Mortar2D.jl

Mortar2D.jl is a Julia package to calculate discrete projections between non-conforming finite element meshes. The resulting "mortar matrices" can be used to tie non-conforming finite element meshes together which are meshed separately to construct bigger models.

Using mortar methods in mesh tie problems results variationally consistent solution. Mathematically, goal is to solve mixed problem with primary field variable and Lagrange multipliers, which have a physical meaning (e.g. contact pressure if unknown field is displacement). The problem arising is a typical saddle point problem with zeros on diagonal.

Installing and testing package

Installing package goes same way like other packages in julia, i.e.

julia> Pkg.add("Mortar2D")

Testing package can be done using Pkg.test, i.e.

julia> Pkg.test("Mortar2D")

Probably the easiest way to test the functionality of package is to use JuliaBox.

Usage example

Let us calculate projection matrices D and M for the following problem:

Problem setup:

Xs = Dict(1 => [0.0, 1.0], 2 => [5/4, 1.0], 3 => [2.0, 1.0])
Xm = Dict(4 => [0.0, 1.0], 5 => [1.0, 1.0], 6 => [2.0, 1.0])
coords = merge(Xm , Xs)
Es = Dict(1 => [1, 2], 2 => [2, 3])
Em = Dict(3 => [4, 5], 4 => [5, 6])
elements = merge(Es, Em)
element_types = Dict(1 => :Seg2, 2 => :Seg2, 3 => :Seg2, 4 => :Seg2)
slave_element_ids = [1, 2]
master_element_ids = [3, 4]

Calculate projection matrices D and M

s, m, D, M = calculate_mortar_assembly(
    elements, element_types, coords,
    slave_element_ids, master_element_ids)

According to theory, the interface should transfer constant without any error. Let's test that:

u_m = ones(3)
u_s = D[s,s] \ (M[s,m]*um)

# output

3-element Array{Float64,1}:
 1.0
 1.0
 1.0

The rest of the story can be read from the documentation. There's also brief review to the theory behind non-conforming finite element meshes.