Stable Generalized Finite Element Method Research Articles

Mass lumping of stable generalized finite element methods for parabolic interface problems

Stable generalized finite element method (SGFEM) for interface problem is based on simple mesh independent of interface curves and possesses merits of optimal convergence rates, conditioning of same order as that of finite element method (FEM), robustness, and freedom from any penalty parameters or stability schemes. This paper investigates mass lumping of SGFEM for parabolic interface problems. The mass lumping is important for computational reduction and maximum principle preserving. The difficulty in the mass lumping of conventional generalized FEM (GFEM) and SGFEM consists in their incorporation of extra enriched functions. The existing mass lumping methods of GFEM cannot work properly due to features of SGFEM that enriched shape functions vanish on mesh nodes, and the nodes are enriched partially rather than totally. In this study, we propose two mass lumping schemes for the SGFEM. The first one is to only lump the FEM part, and enriched parts are kept fixed. This scheme enables us to execute a rigorous optimal convergence analysis. It is noted that a theoretical analysis on the mass lumping of GFEM has not been available yet in the GFEM field. The second scheme is to further compress the enriched part and intersection part of mass matrix separately. The lumped mass matrix is almost diagonal, and there is only a line of nonzero entries in the intersection part of mass matrix. Numerical experiments indicate that two schemes both produce the optimal convergence rates; their errors are almost the same as those of SGFEM without using the mass lumping.

A novel application of recovered stresses in stress intensity factors extraction methods for the generalized/extended finite element method

SummaryAn efficient improvement to compute Stress Intensity Factors (SIFs) in two‐dimensional fracture mechanics problems solved by Generalized/eXtended Finite Element Method and Stable Generalized Finite Element Method is proposed. A stress field obtained by a recovery procedure based on a locally weighted projection of the approximate stresses, is used in the formulation of two energy‐based SIFs extraction methods: the Contour Integral Method (CIM) and the Interaction Integral strategy. The accuracy and the path independence feature of the proposed approach are investigated in three numerical examples, considering crack opening modes I and II, different crack orientations and the presence of holes. The results demonstrate that the recovered stresses significantly improve the accuracy of the CIM and ensure a notable decrease in SIFs variation for different integration paths. On the other hand, an equivalent impact is not observed for the Interaction Integral strategy.

Stable generalized finite element method (SGFEM) for three-dimensional crack problems

This paper proposes a stable generalized finite element method (SGFEM) for the linear 3D elasticity problem with cracked domains. Conventional material-independent branch functions serve as singular enrichments. We prove that the proposed SGFEM with the geometric enrichment scheme yields the optimal order of convergence in the energy norm, O(h), for fully 3D elasticity planar crack problems; h is the mesh parameter. To improve the conditioning of SGFEM, two stability techniques have been employed, namely, (a) a cubic polynomial has been used as the PU (partition of unity), instead of the standard FE hat-functions, to address the possible almost linear dependence between the PU functions and the enrichments, and (b) a local principal component analysis (LPCA) has been implemented to address the local bad conditioning produced by multi-fold enrichments at a node. The scaled condition number for the proposed SGFEM is shown to be O(h^{-2}) (same as that of a standard Finite Element Method), for various relative positions of crack surface and mesh. The robustness of the scaled condition number for the proposed SGFEM, with respect to the relative positions of the crack-surface and the element boundaries, has been observed numerically. The numerical experiments for both the planar and fully 3D planar crack problems are presented to show the efficiency of the proposed SGFEM.

A Stable Generalized Finite Element Method Coupled with Deep Neural Network for Interface Problems with Discontinuities

The stable generalized finite element method (SGFEM) is an improved version of generalized or extended FEM (GFEM/XFEM), which (i) uses simple and unfitted meshes, (ii) reaches optimal convergence orders, and (iii) is stable and robust in the sense that conditioning is of the same order as that of FEM and does not get bad as interfaces approach boundaries of elements. This paper designs the SGFEM for the discontinuous interface problem (DIP) by coupling a deep neural network (DNN). The main idea is to construct a function using the DNN, which captures the discontinuous interface condition, and transform the DIP to an (approximately) equivalent continuous interface problem (CIP) based on the DNN function such that the SGFEM for CIPs can be applied. The SGFEM for the DIP is a conforming method that maintains the features (i)–(iii) of SGFEM and is free from penalty terms. The approximation error of the proposed SGFEM is analyzed mathematically, which is split into an error of SGFEM of the CIP and a learning error of the DNN. The learning dimension of DNN is one dimension less than that of the domain and can be implemented efficiently. It is known that the DNN enjoys advantages in nonlinear approximations and high-dimensional problems. Therefore, the proposed SGFEM coupled with the DNN has great potential in the high-dimensional interface problem with interfaces of complex geometries. Numerical experiments verify the efficiency and optimal convergence of the proposed method.

