CN111539138B - Method for solving time domain response sensitivity of structural dynamics peak based on step function - Google Patents
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Abstract
The invention relates to a method for solving the sensitivity of a structural dynamics time domain response based on a step function, which comprises a time domain response peak value approximation method and an approximation function sensitivity solving method; the time domain response peak value obtains an approximation value by time integration of a step function taking time domain response history as an independent variable; the approximation function sensitivity is achieved by the adjoint vector method. The invention solves the problem that the subjective parameter selection of the peak value approximation method in the prior structure dynamics time domain topology optimization has large influence, has simple structure and easy programming, is easy to realize, and obviously improves the optimization efficiency.
Description
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 sensitivity solving method of a peak value approximation value related to variables.
Background
The structural optimization design is divided into three stages of conceptual design, shape design and parameter design, wherein the most important stage is the conceptual design stage, the conceptual design determines the basic configuration of the structure, and the topological optimization is a tool widely applied in the conceptual design stage. Currently, structural statics topology optimization research is mature day by day, but statics working conditions cannot cover all application scenes in the whole life cycle of the structure, so structural dynamics topology optimization is a research hotspot at present. According to the time-frequency characteristics of design indexes, structural dynamics topological optimization is divided into two branches of a frequency domain and a time domain, the frequency domain topological optimization can avoid large-scale calculation consumption of dynamic analysis and time course sensitivity solution, but can only realize structural dynamic stiffness maximization, and cannot establish visual connection with the real numerical value of structural dynamics response time course.
Dynamic time domain topological optimization takes evaluation of response time histories as a research starting point, and can directly optimize specific indexes of the response histories. At present, response process evaluation indexes comprise an integral and a peak value, the integral can evaluate the total quantity of vibration, and also cannot establish visual connection with the real numerical value of the structure dynamics response time process, the peak value is the most visual index capable of representing the structure dynamics response characteristic, the peak value agent representation is usually carried out by a clustering function method in the current peak value dynamics topology optimization, and the influence of the subjective effect on the optimization result is large due to strong correlation of function parameters.
Disclosure of Invention
The invention solves the technical problems that: in order to overcome the defects of the prior art, a method for solving the time domain response sensitivity of the structural dynamics peak value based on a step function is provided, the method consists of a time domain response peak value approximation equation based on the step function and a sensitivity solving method of the approximation equation about design variables, and the problems that the peak time domain index of the prior structural time domain dynamics topology optimization is greatly influenced by subjective parameters and is difficult to converge are solved.
The solution of the invention is as follows:
the method for solving the time domain response sensitivity of the structural dynamics peak value based on the step function comprises the following steps:
initializing an optimization model, taking a structural design domain as a geometric boundary, establishing a finite element analysis model, and multiplying the elastic modulus of an ith element with the number of the ith element by the pseudo density x i To the power of p, where x i ∈[0,1]P=3, pseudo density x of all cells i Forming a design variable vector x, extracting a structural rigidity matrix K and a quality matrix M according to the finite element model, and determining constraint degrees of freedom;
step (2), determining a load process F (t) carried by the structure, and defining a damping matrix of the finite element model in the step (1) as C=alpha c M+β c K，α c And beta c Is a proportional damping coefficient;
step (3), solving the structural dynamics response of the finite element model constructed in the step (1) under the condition of the load environment and damping defined in the step (2) by a numerical solution method or an approximate analytic solution method of arbitrary differential equation lease to obtain a structural displacement field U (t) and a velocity fieldAcceleration field->t is time;
step (4), according to the structural displacement field U (t) obtained in the step (3), building a structural time domain dynamics topology optimized attention index f (U (t)), combining f (U (t)), U (t) and Max (f (U (t))), solving a fixed integral through an arbitrary numerical integration method, and obtaining an approximation value of a dynamics response peak value Max (f (U (t)))Solving the obtained dynamic time domain response peak value approximation value and returning to the topology optimization main program;
step (5), according to the structural displacement field U (t) obtained in the step (3), solving a first derivative matrix of a structural time domain dynamics topological optimization attention index relative to the structural displacement fieldCombining F (U (t)), U (t) and Max (F (U (t))), and solving the virtual load process F λ (t) defining the virtual damping matrix of the finite element model in the step (1) as-c= - α c M-β c K；
