Nonlinear Finite Elements

Dec 22, 2017

This post builds on the formulations I showed in my previous posts by introducing their nonlinear versions.

In a typical nonlinear problem, the variational setting leads to the weak formulation

Find $u\in V$ such that

for all $v\in V$ where the semilinear form $F$ is nonlinear in terms of $u$ and linear in terms of $v$.

We linearize $F$:

Equating \eqref{eq:femnonlinear4} to zero yields a linear system in terms of $\Var u$

where

We can compute the components of the matrices and vectors according to \eqref{eq:femnonlinear6}

Then the update vector $\Var \Bu = [\Var u^1, \Var u^2, \dots, \Var u^\nnode]\tra$ is obtained by solving

Letting $\Var u$ be the difference between consequent iterates, we obtain the update equation as

Example: Nonlinear Poisson’s Equation

Consider the following nonlinear Poisson’s equation

Find $u\in V$ such that

for all $v\in V$ where $V=H^1_0(\Omega)$.

Applying integration by parts and divergence theorem on the left-hand side

Thus we have the semilinear form

The linearized version of this problem is then with \eqref{eq:femnonlinear5}

and the matrix and vector components are with \eqref{eq:femnonlinear9}

where the previous solution and its gradient are computed as

Nonlinear Time-Dependent Problems

In the case of a nonlinear time-dependent problem, we have the following weak form:

Find $u \in V$ such that

for all $v \in V$ and $t \in [0,\infty)$ where $F$ is a semilinear form.

Discretization yields the following nonlinear system of equations

where

Explicit Euler Scheme

We discretize in time with the finite difference $\dot{u} \approx [u_{n+1}-u_n]/{\Delta t}$ and linearity allows us to write

$$\boxed{ m(\dot{u}, v; t) \approx \frac{1}{\Delta t} [m(u_{n+1}, v; t_{n+1}) - m(u_n, v; t_n)] } \label{eq:discretetimedependent1}$$

We discretize the variational forms in time according to \eqref{eq:discretetimedependent1}, and evaluate the remaining terms at $t_n$:

The corresponding system of equations is

where $\Bf_n = \Bf(u_n, t_n)$. This yields the following update equation

For a time-independent $m$, this becomes

Implicit Euler Scheme

For the implicit scheme, we evaluate the remaining terms at $t_{n+1}$ and let the result be equal to

We will hereon replace $u_{n+1}$ with $u$ for brevity. The update of this nonlinear system requires the linearization of $G(u, v)$:

We thus have the following linear setting for the Newton update $\Var u$:

where

Discretization yields

where

The Newton update is rendered

which is repeated until the solution for the next timestep $\Bu$ converges to a satisfactory value.

Nonlinear Coupled Problems

For a nonlinear coupled problem, the weak formulation is as follows

Find $u\in V_1$, $y\in V_2$ such that

for all $v\in V_1$, $w \in V_2$ where $F(\cdot,\cdot, \cdot)$, $G(\cdot, \cdot, \cdot)$ are nonlinear in terms of $u$ and $y$ and linear in terms of $v$ and $w$.

We linearize the semilinear forms about the nonlinear terms:

where the evaluations take place at $u=\bar{u}$ and $y=\bar{y}$.

Equating the linearized residuals to zero, we obtain a linear system of the form

with the bilinear forms $a$, $b$, $d$, $e$ and the linear forms $c$, $f$ which are defined as

Discretizing as done in the previous section, we obtain the following linear system of equations

whose solution yields the update values $\Var \Bu$ and $\Var \By$. Thus the Newton update equations are

Example: Cahn-Hilliard Equation

The Cahn-Hilliard equation describes the process of phase separation, by which the two components of a binary fluid spontaneously separate and form domains pure in each component. The problem is nonlinear, coupled and time-dependent. The IBVP reads

where

and $t\in I = [0,\infty)$. Here,

• $c$ is the scalar variable for concentration,
• $\mu$ is the scalar variable for the chemical potential,
• $f: c \mapsto f(c)$ is the function representing chemical free energy,
• $\BM$ is a second-order tensor describing the mobility of the chemical,
• $\BLambda$ is a second-order tensor describing both the interface thickness and direction of phase transition.

The fourth-order PDE governing the problem can be formulated as a coupled system of two second-order PDEs with the variables $c$ and $\mu$, as demonstrated in \eqref{eq:cahnhilliard1} and \eqref{eq:cahnhilliard2}.

Find $c \in V_1$, $\mu\in V_2$ such that

for all $v \in V_1$, $w \in V_2$ and $t \in I$.

We discretize in time implicitly with $\del c/\del t \approx (c_{n+1}-c_n)/\Var t$. We also denote the values for the next timestep $c_{n+1}$ and $\mu_{n+1}$ as $c$ and $\mu$ for brevity. Using integration-by-parts, the divergence theorem, and the given boundary conditions, we arrive at the following nonlinear forms

which is a nonlinear coupled system of the form \eqref{eq:nonlinearcoupled1}.

We linearize the forms according to \eqref{eq:nonlinearcoupled2} and obtain the following variations

We substitute basis functions and obtain our system matrix and vectors

which constitute the system

Solution yields the update values $\Var \Bc$ and $\Var \Bmu$. The Newton update equations are then

The system is solved for $c_{n+1}$ and $\mu_{n+1}$ at each $t=t_n$ to obtain the evolutions of the concentration and chemical potential.