Transient dynamics topological optimization method based on inverse Laplacian transformation
By applying the inverse Laplace transform in the field of transient dynamics, the dynamic transient response equation is solved in the frequency domain, and the existing methods have high computational complexity and low resolution efficiency are solved, achieving efficient and accurate transient response analysis and structural topology optimization.
Patent Information
- Application Number
- CN202510168674.0
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-02-17
- Publication Date
- 2025-03-18
- Estimated Expiration
- Not applicable · inactive patent
AI Technical Summary
The existing transient dynamics solution methods, especially the Newmark integral iteration method, have high computational complexity, expensive calculation costs, and are difficult to implement parallel computing, resulting in low solution efficiency.
The transient dynamic topology optimization method based on the inverse Laplace transform is adopted to solve the dynamic transient response equation in the frequency domain, and the fast Fourier transform and algorithm are used to achieve the time domain solution, and the optimized structure is obtained through structural topology optimization design.
Through frequency domain solution, the iterative accumulation of numerical deviation is avoided, parallel calculation is realized, the calculation speed and solution efficiency are significantly improved, and the stability of the solution process and the accuracy of the results are ensured.
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Abstract
Description
Technical Field
[0001] The present application relates to the technical field of structural topology optimization, and in particular to a transient dynamics topology optimization method based on inverse Laplace transform. Background Art
[0002] Vibration is ubiquitous in production and life, and it may have adverse effects on the normal operation of mechanical equipment and people's living environment. With the advancement of science and technology, vibration control is becoming more and more important. Among the many control methods, passive control is favored because of its low cost and simple operation.
[0003] The research on vibration control can be divided into three categories, namely transient response problems, frequency response problems and eigenvalue response problems. The research on transient response problems is to analyze the dynamic response of the structure under the displacement constraint or force constraint boundary conditions that change with time. For the dynamic equations of transient response problems, the input is the time-related displacement boundary conditions, force boundary conditions, etc., and the output is displacement, stress, strain, etc. The difference between the research on transient response problems and static dynamics is that transient dynamics considers the influence of time-related external loads, damping characteristics, stiffness characteristics, etc. on the analysis results, and introduces time-related variables. Therefore, the solution of transient response problems is much more complicated than static problems. In practical applications, for example, in building and bridge engineering, structures may be affected by dynamic loads such as earthquakes, wind loads or mechanical shocks. Transient response analysis can help engineers understand the dynamic behavior of structures under these sudden loads, including possible oscillations, overshoots and other phenomena. This is crucial for evaluating the stability and durability of structures under extreme conditions, because the transient response of structures is directly related to their safety and reliability in actual use.
[0004] Transient response analysis can adjust the material distribution and improve the structural characteristics through topology optimization methods to achieve the best vibration reduction effect. Topology optimization is a very practical and efficient structural design method. In the process of topology optimization, it relies less on engineering experience and can find the best solution for material distribution under given constraints through mathematical optimization algorithms. Therefore, it is widely used in part design and vibration reduction design.
[0005] Transient dynamic optimization design has always been a key research topic in the field of topology optimization. The most commonly used method for solving dynamic equations is the direct integration method. Direct integration methods include central difference method, Houbolt method, Newmark method, Bathe method, etc. Among them, the Newmark method is a relatively common method. The Newmark integral iteration method uses the displacement, velocity, and acceleration data of the previous time point to solve the displacement, velocity, and acceleration of the next time point, and then repeats this process to obtain the solution of the dynamic equation in the entire time domain. The Newmark integral iteration method is an implicit algorithm. When the parameters are selected within a specific range, the algorithm is unconditionally stable, which makes the Newmark method very suitable for solving dynamic transient problems with a long time. However, in the solution process of the Newmark method, the solution of each time period needs to rely on the calculation results of the previous time period, so the solution process needs to be carried out step by step, and parallel calculation cannot be achieved. In addition, for the solution of transient dynamic equations for a long time period, there will be a large deviation due to the accumulation of the number of iterations. Therefore, for the solution of transient dynamics, the calculation complexity is relatively large and the calculation cost is high. In practical applications, this may lead to a large consumption of computing resources and a reduction in solution efficiency. Summary of the invention
[0006] The purpose of the present application is to provide a transient dynamic topology optimization method based on inverse Laplace transform, and solve the nonlinear transient response equation of the material structure in the frequency domain; the inverse Laplace transform method shows significant advantages in solving transient dynamic problems. It converts the time domain problem into the frequency domain, making it possible to apply the reduced-order model and parallel algorithm, thereby greatly improving the calculation speed. Since the calculation in the frequency domain does not depend on the iteration of the time step, it can not only avoid the numerical deviation caused by the iteration accumulation, but also realize parallel calculation and improve the solution efficiency. In addition, the Laplace transform method is more direct when dealing with initial conditions and complex boundary conditions, and is not affected by the selection of the time step, ensuring the stability of the solution process and the accuracy of the results. Therefore, the Laplace transform method is particularly suitable for large-scale and complex transient dynamic problems, and provides an efficient, accurate and stable solution. Therefore, the present invention compares and analyzes the inverse Laplace method with the most commonly used Newmark integral iteration method; and uses the strategy of differentiation first and then discretization to derive the sensitivity, realizes the frequency domain solution of the time domain sensitivity, and is verified by one or more implementation methods.
