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In this paper, we study ultra-weak discontinuous Galerkin methods with generalized numerical fluxes for multi-dimensional high order partial differential equations on both unstructured simplex and Cartesian meshes. The equations we consider as examples are the nonlinear convection-diffusion equation and the biharmonic equation. Optimal error estimates are obtained for both equations under certain conditions, and the key step is to carefully design global projections to eliminate numerical errors on the cell interface terms of ultra-weak schemes on general dimensions. The well-posedness and approximation capability of these global projections are obtained for arbitrary order polynomial space based on a wide class of generalized numerical fluxes on regular meshes. These projections can serve as general analytical tools to be naturally applied to a wide class of high order equations. Numerical experiments are conducted to demonstrate these theoretical results. [ABSTRACT FROM AUTHOR]

This work considers charged systems described by the modified Poisson--Nernst-- Planck (PNP) equations, which incorporate ionic steric effects and the Born solvation energy for dielectric inhomogeneity. Solving the equilibrium of modified PNP equations poses numerical challenges due to the emergence of sharp boundary layers caused by small Debye lengths, particularly when local ionic concentrations reach saturation. To address this, we first reformulate the problem as a constraint optimization, where the ionic concentrations on unstructured Delaunay nodes are treated as fractional particles moving along edges between nodes. The electric fields are then updated to minimize the objective free energy while satisfying the discrete Gauss law. We develop a local relaxation method on unstructured meshes that inherently respects the discrete Gauss law, ensuring curl-free electric fields. Numerical analysis demonstrates that the optimal mass of the moving fractional particles guarantees the positivity of both ionic and solvent concentrations. Additionally, the free energy of the charged system consistently decreases during successive updates of ionic concentrations and electric fields. We conduct numerical tests to validate the expected numerical accuracy, positivity, free-energy dissipation, and robustness of our method in simulating charged systems with sharp boundary layers. [ABSTRACT FROM AUTHOR]

Scale analysis based on coarse‐graining has been proposed recently as an alternative to Fourier analysis. It requires interpolation to a regular mesh for data from unstructured‐mesh models. We propose an alternative coarse‐graining method which relies on implicit filters using powers of discrete Laplacians. This method can work on arbitrary (structured or unstructured) meshes and is applicable to the direct output of unstructured‐mesh models. Illustrations and detailed discussions are provided for discrete fields placed at vertices of triangular meshes. The case with placement on triangles is also briefly discussed. Plain Language Summary: When studying ocean flows scientists are interested in how flow energy is distributed over scales. Ocean circulation models simulate these flows on computational meshes, and some models use highly variable meshes, such as unstructured triangular meshes. To eliminate the effect of mesh inhomogeneity, the output of such models is first interpolated to a regular quadrilateral mesh, which implies additional work and may create uncertainties. The method we propose does not require this interpolation and can be applied to the output of such models directly. Key Points: A coarse‐graining technique for unstructured meshes based on implicit filters is proposed for the analysis of energy distribution over scalesThe technique is well suited for scalar and vector quantities in both flat and spherical geometry [ABSTRACT FROM AUTHOR]

This paper presents a numerical study of the convergence order of the projection-characteristic of the cubic polynomial projection (СРР) method for solving a three-dimensional stationary transport equation on unstructured tetrahedral meshes. The method is based on a characteristic approach to solve the transport equation, has a minimal stencil within a single tetrahedron, and a high (third) order of approximation. Unlike classical grid-characteristic methods, in this method, the final numerical approach is constructed based not on the interpolation operators of some order of approximation but on the orthogonal projection operators on the functional space used to approximate the solution. The base scheme is a one-dimensional scheme referred to as the Hermitian cubic interpolation (СIP) scheme. The use of interpolation operators is often designed to be applied to sufficiently smooth functions. However, even if the exact solution has sufficient smoothness, some types of tetrahedra illumination lead to the appearance of nonsmooth grid solutions. The transition to orthogonal projectors solves two problems: firstly, the problem of the appearance of angular directions that are coplanar with the faces of the cells, and secondly, the problem of the appearance of nonsmooth numerical solutions in the faces of the mesh cell. The convergence result is compared with the theoretical estimates obtained for the first time by one of the authors of this study. The third order of convergence of the method is shown, provided that the solution is sufficiently smooth and the absorption coefficient in the cells is close to constant. [ABSTRACT FROM AUTHOR]