Step (6), solving the structural virtual dynamics response of the finite element model constructed in the step (1) under the condition of the virtual load environment and the virtual damping defined in the step (5) by a numerical solution method or an approximate analytic solution method of any differential equation, obtaining structural virtual dynamics response lambda (tau), and naming a virtual dynamics response vector as an accompanying vector;
step (7), the displacement field U (t) and the velocity field obtained in the step (3)Acceleration field->Solving for the unit node displacement U of the ith unit e,i (t), cell node speed->And unit node is added withSpeed->
Step (8) of solving the unit node adjoint vector lambda of the ith unit according to the adjoint vector lambda (t) obtained in the step (6) e,i (t)；
Step (9) of solving the cell stiffness matrix K when the cell pseudo density is equal to 1 based on the structural stiffness matrix K, the mass matrix M obtained in the step (1) and the damping matrix C obtained in the step (2) e Cell mass matrix M e Cell damping matrix C e ；
Step (10) of obtaining U based on the steps (7) to (9) e,i (t)、λ e,i (t)、K e 、M e And C e Solving a fixed integral by an arbitrary numerical integral method to obtain a dynamic time domain response index approximation value +.>With respect to pseudo density x of the ith cell numbered i Sensitivity of (2);
and (11) repeating the steps (7) to (10) through serial operation or parallel operation until the sensitivity of the structural dynamics time domain response index approximation value about the pseudo density of all units is solved, so that the solution of the structural dynamics peak time domain about the response sensitivity of the structural topology optimization design variable is completed, and the sensitivity of the obtained dynamic time domain response peak approximation value is returned to the topology optimization main program.
Furthermore, the method only relates to the approximation value of the structural dynamics peak time domain response index and the sensitivity solving of the approximation value relative to the pseudo density of all units, and can be embedded into a gradient solving algorithm of any structural topology optimization, and parameters obtained in the step (1) can be realized through any structural dynamics solving method and a program platform.
Further, the structural displacement field U (t) and the velocity field in the step (3)And acceleration field->The method is obtained by solving the following differential equation:
the differential equation set in the above formula is a differential equation set with initial condition and without termination condition, and the initial condition is U (0) =0The above method can be solved by a numerical solution method or an approximate analytic solution method of any differential equation lease.
Further, the approximation of the dynamic peak time domain response f (U (t)) in step (4)Obtained by solving the following equation for the definite integral:
wherein e is a natural index, beta is χ -0.6,chi is far greater than +.>Is a positive real number of (2); Δt is the integral step; t is t f For the moment of termination of the kinetic analysis, the above equation can be solved by any numerical integration method.
Further, the virtual load history F in step (5) λ (t) solving by
Further, the accompanying vector λ (t) in step (6) is obtained by solving the following differential equation set
The differential equation set in the above formula is a differential equation set without initial condition but with termination condition of lambda (t f )＝0The above method can be solved by a numerical solution method or an approximate analytic solution method of any differential equation lease.
Further, the approximation of the time domain response of the dynamic peak in step (10)Pseudo density x for the ith cell numbered i The sensitivity of (2) is obtained by solving the following fixed integral:
the above equation can be solved by any numerical integration method.
Compared with the prior art, the invention has the beneficial effects that:
(1) The invention adopts the peak value approximation method of the time domain response peak value based on step function integration, reduces approximation error through a reference value translation strategy when constructing the peak value approximation function, and can improve the optimization speed under the condition of being suitable for the condition of larger integration step length;
(2) The peak value approximation method of the time domain response peak value based on step function integration has the same monotonicity with the real function value, and can give sensitivity in the same direction as the real value in topology optimization;
(3) The method does not relate to a dynamics solving step, only relates to an approximation value of a structural dynamics peak time domain response index and sensitivity solving of the approximation value about the pseudo density of all units, and can be embedded into a gradient solving algorithm of arbitrary structural topology optimization;
(4) The accuracy of the peak value approximation value of the time domain response peak value based on the step function integration is not strong in response course dependence after the peak value response appears, so that the response time length concerned by dynamics and sensitivity calculation can be reduced, and the optimization time consumption is reduced;
(5) The dynamic response peak value approximation and sensitivity solving method has good portability, can be combined with any numerical integration and differential equation set solving method in solving aspect, and can be combined with any pseudo-density interpolation model and gradient optimization algorithm in application aspect.
