Adaptive Mesh Refinement Algorithm Research Articles

Optimal adaptivity for the SUPG finite element method

For convection dominated problems, the streamline upwind Petrov–Galerkin method (SUPG), also named streamline diffusion finite element method (SDFEM), ensures a stable finite element solution even on coarse meshes, i.e., in the preasymptotic range. Based on some a posteriori error estimators from the literature, we formulate an adaptive mesh-refining algorithm for SUPG. We prove that the adaptively generated SUPG solutions converge at asymptotically optimal rates towards the exact solution. The main focus thus is on the mathematical proof that the SUPG stabilization does not spoil the asymptotic convergence behavior of the adaptive scheme.

Implementation of a Volume-of-Fluid method in a finite element code with applications to thermochemical convection in a density stratified fluid in the Earth’s mantle

We describe the implementation of a second-order accurate Volume-of-Fluid interface tracking algorithm in the open source finite element code ASPECT that is designed to model convection and other processes in the Earth’s mantle. This involves the solution of the incompressible Stokes equations coupled to an advection diffusion equation for the temperature, a Boussinesq approximation that governs the dependence of the density on the temperature, and an advection equation for a marker indicating two initial (constant) density states, that are passively advected in the underlying flow field. The Volume-of-Fluid method in ASPECT is fully parallelized and fully integrated with ASPECT’s adaptive mesh refinement algorithm. We present the results of several interface tracking benchmarks in order to demonstrate the accuracy of the method, as well as the results of several benchmarks commonly used in the computational mantle convection community. Finally, we present the results of computations with and without adaptive mesh refinement of a model problem involving thermochemical convection in a computationally stratified fluid designed to provide insight into how thermal plumes, that eventually reach the Earth’s surface as ocean island basalts, originate at structures near the core-mantle boundary known as Large Low Shear wave Velocity Provinces or “LLSVPs”. LLSVPs are structures in parts of the lowermost portion of the Earth’s mantle, characterized by slow shear wave velocities and higher density than the surrounding mantle, which were discovered by seismic tomography of the deep Earth.

Simulation of liquid jet primary breakup in a supersonic crossflow under Adaptive Mesh Refinement framework

Compressible two-phase flows were simulated based on the five-equation model under the Adaptive Mesh Refinement (AMR) framework to balance the requirements between space resolution and computational cost. And the simulation system was established in an open source software AMROC (Adaptive Mesh Refinement Object-oriented C++). A combination of Godunov method and wave propagation method was introduced to integrate numerical methods with the AMR algorithm. High speed and high liquid-gas density ratio are two main challenges in the simulation of liquid jet in a supersonic crossflow. To enhance the robustness of the simulation system, a MOON-type positivity preserving method was adopted in the development of the codes. Based on the system mentioned above, a liquid jet in a Mach 1.5 supersonic crossflow was simulated as the standard case to study the primary breakup process in the near field. The simulation captured the column and surface breakup which were the results of the development of the unstable waves in two directions respectively. The instabilities causing the surface breakup were found to be generated in the transonic region initially. Crossflow of a higher Mach number (Ma 1.8) was found being able to augment the instable waves along the injection direction and increase the number of instabilities responsible for the surface breakup. While there was no obvious enhancement of the penetration in the condition of periodic injection, extra unstable waves were imposed on both of windward and leeward liquid surface. The introduced unstable waves had an improvement on the column and surface breakup.

Adaptive Mesh Refinement algorithm based on dual trees for cells and faces for multiphase compressible flows

A novel adaptive mesh refinement method is proposed. The novelty of the method lies in using a dual data structure with two trees: A classical one for the computational cells and an extra one dedicated to computational cell faces. This new dual structure simplifies the algorithm, making the method easy to implement. It results in an efficient adaptive mesh refinement method that preserves an acceptable memory cost. This adaptive mesh refinement method is then applied to compressible multiphase flows in the framework of diffuse-interface methods. Efficiency of the method is demonstrated thanks to computational results for different applications: Transport, shock tube, surface-tension flow, cavitation and water-droplet atomization, in one and multi-dimensions. The test cases are performed with the open-source code ECOGEN and with quantitative comparisons regarding non-adaptive mesh refinement methods to analyze benefits. A discussion specific to parallel computing is also presented.

