Questions tagged [adaptive-mesh-refinement]

For questions about AMR, a technique for dynamically updating the precision of a grid to ensure a certain accuracy for a given region.

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numerical integration of integrals in the p-adaptive version of the finite element method

In the p-adaptive version of the finite element method, elements are allowed to have shape functions with arbitrary different polynomial orders. Therefore regarding a 2D problem with quadrilateral ...
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Mesh refinement in the Finite Element Method

I need some good references on how to implement programmatically the hp-refinement of meshes in the Finite Element Method in two/three-dimension. I've searched the web a lot and read many articles and ...
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Comparison on adaptive mesh refinement on finite elements and finite differences

My current work requires using (Adaptive Mesh Refinement) AMR to resolve multi scale physics. I have a general question whether finite element is better than finite difference in this aspect or not. I ...
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Connectivity of octree as grid

What is the algorithm which transforms the octree/quadtree into a grid? i.e, getting the adjacency or connectivity between the cells in order to be able to perform interpolation, gradient,...etc. I am ...
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Proof of R. Verfürth paper on adaptive mesh and bubble functions

I'm studying adaptive meshes, and my professor wrote the following property for a bubble function ( see this scicomp post for the definition I'm using)$b_T$ defined on a triangle $T$. $$||b_T \phi ||_{...
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Computing the residual in a Dual Weighted Residual (DWR) method

I am in the process if computing the Dual-Weighted Residual (DWR) for a linear PDE with a linear functional but I am struggling with the residual part of the calculation. For example suppose we want ...
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Mesh aspect ratio issue with adaptive mesh refinement (AMR)

I am working on implementation of AMR for my finite volume code. Let me use a 2-D mesh to describe my question. Starting with SINGLE initial cell (let the mesh refine level k = 0) as a root of a quad-...
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Finite Difference libray C++

What is the best FD library (or collection of libraries) for C++ codes? I am looking for some data structure implementation that offers the possibility to do parallel computations on adaptively ...
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Recovery of smoothed continuous stresses using the Z^2 error estimator

I have trouble implementing the $Z^2$ error estimator (The superconvergent patch recovery and a posteriori error estimates. Part 1: the recovery technique by Zienkiewicz and Zhu). For this, I am ...
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Implementation of Z^2 error estimator in Abaqus for adaptive mesh refinement

Currently, I am working on a remeshing routine for my simulations (Abaqus 6.14-1) using python scripts. The simulation deals with the Brinell indentation test and as the remeshing software I use Gmsh ...
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Data structures of AMR(Adaptive Mesh Refinement) with quadtree

I am graduate student and recently tried to implement quad-tree based AMR. Well I have implemented simple Poisson Solver on node-based quadtree, but I am not sure that my data structure is right. To ...
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simple example of an adaptive mesh refinement code

I have been writing an adaptive mesh refinement (amr) code. As a prototype for the code, I have been looking at an adaptive mesh refinement code written by my adviser (written in c). I find looking at ...
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Good references for dual-weighted residual (DWR) method for goal-oriented adaptive mesh refinement (AMR)

Can anyone help me with good references (books or papers) where I can learn about dual-weighted residual (DWR) method for goal-oriented adaptive mesh refinement (AMR)?
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Oscillation term in a posteriori error estimator

Assume that in the a (residual type) posteriori error estimator of some PDE is a term of the form $h_T\|g\|_{L^2(\Omega)}$ involved where $h_T$ is the diameter of an element and $g$ is some known data ...
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How can I make sure the flow is divergence-free when I use moving mesh?

I am using projection method and P2/P1 finite element method to solve the incompressible Navier-Stokes equations while the mesh is constantly adapted as the body moves (edge swapping, splitting and ...
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p-refinement in adaptive methods

Some reference in adaptive techniques say that when the solution $u$ is smooth enough we can use p-refinement instead of h-refinement. And when we have for example singularity, we should use h-...
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scalable parallel mesh/amr on unstructured grid

I am trying to code a scalable parallel AMR for unstructured grid. There seems to be three approaches for this a) Store some global grid info on each processor and partition with parmetis (The ...
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Non-cubic blocks in Adaptive Mesh Refinement

