Automatic Mesh Refinement Research Articles

A Parallel Full Geometric Multigrid Solver for Time Harmonic Maxwell Problems

The design of scalable parallel simulation codes for complex phenomena is challenging. For simulations that rely on PDE solution on complex three-dimensional (3D) geometries, all the components ranging from the initialization phases to the ultimate linear system solution must be efficiently parallelized. In this paper we address the solution of 3D harmonic Maxwell's equations using a parallel geometric full multigrid scheme where only a coarse mesh, which resolves the object geometry, has to be supplied by an external mesh generator. The electromagnetic problem is solved on a refined mesh that complies with the discretization constraint that links the mesh size and the wavelength. The mesh hierarchy is built in parallel using an automatic mesh refinement technique, and parallel matrix-free calculations are performed to process all the finer meshes in the multigrid hierarchy. Only the linear system on the coarse mesh is assembled and solved by a parallel sparse direct solver. We illustrate the numerical robustness of the proposed scheme on large 3D complex geometries and assess the parallel scalability of the implementation on large problems with up to 1.3 billion unknowns on a few hundred cores.

Efficient analysis of transient heat transfer problems exhibiting sharp thermal gradients

In this paper, heat transfer problems with sharp spatial gradients are analyzed using the Generalized Finite Element Method with global-local enrichment functions (GFEM gl). With this approach, scale-bridging enrichment functions are generated on the fly, providing specially-tailored enrichment functions for the problem to be analyzed with no a-priori knowledge of the exact solution. In this work, a decomposition of the linear system of equations is formulated for both steady-state and transient heat transfer problems, allowing for a much more computationally efficient analysis of the problems of interest. With this algorithm, only a small portion of the global system of equations, i.e., the hierarchically added enrichments, need to be re-computed for each loading configuration or time-step. Numerical studies confirm that the condensation scheme does not impact the solution quality, while allowing for more computationally efficient simulations when large problems are considered. We also extend the GFEM gl to allow for the use of hexahedral elements in the global domain, while still using tetrahedral elements in the local domain, to allow for automatic localized mesh refinement without the use of constrained approximations. Simulations are run with the use of linear and quadratic hexahedral and tetrahedral elements in the global domain. Convergence studies indicate that the use of a different partition of unity (PoU) in the global (hexahedral elements) and local (tetrahedral elements) domains does not adversely impact the solution quality.

Comparison of hierarchical and non-hierarchical error indicators for adaptive mesh refinement for the Euler equations

This article deals with two automatic mesh refinement methods for the Euler equations discretized by the discontinuous Galerkin finite element method of lowest order. The main purpose is the comparison of an hierarchical error indicator with a non-hierarchical one. The underlying mesh refinements are based on quadrilateral meshes with hanging nodes. The hierarchical indicator is based on the comparison of the solutions of two consecutive globally refined meshes. It therefore includes information of the nonlinear problem on the fine mesh. In contrast to this, the non-hierarchical indicator employs only information on the given mesh in form of jumps of the solution over internal edges. As often done in practice, in both cases, we chose a certain physical quantity such as the density or the Mach number as a sensor variable to which the indicator is applied. Three different standard test problems are considered: •A scramjet test case representing an internal supersonic flow, •A cylinder test case representing an external supersonic flow, •A Naca wing case representing an external transonic flow.

CFD Simulation with Automatic Mesh Refinement for the Flow around Simplified Car Models

CFD Simulation with Automatic Mesh Refinement for the Flow around Simplified Car Models

Crack growth modeling via 3D automatic adaptive mesh refinement based on modified-SPR technique

In this paper, the three-dimensional automatic adaptive mesh refinement is presented in modeling the crack propagation based on the modified superconvergent patch recovery technique. The technique is developed for the mixed mode fracture analysis of different fracture specimens. The stress intensity factors are calculated at the crack tip region and the crack propagation is determined by applying a proper crack growth criterion. An automatic adaptive mesh refinement is employed on the basis of modified superconvergent patch recovery (MSPR) technique to simulate the crack growth by applying the asymptotic crack tip solution and using the collapsed quarter-point singular tetrahedral elements at the crack tip region. A-posteriori error estimator is used based on the Zienkiewicz–Zhu method to estimate the error of fracture parameters and predict the crack path pattern. Finally, the efficiency and accuracy of proposed computational algorithm is demonstrated by several numerical examples.