[0007] In order to achieve the above purpose, the technical solutions adopted are as follows: The present invention provides a transient dynamics topology optimization method based on inverse Laplace transform, the method comprising: Establish the dynamic transient response equation, and transform it into the frequency domain based on Laplace transform and convolution theorem to obtain the transient response equation in the frequency domain; According to the transient response equation, combined with fast Fourier transform and algorithm, and obtain the solution of the transient response equation in the time domain; Based on the solution of the time-domain transient response equation, the structural topology optimization design is carried out to obtain the topology optimized structure.
[0008] Preferably, in the above-mentioned transient dynamic topology optimization method based on inverse Laplace transform, the dynamic transient response equation is a dynamic equation of a linear dynamic response problem, which is expressed as: ; In the formula, and They represent the mass matrix, damping matrix, and stiffness matrix respectively, R represents the load vector of the external action, and They represent acceleration, velocity, and displacement respectively.
[0009] Preferably, in the above-mentioned transient dynamic topology optimization method based on inverse Laplace transform, based on Laplace transform and convolution theorem, conversion into frequency domain includes: Establish the Laplace transform formula, expressed as: ; in, ; In the formula, is the time domain function, for The image function, represents a frequency variable, represents a positive real constant, e represents the natural base, j represents the imaginary unit, t represents the time variable, A ( s )and B ( s ) are the real and imaginary parts of the image function respectively; right A ( s )and B ( s ) in the time period To discretize, t T express f ( t )’s total action time, we get: ; In the formula, , represents the time point sequence number, N is the time number, t n and t n+1 Respectively represent the time points discretized into N The left and right endpoints of each time interval; Performing Laplace transform on the dynamic transient response equation, we get the transient response equation in the frequency domain, which is expressed as: ; In the formula, Represents external force The Laplace transform of , s represents a complex variable, express The image function.
[0010] Preferably, in the above-mentioned transient dynamic topology optimization method based on inverse Laplace transform, according to the transient response equation, combined with fast Fourier transform and Algorithm to obtain the solution of the time-domain transient response equation, including: The inverse Laplace transform formula is established, expressed as: ; The inverse Laplace transform formula satisfies ,in is a positive real number, is a time domain function f ( t ) of the exponential order, ,and satisfy ,Re[ s ] indicates plural s The real part of The speed of solution is met by fast Fourier transform, and The algorithm is used to ensure the accuracy of the solution and discretize the inverse Laplace transform formula: ; In the formula, is the desired relative error of the solution, is the sampling period of the frequency domain, M is the number of time points calculated, N=2M, T is the sampling period of the time domain, k is a variable composed of (0,1,...N-1), ,F N , , The superscript is based on , , , The resulting vector length. Indicates that the matrix elements are multiplied correspondingly, represents a constant vector of M units composed of c, FFT represents the fast Fourier transform, express algorithm; Solve the transient response equation in the frequency domain , , D( s ) represents the coefficient matrix of the frequency domain equation, , and through the inverse Laplace transform, Inversely transform to the time domain to obtain the solution of the time domain transient response equation.
