Gradient Reconstruction Method Research Articles

A deep learning-based spatial gradient reconstruction method for efficient damage identification in composite with high-sparsity Lamb wavefield

The structural integrity and safety of carbon fiber reinforced plastics (CFRP) are vulnerable to delamination, which is often imperceptible to the naked eye. Although the Scanning Laser Doppler Vibrometer (SLDV) has shown promise in damage quantification of CFRP, its time-consuming measurement process limits its application in engineering scenarios. To address this, we introduce a novel damage index, the spatial gradient, which captures the interaction between delamination and the wavefield. We have also developed a neural network capable of reconstructing the spatial gradient directly from high-sparsity Lamb wavefield data obtained at an extremely low spatial sampling rate, thereby significantly reducing measurement time. To enhance the network’s capability to detect wavefield anomalies, we employ the cross-attention technique, allowing for the direct injection of shallow features representing local wavefield distortions caused by damage into the decoder. Additionally, we integrate multiple reconstruction layers to guide the wavefield reconstruction process, ensuring meaningful information is captured at each stage. Our method achieves substantial improvements in reconstruction accuracy, increasing from 70 % to 92 % in single-damage scenario and from 14 % to 72 % in multi-damage scenario compared to the previous state-of-the-art techniques. By using the reconstructed spatial gradient field for damage imaging through spatial covariance analysis, our approach demonstrates its feasibility and generalizability across various damage locations. This suggests its potential as a reliable solution for fast and accurate damage characterization, reducing the measurement burden and enhancing practical applicability.

Validation of a Human Upper Airway Computational Fluid Dynamics Model for Turbulent Mixing

Abstract Validation of a computational fluid dynamics (CFD) model used to simulate turbulent exchange in an anatomically detailed human upper airway with realistic breathing states is provided. Proper model validation is vital in confirming that temporal mixing and species distribution are accurate, therefore making the model useful in generalized turbulent mixing studies of the upper airway. Numerous levels of refinement were tested for time-step and mesh independence. Higher and lower rigor groups of modeling methodologies involved spatial discretization schemes, gradient reconstruction methods, transient formulations, and turbulence frameworks. A dual mesh independence study revealed that the rate of approach to mesh independence is a function of computational rigor and that multiple mesh independence studies should be carried out in parallel. The final validated model consisted of the finest mesh used in this study (8 × 106 cells), a time-step equating to 4000 timesteps per breath cycle, and higher rigor modeling methodologies. While its results were within the acceptable deviation from the experimental data, it was not as close as the model that utilized the coarsest mesh (∼2 × 106 cells), the fewest timesteps per breath cycle (128 timesteps per breath cycle), and lower rigor methodologies. Though the latter model was closer to the experimental data, it was proven to not be numerically independent, highlighting the importance of utilizing a myriad of metrics to prove numerical independence. Restricting independence studies to only using metrics from experimental comparisons is insufficient for proper validation.

A constrained boundary gradient reconstruction method for unstructured finite volume discretization of the Euler equations

We propose a novel boundary gradient and solution reconstruction approach for the second-order cell-centered unstructured finite volume method for solving the inviscid Euler equations. The proposed constrained reconstruction method involving ghost cells is shown to be more accurate than several conventional boundary reconstruction methods and linear-exact in the near-boundary areas. In the present procedure, the solutions at the ghost cell centroids are given by solving constraint equations, which are established based on boundary conditions. At the same time, the solution gradients in boundary cells and the solutions in boundary surfaces are obtained altogether, while guaranteeing the compatibility of the boundary conditions with the reconstructed results. A variety of numerical test cases show that the new method effectively reduces the errors of near-boundary approximations and improves the overall numerical accuracy. Furthermore, because the constrained reconstruction is only used for the boundary cells, the extra computational cost barely changes the overall computational efficiency.

Nonlinear hydrodynamics of freely floating symmetric bodies in waves by three-dimensional fully nonlinear potential-flow numerical wave tank

This study estimated the nonlinear hydrodynamic performance of freely floating structures in nonlinear water waves. For this, a three-dimensional, fully nonlinear numerical wave tank (NWT) was developed based on potential theory and the boundary element method (BEM). The fully nonlinear NWT evaluates the hydrodynamic force acting on the instantaneous body position and free-surface elevation based on the mixed Eulerian and Lagrangian (MEL) method and acceleration-potential approach. The instantaneous nonlinear free surface was traced at each time step, and the far-field outgoing wave condition was satisfied by placing an artificial free-surface damping zone. In addition, the least square gradient reconstruction method, thin-plate spline method, and modified inverse distance weighting method were adopted to re-grid the calculation nodes and evaluate the spatial derivatives of the physical quantities robustly. The developed NWT was used to evaluate the first-order and second-order motion RAOs and excitation forces of a freely floating vertical cylinder. The results were compared systematically with previously published results using perturbation theory. Moreover, the heave-to-pitch Mathieu-instability phenomena on a deep-draft floating cylinder were demonstrated by the fully nonlinear NWT simulations. The results agreed well with the corresponding experimental observations.

