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

  • Dan Wang
  • Jie LiEmail author
Original Paper
  • 53 Downloads

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 equation of motion for a generic structure is typically a second order ordinary differential equation
$$ \varvec{M}_{0} \left(\varvec{\varTheta}\right)\varvec{\ddot{X}} + \varvec{C}_{0} \left(\varvec{\varTheta}\right)\dot{\varvec{X}} + \varvec{f}\left( {\varvec{\varTheta},\varvec{X}} \right) = \varvec{F}\left( {\varvec{\varTheta},t} \right) $$
(1)
$$ \varvec{X}\left( {t_{0} } \right) = \varvec{x}_{0} ,\dot{\varvec{X}}\left( {t_{0} } \right) = \dot{\varvec{x}}_{0} $$
(2)
in which \( \varvec{M}_{0} ,\varvec{C}_{0} \) are the \( m \times m \) mass and damping matrix respectively; \( \varvec{f} \) is the \( m \times 1 \) nonlinear restoring force vector; \( \varvec{F} \) is the \( m \times 1 \) excitation vector; \( \varvec{X,}{\mkern 1mu} {\mkern 1mu} \varvec{\dot{X},}{\mkern 1mu} {\mkern 1mu} \varvec{\ddot{X}} \) are the \( m \times 1 \) displacement, velocity and acceleration vector respectively; \( \varvec{\varTheta} \) is the \( s \times 1 \) random vector which contains all the randomness involved in the excitations and system parameters with the known joint PDF \( p_{\varvec{\varTheta}} \left(\varvec{\theta}\right) \).
For the well-posed problems, the physical solution of (1) can be assumed to take the form
$$ \varvec{X} = \varvec{H}\left( {\varvec{\varTheta},t} \right),\dot{\varvec{X}} = \varvec{h}\left( {\varvec{\varTheta},t} \right) $$
(3)
Consider a set of physical quantities of interest \( \varvec{Z} = \left( {Z_{1} ,Z_{2} , \cdots ,Z_{{m_{z} }} } \right)^{\text{T}} \) related to the system, such as the displacement or velocity of some degrees of freedom, or the strain or stress at some points, etc. \( \varvec{Z} \) can usually be obtained by its connection with the displacement and velocity vector
$$ \dot{\varvec{Z}} = \varvec{f}_{\varvec{Z}} \left( {\varvec{X},\dot{\varvec{X}}} \right) = \varvec{h}_{\varvec{Z}} \left( {\varTheta ,t} \right),\varvec{Z}\left( {t_{0} } \right) = z_{0} $$
(4)
The augmented system \( \left( {\varvec{Z},\varvec{\varTheta}} \right) \) can be regarded as a probability preserved system. Consequently, according to the random event description of the principle of preservation of probability, the GDEE is derived [15]
$$ \frac{{\partial p_{{\varvec{Z\varTheta }}} \left( {\varvec{z},\varvec{\theta},t} \right)}}{\partial t} + \sum\limits_{l = 1}^{{m_{z} }} {h_{{Z_{l} }} \left( {\varvec{\theta},t} \right)\frac{{\partial p_{{\varvec{Z\varTheta }}} \left( {\varvec{z},\varvec{\theta},t} \right)}}{{\partial z_{l} }}} = 0 $$
(5)
where \( p_{{\varvec{Z\varTheta }}} \left( {\varvec{z},\varvec{\theta},t} \right) \) is the joint PDF of \( \varvec{Z} \) and \( \varvec{\varTheta} \). In addition, \( \varvec{z} \) and \( \varvec{\theta} \) can be regarded as the sample realizations of the random vectors \( \varvec{Z} \) and \( \varvec{\varTheta} \), respectively. Note that the dimension of the GDEE is independent to that of the state vector of the original dynamic system. For convenience, the following one-dimensional partial differential equation is considered
$$ \frac{{\partial p_{{Z\varvec{\varTheta}}} \left( {z,\varvec{\theta},t} \right)}}{\partial t} + h_{Z} \left( {\varvec{\theta},t} \right)\frac{{\partial p_{{Z\varvec{\varTheta}}} \left( {z,\varvec{\theta},t} \right)}}{\partial z} = 0 $$
(6)
with initial condition
$$ p_{{Z\varvec{\varTheta}}} \left( {z,\varvec{\theta},t_{0} } \right) = \delta \left( {z - z_{0} } \right)p_{\varvec{\varTheta}} \left(\varvec{\theta}\right) $$
(7)
For some simple problems, the analytical solution of the GDEE can be given directly [1]. However, for most of the practical engineering problems, numerical solving method is necessary. We should first partition the whole probability space \( \Omega_{\varvec{\varTheta}} \), and then a set of probability space subdomains \( \Omega_{{\varvec{\varTheta}_{q} }} {\mkern 1mu} {\mkern 1mu} \left( {q = 1,2, \cdots ,N_{\text{sel}} } \right) \) are obtained. Integrating both sides of (6) with respect to \( \varvec{\theta} \) in each subdomain will give the following equation [18]
$$ \frac{{\partial p_{Z}^{\left( q \right)} \left( {z,t} \right)}}{\partial t} + h_{Z} \left( {\varvec{\theta}_{q} ,t} \right)\frac{{\partial p_{Z}^{\left( q \right)} \left( {z,t} \right)}}{\partial z} = 0 $$
(8)
where \( \varvec{\theta}_{q} \left( {q = 1,2, \cdots ,N_{\text{sel}} } \right) \) are the representative points corresponding to the subdomains, and
$$ p_{Z}^{\left( q \right)} \left( {z,t} \right) = \int_{{\Omega_{{\varvec{\varTheta}_{q} }} }} {p_{{Z\varvec{\varTheta}}} \left( {z,\varvec{\theta},t} \right)d}\varvec{\theta} $$
(9)
In the same way, integrate both sides of (7) with respect to \( {\varvec{\theta}} \) in each subdomain
$$ p_{Z}^{\left( q \right)} \left( {z,t_{0} } \right) = \delta \left( {z - z_{0} } \right)P_{q} $$
(10)
$$ P_{q} = \int_{{\Omega_{{\varvec{\varTheta}_{q} }} }} {p_{\varvec{\varTheta}} \left(\varvec{\theta}\right)} d\varvec{\theta} $$
(11)
where \( P_{q} \) is the corresponding assigned probability.
Solving (8) with (10), \( p_{Z}^{\left( q \right)} \left( {z,t} \right)\left( {q = 1,2, \cdots ,N_{\text{sel}} } \right) \) are obtained. Finally, get the target PDF result by summation
$$ p_{Z} \left( {z,t} \right) = \sum\limits_{q = 1}^{{N_{\text{sel}} }} {p_{Z}^{\left( q \right)} \left( {z,t} \right)} $$
(12)