Fully implicit adaptive method using discontinuous Galerkin finite elements for high speed flows

In this paper, we present automatic and effective mesh refinement techniques for discontinuous Galerkin finite elements methods (DGFEMs) applied to high speed inviscid flows, and based on the CONCHA finite element Library. CONCHA is a C++ software designed for the simulation of complex flow problems. The structure of its finite element library is described in this publication. It also provides different implicit methods for numerical resolution, a posteriori error estimation, and local mesh refinement. The refinement techniques presented are built on hierarchical non-conforming meshes. Then, two test cases of supersonic flows are studied in order to emphasise CONCHA’s efficiency, robustness and flexibility. Perspectives as well as limitations of such refinement techniques are finally discussed.

Numerical evaluation of the transient response due to non-smooth rolling contact using an arbitrary Lagrangian–Eulerian formulation

A theoretical and numerical framework to evaluate rolling contact using an arbitrary Lagrangian–Eulerian (ALE) formulation is established. A finite element formulation is implemented featuring cylinder–plate contact, automated mesh refinement, non-reflecting boundary conditions, and the ability to incorporate surface roughness through user-defined gap functions. Presented examples include rolling contact on a corrugated surface and negotiation of a surface discontinuity. Sensitivity and validation analyses are presented and show the model to be robust and the trends in parametric responses to be reasonable as compared to results in literature. Owing to the ALE formulation, the model can be kept very compact and the computational demands very modest.

Particle Simulations of Planetary Probe Flows Employing Automated Mesh Refinement

Particle Simulations of Planetary Probe Flows Employing Automated Mesh Refinement

Multi-level adaptive solution of anisotropic porous media flow with strong discontinuous jumps in permeability

A fast and robust multi-grid algorithm for the efficient solution of diffusion-like, elliptic problems which exhibit strong discontinuous jumps in diffusivity is presented. Although generally applicable to this class of problem, the focus for illustrative purposes is that of porous media flow; in particular, such flows for which accurate solutions can only be achieved if the full permeability tensor is taken into consideration. The merits of adopting one or the other of two different approaches to deriving a discrete analogue to the steady-state Darcy equation, namely a novel weighted average of permeability formulation and a continuity of flux preservation method, are explored. In addition, automatic mesh refinement is incorporated seamlessly via a multi-level adaptive technique, making full use of the local truncation error estimates available from the inclusive full approximation storage scheme. Adaptive cell- and patch-wise mesh refinement strategies are developed and investigated for this purpose and used to solve a sequence of benchmark problems of increasing complexity. The results obtained reveal: (a) the ease with which the overall approach deals with generating accurate solutions for flows involving both distributed anisotropy and strong discontinuous jumps in permeability; (b) that both discrete analogues produce equivalent results in comparable execution times; and (c) the significant reductions in computing resource, memory, and CPU, to accrue from employing automatic adaptive mesh refinement.

Multigrid solver with automatic mesh refinement for transient elastoplastic dynamic problems

AbstractThis paper presents an adaptive refinement strategy based on a hierarchical element subdivision dedicated to modelling elastoplastic materials in transient dynamics. At each time step, the refinement is automatic and starts with the calculation of the solution on a coarse mesh. Then, an error indicator is used to control the accuracy of the solution and a finer localized mesh is created where the user‐prescribed accuracy is not reached. A new calculation is performed on this new mesh using the non‐linear ‘Full Approximation Scheme’ multigrid strategy. Applying the error indicator and the refinement strategy recursively, the optimal mesh is obtained. This mesh verifies the error indicator on the whole structure. The multigrid strategy is used for two purposes. First, it optimizes the computational cost of the solution on the finest localized mesh. Second, it ensures information transfer among the different hierarchical meshes. A standard time integration scheme is used and the mesh is reassessed at each time step. Copyright © 2010 John Wiley & Sons, Ltd.

