CN116956498A - Rapid robustness design technology for weapon multi-body system - Google Patents
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Abstract
Description
技术领域:Technical areas:
本发明涉及武器多体系统鲁棒性设计领域,特别涉及考虑偶然不确定性因素影响的武器多体系统减振鲁棒性设计。The present invention relates to the field of robust design of multi-body weapons systems, and in particular to the robust design of vibration reduction of multi-body weapons systems that takes into account the influence of accidental uncertainty factors.
背景技术:Background technique:
武器装备领域中大多数机械系统都可视为由“体单元”和“铰单元”以一定连接方式组成的多体系统。保证多体机械系统动力学(振动)特性分析的精度和效率,并根据系统振动特性进行高效、合理的减振设计,对武器装备性能提升至关重要。多体系统动力学方法和有限元法是武器多体系统动力学研究的主要方法。但所有这些方法均需建立系统总体动力学方程,涉及的系统矩阵阶次正比于系统自由度数。随着系统矩阵阶次的提高,多体系统动力学的计算速度成指数快速降低,因为复杂武器系统总体动力学方程涉及矩阵阶次高而导致计算速度慢,无法满足复杂武器系统动力学动态设计快速计算需要。Most mechanical systems in the field of weapons and equipment can be regarded as multi-body systems composed of "body units" and "hinge units" connected in a certain way. Ensuring the accuracy and efficiency of dynamic (vibration) characteristics analysis of multi-body mechanical systems and conducting efficient and reasonable vibration reduction designs based on system vibration characteristics are crucial to improving the performance of weapons and equipment. The multi-body system dynamics method and the finite element method are the main methods for studying the dynamics of weapon multi-body systems. However, all these methods need to establish the overall dynamics equation of the system, and the order of the system matrix involved is proportional to the number of degrees of freedom of the system. As the order of the system matrix increases, the calculation speed of multi-body system dynamics decreases exponentially. Because the overall dynamics equation of the complex weapon system involves a high matrix order, the calculation speed is slow and cannot meet the dynamic design of complex weapon system dynamics. Fast calculations are needed.
由于在设计、制造和使用过程中不可避免地存在一些不确定性因素,例如材料力学性能的随机性,几何尺寸的误差,外载荷的分散性。这些不确定性因素往往进一步影响着武器多体系统的性能和安全性,成为现阶段武器装备性能提升的极大阻碍。因此,开发一种高效的不确定武器多体系统快速鲁棒性设计方法,对现阶段提高武器装备性能,保障装备安全性、节约装备成本、保障其战斗力等,都具有重要的理论和实际意义。Because there are inevitably some uncertainties in the design, manufacturing and use processes, such as the randomness of material mechanical properties, geometric size errors, and the dispersion of external loads. These uncertain factors often further affect the performance and safety of multi-body weapons systems, becoming a great obstacle to improving the performance of weapons and equipment at this stage. Therefore, developing an efficient and fast robust design method for uncertain weapon multi-body systems has important theoretical and practical significance for improving the performance of weapons and equipment, ensuring equipment safety, saving equipment costs, and ensuring its combat effectiveness at this stage. .
多体系统动力学是当今力学研究热点之一,是兵器、船舶、航空、航天、车辆、通用机械等工业大量产品动态性能设计与试验的重要基础。但近50年迅速发展起来的各类多体系统动力学方法普遍存在如下特征:(1)必须建立系统总体动力学方程;(2)系统拓扑结构一旦改变,系统总体动力学方程需重新推导;(3)复杂系统总体动力学方程涉及矩阵阶次高(通常不小于系统的自由度数),计算速度随系统规模增大而明显减慢。多体系统传递矩阵法无需系统总体动力学方程并使系统矩阵阶次远低于系统自由度数,可实现多刚体系统动力学、多刚柔体系统动力学、受控多体系统动力学、特别是复杂发射系统动力学的快速计算。Multi-body system dynamics is one of the hot topics in mechanical research today and is an important basis for the dynamic performance design and testing of a large number of industrial products such as weapons, ships, aviation, aerospace, vehicles, and general machinery. However, various multi-body system dynamics methods that have developed rapidly in the past 50 years generally have the following characteristics: (1) the overall dynamics equation of the system must be established; (2) once the system topology changes, the overall dynamics equation of the system needs to be re-derived; (3) The overall dynamics equation of a complex system involves a high matrix order (usually not less than the number of degrees of freedom of the system), and the calculation speed slows down significantly as the system scale increases. The multi-body system transfer matrix method does not require the overall system dynamics equation and makes the system matrix order much lower than the system degrees of freedom. It can realize multi-rigid body system dynamics, multi-rigid and flexible body system dynamics, controlled multi-body system dynamics, special It is a fast calculation of the dynamics of complex launch systems.
对于不确定性多体动力学,现有国内外的相关研究工作均较少,多采用区间方法处理多体系统中偶然和认知混合的多源不确定性,而针对武器多体系统的相关研究工作更是乏善可陈。区间方法的优点是只需确定随机参数上下界,所需先验统计信息较少,但缺点是精度较差,容易“过度估计”随机性对多体系统动力学响应的影响。因此,该类方法不适用于考虑多源不确定性的武器多体系统振动分析问题。Regarding uncertain multi-body dynamics, there are currently few related research works at home and abroad. Interval methods are mostly used to deal with the multi-source uncertainties mixed by chance and cognition in multi-body systems. However, for the correlation of multi-body weapons systems, Research work is even more lackluster. The advantage of the interval method is that it only needs to determine the upper and lower bounds of the random parameters and requires less prior statistical information. However, the disadvantage is that the accuracy is poor and it is easy to "overestimate" the impact of randomness on the dynamic response of the multi-body system. Therefore, this type of method is not suitable for vibration analysis problems of weapon multi-body systems that consider multi-source uncertainties.
在不确定性多体系统动力学分析研究的基础上,针对多体系统鲁棒性设计的研究工作也基本以区间方法处理多源不确定性参数,并采用双循环方法求解鲁棒性设计优化问题,且目前还没有针对不确定性武器多体系统鲁棒性设计的相关研究工作。需要指出:鲁棒性设计为内外嵌套的双循环,内部需要进行考虑多源不确定性随机输入的多体系统不确定性传播分析(同时还需考虑多体系统动力学分析本身的精度和效率),外部需要进行优化求解。因此,考虑多源不确定性的多体系统鲁棒性设计目前面临着多源不确定性动力学建模、分析的精度和效率较低和鲁棒性设计求解效率较低等难题。在包含典型多体系统的武器装备领域更是如此,这严重制约了现阶段我国武器装备性能的快速提升。Based on the dynamic analysis and research of uncertain multi-body systems, the research work on the robust design of multi-body systems also basically uses the interval method to deal with multi-source uncertainty parameters, and uses the double-loop method to solve the robust design optimization problem, and there is currently no relevant research work on the robust design of uncertain weapon multi-body systems. It should be pointed out that the robustness design is a double loop nested inside and outside, and the multi-body system uncertainty propagation analysis that considers the random input of multi-source uncertainties needs to be carried out internally (it also needs to consider the accuracy and accuracy of the multi-body system dynamics analysis itself). efficiency), external optimization solution is required. Therefore, the robust design of multi-body systems that considers multi-source uncertainties currently faces problems such as low accuracy and efficiency of multi-source uncertainty dynamics modeling and analysis, and low efficiency of robust design solutions. This is especially true in the field of weapons and equipment containing typical multi-body systems, which seriously restricts the rapid improvement of the performance of my country's weapons and equipment at this stage.
