CN111539138A - Method for solving time domain response sensitivity of structural dynamics peak value based on step function - Google Patents

Abstract

The invention relates to a time-domain response sensitivity solution method for structural dynamics based on a step function, including a time-domain response peak approximation method and an approximation function sensitivity solution method; The function is time-integrated to obtain its approximation value; the approximation function sensitivity is realized by the adjoint vector method. The invention solves the problem that the subjective parameter selection of the peak approximation method in the existing structural dynamics time-domain topology optimization has a large influence, and has simple structure, easy programming and easy implementation, and significantly improves the optimization efficiency.

Description

A Step Function-Based Method for Solving the Sensitivity of Structural Dynamics Peak Time Domain Response

technical field

The invention belongs to the field of structural dynamics topology optimization, and particularly relates to an integration method for approximating a time domain response peak value through a step function and a method for solving the sensitivity of the peak approximation value with respect to involved variables.

Background technique

Structural optimization design is divided into three stages: conceptual design, shape design and parameter design. Among them, the most important stage is the conceptual design stage. Conceptual design determines the basic configuration of the structure. Topology optimization is a widely used tool in the conceptual design stage. . At present, the research on topology optimization of structural statics has become more and more mature, but statics conditions cannot cover all application scenarios in the complete life cycle of the structure. Therefore, topology optimization of structural dynamics has become a research hotspot. According to the time and frequency characteristics of design indicators, topology optimization of structural dynamics is divided into two branches: frequency domain and time domain. Frequency domain topology optimization can avoid the large-scale computational consumption of dynamic analysis and time-domain history sensitivity solution, but only Achieving the maximization of structural dynamic stiffness cannot establish an intuitive connection with the real numerical value of the structural dynamic response time history.

Dynamic time-domain topology optimization takes the evaluation of the response time history as the research starting point, and can directly optimize the specific indicators of the response history. At present, there are two types of response history evaluation indicators: integral and peak value. Integral can evaluate the total amount of vibration, but also cannot establish an intuitive connection with the real value of the structural dynamic response time history. Intuitive indicators, the current peak dynamics topology optimization often uses the method of clustering function to represent the peak proxy, and the strong correlation of function parameters leads to subjective influence on the optimization results.

SUMMARY OF THE INVENTION

The technical problem solved by the present invention is: in order to overcome the deficiencies of the prior art, a method for solving the peak time-domain response sensitivity of structural dynamics based on a step function is proposed. It consists of a variable sensitivity solution method, which solves the problem that the peak time domain index of the existing structural time domain dynamics topology optimization is greatly affected by subjective parameters and the convergence is difficult.

The technical solution of the present invention is:

A method for solving the peak time-domain response sensitivity of structural dynamics based on a step function, the steps of the method include:

Step (1), initialize the optimization model, take the structural design domain as the geometric boundary, establish a finite element analysis model, and multiply the elastic modulus of the i-th element by the p power of the pseudo-density x _i , where x _i ∈[ 0,1], p=3, the pseudo-density x _i of all elements constitutes the design variable vector x, the structural stiffness matrix K and the mass matrix M are extracted according to the finite element model, and the constraint degrees of freedom are determined;

Step (2), determine the load history F(t) carried by the structure, define the damping matrix of the finite element model in step (1) as C=α _c M+β _c K, and α _c and β _c are proportional damping coefficients;

加速度场

t为时间；Step (3), solve the structural dynamic response of the finite element model constructed in step (1) under the load environment and damping conditions defined in step (2) by using the numerical solution method or approximate analytical solution method of any differential equation, Obtain structural displacement field U(t), velocity field

acceleration field

t is time;

求解所得动力学时域响应峰值逼近值返回拓扑优化主程序；In step (4), according to the structural displacement field U(t) obtained in step (3), establish the attention index f(U(t)) of structural time-domain dynamics topology optimization, and combine f(U(t)), U (t) and Max(f(U(t))), solve the definite integral by any numerical integration method, and obtain the approximate value of the dynamic response peak Max(f(U(t)))