Residual-based a posteriori error analysis for symmetric mixed Arnold\u2013Winther FEM

This paper introduces an explicit residual-based a posteriori error analysis for the symmetric mixed finite element method in linear elasticity after Arnold–Winther with pointwise symmetric and H({text {div}})-conforming stress approximation. The residual-based a posteriori error estimator of this paper is reliable and efficient and truly explicit in that it solely depends on the symmetric stress and does neither need any additional information of some skew symmetric part of the gradient nor any efficient approximation thereof. Hence, it is straightforward to implement an adaptive mesh-refining algorithm. Numerical experiments verify the proven reliability and efficiency of the new a posteriori error estimator and illustrate the improved convergence rate in comparison to uniform mesh-refining. A higher convergence rate for piecewise affine data is observed in the L^2 stress error and reproduced in non-smooth situations by the adaptive mesh-refining strategy.

Goal-oriented h-type adaptive finite elements for micromorphic elastoplasticity

Generalized continuum theories become necessary for many scenarios like strain localization phenomena or size effects. In this work, we consider a class of higher order continua, namely micromorphic continua, where the kinematics is enhanced by means of a microstructure undergoing an affine micro deformation. The increasing number of the degrees of freedom in such a theory clearly motivates an application of adaptive methods. For linear elastic micromorphic problems, we have shown the consistency of the resulted dual problem, ensuring an optimal convergence order, in Ju and Mahnken (2017). In this work, we extend the framework of goal-oriented adaptivity to time-dependent problems, i.e. plasticity problems, where a backwards-in-time dual problem is crucial to account for the error accumulation over time caused by both error generation and error transport. Our focus is limited to an adaptive control of spatial discretization errors of the finite element method (FEM), while the temporal discretization errors are not considered for simplicity. Based on duality techniques, we derive exact error representations. For practice, four computable error estimators are proposed, where two different ways to obtain enhanced solutions are considered, and where additionally an approximate forwards-in-time dual problem neglecting error transport is introduced. By means of certain localization techniques, these estimators are used to drive an adaptive mesh refinement algorithm. Their effectiveness is shown by several numerical examples based on a prototype model.

A fourth-order adaptive mesh refinement algorithm for the multicomponent, reacting compressible Navier–Stokes equations

In this paper, we present a fourth-order in space and time block-structured adaptive mesh refinement algorithm for the compressible multicomponent reacting Navier–Stokes equations. The algorithm uses a finite-volume approach that incorporates a fourth-order discretisation of the convective terms. The time-stepping algorithm is based on a multi-level spectral deferred corrections method that enables explicit treatment of advection and diffusion coupled with an implicit treatment of reactions. The temporal scheme is embedded in a block-structured adaptive mesh refinement algorithm that includes subcycling in time with spectral deferred correction sweeps applied on levels. Here we present the details of the multi-level scheme paying particular attention to the treatment of coarse–fine boundaries required to maintain fourth-order accuracy in time. We then demonstrate the convergence properties of the algorithm on several test cases including both non-reacting and reacting flows. Finally we present simulations of a vitiated dimethyl ether jet in 2D and a turbulent hydrogen jet in 3D, both with detailed kinetics and transport.

Nonconforming Mesh Refinement for High-Order Finite Elements

We propose a general algorithm for nonconforming adaptive mesh refinement (AMR) of unstructured meshes in high-order finite element codes. Our focus is on $h$-refinement with a fixed polynomial ord...

Thin and sharp edges bodies-fluid interaction simulation using cut-cell immersed boundary method

This study aims to develop an adaptive mesh refinement (AMR) algorithm combined with Cut-Cell IBM using two-stage pressure–velocity corrections for thin-object FSI problems. To achieve the objective of this study, the AMR-immersed boundary method (AMR-IBM) algorithm discretizes and solves the equations of motion for the flow that involves rigid thin structures boundary layer at the interface between the structure and the fluid. The body forces are computed in proportion to the fraction of the solid volume in the IBM fluid cells to incorporate fluid and solid motions into the boundary. The corrections of the velocity and pressure is determined by using a novel simplified marker and cell scheme. The new developed AMR-IBM algorithm is validated using a benchmark data of fluid past a cylinder and the results show that there is good agreement under laminar flow. Simulations are conducted for three test cases with the purpose of demonstration the accuracy of the AMR-IBM algorithm. The validation confirms the robustness of the new algorithms in simulating flow characteristics in the boundary layers of thin structures. The algorithm is performed on a staggered grid to simulate the fluid flow around thin object and determine the computational cost.

