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We propose a method to obtain super-resolution of turbulent statistics for three-dimensional ensemble particle tracking velocimetry (EPTV). The method is "meshless" because it does not require the definition of a grid for computing derivatives, and it is "binless" because it does not require the definition of bins to compute local statistics. The method combines the constrained radial basis function (RBF) formalism introduced Sperotto et al. (Meas Sci Technol 33:094005, 2022) with an ensemble trick for the RBF regression of flow statistics. The computational cost for the RBF regression is alleviated using the partition of unity method (PUM). Three test cases are considered: (1) a 1D illustrative problem on a Gaussian process, (2) a 3D synthetic test case reproducing a 3D jet-like flow, and (3) an experimental dataset collected for an underwater jet flow at Re = 6750 using a four-camera 3D PTV system. For each test case, the method performances are compared to traditional binning approaches such as Gaussian weighting (Agüí and Jiménez in JFM 185:447–468, 1987), local polynomial fitting (Agüera et al. in Meas Sci Technol 27:124011, 2016), as well as binned versions of RBFs. [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]

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]

Within the context of composite beam structure modeling, a refined model is combined with a simpler one solely within a region of interest, aiming to enhance result accuracy. The purpose is to gain computational cost without loss of precision. For that, the Partition Unity Method (PUM) is used to ensure the continuity condition on the displacement field. A 1D refined Sinus model and a 2D approach are involved to demonstrate the capability of the method. It is assessed on different numerical examples involving both laminated and sandwich beams. The accuracy of the results is shown using reference solutions and computations using a mono-model. The interest of the method is further demonstrating by estimating the computational cost with or without the coupling. The results are rather very promising. • A kinematics coupling is considered for the modeling of 2D laminated composite and sandwich beams. • The method is based on the Partition of Unity Method. • An enriched model is only localized in the region of interest. • The possibilities, the limitations and the accuracy of the method are highlighted. • The approach can be applied to different dimensionalities and FE approximations. [ABSTRACT FROM AUTHOR]

The present study proposes a novel approach for efficiently solving an anisotropic transient diffusion problem using an enriched finite element method. We develop directional enrichment for the finite elements in the spatial discretization and a fully implicit scheme for the temporal discretization of the governing equations. Within this comprehensive framework, the proposed class of exponential functions as enrichment enhance the approximation of the finite element method by capturing the directional based behaviour of the solution. The incorporation of these enrichment functions leverages a priori knowledge about the anisotropic problem using the partition of unity technique, resulting in significantly improved approximation efficiency while retaining all the advantages of the standard finite element method. Consequently, the proposed approach yields accurate numerical solutions even on coarse meshes and with significantly fewer degrees of freedom compared to the standard finite element methods. Moreover, the choice of mesh coarseness remains independent of the anisotropy in the problem, enabling the use of the same mesh regardless of changes in the anisotropy. Using extensive numerical experiments, we consistently demonstrate the efficiency of the proposed method in attaining the desired levels of accuracy. Our approach not only provides reliable and precise solutions but also extends the capabilities of the finite element method to effectively address aspects of the heterogeneous anisotropic transient diffusion problems that were previously considered ineffective when using this method. [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]

In the recent decade, several numerical procedures are developed based on the radial basis functions (RBFs) in the strong and the weak form of the mathematical model. These numerical algorithms have been used to solve a wide range of differential equations. However, the accuracy of RBFs collocation method for some partial differential equations (PDEs) is low thus a modification of RBFs collocation plan is required. The main aim of the current paper is to employ an improvement of RBFs collocation algorithm i.e. rational RBFs (RRBFs) collocation method based on the partition of unity (PU) idea to get the numerical solution of the multi-dimensional Ginzburg–Landau equation. It is clear that the RBFs collocation approach is an important numerical procedure for solving PDEs in non-rectangular physical regions. For differential equations with sufficiently smooth solutions, the RBF collocation technique generates acceptable and efficient accuracy. The RBF collocation method may produce solutions with non-physical oscillations for the underlying functions which have steep gradients or discontinuities. Using a fourth-order time-split approach and rational RBFs (RRBFs) collocation method, a new numerical procedure is proposed. First, a fourth-order time-split approach is used to discrete the time variable. Then, a combination of RBFs-PU collocation technique has been developed to get a full-discrete scheme. At the end, several examples have been studied to show the stability, convergence, and accuracy of the proposed numerical algorithm. [ABSTRACT FROM AUTHOR]

