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A gallery of nice pictures generated by deal.II
Table of contents:
This page is essentially a collection of images put here for purely esthetic reasons, and a little bit in order to show what can be done with deal.II. Most of the pictures also contain a brief summary of what they show, in order to give an idea of the kind of problem they are meant to solve.
Feel free to add your own pictures here, but please add the name of the person who did the simulation -- others may be so excited by it that they would want to contact you!
Flow through the blades of a turbine
High-Reynolds number computations lead to instationary solutions that are, however, often enormously important in practice. This picture is by Manuel Quezada de Luna (Texas A&M University) and was taken from a simulation of flow between the blades of a compressor or turbine.
These pictures are from a semester project by Yuhan Zhou at Texas A&M University. The goal was to compute parasitic conducivities and impedances from multilayer chip layouts. Other than generating data, this project also produced a number of pretty pictures!
Thermally driven convection
The images on the left show results obtained with a variant of the step-22 [tutorial program that computes thermally driven convection with Stokes flow. The images show three non-equidistantly space heat sources at the bottom and the flow field that emanates at various times from the rising plumes of hot air.
The flow pattern becomes unstable at various times switching from one configuration to another. The full dynamics only really become visible when watching these images as a movie. Several movies of various configurations are posted on Wolfgang Bangerth's webpage http://www.math.tamu.edu/~bangerth/pictures.html#convection.
(Origin: Wolfgang Bangerth, December 2007)
These are a couple of pictures created for an application in biomedical imaging. The exact meaning of the various quantities shown are explained in more detail at this linke [In short, the iso-contours on the left show the intensity of incident and fluorescent light, whereas the volume image on the right depicts the reconstructed tumor location.
(Origin: Wolfgang Bangerth, December 2007)
Singular and Hypersingular Source Terms
These images show a Laplacian with singular and hypersingular right hand sides, supported on a co-dimension one circle.
(Origin: Luca Heltai, 16 May 2007)
These images result from a constrained topology optimization problem related to the minimization of steady state thermal compliance on a 2D square plate. The optimization is performed on a hierarchically adapted grid: After completing optimization on a coarse grid level, the grid is adapted based on topology indicator derivatives and the optimization loop is restarted with interpolated topology indicator field on the (adaptively) refined grid as initial guess. The figure shows the global optimized topology and a selected zoomed part.
(Origin: Rohallah Tavakoli, 9 Feb. 2007)
These images show the saturation at one time step of a simulation of how a mixture of two fluids moves through a random medium in 2d (left) and 3d (right). The full description as well as movies of this problem can be found as step-21 in the tutorial of deal.II.
(Origin: Wolfgang Bangerth, 2006)
Phase change/crystal growth
Dendritic growth of crystals. (Origin: Denis Danilov)
Phase field modeling of the normal grain growth.
-Slawa, 07:49, 6 September 2010 (UTC)
Here are two images of elastic waves propagating outward from their origin. One can think of this simulation as a two-dimensional solid where we tuck at the center, pull it to the right, then let it snap. What you get is a fast pressure (P-)wave travelling to the left and right, and aslower shear (S-)wave travelling up and downward. After a while, these waves hit the boundary of the domain and are reflected from there. A closer look at the data would reveal that the reflections bring conversions from P- to S-waves with them, but that would require taking a look at the curl and divergence of the vector fields separately. (Origin: Wolfgang Bangerth, 1999)
This image shows waves in an inhomogenous medium after circling around the ring-shaped domain several times. Such waves occur in earthquake simulations, where shear waves only propagate in the the (solid) earth crust, but not in the (liquid) earth mantle and core below the crust. For strong earthquakes, these waves can travel around the entire globe several times before the are too attenuated to be detected. The resolution of this simulation is not high enough to be realistic, and neither are the geometry, dimensions, or physical constants realistic, but it gives an idea. (Origin: Wolfgang Bangerth, 1999)
Visualization of the analytic solution to the Navier-Stokes equations by L. I. G. Kovasznay (Laminar flow behind a two-dimensional grid, Proc. Camb. Philos. Soc. 44, pp. 58-62, 1948). Displayed is the stationary concentration of a tracer entering from the left. Therefore, the lines of same color are stream lines. The nice thing about this solution is its existence for any Reynolds number (even if the stationary solution shown here may be unstable and therefore unphysical for higher Reynolds numbers).
