Discretization and numerical integration of projection matrices

[//]: # (This may be the most platform independent comment, and also 80 chars.)

In theoretical studies, we have a continuum setting where we evaluate the mandatory line integral. When going towards a numerical implementation, this integral must be calculated in discretized setting, which is arising some practical questions. For example, displacement is defined using local coordinate system for each element. Thus, we need to find all master-slave pairs, contact segments, which are giving contribution to the contact virtual work. Discretization of the contact interface follows standard isoparametric approach. First we define finite dimensional subspaces $\boldsymbol{\mathcal{U}}_{h}^{\left(i\right)}$ and $\boldsymbol{\mathcal{V}}_{h}^{\left(i\right)}$, which are approximations of $\boldsymbol{\mathcal{U}}^{\left(i\right)}$ and $\boldsymbol{\mathcal{V}}^{\left(i\right)}$. Geometry and displacement interpolation then goes as a following way: \begin{align} \boldsymbol{x}{h}^{\left(1\right)} & =\sum{k=1}^{n^{\left(1\right)}}N{k}^{\left(1\right)}\boldsymbol{x}{k}^{\left(1\right)}, & \boldsymbol{x}{h}^{\left(2\right)} & =\sum{l=1}^{n^{\left(2\right)}}N{l}^{\left(2\right)}\boldsymbol{x}{l}^{\left(2\right)}, \\ \boldsymbol{u}{h}^{\left(1\right)} & =\sum{k=1}^{n^{\left(1\right)}}N{k}^{\left(1\right)}\boldsymbol{d}{k}^{\left(1\right)}, & \boldsymbol{u}{h}^{\left(2\right)} & =\sum{l=1}^{n^{\left(2\right)}}N{l}^{\left(2\right)}\boldsymbol{d}{l}^{\left(2\right)}. \end{align}

Moreover, we need also to interpolate Lagrange multipliers in the interface: \begin{equation} \boldsymbol{\lambda}{h}=\sum{j=1}^{m^{\left(1\right)}}\Phi{j}\boldsymbol{\lambda}{j}. \end{equation}

function f(x)
    return x
end

Substituting interpolation polynomials to contact virtual work $\delta\mathcal{W}_{\mathrm{co}}$ yields \begin{align} -\delta\mathcal{W}{\mathrm{co}} & =\int{\gamma{\mathrm{c}}^{\left(1\right)}}\boldsymbol{\lambda}\cdot\left(\delta\boldsymbol{u}^{\left(1\right)}-\delta\boldsymbol{u}^{\left(2\right)}\circ\chi\right)\,\mathrm{d}a\nonumber \ & \approx\int{\gamma{\mathrm{c},h}^{\left(1\right)}}\boldsymbol{\lambda}{h}\cdot\left(\delta\boldsymbol{u}{h}^{\left(1\right)}-\delta\boldsymbol{u}{h}^{\left(2\right)}\circ\chi{h}\right)\,\mathrm{d}a\nonumber \ & =\int{\gamma{\mathrm{c},h}^{\left(1\right)}}\left(\sum{j=1}^{m^{\left(1\right)}}\Phi{j}\boldsymbol{\lambda}{j}\right)\cdot\left(\sum{k=1}^{n^{\left(1\right)}}N{k}^{\left(1\right)}\delta\boldsymbol{d}{k}^{\left(1\right)}-\sum{l=1}^{n^{\left(2\right)}}\left(N{l}^{\left(2\right)}\circ\chi{h}\right)\delta\boldsymbol{d}{l}^{\left(2\right)}\right)\,\mathrm{d}a\nonumber \ & =\sum{j=1}^{m^{\left(1\right)}}\sum{k=1}^{n^{\left(1\right)}}\boldsymbol{\lambda}{j}\cdot\int{\gamma{\mathrm{c},h}^{\left(1\right)}}\Phi{j}N{k}^{\left(1\right)}\delta\boldsymbol{d}{k}^{\left(1\right)}\,\mathrm{d}a-\sum{j=1}^{m^{\left(1\right)}}\sum{l=1}^{n^{\left(2\right)}}\boldsymbol{\lambda}{j}\cdot\int{\gamma{\mathrm{c},h}^{\left(1\right)}}\Phi{j}\left(N{l}^{\left(2\right)}\circ\chi{h}\right)\delta\boldsymbol{d}{l}^{\left(2\right)}\,\mathrm{d}a\nonumber \ & =\sum{j=1}^{m^{\left(1\right)}}\sum{k=1}^{n^{\left(1\right)}}\boldsymbol{\lambda}{j}^{\mathrm{T}}\left(\int{\gamma{\mathrm{c},h}^{\left(1\right)}}\Phi{j}N{k}^{\left(1\right)}\,\mathrm{d}a\right)\delta\boldsymbol{d}{k}^{\left(1\right)}-\sum{j=1}^{m^{\left(1\right)}}\sum{l=1}^{n^{\left(2\right)}}\boldsymbol{\lambda}{j}^{\mathrm{T}}\left(\int{\gamma{\mathrm{c},h}^{\left(1\right)}}\Phi{j}\left(N{l}^{\left(2\right)}\circ\chi{h}\right)\,\mathrm{d}a\right)\delta\boldsymbol{d}{l}^{\left(2\right)}.\label{2dprojection:eq:discretizedcontactvirtual_work} \end{align}