An explicit a posteriori error estimations for the cell functional minimization scheme of elliptic problems

This paper presents an explicit form of a posteriori error estimator for the cell functional minimization scheme of elliptic problems. It is shown that this estimator is reliable with respect to an energy-type error. Numerical experiments are conducted to demonstrate the performance of the error estimator in the adaptive mesh refinement algorithm.

High-Order Finite-Volume Method with Block-Based AMR for Magnetohydrodynamics Flows

A high-order central essentially non-oscillatory (CENO) finite volume scheme combined with a block-based adaptive mesh refinement (AMR) algorithm is proposed for the solution of the ideal magnetohydrodynamics equations. The high-order CENO finite-volume scheme is implemented with fourth-order spatial accuracy within a flexible multi-block, body-fitted, hexahedral grid framework. An important feature of the high-order adaptive approach is that it allows for anisotropic refinement, which can lead to large computational savings when anisotropic flow features such as isolated propagating fronts and/or waves, shocks, shear surfaces, and current sheets are present in the flow. This approach is designed to handle complex multi-block grid configurations, including cubed-sphere grids, where some grid blocks may have degenerate edges or corners characterized by missing neighboring blocks. A procedure for building valid high-order reconstruction stencils, even at these degenerate block edges and corners, is proposed, taking into account anisotropic resolution changes in a systematic and general way. Furthermore, a non-uniform or heterogeneous block structure is used where the ghost cells of a block containing the solution content of neighboring blocks are stored directly at the resolution of the neighbors. A generalized Lagrange multiplier divergence correction technique is applied to achieve numerically divergence-free magnetic fields while preserving high-order accuracy on the anisotropic AMR grids. Parallel implementation and local grid adaptivity are achieved by using a hierarchical block-based domain partitioning strategy in which the connectivity and refinement history of grid blocks are tracked using a flexible binary tree data structure. Physics-based refinement criteria as well as the CENO smoothness indicator are both used for directing the mesh refinement. Numerical results, including solution-driven anisotropic refinement of cubed-sphere grids, are presented to demonstrate the accuracy and efficiency of the approach.

A Phase-Field Lattice-Boltzmann Study on Dendritic Growth of Al-Cu Alloy Under Convection

Effects of convection (forced and natural) on dendritic evolution of the Al-Cu alloy were investigated using a phase-field lattice-Boltzmann approach. The non-linear coupled equations were solved by applying a parallel and adaptive mesh refinement algorithm. Important physical aspects including dendritic fragmentation, splitting, and formation of solute plumes were simulated. Results showed that the dendritic growth patterns under convection exhibited remarkable difference from those without convection. The presence of flow led to variation of solute diffusion and upstream–downstream dendritic growth difference, which further influenced the development of dendritic arms and multi-dendritic competitive growth. When the convection intensity was magnified, the convection-induced anisotropy became dominated, and the growth patterns changed accordingly to accommodate the local thermodynamic variation.

Afivo: A framework for quadtree/octree AMR with shared-memory parallelization and geometric multigrid methods

Afivo is a framework for simulations with adaptive mesh refinement (AMR) on quadtree (2D) and octree (3D) grids. The framework comes with a geometric multigrid solver, shared-memory (OpenMP) parallelism and it supports output in Silo and VTK file formats. Afivo can be used to efficiently simulate AMR problems with up to about 108 unknowns on desktops, workstations or single compute nodes. For larger problems, existing distributed-memory frameworks are better suited. The framework has no built-in functionality for specific physics applications, so users have to implement their own numerical methods. The included multigrid solver can be used to efficiently solve elliptic partial differential equations such as Poisson’s equation. Afivo’s design was kept simple, which in combination with the shared-memory parallelism facilitates modification and experimentation with AMR algorithms. The framework was already used to perform 3D simulations of streamer discharges, which required tens of millions of cells. Program summaryProgram Title: AfivoProgram Files doi:http://dx.doi.org/10.17632/5y43rjdmxd.1Licensing provisions: GPLv3Programming language: Fortran 2011External routines/libraries: Silo (LLNL)Nature of problem: Performing multiscale simulations, especially those requiring a fast elliptic solver.Solution method: Provide a framework for parallel simulations on adaptively refined quadtree/octree grids, including a geometric multigrid solver.Unusual features: The framework uses shared-memory parallelism (OpenMP) instead of MPI.

