# A reproducing kernel particle method for solving generalized probability density evolution equation in stochastic dynamic analysis

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## Abstract

Analysis of stochastic dynamic system is still an open research issue. Recently a family of generalized probability density evolution equation, which provides an available way for general nonlinear systems, is put forward. In this paper, a numerical method based on reproducing kernel particle method (RKPM) for the solution of generalized probability density evolution equation, named the refined algorithm based on RKPM, is developed. Besides, the corresponding implementation procedure is elaborated. In this method, the time dependent probability distributions of the responses of interest can be obtained with less computational efforts. In addition, the mesh sensitivity problem in traditional probability density evolution method is settled well. Some details of parameter analysis are also discussed. To verify both the efficiency and accuracy of the method, a single-degree-of-freedom example and a 10-story frame structure are investigated. The refined algorithm based on RKPM can be applied to uni-variable and multi-variable, one-dimensional and multi-dimensional systems.

## Keywords

Stochastic dynamic system Probability density evolution method Refined algorithm Reproducing kernel particle method## 1 Introduction

In many practical engineering problems, there exits unavoidable randomness [1, 2]. For instance, the civil engineering structures will encounter the random external excitations such as earthquakes and strong winds during its service period. At the same time, the properties of the structure itself are often of significantly random. In order to reflect the influence of the above randomness on the structural response, it is necessary to study the theory for analysis of structural stochastic dynamic system.

Various approaches are successfully applied to the stochastic dynamic analysis of linear systems. However, for nonlinear systems particularly for systems with high dimensionality and strong nonlinearity, most of these approaches are not available [3]. For instance, the Monte Carlo simulation (MCS) [4], the random perturbation approach [5] and the orthogonal expansion method [6] are general methods in the stochastic structural analysis [7, 8]. Nevertheless, the MCS is mainly limited by its heavy computational burden of simulation experiments. In addition, the random perturbation approach is more suitable for static problems with small parameter variation; and calculation amount of the orthogonal expansion method is effected by the number of random variables. In the random vibration theory [9], the optional approaches which capture evolution of the probability densities are developed. The classical equations such as the Liouville equation, the Fokker–Planck–Kolmogorov (FPK) equation and the Dostupov–Pugachev equation were established in this framework. Taking the FPK equation as an example, the FPK equation governs evolution of the joint probability density function (PDF) of system state variables. However, the analytical solution of the FPK equation is difficult to obtain, except for linear systems and some specific nonlinear systems [1, 10]. Moreover, the corresponding numerical approaches [11, 12] can not solve the FPK equation of high dimension because of huge computational cost.

In the past decade, Li and Chen [13, 14] developed a family of probability density evolution method (PDEM), which is available for both linear and nonlinear stochastic dynamic systems. In this work, the generalized probability density evolution equation (GDEE), which governs evolution of the joint PDF of response of interest and randomness, can be derived [15]. The GDEE provides a unified way for treatment of randomness. In contrast to the above classical equations, GDEE is the uncoupled one which can be solved in a simplified way. More specially, the dimension of the GDEE is only dependent on the dimension of the response of interest, and nothing to do with the dimension of the system state. The PDEM has been systematically developed in many areas, e.g., structural system reliability evaluation [16] and stochastic optimal control [17], etc.

In short, the solving process of GDEE (referred to as PDEM) is to include partition of probability assigned space and numerical solving of partial differential equations [18]. The former is achieved by selecting the representative points in the random variate space and adopting the corresponding Voronoi cells [19]. The number theoretical method [20] and the GF-discrepancy based method [21] are two common selection strategies. By the number theoretical method, the representative point set is determined uniformly in a high dimensional hypercube, and then cut by a hyperball to reduce the number of selected points. The GF-discrepancy based method is a strategy of representative point set rearrangement via adopting the GF-discrepancy as the objective function. It is worth mentioning that deterministic analysis associated with the representative points is extremely time-consuming, especially for a complex structure involving strong nonlinearity. Moreover, finite difference method (FDM) is often adopted in the latter part, while the result obtained by FDM is sensitive to the mesh size [22]. These issues limit the efficiency and accuracy of PDEM. To overcome these limitations, it is necessary to develop a refined algorithm combing a kind of surrogate model method with PDEM. For example, Jiang and Li [23] proposed the K-PDEM which utilizes the kriging method for building the surrogate model. However, different surrogate models are needed for different time points in this method. Thus the K-PDEM suffers from the computation complexity, particularly when the dimension of the random variable is high. In 2016, Li and Sun [24] firstly introduced the particle approximation (PA) and the reproducing kernel particle method (RKPM) [25, 26, 27] in the analysis process. The accuracy of RKPM is demonstrated better than that of PA when utilized in the refined algorithm. In the RKPM, the kernel function is modified by multiplying a correction function and then used for approximating scalar functions, e.g., the response of dynamic system to reduce the computational cost of the deterministic dynamic analysis in PDEM.

