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In this article, we mainly propose and analyze a parallel stabilized finite element algorithm based upon two-grid discretization for the Navier-Stokes problem. The lowest equal-order finite element pairs are considered for the finite element discretization and a stabilized term based on two local Gauss integrations is introduced to circumvent the discrete inf-sup condition. The main idea is to utilize a partition of unity to assemble weighted local corrections of solutions on sub-domains. A further coarse grid correction is carried out to derive the optimal error estimates both in H 1 norm and L 2 norm. Theoretical results are rigorously established and some numerical experiments are reported to verify the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the differential topological properties of a large class of singular spaces: subcarteisan space. First, a minor further result on the partition of unity for differential spaces is derived. Second, the tubular neighborhood theorem for subcartesian spaces with constant structural dimensions is established. Third, the concept of Morse functions on smooth manifolds is generalized to differential spaces. For subcartesian space with constant structural dimension, a class of examples of Morse functions is provided. With the assumption that the subcartesian space can be embedded as a bounded subset of an Euclidean space, it is proved that any smooth bounded function on this space can be approximated by Morse functions. The infinitesimal stability of Morse functions on subcartesian spaces is studied. Classical results on Morse functions on smooth manifolds can be treated directly as corollaries of our results here. [ABSTRACT FROM AUTHOR]

Kalai conjectured that if P is a simplicial d-polytope that has no missing faces of dimension d - 1 , then the graph of P and the space of affine 2-stresses of P determine P up to affine equivalence. We propose a higher-dimensional generalization of this conjecture: if 2 ≤ i ≤ d / 2 and P is a simplicial d-polytope that has no missing faces of dimension ≥ d - i + 1 , then the space of affine i-stresses of P determines the space of affine 1-stresses of P. We prove this conjecture for (1) k-stacked d-polytopes with 2 ≤ i ≤ k ≤ d / 2 - 1 , (2) d-polytopes that have no missing faces of dimension ≥ d - 2 i + 2 , and (3) flag PL (d - 1) -spheres with generic embeddings (for all 2 ≤ i ≤ d / 2 ). We also discuss several related results and conjectures. For instance, we show that if P is a simplicial d-polytope that has no missing faces of dimension ≥ d - 2 i + 2 , then the (i - 1) -skeleton of P and the set of sign vectors of affine i-stresses of P determine the combinatorial type of P. Along the way, we establish the partition of unity of affine stresses: for any 1 ≤ i ≤ (d - 1) / 2 , the space of affine i-stresses of a simplicial d-polytope as well as the space of affine i-stresses of a simplicial (d - 1) -sphere (with a generic embedding) can be expressed as the sum of affine i-stress spaces of vertex stars. This is analogous to Adiprasito's partition of unity of linear stresses for Cohen–Macaulay complexes. [ABSTRACT FROM AUTHOR]

The development of computer technology and communication systems provides new opportunities for learning in an electronic environment. Remote education in virtual electronic classrooms has been widely used in recent decades, especially during the pandemic. Teaching in an electronic environment makes it possible to conduct lectures with thousands of participants. This article is devoted to a new look at teaching improper integrals at the undergraduate and postgraduate levels. A new partition of unity method is applied in a deep recursive e-learning process. A thorough list of subproblems of a real model problem is demonstrated. Additionally, other appropriate improper integrals are considered. [ABSTRACT FROM AUTHOR]

A collocation method is developed for discrete fracture models in the context of the partition‐of‐unity method. Spline technologies used in isogeometric analysis (IGA) are exploited to provide a smooth inter‐element transition of gradients, thus allowing to get rid of extra flux terms at element boundaries which are generated by Lagrange polynomials. Bézier extraction is utilised to formulate IGA commensurate with a standard finite element data‐structure. The efficacy of the proposed approach is examined through different numerical examples and is compared with other discrete methods for fracture analysis. The proposed approach is competitive in terms of accuracy with the least computational cost, rendering it a suitable candidate for superseding available collocation approaches for fracture simulation. Moreover, the approach naturally assesses the possibility of physics informed neural networks for fracture simulation, to which collocation is central. [ABSTRACT FROM AUTHOR]

