Adaptive Mesh Refinement Algorithm Research Articles

Adaptive hybrid high-order method for guaranteed lower eigenvalue bounds

Adaptive hybrid high-order method for guaranteed lower eigenvalue bounds

Scalable implicit solvers with dynamic mesh adaptation for a relativistic drift-kinetic Fokker–Planck–Boltzmann model

In this work we consider a relativistic drift-kinetic model for runaway electrons along with a Fokker–Planck operator for small-angle Coulomb collisions, a radiation damping operator, and a secondary knock-on (Boltzmann) collision source. We develop a new scalable fully implicit solver utilizing finite volume and conservative finite difference schemes and dynamic mesh adaptivity. A new data management framework in the PETSc library based on the p4est library is developed to enable simulations with dynamic adaptive mesh refinement (AMR), distributed memory parallelization, and dynamic load balancing of computational work. This framework and the runaway electron solver building on the framework are able to dynamically capture both bulk Maxwellian at the low-energy region and a runaway tail at the high-energy region. To effectively capture features via the AMR algorithm, a new AMR indicator prediction strategy is proposed that is performed alongside the implicit time evolution of the solution. This strategy is complemented by the introduction of computationally cheap feature-based AMR indicators that are analyzed theoretically. Numerical results quantify the advantages of the prediction strategy in better capturing features compared with nonpredictive strategies; and we demonstrate trade-offs regarding computational costs. The robustness with respect to model parameters, algorithmic scalability, and parallel scalability are demonstrated through several benchmark problems including manufactured solutions and solutions of different physics models. We focus on demonstrating the advantages of using implicit time stepping and AMR for runaway electron simulations.

An adaptive h-refinement method for the boundary element fast multipole method for quasi-static electromagnetic modeling

Objective. In our recent work pertinent to modeling of brain stimulation and neurophysiological recordings, substantial modeling errors in the computed electric field and potential have sometimes been observed for standard multi-compartment head models. The goal of this study is to quantify those errors and, further, eliminate them through an adaptive mesh refinement (AMR) algorithm. The study concentrates on transcranial magnetic stimulation (TMS), transcranial electrical stimulation (TES), and electroencephalography (EEG) forward problems. Approach. We propose, describe, and systematically investigate an AMR method using the boundary element method with fast multipole acceleration (BEM-FMM) as the base numerical solver. The goal is to efficiently allocate additional unknowns to critical areas of the model, where they will best improve solution accuracy. The implemented AMR method’s accuracy improvement is measured on head models constructed from 16 Human Connectome Project subjects under problem classes of TES, TMS, and EEG. Errors are computed between three solutions: an initial non-adaptive solution, a solution found after applying AMR with a conservative refinement rate, and a ‘silver-standard’ solution found by subsequent 4:1 global refinement of the adaptively-refined model. Main results. Excellent agreement is shown between the adaptively-refined and silver-standard solutions for standard head models. AMR is found to be vital for accurate modeling of TES and EEG forward problems for standard models: an increase of less than 25% (on average) in number of mesh elements for these problems, efficiently allocated by AMR, exposes electric field/potential errors exceeding 60% (on average) in the solution for the unrefined models. Significance. This error has especially important implications for TES dosing prediction—where the stimulation strength plays a central role—and for EEG lead fields. Though the specific form of the AMR method described here is implemented for the BEM-FMM, we expect that AMR is applicable and even required for accurate electromagnetic simulations by other numerical modeling packages as well.

A cut-cell method for the numerical simulation of 3D multiphase flows with strong interfacial effects

We present a numerical model for the approximation of multiphase flows with free surfaces and strong interfacial effects. The model relies on the multiphase incompressible Navier-Stokes equations, and includes surface tension effects on the interfaces between phases, and contact angles. The volume-of-fluid approach is used to track the interfaces and the free surfaces between the various phases and the ambient air. The numerical method relies on an operator splitting strategy. The space discretization relies on a two-grid approach that uses an unstructured finite element mesh for diffusion phenomena and a structured Cartesian grid for advection phenomena. An adaptive mesh refinement algorithm is incorporated to better track the interfaces and free surfaces on the finite element mesh, and approximate more accurately surface forces. The model is validated through numerical experiments, in particular for emulsion problems.

