Narrow Results

Meshless Shepard interpolation method (MSIM) is used to simulate the destruction process within the arch reinforcement area of tunnel lining. In this method, the shape functions are formed by the partition of unity and the finite cover technology, which is not affected by discontinuous domains, with delta property at any desired node, and applies boundary conditions in an easy and correct way. The MSIM method combines the advantages of both the conventional meshless method and the numerical manifold method. The finite cover technology enables the shape functions to not be affected by the discontinuities in the solution domain, which overcomes the difficulty resulting from the conventional meshless method. The finite covers and the partition of unity functions are formed using the influence domains of a series of nodes, which removes the obstacle from conventional numerical manifold method and has simpler formation of finite covers than numerical manifold method. Virtual crack closure technique is used to calculate the intensity factor of crack-tip stress. The results of progressive destruction process simulation on the cracking patterns within the arch reinforcement area indicate that the method is suitable for tracking the crack propagation in complex stress conditions.

This is the second of a series devoted to the direct and inverse approximation theorems of the p-version of the finite element method in the framework of the weighted Besov spaces. In this paper, we combine the approximability of singular solutions in the Jacobi-weighted Besov spaces, which were analyzed in the previous paper,⁴ with the technique of partition of unity in order to prove the optimal rate of convergence of the p-version of the finite element method for elliptic boundary value problems on polygonal domains.

T-splines are an important tool in IGA since they allow local refinement. In this paper we define analysis-suitable T-splines of arbitrary degree and prove fundamental properties: Linear independence of the blending functions and optimal approximation properties of the associated T-spline space. These are corollaries of our main result: A T-mesh is analysis-suitable if and only if it is dual-compatible. Indeed, dual compatibility is a concept already defined and used in L. Beirão da Veiga et al.⁵ Analysis-suitable T-splines are dual-compatible which allows for a straightforward construction of a dual basis.

In this paper, we consider isogeometric discretizations of the Poisson model problem, focusing on high polynomial degrees and strong hierarchical refinements. We derive a posteriori error estimates by equilibrated fluxes, i.e. vector-valued mapped piecewise polynomials lying in the $H(\mathrm{\text{div}})$ space which appropriately approximate the desired divergence constraint. Our estimates are constant-free in the leading term, locally efficient, and robust with respect to the polynomial degree. They are also robust with respect to the number of hanging nodes arising in adaptive mesh refinement employing hierarchical B-splines, though not with respect to the smoothness and support overlaps. Two partitions of unity are designed, one with larger supports corresponding to the mapped splines, and one with small supports corresponding to mapped piecewise multilinear finite element hat basis functions. The equilibration is only performed on the small supports, avoiding the higher computational price of equilibration on the large supports or even the solution of a global system. Thus, the derived estimates are also as inexpensive as possible. An abstract framework for such a setting is developed, whose application to a specific situation only requests a verification of a few clearly identified assumptions. Numerical experiments illustrate the theoretical developments.

The need to analyze aircraft noise over ground with general properties occurs in various applications, most notably in environmental engineering, where the analysis of the sound pressure level (SPL) distribution near the ground due to aircraft noise is desired. Since the human hearing range is very wide, the determination of the SPL distribution for a given source spectrum is not trivial and may be regarded as a multiscale problem. One has to solve repeatedly, for many different wave numbers, the Helmholtz equation in the upper half space while imposing the given impedance boundary condition on the possibly non-flat ground. An efficient Helmholtz solver must therefore be incorporated in the scheme that finds the SPL distribution. Three totally different computational methods that may be used to solve this problem are considered here: A fictitious sources method, a parabolic approximation method and a wave-enriched finite element method (with two versions: PUM and GFEM). The three methods are compared in terms of their computational properties, and numerical examples are presented to demonstrate their performance.

The windowed-Shannon wavelet is not recommended generally as the window function will destroy the partition of unity of Shannon mother wavelet. A novel windowing scheme is proposed to overcome the shortcoming of the general windowed-Shannon function, and then, a novel and efficient Shannon–Cosine wavelet spectral method is provided for solving the fractional PDEs. Taking full advantage of the waveform of sinc function to hold the partition of unity, Shannon–Cosine wavelet is constructed, which is composed of Shannon wavelet and the trigonometric polynomials. It was proved that the proposed wavelet function meets the requirements of being a trial function and possesses many other excellent properties such as normalization, interpolation, two-scale relations, compact support domain, and so on. Therefore, it is a real wavelet function instead of a general Shannon–Gabor wavelet which is a kind of quasi-wavelet. Next, by means of the Shannon–Cosine wavelet collocation method, the corresponding algebraic equation system of the fractional Fokker–Planck equation can be obtained. Approximate solutions of the fractional Fokker–Plank equations are compared with the exact solutions. These calculations illustrate that the accuracy of the Shannon–Cosine wavelet collocation solutions is quite high even using a small number of grid points.

