Partition Of Unity Functions Research Articles

Improved modal analyses using the novel quadrilateral overlapping elements

In this work, the novel quadrilateral overlapping element incorporated with the harmonic trigonometric functions is presented to reduce the efforts in mesh division for modal analysis. The new partition of unity functions in the overlapping element are cubic polynomials. Therefore, low-order polynomials can be utilized to augment local approximation space without the linear dependence problem. Moreover, the original polynomial approximation spaces are further enriched by the harmonic trigonometric functions without introducing additional rank deficiencies in the system matrices. Thus, the presented enriched overlapping finite element method (EOFEM) provides very precise eigenfrequencies and eigenmodes by using a very coarse mesh. However, the shape functions in the presented EOFEM do not possess the Kronecker delta function property. Hence, a conforming method that modifies the shape functions rather than the variational principle is proposed to impose the essential boundary conditions. Several numerical examples in two dimensions are then investigated to elaborate on the performance of the presented EOFEM in modal analyses. Besides, a block-diagonal lumped mass matrix is presented and further extended to the structure vibration analysis by incorporating the explicit time integration method. It is proven that the enriched overlapping element is computationally more efficient than the quadratic finite element and the original overlapping element.

Exponentially Convergent Multiscale Finite Element Method

We provide a concise review of the exponentially convergent multiscale finite element method (ExpMsFEM) for efficient model reduction of PDEs in heterogeneous media without scale separation and in high-frequency wave propagation. The ExpMsFEM is built on the non-overlapped domain decomposition in the classical MsFEM while enriching the approximation space systematically to achieve a nearly exponential convergence rate regarding the number of basis functions. Unlike most generalizations of the MsFEM in the literature, the ExpMsFEM does not rely on any partition of unity functions. In general, it is necessary to use function representations dependent on the right-hand side to break the algebraic Kolmogorov n-width barrier to achieve exponential convergence. Indeed, there are online and offline parts in the function representation provided by the ExpMsFEM. The online part depends on the right-hand side locally and can be computed in parallel efficiently. The offline part contains basis functions that are used in the Galerkin method to assemble the stiffness matrix; they are all independent of the right-hand side, so the stiffness matrix can be used repeatedly in multi-query scenarios.

Simulation of Destructive Test for Arch Reinforcement Area of Tunnel Lining by MSIM

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.

A convergence analysis of Generalized Multiscale Finite Element Methods

In this paper, we consider an approximation method, and a novel general analysis, for second-order elliptic differential equations with heterogeneous multiscale coefficients. We obtain convergence of the Generalized Multi-scale Finite Element Method (GMsFEM) method that uses local eigenvectors in its construction. The analysis presented here can be extended, without great difficulty, to more sophisticated GMsFEMs. For concreteness, the obtained error estimates generalize and simplify the convergence analysis of Y. Efendiev et al. (2011) [22]. The GMsFEM method construct basis functions that are obtained by multiplication of (approximation of) local eigenvectors by partition of unity functions. Only important eigenvectors are used in the construction. The error estimates are general and are written in terms of the eigenvalues of the eigenvectors not used in the construction. The error analysis involve local and global norms that measure the decay of the expansion of the solution in terms of local eigenvectors. Numerical experiments are carried out to verify the feasibility of the approach with respect to the convergence and stability properties of the analysis.

New enriched 3D solid finite elements: 8-node hexahedral, 6-node prismatic, and 5-node pyramidal elements

Herein, we propose new 8-node hexahedral, 6-node prismatic, and 5-node pyramidal 3D solid finite elements enriched by cover functions. To resolve the linear dependence problem, piecewise linear shape functions are adopted as the partition of unity functions. Various mesh patterns are considered for investigation of the linear dependence problem, and no rank deficiency is observed. We demonstrate convergence behavior and effectiveness of the new enriched elements, and adaptive use of cover functions through six problems: an ad hoc problem, a tool jig problem, a slender beam problem, a gear problem, and a connecting rod problem.

Domain Decomposition Preconditioners for Mixed Finite-Element Discretization of High-Contrast Elliptic Problems

In this paper, we design an efficient domain decomposition (DD) preconditioner for the saddle-point problem resulting from the mixed finite-element discretization of multiscale elliptic problems. By proper equivalent algebraic operations, the original saddle-point system can be transformed to another saddle-point system which can be preconditioned by a block-diagonal matrix efficiently. Actually, the first block of this block-diagonal matrix corresponds to a multiscale $$H(\mathrm {div})$$ problem, and thus, the direct inverse of this block is unpractical and unstable for the large-scale problem. To remedy this issue, a two-level overlapping DD preconditioner is proposed for this $$H(\mathrm {div})$$ problem. Our coarse space consists of some velocities obtained from mixed formulation of local eigenvalue problems on the coarse edge patches multiplied by the partition of unity functions and the trivial coarse basis (e.g., Raviart–Thomas element) on the coarse grid. The condition number of our preconditioned DD method for this multiscale $$H(\mathrm {div})$$ system is bounded by $$C(1+\frac{H^2}{{\hat{\delta }}^2})(1+\log ^4(\frac{H}{h}))$$ , where $$\hat{\delta }$$ denotes the width of overlapping region, and H, h are the typical sizes of the subdomain and fine mesh. Numerical examples are presented to confirm the validity and robustness of our DD preconditioner.