Here, Couette flow coming in from the far end of the brick flows around a cube standing on the ground. Imagine a very slow wind flow around a house. Reynolds number is 10 on the left and 100 on the right. You clearly see the wake behind the cube extending. Does anybody know if there is a loop in the wake? I could not find it putting in streamlines. Guido Kanschat 19:01, 17 May 2005 (CEST)
The pressure of an incompressible flow around a backward facing step is visualized. The Reynolds number is 50. The question is, if the positive singularity (the left one, on the inflow side) is a real physical phenomenon. Indeed, it is numerically stable. (Andre Große-Wöhrmann, 2005)
Convection in the earth core
Convection in itself is a hard problem (see the entry further up this page on thermally driven convection]](http://www.math.tamu.edu/~bangerth/pictures.html#optical].), but if it is between moving surfaces or with nonvertical gravity, it is even more challenging. This is what makes simulations of the earth's interior, both of the liquid rock earth mantle as well of the liquid metal outer core, so complicated. The picture to the right shows streamlines of a simulation of a convecting fluid in a radial gravity field between a hotter inner sphere and a cooler outer sphere, both of which are rotating and dragging the fluid along. Under such conditions, very complicated flow patterns develop after a while - among which are those that are responsible for the earth's magnetic field. (Origin: Andre Grosse-Woehrmann, 2005)
The step-25 tutorial program demonstrates the solution of the nonlinear, wave equation-type sine-Gordon equation, from which the pictures at the right are taken. Full movies of these solutions can be found in the results section of that program as well.
(Origin: Ivan Christov, 2006)
Modelling transport via the wavelike approach, numerical waves occur. (Origin: Andre Große-Wöhrmann, 2005)
The rotating cone problem with some diffusion on a spacetime mesh (Origin: Guido Kanschat, 2007)
Modeling nonlinear transport equations is slightly more complicated since they can develop shocks even if the initial conditions are smooth. This isn't the case for the initial conditions that produced this solution of the KPP equations, but the difficulty of capturing the solution's features is still apparent. (Origin: Orhan Mamedov, Vladimir Tomov, Abner Salgado, as part of a student project, 2011)
Fictitious Domain Method
This is the boundary supported fictitious domain method applied for the problem of Stokes.`You can see the pressure and the velocity field. The velocity is enforced on the circle via a weak condition. The key feature of this method is that this curve may be independent of the discretization mesh. (Origin: Andre Große-Wöhrmann, 2005)
This is the boundary supported fictitious domain method applied to the potential equation. Again the curve may be independent of the mesh. Nevertheless the mesh is refined in the vicinity of the curve. (Origin: Andre Große-Wöhrmann, 2005)
Immersed Boundary Method
This is a messy representation of the hydrostatic pressure of a fluid-structure interaction system composed of an elastic shell immersed in an incompressible fluid. The simulation was done using the immersed boundary method and a "net-like" elasticity which results in the interior pressure of the shell being a mess. --Luca Heltai 19:02, 17 January 2006 (CET)
Plastic and quasistatic deformation
Deformation can be described in many ways. step-42 is a tutorial program (written by Joerg Frohne, 2013) that deals with plasticity and the picture above shows the displacement by pressing a printing letter in the shape of the Chinese character for "force" into a metal block. (The displacement at every node is exaggerated by a factor of 100.)
