Coarse Mesh Research Articles

Numerical research of functionally graded magneto-electro-elastic structures under thermal environment via the thermo-mechanical-electro-magnetic enriched finite element method

Abstract In this work, the functionally graded magneto-electro-elastic (FG-MEE) structures were used to study mechanical performance under a thermal environment using the thermo-mechanical-electro-magnetic enriched finite element method (TMEM-EFEM). Through enhancing the traditional solution space of the finite element method (FEM), additional degrees of freedom are introduced without modifying the mesh structure. Therefore, as a class of higher-order finite element methods, field variables with high gradients can be captured without a relatively finer mesh. The accuracy of TMEM-EFEM is improved by using interpolation covers. The feasibility of TMEM-EFEM is proven by numerical examples. The proposed TMEM-EFEM is proven to be accurate under the relatively coarse mesh. Through numerical examples, the high accuracy of TMEM-EFEM is demonstrated. Therefore, TMEM-EFEM can be a good alternative for the statics study of FG-MEE structures.

Improved Delayed Detached-Eddy Simulation of Turbulent Vortex Shedding in Inert Flow over a Triangular Bluff Body

The Improved Delayed Detached-Eddy Simulation (IDDES) is a modification of the original Detached-Eddy Simulation (DES) design to incorporate Wall Modeled Large Eddy Simulation (WMLES) capabilities and to extend the class of flows suitable for this methodology. For thin attached boundary layers, typically seen in external aerodynamic flows, the DES branch of the model is active, whereas with thick boundary layers, typically seen in internal flows and also wake flows, the WMLES branch is active, thus providing a numeric method suited to handling most flow cases automatically. The flow over a triangular bluff body is used to validate the suitability of the IDDES model and compare the results with experimental, DDES, and LES data. The IDDES model is found to be relatively accurate when compared with the experimental results, with recirculation length, streamwise velocity, and Reynolds stresses all showing good agreement with the experimental data. However, when compared with the DDES model, there is a ~4% overprediction of the recirculation length using the same mesh and numerical scheme. The code, with its extra complexity, is also ~3% slower to solve. The IDDES model has also been tested against different meshes, and the results show that even for a coarse mesh, there is still good agreement with the experimental data.

A three-dimensional curve interface reconstruction algorithm for two-phase fluid flow

The curve interface reconstruction algorithm has received significant attention in the context of two-dimensional two-phase flow. However, it remains absent in the three-dimensional scenario. This paper proposes a novel three-dimensional curve interface reconstruction (CIR) algorithm to address this challenge within structured meshes for the first time. Specifically, a portion of the spherical surface is employed to reconstruct the three-dimensional curve interface segment, with the radius and center coordinates determined by curvature and mass conservation constraints, respectively. To enhance curvature accuracy, a sphere-based iterative reconstruction (SIR) algorithm is proposed to calculate the reconstructed distance function (RDF) for the three-dimensional curve interface. Various tests involving the interface reconstruction of spherical, ellipsoidal, and cubic objects demonstrate that the coupled SIR and CIR (SIR-CIR, simplified by SCIR) method achieves higher accuracy than many popular methods, particularly with coarse mesh resolutions. Additionally, the SCIR method offers the advantages of straightforward implementation and coding for interface reconstruction in two-phase flow research. This advantage results in reduced computational costs compared to the coupled volume-of-fluid and level set (VOSET) method, which also utilizes an iterative method to solve RDF.

