Condition Numbers Of Stiffness Matrices Research Articles

Wavelet Numerical Solutions for a Class of Elliptic Equations with Homogeneous Boundary Conditions

This paper focuses on a method to construct wavelet Riesz bases with homogeneous boundary condition and use them to a kind of second-order elliptic equation. First, we construct the splines on the interval [0,1] and consider their approximation properties. Then we define the wavelet bases and illustrate the condition numbers of stiffness matrices are small and bounded. Finally, several numerical examples show that our approach performs efficiently.

A new over-penalized weak galerkin method. Part Ⅰ: Second-order elliptic problems

<p style='text-indent:20px;'>In this article, we propose a new over-penalized weak Galerkin (OPWG) method with a stabilizer for second-order elliptic problems. This method employs double-valued functions on interior edges of elements instead of single-valued ones and elements <inline-formula><tex-math id="M1">\begin{document}$ (\mathbb{P}_{k}, \mathbb{P}_{k}, [\mathbb{P}_{k-1}]^{d}) $\end{document}</tex-math></inline-formula>, or <inline-formula><tex-math id="M2">\begin{document}$ (\mathbb{P}_{k}, \mathbb{P}_{k-1}, [\mathbb{P}_{k-1}]^{d}) $\end{document}</tex-math></inline-formula>, with dimensions of space <inline-formula><tex-math id="M3">\begin{document}$ d = 2, \; 3 $\end{document}</tex-math></inline-formula>. The method is absolutely stable with a constant penalty parameter, which is independent of mesh size and shape-regularity. We prove that for quasi-uniform triangulations, condition numbers of the stiffness matrices arising from the OPWG method are <inline-formula><tex-math id="M4">\begin{document}$ O(h^{-\beta_{0}(d-1)-1}) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ \beta_{0} $\end{document}</tex-math></inline-formula> being the penalty exponent. Therefore we introduce a new <i>mini-block diagonal</i> preconditioner, which is proven to be theoretically and numerically effective in reducing the condition numbers of stiffness matrices to the magnitude of <inline-formula><tex-math id="M6">\begin{document}$ O(h^{-2}) $\end{document}</tex-math></inline-formula>. Optimal error estimates in a discrete <inline-formula><tex-math id="M7">\begin{document}$ H^1 $\end{document}</tex-math></inline-formula>-norm and <inline-formula><tex-math id="M8">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-norm are established, from which the optimal penalty exponent can be easily chosen. Several numerical examples are presented to demonstrate flexibility, effectiveness and reliability of the new method.

DOF‐gathering stable generalized finite element methods for crack problems

AbstractGeneralized or eXtended finite element methods (GFEM/XFEM) have been studied extensively for crack problems. Most of the studies were concentrated on localized enrichment schemes where nodes around the crack tip are enriched by products of singular and finite element shape functions. To attain the optimal convergence rate O(h) (h is the mesh‐size), nodes in a fixed domain containing the tip have to be enriched. This results in many extra degrees of freedom (DOF) and stability issues. A so‐called DOF‐gathering GFEM/XFEM can avoid the increase of DOF, by collecting the singular enriched DOF together. Various novel modifications were designed for the DOF‐gathering GFEM/XFEM to get the optimal convergence O(h). However, they could not improve the stability, namely, condition numbers of stiffness matrices of the DOF‐gathering GFEM/XFEM could be much larger than that of the standard FEM. Motivated from the idea of stable GFEM, we propose in this paper a DOF‐gathering stable GFEM (d.g.SGFEM) for the Poisson problem with crack singularities. The main idea is to modify the singular and Heaviside enrichments by subtracting their finite element interpolants. The optimal convergence O(h) of the proposed d.g.SGFEM is proven theoretically. Moreover, the condition number of stiffness matrices of d.g.SGFEM, utilizing a local orthgonalization technique, is shown to be of same order as that of the standard FEM. Two kinds of commonly used cut‐off functions used to gather the DOF are analyzed in a unified approach. Theoretical convergence and the conditioning results of d.g.SGFEM are verified by numerical experiments.