Drawings
FIG. 1 is a flow chart of the present invention;
FIG. 2 is a structural dimension, load and constraint environment in an embodiment of the present invention;
FIG. 3 is a time-varying load history experienced by a structure in an embodiment of the invention;
FIG. 4 is a structural statics topology optimization result of the present invention;
FIG. 5 is an optimization result of the structural time domain dynamic topology optimization embedded in the present invention with termination times of 0.12s, 0.15s, 0.18s and 0.58s, respectively;
FIG. 6 is a graph of the convergence history of the structural time domain dynamic topology optimization embedded in the present invention at termination times of 0.12s, 0.15s, 0.18s and 0.58s, respectively;
FIG. 7 is a comparison of the dynamic response of a structural time domain dynamic topology optimization optimal solution embedded in the present invention with a hydrostatic topology optimization optimal solution.
Detailed Description
The invention is further illustrated below with reference to examples.
As shown in fig. 1, the invention provides a method for solving the time domain response sensitivity of a structural dynamics peak value based on a step function, which comprises the following steps:
initializing an optimization model, taking a structural design domain as a geometric boundary, establishing a finite element analysis model, and multiplying the elastic modulus of an ith element with the number of the ith element by the pseudo density x i To the power of p, where x i ∈[0,1]P=3, pseudo density x of all cells i Forming a design variable vector x, extracting a structural rigidity matrix K and a quality matrix M according to the finite element model, and determining constraint degrees of freedom;
step (2), determining a load process F (t) carried by the structure, and defining a damping matrix of the finite element model in the step (1) as C=alpha c M+β c K，α c And beta c Is a proportional damping coefficient;
step (3), obtaining a structural displacement field U (t) and a velocity fieldAcceleration field->The structural dynamics response of the finite element model constructed in the step (1) under the condition of the load environment and damping defined in the step (2) is needed to be solved, namely the following differential equation set is solved
The differential equation set in the above formula is a differential equation set with initial condition and without termination condition, and the initial condition is U (0) =0The method can be solved by a numerical solution method or an approximate analytic solution method of arbitrary differential equation lease;
step (4), building a concerned index f (U (t)) of structural time domain dynamics topological optimization according to the structural displacement field U (t) obtained in the step (3), and combining the approximated value of the dynamics response peak value Max (f (U (t))) with the approximated value of f (U (t)), U (t) and Max (f (U (t)))Can be obtained by solving the following formula
Wherein e is a natural index, beta is χ -0.6,chi is far greater than +.>Is a positive real number of (2); Δt is the integral step; t is t f The moment of termination of the kinetic analysis. The above formula can be carried out by any numerical integration method, and the obtained dynamic time domain response peak value approximation value is solved and returned to the topology optimization main program.
Step (5), according to the structural displacement field U (t) obtained in the step (3), solving a first derivative matrix of a structural time domain dynamics topological optimization attention index relative to the structural displacement fieldCombining F (U (t)), U (t), and Max (F (U (t))), virtual load history F λ (t) can be solved by
After the virtual load history is solved, defining the virtual damping matrix of the finite element model in the step (1) as-C= -alpha c M-β c K；
Step (6), obtaining an accompanying vector lambda (tau), and requiring to solve the structural virtual dynamics response of the finite element model constructed in the step (1) under the condition of the virtual load environment and the virtual damping defined in the step (5), namely solving the following differential equation set
The differential equation set in the above formula is a differential equation set without initial condition but with termination condition of lambda (t f )＝0The method can be solved by a numerical solution method or an approximate analytic solution method of arbitrary differential equation lease;
step (7), the displacement field U (t) and the velocity field obtained in the step (3)Acceleration field->Extracting the cell node displacement U of the ith cell e,i (t), cell node speed->Acceleration +.>
Step (8) of extracting the unit node companion vector lambda (t) numbered as the ith unit from the companion vector lambda (t) obtained in step (6) e,i (t)；
Step (9) of solving the cell stiffness matrix K when the cell pseudo density is equal to 1 based on the structural stiffness matrix K, the mass matrix M obtained in the step (1) and the damping matrix C obtained in the step (2) e Cell mass matrix M e Cell damping matrix C e ；
Step (10) of obtaining U based on the steps (7) to (9) e,i (t)、λ e,i (t)、K e 、M e And C e Kinetic peakApproximation of the value-time-domain response +.>Pseudo density x for the ith cell numbered i The sensitivity of (2) is obtained by solving the following fixed integral
The above formula can be solved by any numerical integration method;
and (11) repeating the steps (7) to (10) through serial operation or parallel operation until the sensitivity of the structural dynamics time domain response index approximation value about the pseudo density of all units is solved, and returning the sensitivity of the obtained dynamics time domain response peak value approximation value to a topology optimization main program.
