CN116882195A - Single-step explicit gradual integration method for solving second-order nonlinear dynamics problem, application and system - Google Patents
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Abstract
The invention provides a single-step explicit gradual integration method for solving a second-order nonlinear dynamics problem, application and a system. The method aims to solve the problem that the existing single-step explicit gradual integration method is difficult to realize the uniform second-order precision and the full course change of controllable numerical value dissipation simultaneously under the condition that external force load interpolation calculation is not needed. Selecting a time step and setting a user-specified parameter ρ b And ρ s According to ρ b And ρ s Calculating parameters χ, beta and gamma, calculating an initial acceleration vector again, setting an initial auxiliary acceleration vector as the initial acceleration vector, judging whether the moment is the last moment, if not, repeatedly and sequentially calculating the next momentAcceleration vector, velocity vector, displacement vector and auxiliary acceleration vector, until the final moment the calculation is finished. The method can simultaneously realize uniform second-order precision and full process variation of controllable numerical value dissipation under the condition of no need of external force load interpolation calculation, reduce calculation errors and provide accurate numerical value solution for dynamic response prediction of actual engineering problems.
Description
Technical Field
The invention relates to an explicit time step-by-step integration method for solving a second-order nonlinear dynamics problem.
Background
Vibration generated by the structural body under the action of dynamic load or under the action of initial disturbance, multi-body system motion formed by interconnecting a plurality of rigid bodies or flexible bodies, and dynamic behavior of solid and fluid under the action of mutual coupling are all included in the category of dynamic problems. Such problems are common in scientific research, engineering applications, and even in everyday life. In order to accurately predict the vibration condition of each structure under different actions, modern science and technology has developed a large number of numerical analysis tools, such as a finite element method, an isogeometric and virtual unit method, which implement spatial dispersion, a center difference method, a Newmark method, a generalized a method and a ringe-Kutta method, which implement time dispersion.
For the development and application of time-discrete techniques, time-discrete techniques can be generally divided into explicit and implicit step-by-step integration methods. Currently, the explicit step-by-step integration method is one concern for computing nonlinear dynamics problems. It is superior to implicit methods in terms of computational cost, but it is generally condition stable. Thus, the integration step size is limited by stability rather than accuracy. The central differencing method is, of course, a well-known explicit integration method. Because of its simplicity, it has been implemented in a variety of commercial and open source finite element programs, such as Abaqus and FEAPpv. The initial central difference method is proposed by replacing the first and second differential derivatives with finite differences, so the central difference method is not self-starting. The other two main disadvantages of the central difference method are: (1) It is zero dissipative at the bifurcation point and therefore cannot effectively eliminate some false numerical oscillations; (2) It needs to diagonalize the global damping matrix, otherwise the superiority of the explicit gradual integration method cannot be reflected. Based on the above analysis, a competitive explicit stepwise integration method should have the following numerical characteristics: (1) as great a conditional stability as possible; (2) explicit processing of velocity items; (3) second order of displacement, velocity and acceleration; (4) controllable numerical dissipation at the bifurcation point; (5) self-starting feature.
The single-step explicit step-by-step integration method, which has been proposed up to now, is proposed by analyzing the prediction-correction format corresponding to each implicit method, so that the obtained explicit method cannot well realize the above five properties, such as EG-a method and EHHT method. In addition to the prediction-correction format using implicit methods, there are also single-step explicit integration methods that have been investigated reporting other strategies. Tamma and Namburu propose an explicit integration method (TN) of direct self-starting and implicit processing speed based on Lax-Wendroff/Taylor-Galerkin technology. The TN method involves the first derivative of the external load, so it may be difficult to calculate the first derivative accurately when solving certain problems. The TN method realizes the maximum stable domain, second-order precision and zero dissipation when g=1/2 is taken; otherwise it is only of first order accuracy. Tcamwa et al propose a single step explicit integration method (TW). The method achieves first-order precision and has controllable numerical dissipation when solving general dynamics problems. Chung and Lee will t n The acceleration at the moment is directly output as t n+1 Acceleration at time and displacement and velocity update formats using Newmark method an explicit integration method (CL) is proposed. The CL method can achieve second order accuracy of displacement and velocity and first order accuracy of acceleration, and can also achieve controllable numerical dissipation at the bifurcation point but cannot achieve full history change of dissipation control. In order to improve the acceleration accuracy of the CL method, kim provides an improved CL method (ICL). The ICL method provides exactly the same spectral characteristics as the CL method but achieves consistent second order accuracy, but it still fails to achieve the full history of the value dissipation. For the current explicit gradual integration method, it is difficult to realize the uniform second-order precision and the full course change of the numerical dissipation at the same time without the need of external force load interpolation calculation.