[0011] Preferably, in the above-mentioned transient dynamic topology optimization method based on inverse Laplace transform, a structural topology optimization design is performed based on the solution of the time domain transient response equation to obtain a topology optimized structure, including: Determine a structural model, apply x- and y-direction displacement constraints on both ends of the structural model, and add a concentrated load of magnitude to the middle position of the upper surface of the structural model; Setting the design domain, mesh number and design variables for topology optimization; wherein the design variable is the cell density; The adjoint method is used to derive the sensitivity formula, and the objective function and viscoelastic dynamics equation are obtained, which are expressed as: ; In the formula, f ( x ) represents the objective function, F represents the external force, T represents the matrix transpose, N C It represents the degree of freedom indicator matrix of the node where the optimization target is located. It is a diagonal matrix. The corresponding degree of freedom number of the node where the optimization target is located is 1, and the rest are 0. Using the augmented Lagrangian function to derive the design variable to obtain a first-order derivative; According to the actual working conditions, boundary conditions are added, the unit stiffness matrix and unit mass matrix are calculated, and the topology optimization model is established by assembling using SIMP interpolation format. The stiffness penalty coefficient in SIMP interpolation is , the established topology optimization model is as follows: ; In the formula, N e Indicates the total number of units, e Indicates the unit number, V represents the total volume, represents the cell density, represents the density variable after penalty, d is the penalty coefficient, represents the target volume, represents the Lagrange multiplier; According to the first-order derivative, the design variables are iteratively updated to complete the optimization design of the structural density and obtain the topological optimization structure under the specified force.
[0012] Preferably, in the above-mentioned transient dynamics topology optimization method based on inverse Laplace transform, the augmented Lagrangian function is expressed as: ; In the formula, f 1 represents the augmented Lagrangian function, f represents the objective function, represents the adjoint vector.
[0013] Preferably, in the above-mentioned transient dynamic topology optimization method based on inverse Laplace transform, the design variable is differentiated by using augmented Lagrangian function to obtain a first-order derivative, which is expressed as: .
[0014] Preferably, in the above-mentioned transient dynamics topology optimization method based on inverse Laplace transform, error analysis is performed on the solution of the time-domain transient response equation.
[0015] Preferably, in the above-mentioned transient dynamic topology optimization method based on inverse Laplace transform, error analysis is performed on the solution of the time domain transient response equation, including: The design domain is set, the finite element model is determined, and a transient response analysis is performed; wherein, when performing the transient response analysis, displacement curves at different time points are obtained, and the displacement transient response curves at the target time points are used as reference displacement curves, and velocity curves and acceleration curves at different time points are obtained, and the velocity transient response curves at the target time points are used as reference curves; Based on the finite element model, transient response analysis is performed using the inverse Laplace transform method and the Newmark method respectively. The solution of the Newmark method at ten times the time point is used as the reference solution. The relative error is obtained between the solution results of the two methods and the reference solution. The relative error comparison is performed on the transient response solution results of the force application point at different time point densities.
[0016] The beneficial effects of this application are: (1) One or more embodiments of the present invention verify the correctness of the numerical Laplace transform algorithm and the numerical inverse Laplace transform algorithm, and use the built-in fast Fourier transform algorithm of MATLAB to improve the transformation speed. By comparing with the existing Newmark method, the feasibility of solving the transient response problem in the frequency domain is verified.
[0017] (2) One or more embodiments of the present invention show through numerical examples that the errors of the two methods are in the same order of magnitude. When the time step is large, the inverse Laplace method is more accurate than the Newmark method, which proves the stability and versatility of the inverse Laplace method.
