Coarse Grid Research Articles

Domain sampling methods for an inverse boundary value problem of the heat equation

Abstract We investigate the reconstruction of an unknown cavity inside a heat conductor with only single boundary measurement, which is a typical non-destructive testing of defects in material science using thermography. Mathematically, it can be formulated as an inverse boundary value problem for the heat equation. Two domain sampling methods, i.e., the range test (RT) and no-response test (NRT), are developed to realize the reconstruction. We first justify the convergence of the RT, which is based on the analytical extension property of solutions to the heat equation. Then we prove the duality of the RT and NRT by looking at the relation between their indicator functions. Consequently, the unknown cavity can be reconstructed by using either the RT or NRT if the solution associated to the single measurement cannot be analytically extended across its boundary. Next, we design two efficient algorithms for the RT and NRT to reconstruct the unknown cavity numerically. Each algorithm consists of two steps. The first step is to determine an approximation of the unknown cavity by implementing the sampling methods in a coarse grid, while the second step is to improve the reconstruction by using a fine grid in the approximate domain. The numerical results suggest that both algorithms can generate some rough information on the cavity with only one boundary measurement. Finally, by using this information as an initial guess for Newton's method, we improve the accuracy of the mentioned reconstruction results.

High order recovery of geometric interfaces from cell-average data

We consider the problem of recovering multivariate characteristic functions u := χΩ from cell-average data on a coarse grid, motivated in particular by the accurate treatment of interfaces in finite volume schemes. While linear recovery methods are known to perform poorly, nonlinear strategies based on local reconstructions of the jump interface Γ := ∂Ω by geometrically simpler interfaces may offer significant improvements. We study two main families of local reconstruction schemes the first one based on nonlinear least-squares fitting, the second one based on the explicit computation of a polynomial-shaped curve fitting the data, which yields simpler numerical computations and high order geometric fitting. For each of them, we derive a general theoretical framework which allows us to control the recovery error by the error of best approximation up to a fixed multiplicative constant. Numerical tests in 2d illustrate the expected approximation order of these strategies. Several extensions are discussed, in particular the treatment of piecewise smooth interfaces with corners.

Multigrid Reduction‐In‐Time Convergence for Advection Problems: A Fourier Analysis Perspective

ABSTRACTA long‐standing issue in the parallel‐in‐time community is the poor convergence of standard iterative parallel‐in‐time methods for hyperbolic partial differential equations (PDEs), and for advection‐dominated PDEs more broadly. Here, a local Fourier analysis (LFA) convergence theory is derived for the two‐level variant of the iterative parallel‐in‐time method of multigrid reduction‐in‐time (MGRIT). This closed‐form theory allows for new insights into the poor convergence of MGRIT for advection‐dominated PDEs when using the standard approach of rediscretizing the fine‐grid problem on the coarse grid. Specifically, we show that this poor convergence arises, at least in part, from inadequate coarse‐grid correction of certain smooth Fourier modes known as characteristic components, which was previously identified as causing poor convergence of classical spatial multigrid on steady‐state advection‐dominated PDEs. We apply this convergence theory to show that, for certain semi‐Lagrangian discretizations of advection problems, MGRIT convergence using rediscretized coarse‐grid operators cannot be robust with respect to CFL number or coarsening factor. A consequence of this analysis is that techniques developed for improving convergence in the spatial multigrid context can be re‐purposed in the MGRIT context to develop more robust parallel‐in‐time solvers. This strategy has been used in recent work to great effect; here, we provide further theoretical evidence supporting the effectiveness of this approach.

