Vectorial Finite Elements

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}{\boldsymbol{\delta} } \newcommand{\Bepsilon}{\boldsymbol{\epsilon} } \newcommand{\Bvareps }{\boldsymbol{\varepsilon} } \newcommand{\Bvarepsilon}{\boldsymbol{\varepsilon}} \newcommand{\Bzeta }{\boldsymbol{\zeta} } \newcommand{\Beta }{\boldsymbol{\eta} } \newcommand{\Btheta }{\boldsymbol{\theta} } \newcommand{\Bvarthe }{\boldsymbol{\vartheta} } \newcommand{\Biota }{\boldsymbol{\iota} } \newcommand{\Bkappa }{\boldsymbol{\kappa} } \newcommand{\Blambda }{\boldsymbol{\lambda} } \newcommand{\Bmu }{\boldsymbol{\mu} } \newcommand{\Bnu }{\boldsymbol{\nu} } \newcommand{\Bxi }{\boldsymbol{\xi} } \newcommand{\Bpi }{\boldsymbol{\pi} } \newcommand{\Brho }{\boldsymbol{\rho} } \newcommand{\Bvrho }{\boldsymbol{\varrho} } \newcommand{\Bsigma }{\boldsymbol{\sigma} } \newcommand{\Bvsigma }{\boldsymbol{\varsigma} } \newcommand{\Btau }{\boldsymbol{\tau} } \newcommand{\Bupsilon}{\boldsymbol{\upsilon} } \newcommand{\Bphi }{\boldsymbol{\phi} } \newcommand{\Bvarphi }{\boldsymbol{\varphi} } \newcommand{\Bchi 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% $

Many initial boundary value problems require solving for unknown vector fields, such as solving for displacements in a mechanical problem. Discretization of weak forms of such problems leads to higher-order linear systems which need to be reshaped to be solved by regular linear solvers. There are also more indices involved than a scalar problem, which can be confusing. In this post, I’ll try to elucidate the procedure by deriving for a basic higher-order system and giving an example.

The weak formulation of a linear vectorial problem reads

Find $\Bu\in V$ such that
\[\begin{equation} a(\Bu, \Bv) = b(\Bv) \end{equation}\]
for all $\Bv\in V$.

Discretizations of vectorial problems requires the expansion of vectorial quantities as linear combinations of the basis vectors $\Be_i$:

\[\begin{equation} \Bu = \suml{i=1}{\ndim} u_i \,\Be_i \label{eq:discrete6} \end{equation}\]

where $\cbr{u_i}_{i=1}^{\ndim}$ are the components corresponding to the basis vectors and $\ndim=\dim V$. Here, we chose Cartesian basis vectors for simplicity.

We can therefore express its discretization as

\[\begin{equation} \Bu_h = \suml{I=1}{\nnode} \Bu^I N^I = \suml{I=1}{\nnode}\suml{i=1}{\ndim} u^I_i \Be_i N^I. \label{eq:discrete7} \end{equation}\]

Substituting discretized functions in the weak formulation, we obtain

\[\begin{equation} \begin{aligned} a(\Bu_h, \Bv_h) &= \suml{i=1}{\ndim}\suml{j=1}{\ndim} \, a(u_{h,j}\,\Be_j, v_{h,i}\,\Be_i)\\ &= \suml{I=1}{\nnode}\suml{J=1}{\nnode} \suml{i=1}{\ndim}\suml{j=1}{\ndim} u^J_j v^I_i \,a(\Be_j N^J, \Be_i N^I) \end{aligned} \end{equation}\]

and

\[\begin{equation} b(\Bv_h) = \suml{i=1}{\ndim} b(v_{h,i}\,\Be_i) = \suml{I=1}{\nnode} \suml{i=1}{\ndim} v^I_i b(\Be_i N^I). \end{equation}\]

We define the following arrays

\[\begin{equation} \boxed{ \begin{aligned} A^{I\!J}_{ij} &= a(\Be_j N^J, \Be_i N^I) \\ b^{I}_{i} &= b(\Be_i N^I). \end{aligned} } \end{equation}\]

Hence we can express the linear system

\[\begin{equation} a(\Bu_h,\Bv_h) = b(\Bv_h) \end{equation}\]

\[\begin{equation} \suml{I=1}{\nnode}\suml{J=1}{\nnode} \suml{i=1}{\ndim}\suml{j=1}{\ndim} u^J_j v^I_i \,A^{I\!J}_{ij} = \suml{I=1}{\nnode} \suml{i=1}{\ndim} v^I_i b^{I}_{i}. \end{equation}\]

For arbitrary $\Bv_h$, this yields the following system of equations

\[\begin{equation} \boxed{ \suml{J=1}{\nnode} \suml{j=1}{\ndim} A^{I\!J}_{ij} \,u^J_j = b^{I}_{i} } \label{eq:discrete8} \end{equation}\]

for $i=1,\dots,\ndim$ and $I=1,\dots,\nnode$.