Application of a finite element method variant in nonconvex domains to parabolic problems

In this paper we address one of the major difficulties which is the nonconvex behavior of the domains while finding the solution of the problems. The part of the domain where the nonsmoothness appears is where the challenge arises and the way that area is handled using different numerical methods reveals the effectiveness of these techniques. Here in this article, we study the semilinear parabolic problem in nonconvex polygonal domain. For the approximation of the solution we use the Composite Finite Element (CFE) method, which is a classification of the Finite Element Method. CFE discusses the two-scale discretization — the larger mesh also known as the coarse mesh with the size H and the smaller mesh, also known as the fine mesh with the size h. It helps in reducing the dimension of the domain space of consideration. The fine scale grid is used to resolve the nonconvexity of the boundary whereas the coarse scale grid is comprised of larger grids at an appropriate distance from the boundary. The degrees of freedom depends on the coarse grid. This is the precedence of CFE over other methods, i.e., it eases the task of reducing the domain complexity. In this article, we consider two approaches — the semi discrete analysis where only space discretization is carried out, and the fully discrete analysis where both the time and space discretization is done using both backward Euler and Crank–Nicolson method. We study the error analysis in the L∞(L2)-norm and in the L∞(H1)-norm for the semidiscrete case whereas for the fully discrete case, we study the error analysis in the L∞(L2)-norm. Also, we check for the optimal results. For the CFE technique in the L∞(L2)-norm, we derive the convergence having optimal order in time and almost optimal order in space even if the domain is nonconvex. We consider a T-shaped domain and another star shaped domain to carry out the theoretical findings. Thereafter, numerical computations are implemented to validate the theoretical results.

Adaptative scheme for diffuse field response of multilayered structures

The computation of the diffuse field response of multilayered homogeneous structures leads to integrals which should be computed efficiently. This is all the more true as the frequency increases, since the integrand has localized peaks. The objective of this communication is to propose an adaptive integration scheme that aims to limit the number of resolutions of the physical problem and to maximize the reuse of previous evaluations in the course of the method. After discussing the usual integration methods, the proposed method will be presented. A first approximation of the integral is performed with two nested integration schemes with a coarse mesh of the initial integration domain. A local error indicator is derived and leads to a first partition. On each set of the latter, and depending on the local error estimated previously, an adequate numerical scheme is chosen. The error is controlled during all the process and it is iterated until convergence. The method is tested first on academic cases and then on industrial cases. The performance of the method is discussed, taking into account not only the number of evaluations of the physical function, but also the additional numerical cost generated by the adaptation process.

A novel post-processed finite element method and its convergence for partial differential equations

In this article, by combining high-order interpolation on coarse meshes and low-order finite element solutions on fine meshes, we propose a novel approach to improve the accuracy of the finite element method. The new method is in general suitable for most partial differential equations. For simplicity, we use the second-order elliptic problem as an example to show how the novel approach improves the accuracy of the finite element method. Numerical tests are also conducted to validate the main theoretical results.

Spatially Resolved Analysis of the Influence of Three-Dimensional Flow Field Structures on Loss Processes in PEMFCs

This study introduces a novel measurement device and technique to spatially resolve the influence of 3D flow field structures consisting of extended metal meshes on the performance distribution in PEMFCs. The newly designed segmented cell enables not only the measurement of the current distribution but also local electrochemical impedance spectra, subsequently analyzed by the distribution of relaxation times. The 3D flow field structures provide, despite of an increased HFR, an advantage at high current densities. The spatially resolved DRT analysis enabled the identification of polarization processes revealing that the gas diffusion process is greatly reduced compared to a conventional channel-rib design. In particular, the 3D coarse mesh shows promising advantages for application under high current densities. The findings from this study provide valuable insights into the local power and resistance distribution, enabling optimization of the flow field design and improvement of the performance of fuel cells with large active areas.

Spatially Resolved Analysis of the Influence of Three-Dimensional Flow Field Structures on Loss Processes in PEMFCs

This study introduces a novel measurement device and technique to spatially resolve the influence of 3D flow field structures consisting of extended metal meshes on the performance distribution in PEMFCs. The newly designed segmented cell enables not only the measurement of the current distribution but also local electrochemical impedance spectra, subsequently analyzed by the distribution of relaxation times. The 3D flow field structures provide, despite of an increased HFR, an advantage at high current densities. The spatially resolved DRT analysis enabled the identification of polarization processes revealing that the gas diffusion process is greatly reduced compared to a conventional channel-rib design. In particular, the 3D coarse mesh shows promising advantages for application under high current densities. The findings from this study provide valuable insights into the local power and resistance distribution, enabling optimization of the flow field design and improvement of the performance of fuel cells with large active areas.

