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Making a mesh watertight
There are several efficient algorithms to make a mesh watertight, historically, in Stanford, the pionneers of 3D Scanning developed the Zipper algorithm:
https://graphics.stanford.edu/papers/zipper/
Then many algorithms were developed, you may refer for instance to the following website maintained by friends of mine:
http://www....
10
I would recommend gmsh. I have just started working with this program actually only a few days ago. But it is straight-forward to use. You can create various 2D and even 3D-geometries and it offers a ton of information, boundary nodes, etc..
Here is a link to the website: http://geuz.org/gmsh/
They have many useful references, there is a manual of course ...
9
One of the authors of fenics, A. Logg, have written a very good paper on datastructures of storing meshes. The paper is
A. Logg (2009). Efficient Representation of Computational Meshes
http://arxiv.org/abs/1205.3081
In fact it's always a tradeoff between storing all the topological informations (nodes around nodes, faces around nodes, etc...) OR having to ...
9
First, create a list of faces in the mesh. From there you should be able to create a map from faces to tets, as each face must belong to either one or two tets. The faces that belong to only one tet are your boundary faces.
8
deal.II keeps the entire coarse mesh on every processor, but of course we cannot do that with the actual mesh after many refinement steps. It is true that this somewhat limits the size of problems we can solve to maybe a few hundred thousand coarse mesh cells on typically-sized cluster nodes. However, this is plenty for most realistic cases for a reasonable ...
8
In deal.II, we basically only use vectors. Maps are too slow and scatter data all around memory, so we typically don't use them if the keys are integers and within a given range. For example, for the connectivity between cells, you can do arrays (STL vectors) in which you store neighbor indices and so that neighbor indices $4i\ldots 4i+3$ correspond to cell $...
8
The short answer is no, there is not a standard format. But there are some common ones, like Gmsh for input/output and VTK for output.
Before making a decision you need to find out what do you want to do. If you want to have your (small) program for a while, then you can pick the format that best suit to your taste and needs. If you are planning to change ...
8
The idea of "ordering the nodes" in a finite element mesh to improve the
computational time of the sparse solver originated in the large structural
analysis FE codes of the 70's. Those codes typically used banded or variable-band
storage schemes for the sparse matrices so reducing the bandwidth was the main
criterion. That is the origin of the old Cuthill-...
7
I think you could use the "marching cubes" algorithm. If memory serves, it requires a grid of samples as input, so at the very least you should be able to sample your function and run the algorithm as-is. You also might be able to modify the algorithm to callback to f directly. There's a popular implementation at http://paulbourke.net/geometry/polygonise/ ...
7
Yes. The constants that appear in the interpolation estimates upon which finite element error estimates are based contain minimum and maximum angles of triangles/tetrahedra (or similar geometric measures for quadrilaterals/hexahedra). These constants are smallest whenever you have equilateral triangles or square quadrilaterals.
6
To complement the two answers from Daniel Shapero and Nicoguaro:
Basically, there are two ways of smoothing a mesh, subdivision (generate new vertices) and smoothing (move the points in such a way that the obtained shape is smoother).
Subdivision
To grasp the intuition, imagine you want to "smoothen" a 2D square. The 2D square is not smooth because it has ...
6
Domain decomposition was developed in the late 1990s and early 2000s because it allowed the re-use of sequential PDE solvers: You only have to write a wrapper around it that sends the computed solution to other processors, receives other processors' solutions, and uses these as boundary values for the next iteration. This works reasonably well for the small ...
5
Hashing floating-point numbers can indeed lead to weird results, especially if the node positions can be perturbed by some small amount or if there are denormalized values.
You included the Python tag, so I assume that's what you're using and that you have scipy. A quick and dirty solution would be to construct a kd-tree of the fine grid points, then for ...
5
As mentioned in the answer by @DanielShapero, you can follow an approach based on local approximations of the curvature for your nodes. In the post he suggest, there is an article by Desbrun. I would suggest to check another article by him: Anisotropic Polygonal Remeshing [1].
Another option that comes to my mind is to use Catmull-Clark subdivision ...
5
As @Nicoguaro and @Paul have said in the comments to the question post, there are a great many ways to do this kind of thing, and I'm not sure if there is a single "best" approach.
From a review study of Jonathan Richard Shewchuck at Berkley, an answer is:
Please refer to the original document (version 31/12/2002) for symbology, terminology, special ...
5
A hexahedron with straight edges is the image of the unit cube under a trilinear mapping. So, if you have values on the eight vertices of a hexahedron, and you are asking to interpolate between them to a point $x$ somewhere inside the hexahedron, then the correct approach is as follows:
Invert the trilinear mapping to find the corresponding point $\hat x = \...
