Adaptive Unstructured Grids Research Articles

Simulation of planar ionization wave front propagation on an unstructured adaptive grid

The paper deals with the numerical solution of a basic 2D model of the propagation of an ionization wave. The system of equations describing this propagation consists of a coupled set of reaction–diffusion-convection equations and a Poissons equation. The transport equations are solved by a finite volume method on an unstructured triangular adaptive grid. The upwind scheme and the diamond scheme are used for the discretization of the convection and diffusion fluxes, respectively. The Poisson equation is also discretized by the diamond scheme. Numerical results are presented. We deal in more detail with numerical tests of the grid adaptation technique and its influence on the numerical results. An original behavior is observed. The grid refinement is not sufficient to obtain accurate results for this particular phenomenon. Using a second order scheme for convection is necessary.

Unstructured grid based 2-D inversion of VLF data for models including topography

We present a 2-D inversion code incorporating a damped least-squares and a minimum-model approach for plane wave electromagnetic (EM) methods using an adaptive unstructured grid finite element forward operator. Unstructured triangular grids permit efficient discretization of arbitrary 2-D model geometries and, hence, allow for modeling arbitrary topography. The inversion model is parameterized on a coarse parameter grid which constitutes a subset of the forward modeling grid. The mapping from parameter to forward modeling grid is obtained by adaptive mesh refinement. Sensitivities are determined by solving a modified sensitivity equation system arising from the derivative of the finite element equations with respect to the model parameters. Firstly, we demonstrate that surface topography may induce significant effects on the EM response and in the inversion result, and that it cannot be ignored when the scale length of topographic variations is in the order of magnitude of the skin depth. Secondly, the dependency of the inversion on the starting model is discussed for VLF and VLF-R data. Thirdly, we demonstrate the inversion of a synthetic data set obtained from a model with topography. Finally, the inversion approach is applied to field data collected in a region with undulating topography. ► The topography effect is demonstrated by forward modeling studies. ► We use an unstructured grid forward operator to invert 2-D plane-wave EM data. ► Topography correction or its special consideration during inversion is unnecessary. ► VLF inversion results are strongly dependent on the starting model. ► Inversion of VLF-R data is able to recover the background resistivity well.

CFD by first order PDEs

This research originally was aimed at modeling all flows (except free-molecular) by systems of hyperbolic-relaxation equations (moments of the Boltzmann equation), and developing efficient numerical methods for these. Such systems have many potential numerical advantages, mainly because there are no second or higher derivatives to be approximated. This avoids accuracy problems on adaptive unstructured grids, and the source terms, though often stiff, are only local; the compact stencils facilitate code parallelization. A single code could simulate flows up to intermediate Knudsen numbers, and be hybridized with DSMC where needed. In this project, one major problem arose that we have not yet solved: the accurate representation of shock structures. This makes the methodology currently unsuited for, e.g., re-entry flows. We have validated it for subsonic and transonic flows and are concentrating on applications to MEMS-related flows. We discuss the challenges of our approach, present numerical algorithms and results based on the 10-moment model, and report progress in our latest research topic: formulating accurate solid-boundary conditions.

Three-Dimensional Adaptive Central Schemes on Unstructured Staggered Grids

We present an explicit second-order finite volume generalization of the one-dimensional (1D) Nessyahu–Tadmor schemes for hyperbolic equations on adaptive unstructured tetrahedral grids. The nonoscillatory central difference scheme of Nessyahu and Tadmor, in which the resolution of the Riemann problem at the cell interfaces is bypassed thanks to the use of the staggered Lax–Friedrichs scheme, is extended here to a two-steps scheme. In order to reduce artificial viscosity, we start with an adaptively refined primal grid in three dimensions (3D), where the theoretical a posteriori result of the first-order scheme is used to derive appropriate refinement indicators. We apply those methods to solve Euler's equations. Numerical experimental tests on classical problems are obtained by our method and by the computational fluid dynamics software Fluent. These tests include results for the 3D Euler system (shock tube problem) and flow around an NACA0012 airfoil.

Adaptive unstructured grid generation for engineering computation of aerodynamic flows

A unified framework is presented for automatic unstructured grid generation and grid flow adaptation. The method can simultaneously refine and coarsen the grid cells, a capability that is heavily required in transient flow problems. The proposed method includes a Cartesian grid generation approach in the first stage that enables an automatic field discretization without need to explicitly define the surface grid. The Cartesian grid cells are then subdivided in such a way that prevents the existence of hanging nodes. This allows the application of efficient fully unstructured flow solvers. The capabilities of the method are demonstrated by flow computation around a maneuver wing-flap geometry (SKF 1.1) at transonic flow conditions. An explicit finite volume cell-centered scheme is used for numerical solution of compressible inviscid flow equations. Results show the efficiency and applicability of the method.