The article considers the problems in the numerical stress-strain analysis of heterogeneous models of rock samples and biological tissues. The information about the geometric structure of such models is usually obtained using computed tomography (CT). The construction of the unstructured hexahedral meshes (based on CT scan data) is used to reduce the problem dimension. Examples of estimating the effective properties of a core sample and modeling a human lower jaw are given in the article. [ABSTRACT FROM AUTHOR]

Abstract Scale analysis based on coarse‐graining has been proposed recently as an alternative to Fourier analysis. It requires interpolation to a regular mesh for data from unstructured‐mesh models. We propose an alternative coarse‐graining method which relies on implicit filters using powers of discrete Laplacians. This method can work on arbitrary (structured or unstructured) meshes and is applicable to the direct output of unstructured‐mesh models. Illustrations and detailed discussions are provided for discrete fields placed at vertices of triangular meshes. The case with placement on triangles is also briefly discussed.

We present a novel approach for high-order accurate numerical differentiation on unstructured meshes of quadrilateral elements. To differentiate a given function, an auxiliary function with greater smoothness properties is defined which when differentiated provides the derivatives of the original function. The method generalises traditional finite difference methods to meshes of arbitrary topology in any number of dimensions for any order of derivative and accuracy. We demonstrate the accuracy of the numerical scheme using dual quadrilateral meshes and a refinement method based on subdivision surfaces. The scheme is applied to the solution of a range of partial differential equations, including both linear and nonlinear, second and fourth order equations, and a time-dependent first order equation.

Flood-induced disasters can cause significant harm and economic losses. Using numerical simulations to provide real-time predictions of flood events is an effective method to address this issue. To develop a high-efficiency and adaptable tool for fast flood prediction in complex terrains, this work utilizes Graphic Processing Units (GPUs) to accelerate a full 2D shallow water model on unstructured meshes. Furthermore, a novel Edge Loop Method (ELM) based on the winged-edge data structure is applied to the model to improve the computational efficiency of solving fluxes. A benchmark test and a real-world dam-break case were simulated to verify the accuracy and performance of the current model. The results demonstrate that the ELM accelerates the model by 2.51 and 4.08 times compared to the eight-core CPU-based model, and 14.97 and 19.84 times compared to the single-core CPU-based model in two cases. Notably, when compared to the GPU-based model using the Cell Loop Method (CLM), the computational efficiency of the ELM is improved by 18.34% and 24.29%, respectively. In particular, a quantitative analysis of the performance explains the advantage of the ELM from the perspective of its implementation mechanism, further demonstrating that the ELM exhibits higher computational efficiency as the total number of cells increases. Based on the advantages of high efficiency in the GPU-based model using the ELM, the proposed model can effectively forecast real-world flood events in regions characterized by complex terrains. [ABSTRACT FROM AUTHOR]

This paper is focused on the approximation of the Euler equations of compressible fluid dynamics on a staggered mesh. With this aim, the flow parameters are described by the velocity, the density, and the internal energy. The thermodynamic quantities are described on the elements of the mesh, and thus the approximation is only in L2, while the kinematic quantities are globally continuous. The method is general in the sense that the thermodynamic and kinetic parameters are described by an arbitrary degree of polynomials. In practice, the difference between the degrees of the kinematic parameters and the thermodynamic ones is set to 1. The integration in time is done using the forward Euler method but can be extended straightforwardly to higher-order methods. In order to guarantee that the limit solution will be a weak solution of the problem, we introduce a general correction method in the spirit of the Lagrangian staggered method described in [R. Abgrall and S. Tokareva, SIAM J. Sci. Comput., 39 (2017), pp. A2345--A2364; R. Abgrall, K. Lipnikov, N. Morgan, and S. Tokareva, SIAM J. Sci. Comput., 2 (2020), pp. A343--A370; V. A. Dobrev, T. V. Kolev, and R. N. Rieben, SIAM J. Sci. Comput., 34 (2012), pp. B606--B641], and we prove a Lax--Wendroff theorem. The proof is valid for multidimensional versions of the scheme, even though most of the numerical illustrations in this work, on classical benchmark problems, are one-dimensional because we have easy access to the exact solution for comparison. We conclude by explaining that the method is general and can be used in different settings, for example, finite volume or discontinuous Galerkin method, not just the specific one presented in this paper. [ABSTRACT FROM AUTHOR]