[0018] (3) One or more embodiments of the present invention realize topology optimization of transient response problems, and use the inverse Laplace method to solve the transient response equation and the frequency domain solution of the adjoint vector. BRIEF DESCRIPTION OF THE DRAWINGS
[0019] Figure 1 A flowchart of a transient dynamics topology optimization method based on inverse Laplace transform provided in an embodiment of the present application; Figure 2 A schematic diagram of a design domain model provided for an embodiment of the present application; Figure 3 Reference response curves of displacement, velocity, and acceleration of the action point obtained by the Newmark method provided in the embodiment of the present application; wherein, (a) is a displacement reference response curve; (b) is a velocity reference response curve; (c) is an acceleration reference response curve; Figure 4 Comparison diagram of response curves using the Newmark method at different time points provided in the embodiments of the present application; wherein (a) is the displacement response curve of the Newmark method; (b) is the velocity response curve of the Newmark method; (c) is the acceleration response curve of the Newmark method; Figure 5 Response curves of the inverse Laplace method at different time points provided in the embodiments of the present application; wherein, (a) the displacement response curve of the inverse Laplace method; (b) the velocity response curve of the inverse Laplace method; (c) the acceleration response curve of the inverse Laplace method; Figure 6 The relative error diagram of the action point using the Newmark method at different time points provided in the embodiment of the present application; wherein, (a) the relative error of displacement using the Newmark method at different time points; (b) the relative error of velocity using the Newmark method at different time points; (c) the relative error of acceleration using the Newmark method at different time points; Figure 7 A graph of relative errors of the points of action using the inverse Laplace method at different time points provided in the embodiment of the present application; wherein (a) the relative error of displacement using the inverse Laplace method at different time points; (b) the relative error of velocity using the inverse Laplace method at different time points; (c) the relative error of acceleration using the inverse Laplace method at different time points; Figure 8 A topology optimization model diagram provided for an embodiment of the present application; Fig. 9 A topology optimization structure diagram obtained by using the Newmark method and the inverse Laplace method provided in the embodiment of the present application; Fig.10 The present invention provides a displacement velocity acceleration response diagram obtained by using the Newmark method and the inverse Laplace method in an embodiment of the present invention; wherein, (a) is a displacement response diagram obtained by using the Newmark method and the inverse Laplace method; (b) is a velocity response diagram obtained by using the Newmark method and the inverse Laplace method; (c) is an acceleration response diagram obtained by using the Newmark method and the inverse Laplace method. DETAILED DESCRIPTION
[0020] The following describes the embodiments of the present application through specific examples, and those skilled in the art can easily understand other advantages and effects of the present application from the contents disclosed in this specification. The present application can also be implemented or applied through other different specific embodiments, and the details in this specification can also be modified or changed in various ways based on different viewpoints and applications without departing from the spirit of the present application. It should be noted that the following embodiments and features in the embodiments can be combined with each other without conflict.
[0021] The specific implementation of the present application is further described in detail below in conjunction with the drawings and examples.
[0022] See also Figure 1 , which is a flow chart of a transient dynamic topology optimization method based on inverse Laplace transform provided in an embodiment of the present application. An embodiment of the present application provides a transient dynamic topology optimization method based on inverse Laplace transform, and the method includes the following steps S1 to S3.
[0023] S1: Establish the dynamic transient response equation, and transform it into the frequency domain based on Laplace transform and convolution theorem to obtain the transient response equation in the frequency domain.
[0024] In an exemplary embodiment, step S1 specifically includes: S11: Establish the kinetic equation for the linear dynamic response problem, expressed as: ; In the formula, and They represent the mass matrix, damping matrix, and stiffness matrix respectively, R represents the load vector of the external action, and They represent acceleration, velocity, and displacement respectively.
[0025] S12: Establishing the mathematical formula for Laplace transform ,make , which can be expressed as , we get the Laplace transform formula, which can be expressed as ; in, ; In the formula, is a time domain function, for The image function, represents a frequency variable, represents a positive real constant, e represents the natural base, j represents the imaginary unit, t represents the time variable, A ( s )and B ( s ) are the real and imaginary parts of the image function respectively; right A ( s )and B ( s ) in the time period To discretize, t T express f ( t )’s total action time, we get: ; In the formula, , indicating the time point sequence number, N is the time number, t n and t n+1 Respectively represent the time points discretized into N The left and right endpoints of each time interval; S13: Perform Laplace transform on the transient response equation to obtain the transient response equation in the frequency domain: ; In the formula, Represents external force The Laplace transform of , similarly .
[0026] S2: According to the transient response equation, combined with fast Fourier transform and The algorithm is used to obtain the solution of the time-domain transient response equation.