Physically consistent immersed boundary method: A framework for predicting hydrodynamic forces on particles with coarse meshes

In the present paper, a fluid-particle coupling method is directly derived from the Navier-Stokes equations (NSE) by applying the concept of volume-filtering, yielding a physically consistent methodology to incorporate solid wall boundary conditions in the volume-filtered flow solution, thereby allowing to solve the governing flow equations on non-body conforming meshes. The resulting methodology possesses similarities with a continuous forcing immersed boundary method (IBM) and is, therefore, termed physically consistent IBM (PC-IBM). Based on the recent findings of Hausmann et al. [1], the closures arising in the volume-filtered NSE are closed by suitable models or even expressed analytically. The PC-IBM is fully compatible with the large eddy simulation framework, as the volume-filtered NSE converge to the filtered NSE away from solid boundaries. The potential of the PC-IBM is demonstrated by means of different particle-laden flow applications, including a dense packing of fixed particles and periodic settling of 500 particles. The new methodology turns out to be capable of accurately predicting the fluid forces on particles with relatively coarse mesh resolutions. For the flow around isolated spheres, a spatial resolution of six fluid mesh cells per diameter is sufficient to predict the drag force with less than 10% deviation from highly resolved simulation for the investigated particle Reynolds numbers ranging from 10 to 250.

Super-resolution on unstructured coastal wave computations with graph neural networks and polynomial regressions

Accurate high-resolution wave forecasts are essential for coastal communities, but local and even coastal coverage is often still missing due to the heavy computational load of modern state-of-the-art wave models. This study presents a machine learning super-resolution approach that drastically reduces the computational effort, while keeping errors negligible for the majority of forecasting applications. The method consists of first computing a wave forecast on a coarse mesh which is then converted to a forecast of finer resolution with the help of machine learning. To demonstrate the feasibility and the potential for practical applications of this approach, we present a case study of a 44-year hindcast along the French Basque coast over an unstructured mesh. We introduce two machine learning approaches, a graph neural network and a polynomial ridge regression and compare their performances in different sea states and spatial environments. Both models exhibit very small prediction errors for the significant wave heights, with Root Mean Square Errors (RMSEs) ranging from 0.3cm to 2cm, depending on the study region, while being up to 80 times faster than a direct computation of a numerical wave model at the corresponding spatial resolution. To the best of our knowledge, this is the first time that a super-resolution approach is extended to unstructured meshes in the field of coastal sciences.

Linear Sources and Anisotropic Scattering in the Random Ray Method

An enhanced implementation of the random ray method (TRRM), a stochastic adaptation of the method of characteristics, is presented. This implementation generalizes from a traditional flat source (FS) approximation to a linear source (LS) approximation, and from the standard isotropic scattering approximation to an anisotropic approximation, up to P3. On the two-dimensional C5G7 benchmark, LS alone enables a 1.4× and 1.7× enhancement in run time for parallel and serial executions, respectively, without compromising accuracy. Further testing on the three-dimensional C5G7 Rodded B benchmark demonstrates that LS enables significant axial coarsening and reduction in ray populations. This results in a 5.3× speedup in run time while still offering comparable accuracy. On the Babcock and Wilcox 1484 Core II benchmark, incorporating anisotropic scattering in TRRM up to P3 with a FS decreased keff errors (~110 pcm) and maximum and average pin power errors. LS with anisotropic scattering (LSPN), showed similarly improved keff errors while benefiting from a coarser mesh. The linear isotropic flat anisotropic (LIFA) method, relative to LSPN, offers further run time and memory savings of 4.1× and 1.6×, respectively, for P3, while maintaining comparable accuracy to anisotropic FS on more refined meshes.