In this study, we address the challenge of accurately solving physical problems involving complex partial differential equations using deep learning methods. Traditional approaches typically employ uniformly sampled points from the entire domain as training data. In practical PDE problems, achieving uniform precision across the entire domain is often unnecessary. Instead, it is more advantageous to employ random sampling during neural network training, provided that areas demanding precision can be identified and appropriately refined, akin to the concept of mesh refinement in classical numerical methods. To overcome this limitation, we propose a novel method based on the partition of unity approach. Our method introduces an approximation of the integral in the loss function, enabling localized calculation of the loss function within each subdomain. This innovative approach allows us to focus on areas of the domain that require higher precision while maintaining a fully automated process for selecting random samples. To validate the effectiveness of our proposed method, we conduct extensive testing on various physical problems, including Poisson, heat, and wave equations representing elliptic, parabolic, and hyperbolic problems, respectively. Additionally, we consider Burgers' equation as a nonlinear example. Through benchmarking against the standard Deep Ritz Method (DRM) and Deep Galerkin Method (DGM), we demonstrate that our method consistently outperforms these existing approaches, producing superior results. This approach offers researchers and practitioners the opportunity to significantly improve the accuracy of deep learning models when tackling complex physical problems. [ABSTRACT FROM AUTHOR]

This paper proposes a method to solve a 2D time-fractional parabolic equation on arbitrary domains using an RBF partition of unity procedure based on a non-uniform time-stepping framework. The proposed method involves partitioning the main domain into overlapping subdomains, computing local approximations on each subdomain, and joining them together by PU weights to obtain a global approximation. This approach enhances stability, reduces computational costs, and replaces an ill-conditioned linear system with several small well-conditioned linear systems. To handle the weak singularity near the initial time, a time-graded mesh is used to obtain a non-uniform time-stepping scheme, which increases the temporal convergence rate. The convergence analysis of the method is examined to obtain an error bound for the local approximation. Numerical tests are conducted to validate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Tabulated chemistry models are widely used to simulate large-scale turbulent fires in applications including energy generation and fire safety. Tabulation via piecewise Cartesian interpolation suffers from the curse-of-dimensionality, leading to a prohibitive exponential growth in parameters and memory usage as more dimensions are considered. Artificial neural networks (ANNs) have attracted attention for constructing surrogates for chemistry models due to their ability to perform high-dimensional approximation. However, due to well-known pathologies regarding the realization of suboptimal local minima during training, in practice they do not converge and provide unreliable accuracy. Partition of unity networks (POUnets) are a recently introduced family of ANNs which preserve notions of convergence while performing high-dimensional approximation, discovering a mesh-free partition of space which may be used to perform optimal polynomial approximation. We assess their performance with respect to accuracy and model complexity in reconstructing unstructured flamelet data representative of nonadiabatic pool fire models. Our results show that POUnets can provide the desirable accuracy of classical spline-based interpolants with the low memory footprint of traditional ANNs while converging faster to significantly lower errors than ANNs. For example, we observe POUnets obtaining target accuracies in two dimensions with 40 to 50 times less memory and roughly double the compression in three dimensions. We also address the practical matter of efficiently training accurate POUnets by studying convergence over key hyperparameters, the impact of partition/basis formulation, and the sensitivity to initialization. [ABSTRACT FROM AUTHOR]

We present conditions under which the Gel’fand transform E ∧ , of locally m-convex algebra E, is a dense subalgebra of C c (M (E)) . The partition of unity and the local theorem are given for a commutative unital locally m-convex algebra. [ABSTRACT FROM AUTHOR]

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 (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. [ABSTRACT FROM AUTHOR]

In this article, a two grid partition of unity finite element method is proposed and investigated for electrically conducting incompressible fluid flows. This algorithm involves solving a much smaller nonlinear problem on a coarse grid, utilizing a partition of unity to decompose reasonably the residual problem into a series of independent subproblems on a fine grid, and carrying out a further coarse correction on the coarse grid. Rigorously theoretical analysis is presented and convergence results indicate that the method could reach the optimal convergence orders with proper configurations between the coarse mesh size H and the fine mesh size h. Finally, some numerical results are reported to verify our theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, one non-iterative two-grid parallel stabilized finite element method based on overlapping domain decomposition is developed and investigated for the Stokes problem. The lowest-order finite element pairs P 1 − P 1 are utilized to approximate the velocity and pressure, respectively. The difference between a consistent and an under-integrated mass matrices is chosen as the stabilized term to circumvent the so-called inf-sup condition. Algorithm distributes legitimately residuals into local domains with a partition of unity. Error estimates are rigorously established. Theoretical results indicate that patches of diameter H | l n H | (| l n H | − l n | l n H |) or H | l n H | are sufficient to preserve optimal convergence rates with a proper configuration between the coarse mesh size H and the fine mesh size h in 2-D or 3-D case, respectively. Finally, some numerical experiments are reported to support our theoretical results. [ABSTRACT FROM AUTHOR]