Modeling the Impact of Wind Drag Coefficient on Wind-Driven Currents in Lake Taihu, China

The wind drag coefficient, Cd, has a great influence on the numerical results obtained from shallow lakes. To analyze the modeling impacts of Cd on wind-driven currents, a series of numerical simulations of Lake Taihu were conducted at three grid resolutions (800 m × 800 m, 400 m × 400 m, and 100 m × 100 m) using the empirical formulae of Flather (F76), Large and Pond (LP81), Large and Yeager (LY04), Andreas (A12), and Gao (G20). The G20 formula produced the optimum results of all the formulae for both the water level and velocity simulations; however, the grid resolution was found to have a significant influence on simulation in G20 cases. Thus, the G20 formula is only recommended when using a high-resolution grid to meet the accuracy requirements of analyzing wind-driven currents in the numerical modeling of Lake Taihu. A combination of the A12 formula and a coarse grid is preferred when taking computational efficiency into consideration.

Generating more constrained seeds for seismic horizon extraction

Enhancing accuracy remains the main challenge for the current horizon extraction algorithms. Adding more manually interpreted seeds helps to improve the accuracy of horizon extraction. We intend to generate more seeds prior to the 3D horizon extraction based on seeds that are manually interpreted on the coarse grid of the seismic survey. First, we compute four horizons for the same seismic sequence boundary using deep learning and constrained least-squares algorithms under the constraints of manually interpreted horizon seeds. Next, we generate more seeds by checking the consistency among those four horizons for each seismic trace, and this process is implemented in the same way as the tying loops of manual horizon interpretation. Finally, we extract horizons using constrained least-squares algorithms with the newly computed seeds. The applications of two real seismic surveys demonstrate that our method can generate horizons with higher accuracy when compared with the horizons computed with only manually interpreted seeds.

Changing windstorm characteristics over the US Northeast in a single model large ensemble

Abstract Extreme windstorms pose a significant hazard to infrastructure and public safety, particularly in the highly populated US Northeast (NE). However, the influence climate change and changing land use will have on these events remains unclear. A large ensemble generated using the Max-Planck Institute (MPI) Earth system model is used to generate projections of NE windstorms under different shared socioeconomic pathways (SSPs) and to attribute changes to projected land use land cover (LULC) change, externally forced changes and internal climate variability. To reduce the influence of coarse grid cell resolution and uncertainties in surface roughness lengths, windstorms are identified using simultaneous widespread exceedance of local 99th percentile 10 m wind speeds (U99). Projected declines in forest cover in the NE and the resulting reductions in surface roughness length under SSP3-7.0 lead to projections of large increases in U99 and derived windstorm intensity and scale. However, these projected changes in regional LULC under SSP3-7.0 are unprecedented in a historical context and may not be realistic. After corrections are applied to remove the influence of LULC on wind speeds, regionally averaged U99 exhibit declines for most of the single model initial-condition large ensemble (SMILE) members which are broadly proportional to the radiative forcing and global air temperature increase in the SSPs, with a median value of −0.15 ms−1 °C−1. While weak cyclones are projected to decline in frequency in the NE, intense cyclones and the resulting windstorms and indices of socioeconomic loss do not. Where present, significant trends in these loss indices are positive, and some MPI SMILE members generate future windstorms that are unprecedented in the historical period.

Reactive transport model of the long-term geochemical evolution in a HLW repository in granite at the disposal cell scale: Variants, sensitivities, and model simplifications

