Finite Element Methods - Sccswiki · PDF fileAlgorithms for Scientiﬁc Computing Finite Element Methods Michael Bader Technical University of Munich Summer 2016

Algorithms for Scientific Computing

Finite Element Methods

Michael BaderTechnical University of Munich

Summer 2016

Part I

Looking Back: Discrete Models forHeat Transfer and the Poisson

Equation

Modelling of Heat Transfer• objective: compute the temperature distribution of some object• under certain prerequisites:

• temperature T at object boundaries given• heat sources• material parameters k , . . .

• observation from physical experiments: q ≈ k · δT(heat flow proportional to temperature differences)

Michael Bader | Algorithms for Scientific Computing | Finite Element Methods | Summer 2016 2

A Finite Volume Model

• object: a rectangular metal plate (again)• model as a collection of small connected rectangular cells

hx

hy

• examine the heat flow across the cell edgesMichael Bader | Algorithms for Scientific Computing | Finite Element Methods | Summer 2016 3

Heat Flow Across the Cell Boundaries

• Heat flow across a given edge is proportional to• temperature difference (T1 − T0) between the adjacent cells• length h of the edge

• e.g.: heat flow across the left edge:

q(left)ij = kx

(Tij − Ti−1,j

)hy

kx depends on material• heat flow across all edges determines change of heat energy:

qij = kx(Tij − Ti−1,j

)hy + kx

(Tij − Ti+1,j

)hy

+ ky(Tij − Ti,j−1

)hx + ky

(Tij − Ti,j+1

)hx

• equilibrium with source term Fij = fijhxhy (fij heat flow per area) requiresqij + Fij = 0:

fijhxhy = −kxhy(2Tij − Ti−1,j − Ti+1,j

)−ky hx

(2Tij − Ti,j−1 − Ti,j+1

)Michael Bader | Algorithms for Scientific Computing | Finite Element Methods | Summer 2016 4

Discrete and Continuous Model

• system of equations derived from the discrete model:

fij = −kx

hx

(2Tij − Ti−1,j − Ti+1,j

)−

ky

hy

(2Tij − Ti,j−1 − Ti,j+1

)• result: average temperature in each cell• corresponds to partial differential equation (PDE):

−k(∂2T (x , y)

∂x2 +∂2T (x , y)

∂y2

)= f (x , y)

• wanted: approximate T (x , y) as a function!→ solution possible using “coefficients and basis functions”?


Part II

Outlook: Finite Element Methods

For Model Problem:• 2D Poisson equation:

∂2T (x , y)

∂x2 +∂2T (x , y)

∂y2 = f (x , y)

• first, however, we consider the 1D case:

u′′(x) = f (x) for x ∈ (0,1)

with u(0) = u(1) = 0.


Finite Elements – Main Idea

• we consider the residual of the (1D) PDE:

u′′(x) = f (x) u′′(x)− f (x) = 0

• represent the functions u and f in our “favorite” form:(∑ujφj (x)

)′′−∑

fjφj (x) = 0

• however: we will usually not find uj that solve this equation exactly (assolution u can not be represented as

∑ujφj (x))

• remedy?→ find “best approximation”, given by orthogonality:⟨

w(x),(∑

ujφj (x))′′−∑

fjφj (x)

⟩= 0 “for all w(x)”

• remember that < g, f >=∫

g · f dx


Finite Elements – Main Ingredients

1. compute a function as numerical solution;search in a function space Wh:

uh =∑

j

ujϕj (x), spanϕ1, . . . , ϕJ = Wh

2. solve weak form of PDE to reduce regularity properties

u′′ = f −→ −∫

v ′u′ dx =

∫vf dx

3. choose basis functions with local support, e.g.:

ϕj (xi ) = δij

(such as the hat functions)


Choose Test and Ansatz Space

• search for solution functions uh of the form

uh =∑

j

ujϕj (x)

• the basis (“shape”, “ansatz”) functions ϕj (x) build a vector space (orfunction space) Wh

spanϕ1, . . . , ϕJ = Wh

• the “best” solution uh in this function space is wanted


Example: Nodal Basis

ϕi (x) :=

1h (x − xi−1) xi−1 < x < xi1h (xi+1 − x) xi < x < xi+1

0 otherwise

0,60,4 0,80,20

1

1

0,4

0,2

x

0

0,8

0,6


Or Better A Hierarchical Basis?

.x1,1

.