The field of distributed-order fractional calculus is rapidly expanding, and it has significant applications in the design of complex systems. In this work, we propose a novel approach to simulate the non-homogeneous two-dimensional distributed-order time-fractional Cable equation. Our proposed approach involves using the RBF partition of unity collocation method, which is based on the differential quadrature technique, in conjunction with the Composite M-point Gauss–Legendre quadrature rule. To obtain the numerical solution, we adopt a Crank–Nicolson type discretization for the temporal variable. The unconditionally stability and second-order rate of convergence of the time-discrete technique are demonstrated. We demonstrate the flexibility and effectiveness of our approach by applying it to several examples with varying geometric domains, including regular domains with Halton points and irregular domains, in both one- and two-dimensional scenarios. Through these examples, we showcase the capability of our proposed method to handle a wide range of problems with different geometric configurations. [ABSTRACT FROM AUTHOR]

This paper presents and analyzes a parallel finite element post-processing algorithm for the simulation of Stokes equations with a nonlinear damping term, which integrates the algorithmic advantages of the two-level approach, the partition of unity method and post-processing technique. The most valuable highlights of the present algorithm are that (1) a global continuous approximate solution is generated via the partition of unity method; (2) by adding an extra coarse grid correction step, the smoothness of the approximate solution is improved; (3) it has a good parallel performance since there requires little communication in solving a series of residual problems in the subdomain of interest. We theoretically derive the L 2 -error estimates both for the approximate velocity and pressure and H 1 -error estimate for the velocity under some necessary conditions. Meanwhile, we numerically perform various test examples to validate the theoretically predicted convergence rate and illustrate the high efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]

PHT-splines are defined as polynomial splines over hierarchical T-meshes with very efficient local refinement properties. The original PHT-spline basis functions constructed by the truncation mechanism have a decay phenomenon, resulting in numerical instability. The non-decay basis functions are constructed as the B-splines that are defined on the 2 × 2 tensor product meshes associated with basis vertices in Kang et al., but at the cost of losing the partition of unity. In the field of finite element analysis and topology optimization, forming the partition of unity is the default ingredient for constructing basis functions of approximate spaces. In this paper, we will show that the non-decay PHT-spline basis functions proposed by Kang et al. can be appropriately modified to form a partition of unity. Each non-decay basis function is multiplied by a positive weight to form the weighted basis. The weights are solved such that the sum of weighted bases is equal to 1 on the domain. We provide two methods for calculating weights, based on geometric information of basis functions and the subdivision of PHT-splines. Weights are given in the form of explicit formulas and can be efficiently calculated. We also prove that the weights on the admissible hierarchical T-meshes are positive. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the differential topological properties of a large class of singular spaces: subcarteisan space. First, a minor further result on the partition of unity for differential spaces is derived. Second, the tubular neighborhood theorem for subcartesian spaces with constant structural dimensions is established. Third, the concept of Morse functions on smooth manifolds is generalized to differential spaces. For subcartesian space with constant structural dimension, a class of examples of Morse functions is provided. With the assumption that the subcartesian space can be embedded as a bounded subset of an Euclidean space, it is proved that any smooth bounded function on this space can be approximated by Morse functions. The infinitesimal stability of Morse functions on subcartesian spaces is studied. Classical results on Morse functions on smooth manifolds can be treated directly as corollaries of our results here.

The development of computer technology and communication systems provides new opportunities for learning in an electronic environment. Remote education in virtual electronic classrooms has been widely used in recent decades, especially during the pandemic. Teaching in an electronic environment makes it possible to conduct lectures with thousands of participants. This article is devoted to a new look at teaching improper integrals at the undergraduate and postgraduate levels. A new partition of unity method is applied in a deep recursive e-learning process. A thorough list of subproblems of a real model problem is demonstrated. Additionally, other appropriate improper integrals are considered.