Examples
In order to fully understand the characteristics of the invention and the applicability of the invention to engineering practice, the invention establishes a structure and load constraint environment as shown in the figure, and embeds the structure and load constraint environment into dynamic time domain topology optimization of a continuum structure, the optimization adopts a gradient algorithm moving asymptote method MMA, a pseudo-density interpolation model adopts a p=6 SIMP model, an optimization target is a minimum time domain response peak value, and an optimization constraint is an area ratio not less than 0.3, and the method is embedded into a topology optimization main program framework to solve the optimization target and the optimization target sensitivity. The specific values of the parameters in each real-time step are as follows:
in the step (1), the design domain is shown in fig. 2, the size of the design domain is 100mm multiplied by 70mm, the structure finite element model is 100 units in the longitudinal direction, 70 units in the transverse direction, and the longitudinal direction and the transverse direction are numbered from the upper left corner; young's modulus of 2X 10 5 The density is 7.8X10 Mpa -6 Kg/mm3, poisson's ratio of 0.3, the cell type uses planar bilinear cells.
Proportional damping coefficient α in step (2) C =10 and β C ＝1×10 -5 The method comprises the steps of carrying out a first treatment on the surface of the The structure is constrained at the lower two ends, the structure being in the upper right corner of row 1, 17 cellsThe node receives two time-varying loads along the X direction and the Y direction, a 0.1Kg counterweight is attached to the loaded 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 the step (3), a New Mark-beta method is adopted to solve a kinetic differential equation, the calculation step length delta t is 0.001s, and in order to verify that the dependency of the method on analysis duration is not strong, in this embodiment, kinetic analysis and corresponding topological optimization under 4 groups of termination time of 0.12s, 0.15s, 0.18s and 0.58s are respectively carried out.
The dynamic time domain index f (U (t)) of interest in step (4) is the sum of the X-direction maximum displacement and the Y-direction maximum displacement of the loaded node of the structure, namelyWherein->The output vector is a column vector with the same degree of freedom, the corresponding position of the translational degree of freedom of the X direction of the loaded node is 1, the other positions are 0,the output vector is a column vector with the same degree of freedom, the corresponding position of the translational degree of freedom of the loaded node Y direction is 1, and the other positions are 0; beta is 20, χ is 10000, then +.>The fixed integral is solved by adopting a trapezoidal method, and the integral step length delta t is 0.001s and t corresponding to the step (3) f 0.12s, 0.15s, 0.18s and 0.58s, respectively.
And (5) solving according to the formula provided by the invention without parameter input.
Step (6) adopts New Mark-beta method to solve dynamic differential equation, corresponding to step (3), delta t is 0.001s, t f 0.12s, 0.15s, 0.18s and 0.58s, respectively.
Step (7) does not need parameter input, and extracts according to unit and node numbering logic and sequence;
step (8) does not need parameter input, and extracts according to unit and node numbering logic and sequence;
and (9) a planar bilinear unit is adopted as the unit type without parameter input.
The fixed integral in the step (10) is solved by adopting a trapezoidal method, and the integral step delta t is 0.001s and t corresponding to the step (3) f 0.12s, 0.15s, 0.18s and 0.58s, respectively.
And (11) repeating the steps (7) to (10) until all units are calculated, and returning the sensitivity of the obtained dynamic time domain response peak value approximation value to the topology optimization main program.