The obtained dynamic time-domain response peak approximation value is returned to the main program of topology optimization;

结合f(U(t))、U(t)与Max(f(U(t)))，求解虚拟载荷历程F_λ(t)，定义步骤(1)中有限元模型的虚拟阻尼矩阵为-C＝-α_cM-β_cK；Step (5), according to the structural displacement field U(t) obtained in step (3), solve the first-order derivative matrix of the structural displacement field of the attention index of the structural time-domain dynamic topology optimization

Combine f(U(t)), U(t) and Max(f(U(t))) to solve the virtual load history F _λ (t), define the virtual damping matrix of the finite element model in step (1) as - C=-α _c M-β _c K;

Step (6), through the numerical solution method or approximate analytical solution method of any differential equation, to solve the virtual structure of the finite element model constructed in step (1) under the virtual load environment and virtual damping situation defined in step (5). Dynamic response, obtain the virtual dynamic response λ(τ) of the structure, and name the virtual dynamic response vector as the adjoint vector;

加速度场

求解编号为第i个单元的单元节点位移U_e,i(t)、单元节点速度

与单元节点加速度

Step (7), according to the displacement field U(t) and velocity field obtained in step (3)

acceleration field

Solve the element node displacement U _e,i (t) and the element node velocity of the i-th element

with element node acceleration

Step (8), according to the accompanying vector λ(t) obtained in step (6), solve the unit node accompanying vector λ _e,i (t) numbered as the i-th unit;

Step (9), according to the structural stiffness matrix K obtained in step (1), the mass matrix M, and the damping matrix C obtained in step (2), solve the element stiffness matrix K _e and the element when the pseudo-density of the element is equal to 1. mass matrix _{Me, element damping matrix C e ;}

λ_e,i(t)、K_e、M_e与C_e，通过任意数值积分方法求解定积分，获取动力学时域响应指标逼近值

关于编号为第i单元的伪密度x_i的灵敏度；Step (10), U _e,i (t) obtained based on steps (7) to (9),

λ _e,i (t), _{Ke , Me and C e , solve} the definite integral by any numerical integration method, and obtain the approximation value of the dynamic time domain response index

the sensitivity with respect to the pseudo-density x _i numbered in the i-th unit;

Step (11), through serial operation or parallel operation, repeat steps (7) to (10) until the sensitivities of the approximation value of the structural dynamics time domain response index with respect to the pseudo-density of all elements are solved, and the structural dynamics peak time domain is completed. Regarding the solution of the response sensitivity of structural topology optimization design variables, the sensitivity of the obtained dynamic time-domain response peak approximation value is returned to the main program of topology optimization.

Further, this method only involves the approximation value of the peak time-domain response index of structural dynamics and its sensitivity to the pseudo-density of all elements, and can be embedded in the gradient solution algorithm for topology optimization of any structure. The parameters obtained in step (1) It can be realized by any structural dynamics solution method and program platform.

与加速度场

需通过求解如下微分方程组获得：Further, in step (3), the structural displacement field U(t), the velocity field

with the acceleration field

It needs to be obtained by solving the following differential equations:

上式可以通过任意微分方程租的数值求解方法或者近似解析求解方法进行求解。The differential equation system in the above formula is a differential equation system with initial conditions and no termination conditions, and its initial condition is U(0)=0

The above formula can be solved by numerical solution method or approximate analytical solution method of any differential equation.

通过求解如下式定积分获得：Further, the approximate value of the dynamic peak time-domain response f(U(t)) in step (4)

Obtained by solving the definite integral as follows:

χ为远大于

的正实数；Δt为积分步长；t_f为动力学分析终止时刻，上式可以通过任意数值积分方法进行求解。where e is the natural exponent, β is χ-0.6,

χ is much larger than

Δt is the integration step size; t _f is the termination time of the dynamic analysis, the above formula can be solved by any numerical integration method.

