m×0.48 m.It is clamped at the left side as shown in Fig.1,and is meshed into 100×48 4-node plane stress elements with 9898 DOFs. The white-noise acceleration excitation with the PSD value of 1 g2/(rad/s) is applied to the foundation in the vertical direction.The middle point of the right edge is regarded as the concerned response point.The excitation symbol g equals 9.8 m/s2in this paper.

2.4.2. 3D block

The block structure has a size of 0.6 m×0.3 m×0.24 m. It is supported at the four corners at the left side, as shown in Fig. 2. The design domain is meshed into 60×30×24 8-node hexahedron solid elements with 141825 DOFs. The base is also subjected to the white-noise acceleration excitation with the PSD value of 1 g2/(rad/s) along the vertical direction. The middle point of the lower right edge is regarded as the concerned response point.

The same Rayleigh damping is adopted with α=10-2and β=10-5and the large mass values are 107times of the structures in all cases in this paper.Here,the f irst l=30 modes are employed in both examples. Notice that the 2D cantilever beam structure represents the common problem studied previously with a discretization of about thousands of DOFs.27,29,32The 3D block structure is adopted to illustrate large-scale problems of huge numbers of DOFs that are rarely studied in dynamic topology optimization,especially under base acceleration excitations. According to Eq. (25), the PSD value of the relative random displacement at the response point along the excitation direction can be written as

where c denotes the concerned DOF number. The solution of Eq. (29) is computed by the conventional PEM, the modif ied PEM and MAM and the exact FM,and compared in logarithmic forms. In order to simplify the expression, we suppose

The results are compared in Figs. 3 and 4. Notice that the excitation frequency in each problem is always lower than the ltheigenfrequency by default.

Fig. 1 2D cantilever beam (9898 DOFs).

Fig. 2 3D block (141825 DOFs).

Fig. 3 PSD curves in logarithmic form obtained by three methods for 2D cantilever beam.

Fig. 4 PSD curves in logarithmic form obtained by three methods for 3D block.

It can be seen that the PSD values using the conventional PEM and the modif ied PEM are always close to the exact solution obtained by the FM. The largest relative errors of PSD values of the conventional PEM and the modif ied PEM related to the exact solution obtained by the FM are 5.9% and 7.0%in Fig.3,and the values are both 8.1%in Fig.4.It means that the correction term in MAM corresponding to the modif ied PEM has little contribution to the PSD,which is a very different phenomenon from the conclusion drawn by Zhang et al.25.This is mostly due to the large mass used in LMM and the positions of excited and concerned DOFs.As a matter of fact,as the computing of random displacement responses with CQC are equally transformed into the solving of pseudo harmonic responses with MDM or MAM according to PEM, the accuracy and computing of MDM and MAM will decide the accuracy of conventional PEM and the modif ied PEM.Zhu et al.33has found that the computing accuracies of MDM and MAM are both acceptable for different optimization problems under base acceleration excitations based on LMM. The phenomenon that the computing accuracies of the PEM and the modif ied PEM are both acceptable here is thus reasonable.What's more, Zhu et al.33also compared the computing time of MDM and MAM in detail, including their computing time of sensitivity analysis. The result indicated that the computing eff iciencies of MDM and MAM are almost the same.It means that the computing time of the conventional PEM and the modif ied PEM keep almost the same.

3. Formulation of topology optimization problem

3.1. Topology optimization model

In a dynamic response topology optimization problem under stationary random excitations, the Root Mean Square(RMS) of relative random displacement response of the concerned cthDOF is chosen as the objective function with a prescribed material volume constraint. The design variables are the pseudo-densities describing the structural distribution.The optimization model can be mathematically expressed as

where [ωa, ωb] refers to the frequency band of random excitation; ηLis the lower bound of the set of design variables def ined by element pseudo-densities, and here ηL=0.001 is used to prevent the mass,stiffness and damping matrices from becoming singular; neand v denote the element number and the solid volume, respectively; vUis the upper bound of the latter.

In topology optimization of dynamic problems, it is recognized that the interpolation scheme of Solid Isotropic Microstructure with Penalization(SIMP)will lead to localized modes because of the mismatch between element stiffness and mass especially when the pseudo-densities become low.In fact,there exist a variety of interpolation schemes which are believed to be able to eliminate the localized modes, such as Rational Approximation of Material Properties (RAMP),44Modif ied SIMP interpolation models (MSIMP)9and Polynomial Interpolation Scheme.45Here, we adopt the Polynomial Interpolation which is given below:

where Mhand Khare the mass matrix and stiffness matrix of element h, respectively; Mh0and Kh0denote the mass matrix and stiffness matrix of solid element h, respectively.