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

Consider the following integral transformation of a function \( h_{Z} \left(\varvec{\theta}\right) \) through a Dirac delta function [28]
$$ h_{Z} \left(\varvec{\theta}\right) = \int_{\Omega } {\delta \left( {\varvec{\theta}- \varvec{\theta^{\prime}}} \right)} h_{Z} \left( {\varvec{\theta^{\prime}}} \right)d\varvec{\theta^{\prime}} $$
(13)
where \( \varvec{\theta}= \left( {\theta_{1} ,\theta_{2} , \cdots ,\theta_{s} } \right)^{\text{T}} \) and \( \Omega \) is the domain which is finite in practical structural problems. If \( h_{Z} \left(\varvec{\theta}\right) \) is continuous in the domain, then (13) exactly holds.
Generally, a certain kind of kernel function \( W\left( {\varvec{\theta}- \varvec{\theta^{\prime}},\sigma } \right) \) with compact supports can be used to approximate the Dirac delta function, and \( \sigma \) is called as the smoothing length which controls the spread of the kernel function. The functions such as Gaussian function and spline function are usually used. For example, the cubic B-spline function is introduces, that is [29]
$$ W\left( {\varvec{\theta}- \varvec{\theta^{\prime}},\sigma } \right) = \alpha_{d} \left\{ {\begin{array}{*{20}l} {\frac{2}{3} - \left| {\frac{{\varvec{\theta}- \varvec{\theta^{\prime}}}}{\sigma }} \right|^{2} + \frac{1}{2}\left| {\frac{{\varvec{\theta}- \varvec{\theta^{\prime}}}}{\sigma }} \right|^{3} {\mkern 1mu} {\mkern 1mu} 0 \le \left| {\frac{{\varvec{\theta}- \varvec{\theta^{\prime}}}}{\sigma }} \right| < 1} \hfill \\ {\frac{1}{6}\left( {2 - \left| {\frac{{\varvec{\theta}- \varvec{\theta^{\prime}}}}{\sigma }} \right|} \right)^{3} {\mkern 1mu} \;\;\;\;\;\;\;\;\;\;\;\;{\mkern 1mu} 1 \le \left| {\frac{{\varvec{\theta}- \varvec{\theta^{\prime}}}}{\sigma }} \right| < 2 \, } \hfill \\ {0 \, \qquad\qquad\qquad\qquad\left| {\frac{{\varvec{\theta}- \varvec{\theta^{\prime}}}}{\sigma }} \right| \ge 2} \hfill \\ \end{array} } \right. $$
(14)
in which \( \alpha_{d} \) is the normalized constant.