The assessment of the long-term performance of the engineered barrier systems of high-level radioactive waste (HLW) repositories requires the use of reactive transport models. Montenegro et al. (2023) presented a non-isothermal reactive transport model of the long-term geochemical evolution of a HLW disposal cell in a granitic host rock corresponding to a generic reference concept. The model accounted for the vitrified waste, the carbon-steel canister, the bentonite buffer and the reference granitic rock. Here we extend their model by considering model variants (V), sensitivity cases (SC) and model abstractions (MA). Variants V1, V2 and V3 consist of considering MX-80 bentonite instead of FEBEX bentonite (V1), a larger groundwater flux through the granite (V2) and the Czech reference crystalline rock as a host rock (V3). Cases SC1 and SC2 consider a decrease of the silica concentration threshold value in the glass dissolution rate (SC1) and an earlier canister failure time (SC2), respectively. Runs MA1 to MA4 consider smectite as an unreactive mineral phase (MA1), the porosity feedback effect on chemical and transport parameters (MA2), a time-varying corrosion rate (MA3), and a coarser finite element grid (MA4), respectively. Model results of V1 show a larger pH, a smaller precipitation of magnetite, siderite and greenalite and a slightly smaller dissolution of ISG and smectite than the base run of Montenegro et al. (2023). Model predictions are very sensitive to the increase in the groundwater flow through the granitic host rock (V2). However, predictions are not sensitive to the chemical composition of the granite porewater (V3). The decrease in the silica saturation threshold from 1·10−3 to 5·10−4 mol/L in SC1 leads to a significant decrease in glass dissolution. Glass dissolution after 50,000 years in SC2 (earlier canister failure) is much larger than that of the base run. Model results are not sensitive to considering smectite as an unreactive mineral phase (MA1). However, model results are very sensitive to the porosity feedback effect (MA2). A 60% volume fraction of Fe(s) remains uncorroded after 50,000 years when a variable corrosion rate is considered in MA3. In this case the precipitation of corrosion products is much smaller than that of the base run. The general patterns of the numerical results in MA4 (coarser grid) are similar to those of the base case.

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.

Hybrid TBETI domain decomposition for huge 2D scalar variational inequalities

AbstractThe unpreconditioned H‐TFETI‐DP (hybrid total finite element tearing and interconnecting dual‐primal) domain decomposition method introduced by Klawonn and Rheinbach turned out to be an effective solver for variational inequalities discretized by huge structured grids. The basic idea is to decompose the domain into non‐overlapping subdomains, interconnect some adjacent subdomains into clusters on a primal level, and enforce the continuity of the solution across both the subdomain and cluster interfaces by Lagrange multipliers. After eliminating the primal variables, we get a reasonably conditioned quadratic programming (QP) problem with bound and equality constraints. Here, we first reduce the continuous problem to the subdomains' boundaries, then discretize it using the boundary element method, and finally interconnect the subdomains by the averages of adjacent edges. The resulting QP problem in multipliers with a small coarse grid is solved by specialized QP algorithms with optimal complexity. The method can be considered as a three‐level multigrid with the coarse grids split between primal and dual variables. Numerical experiments illustrate the efficiency of the presented H‐TBETI‐DP (hybrid total boundary element tearing and interconnecting dual‐primal) method and nice spectral properties of the discretized Steklov–Poincaré operators as compared with their finite element counterparts.

Non-Intrusive Reduced Basis two-grid method for flow and transport problems in heterogeneous porous media

Due to its non-intrusive nature and ease of implementation, the Non-Intrusive Reduced Basis (NIRB) two-grid method has gained significant popularity in numerical computational fluid dynamics simulations. The efficiency of the NIRB method hinges on separating the procedure into offline and online stages. In the offline stage, a set of high-fidelity computations is performed to construct the reduced basis functions, which is time-consuming but is only executed once. In contrast, the online stage adapts a coarse-grid model to retrieve the expansion coefficients of the reduced basis functions. Thus it is much less costly than directly solving a high-fidelity model. However, coarse grids in heterogeneous porous media of flow models are often accompanied by upscaled hydraulic parameters (e.g. hydraulic conductivity), thus introducing upscaling errors. In this work, we introduce the two-scale idea to the existing NIRB two-grid method: when dealing with coarse-grid models, we also employ upscaled model parameters. Both the discretization and upscaling errors are compensated by the rectification post-processing. The numerical examples involve flow and heat transport problems in heterogeneous hydraulic conductivity fields, which are generated by self-affine random fields. Our research findings indicate that the modified NIRB method can effectively capture the large-scale features of numerical solutions, including pressure, velocity, and temperature. However, accurately retrieving velocity fields with small-scale features remains highly challenging.