After the operation of all the steps is finished, returning the approximation value of the time domain response peak value calculated in the step (4) and the sensitivity of the approximation value of the time domain response peak value calculated in the step (11) to a topological optimization group program, adjusting the pseudo density distribution of the next step by moving an asymptote algorithm MMA, carrying out the real-time step of the invention again after the adjustment, and repeating the steps (1) to (11) until the topological optimization convergence of the structural time domain dynamics.
In order to verify that the approximation method of the time domain response peak value and the corresponding sensitivity solving method provided by the invention can effectively operate in the structural dynamics time domain topology optimization frame and can reduce the structural dynamics time domain response peak value, the embodiment adopts static topology optimization aiming at the same model, the load environment is static load, the X-direction environmental load and the Y-direction static load are peak values of load processes in the step (2), and other constraint environments and optimization algorithms are consistent with the structural dynamics topology optimization.
By comparing fig. 4 with fig. 5, it can be seen that the optimal solution topology configuration of the static topology optimization is different from the optimal solution topology configuration of the dynamic topology optimization, and the optimal solution configuration of the dynamic topology optimization based on the structure of the invention does not depend on the dynamic analysis duration, and by fig. 6, the optimization can be converged quickly no matter how long the dynamic analysis duration is, and the convergence values are identical. As can be seen from fig. 7, the dynamic response peak value of the optimal configuration of the static topology optimization is larger than the dynamic response peak value of the optimal configuration of the dynamic topology optimization embedded in the invention, which indicates that the method for solving the time domain response sensitivity of the structural dynamic peak value based on the step function can be very effectively embedded in the gradient topology optimization method, and the peak value of the time domain response can be effectively reduced.
The invention adopts the peak value approximation method of the time domain response peak value based on the step function integration, reduces approximation error through the reference value translation strategy when constructing the peak value approximation function, and can improve the optimization speed under the condition of being suitable for the condition of larger integration step length.
The method for approximating the peak value of the time domain response peak value based on step function integration has the same monotonicity as the real function value, and can give sensitivity in the same direction as the real value in topology optimization.
While the preferred embodiments of the present invention have been described above, it is not intended to limit the invention, and any person skilled in the art may make possible variations and modifications to the solution of the present invention using the methods and techniques disclosed above without departing from the spirit and scope of the invention. Therefore, any simple modification equivalent to the above embodiments according to the technical substance of the present invention falls within the scope of the technical solution of the present invention.
Claims (7)
1. The method for solving the time domain response sensitivity of the structural dynamics peak value based on the step function is characterized by comprising the following steps:
initializing an optimization model, taking a structural design domain as a geometric boundary, establishing a finite element analysis model, and multiplying the elastic modulus of an ith element with the number of the ith element by the pseudo density x i To the power of p, where x i ∈[0,1]P=3, pseudo density x of all cells i Forming a design variable vector x, extracting a structural rigidity matrix K and a quality matrix M according to the finite element model, and determining constraint degrees of freedom;
step (2), determining the load process F (t) borne by the structureThe damping matrix of the finite element model in the step (1) is C=alpha c M+β c K，α c And beta c Is a proportional damping coefficient;
step (3), solving the structural dynamics response of the finite element model constructed in the step (1) under the condition of the load environment and damping defined in the step (2) by a numerical solution method or an approximate analytic solution method of arbitrary differential equation lease to obtain a structural displacement field U (t) and a velocity fieldAcceleration field->t is time;
step (4), according to the structural displacement field U (t) obtained in the step (3), building a structural time domain dynamics topology optimized attention index f (U (t)), combining f (U (t)), U (t) and Max (f (U (t))), solving a fixed integral through an arbitrary numerical integration method, and obtaining an approximation value of a dynamics response peak value Max (f (U (t)))Solving the obtained dynamic time domain response peak value approximation value and returning to the topology optimization main program;
