Recovery-based Error Estimator Research Articles

An efficient and easy-to-implement recovery-based a posteriori error estimator for isogeometric analysis of the Stokes equation

In this article, we present an efficient recovery-based a posteriori error estimator for adaptive isogeometric analysis (IGA) of the Stokes equation, which is straightforward to implement. The increased regularity of splines in IGA versus Lagrange polynomials used in the classical finite element method plays a significant role for the success of the presented recovery technique. We consider LR B-splines (Johannessen et al., 2014) for the adaptive mesh refinement, and the isogeometric Taylor–Hood element, Sub-Grid element (Buffa et al., 2011), and Raviart–Thomas element (divergence-conforming) (Buffa et al., 2010; Evans and Hughes, 2013; Johannessen et al., 2015) for the mixed discretization of the Stokes equation. Three benchmark problems with analytical solutions are tested, and they demonstrate clearly the great performance of the recovery-based error estimator compared to classical residual-based error estimator.

Adaptive workflow for simulation of RF heaters

Accurate RF (Radio Frequency) simulations of fusion systems like ITER require the definition of high-fidelity analysis geometries that include detailed antenna, reactor wall, and physics regions. This paper will describe a workflow for the execution of adaptive high-performance simulations of RF fusion systems. In this workflow, the simulation input consists of a CAD model attributed with the needed analysis attributes. The analysis mesh is automatically generated and the analysis steps are executed using the time-harmonic Maxwell's equations solved using high-order Nédélec finite elements. A patch recovery-based error estimator is used to drive a conforming mesh adaptation procedure.

Recovery-Based Error Estimator for Natural Convection Equations Based on Defect-Correction Methods.

In this paper, we propose an adaptive defect-correction method for natural convection (NC) equations. A defect-correction method (DCM) is proposed for solving NC equations to overcome the convection dominance problem caused by a high Rayleigh number. To solve the large amount of computation and the discontinuity of the gradient of the numerical solution, we combine a new recovery-type posteriori estimator in view of the gradient recovery and superconvergent theory. The presented reliability and efficiency analysis shows that the true error can be effectively bounded by the recovery-based error estimator. Finally, the stability, accuracy and efficiency of the proposed method are confirmed by several numerical investigations.

A hierarchically enhanced recovery-based error estimator for bidimensional elastoplastic modelling by generalized finite element approach

This paper aims to contribute with a hierarchically enhanced recovery-based error estimator by using the Zienkiewicz–Zhu (ZZ) recovery procedure and a generalized finite element formulation. The proposed stress recovery consists in an approximation procedure that is enriched hierarchically by using partition of unity (PU) and an enrichment function. In other words, the approximation matrix used for stress field recovery employs the enriched finite element (FE) shape function. Furthermore, the edge-bubble hierarchically enriched strategy in the context of a generalized finite element approach is adopted to develop the enriched $$\hbox {C}^{0}$$ element and approximation the matrix. The error estimator based on this hierarchically enhanced recovery procedure and the generalized finite element approach is employed to solve several elastic and elastoplastic hardening benchmark problems. Moreover, the effectivity index, refinement index, recovered stress field, global error and local error, both in energy norm, are evaluated in several applications. In addition, the sensitivity of the proposed recovery procedure to a severely distorted mesh is also analysed. The results obtained by the proposed procedure is compared to other well-established recovery procedures and finite element approaches. By comparing the results, the robustness in the numerical performance, the versatility in the computational implementation and the competitivity of the proposed procedure are demonstrated in the applications.

Recovery-based error estimators for the VEM and BEM-based FEM

In this article, we consider recovery-based a posteriori error estimators for the Virtual Element Method (VEM) and the Boundary Element Method based Finite Element Method (BEM-based FEM). Both methods are Galerkin methods on polygonal and polyhedral grids for the numerical solution of partial differential equations. Thus, they are highly flexible and particularly efficient in combination with adaptive refinement. Our error estimator computes the distance between the finite element gradient and its post-processed version to obtain an accurate approximation of the true error. The post-processing is realised by local averaging and is therefore easy to implement and fast. Furthermore, we have found points of extraordinary accuracy, so called stress points, on specific regular elements for the BEM-based FEM. We demonstrate that, when sampling these stress points, the recovered gradient becomes superconvergent, which means that it converges at a greater rate than the unprocessed gradient.