In the present work, a numerical method is adopted and applied for simulating the inviscid cavitating flows around two- and three-dimensional geometries on unstructured meshes. The algorithm uses the preconditioned multiphase Euler equations discretized by a cell-centered central difference finite volume scheme with suitable dissipation terms. The interface capturing method with three transport equation-based cavitation models, namely the Merkle et al., Singhal et al. and Kunz et al. models are employed for the mass transfer between the liquid and vapor phases to be calculated. The simulations of the steady inviscid cavitating flows are performed around different two- and three-dimensional geometries, namely the NACA0012, NACA66(MOD) and two-element NACA4412-4415 hydrofoils, the hemispherical head shape body and the twisted NACA0009 hydrofoil, and the results are obtained over these geometries with the three cavitation models used for different flow conditions. The effects of different numerical parameters on the accuracy of the solution are also examined by a sensitivity study. The present results are compared with those of performed by other researchers which exhibit good agreement. It is indicated that the solution method adopted based on the preconditioned finite volume multiphase Euler flow solver on unstructured meshes is capable of accurately predicting the surface pressure distribution and the cavity shape over the arbitrary two- and three-dimensional geometries. [ABSTRACT FROM AUTHOR]

This work aims to supplement the realization and validation of a higher-order well-balanced unstructured finite volume (FV) scheme, that has been relatively recently presented, for numerically simulating weakly non-linear weakly dispersive water waves over varying bathymetries. We investigate and develop solution strategies for the sparse linear system that appears during this FV discretisation of a set of extended Boussinesq-type equations on unstructured meshes. The resultant linear system of equations must be solved at each discrete time step as to recover the actual velocity field of the flow and advance in time. The system's coefficient matrix is sparse, un-symmetric and often ill-conditioned. Its characteristics are affected by physical quantities of the problem to be solved, such as the undisturbed water depth and the mesh topology. To this end, we investigate the application of different well-known iterative techniques, with and without the usage of preconditioners and reordering, for the solution of this sparse linear system. The iiterative methods considered are the GMRES and the BiCGSTAB, three preconditioning techniques, including different ILU factorizations and two different reordering techniques are implemented and discussed. An optimal strategy, in terms of computational efficiency and robustness, is finally proposed which combines the use of the BiCGSTAB method with the ILUT preconditioner and the Reverse Cuthill–McKee reordering. [ABSTRACT FROM AUTHOR]

The welding process is the most used technique for metal joining. Understanding the temperature variation along the welded part during the process can prevent the appearance of failures. The experimental investigation process is quite time-consuming and costly. Therefore, numerical simulation processes based on the finite difference method, finite element method, finite volume method, or meshless Element-Free Galerkin (EFG) methods are important tools to optimize the welding process. The main goal of the present study is to show the feasibility of the Element-based Finite-Volume Method (EbFVM) approach for actual engineering applications. To solve the unsteady 2D and 3D thermal energy equation using enthalpy as an independent variable, an in-house Fortran code has been developed based on the EbFVM approach in conjunction with unstructured and structured meshes. The numerical simulations, with four types of different heat sources, were performed for applications of real welding processes with variations in density and enthalpy as a function of temperature. The results are presented in terms of thermal cycles and temperature fields. Furthermore, the developed code was confronted against experimental works from the literature, simulated and lab-controlled experiments with AISI 409 ferritic workpieces, and exact analytical solutions with thermocouples fixed in different positions. In general, the numerical results from the current investigation are in close agreement with the results from the literature and the experimental results performed by the authors. The numerical results also highlighted the differences between the 2D and 3D models for thermal cycles near the bead weld. [ABSTRACT FROM AUTHOR]

