Gridap.jl: Grid-based Approximation of PDEs in Julia

In this tutorial, we'll learn about Gridap.jl: Grid-based approximation of partial differential equations (PDEs) in Julia. A new feature of Gridap.jl focusing on the solution of transient Partial Differential Equations (PDEs)

What

Gridap provides a set of tools for the grid-based approximation of partial differential equations (PDEs) written in the Julia programming language. The library currently supports linear and nonlinear PDE systems for scalar and vector fields, single and multi-field problems, conforming and nonconforming finite element (FE) discretizations, on structured and unstructured meshes of simplices and n-cubes. It also provides methods for time integration. Gridap is extensible and modular. One can implement new FE spaces, new reference elements, use external mesh generators, linear solvers, post-processing tools, etc. See, e.g., the list of available Gridap plugins.

Gridap has a very expressive API allowing one to solve complex PDEs with very few lines of code. The user can write the underlying weak form with a syntax almost 1:1 to the mathematical notation, and Gridap generates an efficient FE assembly loop automatically by leveraging the Julia JIT compiler. For instance, the weak form for an interior penalty DG method for the Poisson equation can be simply specified as:

a(u,v) =
  ∫( ∇(v)⋅∇(u) )*dΩ +
  ∫( (γ/h)*v*u - v*(n_Γ⋅∇(u)) - (n_Γ⋅∇(v))*u )*dΓ +
  ∫(
    (γ/h)*jump(v*n_Λ)⋅jump(u*n_Λ) -
    jump(v*n_Λ)⋅mean(∇(u)) -
    mean(∇(v))⋅jump(u*n_Λ)
    )*dΛ

l(v) =
  ∫( v*f )*dΩ +
  ∫( (γ/h)*v*u - (n_Γ⋅∇(v))*u )*dΓ

See the complete code here. As an example for multi-field PDEs, this is how the weak form for the Stokes equation with Neumann boundary conditions can be specified:

a((u,p),(v,q)) =
  ∫( ∇(v)⊙∇(u) - (∇⋅v)*p + q*(∇⋅u) )*dΩ

l((v,q)) =
  ∫( v⋅f + q*g )*dΩ +
  ∫( v⋅(n_Γ⋅∇u) - (n_Γ⋅v)*p )*dΓ

See the complete code here.

Documentation

• STABLE — Documentation for the most recently tagged version of Gridap.jl.
• DEVEL — Documentation for the in-development version of Gridap.

Tutorials

A hands-on user-guide to the library is available as a set of tutorials. They are available as Jupyter notebooks and html pages.

Installation

Gridap is a registered package in the official Julia package registry. Thus, the installation of Gridap is straight forward using the Julia's package manager. Open the Julia REPL, type ] to enter package mode, and install as follows

pkg> add Gridap

Plugins

• GridapDistributed Distributed-memory extension of Gridap.
• GridapEmbedded Embedded finite elements in Julia.
• GridapGmsh Generate a FE mesh with GMSH and use it in Gridap.
• GridapMakie Makie plotting recipies for Gridap.
• GridapPardiso Use the Intel Pardiso MKL direct sparse solver in Gridap.
• GridapPETSc Use PETSc linear and nonlinear solvers in Gridap.

Examples

These are some popular PDEs solved with the Gridap library. Examples taken from the Gridap Tutorials.

Poisson equation	Linear elasticity	Hyper-elasticity	p-Laplacian
Poisson eq. with DG	Darcy eq. with RT	Incompressible Navier-Stokes	Isotropic damage

Known issues

Since Julia 1.6 ownwards we have noticed large first call latencies of Gridap.jl codes with the default compiler optimization level (i.e., -O2). In general, while developing code, but specially if you are noting high first call latencies, we recommend to run julia with the -O1 flag. For production runs use -O2 or -O3.

Gridap community

You can ask questions and interact with the Gridap community on the Julia Slack channel #gridap (see here how to join). or our gitter.