On the other hand, the images below correspond to the step-18 tutorial program. It shows the gradual deformation of a cylinder under pressure from above, until it essentially fails. The color coding indicates stress levels in the material. Because it looks nice, at the very bottom, there is also an isosurface view of the x-displacement early on in the simulation. More details about this simulation can be found in the step-18 tutorial. (Origin: Wolfgang Bangerth, 2005)
Here is an image of a mesh of a rather complicated domain. It shows a mesh created from CT data of the left lung (Origin: Li Pan, 2006)
Euler flow; simulating a dense blob sliding downhill. Using DG and a Lax-Friedrichs type flux; Full Newton solve of the nonlinear system, using Trilinos/Aztec solvers. (Origin: David Neckels 2007)
Here are some images of propagated cracks in anisotropic materials. (Origin: Martin Steigemann 2012)
Quasistatic crack propagation in brittle materials can be simulated using the energy principle, where a crack grows in the direction of maximal energy release. Here, the direction of the crack is computed by an asymptotic formula for the change of energy in two dimensions using (classical) stress intensity factors (linear elasticity). The crack is elongated step-by-step and the domain is remeshed in each step.
The pictures show crack paths in an orthotropic material with two axes of elastic symmetry in a compact-tension specimen. Both pictures show the crack under the same Mode-II shear loading, but with different orientation of the initial crack to the planes of elastic symmetry.
The third picture shows a crack propagating in an isotropic base material with an anisotropic local inhomogeneity. This numerical experiment demonstrates how a local perturbation in the material can influence a crack path.
Originally, I only wanted to test a discretization of a sphere, so I created one, attached a boundary object, put it into a Laplace solver where the boundary values are given by |r|, and solved. Out came this picture, which was not quite what I had expected. What had happened is that I created the sphere with a radius 0.8, but the boundary object assumed a radius 1, so all points that were introduced by refining faces on the surface went to a different radius - only the original ones remained at the smaller radius and can now be seen as dimples. That makes for a botched simulation, but a nice picture anyway. (Origin: Wolfgang Bangerth, 2000)
This is slightly more what I had mind for the sphere (or here, a spherical shell, thresholded by one of the variables in my simulation), some 11 years later. This is essentially output from step-32. (Origin: Wolfgang Bangerth, 2011)
Not an accident, but not very realistic either: this is from a program that computes large deformations of objects. At the time when this picture was made, it could only handle some sort of elastic deformation, although it is nonlinear since the mesh is moved in each time step according to the incremental displacement computed. However, it has no notion of parts of the body coming into contact with each other, and the mesh is also way too coarse to resemble anything useful. Nevertheless, it shows something that looks physical, namely a pipe under vertical compression, where the bottom surface is completely clamped whereas the top surface can move horizontally, but is subjected to a prescribed vertical compression. The result is buckling. Pictures of more realistic buckling are shown further up on this page. (Origin: Wolfgang Bangerth, 2004)
Similar direction, different problem: With a second generation of the program that had already computed above cylinder (this time using a displacement-based formulation like in the step-18 example program, but also using pressure stabilization), I wanted to compute the deformation of a cube under extension. However, the tolerance of the linear solver was not tight enough, rendering the solution useless. The reason why the picture has the mottled look is that we deform the mesh based on the computed (wrong) solution, which makes some of the interior cells protrude through the surface we are looking on. In that case, GMV apparently doesn't quite know any more which cells to plot (the cells do intersect and overlap, after all), and decides on a somewhat random, if appealing, pattern. The quantity shown is the norm of the average stress inside each cell. (Origin: Wolfgang Bangerth, 2005)
When changing the numbering to something regular, some of the structures in DataOutRotation were forgotten. Instead of iso-surfaces symmetric to the center, we got this nice picture looking a bit like a complicated turbine. Guido Kanschat 08:10, 17 January 2006 (CET)
Poisson equation with homogeneous Dirichlet boundary conditions solved using Rvachev R-functions method combined with finite elements. Mesh is nonconforming, boundary conditions are enforced exactly. Pacman domain is considered as a benchmark because it contains curved parts as well as reentrant corner :)
For those who are interested to learn a bit about theory of R-functions, here is a link to my old 2-pages conference paper. More detailed and extensive info in English is available at Spatial Automation Lab.
-Slawa, 07:21, 6 September 2010 (UTC)
This had to be a simulation of directional solidification/dendritic growth. But something was wrong with the model and it produced such a nice tree in the end :)
-Slawa, 08:08, 6 September 2010 (UTC)