5
The convergence rate that is mentioned here is in the sense that the error in iteration $k$ and $k-1$ are related by
$$
\| x^{(k)} - x^\ast\| \le r \| x^{(k-1)} - x^\ast\|,
$$
which implies that
$$
\| x^{(k)} - x^\ast\| \le r^k \| x^{(0)} - x^\ast\|.
$$
For this to converge at all, we need that $r<1$, which the statement you quote provides. But for ...
5
In addition to the voxel-based approach that rchilton suggests, you could also look at Delaunay-type algorithms. For example, the Computational Geometry Algorithms Library (CGAL) has some built-in functionality for surface mesh generation with examples here. You could also try distmesh, the essential idea of which has been ported to a number of other ...
4
I've used examples from:
INRIA: huge number of meshes in many file formats
Large Geometric Models archive: handful of very big meshes in .ply format
Stellar: a program for improving tetrahedral meshes; has a few examples on its website
Purely a matter of preference, but I like Tetgen for 3D tetrahedral mesh generation more than gmsh. While gmsh has more ...
4
If you have access to MATLAB, you might consider using PDE Toolbox to generate your geometry and mesh:
http://www.mathworks.com/help/pde/index.html
It is very easy to generate simple geometries like the ones you describe by doing
boolean operations on primitive shapes. The output mesh is described by three MATLAB arrays: node locations, element ...
4
If you want the simplest possible numerical scheme working for Burger's equation that has your suggested form then you should prefer the so called Lax-Friedrichs method.
If you have the book of LeVeque on Finite Volume Methods for Hyperbolic Problems, look for a very simple formula 4.20 (or a little bit more complex 4.21, but in your suggested form). With ...
4
Meshing algorithms can place the vertices of triangles and tetrahedra in a wide variety of ways, but they are us usually essentially constructive (i.e new vertices are introduced, existing vertices stay where they are). Disappointingly, this can cause the meshes which are generated to be unsuitable for numerical calculation. Mesh smoothing, at least in the ...
4
For just mesh smoothing, you can start by looking at Laplacian smoothing and some of the references therein. The idea is to update the position of every vertex in the interior of the mesh by replacing it with the average of its neighbors. There are loads of more sophisticated ways of doing this by using different operators.
If you're doing both surface mesh ...
4
I cannot visualize your geometry properly using Gmsh, or export it. I generated something similar using FreeCAD. Maybe you can modify this script for your purposes.
from __future__ import division, print_function
import FreeCAD as FC
import Draft
from numpy import sin, cos, pi
nturns = 1
nslices = 20
length = 10
width = 20
height = 60
dz = height/nslices
...
4
I do not think that there exists an answer to this question in general, because it all depends on the intended use for the mesh. For instance, if you are doing computational fluid dynamics, you may want to have a mesh that is extremely anisotropic near the boundary layer. Now if you are doing computational electromagnetics, the best mesh will be probably ...
4
The determinant of the Jacobian, as a determinant changes its sign when odd permutations of columns (or rows) are applied.
Imagine, for simplicity a two dimensional case in which the reference element is a triangle with vertices $V_1$, $V_2$ and $V_3$ (chosen counter-clockwise), and basis functions $1-\lambda-\mu$, $\lambda$ and $\mu$ respectively. The pair ...
4
For the discontinuous Galerkin method, the choice of control points is entirely unimportant because they conceptually lie in the interior of the cell (even if they are physically on the boundary of the cell). We think of a control point that is conceptually located at a vertex to be one that all cells adjacent to that vertex share; and of one on an edge that ...
4
To create a coarser mesh, you can set the characteristic length globally to a larger value, e.g.,
SetFactory("OpenCASCADE");
Mesh.CharacteristicLengthFactor = 2;
Circle(1) = {0, 0, 0, 1, 0, 2*Pi};
Line Loop(1) = {1};
Surface(1) = {1};
Increasing the value of Mesh.CharacteristicLengthFactor results in a coarser mesh; decreasing the value results in a finer ...
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You can use a stable sorting algorithm. Take the array $[1,\ldots,3n]$, and sort it according to a rule like
$$ \text{sorts-before}(i, j) = \text{in-front-of}\big(\text{triangle}(\lfloor\tfrac{i}{3}\rfloor), \text{triangle}(\lfloor\tfrac{j}{3}\rfloor)\big). $$
(Here $\text{sorts-before}$ is the comparator function $i\prec j$ you pass to the sort function.)
...
4
What you're looking for is an a-posteriori error estimate for your mesh study. Normally, these quantitative measures are line/area/volume- and/or time- averaged nodal or element quantities (e.g. avg. temperature of some body integrated over time). In structural analysis it is also customary to use the energy norm as a quantitative measure.
A line/area or ...
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