Multi-Fidelity Learned Emulator for Waves and Porous Coastal Structures Interaction Modelling

A thorough understanding and treatment of wave-structure interaction (WSI) mechanics is essential for the rigorous engineering design of coastal protections. Conventional numerical analysis methods are accurate and generalize well, but are heavily dependent on the adopted mesh resolution and frequently incur substantial computational costs. To bypass these limitations, a meshless multi-fidelity residual neural network (MRNN) emulator is introduced in this study to infer the spatio-temporal responses arising from WSI. MRNN first employs a ‘low-fidelity’ simulator to learn basic WSI relationships by training on simulations obtained using a coarse numerical mesh. Subsequently, a ‘high-fidelity’ (HF) simulator is then employed to learn the mapping between numerical simulations performed using the coarse mesh and additional detailed fine meshes. The results indicate that MRNN is a highly robust emulator which requires significantly less HF data through its hierarchical framework compared to conventional single-fidelity data-driven strategies. By way of example, the MRNN emulator is applied to the cases of a porous dam break and breakwater. A broad spectrum of WSI responses, such as water through the porous dam medium, can be accurately captured using the MRNN emulator which is benchmarked against a conventional numerical modelling with a fine mesh. The computational efficiency of the MRNN is shown to be independent of the mesh resolution and complexity of the studied partial differential equations. It provides a generic and utilitarian emulator for any engineering problem of interest.

Accurate power spectrum estimation toward Nyquist limit

The power spectrum, as a statistic in Fourier space, is commonly numerically calculated using the fast Fourier transform method to efficiently reduce the computational costs. To alleviate the systematic bias known as aliasing due to the insufficient sampling, the interlacing technique was proposed. We derive the analytical form of the shot noise under the interlacing technique, which enables the exact separation of the Poisson shot noise from the signal in this case. Thanks to the accurate shot noise subtraction, we demonstrate an enhancement in the accuracy of power spectrum estimation. For the three dark matter samples with number density ranging from 10-2 Mpc-3 h3 to 10-4 Mpc-3 h3 analyzed on a coarse mesh of size 3 h-1 Mpc with Cloud-in-Cell mass assignment scheme, the systematic bias is well under control up to the Nyquist frequency. On the contrary, the bias induced by the estimator that ignore the aliasing on the shot noise typically exceeds the statistical uncertainty on the frequency beyond 0.85 times the Nyquist frequency. The good performance of our estimation allows an abatement in the computational cost by using low resolution and low order mass assignment scheme in the analysis for huge surveys and mocks.

Finite element mesh transition for local–global modeling of composite structures

This study presents an automatic mesh generation algorithm designed to address computational challenges in simulating small-scale defects within large composite structures. The algorithm seamlessly transitions from a coarse mesh, corresponding to the global structure, to a highly refined mesh in targeted local regions of interest. The transition element number and shape can be adjusted by the specified parameters. Tailored to complement this method for non-homogeneous composite models, which include multiple materials such as cohesive layers representing interlayer properties, a volume fraction calculator is integrated to automatically assign the mixture material property in each transition element. Entire processes are fully automated using a MATLAB script, eliminating the need to open the FEA software interface. The validation studies of the reconstructed two-dimensional models, assembled with the wrinkle-defect model, demonstrate their feasibility. The performance of the model is examined in terms of strain and displacement at the connecting boundaries, load–displacement curve, and interlayer failure prediction. The mesh transition model achieves agreeable results compared to a fully fine mesh model, and a 92% reduction in computational time in stress analysis, showing the efficiency of the mesh transition for local–global modeling of composite structures.

