## Damped Wave Equation with Dynamic Boundary ... - Nicolas Fourrier

2. Improvements of the approximation. 3. Damped Wave Equation - 2D. Domain. Problem definition. Pictures & Movies. Nicolas Fourrier (U.V.A.). P.D.E. Seminar.
Damped Wave Equation with Dynamic Boundary Conditions Approximate Solution Nicolas Fourrier

U NIVERSITY

OF

V IRGINIA

[email protected]

April 10, 2011

Nicolas Fourrier (U.V.A.)

P.D.E. Seminar

April 10, 2011

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Overview

1

Finite Element

2

Improvements of the approximation

3

Damped Wave Equation - 2D

Nicolas Fourrier (U.V.A.)

P.D.E. Seminar

April 10, 2011

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Finite Element

Overview

1

Finite Element Quick Recall Interpolation functions

2

Improvements of the approximation

3

Damped Wave Equation - 2D

Nicolas Fourrier (U.V.A.)

P.D.E. Seminar

April 10, 2011

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Finite Element

Quick Recall

Domain Definition - 2 Dimension Ω and Ωh Ω: convex domain with smooth boundary δΩ Th : partition of Ω into disjoint polygons. No vertex of any polygons lies on the interior of another polygon. Ωh : union of the polygons ⊂ Ω Size of the partition - h Let h denote the maximal length of the sides of the polygons of the partition Th . Thus h is a parameter which decreases as the partition is made finer. The elements of Th are essentially the same size and the angles each element is bounded below by a positive constant, indepently of h.

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P.D.E. Seminar

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Finite Element

Quick Recall

Finite Element Model

Finite dimensional subspace of H01 (Ω): Shr (Ω) Continuous functions on the closure Ω of Ω Piecewise polynomial of degree at most r − 1 in each polygon of Th Vanish outside Ωh

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P.D.E. Seminar

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Finite Element

Quick Recall

Finite Element Model

Strong form to Variational Form ∆u = ut on Ω

(1)

u = 0 on δΩ gives Z

Nicolas Fourrier (U.V.A.)

∇u∇vd Ω +

Z

ut vd Ω = 0 ∀v ∈ H01 (Ω)

P.D.E. Seminar

(2)

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Finite Element

Quick Recall

Finite Element Model On Shr (Ω) Let uje be the value of uhe at the j th node of the element e and Nje are the the interpolations function. u ≈ uh =

n X

uje Nje

(3)

j=1

Then the i th algebraic equation of the finite element model is obtained by substituting v = Nie n X

uje

j=1

Nicolas Fourrier (U.V.A.)

Z

Ωe

∇Nie ∇Nje d Ωe

+

P.D.E. Seminar

Z

Ωe

Nie Nje d Ωe



=0

April 10, 2011

(4)

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Finite Element

Interpolation functions

Construction of the interpolation functions 1-Dimensional Case One interpolation function per node per element. Two nodes per element. At the node, the corresponding shape function is 1, i.e., uhe = uje × 1 At the other node the shape function is 0.

N1e (x) = 1 − N2e (x) =

Nicolas Fourrier (U.V.A.)

x h

P.D.E. Seminar

x h

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Finite Element

Interpolation functions

1 0.8 0.6 0.4 0.2 0 −1

0

1 Nicolas Fourrier (U.V.A.)

−1

−0.5

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0

0.5 April 10, 2011

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Finite Element

Interpolation functions

Finite Element Model

2-Dimensional Case Similarly, we can build the shape functions for square elements using the above: x y )(1 − ) h h x y N2e (x) = (1 − ) h h x y e N3 (x) = (1 − ) h h xy e N4 (x) = hh N1e (x) = (1 −

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P.D.E. Seminar

(6)

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Finite Element

Interpolation functions

1

0.8

0.6

0.4

0.2

0 2 1.5 1 0.5 0 −0.5

2 1.5 1

−1 0.5 0 −1.5

−0.5 −1 −1.5

−2 −2

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P.D.E. Seminar

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Finite Element

Interpolation functions

Mass and stiffness matrix Now we are able to compute the integrals a : K →

Z

Ωe

M→

∇Nie ∇Nje d Ωe =

Z

h

Z

Z

Nie Nje d Ωe =

Z

h

Z

=

Z

hZ

Ωe

0

0

0

a

h 0 h 0 h 0

∇Nie (x, y)∇Nje (x, y)dx dy Nie (x, y)Nje (x, y)dx dy

(7)

x y xy (1 − ) dx dy hh h h for the entry (3,4) of M

We assume that each side of the element is the same

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P.D.E. Seminar

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Finite Element

Interpolation functions

Mass and stiffness matrix II

This provides the element mass (M) and stiffness (K) matrices for each element. Then, incorporating these element matrices into the global stiffness and mass matrices allows to solve the problem.     4 2 1 2 4 −1 −2 −1  1 −1 4 −1 −2 h2  2 4 3 1  (8) K =  M=   36 1 2 4 2 6 −2 −1 4 −1 2 1 2 4 −1 −2 −1 4

Nicolas Fourrier (U.V.A.)