发明内容:Contents of the invention:
本发明的技术解决问题:综合国内外研究现状,本发明旨在提高不确定性武器多体系统减振鲁棒性设计的计算效率,通过构建武器多体系统自适应刚-柔耦合动力学分析模型,实现基于传递矩阵法的武器多体系统动力学建模。同时,本发明提出整合自适应混合代理模型、多目标子集模拟优化方法、加权单目标优化方法及单循环求解策略的武器多体系统减振鲁棒性设计高效求解方法,提高整个武器多体系统减振鲁棒性设计过程的效率。The technical problem solved by this invention: Based on the current research status at home and abroad, this invention aims to improve the calculation efficiency of the vibration reduction robustness design of the uncertain weapon multi-body system, by constructing adaptive rigid-soft coupling dynamics analysis of the weapon multi-body system model to implement dynamic modeling of weapon multi-body systems based on the transfer matrix method. At the same time, the present invention proposes an efficient solution method for the vibration reduction robustness design of a multi-body weapon system that integrates an adaptive hybrid agent model, a multi-objective subset simulation optimization method, a weighted single-objective optimization method and a single-loop solution strategy, thereby improving the overall multi-body weapon system. Efficiency of system vibration damping robustness design process.
本发明的技术解决方案:一种武器多体系统快速鲁棒性设计技术,其分析过程可分为三个步骤:首先,快速求解武器多体系统振动响应;然后,得出最优代理模型;最后,基于最优代理模型,高效求解武器多体系统鲁棒性设计问题;整个快速分析与设计方法包含三个模块,即:多体动力学高效建模与分析模块、自适应混合代理模块、鲁棒性设计求解策略模块。The technical solution of the present invention: a rapid robustness design technology for a multi-body weapon system. The analysis process can be divided into three steps: first, quickly solve the vibration response of the multi-body weapon system; then, obtain the optimal surrogate model; Finally, based on the optimal agent model, the robustness design problem of the weapon multi-body system is efficiently solved; the entire rapid analysis and design method includes three modules, namely: multi-body dynamics efficient modeling and analysis module, adaptive hybrid agent module, Robust design solution strategy module.
多体动力学高效建模与分析模块:求解刚性身管及柔性身管刚柔耦合动力学模型,以大运动小变形柔性直梁动力学方程推导传递矩阵方程。Multi-body dynamics efficient modeling and analysis module: solve the rigid-flexible coupling dynamic model of rigid body tube and flexible body tube, and derive the transfer matrix equation based on the dynamic equation of flexible straight beam with large motion and small deformation.
由线性多体系统传递矩阵法建立舰炮动力学响应求解模型,如The naval gun dynamic response solution model is established by the linear multi-body system transfer matrix method, such as
图1所示,其中元件1为炮座,元件3为回转体,元件5为俯仰体,元件7为炮尾,元件8为身管,元件6为滑移铰,元件2、4为弹性铰,其中元件3、5和7为刚体,元件8为弹性体,俯仰体5的射角为θ。As shown in Figure 1, element 1 is the gun base, element 3 is the rotating body, element 5 is the pitching body, element 7 is the breech, element 8 is the barrel, element 6 is the sliding hinge, and elements 2 and 4 are elastic hinges. , where elements 3, 5 and 7 are rigid bodies, element 8 is an elastic body, and the angle of incidence of the pitching body 5 is θ.
(1)刚性身管(1) Rigid body tube
模型每个元件输入输出状态矢量均为:The input and output state vectors of each component of the model are:
每个元件传递方程均为The transfer equation for each element is
ZO=UZI (2)Z O =UZ I (2)
在新版多体系统传递矩阵法中,将弹性铰进行切断,分成两个子系统,弹性铰将当成外力和外力矩加入刚体传递矩阵中。总的传递矩阵为:In the new version of the multi-body system transfer matrix method, the elastic hinges are cut off and divided into two subsystems. The elastic hinges will be added to the rigid body transfer matrix as external forces and external moments. The total transfer matrix is:
两个传递方程所有的边界均为自由边界条件,形式如公式(4)All boundaries of the two transfer equations are free boundary conditions, in the form of formula (4)
(2)大运动小变形柔性直梁(2) Flexible straight beam with large movement and small deformation
以图2所示的一个单端输入单端输出空间运动直梁i为例,对于小变形空间运动直梁i,其整体节点位移列阵δi可用模态叠加形式表示为Taking a single-end input single-end output space motion straight beam i shown in Figure 2 as an example, for a small deformation space motion straight beam i, its overall node displacement array δ i can be expressed in the form of modal superposition as
式中 in the formula
为柔性梁i第j阶模态振型,/>为与其对应的第j阶变形广义坐标,M为选取的模态振型的个数。 is the jth-order mode shape of flexible beam i,/> is the corresponding generalized coordinate of the j-th order deformation, and M is the number of selected mode shapes.
1)直梁运动学方程1)Kinematic equation of straight beam
计及轴向缩短效应的空间运动直梁i节点k在全局惯性坐标系中的位置矢量可表示为Taking into account the axial shortening effect, the position vector of the spatially moving straight beam i node k in the global inertial coordinate system can be expressed as
式中为柔性梁i节点k线位移变形模态振型,lp为空间运动直梁i单元p的初始长度,v和w分别为梁单元中性轴上的点沿浮动坐标系y轴和z轴的变形。H1=[1 0 0]T为常数矩阵。in the formula is the displacement deformation mode shape of the flexible beam i node k line, l p is the initial length of the space motion straight beam i unit p, v and w are the points on the neutral axis of the beam unit along the y and z axes of the floating coordinate system respectively. of deformation. H 1 =[1 0 0] T is a constant matrix.
利用单元形函数矩阵可以将式(8)中的最后一项改写为Using the unit shape function matrix, the last term in equation (8) can be rewritten as
式中in the formula
δip为柔性梁i单元p的节点变形位移列阵,Nv(x)和Nw(x)分别为与v和w对应的形函数矩阵,N′·(x)表示形函数矩阵N·(x)对柔性梁纵坐标x的一阶导数。从式(10)不难发现为对称矩阵。δ ip is the node deformation displacement array of unit p of flexible beam i, N v (x) and N w (x) are the shape function matrices corresponding to v and w respectively, N′ · (x) represents the shape function matrix N · (x) The first derivative with respect to the ordinate x of the flexible beam. It is not difficult to find from equation (10) is a symmetric matrix.
