Irregular Grid Research Articles

On the Issue of Constructing Spatial Isolines for Irregular Monitoring Networks

The article examines spatial isolines construction based on data on an irregular grid. In particular, a possible solution to the problem of contour breaks is explored. Measurements of the magnetic field parameters taken from variation stations are used as data. The paper presents a description of the algorithm for constructing isolines and describes the software system that implements this algorithm. The results of system performance testing are given in the last chapter.

Robust unsupervised 5D seismic data reconstruction on regular and irregular grids

Seismic data reconstruction has become a central focus in seismic data processing, addressing challenges posed by sparse sampling due to physical and budgetary constraints. The advent of 5D acquisition methodologies marks a significant advancement in the quality and completeness of seismic data sets. Most traditional 5D reconstruction methods commonly use the fast Fourier transform (FFT), requiring regular grids and preliminary 4D binning before 5D interpolation. Discrete Fourier transform and nonequidistant FFT can honor the original irregular coordinates. However, when using exact locations, these methods become computationally expensive. We introduce an unsupervised deep-learning methodology to learn a continuous function across the sampling points in seismic data, facilitating reconstruction on regular and irregular grids. The network comprises a multilayer perceptron with linear layers and element-wise periodic activation functions. It excels at mapping the input coordinates to the corresponding seismic data amplitudes without relying on external training sets. The network’s intrinsic low-frequency bias is crucial in prioritizing acquiring self-similar features over high-frequency and incoherent ones during training. This characteristic mitigates incoherent noise in seismic data, such as random and erratic components. To assess the robustness of the unsupervised reconstruction technique, we conduct comprehensive evaluations using synthetic data examples sampled regularly and irregularly, as well as field-data examples with and without binning. The findings demonstrate the efficacy of our deep-learning framework in achieving resilient and accurate seismic data reconstruction across diverse sampling scenarios.

MultIHeaTS: A Fast and Stable Thermal Solver for Multilayered Planetary Surfaces

A fully implicit scheme is proposed for solving the heat equation in 1D heterogeneous media, available as a computationally efficient open-source Python code. The algorithm uses finite differences on an irregular grid and is unconditionally stable due to the implicit formulation. The thermal solver is validated against a stiff analytical solution, demonstrating its robustness in handling stiff initial conditions. Its general applicability for heterogeneous cases is demonstrated through its use in a planetary surface scenario with nonlinear boundary conditions induced by blackbody thermal emission. MultIHeaTS's advantageous stability allows for computation times up to 100 times faster than Spencer’s explicit solver, making it ideal for simulating processes on large timescales. This solver is used to compare the thermal signatures of homogeneous and bilayer profiles on Europa. Results show that homogeneous materials cannot reproduce the thermal signature observed in bilayer profiles, emphasizing the need for multilayer solvers. In order to optimize the scientific return of a space mission, we propose a strategy made of three local time observations that is enough to identify bilayer media, for instance, for the next missions to the Jovian system. A second application of the solver is the estimation of the temperature profile of Europa’s near surface (first 10s m) throughout a 1 million yr simulation with varying orbital parameters. The probability distribution of temperature through depth is obtained. Among its various applications, MultIHeaTS serves as the core thermal solver in a multiphysics simulation model detailed in the companion article by C. Mergny & F. Schmidt.

An integrated dataset of near-surface Eulerian fields and Lagrangian trajectories from an ocean model

A dataset consisting of numerically simulated oceanic velocities and sea surface height changes, provided conjointly from Eulerian and Lagrangian points of view, is made available as cloud-optimized archives on a cloud storage platform for unrestricted access. The Eulerian component of the dataset comprises oceanic velocity components at 0 m and 15 m depth, as well as total and steric sea surface height changes, obtained at hourly time steps for one year, with an approximate horizontal resolution of 1/25 degree on an irregular global geographical spatial grid, from the HYbrid Coordinate Ocean Model. The Lagrangian component of the dataset comprises the trajectories of particles advected in the Eulerian velocity field of the model. The particles were advected forward and backward for 30 days from a regular 1/4 degree grid in order to achieve 60-day long trajectories at 0 m and 15 m depths, with start times separated by 30 days, in 11 releases. This integrated dataset may help to link Eulerian and Lagrangian observational perspectives.