In this paper, the refined algorithm based on RKPM for solving GDEE is developed. The details and numerical implementation of GDEE are given in Sect. 2. Then the basic concept of RKPM is outlined in Sect. 3. The advantage and framework of the refined algorithm based on RKPM are elaborated in Sect. 4. In Sect. 5, several numerical examples are investigated to verify the efficiency and accuracy of the method. Parameter analysis and convergence study are also carried out.

## 2 The generalized density probability evolution equation

The detailed investigation [22] shows that the distribution of the representative points and the mesh size have a significantly influence on the accuracy of the results in solving procedure. Meanwhile, solving the representative time histories of deterministic dynamic system are extremely time-consuming. In this paper, the refined algorithm based on RKPM is introduced to improve these problems.

## 3 The reproducing kernel particle method

### 3.1 Kernel estimate

Compared with a Gaussian function, the B-spline function has a narrower compact support, which leads to the greater computational efficiency [30]. Note that for the refined algorithm of GDEE, there is no derivative calculation of the kernel function involved, then the order of this cubic spline function is sufficient. Hence, in this paper, the cubic spline function is chosen as the kernel function.

### 3.2 Reproducing kernel approximation

Unfortunately, most of the traditional discrete kernel estimate applications do not satisfy the reproducing conditions. Moreover, even the zero-th order reproducing conditions are failed to meet near the boundaries since the supports are cut off [28].

*, and \( d \) is the spatial dimension. In particular, for the linear basis function \( n = 1 \), \( N_{0} \left(\varvec{\theta}\right) \ge d + 1 \). Besides, the error analysis of the RKPM has been investigated [33, 34]. For the linear basis function, the optimal rate of convergence is 2 in the \( L_{2} \) norm.*

**θ**## 4 The refined algorithm based on RKPM

In the FDM which has been mentioned in Sect. 1, the PDF result is sensitive to the mesh size. For the specific number of representative points, a relatively small mesh size may lead to the oscillation of the result while a large one may result in the dissipation. A feasible way is to put more representative points in the space. However, for a complex structural model involving strong nonlinearities, deterministic analysis associated with the representative points is extremely time-consuming. Consequently, the refined algorithm is computationally expensive to implement. To overcome this disadvantage, the RKPM is introduced as a surrogate model instead of the deterministic dynamic system analysis.

In this way, the refined algorithm has greater accuracy than traditional PDEM without a significant increase in the calculation cost of the deterministic dynamic system analysis.

- (1)
Select the basic points \( \varvec{\theta}_{J} \left( {J = 1,2, \cdots ,N} \right) \) in the domain \( \Omega_{\varvec{\varTheta}} \) via the numerical strategies, e.g., the number theoretical method or the GF-discrepancy based method, etc., where \( N \) is the total number of basic points.

- (2)
Carry out deterministic analysis on the dynamic system for each basic point \( \varvec{\theta}_{J} \) and obtain the corresponding velocity time history \( h_{Z} \left( {\varvec{\theta}_{J} ,t} \right) \).

- (3)
Select the densified representative points \( \varvec{\theta}_{I} \left( {I = 1,2, \cdots ,N_{\text{sel}} } \right) \) in the domain \( \Omega_{\varvec{\varTheta}} \) where \( N_{\text{sel}} \) is the total number of densified points. Partition the domain \( \Omega_{\varvec{\varTheta}} \) into subdomains \( \Omega_{{\varvec{\varTheta}_{q} }} {\mkern 1mu} {\mkern 1mu} \left( {q = 1,2, \cdots ,N_{\text{sel}} } \right) \) by the Voronoi cells, and calculate the corresponding assigned probability \( P_{q} \). Note that \( N_{\text{sel}} \) is far more than \( N \) and the densified points are added close to the basic points. The number theoretical method and the GF-discrepancy based method are usually employed.

- (4)
For each densified point \( \varvec{\theta}_{I} \), obtain the reproducing kernel function and calculate the corresponding velocity time history \( h_{Z}^{\text{R}} \left( {\varvec{\theta}_{I} ,t} \right) \) by (31).