In this study, mixed-mode stress intensity factor (MMSIF) and crack growth analysis in functionally graded materials (FGMs) with discontinuities (crack, inclusion and void) are presented. The extended finite element method (XFEM) based study with the partition of unity method is utilized under thermo-mechanical loadings. The effect of mechanical and thermal loadings acting individually and combined on cracked FGM plate with other discontinuities is carried out. In mechanical loading, uniform tensile, shear, combination (tensile and shear), point, uniformly varying, and exponential loading are considered, whereas in thermal loadings, isothermal and adiabatic loadings are utilized. The objective of this present study is to find the MMSIF and crack growth analysis of FGM plate with discontinuities under different thermo-mechanical loadings, where the interaction study of discontinuities and different thermo-mechanical loadings are investigated. The present problem is modeled in MATLAB (R 2015a) environment and the results of the present model are compared with the results from the literature. [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 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, a generalized finite element (FE) method with optimal local approximation spaces for solving high-frequency heterogeneous Helmholtz problems is systematically studied. The local spaces are built from selected eigenvectors of carefully designed local eigenvalue problems defined on generalized harmonic spaces. At both continuous and discrete levels, (i) wavenumber explicit and nearly exponential decay rates for local and global approximation errors are obtained without any assumption on the size of subdomains and (ii) a quasi-optimal convergence of the method is established by assuming that the size of subdomains is O(1/k) (k is the wavenumber). A novel resonance effect between the wavenumber and the dimension of local spaces on the decay of error with respect to the oversampling size is implied by the analysis. Furthermore, for fixed dimensions of local spaces, the discrete local errors are proved to converge as h → 0 (h denoting the mesh size) toward the continuous local errors. The method at the continuous level extends the plane wave partition of unity method [I. Babuška and J. M. Melenk, Internat. J. Numer. Methods Engrg., 40 (1997), pp. 727-758] to the heterogeneous-coefficients case, and at the discrete level, it delivers an efficient noniterative domain decomposition method for solving discrete Helmholtz problems resulting from standard FE discretizations. Numerical results are provided to confirm the theoretical analysis and to validate the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, a new numerical simulation tool named TOUGH-FEMM is presented and applied to model three-dimensional (3D) hydraulic fracturing in porous rock. The fluid flow in both fractures and porous rock is modeled using TOUGH2, which is a well-established code for analysis of multiphase and multi-component fluid flow. Rock deformations associated with fracture propagation are modeled using finite element-meshfree method (FEMM). FEMM is an approach to simulate fracture propagation without remeshing, in which the fracture path does not need to be predetermined. Fracture mechanics with mixed-mode stress intensity factors are employed to detect fracture instability and determine the direction of fracture propagation. TOUGH-FEMM is verified for modeling fluid-driven fracture propagation in 3D through a number of simulation examples, including modeling of hydraulic fracturing laboratory experiments and by comparison to independent numerical simulation results for multiple interacting hydraulic fractures at ten to hundred meter scale.

In this work we develop the standard Hermite interpolation based RBF-generated finite difference (RBF-HFD) method into a new faster and more accurate technique based on partition of unity (PU) method. In the new approach, much fewer number of local linear systems needs to be solved for calculating the stencil weights. This reduces the computational cost of the method, remarkably. In addition, the method is flexible in using different types of PU weight functions, smooth or discontinuous, each results in a different scheme with additional nice properties. We also investigate the scaling property of polyharmonic spline (PHS) kernels to develop a simple and stable algorithm for computing local approximants in PU patches. Experimental results confirm the efficiency and applicability of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, by combining the two-grid method with a partition of unity, a local and parallel partition of unity scheme is designed and investigated for the mixed Navier-Stokes-Darcy problem. The main features of the present method are the following: (1) Once a coarse solution is derived, both the linearized Navier-Stokes and Darcy subproblems with a finer grid are independent, which allows a parallel computing with little communication; (2) a partition of unity is considered to assemble the solutions obtained from the local subdomains to arrive at a global continuous approximation; (3) a further coarse grid correction is carried out to derive optimal error bounds for the fluid velocity and piezometric head in L2-norm. Moreover, the convergence of the proposed method is shown. Some numerical experiments are reported to demonstrate the theoretical results. [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]

