Adaptive Mesh Refinement Framework Research Articles

Eulerian adaptive finite-difference method for high-velocity impact and penetration problems

Owing to the complex processes involved, faithful prediction of high-velocity impact events demands a simulation method delivering efficient calculations based on comprehensively formulated constitutive models. Such an approach is presented herein, employing a weighted essentially non-oscillatory (WENO) method within an adaptive mesh refinement (AMR) framework for the numerical solution of hyperbolic partial differential equations. Applied widely in computational fluid dynamics, these methods are well suited to the involved locally non-smooth finite deformations, circumventing any requirement for artificial viscosity functions for shock capturing. Application of the methods is facilitated through using a model of solid dynamics based upon hyper-elastic theory comprising kinematic evolution equations for the elastic distortion tensor. The model for finite inelastic deformations is phenomenologically equivalent to Maxwell’s model of tangential stress relaxation. Closure relations tailored to the expected high-pressure states are proposed and calibrated for the materials of interest. Sharp interface resolution is achieved by employing level-set functions to track boundary motion, along with a ghost material method to capture the necessary internal boundary conditions for material interactions and stress-free surfaces. The approach is demonstrated for the simulation of high velocity impacts of steel projectiles on aluminium target plates in two and three dimensions.

A Block-Structured Adaptive Mesh Refinement Solver for Morphodynamic Modeling

ABSTRACT Cowles, G.W., 2013. A block-structured adaptive mesh refinement solver for morphodynamic modeling. An adaptive solver for two-dimensional horizontal morphodynamic modeling studies is presented. The method employs an established augmented Riemann solver to compute unsteady, two-dimensional flows over arbitrary topography. An Exner equation is used to model the evolution of the bed using several common load formulas. The morphodynamic coupling is stabilized using an upwind approach. The scheme is implemented in a highly scalable block-structured adaptive mesh refinement framework. The resulting solver is well balanced and resolves the dry state while maintaining nonnegative depth. Idealized tests include comparison with exact Riemann solutions for dam break problems and an analytical solution for the migration of a one-dimensional sinusoidal bedform. Practical tests include the migration of a Gaussian perturbation and the evolution of flood and ebb shoals in a tidal inlet. The solver is benchmarked...

A semi-Lagrangian AMR scheme for 2D transport problems in conservation form

In this paper, we construct a semi-Lagrangian (SL) Adaptive-Mesh-Refinement (AMR) solver for 1D and 2D transport problems in conservation form. First, we describe the à-la-Harten AMR framework: the adaptation process selects a hierarchical set of grids with different resolutions depending on the features of the integrand function, using as criteria the point value prediction via interpolation from coarser meshes, and the appearance of large gradients. We integrate in time by reconstructing at the feet of the characteristics through the Point-Value Weighted Essentially Non-Oscillatory (PV-WENO) interpolator. We propose, then, an extension to the 2D setting by making the time integration dimension-by-dimension thanks to a Strang splitting. We discuss the quality of the results and the speedup with respect to a Fixed Mesh (FM) strategy through the following benchmark tests: in 1D, constant and variable-coefficient advections; in 2D, the 1D Vlasov–Poisson system (2D in the phase space) for the case of constant-coefficient advections, and, for the case of variable-coefficient advections, the deformation flow and the guiding-center model.

Discontinuous Galerkin methods for problems with Dirac delta source

In this article we study discontinuous Galerkin finite element discretizations of linear second-order elliptic partial differential equations with Dirac delta right-hand side. In particular, assuming that the underlying computational mesh is quasi-uniform, we derive an a priori bound on the error measured in terms of the L^2-norm. Additionally, we develop residual-based a posteriori error estimators that can be used within an adaptive mesh refinement framework. Numerical examples for the symmetric interior penalty scheme are presented which confirm the theoretical results.