Improved representation of topographic effects by a vertical adaptive grid in Vector-Ocean-Model (VOM). Part II: Simulations in unstructured adaptive grids

A static adaptive grid approximates the topography and defines the vertical resolution in Vector-Ocean-Model (VOM). The adaptation to topography creates unstructured grids, which are organised in a one-dimensional vector by column-wise storage of only wet cells. The model’s name reflects this data structure. The intention of VOM is better resolving flow and stratification near topographic boundaries in Z-coordinates. This is the second part of a publication that describes the generation of adaptive grids (part I), and simulations with VOM in unstructured grids (this part). Adaptive grids generated for a synthetic topography in part I include shelf, continental slope, and ocean. Three of those grids are here utilised in upwelling simulations. Under the same forcing increased vertical resolution at seabed and slopes yields a significant increase in flow energy as compared to coarser grids. Results allow explaining the surface intensification of a continental slope jet by vertical displacements of water masses in the seabed Ekman layer. Results in unstructured grids are almost identical to reference simulations in equidistant grids where the respective smallest grid size of unstructured grids was used. Negative effects of grids on predicted flow and stratification are absent also over particularly rough topography, as demonstrated by using vertical velocity as most sensitive indicator. In a further simulation an overflow governed by the advection of water mass properties is presented to demonstrate the conservation properties of the model. After 5 months of simulation the predicted domain average temperature deviated by 10 −8 from the initial temperature field. Compared to equidistant grids the advection/diffusion scheme looses about one order of magnitude in accuracy when used in an unstructured grid. The results of VOM, being defined in Z-coordinates, are void of coordinate transformation errors. In an arbitrary topography unforced zero-flow remains quiescent in a stratification that only varies in the vertical. VOM due to its depth-independent vertical resolution appears particularly suitable for simulations of ocean-shelf exchange.

Adaptive unstructured grid finite element simulation of two-dimensional magnetotelluric fields for arbitrary surface and seafloor topography

We present an adaptive unstructured triangular grid finite element approach for effectively simulating plane-wave diffusive electromagnetic fields in 2-D conductivity structures. The most striking advantage of irregular grids is their potential to incorporate arbitrary geometries including surface and seafloor topography. Adaptive mesh refinement strategies using an a posteriori error estimator yield most efficient numerical solutions since meshes are only refined where required. We demonstrate the robustness of this approach by comparison with analytical solutions and previously published numerical simulations. Maximum errors may systematically be reduced to, for example, 0.8 per cent for the apparent resistivity and 0.2° in the phase. An additional accuracy study of the thickness of the air layer in E-polarization suggests to keep a minimum thickness depending on lateral conductivity contrasts within the earth. Furthermore, we point out the new quality and flexibility of our simulation technique by addressing two marine magnetotelluric applications. In the first case, we discuss topographic effects associated with a synthetic sinusoidal sea bottom model and in the second case, we show a close-to-reality scenario using real bathymetry data from the East Pacific Rise at 17°S.

Local Rectangular Refinement in Three Dimensions (LRR3D): Development of a Solution-Adaptive Gridding Technique with Application to Convection-Diffusion Problems

The local rectangular refinement (LRR) solution-adaptive gridding method, developed a decade ago to solve coupled nonlinear elliptic partial differential equations in two dimensions, has been extended to three dimensions. Like LRR2D, LRR3D automatically generates orthogonal unstructured adaptive grids, discretizes governing equations using multiple-scale finite differences, and solves the discretized system at all points simultaneously using Newton's method. The computational/programming challenges overcome in developing LRR3D are described. Next, a three-dimensional convection-diffusion-reaction problem with a known solution is used to demonstrate the accuracy and efficiency of LRR3D versus a Newton solver on a structured grid. Finally, natural convection within a tilted differentially heated cubic cavity with perfectly conducting nonisothermal walls is examined for a range of Rayleigh numbers (103 to 4 × 104) and for two inclinations (45° and 90°) of the isothermal walls. Excellent agreement with published computational and experimental results is observed.

