scholarly journals A Numerical Method for Solving Elliptic Interface Problems Using Petrov-Galerkin Formulation with Adaptive Refinement

2018 ◽  
Vol 2018 ◽  
pp. 1-12
Author(s):  
Liqun Wang ◽  
Songming Hou ◽  
Liwei Shi
Keyword(s):  
Mesh Generation ◽  
Adaptive Mesh Refinement ◽  
Adaptive Mesh ◽  
Previous Method ◽  
Interface Problems ◽  
Adaptive Refinement ◽  
Elliptic Interface Problems ◽  
Cpu Time ◽  
Galerkin Formulation ◽  
The Cost

Elliptic interface problems have wide applications in engineering and science. Non-body-fitted grid has the advantage of saving the cost of mesh generation. In this paper, we propose a Petrov-Galerkin formulation using non-body-fitted grid for solving elliptic interface problems. In this method, adaptive mesh refinement is employed for cells with large errors. The new mesh still has all triangles being right triangles of the same shape. Numerical experiments show side-by-side comparison that to obtain the same accuracy, our new method has much less overall CPU time compared with the previous method even with some cost on mesh generation.

Development of an Object-Oriented Program With Adaptive Mesh Refinement for Heat Transfer Simulations

2007 ◽  
Author(s):  
Jianhu Nie ◽  
David A. Hopkins ◽  
Yitung Chen ◽  
Hsuan-Tsung Hsieh
Keyword(s):  
Heat Transfer ◽  
Adaptive Mesh Refinement ◽  
Large Scale ◽  
Mesh Refinement ◽  
Object Oriented ◽  
Adaptive Mesh ◽  
Time Cost ◽  
Test Bed ◽  
Element Analysis ◽  
Cpu Time

A 2D/3D object-oriented program with h-type adaptive mesh refinement method is developed for finite element analysis of the multi-physics applications including heat transfer. A framework with some basic classes that enable the code to be built accordingly to the type of problem to be solved is proposed. The program consists of different modules and classes, which ease code development for large-scale complex systems, code extension and program maintenance. The developed program can be used as a “test-bed” program for testing new analysis techniques and algorithms with high extensibility and flexibility. The overall mesh refinement causes the CPU time cost to greatly increase as the mesh is refined. However, the CPU time cost does not increase very much with the increase of the level of h-adaptive mesh refinement. The CPU time cost can be saved by up to 90%, especially for the simulated system with a large number of elements and nodes.

Logically rectangular finite volume methods with adaptive refinement on the sphere

2009 ◽  
Vol 367 (1907) ◽  
pp. 4483-4496 ◽  
Author(s):  
Marsha J. Berger ◽  
Donna A. Calhoun ◽  
Christiane Helzel ◽  
Randall J. LeVeque
Keyword(s):  
Finite Volume ◽  
Shallow Water Equations ◽  
Adaptive Mesh Refinement ◽  
Three Dimensional ◽  
Adaptive Mesh ◽  
Finite Volume Methods ◽  
Adaptive Refinement ◽  
Two Dimensional ◽  
Equilibrium Solutions ◽  
Courant Number

The logically rectangular finite volume grids for two-dimensional partial differential equations on a sphere and for three-dimensional problems in a spherical shell introduced recently have nearly uniform cell size, avoiding severe Courant number restrictions. We present recent results with adaptive mesh refinement using the G eo C law software and demonstrate well-balanced methods that exactly maintain equilibrium solutions, such as shallow water equations for an ocean at rest over arbitrary bathymetry.