Given the wide range of scales present in fluid dynamics, highly anisotropic meshes are often employed to resolve boundary layer flow at high Reynolds numbers. However, such meshes result in distorted elements, which can deteriorate the convergence of the preconditioned conjugate gradient solver (PCG) used to solve the discrete pressure equations. To address this issue, the Linelet Preconditioner is commonly used to accelerate the convergence rate of the PCG by building line segments (linelets) along the direction of the strongest couplings and applying a specific operation to each linelet. Under the current state of the art, linelets are constrained to operate independently within each domain partition. In these scenarios, PCG convergence was observed to deteriorate when the number of domain partitions is increased. This work proposes a communication step to be integrated into the preconditioning step, enabling different linelets to exchange information. The proposed method is observed to improve the convergence rate of linear systems with high stiffness due to highly stretched elements. Also, an algorithm is developed to generate the preconditioning matrix by purely algebraic considerations, in contrast to the usual approach of building linelets based on geometrical considerations. Furthermore, since the current approach is agnostic to the domain partition, it does not impose any constraint on the domain decomposition. Hence, the proposed method is a robust preconditioner with a numerical performance almost independent of the quality of the domain decomposition, providing a way for a more balanced load distribution. [ABSTRACT FROM AUTHOR]

Variables interpolation is one of the key concepts of MUSCL schemes. Originally developed for one-dimensional frameworks, many improvements have been made over the last decades to extend these methods to general unstructured multi-dimensional meshes. It results that scalar interpolation in a finite volume cell-centered framework is already quite well understood. It is known that linear reconstructions have to be limited in order to prevent non-physical oscillations of the solutions while ensuring a spatial second-order accuracy for smooth solutions. This is done thanks to a limiting function which allows the reconstruction to satisfy a monotonicity property, which then ensure the stability of the scheme. Nevertheless, some difficulties arise when we try to extend this process to vectorial variables. Generally, vectorial reconstructions are done componentwise, but this process reveals to be frame-dependent and leads to a loss of precision due to false detection of extrema. In this paper, we present a new method dealing with vectorial reconstructions in a multislope MUSCL context. • The cell-centered finite volume method on general unstructured meshes is considered. • Alternative formulations of the scalar multislope MUSCL method are introduced. • The flaws of componentwise vectorial reconstructions are illustrated. • A new vectorial multislope MUSCL method is introduced from the scalar procedure. • Numerical results show its accuracy, frame-independence, efficiency and robustness. [ABSTRACT FROM AUTHOR]

A full approximation scheme (FAS) nonlinear multigrid solver for two-phase flow and transport problems driven by wells with multiple perforations is developed. It is an extension to our previous work on FAS solvers for diffusion and transport problems. The solver is applicable to discrete problems defined on unstructured grids as the coarsening algorithm is aggregation-based and algebraic. To construct coarse basis that can better capture the radial flow near wells, coarse grids in which perforated well cells are not near the coarse-element interface are desired. This is achieved by an aggregation algorithm proposed in this paper that makes use of the location of well cells in the cell-connectivity graph. Numerical examples in which the FAS solver is compared against Newton's method on benchmark problems are given. In particular, for a refined version of the SAIGUP model, the FAS solver is at least 35% faster than Newton's method for time steps with a CFL number greater than 10. [ABSTRACT FROM AUTHOR]

