Partition Of Unity Finite Element Method Research Articles

A modal-based Partition of Unity Finite Element Method for elastic wave propagation problems in layered media

The time-harmonic propagation of elastic waves in layered media is simulated numerically by means of a modal-based Partition of Unity Finite Element Method (PUFEM). Instead of using the standard plane waves or the Bessel solutions of the Helmholtz equation to design the discretization basis, the proposed modal-based PUFEM explicitly uses the tensor-product expressions of the eigenmodes (the so-called Love and interior modes) of a spectral elastic transverse problem, which fulfil the coupling conditions among layers. This modal-based PUFEM approach does not introduce quadrature errors since the coefficients of the discrete matrices are computed in closed-form. A preliminary analysis of the high condition number suffered by the proposed method is also analyzed in terms of the mesh size and the number of eigenmodes involved in the discretization. The numerical methodology is validated through a number of test scenarios, where the reliability of the proposed PUFEM method is discussed by considering different modal basis and source terms. Finally, some indicators are introduced to select a convenient discrete PUFEM basis taking into account the observability of cracks located on a coupling boundary between two adjacent layers.

Partition of Unity Finite Element Method for 2D Vibro-Acoustic Modeling

The Partition of Unity Finite Element Method (PUFEM) shows promise for modeling wave-like problems in the mid-to-high frequency range, allowing to capture several wavelengths in a single element. Despite the increasing attention it received in acoustics and in structural dynamics, its efficacy to deal with coupled problems has not been addressed. The main challenge in this case is to be able to represent different types of physical waves accurately, knowing that the wavelengths can be very different and vary differently, exemplified by the dispersion of flexural waves in a solid. Without a proper handling of the coupling between the coupled media, at best the number of degrees of freedom (DoF) will not be optimal, at worst the coupled model will not converge. Techniques like mesh refinement, wave enrichment and compatible or incompatible meshes might offer a potential solution to the problem, but the model usually needs to be adjusted through a time consuming trial-and-error procedure. To tackle the problem, this paper considers a 2D coupled vibro-acoustic problem, in which the structural and acoustic domains, modeled with PUFEM, are coupled using compatible and incompatible meshes based on different coupling strategies. Numerical analyses show that the proposed method outperforms the classical finite element method by several orders of magnitude in terms of number of DoF. Recommendations are proposed on the technique to choose depending on the frequency range of interest in relation to the critical frequency of the structure to ensure the best convergence rate. Finally, an application example is presented to highlight the performance of the proposed method.

Iterative solution with shifted Laplace preconditioner for plane wave enriched isogeometric analysis and finite element discretization for high-frequency acoustics

In this paper we investigate the iterative solution of enriched finite element methods for solving a frequency domain wave problem. The considered methods are partition of unity isogeometric analysis (PUIGA) and the partition of unity finite element method (PUFEM). We study the performance of an operator based preconditioner, namely, the shifted Laplace preconditioner against a complex shifted ILU preconditioner. Compared to complex shifted ILU, the shifted Laplace preconditioner leads to a significant performance improvement in terms of reduced number of GMRES iterations. Through numerical examples, we show that the shifted Laplace preconditioner results in a wavenumber independent GMRES convergence for the frequency range considered. We also show in general that preconditioned GMRES performs better with PUIGA than for PUFEM. The improvement is explained in terms of the lower condition numbers with PUIGA at higher frequencies. This work is one of the first attempt to evaluate the performance of operator based preconditioners on a Trefftz type method for solving frequency domain wave problems.

Modeling of Thin Plate Flexural Vibrations by Partition of Unity Finite Element Method

This paper presents a conforming thin plate bending element based on the Partition of Unity Finite Element Method (PUFEM) for the simulation of steady-state forced vibration. The issue of ensuring the continuity of displacement and slope between elements is addressed by the use of cubic Hermite-type Partition of Unity (PU) functions. With appropriate PU functions, the PUFEM allows the incorporation of the special enrichment functions into the finite elements to better cope with plate oscillations in a broad frequency band. The enrichment strategies consist of the sum of a power series up to a given order and a combination of progressive flexural wave solutions with polynomials. The applicability and the effectiveness of the PUFEM plate elements are first verified via the structural frequency response. Investigation is then carried out to analyze the role of polynomial enrichment orders and enriched plane wave distributions for achieving good computational performance in terms of accuracy and data reduction. Numerical results show that the PUFEM with high-order polynomials and hybrid wave-polynomial combinations can provide highly accurate prediction results by using reduced degrees of freedom and improved rate of convergence, as compared with the classical FEM.

