Quasi-uniform Meshes Research Articles

Meshfree Variational-Physics-Informed Neural Networks (MF-VPINN): An Adaptive Training Strategy

In this paper, we introduce a Meshfree Variational-Physics-Informed Neural Network. It is a Variational-Physics-Informed Neural Network that does not require the generation of the triangulation of the entire domain and that can be trained with an adaptive set of test functions. In order to generate the test space, we exploit an a posteriori error indicator and add test functions only where the error is higher. Four training strategies are proposed and compared. Numerical results show that the accuracy is higher than the one of a Variational-Physics-Informed Neural Network trained with the same number of test functions but defined on a quasi-uniform mesh.

Finite Element Method on locally refined composite meshes for Dirichlet fractional Laplacian

It is known that the solution of the Dirichlet fractional Laplacian in a bounded domain exhibits singular behavior near the boundary. Consequently, numerical discretizations on quasi-uniform meshes lead to low accuracy and nonphysical solutions. We adopt a finite element discretization on locally refined composite meshes, which consist in a combination of graded meshes near the singularity and uniform meshes where the solution is smooth. We also provide a reference strategy on parameter selection of locally refined composite meshes. Numerical tests confirm that finite element method on locally refined composite meshes has higher accuracy than uniform meshes, but the computational cost is less than that of graded meshes. Our method is applied to discrete the fractional-in-space Allen–Cahn equation and the fractional Burgers equation with Dirichlet fractional Laplacian, some new observations are discovered from our numerical results.

Interactive Mesh Sculpting with Arbitrary Topologies in Head-Mounted VR Environments

Shape modeling is a dynamic area in computer graphics with significant applications in computer-aided design, animation, architecture, and entertainment. Virtual sculpting, a key paradigm in free-form modeling, has traditionally been performed on desktop computers where users manipulate meshes with controllers and view the models on two-dimensional displays. However, the advent of Extended Reality (XR) technology has ushered in immersive interactive experiences, expanding the possibilities for virtual sculpting across various environments. A real-time virtual sculpting system implemented in a Virtual Reality (VR) setting is introduced in this paper, utilizing quasi-uniform meshes as the foundational structure. In our innovative sculpting system, we design an integrated framework encompassing a surface selection algorithm, mesh optimization technique, mesh deformation strategy, and topology fusion methodology, which are all tailored to meet the needs of the sculpting process. The universal, user-friendly sculpting tools designed to support free-form topology are offered in this system, ensuring that the meshes remain watertight, manifold, and free from self-intersections throughout the sculpting process. The models produced are versatile and suitable for use in diverse fields such as gaming, art, and education. Experimental results confirm the system’s real-time performance and universality, highlighting its user-centric design.

The Impact of the Explicit Representation of Convection on the Climate of a Tidally Locked Planet in Global Stretched-mesh Simulations

Convective processes are crucial in shaping exoplanetary atmospheres but are computationally expensive to simulate directly. A novel technique of simulating moist convection on tidally locked exoplanets is to use a global 3D model with a stretched mesh. This allows us to locally refine the model resolution to 4.7 km and resolve fine-scale convective processes without relying on parameterizations. We explore the impact of mesh stretching on the climate of a slowly rotating TRAPPIST-1e-like planet, assuming it is 1:1 tidally locked. In the stretched-mesh simulation with explicit convection, the climate is 5 K colder and 25% drier than that in the simulations with parameterized convection(with both stretched and quasi-uniform meshes). This is due to the increased cloud reflectivity—because of an increase in low-level cloudiness—and exacerbated by the diminished greenhouse effect due to less water vapor. At the same time, our stretched-mesh simulations reproduce the key characteristics of the global climate of tidally locked rocky exoplanets, without any noticeable numerical artifacts. Our methodology opens an exciting and computationally feasible avenue for improving our understanding of 3D mixing in exoplanetary atmospheres. Our study also demonstrates the feasibility of a global stretched-mesh configuration for LFRic-Atmosphere, the next-generation Met Office climate and weather model.