Contributing to Gridap

Gridap is a collaborative project open to contributions. If you want to contribute, please take into account:

• Before opening a PR with a significant contribution, contact the project administrators, e.g., by writing a message in our gitter chat or by opening an issue describing what you are willing to implement. Wait for feed-back.
• Carefully read and follow the instructions in the CONTRIBUTING.md file.
• Carefully read and follow the instructions in the CODE_OF_CONDUCT.md file.
• Open a PR with your contribution.

Want to help? We have a number of issues waiting for help. You can start contributing to the Gridap project by solving some of those issues.

How to cite Gridap

In order to give credit to the Gridap contributors, we simply ask you to cite the references below in any publication in which you have made use of the Gridap project. If you are using other Gridap sub-packages, please cite them as indicated in their repositories.

@article{Badia2020,
  doi = {10.21105/joss.02520},
  url = {https://doi.org/10.21105/joss.02520},
  year = {2020},
  publisher = {The Open Journal},
  volume = {5},
  number = {52},
  pages = {2520},
  author = {Santiago Badia and Francesc Verdugo},
  title = {Gridap: An extensible Finite Element toolbox in Julia},
  journal = {Journal of Open Source Software}
}

@article{Verdugo2022,
  doi = {10.1016/j.cpc.2022.108341},
  url = {https://doi.org/10.1016/j.cpc.2022.108341},
  year = {2022},
  month = jul,
  publisher = {Elsevier {BV}},
  volume = {276},
  pages = {108341},
  author = {Francesc Verdugo and Santiago Badia},
  title = {The software design of Gridap: A Finite Element package based on the Julia {JIT} compiler},
  journal = {Computer Physics Communications}
}

Download Details: 
Author: gridap
Source Code: https://github.com/gridap/Gridap.jl 
License: MIT
#julia

What is GEEK

Buddha Community

Gridap.jl: Grid-based Approximation of PDEs in Julia

In this tutorial, we'll learn about Gridap.jl: Grid-based approximation of partial differential equations (PDEs) in Julia. A new feature of Gridap.jl focusing on the solution of transient Partial Differential Equations (PDEs)

What

Gridap provides a set of tools for the grid-based approximation of partial differential equations (PDEs) written in the Julia programming language. The library currently supports linear and nonlinear PDE systems for scalar and vector fields, single and multi-field problems, conforming and nonconforming finite element (FE) discretizations, on structured and unstructured meshes of simplices and n-cubes. It also provides methods for time integration. Gridap is extensible and modular. One can implement new FE spaces, new reference elements, use external mesh generators, linear solvers, post-processing tools, etc. See, e.g., the list of available Gridap plugins.

Gridap has a very expressive API allowing one to solve complex PDEs with very few lines of code. The user can write the underlying weak form with a syntax almost 1:1 to the mathematical notation, and Gridap generates an efficient FE assembly loop automatically by leveraging the Julia JIT compiler. For instance, the weak form for an interior penalty DG method for the Poisson equation can be simply specified as:

a(u,v) =
  ∫( ∇(v)⋅∇(u) )*dΩ +
  ∫( (γ/h)*v*u - v*(n_Γ⋅∇(u)) - (n_Γ⋅∇(v))*u )*dΓ +
  ∫(
    (γ/h)*jump(v*n_Λ)⋅jump(u*n_Λ) -
    jump(v*n_Λ)⋅mean(∇(u)) -
    mean(∇(v))⋅jump(u*n_Λ)
    )*dΛ

l(v) =
  ∫( v*f )*dΩ +
  ∫( (γ/h)*v*u - (n_Γ⋅∇(v))*u )*dΓ

See the complete code here. As an example for multi-field PDEs, this is how the weak form for the Stokes equation with Neumann boundary conditions can be specified:

a((u,p),(v,q)) =
  ∫( ∇(v)⊙∇(u) - (∇⋅v)*p + q*(∇⋅u) )*dΩ

l((v,q)) =
  ∫( v⋅f + q*g )*dΩ +
  ∫( v⋅(n_Γ⋅∇u) - (n_Γ⋅v)*p )*dΓ

See the complete code here.