A Partition of unity finite element method for valuation American option under Black-Scholes model

Abstract In this paper, we present an intelligent combination of partition of unity (PU) and finite element (FE) methods for valuing American option pricing problems governed by the Black-Scholes (BS) model. The model is based on a partial differential equation (PDE) from which one can deduce the Black-Scholes formula, which gives a theoretical estimated value of options using current stock prices, expected dividends, the option’s strike price, expected interest rates, time to expiration and expected volatility. Although the finite element method (FEM) seems to be an alternative tool for pricing options with a few applications reported in the literature, this combination called the Partition of Unity Finite Element Method (PUFEM) appears to offer many of the desired properties. The main advantage of the proposed approach is its ability to locally refine the solution by adapting an incorporated specific class of enrichment in the finite element space instead of generating a new fine mesh for the problem under study. Numerical computations are carried out to show a huge reduction in the number of degrees of freedom required to achieve a fixed accuracy which confirms that the PUFE method used is very efficient and gives better accuracy than the conventional FE method.

P.831Trajectories of suicidal ideation after a first-episode psychosis: a growth mixture modeling approach

Singular mode shapes around the sharp frictional contact interfaces of different materials with linear elastic behavior are derived semi-analytically using the asymptotic eigen-function expansion method. The characteristic equation which is not proposed in a short explicit expression in the literature is extracted in symbolic terms; then solved numerically and the singularity order is derived using Newton's iterative method for the first time in the literature.The extracted eigen-shapes of the high gradient stress fields and the corresponding displacement fields satisfy the compatibility of deformations and tractions on the contact interface with arbitrary frictional interface equations. These asymptotic mode shapes are then implemented into the finite element interpolation using Partition of Unity Finite Element Method (PUFEM). Since the additional enrichment degrees of freedom on the singular contact nodes have the physical meaning of generalized stress intensity factor (GSIF), the direct GSIFs obtained are more accurate and the contour integral manipulations need not be maintained in the post processing phase of the contact analysis.Finally, some engineering problems are solved which demonstrate the capacities of the approach concerning the accuracy of GSIF obtained from the direct-PUFEM proposed compared with that derived by reciprocal work contour integral method used in traditional finite element analyses.

Development of Partition of Unity Finite Element Method for the Modelling of Acoustic Waves Propagating through Porous Domain

The numerical simulation of propagating sound fields in two dimensions using Partition of Unity Finite Element Method(PUFEM) technique is implemented in this work. PUFEM is one of the most practical methods developed to alleviate high computational cost of conventional Finite Element Method when we deal with the medium and high frequency problem. The main idea of PUFEM is to expand the finite element approximation space by incorporating a group of wave functions which are solutions of the governing equation into the target problem. With the intention of further accelerating the computational process, an advanced 2D Exact Integration Technique(EIT) for accelerating the computation process of highly oscillating integrals is proposed and the related PUFEM triangular element is created. The performance and practical interest of this new PUFEM element combined with EIT are evaluated for the wave propagation problem where a porous absorber is included.

Subject-specific FE models of the human femur predict fracture path and bone strength under single-leg-stance loading

Hip fractures are a major health problem with high socio-economic costs. Subject-specific finite element (FE) models have been suggested to improve the fracture risk assessment, as compared to clinical tools based on areal bone mineral density, by adding an estimate of bone strength. Typically, such FE models are limited to estimate bone strength and possibly the fracture onset, but do not model the fracture process itself. The aim of this study was to use a discrete damage approach to simulate the full fracture process in subject-specific femur models under stance loading conditions. A framework based on the partition of unity finite element method (PUFEM), also known as XFEM, was used. An existing PUFEM framework previously used on a homogeneous generic femur model was extended to include a heterogeneous material description together with a strain-based criterion for crack initiation. The model was tested on two femurs, previously mechanically tested in vitro. Our results illustrate the importance of implementing a subject-specific material distribution to capture the experimental fracture pattern under stance loading. Our models accurately predicted the fracture pattern and bone strength (1% and 5% error) in both investigated femurs. This is the first study to simulate complete fracture paths in subject-specific FE femur models and it demonstrated how discrete damage models can provide a more complete picture of fracture risk by considering both bone strength and fracture toughness in a subject-specific fashion.