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.

Consequently, in computation (13) is approximated by
$$h_{Z}^{\text{K}} \left( {\varvec{\theta}} \right) = \int_{\Omega } {W\left( {{\varvec{\theta}} - {\mathbf{\varvec{\theta^{\prime}}}},\sigma } \right)} h_{Z} \left( {{\mathbf{\varvec{\theta^{\prime}}}}} \right)d{\mathbf{\varvec{\theta^{\prime}}}}$$
(15)
where \( h_{Z}^{\text{K}} \left( {\varvec{\theta}} \right) \) is known as the kernel estimate of \( h_{Z} \left( {\varvec{\theta}} \right) \), and we use the superscript \( \text{K} \) to denote it.
Further, suppose that the domain \( \Omega \) is discretized by a set of basic points \( \varvec{\theta}_{J} \left( {J = 1,2, \cdots ,N} \right) \) where \( N \) is the total number of basic points. By performing numerical integration, the discrete kernel estimate of \( h_{Z} \left(\varvec{\theta}\right) \) is expressed as
$$ h_{Z}^{\text{K}} \left(\varvec{\theta}\right) = \sum\limits_{J = 1}^{N} {W\left( {\varvec{\theta}-\varvec{\theta}_{J} ,\sigma } \right)h_{Z} \left( {\varvec{\theta}_{J} } \right)} \Delta\varvec{\theta}_{J} $$
(16)
where \( \Delta\varvec{\theta}_{J} \) is the domain measure related to the basic point set.
The conditions which the above discrete approximation needs to satisfy can be given as
$$ \sum\limits_{J = 1}^{N} {W\left( {\varvec{\theta}-\varvec{\theta}_{J} ,\sigma } \right)\varvec{K}\left( {\varvec{\theta}-\varvec{\theta}_{J} } \right)} \Delta\varvec{\theta}_{J} = \varvec{K}\left( 0 \right) $$
(17)
where
$$ \begin{aligned} \varvec{K}\left( {\varvec{\theta}-\varvec{\theta}_{J} } \right) &= \left[ {1,\theta_{1} - \theta_{J,1} ,\theta_{2} - \theta_{J,2} , \cdots ,\left( {\theta_{1} - \theta_{J,1} } \right)^{2} , \cdots ,}\right.\\&\quad\left.{\left( {\theta_{1} - \theta_{J,1} } \right)^{n} , \cdots } \right]^{\text{T}} \end{aligned} $$
(18)
and
$$ \varvec{K}\left( 0 \right) = \left[ {1,0, \cdots ,0} \right]^{\text{T}} $$
(19)
which are called the reproducing conditions. In addition, \( n \) is the highest order of these conditions, which is known as a reproducing order.

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].

