Adaptive Mesh Refinement Algorithm Research Articles

Spatial adaptivity of the SAAF and Weighted Least Squares (WLS) forms of the neutron transport equation using constraint based, locally refined, isogeometric analysis (IGA) with dual weighted residual (DWR) error measures

This paper describes a methodology that enables NURBS (Non-Uniform Rational B-spline) based Isogeometric Analysis (IGA) to be locally refined. The methodology is applied to continuous Bubnov-Galerkin IGA spatial discretisations of second-order forms of the neutron transport equation. In particular this paper focuses on the self-adjoint angular flux (SAAF) and weighted least squares (WLS) equations. Local refinement is achieved by constraining degrees of freedom on interfaces between NURBS patches that have different levels of spatial refinement. In order to effectively utilise constraint based local refinement, adaptive mesh refinement (AMR) algorithms driven by a heuristic error measure or forward error indicator (FEI) and a dual weighted residual (DWR) or goal-based error measure (WEI) are derived. These utilise projection operators between different NURBS meshes to reduce the amount of computational effort required to calculate the error indicators. In order to apply the WEI to the SAAF and WLS second-order forms of the neutron transport equation the adjoint of these equations are required. The physical adjoint formulations are derived and the process of selecting source terms for the adjoint neutron transport equation in order to calculate the error in a given quantity of interest (QoI) is discussed. Several numerical verification benchmark test cases are utilised to investigate how the constraint based local refinement affects the numerical accuracy and the rate of convergence of the NURBS based IGA spatial discretisation. The nuclear reactor physics verification benchmark test cases show that both AMR algorithms are superior to uniform refinement with respect to accuracy per degree of freedom. Furthermore, it is demonstrated that for global QoI the FEI driven AMR and WEI driven AMR produce similar results. However, if local QoI are desired then WEI driven AMR algorithm is more computationally efficient and accurate per degree of freedom.

An octree-based immersogeometric approach for modeling inertial migration of particles in channels

In this paper, we develop a scalable, adaptively refined, octree-based finite element approach with immersogeometric analysis to track inertial migration of particles in microchannels. Fluid physics is modeled using a residual-based variational multiscale method, and the particle movement is modeled as rigid body motion. A parallel, hierarchically refined octree mesh is employed as the background mesh, on which a variationally consistent immersogeometric formulation is adopted for tracking the particle motion in the fluid. Adaptations of immersogeometric analysis on an octree background mesh are developed to enable efficient searching of background element for a given surface quadrature point, as well as a distribution of surface quadrature points over processors to reduce memory overhead and better parallelize the surface assembly. An octree-based adaptive mesh refinement algorithm adapted to in-out test in the immersogeometric approach is also developed. The validation of our octree-based immersogeometric approach is carried out using a benchmark case of a sphere settling in quiescent fluid, with good agreement presented. In addition, good strong (and weak) scalability on supercomputing resources for this benchmark case up to 16,384 processes is demonstrated. The proposed method is further deployed for exploring particle migration in straight and converging-diverging channels. This example illustrates the potential of the octree-based immersogeometric approach for efficiently tracking particle motion in complex channel flows – a problem with a diverse array of applications.

GPU-native gas dynamic solver on octree-based AMR grids

GPU implementation of the Godunov-type gas dynamic solver on three-dimensional octree-based grids is proposed. In this implementation, all dynamic adaptive mesh refinement (AMR) procedures are performed entirely on GPU as opposed to most of other implementations where grid metadata is transferred to CPU for further updating. Conducted bench tests show that eliminating regular CPU–GPU grid data transferring in the proposed parallel AMR algorithms makes it possible to reduce the grid modification overhead to lower than 2% on Nvidia Tesla V100. The results concerned with both CUDA and OpenMP implementation performance are also presented.

Adaptive Mesh Refinement Simulations of Gas Dynamic Flows on Hybrid Meshes

An adaptive mesh refinement algorithm for simulations of the Navier–Stokes equations on unstructured hybrid meshes is presented. Numerical results for compressible supersonic flow around a sphere are given.