Further, the virtual load history F _λ (t) in step (5) is solved by the following formula

Further, the adjoint vector λ(t) in step (6) needs to be obtained by solving the following differential equations:

上式可以通过任意微分方程租的数值求解方法或者近似解析求解方法进行求解。The system of differential equations in the above formula is a system of differential equations without initial conditions but with termination conditions, and its termination condition is λ(t _f )=0

The above formula can be solved by numerical solution method or approximate analytical solution method of any differential equation.

关于编号为第i个单元的伪密度x_i的灵敏度需要求解如下定积分获得：Further, the approximate value of the dynamic peak time domain response in step (10)

The sensitivity of the pseudo-density x _i of the i-th element needs to be obtained by solving the following definite integral:

The above equation can be solved by any numerical integration method.

The beneficial effects of the present invention compared with the prior art are:

(1) The present invention adopts the peak approximation method based on the time-domain response peak value of the step function integration, and when constructing the peak approximation function, the approximation error is reduced through the reference value translation strategy, which can be used in applications with larger integration steps. situation, can improve the optimization speed;

(2) The peak approximation method of the time domain response peak value based on the step function integration adopted in the present invention has the same monotonicity as the approximation value and the real function value. In topology optimization, the sensitivity in the same direction as the real value can be given. ;

(3) The present invention does not involve a dynamic solution step, but only involves an approximation value of the structural dynamic peak time-domain response index and its sensitivity solution with respect to the pseudo-density of all units, which can be embedded in a gradient solution algorithm for topology optimization of any structure;

(4) The accuracy of the peak approximation value of the time-domain response peak based on the step function integration proposed by the present invention is not strongly dependent on the response history after the peak response occurs, which can reduce the response time length concerned by the dynamics and sensitivity calculations. , reduce optimization time;

(5) The dynamic response peak approximation and sensitivity solution method of the present invention has good portability. In terms of solution, it can be combined with any numerical integration and differential equation solution methods. In application, it can be combined with any A pseudo-density interpolation model is combined with a gradient optimization algorithm.

Description of drawings

Fig. 1 is the flow chart of the present invention;

Fig. 2 is the structural dimension, load and restraint environment in the embodiment of the present invention;

Fig. 3 is the time-varying load history that the structure receives in the embodiment of the present invention;

Fig. 4 is the structural statics topology optimization result of the present invention;

Fig. 5 is the optimization result of the structure time-domain dynamics topology optimization embedded in the present invention when the termination time is 0.12s, 0.15s, 0.18s and 0.58s respectively;

6 is the convergence history of the structure time-domain dynamics topology optimization embedded in the present invention when the termination time is 0.12s, 0.15s, 0.18s and 0.58s respectively;

FIG. 7 is a dynamic response comparison of the optimal solution of the structural time-domain dynamic topology optimization embedded in the present invention and the optimal solution of the static topology optimization.

Detailed ways

The present invention will be further elaborated below in conjunction with the examples.

As shown in FIG. 1 , the present invention proposes a method for solving the peak time-domain response sensitivity of structural dynamics based on a step function, which includes the following steps:

Step (1), initialize the optimization model, take the structural design domain as the geometric boundary, establish a finite element analysis model, and multiply the elastic modulus of the i-th element by the p power of the pseudo-density x _i , where x _i ∈[ 0,1], p=3, the pseudo-density x _i of all elements constitutes the design variable vector x, the structural stiffness matrix K and the mass matrix M are extracted according to the finite element model, and the constraint degrees of freedom are determined;

Step (2), determine the load history F(t) carried by the structure, define the damping matrix of the finite element model in step (1) as C=α _c M+β _c K, and α _c and β _c are proportional damping coefficients;

加速度场

需求解步骤(1)所构建的有限元模型在步骤(2)定义的载荷环境与阻尼情况下的结构动力学响应，即求解如下微分方程组Step (3), obtain the structural displacement field U(t), velocity field

acceleration field

It is required to solve the structural dynamic response of the finite element model constructed in step (1) under the load environment and damping conditions defined in step (2), that is, to solve the following differential equations