Adaptive Numerical Modeling of Natural Convection, Conduction, and Solidification within Mold Cavities

Time-dependent solidification of an aluminum alloy within a graphite mold cavity is adaptively modeled, including effects of latent heat convection, temperature-dependent variation in viscosity during phase change, and mold wall conduction. Three other simpler (but related) applications are also examined. The coupled, nonlinear conservation equations, written using the vorticity–velocity formulation, are solved simultaneously using Newton's method within the context of the local rectangular refinement (LRR) solution-adaptive gridding method. The LRR method automatically generates orthogonal unstructured adaptive grids and incorporates multiple-scale finite differences, producing considerable computational savings without loss of accuracy, compared to similar solution methods on structured grids. For the two (steady) applications for which published results exist, excellent agreement with published computational and experimental results is observed.

Study of the effects of wing sweep on the aerodynamic performance of a blended wing body aircraft

This article presents a study of the effects of wing sweep on the aerodynamic performance of a blended wing body (BWB) aircraft that is based on an aerodynamically optimized design with a fixed planform and pitching moment constraint. Sixteen BWB geometries with varying wing sweep angles ranging from −40° (forward sweep) to 55° sweep were evaluated, while keeping the aerofoil profiles and twist distribution unchanged from the original optimized geometry. This gives some insight into the effects of one of the key planform design parameters. Numerical simulations were carried out using Euler solutions of the flow field with adaptive unstructured grids. Grid sensitivity studies were carried out, along with grid adaptation, on all geometries for solution accuracy. Results show that within 10° −55° sweep, there is significant variation in the lift-to-drag ratio ( LID), whereas for sweep values from −40° to 10°, the L/D ratio remained relatively constant. To maintain the design lift for varying wing sweep, the angle of attack of the aircraft needs to be changed. It was found that this variation followed an approximately parabolic curve, with a minimum incidence of 4.91° at 25° wing sweep. Furthermore, it was seen that the coefficient of moments versus the sweep angle varied almost linearly for all geometries under investigation.

A Godunov-Type Scheme for Atmospheric Flows on Unstructured Grids: Euler and Navier-Stokes Equations

In recent years there has been a growing interest in using Godunov-type methods for atmospheric flow problems. Godunov's unique approach to numerical modeling of fluid flow is characterized by introducing physical reasoning in the development of the numerical scheme (van Leer, 1999). The construction of the scheme itself is based upon the physical phenomenon described by the equation sets. These finite volume discretizations are conservative and have the ability to resolve regions of steep gradients accurately, thus avoiding dispersion errors in the solution. Positivity of scalars (an important factor when considering the transport of microphysical quantities) is also guaranteed by applying the total variation diminishing condition appropriately. This paper describes the implementation of a Godunov-type finite volume scheme based on unstructured adaptive grids for simulating flows on the meso-, micro- and urban-scales. The Harten-Lax-van Leer-Contact (HLLC) approximate Riemann solver used to calculate the Godunov fluxes is described in detail. The higher-order spatial accuracy is achieved via gradient reconstruction techniques after van Leer and the total variation diminishing condition is enforced with the aid of slope-limiters. A multi-stage explicit Runge-Kutta time marching scheme is used for maintaining higher-order accuracy in time. The scheme is conservative and exhibits minimal numerical dispersion and diffusion. The subgrid scale diffusion in the model is parameterized via the Smagorinsky-Lilly turbulence closure. The scheme uses a non-staggered mesh arrangement of variables (all quantities are cell-centered) and requires no explicit filtering for stability. A comparison with exact solutions shows that the scheme can resolve the different types of wave structures admitted by the atmospheric flow equation set. A qualitative evaluation for an idealized test case of convection in a neutral atmosphere is also presented. The scheme was able to simulate the onset of Kelvin-Helmholtz type instability and shows promise in simulating atmospheric flows characterized by sharp gradients without using explicit filtering for numerical stability.

1305 そろばん格子CIP法を用いたBGK方程式の数値解法(2)(OS13-1 高クヌッセン数流れ,OS13 高クヌッセン数流れ,オーガナイズドセッション)

A new numerical scheme to solve the Boltzmann equation (the BGK equation) in phase space for the computations of rarefied gas is described on the basis of the Cubic Interpolated Propagation (CIP) method. The CIP procedure is extended to adaptive unstructured grid system by the Soroban grid. The grid points in velocity space can move dynamically following the spread of velocity space in a spatially localized manner. Such adaptively moving points in velocity space are similar to the particle codes but can provide higher-order-accurate solutions. Numerical solutions obtained by the Soroban-grid CIP are examined and the validity in various Knudsen numbers is proved.