On Some Properties of Vector Measures

We study the properties of a parameterized sequence of countably additive vector measures with densities defined on a compact space T with a nonnegative nonatomic Radon measure μ and taking values in a separable Banach space. Each vector measure of this sequence depends continuously on a parameter belonging to a metric space. We assume that a countable locally finite open cover and a partition of unity inscribed into this cover are given in the metric space of parameters. We prove that, for each value of the parameter, there exists a sequence of μ-measurable subsets of the space T which is a partition of T. In addition, this sequence depends uniformly continuously on the parameter and, for each value of the parameter and each element of the initial parameterized sequence of vector measures, the relative value of the measure of the corresponding subset in the partition of T can be uniformly approximated by the corresponding value of the corresponding partition of unity function.

Local and parallel finite element algorithm based on the partition of unity method for the incompressible MHD flow

Based on the partition of unity method (PUM), a local and parallel finite element method is designed and analyzed for solving the stationary incompressible magnetohydrodynamics (MHD). The key idea of the proposed algorithm is to first solve the nonlinear system on a coarse mesh, divide the globally fine grid correction into a series of locally linearized residual problems on some subdomains derived by a class of partition of unity, then compute the local subproblems in parallel, and obtain the globally continuous finite element solution by assembling all local solutions together by the partition of unity functions. The main feature of the new method is that the partition of unity provide a flexible and controllable framework for the domain decomposition. Finally, the efficiency of our theoretical analysis is tested by numerical experiments.

Bending Analysis of Moderately Thick Arbitrarily Shaped Plates with Point Supports Using Simple Hp Cloud Method

In this paper, a simple hp cloud method is developed for the static analysis of Mindlin plates with various shapes and different boundary conditions. A new application of the hp cloud method, “simple hp cloud method,” is developed based on Kronecker delta property, so that the essential boundary conditions can be imposed directly and as easily as finite element method. Contrary to the hp cloud method, the simple hp cloud method does not require utilizing Lagrange multipliers; hence, the cost of solving algebraic equations is decreased. Shepard functions are used for the partition of unity functions and complete polynomials of order less than or equal to 3 for the enrichment functions part. By using enough order of complete polynomials, the shear locking can be properly controlled. Numerical results are compared against some other solutions to illustrate the accuracy and efficiency of the present method. To show the applications of the simple hp cloud method, deflections and bending moments of quadrilateral, triangular and circular plates with different boundary conditions subjected to transversal distributed loading are considered. In addition, deflections of rectangular and skew plates with point supports are analyzed.

Stress intensity factor extraction from the enriched partition of unity solution for a cracked linear elastic medium

The stress intensity factor is a useful tool for predicting material failure and describing the stress states of brittle materials. We present a technique to calculate the stress intensity factor for a linear elasticity problem on a cracked domain with an enriched partition of unity method. We use a particular partition of unity function, which is piecewise polynomial and has wide flat-top region. The flat-top area in the partition of unity function helps the displacements and the stress fields in the vicinity of the crack tip to be accurately represented, even with a coarse background mesh. Among other methods for calculating the stress intensity factor, we find that the direct extraction method is the most accurate and efficient one given a relatively coarse background mesh for the enriched partition of unity method.

Convergence Rate of Overlapping Domain Decomposition Methods for the Rudin--Osher--Fatemi Model Based on a Dual Formulation

This paper is concerned with overlapping domain decomposition methods (DDMs), based on successive subspace correction (SSC) and parallel subspace correction (PSC), for the Rudin--Osher--Fatemi (ROF) model in image restoration. In contrast to recent attempts, we work with a dual formulation of the ROF model, where one significant difficulty resides in the decomposition of the global constraint of the dual variable. We introduce a stable “unity decomposition” using a set of “partition of unity functions,” which naturally leads to overlapping DDMs based on the dual formulation. The main objective of this paper is to rigorously analyze the convergence of the SSC and PSC algorithms and derive the rate of convergence $O(n^{-1/2})$, where $n$ is the number of iterations. Moreover, we characterize the explicit dependence of the convergence rate on the subdomain overlapping size and other important parameters. To the best of our knowledge, such a convergence rate has not yet been claimed for domain decomposition related algorithms for the ROF model.