Higher-order mesh generation using CISAMR: A case study on bias in presentation and interpretation of results

A modified version of the Conforming to Interface Structured Adaptive Mesh Refinement (CISAMR) algorithm is presented for the construction of higher-order Lagrangian and NURBS-enhanced (NE) finite element meshes. CISAMR non-iteratively transforms a structured grid into a conforming mesh with an upper bound of three on elements aspect ratio. In this work, we introduce new algorithmic aspects for generating higher-order Lagrangian and NE meshes using CISAMR. For each type of element, a comprehensive study is then provided on the performance of first, second, and third-order meshes for solving linear elastic problems with smooth and oscillatory curvilinear edges. Local gradient recovery error and computational cost are used as performance metrics in these examples. Outcomes are then used as a case study to elucidate common sources of bias in presenting and interpreting results. In particular, we show how bias, whether purposeful or not, could lead to misleading conclusions regarding the performance of a method.

An efficient Pseudo-spectral based phase field method for dendritic solidification

In the current study, Fourier Pseudo-spectral based dealiasing scheme Random Phase Shift Method (RPSM) using the RK4 temporal scheme is implemented for simulating some benchmark problems of Phase Field based pure metal solidification. This high-resolution scheme is capable of resolving the thin solid-liquid interface accurately without using Adaptive Mesh Refinement (AMR) algorithm even using just adequate number of grid points. Both the conservative and non-conservative forms of the Phase Field equations are studied. It is found that the conservative RPSM scheme gives better computational performances as indicated by the lower values of average CPU time and different error norms calculated for the Phase Field variable. The RPSM based Phase Field model is applied for simulating dendritic solidification and it is found that dendrite shape and dimensionless tip velocity agree well with the existing literature. The RK4 based RPSM scheme is also compared with a 10th order exponential Fourier smoothing Filter in the evolution of four arm dendritic structure and it is observed that the conservative Phase Field equations with filtering scheme display numerical instabilities at later simulation time, while both forms of the RPSM scheme show stable and accurate solutions. Subsequently, the capability of the RPSM based Phase Field model in simulating microstructure evolution consisting of multiple dendrites with random location and orientation is demonstrated. The present study shows that the RPSM based Phase Field model computes dendritic growth accurately and efficiently.

Direct numerical simulation of multiscale flow physics of binary droplet collision

Direct numerical simulations are performed to investigate the multiscale flow physics of binary droplet collision over a wide range of Weber numbers and impact factors. All possible collision outcomes, including bouncing (both head-on and off-center), coalescence, reflexive separation, and stretching separation, are considered. The theoretical formulation is based on a complete set of conservation equations for both the liquid and gas phases. An improved volume-of-fluid technique, which is augmented by an adaptive mesh refinement algorithm, is used to track the liquid/gas interface. Several local refinement criteria are validated and employed to improve the computational accuracy and efficiency substantially. In particular, a thickness-based refinement technique is implemented for treating cases involving extremely thin gas films between droplets. The smallest numerical grid is ∼10 nm, which is on the order of 10−5 times the initial droplet diameter. A photorealistic visualization technique is employed to gain direct insights into the detailed collision dynamics, including both the shape evolution and mass relocation. The numerical framework allows us to systematically investigate the underlying mechanisms and processes, such as gas-film drainage and energy and mass transfer, at scales sufficient to resolve the near-field dynamics during droplet collision. The nonmonotonic transition of bouncing and merging outcomes for head-on collision is identified by varying the Weber number over two orders of magnitude. A geometric relation defining the droplet interactions is developed. Analytical models are also established to predict the mass transfer between colliding droplets.

A Posteriori Error Estimates for Hughes Stabilized SUPG Technique and Adaptive Refinement for a Convection-Diffusion Problem

The motive of the present work is to propose an adaptive numerical technique for singularly perturbed convection-diffusion problem in two dimensions. It has been observed that for small singular perturbation parameter, the problem under consideration displays sharp interior or boundary layers in the solution which cannot be captured by standard numerical techniques. In the present work, Hughes stabilization strategy along with the streamline upwind/Petrov-Galerkin (SUPG) method has been proposed to capture these boundary layers. Reliable a posteriori error estimates in energy norm on anisotropic meshes have been developed for the proposed scheme. But these estimates prove to be dependent on the singular perturbation parameter. Therefore, to overcome the difficulty of oscillations in the solution, an efficient adaptive mesh refinement algorithm has been proposed. Numerical experiments have been performed to test the efficiency of the proposed algorithm.