Stability and a posteriori error analysis of discontinuous Galerkin methods for linearized elasticity

We consider discontinuous Galerkin finite element methods for the discretization of linearized elasticity problems in two space dimensions. Inf–sup stability results on the continuous and the discrete level are provided. Furthermore, we derive upper and lower a posteriori error bounds that are robust with respect to nearly incompressible materials, and can easily be implemented within an automatic mesh refinement procedure. The theoretical results are illustrated with a series of numerical experiments.

Finite-element analysis of thermal-induced stresses around a cased injection well

Injection of surface fluids (sea water, CO2, steam) into hydrocarbon reservoirs induces thermal stresses in the wellbore structures and in the near-well area. These stresses may endanger the integrity of the casing, the cement and the surrounding formation. Therefore, an accurate assessment of injection-induced thermal stresses and the associated risk of failure are of utmost importance for a safe and environmentally secure oil production. A coupled finite-element model has been developed and tested as a tool for assessing the probability and extent of failure caused by thermal-induced stresses around a cased wellbore. A feature of the model is an option for an automatic mesh refinement using Nikishkov elements and a quad-tree data structure. The refinement is automatically provided in the regions of rapid change of the nodal variables (displacements, temperature).

An adaptive method for computing invariant manifolds in non-autonomous, three-dimensional dynamical systems

We present a computational method for determining the geometry of a class of three-dimensional invariant manifolds in non-autonomous (aperiodically time-dependent) dynamical systems. The presented approach can be also applied to analyse the geometry of 3D invariant manifolds in three-dimensional, time-dependent fluid flows. The invariance property of such manifolds requires that, at any fixed time, they are given by surfaces in R 3 . We focus on a class of manifolds whose instantaneous geometry is given by orientable surfaces embedded in R 3 . The presented technique can be employed, in particular, to compute codimension one (invariant) stable and unstable manifolds of hyperbolic trajectories in 3D non-autonomous dynamical systems which are crucial in the Lagrangian transport analysis. The same approach can also be used to determine evolution of an orientable ‘material surface’ in a fluid flow. These developments represent the first step towards a non-trivial 3D extension of the so-called lobe dynamics — a geometric, invariant-manifold-based framework which has been very successful in the analysis of Lagrangian transport in unsteady, two-dimensional fluid flows. In the developed algorithm, the instantaneous geometry of an invariant manifold is represented by an adaptively evolving triangular mesh with piecewise C 2 interpolating functions. The method employs an automatic mesh refinement which is coupled with adaptive vertex redistribution. A variant of the advancing front technique is used for remeshing, whenever necessary. Such an approach allows for computationally efficient determination of highly convoluted, evolving geometry of codimension one invariant manifolds in unsteady three-dimensional flows. We show that the developed method is capable of providing detailed information on the evolving Lagrangian flow structure in three dimensions over long periods of time, which is crucial for a meaningful 3D transport analysis.

Thin film flow over flexible membranes containing surface texturing: Bio-inspired solutions

Gravity-driven continuous thin liquid film flow over a flexible membrane containing surface topography is modelled using lubrication theory. The associated coupled non-linear equation set, for the film thickness, pressure and membrane deflection, is solved using a state-of-the-art full approximation storage multi-grid algorithm with automatic mesh refinement and adaptive time-stepping, in order to maximize computational efficiency when fine-scale resolution is required while ensuring accurate mesh-independent solutions at the micro-scale. The robustness of the approach is demonstrated through the solution of a series of problems and comparisons drawn with the same flow on an equivalent completely rigid membrane. It is shown that the former differs considerably from the latter in that the film thickness affects the shape of the flexible membrane, the compliance of which in turn impinges on the profile of the resulting free-surface disturbance.

Modeling of crack propagation via an automatic adaptive mesh refinement based on modified superconvergent patch recovery technique

In this paper, an automated adaptive remeshing procedure is presented for simulation of arbitrary shape crack growth in a 2D finite element mesh. The Zienkiewicz–Zhu error estimator is employed in conjunction with a modified SPR technique based on the recovery of gradients using analytical crack-tip fields in order to obtain more accurate estimation of errors. The optimization of crack-tip singular finite element size is achieved through the adaptive mesh strategy. Finally, several numerical examples are illustrated to demonstrate the effectiveness, robustness and accuracy of computational algorithm in calculation of fracture parameters and prediction of crack path pattern.