A systematic numerical study of drag force and mass transfer in gravity-driven bubbles in a vertical pipe is performed at Re ∼ O (100 − 1000). This research employs a parallel multi-marker unstructured conservative level set method for interface capturing, avoiding the numerical coalescence in bubble swarms. The finite volume method discretizes transport equations on 3D collocated unstructured meshes. Unstructured flux limiter schemes solve the convective term of transport equations, preserving the numerical stability at high Reynolds numbers and high-density ratios. Thermodynamic equilibrium is assumed at the interface for the concentration of chemical species. The hydrodynamics and mass transfer of bubbles are validated against classical correlations from the literature. Finally, direct simulations are executed to develop new correlations for the drag force, normalized bubble surface and Sherwood number for bubbles in a vertical pipe. [Display omitted] • Validation of the multi-marker UCLS in bubbles in a vertical pipe. • DNS of the drag force, interfacial area and Sherwood number of bubbles. • Novel correlation for interfacial area of bubbles A*=A*(Eo,Re,CR). • Novel correlations ShPe−1/2=f(Re,Mo,CR), ShPe−1/2=f(Re,Eo,CR), ShPe−1/2=f(Gr,CR). • DNS of drag force, A* and mass transfer in bubble swarms in a vertical pipe. [ABSTRACT FROM AUTHOR]

A Galilean-invariant generalization of the Brinkman volume penalization method (BVPM) for compressible flows, which extends the applicability of the method to problems of the flow around moving obstacles, is proposed. The developed method makes it possible to carry out simulations on non-body fitted meshes of arbitrary structure, including completely unstructured computational grids. The efficiency of the Galilean-invariant generalization of the BVPM for compressible flows around moving obstacles is demonstrated for a number of test problems of the direct reflection of a one-dimensional acoustic pulse from a stationary and moving plane surface, scattering of an acoustic wave by a stationary cylinder, and the subsonic flow of a viscous gas around an oscillating cylinder. The numerical results agree closely with the reference solutions and theoretical estimates of the convergence of the method and they confirm the invariance of the proposed formulation with respect to the Galilean transformations. [ABSTRACT FROM AUTHOR]

A new energy and enstrophy conserving scheme (EEC) for the shallow water equations is proposed and evaluated using a suite of test cases over the global spherical or bounded domain. The evaluation is organized around a set of pre-defined properties: accuracy of individual operators, accuracy of the whole scheme, conservation of key quantities, control of the divergence variable, representation of the energy and enstrophy spectra, and simulation of nonlinear dynamics. The results confirm that the scheme is between the first and second order accurate, and conserves the total energy and potential enstrophy up to the time truncation errors. The scheme is capable of producing more physically realistic energy and enstrophy spectra, indicating that it can help prevent the unphysical energy cascade towards the finest resolvable scales. With an optimal representation of the dispersive wave relations, the scheme is able to keep the flow close to being non-divergent, and maintain the geostrophically balanced structures with large-scale geophysical flows over long-term simulations. [ABSTRACT FROM AUTHOR]

We propose a cell-centered Lagrangian scheme for solving the three dimensional ideal magnetohydrodynamics (MHD) equations on unstructured meshes. The physical conservation laws are compatibly discretized on the unstructured meshes to satisfy the geometric conservation law (GCL). By introducing a generalized Lagrange multiplier, the magnetic divergence constraint is coupled with the conservation laws hence the magnetic divergence errors can dissipate and transport to the domain boundaries. Invoking the Galilean invariance, magnetic flux conservation and the thermodynamic consistency, the nodal approximate Riemann solver is derived and the corresponding first order finite volume scheme is then constructed. The piecewise linear spatial reconstruction and two step predictor corrector time integration are then adopted to increase the accuracy of the scheme. Various numerical tests are presented to assert the robustness and accuracy of our scheme. [ABSTRACT FROM AUTHOR]