Modeling gaseous non-reactive flow in a lean direct injection gas turbine combustor through an advanced mesh control strategy

Fuel efficiency improvement and harmful emissions reduction are the main motivations for the development of gas turbine combustors. Numerical computational fluid dynamics (CFD) simulations of these devices are usually computationally expensive since they imply a multi-scale problem. In this work, gaseous non-reactive unsteady Reynolds-Averaged Navier–Stokes and large eddy simulations of a gaseous-fueled radial-swirled lean direct injection combustor have been carried out through CONVERGE™ CFD code by solving the complete inlet flow path through the swirl vanes and the combustor. The geometry considered is the gaseous configuration of the CORIA lean direct injection combustor, for which detailed measurements are available. The emphasis of the work is placed on the demonstration of the CONVERGE™ applicability to the multi-scale gas turbine engines field and the determination of an optimal mesh strategy through several grid control tools (i.e., local refinement, adaptive mesh refinement) allowing the exploitation of its automatic mesh generation against traditional fixed mesh approaches. For this purpose, the normalized mean square error has been adopted to quantify the accuracy of turbulent numerical statistics regarding the agreement with the experimental database. Furthermore, the focus of the work is to study the behavior when coupling several large eddy simulation sub-grid scale models (i.e., Smagorinsky, Dynamic Smagorinsky, and Dynamic Structure) with the adaptive mesh refinement algorithm through the evaluation of its specific performances and predictive capabilities in resolving the spatial-temporal scales and the intrinsically unsteady flow structures generated within the combustor. This investigation on the main non-reacting swirling flow characteristics inside the combustor provides a suitable background for further studies on combustion instability mechanisms.

Goal-Oriented Error Estimation for the Method of Moments to Compute Antenna Impedance

A posteriori error estimators typically approximate the locally measured error in a numerical solution. These error distributions can be used to drive adaptive mesh refinement algorithms. Goal-oriented error estimation is concerned with deriving error estimates for chosen goal quantities, based on solving dual (adjoint) problems. A goal-oriented error estimator is formulated for the electric field integral equation (EFIE)-based method of moments (MoM), for computation of antenna array multiport impedance parameters. Estimated error distributions are presented and used to drive a single $h$ -refinement cycle. Performance is assessed against uniformly refined reference results. It is demonstrated that the scheme can significantly increase the efficiency of the MoM for computing the goal quantities. The estimator can be formulated for other goals, such as observed electromagnetic fields.

A posteriori error analysis and adaptive non-intrusive numerical schemes for systems of random conservation laws

This article considers one-dimensional random systems of hyperbolic conservation laws. Existence and uniqueness of random entropy admissible solutions for initial value problems of conservation laws, which involve random initial data and random flux functions, are established. Based on these results an a posteriori error analysis for a numerical approximation of the random entropy solution is presented. For the stochastic discretization, a non-intrusive approach, namely the Stochastic Collocation method is used. The spatio-temporal discretization relies on the Runge–Kutta Discontinuous Galerkin method. The a posteriori estimator is derived using smooth reconstructions of the discrete solution. Combined with the relative entropy stability framework this yields computable error bounds for the entire space-stochastic discretization error. The estimator admits a splitting into a stochastic and a deterministic (space-time) part, allowing for a novel residual-based space-stochastic adaptive mesh refinement algorithm. The scaling properties of the residuals are investigated and the efficiency of the proposed adaptive algorithms is illustrated in various numerical examples.

Optimized wavelet‐based adaptive mesh refinement algorithm for numerical modeling of three‐dimensional global‐scale atmospheric chemical transport

AbstractSubstantial numerical difficulties associated with the computational modeling of multiscale global atmospheric chemical transport impose severe limitations on the spatial resolution of nonadaptive fixed grids. The crude spatial discretization introduces a large amount of numerical diffusion into the system, which, in combination with strong flow stretching, causes large numerical errors. To resolve this issue, we have developed an optimized wavelet‐based adaptive mesh refinement (OWAMR) method. The OWAMR is a three‐dimensional adaptive method that introduces a fine grid dynamically only in the regions where small spatial structures occur. The algorithm uses a new two‐parameter adaptation criterion that significantly (by factors between 1.5 and 2.7) reduces the number of grid points compared with the more conventional one‐parameter grid adaptation used by wavelet‐based adaptive techniques and high‐order upwind schemes, which enable one to increase the accuracy of approximation of the advection operator substantially. It has been shown that the method simulates the dynamics of a pollution plume that travels on a global scale, producing less than 3% error. To achieve such accuracy, conventional three‐dimensional nonadaptive techniques would require five orders of magnitude more computational resources. Therefore, the method provides a realistic opportunity to model accurately a variety of the most demanding multiscale problems in the area of atmospheric chemical transport, which are difficult or impossible to simulate on existing computational facilities with conventional fixed‐grid techniques.