Hurricane track forecasting with OMEGA

In the early days of computing, geophysical fluid dynamics (GFD), predominately numerical weather prediction (NWP), was a dominant factor in the design of computer architecture and algorithms. This early work focused initially on finite difference algorithms on rectangular computational grids and later on spectral methods. After the initial work of von Neumann, Charney, and Arakawa however, the focus shifted from the basic algorithms to improvements in the physical models. Further work on fundamental numerical algorithms shifted to other disciplines—predominately the then emerging aerospace community. As a result, for 40 years, the GFD community has been using numerical techniques that are virtually unchanged. The two primary numerical methodologies that have been used for modeling the atmosphere and the ocean have been spectral methods for global mod- eling and structured rectilinear grids for regional modeling. In 1992, work began on the Operational Multiscale Environment model with Grid Adaptivity (OMEGA), a new atmospheric simulation tool based upon an adaptive unstructured grid. Over the past 14 years, this model has found application to atmospheric dispersion, point weather forecasting, and, the topic of this paper, hurricane track forecasting. The unstructured grid paradigm employed in OMEGA has the advantage of flexibility in providing high resolution where required by either static physical properties (terrain elevation, coastlines, land use) or the changing dynamical situation. This paper provides a description of this new paradigm and presents its use in atmospheric simulation.

Godunov-type solution of the shallow water equations on adaptive unstructured triangular grids

A Godunov-type upwind finite volume solver of the non-linear shallow water equations is described. The shallow water equations are expressed in a hyperbolic conservation law formulation for application to cases where the bed topography is spatially variable. Inviscid fluxes at cell interfaces are computed using Roe's approximate Riemann solver. Second-order accurate spatial calculations of the fluxes are achieved by enhancing the polynomial approximation of the gradients of conserved variables within each cell. Numerical oscillations are curbed by means of a non-linear slope limiter. Time integration is second-order accurate and implicit. The numerical model is based on dynamically adaptive unstructured triangular grids. Test cases include an oblique hydraulic jump, jet-forced flow in a flat-bottomed circular reservoir, wind-induced circulation in a circular basin of non-uniform bed topography and the collapse of a circular dam. The model is found to give accurate results in comparison with published analytical and alternative numerical solutions. Dynamic grid adaptation and the use of a second-order implicit time integration scheme are found to enhance the computational efficiency of the model.

Towards a more efficient dynamic mesh adaptation methodology for continuum discretization in complex engineering problems

A novel and efficient method of adaptive mesh generation, for dynamically adaptive unstructured grids, is proposed. A locally refined triangulation is constructed on a coarse background mesh, subdividing each triangle in the refinement region R into four congruent sub-triangles iteratively, by connecting edge midpoints, until triangles of a prescribed lengthscale are obtained. The unavoidable propagation outside the refinement region R is restricted to a single triangle in the coarse background mesh. The triangles, in the immediate vicinity of region R, are broken down using the concept of iterated function systems, widely used in fractal modeling, by recursive generation of sub-triangles with a gradation towards the region R triangles. A quantitative assessment of the present algorithm proves its superiority over other comparable models reported in the literature. The time cost of the algorithm is linear, and the method can be easily extended to three dimensions.

Application of the multidimensional positive definite advection transport algorithm (MPDATA) to environmental modelling on adaptive unstructured grids

AbstractTwenty years ago, the multidimensional, positive definite, advection transport algorithm was introduced by Smolarkiewicz. Over the two decades since, it has been applied countless times to numerous problems, however almost always on rectilinear grids. One of the few exceptions is the Operational Multiscale Environment model with Grid Adaptivity (OMEGA), an atmospheric simulation system originally designed to simulate atmospheric dispersion in the planetary boundary layer, but since then used for both mesoscale (from meso‐α to meso‐γ) dispersion and weather forecasting. One of the unique aspects of OMEGA is the triangular unstructured grid geometry which leads in a natural way to the creation of a global grid with continuously variable resolution from roughly 100 km over the oceans to less than 10 km over regions of interest. Another unique aspect is the concept of dynamically adapting grid resolution—sometimes also called solution‐adaptive grid resolution. A central element of the modelling system, however, is its advection solver—MPDATA. This paper presents the implementation of MPDATA on an unstructured grid and demonstrates its accuracy and efficiency using analytic and idealized test cases. Copyright © 2005 John Wiley & Sons, Ltd.