Convergence Acceleration for Heat Transfer and Structural Simulations Using Adaptive Mesh Refinement

2006 ◽  
Author(s):  
Jianhu Nie ◽  
Yitung Chen ◽  
David A. Hopkins ◽  
Lijian Sun ◽  
Hsuan-Tsung Hsieh
Keyword(s):  
Heat Transfer ◽  
Adaptive Mesh Refinement ◽  
Mesh Refinement ◽  
Adaptive Mesh ◽  
Convergence Acceleration ◽  
Heat Transfer Fluid ◽  
Time Cost ◽  
Computational Accuracy ◽  
Finite Element Program ◽  
Cpu Time

A finite element program with h-type mesh adaptation is developed and several test cases for heat transfer, fluid mechanics and structural mechanics are selected for code validations. The element division method is used because of its advantage of avoiding overly twisted elements during mesh refinement and recovery. The adaptive mesh is refined only in the localization region where the feature gradient is high. The overall mesh refinement and the h-adaptive mesh refinement are justified with respect to the computational accuracy and the CPU time cost. Both can improve the computational accuracy. The overall mesh refinement causes the CPU time to greatly increase. However, the CPU time does not increase very much with the increase of the level of h-adaptive mesh refinement. The CPU time cost can be saved using the developed program by orders of magnitude, especially for the system with a large number of elements and nodes.

scholarly journals D2.2 First release of the octree mesh-generation capabilities and of the parallel mesh adaptation kernel

2021 ◽  
Author(s):  
A. Martín ◽  
L. Cirrottola ◽  
A. Froehly ◽  
R. Rossi ◽  
C. Soriano
Keyword(s):  
Finite Element ◽  
Mesh Generation ◽  
Adaptive Mesh Refinement ◽  
Adaptive Mesh ◽  
Mesh Adaptation ◽  
The Other ◽  
Project Proposal ◽  
Parallel Mesh Adaptation ◽  
Algebraic Constraints ◽  
Octree Mesh

This document presents a description of the octree mesh-generation capabilities and of the parallel mesh adaptation kernel. As it is discussed in Section 1.3.2 of part B of the project proposal there are two parallel research lines aimed at developing scalable adaptive mesh refinement (AMR) algorithms and implementations. The first one is based on using octree-based mesh generation and adaptation for the whole simulation in combination with unfitted finite element methods (FEMs) and the use of algebraic constraints to deal with non-conformity of spaces. On the other hand the second strategy is based on the use of an initial octree mesh that, after make it conforming through the addition of templatebased tetrahedral refinements, is adapted anisotropically during the calculation. Regarding the first strategy the following items are included:

Adaptive Mesh Refinement and Superconvergence for Two-Dimensional Interface Problems

2014 ◽  
Vol 36 (4) ◽  
pp. A1478-A1499 ◽  
Author(s):  
Huayi Wei ◽  
Long Chen ◽  
Yunqing Huang ◽  
Bin Zheng
Keyword(s):  
Adaptive Mesh Refinement ◽  
Mesh Refinement ◽  
Adaptive Mesh ◽  
Interface Problems ◽  
Two Dimensional

scholarly journals A sharp interface method using enriched finite elements for elliptic interface problems

2021 ◽  
Author(s):  
Susanne Höllbacher ◽  
Gabriel Wittum
Keyword(s):  
Adaptive Mesh Refinement ◽  
Order Approximation ◽  
Interface Problems ◽  
Stability And Convergence ◽  
Optimal Order ◽  
Bilinear Operator ◽  
Optimal Order Of Convergence ◽  
Elliptic Interface Problems ◽  
Integration Schemes ◽  
Cut Elements

AbstractWe present an immersed boundary method for the solution of elliptic interface problems with discontinuous coefficients which provides a second-order approximation of the solution. The proposed method can be categorised as an extended or enriched finite element method. In contrast to other extended FEM approaches, the new shape functions get projected in order to satisfy the Kronecker-delta property with respect to the interface. The resulting combination of projection and restriction was already derived in Höllbacher and Wittum (TBA, 2019a) for application to particulate flows. The crucial benefits are the preservation of the symmetry and positive definiteness of the continuous bilinear operator. Besides, no additional stabilisation terms are necessary. Furthermore, since our enrichment can be interpreted as adaptive mesh refinement, the standard integration schemes can be applied on the cut elements. Finally, small cut elements do not impair the condition of the scheme and we propose a simple procedure to ensure good conditioning independent of the location of the interface. The stability and convergence of the solution will be proven and the numerical tests demonstrate optimal order of convergence.

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