A flow control strategy for a near-wall square cylinder using porous media: A direct numerical simulation study

The topic of flows around a near-wall square cylinder has garnered increasing attention in recent decades. However, there are a few publications that have focused on mitigating the occurrence of a substantial negative lift in near-wall flows. In light of this, the present study has developed a novel flow control strategy that covers porous media at inward corners of a near-wall square cylinder to address this problem. We achieve such a control strategy with the aid of a high-fidelity computational framework at Re = 1000. Direct numerical simulations are employed to account for accurate flow behaviors, and the Cartesian cut-cell method as well as an adaptive mesh refinement algorithm are advocated to simplify grid generation and reduce computational costs. Additionally, a quasi-microscopic flow model is introduced to model the porous medium pore structure, providing an intuitive and accurate description of internal flows within the porous medium. Six porous medium layouts are first designed, and their influences and mechanisms on flow control are assessed using the presented computational framework to identify an optimal strategy. The optimal strategy yields a notable reduction of 52.472% in the lift coefficient. The identified strategy is then applied to a case involving a near-wall square cylinder with a substantial negative lift, where a gap ratio of 0.6 is determined via parameterization. The capacity of the presented strategy in flow control of the near-wall square cylinder is fully explored and demonstrated via the consideration of different porosities. The results indicate that the most effective flow control is achieved when the porosity exceeds 90%, leading to a near-zero lift coefficient. Furthermore, the underlying mechanism contributing to the variations in flow control effectiveness due to different porosities is analyzed.

3D Simulations of Nucleate Boiling with Sharp Interface Vof and Localized Adaptive Mesh Refinement in Ansys-Fluent

Abstract The present work demonstrates the use of customized Ansys-Fluent in performing 3D numerical simulations of nucleate boiling with a sharp interface and adaptive mesh refinement. The developed simulation approach is a reliable and effective tool to investigate 3D boiling phenomena by accurately capturing the thermal and fluid dynamic interfacial vapor-liquid interaction and reducing computational time. These methods account for 3D sharp interface and thermal conditions of saturation temperature refining the mesh around the bubble edge. User-Defined-Functions (UDFs) were developed to customize the software Ansys-Fluent to preserve the interface sharpness, maintain saturation temperature conditions, and perform effective adaptive mesh refinement in a localized region around the interface. Adaptive mesh refinement is accomplished by a UDF that identifies the cells near the contact line and the liquid-vapor interface and applies the adaptive mesh refinement algorithms only at the identified cells. Validating the approach considered spherical bubble growth with an observed acceptable difference between theoretical and simulation bubble growth rates of 10%. Bubble growth simulations with water reveal an influence region of 2.7 times the departure bubble diameter, and average heat transfer coefficient of 15000 W/m2-K. In addition, the results indicate a reduced computational time of 75 hours using adaptive mesh compared to uniform mesh.

Automated Reconstruction and Conforming Mesh Generation for Polycrystalline Microstructures from Imaging Data

A new algorithm named PolyCISAMR is introduced to automatically generate high-fidelity conforming finite element (FE) meshes for two-dimensional polycrystalline microstructures. PolyCISAMR extends the capabilities of the Conforming to Interface Structured Adaptive Mesh Refinement (CISAMR) algorithm, which transforms a structured grid overlaid on the domain geometry into a high-quality conforming mesh. The PolyCISAMR approach uses a segregated meshing strategy, where CISAMR is used to discretize each grain independently and the resulting matching meshes are merged to form the final FE model. In addition, this article presents a set of integrated algorithms for processing low-resolution images of a polycrystal, reconstructed using DREAM.3D software (Version 6.5.121), to generate NURBS characterizations for each grain prior to mesh generation. Example problems demonstrate the effectiveness of PolyCISAMR in creating high-quality meshes for various polycrystalline metallic microstructures along with corresponding crystal plasticity finite element (CPFE) simulations.