Higher-order diffraction forces on vertical circular cylinders by using a three-dimensional fully-nonlinear potential numerical wave tank

The aim of this study is to evaluate higher-order diffraction forces on vertical circular cylinders by using an author-developed three-dimensional fully-nonlinear numerical wave tank (NWT). The nonlinear NWT is based on the potential theory, boundary element method, and mixed Eulerian Lagrangian (MEL) approach. The acceleration-potential method is used to calculate the hydrodynamic pressure accurately. The artificial damping zone is adopted to prevent reflected waves from bodies and side walls. The least square gradient reconstruction method and the thin plate spline method are applied to evaluate the spatial derivatives of physical quantities and updated locations of interpolation points. The third-order stokes monochromatic or bichromatic waves are inputted as incident-wave boundary condition. From the nonlinear NWT simulations with FFT, first-order, second-order, and third-order diffraction forces are simultaneously calculated and the results are systematically compared with published experimental and perturbation-based numerical results. The sensitivity of the nonlinear NWT results with the respective orders of Stokes monochromatic or bichromatic input waves is also investigated. The increase of third-order sum-frequency force in shallow water long wave is noticed.

Iterative reconstruction algorithm comparison using Poisson noise distributed sinogram data in passive gamma emission tomography

ABSTRACT Gamma emission tomography (GET) detects fuel rods that emit gamma rays for potential use as verification tools in nuclear safeguards. In GET, iterative reconstruction algorithms are often used to reconstruct passive gamma-ray emitter source distributions. Generally, using noisy sinogram data, the iterative reconstruction algorithms increase the noise level on a reconstruction image with iterations. Thus, it is important to evaluate how performances of representative iterative reconstruction algorithms change with noisy sinogram data. The passive gamma-ray source distributions inside the mock-up of the water-water energetic reactor (WWER) fuel assembly having missing fuel rods were reconstructed by using the following algorithms: gradient method, steepest descent method, conjugate gradient reconstruction method, algebraic reconstruction technique, simultaneous iterative reconstruction technique, and maximum likelihood expectation maximization (MLEM). Consequently, MLEM algorithm yielded a higher contrast reconstruction image and was regarded as higher reliable algorithm to discriminate the fuel rods from the passive gamma-ray emitter source distribution.

MIN3P-HPC: A High-Performance Unstructured Grid Code for Subsurface Flow and Reactive Transport Simulation

The numerical simulation of flow and reactive transport in porous media with complex domains is nontrivial. This paper presents a method to implement fully unstructured grid capabilities into the well-established software ParMIN3P-THCm, a process-based numerical model designed for the investigation of subsurface fluid flow and multicomponent reactive transport in variably saturated porous media with parallelization capability. The enhanced code, MIN3P-HPC, is modularized to support different cell types, spatial discretization methods and gradient reconstruction methods. MIN3P-HPC uses a vertex-centered control volume method with consideration of both vertex-based and cell-based material properties (e.g., permeability). A flexible parallelization scheme based on domain decomposition and thread acceleration was implemented, which allows the use of OpenMP, MPI and hybrid MPI-OpenMP, making optimized use of computer resources ranging from desktop PCs to distributed memory supercomputers. The code was verified by comparing the results obtained with the unstructured grid version to those produced by the structured grid version. Numerical accuracy was also verified against analytical solutions for 2D and 3D solute transport, and by comparison with third-party software using different cell types. Parallel efficiency of OpenMP, MPI and hybrid MPI-OpenMP versions was examined through a series of solute transport and reactive transport test cases. The results demonstrate the versatility and enhanced performance of MIN3P-HPC for reactive transport simulation.