DRRGlobal: Uncovering the weak phases from global seismograms using the damped rank-reduction method

Some target seismic signals in the earthquake data can be very weak compared with interfering phases, and are thus difficult to detect, which further hinders the effective usage of these weak phases for subsequent high-resolution imaging of earth interiors. The strong ambient noise makes this situation even more troublesome since the weak signals can be mostly buried in the noise. Here, we present an open-source package for uncovering the weak phases from global seismograms. We adopt a two-step scheme to reconstruct and denoise array data. The first step is weighted average interpolation which puts the data into irregular grids. The second step adopts the weighted projection-onto-convex sets based on damped rank-reduction to further interpolate and denoise for the binned data. Taking the complexity of the weak signal into consideration, we adopt the automatic strategy to select an appropriate rank in different localized windows. We conduct several synthetic tests to carefully investigate the performance regarding effectiveness, robustness, and efficiency, and compare the algorithm with the frequency–wavenumber-domain projection onto convex sets method that is already used in the global seismology literature. Finally, the proposed framework is validated via a recorded array data set of the 1995 May 5 Philippines earthquake.

Stable neural network-based traveltime tomography using hard-constrained measurements

Traveltime tomography, or travel time inversion, has been one of the primary seismological tools for decades and has been used to understand Earth's properties and dynamic processes. An accurate, preferably flexible, eikonal solver to compute the travel time field is at the heart of the inversion process. However, most conventional eikonal solvers suffer from first-order convergence errors and difficulties dealing with irregular computational grids. Physics-informed neural networks (PINNs) have been introduced to tackle these problems and have successfully addressed those challenges. Nevertheless, these approaches still suffer from slow convergence and unstable training dynamics due to the multi-term nature of the loss function. To improve this, we propose a new formulation for the isotropic eikonal equation, which imposes boundary conditions as hard constraints. We employ the theory of functional connections to the traveltime tomography problem, which allows for the use of a single loss term for training the PINN model. Its efficiency, stability, and flexibility in tackling various cases, including topography-dependent and 3D models, are attested through rigorous numerical tests, thus providing an efficient and stable PINN-based traveltime tomography.

On the impact of spatial covariance matrix ordering on tile low‐rank estimation of Matérn parameters

AbstractSpatial statistical modeling involves processing an symmetric positive definite covariance matrix, where denotes the number of locations. However, when is large, processing this covariance matrix using traditional methods becomes prohibitive. Thus, coupling parallel processing with approximation can be an elegant solution by relying on parallel solvers that deal with the matrix as a set of small tiles instead of the full structure. The approximation can also be performed at the tile level for better compression and faster execution. The tile low‐rank (TLR) approximation has recently been used to compress the covariance matrix, which mainly relies on ordering the matrix elements, which can impact the compression quality and the efficiency of the underlying solvers. This work investigates the accuracy and performance of location‐based ordering algorithms. We highlight the pros and cons of each ordering algorithm and give practitioners hints on carefully choosing the ordering algorithm for TLR approximation. We assess the quality of the compression and the accuracy of the statistical parameter estimates of the Matérn covariance function using TLR approximation under various ordering algorithms and settings of correlations through simulations on irregular grids. Our conclusions are supported by an application to daily soil moisture data in the Mississippi Basin area.

