A Mesh Adaptive Procedure for Large Increment Method

2015 ◽  
Vol 07 (04) ◽  
pp. 1550061 ◽  
Author(s):  
Zaoyang Guo ◽  
Yujie Zhao ◽  
Zhaohui Chen ◽  
Minmao Liao ◽  
Zhengliang Li ◽  
...  
Keyword(s):  
Adaptive Mesh Refinement ◽  
Adaptive Mesh ◽  
Adaptive Strategy ◽  
Original Method ◽  
Initial Solution ◽  
Error Estimator ◽  
Adaptive Procedure ◽  
Minimum Norm Solution ◽  
Large Increment ◽  
Refinement Procedure

A posteriorih-version mesh adaptive procedure is presented in the framework of large increment method (LIM) for elastic problems. In this mesh adaptive strategy, the classical Zienkiewicz–Zhu (ZZ) error estimator is adopted and a first class h-adaptive mesh refinement procedure is implemented. A major advantage of the proposed mesh adaptive procedure is that the numerical results from the previous mesh can be utilized to obtain the initial solution for the new mesh. Two-dimensional (2D) examples show that this initial solution is much closer to the real solution than the minimum norm solution used in the original LIM and the revised method can converge faster than the original method.

Efficiency and Optimality of Some Weighted-Residual Error Estimator for Adaptive 2D Boundary Element Methods

2013 ◽  
Vol 13 (3) ◽  
pp. 305-332 ◽  
Author(s):  
Markus Aurada ◽  
Michael Feischl ◽  
Thomas Führer ◽  
Michael Karkulik ◽  
Dirk Praetorius
Keyword(s):  
Boundary Element ◽  
Adaptive Mesh Refinement ◽  
Mesh Refinement ◽  
Adaptive Mesh ◽  
Weak Form ◽  
Adaptive Strategy ◽  
Optimal Choice ◽  
Residual Error ◽  
Boundary Element Methods ◽  
Error Estimator

Abstract. We prove convergence and quasi-optimality of a lowest-order adaptive boundary element method for a weakly-singular integral equation in 2D. The adaptive mesh-refinement is driven by the weighted-residual error estimator. By proving that this estimator is not only reliable, but under some regularity assumptions on the given data also efficient on locally refined meshes, we characterize the approximation class in terms of the Galerkin error only. In particular, this yields that no adaptive strategy can do better, and the weighted-residual error estimator is thus an optimal choice to steer the adaptive mesh-refinement. As a side result, we prove a weak form of the saturation assumption.

A h-Adaptive Algorithm Using Residual Error Estimates for Fluid Flows

2013 ◽  
Vol 13 (2) ◽  
pp. 461-478 ◽  
Author(s):  
N. Ganesh ◽  
N. Balakrishnan
Keyword(s):  
Adaptive Algorithm ◽  
Adaptive Mesh Refinement ◽  
Fluid Flows ◽  
Adaptive Mesh ◽  
Viscous Flows ◽  
Adaptive Strategy ◽  
Residual Error ◽  
Error Estimator ◽  
Flow Problems ◽  
Error Indicators

AbstractAlgorithms for adaptive mesh refinement using a residual error estimator are proposed for fluid flow problems in a finite volume framework. The residual error estimator, referred to as the ℜ-parameter is used to derive refinement and coarsening criteria for the adaptive algorithms. An adaptive strategy based on the ℜ-parameter is proposed for continuous flows, while a hybrid adaptive algorithm employing a combination of error indicators and the ℜ-parameter is developed for discontinuous flows. Numerical experiments for inviscid and viscous flows on different grid topologies demonstrate the effectiveness of the proposed algorithms on arbitrary polygonal grids.

scholarly journals Implementation and performance of adaptive mesh refinement in the Ice Sheet System Model (ISSM v4.14)

2019 ◽  
Vol 12 (1) ◽  
pp. 215-232 ◽  
Author(s):  
Thiago Dias dos Santos ◽  
Mathieu Morlighem ◽  
Hélène Seroussi ◽  
Philippe Remy Bernard Devloo ◽  
Jefferson Cardia Simões
Keyword(s):  
Adaptive Mesh Refinement ◽  
Mesh Refinement ◽  
Computational Cost ◽  
Ice Sheet ◽  
Adaptive Mesh ◽  
System Model ◽  
Computational Time ◽  
Error Estimator ◽  
Fine Mesh ◽  
Grounding Line