Numerical investigation of flow instabilities using fully unstructured discretization for variably saturated flow problems

Richards equation for variably saturated flow can exhibit stability problems due to its nonlinearity. The challenge is to resolve sharp wetting fronts without introducing spurious oscillations, especially for simulations with very dry initial conditions. Flow instabilities at sharp wetting fronts may arise particularly in anisotropic and heterogeneous media, leading to oscillations or convergence problems. The focus of this work is to evaluate and minimize instability problems for simulations with unstructured meshes. To this end, numerical experiments were performed to investigate the accuracy, monotonicity and convergence behavior of numerical solutions for variably saturated flow based on different control volume methods, piecewise gradient reconstruction methods, and flux approximation methods. In particular, nonphysical oscillations at the wetting front were investigated. A novel multi-point flux approximation with multi-point upstream weighting based on piecewise gradient reconstruction was developed. Numerical simulations in both homogeneous and heterogeneous domains with isotropic and anisotropic conductivity tensors demonstrate that the proposed multi-point flux approximation with multi-point upstream weighting avoids spurious oscillations and improves the results for challenging sharp wetting front problems using general unstructured meshes as well as meshes with distortion. The revised Voronoi-dual control volume method has also been found to provide more flexibility than commonly used center-dual and median-dual control volume methods.

An iterative near-boundary reconstruction strategy for unstructured finite volume method

Near-boundary approximation of the second-order cell-centered unstructured finite volume method is investigated and improved to avoid accuracy loss. In this work, an iterative strategy is proposed, combining with a vertex-based weighted least squares gradient reconstruction method, which is more accurate and efficient than several conventional methods. In the present approach, the solutions at boundary surfaces and the gradients in boundary cells are iteratively updated, to minimize overall numerical errors. In the meantime, a vectorial limiter is implemented to guarantee the correctness of the iterative process, and physical properties of boundaries are taken into account to correctly attain physical results. A series of numerical test cases show that the present method significantly reduces the error of near-boundary approximations. Moreover, since the iterative scheme is only activated for near-boundary solutions, the minor extra computational effort does not damage the overall performance of the solver.

Gradient Based Elastic Property Reconstruction in Digital Image Correlation Elastography.

Aim:A Conjugate Gradient implementation of the Digital Image Correlation Elastography method is presented.Method:The gradient is calculated using the adjoint method, requiring only two forward solutions regardless of the number of mechanical properties to reconstruct. A power-law based multi-frequency viscoelastic model is used to relate the reconstructed mechanical properties and the Digital Image Correlation surface displacement measurements.Result:The method is tested against harmonic surface motion fields generated through numerical simulation and measured on a silicon phantom.Conclusion:Reconstruction results show the ability of the gradient reconstruction method to detect stiff internal inclusions in simulated and phantom data.

Development of a Three-Dimensional Fully Nonlinear Potential Numerical Wave Tank for a Heaving Buoy Wave Energy Converter

The hydrodynamic performance of a vertical cylindrical heaving buoy-type floating wave energy converter under large-amplitude wave conditions was calculated. For this study, a three-dimensional fully nonlinear potential-flow numerical wave tank (3D-FN-PNWT) was developed. The 3D-FN-PNWT was based on the boundary element method with Rankine panels. Using the mixed Eulerian–Lagrangian (MEL) method for water particle movement, nonlinear waves were produced in the PNWT. The PNWT can calculate the wave forces acting on the buoy accurately using an acceleration potential approach. The constant panels and least-square gradient reconstruction method were applied to regridding of computational boundaries. An artificial damping zone was employed to satisfy the open-sea conditions at the end free surface boundaries. The diffraction and radiation problems were solved, and their solutions were confirmed by a comparison with previous studies. The interaction of the incident wave, floating body, and power take-off (PTO) behavior was examined in the time domain using the developed 3D-FN-PNWT. From comparison, the difference between the conventional linear analysis and the nonlinear analysis in large-amplitude waves was examined.

Accuracy analysis of gradient reconstruction on isotropic unstructured meshes and its effects on inviscid flow simulation

The accuracy of gradient reconstruction methods on unstructured meshes is analyzed both mathematically and numerically. Mathematical derivations reveal that, for gradient reconstruction based on the Green-Gauss theorem (the GG methods), if the summation of first-and-lower-order terms does not counterbalance in the discretized integral process, which rarely occurs, second-order accurate approximation of face midpoint value is necessary to produce at least first-order accurate gradient. However, gradient reconstruction based on the least-squares approach (the LSQ methods) is at least first-order on arbitrary unstructured grids. Verifications are performed on typical isotropic grid stencils by analyzing the relationship between the discretization error of gradient reconstruction and the discretization error of the face midpoint value approximation of a given analytic function. Meanwhile, the numerical accuracy of gradient reconstruction methods is examined with grid convergence study on typical isotropic grids. Results verify the phenomenon of accuracy degradation for the GG methods when the face midpoint value condition is not satisfied. The LSQ methods are proved to be at least first-order on all tested isotropic grids. To study gradient accuracy effects on inviscid flow simulation, solution errors are quantified using the Method of Manufactured Solutions (MMS) which was validated before adoption by comparing with an exact solution case, i.e., the 2-dimensional (2D) inviscid isentropic vortex. Numerical results demonstrate that the order of accuracy (OOA) of gradient reconstruction is crucial in determining the OOA of numerical solutions. Solution accuracy deteriorates seriously if gradient reconstruction does not reach first-order.