.

x2,1 x2,3

x3,1 x3,3 x3,5 x3,7

Φ1,1

Φ2,1 Φ2,3

Φ3,1 Φ3,3 Φ3,5 Φ3,7


Weak Forms and Weak Solutions

• consider a PDE Lu = f (e.g. Lu = ∆u)• transformation to the weak form:

〈v ,Lu〉 =

∫vLu dx =

∫vf dx = 〈f , v〉 ∀v ∈ V

V a certain class of functions• “real solution” u also solves the weak form

(but additional, approximate solutions accepted . . . )• motivation for weak form:

– we cannot test Lu(x) = f (x) for all x ∈ (0,1) on a computer(infinitely many x)

– frequent choice V = Wh, so check whether Lu and f have the “samebehaviour” w.r.t. scalar product

– approximate solution u might not solve PDE: Lu 6= fthus: additional functions need to be “acceptable” as solution→ “orthogonality” idea


Weak Form of the Poisson Equation – 1D

• Poisson equation with Dirichlet conditions:

−u′′(x) = f (x) in Ω = (0,1), u(0) = u(1) = 0

• weak form:−∫

Ω

v(x)u′′(x) dx =

∫Ω

v(x)f (x) dx ∀v

• integration by parts:

−∫

Ω

v(x)u′′(x) dx = −v(x) · u′(x)

∣∣∣∣10

+

∫Ω

v ′(x) · u′(x) dx

• choose functions v such that v(0) = v(1) = 0:∫Ω

v ′(x) · u′(x) dx =

∫Ω

v(x)f (x) dx ∀v


Weak Form of the Poisson Equation – 2D/3D


−∆u = f in Ω, u = 0 on δΩ

• weak form:−∫

Ω

v∆u dΩ =

∫Ω

vf dΩ ∀v

• apply Green’s formula:

−∫

Ω

v∆u dΩ =

∫Ω

∇v · ∇u dΩ−∫∂Ω

v · ∇u ds

• choose functions v such that v = 0 on ∂Ω:∫Ω

∇v · ∇u dΩ =

∫Ω

vf dΩ ∀v


Weak Form of the Poisson Equation – Summary


−∆u = f in Ω,u = 0 on δΩ

• transformed into weak form:∫Ω

∇v · ∇u dΩ =

∫Ω

vf dΩ ∀v

• weaker requirements for a solution u:twice differentiabale→ first derivative integrable

• remember use of nodal basis: availability of first vs. second derivative!


Choose Test and Ansatz Space

• search for solutions uh in a function space Wh:

uh =∑

j

ujϕj (x)

where spanϕj = Wh (“ansatz space”)• insert into weak solution∫

vL(∑

j

ujϕj (x)

)dx =

∫vf dx ∀v ∈ V


Choose Test and Ansatz Space (2)

• choose a basis ψi of the test space V• then: if all basis functions ψi satisfy∫

ψiL(∑

j

ujϕj (x)

)dx =

∫ψi f dx ∀ψi

then all v ∈ V satisfy the equation• leads to system of equations for unknowns uj

(one equation per test basis function ψi )• V is often chosen to be identical to Wh (Ritz-Galerkin method)


Discretisation – Finite Elements

• L linear⇒ system of linear equations∑j

(∫ψiLϕj (x) dx︸︷︷︸

=:Aij

)uj =

∫ψi f dx ∀ψi

• aim: make system of equations easy to solve!→ thus, make matrix A sparse→ most Aij = 0

• approach: local basis functions on a discretisation grid• consider hat functions, e.g.: ψj , ϕj zero everywhere, except in grid cells

adjacent to grid point xj

• then Aij = 0, if ψi and ϕj don’t overlap


Example Problem: Poisson 1D

• in 1D: u′′(x) = f (x) on Ω = (0,1),hom. Dirichlet boundary cond.: u(0) = u(1) = 0

• weak form: ∫ 1

0v ′(x) · u′(x) dx =

∫ 1

0v(x)f (x) dx ∀v

• computational grid:xi = ih, (for i = 1, . . . ,n − 1); mesh size h = 1/n

• V = W : piecewise linear functions(on intervals [xi , xi+1])


Nodal Basis

ϕi (x) :=

1h (x − xi−1) xi−1 < x < xi1h (xi+1 − x) xi < x < xi+1

0 otherwise

0,60,4 0,80,20

1

1

0,4

0,2

x

0

0,8

0,6


Nodal Basis – System of Equations

• stiffness matrix:

1h

2 −1

−1 2. . .