Disclosure of Invention
The invention aims to solve the technical problems that:
the method aims to solve the problem that the existing single-step explicit gradual integration method is difficult to realize the uniform second-order precision and the full course change of controllable numerical value dissipation simultaneously under the condition that external force load interpolation calculation is not needed.
The invention adopts the technical scheme for solving the technical problems:
the invention provides a single-step explicit gradual integration method for solving a second-order nonlinear dynamics problem, which comprises the following steps:
step one, selecting a time step Deltat, and setting a user-specified parameter ρ b And ρ s According to ρ b And ρ s The undetermined parameters χ, β and γ are calculated, the calculation method is as follows,
step two, solving an initial acceleration vector by utilizing a second-order motion differential equation at the initial moment
Wherein M represents a global quality matrix; f is the resultant of the internal force and the external force applied to the structure;is an initial acceleration vector; />For the beginningA start speed vector; u (U) 0 Is an initial displacement vector; t is t 0 Is the initial time;
at the time of calculation, the initial acceleration is obtainedAfter that, the initial auxiliary acceleration variable is set +.>
Step three, known t n Time of day displacement U n Speed and velocity ofAcceleration->And auxiliary acceleration A n Determining whether the time N is the last time N, if not, repeating the following calculation,
solving by using equation (6) to obtain t n+1 Acceleration at timeThe calculation method comprises the following steps:
in the method, in the process of the invention,for calculating t n+1 A displacement vector of the moment; />For calculating t n+1 A velocity vector of the moment;
in the process of calculating and obtaining the accelerationThen, t is calculated by using an equation n+1 Speed of moment->And displacement U n+1 The calculation method is as follows:
t n+1 auxiliary acceleration A at time n+1 The calculation is updated by the following equation:
and ending the calculation until the moment N is the last moment N.
Further, in step one, ρ b And ρ s The condition should be satisfied and,
|ρ s |≤ρ b ∈0,1 ⑷
wherein ρ is b And ρ s Two parameters specified for the user.
Further, the parameter ρ b The spectrum radius value of the algorithm at the bifurcation point is controlled, and the numerical value dissipation quantity at the bifurcation point is further controlled; parameter ρ s The amount of numerical dissipation of the algorithm in the low frequency region is controlled.
Further, by default, the user is recommended to use ρ b ＝ρ s To reduce one user-specified parameter.
Further, when maximum value dissipation is achieved, ρ is made b ＝ρ s =0 for one-step cancellation of spurious high frequency responses in the structural response.
Further, in step two, the partial differential equation of the dominant structure is degenerated to a second order differential equation of motion of the form after the application of the spatial discrete technique:
wherein U (t),And->Displacement, velocity and acceleration vectors, respectively.
Further, to determine a unique set of numerical solutions, the initial displacement U (t 0 ) And velocity vectorMust be given, i.e. U (t 0 )＝U 0 And->
The application of the single-step explicit step-by-step integration method for solving the second-order nonlinear dynamics problem can be used for dynamic calculation of a folding control surface with gap nonlinearity, and the formula (4) needs to be adjusted according to actual conditions.
A system for a single-step explicit step-by-step integration method for solving a second-order nonlinear dynamics problem, the system having program modules corresponding to the steps described above, the steps in the single-step explicit step-by-step integration method for solving a second-order nonlinear dynamics problem described above being executed at run-time.
A computer readable storage medium storing a computer program configured to implement the steps of a single-step explicit step-by-step integration method of solving a second order nonlinear dynamics problem when called by a processor.
Compared with the prior art, the invention has the beneficial effects that:
the invention relates to a single-step explicit gradual integration method for solving a second-order nonlinear dynamics problem, application and a system thereof, which are used for selecting a time step and setting a user-specified parameter rho b And ρ s According to ρ b And ρ s Calculating undetermined parameters χ, β and γ, calculating initial acceleration vectors,setting an initial auxiliary acceleration vector as an initial acceleration vector, judging whether the moment is the last moment, if not, repeatedly and sequentially calculating the acceleration vector, the speed vector, the displacement vector and the auxiliary acceleration vector at the next moment until the last moment, and ending the calculation;
the single-step explicit gradual integration method for solving the second-order nonlinear dynamics problem can simultaneously realize the consistent second-order precision and the full-process change of controllable numerical dissipation, can provide an accurate numerical solution for the dynamic response prediction of the actual engineering problem, and further accelerates the conversion of the structure from theoretical design to actual application.