P.D.E. Seminar

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Improvements of the approximation

Overview

1

Finite Element

2

Improvements of the approximation Mesh refinement vs. Higher order shape function Example Lobatto Polynomials Chebyshev Points 1-D shape functions using the Chebychev points 2-D shape function

3

Damped Wave Equation - 2D

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P.D.E. Seminar

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Improvements of the approximation

Mesh refinement vs. Higher order shape function

Estimate Error estimate for the finite methods are of the form: |u − uh |L2 (Ω) ≤ Chr Mesh refinement More natural.

Higher order

Lot of work necessary.

Computation time increase dramatically.

Nicolas Fourrier (U.V.A.)

(9)

P.D.E. Seminar

Much better solution for smooth solution.

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Improvements of the approximation

Example

1.2

1

½

−∆u(x) = u=0

π2 πx 4 cos( 2 )

in Ω = [−1; 1] on δΩ

p−refinement

0.8

0.6

exact solution 0.4

h−refinement 0.2

0

−0.2 −1.5

Nicolas Fourrier (U.V.A.)

−1

−0.5

0

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0.5

1

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1.5

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Improvements of the approximation

Lobatto Polynomials

Non Uniform distributed nodes

Recall that shape functions are polynomials of degree at most r − 1 in space and distribute the solution uh computed at its corresponding node through the whole element. It is common to use uniformly distributed nodes, but it has been shown that the non-uniform distribution (with high density toward the endpoints) gives the least error in the L2 -norm. The Gauss-Lobatto points (and Chebyshev points) are a way to define this non-uniform distribution. The drawback is that there is no explicit formula to determine them. The following picture shows the GaussLobatto points on a 1-dimension element [−1, 1].

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P.D.E. Seminar

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Improvements of the approximation

16

Chebyshev Points

Chebyshev points for p = 1, 2, ...15

14

12

10

8

6

4

2

0

−1

−0.8

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−0.6

−0.4

−0.2

0

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0.2

0.4

0.6

0.8

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1

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Improvements of the approximation

1-D shape functions using the Chebychev points

1

(7)

(7)

N1 (x)

N5 (x)

0

−1 (7) (7)

l0 l1

Nicolas Fourrier (U.V.A.)

(7)

l2

(7)

l3

P.D.E. Seminar

1 (7)

l4

(7)

l5

(7) (7)

l6 l7

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Improvements of the approximation

2-D shape function

1.2

1

0.8

0.6

0.4

0.2

−1

0

−0.8 −0.6 −0.4

−0.2

−0.2 0 −0.4 −1

0.2 −0.8

0.4 −0.6

Nicolas Fourrier (U.V.A.)

−0.4

−0.2

0.6 0

0.2

0.4

P.D.E. Seminar

0.8 0.6

0.8

1

1

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Damped Wave Equation - 2D

Overview

1

Finite Element

2

Improvements of the approximation

3

Damped Wave Equation - 2D Domain Problem definition Pictures & Movies

Nicolas Fourrier (U.V.A.)

P.D.E. Seminar

April 10, 2011

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Damped Wave Equation - 2D

21

Domain

22

11 16

23

12 17

7 11

8

14

9

1

10

9

3 3

15

6

8

2 2

20

14

Γ0 7

25

19

13

5

1

13 18

12

6

24

10

4 4

5

Γ

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P.D.E. Seminar

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Damped Wave Equation - 2D

Problem definition

Problem definition

 on Ω  utt + cΩ ut = ∆u − kΩ u du − kΓ ∆Γ u = 0 on Γ = Exterior Boundary utt + cΓ ut + dη  u=0 on Γ0 =Interior Boundary

(10)

There are four damping coefficients (cΩ , kΩ , cΓ , kΓ ): Today, kΩ = 0 and kΓ = 1, so that we can focus on the dynamic coefficients cΩ and cΓ . Subcript Ω and Γ denote coefficients acting respectively on the interior and on the boundary.

