Winter 2016 / 17

(1)

Exercises to Wissenschaftliches Rechnen I/Scientific Computing I (V3E1/F4E1)

Winter 2016 / 17

Prof. Dr. Martin Rumpf

Alexander Effland — Stefanie Heyden — Stefan Simon — Sascha Tölkes

Problem sheet 3

Please hand in the solutions of exercise 8 and 9 on Tuesday November 15 ! Please prepare the solution of programming task 1 for Thursday November 24 !

Exercise 8 2+4 Points

(i.) Show that ∆ log (| x |) = 0 if x ∈ _R

²

\{ 0 } .

(ii.) Determine the weak solution of the following boundary value problem in R

²

:



 



 



− div ( a ∇ u ) = 0 in B

2

( 0 ) \ B

1 2

( 0 ) , u = ₁ _on ∂B

1

2

( ₀ ) _, u = 0 on ∂B

2

( 0 ) with

a ( x ) =

( 1 if | x | ≤ 1 , 2 if | x | > 1 .

Exercise 9 4+2+2 Points

Consider the Argyris finite element ( T, P

₅

, Γ ) with the following 21 degrees of free- dom

Γ ( p ) = ( _Γ

_α

( P ))

_α₌_1,...,21

= { p ( x

_i

) , ∂

k

p ( x

_i

) , ∂

k

∂

l

p ( x

_i

) , ∂

n_i

p ( m

_i

)}

_i_∈{_0,1,2_}_,k,l_∈{_1,2_}

, where n

_i

denotes the outer normal associated with the edge E

_i

at the edge midpoint m

_i

.

(i.) Show that any function in P

₅

( T ) is uniquely determined by an Argyris finite

element function on T.

(2)

x

₀

x

₁

x

2

E

₀

E

₁

E

₂

Let T

_h

be any triangulation of a polygonal domain Ω ⊂ _R

²

, V

_h

be the space of Argyris finite elements on T

_h

.

(ii.) Show that a function v ∈ V

_h

is continuous.

(iii.) Show that a function v ∈ V

_h

restricted to an edge has continuous normal and

tangential derivatives and conclude that v is differentiable.

(3)

Programming task 1

Consider Poisson’s problem

− 4 u = f on Ω , ( 1 )

u = u

^∂

on ∂Ω .

We assume u

^∂

≡ 0. Then the weak formulation of ( 1 ) is

Z

Ω

∇ _u · ∇ _ϕ _dy =

Z

Ω

f ϕ dy ( 2 )

for f ∈ L

²

( _Ω ) .

Consider two different functions f : 1 . f ≡ 1 and

2 . a function f that is obtained by mapping [ 0, 1 ]

²

to [− 1, 1 ]

²

, interpreting that domain as the complex plain and then taking the imaginary part of z

²³

. For details on the implementation and a simplified formula, we refer to ( 4 ) below.

In this programming task, problem ( 2 ) should be solved using linear ( P

¹

) finite elements. On the lecture web site, you can download a code framework written in C++ which implements a general two-dimensional triangular mesh for 2 D problems and surfaces in 3 D using P

¹

finite elements. All computations are performed on the unit triangle and then mapped to the respective element using a transformation X.

x ˆ

0

x ˆ

1

x ˆ

2

E

2

E

0

E

₁

X ( x ˆ ) = A x ˆ + b

x

0

x

1

x

2

x

i

= X ( x ˆ

i

)

We denote the unit triangle by ˆ T and set ˆ u = u ◦ X and ˆ v = v ◦ X as the pullback of u and v. Using the transformation formula we derive

Z

T

∇ u · ∇ v dy =

Z

Tˆ

√

det DX

^T

DX ∇ u ˆ

^T

DX

^T

DX

−1

∇ v ˆ d ˆ y ( 3 ) for a triangle T ∈ T

_h

, where T

_h

is a triangulation of Ω. The transformation formula also has to be applied to the right hand side integral.

To assemble the system matrix, we use the common scheme of iterating over all elements, assembling local matrices there and then mapping local indices on the element to global node indices.

The following code fragments have to be filled in:

lib/triangleMesh/unityTriangleIntegratorShellFE.h The local assembly of

Z

Tˆ

∑

k,l

∇( φ

_i

)

_k

a

_kl

∇( φ

_j

)

_l

d ˆ y

for basis functions φ

_i

and φ

_j

and a given matrix A = ( a

_kl

)

_k,l

.