CESAM: a free code for stellar evolution calculations

The cesam code is a consistent set of programs and routines which perform calculations of 1D quasi-hydrostatic stellar evolution including microscopic diffusion of chemical species and diffusion of angular momentum. The solution of the quasi-static equilibrium is performed by a collocation method based on piecewise polynomials approximations projected on a B-spline basis; that allows stable and robust calculations, and the exact restitution of the solution, not only at grid points, even for the discontinuous variables. Other advantages are the monitoring by only one parameter of the accuracy and its improvement by super-convergence. An automatic mesh refinement has been designed for adjusting the localisations of grid points according to the changes of unknowns. For standard models, the evolution of the chemical composition is solved by stiffly stable schemes of orders up to four; in the convection zones mixing and evolution of chemical are simultaneous. The solution of the diffusion equation employs the Galerkin finite elements scheme; the mixing of chemicals is then performed by a strong turbulent diffusion. A precise restoration of the atmosphere is allowed for.

A posteriori error estimates for a nonconforming finite element discretization of the time-dependent Stokes problem

We present aposteriori error estimators for the time-dependent Stokes problem in d , d 2 or 3. Our analysis covers non-conforming finite element approximation (Crouzeix-Raviart's elements) in space and backward Euler's scheme in time. For this discretization, we derive a residual indicator, which uses a spatial residual indicator based on the jumps of normal and tangential deriva- tives of the nonconforming approximation and a time residual indicator based on the jump of broken gradients at each time step. Lower and upper bounds form the main results with minimal assumptions on the mesh. Numerical experiments confirm the theoretical predictions and show the usefulness of these estimators on automatic mesh refinement.

MLS (moving least square)-based finite elements for three-dimensional nonmatching meshes and adaptive mesh refinement

With the aid of moving least square (MLS) approximation, a new class of three-dimensional finite elements are proposed for treating nonmatching meshes and adaptive mesh refinement, for which the existing finite elements are hardly efficient. With a special choice of the weight-function supports and the base functions, the method results in useful elements with the polynomial shape function, for which the C 1 continuity breaks down on the boundaries between the neighboring subdomains comprising one element. The effectiveness of the new elements in handling the discontinuities due to nonmatching interfaces and automatic mesh refinement is demonstrated via three-dimensional examples.

Refinement Indicator for Mesh Adaption in Shallow-Water Modeling

Automatic mesh refinement can create suitable resolution for a hydrodynamic simulation in a computationally efficient manner. Development of an automatic adaptive procedure will rely on estimating and/or controlling computational error by adapting the mesh parameters with respect to a particular measurement. Since a primary source of error in a discrete approximation of the shallow-water equations is inadequate mesh resolution, an adaptive mesh can be an efficient approach to increase accuracy. This paper introduces a simple indicator for the shallow water equations that measures the error in a norm of mass conservation to determine which elements require refinement or coarsening. The resulting adaptive grid gives results comparable to a much higher resolution (uniformly refined) mesh with less computational expense.

Fully automatic hp-adaptivity in three dimensions

The paper presents a three-dimensional implementation of the fully automatic hp-adaptive mesh refinement strategy proposed in [L. Demkowicz, W. Rachowicz, Ph. Devloo, A fully automatic hp-adaptivity, J. Sci. Comput. 17 (1–3) (2002) 127–155]. The methodology is based on a minimization of the projection-based interpolation error with a fixed number of degrees-of-freedom. A (coarse) grid is first uniformly refined in both h and p to yield a corresponding fine grid. A two-grid solver is used to solve the problem on the fine grid or iterate towards the fine grid solution. The fine grid solution (possibly partially converged) serves then as a reference solution to be interpolated on the original coarse mesh, and the next, optimal coarse grid to be determined. The difference between the coarse and fine grid solutions provides an excellent error estimate for the coarse grid error. The procedure is repeated, with the coarse and fine meshes generated during the adaptive process simultaneously, until a prescribed error tolerance for the coarse grid error is met. The solution obtained on the last fine grid is considered to be the final product delivered by the adaptive procedure.