Fractal properties of 4-point interpolatory subdivision schemes and wavelet scattering transform for signal classification

Wavelet scattering is a recent time-frequency transform that shares the convolutional architecture with convolutional neural networks, but it allows for a faster training and it often requires smaller training sets. It consists of a multistage non-linear transform that allows us to compute the deep spectrum of a signal by cascading convolution, non-linear operator and pooling at each stage, resulting a powerful tool for signal classification when embedded in machine learning architectures. One of the most delicate parameters in convolutional architectures is the temporal sampling that strongly affects the computational load as well as the classification rate. In this paper the role of sampling in the wavelet scattering transform is studied for signal classification purposes. In particular, the role of subdivision schemes in properly compensating the information lost when using sampling at each stage of the transform is investigated. Preliminary experimental results show that, starting from coarse grids, interpolatory subdivision schemes reproduce copies of the original scattering coefficients at a fixed full grid that still represent distinctive features for signal classes. In fact, thanks to the ability of the scheme in reproducing similar fractal properties of the transform through an efficient iterative refinement procedure, the reproduced coefficients enable to obtain classification rates similar to those provided by the native wavelet scattering transform. The relationships between the tension parameter of the scheme and the fractal dimension of its limit curve are also investigated.

Eddy viscosity enhanced temporal direct deconvolution model for temporal large-eddy simulation of turbulent channel flow

In this work, a novel eddy viscosity enhanced temporal direct deconvolution model (TDDM) is proposed for temporal large-eddy simulation (TLES) of turbulent channel flow at large filter widths. To improve the accuracy of the constant eddy viscosity (CEV) model, particularly in the near-wall region, a damping function is incorporated to refine its performance. Moreover, a spatial filtering strategy is introduced to reduce the aliasing errors associated with the computation of subfilter-scale (SFS) stress, thereby enhancing numerical stability. In the a posteriori study, the accuracy of the CEV model is assessed comprehensively by comparing the TLES results with corresponding temporally filtered direct numerical simulation data. The results demonstrate that the CEV-enhanced TDDM provides accurate predictions across various statistical properties of velocity, instantaneous flow structures, kinetic energy spectra, and SFS energy fluxes. The coefficient sensitivity analysis of the CEV model reveals that the model coefficient significantly influences low Reynolds number flows, while its impact on high Reynolds number flows is relatively small. TLES on coarse grids demonstrate that the CEV-enhanced TDDM exhibits strong robustness and accuracy at different grid resolutions. Additionally, the CEV-enhanced TDDM in high Reynolds number flows is stable and accurate at remarkably large filter widths.

An improved detached eddy simulation method for cavitation multiphase flow

The evolution of cavitation flow involves intense transient characteristics and complex multi-scale motions, significantly increasing the difficulty and computational cost of solving in separation zones. This study proposes an Improved Detached Eddy Simulation (IDES) method, based on a single-equation eddy-viscosity model, aimed at enhancing the resolution of small-scale cavitation flows. By incorporating the broad-spectrum von Karman scale concept, the method effectively improves the resolution capability for small-scale flows in separated zones. The performance of the IDES model was validated through three computational cases, and the main conclusions are as follows: The results of the triangular cylinder flows show that the IDES model demonstrates a resolution capability for large-scale turbulence comparable to the Large Eddy Simulation (LES) model. The orifice flow results indicate that activating the LES method with insufficient grid resolution is highly discouraged, whereas the IDES model performs more robustly under such conditions. For hydrofoil flows, the IDES model more accurately captures the temporal evolution of cavitation, avoiding the premature cavity shedding predicted by LES under coarse grid conditions. The decomposition results of the vorticity transport equation show that both the dilatation term and viscous term are significant contributors in the vorticity transport process, with their average contributions to vorticity reaching 49.25% and 39.99%, respectively. Vortices can be considered the primary controllers of cavitation dynamics, while the dilatation term and viscous term can be regarded as the main controllers of vorticity. Overall, the energy compensation mechanism of the IDES model, based on local flow information, gives it unique advantages in handling small-scale turbulence and cavitation phenomena, providing both high computational efficiency and accuracy.

