This paper proposes a non-intrusive uncertainty analysis method for artillery dynamics involving hybrid uncertainty using polynomial chaos expansion (PCE). The uncertainty parameters with sufficient information are regarded as stochastic variables, whereas the interval variables are used to treat the uncertainty parameters with limited stochastic knowledge. In this method, the PCE model is constructed through the Galerkin projection method, in which the sparse grid strategy is used to generate the integral points and the corresponding integral weights. Through the sampling in PCE, the original dynamic systems with hybrid stochastic and interval parameters can be transformed into deterministic dynamic systems, without changing their expressions. The yielded PCE model is utilized as a computationally efficient, surrogate model, and the supremum and infimum of the dynamic responses over all time iteration steps can be easily approximated through Monte Carlo simulation and percentile difference. A numerical example and an artillery exterior ballistic dynamics model are used to illustrate the feasibility and efficiency of this approach. The numerical results indicate that the dynamic response bounds obtained by the PCE approach almost match the results of the direct Monte Carlo simulation, but the computational efficiency of the PCE approach is much higher than direct Monte Carlo simulation. Moreover, the proposed method also exhibits fine precision even in high-dimensional uncertainty analysis problems.

In practical dynamic systems, a variety of uncertainties associated with material properties, environmental factors, external loads, dimensional tolerances, and boundary conditions are ubiquitous. These uncertainties will inevitably affect the final system performances, and small variations associated with uncertainties might result in significant changes in the dynamic responses. The traditional way to investigate the dynamic response problems with uncertainty parameters is the probabilistic method, in which the uncertainty parameters are described as stochastic variables or processes with precise probability distributions. Among these probabilistic methods, Monte Carlo simulation (MCS) [

Interval methods depict uncertainty through the upper and lower bounds of uncertain-but-bounded parameters rather than other stochastic information, therefore, they are considered to be a powerful supplement to the classical probabilistic approaches and have been perfectly applied in dynamic structural response analysis. The interval arithmetic operation [

As aforementioned, the probabilistic methods and interval methods both have their own merits, deficiencies, and limited scope of application. In practical structures, the stochastic and interval parameters may exist simultaneously due to the different availability of uncertainty information. In the early researches [

These aforementioned methods are very heuristic, and have laid a solid methodological foundation for uncertainty analysis of artillery dynamics. However, it is frankly to say that few literatures considered uncertainties in artillery dynamics. Among them, Wang et al. [

The dynamics systems described by ordinary differential equations (ODEs) which are quite common in artillery dynamics are considered in the present paper. To overcome the potential limitations for current research, the focus of this paper is to develop an accurate, efficient and generally applicable method for estimation of the response bounds of artillery dynamics with hybrid uncertainty parameters. Thus, a non-intrusive polynomial chaos expansion framework will be adopted due to its fine capacity and strong mathematical basis in quantifying the uncertainty of the dynamic systems. The PCE model is constructed through the Galerkin projection method, in which the sparse grid numerical integration method is used to generate the integral points and the corresponding integral weights. Through the sampling in PCE model, the original dynamic systems with hybrid stochastic and interval parameters can be transformed into ones with deterministic parameters, without changing their expressions. The yielded PCE model is utilized as a computationally efficient, surrogate model, and the supremum and infimum of the dynamic responses at each iteration time step can be easily approximated through MCS and percentile difference.

The remainder of this paper is organized as follows. A statement of the dynamic systems with hybrid uncertainty parameters is given in Section 2. Then in Sections 3 and 4, the proposed method and its solving process are introduced in detail, respectively. In Section 5, an artillery exterior ballistic dynamics model is given to show the effectiveness and feasibility of the proposed method in comparison with MCS. Finally, the conclusions are drawn in Section 6.

The dynamics systems expressed as a set of ODEs are quite common in artillery dynamics. Suppose that there are

where

Assuming that there are uncertainties in the structure, of which the uncertainty parameters with precise probability distributions are regarded as stochastic variables, given by

where superscript _{i}

where superscript

A dynamic system with hybrid stochastic and interval uncertainty parameters can be rewritten as

Considering the transitivity of uncertainty parameters, the solution of

In

Polynomial chaos employs a series of orthogonal polynomials with specific distribution in the uncertainty space to approximate the uncertainty processes. The original polynomial chaos proposed by Wiener [

The polynomial chaos expansion is essentially a representation of a function

where

where _{i}

Note that

where (

The most important aspect of the above expansions is that an uncertainty process has been decomposed into a set of deterministic functions, and the truncated expansion series,

The polynomials

where

Here,

In

where

For the stochastic parameters satisfying normal distribution, the one-dimensional polynomial

