CN111539138B - Method for solving time domain response sensitivity of structural dynamics peak based on step function - Google Patents

Abstract

The invention relates to a method for solving the time-domain response sensitivity of structural dynamics based on step functions, including a time-domain response peak approximation method and an approximation function sensitivity solution method; the time-domain response peak value is determined by calculating the step value of the time-domain response history as an independent variable. The function is integrated over time to obtain its approximation value; the sensitivity of the approximation function is achieved through 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 great influence. It has a simple structure, easy programming and implementation, and significantly improves the optimization efficiency.

Description

Structural dynamics peak time domain response sensitivity solution method based on step function

Technical field

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

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 structural static topology optimization has become increasingly mature, but the static working conditions cannot cover all application scenarios in the complete life cycle of the structure. Therefore, structural dynamics topology optimization has become a research hotspot. According to the time and frequency characteristics of design indicators, structural dynamics topology optimization is divided into two branches: frequency domain and time domain. Frequency domain topology optimization can avoid the large-scale calculation consumption of dynamic analysis and time domain history sensitivity solution, but it can only Maximizing the dynamic stiffness of the structure cannot establish an intuitive connection with the real numerical value of the structural dynamic response time history.

Dynamic time-domain topology optimization starts from the evaluation of response time history and can directly optimize specific indicators of the response history. Currently, there are two types of response history evaluation indicators: integral and peak. The integral can evaluate the total amount of vibration, but it also cannot establish an intuitive connection with the real numerical value of the structural dynamic response time history. The peak is the most representative value that can characterize the structural dynamic response characteristics. Intuitive indicators. Currently, peak dynamics topology optimization often uses clustering functions to characterize peak agents. The strong correlation between function parameters leads to a great subjective impact on the optimization results.

Contents of the invention

The technical problem solved by this invention is: in order to overcome the shortcomings of the existing technology, a method for solving the peak time domain response sensitivity of structural dynamics based on a step function is proposed. The time domain response peak approximation equation and the approximation equation based on the step function are related to the design It consists of a variable sensitivity solution method to solve the problem that the peak time domain index of the existing structural time domain dynamics topology optimization is greatly affected by subjective parameters and has difficulty in convergence.

The technical solution of the present invention is:

A method for solving the peak time domain response sensitivity of structural dynamics based on step functions. The steps of this method include:

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

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, α _c and β _c are proportional damping coefficients;

Step (3), use the numerical solution method of any differential equation or the approximate analytical solution method 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), Obtain the structural displacement field U(t) and velocity field Acceleration field/> t is time;

Step (4), based on the structural displacement field U(t) obtained in step (3), establish the index of concern f(U(t)) for structural time-domain dynamics topology optimization, combining f(U(t)), U (t) and Max(f(U(t))), solve the definite integral through any numerical integration method to obtain the approximate value of the dynamic response peak Max(f(U(t))) The peak approximation value of the obtained dynamic time domain response is returned to the main program of topology optimization;

Step (5). Based on the structural displacement field U(t) obtained in step (3), solve the first-order derivative matrix of the structural time-domain dynamics topology optimization concern index with respect to the structural displacement field. Combining f(U(t)), U(t) and Max(f(U(t))), solve the virtual load history F _λ (t), and define the virtual damping matrix of the finite element model in step (1) as - C=-α _c M-β _c K;

Step (6): Use the numerical solution method of any differential equation or the approximate analytical solution method to solve the virtual structure of the finite element model constructed in step (1) under the virtual load environment and virtual damping 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), the displacement field U(t) and velocity field obtained according to step (3) Acceleration field/> Solve for the unit node displacement U _e,i (t) and unit node velocity of the i-th unit/> and unit node acceleration/>

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

Step (9), based on the structural stiffness matrix K, mass matrix M, and damping matrix C obtained in step (1), solve the unit stiffness matrix K _{e and} unit when the unit pseudo-density is equal to 1 Mass matrix _Me , unit damping matrix C _e ;

Step (10), based on U _e,i (t) obtained from steps (7) to (9), λ _e,i (t), K _e , M _e and C _e , solve the definite integral through any numerical integration method, and obtain the approximate value of the dynamic time domain response index/> Regarding the sensitivity of the pseudo density x _i numbered as the i-th unit;

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

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

Further, the structural displacement field U(t) and velocity field in step (3) and acceleration field/> It needs to be obtained by solving the following system of differential equations:

The system of differential equations in the above formula is a system of differential equations with initial conditions and no termination conditions, and its initial condition is U(0)=0 The above equation can be solved by numerical solution method or approximate analytical solution method of any differential equation.