Substantial Cold Bias During Wintertime Cold Extremes in the Southern Cascadia Region in Historical CMIP6 Simulations

AbstractGlobal climate models often simulate atmospheric conditions incorrectly due to their coarse grid resolution, flaws in their dynamics, and biases resulting from parameterization schemes. Here we document a bias in the magnitude and extent of minimum temperature extremes in the CMIP6 model ensemble, relative to ERA5. The bias is present in the southern Cascadia region (i.e., Pacific Northwestern United States and southwestern British Columbia, Canada, spanning from the coast to the Rocky Mountains), with some models showing a bias magnitude in excess of −10°C in the first percentile of daily winter minimum temperature. The sea level pressure pattern for these events is similar in CMIP6 models and ERA5, showing high anomalies in the Northeast Pacific that are indicative of an atmospheric blocking pattern and consequently more northerly flow. Though this atmospheric blocking pattern is typically concurrent with cold winter temperatures across much of North America, Rocky and Cascade mountain ranges prevent the cold air from reaching the southern Cascadia region as confirmed by the observations and reanalysis. Our results suggest that the bias in CMIP6 minimum temperatures is a result of unresolved topography in the Rockies and Cascade mountain ranges, such that the terrain does not adequately block cold air advection from reaching the southern Cascadia region.

An efficient approach for searching three-body periodic orbits passing through Eulerian configuration

A new efficient approach for searching three-body periodic equal-mass collisionless orbits passing through Eulerian configuration is presented. The approach is based on a symmetry property of the solutions at the half period. Depending on two previously established symmetry types on the shape sphere, each solution is presented by one or two distinct initial conditions (one or two points in the search domain). A numerical search based on Newton’s method on a relatively coarse search grid for solutions with relatively small scale-invariant periods T∗<70 is conducted. The linear systems at each Newton’s iteration are computed by high order high precision Taylor series method. The search produced 12,431 initial conditions (i.c.s) corresponding to 6333 distinct solutions. In addition to passing through the Eulerian configuration, 35 of the solutions are also free-fall ones. Although most of the found solutions are new, all linearly stable solutions among them (only 7) are old ones. Particular attention is paid to the details of the high precision computations and the analysis of accuracy. All i.c.s are given with 100 correct digits.

Behavior of a phase field model for wetting on structured surfaces

AbstractTechnical surfaces are generally not perfectly smooth, but usually exhibit roughness. Additionally, as micro production techniques continue to evolve, geometrical surface structures at the micro scale can be manufactured, which presents a challenge for wetting models that are commonly formulated for smooth surfaces. A phase field model for surface wetting, proposed by Diewald et al., serves as a basis for this investigation. On smooth surfaces, the width of the gas‐liquid interface can be widened for scale bridging purposes, as this allows for accurate computations on coarse grids. However, it was observed that this interface scaling impacts the results when the model is applied to rough surfaces, and therefore a free choice of the interface width is not always sensible. In this work, the ability of the model to deal with sinusoidally shaped surfaces is investigated. An Allen–Cahn evolution equation is used to determine static equilibrium configurations for droplets on such surfaces.

Physics-informed multi-grid neural operator: Theory and an application to porous flow simulation

We propose a physics-informed multi-grid finite neural operator as an efficient alternative for PDE solvers. The operator aims to map a random parameter function to the corresponding solution function of a linear PDE, both distributed on a fine grid (called original grid). The training of the operator involves using the physics-informed method to train a neural network on the original grid to approximate the PDE solution. The grid is then coarsened, and a new neural network is trained on the coarse grid to approximate the prediction error of the previous network. This process is iteratively repeated until the final grid's neural network achieves adequate accuracy. The combination of all neural networks from coarse to fine grids forms the multi-grid neural operator capable of mapping random parameters to solutions on the original grid. The above process involving fine-coarse-fine grid is called one V cycle (following the traditional multi-grid PDE literature), and multiple V cycles can be included into the operator to enhance its performance. Additionally, the operator integrates the idea of using coarse-grid predictions as initial guesses for fine-grid predictions. Validation is performed using three porous flow examples of varying dimensions. The operator's prediction accuracy improves with more V cycles included, and its predictions closely match results obtained from a numerical solver while executing 10 to 100 times faster.