Abstract. Accurate projections of the evolution of ice sheets in a changing climate require a fine mesh/grid resolution in ice sheet models to correctly capture fundamental physical processes, such as the evolution of the grounding line, the region where grounded ice starts to float. The evolution of the grounding line indeed plays a major role in ice sheet dynamics, as it is a fundamental control on marine ice sheet stability. Numerical modeling of a grounding line requires significant computational resources since the accuracy of its position depends on grid or mesh resolution. A technique that improves accuracy with reduced computational cost is the adaptive mesh refinement (AMR) approach. We present here the implementation of the AMR technique in the finite element Ice Sheet System Model (ISSM) to simulate grounding line dynamics under two different benchmarks: MISMIP3d and MISMIP+. We test different refinement criteria: (a) distance around the grounding line, (b) a posteriori error estimator, the Zienkiewicz–Zhu (ZZ) error estimator, and (c) different combinations of (a) and (b). In both benchmarks, the ZZ error estimator presents high values around the grounding line. In the MISMIP+ setup, this estimator also presents high values in the grounded part of the ice sheet, following the complex shape of the bedrock geometry. The ZZ estimator helps guide the refinement procedure such that AMR performance is improved. Our results show that computational time with AMR depends on the required accuracy, but in all cases, it is significantly shorter than for uniformly refined meshes. We conclude that AMR without an associated error estimator should be avoided, especially for real glaciers that have a complex bed geometry.

scholarly journals Implementation and Performance of Adaptive Mesh Refinement in the Ice Sheet System Model (ISSM v4.14)

2018 ◽  
Author(s):  
Thiago Dias dos Santos ◽  
Mathieu Morlighem ◽  
Hélène Seroussi ◽  
Philippe Remy Bernard Devloo ◽  
Jefferson Cardia Simões
Keyword(s):  
Adaptive Mesh Refinement ◽  
Mesh Refinement ◽  
Computational Cost ◽  
Ice Sheet ◽  
Adaptive Mesh ◽  
System Model ◽  
Computational Time ◽  
Error Estimator ◽  
Grounding Line ◽  
Refined Meshes

Abstract. Accurate projections of the evolution of ice sheets in a changing climate require a fine mesh/grid resolution to correctly capture fundamental physical processes, such as the evolution of the grounding line, the region where grounded ice starts to float. The evolution of the grounding line indeed plays a major role in ice sheet dynamics, as it is a fundamental control on marine ice sheet stability. Numerical modeling of grounding line requires significant computational resources since the accuracy of its position depends on grid or mesh resolution. A technique that improves accuracy with reduced computational cost is the adaptive mesh refinement approach, AMR. We present here the implementation of the AMR technique in the finite element Ice Sheet System Model (ISSM) to simulate grounding line dynamics under two different benchmarks, MISMIP3d and MISMIP+. We test different refinement criteria: (a) distance around grounding line, (b) a posteriori error estimator, the Zienkiewicz-Zhu (ZZ) error estimator, and (c) different combinations of (a) and (b). We find that for MISMIP3d setup, refining 5 km around the grounding line, both on grounded and floating ice, is sufficient to produce AMR results similar to the ones obtained with uniformly refined meshes. However, for the MISMIP+ setup, we note that there is a minimum distance of 30 km around the grounding line required to produce accurate results. We find this AMR mesh-dependency is linked to the complex bedrock topography of MISMIP+. In both benchmarks, the ZZ error estimator presents high values around the grounding line. Particularly for MISMIP+ setup, the estimator also presents high values in the grounded part of the ice sheet, following the complex shape of the bedrock geometry. This estimator helps guide the refinement procedure such that AMR performance is improved. Our results show that computational time with AMR depends on the required accuracy, but in all cases, it is significantly shorter than for uniformly refined meshes. We conclude that AMR without an associated error estimator should be avoided, especially for real glaciers that have a complex bed geometry.

scholarly journals Residual-based a posteriori error analysis for the coupling of the Navier-Stokes and Darcy-Forchheimer equations

2021 ◽  
Author(s):  
Sergio Caucao ◽  
Gabriel Gatica ◽  
Ricardo Oyarzúa ◽  
Felipe Sandoval
Keyword(s):  
Adaptive Mesh Refinement ◽  
Adaptive Mesh ◽  
Posteriori Error ◽  
Local Approximation ◽  
Navier Stokes ◽  
Error Estimator ◽  
A Posteriori ◽  
Approximation Properties ◽  
A Posteriori Error ◽  
Posteriori Error Analysis

In this paper we consider a mixed variational formulation that have been recently proposed for the coupling of the Navier--Stokes and Darcy--Forchheimer equations, and derive,  though in a non-standard sense,  a reliable and efficient residual-based a posteriori error estimator suitable for an adaptive mesh-refinement method.  For the reliability estimate, which holds with respect to the square root of the error estimator, we make use of the inf-sup condition and the strict monotonicity of the operators involved, a suitable Helmholtz decomposition in non-standard Banach spaces in the porous medium, local approximation properties of the Cl\'ement interpolant and Raviart--Thomas operator, and a smallness assumption on the data.   In turn, inverse inequalities, the localization technique based on triangle-bubble and edge-bubble functions in local $\L^\rp$ spaces, are the main tools for developing the effi\-ciency analysis, which is valid for the error estimator itself up to a suitable additional error term. Finally, several numerical results confirming the properties of the estimator and illustrating the performance of the associated adaptive algorithm are reported.

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