To solve these problems, Liu et al. [26, 31] introduced the RKPM in which the kernel function is multiplied by the correction function. Using the RKPM, the function \( h_{Z} \left(\varvec{\theta}\right) \) is approximated by
$$ h_{Z}^{\text{R}} \left(\varvec{\theta}\right) = \sum\limits_{J = 1}^{N} {\bar{W}\left( {\varvec{\theta}-\varvec{\theta}_{J} ,\sigma } \right)h_{Z} \left( {\varvec{\theta}_{J} } \right)} \Delta\varvec{\theta}_{J} $$
(20)
where \( h_{Z}^{\text{R}} \left(\varvec{\theta}\right) \) is called the reproducing kernel particle approximation which is denoted by the superscript \( \text{R} \) and \( \bar{W}\left( {\varvec{\theta}-\varvec{\theta}_{J} ,\sigma } \right) \) is the correction kernel function
$$ \bar{W}\left( {\varvec{\theta}-\varvec{\theta}_{J} ,\sigma } \right) = C\left( {\varvec{\theta};\varvec{\theta}-\varvec{\theta}_{J} } \right)W\left( {\varvec{\theta}-\varvec{\theta}_{J} ,\sigma } \right) $$
(21)
in which \( C\left( {\varvec{\theta};\varvec{\theta}-\varvec{\theta}_{J} } \right) \) is the correction function which is typically given by a linear combination of the polynomial basis functions as follows:
$$ C\left( {\varvec{\theta};\varvec{\theta}-\varvec{\theta}_{J} } \right) = \varvec{K}^{\text{T}} \left( {\varvec{\theta}-\varvec{\theta}_{J} } \right)\varvec{b}\left(\varvec{\theta}\right) $$
(22)
where \( \varvec{b}\left(\varvec{\theta}\right) \) is a vector of coefficients which can be determined by imposing the reproducing conditions
$$ \sum\limits_{J = 1}^{N} {\bar{W}\left( {\varvec{\theta}-\varvec{\theta}_{J} ,\sigma } \right)\varvec{K}\left( {\varvec{\theta}-\varvec{\theta}_{J} } \right)} \Delta\varvec{\theta}_{J} = \varvec{K}\left( 0 \right) $$
(23)
Substituting (21) into (23) leads to
$$ \varvec{B}\,\left(\varvec{\theta}\right)\, \varvec{b}\left(\varvec{\theta}\right) = \varvec{K}\left( 0 \right) $$
(24)
where
$$ \varvec{B}\left(\varvec{\theta}\right) = \sum\limits_{J = 1}^{N} {\varvec{K}\left( {\varvec{\theta}-\varvec{\theta}_{J} } \right)\varvec{K}^{\text{T}} \left( {\varvec{\theta}-\varvec{\theta}_{J} } \right)} W\left( {\varvec{\theta}-\varvec{\theta}_{J} ,\sigma } \right)\Delta\varvec{\theta}_{J} $$
(25)
Therefore, \( \varvec{b}\left(\varvec{\theta}\right) \) can be solved by
$$ \varvec{b}\left(\varvec{\theta}\right) = \varvec{B}^{ - 1} \left(\varvec{\theta}\right)\,\varvec{K}\left( 0 \right) $$
(26)
Finally, the following reproducing kernel particle approximation are obtained as
$$ h_{Z}^{\text{R}} \left(\varvec{\theta}\right) = \sum\limits_{J = 1}^{N} {\varPsi_{J} \left(\varvec{\theta}\right)} h_{Z} \left( {\varvec{\theta}_{J} } \right) $$
(27)
and
$$ \varPsi_{J} \left(\varvec{\theta}\right) = \varvec{K}^{\text{T}} \left( 0 \right)\varvec{B}^{ - 1} \left(\varvec{\theta}\right)\varvec{K}\left( {\varvec{\theta}-\varvec{\theta}_{J} } \right)W\left( {\varvec{\theta}-\varvec{\theta}_{J} ,\sigma } \right)\Delta\varvec{\theta}_{J} $$
(28)
Enough basic points should be contained in the support of \( \varvec{\theta} \) to make the matrix defined in (25) not singular [32, 33]. Actually, the necessary condition is
$$ N_{0} \left(\varvec{\theta}\right) \ge \left( {\begin{array}{*{20}c} {n + d} \\ d \\ \end{array} } \right) = \frac{{\left( {n + d} \right)!}}{n!d!} $$
(29)
where \( N_{0} \left(\varvec{\theta}\right) \) is the number of basic points contained in the support of θ, 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.

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.