Documentation

• STABLE — Documentation for the most recently tagged version of Gridap.jl.
• DEVEL — Documentation for the in-development version of Gridap.

Tutorials

A hands-on user-guide to the library is available as a set of tutorials. They are available as Jupyter notebooks and html pages.

Installation

Gridap is a registered package in the official Julia package registry. Thus, the installation of Gridap is straight forward using the Julia's package manager. Open the Julia REPL, type ] to enter package mode, and install as follows

pkg> add Gridap

Plugins

• GridapDistributed Distributed-memory extension of Gridap.
• GridapEmbedded Embedded finite elements in Julia.
• GridapGmsh Generate a FE mesh with GMSH and use it in Gridap.
• GridapMakie Makie plotting recipies for Gridap.
• GridapPardiso Use the Intel Pardiso MKL direct sparse solver in Gridap.
• GridapPETSc Use PETSc linear and nonlinear solvers in Gridap.

Examples

These are some popular PDEs solved with the Gridap library. Examples taken from the Gridap Tutorials.

Poisson equation	Linear elasticity	Hyper-elasticity	p-Laplacian
Poisson eq. with DG	Darcy eq. with RT	Incompressible Navier-Stokes	Isotropic damage

Known issues

Since Julia 1.6 ownwards we have noticed large first call latencies of Gridap.jl codes with the default compiler optimization level (i.e., -O2). In general, while developing code, but specially if you are noting high first call latencies, we recommend to run julia with the -O1 flag. For production runs use -O2 or -O3.

Gridap community

You can ask questions and interact with the Gridap community on the Julia Slack channel #gridap (see here how to join). or our gitter.

Contributing to Gridap

Gridap is a collaborative project open to contributions. If you want to contribute, please take into account:

• Before opening a PR with a significant contribution, contact the project administrators, e.g., by writing a message in our gitter chat or by opening an issue describing what you are willing to implement. Wait for feed-back.
• Carefully read and follow the instructions in the CONTRIBUTING.md file.
• Carefully read and follow the instructions in the CODE_OF_CONDUCT.md file.
• Open a PR with your contribution.

Want to help? We have a number of issues waiting for help. You can start contributing to the Gridap project by solving some of those issues.

How to cite Gridap

In order to give credit to the Gridap contributors, we simply ask you to cite the references below in any publication in which you have made use of the Gridap project. If you are using other Gridap sub-packages, please cite them as indicated in their repositories.

@article{Badia2020,
  doi = {10.21105/joss.02520},
  url = {https://doi.org/10.21105/joss.02520},
  year = {2020},
  publisher = {The Open Journal},
  volume = {5},
  number = {52},
  pages = {2520},
  author = {Santiago Badia and Francesc Verdugo},
  title = {Gridap: An extensible Finite Element toolbox in Julia},
  journal = {Journal of Open Source Software}
}

@article{Verdugo2022,
  doi = {10.1016/j.cpc.2022.108341},
  url = {https://doi.org/10.1016/j.cpc.2022.108341},
  year = {2022},
  month = jul,
  publisher = {Elsevier {BV}},
  volume = {276},
  pages = {108341},
  author = {Francesc Verdugo and Santiago Badia},
  title = {The software design of Gridap: A Finite Element package based on the Julia {JIT} compiler},
  journal = {Computer Physics Communications}
}

Download Details: 
Author: gridap
Source Code: https://github.com/gridap/Gridap.jl 
License: MIT
#julia

Grid-based Approximation of Partial Differential Equations in Julia

What

Gridap provides a set of tools for the grid-based approximation of partial differential equations (PDEs) written in the Julia programming language. The library currently supports linear and nonlinear PDE systems for scalar and vector fields, single and multi-field problems, conforming and nonconforming finite element (FE) discretizations, on structured and unstructured meshes of simplices and n-cubes. Gridap is extensible and modular. One can implement new FE spaces, new reference elements, use external mesh generators, linear solvers, post-processing tools, etc. See, e.g., the list of available Gridap plugins.

