Defect Correction Method Research Articles

Corrigendum to “Improvements to a class of hybrid methods for radiation transport: Nyström reconstruction and defect correction methods”

Corrigendum to “Improvements to a class of hybrid methods for radiation transport: Nyström reconstruction and defect correction methods”

Improved defect-correction algorithms for the Navier–Stokes equations at small viscosity

In this article, based on finite element discretization, we propose some improved defect-correction algorithms for solving the stationary Navier–Stokes equations with small viscosity. The proposed algorithms are mainly inspired by the idea of the grad-div stabilized method and error correction technique. Maintaining the benefit of the usual defect-correction method, the proposed algorithms further improve the ability to solve problems with small viscosity and have a fast convergence rate. Moreover, stability analysis and error estimation of these algorithms are provided under the uniqueness requirement. Finally, some numerical experiments are tested to illustrate the effectiveness of the presented algorithms for small viscosity problem.

A Posteriori Error Analysis of Defect Correction Method for Singular Perturbation Problems With Discontinuous Coefficient and Point Source

Abstract The paper presents a defect correction method to solve singularly perturbed problems with discontinuous coefficient and point source. The method combines an inexpensive, lower-order stable, upwind difference scheme and a higher-order, less stable central difference scheme over a layer-adapted mesh. The mesh is designed so that most mesh points remain in the regions with rapid transitions. A posteriori error analysis is presented. The proposed numerical method is analyzed for consistency, stability, and convergence. The error estimates of the proposed numerical method satisfy parameter-uniform second-order convergence on the layer-adapted grid. The convergence obtained is optimal because it is free from any logarithmic term. The numerical analysis confirms the theoretical error analysis.

A defect correction weak Galerkin finite element method for the Kelvin–Voigt viscoelastic fluid flow model

In this paper, we present a defect correction weak Galerkin finite element method for solving the Kelvin–Voigt viscoelastic fluid flow model, which can guarantee that the discrete velocity is globally divergence-free. The second-order Crank–Nicolson difference scheme is considered for temporal discretization. In particular, combined with the defect correction method, the numerical oscillation effect is reduced at high Reynolds number. Our algorithm not only gives the stability and convergence of the numerical solutions of velocity and pressure but also is uniformly valid at the retardation time κ→0. Several numerical experiments are performed to verify the method has good behaviors.

In-situ process monitoring and adaptive quality enhancement in laser additive manufacturing: A critical review

Laser Additive Manufacturing (LAM) presents unparalleled opportunities for fabricating complex, high-performance structures and components with unique material properties. Despite these advancements, achieving consistent part quality and process repeatability remains challenging. This paper provides a comprehensive review of various state-of-the-art in-situ process monitoring techniques, including optical-based monitoring, acoustic-based sensing, laser line scanning, and operando X-ray monitoring. These techniques are evaluated for their capabilities and limitations in detecting defects within Laser Powder Bed Fusion (LPBF) and Laser Directed Energy Deposition (LDED) processes. Furthermore, the review discusses emerging multisensor monitoring and machine learning (ML)-assisted defect detection methods, benchmarking ML models tailored for in-situ defect detection. The paper also discusses in-situ adaptive defect remediation strategies that advance LAM towards zero-defect autonomous operations, focusing on real-time closed-loop feedback control and defect correction methods. Research gaps such as the need for standardization, improved reliability and sensitivity, and decision-making strategies beyond early stopping are highlighted. Future directions are proposed, with an emphasis on multimodal sensor fusion for multiscale defect prediction and fault diagnosis, ultimately enabling self-adaptation in LAM processes. This paper aims to equip researchers and industry professionals with a holistic understanding of the current capabilities, limitations, and future directions in in-situ process monitoring and adaptive quality enhancement in LAM.