Adaptive guaranteed lower eigenvalue bounds with optimal convergence rates

AbstractGuaranteed lower Dirichlet eigenvalue bounds (GLB) can be computed for the m-th Laplace operator with a recently introduced extra-stabilized nonconforming Crouzeix–Raviart ($$m=1$$ m = 1 ) or Morley ($$m=2$$ m = 2 ) finite element eigensolver. Striking numerical evidence for the superiority of a new adaptive eigensolver motivates the convergence analysis in this paper with a proof of optimal convergence rates of the GLB towards a simple eigenvalue. The proof is based on (a generalization of) known abstract arguments entitled as the axioms of adaptivity. Beyond the known a priori convergence rates, a medius analysis is enfolded in this paper for the proof of best-approximation results. This and subordinated $$L^2$$ L 2 error estimates for locally refined triangulations appear of independent interest. The analysis of optimal convergence rates of an adaptive mesh-refining algorithm is performed in 3D and highlights a new version of discrete reliability.

A Posterior h/p Adaptive Refinement Algorithm for 3-D Goal-Oriented Electromagnetic Problems

We propose a posterior h/p adaptive mesh refinement algorithm for 3-D goal-oriented electromagnetic wave modeling in complex media. This new algorithm allows both <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$h$</tex-math> </inline-formula> -version refinement—mesh size adjustment—and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p$</tex-math> </inline-formula> -version refinement—polynomial order adjustment—simultaneously. To be specific, we use the adjoint system on the dual space of the forward solver; as a result, error generation, propagation, and accumulation can be evaluated, even without the analytical solutions. In particular, the posterior error is explicitly indicated by postprocessing of original forward solver. Compared with conventional mesh adaptation strategies, such an algorithm is goal-oriented and auxiliary-system-free; as a result, it does not require pay extensive attention to entire computational domain and is complimentary in solving adjoint state equation. Numerical examples demonstrate the superior performance of the proposed mesh refinement algorithms for real-world applications, compared to uniform refinement, nongoal-oriented adaptation, and other goal-oriented adaptation.

Stabilization-free HHO a posteriori error control

The known a posteriori error analysis of hybrid high-order methods treats the stabilization contribution as part of the error and as part of the error estimator for an efficient and reliable error control. This paper circumvents the stabilization contribution on simplicial meshes and arrives at a stabilization-free error analysis with an explicit residual-based a posteriori error estimator for adaptive mesh-refining as well as an equilibrium-based guaranteed upper error bound (GUB). Numerical evidence in a Poisson model problem supports that the GUB leads to realistic upper bounds for the displacement error in the piecewise energy norm. The adaptive mesh-refining algorithm associated to the explicit residual-based a posteriori error estimator recovers the optimal convergence rates in computational benchmarks.

DetonationFoam: An open-source solver for simulation of gaseous detonation based on OpenFOAM