As has been mentioned in Sect. 3, using the RKPM, the response of dynamic system \( h_{Z} \left( {\varvec{\theta},t} \right) \) can be approximated by
$$ h_{Z}^{\text{R}} \left( {\varvec{\theta},t} \right) = \sum\limits_{J = 1}^{N} {\bar{W}\left( {\varvec{\theta}-\varvec{\theta}_{J} ,\sigma } \right)h_{Z} \left( {\varvec{\theta}_{J} ,t} \right)} \Delta\varvec{\theta}_{J} $$
(30)
Using the calculation framework of the traditional PDEM, the basic points \( \varvec{\theta}_{J} \) are first obtained, with the corresponding time histories \( h_{Z} \left( {\varvec{\theta}_{J} ,t} \right) \). Then the time histories \( h_{Z} \left( {\varvec{\theta}_{I} ,t} \right) \) corresponding to the densified representative points \( \varvec{\theta}_{I} \) are derived by the RKPM
$$ h_{Z}^{\text{R}} \left( {\varvec{\theta}_{I} ,t} \right) = \sum\limits_{J = 1}^{N} {\varPsi_{J} \left( {\varvec{\theta}_{I} } \right)h_{Z} \left( {\varvec{\theta}_{J} ,t} \right)} $$
(31)

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.

Therefore, the procedure of the refined algorithm based on RKPM is listed below:
  1. (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. (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. (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. (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. (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. (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].

In the refined algorithm based on RKPM, the distributions of 50 basic points and 800 densified representative points are selected. The linear basis function and the cubic spline kernel function are employed. Furthermore, the size of smoothing length is chosen as
$$ \sigma = \lambda \Delta \theta_{J} $$
(33)
where \( \Delta \theta_{J} \) is the average domain measure related to the basic point and \( \lambda \) is the constant which takes the value 0.75 in the flowing example. For comparison, the distributions of 50 and 200 representative points are employed in the traditional PDEM to demonstrate the advantage of the proposed method in accuracy. Meanwhile, 800 representative points are selected in the traditional PDEM to show the superiority of the refined algorithm based on RKPM in computational cost.

Example 1

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

Consider the following one-dimensional Helmholtz oscillator
$$ \ddot{x}\left( t \right) + 2\dot{x}\left( t \right) + \frac{24}{25}x\left( t \right) - \theta x^{2} \left( t \right) = 0 $$
(34)
with initial condition
$$ x\left( {t = 0} \right) = \frac{6}{25\theta },\dot{x}\left( {t = 0} \right) = - \frac{12}{125\theta } $$
(35)
where the analytical solution of the velocity response is
$$ \dot{x}\left( {\theta ,t} \right) = - \frac{96}{125\theta }\left( {1 + e^{{{{2t} \mathord{\left/ {\vphantom {{2t} 5}} \right. \kern-0pt} 5}}} } \right)^{ - 3} e^{2t/5} $$
(36)
Assume that the random variable \( \theta \) obeys standard normal distribution. The PDF is given by
$$ p_{\varTheta } \left( \theta \right) = {{e^{{ - \frac{1}{2}\theta^{2} }} } \mathord{\left/ {\vphantom {{e^{{ - \frac{1}{2}\theta^{2} }} } {\sqrt {2\pi } }}} \right. \kern-0pt} {\sqrt {2\pi } }} $$
(37)
The analytical solution to the time dependent PDF of displacement is
$$ p_{X} \left( {x,t} \right) = {{p_{\varTheta } \left[ {\frac{24}{{25\left( {1 + e^{{{{2t} \mathord{\left/ {\vphantom {{2t} 5}} \right. \kern-0pt} 5}}} } \right)^{2} x}}} \right]} \mathord{\left/ {\vphantom {{p_{\varTheta } \left[ {\frac{24}{{25\left( {1 + e^{{{{2t} \mathord{\left/ {\vphantom {{2t} 5}} \right. \kern-0pt} 5}}} } \right)^{2} x}}} \right]} {\left| { - \frac{25}{24}\left( {1 + e^{{{{2t} \mathord{\left/ {\vphantom {{2t} 5}} \right. \kern-0pt} 5}}} } \right)^{2} x^{2} } \right|}}} \right. \kern-0pt} {\left| { - \frac{25}{24}\left( {1 + e^{{{{2t} \mathord{\left/ {\vphantom {{2t} 5}} \right. \kern-0pt} 5}}} } \right)^{2} x^{2} } \right|}} $$
(38)
The PDF results of Example 1 are shown in Fig. 1. It can be seen in Fig. 1a–c that at \( t = 0.5 \), the probability distributions obtained by both traditional PDEM and refined algorithm with \( \Delta x = 0.020 \) are greatly smoothed in the region where the sudden change exits. At the same time, a tiny high-frequency oscillation occurs in the tail when the traditional PDEM with 50 representative points is employed. Hence the mesh size is reduced to \( \Delta x = 0.005 \) for improving the accuracy. In Fig. 1d, the PDF obtained by traditional PDEM with 50 representative points oscillates heavily. However, in Fig. 1e–f, the traditional PDEM with 200 and 800 points and the refined algorithm results are in good agreement with the analytical solutions.
Fig. 1