Assessment of the flat-top stable GFEM for free vibration analysis

The stable Generalized Finite Element Method, SGFEM, has been used for numerical stabilization of the generalized finite element method during enrichment process of shape functions. However, some troubles persist in dynamic analysis. The flat-top strategy is an alternative to hold up the SGFEM but, despite its good performance in quasi-static problems, for dynamic problems some points remain opened, such as the accuracy of natural frequencies, the computational effort to reach a desired degree of precision (for each natural frequency), and the conditioning of mass and stiffness matrices. The main objective of this work is to assess the performance of the flat-top SGFEM in the context of free vibration analysis Based in many numerical simulations, the sensitivity of the method is verified for different parameters, such as, the size of the flat-top partition of unity (PU) intervals, the smoothness of enriched functions, and the stabilization parameter. Despite the flat-top SGFEM improves the numerical stability for modal analysis, it is shown that the convergence rates can be degraded.

Topology Optimization – unconventional approaches using the Generalized Finite Element Method and the Stable Generalized Finite Element Method

The Structural Optimization process has an increasing importance in industry and academic fields, assisting in the development of designs at the initial stages of a project. Nowadays, the structural optimization methodology can be conducted by Topology Optimization Method (TOM), which is an efficiently combination of the Finite Element Method (FEM) with an optimization algorithm, in order to find the optimized material distribution inside a given design domain subjected to a set of constraints. Application of the FEM in TOM suffers from a series of instability problems, being one of them the checkerboard pattern. This paper investigates the impact of the Generalized Finite Element Method (GFEM) and Stable Generalized Finite Element Method (SGFEM) in the implementation of the TOM algorithm. This work shows that these unconventional FEM formulations are able to solve most of the checkerboard pattern problem when combined with an enriched mesh designed specifically to each example evaluated. Significant improvement in results of the topology optimization is achieved when compared to the conventional formulation of TOM.

On the Application of Stable Generalized Finite Element Method for Quasilinear Elliptic Two-Point BVP

In this paper, we discuss an application of the generalized finite element method (GFEM) to approximate the solutions of quasilinear elliptic equations with multiple interfaces in one dimensional space. The problem is characterized by spatial discontinuity of the elliptic coefficient that depends on the unknown solution. It is known that unless the partition of the domain matches the discontinuity configuration, accuracy of standard finite element techniques significantly deteriorates and standard refinement of the partition may not suffice. The GFEM is a viable alternative to overcome this predicament. It is based on the construction of certain enrichment functions supplied to the standard finite element space that capture effects of the discontinuity. This approach is called stable (SGFEM) if it maintains an optimal rate of convergence and the conditioning of GFEM is not worse than that of the standard FEM. A convergence analysis is derived and performance of the method is illustrated by several numerical examples. Furthermore, it is known that typical global formulations such as FEMs do not enjoy the numerical local conservation property that is crucial in many conservation law-based applications. To remedy this issue, a Lagrange multiplier technique is adopted to enforce the local conservation. A numerical example is given to demonstrate the performance of proposed technique.

High-order stable generalized/eXtended finite element approximations for accurate stress intensity factors

This paper investigates the accuracy and robustness of stress intensity factor (SIF) extraction using the p-Hierarchical Discontinuous Interpolant Stable Generalized Finite Element Method (pFEM-DSGFEM). This method is well-conditioned and optimally convergent for fracture mechanics problems if a proper enrichment strategy and mesh refinement are used. The method performs well when calculating global quantities such as strain energy, but its ability to capture local quantities such as SIFs has not yet been explored. pFEM-DSGFEM spaces are used to solve a variety of fracture mechanics problems and their solutions are used to extract SIFs with the Displacement Correlation Method (DCM). These numerical experiments show that pFEM-DSGFEM spaces provide highly accurate SIFs, even better than standard GFEM spaces, while being well-conditioned.