The invention discloses a single-step explicit step-by-step integration method for solving a second-order nonlinear dynamics problem, which constructs, analyzes and develops a single-step explicit integration algorithm for the first time by using auxiliary variables. The introduction of the auxiliary acceleration variable not only does not require additional computer resources, but also solves the problem that the single-step explicit method cannot realize consistent second-order precision.
Compared with the existing CL method, the single-step explicit gradual integration method for solving the second-order nonlinear dynamics problem not only realizes the full-process change of controllable numerical dissipation, but also provides the control of the numerical damping rate in a larger range of the CL method, which is extremely beneficial to solving the spatial discrete model with false high-frequency modes; the method realizes the second-order acceleration which can not be predicted by the CL method and other explicit methods, namely, the consistent second-order precision is realized.
The single-step explicit gradual integration method for solving the second-order nonlinear dynamics problem does not need to consider interpolation calculation of external force load, can directly apply the external force load value at discrete moment to update iterative calculation, avoids numerical errors introduced by the existing method during the interpolation calculation of the external force load, further improves the solving precision, has flexible dissipation control capability to cope with more complex engineering practical problems, such as a folding control surface problem containing gap nonlinearity, and is suitable for solving and analyzing any second-order dynamics problem, and particularly, is particularly efficient for solving a large-scale practical structure.
Drawings
FIG. 1 is a flow chart of a single-step explicit step-by-step integration method for solving a second-order nonlinear dynamics problem in an embodiment of the present invention;
FIG. 2 is a graph comparing spectral radii and numerical damping rates with the CL approach in an embodiment of the invention;
FIG. 3 is a graph comparing convergence rates with the CL method in solving a standard single degree of freedom system in an embodiment of the present invention.
Detailed Description
In the description of the present invention, it should be noted that the terms "first," "second," and "third" mentioned in the embodiments of the present invention are used for descriptive purposes only and are not to be construed as indicating or implying relative importance or implicitly indicating the number of technical features indicated. Thus, a feature defining "a first", "a second", or a third "may explicitly or implicitly include one or more such feature.
In order that the above objects, features and advantages of the invention will be readily understood, a more particular description of the invention will be rendered by reference to specific embodiments thereof which are illustrated in the appended drawings.
The specific embodiment I is as follows: referring to fig. 1, the present invention provides a single-step explicit step-by-step integration method for solving a second-order nonlinear dynamics problem, comprising the steps of:
step one, selecting a time step Deltat, and setting a user-specified parameter ρ b And ρ s According to ρ b And ρ s The undetermined parameters χ, β and γ are calculated, the calculation method is as follows,
wherein ρ is b And ρ s Parameters specified for two users, and both should satisfy the condition:
|ρ s |≤ρ b ∈0,1 ⑷
step two, solving an initial acceleration vector by utilizing a second-order motion differential equation at the initial moment
At the time of calculation, the initial acceleration is obtainedAfter that, the initial auxiliary acceleration variable is set +.>
Step three, known t n Time of day displacement U n Speed and velocity ofAcceleration->And auxiliary acceleration A n Judging whether the time N is the last time N; if yes, ending calculation;
the repeated calculation method is that,
solving by using equation (6) to obtain t n+1 Acceleration at timeThe calculation method comprises the following steps:
in the method, in the process of the invention,for calculating t n+1 A displacement vector of the moment; />For calculating t n+1 A velocity vector of the moment;
it should be noted that the speedAnd (3) displacement->Are not equal to the true speed +.>And displacement U n+1 The former is used only in the equation to calculate t n+1 Acceleration of time of day->
In the process of calculating and obtaining the accelerationThen, t is calculated by using an equation n+1 Speed of moment->And displacement U n+1 The calculation method is as follows:
t n+1 auxiliary acceleration A at time n+1 The calculation is updated by the following equation:
for the method, no single-step explicit integration method which does not need external load interpolation calculation and simultaneously realizes consistent second-order precision and numerical dissipation full-process change is proposed in the prior art. The method does not need interpolation calculation of external force, reduces calculation error of external force load, and shows obvious programming and calculation advantages when solving the actual problem; the method fills the gap under a single-step framework, not only improves the solving precision, but also has flexible dissipation control capability to cope with more complex engineering practical problems.