Comparison of the PDFs at \( t = 0.5 \) with two different mesh sizes: \( \Delta x = 0.020 \); \( \Delta x = 0.005 \)

The Kullback–Leibler (KL) divergence, also known as relative entropy, is a measure for describing the difference between two probability distributions \( P \) and \( Q \) over the same variable \( x \) [37]
$$ D_{KL} \left( {P\parallel Q} \right) = \sum\limits_{i} {\ln \left( {\frac{P\left( i \right)}{Q\left( i \right)}} \right)} P\left( i \right) $$
(39)
Comparing the data in Table 1, it can be inferred that the KL divergence reduces as \( N_{\text{sel}} \) increases. The KL divergence of traditional PDEM with 200 representative points is identical to that of 800 points, and thus \( N_{\text{sel}} = 200 \) is sufficient for the accuracy in Example 1. Meanwhile, the accuracy of the refined algorithm based on RKPM is almost identical to the traditional PDEM with 800 representative points. Therefore, we can conclude that the problem of mesh sensitivity is largely alleviated in the refined algorithm based on RKPM. However, the refined algorithm based on RKPM requires less calculation cost of the deterministic dynamic system analysis. The comparison time is provided later in next example.
Table 1

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

Next, the influences of the smoothing length and the distribution of basic points on the accuracy of the refined algorithm based on RKPM are discussed. Consider a 2-span 10-story shear frame structure as shown in Fig. 2. The seismic ground motion is assumed to be the El Centro N-S wave with the peak acceleration of 0.2 g. The cross section size and the height of the columns of the ground floor are \( 500{\text{mm}} \times 500{\text{mm}} \) and \( 4{\text{m}} \) while those of the other floors are \( 400{\text{mm}} \times 400{\text{mm}} \) and \( 3{\text{m}} \); the lumped mass of each floor is \( 2.6 \times 10^{5} {\text{kg}} \); the Rayleigh damping model is adopted where the damping ratios of the first two modes are both \( 5\% \). The Bouc–Wen model is employed to describe the nonlinear hysteresis behavior of the structure [38].
Fig. 2

A 2-span 10-story frame subjected to earthquake

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.

The root of mean square (RMS) error and the maximum error measure at the time point \( t \) are computed by
$$ {\text{RMS err}} = \frac{1}{{\left| {h_{Z} \left( {\varvec{\theta}_{I} ,t} \right)} \right|_{\hbox{max} } }}\sqrt {\frac{1}{{N_{\text{sel}} }}\sum\limits_{I = 1}^{{N_{\text{sel}} }} {\left[ {h_{Z}^{\text{E}} \left( {\varvec{\theta}_{I} ,t} \right) - h_{Z}^{\text{R}} \left( {\varvec{\theta}_{I} ,t} \right)} \right]^{2} } } $$
(40)
$$ \begin{aligned} {\text{MAX err}} &= \frac{1}{{\left| {h_{Z} \left( {\varvec{\theta}_{I} ,t} \right)} \right|_{\hbox{max} } }}\\&\quad \times \left| {h_{Z}^{\text{E}} \left( {\varvec{\theta}_{I} ,t} \right) - h_{Z}^{\text{R}} \left( {\varvec{\theta}_{I} ,t} \right)} \right|_{\hbox{max} } ,I = 1,2, \cdots ,N_{\text{sel}} \end{aligned} $$
(41)
where the superscripts \( {\text{E}} \) and \( {\text{R}} \) denote the exact and the RKPM solution, respectively.
Then the combined error at the time point \( t \) is given as
$$ {\text{err}}\left( t \right) = 0.5 \times {\text{RMS err}} + 0.5 \times {\text{MAX err}} $$
(42)

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.

