A Zienkiewicz-type finite element applied to fourth-order problems

AbstractThis paper deals with convergence analysis and applications of a Zienkiewicz-type (Z-type) triangular element, applied to fourth-order partial differential equations. For the biharmonic problem we prove the order of convergence by comparison to a suitable modified Hermite triangular finite element. This method is more natural and it could be applied to the corresponding fourth-order eigenvalue problem. We also propose a simple postprocessing method which improves the order of convergence of finite element eigenpairs. Thus, an a posteriori analysis is presented by means of different triangular elements. Some computational aspects are discussed and numerical examples are given

Recovery Techniques For Finite Element Methods And Their Applications

Recovery techniques are important post-processing methods to obtain improved approximate solutions from primary data with reasonable cost. The practical us- age of recovery techniques is not only to improve the quality of approximation, but also to provide an asymptotically exact posteriori error estimators for adaptive meth- ods. This dissertation presents recovery techniques for nonconforming finite element methods and high order derivative as well as applications of gradient recovery. Our first target is to develop a systematic gradient recovery technique for Crouzeix- Raviart element. The proposed method uses finite element solution to build a better approximation of the exact gradient based on local least square fittings. Due to poly- nomial preserving property of least square fitting, it is easy to show that the new proposed method preserves quadratic polynomials. In addition, the proposed gra- dient recovery is linearly bounded. Numerical tests indicate the recovered gradient is superconvergent to the exact gradient for both second order elliptic equation and Stokes equation. The gradient recovery technique can be used in a posteriori error estimates for Crouzeix-Raviart element, which is relatively simple to implement and problem independent. Our second target is to propose and analyze a new effective Hessian recovery for continuous finite element of arbitrary order. The proposed Hessian recovery is based on polynomial preserving recovery. The proposed method preserves polynomials of degree (k + 1) on general unstructured meshes and polynomials of degree (k + 2) on translation invariant meshes. Based on it polynomial preserving property, we can able to prove superconvergence of the proposed method on mildly structured meshes. In addition, we establish the ultraconvergence result for the new Hessian recovery technique on translation invariant finite element space of arbitrary order. Our third target is to demonstrate application of gradient recovery in eigenvalue computation. We propose two superconvergent two-grid methods for elliptic eigen- value problems by taking advantage of two-gird method, two-space method, shifted- inverse power method, and gradient recovery enhancement. Theoretical and numer- ical results reveal that the proposed methods provide superconvergent eigenfunction approximation and ultraconvergent eigenvalue approximation. In addition, two mul- tilevel adaptive methods based recovery type a posterior error estimate are proposed

Polynomial Preserving Recovery For Weak Galerkin Methods And Their Applications

Gradient recovery technique is widely used to reconstruct a better numerical gradient from a finite element solution, for mesh smoothing, a posteriori error estimate and adaptive finite element methods. The PPR technique generates a higher order approximation of the gradient on a patch of mesh elements around each mesh vertex. It can be used for different finite element methods for different problems. This dissertation presents recovery techniques for the weak Galerkin methods and as well as applications of gradient recovery on various of problems, including elliptic problems, interface problems, and Stokes problems. Our first target is to develop a boundary strategy for the current PPR algorithm. The current accuracy of PPR near boundaries is not as good as that in the interior of the domain. It might be even worse than without recovery. Some special treatments are needed to improve the accuracy of PPR on the boundary. In this thesis, we present two boundary recovery strategies to resolve the problem caused by boundaries. Numerical experiments indicate that both of the newly proposed strategies made an improvement to the original PPR. Our second target is to generalize PPR to the weak Galerkin methods. Different from the standard finite element methods, the weak Galerkin methods use a different set of degrees of freedom. Instead of the weak gradient information, we are able to obtain the recovered gradient information for the numerical solution in the generalization of PPR. In the PPR process, we are also able to recover the function value at the nodal points which will produce a global continuous solution instead of piecewise continuous function. Our third target is to apply our proposed strategy and WGPPR to interface problems. We treat an interface as a boundary when performing gradient recovery, and the jump condition on the interface can be well captured by the function recovery process. In addition, adaptive methods based on WGPPR recovery type a posteriori error estimator is proposed and numerically tested in this thesis. Application on the elliptic problem and interface problem validate the effectiveness and robustness of our algorithm. Furthermore, WGPPR has been applied to 3D problem and Stokes problem as well. Superconvergent phenomenon is again observed

Extended Finite Element Methods for Brittle and Cohesive Fracture

The safety of engineering structures depends heavily on the presence of cracks, which may propagate and lead eventually to structural failure. This dissertation aims to advance the computational modeling of fracture, within the context of linear elastic fracture mechanics (LEFM) and cohesive zone models (CZMs). The extended finite element method (XFEM) is employed as the discretization method and cracks in both homogeneous and bimaterial solids are considered in this work. First, a novel set of enrichment functions within the framework of XFEM is proposed for the LEFM analysis of interface cracks in bimaterials. The motivation for the new enrichment set stems from the revelation that the accuracy of the widely accepted 12-fold bimaterial enrichment functions significantly deteriorates with the increase in material mismatch. To this end, we propose an 8-fold material-dependent enrichment set, derived from the analytical asymptotic displacement field, that well captures the near-tip oscillating singular fields of interface cracks, including the transition to weak discontinuities of bimaterials. The new enrichment set is tested on various examples and found to outperform the 12-fold set in terms of accuracy, conditioning, and total number of degrees of freedom (DOFs). The formulation is then extended to include high-order enrichment functions and accurate stress and displacement fields are obtained. The complex stress intensity factors (SIFs) of interface cracks are evaluated by employing Irwin's crack closure integral. To this end, a closed-form SIF formulation in terms of the enriched DOFs is derived by matching the leading term in the XFEM with an analytical expression of Irwin's integral. Hence, the SIFs of interface cracks can be directly obtained upon the solution of the XFEM discrete system without cumbersome post-processing requirements. The proposed method is shown to work well on several benchmark examples involving straight and curved interface cracks, giving accurate SIF results. Another contribution of the work is the application of Irwin's integral to the estimation of SIFs for curved homogeneous cracks. At the core, the proposed approach employs high-order enrichment functions to accurately capture the near-tip fields and evaluates the original definition of Irwin's integral through closed-form formulations in terms of enriched DOFs. An improved quadrature scheme using high-order isoparametric mapping together with a generalized Duffy transformation is proposed to integrate singular fields in tip elements with curved cracks. The proposed extraction approach is shown to yield decomposed SIFs with excellent accuracy and avoid the need for auxiliary fields as in J-integral method. Second, with respect to cohesive fracture, a discrete damage zone model (DDZM) is proposed following a rigorous thermodynamic framework similar to that of continuum damage mechanics (CDM). For the modeling of mixed-mode delamination, a novel damage evolution law is proposed to account for the coupled interaction between opening and sliding modes of interface deformations. A comprehensive comparison made with several popular CZMs in the literature demonstrates the thermodynamic consistency of the DDZM. The proposed interface model is integrated with the XFEM and the effectiveness of this framework has been validated on various benchmark problems. Finally, a novel continuous/discontinuous method is proposed to simulate the entire failure process of quasi-brittle materials: from the nucleation of diffuse damage to the development of discrete cracks . An integral-type nonlocal continuum damage model is coupled in this framework with DDZM with a new numerical energetic coupling scheme. The transition from the continuous (CDM) to the discontinuous approach (DDZM) can be triggered at any damage level with a weak energetic equivalence preserved. A few benchmark problems involving straight and curved cracks are investigated to demonstrate the applicability and robustness of the coupled XFEM cohesive-damage approach

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described