Performance of Heat Transfer in Micropolar Fluid with Isothermal and Isoflux Boundary Conditions Using Supervised Neural Networks

The current study delivers a numerical investigation on the performance of heat transfer and flow of micropolar fluid in porous Darcy structures with isothermal and isoflux walls (boundary conditions) of a stretching sheet. The dynamics and mechanism of such fluid flows are modelled by nonlinear partial differential equations that are reduced to a system of nonlinear ordinary differential equations by utilizing the porosity of medium and similarity functions. Generally, the explicit or analytical solutions for such nonlinear problems are hard to calculate. Therefore, we have designed a computer or artificial intelligence-based numerical technique. The reliability of neural networks using the machine learning (ML) approach is used with a local optimization technique to investigate the behaviours of different material parameters such as the Prandtl number, micropolar parameters, Reynolds number, heat index parameter, injection/suction parameter on the temperature profile, fluid speed, and spin/rotational behaviour of the microstructures. The approximate solutions determined by the efficient machine learning approach are compared with the classical Runge–Kutta fourth-order method and generalized finite difference approximation on a quasi-uniform mesh. The accuracy of the errors lies around 10−8 to 10−10 between the traditional analytical solutions and machine learning strategy. ML-based techniques solve different problems without discretization or computational work, and are not subject to the continuity or differentiability of the governing model. Moreover, the results are illustrated briefly to help implement microfluids in drug administering, elegans immobilization, and pH controlling processes.

A high-order stabilizer-free weak Galerkin finite element method on nonuniform time meshes for subdiffusion problems

<abstract><p>We present a stabilizer-free weak Galerkin finite element method (SFWG-FEM) with polynomial reduction on a quasi-uniform mesh in space and Alikhanov's higher order L2-$ 1_\sigma $ scheme for discretization of the Caputo fractional derivative in time on suitable graded meshes for solving time-fractional subdiffusion equations. Typical solutions of such problems have a singularity at the starting point since the integer-order temporal derivatives of the solution blow up at the initial point. Optimal error bounds in $ H^1 $ norm and $ L^2 $ norm are proven for the semi-discrete numerical scheme. Furthermore, we have obtained the values of user-chosen mesh grading constant $ r $, which gives the optimal convergence rate in time for the fully discrete scheme. The optimal rate of convergence of order $ \mathcal{O}(h^{k+1}+M^{-2}) $ in the $ L^\infty(L^2) $-norm has been established. We give several numerical examples to confirm the theory presented in this work.</p></abstract>

A C0 finite element method for the biharmonic problem with Navier boundary conditions in a polygonal domain

Abstract In this paper we study the biharmonic equation with Navier boundary conditions in a polygonal domain. In particular, we propose a method that effectively decouples the fourth-order problem as a system of Poisson equations. Our method differs from the naive mixed method that leads to two Poisson problems but only applies to convex domains; our decomposition involves a third Poisson equation to confine the solution in the correct function space, and therefore can be used in both convex and nonconvex domains. A $C^0$ finite element algorithm is in turn proposed to solve the resulting system. In addition, we derive optimal error estimates for the numerical solution on both quasi-uniform meshes and graded meshes. Numerical test results are presented to justify the theoretical findings.

MPAS-Seaice (v1.0.0): sea-ice dynamics on unstructured Voronoi meshes

Abstract. We present MPAS-Seaice, a sea-ice model which uses the Model for Prediction Across Scales (MPAS) framework and spherical centroidal Voronoi tessellation (SCVT) unstructured meshes. As well as SCVT meshes, MPAS-Seaice can run on the traditional quadrilateral grids used by sea-ice models such as CICE. The MPAS-Seaice velocity solver uses the elastic–viscous–plastic (EVP) rheology and the variational discretization of the internal stress divergence operator used by CICE, but adapted for the polygonal cells of MPAS meshes, or alternatively an integral (“finite-volume”) formulation of the stress divergence operator. An incremental remapping advection scheme is used for mass and tracer transport. We validate these formulations with idealized test cases, both planar and on the sphere. The variational scheme displays lower errors than the finite-volume formulation for the strain rate operator but higher errors for the stress divergence operator. The variational stress divergence operator displays increased errors around the pentagonal cells of a quasi-uniform mesh, which is ameliorated with an alternate formulation for the operator. MPAS-Seaice shares the sophisticated column physics and biogeochemistry of CICE and when used with quadrilateral meshes can reproduce the results of CICE. We have used global simulations with realistic forcing to validate MPAS-Seaice against similar simulations with CICE and against observations. We find very similar results compared to CICE, with differences explained by minor differences in implementation such as with interpolation between the primary and dual meshes at coastlines. We have assessed the computational performance of the model, which, because it is unstructured, runs with 70 % of the throughput of CICE for a comparison quadrilateral simulation. The SCVT meshes used by MPAS-Seaice allow removal of equatorial model cells and flexibility in domain decomposition, improving model performance. MPAS-Seaice is the current sea-ice component of the Energy Exascale Earth System Model (E3SM).