Two-level Arrow–Hurwicz iteration methods for the steady bio-convection flows

To avoid solving a saddle-point system, in this paper, we study two-level Arrow–Hurwicz finite element methods for the steady bio-convection flows problem which is coupled by the steady Navier–Stokes equations and the steady advection–diffusion equation. Using the mini element to approximate the velocity, pressure, and the piecewise linear element to approximate the concentration, we use the linearized Arrow–Hurwicz iteration scheme to obtain the coarse mesh solution and use three different one-step Stokes/Oseen/Newton linearized scheme to obtain the fine mesh solution. The optimal error estimate O(h+H2+χm/2) of the velocity and concentration in the H1-norm and the pressure in the L2-norm are derived, where h and H are fine and coarse mesh sizes, respectively, and χm/2 denotes the iteration error with 0<χ<1. Numerical results are given to support the theoretical analysis and confirm the efficiency of the proposed two-level methods.

Approximation of acoustic black holes with finite element mixed formulations and artificial neural network correction terms

Wave propagation in elastodynamic problems in solids often requires fine computational meshes. In this work we propose to combine stabilized finite element methods (FEM) with an artificial neural network (ANN) correction term to solve such problems on coarse meshes. Irreducible and mixed velocity–stress formulations for the linear elasticity problem in the frequency domain are first presented and discretized using a variational multiscale FEM. A non-linear ANN correction term is then designed to be added to the FEM algebraic matrix system and produce accurate solutions when solving elastodynamics on coarse meshes. As a case study we consider acoustic black holes (ABHs) on structural elements with high aspect ratios such as beams and plates. ABHs are traps for flexural waves based on reducing the structural thickness according to a power-law profile at the end of a beam, or within a two-dimensional circular indentation in a plate. For the ABH to function properly, the thickness at the termination/center must be very small, which demands very fine computational meshes. The proposed strategy combining the stabilized FEM with the ANN correction allows us to accurately simulate the response of ABHs on coarse meshes for values of the ABH order and residual thickness outside the training test, as well as for different excitation frequencies.

Progressive failure analysis of laminar composites under compression using smeared crack-band damage model and full layerwise theory

This paper presents an original finite element (FE) model that integrates the smeared crack band (SCB) approach and full layerwise plate theory (FLWT). The model enhances the computational efficiency of progressive failure analysis (PFA) of laminar composites in compression, by utilizing the layerwise approach which reduces a 3D model to a 2D one. The model distributes damage throughout the FE domain, with fracture mechanisms represented by material stiffness degradation controlled by damage variables (based on equivalent strains specifically defined for each failure mode). Mesh dependency issues are addressed by scaling fracture energy using a characteristic element length, and failure initiation and modes are determined using the 3D Hashin failure criterion.Accurately describing lamina response in fiber direction under compression requires linear-brittle softening with a stress plateau. The study showed that a model considering 30 % of residual stress accurately predicts maximum stress regardless of mesh refinement, demonstrating results’ minor dependence from the selected element size.The model accuracy has been confirmed by comparing the obtained results against experimental and benchmark data from the literature. The size effect study demonstrated a decrease in maximum stress of the open-hole laminates in compression with increasing specimen in-plane size. This trend is consistent with experimental and reference numerical observations, confirming the model accuracy and applicability even for relatively coarse meshes. Therefore, computational efficiency is improved, with preserved accuracy of conventional solid finite element models.