3D large-scale forward modeling of gravitational fields using triangular spherical prisms with polynomial densities in depth

To take the sphericity of the Earth into account, tesseroids are often utilized as grid elements in large-scale gravitational forward modeling. However, such elements in a latitude–longitude mesh suffer from degenerating into poorly shaped triangles near poles. Moreover, tesseroids have limited flexibility in describing laterally variable density distributions with irregular boundaries and also face difficulties in achieving completely equivalent division over a spherical surface that may be desired in a gravity inversion. We develop a new method based on triangular spherical prisms (TSPs) for 3D gravitational modeling in spherical coordinates. A TSP is defined by two spherical surfaces of triangular shape, with one of which being the radial projection of the other. Due to the spherical triangular shapes of the upper and lower surfaces, TSPs enjoy more advantages over tesseroids in describing mass density with different lateral resolutions. In addition, such an element also allows subdivisions with nearly equal weights in spherical coordinates. To calculate the gravitational effects of a TSP, we assume the density in each element to be polynomial along radial direction so as to accommodate a complex density environment. Then, we solve the Newton’s volume integral using a mixed Gaussian quadrature method, in which the surface integral over the spherical triangle is calculated using a triangle-based Gaussian quadrature rule via a radial projection that transforms the spherical triangles into linear ones. A 2D adaptive discretization strategy and an extension technique are also combined to improve the accuracy at observation points near the mass sources. The numerical experiments based on spherical shell models show that the proposed method achieves good accuracy from near surface to a satellite height in the case of TSPs with various dimensions and density variations. In comparison with the classical tesseroid-based method, the proposed algorithm enjoys better accuracy and much higher flexibility for density models with laterally irregular shapes. It shows that to achieve the same accuracy, the number of elements required by the proposed method is much less than that of the tesseroid-based method, which substantially speeds up the calculation by more than 2 orders. The application to the tessellated LITHO1.0 model further demonstrates its capability and practicability in realistic situations. The new method offers an attractive tool for gravity forward and inverse problems where the irregular grids are involved.

Influence of geological uncertainty on longitudinal deformation of tunnel based on improved coupled Markov chain

The complex geological condition encountered during excavation is the key challenge for tunneling projects. Therefore, tunnel design and construction require a comprehensive understanding of the subsurface geology. This study aims to reveal the effect of geological uncertainty on tunnel longitudinal deformation based on a reconstruction geological model using the improved coupled Markov chain method. The improved coupled Markov chain method makes the predicted results more suitable for the possible stratigraphic distribution by optimizing the transition probability matrix, and breaks through the limitations of non-coaxial predicted cell updating in traditional methods. A mapping method was proposed to establish a matching relationship between the irregular grids of the geological model and the numerical tunnel model. Then, a long tunnel in Shenzhen was selected as a case study to demonstrate the effectiveness of the proposed framework. The correlation coefficient was used to verify the relationship between tunnel deformation and the surrounding stratum with different tunnel cover depths. The geological uncertainty of various strata was quantitatively measured based on the proposed geological uncertainty index. The three-dimensional strata surrounding the tunnel were reconstructed based on exceedance probabilities for adverse geological conditions. Quantitative analysis was conducted on the tunnel vertical convergence during the excavation process.

LBM modeling of three-dimensional mixed convection in CZ crystal growth on curvilinear coordinates system

A three-dimensional (3D) thermal lattice Boltzmann model of crystal growth is presented to study the magneto-hydrodynamics (MHD) mixed convection in the melt of Czochralski silicon single crystal growth in this paper. In order to solve the problem of lattice Boltzmann method’s difficulty in solving the flow of 3D cylindrical cavity, the 3D physical model of crystal growth is transformed from the Cartesian coordinate system to the Curvilinear coordinate system. And the irregular grids in the Cartesian coordinate system are transformed into regular grids in the Curvilinear coordinate system to facilitate the establishment of the Lattice Boltzmann model. Under the above operations, two coupled distribution functions are established to solve the governing equations. Two D3Q19 models are adopted to solve the melt velocity and temperature. And the 3D lattice Boltzmann modeling of crystal growth with cylinder boundary is realized by interpolation in Curvilinear coordinate system. The comparison with the experiment and other references proves that the proposed model and boundary treatment scheme is correct and it proposes a new way for studying the 3D crystal growth or other similar 3D cylindrical cavity simulations. Furthermore, the unstable melt flow for two cases are also simulated in this paper and the critical Reynolds number are also obtained. Results show that the introduction of the transverse magnetic field enhances the stability of the crystal growth system and increases the adjustable range of crystal rotation parameter. The obtained conclusion has an important reference for adjusting the process parameters during the CZ crystal growth.