[0027] In an exemplary embodiment, step S2 specifically includes: S21: Establish the inverse Laplace transform formula, expressed as: ; The inverse Laplace transform formula satisfies ,in is a positive real number, is a time domain function f ( t ) of the exponential order, ,and satisfy ,Re[ s ] indicates plural s The real part of S22: The inverse Laplace transform formula is discretized by using the fast Fourier transform to meet the speed of solution and the ε algorithm to ensure the accuracy of solution: ; In the formula, is the desired relative error of the solution, is the sampling period of the frequency domain, M is the number of time points calculated, N=2M, T is the sampling period of the time domain, k is a variable composed of (0,1,...N-1), ,F N , , The superscript is based on , , , The resulting vector length. Indicates that the matrix elements are multiplied correspondingly, represents a constant vector of M units composed of c, FFT represents the fast Fourier transform, express algorithm; S23: Solve the frequency domain solution of the transient response equation , , D(s) represents the coefficient matrix of the frequency domain equation, , and through the inverse Laplace transform, Inversely transform to the time domain to obtain the solution of the time domain transient response equation.
[0028] S3: Based on the solution of the time-domain transient response equation, the structural topology optimization design is carried out to obtain the topology optimized structure.
[0029] In an exemplary embodiment, step S3 specifically includes: S31: Determine a structural model, apply x- and y-direction displacement constraints to both ends of the structural model, and add a concentrated load of a magnitude to the middle position of the upper surface of the structural model.
[0030] In this embodiment, taking the two-end clamped beam as an example, its material is divided into material 1 (elastic modulus ) and Material 2 (elastic modulus ). After design, the material usage of material 1 is 40%, displacement constraints in the x and y directions are applied at both ends, and a size of N concentrated load, design domain and boundary conditions are as follows Figure 8 shown.
[0031] S32: Setting the design domain, number of grids and design variables for topology optimization; wherein the design variable is unit density.
[0032] In this embodiment, the design domain of topology optimization is defined as The number of grids is 200*50, and the cell density is defined is the design variable.
[0033] S33: The sensitivity formula is derived using the adjoint method to obtain the objective function and viscoelastic dynamics equation, which are expressed as: ; In the formula, f ( x ) represents the objective function, F represents the external force, T represents the matrix transpose, N C It represents the degree of freedom indicator matrix of the node where the optimization target is located. It is a diagonal matrix. The corresponding degree of freedom number of the node where the optimization target is located is 1, and the rest are 0. S34: using the augmented Lagrangian function to derive the design variable to obtain a first-order derivative.
[0034] In this embodiment, the augmented Lagrangian function is expressed as: ; In the formula, f 1 represents the augmented Lagrangian function, f represents the objective function, represents the adjoint vector.
[0035] The augmented Lagrangian function is used to differentiate the design variables to obtain the first-order derivative, which is expressed as: .
[0036] S35: Add boundary conditions according to the actual working conditions, calculate the unit stiffness matrix and unit mass matrix, and assemble and establish the topology optimization model using the SIMP interpolation format. The stiffness penalty coefficient in the SIMP interpolation is , the established topology optimization model is as follows: ; In the formula, N e Indicates the total number of units, e Indicates the unit number, V represents the total volume, represents the cell density, represents the density variable after penalty, d is the penalty coefficient, represents the target volume, represents the Lagrange multiplier; S36: According to the first-order derivative, the design variables are iteratively updated to complete the optimization design of the structural density and obtain a topologically optimized structure under a specified force.
[0037] In some embodiments, the transient dynamic topology optimization method based on inverse Laplace transform further includes step S4: performing error analysis on the solution of the time-domain transient response equation. Step S4 is an error analysis step, and the execution of step S4 can be between step S2 and step S3, or after step S3, which is not specifically limited in this embodiment.
[0038] In an exemplary embodiment, step S4 specifically includes: S41: setting the design domain, determining the finite element model, and performing transient response analysis; wherein, when performing the transient response analysis, obtaining displacement curves at different numbers of time points, and using the displacement transient response curves at the target number of time points as reference displacement curves, obtaining velocity curves and acceleration curves at different numbers of time points, and using the velocity transient response curves at the target number of time points as reference curves.
[0039] In this embodiment, the design domain of topology optimization is defined as A finite element model with a mesh size of 100*25 is selected. The design domain and boundary conditions are as follows Figure 2 As shown. For transient response analysis, the displacement curves when the number of time points is 4, 8, 16, 32 and 512 are taken, and the displacement transient response curve when the number of time points is 512 is used as a reference. The velocity and acceleration curves when the number of time points is 4, 8, 16, 32, 128 and 512 are taken, and the velocity transient response curve when the number of time points is 512 is taken as a reference.