In this article, two kinds of local and parallel stabilized finite element methods based upon two grid discretizations are proposed and investigated for the Stokes–Darcy model. The lowest equal-order finite element pairs ( P 1 - P 1 - P 1 ) are taken into account to approximate the velocity, pressure, and piezometric head, respectively. To circumvent the inf-sup condition, the stabilized term is chosen as the difference between a consistent and an under-integrated mass matrix. The proposed algorithms consist of approximating the low-frequency component on the global coarse grid and the high-frequency component on the local fine grid and assembling them to obtain the final approximation. To obtain a global continuous solution, the technique tool of the partition of unity is used. A rigorous theoretical analysis for the algorithms was conducted and numerical experiments were carried out to indicate the validity and efficiency of the algorithms. [ABSTRACT FROM AUTHOR]

Partial differential equations are a vital component of the study of mathematical models in science and engineering. There are various tools and techniques developed by the researchers to solve the differential equations. The radial basis functions have proven to be an efficient basis function for approximating the solutions to ordinary and partial differential equations. There are different types of radial basis function methods that have been developed by the researchers to solve various well known differential equation. It has been developed for approximation of the solution with various approaches that lead to the development of hybrid methods. Radial basis function methods are widely used in numerical analysis and statistics because of their ability to deal with meshless domain. In this work, the different radial basis function approaches were investigated along with the focus on the strategies being addressed to find the shape parameter value. The mathematical formulations of the various radial basis function methods are discussed along with the available shape parameters to get the optimal value of the numerical solutions. The present work will lay a foundation to understand the development of the radial basis functions that could lead to a play an important role in development of method thereafter. [ABSTRACT FROM AUTHOR]

In this paper, we present a scattered data approximation method for detecting and approximating the discontinuities of a bivariate function and its gradient. The new algorithm is based on partition of unity, polyharmonic kernel interpolation, and principal component analysis. Localized polyharmonic interpolation in partition of unity setting is applied for detecting a set of fault points on or close to discontinuity curves. Then a combination of partition of unity and principal component regression is used to thinning the detected points by moving them approximately on the fault curves. Finally, an ordered subset of these narrowed points is extracted and a parametric spline interpolation is applied to reconstruct the fault curves. A selection of numerical examples with different behaviors and an application for solving scalar conservation law equations illustrate the performance of the algorithm. [ABSTRACT FROM AUTHOR]

In this article, some local and parallel finite element algorithms are proposed and investigated for the magnetohydrodynamic flows with low electromagnetic Reynolds number. For a solution to this problem, it comprises of two main components, the low-frequency components and the high-frequency components. Motivated by this, we obtain the low-frequency components globally via some relatively coarse grid and catch the high-frequency components locally using a fine grid by some local and parallel procedures. Some local a priori estimates that are crucial for our theoretical analysis are derived. The optimal error estimates are rigorously derived and some numerical tests are reported to support our theoretical findings. [ABSTRACT FROM AUTHOR]

We propose the Convolution Hierarchical Deep-learning Neural Network (C-HiDeNN) that can be tuned to have superior accuracy, higher smoothness, and faster convergence rates like higher order finite element methods (FEM) while using only linear element's degrees of freedom. This is based on our newly developed convolution interpolation theory (Lu et al. in Comput Mech, 2023) and this article focuses on the deep-learning interpretation of C-HiDeNN with graphics processing unit (GPU) programming using JAX library in Python. Instead of increasing the degrees of freedom like higher order FEM, C-HiDeNN takes advantage of neighboring elements to construct the so-called convolution patch functions. The computational overhead of C-HiDeNN is reduced by GPU programming and the total solution time is brought down to the same order as commercial FEM software running on a CPU, however, with orders of magnitude better accuracy and faster convergence rates. C-HiDeNN is locking-free regardless of element types (even with 3-node triangular elements or 4-node tetrahedral elements). C-HiDeNN is also capable of r-h-p-mesh adaptivity like its predecessor HiDeNN (Zhang et al. in Comput Mech 67:207–230, 2021) with additional "a" (dilation parameter) adaptivity that stems from the convolution patch function and "p" adaptivity with higher accuracy and with the same degrees of freedom as that of the linear finite elements. C-HiDeNN potentially has myriad future applications in multiscale analysis, additive and advanced manufacturing process simulations, and high-resolution topology optimization. Details on these applications can be found in the companion papers (Lu et al. 2023; Saha et al. in Comput Mech, 2023; Li et al. in Comput Mech, 2023) published in this special issue. [ABSTRACT FROM AUTHOR]