Detonation has promising applications in advanced propulsion systems, and numerical simulation is widely used to gain insights into the complex interaction between the hydrodynamic flow and chemical reactions involved in detonation. In this work, detonationFoam, an open-source solver for accurate and efficient simulation of compressible reactive flow and detonation are developed based on OpenFOAM. detonationFoam can simulate compressible, multi-component, reactive flow and it can accurately evaluate the detailed transport coefficients using the mixture-averaged transport model. Compared to rhoCentralFoam, the improved HLLC-P approximate Riemann solver is used in detonationFoam and it helps to accurately resolve shock waves appearing in detonation. Besides, the adaptive mesh refinement and dynamic load balancing algorithms are used in detonationFoam, which greatly improves the computational efficiency. Validation tests including homogenous ignition, unsteady diffusion, shock tube problem, premixed flame, planar detonation, double Mach reflection, detonation cellular structure and oblique detonation wave are conducted. These tests demonstrate that detonationFoam can be used to accurately and efficiently to simulate the compressible, multi-component reactive flow and detonation processes. Program summaryProgram Title: detonationFoamCPC Library link to program files:https://doi.org/10.17632/x45zh4nz28.1Developer's repository link:https://github.com/JieSun-pku/detonationFoamLicensing provisions: GPLv3Programming language: C++Nature of problem: Gaseous detonation involves different length scales and complicated chemistry. To accurately and efficiently simulate the gaseous detonation, adaptive mesh refinement needs to be conducted and detailed chemistry should be considered. Besides, the severe load imbalance caused by the chemical source term evaluation may greatly reduce the computation efficiency.Solution method: An open-source solver, detonationFoam is developed based on OpenFOAM. The species equations considering detailed chemistry are solved in detonationFoam and thereby detonation in a compressible, multi-component, reactive flow can be simulated. The adaptive mesh refinement technique and the dynamic load balancing algorithm are incorporated into detonationFoam. It is demonstrated that detonationFoam can accurately and efficiently simulate gaseous detonation.

A continuous field adaptive mesh refinement algorithm for isogeometric topology optimization using PHT-Splines

This study proposes a continuous density-field based isogeometric topology optimization (ITO) using Polynomial splines over Hierarchical T-meshes (PHT-Splines). Taking the benefit of isogeometric methods, the design density values for the mesh are defined using spline basis functions as a continuous density function (CDF) through the control points. The material density at the control points will then appear as a facade over the domain, which is further employed to generate a very smooth topology and also to propose a marking criterion for an adaptive mesh refinement (AMR) strategy during the optimization process. During this adaptive meshing, the isogeometric domain mesh is refined during each adaptive step, properly tracking the density variation over the elements and sharpening the boundaries of the optimized topologies. By utilizing the rationale of the Geometry Independent Field approximaTion (GIFT) framework, any complex multi-patch geometry constructed in an industry-standard CAD package using Non-Uniform Rational B-Splines (NURBS) may be imported and discretized using PHT-Splines, including higher dimensions. This helps to maintain the geometrical accuracy through an initial coarse mesh and computational accuracy of the optimization iterations through the local refinement feature of the PHT-Splines. An adaptive first-neighbourhood smoothening algorithm based on the Shepard function is also proposed to obtain manufacturable topologies and to circumvent post-processing stages. The proposed method was observed to provide remarkably smooth topologies with an initial coarse mesh, thereby obtaining a computationally efficient domain to be solved. Compared to existing element density based methods, the proposed method demonstrates a substantial minimization in Degree’s-of-Freedom (DoF) requirement to arrive at clear and realistic solutions. Also, when compared to globally refined solutions, a 50%–90% reduction in DoF is achieved using the adaptive refinement strategy.

3D Transient CFD Simulation of an In-Vessel Loss-of-Coolant Accident in the EU DEMO WCLL Breeding Blanket

The in-vessel Loss-of-Coolant Accident (LOCA) is one of the design basis accidents in the design of the EU DEMO tokamak fusion reactor. System-level codes are typically employed to analyse the evolution of these transients. However, being based on a lumped approach, they are unable to quantify localised quantities of interest, such as local pressure peaks on the vacuum vessel walls, to which the failure criteria are linked. To calculate local quantities, the 3D nature of the phenomenon needs to be considered. In this work, a 3D transient model of the in-vessel LOCA from a water-cooled blanket is developed. The model is implemented in the commercial CFD software STAR-CCM+. It simulates the propagation of the water jet in the vessel from the beginning of the accident, thus accounting for the phase change of the water, i.e., from the pressurised liquid phase to the vapour phase inside the vessel, being the latter at a much lower pressure than in the blanket coolant pipes. Due to the large pressure ratio (&gt;1000), shocks are expected; therefore, an Adaptive Mesh Refinement (AMR) algorithm is employed. The physical models (in particular, the multiphase model) are benchmarked to a 2D reference problem before being applied to the 3D EU DEMO-relevant problem. The simulation results show that the pressure peaks in front of the vessel walls are not dangerous as they are below the design limit. The entire evolution of the water jet is followed up to the opening of the burst disks, in order to compare the average pressure evolution with that computed with system-level codes. A comparison with the in-vessel LOCA from a helium-cooled blanket is also carried out, showing that the accident evolution in the water case is less violent than in the helium case.