[0040] S42: Based on the finite element model, transient response analysis is performed using the inverse Laplace transform method and the Newmark method respectively. The solution of the Newmark method at ten times the time point is used as a reference solution. The relative error between the solution results of the two methods and the reference solution is obtained, and the relative error of the transient response solution results of the force application point at different time point densities is compared.
[0041] In this embodiment, the model constructed in step S41 is used to perform transient response analysis on the structure using the inverse Laplace method and the Newmark method respectively, and the solution of the Newmark method at ten times the time point is used as the reference solution. The relative error is obtained by comparing the solution results of the two methods with the reference solution, and the relative error of the transient response solution results of the force application point at different time point densities is compared.
[0042] The following examples will further illustrate the feasibility and progress of the present application in combination with specific experimental comparison data.
[0043] It should be noted that the Newmark method is an existing method and will not be described in detail in this embodiment. The inverse Laplace method is a method provided in the embodiment of this application.
[0044] like Figure 3 As shown in the figure, the reference response curves of displacement, velocity and acceleration of the action point obtained by the Newmark method are shown in the figure. Figure 4 As shown in the figure, the response curve of the Newmark method with different time points is shown. Figure 5 As shown in the figure, the response curves of the inverse Laplace method with different time points are shown. Figure 6 As shown in the figure, the relative error diagram of the action point using the Newmark method and the inverse Laplace method at different time points. Figure 7 As shown in the figure, the relative error diagram of the action point using the inverse Laplace method at different time points. Fig. 9 As shown in the figure, the topology optimization structure diagram is obtained by using the Newmark method and the inverse Laplace method. Fig.10 As shown in Figure 1, the displacement velocity acceleration response diagram obtained by using the Newmark method and the inverse Laplace method. Figures 3 to 7 as well as Fig. 9 and Fig.10 ,The numerical examples show that the errors of the two methods are in the same order of magnitude. When the time step is large, the inverse Laplace method is more accurate than the Newmark method, which proves the stability and versatility of the inverse Laplace method.
[0045] The above implementation modes are only used to illustrate the present application, and are not intended to limit the present application. Ordinary technicians in the relevant technical field may make various changes and modifications without departing from the spirit and scope of the present application. Therefore, all equivalent technical solutions also belong to the scope of the present application, and the scope of patent protection of the present application shall be limited by the claims.
Claims
1. A transient dynamic topology optimization method based on inverse Laplace transform, characterized in that: The method comprises: Establish the dynamic transient response equation, and transform it into the frequency domain based on Laplace transform and convolution theorem to obtain the transient response equation in the frequency domain; According to the transient response equation, combined with fast Fourier transform and algorithm, and obtain the solution of the time-domain transient response equation; Based on the solution of the time-domain transient response equation, the structural topology optimization design is carried out to obtain the topology optimized structure.
2. The transient dynamic topology optimization method based on inverse Laplace transform according to claim 1, characterized in that: The dynamic transient response equation is a dynamic equation for a linear dynamic response problem, which is expressed as: ； In the formula, and They represent the mass matrix, damping matrix, and stiffness matrix respectively, R represents the load vector of the external action, and They represent acceleration, velocity, and displacement respectively.
3. The transient dynamic topology optimization method based on inverse Laplace transform according to claim 2, characterized in that: Based on Laplace transform and convolution theorem, conversion to frequency domain includes: Establish the Laplace transform formula, expressed as: ； in, ; In the formula, is a time domain function, for The image function, represents a frequency variable, represents a positive real constant, e represents the natural base, j represents the imaginary unit, t represents the time variable, A ( s )and B ( s ) are the real and imaginary parts of the image function respectively; right A ( s )and B ( s ) in the time period To discretize, t T express f ( t )’s total action time, we get: ； In the formula, , represents the time point sequence number, N is the time number, t n and t n+1 Respectively represent the time points discretized into N The left and right endpoints of each time interval; Performing Laplace transform on the dynamic transient response equation, we get the transient response equation in the frequency domain, which is expressed as: ； In the formula, Represents external force The Laplace transform of , s represents a complex variable, express The image function.
