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Technische Universiteit Delft (TU Delft)
Finite Elements (AP3001FE)
Summary
Summary [MSc AP] Formula Sheet Finite Elements
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Course
Finite Elements (AP3001FE)
Institution
Technische Universiteit Delft (TU Delft)
Book
Numerical methods in scientific computing
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This formula sheet covers the essentials of the 1st year's Applied Physics course "AP3001-FE - Finite Elements". This is kept to one page only, so it can be brought to the final exam (this is allowed at the time of writing this).
Finite Element Analysis Formula Sheet by Ruben Tol
Minimization Problems Construct the approximate functional, Galerkin Method
compute the stationary point condition
u = arg min J(v) Derive the Weak Form of the PDE
v∈Σ Use the form of J(u) with the small-
Reduce the amount of derivatives
est order derivatives (original equation
Derive Euler-Lagrange Equations present as much as possible using IBP.
or Euler-Lagrange equation), and sum
Then, the weak form is given by
Identify Solution Space and Boundaries over all j’s for which ui is unknown: Z Z
Σg := {u ∈ C n (Ω) : u|δΩ = g} ∂J ˜ G(u)η dΩ = f η dΩ ∀η ∈ V0 ,
˜
J(u) := J(ũ), = 0, j = 0, . . . , m−1 Ω Ω
g(x) a function, g a variable, or just 0. ∂uj
V0 := {η ∈ C s (Ω) : η|δΩ = 0}.
Identify Test Function Useful identities:
Galerkin Equations
n
η ∈ Σ0 := {η ∈ C (Ω) : η|δΩ = 0} ∂ ∂
∇ũ = ∇φj , ũ = φj . Perform the Ritz method on the weak
Notation implies what holds for u holds ∂uj ∂uj form of the PDE to obtain the Galerkin
for η, now with η|δΩ = 0. equations.
Move any summation signs outside the
Perform Variational Analysis integral signs, move any boundary con-
ditions to the right-hand side, and inter- Non-Linear Problems
d
J(u + ϵη) = 0, ∀η ∈ Σ0 change η = φj if needed: {xk }∞
dϵ ϵ=0 k=0 , lim xk = x
k→∞
n−1
For multivariable functions: X Z Z
Brouwer Fixed Point Theorem
d ∂F ∂a ∂F ∂b ui G(φi , φj ) dΩ = f φj dΩ,
F (a(ϵ), b(ϵ)) = + i=0 Ω Ω Given g(x) ∈ C(I), ∀x ∈ Ω, with I
dϵ ∂a ∂ϵ ∂b ∂ϵ a compact and convex domain (so I is
j = 0, . . . , m − 1.
Obtain Euler-Lagrange Equations closed and bounded), then x has at least
one fixed point in Ω.
Z
Express in matrix form Lu = f, with:
G(u)η dΩ = 0, ∀η ∈ Σ0
Ω Z
L = G(φi , φj ) dΩ, Banach Fixed Point Theorem
Use integration by parts (IBP): ij
Z Z I Ω If a contraction mapping g(x) : I → I
u·∇v dΩ = − ∇u·v dΩ+ uv·n dΓ. u = (u0 , . . . , un−1 )T , for γ ∈ [0, 1] such that
Ω Ω δΩ=Γ
Z
Dubois-Reymond theorem then tells us: fj = f φj dΩ. d(g(x), g(y)) ≤ γ d(x, y) , ∀x, y ∈ I
Ω
→ |g(x) − g(y)| ≤ γ|y − x|, ∀x, y ∈ I
Z I
f (x)η dΩ + h(x)η dΓ = 0,
Ω Γ PDE to Minimization Problem
exists, then the fixed point x is the only
→ f (x) = 0 on Ω ∨ h(x) = 0 on Γ, Check Differential Operator fixed point in I.
with boundary conditions (BC’s) First, identify the solution space and
u|δΩ = g(x). boundaries. Then, define the differ- Picard Method
ential operator L and prove its lin-
If Brouwer’s and Banach’s fixed point
Ritz Method earity, self-adjointness and positive-
theorems hold, then the Picard iteraton
definiteness for homogeneous boundary
Define a discrete space Σ̃ for approxi- conditions u, v ∈ Σ0 : xk+1 = g(xk ), k = 0, 1, . . .
mate solution ũ
L(αu + βv) = αLu + βL + v,
V := C s → Ṽn := span{φi }n−1 i=0 ⊂ V, Z Z converges to the fixed point x for all ini-
u(Lv) dΩ = (Lu)v dΩ, tial guesses x0 ∈ I.
with smallest s so C is smooth enough,
Ω Ω
depending on the order of derivatives in Z Z
Newton Method
the Euler-Lagrange equations. uLu dΩ ≥ γ u2 dΩ, γ > 0.
Ω Ω
Write any element of Ṽ n as linear com- If Brouwer’s and Banach’s fixed point
2 df
binations of the basis: To prove self-adjointness, use IBP; to theorems hold, f (x) ∈ C (I), dx > 0
n−1
X prove positive-definiteness, use homoge- (Jf invertible), then f (x) = 0 is approx-
ũ = ui φi ∈ Ṽ n . neous boundary conditions. imated for x by any initial guess x0 ∈ I
i=0
Then, for any homogeneous PDE (Σ0 ) by using Newton-Rapshon’s method:
Prescribe essential boundary conditions of the form Lu = f , u is given by solv-
on ũ according to any boundaries B: xk+1 = xk − (Jf (xk ))−1 f(xk ),
ing the minimization problem for
u|δΩ = g(x), Ṽ = Ṽ0 ⊕ B, Z
1 where f(xk ) = (f (x1 ), . . . , f (xn ))T , and
m−1 n−1 J(u) = uLu − f u dΩ,
Ω 2
X X ∂f1 ∂f1
→ ũ = ui φi + g(xi )φi ∈ Ṽ n . ∂x1 . . . ∂x n
i=0 i=m and for any non-homogeneous PDE (Σg ) Jf = ... .. .. .
. .
Essential BC’s (u|δΩ = g(x)), explicitly Z ∂f n ∂f
. . . ∂xnn
need their own series to be solved; nat- 1 ∂x1
J(u) = (u − v)L(u + v) − f u dΩ.
ural BC’s (∇u · n|δΩ = h(x)) are ”natu- Ω 2
rally” satisfied and grouped with u ∈ Ω.
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