Comparison of In-situ Nondestructive Testing and Ex-situ Methods in Additive Manufactured Specimens for Internal Feature Detection

ABSTRACT The effectiveness and repeatability of Additive Manufacturing (AM) technologies has been often measured through rigorous testing, both in-situ and ex-situ. In-situ nondestructive testing (NDT) has been used to understand the AM process and detect anomalies during the build. Ex-situ testing has been used for material characterization and internal defect detection. In this work, in-situ NDT and ex-situ testing methods are compared as applied to a Fused Filament Fabrication (FFF) process to detect defects that have formed internally in the part. Intentionally embedded features and defects are monitored and analyzed in-situ and ex-situ. Results from an automated image-based real-time defect detection system, an ultrasound ex-situ system, and a tensile testing system are compared to detect defects and their effects on mechanical properties. Real-time image-based defect detection is found to effectively spot defects as they form and allow the potential for implementing defect correction methods, while ex-situ tensile testing can be used as a form of validation to in-situ results.

A New Approach to Increase the Accuracy of a Numerical Scheme Using Modified Partial Differential Equation : Application to Meshless Methods

Meshless methods are gaining popularity for numerically solving the governing equations of Fluid Dynamics primarily because they do not require the generation of grid around the given geometry in the computational domain. They just require a point distribution with a proper connectivity, which is extremely simple to generate as compared to the grid around complex geometries. However, obtaining a higher order accurate numerical solution is difficult in case of meshless methods as it requires either a solution of a larger system of linear algebraic equations, or defect correction method coupled with inner iterations. This method becomes too much complicated if still further higher order accuracy is desired. Here an alternate method based on Modified Partial Differential Equations has been used to obtain higher order accurate solution using the meshless methods. The basic idea behind the new approach is to explicitly remove the truncation error from the Modified Partial Differential Equation (formed from the governing equations) and numerically solve the resulting equation with the meshless technique. This technique has been successfully applied for the simulation of subsonic, weak transonic, strong transonic and supersonic flow past NACA0012 aerofoil. A good match of the numerical results with the standard test results of AGARD/GAMM workshop has been.

A Higher-Order Defect Correction Method Over an Adaptive Bakhvalov–Shishkin Mesh for Advection–Diffusion Equations

The paper presents a defect correction method based on finite difference discretizations over a Bakhvalov–Shishkin mesh for a class of convection-diffusion equations. The technique combines a stable, low-order accurate, computationally inexpensive upwind difference scheme with a higher-order less stable modified central difference operator at several grid points. We prove that the method is second-order uniformly convergent at all mesh points. We do not use asymptotic expansions of discretization errors in our analysis. The method has no directional bias, and it can be used in an adaptive procedure to refine the mesh in the non-smooth parts. Numerical results are presented, which indicate that defect correction on layer-adapted meshes is an excellent method to achieve higher-order accuracy for singular perturbation problems.

Defect deferred correction method for two domain natural convection problem

This paper concerns defect deferred correction method used to resolve interactions between the atmosphere and ocean for applications like climate research, hurricane modelling and regional weather forecasting. The defect deferred correction method is obtained by combining the defect correction method and the deferred correction method. There are two advantages of using the defect deferred correction method to solve a two domain natural convection problem, namely, reaching the second-order time accuracy and solving small viscosity or diffusion coefficient problems. Finally, we will verify the conclusion from both theoretical analysis and numerical experiments.

Parallel defect-correction methods for incompressible flows with friction boundary conditions

We study three parallel defect-correction methods based on finite element approximations for the incompressible Navier–Stokes problem with friction boundary conditions and high Reynolds numbers in this work, where a fully overlapping domain decomposition is considered for parallelization. In the proposed methods, with a global multiscale grid that builds a fine grid around its own subdomain and coarse elsewhere, we iteratively solve an artificial viscosity nonlinear variational inequality problem in a defect step, and then compute the residual by the linearized variational inequality problems in the r-step corrections. The studied methods are easy to implement on the basis of the existing Navier–Stokes solver and possess less communication complexity. We provide a rigorously theoretical derivation for the error estimates of the one-step correction solutions from the proposed methods under some stable conditions, and derive scalings of the algorithmic parameters. We demonstrate by a series of numerical experiments that the velocity and pressure errors computed by our parallel defect-correction methods are comparable to those of the standard defect-correction method, while our present methods reduce the computational cost.