In the first part of the paper, we present a theorem of type "partition of unity", and three of its consequences, i.e., for algebras, linear subspaces and convex cones. In the second part, some theorems of localizability (or density) in a weighted space are presented. We mention that the weighted spaces are classes of continuous scalar functions on a locally compact space (for example, the space of function with compact support, the space of bounded functions, the space of functions vanishing at infinity, the space of functions rapidly decreasing at infinity). [ABSTRACT FROM AUTHOR]

In this paper, a truly meshless method based on the partition of unity method (PUM) is developed for the numerical solution of the two-dimensional forward–backward heat equations. We propose a novel method according to the domain decomposition scheme and RBF-PUM technique. Particularly, the physical domain needs to be separated into two subdomains each defining a forward or a backward subproblem. The subproblems have been treated by a radial basis function meshfree method based on partition of unity for spatial dimension and a finite difference scheme for the time derivative. In addition, we prove that the time discrete scheme is stable and convergent. Some numerical experiments are going to be presented to show the performance of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, by combining a partition of unity with backtracking technique, a modified local and parallel finite element method (MLPFEM) is proposed and investigated for the coupled Stokes-Darcy problem with the Beavers-Joseph (BJ) interface condition. The well-posedness of the coupled Stokes-Darcy model with BJ interface condition is established when the parameter αBJ is small enough in Cao et al. (Commun. Math. Sci.8(1), 1–25 2010). The MLPFEM is adopted based on its significant advantages that we only need to solve a series of local subproblems once the coarse approximation is derived. Compared with the global discontinuous solution by the algorithm in Du and Zuo (2017), the main features of our algorithm are as follows: (1) partition of unity functions are utilized to gather local approximations computed on a fine grid by local and parallel procedures to generate a global continuous solution; (2) a further global coarse correction, namely the backtracking technique, is considered to obtain optimal error bounds of the velocity field and the piezometric head in L2 norm. Optimal error bounds are derived and numerical tests are carried out to support the theoretical analysis. [ABSTRACT FROM AUTHOR]

Based on two-grid discretizations, some local and parallel stabilized finite element methods are proposed and investigated for the Stokes problem in this paper. For the finite element discretization, the lowest equal-order finite element pairs are chosen to circumvent the discrete inf-sup condition. In these algorithms, we derive the low-frequency components of the solution for the Stokes problem on a coarse grid and catch the high-frequency components on a fine grid using some local and parallel procedures. Optimal error bounds are demonstrated and some numerical experiments are carried out to support theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, strategies are provided for a powerful numerical manifold method (NMM) with h and p refinement in analyses of elasticity and elasto-plasticity. The new NMM is based on the concept of the independent cover, which gets rid of NMM's important defect of rank deficiency when using higher-order local approximation functions. Several techniques are presented. In terms of mesh generation, a relationship between the quadtree structure and the mathematical mesh is established to allow a robust h-refinement. As to the condition number, a scaling based on the physical patch is much better than the classical scaling based on the mathematical patch; an overlapping width of 1%–10% can ensure a good condition number for 2nd, 3rd, and 4th order local approximation functions; the small element issue can be overcome after the local approximation on small patch is replaced by that on a regular patch. On numerical accuracy, local approximation using complete polynomials is necessary for the optimal convergence rate. Two issues that may damage the convergence rate should be prevented. The first is to approximate the curved boundary of a higher-order element by overly few straight lines, and the second is excessive overlapping width. Finally, several refinement strategies are verified by numerical examples. [ABSTRACT FROM AUTHOR]