3.2. Sensitivity analysis

For a random problem, design sensitivity of the RMS is derived with respect to the pseudo-density variable ηhin this section. The design sensitivity of the RMS here belongs to eigenvector-based sensitivity analysis for steady state response of dynamic systems which could be found in some previous literatures such as Ref.46. The differentiation of objective function can be written as

According to Eq. (29), the following equation can be obtained

Clearly, the sensitivity of pseudo harmonic displacement is the basic calculation for Eq. (34). With the implementation of the MDM, the sensitivity of pseudo harmonic displacement can be obtained by directly differentiating Eq. (24).

With the implementation of the MAM, the sensitivity of pseudo harmonic displacement can be obtained by directly differentiating Eq. (26)

where

Obviously,sensitivities of eigenfrequencies and eigenvectors are basic calculations for solutions of Eqs. (35) and (36). The sensitivity analysis of eigenfrequencies is discussed in Ref.47.For a simple eigenfrequency ωi, its sensitivity is given by the following equation

The derivatives of the eigenvectors hold the following form48

where βiris calculated as

In this way, the sensitivities for Eq. (35) and the second term in Eq. (36) are obtained. But for the f irst term in Eq.(36),we suppose that a is a column vector with all terms being zero except term c being 1.So the sensitivity of response at the concerned DOF c can be expressed as

Thus, Eq. (41) can be calculated by adjoint method. The derivative of (Xq)staticcan be solved as

The following equation is then established

where λ is the adjoint vector obtained by

Then we have

When the sensitivity of pseudo harmonic displacement is obtained, the sensitivity of relative displacement amplitude of pseudo harmonic response can be derived through the chain rule as

Finally, Eq. (33) can be calculated.

It should be noticed that the eigenvalue sensitivity analysis method in this paper is valid only for the cases of simple eigenpairs. Repeated eigenvalues maybe appear in some cases. The appearance of repeated eigenvalues is a particular case leading to some unwieldy problems. Thus, additional efforts must be made for repeated eigenvalues, such as the derivatives based on perturbation technique and directional derivatives which have been provided by Hu,49Seyranian50and Du9et al.

4. Numerical examples

In this section, several numerical examples under stationary random base acceleration excitations with specif ic frequency band are tested to illustrate the proposed method. Checkerboard pattern is avoided by means of the f iltering technique.51The Globally Convergent Method of Moving Asymptotes(GCMMA) algorithm52is used as the optimizer.

Fig. 5 Optimized conf igurations of 2D cantilever beam by conventional PEM with a mesh of 100×48 elements.

Fig. 6 Iteration curves of objective function in logarithmic form and volume constraint for optimized conf iguration in Fig. 5.

4.1. 2D cantilever beam

The f irst test is a 2D problem as stated in Fig.1.It is supposed that the RMS of vertical relative displacement of the middle node of the right edge is minimized as the objective function.A concentrated mass with a value of 0.5 kg is added at this node. The volume fraction of the solid material is constrained to be less than 50%. Initial values of all pseudo-densities are set to be 0.5 with the f irst eigenfrequency of the initial structure being 108 Hz. The white-noise base acceleration excitation of PSD value 1 g2/(rad/s) is loaded on the f ixed side. Two frequency bands are considered with f=[0,100]Hz and [0,500]Hz. l=30 modes are employed by the conventional PEM and the modif ied PEM.

Fig.5 gives the optimized results by the conventional PEM.Iteration curves of the objective function and constraint related to optimized conf igurations given in Fig. 5 are shown in Fig. 6. Fig. 7 gives the optimized results by the modif ied PEM.Iteration curves of the objective function and constraint related to optimized conf igurations given in Fig. 7 are shown in Fig. 8. The conf igurations are quite clear and almost the same by two methods for each prescribed frequency band.The iteration curves demonstrate that the objective is successfully minimized in each case. To verify the effect of optimization,the exact solutions of PSD curves at the concerned point for the optimized conf igurations related to Fig.7 are shown in Fig.9.It can be seen that the PSD curves within the prescribed frequency bands decrease obviously. Here, the DOFs of the structure are small, and other problems with the similar scale of DOFs can also be found in other works.27,29,32

Fig. 9 Exact solutions of PSD curves in logarithmic form of initial and optimized conf igurations shown in Fig. 7.

Then the structure is further optimized with ref ined meshes of large numbers of DOFs.Figs.10 and 11 give the optimized results.Clear conf igurations are found in all cases and the optimized structural patterns by two methods are quite similar with the results in Figs. 5 and 7 for each prescribed frequency band. It demonstrates the phenomenon that the computing accuracies of the conventional PEM and the modif ied PEM drawn in Figs. 3 and 4 are both acceptable.

4.2. 3D block

Fig. 7 Optimized conf igurations of 2D cantilever beam by modif ied PEM with a mesh of 100×48 elements.

Fig. 8 Iteration curves of objective function in logarithmic form and volume constraint for optimized conf iguration in Fig. 7.