And a specific embodiment II: this embodiment differs from the specific embodiment in that the partial differential equation of the dominant structure is degenerated to a second order differential equation of motion of the form after the application of the spatial discrete technique:
wherein M represents a global quality matrix; f is the resultant of the internal force and the external force applied to the structure; u (t),and->Displacement, velocity and acceleration vectors, respectively;
to determine a unique set of numerical solutions, an initial displacement U (t 0 ) And velocity vectorMust be given, i.e. U (t 0 )＝U 0 And->
In the computer programming implementation process, the auxiliary acceleration vector A does not need to allocate the memory amount of the whole time process, and only needs to be updated iteratively in the gradual calculation process, so that the calculation resources can be saved.
Other steps and parameters are the same as in the first embodiment.
And a third specific embodiment: the present embodiment differs from the one or two embodiments in that the parameter ρ is b The spectrum radius value of the algorithm at the bifurcation point is controlled, and the numerical value dissipation quantity at the bifurcation point is further controlled; and parameter ρ s The amount of numerical dissipation of the algorithm in the low frequency region is controlled. By default, recommended user usage ρ b ＝ρ s To reduce one user-specified parameter. Other steps and parameters are the same as in one or both embodiments.
And a specific embodiment IV: the invention provides an application of a single-step explicit step-by-step integration method for solving a second-order nonlinear dynamics problem, which can be used for dynamic calculation of a gap-containing nonlinear folding control surface, and a formula (4) needs to be adjusted according to actual conditions. Other steps and parameters are the same as those of the first, second or third embodiments.
Fifth embodiment: the invention provides a system for solving a single-step explicit step-by-step integration method of a second-order nonlinear dynamics problem, which is provided with a program module corresponding to the steps, and the steps of the single-step explicit step-by-step integration method for solving the second-order nonlinear dynamics problem are executed in running. Other steps and parameters are the same as those of the first, second or third embodiments.
Specific embodiment six: the present invention provides a computer readable storage medium storing a computer program configured to implement the steps of a single-step explicit step-by-step integration method of solving a second order nonlinear dynamics problem when called by a processor. Other steps and parameters are the same as those of the first, second or third embodiments.
Examples
The method is compared with the existing CL method in terms of spectral radius, numerical damping rate and convergence rate when solving a standard single-degree-of-freedom system.
Results such asFIG. 2 shows that the spectral radius curve of the method can be obtained by the methodInternal continuous change, and the spectrum radius value can be taken to be ρ b =0, i.e. the method achieves a full history of controllable value dissipation. On the other hand, the existing CL method cannot make the spectral radius value reach ρ b =0, so it is only moderately dissipative. Furthermore, fig. 2 also shows that the present method provides a larger range of numerical damping rate control than the CL method, which is extremely advantageous for solving spatially discrete models with spurious high frequency modes. Other existing methods, such as the ICL method, suffer from the same disadvantage that they do not achieve the full history of controlled value dissipation.
As shown in fig. 3, both methods predict the displacement and velocity solutions of the second order, but only the present method predicts the second order acceleration. In other words, the present method achieves consistent second order accuracy, whereas the CL method does not achieve this feature. Other existing explicit methods, such as the EG-a method, suffer from the disadvantage that second order accurate acceleration solutions cannot be predicted.
In order to compare the numerical characteristics of the method with those of the existing single-step explicit step-by-step integration method, performance comparison is carried out, and the results are shown in table 1. Table 1 shows that of all explicit step-wise integration methods that achieve consistent second order accuracy, only the present method achieves the full history of controllable value dissipation.
Table 1: performance comparison of single step explicit step-by-step integration method
For table 1, it should be noted that:
(1) The speed processing mode determines the calculation cost of the method and the size of the stable domain under the damping condition. For example, the central difference method uses implicit processing speed, so its calculation cost is larger than other explicit processing speed methods but the stability domain in damping is not less than 2. The invention mainly focuses on the development and application of a speed explicit processing method.
(2) The value dissipation is ρ b ∈[0，1]The corresponding method is said to realize the full process variation of controllable numerical dissipation; otherwise, the corresponding method does not achieve this feature. Meanwhile, "-" means that the corresponding method has no numerical dissipation.