Effects of arbitrary holes/voids on crack growth using local mesh refinement adaptive XIGA

This paper is devoted to numerical investigation of holes/voids effects on crack growth in solids using a locally refined (LR) B-splines extended isogeometric analysis (XIGA). We particularly focus our attention on crack-hole interaction analysis. Special enrichment functions captured discontinuity and singularity are used, by which the computational mesh is thus independent of both cracks and holes, eliminating re-meshing when modeling crack growth. Geometries of arbitrary shaped cracks and holes are accurately described with the NURBS curves, and the coordinates of control points and weights of the NURBS curves are produced from the iges file of Rhino. An efficient stress recovery based error estimator in combination with the structured mesh refinement strategy is adopted to conduct local mesh refinement. The interaction integral method is applied to evaluate the stress intensity factors (SIFs), and the direction of crack growth is determined with the maximum circumferential stress criterion. Analysis of crack-hole interaction is examined through several numerical examples, and computed results are validated with respect to reference solutions.

Application of PHT-splines in bending and vibration analysis of cracked Kirchhoff–Love plates

In this work, we present an eXtended Geometry Independent Field approximaTion (X–GIFT) formulation for cracked Kirchhoff–Love plates. The plate geometry is modeled by Non-Uniform Rational B-Splines (NURBS) while the solution is approximated by Polynomial Splines over Hierarchical T-meshes (PHT-splines) and enriched by the Heaviside function and crack tip asymptotic expansions. The adaptive refinement is driven by a recovery-based error estimator. The formulation is employed for bending and vibration analysis. We compare different strategies for refinement, enrichment and evaluation of fracture parameters. The obtained results are shown to be in a good agreement with the reference solutions.

Recovery-based error estimator for the natural-convection problem based on penalized finite element method

Purpose This paper aims to proposes and analyzes a novel recovery-based posteriori error estimator for the stationary natural-convection problem based on penalized finite element method. Design/methodology/approach The optimal error estimates of the penalty FEM are established by using the lower-order finite element pair P1-P0-P1 which does not satisfy the discrete inf-sup condition. Besides, a new recovery type posteriori estimator in view of the gradient recovery and superconvergent theory to deal with the discontinuity of the gradient of numerical solution. Findings The stability, accuracy and efficiency of the proposed method are confirmed by several numerical investigations. Originality/value The provided reliability and efficiency analysis is shown that the true error can be effectively bounded by the recovery-based error estimator.

A Comparative Study of Mesh-Free Radial Point Interpolation Method and Moving Least Squares Method-Based Error Estimation in Elastic Finite Element Analysis

The mesh-free interpolation method-based recovery of finite element discretization error is presented in this study. Two interpolation schemes, namely radial point interpolation method and moving least squares method, are taken for the recovery of finite element solution error. The global and elemental errors of finite element solution are evaluated in energy norm. The two-dimensional benchmark elastic problems are analysed using triangular/quadrilateral meshing schemes to prove the proposed mesh-free error estimation techniques validity and the effectiveness. The results of the mesh-less interpolation recovery-based error estimation are also compared with mesh-dependent least squares interpolation method-based error estimation. The mesh-free recovery is based on the fitting of a higher-order polynomial to the field variable over a mesh-less patch (radial support domain) using RPI and MLS interpolation method, while the mesh-dependent recovery is based on recovery of the field variable over a patch of nodes surrounding the particular given node using RPI, MLS and LS interpolation method. The study presents the effect of polynomial basis function (LS), polynomial basis function along with radial basis function (RPI)-based interpolation method and weighted polynomial basis function (MLS) on the recovery of finite element solution error. The quality of error estimation under different interpolation schemes is compared in terms of convergence characteristics, local/global effectivity, error distribution patterns and adaptively refined meshes. It can be concluded that weighting function of least squares interpolation affects considerably the mesh-free error estimation.

Formula omitted]- and [formula omitted]-adaptivity driven by recovery and residual-based error estimators for PHT-splines applied to time-harmonic acoustics

In this work, we demonstrate the application of PHT-splines for time-harmonic acoustic problems, modeled by the Helmholtz equation. Solutions of the Helmholtz equation have two features: global oscillations associated with the wave number and local gradients caused by geometrical irregularities. We show that after a sufficient number of degrees of freedom is used to approximate global oscillations, adaptive refinement can capture local features of the solution. We compare residual-based and recovery-based error estimators and investigate the performance of p-refinement. The simulations are done in the context of recently introduced Geometry Independent Field approximaTion (GIFT), where PHT-splines are only used to approximate the solution, while the computational domain is parameterized with NURBS. This approach builds on the natural adaptation ability of PHT-splines and avoids the re-parameterization of the NURBS geometry during the solution refinement process.