Tabulated chemistry approaches for laminar flames: Evaluation of flame-prolongation of ILDM and flamelet methods

The present study considers the performance of tabulation methods for numerical simulation of complex chemical kinetics in laminar combusting flows and compares their predictions to results obtained by direct calculation. Two tabulation methods are considered: the Flame Prolongation of Intrinsic low-dimensional manifold (FPI) method and Steady Laminar Flamelet Model (SLFM). The FPI method is of current interest as it is a potentially unifying approach capable of dealing with both premixed and non-premixed flames for gaseous fuels. SLFM tabulation methods are popular for non-premixed flames and form a good basis for comparing the performance of the FPI approach. The performance of each method is also evaluated by comparing the results to the direct simulation of the laminar flames using two chemical kinetic schemes: simplified chemistry involving five species and one reaction and detailed chemistry involving 53 species and 325 reaction steps. As part of the evaluation process, the computational cost of each method is also assessed. The laminar flames considered in this study include: freely propagating laminar premixed flames, a two-dimensional axisymmetric methane–air opposed-jet diffusion flame, and a two-dimensional axisymmetric methane–air co-flow diffusion flame. Both tabulation methods are implemented in a parallel adaptive mesh refinement (AMR) framework for solving the complete set of governing partial differential equations. These equations are solved using a fully-coupled finite-volume formulation on body-fitted multi-block quadrilateral mesh. Significant improvements in terms of reduced computational requirements, as measured by both storage and processing time, are demonstrated for the tabulated methods.

An adaptive mesh refinement algorithm for compressible two-phase flow in porous media

We describe a second-order accurate sequential algorithm for solving two-phase multicomponent flow in porous media. The algorithm incorporates an unsplit second-order Godunov scheme that provides accurate resolution of sharp fronts. The method is implemented within a block structured adaptive mesh refinement (AMR) framework that allows grids to dynamically adapt to features of the flow and enables efficient parallelization of the algorithm. We demonstrate the second-order convergence rate of the algorithm and the accuracy of the AMR solutions compared to uniform fine-grid solutions. The algorithm is then used to simulate the leakage of gas from a Liquified Petroleum Gas (LPG) storage cavern, demonstrating its capability to capture complex behavior of the resulting flow. We further examine differences resulting from using different relative permeability functions.

CONSTRAINED-TRANSPORT MAGNETOHYDRODYNAMICS WITH ADAPTIVE MESH REFINEMENT IN CHARM

We present the implementation of a three-dimensional, second order accurate Godunov-type algorithm for magneto-hydrodynamic (MHD), in the adaptive-mesh-refinement (AMR) cosmological code {\tt CHARM}. The algorithm is based on the full 12-solve spatially unsplit Corner-Transport-Upwind (CTU) scheme. The fluid quantities are cell-centered and are updated using the Piecewise-Parabolic-Method (PPM), while the magnetic field variables are face-centered and are evolved through application of the Stokes theorem on cell edges via a Constrained-Transport (CT) method. The multidimensional MHD source terms required in the predictor step for high-order accuracy are applied in a simplified form which reduces their complexity in three dimensions without loss of accuracy or robustness. The algorithm is implemented on an AMR framework which requires specific synchronization steps across refinement levels. These include face-centered restriction and prolongation operations and a {\it reflux-curl} operation, which maintains a solenoidal magnetic field across refinement boundaries. The code is tested against a large suite of test problems, including convergence tests in smooth flows, shock-tube tests, classical two- and three-dimensional MHD tests, a three-dimensional shock-cloud interaction problem and the formation of a cluster of galaxies in a fully cosmological context. The magnetic field divergence is shown to remain negligible throughout.

Adaptive mesh refinement based on high order finite difference WENO scheme for multi-scale simulations