A uniformly convergent defect correction method for parabolic singular perturbation problems with a large delay

A uniformly convergent defect correction method for parabolic singular perturbation problems with a large delay

Multiscale coupling of FFT-based simulations with the LDC approach

FFT based solvers have proven to be efficient computational methods for the numerical simulations of composite materials with complex microstructures and sophisticated material behavior. Nevertheless, these solvers require a regular discretization of the investigated domain and periodic boundary conditions.Local multi-grid methods appear as an appropriate framework to perform simulations involving local refinement with FFT solvers. A coarse (global) discretization is improved by one or several local refined discretizations defined in subdomains overlapping the global grid. The Local Defect Correction (LDC) method is adapted to couple the resolution on the global coarse grid problem with the resolution of the local grids problems. Numerical tests demonstrate a significant reduction of CPU time and memory consumption with a relatively low loss in accuracy in comparison to the computation on one fully refined global grid.This approach also provides a baseline for extensions to the coupling of FFT codes with other solvers in a multi-scale and multi-physics framework.

A higher-order uniformly convergent defect correction method for singularly perturbed convection-diffusion problems on an adaptive mesh

This paper presents a defect correction scheme based on finite-difference discretization for a singularly perturbed convection-dominated diffusion problem. The solution of this class of problems exhibits a multiscale character. There are narrow regions in which the solution grows exponentially and reveals layer behaviour. The defect correction method that we propose improves the efficiency of a numerical solution through iterative improvement and generates a stable higher-order method over an adaptive non-uniform polynomial-Shishkin mesh. An extensive theoretical analysis is presented, which establishes that the method is second-order uniformly convergent and highly stable. The convergence obtained is optimal because it is free from any logarithmic term. The numerical result for two model problems is presented, agreeing with the theoretical estimates. Furthermore, we compare the results with those of other non-uniform mesh found in the literature.

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.

Local defect-correction method based on multilevel discretization for Steklov eigenvalue problem

In this paper, we propose a local defect-correction method for solving the Steklov eigenvalue problem arising from the scalar second order positive definite partial differential equations based on the multilevel discretization. The objective is to avoid solving large-scale equations especially the large-scale Steklov eigenvalue problem whose computational cost increases exponentially. The proposed algorithm transforms the Steklov eigenvalue problem into a series of linear boundary value problems, which are defined in a multigrid space sequence, and a series of small-scale Steklov eigenvalue problems in a coarse correction space. Furthermore, we use the local defect-correction technique to divide the large-scale boundary value problems into small-scale subproblems. Through our proposed algorithm, we avoid solving large-scale Steklov eigenvalue problems. As a result, our proposed algorithm demonstrates significantly improved the solving efficiency. Additionally, we conduct numerical experiments and a rigorous theoretical analysis to verify the effectiveness of our proposed approach.

A unified framework for the computational comparison of adaptive mesh refinement strategies for all-quadrilateral and all-hexahedral meshes: Locally adaptive multigrid methods versus h-adaptive methods

This paper provides a detailed comparison in a solids mechanics context of adaptive mesh refinement methods for all-quadrilateral and all-hexahedral meshes. The adaptive multigrid Local Defect Correction method and the well-known hierarchical h-adaptive refinement techniques are placed into a generic algorithmic setting for an objective numerical comparison. Such a comparison is of great interest as local multigrid AMR approaches are from now rarely employed to adaptively solve implicit systems in solid mechanics. However they present various interesting features mainly related to their intrinsic idea of partitioning the degrees of freedom on different mesh levels. For this study, we rely on a fully-automatic mesh refinement algorithm providing the desired refined mesh directly from the user-prescribed accuracy. The refinement process is driven by an a posteriori error estimator combined to mesh optimality criteria. In this study, the most efficient strategy based on mesh optimality criterion and refinement ratio is identified for all-quadrilateral and all-hexahedral finite elements meshes. The quality of refined meshes is finally appreciated in term of number of nodes but also through the verification of final solution's accuracy. A special attention is devoted to the fulfillment of local precisions which are of great importance from an engineering point of view. Numerical 2D and 3D experiments of different complexities revealing local phenomena enable to highlight the essential features of the considered mesh refinement methods within an elastostatic framework. This study points out the great potentialities of locally adaptive multigrid method, which clearly appears to be the most powerful strategy in terms of standard metrics of efficiency (dimension of systems to be solved, storage requirements, CPU time).