Optimal maximum-norm estimate of the LDG method for singularly perturbed convection–diffusion problem

We consider the local discontinuous Galerkin (LDG) method with a generalized alternating numerical flux for a one-dimensional singularly perturbed convection–diffusion problem. The double-optimal local maximum-norm error estimate is derived on the quasi-uniform meshes for the first time. Here, “double-optimal” refers to the maximum-norm error estimate out of the boundary layer region being optimal and the width of the boundary layer being quasi-optimal. Furthermore, we improve the discrete maximum principle and global L1-error estimate established in the literature.

Analysis of stability and convergence for L-type formulas combined with a spatial finite element method for solving subdiffusion problems

A time-fractional diffusion equation with the Caputo fractional derivative of order α∈(0,1) is considered on a bounded polygonal domain. Some numerical methods are presented based on the finite element method (FEM) in space on a quasi-uniform mesh and L-type discretizations (i.e., L1, L1-2, and L1-2-3 formulas) to approximate the Caputo derivative. Stability and convergence of the L1-2-3 FEM as well as L1-2 FEM are proved rigorously. The lack of positivity of the coefficients of these formulas is the main difficulty in the analysis of the proposed methods. This has hampered the analysis of methods using finite elements mixed with L1-2 and L1-2-3 discretizations. Our proofs are based on the concept of a special kind of discrete Grönwall's inequality and the energy method. Numerical examples confirm the theoretical analysis.

A finite element method for Dirichlet boundary control of elliptic partial differential equations

This paper introduces a new variational formulation for Dirichlet boundary control problem of elliptic partial differential equations, based on observations that the state and adjoint state are related through the control on the boundary of the domain, and that such a relation may be imposed in the variational formulation of the adjoint state. Well-posedness (unique solvability and stability) of the variational problem is established in the $H^{1}(\Omega)\times H_{0}^{1}(\Omega)$ space for the respective state and adjoint state. A finite element method based on this formulation is analyzed. It is shown that the conforming $k-$th order finite element approximations to the state and the adjoint state, in the respective $L^{2}$ and $H^{1}$ norms converge at the rate of order $k-1/2$ on quasi-uniform mesh for conforming element of order $k$. Numerical examples are presented to validate the theory.

Difference finite element method for the 3D steady Stokes equations

In this paper, a difference finite element (DFE) method is presented for the 3D steady Stokes equations. This new method consists of transmitting the finite element solution (uh,ph) of the 3D steady Stokes equations in the direction of (x,y,z) into a series of the finite element solution (uhk,phk) of the 2D steady Stokes equations. Here the 2D steady Stokes equations are solved by the finite element space pair (P1b,P1b,P1)×P1, where the 2D finite element pair (P1b,P1b)×P1 satisfies the discrete inf-sup condition in a 2D domain ω. Here we design the weak formulation of the DFE method based on the 3D finite element pair ((P1b,P1b,P1)×P1)×(P1×P0) under the quasi-uniform mesh condition, prove that the 3D finite element pair satisfies the discrete inf-sup condition in a 3D domain Ω and provide the existence, uniqueness and stability of the DFE solution (uh,ph)=(∑k=0l3uhkϕk(z),∑k=1l3phkψk(z)) and deduce the first order convergence of the DFE solution (uh,ph) with respect to the exact solution (u,p) of the 3D steady Stokes equations. Finally, some numerical tests are presented to show the accuracy and efficiency for the proposed method.

Strength and Challenges of global model MPAS with regional mesh refinement for mid-latitude storm forecasting: a case study

Abstract. With the rising share of renewable energy sources like wind energy in the energy mix, high-impact weather events like mid-latitude storms increasingly affect energy production, grid stability and safety and reliable forecasting becomes very relevant for e.g. transmission system operators to allow for actions to reduce imbalances. Traditionally, meteorological forecasts are provided by limited-area weather prediction models (LAMs), which can use high enough model resolution to represent the range of atmospheric scales of motions associated with such storm structures. While generally satisfactory, deterioration and insufficient deepening of large-scale storm structures are observed when they are introduced near the lateral boundaries of the LAM due to inadequate spatial and temporal interpolation. Global models with regional mesh refinement capabilities like the Model for Prediction Across Scales (MPAS) have the potential to provide an alternative, while avoiding sharp resolution jumps and lateral boundaries. In this study, MPAS' capabilities of simulating key evaluation metrics like storm intensity, storm location and storm duration are investigated based on a case study and assessed in comparison with buoy measurements, forecast products from the Climate Forecast System (CFSv2) and simulations with the Weather Research and Forecasting (WRF) LAM. Quasi-uniform and variable-resolution MPAS mesh configurations with different model physics settings are designed to analyze the impact of the mesh refinement and model physics on the model performance. MPAS shows good performance in predicting storm intensity based on the local minimum sea level pressure, while time of local minimum sea level pressure (storm duration) was generally estimated too late (too long) in comparison with the buoy measurements in part due to an early west-wards shift of the storm center in MPAS. The variable-resolution configurations showed a combination of an additional south-westwards shift and deviations in the sea level pressure field south-west of the storm center that introduced additional bias to the time of local minimum sea level pressure at some locations. The study highlights the need for a more detailed analysis of applied mesh refinements for particular applications and emphasizes the importance of methods like data assimilation techniques to prevent model drifts.