The locust slice images have all the features such as strong self-similarity, piecewise smoothness and nonlinear texture structure. Multi-scale interpolation operator is an effective tool to describe such structures, but it cannot overcome the influence of noise on images. Therefore, this research designed the Shannon–Cosine wavelet which possesses all the excellent properties such as interpolation, smoothness, compact support and normalization, then constructing multi-scale wavelet interpolative operator, the operator can be applied to decompose and reconstruct the images adaptively. Combining the operator with the local filter operator (mean and median), a multi-scale Shannon–Cosine wavelet denoising algorithm based on cell filtering is constructed in this research. The algorithm overcomes the disadvantages of multi-scale interpolation wavelet, which is only suitable for describing smooth signals, and realizes multi-scale noise reduction of locust slice images. The experimental results show that the proposed method can keep all kinds of texture structures in the slice image of locust. In the experiments, the locust slice images with mixture noise of Gaussian and salt–pepper are taken as examples to compare the performances of the proposed method and other typical denoising methods. The experimental results show that the Peak Signal-To-Noise Ratio (PSNR) of the denoised images obtained by the proposed method is greater 27.3%, 24.6%, 2.94%, 22.9% than Weiner filter, wavelet transform method, median and average filtering, respectively; and the Structural Similarity Index (SSIM) for measuring image quality is greater 31.1%, 31.3%, 15.5%, 10.2% than other four methods, respectively. As the variance of Gaussian white noise increases from 0.02 to 0.1, the values of PSNR and SSIM obtained by the proposed method only decrease by 11.94% and 13.33%, respectively, which are much less than other 4 methods. This shows that the proposed method possesses stronger adaptability.

In this paper efforts are made to enrich Reproducing Kernel Particle Method (RKPM). Firstly, the RKPM shape functions are expressed explicitly in terms of kernel function moments. This avoids numerical matrix inversions and solutions of linear algebraic equations which are involved in classical RKPM, and thus makes RKPM more accurate, faster and more efficient. Then, a recently developed truly meshless body integration technique is introduced into RKPM. It is based on a partition of unity by a set of overlapping patches covering the domain and eliminates background cells completely. We borrow an idea from computational geometry and propose a sweeping-line method to determine the quadrature points inside the domain. The method is robust and effective even for domain with complicated shapes. The truly meshless integration technique in combination with this sweeping-line method make implementation of RKPM quite simple, smart and especially very advantageous when nodes are irregularly scattered. Numerical results presented herein demonstrate that these enrichments make RKPM more efficient, versatile and particularly truly meshless.

A meshfree multiscale method is presented for efficient analysis of solids with strain gradient plastic effects. In the analysis of strain gradient plastic solids, localization due to increased hardening of strain gradient effect appears. Chen-Wang theory is adopted, as a strain gradient plasticity theory. It represents strain gradient effects as an internal variable and retains the essential structure of classical plasticity theory. In this work, the scale decomposition is carried out based on variational form of the problem. Coarse scale is designed to represent global behavior and fine scale to represent local behavior and gradient effect by using the intrinsic length scale. From the detection of high strain gradient region, fine scale region is adopted. Each scale variable is approximated using meshfree method. Meshfree approximation is well suited for adaptivity. As a method of increasing resolution, partition of unity based extrinsic enrichment is used. Each scale problem is solved iteratively. The proposed method is applied to bending of a thin beam and bimaterial shear layer and micro-indentation problems. Size effects can be effectively captured in the results of the analysis.

A 4-node, quadratic tetrahedral element is presented. The development of the element is based on a partition-of-unity (PU) approximation. The local function of the PU approximation is constructed using the basic deformation components of rigid body movements, node-centered rotation, and locally defined linear strain fields. The linear dependence issue of the three-dimensional PU approximation is investigated through numerical experiments. It is found that the global matrix associated with the PU approximation constructed has a constant number of zero eigenvalues independent of mesh subdivision. A method of suppressing the zero eigenvalues is proposed. The element developed does not contain unsuppressed spurious zero energy modes and the condition number of the stiffness matrix of the element is upper bounded. Good accuracy and robustness of the 4-node tetrahedron are demonstrated through numerical examples.

In this paper we consider the problem of reconstructing separatrices in dynamical systems. In particular, here we aim at partitioning the domain approximating the boundaries of the basins of attraction of different stable equilibria. We start from the 2D case sketched in Cavoretto et al. [2011] and the approximation scheme presented in [Cavoretto et al. (2011); Cavoretto et al. (2015)], and then we extend the reconstruction scheme of separatrices in the cases of three-dimensional models with two and three stable equilibria. For this purpose we construct computational algorithms and procedures for the detection and the refinement of points located on the separatrix manifolds that partition the phase space. The use of the so-called Radial Basis Function (RBF) Partition of Unity (PU) method is used to reconstruct the separatrices.

We explain and motivate Stefan–Sussmann singular foliations, and by replacing the tangent bundle of a manifold with an arbitrary Lie algebroid, we introduce singular subalgebroids. Both notions are defined using compactly supported sections. The main results of this note are an equivalent characterization, in which the compact support condition is removed, and an explicit description of the sheaf associated to any Stefan–Sussmann singular foliation or singular subalgebroid.

In this paper, a new methodology to analyze three dimensional crack problems with flexible modeling by means of overlaying mesh method and extended finite element method (X-FEM) is presented. The overlaying mesh method increases the accuracy of analysis locally by superimposing additional mesh of higher resolution on the global mesh which represents rough deformation of structures. In this method the boundaries and nodes in the two meshes do not have to coincide with each other. It makes modeling process becomes very flexible. In X-FEM, discontinuous function and the two-dimensional asymptotic crack-tip displacement fields are added to the finite element approximation to account for the crack using the notion of partition of unity. This technique allows the entire crack to be represented independently of the mesh. As numerical example, an inclined semi-circle surface crack under tension is analyzed.