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Differential Quadrature and Differential Quadrature Based Element Methods: Theory and Applications is a comprehensive guide to these methods and their various applications in recent years.

In this paper, a new method is put forward for simultaneous reliability and reliability-sensitivity analyses based on the information-reuse of sparse grid numerical integration (SGNI). First, the reliability analysis is conducted on the basis of fractional exponential moments-based maximum entropy method (FEM-MEM), where the SGNI is employed for FEM assessments. The reliability index can be evaluated by integrating over the distribution derived by FEM-MEM. Then, the reliability-sensitivity analysis is carried out by reusing the output samples in previous reliability analysis and updating the corresponding weights, where no additional model evaluations are required. Then, the FEM-MEM is applied again to derive the conditional distribution and reliability index. By comparing the conditional and original reliability indexes, one can define the reliability-sensitivity index to identify the importance of each random variable to reliability. Since only one-round of model evaluations are necessary in the proposed method, the computational efficiency is highly desirable. Four numerical examples are investigated to check the effectiveness of the proposed method, where pertinent results obtained from Monte Carlo simulations (MCS) and Sobol’s index are compared. The results demonstrate the proposed method is accurate and efficient for simultaneous reliability and reliability-sensitivity analyses.

A wide variety of uncertainty propagation methods exist in literature; however, there is a lack of good understanding of their relative merits. In this paper, a comparative study on the performances of several representative uncertainty propagation methods, including a few newly developed methods that have received growing attention, is performed. The full factorial numerical integration, the univariate dimension reduction method, and the polynomial chaos expansion method are implemented and applied to several test problems. They are tested under different settings of the performance nonlinearity, distribution types of input random variables, and the magnitude of input uncertainty. The performances of those methods are compared in moment estimation, tail probability calculation, and the probability density function construction, corresponding to a wide variety of scenarios of design under uncertainty, such as robust design, and reliability-based design optimization. The insights gained are expected to direct designers for choosing the most applicable uncertainty propagation technique in design under uncertainty.

The peridynamic theory is a nonlocal formulation of continuum mechanics based on integro-differential equations, devised to deal with fracture in solid bodies. In particular, the forces acting on each material point are evaluated as the integral of the nonlocal interactions with all the neighboring points within a spherical region, called “neighborhood”. Peridynamic bodies are commonly discretized by means of a meshfree method into a uniform grid of cubic cells. The numerical integration of the nonlocal interactions over the neighborhood strongly affects the accuracy and the convergence behavior of the results. However, near the boundary of the neighborhood, some cells are only partially within the sphere. Therefore, the quadrature weights related to those cells are computed as the fraction of cell volume which actually lies inside the neighborhood. This leads to the complex computation of the volume of several cube–sphere intersections for different positions of the cells. We developed an innovative algorithm able to accurately compute the quadrature weights in 3D peridynamics for any value of the grid spacing (when considering fixed the radius of the neighborhood). Several examples have been presented to show the capabilities of the proposed algorithm. With respect to the most common algorithm to date, the new algorithm provides an evident improvement in the accuracy of the results and a smoother convergence behavior as the grid spacing decreases.

The stability prediction is usually used to avoid the unstable machining in milling process. According to the Lagrange-Simpson hybrid interpolation scheme, this paper improves a numerical integration method (NIM) to perform the milling chatter prediction accurately and efficiently. Firstly, the higher-order numerical integral formulas (NIFs) are constructed based on the Lagrange polynomial. Thus, the third-order and fourth-order NIMs are built and investigated respectively. Then, to improve the calculated performance of the NIMs, the Simpson scheme is introduced to decrease the local discretization error. Finally, the comparisons among the built NIMs and the existing discretization methods are carried out by calculating the convergence rate and the stability boundaries. Compared to the third-order NIM and the third-order full-discretization method, the proposed second-order Lagrange-Simpson NIM shows the better computational performance.