Interpolation Type Stress Recovery Technique Based Error Estimator for Elasticity Problems

A finite element method coupled with error estimation has gained considerable prominence in industry. However, effective and reliable error control of finite element solution is always a challenging task particularly for incompressible and large deformations problems. Effective interpolation type gradient recovery based error estimation procedure is proposed in the present study. The recovery is based on Moving Least Squares approximation (mesh free approach) of the displacement field or their derivatives by a higher order polynomial over a patch of nodes in a circular boundary. The performance of error estimation scheme in terms of its effectivity and convergence has been compared with that of Zienkiewicz-Zhu (ZZ) super-convergent recovery scheme by applying the scheme to benchmark elastic problems. Error estimators compute the error in energy norm of the recovered solution both at local and global levels. The adaptive meshing based on guidance of the local error predicted by ZZ and proposed interpolation type error estimators, is also used to study the error distribution in the domain. The proposed mesh independent node patch based recovery scheme is found to be better than that for the mesh dependent node patch based ZZ super convergent recovery scheme.DOI: http://dx.doi.org/10.5755/j01.mech.24.5.19937

An adaptive isogeometric analysis collocation method with a recovery-based error estimator

In this paper, we propose an enhanced isogeometric analysis (IGA) collocation method. It is well known that the location of the collocation points plays an important role in the accuracy and stability of IGA collocation methods. This is particularly true for non-uniform meshes and domains generated from multi-patch geometries. We present an enhanced collocation method based on Gauss points, which has improved accuracy as compared to using C1 splines and a recovery-based error estimator that can be derived by sampling the solution at particular points in the domain. Adaptivity is implemented using a hierarchical spline basis, which satisfies the C1 continuity requirement. The proposed approach has been tested by several benchmark problems, including multipatch domains and geometries with re-entrant corners.

基于亏量校正求解自然对流问题的后验误差估计子

基于亏量校正求解自然对流问题的后验误差估计子

An h-adaptive numerical manifold method for solid mechanics problems

The numerical manifold method (NMM) features its dual cover systems, namely the mathematical cover and physical cover, which provide a unified framework for mechanics problems involving continuum and discontinuum deformation. Uniform finite element meshes can be and are usually used to construct the mathematical cover. Though this strategy can handle different kinds of problems in a unified way, it is not economical for cases with steep deformation gradients or singularities. In this paper, using the recovery-based error estimator, an h -adaptive NMM on quadtree meshes is proposed to deal with such cases. The quadtree meshes serve as the mathematical meshes, on which the local refinement is carried out. When the quadtree meshes are refined, the corresponding mathematical cover, physical cover and manifold elements are updated accordingly. To handle the hanging nodes in the quadtree meshes, we resort to mean value coordinates. Comparing to the refinement based on manifold elements, the proposed strategy guarantees consistent structured meshes throughout the adaptive process, thus retaining the unique feature of original NMM. In contrast with polygonal finite element method, an advantage of the proposed method is that the meshes do not need to conform to the crack face and material boundary, which means the adaptive NMM is very suitable for problems with complex geometric boundary. Several representative mechanics problems, including crack problems, are analyzed to investigate the effectiveness of the proposed method. It is demonstrated that the proposed adaptive NMM has higher accuracy and better performance comparing to uniform refinement strategy.

Recovery-based error estimation and adaptivity using high-order splines over hierarchical T-meshes

An adaptive higher-order method based on a generalization of polynomial/rational splines over hierarchical T-meshes (PHT/RHT-splines) is introduced. While most problems considered in isogeometric analysis can be solved efficiently when the solution is smooth, many non-trivial simulations have rough solutions. This can be caused, for example, by the presence of re-entrant corners in the domain. For such problems, a tensor-product basis is less suitable for resolving the singularities that appear, as refinement propagates throughout the computational domain. Hierarchical bases and adaptivity allow for a more efficient way of dealing with singularities, by adding more degrees of freedom only where they are necessary to improve the approximation. In order to drive the adaptive refinement, an efficient recovery-based error estimator is proposed in this work. The estimator produces a “recovered solution” which is a more accurate approximation than the computed numerical solution. Several 2D and 3D numerical investigations with PHT-splines of higher order and greater continuity show good performance compared to uniform refinement in terms of degrees of freedom and computational cost.