2-D Crack propagation analysis using stable generalized finite element method with global-local enrichments

In this paper, the technique of the Stable Generalized Finite Element Method (SGFEM) is applied to the numerically constructed functions of the Generalized Finite Element Method with Global-Local Enrichments (GFEMgl). The application of the resulting approach, named SGFEMgl, is expanded here to 2-D quasi-static crack propagation problems. Crack growth is performed by a two-scale strategy, using local problems generated at each propagation step – whose solutions enrich a single global problem defined on a coarse mesh. Stress Intensity Factors (SIFs) computed along crack growth, strain energy measures, performance in blending elements and the condition number are used to study the accuracy and conditioning of SGFEMgl. The method is compared with the standard GFEMgl. Numerical experiments demonstrate remarkable accuracy of SGFEMgl in linear elastic fracture mechanics problems, considering crack opening modes I and II. Convergence rates analyses also show the superiority of the method, especially with the use of geometrical enrichments.

A Stable Generalized/eXtended [formula omitted]-hierarchical FEM for three-dimensional linear elastic fracture mechanics

In this paper, the quadratic Stable Generalized Finite Element Method (SGFEM) proposed in Sanchez-Rivadeneira and Duarte (2019) is extended to 3-D fracture problems with non-planar crack surfaces and curved crack fronts. This SGFEM is based on p-hierarchical FEM enrichments and its approximation space is the same as in its GFEM counterpart. Singular enrichments are modified using a discontinuous finite element interpolant for conditioning control. This leads, with the proper choice of enrichments, to stiffness matrices with a scaled condition number of O(h−2) — the same order as in the standard FEM. The robustness of the method with respect to the position of a crack relative to the mesh is demonstrated. A scalar implementation of singular enrichments that can exactly reproduce the first term of the Mode I, II, and III of the asymptotic expansion of the elasticity solution is presented. Improved approximations of signed distance functions based on integration sub-elements are also presented. Convergence studies of a fully 3-D fracture problem with a non-planar crack surface and a curved crack front are presented. They show that the convergence rate of the proposed method on a sequence of uniform meshes is three times higher than quadratic quarter-point finite elements.

Strongly Stable Generalized Finite Element Method (SSGFEM) for a non-smooth interface problem II: A simplified algorithm

The solution of interface problems with non-smooth interfaces, having corners connected by straight lines, exhibit weak singularity along the interface and may have singularities at the corners. In an earlier paper Zhang et al. (2019), a Strongly Stable Generalized Finite Element Method (SSGFEM) was proposed to approximate solution of such problems. The method required two separate enrichment functions at O(h−1) number of nodes (h is the mesh parameter) and a local orthogonalization technique, requiring additional operations, was used to ensure its stability and robustness. In this paper, we present a simplified SSGFEM that does not require two enrichment functions at any of the nodes. As a result, we obtain an optimally converging GFEM that is stable, i.e., its scaled condition number is of the same order that of a standard FEM, and robust, i.e., the conditioning is independent of the position of meshes relative to the interface; these properties are illuminated by well-designed experiments. The simplified SSGFEM does not require LOT and requires less degrees of freedom, and is easier to implement, in contrast to the SSGFEM proposed in (Zhang et al., 2019). We also proved the optimal convergence for the simplified SSGFEM in this paper that required a more involved analysis based on a smart observation on approximation of weak singularities near the corner.

A stable generalized finite element method (SGFEM) of degree two for interface problems

Generalized or Extended Finite Element Methods (GFEM/XFEM) of degree 1 for interface problems have been reported in the literature; they (i) yield optimal order of convergence, i.e., O(h) (h is a discretization parameter), (ii) are stable in a sense that conditioning is not worse than that of the standard FEM, and (iii) are robust in that the conditioning does not deteriorate as interface curves are close to boundaries of underlying elements. However, higher order GFEM/XFEM with the properties (i)–(iii) have not been successfully addressed yet. Various enrichment schemes for GFEM/XFEM based on D or DPk (D is a distance function or the absolute value of level set function, and Pk is the polynomial basis of degree k) have been reported recently to obtain higher order convergence, but they are not stable or robust in general; in fact, they even may not yield the optimal orders of convergence. In this paper, we propose a stable GFEM/XFEM of degree 2 (SGFEM2) for the interface problems, where we used the enrichment scheme based on D{1,x,y}, instead of D or D{1,x,y,x2,xy,y2} in the literature. We prove that the SGFEM2 yields the optimal order of convergence, i.e., O(h2), for the interface problems with curved (smooth) interfaces. A local principal component analysis technique has been proposed, which ensures that the SGFEM2 is stable and robust. Numerical experiments for straight and curved interfaces have been presented to illuminate these properties.