Gridap has a very expressive API allowing one to solve complex PDEs with very few lines of code. The user can write the underlying weak form with a syntax almost 1:1 to the mathematical notation, and Gridap generates an efficient FE assembly loop automatically by leveraging the Julia JIT compiler. For instance, the weak form for an interior penalty DG method for the Poisson equation can be simply specified as:

a(u,v) =
  ∫( ∇(v)⋅∇(u) )*dΩ +
  ∫( (γ/h)*v*u - v*(n_Γ⋅∇(u)) - (n_Γ⋅∇(v))*u )*dΓ +
  ∫(
    (γ/h)*jump(v*n_Λ)⋅jump(u*n_Λ) -
    jump(v*n_Λ)⋅mean(∇(u)) -
    mean(∇(v))⋅jump(u*n_Λ)
    )*dΛ

l(v) =
  ∫( v*f )*dΩ +
  ∫( (γ/h)*v*u - (n_Γ⋅∇(v))*u )*dΓ

See the complete code here. As an example for multi-field PDEs, this is how the weak form for the Stokes equation with Neumann boundary conditions can be specified:

a((u,p),(v,q)) =
  ∫( ∇(v)⊙∇(u) - (∇⋅v)*p + q*(∇⋅u) )*dΩ

l((v,q)) =
  ∫( v⋅f + q*g )*dΩ +
  ∫( v⋅(n_Γ⋅∇u) - (n_Γ⋅v)*p )*dΓ

See the complete code here.

Documentation

• STABLE — Documentation for the most recently tagged version of Gridap.jl.
• DEVEL — Documentation for the in-development version of Gridap.

Tutorials

A hands-on user-guide to the library is available as a set of tutorials. They are available as Jupyter notebooks and html pages.

Installation

Gridap is a registered package in the official Julia package registry. Thus, the installation of Gridap is straight forward using the Julia's package manager. Open the Julia REPL, type ] to enter package mode, and install as follows

pkg> add Gridap

Plugins

• GridapGmsh Generate a FE mesh with GMSH and use it in Gridap.
• GridapMakie Makie plotting recipies for Gridap.
• GridapPardiso Use the Intel Pardiso MKL direct sparse solver in Gridap.
• GridapEmbedded Embedded finite elements in Julia.
• GridapODEs Gridap support for time-dependent PDEs.
• GridapDistributed Distributed-memory extension of Gridap.

Examples

These are some popular PDEs solved with the Gridap library. Examples taken from the Gridap Tutorials.

Poisson equation	Linear elasticity	Hyper-elasticity	p-Laplacian
Poisson eq. with DG	Darcy eq. with RT	Incompressible Navier-Stokes	Isotropic damage

Gridap community

You can ask questions and interact with the Gridap community on the Julia Slack channel #gridap (see here how to join). or our gitter.

Contributing to Gridap

Gridap is a collaborative project open to contributions. If you want to contribute, please take into account:

• Before opening a PR with a significant contribution, contact the project administrators, e.g., by writing a message in our gitter chat or by opening an issue describing what you are willing to implement. Wait for feed-back.
• Carefully read and follow the instructions in the CONTRIBUTING.md file.
• Carefully read and follow the instructions in the CODE_OF_CONDUCT.md file.
• Open a PR with your contribution.

Want to help? We have a number of issues waiting for help. You can start contributing to the Gridap project by solving some of those issues.