将式(5)代入式(9),进一步整理可得Substitute equation (5) into equation (9) and further organize it to get
式中in the formula
将式(11)代入式(8)可得Substituting equation (11) into equation (8), we can get
式(13)对时间求一阶导数可得空间运动直梁i节点k在全局惯性坐标系中的绝对速度矢量,即Calculating the first derivative of equation (13) with respect to time can obtain the absolute velocity vector of node k of the spatially moving straight beam i in the global inertial coordinate system, that is
式中为柔性梁i浮动坐标系的绝对角速度矢量在全局惯性坐标系中的投影。由角速度叠加定理可得节点坐标系/>相对于全局惯性坐标系的角速度矢量为in the formula is the projection of the absolute angular velocity vector of the floating coordinate system of flexible beam i in the global inertial coordinate system. The node coordinate system can be obtained from the angular velocity superposition theorem/> The angular velocity vector relative to the global inertial coordinate system is
式中为柔性梁i节点k角位移变形模态振型。in the formula is the angular displacement deformation mode shape of node k of the flexible beam.
式(14)和(15)可整合为Equations (14) and (15) can be integrated into
式中in the formula
式(18)的若丹变分形式为The Jordan variation form of equation (18) is
式(19)对时间求一阶导数可得空间运动直梁i节点k的绝对加速度(包括角加速度),即Calculating the first derivative of equation (19) with respect to time can obtain the absolute acceleration (including angular acceleration) of node k of the straight beam i in space, that is
式中的具体表达式为in the formula The specific expression of
2)直梁动力学方程2) Straight beam dynamic equation
利用若丹变分原理列写的空间运动直梁i的虚功率方程为The virtual power equation of the space moving straight beam i written using the Rodin variation principle is:
式中为在浮动坐标系中表示的柔性梁i节点k的节点质量矩阵,/>为在浮动坐标系中的表示的节点转动惯量矩阵,/>和/>分别为柔性梁i节点k受到的节点外力和节点外力矩在全局惯性坐标系中的投影,/>和/>分别为柔性梁i广义变形质量矩阵和广义变形刚度矩阵,α和β为广义变形阻尼系数。/>和Ωi,I,/>和Ωi,O分别为单端输入单端输出空间运动柔性直梁i输入输出端的绝对速度和绝对角速度;qi,I和mi,I,qi,O和mi,O分别为输入输出端的内力和内力矩。qi,I沿坐标轴正向为正,mi,I沿坐标轴反向为正,qi,O和mi,O的正方向与qi,I和mi,I相反。in the formula is the node mass matrix of node k of flexible beam i represented in the floating coordinate system, /> is the node moment of inertia matrix represented in the floating coordinate system,/> and/> are respectively the projection of the nodal external force and the nodal external moment on the global inertial coordinate system of the flexible beam i and node k,/> and/> are the generalized deformation mass matrix and generalized deformation stiffness matrix of the flexible beam i, respectively, α and β are the generalized deformation damping coefficients. /> andΩi ,I ,/> and Ω i, O are respectively the absolute velocity and absolute angular velocity of the input and output ends of the single-end input single-end output space motion flexible straight beam i; q i, I and m i, I , q i, O and m i, O are the inputs respectively. Internal force and internal moment at the output end. q i, I is positive along the forward direction of the coordinate axis, m i, I is positive along the reverse coordinate axis, and the positive directions of q i, O and m i, O are opposite to q i, I and m i, I.
将式(19)和(20)代入式(23),整理可得Substituting equations (19) and (20) into equation (23), we can get
式中Mi为柔性梁i广义质量矩阵,Qi,I和Qi,O为柔性梁i输入输出端内力列阵,为由柔性梁i离心惯性力、科氏惯性力、广义外力和广义弹性力组成的广义力列阵。各矩阵的具体形式分别为/> In the formula, M i is the generalized mass matrix of flexible beam i, Q i, I and Q i, O is the internal force array at the input and output ends of flexible beam i, It is a generalized force array composed of centrifugal inertial force, Coriolis inertial force, generalized external force and generalized elastic force of the flexible beam i. The specific forms of each matrix are/>
其中各分块矩阵的具体表达式为The specific expression of each block matrix is
Mrr=I1 M rr =I 1
MFF=I6 M FF =I 6
式中in the formula
表示柔性体i浮动坐标系相对于全局惯性坐标系的绝对角速度矢量在浮动坐标系中的投影。Represents the projection of the absolute angular velocity vector of the floating coordinate system of the flexible body i relative to the global inertial coordinate system in the floating coordinate system.
分块矩阵表达式中的23个常矩阵分别为The 23 constant matrices in the block matrix expression are respectively
式中in the formula
H1=[1 0 0]T,H2=[0 1 0]T,H3=[0 0 1]T (56)H 1 = [1 0 0] T , H 2 = [0 1 0] T , H 3 = [0 0 1] T (56)
由空间运动直梁i的虚功率方程(24)可得空间运动直梁i的动力学方程,即From the virtual power equation (24) of the space-moving straight beam i, the dynamic equation of the space-moving straight beam i can be obtained, that is
3)直梁传递方程3) Straight beam transfer equation
与空间运动刚体类似,在新版多体系统传递矩阵法中,仅仅通过改写空间运动直梁的动力学和运动学方程的形式即可导出相应的元件传递方程,具体推导过程如下。Similar to the rigid body moving in space, in the new version of the multi-body system transfer matrix method, the corresponding component transfer equation can be derived simply by rewriting the dynamics and kinematic equations of the straight beam moving in space. The specific derivation process is as follows.
将空间运动柔性直梁i的运动学方程(20)的考查对象分别设为其输入和输出端,即可得到空间运动柔性直梁i输入输出端的加速度列阵,即By setting the test objects of the kinematics equation (20) of the flexible straight beam i in space as its input and output terminals respectively, the acceleration array of the input and output terminals of the flexible straight beam i in space can be obtained, that is
由空间运动直梁i的动力学方程(57)可得From the dynamic equation (57) of the space moving straight beam i, we can get
将式(60)分别代入式(58)和式(59)可得Substituting equation (60) into equation (58) and equation (59) respectively, we can get
式(61)和(62)可整理为Equations (61) and (62) can be organized as
式中in the formula
定义单端输入单端输出空间运动柔性直梁输入输出端的状态矢量/>(/>可以为输入端I或输出端O)形式为 Define the state vector of the input and output ends of the single-end input single-end output spatial motion flexible straight beam/> (/> It can be the input terminal I or the output terminal O). The form is
也即That is to say
则式(63)可改写为传递方程的形式,即Then equation (63) can be rewritten in the form of a transfer equation, that is
zi,O=Uizi,I (67)z i, O = U i z i, I (67)
式中Ui即为单端输入单端输出空间运动柔性直梁i的传递矩阵,具体形式为In the formula, U i is the transfer matrix of the single-end input single-end output spatial motion flexible straight beam i. The specific form is:
从式(67)可以看出单端输入单端输出空间运动直梁的状态矢量与单端输入单端输出刚体的状态矢量形式完全相同。在利用系统拓扑结构拼装系统总传递方程的过程中,可以简单地将单端输入单端输出空间运动直梁的传递矩阵像刚体一样与其它元件的传递矩阵进行运算。利用系统总传递方程以及系统边界条件可以求得系统边界点状态矢量中未知的状态变量,进而再次利用各元件传递方程求得系统中各联接端的状态矢量。由于单端输入单端输出空间运动直梁的状态矢量中不包含其柔性变形广义加速度因此在求得单端输入单端输出空间运动直梁输入输出端状态矢量zi,j和zi,O后需要借助于式(60)来求得它的柔性变形广义加速度/> It can be seen from Equation (67) that the state vector of a single-end input single-end output space motion straight beam is exactly the same as the state vector form of a single-end input single-end output rigid body. In the process of assembling the total transfer equation of the system using the system topology, the transfer matrix of the single-end input single-end output spatial motion straight beam can be simply calculated with the transfer matrices of other components like a rigid body. The unknown state variables in the system boundary point state vector can be obtained by using the total transfer equation of the system and the system boundary conditions, and then the state vectors of each connection end of the system can be obtained by using the transfer equation of each component. Since the state vector of the single-end input single-end output spatial motion straight beam does not include its flexible deformation generalized acceleration Therefore, after obtaining the single-end input single-end output spatial motion straight beam input and output end state vectors z i, j and z i, O, it is necessary to use equation (60) to obtain its flexible deformation generalized acceleration/>
自适应混合代理模块:Adaptive hybrid agent module:
(1)响应面方法(1) Response surface method
本模型中将高阶响应面法中切比雪夫(Chebyshev)多项式替换为埃尔米特(Hermite)多项式,并设计新的抽样方法、高阶响应面法的多项式最高阶数和交叉项的确定方法。这种响应面方法能够提高识别高阶多项式中各随机变量的最高阶数的准确性,并有效减少不必要的附加样本点进而提高高阶响应面法的计算效率。In this model, the Chebyshev polynomials in the high-order response surface method are replaced by Hermite polynomials, and a new sampling method is designed to determine the highest polynomial order and cross terms of the high-order response surface method. method. This response surface method can improve the accuracy of identifying the highest order of each random variable in a high-order polynomial, and effectively reduce unnecessary additional sample points to improve the computational efficiency of the high-order response surface method.