This paper deals with the geometric numerical integration of gradient flow and its application to optimization. Gradient flows often appear as model equations of various physical phenomena, and their dissipation laws are essential. Therefore, dissipative numerical methods, which are numerical methods replicating the dissipation law, have been studied in the literature. Recently, Cheng, Liu, and Shen proposed a novel dissipative method, the Lagrange multiplier approach, for gradient flows, which is computationally cheaper than existing dissipative methods. Although their efficacy is numerically confirmed in existing studies, the existence results of the Lagrange multiplier approach are not known in the literature. In this paper, we establish some existence results. We prove the existence of the solution under a relatively mild assumption. In addition, by restricting ourselves to a special case, we show some existence and uniqueness results with concrete bounds. As gradient flows also appear in optimization, we further apply the latter results to optimization problems.

Chatter is not conducive to machining efficiency and surface quality. One of the essential types of chatter in the machining process is regenerative chatter. This study presents the numerical integration scheme–based semi-discretization methods (NISDMs) for milling stability prediction. Firstly, the dynamic model of the milling process is represented by the delay differential equation (DDE). The forced vibration period is discretized into many small-time intervals. After integrating on the small-time interval, only the time-delay term–related part is approximated by different order numerical integration schemes. Both the free and forced vibration processes are considered in the derivation process. The state transition matrix is constructed by mapping the dynamic displacement between the current and previous time periods. The NISDMs are compared with the benchmark methods in terms of the rate of convergence and computational time. The comparison results show that the NISDMs converge faster than the benchmark methods. To improve the computational efficiency of the NISDMs, the precise integration method is used in the calculation process. The computational time consumed by the NISDMs is much less than that consumed by the benchmark methods. The NISDMs are proved to be more accurate and efficient methods for stability prediction in milling than the other considered methods.

Different from the traditional watermarking schemes, zero-watermarking schemes are lossless embedding methods, which are applicable to be used in medical, military, remote sensing and other fields requiring high-integrity image copyright protection. However, most of the existing zero-watermarking schemes only provide copyright protection for one image at a time, which has certain limitations. This paper proposes a novel zero-watermarking scheme for protecting the copyright of two similar medical images simultaneously. Firstly, an accurate polar complex exponential transform (APCET) is designed using Gaussian numerical integration (GNI) method, which effectively improves the computation accuracy of polar complex exponential transform (PCET). Then, ternary accurate polar complex exponential transform (TAPCET) is constructed based on ternary number theory and APCET, which describes two similar medical images simultaneously. Finally, a robust zero-watermarking algorithm for two similar medical images is proposed based on TAPCET and chaotic mapping. The experimental results show that the proposed scheme can resist common image processing attacks and geometric attacks, and is superior to other zero-watermarking algorithms, being applicable for the copyright protection of two similar medical images simultaneously.

To predict the milling stability accurately and efficiently, an improved numerical integration method (INIM) is proposed based on the Lagrange interpolation scheme. First, the milling dynamic model considering the regenerative chatter can be described as a delay linear differential equation. The tooth passing period of the milling cutter is divided into the free and forced vibration stages. Then, the forced vibration stage is equally discretized, and the INIMs are built based on the Lagrange interpolation scheme within the discretized intervals to construct the state transition matrix. Finally, the convergence rates and the stability lobes for the benchmark milling systems are calculated and discussed by using the proposed INIMs and the existing methods, respectively. The comparison results reveal that the proposed second-order INIM shows higher computational efficiency and accuracy compared with the related discretization methods, and in the meantime, it is more accurate under just slightly loss of the time cost compared with the existing NIMs.

Uncertainty propagation analysis, which assesses the impact of the uncertainty of input variables on responses, is an important component in risk assessment or reliability analysis of structures. This paper proposes an uncertainty propagation analysis method for structures with parameterized probability-box (p-box) representation, which could efficiently compute both the bounds on statistical moments and also the complete probability bounds of the response function. Firstly, based on the sparse grid numerical integration (SGNI) method, an optimized SGNI (OSGNI) is presented to calculate the bounds on the statistical moments of the response function and the cumulants of the cumulant generating function (CGF), respectively. Then, using the bounds on the first four cumulants, an optimization procedure based on the saddlepoint approximation is proposed to obtain the whole range of probability bounds of the response function. Through using the saddlepoint approximation, the present approach can achieve a good accuracy in estimating the tail probability bounds of a response function. Finally, two numerical examples and an engineering application are investigated to demonstrate the effectiveness of the proposed method.