Convergence and error analysis of an automatically differentiated finite volume based heat conduction code

Purpose This paper aims to investigate the convergence and error properties of a finite volume-based heat conduction code that uses automatic differentiation to evaluate derivatives of solutions outputs with respect to arbitrary solution input(s). A problem involving conduction in a plane wall with convection at its surfaces is used as a test problem, as it has an analytical solution, and the error can be evaluated directly. Design/methodology/approach The finite volume method is used to discretize the transient heat diffusion equation with constant thermophysical properties. The discretized problem is then linearized, which results in two linear systems; one for the primary solution field and one for the secondary field, representing the derivative of the primary field with respect to the selected input(s). Derivatives required in the formation of the secondary linear system are obtained by automatic differentiation using an operator overloading and templating approach in C++. Findings The temporal and spatial discretization error for the derivative solution follows the same order of accuracy as the primary solution. Second-order accuracy of the spatial and temporal discretization schemes is confirmed for both primary and secondary problems using both orthogonal and non-orthogonal grids. However, it has been found that for non-orthogonal cases, there is a limit to the error reduction, which is concluded to be a result of errors in the Gauss-based gradient reconstruction method. Originality/value The convergence and error properties of derivative solutions obtained by forward mode automatic differentiation of finite volume-based codes have not been previously investigated.

Automatic Depth Prediction from 2D Videos Based on Non-parametric Learning and Bi-directional Depth Propagation

With progress in 3D video technology, 2D to 3D conversion has drawn great attention in recent years. Predicting perceptually reasonable depth information from traditional monocular videos is a challenging task in 2D to 3D conversion. For the purpose of generating convincing depth maps, an efficient depth estimation algorithm utilizing non-parametric learning and bi-directional depth propagation is proposed. First, global depth maps on key frames are generated on the basis of gradient samples and the gradient reconstruction method. Then, foreground objects are extracted and employed to refine global depth maps for producing more local depth details. Next, the depth information of key frames is propagated utilizing bi-directional motion compensation to recover the forward and backward depth information of non-key frames. In the end, a weighting fusion strategy is designed to integrate forward and backward depths for predicting the depth information of each non-key frame. The quality of estimated depth maps is assessed by leveraging objective and subjective quality evaluation criteria. The experimental results verify that the proposed depth prediction framework outperforms some depth estimation methods and is efficient at producing 3D views.

A novel stencil selection method for the gradient reconstruction on unstructured grid based on OpenFOAM

For the unstructured finite volume algorithms, the stencil utilized for gradient reconstruction is important for the accuracy, efficiency and robustness. In this paper, we present a novel stencil selection method for the gradient reconstruction of the cell-centered finite volume methodology. In the novel stencil selection method, two local directions are generated for each control volume, and the stencil is generated based on the local directions. In this way, the characteristics of the structured grid are able to be reflected on the unstructured grid. To obtain the suitable local directions, an advancing front technique (AFT) is utilized to guide their generation. Through AFT, the direction of the boundary normal vector, along which the flow fields change dramatically, is obtained and propagated from the boundaries to the internal regions. Finally, the novel stencil extends in two directions, one of which is approximately aligned with the direction of the boundary normal vector. Hence, the gradient reconstruction methods with the new stencil are able to use the information along the boundary normal. In order to verify the validity of the new stencil, we present a new gradient reconstruction method, the structured least squares (struLSQR) method, by applying the new stencil to the least squares (LSQR) method. The performance of struLSQR, and two existing gradient reconstruction methods in OpenFOAM is tested by adopting the 2D channel flow and the 2D circular cylinder flow on both the quadrilateral and the triangular grids. A laminar flow past a flat plate is also adopted to verify the applicability of struLSQR to the viscous flow. The numerical results illustrate that the new stencil is able to solve the blow-up problem of LSQR. It can also obtain a higher order of accuracy compared to extLSQR. Moreover, the convergence speed of struLSQR can be up to 1.64 ∼ 2.60x of that of extLSQR.