This paper presents the first classical Convolutional Neural Network (CNN) that can be applied directly to data from unstructured finite element meshes or control volume grids. CNNs have been hugely influential in the areas of image classification and image compression, both of which typically deal with data on structured grids. Unstructured meshes are frequently used to solve partial differential equations and are particularly suitable for problems that require the mesh to conform to complex geometries or for problems that require variable mesh resolution. Central to our approach are space-filling curves, which traverse the nodes or cells of a mesh tracing out a path that is as short as possible (in terms of numbers of edges) and that visits each node or cell exactly once. The space-filling curves (SFCs) are used to find an ordering of the nodes or cells that can transform multi-dimensional solutions on unstructured meshes into a one-dimensional (1D) representation, to which 1D convolutional layers can then be applied. Although developed in two dimensions, the approach is applicable to higher dimensional problems. To demonstrate the approach, the network we choose is a convolutional autoencoder (CAE), although other types of CNN could be used. The approach is tested by applying CAEs to data sets that have been reordered with a space-filling curve. Sparse layers are used at the input and output of the autoencoder, and the use of multiple SFCs is explored. We compare the accuracy of the SFC-based CAE with that of a classical CAE applied to two idealised problems on structured meshes, and then apply the approach to solutions of flow past a cylinder obtained using the finite-element method and an unstructured mesh. • Use of space-filling curves (SFCs) to apply CNNs to data on unstructured meshes. • Introduction of multiple SFCs in order to improve the accuracy of the new CNNs. • Use of sparse smoothing layers at the input and output of the CNN to reduce noise. • Edge weights are included in a recurrent neural network to generate multiple SFCs. • Application to fluid flow data on an unstructured mesh. [ABSTRACT FROM AUTHOR]

In the previous studies, the high-order gas-kinetic schemes (HGKS) have achieved successes for unsteady flows on three-dimensional unstructured meshes. In this paper, to accelerate the rate of convergence for steady flows, the implicit non-compact and compact HGKSs are developed. For non-compact scheme, the simple weighted essentially non-oscillatory (WENO) reconstruction is used to achieve the spatial accuracy, where the stencils for reconstruction contain two levels of neighboring cells. Incorporate with the nonlinear generalized minimal residual (GMRES) method, the implicit non-compact HGKS is developed. In order to improve the resolution and parallelism of non-compact HGKS, the implicit compact HGKS is developed with Hermite WENO (HWENO) reconstruction, in which the reconstruction stencils only contain one level of neighboring cells. The cell averaged conservative variable is also updated with GMRES method. Simultaneously, a simple strategy is used to update the cell averaged gradient by the time evolution of spatial-temporal coupled gas distribution function. To accelerate the computation, the implicit non-compact and compact HGKSs are implemented with the graphics processing unit (GPU) using compute unified device architecture (CUDA). A variety of numerical examples, from the subsonic to supersonic flows, are presented to validate the accuracy, robustness and efficiency of both inviscid and viscous flows. • To accelerate the convergence for steady flows, the implicit non-compact and compact HGKSs are developed for compressible flows. • For non-compact HGKS, the third-order WENO method is adopted, and the GMRES method with Jacobian matrix is used for temporal evolution. • For the compact HGKS, the third-order HWENO method is used, a simple strategy is used to update the cell averaged gradient with GMRES method. • To accelerate the computation, the current schemes are implemented with GPU. [ABSTRACT FROM AUTHOR]

This paper introduces a high-order finite volume method for solving solid dynamics problems on three-dimensional unstructured meshes. The method is based on truncated Taylor series constructed about each control volume face using the least squares method, extending the classical finite volume method to arbitrary interpolation orders. As verification tests, a static analytical example for small deformations, a hyperelastic cantilever beam with large deformations, and a cantilever beam subject to a dynamic load are analyzed. The results provide an optimal set of parameters for the interpolation method and allow a comparison with other classic schemes, yielding to improved results in terms of accuracy and computational cost. The final test consists in the simulation of a compressor reed valve in a dynamic scenario mimicking real-life conditions. Numerical results are compared against experimental data in terms of displacements and velocity; then, a comprehensive physical analysis of stresses, caused by bending and impact of the valve, is carried out. Overall, the method is demonstrated to be accurate and effective in handling shear locking, stress concentrations, and complex geometries and improves the effectiveness of the finite volume method for solving structural problems. • New high-order FVM for hyperelastic solid dynamics on unstructured meshes. • Arbitrary interpolation order using Local Regression Estimators. • Verification: static and dynamic cases considering small and large deformations. • Detailed analysis of stresses on a real compressor reed valve in dynamic scenario. • Efficient resolution of 3D solid problems through the finite volume method. [ABSTRACT FROM AUTHOR]