上式可以通过任意微分方程租的数值求解方法或者近似解析求解方法进行求解；The differential equation system in the above formula is a differential equation system with initial conditions and no termination conditions, and its initial condition is U(0)=0

The above formula can be solved by numerical solution method or approximate analytical solution method of any differential equation;

可以通过求解下式获得In step (4), according to the structural displacement field U(t) obtained in step (3), establish the attention index f(U(t)) of structural time-domain dynamics topology optimization, and combine f(U(t)), U (t) and Max(f(U(t))), the approximate value of the dynamic response peak Max(f(U(t)))

can be obtained by solving the following equation

χ为远大于

的正实数；Δt为积分步长；t_f为动力学分析终止时刻。上式可以通过任意数值积分方法进行，求解求解所得动力学时域响应峰值逼近值返回拓扑优化主程序。where e is the natural exponent, β is χ-0.6,

χ is much larger than

is a positive real number; Δt is the integration step; t _f is the termination time of the kinetic analysis. The above formula can be carried out by any numerical integration method, and the obtained dynamic time-domain response peak approximation value is returned to the main program of topology optimization.

结合f(U(t))、U(t)与Max(f(U(t)))，虚拟载荷历程F_λ(t)可以通过下式进行求解Step (5), according to the structural displacement field U(t) obtained in step (3), solve the first-order derivative matrix of the structural displacement field of the attention index of the structural time-domain dynamic topology optimization

Combining f(U(t)), U(t) and Max(f(U(t))), the virtual load history F _λ (t) can be solved by the following formula

After the virtual load history is solved, the virtual damping matrix of the finite element model in step (1) is defined as -C=-α _c M-β _c K;

Step (6), obtain the adjoint vector λ(τ), and need to solve the virtual dynamic response of the structure of the finite element model constructed in step (1) under the virtual load environment and virtual damping defined in step (5), that is, to solve system of differential equations as follows

上式可以通过任意微分方程租的数值求解方法或者近似解析求解方法进行求解；The system of differential equations in the above formula is a system of differential equations without initial conditions but with termination conditions, and its termination condition is λ(t _f )=0

The above formula can be solved by numerical solution method or approximate analytical solution method of any differential equation;

加速度场

提取编号为第i个单元的单元节点位移U_e,i(t)、单元节点速度

与单元节点加速度

Step (7), according to the displacement field U(t) and velocity field obtained in step (3)

acceleration field

Extract the element node displacement U _e,i (t) and the element node velocity of the i-th element

with element node acceleration

Step (8), according to the accompanying vector λ(t) obtained in step (6), extract the unit node accompanying vector λ _{e, i} (t) numbered as the i-th unit;

Step (9), according to the structural stiffness matrix K obtained in step (1), the mass matrix M, and the damping matrix C obtained in step (2), solve the element stiffness matrix K _e and the element when the pseudo-density of the element is equal to 1. mass matrix _{Me, element damping matrix C e ;}

λ_e,i(t)、K_e、M_e与C_e，则动力学峰值时域响应的逼近值

关于编号为第i个单元的伪密度x_i的灵敏度需要求解如下定积分获得Step (10), U _e,i (t) obtained based on steps (7) to (9),

λ _e,i (t), _{Ke , Me and C e , then} the approximation of the dynamic peak time-domain response

The sensitivity of the pseudo-density x _i of the i-th element needs to be obtained by solving the following definite integral

The above equation can be solved by any numerical integration method;

In step (11), through serial operation or parallel operation, repeat steps (7) to (10) until the sensitivities of the approximate value of the structural dynamics time-domain response index with respect to the pseudo-density of all elements are solved, and the obtained dynamic time-domain response peak value is approximated. The sensitivity of the value is returned to the main topology optimization routine.