. . . . . . −1−1 2

• right hand sides (assume f (x) = α ∈ R):∫ 1

0ϕi (x)f (x) dx =

∫ 1

0ϕi (x)α dx = αh

• system of equations very similar to finite differences


Hierarchical Basis

0,60,4 0,80,20

1

1

0,4

0,2

x

0

0,8

0,6

• leads to diagonal stiffness matrix!(for 1D Poisson)

• solution function identical to that with nodal basis (same function space)


Part III

Finite Element Methods – BasisFunctions for 2D

Hierarchical Basis in 2DQuadtrees and Hierarchical Bases

Quadtrees to Represent ObjectsHierarchical Basis vs. Quadtree


2D Hierarchical Basis – Tensor Product

• define 2D basis functions via tensor product:

φi,j (x , y) := φi (x) · φj (y)

• remember multi-index for 2D hierarchical basis:

φ~l,~k (x1, x2) := φl1,l2,k1,k2 (x1, x2) := φl1,k1 (x1) · φl2,k2 (x2)

• illustrate via support of the basis functions:


Illustrate via Location of Hat Functions

l1=1 l1=2 l1=3 l1

l2=1

l2=2

l2=3

l2


Adding Adaptivity: Quadtrees

Quadtrees to Represent Objects:

• start with an initial square (covering the entire domain)• recursive substructuring into four subsquares• adaptive refinement?


Quadtrees for Adaptive Simulations

Adaptively Refined Meshes for Finite Elements:

• refine, unless squares entirely within or outside domain• also: refine, if solution not exact enough!• question: can we build a hierarchical basis on such a quadtree?


Hierarchical Basis vs. Quadtree

Use hierarchical basis as in 2D sparse grids?

⇒ stretched tensor basis functions do not match quadtree cells⇒ use basis functions with “square” domain (cover 4 siblings→ to solve)


Hierarchical Basis for Quadtrees

Switch to hierarchical “multilevel” basis:

hierarchical concept (again): skip basis functions that exist on previous level!


Illustrate via Location of Hat Functions

l1=1 l1=2 l1=3 l1

l2=1

l2=2

l2=3

l2


Quadtree-Compatible Hierarchical BasisBasis Functions

Similar to tensor-product basis:• Level-wise hierarchical increments

W~l := spanφ~l,~i~i∈I~l

• Only use “diagonal” levels:

~l := l , . . . , l

• Omit grid points for which all indices areeven:

I~l := ~i : ~1 ≤~i < 2~n, any ij odd

l1=1 l1=2 l1=3 l1

l2=1

l2=2

l2=3

l2


Part IV

Finite Element Methods – TowardsImplementation

FEM and Hierarchical Basis TransformHierarchical Basis TransformationFEM and Hierarchical Basis TransformElement Stiffness MatricesWorkflow


Project: 2D Adaptive Hierarchical Basis

Consider:• 2D Poisson problem• FEM with quadtree-compatible hierarchical basis• adaptive quadtree-based hierarchical basis

Discuss (again):• how to compute the stiffness matrix?• what do you need to compute, if you add a hierarchical basis function?• how do you know when to add a basis function?

Idea: move from node-oriented to element-oriented approach


Recall: Hierarchical Basis Transformation

0,60,4 0,80,20

1

1

0,4

0,2

x

0

0,8

0,6

−→

0,60,4 0,80,20

1

1

0,4

0,2

x

0

0,8

0,6

• represent “wider” hat function φ1,1(x) via basis functions φ2,j (x)

φ1,1(x) = 12φ2,1(x) + φ2,2(x) + 1

2φ2,3(x)

• consider vector of hierarchical/nodal basis functionsand write transformation as matrix-vector product:ψ2,1(x)

ψ2,2(x)ψ2,3(x)

:=

φ2,1(x)φ1,1(x)φ2,3(x)

=

1 0 012 1 1

20 0 1

φ2,1(x)φ2,2(x)φ2,3(x)


Recall: Hierarchical Basis Transformation (2)

• hierarchical basis transformation: ψn,i (x) =∑

jHi,jφn,j (x)

• written as matrix-vector product: ~ψn = Hn~φn

• H can be written as a sequence of level-wise transforms:

Hn = H(n−1)n H(n−2)

n . . . H(2)n H(1)

n

• where each transform has a shape similar to

H(1)3 =

1 0 0 0 0 0 012 1 1

2 0 0 0 00 0 1 0 0 0 00 0 1

2 1 12 0 0

0 0 0 0 1 0 00 0 0 0 1

2 1 12

0 0 0 0 0 0 1


Recall: Hierarchical Coordinate Transformation

• consider function f (x) ≈∑

iaiψn,i (x) represented via hier. basis

• wanted: corresponding representation in nodal basis∑j

bjφn,j (x) =∑

i

aiψn,i (x) ≈ f (x)

• with ψn,i (x) =∑

jHi,jφn,j (x) we obtain

∑j

bjφn,j (x) =∑

i

ai

∑j

Hi,jφn,j (x) =∑

j

∑i

aiHi,jφn,j (x)

• compare coordinates and get

bj =∑

i

Hi,jai =∑

i

(HT )

j,i ai

• written in vector notation: b = HT a


FEM and Hierarchical Basis Transform

• FEM discretisation with hierarchical test and shape functions:∫ψi (x)L

(∑j

ujψj (x)