(4)

lib/triangleMesh/unityTriangleIntegratorShellFE.h

The assembly of the global system matrix from local matrices.

course/stiffnessMatrixIntegrator.h Implement the transformation of R

T

∇ u · ∇ v dy using formula ( 3 ).

course/rhs.h

The assembly of the right hand sides. Implement f ≡ _{1 and}

f ( y ) = (− 1 + 2y

₁

)

²

+ (− 1 + 2y

2

)

²

1

3

sin

²₃

atan2(1 + 2y

2

, − 1 + 2y

₁

) . ( 4 ) atan2(y

2

, y

₁

) is a C++ function computing the angle in radians.

In the operator structure given, you only have to implement the evaluation of the numerical quadrature of f on a triangle T (given as element el in the code).

Further information: The code can be downloaded from the lecture web site, where you can also find input files containing discretized computational domains of varying grid size, additional information and a documentation of the classes needed to solve this task. A configuration file for CMake , which can be used to create Makefiles and project files for different IDEs, is provided. To compile, you need to download and install the Eigen library.

Winter 2016 / 17

Exercises to Wissenschaftliches Rechnen I/Scientific Computing I (V3E1/F4E1)

Winter 2016 / 17

Prof. Dr. Martin Rumpf

Alexander Effland — Stefanie Heyden — Stefan Simon — Sascha Tölkes

Problem sheet 3

Please hand in the solutions of exercise 8 and 9 on Tuesday November 15 ! Please prepare the solution of programming task 1 for Thursday November 24 !

Exercise 8 2+4 Points

(i.) Show that ∆ log (| x |) = 0 if x ∈ R

\{ 0 } .

(ii.) Determine the weak solution of the following boundary value problem in R

:



 



 



− div ( a ∇ u ) = 0 in B

( 0 ) \ B

( 0 ) , u = 1 on ∂B

( 0 ) , u = 0 on ∂B

( 0 ) with

a ( x ) =

( 1 if | x | ≤ 1 , 2 if | x | > 1 .

Exercise 9 4+2+2 Points

Consider the Argyris finite element ( T, P

, Γ ) with the following 21 degrees of free- dom

Γ ( p ) = ( Γ

( P ))

= { p ( x

) , ∂

p ( x

) , ∂

∂

p ( x

) , ∂

p ( m

)}

, where n

denotes the outer normal associated with the edge E

at the edge midpoint m

.

(i.) Show that any function in P

( T ) is uniquely determined by an Argyris finite

element function on T.

x

x

x

E

E

E

Let T

be any triangulation of a polygonal domain Ω ⊂ R

, V

be the space of Argyris finite elements on T

.

(ii.) Show that a function v ∈ V

is continuous.

(iii.) Show that a function v ∈ V

restricted to an edge has continuous normal and

tangential derivatives and conclude that v is differentiable.

Programming task 1

Consider Poisson’s problem

− 4 u = f on Ω , ( 1 )

u = u

on ∂Ω .

We assume u

≡ 0. Then the weak formulation of ( 1 ) is

∇ u · ∇ ϕ dy =

f ϕ dy ( 2 )

for f ∈ L

( Ω ) .

Consider two different functions f : 1 . f ≡ 1 and

2 . a function f that is obtained by mapping [ 0, 1 ]

to [− 1, 1 ]

, interpreting that domain as the complex plain and then taking the imaginary part of z

. For details on the implementation and a simplified formula, we refer to ( 4 ) below.

In this programming task, problem ( 2 ) should be solved using linear ( P

) finite elements. On the lecture web site, you can download a code framework written in C++ which implements a general two-dimensional triangular mesh for 2 D problems and surfaces in 3 D using P

finite elements. All computations are performed on the unit triangle and then mapped to the respective element using a transformation X.

(i.) Show that ∆ log (| x |) = 0 if x ∈ _R

( 0 ) , u = ₁ _on ∂B

( ₀ ) _, u = 0 on ∂B

Γ ( p ) = ( _Γ

be any triangulation of a polygonal domain Ω ⊂ _R

∇ _u · ∇ _ϕ _dy =

( _Ω ) .

The assembly of the right hand sides. Implement f ≡ _{1 and}