Higher order stable generalized finite element method for the elliptic eigenvalue and source problems with an interface in 1D

We study the generalized finite element methods (GFEMs) for the second-order elliptic eigenvalue problem with an interface in 1D. The linear stable generalized finite element methods (SGFEM) were recently developed for the elliptic source problem with interfaces. We first generalize SGFEM to arbitrary order elements and establish the optimal error convergence of the approximate solutions for the elliptic source problem with an interface. We then apply the abstract theory of spectral approximation of compact operators to establish the error estimation for the eigenvalue problem with an interface. The error estimations on eigenpairs strongly depend on the estimation of the discrete solution operator for the source problem. We verify our theoretical findings in various numerical examples including both source and eigenvalue problems.

Stable generalized finite element method (SGFEM) for parabolic interface problems

We study a stable generalized finite element method (SGFEM) for parabolic interface problems with smooth interfaces. The standard finite element (FE) space based on a simple mesh is enriched by a one-side distance function to simulate gradient jumps of exact solutions on the interfaces. The mesh is (quasi)uniform and independent of the interfaces so that mesh matching or refinement along the interfaces is not needed. The main idea to derive the SGFEM is to modify the one-side distance function by subtracting its FE interpolant. The SGFEM approximates the parabolic interface problem with an optimal convergence rate and is stable in that its scaled condition number is of the same order as the FEM (O(h−2), h is a mesh-size parameter) and does not depend on the relative positions of the mesh and interfaces. The proposed SGFEM is a conforming method and is free from any penalty parameters or stability schemes. Moreover, the components of the approximation functions of the SGFEM are independent when extended to vector-valued interface problems. The optimal convergence rates of SGFEM for curved interfaces are proven theoretically for the backward Euler and Crank–Nicolson formulations. The convergence and conditioning are tested by numerical experiments.

2-D fracture mechanics problems by SGFEM

In this paper, an extensive investigation is done on Stable Generalized Finite Element Method (SGFEM) performance through the analysis of 2-D fracture mechanics problems. Condition number, Stress Intensity Factors (SIFs), global and local measures of the energy norm are used to study SGFEM conditioning and accuracy. Computational time is also briefly discussed. The method is compared with the standard Generalized/eXtended Finite Element Method (G/XFEM). Numerical experiments corroborate and complement the knowledge available in the literature so far, and demonstrate SGFEM accuracy in 2-D cracked problems. Modified Heaviside Functions, combined with other enrichment functions, are also studied in the simulations. A simple and yet generic implementation for SGFEM is described, under the Object Oriented strategy, in a open source software. The implementation can be used in 2-D and 3-D problems, and it allows to generalize the implementation of any type of enrichment function under the SGFEM approach.

Stable Generalized Iso-Geometric Analysis (SGIGA) for problems with discontinuities and singularities

Numerical analysis of physical/mathematical problems based on generalized/extended isogeometric analysis suffers from the major drawbacks of sub optimal convergence rates and ill conditioning of system matrices. Blending elements and linear dependency of basis functions are some of the causes attributed to these drawbacks. The presence of blending elements reduces the overall convergence rates and the ill conditioning of system matrices results in either increasing computational time when iterative solvers are used or erroneous results when direct solvers are employed. In order to alleviate these drawbacks, three different Stable Generalized IsoGeometric Analysis (SGIGA) methods are proposed in this paper. In SGIGA, the enrichment functions are modified by shifting the enrichment function using linear or least square interpolant of the enrichment function. Problems with weak and strong discontinuities, singularities and combination of both discontinuities and singularities are considered as case studies to illustrate the performance of the proposed SGIGA methods. From the results, it is observed that SGIGA yields optimal convergence rates as well as better conditioning of system matrices. The results obtained from the proposed SGIGA methods are also compared with the results from the established methods, Stable Generalized Finite Element Method (SGFEM) and eXtended IsoGeometric Analysis (XIGA), to study the relative performances with respect to accuracy and conditioning.