Formula omitted]-[formula omitted]: Computationally efficient adaptive mesh refinement of Monte Carlo mesh based tallies

The reactor physics community is always focused on reducing the computational time and memory required for simulations. χ-MeRA, which stands for flux-based-(χ)-Mesh tally Refinement Adaptively, was built to reduce the computational time and memory required to solve the neutronics side of a multiphysics problem when compared to traditional methods for mesh based tallies in Monte Carlo (MC) simulations. χ-MeRA couples a MC code with an adaptive mesh refinement (AMR) algorithm to take advantage of the accuracy of a MC code and the efficiency of an AMR algorithm. Also developed within χ-MeRA was a set of metrics to assess the effects of the refinement on various parameters in the simulation space. For a plutonium sphere, χ-MeRA shows a reduction in memory usage and computation time when compared to a fully refined mesh by a factor of 14.7 and 6.7, respectively. When compared to an unstructured mesh, improvement of 1.3 and 4.8 was achieved for memory usage and computation time. The development of χ-MeRA helps solve the neutronics side of a multiphysics problem in a faster, more computationally efficient manner than traditional methods, and the final mesh created contains accurate results that can be passed onto the next physics code.

Vortexlet formation in Schardin's problem

The present study focuses on the shock diffraction problem over a triangle wedge for Mach numbers of M=1.3, 1.5, 1.7, and 2.0 by using a two-dimensional, high-order, in-house Euler solver. The solver is based on a family of advection upstream splitting method in combination with a central essentially non-oscillatory scheme and benefits a block-based adaptive mesh refinement algorithm to resolve the regions that contain discontinuities. High accuracies in time and space, and adaptive mesh refinement capabilities of the solver allow us to investigate vortexlet formation mechanism in detail. Our results reveal that there are two different types of vortexlet formation mechanisms. While the first type of formation is observed at all Mach numbers considered here, the second type arises when the Mach number is greater than 1.3. This difference results from their driving mechanisms, which are the upward moving accelerated shock and embedded shock in the primary vortex. In addition to their driving mechanisms, two types are also different in terms of their locations.

Fully automatic multigrid adaptive mesh refinement strategy with controlled accuracy for nonlinear quasi-static problems

We propose an adaptive mesh refinement (AMR) algorithm dedicated to the simulation of nonlinear quasi-static solid mechanics problems with complex local phenomena at the structural scale. The proposed method allows us to follow in time the evolution of studied phenomena in a fully-automatic (based on error estimators), precise (respecting user-prescribed accuracies) and efficient (in terms of memory space and computational time) way. This algorithm is based on the multilevel Local Defect Correction (LDC) refinement approach. We first introduce an algorithmic extension of the LDC method to nonlinear quasi-static problems and provide key aspects associated with its practical implementation. Generic still open AMR-related questions associated with dynamic mesh adaptation, such as fields transfer between time steps and discretization error control over time, are then addressed. We propose a straightforward and efficient error non accumulation strategy lying on the introduction of the unbalance residual as an initial source term of the problem. Moreover, a reliable remeshing algorithm is introduced, aiming to limit the number of mesh regenerations over time while guaranteeing the fulfillment of user-prescribed errors. The efficiency of the proposed algorithm is demonstrated on several numerical experiments, in 2D and 3D, with different types of material behavior as well as evolving loads. Thanks to its natural ability to generate a hierarchy of meshes of limited sizes that dynamically follow the evolution over time of studied phenomena, the proposed extension of the LDC method clearly appears to be of great potential for many challenging applications.