Example

In order to fully understand the characteristics of the present invention and its applicability to engineering practice, the present invention establishes the structure and load constraint environment as shown in the figure, and embeds it in the time-domain topology optimization of continuum structure dynamics, and the optimization adopts the gradient algorithm to move The asymptotic line method MMA, the pseudo-density interpolation model adopts the SIMP model of p=6, the optimization objective is the minimum response peak value in the time domain, and the optimization constraint is that the area ratio is not less than 0.3. The method proposed in the present invention is embedded in the framework of the main program of topology optimization. It is used to solve the optimization objective and the optimization objective sensitivity. The specific values of the parameters in each real-time step are as follows:

In step (1), the design domain is shown in Figure 2. The size of the design domain is 100mm×70mm. The structural finite element model has 100 vertical elements and 70 horizontal elements. The vertical and horizontal elements start from the upper left corner; The amount is 2 × 10 ⁵ Mpa, the density is 7.8 × 10 ^-6 Kg/mm3, the Poisson's ratio is 0.3, and the element type is a plane bilinear element.

The proportional damping coefficients α _C =10 and β _C =1×10 ^-5 in step (2); the structure is constrained at the lower two endpoints, and the structure is constrained along the X direction at the upper right node of the 17th element in the first row. For two time-varying loads in the Y direction, a 0.1Kg counterweight is attached to the load point. The load history is shown in Figure 3. The maximum load in the X direction appears at 0.062s, and the maximum load in the Y direction appears at 0.082s.

In step (3), the New Mark-β method is used to solve the dynamic differential equation, and the calculation step Δt is 0.001s. In order to verify that the present invention is not strongly dependent on the analysis time, in this example, 0.12s, 0.15s, 0.18s were carried out respectively. The kinetic analysis and the corresponding topology optimization under the four groups of termination time s and 0.58s.

其中

为输出向量，其为一个与自由度数相同的列向量，受载节点X方向平动自由度对应位置为1，其他位置为0，

为输出向量，其为一个与自由度数相同的列向量，受载节点Y方向平动自由度对应位置为1，其他位置为0；β为20，χ为10000，则

定积分采用梯形法进行求解，对应于步骤(3)，积分步长Δt为0.001s，t_f分别为0.12s、0.15s、0.18s与0.58s。The dynamic time domain index f(U(t)) concerned in step (4) is the sum of the maximum displacement in the X direction and the maximum displacement in the Y direction of the node under load of the structure, namely

in

is the output vector, which is a column vector with the same number of degrees of freedom. The position corresponding to the translational degree of freedom in the X direction of the loaded node is 1, and the other positions are 0.

is the output vector, which is a column vector with the same number of degrees of freedom. The position corresponding to the translation degree of freedom in the Y direction of the loaded node is 1, and the other positions are 0; β is 20, and χ is 10000, then

The definite integral is solved by the trapezoidal method, corresponding to step (3), the integral step Δt is 0.001s, and t _f is 0.12s, 0.15s, 0.18s and 0.58s, respectively.

Step (5) does not require parameter input, and solves according to the formula proposed in the present invention.

Step (6) uses the New Mark-β method to solve the kinetic differential equation, corresponding to step (3), Δt is 0.001s, and t _f is 0.12s, 0.15s, 0.18s and 0.58s, respectively.

Step (7) does not require parameter input, and extracts according to the logic and order of unit and node numbering;

Step (8) does not require parameter input, and extracts according to the logic and order of element and node numbering;

Step (9) does not require parameter input, and the element type adopts a plane bilinear element.

In step (10), the definite integral is solved by the trapezoidal method. Corresponding to step (3), the integration step Δt is 0.001s, and t _f is 0.12s, 0.15s, 0.18s and 0.58s, respectively.

Step (11) Repeat steps (7) to (10) until all units are calculated, and the sensitivity of the obtained dynamic time-domain response peak approximation value is returned to the main program of topology optimization.