)dx =

∫ψi (x)f (x) dx ∀ψi

• leads to respective stiffness matrix AHBi,j :∫

ψi (x)L(∑

j

ujψj (x)

)dx =

∑j

uj

∫ψi (x)Lψj (x) dx =

∑j

ujAHBi,j

• vs. stiffness matrix with nodal basis as shape functions:∫ψi (x)L

(∑j

vjφj (x)

)dx =

∑j

vj

∫ψi (x)Lφj (x) dx =

∑j

vjA∗i,j

• note that (AHBu)i =∑

j ujAHBi,j =

∑j vjA∗i,j = (A∗v)i and v = HT u


FEM and Hierarchical Basis Transform (2)

• status: FEM with hierarchical test and nodal shape functions∫ψi (x)L

(∑j

vjφj (x)

)dx =

∫ψi (x)f (x) dx

• represent test functions via nodal basis:∫ ∑k

Hi,kφk (x)L(∑

j

vjφj (x)

)dx =

∫ ∑k

Hi,kφk (x)f (x) dx

∑k

Hi,k

∫φk (x)L

(∑j

vjφj (x)

)dx =

∑k

Hi,k

∫φk (x)f (x) dx

• leads to new system of equations: HANB v = H bNB

where ANB and bNB stem from nodal-basis FEM discretisation!• with v = HT u we obtain H ANB HT u = H b as system of equations, thus:

AHB = H ANB HT ( Galerkin coarsening)


Element Stiffness Matrices

• domain Ω splitted into finite elements Ω(k):

Ω = Ω(1) ∪ Ω(2) ∪ · · · ∪ Ω(n)

• observation: basis functions are defined element-wise• use:

∫ ba f (x)dx =

∫ ca f (x)dx +

∫ bc f (x)dx

• element-wise evaluation of the integrals:∫Ω

∇v · ∇u dx =∑

k

∫Ω(k)

∇v · ∇u dx∫Ω

vf dx =∑

i

∫Ω(i)

vf dx


Element Stiffness Matrices (2)

• leads to local stiffness matrices for each element:∫Ω(k)

∇φi · ∇φj dx︸︷︷︸=:A(k)

ij

• and respective element systems:

A(k)x = b(k)

• accumulate to obtain global system:∑k

A(k)

︸︷︷︸=:A

x =∑

k

b(k)


Element Stiffness Matrices (3)

Some comments on notation:• assume: 1D problem, n elements (i.e. intervals)• in each element only two basis functions are non-zero!

• hence, almost all A(k)ij are zero:

A(k)ij =

∫Ω(k)

∇φi · ∇φj dx

• only 2× 2 elements of A(k) are non-zero• therefore convention to omit zero columns/rows⇒ leaves only unknowns that are in Ω(k)


Typical workflow

1. choose elements:• quadratic or cubic cells• triangles (structured, unstructured)• tetrahedra, etc.

2. set up basis functions for each element Ω(k);for example, at all nodes xi ∈ Ω(k)

ϕi (xi ) = 1ϕi (xj ) = 0 for all j 6= i

3. for element stiffness matrix, compute all

A(k)ij =

∫Ω(k)

ϕiLϕj dΩ

4. accumulate global stiffness matrix


Example: 1D Poisson

• Ω = [0,1] splitted into Ω(k) = [xk−1, xk ]

• nodal basis; leads to element stiffness matrix:

A(k) =

(1 −1−1 1

)• consider only two elements:

A(1) + A(2) =

1 −1 0−1 1 00 0 0

+

0 0 00 1 −10 −1 1

• in stencil notation:

[−1 1∗ ] + [ 1∗ − 1 ] → [−1 2 − 1 ]


Project: Adaptive Hierarchical Basis

Consider:• 1D Poisson problem• FEM with hierarchical basis• however: not all basis functions used on each grid→ adaptive hierarchical basis

Discuss:• how to compute the stiffness matrix?• what do you need to compute, if you add a hierarchical basis function?• how do you know when to add a basis function?


Example: 2D Poisson

• −∆u = f on domain Ω = [0,1]2

• splitted into Ω(i,j) = [xi−1, xi ]× [xj−1, xj ]• bilinear basis functions

ϕij (x , y) = ϕi (x)ϕj (y)

• “pagoda” functions


Example: 2D Poisson (2)

• leads to element stiffness matrix:

A(k) =

2 − 1

2 − 12 −1

− 12 2 −1 − 1

2

− 12 −1 2 − 1

2

−1 − 12 − 1

2 2

• accumulation leads to 9-point stencil −1 −1 −1

−1 8 −1−1 −1 −1


Documents

Finite Element Methods - Sccswiki · PDF fileAlgorithms for Scientiﬁc Computing Finite Element Methods Michael Bader Technical University of Munich Summer 2016