Further, the approximate value of the dynamic peak time domain response f(U(t)) in step (4) It is obtained by solving the definite integral of the following equation:

where e is the natural index, β is χ-0.6, χ is much larger than/> is a positive real number; Δt is the integration step size; t _f is the termination moment of the dynamic analysis. The above equation 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:

The system of differential equations in the above formula is a system of differential equations without initial conditions but with termination conditions. The termination condition is λ(t _f )=0 The above equation can be solved by numerical solution method or approximate analytical solution method of any differential equation.

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 unit needs to be obtained by solving the following definite integral:

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

Compared with the prior art, the beneficial effects of the present invention are:

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

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

(3) The present invention does not involve dynamics solution steps, but only involves the approximation value of the structural dynamics peak time domain response index and its sensitivity solution regarding the pseudo-density of all units, and can be embedded in the gradient solution algorithm of any structural topology optimization;

(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 does not depend strongly on the response history after the peak response occurs, and can reduce the response time length that is focused on in the dynamics and sensitivity calculations. , reduce optimization time;

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

Description of the drawings

Figure 1 is a flow chart of the present invention;

Figure 2 shows the structural dimensions, loads and constraint environment in the embodiment of the present invention;

Figure 3 is the time-varying load history of the structure in the embodiment of the present invention;

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

Figure 5 is the optimization results of the structural time-domain dynamics topology optimization embedded in the present invention when the termination times are 0.12s, 0.15s, 0.18s and 0.58s respectively;

Figure 6 is the convergence process of the structural time-domain dynamics topology optimization embedded in the present invention when the termination times are 0.12s, 0.15s, 0.18s and 0.58s respectively;

Figure 7 is a comparison of the dynamic responses between the optimal solution of time domain dynamic topology optimization and the optimal solution of static topology optimization embedded in the structure of the present invention.

Detailed ways

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

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

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

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, α _c and β _c are proportional damping coefficients;

Step (3), obtain the structural displacement field U(t) and velocity field Acceleration field/> It is necessary 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, solve the following system of differential equations

The system of differential equations in the above formula is a system of differential equations with initial conditions and no termination conditions, and its initial condition is U(0)=0 The above equation can be solved by numerical solution method or approximate analytical solution method of any differential equation;

Step (4), based on the structural displacement field U(t) obtained in step (3), establish the index of concern f(U(t)) for structural time-domain dynamics topology optimization, combining f(U(t)), U (t) and Max(f(U(t))), the approximate value of the dynamic response peak Max(f(U(t))) It can be obtained by solving the following equation

where e is the natural index, 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 moment of the dynamic analysis. The above equation can be performed by any numerical integration method, and the approximate value of the peak value of the obtained dynamic time domain response 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 time-domain dynamics topology optimization concern index with respect to the structural displacement field. 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, define the virtual damping matrix of the finite element model in step (1) 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 when the finite element model constructed in step (1) is under the virtual load environment and virtual damping defined in step (5), that is, solve The following system of differential equations

The system of differential equations in the above formula is a system of differential equations without initial conditions but with termination conditions. The termination condition is λ(t _f )=0 The above equation can be solved by numerical solution method or approximate analytical solution method of any differential equation;

Step (7), the displacement field U(t) and velocity field obtained according to step (3) Acceleration field/> Extract the unit node displacement U _e,i (t) and unit node velocity of the i-th unit/> and unit node acceleration/>

Step (8): According to the adjoint vector λ(t) obtained in step (6), extract the unit node adjoint vector λ _e,i (t) numbered as the i-th unit;

Step (9), based on the structural stiffness matrix K, mass matrix M, and damping matrix C obtained in step (1), solve the unit stiffness matrix K _{e and} unit when the unit pseudo-density is equal to 1 Mass matrix _Me , unit damping matrix C _e ;

Step (10), based on U _e,i (t) obtained from steps (7) to (9), λ _e,i (t), K _e , M _e and C _e , then the approximate value of the dynamic peak time domain response/> The sensitivity of the pseudo-density x _i of the i-th unit needs to be obtained by solving the following definite integral

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

Step (11), through serial operation or parallel operation, repeat steps (7) to (10) until the structural dynamics time domain response index approximation value with respect to the sensitivity of all unit pseudo-densities is solved, and the obtained dynamic time domain response peak value is approximated The value of sensitivity is returned to the topology optimization main program.