(2)自适应Kriging模型(2)Adaptive Kriging model
1)Kriging模型基本原理1) Basic principles of Kriging model
Kriging代理模型技术是一种基于已知样本点信息的一种空间插值技术,预测值可以表示为回归模型F(β,x)和随机过程Z(x)之和。其数学表达式为:Kriging surrogate model technology is a spatial interpolation technology based on known sample point information. The predicted value can be expressed as the sum of the regression model F (β, x) and the random process Z (x). Its mathematical expression is:
G(x)=F(β,x)+Z(x)=h(x)Tβ+Z(x) (69)G(x)=F(β,x)+Z(x)=h(x) T β+Z(x) (69)
其中,F(β,x)为Kriging代理模型的回归函数,h(x)T为Kriging代理模型的基函数,其形式决定了Kriging代理模型的回归形式,主要包括常数、线性函数以及二次函数模型,本文中选取常数项作为代理模型的回归形式。相关系数β为回归参数。Z(x)为随机过程,其均值为0,不同样本点预测值之间的协方差矩阵可以定义为:Among them, F (β, x) is the regression function of the Kriging surrogate model, h (x) T is the basis function of the Kriging surrogate model, and its form determines the regression form of the Kriging surrogate model, which mainly includes constants, linear functions and quadratic functions. Model, in this article, the constant term is selected as the regression form of the proxy model. The correlation coefficient β is the regression parameter. Z(x) is a random process with a mean value of 0. The covariance matrix between the predicted values of different sample points can be defined as:
COV(Z(xi),Z(xj))=σZ 2R(xi,xj,θ) (70)COV(Z(x i ), Z(x j ))=σ Z 2 R(x i , x j , θ) (70)
其中,xi和xj代表测试样本点,σZ 2为过程方差,R(xi,xj,θ)为xi和xj之间的相关性函数,其主要形式包括:高斯型函数、指数型函数以及Cubic函数等,本文中选取高斯型函数作为不同样本点之间的相关性函数:Among them, x i and x j represent test sample points, σ Z 2 is the process variance, R( xi , x j , θ) is the correlation function between x i and x j . Its main forms include: Gaussian function , exponential function and Cubic function, etc. In this article, a Gaussian function is selected as the correlation function between different sample points:
其中,n为随机变量的维度,xi k,xj k和θk为xi,xj和θ的k维分量。其参数θ可以通过最大似然估计方法得到:Among them, n is the dimension of the random variable, x i k , x j k and θ k are the k-dimensional components of x i , x j and θ. Its parameter θ can be obtained through the maximum likelihood estimation method:
其中,m为测试样本点数量。当θ确定后,回归参数β以及过程方差σZ 2可以表示为:Among them, m is the number of test sample points. When θ is determined, the regression parameter β and process variance σ Z 2 can be expressed as:
β*=(FTR-1F)-1FTR-1Y (73)β * =(F T R -1 F) -1 F T R -1 Y (73)
则Kriging代理模型在预测点集的均值和方差可以定义为:Then the mean and variance of the Kriging surrogate model in the prediction point set can be defined as:
其中,u=FTR-1r(x)-f(x),Y为训练样本点集的真实响应集,r为预测样本点集与训练样本点集之间的相关系数矩阵。综上所述,预测样本点集合x利用Kriging代理模型的预测值服从高斯分布:Among them, u=F T R -1 r(x)-f(x), Y is the true response set of the training sample point set, and r is the correlation coefficient matrix between the predicted sample point set and the training sample point set. To sum up, the predicted value of the predicted sample point set x using the Kriging surrogate model obeys the Gaussian distribution:
2)自适应方案2) Adaptive solution
自适应过程是在探索未知区域模型特征时,根据已知样本点的数据特征自动的调整、补充其模型处理参数及其处理方法,使其与所需处理目标数据的统计分布特征或其结构特征一致;充分利用已知样本点信息,不断地逼近所需目标,取得最佳效果的过程。相比于传统的单步建模方法,自适应建模方法可以利用更少的试验方案得到相对更加准确的结果。在小样本可靠性分析领域具有重要的研究价值。自适应过程的流程如The adaptive process is to automatically adjust and supplement the model processing parameters and processing methods according to the data characteristics of known sample points when exploring the model characteristics of unknown areas, so that they are consistent with the statistical distribution characteristics or structural characteristics of the target data to be processed. Consistent; the process of making full use of known sample point information to continuously approach the required target and achieve the best results. Compared with the traditional single-step modeling method, the adaptive modeling method can use fewer test plans to obtain relatively more accurate results. It has important research value in the field of small sample reliability analysis. The flow of the adaptive process is as follows:
图3所示。As shown in Figure 3.
3)U-H混合加点准则3) U-H mixed point addition criteria
U学习函数识别符号误判概率最高的样本更新代理模型,H学习函数选择预测不确定性最大的样本对模型进行细化。因此,本模块更新策略将结合U学习函数和H学习函数,提高了代理模型的准确性和更新效率。The U learning function identifies samples with the highest probability of symbol misjudgment to update the surrogate model, and the H learning function selects samples with the highest prediction uncertainty to refine the model. Therefore, the update strategy of this module will combine the U learning function and the H learning function to improve the accuracy and update efficiency of the agent model.