The traditional finite element method (FEM) could only provide acceptable numerical solutions for the Helmholtz equation in the relatively small wave number range due to numerical dispersion errors. For the relatively large wave numbers, the corresponding FE solutions are never adequately reliable. With the aim to enhance the numerical performance of the FEM in tackling the Helmholtz equation, in this work an extrinsic enriched FEM (EFEM) is proposed to reduce the inherent numerical dispersion errors in the standard FEM solutions. In this extrinsic EFEM, the standard linear approximation space in the linear FEM is enriched extrinsically by using the polynomial and trigonometric functions. The construction of this enriched approximation space is realized based on the partition of unity concept and the highly oscillating features of the Helmholtz equation in relatively large wave numbers can be effectively captured by the employed specially-designed enrichment functions. A number of typical numerical examples are considered to examine the ability of this extrinsic EFEM to control the dispersion error for solving Helmholtz problems. From the obtained numerical results, it is found that this extrinsic EFEM behaves much better than the standard FEM in suppressing the numerical dispersion effects and could provide much more accurate numerical results. In addition, this extrinsic EFEM also possesses higher convergence rate than the conventional FEM. More importantly, the formulation of this extrinsic EFEM can be formulated quite easily without adding the extra nodes. Therefore, the present extrinsic EFEM can be regarded as a competitive alternative to the traditional finite element approach in dealing with the Helmholtz equation in relatively high frequency ranges. [ABSTRACT FROM AUTHOR]

In this paper we obtained generalisations of the L. V. Taikov’s and N. Ainulloev’s sharp inequalities, which estimate a norm of function's first-order derivative (L. V. Taikov) and a norm of function's second-order derivative (N. Ainulloev) via the modulus of continuity or the modulus of smoothness of the function itself and the modulus of continuity or the modulus of smoothness of the function's second-order derivative. The generalisations are obtained on the power of unbounded self-adjoint operators which act in a Hilbert space. The moduli of continuity or smoothness are defined by a strongly continuous group of unitary operators.

In this article, two kinds of local and parallel stabilized finite element methods based upon two grid discretizations are proposed and investigated for the Stokes–Darcy model. The lowest equal-order finite element pairs (P1-P1-P1) are taken into account to approximate the velocity, pressure, and piezometric head, respectively. To circumvent the inf-sup condition, the stabilized term is chosen as the difference between a consistent and an under-integrated mass matrix. The proposed algorithms consist of approximating the low-frequency component on the global coarse grid and the high-frequency component on the local fine grid and assembling them to obtain the final approximation. To obtain a global continuous solution, the technique tool of the partition of unity is used. A rigorous theoretical analysis for the algorithms was conducted and numerical experiments were carried out to indicate the validity and efficiency of the algorithms.

In this paper, a spatial high-order and extra-dof-free generalized finite element method (GFEM), which is based on partition of unity, is investigated for solving a class of fractional diffusion equations. An L1 scheme is used for time discretization. To resort to orthogonal projection, we first derive the error estimate of stationary problem by a local interpolation operator. Then we further study the well-posedness, stability and convergence of a discrete scheme of time-fractional diffusion problem. The offered numerical scheme demonstrates O (Δ t 1 + δ - α + h k) convergence rates, where α ∈ 0 , 1 , 0 ≤ δ ≤ 1 and k is 3, 4, 5. Finally, ample numerical examples are given to illustrate our theoretical achievements. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Surface reconstruction from a set of scattered points, or a point cloud, has many applications ranging from computer graphics to remote sensing. We present a new method for this task that produces an implicit surface (zero-level set) approximation for an oriented point cloud using only information about (approximate) normals to the surface. The technique exploits the fundamental result from vector calculus that the normals to an implicit surface are curl-free. By using curl-free radial basis function (RBF) interpolation of the normals, we can extract a potential for the vector field whose zero-level surface approximates the point cloud. We use curl-free RBFs based on polyharmonic splines for this task, since they are free of any shape or support parameters. To make this technique efficient and able to better represent local sharp features, we combine it with a partition of unity method. The result is the curl-free partition of unity (CFPU) method. We show how CFPU can be adapted to enforce exact interpolation of a point cloud and can be regularized to handle noise in both the normals and the point positions. Numerical results are presented that demonstrate how the method converges for a known surface as the sampling density increases, how regularization handles noisy data, and how the method performs on various problems found in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we propose a radial basis function partition of unity (RBF-PU) method to solve sparce optimal control problem governed by the elliptic equation. The objective function, in addition to the usual quadratic expressions, also includes an L1-norm of the control function to compute its spatio sparsity. Meshless methods based on RBF approximation are widely used for solving PDE problems but PDE-constrained optimization problems have been barely solved by RBF methods. RBF methods have the benefits of being versatile in terms of geometry, simple to use in higher dimensions, and also having the ability to give spectral convergence. In spite of these advantages, when globally RBF collocation methods are used, the interpolation matrix becomes dens and computational costs grow with increasing size of data set. Thus, for overcome on these problemes RBF-PU method will be proposed. RBF-PU methods reduce the computational effort. The aim of this paper is to solve the first-order optimality conditions related to original problem. [ABSTRACT FROM AUTHOR]