BackgroundThe high-fidelity neutron transport calculation requires refined geometric modeling whilst the unstructured meshes have strong adaptability to copy with the changes bring by complex geometry structure, and overcome the deficiencies of structured meshes in modeling capability.PurposeThis study aims to develop and validate a two-dimensional shielding calculation code ThorSNIPE which can be used to improve the modeling ability for analysis complex problems.MethodsFirst of all, problem solving model was established with discrete ordinates method and finite element method on the basis of the first order Boltzmann transport equation. The computational performance of continuous finite element method and discontinuous finite element method were compared and analyzed. Mass-matrix lumping technique was further applied to improve the reliability of solving model. Then, a two-dimensional discrete ordinate-finite element shielding calculation program ThorSNIPE was developed on the basis of above model. Finally, the code was validated by BWR cell critical benchmark, Argonne-5-A1 fixed source benchmark and Dog leg duct benchmark.Results & ConclusionsThe numerical results show that calculation value provided by ThorSNIPE is in good agreement with reference value, indicating that ThorSNIPE is suitable for complex shielding calculation, and Mass-matrix lumping technique can effectively suppress the non-physical spatial oscillations without reducing the calculation accuracy.

• Numerical method for the solution of non-isothermal incompressible flows for natural/mixed convection in irregular domains. • Incompressible Navier-Stokes equations and energy conservation equation discretized over unstructured meshes. • Two fractional time step procedure are sequentially solved inside a time step. • Numerical stability for CFL > 1 and "M-property" of the matrices of the corrective diffusive steps. • Several "real world" applications with measured data are presented. In the present paper we propose a new numerical methodology for the solution of 2D non-isothermal incompressible flows for natural and mixed convection in irregular geometries. The governing equations are the Incompressible Navier-Stokes equations and the energy conservation equation. Fluid velocity and temperature are coupled in the buoyancy term of the momentum equations according to the Oberbeck-Boussinesq approximation. The governing equations are discretized over unstructured triangular meshes satisfying the Delaunay property. Thanks to the Oberbeck-Boussinesq hypothesis, the flow and energy problems are solved in an uncoupled way, and two fractional time step procedures are sequentially applied to solve each problem. The prediction steps of both procedures are solved applying a Marching in Space and Time (MAST) numerical Eulerian scheme, which explicitly handles the non-linear terms in the momentum equations of the fluid problem, and allows numerical stability for Courant numbers greater than one. An analytical solution is applied for the prediction thermal problem. The correction steps of the two fractional time step procedures involve the solution of large linear systems, whose matrices are sparse and symmetric and have the " M " property if the mesh satisfies the Delaunay condition. This allows a well performing condition number. The matrix coefficients are constant in time, so that they are calculated and factorized only once, before the simulations loop starts, saving a lot of computational time. We present four numerical applications. For the first test, an analytical solution is available, and this makes it possible to analyze the spatial convergence order of the numerical solver. In the other applications, we investigate the capability of the proposed algorithm to handle irregular geometries, and we compare the computed results with experimental data and the outputs of literature models. A study of the required computational costs is also presented. [ABSTRACT FROM AUTHOR]

An unstructured finite volume method of third-order accuracy in space is considered. A detailed description of the method is given as applied to the continuity equation. The method is used to compute the unsteady viscous compressible gas flow past a sphere at small Reynolds numbers. [ABSTRACT FROM AUTHOR]

In this paper, we generalize the compact subcell weighted essentially non oscillatory (CSWENO) limiting strategy for Runge-Kutta discontinuous Galerkin method developed recently in [1] for structured meshes to unstructured triangular meshes. The main idea of the limiting strategy is to divide the immediate neighbors of a given cell as subcells into the required stencil and to use a WENO reconstruction for limiting. This strategy can be applied for any type of WENO reconstruction. We have used the WENO reconstruction proposed in [2] and provided accuracy tests and results for two-dimensional Burgers’ equation and two dimensional Euler equations to illustrate the performance of this limiting strategy. [ABSTRACT FROM AUTHOR]