4)基于U-H混合加点准则的自适应Kriging模型4) Adaptive Kriging model based on U-H mixed point addition criterion
基于自适应方案和U-H混合加点准则,自适应Kriging模型的算法流程如下:Based on the adaptive scheme and the U-H mixed point addition criterion, the algorithm flow of the adaptive Kriging model is as follows:
①.参数设置。设置初始样本数N来构建初始Kriging模型。①.Parameter setting. Set the initial number of samples N to build the initial Kriging model.
②.随机过程离散化。时间间隔为[0,tf],首先离散为s=tf/Δt+1,时间节点为ti,i=1,2...s其时间步长为Δt。用EOLE方法将随机过程Y(t)转化为标准正规变量Z的函数。②. Discretization of random processes. The time interval is [0, t f ], and the first discretization is s=t f /Δt+1. The time node is t i , i=1, 2...s, and its time step is Δt. Use the EOLE method to transform the random process Y(t) into a function of the standard normal variable Z.
③.初始训练样本集生成。生成样本池D。然后,从D中选择N个样本W并评价其相应的性能函数的瞬时响应Gn=g(W,ti),i=1,2,...,s。并由W和构成初始训练样本集 ③.Generate initial training sample set. Generate sample pool D. Then, select N samples W from D and evaluate the instantaneous responses of their corresponding performance functions G n =g (W, ti ), i = 1, 2, ..., s. And by W and Constitute the initial training sample set
④.构建Kriging模型。通过使用MATLAB工具箱UQLAB建立基于T的Kriging模型。这里使用高斯相关函数。④.Construct the Kriging model. Create a T-based Kriging model by using the MATLAB toolbox UQLAB. Gaussian correlation function is used here.
⑤.判断样本符号的正确性。根据等式中表示的U-学习函数的停止准则,检验D中样本的预测符号是否都可信(23)。如果满足等式(23),转到步骤6。否则样品D中满足等式(23)的样品被组成到样本池然后转到步骤9。⑤. Determine the correctness of the sample symbols. According to the stopping criterion of the U-learning function expressed in Eq., check whether the predicted symbols of the samples in D are all credible (23). If equation (23) is satisfied, go to step 6. Otherwise, the samples in sample D that satisfy equation (23) are composed into the sample pool Then go to step 9.
⑥.通过H-学习标准来识别最佳样本。基于H-学习准则寻求在D中的最佳样本 ⑥.Identify the best samples through the H-learning criterion. Find the best sample in D based on H-learning criterion
⑦.H-学习函数的停止准则。检查H-学习函数的停止准则是否满足。如果没有,则将转到步骤8,否则转到步骤12。⑦.H-stop criterion of learning function. Check whether the stopping criterion of the H-learning function is satisfied. If not, you will go to step 8, otherwise go to step 12.
⑧.评估在最佳样本上的真实瞬时响应。在样本计算性能函数的真实瞬时响应增加/>和/>到训练集W和/>分别表示为/>和/>然后,返回至步骤3。⑧.Evaluate the true instantaneous response on the best sample. in sample Calculate the true instantaneous response of the performance function Add/> and/> to the training set W and/> Respectively expressed as/> and/> Then, return to step 3.
⑨.通过U-学习标准和H-学习标准识别两个最佳样本。基于U-学习标准和H-学习标准找到两个最佳样本值得注意的是,/>在D中被找到,/>在D1中被找到。⑨.Identify the two best samples through U-learning criteria and H-learning criteria. Find the two best samples based on U-learning criterion and H-learning criterion It is worth noting that/> Found in D,/> Found in D 1 .
⑩.提出的停止准则。检查提出的停止准则是否满足。如果不满足,则转到步骤11,否则转到步骤12。⑩. Proposed stopping criteria. Check whether the proposed stopping criteria are met. If not satisfied, go to step 11, otherwise go to step 12.
评估在两个最佳样本上的真实瞬时响应。在样本/>和/>处计算性能函数的真实瞬间响应/>将/>和/>添加到初始集W和/>分别地,表示为和/>然后返回步骤3。 Evaluate the true instantaneous response on the two best samples. In sample/> and/> Calculate the true instantaneous response of the performance function/> Will/> and/> Add to initial set W and /> respectively, expressed as and/> Then return to step 3.
完成Kriging模型构建。 Complete the Kriging model construction.
(3)基于U-H混合加点准则的PCE-Kriging(PCK)代理模型(3) PCE-Kriging (PCK) agent model based on U-H mixed point addition criterion
本模块还结合多项式混沌展开(PCE)和Kriging模型的各自优点的改进代理建模技术,提出PCE-Kriging(PCK)代理模型,。PCE优点在于能够更好地捕获武器多体系统振动品质计算模型的全局行为,Kriging模型优点在于其服从正态分布,能够同时计算出计算预测点的均值和均方误差,有利用模型更新。PCK模型中两者结合方式则为PCE代替原Kriging的回归基函数部分来提高全局逼近精度,而局部变异性的近似仍然由原Kriging的随机过程提供。因此,PCK模型表达式为This module also combines the improved agent modeling technology with the respective advantages of polynomial chaos expansion (PCE) and Kriging model, and proposes the PCE-Kriging (PCK) agent model. The advantage of PCE is that it can better capture the global behavior of the vibration quality calculation model of the weapon multi-body system. The advantage of the Kriging model is that it obeys the normal distribution and can simultaneously calculate the mean and mean square errors of the calculation prediction points, making use of model updates. In the PCK model, the combination of the two is that PCE replaces the regression basis function part of the original Kriging to improve the global approximation accuracy, while the approximation of local variability is still provided by the stochastic process of the original Kriging. Therefore, the PCK model expression is
PCK模型继承了原始Kriging模型能够提供预测值及其均方误差特点,同时又增强了对全局近似的能力,因而在本项目中更具有实际应用价值。The PCK model inherits the original Kriging model's ability to provide predicted values and its mean square error characteristics, and at the same time enhances its global approximation capabilities, so it has more practical application value in this project.
基于U-H混合加点准则的武器多体系统动力学响应均方根的PCE-Kriging模型计算流程与相应自适应Kriging模型一致,仅需把模型构建部分替换即可。并在第一步输入PCE所需的最高阶数P。The PCE-Kriging model calculation process for the root mean square of the dynamic response of the weapon multi-body system based on the U-H mixed point criterion is consistent with the corresponding adaptive Kriging model, and only the model construction part is replaced. And enter the highest order P required by PCE in the first step.
(4)自适应混合代理模型(4)Adaptive hybrid agent model
基于三种代理模型方法:响应面方法、自适应Kriging方法及自适应PCE-Kriging方法,形成面向武器多体系统振动响应的自适应混合代理模型库,如Based on three surrogate model methods: response surface method, adaptive Kriging method and adaptive PCE-Kriging method, an adaptive hybrid surrogate model library for the vibration response of weapon multi-body systems is formed, such as
图4所示,然后使用构造简单的代理模型来替代复杂的武器多体系统结构分析模型,即模块一中所得的多体系统动力学方程。针对不同问题,通过面向武器多体系统动力学响应的拟合精度及效率进行综合对比,选用最优代理模型,具体分析过程将在后文具体实施方式中展示。As shown in Figure 4, a simple surrogate model is then used to replace the complex weapon multi-body system structural analysis model, that is, the multi-body system dynamics equation obtained in module one. For different problems, the optimal surrogate model is selected through a comprehensive comparison of the fitting accuracy and efficiency of the dynamic response of the weapon multi-body system. The specific analysis process will be shown in the specific implementation methods later.
