Partition Of Unity Method Research Articles

Some Aspects of Teaching Improper Integrals in an Electronic Environment

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.

A Meshless And Binless Approach To Compute Statistics In 3D Ensemble PTV

We propose a method to obtain super-resolution of turbulent statistics for three-dimensional ensemble particle tracking velocimetry (EPTV). Our approach is “meshless” because it does not require a grid to compute derivatives, and it is “binless” because it does not require bins to compute local statistics. The method combines the constrained radial basis function (RBF) formalism introduced in Sperotto et al. (Meas Sci Technol, 33:094005, 2022) with a kernel estimate approach for the ensemble average of the RBF regressions. The computational cost for the RBF regression is alleviated with the partition of unity method (PUM). We compare the newly proposed method with traditional binning-based approaches such as Gaussian weighting (Agüí and Jiménez, JFM, 185:447-468, 1987) and local polynomial fitting (Agüera et al, Meas Sci Technol, 27:124011, 2016). The results on experimental data of an underwater jet show a better resolution of spatial gradients of turbulent stresses while at the same time yielding an analytical expression which results in super-resolution of statistical quantities.

A Partition of Unity construction of the stabilization function in Nitsche’s method for variational problems

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.

A multiscale fracture model using peridynamic enrichment of finite elements within an adaptive partition of unity: Experimental validation

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.

Partition of unity method and smooth approximation

Partition of unity method and smooth approximation

Exploration of Kernel Parameters in Signal GBF-PUM Approximation on Graphs

Abstract The application of the Partition of Unity Method (PUM) to signal approximation on graphs represents a recent advancement of this versatile and efficient interpolation technique. Given the novelty of this approach, little is yet known regarding the role of kernel parameters employed in constructing the associated Graph Basis Functions (GBFs). In order to shed light on this aspect, this study proposes several numerical tests obtained using GBFs generated by heat kernels and variational spline kernels.

Node-bound communities for partition of unity interpolation on graphs

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 parallel finite element post-processing algorithm for the damped Stokes equations

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 L2-error estimates both for the approximate velocity and pressure and H1-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.

Simulation of the coupled Schrödinger–Boussinesq equations through integrated radial basis functions-partition of unity method

In this paper, integrated radial basis functions-partition of unity (IRBF-PU) method is presented for the numerical solution of the one- and two-dimensional coupled Schrödinger–Boussinesq equations. The IRBF-PU method is a local mesh-free method that prepares flexibility and high orders of accuracy for PDEs problems with adequately smooth initial conditions and also has a moderate condition number. First, the temporal direction is discretized using the Runge–Kutta method with non-decreasing abscissas and nine stages, that allows for greater flexibility in the temporal step width. The integrated radial basis function based on partition of unity method (IRBF-PUM) then makes an approximation of the spatial direction. Numerical simulations are also used to track conserved quantities to determine how well the suggested approach keeps them. IRBF-PU simulates solitary waves for an extended period of time while effectively preserving conservation laws, according to numerical experiments. To confirm the effectiveness and dependability of the suggested method, the obtained results are contrasted with those obtained using other methods found in the literature.

Numerical modeling of the ion‐acoustic solitary waves arising in nonlinear dispersive system

This paper proposes a numerical scheme for the (2 + 1)‐dimensional nonlinear Zakharov–Kuznetsov–Benjamin–Bona–Mahony equation (ZK‐BBME). The ZK‐BBME represents a long‐wave model with large wavelength that explains the water wave phenomena in nonlinear dispersive system. The solution of the ZK‐BBME is discretized by using hybridization of the finite difference and localized radial basis function based on partition of unity method. The association of this hybridization leads to a meshless method and it does not requires any linearization. In the initial step, the PDE is converted into a nonlinear ODE system by making use of radial kernels. In the next step, the obtained nonlinear ODE system is solved based on an ODE solver of high order. The global collocation technique imposes a large computational cost because a dense algebraic system must be calculated. This proposed strategy is based on decomposing the initial domain into a number of sub‐domains using kernel approximation on each sub‐domain. Three numerical examples are illustrated to clarify the efficiency and accuracy of the proposed strategy.

An extended GBT formulation for elastoplastic analyses of perforated thin-walled members

A physically and geometrically nonlinear beam finite element modelling of thin-walled members with arbitrarily complex holes is presented. The beam finite element approximation of the member configuration space is achieved by using the extended generalized beam theory kinematic description, which covers discontinuities in member cross-sections due to holes or cut-outs of various shapes. The proposed beam finite element deals with perforated members in a manner of treating them as non-perforated ones by filling the perforations with much softer materials. Then the level-set functions are used to represent the bi-material interfaces and also used to construct the enrichment functions which describe the discontinuities in the displacement fields. The enrichment functions are then integrated into the standard GBT-based beam finite element approximation by the Partition of Unity Method. The J2-plasticity model combined with isotropic hardening and associated flow rule for isotropic materials (e.g., soft steel) is used in the model. To show the potential of the proposed approach, four illustrative examples are provided, where results obtained by the presented model are compared with the shell finite element results. The proposed GBT model shows significant superiorities in the computational efficiency and structural clarity over the shell model.

Implicit Surface Reconstruction With a Curl-Free Radial Basis Function Partition of Unity Method

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.