After all steps of the present invention are finished, return the sensitivity of the approximation value of the time domain response peak calculated in step (4) and the approximation value of the time domain response peak calculated in step (11) to the topology optimization group program, and by moving The asymptotic algorithm MMA adjusts the pseudo-density distribution in the next step. After the adjustment, the real-time steps of the present invention are performed again, and steps (1) to (11) are repeated until the structure time-domain dynamics topology optimization converges.

In order to verify that the approximation method of the time-domain response peak and the corresponding sensitivity solution method proposed in the present invention can operate effectively in the structural dynamics time-domain topology optimization framework, and can reduce the structural dynamic time-domain response peak, this embodiment also aims at the same The model adopts static topology optimization, the load environment is static load, the X-direction environmental load and the Y-direction static load are the peaks of the load history in step (2), and other constraint environments and optimization algorithms are consistent with the structural dynamics topology optimization.

By comparing Fig. 4 and Fig. 5, it can be seen that the optimal solution topology of static topology optimization is different from the optimal solution topology of dynamic topology optimization, and the optimal solution topology of structural dynamics topology optimization based on the present invention is different. Depending on the duration of the kinetic analysis, it can also be seen from Figure 6 that the optimization can converge quickly and the convergence values are exactly the same regardless of the duration of the kinetic analysis. It can be seen from FIG. 7 that the dynamic response peak value of the optimal configuration of static topology optimization is larger than the dynamic response peak value of the optimal configuration of dynamic topology optimization embedded in the present invention, indicating that the step function-based method proposed by the present invention The structural dynamics peak time-domain response sensitivity solution method can be embedded in the gradient topology optimization method very effectively, and can effectively reduce the peak time-domain response.

The invention adopts the peak approximation method based on the time-domain response peak value of the step function integration, and when constructing the peak approximation function, the approximation error is reduced by the reference value translation strategy, which can be applied to the situation with a large integration step, and can Improve optimization speed.

The peak approximation method of the time domain response peak value based on the step function integration adopted in the present invention has the same monotonicity as the real function value, and can give the sensitivity in the same direction as the real value in topology optimization.

Although the preferred examples of the present invention have been disclosed above, they are not intended to limit the present invention. Any person skilled in the art can make use of the methods and techniques disclosed above to make technical solutions of the present invention within the spirit and scope of the present invention. possible changes and modifications. Therefore, any simple modifications, equivalent changes and modifications made to the above embodiments according to the technical essence of the present invention without departing from the content of the technical solution of the present invention all belong to the protection scope of the technical solution of the present invention.

Claims (7)

1. based on the structural dynamics peak time-domain response sensitivity solution method of step function, it is characterized in that, the step of this method comprises: Step (1), initialize the optimization model, take the structural design domain as the geometric boundary, establish a finite element analysis model, and multiply the elastic modulus of the i-th element by the p power of the pseudo-density x _i , where x _i ∈[ 0,1], p=3, the pseudo-density x _i of all elements constitutes the design variable vector x, the structural stiffness matrix K and the mass matrix M are extracted according to the finite element model, and the constraint degrees of freedom are determined; Step (2), determine the load history F(t) carried by the structure, define the damping matrix of the finite element model in step (1) as C=α _c M+β _c K, and α _c and β _c are proportional damping coefficients; Step (3), solve the structural dynamic response of the finite element model constructed in step (1) under the load environment and damping conditions defined in step (2) by using the numerical solution method or approximate analytical solution method of any differential equation, Obtain structural displacement field U(t), velocity field

acceleration field

t is time; In step (4), according to the structural displacement field U(t) obtained in step (3), establish the attention index f(U(t)) of structural time-domain dynamics topology optimization, and combine f(U(t)), U (t) and Max(f(U(t))), solve the definite integral by any numerical integration method, and obtain the approximate value of the dynamic response peak Max(f(U(t)))

The obtained dynamic time-domain response peak approximation value is returned to the main program of topology optimization; Step (5), according to the structural displacement field U(t) obtained in step (3), solve the first-order derivative matrix of the structural displacement field of the attention index of the structural time-domain dynamic topology optimization

