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

A new adaptive local mesh refinement algorithm and its application on fourth order thin film flow problem

Towards adaptive kinetic-fluid simulations of weakly ionized plasmas

HiFlow3 Version 2.0 continues the path as a multi-purpose finite element software, which provides powerful tools for efficient and accurate solution of a wide range of problems modeled by partial differen- tial equations (PDEs). New features and functionalities, which allow to run numerical simulations with more advanced solution algorithms and discretizations in comparison to previous releases of HiFlow3, have been implemented. These comprise Uncertainty Quantification (UQ), energy-efficient multigrid techniques, Schur complement solvers for saddle-point problems, extended third-party library support, and adaptive local mesh refinement in a parallel computing environment. Furthermore, HiFlow3 has been successfully integrated into advanced and state-of-the-art simulation environments by means of the Medical Simulation Markup Language (MSML), for example.The presented new algorithms and features as well as general under-the-hood improvements have enabled excellent and relevant research activities in the fields of both medical engineering and meteorol- ogy and environmental sciences. The described show cases demonstrate the potential and advantages, which HiFlow3 can offer in performing a numerical simulation by means of finite element methods. Especially, the high performance computing capabilities of HiFlow3 – not only in the mentioned fields of applications, but also in general – have been significantly improved in Version 2.0.

High-precision computational fluid dynamics-discrete element method (CFD-DEM) simulation serves as a foundation for the detailed design of supercritical water fluidized bed reactors (SCWFBRs). However, as the scale of these reactors increases, the computational domain and reacting particles expand exponentially. Traditional adaptive mesh refinement is constrained in dealing with dense phase zones in SCWFBRs, as the nonlinear characteristics of the dense reacting particle flow render local-field error estimation ineffective. To address this limitation, an efficient and accurate CFD-DEM framework is established, integrating adaptive mesh refinement and heterogeneous computing. The mesh adaptation strategy is based on whole-field error estimation and is defined by a set of cell-marking criteria. The effect of the criteria on the accuracy and efficiency of the computational process is evaluated. This method reduces the demand for computational resources, thereby enabling the simulation of large-scale reactors at the particle level. The optimized adaptation strategy maintains an error of 1.67% while saving 51.59% of mesh cells required and 28.4% computation time. The primary factors influencing computational accuracy are identified as fluid velocity gradient and the particle reaction rate. The scalability of the adaptation strategy is also validated. For a fivefold radial scaled-up reactor, the cell count is reduced by 46.86% and the computation time by 22.2%, while the maximum error is limited to 2.79%. This work provides a solution that can consider both the computational scale and accuracy, thereby enabling the detailed design of large-scale SCWFBRs.

A stochastic computational method based on goal-oriented error estimation for heterogeneous geological materials

Summary An adaptive three-dimensional mesh refinement method based on the law of mass conservation has been introduced and tested using some analytical velocity fields as accurate in identifying singular point, asymptotic plane and drawing closed streamlines. This paper further investigates the adaptive mesh refinement method using a velocity field that is due to a uniform strain and a point vortex. Similar results have been obtained.

We perform stability analysis on the finite-difference time domain\nmethod (FDTD) when extended to incorporate local space-time adaptive\nmesh refinement (AMR). The neutrally stable Yee algorithm becomes\nextremely sensitive to perturbations introduced by the interpolation\nschemes employed at grid refinement interfaces. In this paper we\ninvestigate the stability of a range of interpolation schemes using\nGustafsson-Kreiss-Sundstrom-Trefethen (GKS-T) mode and\nreflection/transmission coefficients analysis on the infinite domain\nwith a single interface. This analysis allows detection of trapping\ninstabilities, exponentially growing modes, mode resonances with the\ninterface and mode-mode resonances. We also apply matrix stability\nanalysis for more complicated computational domains containing\nmultiple grid refinement interfaces.

Mesh quality strongly affects the solution accuracy in electromagnetic finite-element analysis. Hence, the realization of adequate mesh generation becomes a very important task. Several adaptive meshing methods for automatic adjustments of the mesh density in accordance with the shape and complexity of the analyzed problem have been proposed. However, the most of them are not enough robust, some are quite laborious and could not be universally used for adaptive meshing of complex analysis models. In this paper, a new adaptive mesh refinement method based on magnetic field conservation at the border between finite elements is proposed. The proposed error estimation method provides easy mesh refinements, and generates smaller element within regions with large curvature of the magnetic flux lines. The proposed adaptive mesh refinement method based on non-conforming edge finite elements, which could avoid generation of flat or ill-shaped elements, was applied to a simple magnetostatic permanent magnet model. To confirm the validity and accuracy, the obtained results were compared with those obtained by means of the Zienkiewicz-Zhu (ZZ) error estimator. The results show that the computational error using the proposed method was reduced down to 1% compared with that of the ZZ method, which yields error of 8.6%, for the same model.