TOWARDS UNDERSTANDING THE INFLUENCE OF GRADIENT RECONSTRUCTION METHODS ON UNSTRUCTURED FLOWSIMULATIONS

In this paper, the formal order of accuracy of three commonly used gradient reconstruction methods is derived. The analysis showed that the Green–Gauss cell based (GGCB) method is intrinsically inconsistent, due to the leading error term that is independent of the mesh spacing. On the other hand, the Green–Gauss node based (GGNB) and the Least Squares cell based (LSCB) methods achieved a minimum of 1st order accuracy regardless of the mesh geometric properties. Implications of the former results were practically tested on four CFD applications to show that in three out of four cases, the LSCB method achieved the highest order of accuracy. In terms of the computational expenses, the GGNB method consumed 9–34% additional time when compared to the fastest converging method in each test case. Both the GGCB and the LSCB methods consumed nearly the same computational time to reach convergence.

Restoration of multispectral images by the gradient reconstruction method and estimation of the blur parameters on the basis of the multipurpose matching model

In this study, an original method for restoration of multispectral images by means of gradient reconstruction is proposed. The method uses a simple relationship between the gradient at the neighboring points in the distorted image and the gradient at distant points in the initial image. The result of restoration by the proposed algorithm excels the result obtained by the standard method based on the Wiener filtering. A new method for estimation of the parameters of the distorting motion-blur operator is also proposed. In this case, the blurred image is considered as a superposition of M shifted original images, and the autocorrelation convolution of the distorted image can be represented as a linear combination of M2 mutual convolutions of several identical shifted images. Thus, the autocorrelation function of the distorted image is a straight line passing through the center, and the direction of this line and its length coincides with the parameters of the distorting operator. As compared to the best present-day algorithms, the proposed method exhibits higher accuracy of parameter estimation. In addition, computation of parameters with the use of this method takes much less time than with the use of popular estimation algorithms based on the Radon transform.

Analysis of the Numerical Diffusion in Anisotropic Mediums: Benchmarks for Magnetic Field Aligned Meshes in Space Propulsion Simulations

This manuscript explores numerical errors in highly anisotropic diffusion problems. First, the paper addresses the use of regular structured meshes in numerical solutions versus meshes aligned with the preferential directions of the problem. Numerical diffusion in structured meshes is quantified by solving the classical anisotropic diffusion problem; the analysis is exemplified with the application to a numerical model of conducting fluids under magnetic confinement, where rates of transport in directions parallel and perpendicular to a magnetic field are quite different. Numerical diffusion errors in this problem promote the use of magnetic field aligned meshes (MFAM). The generation of this type of meshes presents some challenges; several meshing strategies are implemented and analyzed in order to provide insight into achieving acceptable mesh regularity. Second, Gradient Reconstruction methods for magnetically aligned meshes are addressed and numerical errors are compared for the structured and magnetically aligned meshes. It is concluded that using the latter provides a more correct and straightforward approach to solving problems where anisotropicity is present, especially, if the anisotropicity level is high or difficult to quantify. The conclusions of the study may be extrapolated to the study of anisotropic flows different from conducting fluids.

An accurate gradient and Hessian reconstruction method for cell‐centered finite volume discretizations on general unstructured grids

AbstractIn this paper, a novel reconstruction of the gradient and Hessian tensors on an arbitrary unstructured grid, developed for implementation in a cell‐centered finite volume framework, is presented. The reconstruction, based on the application of Gauss' theorem, provides a fully second‐order accurate estimate of the gradient, along with a first‐order estimate of the Hessian tensor. The reconstruction is implemented through the construction of coefficient matrices for the gradient components and independent components of the Hessian tensor, resulting in a linear system for the gradient and Hessian fields, which may be solved to an arbitrary precision by employing one of the many methods available for the efficient inversion of large sparse matrices. Numerical experiments are conducted to demonstrate the accuracy, robustness, and computational efficiency of the reconstruction by comparison with other common methods. Copyright © 2009 John Wiley & Sons, Ltd.

Accurate and Efficient Discretization of Navier-Stokes Equations on Mixed Grids

The discretization of Navier-Stokes equations on mixed unstructured grids is discussed. Because mixed grids consist of different cell types, the question arises as to how the discretization should treat these cell types in order to result in a stable and accurate solution method. This issue is addressed in relation to the discretization of inviscid and viscous fluxes. The discretization of the inviscid fluxes is carried out with both a centered and an upwind scheme. For the centered scheme, problems exist on mixed grids with the damping properties of the fourth-difference operator. The problems with the fourth-difference operator are investigated for two stencils by both truncation error and Fourier analysis. It is shown that one form exhibits superior damping for high frequencies, which is corroborated using numerical experiments. For the upwind scheme, a commonly used gradient-reconstruction method based on the Green-Gauss theorem does not show satisfactory behavior on mixed grids. Another gradient reconstruction method based on a Taylor series expansion previously derived in two dimensions is extended to three dimensions. This method is more accurate on mixed grids but requires more storage