Two-level defect-correction stabilized algorithms for the simulation of 2D/3D steady Navier-Stokes equations with damping

By combining the defect-correction method with the two-level discretization strategy and the local pressure projection stabilized method, this paper presents and studies two kinds of two-level defect-correction stabilized algorithms for the simulation of 2D/3D steady Navier-Stokes equations with damping, where the lowest equal-order P1−P1 finite elements are used for the velocity and pressure approximations. In the proposed algorithms, an artificial viscosity stabilized nonlinear Navier-Stokes problem with damping is first solved in the coarse grid defect step, and then corrections are computed in the fine grid correction step by solving a linear problem based on Oseen-type and Newton-type iterations, respectively. Under the uniqueness condition, stability of the proposed algorithms is analyzed, and optimal error estimates of the approximate solutions are deduced. The correctness of the theoretical predictions and the effectiveness of the proposed algorithms are illustrated by some 2D and 3D numerical results.

Improvements to a class of hybrid methods for radiation transport: Nyström reconstruction and defect correction methods

In this paper, two modifications are introduced for improving the accuracy, versatility, and robustness of a class of hybrid methods for radiation transport. In general, such methods are constructed by splitting the radiative flux into collided and uncollided components to which low- and high-resolution angular approximations are applied, respectively. In this work we focus on discrete ordinates discretizations of high and low order. The first modification we introduce changes the way in which the collided component is mapped into the uncollided component at the end of each time step in a simulation. The new mapping is a Nyström-type reconstruction that is applicable to arbitrary discrete ordinates quadratures, is guaranteed to preserve positivity of the solution provided that all ordinate weights are positive, is significantly more accurate than previous methods, and can be readily extended to other discretizations such as moment methods, finite element methods, and diffusion approximations. The second modification leverages integral deferred correction (IDC) to iteratively correct for the splitting error introduced by the inconsistency in angular discretization between the collided and uncollided components, in addition to improving the accuracy of the low-order temporal error that is treated by traditional IDC methods. Numerical tests in one- and two-dimensional geometries are used to demonstrate the increased accuracy and efficiency of the proposed modifications. It is found that the two techniques combined yield methods with solution accuracy and memory requirements comparable to that of monolithic discrete ordinates methods while reducing runtime by as much as a factor of between two and ten, depending on the problem.

Defect correcting extrapolation technique for Oseen and Navier–Stokes Flows

This report presents defect correcting extrapolation technique (DCE), an alternative approach for defect correction (DC). In contrast to conventional DC methods, which always solve for separate correction steps, DCE instantly assumes weighted linear combinations of several artificial viscosity approximations as a corrected solution. This way, it merely requires solving an identical problem with varying viscosities. The theory and computations show that DCE uncovers ”hidden turbulence” in artificially-diffused laminar solutions, and recovers their energy and turbulent characteristics.

A new two-level defect-correction method for the steady Navier–Stokes equations

A new defect-correction method based on subgrid stabilization for the simulation of steady incompressible Navier–Stokes equations with high Reynolds numbers is proposed and studied. This method uses a two-grid finite element discretization strategy and consists of three steps: in the first step, a small nonlinear coarse mesh system is solved, and then, in the following two steps, two Newton-linearized fine mesh problems which have the same stiffness matrices with only different right-hand sides are solved. The nonlinear coarse mesh system incorporates an artificial viscosity term into the Navier–Stokes system as a stabilizing factor, making the nonlinear system easier to resolve. While the linear fine mesh problems are stabilized by a subgrid model defined by an elliptic projection into lower-order finite element spaces for the velocity. Error bounds of the approximate solutions are estimated. Algorithmic parameter scalings are derived from the analysis. Effectiveness of the proposed method is also illustrated by some numerical results.