step (5), according to the structural displacement field U (t) obtained in the step (3), solving a first derivative matrix of a structural time domain dynamics topological optimization attention index relative to the structural displacement fieldCombining F (U (t)), U (t) and Max (F (U (t))), and solving the virtual load process F λ (t) defining the virtual damping matrix of the finite element model in the step (1) as-c= - α c M-β c K；
Step (6), solving the structural virtual dynamics response of the finite element model constructed in the step (1) under the condition of the virtual load environment and the virtual damping defined in the step (5) by a numerical solution method or an approximate analytic solution method of any differential equation, obtaining structural virtual dynamics response lambda (tau), and naming a virtual dynamics response vector as an accompanying vector;
step (7), the displacement field U (t) and the velocity field obtained in the step (3)Acceleration field->Solving for the unit node displacement U of the ith unit e,i (t), cell node speed->Acceleration +.>
Step (8) of solving the unit node adjoint vector lambda of the ith unit according to the adjoint vector lambda (t) obtained in the step (6) e,i (t)；
Step (9) of solving the cell stiffness matrix K when the cell pseudo density is equal to 1 based on the structural stiffness matrix K, the mass matrix M obtained in the step (1) and the damping matrix C obtained in the step (2) e Cell mass matrix M e Cell damping matrix C e ；
Step (10) of obtaining U based on the steps (7) to (9) e,i (t)、λ e,i (t)、K e 、M e And C e Solving a fixed integral by an arbitrary numerical integral method to obtain a dynamic time domain response index approximation value +.>With respect to pseudo density x of the ith cell numbered i Sensitivity of (2);
and (11) repeating the steps (7) to (10) through serial operation or parallel operation until the sensitivity of the structural dynamics time domain response index approximation value about the pseudo density of all units is solved, so that the solution of the structural dynamics peak time domain about the response sensitivity of the structural topology optimization design variable is completed, and the sensitivity of the obtained dynamic time domain response peak approximation value is returned to the topology optimization main program.
2. The step function-based structural dynamics peak time domain response sensitivity solving method according to claim 1, wherein: the method only relates to the approximation value of the structural dynamics peak time domain response index and the sensitivity solving of the approximation value about the pseudo density of all units, can be embedded into a gradient solving algorithm of any structural topological optimization, and the parameters obtained in the step (1) can be realized through any structural dynamics solving method and a program platform.
3. The step function-based structural dynamics peak time domain response sensitivity solving method according to claim 1, wherein: a structural displacement field U (t) and a velocity field in the step (3)And acceleration field->The method is obtained by solving the following differential equation:
the differential equation set in the above formula is a differential equation set with initial condition and without termination condition, and the initial condition is U (0) =0The above method can be solved by a numerical solution method or an approximate analytic solution method of any differential equation lease.
4. The step function-based structural dynamics peak time domain response sensitivity solving method according to claim 1, wherein: approximation of the dynamic peak time domain response f (U (t)) in step (4)Obtained by solving the following equation for the definite integral:
wherein e is a natural index, beta is χ -0.6,chi is far greater than +.>Is a positive real number of (2); Δt is the integral step; t is t f For the moment of termination of the kinetic analysis, the above equation can be solved by any numerical integration method.
5. The step function based structural dynamics peak time domain response sensitivity solving method according to claim 4, wherein: virtual load history F in step (5) λ (t) solving by
6. The step function-based structural dynamics peak time domain response sensitivity solving method according to claim 1, wherein: the syndrome vector lambda (t) in step (6) is obtained by solving the differential equation set
The differential equation set in the above formula is a differential equation set without initial condition but with termination condition of lambda (t f )＝0The above method can be solved by a numerical solution method or an approximate analytic solution method of any differential equation lease.
7. The step function-based structural dynamics peak time domain response sensitivity solving method according to claim 1, wherein: approximation of the time domain response of the dynamic peak in step (10)Pseudo density x for the ith cell numbered i The sensitivity of (2) is obtained by solving the following fixed integral:
the above equation can be solved by any numerical integration method.
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| CN109508495A (en) * | 2018-11-12 | 2019-03-22 | 华东交通大学 | A kind of compliant mechanism overall situation stress constraint Topology Optimization Method based on K-S function |
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| CN107942664A (en) * | 2017-11-23 | 2018-04-20 | 中国南方电网有限责任公司 | A kind of hydrogovernor parameter tuning method and system based on sensitivity analysis |
| CN109508495A (en) * | 2018-11-12 | 2019-03-22 | 华东交通大学 | A kind of compliant mechanism overall situation stress constraint Topology Optimization Method based on K-S function |
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