How to cite Gridap

In order to give credit to the Gridap contributors, we simply ask you to cite the reference below in any publication in which you have made use of Gridap packages:

@article{Badia2020,
  doi = {10.21105/joss.02520},
  url = {https://doi.org/10.21105/joss.02520},
  year = {2020},
  publisher = {The Open Journal},
  volume = {5},
  number = {52},
  pages = {2520},
  author = {Santiago Badia and Francesc Verdugo},
  title = {Gridap: An extensible Finite Element toolbox in Julia},
  journal = {Journal of Open Source Software}
}

Download Details:
Author: gridap
The Demo/Documentation: View The Demo/Documentation
Download Link: Download The Source Code
Official Website: https://github.com/gridap/Gridap.jl 
License: MIT
 

#julia #programming #developer

Perftests.jl: Base perftests for Julia

Perftests

A redo of the main Julia test/perf directory. Retooled to use Benchmarks.jl, this repository will spit out a bunch of .csv files into the test/results-$COMMIT directory (Where $COMMIT is the commit hash of the version of Julia you are running) and display small summary statistics as it does so.

To run a specific group of tests, use run_perf_groups(). For example, to run and output the .csv files for the kernel and simd performance tests, one would write:

using Perftests
run_perf_groups(["kernel", "simd"])

Test organization

Tests are organized foremost by group, then name, then variant. Not all tests are required to have variants, but they fit into a natural hierarchy when running the same test across multiple element types, for example. The resultant .csv files are named $group-$name-$variant.csv, for ease of access.

Test environment

Information about the test environment (the commit hash of this repository used to create the results, the word size of the running machine, the number of CPU cores, etc...) is saved in the env.csv file put into the results-$COMMIT directory when testing.

---

Download Details:

Author: Staticfloat
Source Code: https://github.com/staticfloat/Perftests.jl 
License: View license

#julia #base 

Dimers.jl: A Julia Package for Sampling The Dimer Model on A Grid

Dimers

Dimers is a package for simulating the dimer model on a 2D rectangular grid, using an algorithm of Kenyon, Propp, and Wilson. Dimers also provides support for loop erased random walks and Wilson's algorithm on an arbitrary graph.

showgraphics(drawgraph(dimersample(20)))

We can also compute the height function associated with the dimer sample:

dimerheight(dimersample(20))

11x11 Array{Int64,2}:
  0   1   0   1   0  1   0   1   0   1   0
 -1  -2  -1  -2  -1  2  -1  -2  -1  -2  -1
  0  -3  -4  -3   0  1   0   1   0  -3   0
 -1  -2  -1  -2  -1  2  -1  -2  -1  -2  -1
  0  -3   0   1   0  1   0   1   0   1   0
 -1  -2  -1   2  -1  2  -1  -2  -1   2  -1
  0   1   0   1   0  1   0   1   0   1   0
 -1  -2  -1   2  -1  2   3   2  -1  -2  -1
  0   1   0   1   0  1   0   1   0   1   0
 -1   2   3   2   3  2   3   2   3   2  -1
  0   1   0   1   0  1   0   1   0   1   0

Wilson takes a graph as its first argument and an array of true/false values specifying the roots.

showgraphics(drawgraph(Wilson(G,[[true];[false for i=1:length(G.vertices)-1]])))

LERW(G,v0,roots) samples a loop-erased random walk on the graph G starting from the vertex whose index in G.vertices is v0 stopped upon hitting one of the vertices v for which roots[v] is true.

import Graphs
n = 100
G = Graphs.adjlist((Int64,Int64),is_directed=false)

for i=1:n
    for j=1:n
        Graphs.add_vertex!(G,(i,j))
    end
end

roots = Bool[v[1] == 1 || v[1] == n || v[2] == 1 || v[2] == n  for v in
G.vertices];

v0 = find(x->x==(div(n,2),div(n,2)),G.vertices)[1]

lerw = LERW(gridgraph(n),v0,roots)
for i=1:length(lerw)-1
    add_edge!(G,lerw[i],lerw[i+1])
end

showgraphics(drawgraph(G))

Download Details:

Author: sswatson
Source Code: https://github.com/sswatson/Dimers.jl 
License: View license

#julia #grid 

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Vision

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