Error estimates for a fully discrete $\varepsilon-$uniform finite element method on quasi uniform meshes

In this article, we analyze a fully discrete $\varepsilon-$uniformly convergent finite element method for singularly perturbed convection-diffusion-reaction boundary-value problems, on piecewise-uniform meshes. Here, we choose $L-$splines as basis functions. We will concentrate on the convergence analysis of the finite element method which employ the discrete $L-$spline basis functions instead of their continuous counterparts. The $L-$splines are approximated on the piecewise-uniform Shishkin mesh inside each element. These approximations are used as basis functions in the frame of Galerkin FEM on a coarse piecewise-uniform mesh to discretize the domain. Further, we determine the amount of error introduced by the discrete $L-$spline basis functions in the overall numerical method, and explore the possibility of recovering the order of convergence that are consistent with the classical order of convergence for the numerical methods using the exact $L-$splines.

On the topological enrichment for crack modeling via the generalized/extended FEM: a novel discussion considering smooth partitions of unity

Purpose It has been usual to prefer an enrichment pattern independent of the mesh when applying singular functions in the Generalized/eXtended finite element method (G/XFEM). This choice, when modeling crack tip singularities through extrinsic enrichment, has been understood as the only way to surpass the typical poor convergence rate obtained with the finite element method (FEM), on uniform or quasi-uniform meshes conforming to the crack. Then, the purpose of this study is to revisit the topological enrichment strategy in the light of a higher-order continuity obtained with a smooth partition of unity (PoU). Aiming to verify the smoothness' impacts on the blending phenomenon, a series of numerical experiments is conceived to compare the two GFEM versions: the conventional one, based on piecewise continuous PoU's, and another which considers PoU's with high-regularity. Design/methodology/approach The stress approximations right at the crack tip vicinity are qualified by focusing on crack severity parameters. For this purpose, the material forces method originated from the configurational mechanics is employed. Some attempts to improve solution using different polynomial enrichment schemes, besides the singular one, are discussed aiming to verify the transition/blending effects. A classical two-dimensional problem of the linear elastic fracture mechanics (LEFM) is solved, considering the pure mode I and the mixed-mode loadings. Findings The results reveal that, in the presence of smooth PoU's, the topological enrichment can still be considered as a suitable strategy for extrinsic enrichment. First, because such an enrichment pattern still can treat the crack independently of the mesh and deliver some advantage in terms of convergence rates, under certain conditions, when compared to the conventional FEM. Second, because the topological pattern demands fewer degrees of freedom and impacts conditioning less than the geometrical strategy. Originality/value Several outputs are presented, considering estimations for the J–integral and the angle of probable crack advance, this last computed from two different strategies to monitoring blending/transition effects, besides some comments about conditioning. Both h- and p-behaviors are displayed to allow a discussion from different points of view concerning the topological enrichment in smooth GFEM.

Convergence analysis of the hp-version spectral collocation method for a class of nonlinear variable-order fractional differential equations

In this paper, a general class of nonlinear initial value problems involving a Riemann-Liouville fractional derivative and a variable-order fractional derivative is investigated. An existence result of the exact solution is established by using Weissinger's fixed point theorem and Gronwall-Bellman lemma. An hp-version spectral collocation method is presented to solve the problem in numerical frames. The collocation method employs the Legendre-Gauss interpolations to conquer the influence of the nonlinear term and variable-order fractional derivative. The most remarkable feature of the method is its capability to achieve higher accuracy by refining the mesh and/or increasing the degree of the polynomial. The error estimates under the H1-norm for smooth solutions on arbitrary meshes and singular solutions on quasi-uniform meshes are derived. Numerical results are given to support the theoretical conclusions.

