A Comparison Study of Numerical Techniques for Solving Ordinary Differential Equations Defined on a Semi-Infinite Domain Using Rational Chebyshev Functions

A rational Chebyshev (RC) spectral collocation technique is considered in this paper to solve high-order linear ordinary differential equations (ODEs) defined on a semi-infinite domain. Two definitions of the derivative of the RC functions are introduced as operational matrices. Also, a theoretical study carried on the RC functions shows that the RC approximation has an exponential convergence. Due to the two definitions, two schemes are presented for solving the proposed linear ODEs on the semi-infinite interval with the collocation approach. According to the convergence of the RC functions at the infinity, the proposed technique deals with the boundary value problem which is defined on semi-infinite domains easily. The main goal of this paper is to present a comparison study for differential equations defined on semi-infinite intervals using the proposed two schemes. To demonstrate the validity of the comparisons, three numerical examples are provided. The obtained numerical results are compared with the exact solutions of the proposed problems.

Semiparametric maximum likelihood probability density estimation

A comprehensive methodology for semiparametric probability density estimation is introduced and explored. The probability density is modelled by sequences of mostly regular or steep exponential families generated by flexible sets of basis functions, possibly including boundary terms. Parameters are estimated by global maximum likelihood without any roughness penalty. A statistically orthogonal formulation of the inference problem and a numerically stable and fast convex optimization algorithm for its solution are presented. Automatic model selection over the type and number of basis functions is performed with the Bayesian information criterion. The methodology can naturally be applied to densities supported on bounded, infinite or semi-infinite domains without boundary bias. Relationships to the truncated moment problem and the moment-constrained maximum entropy principle are discussed and a new theorem on the existence of solutions is contributed. The new technique compares very favourably to kernel density estimation, the diffusion estimator, finite mixture models and local likelihood density estimation across a diverse range of simulation and observation data sets. The semiparametric estimator combines a very small mean integrated squared error with a high degree of smoothness which allows for a robust and reliable detection of the modality of the probability density in terms of the number of modes and bumps.

An Efficient Mixed Finite Element Perfectly Matched Layer with Optimal Parameters Selection for Two-Dimensional Time Domain Soil-Structure Interaction Analysis

Perfectly matched layer (PML) is known as one of the best methods to simulate infinite domains in many fields such as soil-structure interaction (SSI). The performance of PML is significantly affected by PML parameters selection. However, the way to select PML parameters still remains unclear. This study proposes a method for PML parameters determination for elastic wave propagation in two-dimensional (2D) media. The scaling and attenuation functions are developed in order to increase the accuracy and effectiveness of the PML. The proposed scheme is applied for a mixed PML in time domain. The finite element method (FEM) formulations of the PML are presented so that it can be easily applied to the existing codes. ABAQUS, a popular FEM code, is used for numerical applications in this study. The proposed PML is imported into ABAQUS by using a user-defined element (UEL) written in Fortran language. Six numerical analyses of SSI are implemented to prove the efficiency of the proposed PML. The numerical analyses cover many realistic problems, including free field, surface structure, and embedded structure problems. The results demonstrate the efficiency of the proposed PML in terms of the accuracy and computational cost.

The Combined Basic LP and Affine IP Relaxation for Promise VCSPs on Infinite Domains

Convex relaxations have been instrumental in solvability of constraint satisfaction problems (CSPs), as well as in the three different generalisations of CSPs: valued CSPs, infinite-domain CSPs, and most recently promise CSPs. In this work, we extend an existing tractability result to the three generalisations of CSPs combined: We give a sufficient condition for the combined basic linear programming and affine integer programming relaxation for exact solvability of promise valued CSPs over infinite-domains. This extends a result of Brakensiek and Guruswami (SODA’20) for promise (non-valued) CSPs (on finite domains).

Analysis of vibration characteristics of elastic metamaterial sandwich beam

Elastic metamaterials (EMs) are a new kind of artificial composite medium composed of complex micro-structural elements, which have unique dynamic properties and elastic wave regulation ability that their constituent materials do not possess. The existing researches on EMs mainly focus on wave characteristics in two-dimensional and three-dimensional infinite domains. However, actual EM structures are always in the form of finite structures such as rods, beams and plates, so it is more important for engineering applications to understand and master their natural and forced vibration characteristics. Therefore, it is necessary to establish an effective simplified solution method and framework with certain accuracy for the vibration analysis of such structures. In the early stage, we have studied the natural and forced vibration characteristics of EM beams from this point of view, and presented a simplified solution process. In this paper, a kind of sandwich beam structure with EMs as the core is further constructed, the simplified solution process is extended to such more practical model analysis, and the free and steady forced vibration analysis processes of the finite-size sandwich beam are given. The vibration characteristics different from the traditional sandwich beam are investigated, and some interesting and useful phenomena are revealed, including the absence of natural frequencies within bandgap (BG), the gathering of natural frequencies in the vicinity of band edges, and the particular modal correspondence before and after BG. Then, the corresponding formation mechanisms are explained from the perspective of wave propagation.

Development of an MTF in ADINA and its application to the study of the Yuxi Basin effect

A three-dimensional multitransmitting formula is developed in ADINA to simulate the input of seismic waves and the scattering of infinite domains at the same time, consistent with the progress of the explicit finite element method of lumped mass. A three-dimensional cube model is built, and a delta pulse wave is input to compare the simulation results with the analytical solutions. The simulation results show that the peak error is 0.2% of the input wave, which meets the requirements of the usual numerical simulation. This method has a certain efficiency advantage in site effect analyses of fine models for localized fields. A velocity structure model of the Yuxi Basin is built, and the associated basin effect is studied by numerical simulation. The distribution of the focusing effect is related to the structure of the narrow east-west and wide north-south features in the Yuxi Basin, and the edge effect is related to the slope of the basin base. A distribution map is given of the amplification effect of ground motion in the basin.