Softness-based adaptive mesh refinement algorithm for soft tissue deformation

Soft tissue and organ modeling is the most critical function of any virtual surgical system. This study proposes a softness-based adaptive mesh refinement algorithm to simultaneously ensure realistic and real-time soft tissue simulation. The algorithm was constructed to consider that in a virtual surgery scenario, the surgical sites involve large deformation and thus require high simulation precision, whereas the nonsurgical sites involve small deformations and thus require low simulation precision. This study used the stomach lining as an example, applying mesh refinement in the deformation sites of the stomach lining to enhance the accuracy of the simulations. In addition, low mesh models were adopted for nonsurgical sites to ensure computing efficiency.

Robust and efficient prediction of the collection efficiency in icing accretion simulation for 3D complex geometries using the Lagrangian approach I: an adaptive interpolation method based on the restricted radial basis functions

As a preliminary step in the icing accretion simulation, prediction of the collection efficiency is important for the icing analysis. At the present time, there are two different ways to accomplish this task, namely, the Eulerian and Lagrangian approach. The former could be easily implemented to 3D complex geometries and widely used right now because it can be obtained together with the Eulerian computation of the flow field. But it requires very much computational costs include both memory and CPU time. On the other hand, the latter has great potential to achieve high efficiency. However, its efficiency has been demonstrated only in simple cases. To facilitate the implementation of the Lagrangian algorithm for 3D complicated configurations, some new techniques are proposed or introduced into the Lagrangian framework in the current research. The first is a new δ form of the local collection efficiency. In this new form, four water droplets are placed very close together to ensure that all of them could hit the wall, so that the collection efficiency could be evaluated robustly for 3D cases. The second is that the adaptive mesh refinement (AMR) algorithm is introduced into the computation of droplets trajectories. The AMR method could capture the droplets impingement limits correctly in the Lagrangian approach. The third is that the interpolation algorithm based on the radial basis functions (RBF) is employed in the present paper to transfer the collection efficiency evaluated at the impingement positions to the CFD (computational fluid dynamics) mesh. To prevent the un-impinged area from being immersed by the interpolation, the restricted RBF method developed by the authors for the mesh motion is also adopted here. The above simple but useful techniques are synthetically used in the Lagrangian method and lead to an algorithm which is suitable for 3D complex configurations. Some cases include several 3D complex models are used here to validate the performances of the proposed approach. These examples show that the algorithms developed in our research can be efficiently applied to 3D complicated shapes and provide reliable results.

A Posteriori Error Estimate and Adaptive Mesh Refinement Algorithm for Atomistic/Continuum Coupling with Finite Range Interactions in Two Dimensions

In this paper, we develop the residual based a posteriori error estimates and the corresponding adaptive mesh refinement algorithm for atomistic/continuum (a/c) coupling with finite range interactions in two dimensions. We have systematically derived a new explicitly computable stress tensor formula for finite range interactions. In particular, we use the geometric reconstruction based consistent atomistic/continuum (GRAC) coupling scheme, which is optimal if the continuum model is discretized by $P^1$ finite elements. The numerical results of the adaptive mesh refinement algorithm is consistent with the optimal a priori error estimates.

Effect of Temperature Gradient on the Grain Size Homogeneity of SEED Produced Semi-Solid Slurries by Phase-Field Simulation.

The distribution homogeneity of grain size affects the fluidity of the semi-solid slurry, which in turn affects the properties of the casting. One key factor affecting grain size uniformity resides in the nucleation number, which has been studied thoroughly, while the other factor is temperature gradient which has not been investigated yet. In this study, the microstructure evolutions under certain temperature gradients are investigated by experiment and simulation using a two-dimensional quantitative phase-field (PF) model. A parallel and adaptive mesh refinement algorithm is adopted to solve the nonlinear phase-field equations. The results indicate that temperature gradient can affect the size distribution of microstructure in the semi-solid slurry prepared by the SEED process. A higher temperature gradient (in the range of 0.230~0.657 °C/mm) along the radial direction is beneficial to the homogeneity of the grain size in a slurry.