Most Adaptive Mesh Refinement softwares/logic have cubic blocks. I have explored the BoxLib library and it ONLY supports cubic blocks. To be more precise it supports non-cubic blocks at the coarsest ...
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Algorithm for Adaptive Mesh Refinement

I am trying to implement Adaptive Mesh Refinement. I am not a Mathematics/Computational Science person so I will try to write the algorithm in a simpler way. I will be grateful if experts can comment ...
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1 answer
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fast adaptive quadrature on equispaced 2-D grid

I need to numerically evaluate 2-D integrals of the form: $$ \mathcal{I}(\theta) = \int_{0}^{1} \int_0^1 \varphi_\theta(x,y) dx dy $$ where $\varphi_\theta$ is a family of smooth functions indexed by ...
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Square error estimate for adaptive mesh refinement

In a particular implementation(Finite volume advection using upwind) of adaptive mesh refinement the error square estimate for a cell C is given as $$ \sum_{i = x,y,z} vol * \frac{1}{12} * h^{2} * (\...
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How to deal with transition elements in adaptive fem

It is necessary for me to solve a Poisson equation by adaptive finite element method with transition elements technique to get conforming mesh. For the first local refinement everything is OK, ...
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Non-uniform finite difference Adaptive Mesh Refinement

Assuming that the crosses in the figure below are unknowns in a vertex-centered finite difference scheme in an Adaptive Mesh, how can I calculate the double derivate (Laplacian) at the Red x ? The Red ...
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2 votes
2 answers
229 views

Finite element results by different meshes

There are some technique to generate mesh in a domain. My qustion is that: Is there any difference between the results using different techniques for mesh generation? If yes which one is better. For ...
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Refinement in AMR

Assume I start with an 8x8 coarse mesh (see Fig 1) where the vertices (except boundary vertices) represent the unknown variable. After iterative approximation - I ...
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2 votes
1 answer
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Implementing Finite Difference Adaptive Mesh Refinement code

For a start I need to implement a 2-D, vertex centered, finite difference scheme, Adaptive mesh refinement serial code. I have the following doubts before starting: Is the input to AMR (say 2-D) ...
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2 votes
1 answer
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Strategies for controlling number of new elements in adaptive mesh refinement

I am working on adaptive techniques for solving some elliptic equations. The technique is based on residual on elements. My problem is that when I use a predefined tolerance for refining elements, the ...
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Using finite element error estimators for adaptive mesh refinement

I am in the process of implementing adaptive mesh refinement for a finite element code that solves the Poisson equation. I have had some trouble finding good references on deciding which elements to ...
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5 votes
1 answer
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unstructured grid AMR

Are there libraries for conducting parallel AMR on an unstructured grid ? For a finite volume code, polyhedral cells with arbitrarily shaped faces are as easy to handle as hexahedra, and infact ...
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5 votes
2 answers
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How to implement adaptive mesh refinement using conformal triangles

I am trying to implement adaptive mesh refinement for a finite element code. The code uses (at least for now) linear triangles and so when I do the mesh refinement I want the triangular mesh to remain ...
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4 votes
1 answer
397 views

Finite element mesh software

I'm looking for a program to obtain meshes to finite element codes 2D and 3D as complete as possible, preferably in fortran 90 or C/C++. For example, softwares "Triangle" or "TetGen" generate meshes ...
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How to define fluxes for two dimensional convection-diffusion equation?

I want to solve the following differential equation using control volume approach on a Cartesian mesh: $$\frac{\partial T}{\partial t} + \frac{\partial T}{\partial x} + \frac{\partial T}{\partial y}= \...
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356 views

Adaptive Finite Element Method - Laplace

I'm currently attempting to turn my code for solving the laplace equation using finite element approximations into an adaptive one using the dual weighted residual as my error estimator: i.e. my ...
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1 answer
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Flux at coarse-fine mesh grid interface?

I am trying to solve one dimensional inviscid Burger's equation using adaptive mesh refinement. This is the PDE: $$\frac{\delta U}{\delta t} + \frac{\delta F}{\delta x} = 0$$ where the flux F of the ...
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2 answers
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How to discretize Burger's equation?