NEOPARD-X: Multidimensional highly efficient and scalable modular framework for modelling of electrolysers and fuel cells

A novel framework for modelling of electrolysers and fuel cells is presented. The framework features a modular and flexible structure, which enables wide applicability for a variety of fuel cell and electrolyser technologies. The framework is multi-threaded for shared memory architectures. The generic approach for discretisation allows efficient distribution of tasks between processing cores. The physical model is clearly separated from discretisation and the linear solver. This simplifies the model development and provides high flexibility with respect to the discretisation methods, linear solvers, as well as the dimensionality of the application. Irregular grids and complex geometries can be realised through generic mapping of coordinate systems. Single cell models can be easily scaled to short stacks. The framework features are demonstrated with realistic PEMFC and SOEC models. Whilst the PEMFC example focuses on the differences between different model dimensions and different boundary conditions, the SOEC example demonstrates the extension of a transient single cell model to a transient short stack model. The performance scalability with the number of CPUs is demonstrated.

Workflow of a Preclinical Robotic Magnetic Resonance Imaging-guided Focused Ultrasound Body System

Abstract Background: Establishing an efficient workflow is crucial for the success of magnetic resonance-guided focused ultrasound (MRgFUS) procedures. The current study provides a comprehensive description of the workflow of a customized MRgFUS robotic body device for preclinical use and accompanied software through experiments in excised porcine tissue. Methods: The employed system comprises a single-element spherically focused transducer of 2.6 MHz that can be moved along four PC-controlled axes. A detailed description of essential software functionalities and its integration with a 3T Siemens magnetic resonance imaging (MRI) scanner through Access-I for interactive remote control of the scanner and real-time access to imaging data is provided. Following treatment planning on preoperative MR images, porcine tissue samples were sonicated in rectangular and irregular grid patterns with varying ultrasonic parameters and spatial step under software-based monitoring. Results: MRgFUS ablations of ex vivo porcine tissue were successfully performed utilizing a multimodal monitoring approach combining MRI-based temperature, thermal dose, and necrotic area mapping, thus demonstrating an efficient procedural workflow. The simulated necrotic regions were in excellent agreement with the actual lesions revealed upon tissue dissection and highly consistent with the planned sonication patterns. The software’s ability to accurately identify regions where necrosis did not occur and indicate to the user the specific points to be re-sonicated was demonstrated. Conclusion: Overall, the study highlights critical aspects in accurately planning and executing preclinical MRgFUS protocols within an efficient workflow. The provided data could serve as the basis for other researchers in the field.

Accuracy and sensitivity of soil erosion estimation using 137Cs technology: A statistical perspective

Random spatial variation of the 137Cs inventory is the principal contributor to uncertainty in soil erosion estimation using 137Cs technology. A statistically sound sampling design is imperative for obtaining reliable soil erosion estimations. The objectives of this study are to: 1) characterize the effect of sample size on the estimates of mean inventories; 2) evaluate the sensitivity of the estimated soil redistribution to sample size using a new 137Cs model; 3) compare erosion detection limits among five conversion models; and 4) assess three spatial interpolation methods. Two small watersheds were sampled in an irregular grid design, and 30 samples were taken on a reference site. A moving window scheme was used to compute the 7-point and 11-point means of 137Cs inventories. The Welch’s modified t-test was used to test the mean estimates and to compute the detection limits for the replicated samples. Five models were selected to convert 137Cs inventories to soil redistribution rates. Since most single samples without replication yielded insignificant test results, the 137Cs technique cannot be used to estimate soil erosion at a single point due to random spatial variation. With 30 reference samples, 7 replicate samples on the sampling site provided reliable estimations of 137Cs inventories and soil redistribution. The spatial patterns of the estimated 137Cs inventories and soil redistribution became more regular and systematic as sample size increased, which agreed increasingly well with topography and surface hydrology. Replicate samples can be taken from each landform element or slope position where the erosion rate is expected to be uniform. Alternatively, a systematic grid sampling scheme can be used, and the nearest neighbors can be treated as replicates to calculate moving averages. The Type I and II errors with replicated samples showed that all sampling positions should be included in calculating watershed or area mean erosion rates even though the Type I errors indicated insignificant redistribution rates for some samples. Due to the lack of spatial autocorrelation, spline and inverse distance weighting methods performed better than kriging for interpolation. Detection limits and magnitudes of soil redistribution varied substantially with conversion models, depending on their sensitivities to changes in 137Cs inventories.