Conventional finite element methods are known to be tedious in adaptive refinements due to their conformal regularity requirements. Further, the enrichment functions for adaptive refinements are often not readily available in general applications. This work introduces a novel neural network-enriched Partition of Unity (NN-PU) approach for solving boundary value problems via artificial neural networks with a potential energy-based loss function minimization. The flexibility and adaptivity of the NN function space are utilized to capture complex solution patterns that the conventional Galerkin methods fail to capture. The NN enrichment is constructed by combining pre-trained feature-encoded NN blocks with an additional untrained NN block. The pre-trained NN blocks learn specific local features during the offline stage, enabling efficient enrichment of the approximation space during the online stage through the Ritz-type energy minimization. The NN enrichment is introduced under the Partition of Unity (PU) framework, ensuring convergence of the proposed method. The proposed NN-PU approximation and feature-encoded transfer learning form an adaptive approximation framework, termed the neural-refinement (n-refinement), for solving boundary value problems. Demonstrated by solving various elasticity problems, the proposed method offers accurate solutions while notably reducing the computational cost compared to the conventional adaptive refinement in the mesh-based methods. [ABSTRACT FROM AUTHOR]

The radial basis function partition of unity (RBF-PU) method  was applied to obtain the numerical solution of 2D nonlocal diffusion and peridynamic problems. The main idea is to partition the original domain into several patches, use the RBF approximation on each local domain, and then give weighting to obtain the global approximation of the unknown function. The radial basis function method based on the strong form of the equation has many advantages, such as avoiding an additional layer of integral calculation, no need to deal with intersections of neighborhoods with the mesh, and easiness of implementation. The numerical results show that, this method can solve nonlocal diffusion equations and peridynamic equations accurately and efficiently. [ABSTRACT FROM AUTHOR]

The article presents new sup -sums principles for integral F-divergence for arbitrary convex functions F on the whole real axis and arbitrary (not necessarily positive and normalized) measures. Among applications of these results, we work out a new 'integral' definition for t-entropy explicitly establishing its relation to Kullback–Leibler divergence. [ABSTRACT FROM AUTHOR]

In this paper, the generalized finite element method (GFEM) for solving second order elliptic equations with rough coefficients is studied. New optimal local approximation spaces for GFEMs based on local eigenvalue problems involving a partition of unity are presented. These new spaces have advantages over those proposed in [I. Babuska and R. Lipton, Multiscale Model. Simul., 9 (2011), pp. 373-406]. First, in addition to a nearly exponential decay rate of the local approximation errors with respect to the dimensions of the local spaces, the rate of convergence with respect to the size of the oversampling region is also established. Second, the theoretical results hold for problems with mixed boundary conditions defined on general Lipschitz domains. Finally, an efficient and easy-to-implement technique for generating the discrete A-harmonic spaces is proposed which relies on solving an eigenvalue problem associated with the Dirichlet-to-Neumann operator, leading to a substantial reduction in computational cost. Numerical experiments are presented to support the theoretical analysis and to confirm the effectiveness of the new method. [ABSTRACT FROM AUTHOR]

In this paper, based on two-grid discretizations and superposition principle, a new local and parallel finite element method is presented and studied for the coupled Stokes–Darcy model. Superposition principle is used to generate a series of local and independent subproblems to obtain a global continuous approximation. Optimal error estimates are derived. Numerical results are reported to support the theoretical findings. [ABSTRACT FROM AUTHOR]