The (

hence the first few Hermite polynomials can be expressed as

The inner product of two Hermite polynomials can be formulated as

where

Moreover, the Legendre polynomial

The (

hence the first few Legendre polynomials can be expressed as

The inner product of two Legendre polynomials can be formulated as

where

The Galerkin projection method [_{i}

Owing to the orthogonality, the expansion coefficient _{i}

It is noted that the denominator term of

where

The numerator term of

where

where _{d}_{j}_{j}_{j}

SGNIM performs the tensor product only on the one-dimensional integration corresponding to the multi-indices that meet

Assuming that we obtain a total of _{l}

where

Moreover, the one-dimensional integral node

where

Using the

Then substituting _{i}

After obtaining the PCE model, the statistical moments of the response can be directly derived through the expansion coefficients and the polynomial bases. The mean and standard deviation of

The lower and upper bounds of the response can be determined by the combination of MCS and PCE conveniently. In this process, the first step is to generate the samples of the hybrid uncertainty vector

where

Through the theory of percentile difference, the lower and upper bounds of the response can be regarded as the positions at the left and right tail of its distribution function, satisfying the given CDF

where

In this section, the direct MCS for dynamic response analysis will be reviewed briefly, and the polynomial chaos expansion approach for dynamic systems described by ODEs with hybrid uncertainty parameters will be introduced. A numerical example based on the two methods will then be compared to illustrate the advantages of the polynomial chaos expansion approach.

When using numerical methods to solve the ODEs in _{j}

Step 1: Define the numerical solving method for ODEs, the number of samples for MCS, _{M}_{e}

Step 2: Initialize the cyclic counting indices

Step 3: Generate one group sample of

Step 4: Take the sample into the original ODEs, and use an appropriate numerical method to calculate the real response at time _{j}

Step 5: If _{M}

Step 6: Calculate the statistical moments, and obtain the PDF of the response at time _{j}_{M}

Step 7: Calculate the response bounds at time _{j}

Step 8: If

At time _{j}_{j}_{i}_{j}

Step 1: Define the numerical solving method for ODEs, the expansion series of polynomial chaos, _{M}_{e}

Step 2: Determine the polynomial basis functions according to the distributions of the uncertainty parameters.

Step 3: Utilize the sparse grid collocation strategy to generate the integral collocation nodes and the corresponding integral weights according to

Step 4: Initialize the cyclic counting indices

Step 5: Take the collocation nodes into the original ODEs, and use an appropriate numerical method to calculate the real responses at time _{j}

Step 6: Calculate the expansion coefficient of polynomial chaos through the Galerkin projection method according to _{j}

Step 7: Calculate the statistical moments of the response at time _{j}

Step 8: Generate _{M}

Step 9: Take the _{M}

Step 10: Obtain the PDF of the responses at time _{j}_{M}

Step 11: Calculate the response bounds at time _{j}

Step 12: If

In a brief description, the proposed polynomial chaos expansion algorithm for solving the dynamic systems described by ODEs with hybrid stochastic and interval parameters is similar to a type of sampling method. It transforms the original ODEs with uncertainty parameters into ODEs with deterministic parameters through sampling, without changing their expressions. As a non-intrusive method, any numerical methods can be used to solve the differential equations. Thus, the polynomial chaos expansion approach can be regarded as a computationally convenient method.

A simple numerical example is used here to demonstrate the validity of the present method. The results obtained by the polynomial chaos method which is abbreviated as PCEM are compared with those of the direct MCS. The schematic of a double pendulum system is analyzed as depicted in

where _{1} and _{2} are the masses of the two pendulums, respectively; _{1} and _{2} are the lengths of the pendulum rods, respectively; _{1} and _{2} satisfy Gaussian distribution, with the mean and standard deviation of 1.0 and 0.1 kg, respectively. Note that Gaussian distribution is unbounded distribution with infinite interval, it means that the parameter values will be negative to some extent. Therefore, the modeling of stochastic variables is an approximate situation. _{1} and _{2} are considered as interval parameters, and their interval ranges are [0.45, 0.55] m, and [0.9, 1, 1] m, respectively. The initial conditions

For this problem, the fifth-order polynomial chaos approach is used to solve the differential equations in the time period of 0–10 s. The collocation level