Combine f(U(t)), U(t) and Max(f(U(t))) to solve the virtual load history F _λ (t), define the virtual damping matrix of the finite element model in step (1) as - C=-α _c M-β _c K; Step (6), through the numerical solution method or approximate analytical solution method of any differential equation, to solve the virtual structure of the finite element model constructed in step (1) under the virtual load environment and virtual damping situation defined in step (5). Dynamic response, obtain the virtual dynamic response λ(τ) of the structure, and name the virtual dynamic response vector as the adjoint vector; Step (7), according to the displacement field U(t) and velocity field obtained in step (3)

acceleration field

Solve the element node displacement U _e,i (t) and the element node velocity of the i-th element

with element node acceleration

Step (8), according to the accompanying vector λ(t) obtained in step (6), solve the unit node accompanying vector λ _e,i (t) numbered as the i-th unit; Step (9), according to the structural stiffness matrix K obtained in step (1), the mass matrix M, and the damping matrix C obtained in step (2), solve the element stiffness matrix K _e and the element when the pseudo-density of the element is equal to 1. mass matrix _{Me, element damping matrix C e ;} Step (10), U _e,i (t) obtained based on steps (7) to (9),

λ _e,i (t), _{Ke , Me and C e , solve} the definite integral by any numerical integration method, and obtain the approximation value of the dynamic time domain response index

the sensitivity with respect to the pseudo-density x _i numbered in the i-th unit; Step (11), through serial operation or parallel operation, repeat steps (7) to (10) until the sensitivities of the approximation value of the structural dynamics time domain response index with respect to the pseudo-density of all elements are solved, and the structural dynamics peak time domain is completed. Regarding the solution of the response sensitivity of structural topology optimization design variables, the sensitivity of the obtained dynamic time-domain response peak approximation value is returned to the main program of topology optimization. 2. The method for solving the peak time-domain response sensitivity of structural dynamics based on a step function according to claim 1, characterized in that: the method only involves the approximation value of the peak time-domain response index of structural dynamics and its pseudo-values about all units. The density sensitivity solution can be embedded in the gradient solution algorithm for topology optimization of any structure, and the parameters obtained in step (1) can be realized by any structure dynamics solution method and program platform. 3. the structural dynamics peak time-domain response sensitivity solution method based on step function according to claim 1, is characterized in that: the structural displacement field U(t), velocity field in step (3)

with the acceleration field

It needs to be obtained by solving the following differential equations:

The differential equation system in the above formula is a differential equation system with initial conditions and no termination conditions, and its initial condition is U(0)=0

The above formula can be solved by numerical solution method or approximate analytical solution method of any differential equation. 4. The method for solving the sensitivity of structural dynamics peak time-domain response based on step function according to claim 1, characterized in that: the approximation of the dynamic peak time-domain response f(U(t)) in step (4) value

Obtained by solving the definite integral as follows:

where e is the natural exponent, β is χ-0.6,

χ is much larger than

Δt is the integration step size; t _f is the termination time of the dynamic analysis, the above formula can be solved by any numerical integration method. 5. the structural dynamics peak time-domain response sensitivity solution method based on step function according to claim 4, is characterized in that: the virtual load history F _λ (t) in step (5) is solved by following formula

6. the structural dynamics peak time-domain response sensitivity solution method based on step function according to claim 1, is characterized in that: the adjoint vector λ (t) in step (6) needs to obtain by solving following differential equation system

The system of differential equations in the above formula is a system of differential equations without initial conditions but with termination conditions, and its termination condition is λ(t _f )=0

The above formula can be solved by numerical solution method or approximate analytical solution method of any differential equation. 7. The method for solving the sensitivity of structural dynamics peak time-domain response based on step function according to claim 1, characterized in that: the approximation value of the dynamic peak time-domain response in step (10)

The sensitivity of the pseudo-density x _i of the i-th element needs to be obtained by solving the following definite integral:

The above equation can be solved by any numerical integration method.

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