Coarse mesh finite difference acceleration of the random ray method

This paper presents the development and evaluation of the Coarse Mesh Finite Difference (CMFD) acceleration method to The Random Ray Method (TRRM). We demonstrate its effectiveness in accelerating the convergence of the fission sources and reducing the real statistical error in neutron transport calculations. TRRM treats the angular variable of the neutron flux stochastically and performs batch sampling of characteristic rays to transform the Method of Characteristics (MOC) into a stochastic process, that is capable of making significant strides in memory efficiency and computational performance for some problems. Despite its advantages, TRRM exhibits a potential challenge with a large number of inactive cycles and inherent inter-cycle correlation much like the Monte Carlo method. To address this, the CMFD acceleration method is explored and demonstrated to dramatically reduce the number of required inactive cycles and diminish inter-cycle correlation. Results from the numerical analysis of a 2D C5G7 core problem indicate that the application of CMFD leads to enhanced convergence, with the integration of a CMFD acceleration step every cycle offering the most substantial reduction in statistical noise and error. The study reveals that applying CMFD with every cycle effectively resolves the issue of needing inactive cycles and significantly lowers the inter-cycle correlation, thereby providing a more accurate estimation of standard deviation for pin power distributions. We conclude that using CMFD not only minimizes the number of inactive cycles of TRRM – much like normal Monte Carlo transport – but also lowers real statistical error effectively. For a targeted maximum standard deviation of 0.1% in the pin power, the addition of CMFD can decrease the number of necessary active cycles by 41% compared to standard TRRM, as demonstrated by the 2D C5G7 benchmark analysis.

Assessment of the CTF subchannel code for modeling a large-break loss-of-coolant accident reflood transient

With increased industry interest in extending reactor operating cycles, the neams! (neams!) program has been investigating the behavior of high-burnup fuel during design basis accidents such as the lbloca! (lbloca!) with consideration for risk of ffrd! (ffrd!). As part of that activity, the neams! subchannel th! (th!) code, CTF, is being used for modeling of lbloca! and to determine the impact of subchannel resolution on results. Although CTF includes a wide range of models for lbloca! conditions, the code has not been used for this application while maintained at ornl! (ornl!) until now. Therefore, in this work, a preliminary assessment of several of these models was performed using openly available reflood experimental data from the feba! (feba!) tests. One coarse mesh and one fine mesh model were set up in CTF for high and low flooding rate tests performed in the unblocked feba! facility. A coarse TRACE model was set up to be as consistent as possible with the coarse CTF model to allow for code-to-code benchmarking. The assessment shows a tendency of the codes to over-predict pct! (pct!) near the top of the bundle and to quench early. Advanced spacer grid models were shown to improve upper bundle predictions in CTF. The resolved CTF model over-predicted pct! by a larger degree in the center channels in the low-flooding rate test, and it is believed that the radiative heat transfer model, which was not used in this study, may be needed to correct this over-prediction. Finally, this work demonstrates the importance of the droplet model in determining quench time and vapor temperature and pct! prediction, which necessitates a more in-depth validation of these models in the future.

Two-Level method for blind image deblurring problems

Blind image deblurring (BID) is a procedure for reducing blur and noise in a deteriorated image. In this process, the estimation of the original image, as well as the blurring kernel of the degraded image, is done without or with only partial information about the imaging system and degradation. This is an inverse problem (ill-posed) that corresponds to the direct problem of deblurring. To overcome the ill-posedness of BID and attain useful solutions, the regularization models based on mean curvature (MC) are utilized. The discretization of MC-based models often leads to a large ill-conditioned nonlinear system of equations, which is computationally expensive. Moreover, the existence of MC functionals in the governing equations of the BID model complicates the calculation of the nonlinear system. To overcome these problems, in this paper, we propose the Two-Level blind image deblurring method (TLBID). First, on the coarse-grid, we solve a small nonlinear system (with a small number of pixels) for a mesh size of H, followed by solving a large linear system of equations on the finer grid (with a large number of pixels) of size h (h≤H). On the coarse mesh, we solve the BID problem utilizing the computationally expensive MC regularization functional. After this, we interpolate the results to the finer mesh. On the finer mesh, we solve the BID problem with less computationally expensive regularization functionals such as total variation (TV) or Tikhonov. This approach produces an approximate solution of the BID equations with high accuracy, which is cost-effective. The TLBID algorithm is implemented with MATLAB, and verification and validation are carried out using benchmark problems and medical digital images.