4. The transient dynamic topology optimization method based on inverse Laplace transform according to claim 3, characterized in that: According to the transient response equation, combined with fast Fourier transform and Algorithm to obtain the solution of the time-domain transient response equation, including: The inverse Laplace transform formula is established, expressed as: ； The inverse Laplace transform formula satisfies ,in is a positive real number, is a time domain function f ( t ) of the exponential order, ,and satisfy ,Re[ s ] indicates plural s The real part of The speed of solution is met by fast Fourier transform, and The algorithm is used to ensure the accuracy of the solution and discretize the inverse Laplace transform formula: ； In the formula, is the desired relative error of the solution, is the sampling period of the frequency domain, M is the number of time points calculated, N=2M, T is the sampling period of the time domain, k is a variable composed of (0,1,...N-1), ,F N , , The superscript is based on , , , The resulting vector length is Indicates that the matrix elements are multiplied correspondingly, represents a constant vector of M units composed of c, FFT represents the fast Fourier transform, express algorithm; Solve the transient response equation in the frequency domain , , where D( s ) is the coefficient matrix of the frequency domain equation, , and through the inverse Laplace transform, Inversely transform to the time domain to obtain the solution of the time domain transient response equation.
5. The transient dynamic topology optimization method based on inverse Laplace transform according to claim 4, characterized in that: Based on the solution of the time-domain transient response equation, the structural topology optimization design is carried out to obtain the topology optimized structure, including: Determine a structural model, apply x- and y-direction displacement constraints on both ends of the structural model, and add a concentrated load of magnitude to the middle position of the upper surface of the structural model; Setting the design domain, mesh number and design variables for topology optimization; wherein the design variable is the cell density; The adjoint method is used to derive the sensitivity formula, and the objective function and viscoelastic dynamics equation are obtained, which are expressed as: ； In the formula, f ( x ) represents the objective function, F represents the external force, T represents the matrix transpose, N C It represents the degree of freedom indicator matrix of the node where the optimization target is located. It is a diagonal matrix. The corresponding degree of freedom number of the node where the optimization target is located is 1, and the rest are 0. Using the augmented Lagrangian function to derive the design variable to obtain a first-order derivative; According to the actual working conditions, boundary conditions are added, the unit stiffness matrix and unit mass matrix are calculated, and the topology optimization model is established by assembling using SIMP interpolation format. The stiffness penalty coefficient in SIMP interpolation is , the established topology optimization model is as follows: ； In the formula, N e Indicates the total number of units, e Indicates the unit number, V represents the total volume, represents the cell density, represents the density variable after penalty, d is the penalty coefficient, represents the target volume, represents the Lagrange multiplier; According to the first-order derivative, the design variables are iteratively updated to complete the optimization design of the structural density and obtain the topological optimization structure under the specified force.
6. The transient dynamic topology optimization method based on inverse Laplace transform according to claim 5, characterized in that: The augmented Lagrangian function is expressed as: ； In the formula, f 1 represents the augmented Lagrangian function, f represents the objective function, represents the adjoint vector.
7. The transient dynamic topology optimization method based on inverse Laplace transform according to claim 6, characterized in that: The augmented Lagrangian function is used to differentiate the design variables to obtain the first-order derivative, which is expressed as: 。 8. The transient dynamic topology optimization method based on inverse Laplace transform according to any one of claims 1 to 7, characterized in that: Perform error analysis on the solutions of the time-domain transient response equations.
9. The transient dynamic topology optimization method based on inverse Laplace transform according to claim 8, characterized in that: Perform error analysis on the solutions to the time-domain transient response equations, including: The design domain is set, the finite element model is determined, and a transient response analysis is performed; wherein, when performing the transient response analysis, displacement curves at different time points are obtained, and the displacement transient response curves at the target time points are used as reference displacement curves, and velocity curves and acceleration curves at different time points are obtained, and the velocity transient response curves at the target time points are used as reference curves; Based on the finite element model, transient response analysis was performed using the inverse Laplace transform method and the Newmark method respectively. The solution of the Newmark method at ten times the time point was used as the reference solution. The relative error was obtained between the solution results of the two methods and the reference solution. The relative error of the transient response solution results of the force application point at different time point densities was compared.