We derive stability estimates for three commonly used radial basis function (RBF) methods to solve hyperbolic time-dependent PDEs: the RBF generated finite difference (RBF-FD) method, the RBF partition of unity method (RBF-PUM) and Kansa's (global) RBF method. We give the estimates in the discrete ℓ 2 -norm intrinsic to each of the three methods. The results show that Kansa's method and RBF-PUM can be ℓ 2 -stable in time under a sufficiently large oversampling of the discretized system of equations. The RBF-FD method in addition requires stabilization of the spurious jump terms due to the discontinuous RBF-FD cardinal basis functions. Numerical experiments show an agreement with our theoretical observations. • Stability estimates for RBF methods discretizing linear conservation laws. • Spurious terms in Kansa's method, RBF-PUM, RBF-FD arise from lack of exact quadrature. • The RBF-FD estimate also entails a spurious term due to discontinuous trial space. • Oversampling and jump stabilization are possible stabilizations. [ABSTRACT FROM AUTHOR]

In this paper, we employ Bayesian optimization to concurrently explore the optimal values for both the shape parameter and the radius in the partition of unity interpolation using radial basis functions. Bayesian optimization is a probabilistic, iterative approach that models the error function through a progressively self-updated Gaussian process. Meanwhile, the partition of unity approach harnesses a meshfree method, allowing us to significantly reduce computational expenses, particularly when considering a substantial number of scattered data points. This reduction in computational cost is achieved by decomposing the entire domain into several smaller subdomains, each of them with a variable radius. We provide an estimation of the complexity of our algorithm and carry out numerical experiments to illustrate the effectiveness of our approach, dealing with test and real-world datasets. [ABSTRACT FROM AUTHOR]

In this paper we develop a partition-of-unity construction of the stabilization function required in Nitsche's method, which can be seen as a generalization of the element-wise construction that is widely used in finite element methods. This allows for the use of Nitsche's method within the Partition of Unity Method with a stabilization function that is not simply a constant over the whole boundary. In addition to that, we introduce a patch-aggregation approach designed to avoid arbitrarily large values of the stabilization function and the associated ill-conditioned systems and deteriorated convergence rates. We present numerical results to validate the proposed methods, covering Dirichlet boundary conditions, interface constraints and higher-order problems. These results clearly show that our approach leads to optimal convergence rates. [ABSTRACT FROM AUTHOR]

• Beam finite element-based buckling analysis of arbitrarily stiffened thin-walled members. • Modal approach-based beam model allowing for the cross-section discontinuity. • Combined the beam model with 4-node shell finite elements via a three-field mortar method. • Non-conforming meshes allow for arbitrary arrangement of stiffeners. • Pure buckling mode identification from the coupled buckled shapes of any stiffened member. Thin-walled members with transversal/diagonal stiffeners (e.g., plate girders), from the point of view of the generalized beam theory (GBT), can be treated as beams with discontinuous cross-sections. Generally, it is hard to deal with these members with the classic GBT-based beam finite elements for the presence of stiffeners spoils the continuities of the mode amplitude functions, i.e., one of the essential assumptions in the GBT kinematics. This paper presents an extended GBT (XGBT)-based beam finite element through a combination of the classic GBT with the extended finite element method (XFEM). Specifically, the classic GBT kinematics are used to construct the standard approximation of the cross-section mid-line displacement fields of a beam finite element, and extra enrichment functions which describe the field discontinuities across the beam-stiffener joint edges are added to the standard approximation by using the partition of unity method (PUM). Furthermore, the stiffeners are modelled by shell finite elements, and they are integrated into the extended GBT-based beam model via a three-field mortar method, where the discrete Lagrange multiplier space on each beam-stiffener interface is spanned by the constant functions supported on coarsely partitioned segments over the interface. The implementation of the proposed XGBT formulation is performed via FORTRAN programming and several illustrative examples concerning the linear buckling and statically elastoplastic analyses of stiffened thin-walled beams are employed to accomplish the validation, where the XGBT results are verified by shell finite element results. Results show that the proposed XGBT approach exhibits much lower computational costs for producing equivalently accurate solutions and is much more structurally meaningful than the shell model. [ABSTRACT FROM AUTHOR]

Partition of unity methods (PUM) are of domain decomposition type and provide the opportunity for multiscale and multiphysics numerical modeling. Here, we apply Peridynamic (PD) enrichment to propagate cracks in the PUM global–local enrichment scheme. We apply linear elasticity globally and PD over local zones where fractures occur. The elastic fields provide appropriate boundary data for local PD simulations on a subdomain containing the crack tip to grow the crack. Once the updated crack path is found the elastic field in the body and surrounding the crack is updated using the PUM basis with an elastic field enrichment near the crack. The subdomain for the PD simulation is chosen to include the current crack tip as well as features that influence crack growth. This paper is part II of this series and validates the combined PD/PUM simulator against the experimental results. The results of numerical simulation show that we attain good agreement between experiment and simulation with a local PD subdomain that is moving with the crack tip and adaptively chosen size. • A novel Multiscale Fracture Model using Peridynamic Enrichment of Finite Elements with an Adaptive Partition of Unity • Validation against three experimental data sets. • In addition to the experiments, we validate our hierarchical coupling technique approach with a concurrent coupling approach proposed by Ni et al. using the same experimental data. [ABSTRACT FROM AUTHOR]