In this paper, we propose a finite difference AMR-WENO method for hyperbolic conservation laws. The proposed method combines the adaptive mesh refinement (AMR) framework [4,5] with the high order finite difference weighted essentially non-oscillatory (WENO) method in space and the total variation diminishing (TVD) Runge–Kutta (RK) method in time (WENO-RK) [18,10] by a high order coupling. Our goal is to realize mesh adaptivity in the AMR framework, while maintaining very high (higher than second) order accuracy of the WENO-RK method in the finite difference setting. The high order coupling of AMR and WENO-RK is accomplished by high order prolongation in both space (WENO interpolation) and time (Hermite interpolation) from coarse to fine grid solutions, and at ghost points. The resulting AMR-WENO method is accurate, robust and efficient, due to the mesh adaptivity and very high order spatial and temporal accuracy. We have experimented with both the third and the fifth order AMR-WENO schemes. We demonstrate the accuracy of the proposed scheme using smooth test problems, and their quality and efficiency using several 1D and 2D nonlinear hyperbolic problems with very challenging initial conditions. The AMR solutions are observed to perform as well as, and in some cases even better than, the corresponding uniform fine grid solutions. We conclude that there is significant improvement of the fifth order AMR-WENO over the third order one, not only in accuracy for smooth problems, but also in its ability in resolving complicated solution structures, due to the very low numerical diffusion of high order schemes. In our work, we found that it is difficult to design a robust AMR-WENO scheme that is both conservative and high order (higher than second order), due to the mass inconsistency of coarse and fine grid solutions at the initial stage in a finite difference scheme. Resolving these issues as well as conducting comprehensive evaluation of computational efficiency constitute our future work.

High‐Resolution Numerical Simulation and Analysis of Mach ReflectionStructures in Detonation Waves in Low‐Pressure H2–O2–Ar Mixtures: A Summary of Results Obtained with the Adaptive Mesh Refinement Framework AMROC

Numerical simulation can be key to the understanding of the multidimensional nature of transient detonation waves. However, the accurate approximation of realistic detonations is demanding as a wide range of scales needs to be resolved. This paper describes a successful solution strategy that utilizes logically rectangular dynamically adaptive meshes. The hydrodynamic transport scheme and the treatment of the nonequilibrium reaction terms are sketched. A ghost fluid approach is integrated into the method to allow for embedded geometrically complex boundaries. Large‐scale parallel simulations of unstable detonation structures of Chapman‐Jouguet detonations in low‐pressure hydrogen‐oxygen‐argon mixtures demonstrate the efficiency of the described techniques in practice. In particular, computations of regular cellular structures in two and three space dimensions and their development under transient conditions, that is, under diffraction and for propagation through bends are presented. Some of the observed patterns are classified by shock polar analysis, and a diagram of the transition boundaries between possible Mach reflection structures is constructed.

ALE-AMR: A new 3D multi-physics code for modeling laser/target effects

We have developed a new 3D multi-physics multi-material code, ALE-AMR, for modeling laser/target effects including debris/shrapnel generation. The code combines Arbitrary Lagrangian Eulerian (ALE) hydrodynamics with Adaptive Mesh Refinement (AMR) to connect the continuum to microstructural regimes. The code is unique in its ability to model hot radiating plasmas and cold fragmenting solids. New numerical techniques were developed for many of the physics packages to work efficiency on a dynamically moving and adapting mesh. A flexible strength/failure framework allows for pluggable material models. Material history arrays are used to store persistent data required by the material models, for instance, the level of accumulated damage or the evolving yield stress in J2 plasticity models. We model ductile metals as well as brittle materials such as Si, Be, and B4C. We use interface reconstruction based on volume fractions of the material components within mixed zones and reconstruct interfaces as needed. This interface reconstruction model is also used for void coalescence and fragmentation. The AMR framework allows for hierarchical material modeling (HMM) with different material models at different levels of refinement. Laser rays are propagated through a virtual composite mesh consisting of the finest resolution representation of the modeled space. A new 2nd order accurate diffusion solver has been implemented for the thermal conduction and radiation transport packages. The code is validated using laser and x-ray driven spall experiments in the US and France. We present an overview of the code and simulation results.