Mesh adaptivity for quasi‐static phase‐field fractures based on a residual‐type a posteriori error estimator

In this work, we consider adaptive mesh refinement for a monolithic phase‐field description for fractures in brittle materials. Our approach is based on an a posteriori error estimator for the phase‐field variational inequality realizing the fracture irreversibility constraint. The key goal is the development of a reliable and efficient residual‐type error estimator for the phase‐field fracture model in each time‐step. Based on this error estimator, error indicators for local mesh adaptivity are extracted. The proposed estimator is based on a technique known for singularly perturbed equations in combination with estimators for variational inequalities. These theoretical developments are used to formulate an adaptive mesh refinement algorithm. For the numerical solution, the fracture irreversibility is imposed using a Lagrange multiplier. The resulting saddle‐point system has three unknowns: displacements, phase‐field, and a Lagrange multiplier for the crack irreversibility. Several numerical experiments demonstrate our theoretical findings with the newly developed estimators and the corresponding refinement strategy.

A multi-level adaptive mesh refinement for an integrated finite element/level set formulation to simulate multiphase flows with surface tension

A multi-level adaptive mesh refinement technique for an integrated finite element/level set formulation was implemented to simulate the incompressible interfacial flows with surface tension by using unstructured P2P1 triangular and tetrahedral meshes. Linear/quadratic elements near the interface were dynamically refined as an interface evolves in order to produce an accurate solution with a lower computational cost. An adaptive mesh refinement algorithm for a linear element has been extended to a quadratic element by using a node-reordering. The proposed algorithm was successfully validated by solving several benchmark problems. Simulation of a static bubble problem was found to give an accurate solution with negligible spurious currents compared with existing results. Simulation results of rising bubble problems in 2D and 3D were also compared against existing numerical and experimental results in terms of terminal rising speed and terminal bubble shape. We have found that the present algorithm provided stable and accurate solutions even for a high density-ratio and that the solution accuracy strongly depended on the mesh resolution near the interface. Further, the present algorithm was shown to be very efficient because the computational overhead by mesh-refinement algorithm was very small compared to the elapsed times consumed for solving the incompressible Navier–Stokes equations and the advection/re-initialization equations of level set method. Lastly, the coalescence scenario of two oblique bubbles was well reproduced by the present 3D simulation.

An a posteriori Error Estimate for Scanning Electron Microscope Simulation with Adaptive Mesh Refinement.

The intensity variation in a scanning electron microscope is a complex function of sample topography and composition. Measurement accuracy is improved when an explicit accounting for the relationship between signal and measurand is made. Because the determinants of the signal are many, the theoretical understanding usually takes the form of a simulator. For samples with nonconducting regions that charge, one phase of the simulation is finite element analysis to compute the electric field. The size of the finite element mesh, and consequently computation time, can be reduced through the use of adaptive mesh refinement. We present a new a posteriori local error estimator and adaptive mesh refinement algorithm for the scanning electron microscope simulation. This error estimate is designed to minimize the error in the electron trajectories, rather than the energy norm of the error that traditional error estimators minimize. Using a test problem with a known exact solution, we show that the adaptive mesh can achieve the same error in electron trajectories as a carefully designed hand-graded mesh while using 3.5 times fewer vertices and 2.25 times less computation time.

Adaptive inverse ray-tracing for accurate and efficient modeling of cross beam energy transfer in hydrodynamics simulations

Integrated hydrodynamics simulations of inertial confinement fusion rely on reduced physics models. To reproduce experimental trends, these models often feature tuning parameters, but this comes with a risk: the over-tuning of one model can hide physics inadequacies in another. The ray-based models of cross-beam-energy transfer (CBET) represent this risk. Here, we present an accurate and efficient model of CBET suitable for inline implementation in 3D hydrodynamics simulations. Inverse Ray Tracing (IRT) is used to compute the ray field in a 3D permittivity profile described on an unstructured tetrahedral mesh using the Inline Field Reconstruction and Interaction using Inverse Tracing framework. CBET is accounted for through perturbations to the permittivity associated with ion acoustic waves driven by the overlapped fields. Large gradients in the permittivity are resolved by coupling the IRT to a recursive Adaptive Mesh Refinement (AMR) algorithm. The use of AMR also allows for the resolution of caustics, with accurate field reconstruction performed using the Etalon integral method. Comparisons of the model with wave-based solutions from the Laser Plasma Simulation Environment demonstrate its ability to control energy conservation and gain convergence through the AMR depth only, without the use of ad hoc physical models or artificial tuning parameters.