Elasticity (Ch8.2)
Navier Stokes eq. (Ch8.1)
−νΔu+ρ(u⋅∇)u−∇p=f
divu=0
Maxwell eq (Ch8.3)
in more dimensions
\( -\mathrm{div\,}D\epsilon(u)=f \)
structural mechanics
setup in 1D
\( -(Eu')'=f \)
proving LBB for Stokes eq. (Supp)
non-linear
variables
types
finite elements
\( H(\mathrm{curl})= \{v \in [L_2(\Omega)]^3:\mathrm{curl\,}v\in [L_2(\Omega)]^3\}\)
\( \Vert v \Vert_{H(\mathrm{curl})}= \sqrt{ \Vert v \Vert_{L_2}^2+ \Vert \mathrm{curl\,}v \Vert_{L_2}^2}\)
Finite elements in \( H(\mathrm{curl}) \)
Navier Stokes: Modeling & first numerical methods (Supp)
Trace theorem (Th106)
\( \exists \) tangential trace operator \( \mathrm{tr}_{\tau} v : H(\mathrm{curl}) \rightarrow W(\partial\Omega),\,\mathrm{tr}_{\tau} v=(v\vert_{\partial\Omega})_{\tau}\)
for \( v\in [C(\overline{\Omega})]^3 \)
setup in \( \mathbb{R} \)
gradient operator \( \nabla:H^1 \rightarrow H(\mathrm{curl}) \)
de'Rham complex
complete sequences of spaces
kernel space \( H^0(\mathrm{curl})=\{ v\in H(\mathrm{curl}): \mathrm{curl\,}v=0 \} \)
weak form through Coloumb-Gauging
Oseen iteration/eq.
mixed system is well posed on \( H(\mathrm{curl})\times H^1/\mathbb{R} \)
in general: no unique solution
\( u \in H(\mathrm{curl},\Omega=\cup\Omega_i) \) (Th107)
eq. of elasticity in weak form
\( u \in V= [H^1_{0,D}(\Omega)]^d\)
\( \int_{\Omega} D\epsilon(u):\epsilon(v)\mathrm{\, dx}=\int_{\Omega}f\cdot v\mathrm{\, dx},\, \forall v \in V \)
rigid body motions (=translation or linearized rotation)
discretization is straight forward
weak form well posed in \( [H^1]^d \) (Th103)
problem: sim results on thin structures worse
semi discretizationgoal: derive system of 1D eq. for 2D deformation w/ semi-discretization
Discrete LBB (Ch2)
trafo of 1D setup
flow of a fluid
BC
approx. by Galerkin
stationary = steady state (no change in \(t\))
spaces for convergence
stability of cont. eq (Ch1)
Example sim: Timoshenko beam
proving discrete-LBB w/ Fortin operator
force density \( f \)
strain \( \epsilon = u'\)
2nd order eq. for displacement \( u \)
incompressible fluids
Hooks Law \( \sigma = E \epsilon \)
simplest elements not work (LBB not fulfilled)
velocity \( u=(u_x, u_y, u_z) \)
current density \( j \)
pressure \( p\)
model forces \( f\)
density \( \rho \)
tetrahedral
error of same order as approx. err
\( W_h\overset{\nabla}{\longrightarrow}V_h\overset{\textrm{curl}}{\longrightarrow}Q_h\overset{\textrm{div}}{\longrightarrow}S_h \) (Th110)
\( H^{1}\overset{\nabla}{\longrightarrow}H(\textrm{curl})\overset{\textrm{curl}}{\longrightarrow}H(\textrm{div})\overset{\textrm{div}}{\longrightarrow}L^{2} \)
stress \( \sigma' = -f\)
Henrys Law + Stokes
electromagnetic fields
permeability \( \mu \)
cheaper Nedelec / edge-element
var. problem: mixed form
magnetic field intensity \( H \)
BC
Lagrangian vs. Eulerian (Ch1)
system of eq.s
w/ Brezzi and conditions \( \Rightarrow \) unique solution
The N-S.eq (Ch2)
viscosity \( \nu \)
magnetic flux \( B\)
finite elements in 3D
basis function \( \varphi_{E_i}\)
vertical displacement \( w(x) \)
in principle well posed (Le104)
1st FEM discretization & time stepping (Ch3)
Elements w/ discont. pressure (Ch2.1)
de'Rham complex diagram commutes (Th111)
\( \mathrm{curl}=\mathrm{rot}=\nabla\times\)
compare to diffusion eq.
lin. rotation of normal vector \( \beta \)
small thickness
\( \Rightarrow \) large cont./coerc. ratio
\( \Rightarrow \) error increase
piece-wise constant for \( p \)
Elements w/ cont. pressure (Ch2.2)
Dirichlet (displacement at boundary)
stokes eq. = N-S-eq - viscos. & density terms
inflow \( \Gamma_i \) and outflow boundaries \( \Gamma_o \)
solvability by Brezzi
variables
Langrangian
Reynolds transport Theorem (Th1)
average longitudinal displacing \( U(x) \)
unknown vars
new var. form
Neumann (stress at boundary )
non-conf. \( P_1 \) for \( u \)
\( \Rightarrow \) 2 decoupled problems
\( V_h = P^{1,nc}, \, Q_h = P^{0,dc} \)
solution = pairing
\( V_h = P^{2+}, \, Q_h = P^{1,nc} \)
basis: conservation eq.s (e.g. mass)
mixed method
Eulerian
conserved quants.
implicit + explicit Euler method (mixture used)
extended Brezzi (Th105)
\( V_h = P^{2}, \, Q_h = P^{0,dc} \)
Fortin operator (preserving mean values)
goal: rewrite terms to solve per time step
idea: weaken term w/ large coefficient
\( O(h) \) convergent
space-discretized system
Proof1: ignore Dirichlet and use extension operator
Proof2: on star shaped \( \Omega \)
clement operator
\( \Omega \in \mathbb{R} \)
position \( x \in \Omega \)
mini element \( V_h = P^{1+}, \, Q_h = P^{1,cont} \)
Alternative: Taylor-Hood element
\( \frac{d}{dt}\int_{V(t)} c(x,t)\mathrm{\,dx}= \int_{V(t)} \frac{\partial c}{\partial t}(x,t)\mathrm{\,dx}+ \int_{\partial V(t)} n\cdot uc\mathrm{\,ds} \)
Interpretation
solids
particle \( X \in \Omega\)
velocity of particle \( u(x,t) \)
LBB-cond. non-trivial
correction operator
quantity preserved
Fortin operator
evolution of particles \( \varphi : \Omega\times [0,T] \rightarrow \Omega \)
material derivative
\( \frac{Dc}{Dt} := \frac{\partial c}{\partial t} + \mathrm{div\,}(uc) \)
2nd order space w/ cubic bubbles
\( P^{2+}(\mathcal{T})=\{ v_h \in H^1:v_{h\vert T}\in P^3(T),v_{h\vert E} \in P^2(E) \} \)
vector valued velocity field \( u \)
error-estimate
2nd order convergence
\( x \) variables
scalar-valued pressure field \( p \)
momentum
fluids
Fortin + Clement operator
coordinates \( X \)
1st order convergence
error-estimates
requires add. assumptions
cost low for add. bubbles
stable pair \( P^2 \times P^{1,cont} \)
convergence rate \( O(h) \) optimal
mass
correction operator
conv. rate \( O(h^2) \)
this element sub-optimal
longitdunial displacement
bending & shearing