We reshape this higher-order system as shown in the previous post Reshaping Higher Order Linear Systems:

\[\begin{equation} \BAhat \Buhat = \Bbhat \end{equation}\]

by defining a map $i_d$ that maps original indices to the reshaped indices

\[\begin{equation} i_d := \left\{ \begin{array}{rl} [1,\nnode]\times[1,\ndim] & \to [1,\nnode\ndim]\\[1ex] (I,i) & \mapsto \ndim(I-1)+i\\ \end{array} \right. \end{equation}\]

where we used 1-based indexing of the arrays. We set

\[\begin{equation} \boxed{ \begin{alignedat}{3} \alpha &:= i_d(I,i) &&= \ndim(I-1) + i \\ \beta &:= i_d(J,j) &&= \ndim(J-1) + j \\ \end{alignedat} } \end{equation}\]

and write

\[\begin{equation} \hat{A}_{\alpha\beta} = A^{I\!J}_{ij} \quad,\quad \hat{u}_{\beta} = u^{J}_{j} \eqand \hat{b}_{\alpha} = b^{I}_{i} \end{equation}\]

The inverse index mapping can be obtained as shown in the previous post.

Example: Linear Elasticity

Our initial-boundary value problem is

\[\begin{equation} \begin{alignedat}{4} -\div\Bsigma &= \rho\Bgamma \qquad&& \text{in} \qquad&& \Omega\\ \Bu &= \bar{\Bu} && \text{on} && \del\Omega_u \\ \Bt &= \bar{\Bt} && \text{on} && \del\Omega_t \\ \end{alignedat} \end{equation}\]

The weak formulation reads

Find $\Bu\in V$ such that
\[\begin{equation} -\int_\Omega \div\Bsigma \dtp \Bv \dv = \int_\Omega \rho \Bgamma\dtp\Bv \dv \end{equation}\]
for all $\Bv\in V$ where $V=H^1(\Omega)$.

We apply integration by parts on the left-hand side

\[\begin{equation} \int_\Omega \div\Bsigma\dtp\Bv \dv = \int_\Omega \div(\Bsigma\Bv) \dv - \int_\Omega \Bsigma : \nabla\Bv \dv \end{equation}\]

and apply the divergence theorem to the first resulting term:

\[\begin{equation} \int_\Omega \div(\Bsigma\Bv) \dv = \int_{\del\Omega_t} \bar{\Bt}\dtp\Bv \da \end{equation}\]

Substituting the linear stress $\Bsigma=\IC:\Bvareps=\IC:\nabla\Bu$, we obtain the following variational forms:

\[\begin{align} \label{eq:linelastdiscretebilinear} a(\Bu,\Bv) &= \int_\Omega \nabla\Bv:\IC:\nabla\Bu \dv \\ \label{eq:linelastdiscretelinear} b(\Bv) &= \int_\Omega \rho \Bgamma\dtp\Bv \dv + \int_{\del\Omega_t} \bar{\Bt}\dtp\Bv \da \end{align}\]

We have the following discretizations of the unknown function and test function

\[\begin{equation} \Bu_h = \suml{J=1}{\nnode} \Bu^J N^J \eqand \Bv_h = \suml{I=1}{\nnode} \Bv^I N^I. \end{equation}\]

With the given discretizations, the matrix corresponding to \eqref{eq:linelastdiscretebilinear} can be calculated as

\[\begin{equation} \begin{aligned} A^{I\!J}_{ij} = a(\Be_j N^J, \Be_i N^I) &= \int_\Omega \nabla(\Be_iN^I) : \IC : \nabla(\Be_jN^J) \dv \\ &= \int_\Omega (\Be_i\dyd \nabla N^I) : \IC : (\Be_j \dyd \nabla N^J) \dv \\ &= \int_\Omega \partd{N^I}{x_k} \, C_{ikjl} \, \partd{N^J}{x_l} \dv, \end{aligned} \end{equation}\]

and finally obtain

\[\begin{equation} \boxed{ A^{I\!J}_{ij} = \int_\Omega B^I_k \, C_{ikjl} \, B^J_l \dv \,. } \end{equation}\]

The vector corresponding to \eqref{eq:linelastdiscretelinear} is calculated as

\[\begin{equation} b^{I}_{i} = b(\Be_i N^I) = \int_{\del\Omega_t} \bar{\Bt}\dtp(\Be_iN^I) \da + \int_\Omega\rho\Bgamma\dtp(\Be_iN^I) \dv \end{equation}\]

which yields

\[\begin{equation} \boxed{ b^{I}_{i} = \int_{\del\Omega_t} \bar{t}_i N^I \da + \int_\Omega \rho \gamma_i N^I \dv } \end{equation}\]