Potential of Room Acoustic Solver with Plane-Wave Enriched Finite Element Method

Predicting room acoustics using wave-based numerical methods has attracted great attention in recent years. Nevertheless, wave-based predictions are generally computationally expensive for room acoustics simulations because of the large dimensions of architectural spaces, the wide audible frequency ranges, the complex boundary conditions, and inherent error properties of numerical methods. Therefore, development of an efficient wave-based room acoustic solver with smaller computational resources is extremely important for practical applications. This paper describes a preliminary study aimed at that development. We discuss the potential of the Partition of Unity Finite Element Method (PUFEM) as a room acoustic solver through the examination with 2D real-scale room acoustic problems. Low-order finite elements enriched by plane waves propagating in various directions are used herein. We examine the PUFEM performance against a standard FEM via two-room acoustic problems in a single room and a coupled room, respectively, including frequency-dependent complex impedance boundaries of Helmholtz resonator type sound absorbers and porous sound absorbers. Results demonstrated that the PUFEM can predict wideband frequency responses accurately under a single coarse mesh with much fewer degrees of freedom than the standard FEM. The reduction reaches O ( 10 − 2 ) at least, suggesting great potential of PUFEM for use as an efficient room acoustic solver.

Partition of Unity Finite Element Method for the modelling of Acoustic Black Hole wedges

The Acoustic Black Hole (ABH) phenomenon can be exploited to manipulate and mitigate flexural wave propagation in thin-walled structures. ABH structures feature unique space-dependent wavenumber variation and wave celerity reduction in the tapered ABH area, thus posing challenges to the existing modelling techniques. In this work, the Partition of Unity Finite Element Method (PUFEM) is revamped to simulate the structural response of an ABH wedge subject to a harmonic loading. This method allows the incorporation of auxiliary enrichment functions into the finite element framework in order to cope with the ABH-induced wave oscillating behaviour, exemplified by the varying wavenumber and amplitude in space. The PUFEM tapered Timoshenko beam elements are constructed by employing wave enrichment functions with the Wentzel-Kramers-Brillouin (WKB) approximation method. A wavelet enrichment is also investigated as hierarchic refinement. Using these enriched elements, the frequency responses of an ABH wedge and the convergence of numerical solutions are computed and compared with the classical linear FEM and the elements enriched with ‘local’ wave solutions. An adaptive meshing scheme is designed and implemented to further accelerate the solution convergence. It is shown that the PUFEM offers a good computational accuracy and drastic reduction of degrees of freedom for solving the broadband ABH problems, outperforming the classical FEM.

A residual a posteriori error estimate for partition of unity finite elements for three‐dimensional transient heat diffusion problems using multiple global enrichment functions

SUMMARYIn this article, a study of residual based a posteriori error estimation is presented for the partition of unity finite element method (PUFEM) for three‐dimensional (3D) transient heat diffusion problems. The proposed error estimate is independent of the heuristically selected enrichment functions and provides a useful and reliable upper bound for the discretization errors of the PUFEM solutions. Numerical results show that the presented error estimate efficiently captures the effect of h‐refinement and q‐refinement on the performance of PUFEM solutions. It also efficiently reflects the effect of ill‐conditioning of the stiffness matrix that is typically experienced in the partition of unity based finite element methods. For a problem with a known exact solution, the error estimate is shown to capture the same solution trends as obtained by the classical L2 norm error. For problems with no known analytical solutions, the proposed estimate is shown to be used as a reliable and efficient tool to predict the numerical errors in the PUFEM solutions of 3D transient heat diffusion problems.

Partition of unity finite element for analysis of contact interface stress singularity

Singular mode shapes around the sharp frictional contact interfaces of different materials with linear elastic behavior are derived semi-analytically using the asymptotic eigen-function expansion method. The characteristic equation which is not proposed in a short explicit expression in the literature is extracted in symbolic terms; then solved numerically and the singularity order is derived using Newton's iterative method for the first time in the literature.The extracted eigen-shapes of the high gradient stress fields and the corresponding displacement fields satisfy the compatibility of deformations and tractions on the contact interface with arbitrary frictional interface equations. These asymptotic mode shapes are then implemented into the finite element interpolation using Partition of Unity Finite Element Method (PUFEM). Since the additional enrichment degrees of freedom on the singular contact nodes have the physical meaning of generalized stress intensity factor (GSIF), the direct GSIFs obtained are more accurate and the contour integral manipulations need not be maintained in the post processing phase of the contact analysis.Finally, some engineering problems are solved which demonstrate the capacities of the approach concerning the accuracy of GSIF obtained from the direct-PUFEM proposed compared with that derived by reciprocal work contour integral method used in traditional finite element analyses.