Higher-order displacement-based inite element methods are useful for simulating bending problems and potentially addressingmesh-locking associated with nearly-incompressible elasticity, yet are computationally expensive. To address the computational expense, the paper presents a matrix-free, displacement-based, higher-order, hexahedral inite element implementation of compressible and nearly-compressible (v→0.5) linear isotropic elasticity at small strain with p-multigrid preconditioning. The cost, solve time, and scalability of the implementation with respect to strain energy error are investigated for polynomial order p =1, 2, 3, 4 for compressible elasticity, and p=2, 3, 4 for nearly-incompressible elasticity, on different number of CPU cores for a tube bending problem. In the context of this matrix-free implementation, higher-order polynomials (p = 3, 4) generally are faster in achieving better accuracy in the solution than lower-order polynomials (p = 1, 2). However, for a beam bending simulation with stress concentration (singularity), it is demonstrated that higher-order inite elements do not improve the spatial order of convergence, even though accuracy is improved. [ABSTRACT FROM AUTHOR]

The problem of paralleling computations is considered when solving the integro-differential equation of radiation transport in strongly scattering media. Parallelization is performed for a two-step iterative algorithm for solving a system of mesh equations. The first step is a simple iteration. At the second step, a correction accelerating the convergence of iterations is added to the mesh values obtained at the first step. The equation for the correction is solved by the Krylov subspace method. The efficiency of the two methods of parallelization of the two-step iterative algorithm is compared. In the Block Jacobi (BJ) method, a simple iteration calculation is performed locally in each spatial subregion. The Block Seidel (BS) method performs an end-to-end analysis over the entire region. Both methods are implemented in the RADUGA T program for solving the transport equation on unstructured meshes. The effectiveness of the methods was studied on the model of a light-water reactor. [ABSTRACT FROM AUTHOR]

Three-dimensional structural geomodels are increasingly being used for a wide variety of scientific and societal purposes. Most advanced methods for generating these models are implicit approaches, but they suffer limitations in the types of interpolation constraints permitted, which can lead to poor modeling in structurally complex settings. A geometric deep learning approach, using graph neural networks, is presented in this paper as an alternative to classical implicit interpolation that is driven by a learning through training paradigm. The graph neural network approach consists of a developed architecture utilizing unstructured meshes as graphs on which coupled implicit and discrete geological unit modeling is performed, with the latter treated as a classification problem. The architecture generates three-dimensional structural models constrained by scattered point data, sampling geological units and interfaces as well as planar and linear orientations. The modeling capacity of the architecture for representing geological structures is demonstrated from its application on two diverse case studies. The benefits of the approach are (1) its ability to provide an expressive framework for incorporating interpolation constraints using loss functions and (2) its capacity to deal with both continuous and discrete properties simultaneously. Furthermore, a framework is established for future research for which additional geological constraints can be integrated into the modeling process. [ABSTRACT FROM AUTHOR]

This work aims to supplement the realization and validation of a higher-order well-balanced unstructured finite volume (FV) scheme, that has been relatively recently presented, for numerically simulating weakly non-linear weakly dispersive water waves over varying bathymetries. We investigate and develop solution strategies for the sparse linear system that appears during this FV discretisation of a set of extended Boussinesq-type equations on unstructured meshes. The resultant linear system of equations must be solved at each discrete time step as to recover the actual velocity field of the flow and advance in time. The system’s coefficient matrix is sparse, un-symmetric and often ill-conditioned. Its characteristics are affected by physical quantities of the problem to be solved, such as the undisturbed water depth and the mesh topology. To this end, we investigate the application of different well-known iterative techniques, with and without the usage of preconditioners and reordering, for the solution of this sparse linear system. The iiterative methods considered are the GMRES and the BiCGSTAB, three preconditioning techniques, including different ILU factorizations and two different reordering techniques are implemented and discussed. An optimal strategy, in terms of computational efficiency and robustness, is finally proposed which combines the use of the BiCGSTAB method with the ILUT preconditioner and the Reverse Cuthill–McKee reordering.