High-resolution simulation and characterization of density-driven flow in CO 2 storage in saline aquifers

Simulations are routinely used to study the process of carbon dioxide (CO 2) sequestration in saline aquifers. In this paper, we describe the modeling and simulation of the dissolution–diffusion–convection process based on a total velocity splitting formulation for a variable-density incompressible single-phase model. A second-order accurate sequential algorithm, implemented within a block-structured adaptive mesh refinement (AMR) framework, is used to perform high-resolution studies of the process. We study both the short-term and long-term behaviors of the process. It is found that the onset time of convection follows closely the prediction of linear stability analysis. In addition, the CO 2 flux at the top boundary, which gives the rate at which CO 2 gas dissolves into a negatively buoyant aqueous phase, will reach a stabilized state at the space and time scales we are interested in. This flux is found to be proportional to permeability, and independent of porosity and effective diffusivity, indicative of a convection-dominated flow. A 3D simulation further shows that the added degrees of freedom shorten the onset time and increase the magnitude of the stabilized CO 2 flux by about 25%. Finally, our results are found to be comparable to results obtained from TOUGH2-MP.

In vivo simulation of shock wave lithotripsy: Wave focusing in inhomogeneous materials.

Shock wave generation and focusing in electrohydraulic, electromagnetic, and piezoelectric lithotripters are simulated numerically. Inhomogeneities associated with wave propagation in vivo are modeled using data from The Visible Human Project. The approximate time‐domain relaxation model of Yang and Cleveland (2005) was used to simulate attenuation and dispersion in the tissuelike media. The simulations utilize a MUSCL‐type shock capturing scheme with adaptive mesh refinement (AMR). In vitro focal region pressure measurements compare favorably with available experimental data for each lithotripter type. Preliminary in vivo simulation results show significant dispersion associated with the nonuniform sound speed, including increases of up to 50% in focal pressure amplitudes and the presence of multiple pressure minima/maxima in the focal waveform. We will also discuss the extension of the AMR framework to account for cavitation via a previously developed ensemble‐averaged continuous two‐phase flow model.

Relativistic Hydrodynamic Flows Using Spatial and Temporal Adaptive Structured Mesh Refinement

Astrophysical relativistic flow problems require high-resolution three-dimensional (3D) numerical simulations. In this paper, we describe a new parallel 3D code for simulations of special relativistic hydrodynamics (SRHD) using both spatially and temporally structured adaptive mesh refinement (AMR). We used the method of lines to discretize the SRHD equations spatially and a total variation diminishing (TVD) Runge-Kutta scheme for time integration. For spatial reconstruction, we have implemented piecewise linear method (PLM), piecewise parabolic method (PPM), third-order convex essentially nonoscillatory (CENO) and third- and fifth-order weighted essentially nonoscillatory (WENO) schemes. Flux is computed using either direct flux reconstruction or approximate Riemann solvers including HLL, modified Marquina flux, local Lax-Friedrichs flux formulas, and HLLC. The AMR part of the code is built on top of the cosmological Eulerian AMR code enzo. We discuss the coupling of the AMR framework with the relativistic solvers. Via various test problems, we emphasize the importance of resolution studies in relativistic flow simulations because extremely high resolution is required especially when shear flows are present in the problem. We also present the results of two 3D simulations of astrophysical jets: AGN jets and GRB jets. Resolution study of those two cases further highlights the need of high resolutions to calculate accurately relativistic flow problems.

A high order Godunov scheme with constrained transport and adaptive mesh refinement for astrophysical and geophysical MHD

We propose to extend the well-known MUSCL-Hancock scheme for Euler equations to the induction equation modeling the magnetic field evolution in kinematic dynamo problems. The scheme is based on an integral form of the underlying conservation law which, in our formulation, results in a “finite-surface” scheme for the induction equation. This naturally leads to the well-known “constrained transport” method, with additional continuity requirement on the magnetic field representation. The second ingredient in the MUSCL scheme is the predictor step that ensures second order accuracy both in space and time. We explore specific constraints that the mathematical properties of the induction equations place on this predictor step, showing that three possible variants can be considered. We show that the most aggressive formulations reach the same level of accuracy than the other ones, at a lower computational cost. More interestingly, these schemes are compatible with the Adaptive Mesh Refinement (AMR) framework. It has been implemented in the AMR code RAMSES. It offers a novel and efficient implementation of a second order scheme for the induction equation. The scheme is then adaptated to solve for the full MHD equations using the same methodology. Through a series of test problems, we illustrate the performances of this new code using two different MHD Riemann solvers (Lax–Friedrich and Roe) and the need of the Adaptive Mesh Refinement capabilities in some cases. Finally, we show its versatility by applying it to the ABC dynamo problem and to the collapse of a magnetized cloud core.