Three-Dimensional Modeling and Inversion of Gravity Data Based on Topography: Urals Case Study

In this paper, the derivation of a concise closed form for the gravitational field of a polyhedron is presented. This formula forms the basis of the algorithm for calculating the gravitational field of an arbitrary shape body with high accuracy. Based on this algorithm, a method for gravity data inversion (creating density models of the Earth’s crust) has been developed. The algorithm can accept either regular or irregular polyhedron discretization for density model creation. The models are approximated with dense irregular grids, elements of which are polyhedrons. When performing gravity data inversion, we face three problems: topography with large amplitude, the sphericity of the planet, and a long computation time because of the large amount of data. In our previous works, we have already considered those problems separately but without explaining the details of the computation of the closed-form solution for a polyhedron. In this paper, we present for the first time a performance-effective numerical method for the inversion of gravity data based on topography. The method is based on closed-form expression for the gravity field of a spherical density model of the Earth’s crust with the upper topography layer, and provides great accuracy and speed of calculation. There are no restrictions on the model’s geometry or gravity data grid. As a case study, a spherical density model of the Earth’s crust of the Urals is created.

A residual graph convolutional network for setting initial flow field in computational fluid dynamics simulations

The computational cost of computational fluid dynamics (CFD) simulation is relatively high due to its computational complexity. To reduce the computing time required by CFD, researchers have proposed various methods, including efficient time advancement methods, correction methods for discrete control equations, multigrid methods, reasonable initial field setting methods, and parallel methods. Among these methods, the initial field setting method can provide significant performance improvements, but there is little work on it. Existing CFD industrial software typically uses inflow conditions for the initial flow field or applies empirical methods, which can cause instability in the CFD calculation process and make convergence difficult. With the rapid development of deep learning, researchers are increasingly attempting to replace CFD simulations with deep neural networks and have achieved significant performance improvements. However, these methods still face some challenges. First, they can only predict the computational flow field on regular grids. They cannot directly make predictions for irregular grids such as multi-block grids and unstructured grids, so the final flow field can only be obtained through interpolation and similar methods. Second, although these methods have been claimed to provide high accuracy, there is still a significant gap in performance with CFD and they cannot yet be applied to real scenarios. To address these issues, we propose a Residual Graph Convolutional Network for Initial Flow Field Setting (RGCN-IFS) in CFD simulations. This method converts the grid into a graph structure and uses an improved graph neural network to predict the flow field. In this way, we can predict the flow field on any type of grid. More importantly, this method does not directly replace CFD simulations, but it rather serves in an auxiliary role, providing appropriate initial flow fields for the CFD calculations, improving the convergence efficiency while ensuring calculation accuracy, and directly bridging the accuracy gap between intelligent surrogate models and CFD simulations.