I am trying to solve the very simple one dimensional burgers equation which is: $$\frac{\delta U}{\delta t} + \frac{\delta F}{\delta x} = 0$$ where the flux F of some variable U is defined as$$ F= \...
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9 votes
0 answers
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Time advance in Adaptive Mesh Refinement method

I am working on solving complex system of 2D PDEs governing the behaviour of plasma in a gas lamp during discharge. Recent tests have shown that because of steep gradients in temperature field and ...
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Conservation at grid interface in adaptive mesh refinement

I am using adaptive mesh refinement to solve one dimensional inviscid Burgers equation. However I am facing some difficulty to handle grid interfaces which are not uniform (coarse-fine grid interface)....
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1 answer
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Adaptive mesh refinement basic conceptual problem

I am a beginner in adaptive mesh refinement (AMR). After I am done with the first two papers by Dr. Marsha Berger, I was trying to write my own code for a problem which has a parabolic partial ...
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4 votes
1 answer
91 views

How can I compare errors in PDE solvers with non-uniform grids?

Is there a standard approach to testing codes with refined regions? Specifically, I am interested in testing whether the refinement is working correctly. For the sake of simplicity, let's consider a (...
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4 votes
2 answers
257 views

Adaptive mesh refinement algorithms and the difference between AMR and moving mesh

I'm working on my thesis and a part of it has to do with adaptive mesh refinement. As a computer science major, I'm not too familiar with this field. The best way I can put my knowledge of AMR is: I ...
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3 votes
1 answer
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Fast methods to solve an elliptic PDE if high accuracy is needed only in part of the domain

Does someone know a method to get cheap approximation of harmonic problems (and possibly local approximations)? Let me explain: I need to compute the solution of an harmonic problem \begin{equation} ...
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2 votes
0 answers
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Adaptive Mesh vs Uniform Mesh for multiple source/boundary/initial Data

I'm going to ask some beginners' questions. Adaptive mesh can save many DOFs than a uniform mesh. But it also needs to solve linear systems changing with mesh adaptive process. Is this not problem? ...
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4 votes
1 answer
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Meshes in codes that do AMR

In light of this question and some stuff I read online I am wondering if large FE libs (e.g., deal.ii, libmesh etc.) that do AMR keep the entire mesh or possibly a coarse version of the entire mesh on ...
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9 votes
3 answers
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Finite Element Method vs Extended Finite Element Method (FEM vs XFEM)

What are main differences between FEM and XFEM? When should we (not) use XFEM intead of FEM and vice versa? In other words, when I meet a new problem, how I can know to use which one of them?
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6 votes
1 answer
255 views

Proving convergence of adaptive finite elements - min res FEM?

There's a body of work out there dealing with the discrete convergence of adaptive finite element methods using error estimators. Most deal with proving the property $\|u-u_{k+1}\|_U \leq (1-\alpha) \...
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7 votes
3 answers
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Conforming mesh refinement for quads/hex elements

The context - I'm working with a spectral FE (higher order interpolation at GLL nodes) code on conforming hexahedral meshes, and our PI is interested in improving mesh quality, possibly with adaptive ...
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7 votes
2 answers
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Convergence of adaptive finite elements with inexact solves

I'm working on some adaptive discontinuous Galerkin codes for time harmonic wave propagation, currently just Helmholtz, but will be branching out once I have a working prototype in this case. There ...
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2 votes
1 answer
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Scalable, effective and mesh quality assuring local dynamic tetrahedral mesh refinement algorithm

I have been reading about Tetrahedral Mesh Refinement algorithms, but the literature covering this is very wide. My work involves implementation of different 3D computational geometry algorithms, and ...
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13 votes
3 answers
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What are the basic principles behind generating a moving mesh?

I am interested in implementing an moving mesh for an advection-diffusion problem. Adaptive Moving Mesh Methods gives a good example of how to do this for Burger's equation in 1D using finite-...
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5 votes
0 answers
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Adaptive mesh data structure for Fast Marching Method to overcome RAM limit

On an uniform mesh of positions in space $\ (x_i,y_j,z_k)$: $$\ x_i = x_0 + i\Delta x,\quad i=0,\ldots,n_x$$ $$\ y_j = y_0 + j\Delta y,\quad j=0,\ldots,n_y$$ $$\ z_k = z_0 + k\Delta z,\quad k=0,\...
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