2D1260: Finite Element Methods. Lecturer: Johan ... 2. project presentation (Mon
12 Dec/Tue 13 Dec) ... Space-time FEM, stabilization, Convection-Diffusion. 9.
Finite Element Methods Lecture 1 Johan Hoffman
[email protected]
Johan Hoffman – KTH – p.1
2D1260: Finite Element Methods Lecturer: Johan Hoffman Assistant Professor (Forskarassistent) Research: Adaptive FEM for CFD, turbulence,...
[email protected] Teaching Assistant: Erik von Schwerin PhD student (Doktorand) Research: Adaptive FEM, Stochastic DE,...
[email protected] www.nada.kth.se/kurser/kth/2D1260
Office Hours: Mon 9-10 (JH), Tue 9-10 (ES)
Johan Hoffman – KTH – p.2
2D1260: Finite Element Methods 9 lectures + 6 excercise sessions Book: Computational Differential Equations Examination: 1. project report (Fri 9 Dec) 2. project presentation (Mon 12 Dec/Tue 13 Dec) 3. written exam (Fri 16 Dec) Project starts Week 3 Mon 14 Nov; 2 parts: 1. computer assignement; the same for everyone 2. extension of the first part chosen by the group
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Overview: Lectures Introduction, DE, Galerkin’s method
2.
BVP, FE basis functions
3.
Abstract problem, Lax-Milgram Theorem
4.
Interpolation, error estimation, adaptivity
5.
FE software; assembly, mapping, quadrature
6.
IVP, stability, -method, space-time FEM
7.
Adaptivity, error estimation, duality
8.
Space-time FEM, stabilization, Convection-Diffusion
9.
Navier-Stokes, adaptivity, functional output
1.
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in
(DE)
Differential Equation (DE)
: differential operator : solution : source term : domain where (DE) is valid:
Ordinary Differential Equation (ODE) : Partial Differential Equation (PDE) : u scalar : scalar DE u vector : system of DE
DE : Equation relating derivatives of a function
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BC: specifying equation on boundary
of the domain
on
in
(BC)
(DE)
Boundary conditions
for all
for all
Neumann boundary condition:
Dirichlet boundary condition:
Examples:
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is two
This equation smooths the solution: the solution times more differentiable than the source term
Ex: Poisson’s equation
Modeling: gravitation, ground water flow, electrostatics,...
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for
and
for
BC:
Dirichlet BC
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for
and
for
BC:
Dirichlet + Neumann BC
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Ex: Heat equation
This equation smooth the solution over time:
Modeling: heat conduction, pollution,...
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Ex: Heat equation
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Ex: Heat equation Osmosis; diffusion through cell membrane
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Ex: Schrödinger equation
is the wave function in the quantum mechanical model of the motion of an electron orbiting around one proton at the origin.
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Ex: Black-Scholes equation
This equation is close to the heat equation, with the asset playing the role of the spatial dimension price Modeling: option pricing,...
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Ex: Linear Elasticity
and the stress
Models the displacement elastic bodies
div
div
Cauchy-Navier’s elasticity equations:
for
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):
This equation conserves energy (for
Ex: Wave equation
Modeling: wave phenomena, accoustics,...
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Ex: Wave equation Seismic waves in simulation of California Earthquake
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is transported (convected) by the .
The solution convection field
Ex: Transport equation
Modeling: Pollution,...
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Ex: Convection-Diffusion-Reaction
This equation is a combination of transport (convection) by , diffusion with diffusivity , the convection field , of a spieces and reaction with reaction coefficient . Modeling: chemical reactions, pollution,...
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Ex: Convection-Diffusion-Reaction Chernobyl 1986; simulation by SMHI
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Ex: Maxwell equations
magnetic field,
current density
electric field,
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Ex: Maxwell equations Magnetic field around a coil.
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Ex: Stokes equations
velocity and
pressure
Modeling: low velocity flow phenomena
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Ex: Stokes equations Groundwater flow
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Ex: Navier-Stokes equations
velocity and
pressure
Modeling: flow phenomena, weather prediction, blood flow,...
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Ex: Navier-Stokes equations
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Vorticity
Ex: Navier-Stokes equations around wheel.
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Vorticity
Ex: Navier-Stokes equations around a full car.
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Vorticity
Ex: Navier-Stokes equations around a full car.
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Ex: Navier-Stokes equations Blood flow in artery.
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Function spaces to DE
and
in
)
and all functions
(for all real numbers
in a certain class
is a vector space (linear space):
A function space
look for approximate solutions of functions: FUNCTION SPACE
Often impossible to find exact solutions
Functional Analysis: linear algebra for function spaces
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Hölder inequality:
-norm:
max-norm:
orthogonal:
Cauchy-Schwarz inequality: and
(size of )
-norm:
Inner product (scalar product):
Linear Algebra for Function spaces
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)
so that the error is small:
Determine
in a function space
Seek approximate solution with “simple” basis functions:
Approximation methods
Examples of spaces : trigonometric functions (Fourier), polynomials (Newton, Lagrange), piecewise polynomials (Finite Element Methods)
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such that
the function space
is orthogonal to all
:
The residual of the exact solution functions:
The RESIDUAL: is orthogonal to all testfunctions GALERKIN ORTHOGONALITY
Galerkin method: Find
Galerkin method
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,
for
linear system of equations
Solve for
Galerkin method
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Galerkin method spectral method (trigonomatric b.f.), p-method (global polynomial b.f.), h-method (piecewise polynomial b.f.),...
FEM: piecewise polynomial functions on a mesh in 1d
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Galerkin method
FEM: computational mesh in 3d (www.geuz.org/gmsh)
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Finite Element Methods FEM: piecewise polynomials 1. basis functions are almost orthogonal (local support) 2. basis functions are simple to differentiate and integrate 3. applicable to general geometry ϕ 1
i−1
ϕ
i
ϕ
i+1
x x i−2
x i−1
xi
x i+1
x i+2
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Dirichlet Boundary conditions:
Example
boundary nodes h i=x i−xi−1 interior nodes I i=(x i−1,x i) x x2
x i−1
xi
xM+1 =1
x1
x 0=0
continuous piecewise linear functions on a mesh
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for
,
:
Integration by parts
nodal basis for
Example
Variational form:
weak form of DE
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for
, Vector
Linear system of equations for :
Galerkin (cG(1)): Find
Matrix
Example
:
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Goal of FEM course Construct Finite Element Methods for general DE (ODE and PDE). Analyze stability of DE, and the corresponding FEM solution. Analyze the error in the FEM approximation. Construct adaptive FEM methods. Learn the structure of FEM software.
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