Partition of Unity Finite Element Method applied to exterior problems with Perfectly Matched Layers

The Partition of Unity Finite Element Method (PUFEM) is now a well established and efficient method used in computational acoustics to tackle short-wave problems. This method is an extension of the classical finite element method whereby enrichment functions are used in the approximation basis in order to enhance the convergence of the method whilst maintaining a relatively low number of degrees of freedom. For exterior problems, the computational domain must be artificially truncated and special treatments must be followed in order to avoid or reduce spurious reflections. In recent papers, different Non-Reflecting Boundary Conditions (NRBCs) have been used in conjunction with the PUFEM. An alternative is to use the Perfectly Match Layer (PML) concept which consists in adding a computational sponge layer which prevents reflections from the boundary. In contrast with other NRBCs, the PML is not case specific and can be applied to a variety of configurations. The aim of this work is to show the applicability of PML combined with PUFEM for solving the propagation of acoustic waves in unbounded media. Performances of the PUFEM-PML are shown for different configurations ranging from guided waves in ducts, radiation in free space and half-space problems. In all cases, the method is shown to provide acceptable results for most applications, similar to that of local approximation of NRBCs.

Robustness and dispersion analysis of the Partition of Unity Finite Element Method applied to the Helmholtz equation

The Partition of Unity Finite Element Method (PUFEM) has been widely used for the numerical simulation of the Helmholtz equation in different physical settings. In fact, it is a numerical pollution-free alternative method to the classical piecewise polynomial-based finite element methods. Taking into account a plane wave enrichment of the piecewise linear finite element method, the main goal of this work is focused on the derivation of the numerical dispersion relation and the robustness analysis of the PUFEM discretization when a spurious perturbation is presented in the wave number value used in the enrichment definition. From the one-dimensional Helmholtz equation, the discrete wave number is estimated based on a Bloch’s wave analysis and a priori error estimates are computed explicitly in terms of the mesh size, the wave number, and the perturbation value.

Performance of the Partition of Unity Finite Element Method for the modeling of Timoshenko beams

The Partition of Unity Finite Element Method (PUFEM) is developed and applied to compute the vibrational response of a Timoshenko beam subject to a uniformly distributed harmonic loading. In the proposed method, classical finite elements are enriched with three types of special functions: propagating and evanescent wave functions, a Fourier-type series and a polynomial enrichment. Different formulations are first assessed through comparisons on the frequency response functions at a specific point on the beam. The computational performance, in terms of both accuracy and data reduction, is then illustrated through convergence analyses. It is found that, by using a very limited number of degrees of freedom, the wave enrichment offers highly accurate results with a convergence rate which is much higher than other formulations. Evanescent waves and the constant term in the wave basis are also shown to play an important role. In all cases, the proposed PUFEM formulations clearly outperform classical finite element method in terms of computational efficiency.

A partition of unity approach to adaptivity and limiting in continuous finite element methods

The partition of unity finite element method (PUFEM) proposed in this paper makes it possible to blend space and time approximations of different orders in a continuous manner. The lack of abrupt changes in the local mesh size h and polynomial degree p simplifies implementation and eliminates the need for using sophisticated hierarchical data structures. In contrast to traditional hp-adaptivity for finite elements, the proposed approach preserves discrete conservation properties and the continuity of traces at common boundaries of adjacent mesh cells. In the context of space discretizations, a continuous blending function is used to combine finite element bases corresponding to high-order polynomials and piecewise-linear approximations based on the same set of nodes. In a similar vein, spatially partitioned time discretizations can be designed using weights that depend continuously on the space variable. The design of blending functions may be based on a priori knowledge (e.g., in applications to problems with singularities or boundary layers), local error estimates, smoothness indicators, and/or discrete maximum principles. In adaptive methods, changes of the finite element approximation exhibit continuous dependence on the data. The presented numerical examples illustrate the typical behavior of local H1 and L2 errors.