Algorithm refinement for the stochastic Burgers’ equation

In this paper, we develop an algorithm refinement (AR) scheme for an excluded random walk model whose mean field behavior is given by the viscous Burgers’ equation. AR hybrids use the adaptive mesh refinement framework to model a system using a molecular algorithm where desired while allowing a computationally faster continuum representation to be used in the remainder of the domain. The focus in this paper is the role of fluctuations on the dynamics. In particular, we demonstrate that it is necessary to include a stochastic forcing term in Burgers’ equation to accurately capture the correct behavior of the system. The conclusion we draw from this study is that the fidelity of multiscale methods that couple disparate algorithms depends on the consistent modeling of fluctuations in each algorithm and on a coupling, such as algorithm refinement, that preserves this consistency.

A high order Godunov scheme with constrained transport and adaptive mesh refinement for astrophysical magnetohydrodynamics

Aims. In this paper, we present a new method to perform numerical simulations of astrophysical MHD flows using the Adaptive Mesh Refinement framework and Constrained Transport. Methods. The algorithm is based on a previous work in which the MUSCL-Hancock scheme was used to evolve the induction equation. In this paper, we detail the extension of this scheme to the full MHD equations and discuss its properties. Results. Through a series of test problems, we illustrate the performances of this new code using two different MHD Riemann solvers (Lax-Friedrich and Roe) and the need of the Adaptive Mesh Refinement capabilities in some cases. Finally, we show its versatility by applying it to two completely different astrophysical situations well studied in the past years: the growth of the magnetorotational instability in the shearing box and the collapse of magnetized cloud cores.Conclusions. We have implemented a new Godunov scheme to solve the ideal MHD equations in the AMR code RAMSES. We have shown that it results in a powerful tool that can be applied to a great variety of astrophysical problems, ranging from galaxies formation in the early universe to high resolution studies of molecular cloud collapse in our galaxy.

Kinematic dynamos using constrained transport with high order Godunov schemes and adaptive mesh refinement

We propose to extend the well-known MUSCL-Hancock scheme for Euler equations to the induction equation modeling the magnetic field evolution in kinematic dynamo problems. The scheme is based on an integral form of the underlying conservation law which, in our formulation, results in a “finite-surface” scheme for the induction equation. This naturally leads to the well-known “constrained transport” method, with additional continuity requirement on the magnetic field representation. The second ingredient in the MUSCL scheme is the predictor step that ensures second order accuracy both in space and time. We explore specific constraints that the mathematical properties of the induction equations place on this predictor step, showing that three possible variants can be considered. We show that the most aggressive formulations (referred to as C-MUSCL and U-MUSCL) reach the same level of accuracy as the other one (referred to as Runge–Kutta), at a lower computational cost. More interestingly, these two schemes are compatible with the adaptive mesh refinement (AMR) framework. It has been implemented in the AMR code RAMSES. It offers a novel and efficient implementation of a second order scheme for the induction equation. We have tested it by solving two kinematic dynamo problems in the low diffusion limit. The construction of this scheme for the induction equation constitutes a step towards solving the full MHD set of equations using an extension of our current methodology.

Adaptive Mesh Refinement and Multilevel Iteration for Flow in Porous Media

An adaptive local mesh refinement algorithm originally developed for unsteady gas dynamics by M. J. Berger is extended to incompressible flow in porous media. Multilevel iteration and domain decomposition methods are introduced to accommodate the elliptic/parabolic aspects of the flow equations. The algorithm is applied to a two-phase polymer flooding model consisting of a system of nonlinear hyperbolic mass conservation equations coupled to an elliptic pressure equation. While the various numerical methods used have been presented previously, our emphasis is on their consistent combination within the adaptive mesh refinement framework to treat important problems in porous media flow. To achieve efficient, easily maintainable code, we have exploited the features of object-oriented programming for the overall program structure and data management. Examples of algorithmic performance and computational results are provided.