Implicit-Explicit schemes for decoupling multicontinuum problems in porous media

In this work, we present an efficient way to decouple the multicontinuum problems. To construct decoupled schemes, we propose Implicit-Explicit time approximation in general form and study them for the fine-scale and coarse-scale space approximations. We use a finite-volume method for fine-scale approximation, and the nonlocal multicontinuum (NLMC) method is used to construct a coarse-scale approximation. The NLMC method is a multiscale method for developing an accurate and physically meaningful coarse-scale model based on defining the macroscale variables. The multiscale basis functions are constructed in local domains by solving constraint energy minimization problems and projecting the system to the coarse grid. The resulting basis functions have exponential decay properties and lead to the accurate approximation on a coarse grid. We construct a fully Implicit time approximation for semi-discrete systems arising after fine-scale and coarse-scale space approximations. We investigate the stability of the two and three-level schemes for fully Implicit and Implicit-Explicit time approximations schemes for multicontinuum problems in fractured porous media. We show that combining the decoupling technique with multiscale approximation leads to developing an accurate and efficient solver for multicontinuum problems.

Seismic travel-time tomography based on Ensemble Kalman Inversion

Abstract In this paper, we present a new seismic travel-time tomography approach that combines ensemble Kalman inversion (EKI) with Neural Networks (NNs) to facilitate the inference of complex underground velocity fields. Our methodology tackles the challenges of high-dimensional velocity models through an efficient neural network parameterization, enabling efficient training on coarse grids and accurate output on finer grids. This unique strategy, combined with a reduced-resolution forward solver, significantly enhances computational efficiency. Leveraging the robust capabilities of EKI, our method not only achieves rapid computations but also delivers informative uncertainty quantification for the estimated results. Through extensive numerical experiments, we demonstrate the exceptional accuracy and uncertainty quantification capabilities of our EKI-NNs approach. Even in the face of challenging geological scenarios, our method consistently generates valuable initial models for full wave inversion (FWI).

Coupled nodal integral-immersed boundary method (NI-IBM) for simulating convection-diffusion physics

The Nodal Integral Method (NIM), which is a coarse grid numerical method, works quite efficiently on orthogonal grids. However, there are many challenges to the implementation of this method in the complex shaped domains due to its intricate discretization process. In contrast, the Immersed Boundary Method (IBM) offers the advantage of implementing boundary conditions in an arbitrarily shaped domain using rectilinear, non-conformal body grids. This work presents the NIM, coupled with the IBM methodology, to solve convection-diffusion physics in complex shaped domains. IBM is utilized to enforce the boundary effect on the non-body-conformal Cartesian grid, while the NIM serves as a numerical technique to discretize the governing equations. The study describes the detailed implementation of coupled NI-IBM (Nodal Integral-Immersed Boundary Method) for convection-diffusion physics. The validation and verification of the NI-IBM methodology is done through the comparison of the numerical results with analytical solutions of the diffusion and convection-diffusion test cases. Results demonstrate that the scheme maintains its accuracy in the coarse grid structure as well as second-order accurate. Furthermore, the numerical results produced by NI-IBM are at par or more accurate with respect to the NIM based schemes for arbitrarily shaped domains and the FV-IBM (Finite Volume-Immersed Boundary Method) method for the convection-diffusion physics. The results also indicate that even with coarse grids and higher convection, the NI-IBM method preserves its nature to produce reasonably accurate numerical results.