鲁棒性设计求解策略模块:基于模块二所得的最优代理模型,本模块采用决策模型和优化方法策略对武器多体系统鲁棒性进行优化设计,决策模型与优化方法策略如下。Robust design solution strategy module: Based on the optimal agent model obtained in module two, this module uses decision-making models and optimization methods and strategies to optimize the robustness of the weapon multi-body system. The decision-making model and optimization methods and strategies are as follows.
(1)考虑偶然不确定性情形下的鲁棒性设计优化决策模型(1) Robust design optimization decision-making model considering accidental uncertainty
根据传统确定性优化模型,基于最优代理模型,考虑偶然不确定性情形下的鲁棒性设计优化决策模型如下::According to the traditional deterministic optimization model, based on the optimal agent model, the robust design optimization decision-making model considering accidental uncertainty is as follows::
其中,为上文所述基于最优代理模型的x及y方向的振幅在时间历程上二范数的平方和算术平方根均值,/>为相应标准差。优化目标为两个方向振幅均值及方差在时间历程上均方和最小。x为设计变量向量d对应的随机变量向量。in, is the square and arithmetic square root mean of the two norms of the amplitudes in the x and y directions on the time history based on the optimal surrogate model mentioned above,/> is the corresponding standard deviation. The optimization goal is to minimize the mean square sum of the amplitude mean and variance in the two directions over time. x is the random variable vector corresponding to the design variable vector d.
(2)加权单目标优化方法(2) Weighted single objective optimization method
武器多体系统减振鲁棒性优化设计是一个多目标多参数问题,其多目标数学模型如下:The optimization design of vibration reduction robustness of a multi-body weapon system is a multi-objective and multi-parameter problem. Its multi-objective mathematical model is as follows:
其中,x(t)及y(t)为上文所述x方向及y方向的振幅。n为设计变量d个数。该武器多体系统为双目标优化问题,可以采用多目标子集模拟优化求解,求解流程如Among them, x(t) and y(t) are the amplitudes in the x-direction and y-direction mentioned above. n is the number of design variables d. This weapon multi-body system is a dual-objective optimization problem, which can be solved using multi-objective subset simulation optimization. The solution process is as follows
图5所示。As shown in Figure 5.
直接求解多目标问题的最优解具有较高难度。因此,可以基于一定的工程经验,选取合适的权重系数,考虑通过加权方法将多目标问题转化为单目标问题,根据上述已确定的设计变量可构建出如下武器多体系统确定性减振优化模型。It is difficult to directly solve the optimal solution of multi-objective problems. Therefore, based on certain engineering experience, appropriate weight coefficients can be selected, and the multi-objective problem can be converted into a single-objective problem through the weighting method. Based on the above-identified design variables, the following deterministic vibration reduction optimization model of the weapon multi-body system can be constructed. .
式中ω1和ω2分别为x方向及y方向振幅总体效应的权重系数,考虑x和y方向的振幅均需有效抑制,此处选取ω1=ω2=0.5。由上式可以看出:原始双目标问题可以转化成单目标问题求解。In the formula, ω 1 and ω 2 are the weight coefficients of the overall effect of the amplitude in the x and y directions respectively. Considering that the amplitudes in both the x and y directions need to be effectively suppressed, ω 1 =ω 2 =0.5 is selected here. It can be seen from the above formula that the original dual-objective problem can be transformed into a single-objective problem to solve.
为了进一步缩减目标函数个数,避免目标函数个数太多给鲁棒性设计问题增加额外的计算负担;同时,考虑到设计目标为振动响应最小,可以将x和y方向的振幅在时间历程上的二范数的均方和,即:位移总效应作为优化目标。因此,公式(80)可以改写为:In order to further reduce the number of objective functions and prevent too many objective functions from adding additional computational burden to the robust design problem; at the same time, considering that the design goal is to minimize the vibration response, the amplitudes in the x and y directions can be calculated on the time history The mean square sum of the two norms, that is, the total displacement effect is used as the optimization objective. Therefore, formula (80) can be rewritten as:
(3)单循环求解策略(3)Single loop solution strategy
对于鲁棒性(稳健性)优化设计问题,其求解过程如For robustness (robustness) optimization design problems, the solution process is as follows
图6所示。可以看出:在每一步设计点的迭代时,均需要进行物理模型(仿真或实验,本项目为该舰载火炮多体动力学响应的最优代理模型)的不确定性传播分析。具体来说,需要进行大量模型调用分析,统计分析得到鲁棒性设计所需的均值及方差等特征量。因此,该问题实际上为一个双循环求解问题。外循环为优化过程,内循环为不确定性传播分析,即成千上万次统计计算,该双循环问题往往计算量十分巨大。As shown in Figure 6. It can be seen that at each iteration of the design point, uncertainty propagation analysis of the physical model (simulation or experiment, this project is the optimal proxy model for the multi-body dynamic response of the shipborne gun) is required. Specifically, a large number of model call analyzes are required, and statistical analysis is required to obtain feature quantities such as mean and variance required for robust design. Therefore, the problem is actually a double loop solution problem. The outer loop is an optimization process, and the inner loop is uncertainty propagation analysis, that is, tens of thousands of statistical calculations. This double-loop problem often requires a huge amount of calculations.
针对上述多目标优化问题和双循环求解问题,本模块整合加权单目标优化方法、多目标子集模拟优化方法和单循环鲁棒性求解策略进行减振鲁棒性设计。In view of the above-mentioned multi-objective optimization problems and double-loop solution problems, this module integrates the weighted single-objective optimization method, the multi-objective subset simulation optimization method and the single-loop robust solution strategy to conduct vibration reduction robustness design.
具体流程如图7:首先,初始化设计变量,并利用加权单目标优化方法和多目标子集模拟优化方法进行优化,再计算所得样本点之间的欧氏距离,找到距离任意样本距离最近的点x,由它邻域欧氏距离T内的样本点集(x1,...,xk)得到种子样本矩阵,对它们进行鲁棒性优化设计,根据F=F(x),取响应值最大的样本点作为阈值样本,对应响应值为F(xk)。The specific process is shown in Figure 7: First, initialize the design variables, and use the weighted single-objective optimization method and the multi-objective subset simulation optimization method to optimize, then calculate the Euclidean distance between the obtained sample points, and find the point closest to any sample. x, obtain the seed sample matrix from the sample point set (x 1 ,...,x k ) within the Euclidean distance T of its neighborhood, perform robust optimization design on them, and obtain the response according to F=F(x) The sample point with the largest value is used as the threshold sample, and the corresponding response value is F(x k ).