Pressure-robust error estimate of optimal order for the Stokes equations: domains with re-entrant edges and anisotropic mesh grading

The velocity solution of the incompressible Stokes equations is not affected by changes of the right hand side data in form of gradient fields. Most mixed methods do not replicate this property in the discrete formulation due to a relaxation of the divergence constraint which means that they are not pressure-robust. A recent reconstruction approach for classical methods recovers this invariance property for the discrete solution, by mapping discretely divergence-free test functions to exactly divergence-free functions in the sense of {varvec{H}}({text {div}}). Moreover, the Stokes solution has locally singular behavior in three-dimensional domains near concave edges, which degrades the convergence rates on quasi-uniform meshes and makes anisotropic mesh grading reasonable in order to regain optimal convergence characteristics. Finite element error estimates of optimal order on meshes of tensor-product type with appropriate anisotropic grading are shown for the pressure-robust modified Crouzeix–Raviart method using the reconstruction approach. Numerical examples support the theoretical results.

A Non-Standard Finite Difference Scheme for Magneto-Hydro Dynamics Boundary Layer Flows of an Incompressible Fluid Past a Flat Plate

This paper deals with a non-standard implicit finite difference scheme that is defined on a quasi-uniform mesh for approximate solutions of the Magneto-Hydro Dynamics (MHD) boundary layer flow of an incompressible fluid past a flat plate for a wide range of the magnetic parameter. The proposed approach allows imposing the given boundary conditions at infinity exactly. We show how to improve the obtained numerical results via a mesh refinement and a Richardson extrapolation. The obtained numerical results are favourably compared with those available in the literature.

Properties of High-Order Finite Difference Schemes and Idealized Numerical Testing

Construction of high-order difference schemes based on Taylor series expansion has long been a hot topic in computational mathematics, while its application in comprehensive weather models is still very rare. Here, the properties of high-order finite difference schemes are studied based on idealized numerical testing, for the purpose of their application in the Global/Regional Assimilation and Prediction System (GRAPES) model. It is found that the pros and cons due to grid staggering choices diminish with higher-order schemes based on linearized analysis of the one-dimensional gravity wave equation. The improvement of higher-order difference schemes is still obvious for the mesh with smooth varied grid distance. The results of discontinuous square wave testing also exhibits the superiority of high-order schemes. For a model grid with severe non-uniformity and non-orthogonality, the advantage of high-order difference schemes is inapparent, as shown by the results of two-dimensional idealized advection tests under a terrain-following coordinate. In addition, the increase in computational expense caused by high-order schemes can be avoided by the precondition technique used in the GRAPES model. In general, a high-order finite difference scheme is a preferable choice for the tropical regional GRAPES model with a quasi-uniform and quasi-orthogonal grid mesh.

Configuration and evaluation of a global unstructured mesh atmospheric model (GRIST-A20.9) based on the variable-resolution approach

Abstract. Targeting a long-term effort towards a variable-resolution (VR) global weather and climate model, this study systematically configures and evaluates an unstructured mesh atmospheric model based on the multiresolution approach. The model performance is examined from dry dynamics to simple physics and full physics scenarios. In the dry baroclinic wave test, the VR model reproduces comparable fine-scale structures in the refined regions as a fine-resolution quasi-uniform (QU) mesh model. The mesh transition zone does not adversely affect the wave pattern. Regional kinetic energy spectra show that the fine-scale resolving ability improves as the fine resolution increases. Compared to a QU counterpart that has equivalent degrees of freedom, the VR model tends to increase the global errors, but the errors can be reduced when the resolution of the coarse region is increased. The performance over the coarse region is generally close to that of a low-resolution QU counterpart. Two multi-region refinement approaches, the hierarchical and polycentric refinement modes, further validate the model performance under the multiresolution refinement. Activating hyperdiffusion for horizontal velocity is helpful with respect to VR modeling. An idealized tropical cyclone test is further used to examine its ability to resolve fine-scale structures. In the simple physics environment, the VR model can have the tropical cyclone stably pass the transition zone in various configurations. A series of sensitivity tests examines the model performance in a hierarchical refinement mode. The simulations exhibit consistency even when the VR mesh is slightly perturbed by one of the three parameters that control the density function. The tropical cyclone, starting from the second refinement region and passing through the inner transition zone, gets intensified and covers a smaller area in the refined regions. Such variations are consistent with the behavior that one may observe when uniformly refining the QU mesh. In the full physics environment with a highly variable mesh that reaches sub-10 km resolution, the VR model also produces a reasonable evolution for the tropical cyclone. The explicit diffusion shows its usefulness in terms of suppressing some unrealistic isolated-scale structures that are far away from the initial vortex and does not adversely affect the physically important object. The fine-scale structure is determined mainly by the fine-resolution area, although the systems may have larger differences before they move into the fine-resolution area. Altogether, this work demonstrates that the multiresolution configuration is a reliable and economic alternative to high-resolution global modeling. The adverse impact due to mesh transition and the coarse region can be controlled well.