Time (s) | Lower bound | Upper bound | ||||
---|---|---|---|---|---|---|

MCS | PCEM | Relative error (%) | MCS | PCEM | Relative error (%) | |

1.0 | −3.9829 | −3.9794 | 0.0879 | −1.7749 | −1.7923 | 0.9803 |

2.0 | 2.0924 | 2.1233 | 1.4768 | 3.5345 | 3.4785 | 1.5844 |

3.0 | −8.4598 | −8.2306 | 2.7093 | 8.3898 | 8.4254 | 0.4243 |

4.0 | −2.5410 | −1.8044 | 28.9886 | 6.9487 | 6.6246 | 4.6642 |

5.0 | −8.7390 | −8.7762 | 0.4257 | 3.2666 | 3.0824 | 5.6389 |

6.0 | −4.7024 | −4.7300 | 0.5869 | 7.0896 | 6.8157 | 3.8634 |

7.0 | −7.5115 | −7.4272 | 1.1223 | 6.7837 | 6.5689 | 3.1664 |

8.0 | −8.6512 | −8.6269 | 0.2809 | 3.9777 | 3.7523 | 5.6666 |

9.0 | −3.1432 | −2.5161 | 19.9510 | 7.5285 | 7.3223 | 2.7389 |

10.0 | −10.5592 | −9.7787 | 7.3917 | 7.7989 | 7.1065 | 8.8782 |

Time (s) | Lower bound | Upper bound | ||||
---|---|---|---|---|---|---|

MCS | PCEM | Relative error (%) | MCS | PCEM | Relative error (%) | |

1.0 | −1.1334 | −0.8902 | 21.4623 | 0.9580 | 0.7779 | 18.7933 |

2.0 | 1.0341 | 1.1190 | 8.2070 | 4.7987 | 4.6801 | 2.4709 |

3.0 | −10.0280 | −9.8051 | 2.2228 | −1.9048 | −2.3057 | 21.0494 |

4.0 | 0.9586 | 0.9971 | 4.0143 | 9.1402 | 8.9929 | 1.6111 |

5.0 | −9.7112 | −9.6075 | 1.0682 | 2.8464 | 2.7273 | 4.1838 |

6.0 | −1.4856 | −1.0830 | 27.1017 | 4.2870 | 4.0211 | 6.2030 |

7.0 | −2.7584 | −2.2496 | 18.4463 | 4.0624 | 3.7918 | 6.6594 |

8.0 | −9.2642 | −9.2932 | 0.3123 | 5.3171 | 5.2181 | 1.8621 |

9.0 | −2.9232 | −2.6520 | 9.2765 | 10.4641 | 10.0964 | 3.5141 |

10.0 | −11.3124 | −10.9938 | 2.8164 | 2.2682 | 1.7392 | 23.3230 |

As shown in

Moreover, the computations of PCEM and direct MCS take 128.67 s and 1347.42 s, respectively. Clearly, the computational efficiency of PCEM is much higher than that of the direct MCS. Therefore, the proposed method can be considered as an effective and efficient uncertainty analysis method for the dynamic systems with hybrid stochastic and interval uncertainty parameters.

The schematic of the artillery exterior ballistic dynamics model is analyzed as depicted in

A general six-degree of freedom exterior ballistic model can be formulated as follows:

where _{C}_{A}_{x}_{y}_{z}_{xz}_{zz}_{y}_{z}_{p}_{max}_{cg}_{x2}, _{y2}, and _{z2} are the projections of wind speed on the velocity coordinate, which are given by

where _{x}_{z}

Through the above equations, we can obtain the flight trajectory and motion attitude of the projectile. Assuming that there are uncertain parameters in this dynamics model, the detailed uncertainty types and values are shown in

Uncertainty | Type | Mean value | Standard deviation | Range |
---|---|---|---|---|

Interval | 0.0025 | N/A | [0.0, 0.005] | |

Interval | 0.0025 | N/A | [0.0, 0.005] | |

Interval | 0.005 | N/A | [0.0, 0.01] | |

Interval | 0.005 | N/A | [0.0, 0.01] | |

Interval | 5.0 | N/A | [0.0, 10.0] | |

Interval | 3.0 | N/A | [0.0, 6.0] | |

_{0}) (m/s) |
Gaussian | 930.0 | 5.0 | N/A |

_{p} |
Gaussian | 45.5 | 0.2 | N/A |

Gaussian | 0.15494 | 1.55E-4 | N/A | |

_{A} |
Gaussian | 1.7558 | 0.02 | N/A |

_{C} |
Gaussian | 0.1589 | 0.002 | N/A |

_{cg} |
Gaussian | 0.551 | 0.006 | N/A |

_{x} |
Gaussian | 2.0 | 0.4 | N/A |

_{z} |
Gaussian | 2.0 | 0.4 | N/A |

The second-order polynomial chaos approach is used to solve the differential equations in the time period of 0–107 s. The collocation level