A nodal triangle-based spectral element method for the shallow water equations on the sphere

A nodal triangle-based spectral element (SE) method for the shallow water equations on the sphere is presented. The original SE method uses quadrilateral elements and high-order nodal Lagrange polynomials, constructed from a tensor-product of the Legendre–Gauss–Lobatto points. In this work, we construct the high-order Lagrange polynomials directly on the triangle using nodal sets obtained from the electrostatics principle [J.S. Hesthaven, From electrostatics to almost optimal nodal sets for polynomial interpolation in a simplex, SIAM Journal on Numerical Analysis 35 (1998) 655–676] and Fekete points [M.A. Taylor, B.A. Wingate, R.E. Vincent, An algorithm for computing Fekete points in the triangle, SIAM Journal on Numerical Analysis 38 (2000) 1707–1720]. These points have good approximation properties and far better Lebesgue constants than any other nodal set derived for the triangle. By employing triangular elements as the basic building-blocks of the SE method and the Cartesian coordinate form of the equations, we can use any grid imaginable including adaptive unstructured grids. Results for six test cases are presented to confirm the accuracy and stability of the method. The results show that the triangle-based SE method yields the expected exponential convergence and that it can be more accurate than the quadrilateral-based SE method even while using 30–60% fewer grid points especially when adaptive grids are used to align the grid with the flow direction. However, at the moment, the quadrilateral-based SE method is twice as fast as the triangle-based SE method because the latter does not yield a diagonal mass matrix.

A Scalable Spectral Element Eulerian Atmospheric Model (SEE-AM) for NWP: Dynamical Core Tests

A new dynamical core for numerical weather prediction (NWP) based on the spectral element method is presented. This paper represents a departure from previously published work on solving the atmospheric primitive equations in that the horizontal operators are all written, discretized, and solved in 3D Cartesian space. The advantages of using Cartesian space are that the pole singularity that plagues the equations in spherical coordinates disappears; any grid can be used, including latitude–longitude, icosahedral, hexahedral, and adaptive unstructured grids; and the conversion to a semi-Lagrangian formulation is easily achieved. The main advantage of using the spectral element method is that the horizontal operators can be approximated by local high-order elements while scaling efficiently on distributed-memory computers. In order to validate the 3D global atmospheric spectral element model, results are presented for seven test cases: three barotropic tests that confirm the exponential accuracy of the horizontal operators and four baroclinic test cases that validate the full 3D primitive hydrostatic equations. These four baroclinic test cases are the Rossby–Haurwitz wavenumber 4, the Held–Suarez test, and the Jablonowski–Williamson balanced initial state and baroclinic instability tests. Comparisons with four operational NWP and climate models demonstrate that the spectral element model is at least as accurate as spectral transform models while scaling linearly on distributed-memory computers.

Simulation of the dispersion of nuclear contamination using an adaptive Eulerian grid model

Application of an Eulerian model using layered adaptive unstructured grids coupled to a meso-scale meteorological model is presented for modelling the dispersion of nuclear contamination following the accidental release from a single but strong source to the atmosphere. The model automatically places a finer resolution grid, adaptively in time, in regions were high spatial numerical error is expected. The high-resolution grid region follows the movement of the contaminated air over time. Using this method, grid resolutions of the order of 6 km can be achieved in a computationally effective way. The concept is illustrated by the simulation of hypothetical nuclear accidents at the Paks NPP, in Central Hungary. The paper demonstrates that the adaptive model can achieve accuracy comparable to that of a high-resolution Eulerian model using significantly less grid points and computer simulation time.

Simulation of the dispersion of nuclear contamination using an adaptive Eulerian grid model

Application of an Eulerian model using layered adaptive unstructured grids coupled to a meso-scale meteorological model is presented for modelling the dispersion of nuclear contamination following the accidental release from a single but strong source to the atmosphere. The model automatically places a finer resolution grid, adaptively in time, in regions were high spatial numerical error is expected. The high-resolution grid region follows the movement of the contaminated air over time. Using this method, grid resolutions of the order of 6 km can be achieved in a computationally effective way. The concept is illustrated by the simulation of hypothetical nuclear accidents at the Paks NPP, in Central Hungary. The paper demonstrates that the adaptive model can achieve accuracy comparable to that of a high-resolution Eulerian model using significantly less grid points and computer simulation time.