然后,计算新的样本点xk+1,并得出对应的响应值F(xk+1),若F(xk+1)≤F(xk),则更新其邻域(x1,...,xk,xk+1),以此更新种子样本矩阵,并将样本点xk+1加入信息矩阵。Then, calculate the new sample point x k+1 and obtain the corresponding response value F(x k+1 ). If F(x k+1 )≤F(x k ), update its neighborhood (x 1 ,..., x k , x k+1 ), thereby updating the seed sample matrix, and adding the sample point x k+1 to the information matrix.
最后,如果满足终止准则或计算的样本数达到最大可承受值,则停止并输出当前数值。否则,继续更新样本。Finally, if the termination criterion is met or the calculated number of samples reaches the maximum tolerable value, stop and output the current value. Otherwise, continue updating the sample.
本发明与现有技术相比的优点在于:The advantages of the present invention compared with the prior art are:
(1)采用新型传递矩阵法构建武器多体系统刚柔耦合的动力学模型,在保证输出结果满足精度要求的情况下,大大提高仿真模拟效率(1) The new transfer matrix method is used to construct a rigid-flexible coupling dynamic model of the weapon multi-body system, which greatly improves the simulation efficiency while ensuring that the output results meet the accuracy requirements.
(2)针对不确定武器多体系统快速鲁棒性振动响应多保真要求,整合响应面方法、自适应Kriging方法及自适应PCE-Kriging方法,形成针对不同多体系统的自适应混合代理模型库,综合考虑代理模型的精度和效率,给用最优代理模型方案。(2) In view of the multi-fidelity requirements for fast and robust vibration response of uncertain weapon multi-body systems, the response surface method, adaptive Kriging method and adaptive PCE-Kriging method are integrated to form an adaptive hybrid agent model for different multi-body systems. The library takes into account the accuracy and efficiency of the proxy model and uses the optimal proxy model solution.
(3)构建了考虑偶然不确定性情形下的鲁棒性设计优化决策模型,采用加权单目标求解策略、单循环求解策略以及多目标子集模拟优化方法,形成面向不确定武器多体系统快速鲁棒性设计方法,极大提高了武器多体系统鲁棒性设计的效率。(3) A robust design optimization decision-making model considering accidental uncertainty was constructed, using a weighted single-objective solution strategy, a single-loop solution strategy, and a multi-objective subset simulation optimization method to form a fast multi-body system for uncertain weapons. The robust design method greatly improves the efficiency of robust design of multi-body weapons systems.
附图说明:Picture description:
图1舰炮系统动力学响应计算模型示意图;Figure 1 Schematic diagram of the dynamic response calculation model of the naval gun system;
图2单端输入单端输出空间运动直梁示意图;Figure 2 Schematic diagram of a single-end input single-end output spatial motion straight beam;
图3自适应过程流程图;Figure 3 Adaptive process flow chart;
图4混合代理模型示意图;Figure 4 Schematic diagram of the hybrid agent model;
图5多目标子集模拟优化流程图;Figure 5 Multi-objective subset simulation optimization flow chart;
图6双循环鲁棒性设计流程图;Figure 6 Dual-loop robustness design flow chart;
图7单循环鲁棒性设计流程图;Figure 7 Single-cycle robustness design flow chart;
图8多目标子集模拟偶然不确定性优化帕累托前沿;Figure 8 Multi-objective subset simulation of accidental uncertainty optimization Pareto front;
图9加权单目标子集模拟偶然不确定性优化迭代过程Figure 9 Weighted single-objective subset simulation of accidental uncertainty optimization iterative process
图10多目标子集模拟偶然不确定性优化帕累托前沿Figure 10 Multi-objective subset simulation of accidental uncertainty optimization Pareto front
具体实施方式:Detailed ways:
下面结合附图和实例对本发明做进一步说明。以某舰炮减振鲁棒性分析过程为实例。The present invention will be further described below in conjunction with the accompanying drawings and examples. Take the vibration reduction robustness analysis process of a certain naval gun as an example.
(1)多体系统模型建立(1) Establishment of multi-body system model
由线性多体系统传递矩阵法(MSTMM)建立舰炮动力学响应求解模型,如The naval gun dynamic response solution model is established by the linear multibody system transfer matrix method (MSTMM), such as
图1所示,其中元件1为炮座,元件3为回转体,元件5为俯仰体,元件7为炮尾,元件8为身管,元件6为滑移铰,元件2、4为弹性铰,其中元件3、5和7为刚体,元件8为弹性体,俯仰体5的射角为θ。As shown in Figure 1, element 1 is the gun base, element 3 is the rotating body, element 5 is the pitching body, element 7 is the breech, element 8 is the barrel, element 6 is the sliding hinge, and elements 2 and 4 are elastic hinges. , where elements 3, 5 and 7 are rigid bodies, element 8 is an elastic body, and the angle of incidence of the pitching body 5 is θ.
动力学响应计算需要使用新版多体系统传递矩阵法,每个元件输入输出状态矢量均为:The calculation of dynamic response requires the use of the new version of the multi-body system transfer matrix method. The input and output state vectors of each element are:
每个元件传递方程均为The transfer equation for each element is
ZO=UZI (84)Z O =UZ I (84)
在NVMSTMM中,将弹性铰进行切断,分成两个子系统,弹性铰将当成外力和外力矩加入刚体传递矩阵中。总的传递矩阵为:In NVMSTMM, the elastic hinge is cut off and divided into two subsystems. The elastic hinge will be added to the rigid body transfer matrix as an external force and external moment. The total transfer matrix is:
若是考虑元件8为刚性梁,则U3,U5,U7,U8均为一端输入一段输出刚体传递矩阵,U6为滑移铰传递矩阵,相关的传递矩阵具体形式见附录二。If element 8 is considered to be a rigid beam, then U 3 , U 5 , U 7 , and U 8 are all one-end input and one output rigid body transfer matrices, and U 6 is the slip hinge transfer matrix. The specific form of the relevant transfer matrix is shown in Appendix 2.
两个传递方程所有的边界均为自由边界条件,形式如公式(4)All boundaries of the two transfer equations are free boundary conditions, in the form of formula (4)
在刚性身管舰炮多体动力学拓扑结构的基础上,将刚性身管考虑成柔性结构,采用大运动小变形柔性直梁的传递方程建模,得到炮口处X、Y方向的相对振动响应。Based on the multi-body dynamic topology of a rigid barrel naval gun, the rigid barrel is considered as a flexible structure, and the transfer equation of a flexible straight beam with large motion and small deformation is used to model the relative vibration in the X and Y directions at the muzzle. response.
在上述确定性多体动力学分析后,可以获得炮口在x及y方向在时间历程上的振动响应个及y(t)。为了综合评价炮口的振动特性,按如下相对振动响应的均方根量化其综合振动品质After the above deterministic multi-body dynamics analysis, the vibration responses of the muzzle in the x and y directions on the time history and y(t) can be obtained. In order to comprehensively evaluate the vibration characteristics of the muzzle, its comprehensive vibration quality is quantified according to the root mean square of the relative vibration response as follows:
(2)最优代理模型(2) Optimal agent model
利用自适应混合代理模型库,将三类代理模型方法,即:高阶响应面法、自适应Kriging模型及自适应PCK混合代理模型,面向武器多体系统动力学响应的拟合精度及效率进行综合对比,可给出某舰炮多体系统振动品质的最优代理模型选择方案,做为后续鲁棒性设计优化的数学模型输入,对比结果如下表所示。Using the adaptive hybrid surrogate model library, three types of surrogate model methods, namely: high-order response surface method, adaptive Kriging model and adaptive PCK hybrid surrogate model, are used to improve the fitting accuracy and efficiency of the dynamic response of the weapon multi-body system. Comprehensive comparison can provide the optimal surrogate model selection scheme for the vibration quality of a certain naval gun multi-body system, which can be used as the mathematical model input for subsequent robust design optimization. The comparison results are shown in the table below.