This introductory chapter on the mathematical theory of finite element methods (FEMs) discusses its h‐version for elliptic boundary value problems in the displacement formulation. Topics addressed range froma prioritoa posteriorierror estimates and also include weak forms of elliptic PDEs, Galerkin schemes, finite element spaces, and adaptive local mesh refinement. Nonconformities and variational crimes as well as algorithmic aspects conclude the chapter.

Mesh adaptation is essential for accurate computational fluid dynamics (CFD) simulations, especially when computational resources are limited.In simulating the advection-diffusion equation using the finite volume method (FVM), mesh adaptation is often necessary to accurately capture sharp gradients and complex flow features.Although the metric-based mesh adaptation used in this study is highly effective with true errors, these are typically unavailable, requiring the use of error estimators.However, adjoint-based error estimators, though effective, are computationally expensive as they require solving the larger adjoint system.To address this, an inexpensive error estimator based on the gradient of the numerical solution and the embedded method is proposed.Specifically, the central difference and upwind difference schemes are utilized for error estimation within this framework.Since errors are predominantly influenced by sharp gradients in the scalar boundary layer, a target functional based on the product of the embedded error and the solution gradient to compute the error estimator is proposed.Consequently, these proposed error estimators generate meshes closely resembling those produced using true error values, effectively resolving the scalar boundary layer.As a result, the L 2 and L norms of the error typically decrease with successive adaptation cycles.Furthermore, the L 1 , L 2 , and L norms of error per element also reduce with successive adaptation cycles.In conclusion, the proposed error estimator facilitates mesh adaptation without needing true errors or computationally expensive adjoint-based error estimators, thus ensuring accurate numerical solutions, especially in scenarios with sharp boundary layers.

A posteriori error estimator and AMR for discrete ordinates nodal transport methods

Loaded contact pressure distribution prediction for spiral bevel gear

We present an Adaptive Mesh Refinement benchmark for evaluating programmability and performance of modern parallel programming languages. Benchmarks employed today by language developing teams, originally designed for performance evaluation of computer architectures, do not fully capture the complexity of state-of-the-art computational software systems running on today's parallel machines or to be run on the emerging ones from the multi-cores to the peta-scale High Productivity Computer Systems. This benchmark, extracted from a real application framework, presents challenges for a programming language in both expressiveness and performance. It consists of an infrastructure for finite difference calculations on block-structured adaptive meshes and a solver for elliptic Partial Differential Equations built on this infrastructure. Adaptive Mesh Refinement algorithms are challenging to implement due to the irregularity introduced by local mesh refinement. We describe those challenges posed by this benchmark through two reference implementations (C++ /Fortran/MPI and Titanium) and in the context of three programming models.

An adaptive mesh refinement algorithm has been developed for non-linear computations in geomechanics, based on a smoothed stress–strain finite element formulation. This uses estimates of error in the incremental shear strain invariant to guide the regeneration of unstructured meshes at regular intervals during loading. Following each mesh-update, no re-analysis of previous increments with the new mesh is necessary. Algorithm performance has been investigated by analysing a passive earth pressure problem using a linear elastic-perfectly plastic Mohr–Coulomb soil model. Perfectly drained behaviour has been considered, as have partially drained situations using hydromechanical coupling, while undrained behaviour has been approximated using time steps close to zero. In all cases, mesh adaptivity has been successful in capturing regions of high strain gradient. The results have been compared with analytical solutions. Accurate computations of limit load and shear band orientation have been obtained for a wide range of material dilation angles. Copyright © 2000 John Wiley & Sons, Ltd.

Significant progress has been achieved over the last decade in the realistic animation of garments. However it is still a very costly process in terms of computational resources. Since wrinkles and vast smooth areas co-exist commonly, it is tempting to reduce computational cost by avoiding redundant tessellation at the smooth areas. In this paper we present a method for dynamic adaptation of triangular meshes suitable for the most elaborated cloth simulation approaches, such as finite-element based or alike. We use bottom-up approach to mesh refinement, which does not require precomputation and storage of multiresolution hierarchy. The hierarchy is constructed in runtime using √3-refinement rule. The hierarchy is essential to allow reverting of the refinement locally. Local mesh refinement and simplification are triggered by curvature-induced criterion, where the curvature is estimated using methods of discrete differential geometry. In the existing literature of adaptive meshes only the formulas for estimating the discrete mean curvature at the inner mesh vertices can be found. We extend it to the triangulated 2-manifolds with boundary, such as cloth meshes. The results demonstrated are the realistic animation of garment worn by a walking mannequin generated with Baraff-Witkin type cloth solver enhanced with our mesh adaptation scheme.