Radial Basis Function Regression Of Lagrangian Three-Dimensional Particle Tracking Data

Flow characterization by means of Particle Tracking Velocimetry (PTV) has gained significant importance in recent years. This is especially true in microfluidics, where the limited optical access only allows for using one camera. A commonly used technique is the Astigmatism Particle Tracking Velocimetry (APTV), which provides reliable 3D3C measurements. However, the resulting data is available on scattered points and usually requires interpolation onto regular grids for further processing. In this work, we test Radial Basis Function (RBF) with the Partition of Unity Method (PUM) for the regression and mesh-free derivative evaluation in large velocity fields. The RBF-PUM approach is first benchmarked on a synthetic test case against the classic Gaussian Window interpolation (AGW) and global RBF. Then, we test the RBF-PUM approach on a three-dimensional experimental dataset consisting of 500.000 data points in a vortex flow. The results prove that the RBF-PUM allows for accurate regression at an accessible computational cost.

XFEM based thermo-elastic numerical analysis of FGMs with various discontinuities

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.

A condensed generalized finite element method (CGFEM) for interface problems

Extensive developments on various generalizations of the Finite Element Method (FEM) for the interface problems, with unfitted mesh, have been made in the last few decades. Typical generalizations and techniques include the immersed FEM, the penalized FEM, the generalized or the extended FEM (GFEM/XFEM). The GFEM/XFEM achieves good approximation by augmenting the finite element approximation space with enrichment functions, which require additional degrees of freedom (DOF). However these additional DOFs, in general, deteriorate the conditioning of the GFEM/XFEM. In this study, we propose a condensed GFEM (CGFEM) for the interface problem based on the partition of unity method and a local least square scheme. Compared to the conventional GFEM/XFEM, the CGFEM possesses the same number of DOFs as in the FEM, yields good approximation, and is well-conditioned. In comparison with the immersed FEM, the construction of shape functions of CGFEM is independent of the equation types and the material coefficients of the interface problem. As a result, the CGFEM could be applied to more general interface problems, e.g., problems with anisotropic materials and elasticity systems, in a unified approach. Moreover, the shape functions of CGFEM are continuous, and thus do not need any penalty terms and parameters. The optimal order of convergence of the CGFEM has been analyzed and established theoretically in this paper. Numerical experiments for isotropic and anisotropic interface problems and comparisons with the conventional GFEM/XFEM have also been presented to illuminate the theory and effectiveness of CGFEM.

Generalized beam theory-based advanced beam finite elements for linear buckling analyses of perforated thin-walled members

This paper presents a numerically efficient tool for linear buckling predictions of perforated thin-walled bars within the framework of generalized beam theory (GBT). GBT-based beam finite element methods (GBT-FEM) have been well developed for eigenvalue buckling analyses of non-perforated thin-walled bars. The novelty of this paper consists in an extension of the standard GBT to the scope of thin-walled members with arbitrarily shaped and placed holes. This is achieved by combining the standard GBT and the extended finite element method (X-FEM). More specifically, insert a set of locally supported enrichment functions, accounting for the discontinuities on displacement fields arising from the cross-section cut-outs/holes, into the GBT-based finite element approximations of the member configuration spaces, using the partition of unity method (PUM), where a set of level-set functions are used to describe the geometric profiles of hole edges, i.e., with the hole edges being the zero level sets, and also used to construct the enrichment functions. The proposed approach makes it possible to calculate the deformation mode participations for any perforated thin-walled members as the classic GBT for non-perforated ones. Finally, the proposed approach is calibrated against the shell/solid finite element analysis with four illustrative examples. It can be found that the presented approach is of higher computation efficiency than the shell model.

Partition of Unity Methods for Signal Processing on Graphs

Partition of unity methods (PUMs) on graphs are simple and highly adaptive auxiliary tools for graph signal processing. Based on a greedy-type metric clustering and augmentation scheme, we show how a partition of unity can be generated in an efficient way on graphs. We investigate how PUMs can be combined with a local graph basis function (GBF) approximation method in order to obtain low-cost global interpolation or classification schemes. From a theoretical point of view, we study necessary prerequisites for the partition of unity such that global error estimates of the PUM follow from corresponding local ones. Finally, properties of the PUM as cost-efficiency and approximation accuracy are investigated numerically.

Finite element modelling of traction-free cracks: Benchmarking the augmented finite element method (AFEM)

This paper investigates the accuracy and the convergence properties of the augmented finite element method (AFEM). The AFEM is here used to model strong 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 to represent cracks. Numerical 2D experiments illustrate the performance of the method and draw comparisons with the element deletion method (EDM), the phantom node method (PNM), the finite element method (FEM) and the embedded finite element method (EFEM). The h-convergence in the energy norm of the AFEM is studied for the first time and it is shown to outperform the aforementioned numerical methods when cracks are loaded in Mode I.

Implicit reconstructions of thin leaf surfaces from large, noisy point clouds

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.

On a connection between some classes of mapping on Riemannian manifolds

In this paper we study the connection between $\eta$-quasisymmetric homomorphisms and $K$-quasi\-con\-for\-mal mappings on $n$-dimensional smooth connected Riemannian manifolds. The main result of our research is the Theorem 3.1. For its proof we use a partition of unity method, which subordinate to the locally finite atlas of the manifold. Several results on the boundary behavior of $\eta$-quasisymmetric homomorphisms between two arbitrary domains, QED (uniform) domains and domains with weakly flat boundaries and compact closures on the Riemannian manifolds are also obtained in view of the above relations. The obtained results can be applied to Finsler manifolds with the addition of some conditions, which will take into account the specific of the initial manifold.