Time (s) | Lower bound | Upper bound | ||||
---|---|---|---|---|---|---|

MCS | PCEM | Relative error (%) | MCS | PCEM | Relative error (%) | |

10.0 | 595.9562 | 596.0656 | 0.0184 | 620.0387 | 620.0365 | 0.0004 |

20.0 | 448.8532 | 448.9614 | 0.0241 | 470.2198 | 470.2250 | 0.0011 |

30.0 | 358.9886 | 359.0921 | 0.0288 | 378.6309 | 378.6382 | 0.0019 |

40.0 | 305.5613 | 305.6526 | 0.0299 | 322.6491 | 322.6559 | 0.0021 |

50.0 | 285.7332 | 285.8026 | 0.0243 | 298.8509 | 298.8556 | 0.0016 |

60.0 | 296.9672 | 297.0107 | 0.0146 | 305.9062 | 305.9071 | 0.0003 |

70.0 | 329.0193 | 329.0473 | 0.0085 | 335.5312 | 335.5296 | 0.0005 |

80.0 | 366.4526 | 366.4884 | 0.0098 | 372.9504 | 372.9542 | 0.0010 |

90.0 | 393.2813 | 393.3455 | 0.0163 | 403.1814 | 403.1956 | 0.0035 |

100.0 | 396.7107 | 396.8114 | 0.0254 | 412.5039 | 412.5271 | 0.0056 |

107.0 | 382.9736 | 383.0948 | 0.0316 | 402.4136 | 402.4417 | 0.0070 |

Time (s) | Lower bound | Upper bound | ||||
---|---|---|---|---|---|---|

MCS | PCEM | Relative error (%) | MCS | PCEM | Relative error (%) | |

10.0 | 0.01043 | 0.01038 | 0.4794 | 0.02996 | 0.02990 | 0.2003 |

20.0 | 0.01921 | 0.01914 | 0.3644 | 0.04968 | 0.04961 | 0.1409 |

30.0 | 0.02803 | 0.02794 | 0.3211 | 0.06851 | 0.06841 | 0.1460 |

40.0 | 0.03729 | 0.03718 | 0.2950 | 0.08602 | 0.08591 | 0.1279 |

50.0 | 0.04361 | 0.04352 | 0.2064 | 0.09619 | 0.09609 | 0.1040 |

60.0 | 0.04255 | 0.04250 | 0.1175 | 0.09306 | 0.09300 | 0.0645 |

70.0 | 0.03667 | 0.03666 | 0.0273 | 0.08187 | 0.08188 | 0.0122 |

80.0 | 0.02981 | 0.02986 | 0.1677 | 0.07036 | 0.07045 | 0.1279 |

90.0 | 0.02269 | 0.02283 | 0.6170 | 0.06161 | 0.06179 | 0.2922 |

100.0 | 0.01465 | 0.01488 | 1.5700 | 0.05656 | 0.05688 | 0.5658 |

107.0 | 0.00803 | 0.00832 | 3.6115 | 0.05538 | 0.05580 | 0.7584 |

Furthermore,

In this paper, a hybrid stochastic and interval uncertainty analysis method with polynomial chaos expansion is proposed to evaluate the response bounds of artillery dynamics. In this method, the Hermite polynomial in stochastic space and the Legendre polynomial in interval space are employed, respectively, as the trial basis to expand the Gaussian and interval processes. The polynomial coefficients are calculated through the Galerkin projection method, in which the sparse grid, numerical integration method is used to generate the integral points and the corresponding integral weights. Through the sampling in polynomial chaos expansion, the original artillery dynamics systems with hybrid stochastic and interval parameters can be transformed into ones with deterministic parameters, without changing their expressions. The yielded polynomial chaos expansion model is utilized as a computationally efficient surrogate model, and the supremum and infimum of the dynamic responses at each iteration time step can be easily approximated through Monte Carlo simulation and percentile difference. A numerical example and an artillery exterior ballistic dynamics model were used to illustrate the feasibility and efficiency of this approach. The numerical results indicate that the dynamic response bounds obtained by the polynomial chaos expansion approach almost match the results of the direct Monte Carlo simulation, but the computational efficiency of the polynomial chaos expansion approach is much higher than direct Monte Carlo simulation. Moreover, the proposed method also exhibits fine precision even in high-dimensional uncertainty analysis problems. Another advantage of the polynomial chaos expansion approach is that it is non-intrusive. In a brief description, the proposed algorithm is similar to a type of sampling method. It has no special restrictions on numerical methods for solving the differential equations. Therefore, this method can be potentially applied to other artillery dynamics systems with hybrid uncertainty parameters, such as the artillery multi-body dynamics system described by the differential-algebraic equations (DAEs). However, further research is required to explore the potential of the proposed method in dealing with other artillery dynamics system.