Adaptive phase-field modeling of brittle fracture using a robust combination of error-estimator and markers

An error estimator is proposed to carry out adaptive mesh refinement (AMR) for phase-field fracture (PFF) simulation. The proposed error estimator works with the mesh-induced crack (MIC) method for modeling initial cracks, resulting in significant gains in the AMR algorithm’s computational efficiency. The error estimator is derived using the iterative change in the crack driving energy over each element in the mesh. A sensitivity analysis with variation in the control parameter of the marking methods is also carried out to present the robustness of the AMR algorithm with the proposed error estimator. We also study the accuracy of the proposed algorithm by implementing it to standard problems and comparing the responses to the non-adaptive (NA) algorithm. The algorithm adaptively and accurately refines the mesh along the expected crack path to estimate the system’s global force versus displacement response. Compared to the NA algorithm, a reduction of 84%–98% is obtained in the total CPU time with the proposed AMR algorithm.

An adaptive mesh refinement algorithm for phase-field fracture models: Application to brittle, cohesive, and dynamic fracture

Phase-field models (PFMs) have proven to accurately predict complex crack patterns like crack branching, merging, and crack fragmentation, but they are computationally costly. Adaptive mesh refinement (AMR) algorithms based on nodal damage are recently developed to reduce computational expense. However, these AMR algorithms are unable to simulate the crack initiation accurately without a priori local refinement. To solve this problem, in this work, we have proposed an AMR algorithm based on the effective crack driving energy. A multi-level mark–unmark scheme is developed integrating the effective crack driving energy-based followed by a damage-based scheme, which can capture the crack initiation and propagation very effectively. The proposed AMR algorithm works efficiently on brittle, cohesive, and dynamic fractures. The three popular PFMs, namely AT1, AT2, and PF-CZM are selected from the literature, implemented within a single code, and used to study the AMR algorithm. The proposed AMR algorithm reduces the simulation time by 5–50 times, depending on the type of problem, as compared to simulations that adopt a priori non-adaptively refined meshes.

Inf-sup stability implies quasi-orthogonality

We prove new optimality results for adaptive mesh refinement algorithms for non-symmetric, indefinite, and time-dependent problems by proposing a generalization of quasi-orthogonality which follows directly from the inf-sup stability of the underlying problem. This completely removes a central technical difficulty in modern proofs of optimal convergence of adaptive mesh refinement algorithms and leads to simple optimality proofs for the Taylor-Hood discretization of the stationary Stokes problem, a finite-element/boundary-element discretization of an unbounded transmission problem, and an adaptive time-stepping scheme for parabolic equations. The main technical tools are new stability bounds for the L U LU -factorization of matrices together with a recently established connection between quasi-orthogonality and matrix factorization.

Numerical modeling of solute convection in the system of two porous layers with different permeabilities

The numerical simulation of the dynamics of a liquid with solute of high concentration in two horizontal layers of a porous medium with different permeability coefficients is carried out. The dynamics of the solute is modeled within the framework of a nonlinear MIM model that takes into account the adhesion and separation process at the skeleton of a porous medium. The modeling was carried out in a two-dimensional setting. The mobile (moving together with the carrier fluid) and immobile (deposited on the skeleton) phases are modeled by the finite volume method with adaptive mesh refinement algorithm. At each point in space, functions of volumetric saturation are set, and a kinematic equation is solved that describes the dynamics of adhesion and separation. The effective permeability of a porous medium depends on the saturation function of the immobile phase and is calculated using the Kozeny-Karman formula. The obtained numerical data are compared with the results of a linear analysis of stability, and the nonlinear regimes of concentration convection and the distribution of impurities at high supercriticalities are studied. In the case of high ratio of permeability coefficients of upper and lower layers, convection occurs locally in a more permeable sublayer, even with a decrease in its thickness. With an increase in supercriticality the penetration of the convective flow into the poorly permeable sublayer and the transition from local convection to large-scale convection are observed.