表1三种代理模型对比Table 1 Comparison of three agency models
在目前有限成本模型关系下,对比三个代理模型最大误差发现自适应PCK模型,相对误差最低,同时调用原始仿真模型次数最少,表现最好,用于后续的鲁棒性设计优化研究。Under the current limited cost model relationship, comparing the maximum errors of the three surrogate models, it was found that the adaptive PCK model has the lowest relative error and the least number of calls to the original simulation model, which performs best and can be used for subsequent robust design optimization research.
(3)武器多体系统鲁棒性设计(3) Robust design of weapon multi-body system
在最优代理模型的基础上,采用多目标子集模拟优化,分别给出单循环和双循环鲁棒性设计策略下的某舰载火炮多体系统炮口振动偶然不确定性鲁棒性决策模型及加权多目标偶然不确定性鲁棒性决策模型的计算结果。On the basis of the optimal surrogate model, multi-objective subset simulation optimization is used to provide the robustness decisions for the accidental uncertainty of muzzle vibration of a certain shipborne artillery multi-body system under single-cycle and double-cycle robust design strategies. Model and computational results of a weighted multi-objective contingent uncertainty robust decision-making model.
1)双循环鲁棒性设计方法1)Double cycle robust design method
采用双循环方法求解该鲁棒性设计问题。首先求解双目标优化问题,多目标子集模拟每层样本N(种群数)=100,p0取0.2,共进行迭代(种群数)20次,外层采用1700个样本,内层用Monte Carlo法求解每个样本点对应的振动品质均值及方差,每次需100次抽样,共需1700*100=170000次原始结构系统分析次数,所得帕累托前沿及结果如A double-loop method is used to solve this robust design problem. First, solve the dual-objective optimization problem. The multi-objective subset simulates each layer with N (population number) = 100 samples. p 0 is 0.2. A total of 20 iterations (population number) are performed. 1700 samples are used in the outer layer and Monte Carlo is used in the inner layer. The method is used to solve the vibration quality mean and variance corresponding to each sample point. Each time requires 100 samplings, and a total of 1700*100=170000 times of original structural system analysis is required. The resulting Pareto front and results are as follows
图8所示。As shown in Figure 8.
图中所有“○”构成了考虑振动品质均值及方差双目标下的可行域解集,其中振动品质均值最小为0.4148。为方便与后文加权单目标优化对比,折中选取其中振动品质均值为0.4427,标准差为0.0999的一组(第11组)解集对应设计值为:All "○" in the figure constitute a feasible region solution set considering the dual objectives of vibration quality mean and variance, in which the minimum vibration quality mean is 0.4148. In order to facilitate comparison with the weighted single-objective optimization described later, a compromise was made to select a group (Group 11) with a vibration quality mean value of 0.4427 and a standard deviation of 0.0999. The corresponding design values of the solution set are:
表2多目标子集模拟偶然不确定性优化最优设计Table 2 Multi-objective subset simulation optimal design for accidental uncertainty optimization
然后,求解加权转化后的单目标优化问题,子集模拟每层样本N(种群数)=100,p0取0.2,共进行迭代10次,外层采用820个样本,内层用Monte Carlo法求解每个样本点对应的振动品质均值及方差,每次需100次抽样,共需820*100=82000次原始结构系统分析次数,获得最小振动品质0.3966,迭代过程如Then, solve the single-objective optimization problem after weighted transformation. The subset simulates each layer with N (population number) = 100 samples, p 0 is 0.2, and a total of 10 iterations are performed. 820 samples are used in the outer layer, and the Monte Carlo method is used in the inner layer. To solve for the mean and variance of vibration quality corresponding to each sample point, 100 samplings are required each time. A total of 820*100=82000 times of original structural system analysis is required to obtain the minimum vibration quality of 0.3966. The iterative process is as follows
图9所示,所得最优设计为:As shown in Figure 9, the optimal design obtained is:
表3加权单目标偶然不确定性优化最优设计Table 3 Weighted single-objective accidental uncertainty optimization optimal design
对比两次优化结果,可以看出:对于偶然不确定性减振鲁棒性设计,获得差不多最小振动品质均值(多目标为0.4148,单目标为0.3966),多目标需要170000次原始结构系统分析次数,而加权单目标仅需82000次,对于该双目标问题,仅采用单目标加权法,效率就已提高一倍多。Comparing the two optimization results, it can be seen that for the accidental uncertainty vibration reduction robustness design, almost the minimum mean vibration quality is obtained (0.4148 for multi-objective and 0.3966 for single-objective). Multi-objective requires 170,000 times of original structural system analysis. , while the weighted single-objective method only needs 82,000 times. For this dual-objective problem, the efficiency has been more than doubled by using only the single-objective weighting method.
2)单循环鲁棒性设计方法2) Single cycle robustness design method
采用单循环方法求解上述问题,仅需次结构分析8720次数(样本),所得帕累托前沿及结果如Using the single cycle method to solve the above problem requires only 8720 times (samples) of secondary structure analysis. The resulting Pareto front and results are as follows
图10所示。As shown in Figure 10.
图中所有“○”构成了考虑振动品质均值及方差双目标下的可行域解集,其中振动品质均值最小为0.3027。为方便与后文加权单目标优化对比,折中选取其中振动品质均值为0.3642,标准差为0.0960的一组(第13组)解集对应设计值为:All "○" in the figure constitute a feasible region solution set considering the dual objectives of vibration quality mean and variance, in which the minimum vibration quality mean is 0.3027. In order to facilitate comparison with the weighted single-objective optimization described later, a compromise was made to select a group (Group 13) with a vibration quality mean value of 0.3642 and a standard deviation of 0.0960. The corresponding design values of the solution set are:
表4多目标子集模拟偶然不确定性优化最优设计Table 4 Multi-objective subset simulation optimal design for accidental uncertainty optimization
进一步可以发现:若采用单循环方法求解上述多目标问题,相较于原始双循环方法,考虑偶然不确定性的减振鲁棒性设计问题,计算效率进一步提高约17倍。It can further be found that if the single-loop method is used to solve the above-mentioned multi-objective problem, compared with the original double-loop method, the calculation efficiency of the vibration damping robustness design problem considering accidental uncertainty is further improved by about 17 times.
以上所述的仅为本发明的较佳实例,本发明不仅仅局限于上述实例,凡在本发明的精神和原则之内所做的局部改动、改进等均应包含在本发明的保护范围之内。The above are only preferred examples of the present invention. The present invention is not limited to the above examples. Any local changes, improvements, etc. made within the spirit and principles of the present invention should be included in the protection scope of the present invention. Inside.
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