The combined errors associated with the velocity response at the time point \( t \) are shown in Fig. 3. The error decreases with the decrease of \( \lambda \) which confirms the theoretical prediction. Actually, smaller values of \( \lambda \) can be used provided that enough basic points are contained in the support to make the matrix defined in (25) for every densified point not singular. For \( \lambda = 0.75 \), the minimum number of the basic points contained in the support of the densified point is 2. Besides, the maximum of combined errors for \( \lambda = 0.75 \) in Fig. 3 is \( 2.6 \times 10^{ - 4} \)(\( N = 50 \)). This is why we use \( \sigma = 0.75\Delta \theta_{J} \) for the analysis of Example 1. In Fig. 3, it can also be seen that as we increase the number of basic points, the RKPM solutions approach the exact solutions. The convergence of the proposed method is shown in Fig. 4. The global error measure where the whole time history is considered is computed by
$$ {\text{err}} = \frac{1}{{\left| {h_{Z} \left( {\varvec{\theta}_{I} ,t_{k} } \right)} \right|_{\hbox{max} } }}\sqrt {\frac{1}{{\left( {N_{t} \times N_{\text{sel}} } \right)}}\sum\limits_{k = 1}^{{N_{t} }} {\sum\limits_{I = 1}^{{N_{\text{sel}} }} {\left[ {h_{Z}^{\text{E}} \left( {\varvec{\theta}_{I} ,t_{k} } \right) - h_{Z}^{\text{R}} \left( {\varvec{\theta}_{I} ,t_{k} } \right)} \right]^{2} } } } $$
(43)
Fig. 3

(1D) Combined errors associated with the velocity response of the densified representative points at the time point 2

Fig. 4

(1D) Convergence of the refined algorithm based on RKPM

As \( \Delta \theta_{J} \) decreases (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.
Fig. 5

(1D) Numerical solution and exact solution for some densified points: a\( \theta_{1} \); b\( \theta_{400} \)

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.

Similarly, regular distributions of 50, 100, 150, 200, 250, 300, 350 and 400 basic points are adopted. 800 densified representative points are selected. The linear basis function and the cubic spline kernel function are employed. The constant \( \lambda \) takes the value 1.1. The slope of the convergence plot shown in Fig. 6 is about 2. Shown in Fig. 7 is the velocity time history corresponding to some densified points. We can conclude that there is good agreement between the numerical results for the distribution of 200 basic points and the exact solution.
Fig. 6

(10D) Convergence of the refined algorithm based on RKPM

Fig. 7

(10D) Numerical solution and exact solution for some densified points: a\( \varvec{\theta}_{1} \); b\( \varvec{\theta}_{400} \)

The comparison of the computational time is carried out in Table 2. The computational time of the traditional PDEM with 800 representative points is almost 15 times than that of the proposed method with 50 basic points and 800 densified points. Compared to the deterministic analysis corresponding to the basic points in Step (2), Step (4) takes tiny computational time.
Table 2

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

Further, the GDEE is solved to obtain the probability information of the stochastic response in this case. Figures 89 illustrate some of the typical results for the distribution of 50 basic points with 800 densified points. Figure 8a shows that the mean and the standard deviation by the refined algorithm accord with those of the MCS. The PDFs at different time points are illustrated in Fig. 8b. The PDFs evolving against time construct a PDF surface which is shown in Fig. 9a. Meanwhile, the corresponding contour is plotted in Fig. 9b.
Fig. 8

a The mean and the standard deviation; b the PDFs at different time points

Fig. 9

a The PDFs evolving against time; b the contour of the PDF surface in (a)

6 Conclusions

In this paper, a refined algorithm which combines RKPM with PDEM is proposed to solve the GDEE. The details of the algorithm including the implementation procedure are elaborated. Several numerical examples, uni-variable and multi-variable, one-dimensional and multi-dimensional systems, are investigated in this paper. The conclusions include:
  1. (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. (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. (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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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Civil EngineeringTongji UniversityShanghaiChina
  2. 2.The State Key Laboratory on Disaster Reduction in Civil EngineeringTongji UniversityShanghaiChina

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