In this paper, we propose a plane wave activation based neural network (PWNN) to solve the Helmholtz equation with constant coefficients and relatively large wave number k efficiently. Since the complex activation function e i x → is introduced to be the activation function of the neural network, PWNN significantly improves the computational speed and accuracy as compared to traditional activation based neural networks (TANN) and the finite difference method (FDM), for relatively large wave number problems. We establish a new upper bound for error estimates for homogeneous Helmholtz solutions in two dimensions by plane waves, which is different from the previous estimates by generalized harmonic polynomials. Based on the new error estimates, the theoretical guidance is given for choosing the number of neural network's neurons and the initial value to accelerate network training. The analyses of computational complexity related to the wave number k are given for PWNN with two layers, TANN, the plane wave partition of unity method (PWPUM) and FDM. Numerical experiments in 2D and 3D are performed to demonstrate the efficiency and accuracy of PWNN. Especially for large wave number problems, like k = 500 , PWNN can get the solution with relative error less than 10 − 4 in less than 20 seconds, which is more efficient than other methods. [ABSTRACT FROM AUTHOR]

Aiming at the low efficiency and high energy consumption of unmanned ships traversing the entire area, a complete coverage path planning algorithm based on the improved A-star algorithm is proposed. The positioning and vision systems of unmanned ships are used to digitize the actual water information, and the grid method is used to convert the information into an environmental map that can be planned. Compared to the trapezoidal partition of unity method and the short-side reciprocating traversal algorithm in the traversal process, experiments show that path planning is more efficient with the boustrophedon partition of unity method and the long-side reciprocating traversal algorithm. Aiming at the "dead zone", an improved A-star algorithm is proposed on the basis of the traditional A-star algorithm, that it can shorten about 1/4 path using the proposed algorithm. Simulation shows that the improved A-star algorithm can shorten the traversal path to 40 steps but the traditional A-star algorithm needs 54 steps. Navigation test shows that the proposed algorithm can shorten the traversal path and improve traversal efficiency while ensuring the coverage of unmanned ships. [ABSTRACT FROM AUTHOR]

• Implicit reconstruction can be accurate and efficient for thin leaf surfaces. • Whole plant reconstructions are handled naturally, including clustering of leaves. • Reconstructed leaf surfaces have continuous mean curvature. Thin surfaces, such as the leaves of a plant, pose a significant challenge for implicit surface reconstruction techniques, which typically assume a closed, orientable surface. We show that by approximately interpolating a point cloud of the surface (augmented with off-surface points) and restricting the evaluation of the interpolant to a tight domain around the point cloud, we need only require an orientable surface for the reconstruction. We use polyharmonic smoothing splines to fit approximate interpolants to noisy data, and a partition of unity method with an octree-like strategy for choosing subdomains. This method enables us to interpolate an N -point dataset in O (N) operations. We present results for point clouds of capsicum and tomato plants, scanned with a handheld device. An important outcome of the work is that sufficiently smooth leaf surfaces are generated that are amenable for droplet spreading simulations. [ABSTRACT FROM AUTHOR]

In this paper, based on two-grid discretization, a new local and parallel finite element method is proposed and investigated for the mixed Navier–Stokes–Darcy problem. Compared with the classical local and parallel finite element method, the main features of our method are: (1) partition of unity method is considered to generate local subproblems; (2) global continuous approximations are achieved. We theoretically analyze the proposed method and report on some numerical tests to support the theoretical results. [ABSTRACT FROM AUTHOR]