3-D Adaptive Finite-Element Modeling of Marine Controlled-Source Electromagnetics with Seafloor Topography Based on Secondary Potentials

We present a solution of using adaptive nodal finite-element (FE) method to solve the marine controlled-source electromagnetic (CSEM) problem for 3-D earth models in the frequency domain. The forward problem is solved based on the secondary Coulomb-gauged electromagnetic (EM) potentials. We implement an adaptive mesh refinement algorithm according to an a posteriori error estimator based on a gradient-recovery operator of the secondary EM potentials. To increase the quality of the mesh at the receiver locations, the elements containing the receiver locations are constrained by assigning them a maximum volume for the initial mesh. An unstructured tetrahedral mesh used in our approach can provide an accurate description of complex structures such as dipping layers and rough topography that are not accurately fitted using structured meshes. We first validate the adaptive FE approach and demonstrate the convergence of the adaptive grid refinement procedure using a 1-D layered model. The canonical disc model example illustrates the capability of the adaptive FE approach for 3-D CSEM modeling. The bathymetry model shows that the algorithm is well suited to deal with complex seafloor topography, which needs to be simulated exactly to avoid the misinterpretation of marine CSEM data sets.

The influence of radial entrance width of the circular cavity on incident shock wave focusing

Despite the achievement of shock wave focusing with certain reflectors, the influence of the radial entrance width of a circular cavity on the flow field has yet to be addressed. In this study, we systematically investigated the effects of the shock wave focusing process in a cavity based on the radial entrance widths. An experimental system was installed to research the evolution of the flow field under conditions with different radial entrance widths of 3.0, 11.1, 19.5, and 33.0 mm. A schlieren system was used to photograph the structures of the flow field in the cavity, and a data acquisition system was used to record the dynamic pressure histories of different points. A numerical simulation was carried out to investigate greater details of the shock wave focusing process. A third-order strong stability-preserving Runge-Kutta method, third-order weighed essential non-oscillation scheme, and an adaptive mesh refinement algorithm were adopted to simulate the shock wave reflection, diffraction, and focus process. Good agreement between the experimental and numerical results was observed. By comparing the evolution process of the flow field under the conditions of four different entrance cavity widths, we found that when the entrance width was 19.5 mm, there was the stronger intensity of the shock wave focusing in the focal region, and the larger pressure value at the apex of the cavity than the other three entrance widths, occur. This study improves our understanding of shock wave focusing.

Quantitative phase-field lattice-Boltzmann study of lamellar eutectic growth under natural convection.

The influence of natural convection on lamellar eutectic growth was determined by a comprehensive phase-field lattice-Boltzmann study for Al-Cu and CBr_{4}-C_{2}Cl_{6} eutectic alloys. The mass differences resulting from concentration differences led to the fluid flow and a robust parallel and adaptive mesh refinement algorithm was employed to improve the computational efficiency. By means of carefully designed "numerical experiments", the eutectic growth under natural convection was explored and a simple analytical model was proposed to predict the adjustment of the lamellar spacing. Furthermore, by alternating the solute expansion coefficient, initial lamellar spacing, and undercooling, the microstructure evolution was presented and compared with the classical eutectic growth theory. Results showed that both interfacial solute distribution and average curvature were affected by the natural convection, the effect of which could be further quantified by adding a constant into the growth rule proposed by Jackson and Hunt [Jackson and Hunt, Trans. Metall. Soc. AIME 236, 1129 (1966)].

An efficient Adaptive Mesh Refinement (AMR) algorithm for the Discontinuous Galerkin method: Applications for the computation of compressible two-phase flows