- (5)
For each subdomain \( \Omega_{{\varvec{\varTheta}_{q} }} \), solve the partial differential Eq. (8) by FDM and get the corresponding numerical solution of \( p_{Z}^{\left( I \right)} \left( {z,t} \right) \). Add \( p_{Z}^{\left( I \right)} \left( {z,t} \right) \) together and obtain numerical integration of \( p_{Z} \left( {z,t} \right) \)

- (6)$$ p_{Z} \left( {z,t} \right) = \sum\limits_{I = 1}^{{N_{\text{sel}} }} {p_{Z}^{\left( I \right)} \left( {z,t} \right)} $$(32)

Note that there is only one surrogate model needed for the whole time history in the refined algorithm based on RKPM, which has great advantages in the computational efficiency. Moreover, the error analysis of RKPM mentioned in Sect. 4 is still applicable here.

## 5 Numerical examples

The superiority of the refined algorithm based on RKPM to the traditional PDEM is demonstrated by one numerical example, which is chosen because its analytical solution has been derived by Jiang and Li [35].

### *Example 1*

As a nonlinear differential oscillator with a simple form, Helmholtz oscillator is widely used in many fields [36].

KL divergence between the analytical and the numerical solutions (\( \Delta x = 0.005 \))

KL divergence | Helmholtz oscillator |
---|---|

The traditional PDEM (\( N_{\text{sel}} = 50 \)) | 0.0468 |

The traditional PDEM (\( N_{\text{sel}} = 200 \)) | 0.0002 |

The traditional PDEM (\( N_{\text{sel}} = 800 \)) | 0.0002 |

The refined algorithm based on RKPM | 0.0003 |

### *Example 2*

There are 13 parameters in this model which take the values \( \alpha = 0.04 \), \( A = 1{\text{s}} \), \( c = 1 \), \( \beta = 15{\text{s}}/{\text{m}}\; \), \( \gamma = 1 5 0 {\text{s/m}}\; \), \( d_{v} = d_{\eta } = 1000/{\text{m}}^{2} \; \), \( p = 1000/{\text{m}}^{2} \; \), \( q = 0.25 \), \( d_{\psi } = 5/{\text{m}}\; \), \( \chi = 0.5 \), \( \zeta_{s} = 0.99 \), \( \psi = 0.05{\text{m}} \), where \( \alpha \) is the ratio of linear to nonlinear response, \( A,c,\beta ,\gamma \) are parameters which control the hysteretic displacement, \( d_{v} ,d_{\eta } \) are parameters reflecting degradation of the strength and the stiffness, and \( p,q,d_{\psi } ,\chi ,\zeta_{s} ,\psi \) are related to the effect of pinching. The initial elastic modulus is taken as the random variable. For convenience of parameter analysis, a regular distribution of basic point set is adopted. The physical quantity which we focus on in this example is the top floor displacement \( Z \) of the frame structure.

*Case 1* (one random variable involved)

Assume that the elastic modulus of the 10 floors are completely correlated and thus the structure can be characterized by one random variable.

Regular distributions of 50, 100, 150, 200, 250, 300, 350 and 400 basic points are used to study the effect of changing the number of basic points \( N \) on the results. Meanwhile, 800 densified points are selected in the refined algorithm based on RKPM. The linear basis function and the cubic spline kernel function are employed. The values 0.75, 0.8, 0.85, 0.9, 0.95, 1, 1.2, 1.4, 1.6, 1.8, 2 of \( \lambda \) are taken to investigate.

*N*increases), the global error reduces. The slope of the convergence plot is about 2. For the distribution of 50 and 200 basic points, the velocity time history corresponding to some densified points is shown in Fig. 5. It can be seen that the numerical results obtained by the proposed method are in very good agreement with the exact solution.

*Case 2* (ten random variables involved)

Assume that the elastic modulus of the 10 floors are independent and the structure can be characterized by ten random variables.

Computational time of the traditional PDEM and the refined algorithm based on RKPM

Computational time | Traditional PDEM (\( N_{\text{sel}} = 800 \)) | Refined algorithm based on RKPM (\( N = 50,N_{\text{sel}} = 800 \)) | Refined algorithm based on RKPM (\( N = 200,N_{\text{sel}} = 800 \)) |
---|---|---|---|

(s) | 283.62 | 18.53 | 79.57 |

## 6 Conclusions

- (1)
By the proposed method, the time dependent probability distributions of the quantities of interest can be obtained. This refined algorithm is applicable to uni-variable and multi-variable, one-dimensional and multi-dimensional systems.

- (2)
In this paper, the problem of mesh sensitivity is settled well. Even if only 50 basic points are employed, the accuracy of the method is fairly high provided that the optimal smoothing length \( \sigma \) is used.

- (3)
Using the RKPM as a surrogate model method for the deterministic dynamic system analysis, the computational efforts are considerably reduced.

## Notes

### Acknowledgements

Financial support from the National Natural Science Foundation of China (Grant No. 51538010) is gratefully appreciated.

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