Fig. 10 Optimized conf igurations of 2D cantilever beam by conventional PEM with a mesh of 200×96 elements.

Fig. 11 Optimized conf igurations of 2D cantilever beam by modif ied PEM with a mesh of 200×96 elements.

Fig. 12 Optimized conf igurations of 3D block with different frequency bands.

To further verify the proposed method, we now consider the 3D problem illustrated in Fig. 2 with large numbers of DOFs.It is supposed that the RMS of relative random displacement response at the response point along the excitation direction is regarded as the objective to be minimized. A concentrated mass with a value of 6 kg is added at the response point.The volume fraction of the solid material is constrained to be less than 20% and the initial values of all pseudodensities are uniformly set to be 0.2 correspondingly. The f irst eigenfrequency of the initial structure is 22 Hz.The white-noise base acceleration excitation of PSD value 1 g2/(rad/s) is applied at the base. Two frequency bands are considered with f=[0,100]Hz and[0,350]Hz.l=30 modes are still employed in the modif ied PEM.

Optimized conf igurations are shown in Fig. 12. In order to show the complete picture of the conf igurations, the grey elements whose pseudo-densities are smaller than 0.2 are hidden.Obviously,the conf igurations are all quite clear.The optimized results in Fig. 12 are further analysed. The exact solutions of PSD curves at the concerned point are shown in Fig.13 respectively. It is shown that PSD curves of optimized structures decrease obviously within the prescribed frequency bands.

Fig. 13 Exact solutions of PSD curves in logarithmic form of initial and two optimized conf igurations shown in Fig. 12.

4.3. Cuboid bracket structure

Fig. 14 Cuboid bracket structure (125307 DOFs).

A cuboid bracket structure is considered, as shown in Fig. 14.It is supported on the bottom plate and is meshed into 27600 8-node hexahedron solid elements with 125,307 DOFs in all.The center of the top is regarded as the concerned response point.The vertical four walls are considered as the design domain for which the volume fraction of the solid material is constrained to be less than 30%.Initial values of all pseudo-densities in the design domain are uniformly set to be 0.3. The f irst eigenfrequency of the initial structure is 117 Hz. The white-noise base acceleration excitation of PSD value 1 g2/(rad/s) is f irstly applied at the base along the X direction and the Y direction,respectively. Namely two load cases are considered here. To reduce the dynamic responses, the sum of RMS of the relative random displacement response at the response point along the X direction and the Y direction is regarded as the objective to be minimized. To optimize the structure, two frequency bands are considered with f=[0,300] Hz and [0,500] Hz.l=30 modes are still employed in the modif ied PEM.

Fig. 17 Exact solutions of PSD curves in logarithmic form of initial and two optimized conf igurations in X direction under X direction excitation.

Fig. 15 Optimized conf igurations of cuboid bracket structure with different frequency bands.

Fig. 16 Iteration curves of objective function in logarithmic form and volume constraint for optimized conf iguration in Fig. 15.

Fig. 18 Exact solutions of PSD curves in logarithmic form of initial and two optimized conf igurations in Y direction under Y direction excitation.

Obviously, problems considering the responses in two directions became more complicated. Optimized conf igurations are shown in Fig. 15. Iteration curves of the objective function and constraint related to optimized conf igurations given in Fig. 15(a) and (b) are shown in Fig. 16(a) and (b),respectively.In order to show the complete picture of the conf igurations,the grey elements whose pseudo-densities are smaller than 0.2 are hidden. The conf igurations are all quite clear.The iteration curves demonstrate that the objective is successfully minimized in each case.

To verify the effect of optimization, we also choose the optimized conf igurations in Fig. 15 for further analysis. The exact solutions of PSD curves in the X and Y directions at the concerned response point are shown in Figs. 17 and 18,respectively. It can be seen that the PSD curves of optimized structures in the frequency bands decrease obviously.

5. Conclusion

The present study investigates the topology optimization under stationary random base acceleration excitations. LMM is used to transform the base acceleration excitations into force excitations. The CQC was generally adopted as the dynamic analysis method related to random responses. However, prohibitive computing cost became inevitable when large-scale problems were concerned in practice. The conventional PEM is introduced to improve the computing eff iciency, and the modif ied PEM is adopted to improve the computing accuracy of random response based on conventional PEM. However,during the implementation of the topology optimization, it is found that the computing accuracies of the conventional PEM and the modif ied PEM are both acceptable for different optimization problems, which is different from the conclusion drawn by the previous work. This is mostly due to the large mass used in LMM and the positions of excited and concerned DOFs.

2D and 3D examples are then used to illustrate the effect of the proposed method. The conf igurations are quite clear with the dynamic responses at the concerned positions signif icantly suppressed.

Acknowledgements

This work was supported by the Innovation and Development Foundation of ISE, CAEP (2017 cxj18), and the Presidential Foundation of CAEP (YZJJLX2018008).