Divergence-free (div-free) and curl-free vector fields are pervasive in many areas of science and engineering, from fluid dynamics to electromagnetism. A common problem that arises in applications is that of constructing smooth approximants to these vector fields and/or their potentials based only on discrete samples. Additionally, it is often necessary that the vector approximants preserve the div-free or curl-free properties of the field to maintain certain physical constraints. Div/curl-free radial basis functions (RBFs) are a particularly good choice for this application as they are meshfree and analytically satisfy the div-free or curl-free property. However, this method can be computationally expensive due to its global nature. In this paper, we develop a technique for bypassing this issue that combines div/curl-free RBFs in a partition of unity framework, where one solves for local approximants over subsets of the global samples and then blends them together to form a div-free or curl-free global approximant. The method is applicable to div/curl-free vector fields in R2 and tangential fields on two-dimensional surfaces, such as the sphere, and the curl-free method can be generalized to vector fields in Rd. The method also produces an approximant for the scalar potential of the underlying sampled field. We present error estimates and demonstrate the effectiveness of the method on several test problems. [ABSTRACT FROM AUTHOR]

Graph signal processing benefits significantly from the direct and highly adaptable supplementary techniques offered by partition of unity methods (PUMs) on graphs. In our approach, we demonstrate the generation of a partition of unity solely based on the underlying graph structure, employing an algorithm that relies exclusively on centrality measures and modularity, without requiring the input of the number of subdomains. Subsequently, we integrate PUMs with a local graph basis function (GBF) approximation method to develop cost-effective global interpolation schemes. We also discuss numerical experiments conducted on both synthetic and real datasets to assess the performance of this presented technique. • A new algorithm is implemented for partition of unity interpolation on graphs. • The partition of unity generation is based on the graph structure involving centrality measures and modularity. • The process automatically finds an optimal number of graph communities (or partition of unity subdomains). • Numerical experiments show efficacy and performance of new adaptive method on both test and real-world datasets. [ABSTRACT FROM AUTHOR]

A methodology is presented for the numerical solution of nonlinear elliptic systems in unbounded domains, consisting of three elements. First, the problem is posed on a finite domain by means of a proper nonlinear change of variables. The compressed domain is then discretised, regardless of its final shape, via the radial basis function partition of unity method. Finally, the system of nonlinear algebraic collocation equations is solved with the trust-region algorithm, taking advantage of analytically derived Jacobians. We validate the methodology on a benchmark of computational fluid mechanics: the steady viscous flow past a circular cylinder. The resulting flow characteristics compare very well with the literature. Then, we stress-test the methodology on less smooth obstacles—rounded and sharp square cylinders. As expected, in the latter scenario the solution is polluted by spurious oscillations, owing to the presence of boundary singularities. • Infinite domain compression removes the need for artificial computational domain and the hard-to-quantify associated error. • Nonlinear elliptic PDEs, particularly systems thereof, are very rare in the RBF literature. • RBF-PU allows for handling systems of nonlinear PDEs in strong form on irregular domains without loss of performance. • RBF-PU's accurate derivates lead to a compact, sparse, easy-to-derive analytical Jacobian. • Detailed transition from circular to square cylinder investigated numerically, showing good agreement with published results. [ABSTRACT FROM AUTHOR]

In this work, a refined model is superimposed on a simple one only in a region of interest to improve the accuracy of the modeling of composite beam structures. The purpose is to concentrate the computational effort in a localized zone without loss of local precision. Outside the region of interest is modelized using a simple cheap model. The present approach is based on the Partition Unity Method (PUM) to ensure the continuity condition on the displacement field. Two refined models based on the Sinus model are used to demonstrate the capability of the method. Each of the kinematics needs a suitable Finite Element (FE) approximation. The method is assessed on numerical examples involving both homogeneous and laminated beams. The results are compared with reference solutions and computations using a mono-model. The computational cost is also evaluated to show the interest of the method. The results are very promising. • A local refinement of kinematics is presented for the analysis of 2D laminated composite beams. • The Partition of Unity Method is carried out to superimpose simple models with dedicated refined ones. • The enrichment is only localized in the region of interest. • This finite element yields satisfactory results with a decreasing computational cost. • The approach can be generalized to any refined models and FE approximations. [ABSTRACT FROM AUTHOR]

We propose an efficient inverse solver for identifying the diffusion coefficient based on few random measurements which can be contaminated with noise. We focus mainly on problems involving solutions with steep heat gradients common with sudden changes in the temperature. Such steep gradients can be a major challenge for numerical solutions of the forward problem as they may involve intensive computations especially in the time domain. This intensity can easily render the computations prohibitive for the inverse problems that requires many repetitions of the forward solution. Compared to the literature, we propose to make such computations feasible by developing an iterative approach that is based on the partition of unity finite element method, hence, significantly reducing the computations intensity. The proposed approach inherits the flexibility of the finite element method in dealing with complicated geometries, which otherwise cannot be achieved using analytical solvers. The algorithm is evaluated using several test cases. The results show that the approach is robust and highly efficient even when the input data is contaminated with noise. [ABSTRACT FROM AUTHOR]