Numerical experiments with the Generalized Finite Element Method based on a flat-top Partition of Unity

Abstract The Stable Generalized Finite Element Method (SGFEM) is essentially an improved version of the Generalized Finite Element Method (GFEM). Besides of retaining the good flexibility for constructing local enriched approximations, the SGFEM has the advantage of presenting much better conditioning than that of the conventional GFEM. Actually, bad conditioning is well known as one of the main drawbacks of the GFEM, while affecting severely the precision of the numerical scheme used for solving the linear system associated to the problem. Despite of its consistent mathematical basis, the numerical experiments so far conducted on using SGFEM are not yet clearly conclusive, especially regarding the robustness of the method. Therefore, the main purpose of the present paper is to give a contribution in this direction, through further investigating the SGFEM accuracy and stability. In particular, the so called Flat-Top SGFEM is a recent proposed version of the method hereby considered. As a flat-top Partition of Unit (PoU) is used for constructing the augmented approximation space with polynomial enrichments this version of the method is called SGFEM with flat-top PoU, or simply FT-SGFEM. Some computational aspects are briefly addressed, as the ones related to the implementation and integration of the flat-top for 2-D analysis. The numerical simulations consist essentially of linear analysis of panels presenting edge cracks and reentrant corners on its boundaries. Our findings from the numerical tests done are highly relevant regarding accuracy of the SGFEM versions, which present order of convergence similar to the conventional GFEM. Moreover, the measure of stability given by the scaled condition number presented in particular by the FT-SGFEM is comparable to the conventional FEM order.

Strongly Stable Generalized Finite Element Method (SSGFEM) for a non-smooth interface problem

In this paper, we propose a Strongly Stable generalized finite element method (SSGFEM) for a non-smooth interface problem, where the interface has a corner. The SSGFEM employs enrichments of 2 types: the nodes in a neighborhood of the corner are enriched by singular functions characterizing the singularity of the unknown solution, while the nodes close to the interface are enriched by a distance based function characterizing the jump in the gradient of the unknown solution along the interface. Thus nodes in the neighborhood of the corner and close to the interface are enriched with two enrichment functions. Both types of enrichments have been modified by a simple local procedure of “subtracting the interpolant.” A simple local orthogonalization technique (LOT) also has been used at the nodes enriched with both enrichment functions. We prove that the SSGFEM yields the optimal order of convergence. The numerical experiments presented in this paper indicate that the conditioning of the SSGFEM is not worse than that of the standard finite element method, and the conditioning is robust with respect to the position of the mesh relative to the interface.

Strongly stable generalized finite element method: Application to interface problems

In this paper, we present the theoretical justification for the Stable Generalized Finite Element Method (SGFEM) when applied to smooth interface problems. We prove that a Generalized Finite Element Method (GFEM) is stable, i.e., its conditioning is(a) not worse than that of the Finite Element Method, and (b) robust with respect to the mesh, if the enrichment space of the GFEM satisfies two axioms. We provide element based sufficient conditions on the enrichment space that will guarantee that the axioms are satisfied. We show that the enrichment space of the GFEM, used to address the interface problems in 2D, satisfies the sufficient conditions and thus the two axioms yielding an SGFEM. The idea of a strongly stable GFEM associated with one of the two axioms has been introduced; strong stability of the GFEM is important for designing efficient iterative methods to solve the underlying linear system. A proof of the optimal convergence of the GFEM for the interface problem has also been derived in this paper. The work in this paper is the continuation of Kergrene et al. (2016), where the stability of the GFEM, when applied to interface problems, was established through numerical experiments. The numerical results in Kergrene et al. (2016) indicated that the GFEM is indeed strongly stable.

Robustness in stable generalized finite element methods (SGFEM) applied to Poisson problems with crack singularities

In this paper, we study the performance of the Generalized Finite Element Method (GFEM) applied to the Poisson problem with crack singularities. Recently, a GFEM with modified Heaviside enrichments was proposed in Gupta et al. (2013) to approximate the solution of a 2D elasticity problem with a crack. It was shown that the GFEM is indeed a Stable GFEM (SGFEM), i.e., it yields the optimal order of convergence and its conditioning is not worse than that of the standard finite element method (FEM). However, the robustness of the conditioning of the GFEM with respect to the position of the mesh relative to the crack was not addressed in Gupta et al. (2013). In this paper, we observed that the conditioning of the GFEM with the enrichments used in Gupta et al. (2013) is not robust when applied to approximate the solution of the Poisson problem. Moreover, the order of convergence may not be optimal when the position of the mesh is changed with respect to the crack interfaces. We proposed using additional singular enrichments at the nodes close to the crack near the crack-tip. We proved that the GFEM, with enrichments proposed in this paper, yields optimal order of convergence irrespective of the position of the mesh. Moreover, with a local orthogonalization procedure, we have shown through numerical experiments that the conditioning of this GFEM is not worse than that of the standard FEM and the conditioning is robust with respect to the position of the mesh. Thus the GFEM, with the enrichments suggested in this paper, is indeed an SGFEM when applied to a Poisson problem with the crack singularities.