We present an Adaptive Mesh Refinement (AMR) method suitable for hybrid unstructured meshes that allows for local refinement and de-refinement of the computational grid during the evolution of the flow. The adaptive implementation of the Discontinuous Galerkin (DG) method introduced in this work (ForestDG) is based on a topological representation of the computational mesh by a hierarchical structure consisting of oct- quad- and binary trees. Adaptive mesh refinement (h-refinement) enables us to increase the spatial resolution of the computational mesh in the vicinity of the points of interest such as interfaces, geometrical features, or flow discontinuities. The local increase in the expansion order (p-refinement) at areas of high strain rates or vorticity magnitude results in an increase of the order of accuracy in the region of shear layers and vortices.A graph of unitarian-trees, representing hexahedral, prismatic and tetrahedral elements is used for the representation of the initial domain. The ancestral elements of the mesh can be split into self-similar elements allowing each tree to grow branches to an arbitrary level of refinement. The connectivity of the elements, their genealogy and their partitioning are described by linked lists of pointers. An explicit calculation of these relations, presented in this paper, facilitates the on-the-fly splitting, merging and repartitioning of the computational mesh by rearranging the links of each node of the tree with a minimal computational overhead. The modal basis used in the DG implementation facilitates the mapping of the fluxes across the non conformal faces.The AMR methodology is presented and assessed using a series of inviscid and viscous test cases. Also, the AMR methodology is used for the modelling of the interaction between droplets and the carrier phase in a two-phase flow. This approach is applied to the analysis of a spray injected into a chamber of quiescent air, using the Eulerian–Lagrangian approach. This enables us to refine the computational mesh in the vicinity of the droplet parcels and accurately resolve the coupling between the two phases.

A priori and a posteriori error control of discontinuous Galerkin finite element methods for the von Kármán equations

This paper analyses discontinuous Galerkin finite element methods (DGFEM) to approximate a regular solution to the von Karman equations defined on a polygonal domain. A discrete inf–sup condition sufficient for the stability of the discontinuous Galerkin discretization of a well-posed linear problem is established, and this allows the proof of local existence and uniqueness of a discrete solution to the nonlinear problem with a Banach fixed point theorem. The Newton scheme is locally second-order convergent and appears to be a robust solution strategy up to machine precision. A comprehensive a priori and a posteriori energy-norm error analysis relies on one sufficiently large stabilization parameter and sufficiently fine triangulations. In case the other stabilization parameter degenerates towards infinity, the DGFEM reduces to a novel C0-interior penalty method (IPDG). In contrast to the known C0-IPDG dueto Brenner et al., (2016, A C0 interior penalty method for a von Karman plate. Numer. Math., 1–30), the overall discrete formulation maintains symmetry of the trilinear form in the first two components—despite the general nonsymmetry of the discrete nonlinear problems. Moreover, a reliable and efficient a posteriori error analysis immediately follows for the DGFEM of this paper, while the different norms in the known C0-IPDG lead to complications with some nonresidual-type remaining terms. Numerical experiments confirm the best-approximation results and the equivalence of the error and the error estimators. A related adaptive mesh-refining algorithm leads to optimal empirical convergence rates for a nonconvex domain.

Adaptive mesh refinement for elastic modulus reconstruction in elastography.

Meshes play a crucial role in determining the accuracy of the elastic modulus reconstruction in the elastography when the finite element method is employed. In this article, we propose an adaptive mesh refinement strategy which can ensure the coincidence of the meshes with the shape of the inclusions in the observed tissue. This strategy is based on the intensity distribution of the strain image where the variance of the strain distribution in each element of the meshes is used to measure the homogeneity of the element, that is, the larger the strain variance is the more inhomogeneous the element will be and hence more detailed information will be included in this element. For more accurate reconstruction of such detailed information, mesh refinement procedure is implemented in such elements. Besides, two refinement steps are employed for the reconstruction to improve the fitness of the reconstructed image and the observed tissue. Simulation results show that the two-stage adaptive mesh refinement algorithm performs well without needing any prior information about the internal geometric shape in tissue. Not only Young's moduli of models but also shapes of the inclusions can be reconstructed perfectly and quickly with our proposed method.

A Sufficient Condition for Stability of Sampled–data Model Predictive Control using Adaptive Time–mesh Refinement

In this work, we address through model predictive control (MPC) a constrained nonlinear plant described by a continuous-time dynamical model, which naturally leads to a sampled-data control system. The numerical solution of the optimal control problems involved in MPC must utilize, eventually, some form of discretization. Nevertheless, there are several advantages in maintaining a continuous-time model until later stages. One advantage is that we can devise numerical procedures which, by exploiting additional freedom in selecting the discretization points, are more efficient when continuous-time models are used. Here, we discuss an extension to MPC of an Adaptive Mesh Refinement (AMR) algorithm, which has shown to be efficient in solving nonlinear optimal control problems. We derive a sufficient condition that guarantees that an MPC scheme using an adaptive time-mesh refinement algorithm preserves stability.