Removing mesh bias in mixed‐mode cohesive fracture simulation with stress recovery and domain integral

SummaryTo remove mesh bias and provide an accurate crack path representation in mixed‐mode investigation, a novel stress recovery technique is proposed in conjunction with a domain integral and element splits. Based on a domain integral and stress recovery technique, a maximum strain energy release rate is estimated to determine a crack path direction. Then, for a given crack path direction, continuum elements are split, and a cohesive surface element is adaptively inserted. One notes that the proposed stress recovery technique provides a more accurate stress field than a standard stress evaluation procedure. The proposed computational framework is verified and validated by solving mode‐I and mixed‐mode examples. Computational results demonstrate that the domain integral with the stress recovery accurately evaluates a crack path, even with a lower‐quality mesh and under a biaxial stress state. Furthermore, the cohesive surface element approach, with the element split in conjunction with the stress recovery and the domain integral, predicts mixed‐mode fracture behaviors while removing mesh bias in the crack path representation. Additionally, the condition numbers of stiffness matrices are within the same order of magnitude during cohesive fracture simulation.

Detecting the causes of ill-conditioning in structural finite element models

In 2011, version 8.6 of the finite element-based structural analysis package Oasys GSA was released. A new feature in this release was the estimation of the 1-norm condition number κ1(K)=∥K∥1∥K-1∥1 of the stiffness matrix K of structural models by using a 1-norm estimation algorithm of Higham and Tisseur to estimate ∥K-1∥1. The condition estimate is reported as part of the information provided to engineers when they carry out linear/static analysis of models and a warning is raised if the condition number is found to be large. The inclusion of this feature prompted queries from users asking how the condition number impacted the analysis and, in cases where the software displayed an ill conditioning warning, how the ill conditioning could be “fixed”. We describe a method that we have developed and implemented in the software that enables engineers to detect sources of ill conditioning in their models and rectify them. We give the theoretical background and illustrate our discussion with real-life examples of structural models to which this tool has been applied and found useful. Typically, condition numbers of stiffness matrices reduce from O(1016) for erroneous models to O(108) or less for the corrected model.

On condition numbers in hp-FEM with Gauss–Lobatto-based shape functions

Sharp bounds on the condition number of stiffness matrices arising in hp/spectral discretizations for two-dimensional problems elliptic problems are given. Two types of shape functions that are based on Lagrange interpolation polynomials in the Gauss–Lobatto points are considered. These shape functions result in condition numbers O( p) and O(p ln p) for the condensed stiffness matrices, where p is the polynomial degree employed. Locally refined meshes are analyzed. For the discretization of Dirichlet problems on meshes that are refined geometrically toward singularities, the conditioning of the stiffness matrix is shown to be independent of the number of layers of geometric refinement.

Bounds for eigenvalues and condition numbers in the 𝑝-version of the finite element method

In this paper, we present a theory for bounding the minimum eigenvalues, maximum eigenvalues, and condition numbers of stiffness matrices arising from the p p -version of finite element analysis. Bounds are derived for the eigenvalues and the condition numbers, which are valid for stiffness matrices based on a set of general basis functions that can be used in the p p -version. For a set of hierarchical basis functions satisfying the usual local support condition that has been popularly used in the p p -version, explicit bounds are derived for the minimum eigenvalues, maximum eigenvalues, and condition numbers of stiffness matrices. We prove that the condition numbers of the stiffness matrices grow like p 4 ( d − 1 ) p^{4(d-1)} , where d d is the number of dimensions. Our results disprove a conjecture of Olsen and Douglas in which the authors assert that “regardless of the choice of basis, the condition numbers grow like p 4 d p^{4d} or faster". Numerical results are also presented which verify that our theoretical bounds are correct.