Case Study of Contaminant Transport Using Lagrangian Particle Tracking Model in a Macro-Tidal Estuary

This study presents a comprehensive analysis of contaminant transport in estuarine environments, focusing on the impact of tidal creeks and flats. The research employs advanced hydrodynamic models with irregular grid systems and conducts a detailed residual current analysis to explore how these physical features influence the movement and dispersion of contaminants. The methodology involves simulating residual currents and Lagrangian particle trajectories in both ‘Creek’ and ‘No Creek’ cases, under varying tidal conditions. The results indicate that tidal creeks significantly affect particle retention and transport, with notable differences observed in the dispersion patterns between the two scenarios. The ‘Creek’ case demonstrates enhanced material retention along the creek pathways, while the ‘No Creek’ case shows broader dispersion, potentially leading to increased sedimentation in open sea areas. The discussion highlights the implications of these findings for sediment dynamics, contaminant transport, and estuarine ecology, emphasizing the role of tidal creeks in modulating flow and material transport. The research underlines the necessity of incorporating detailed environmental features in estuarine models for accurate contaminant transport prediction and effective estuarine management. This study contributes to a deeper understanding of estuarine hydrodynamics and offers valuable insights for environmental policy and management in coastal regions.

A CASE STUDY ON HIGH POWER COMPENSATOR OF THE POWER GRID IRREGULARITIES FOR INDUSTRIAL APPLIANCES

The paper presents a case study on development of the high power compensator of the power grid irregularities. The compensator is dedicated for sensitive industrial installations protection against dips, sags and short duration interruptions in public power network. The energy for compensation will be stored in bank of high voltage stacked supercapacitors. The development of the new high power medium voltage compensator will be based on experience gained during the realisation in The Electrotechnical Institute Gdansk Branch a similar medium power low voltage device. The study includes: main power converter topology considerations with some preliminary simulation results including harmonic components discussion, the solution to the problem of fast identification of the grid voltage disturbances, the solution to the problem of compensator waveforms synchronization with the grid and the description of the main operational algorithm of the compensator.

Kinematic Characteristics of Deformed Porous Structures

Experimental and computational methods of studying the stress state in the plastic region are characterized by various methods and accuracy of measurements, methods of mathematical processing of experimental information, and interpretation of results. The experimentally determined kinematics as a starting point is the most widely used method to study the stress-strain state in the plastic region. When studying the process of plastic deformation of porous blanks, the model of a rigid-plastic isotropic-strengthening porous body with a loading surface that has the shape of an ellipsoid with semi-axes. It depends on the amount of porosity and the ratio of the associated flow law as a mechanical model of the material. In the axisymmetric extrusion of porous blanks, the viscoplasticity method was used to determine the field of flow velocities based on the results of experimental studies. R-functions were applied to approximate experimentally obtained values. The problem of finding approximations was formulated in a variational statement. Cubic splines of one argument were used to interpolate functions. As a result, an approach was proposed, which consists of a particular sequence of calculating the derivatives of the coordinates of the nodes in time in combination with the R-functions approach. All the calculations were performed in Euler variables, eliminating the need to switch from Lagrangian variables and simplifying the solution. Additionally, this method allowed for working with an irregular and non-rectangular grid in areas with any shape of boundaries. This approach is more effective from the point of view of the approximation’s accuracy and the speed of calculations. Finally, the equation for calculating the porosity in the volume of the deformable workpiece based on the information about the distortion of the dividing grid elements was obtained. For stationary axisymmetric processes, a technique was developed that allowed for replacing the calculation of the accumulated deformation of the base material along the deformation trajectory by integration over the region. A technique was developed for determining the stress-strain state at unstable and stable stages of axisymmetric plastic deformation of porous blanks. The calculation results were compared based on the proposed experimental and calculation techniques and the finite element method.

Projection-Grid Schemes on Irregular Grids for a Parabolic Equation

A family of projection-grid schemes has been constructed for approximating parabolic equations with a variable diffusion coefficient in tensor form. The schemes are conservative and retain the self-adjointness of the original differential operator and are destined for calculations on 3D irregular difference grids, including tetrahedral, mixed (grids of arbitrary polyhedra), and locally adaptive (octal-tree type).

Projection-Grid Schemes on Irregular Grids for a Parabolic Equation

Projection-Grid Schemes on Irregular Grids for a Parabolic Equation