Bernstein–Bézier based finite elements for efficient solution of short wave problems

In this work, the Bernstein–Bézier Finite Element Method (BBFEM) is implemented to solve short wave problems governed by the Helmholtz equation on unstructured triangular mesh grids. As for the hierarchical Finite Element (FE) approach, this high order FE method benefits from the use of static condensation which is an efficient tool for reducing the total number of degrees of freedom and bandwidth of high order FE global matrices. The performance of BBFEM with static condensation (BBFEMs) is assessed via three benchmark problems and compared to that of the Partition of Unity Finite Element Method (PUFEM) in terms of accuracy, conditioning and memory requirement. Numerical results dealing with problems of Hankel sources interference and wave scattering by a rigid cylinder on quasi-uniform mesh grids indicate that BBFEMs is able to achieve a better accuracy but PUFEM is slightly better conditioned when the wave is not well resolved. However, with a sufficient wave resolution, BBFEMs is better conditioned than PUFEM. Results from L-shaped domain problem, with non quasi-uniform mesh grids, show that the conditioning of BBFEMs remains reasonable while PUFEM with large numbers of enriching plane waves on mesh grids locally well resolved leads to ill-conditioning.

Plane wave enriched Partition of Unity Isogeometric Analysis (PUIGA) for 2D-Helmholtz problems

In general, numerical concerns in simulation of time harmonic acoustic waves are mainly due to the dispersion errors and the demands in modeling the problems with higher number of degrees of freedom per wavelength to obtain acceptable results. As a remedy, plane wave enriched Partition of Unity Finite Element Method (PUFEM) was earlier proposed by Melenk and Babuška (1996). Even though, PUFEM gives accurate results with lesser degrees of freedom and shows exponential convergence, it suffers from the approximate representation of geometry and leads to the boundary representation induced errors. In this regard, plane wave enriched PUFEM under Isogeometric framework named Partition of Unity Isogeometric Analysis (PUIGA) is proposed for accurate geometry representation and smooth solution approximation which can lead to a better solution. Several benchmark problems are considered in the present study, to illustrate the computational effectiveness of PUIGA.

High-order finite elements for the solution of Helmholtz problems

In this paper, two high-order finite element models are investigated for the solution of two-dimensional wave problems governed by the Helmholtz equation. Plane wave enriched finite elements, developed in the Partition of Unity Finite Element Method (PUFEM), and high-order Lagrangian-polynomial based finite elements are considered. In the latter model, the Chebyshev-Gauss-Lobatto nodal distribution is adopted and the approach is often referred to as the Spectral Element Method (SEM). The two strategies, PUFEM and SEM, were developed separately and the current study provides data on how they compare for solving short wave problems, in which the characteristic dimension is a multiple of the wavelength. The considered test examples include wave scattering by a rigid circular cylinder, evanescent wave cases and propagation of waves in a duct with rigid walls. The two approaches are assessed in terms of accuracy for increasing SEM order and PUFEM enrichment. The conditioning, discretization level, total number of storage locations and total number of non-zero entries are also compared.

Partition of unity finite element method for quantum mechanical materials calculations

The current state of the art for large-scale quantum-mechanical simulations is the planewave (PW) pseudopotential method, as implemented in codes such as VASP, ABINIT, and many others. However, since the PW method uses a global Fourier basis, with strictly uniform resolution at all points in space, it suffers from substantial inefficiencies in calculations involving atoms with localized states, such as first-row and transition-metal atoms, and requires significant nonlocal communications, which limit parallel efficiency. Real-space methods such as finite-differences (FD) and finite-elements (FE) have partially addressed both resolution and parallel-communications issues but have been plagued by one key disadvantage relative to PW: excessive number of degrees of freedom (basis functions) needed to achieve the required accuracies. In this paper, we present a real-space partition of unity finite element (PUFE) method to solve the Kohn–Sham equations of density functional theory. In the PUFE method, we build the known atomic physics into the solution process using partition-of-unity enrichment techniques in finite element analysis. The method developed herein is completely general, applicable to metals and insulators alike, and particularly efficient for deep, localized potentials, as occur in calculations at extreme conditions of pressure and temperature. Full self-consistent Kohn–Sham calculations are presented for LiH, involving light atoms, and CeAl, involving heavy atoms with large numbers of atomic-orbital enrichments. We find that the new PUFE approach attains the required accuracies with substantially fewer degrees of freedom, typically by an order of magnitude or more, than the PW method. We compute the equation of state of LiH and show that the computed lattice constant and bulk modulus are in excellent agreement with reference PW results, while requiring an order of magnitude fewer degrees of freedom to obtain.