Space-time mesh adaptation for solute transport in randomly heterogeneous porous media

We assess the impact of an anisotropic space and time grid adaptation technique on our ability to solve numerically solute transport in heterogeneous porous media. Heterogeneity is characterized in terms of the spatial distribution of hydraulic conductivity, whose natural logarithm, Y, is treated as a second-order stationary random process. We consider nonreactive transport of dissolved chemicals to be governed by an Advection Dispersion Equation at the continuum scale. The flow field, which provides the advective component of transport, is obtained through the numerical solution of Darcy's law. A suitable recovery-based error estimator is analyzed to guide the adaptive discretization. We investigate two diverse strategies guiding the (space-time) anisotropic mesh adaptation. These are respectively grounded on the definition of the guiding error estimator through the spatial gradients of: (i) the concentration field only; (ii) both concentration and velocity components. We test the approach for two-dimensional computational scenarios with moderate and high levels of heterogeneity, the latter being expressed in terms of the variance of Y. As quantities of interest, we key our analysis towards the time evolution of section-averaged and point-wise solute breakthrough curves, second centered spatial moment of concentration, and scalar dissipation rate. As a reference against which we test our results, we consider corresponding solutions associated with uniform space-time grids whose level of refinement is established through a detailed convergence study. We find a satisfactory comparison between results for the adaptive methodologies and such reference solutions, our adaptive technique being associated with a markedly reduced computational cost. Comparison of the two adaptive strategies tested suggests that: (i) defining the error estimator relying solely on concentration fields yields some advantages in grasping the key features of solute transport taking place within low velocity regions, where diffusion-dispersion mechanisms are dominant; and (ii) embedding the velocity field in the error estimator guiding strategy yields an improved characterization of the forward fringe of solute fronts which propagate through high velocity regions.

Recovery-Based Error Estimator for Stabilized Finite Element Method for the Stationary Navier--Stokes Problem

A recovery-based error estimator is proposed and analyzed for stabilized $P_1/P_0$ (continuous linear velocity/constant pressure) finite element approximations to the stationary Navier--Stokes problem. We establish the reliability and efficiency of the error estimator. A crucial part of this work is the estimation for the nonlinear term of the Navier--Stokes problem. It turns out such a term can be bounded by the recovery-based error estimator. Numerical results are provided to illustrate the performance of the error estimator.

Recovery-based error estimation in the dynamic analysis of offshore wind turbine monopile foundations

Offshore wind turbine foundations are affected by cyclic loads due to oscillatory kinematic loads, such as those from wind, waves, and earthquakes. Monopiles are often used as a foundation concept for offshore windmill turbines. In this study, coupled dynamic equations with the u-P formulation for low-frequency load are considered for an offshore wind turbine monopile foundation, to present the response in terms of pore water pressure (PWP), stress and strain distribution in an elastic porous medium at regions around the monopile foundation. Different stress recovery techniques based on the Zienkeiwicz–Zhu (ZZ) error estimator namely, super-convergent patch recovery (SPR), weighted super-convergent patch recovery (WSPR), and L2-projection techniques are also investigated to recover the stresses at nodal points in the finite element method. To estimate errors in the time domain when performing transient simulations, three recovery processes are used with different meshes. The convergence of the dynamic problem is also studied. The results are verified with findings in the literature, revealing that the time period of effective stresses follows the applied load frequency. In conclusion, the history of the shear stress can have an important effect on the shear stress distribution, making it asymmetric in the time domain.

Recovery-Based Error Estimator for the Discontinuous Galerkin Method for Nonlinear Scalar Conservation Laws in One Space Dimension

In this paper, we propose and analyze a robust recovery-based error estimator for the original discontinuous Galerkin method for nonlinear scalar conservation laws in one space dimension. The propo...

Adaptive thermo-mechanical finite element formulation based on goal-oriented error estimation

Abstract In this paper, a rh-adaptive thermo-mechanical formulation based on goal-oriented error estimation is proposed. The goal-oriented error estimation relies on different recovery-based error estimators, i.e. the superconvergent patch recovery (SPR), L 2 -projection patch recovery ( L 2 -PR) and weighted superconvergent patch recovery (WSPR). A new adaptive refinement strategy (ARS) is presented that minimizes the error in a quantity of interest and refines the discretization such that the error is equally distributed on the refined mesh. The method is validated by numerous numerical examples where an analytical solution or reference solution is available.