(3) When the time integration method only needs to solve the initial acceleration vectorWhen transient analysis is started, the corresponding method is called to realize the self-starting characteristic; otherwise, the corresponding method is not self-starting. For example, the center difference method is not self-starting. Meanwhile, "-" means that the corresponding method is directly self-initiated.
Although the present disclosure is disclosed above, the scope of the present disclosure is not limited thereto. Various changes and modifications may be made by one skilled in the art without departing from the spirit and scope of the disclosure, and such changes and modifications would be within the scope of the disclosure.
Claims (10)
1. The single-step explicit step-by-step integration method for solving the second-order nonlinear dynamics problem is characterized by comprising the following steps of:
step one, selecting a time step Deltat, and setting a user-specified parameter ρ b And ρ s According to ρ b And ρ s The undetermined parameters χ, β and γ are calculated, the calculation method is as follows,
step two, solving an initial acceleration vector by utilizing a second-order motion differential equation at the initial moment
Wherein M represents a global quality matrix; f is the resultant of the internal force and the external force applied to the structure;is an initial acceleration vector; />Is an initial velocity vector; u (U) 0 Is an initial displacement vector; t is t 0 Is the initial time;
at the time of calculation, the initial acceleration is obtainedAfter that, the initial auxiliary acceleration variable is set +.>
Step three, known t n Time of day displacement U n Speed and velocity ofAcceleration->And auxiliary acceleration A n Determining whether the time N is the last time N, if not, repeating the following calculation,
solving by using equation (6) to obtain t n+1 Acceleration at timeThe calculation method comprises the following steps:
in the method, in the process of the invention,for calculating t n+1 A displacement vector of the moment; />For calculating t n+1 A velocity vector of the moment;
in the process of calculating and obtaining the accelerationThen, t is calculated by using an equation n+1 Speed of moment->And displacement U n+1 The calculation method is as follows:
t n+1 auxiliary acceleration A at time n+1 The calculation is updated by the following equation:
and ending the calculation until the moment N is the last moment N.
2. A single-step explicit step-wise integration method for solving a second order nonlinear dynamics problem according to claim 1, wherein: in step one ρ b And ρ s The condition should be satisfied and,
|ρ s |≤ρ b ∈0,1 ⑷
wherein ρ is b And ρ s Two parameters specified for the user.
3. A single-step explicit step-wise integration method for solving a second order nonlinear dynamics problem according to claim 2, wherein: parameter ρ b The spectrum radius value of the algorithm at the bifurcation point is controlled, and the numerical value dissipation quantity at the bifurcation point is further controlled; parameter ρ s The amount of numerical dissipation of the algorithm in the low frequency region is controlled.
4. A single-step explicit step-wise integration method for solving a second order nonlinear dynamics problem according to claim 3, wherein: by default, recommended user usage ρ b ＝ρ s To reduce one user-specified parameter.
5. The single-step explicit step-wise integration method for solving second-order nonlinear dynamics problems according to claim 4, wherein: when maximum value dissipation is achieved, ρ is made b ＝ρ s =0 for one-step cancellation of spurious high frequency responses in the structural response.
6. The single-step explicit step-by-step integration method for solving second-order nonlinear dynamics problems according to claim 5, wherein: in step two, the partial differential equation of the dominant structure is degenerated to a second order differential equation of motion of the form after the application of the spatial discrete technique:
wherein U (t),And->Displacement, velocity and acceleration vectors, respectively.
7. The single-step explicit step-wise integration method for solving second-order nonlinear dynamics problems according to claim 6, wherein: to determine a unique set of numerical solutions, an initial displacement U (t 0 ) And velocity vectorMust be given, i.e. U (t 0 )＝U 0 And->
8. Use of a single-step explicit step-by-step integration method for solving a second order nonlinear dynamics problem according to any of the claims 1-7, characterized in that: the method can be used for dynamic calculation of the folding control surface with the gap nonlinearity, and the formula (4) is required to be adjusted according to actual conditions.
9. A system for solving a single-step explicit step-by-step integration method of a second-order nonlinear dynamics problem is characterized in that: the system having program modules corresponding to the steps of any of the preceding claims 1-7, the steps of the single-step explicit step-by-step integration method for solving a second order nonlinear dynamics problem being performed at run-time.
10. A computer-readable storage medium, characterized by: the computer readable storage medium stores a computer program configured to implement the steps of the single-step explicit step-by-step integration method of solving a second order nonlinear dynamics problem of any one of claims 1-7 when called by a processor.
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