Option pricing under continuous-time CAPM has been formulated by partial differential equation, which is an extension of the Black–Scholes PDE. The focus of this paper is to present the radial basis function partition of unity method for evaluation of options in which underlying asset price follows the two-state continuous CAPM. Our experiments illustrate the behavior of the market volatility skew and the effect of the asset's beta on option price. The numerical examples show that the proposed scheme is accurate and competitive with finite difference method. [ABSTRACT FROM AUTHOR]

This article investigates the convergence properties of the augmented finite element method (AFEM). The AFEM is here used to model weak discontinuities independently of the underlying mesh. One noticeable advantage of the AFEM over other partition of unity methods is that it does not introduce additional global unknowns. Numerical 2D experiments illustrate the performance of the method and draw comparisons with the finite element method (FEM) and the nonconforming FEM. It is shown that the AFEM converges with an error of 𝒪(h0.5) in the energy norm. The nonconforming FEM shares the same property while the FEM converges at 𝒪(h). Yet, the AFEM is on par with the FEM for certain homogenization problems. [ABSTRACT FROM AUTHOR]

In this paper, a new localized radial basis function (RBF) method based on partition of unity (PU) is proposed for solving boundary and initial-boundary value problems. The new method benefits from a direct discretization approach and is called the "direct RBF partition of unity (D-RBF-PU)" method. Thanks to avoiding all derivatives of PU weight functions as well as all lower derivatives of local approximants, the new method is faster and simpler than the standard RBF-PU method. Besides, the discontinuous PU weight functions can now be utilized to develop the method in a more efficient and less expensive way. Alternatively, the new method is an RBF-generated finite difference (RBF-FD) method in a PU setting which is much faster and in some situations more accurate than the original RBF-FD. The polyharmonic splines are used for local approximations, and the error and stability issues are considered. Some numerical experiments on irregular two- and three-dimensional domains, as well as cost comparison tests, are performed to support the theoretical analysis and to show the efficiency of the new method. [ABSTRACT FROM AUTHOR]

Purpose: The purpose of this paper is to evaluate the capabilities of the generalized finite element method (GFEM) under the context of the geometrically nonlinear analysis. The effect of large displacements and deformations, typical of such analysis, induces a significant distortion of the element mesh, penalizing the quality of the standard finite element method approximation. The main concern here is to identify how the enrichment strategy from GFEM, that usually makes this method less susceptible to the mesh distortion, may be used under the total and updated Lagrangian formulations. Design/methodology/approach: An existing computational environment that allows linear and nonlinear analysis, has been used to implement the analysis with geometric nonlinearity by GFEM, using different polynomial enrichments. Findings: The geometrically nonlinear analysis using total and updated Lagrangian formulations are considered in GFEM. Classical problems are numerically simulated and the accuracy and robustness of the GFEM are highlighted. Originality/value: This study shows a novel study about GFEM analysis using a complete polynomial space to enrich the approximation of the geometrically nonlinear analysis adopting the total and updated Lagrangian formulations. This strategy guarantees the good precision of the analysis for higher level of mesh distortion in the case of the total Lagrangian formulation. On the other hand, in the updated Lagrangian approach, the need of updating the degrees of freedom during the incremental and iterative solution are for the first time identified and discussed here. [ABSTRACT FROM AUTHOR]

Purpose: The purpose of this paper is to evaluate some numerical integration strategies used in generalized (G)/extended finite element method (XFEM) to solve linear elastic fracture mechanics problems. A range of parameters are here analyzed, evidencing how the numerical integration error and the computational efficiency are improved when particularities from these examples are properly considered. Design/methodology/approach: Numerical integration strategies were implemented in an existing computational environment that provides a finite element method and G/XFEM tools. The main parameters of the analysis are considered and the performance using such strategies is compared with standard integration results. Findings: Known numerical integration strategies suitable for fracture mechanics analysis are studied and implemented. Results from different crack configurations are presented and discussed, highlighting the necessity of alternative integration techniques for problems with singularities and/or discontinuities. Originality/value: This study presents a variety of fracture mechanics examples solved by G/XFEM in which the use of standard numerical integration with Gauss quadratures results in loss of precision. It is discussed the behaviour of subdivision of elements and mapping of integration points strategies for a range of meshes and cracks geometries, also featuring distorted elements and how they affect strain energy and stress intensity factors evaluation for both strategies. [ABSTRACT FROM AUTHOR]