Here, $\chi_{h}:\gamma_{\mathrm{c},h}^{\left(1\right)}\mapsto\gamma_{\mathrm{c},h}^{\left(2\right)}$ is a discrete mapping from the slave surface to the master surface. From the equation \ref{2dprojection:eq:discretizedcontactvirtualwork}, we find the so called mortar matrices, commonly denoted as $\boldsymbol{D}$ and $\boldsymbol{M}$: \begin{align} \boldsymbol{D}\left[j,k\right] & =D{jk}\boldsymbol{I}{\mathrm{ndim}}=\int{\gamma{\mathrm{c},h}^{\left(1\right)}}\Phi{j}N{k}^{\left(1\right)}\,\mathrm{d}a\boldsymbol{I}{\mathrm{ndim}}, & j=1,\ldots,m^{\left(1\right)}, & k=1,\ldots,n^{\left(1\right)},\ \boldsymbol{M}\left[j,l\right] & =M{jl}\boldsymbol{I}{\mathrm{ndim}}=\int{\gamma{\mathrm{c},h}^{\left(1\right)}}\Phi{j}\left(N{l}^{\left(2\right)}\circ\chi{h}\right)\,\mathrm{d}a\boldsymbol{I}_{\mathrm{ndim}}, & j=1,\ldots,m^{\left(1\right)}, & l=1,\ldots,n^{\left(2\right)}. \end{align}

The process of discretizing mesh tie virtual work $\delta\mathcal{W}_{\mathrm{mt}}$, is identical to what is now presented, with a difference that integrals are evaluated in reference configuration instead of current configuration. Also, when considering linearized kinematic and small deformations, integration can be done in reference configuration. For mesh tie contact, the discretized contact virtual work is \begin{align} -\delta\mathcal{W}{\mathrm{mt},h}= & \sum{j=1}^{m^{\left(1\right)}}\sum{k=1}^{n^{\left(1\right)}}\boldsymbol{\lambda}{j}^{\mathrm{T}}\left(\int{\Gamma{\mathrm{c},h}^{\left(1\right)}}\Phi{j}N{k}^{\left(1\right)}\,\mathrm{d}A{0}\right)\delta\boldsymbol{d}{k}^{\left(1\right)}\nonumber \ & -\sum{j=1}^{m^{\left(1\right)}}\sum{l=1}^{n^{\left(2\right)}}\boldsymbol{\lambda}{j}^{\mathrm{T}}\left(\int{\Gamma{\mathrm{c},h}^{\left(1\right)}}\Phi{j}\left(N{l}^{\left(2\right)}\circ\chi{h}\right)\,\mathrm{d}A{0}\right)\delta\boldsymbol{d}{l}^{\left(2\right)}.\label{eq:2dprojection:discretizedvirtual_work} \end{align}