Local and Parallel Finite Element Algorithm Based on the Partition of Unity for Incompressible Flows

By combining two-grid method with domain decomposition method, a new local and parallel finite element algorithm based on the partition of unity is proposed for the incompressible flows. The interesting points in this algorithm lie in (1) a class of partition of unity is derived by a given triangulation, which guides the domain decomposition (2) the globally fine grid correction step is decomposed into a series of local linearized residual problems on some subdomains and (3) the global continuous finite element solution is obtained by assembling all local solutions together using the partition of unity functions. Some numerical simulations are presented to demonstrate the high efficiency and flexibility of the new algorithm.

Local and Parallel Finite Element Algorithms Based on the Partition of Unity for the Stokes Problem

By combining the techniques of the two-grid method and the partition of unity, we derive two local and parallel finite element algorithms for the Stokes problem. The most interesting features of these algorithms are (1) the partition of unity technique introduces a framework for domain decomposition; (2) only a series of local residual problems need to be solved on these subdomains in parallel, meanwhile requiring very little communication; (3) a globally continuous finite element solution is constructed by combining all the local solutions via the partition of unity functions. The optimal error estimates in $L^2$ and energy norms are proved under some assumptions. Also, several numerical simulations are presented to demonstrate the effectiveness and flexibility of the new algorithms.

Multiscale failure Modeling of composites using generalized finite element method

AbstractIn this work multiscale failure modeling of composites is made using generalized finite element method (GFEM). In this method the global approximation are constructed by combining the local basis with partition of unity functions. The enrichment functions for the GFEM approximation are computed using a proper orthogonal decomposition (POD) technique. The approximation is then used in a two scale Galerkin scheme for failure modeling of composites. (© 2012 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Surface reconstruction of scanned human body using radial basis functions and adaptive partition of unity

It is important to reconstruct a continuous surface representation of the point cloud scanned from a human body. In this paper a new implicit surface method is proposed to reconstruct the human body surface from the points based on the combination of radial basis functions (RBFs) and adaptive partition of unity (PoU). The whole 3D domain of the scanned human body is firstly subdivided into a set of overlapping subdomains based on the improved octrees. The smooth local surfaces are then computed in the subdomains based on RBFs. And finally the global human body surface is reconstructed by blending the local surfaces with the adaptive PoU functions. This method is robust for the surface reconstruction of the scanned human body even with large or non-uniform point cloud which has a sharp density variation.

A three-dimensional numerical manifold method based on tetrahedral meshes

A three-dimensional NMM based on tetrahedral meshes is developed in this paper. First, the displacement functions of manifold elements are formulated by the partition of unity function. Then, the global equilibrium equations for three-dimensional elasto-statics problem are established by minimizing the total potential energy. The stiffness matrix, loading matrix, displacement resistance matrix are derived for program coding. Thereafter, the computational cost of the global stiffness matrix in the NMM is discussed and compared with that in the classical finite element techniques. Finally, two typical examples are investigated to demonstrate the validity of the proposed method. The results indicate that an enough accuracy of approximation will be obtained with decreasing size of the mathematical mesh. The proposed method can be used as potentially powerful tool for three-dimensional structural analysis.

The reproducing singularity particle shape functions for problems containing singularities

In this paper, we construct particle shape functions that reproduce singular functions as well as polynomial functions. We also construct piecewise polynomial convolution partition of unity functions by taking the convolution of the scaled conical window function with the characteristic functions of quadrangular patches (we provide the computer code for this construction). We demonstrate that the reproducing singular particle shape functions yield highly accurate numerical solutions for the singularity problems with crack singularity or a jump boundary data singularity.

Enriched meshless manifold method for two-dimensional crack modeling

The meshless manifold method is based on the partition of unity method and the finite cover approximation theory which provides a unified framework for solving problems dealing with both continuum with and without discontinuities. The meshless manifold method employs two cover systems. The mathematical cover system provides the nodes for forming finite covers of the solution domain and the partition of unity functions. And the physical cover system describes geometry of the domain and the discontinuous surfaces in the domain. The shape functions are derived by the partition of unity and the finite covers approximation theory. In meshless manifold method, the mathematical finite cover approximation theory is used to model cracks that lead to interior discontinuities in the displacement. Therefore, the discontinuity is treated mathematically instead of empirically by the existing methods. However, one cover of a node is divided into two irregular sub-covers when the meshless manifold method is used to model the discontinuity. As a result, the method sometimes causes numerical errors at the tip of a crack. To improve the precision of the meshless manifold method, the enriched methods are introduced in this work for crack problems.

A mesh-based partition of unity method for discontinuity modeling

This study explores a numerical analysis scheme, the manifold method, within the framework of the partition of unity method. The manifold method is rooted in the discrete element method and has been used in solving discrete-continuum interaction problems. At its core, two meshes are employed in an analysis: the mathematical mesh provides the nodes for building a finite covering of the solution domain and the partition of unity functions; while the physical mesh provides the domain of integration. After providing a geometric interpretation and giving a construct overview, the working of the method is illustrated with a number of examples.