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 into the continuum structural dynamics time domain topology optimization, and uses the gradient algorithm to optimize the movement The asymptote method MMA, the pseudo-density interpolation model adopts the SIMP model of p=6, the optimization goal is to minimize the peak value of the time domain response, and the optimization constraint is that the area ratio is not less than 0.3. The method proposed by the present invention is embedded in the topology optimization main program framework Used to solve the optimization objective and optimization objective sensitivity. The specific values of parameters in each real-time step are as follows:

The design domain in step (1) is shown in Figure 2. The design domain size is 100mm×70mm. The structural finite element model has 100 units in the longitudinal direction and 70 units in the transverse direction. The unit numbers in both the longitudinal and transverse directions start from the upper left corner; Yang's model The mass is 2×10 ⁵ MPa, the density is 7.8×10 ^-6 Kg/mm3, the Poisson’s ratio is 0.3, and the unit type adopts planar bilinear unit.

The proportional damping coefficients α _C =10 and β _C =1×10 ^-5 in step (2); the structure is constrained at the two lower end points, and the structure is restrained along the X direction at the upper right corner node of the 17th unit in the first row. There are two time-varying loads in the Y direction, and 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.

Step (3) uses the New Mark-β method to solve the dynamic differential equation, and the calculation step size Δt is 0.001s. In order to verify that the present invention is not strongly dependent on the analysis time, 0.12s, 0.15s, and 0.18 were performed in this example. Dynamic analysis and corresponding topology optimization under four sets of termination times: s and 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 loaded node of the structure, that is Among them/> is the output vector, which is a column vector with the same number of degrees of freedom. The corresponding position of the translational degree of freedom in the X direction of the loaded node is 1, and other positions are 0. is the output vector, which is a column vector with the same number of degrees of freedom. The corresponding position of the translational degree of freedom in the Y direction of the loaded node is 1, and other positions are 0; β is 20, χ is 10000, then/> The definite integral is solved using the trapezoidal method, corresponding to step (3), the integration step size Δt is 0.001s, and t _f are 0.12s, 0.15s, 0.18s and 0.58s respectively.

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

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

Step (7) does not require parameter input, and is extracted according to the logic and sequence of unit and node numbers;

Step (8) does not require parameter input, and is extracted according to the logic and sequence of unit and node numbers;

Step (9) does not require parameter input, and the unit type adopts planar bilinear unit.

The definite integral in step (10) is solved using the trapezoidal method, which corresponds to step (3). The integration step size Δt is 0.001s, and t _f are 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 solved and returned to the main program of topology optimization.

After all the steps of the present invention are completed, the approximate value of the time domain response peak value calculated in step (4) and the sensitivity of the approximate value of the time domain response peak value calculated in step (11) are returned to the topology optimization group program. By moving The asymptote algorithm MMA adjusts the pseudo-density distribution in the next step. After the adjustment, the real-time step of the present invention is carried out again, and steps (1) to (11) are repeated until the structural time-domain dynamics topology optimization converges.

In order to be able to verify that the approximation method of the time domain response peak and the corresponding sensitivity solution method proposed by the present invention can effectively operate 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 ambient load and Y-direction static load are the peak values of the load history in step (2). The other constraint environments and optimization algorithms are consistent with the structural dynamics topology optimization.

By comparing Figure 4 and Figure 5, it can be seen that there is a difference between the topological configuration of the optimal solution of static topology optimization and the optimal solution of dynamic topology optimization, and the optimal configuration of the structural dynamic topology optimization solution based on the present invention is different. Dependent on the duration of the dynamics analysis, it can also be seen from Figure 6 that no matter how long the dynamics analysis is, the optimization can converge quickly, and the convergence values are exactly the same. It can be seen from Figure 7 that the dynamic response peak value of the optimal configuration of static topology optimization is greater 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 very effectively embedded in the gradient topology optimization method, and can effectively reduce the peak time domain response.

The present invention adopts the peak approximation method of the time domain response peak based on step function integration, and when constructing the peak approximation function, the approximation error is reduced through the reference value translation strategy, which can be applied to situations where the integration step size is large. Improve optimization speed.

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

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 use the methods and techniques disclosed above to make technical solutions to 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 based on the technical essence of the present invention without departing from the content of the technical solution of the present invention shall fall within the protection scope of the technical solution of the present invention.

Claims (7)

1. A method for solving the peak time domain response sensitivity of structural dynamics based on step functions, which is characterized in that the steps of this method include: Step (1), initialize the optimization model, use the structural design domain as the geometric boundary, establish a finite element analysis model, multiply the elastic modulus of the i-th unit by the pth power of the pseudo density x _i , where x _i ∈[ 0,1], p=3, the pseudo-densities x _i of all units constitute the design variable vector x, and the structural stiffness matrix K and mass matrix M are extracted according to the finite element model to determine the constrained degrees of freedom; 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, α _c and β _c are proportional damping coefficients; Step (3), use the numerical solution method of any differential equation or the approximate analytical solution method 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), Obtain the structural displacement field U(t) and velocity field Acceleration field/> t is time; Step (4), based on the structural displacement field U(t) obtained in step (3), establish the index of concern f(U(t)) for structural time-domain dynamics topology optimization, combining f(U(t)), U (t) and Max(f(U(t))), solve the definite integral through any numerical integration method to obtain the approximate value of the dynamic response peak Max(f(U(t))) The peak approximation value of the obtained dynamic time domain response is returned to the main program of topology optimization; Step (5). Based on the structural displacement field U(t) obtained in step (3), solve the first-order derivative matrix of the structural time-domain dynamics topology optimization concern index with respect to the structural displacement field. Combining f(U(t)), U(t) and Max(f(U(t))), solve the virtual load history F _λ (t), and define the virtual damping matrix of the finite element model in step (1) as - C=-α _c M-β _c K; Step (6): Use the numerical solution method of any differential equation or the approximate analytical solution method to solve the virtual structure of the finite element model constructed in step (1) under the virtual load environment and virtual damping 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), the displacement field U(t) and velocity field obtained according to step (3) Acceleration field/> Solve for the unit node displacement U _e,i (t) and unit node velocity of the i-th unit/> and unit node acceleration/> Step (8): According to the adjoint vector λ(t) obtained in step (6), solve the unit node adjoint vector λ _e,i (t) numbered as the i-th unit; Step (9), based on the structural stiffness matrix K, mass matrix M, and damping matrix C obtained in step (1), solve the unit stiffness matrix K _{e and} unit when the unit pseudo-density is equal to 1 Mass matrix _Me , unit damping matrix C _e ; Step (10), based on U _e,i (t) obtained from steps (7) to (9), λ _e,i (t), K _e , M _e and C _e , solve the definite integral through any numerical integration method, and obtain the approximate value of the dynamic time domain response index/> Regarding the sensitivity of the pseudo density x _i numbered as the i-th unit; Step (11), through serial operation or parallel operation, repeat steps (7) to (10) until the structural dynamics time domain response index approximation value with respect to the sensitivity of all unit pseudo-densities is solved, and the structural dynamics peak time domain is completed. Regarding the solution of the response sensitivity of the design variables in the structural topology optimization, the sensitivity of the obtained dynamic time domain response peak approximation value is returned to the main program of the topology optimization. 2. The method for solving the peak time domain response sensitivity of structural dynamics based on the 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 for all units. The sensitivity solution of density can be embedded in the gradient solution algorithm of any structural topology optimization. The parameters obtained in step (1) can be realized through any structural dynamics solution method and program platform. 3. The method for solving peak time domain response sensitivity of structural dynamics based on step function according to claim 1, characterized in that: the structural displacement field U(t), velocity field in step (3) and acceleration field/> It needs to be obtained by solving the following system of differential equations: The system of differential equations in the above formula is a system of differential equations with initial conditions and no termination conditions, and its initial condition is U(0)=0 The above equation can be solved by numerical solution method or approximate analytical solution method of any differential equation. 4. The step function-based structural dynamics peak time domain response sensitivity solution method according to claim 1, characterized in that: the approximation of the dynamic peak time domain response f(U(t)) in step (4) value It is obtained by solving the definite integral of the following equation: where e is the natural index, β is χ-0.6, χ is much larger than/> is a positive real number; Δt is the integration step size; t _f is the termination moment of the dynamic analysis. The above equation can be solved by any numerical integration method. 5. The method for solving the peak time domain response sensitivity of structural dynamics based on step functions according to claim 4, characterized in that: the virtual load history F _λ (t) in step (5) is solved by the following formula 6. The method for solving the peak time domain response sensitivity of structural dynamics based on step functions according to claim 1, characterized in that: the adjoint vector λ(t) in step (6) needs to be obtained by solving the following differential equations The system of differential equations in the above formula is a system of differential equations without initial conditions but with termination conditions. The termination condition is λ(t _f )=0 The above equation can be solved by numerical solution method or approximate analytical solution method of any differential equation. 7. The method for solving the sensitivity of the structural dynamics peak time domain response based on the 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 unit needs to be obtained by solving the following definite integral: The above equation can be solved by any numerical integration method.