Nicolas Fourrier (U.V.A.)

P.D.E. Seminar

April 10, 2011

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Damped Wave Equation - 2D

Problem definition

Problem definition

 on Ω  utt + cΩ ut = ∆u − kΩ u du − kΓ ∆Γ u = 0 on Γ = Exterior Boundary utt + cΓ ut + dη  u=0 on Γ0 =Interior Boundary

(10)

There are four damping coefficients (cΩ , kΩ , cΓ , kΓ ): Today, kΩ = 0 and kΓ = 1, so that we can focus on the dynamic coefficients cΩ and cΓ . Subcript Ω and Γ denote coefficients acting respectively on the interior and on the boundary.

Nicolas Fourrier (U.V.A.)

P.D.E. Seminar

April 10, 2011

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Damped Wave Equation - 2D

Pictures & Movies

Next picture

General behavior of the solution Main result, boundary & interior damping Impact of the boundary damping on the solution Impact of the interior damping on the solution

Nicolas Fourrier (U.V.A.)

P.D.E. Seminar

April 10, 2011

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Damped Wave Equation - 2D

  utt + cΩ ut = ∆u − kΩ u utt + cΓ ut + du dη − kΓ ∆Γ u = 0  u=0

Pictures & Movies

on Ω on Γ on Γ0

1

cΩ =0 cΓ =1 kΩ =0

0.5 0 −0.5

kΓ =1 1 1 0.8

0.5

0.6 0.4 0

Nicolas Fourrier (U.V.A.)

0.2 0

P.D.E. Seminar

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Damped Wave Equation - 2D

Pictures & Movies

Next picture

General behavior of the solution Main result, boundary & interior damping First, let’s observe the spectrum. Impact of the boundary damping on the solution Impact of the interior damping on the solution

Nicolas Fourrier (U.V.A.)

P.D.E. Seminar

April 10, 2011

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Damped Wave Equation - 2D

Pictures & Movies

Imaginary Axis

cΩ =2 cΓ =2 kΩ =0 kΓ =1

8.5i

Real Axis

−0.6

  utt + cΩ ut = ∆u − kΩ u utt + cΓ ut + du dη − kΓ ∆Γ u = 0  u=0

Nicolas Fourrier (U.V.A.)

0

on Ω on Γ on Γ0

P.D.E. Seminar

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Damped Wave Equation - 2D

  utt + cΩ ut = ∆u − kΩ u utt + cΓ ut + du dη − kΓ ∆Γ u = 0  u=0

Pictures & Movies

on Ω on Γ on Γ0

0.4

cΩ =2 0.2 0

cΓ =2 kΩ =0

−0.2 −0.4 −0.6

kΓ =1 −0.8 1 1 0.8

0.5

0.6 0.4 0

Nicolas Fourrier (U.V.A.)

0.2 0

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Damped Wave Equation - 2D

Pictures & Movies

Next picture

General behavior of the solution Main result, boundary & interior damping Impact of the boundary damping on the solution Impact of the interior damping on the solution

Nicolas Fourrier (U.V.A.)

P.D.E. Seminar

April 10, 2011

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Damped Wave Equation - 2D

  utt + cΩ ut = ∆u − kΩ u utt + cΓ ut + du dη − kΓ ∆Γ u = 0  u=0

0.4

cΩ =2

0.4

0.2

cΩ =2

0 −0.2

cΓ =2

Pictures & Movies

cΓ =0

−0.4

kΩ =0 kΓ =1

on Ω on Γ on Γ0

0.2 0 −0.2 −0.4

kΩ =0

−0.6 −0.8

kΓ =1

1

−0.6 −0.8 1

1 0.5 0

Nicolas Fourrier (U.V.A.)

1 0.5

0.5

0.5 0

0

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0

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Damped Wave Equation - 2D

Pictures & Movies

Next picture

General behavior of the solution Main result, boundary & interior damping Impact of the boundary damping on the solution Impact of the interior damping on the solution

Nicolas Fourrier (U.V.A.)

P.D.E. Seminar

April 10, 2011

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Damped Wave Equation - 2D

  utt + cΩ ut = ∆u − kΩ u utt + cΓ ut + du dη − kΓ ∆Γ u = 0  u=0

0.4

cΩ =2

0.4

0.2

cΩ =0

0 −0.2

cΓ =2

Pictures & Movies

cΓ =2

−0.4

kΩ =0 kΓ =1

on Ω on Γ on Γ0

0.2 0 −0.2 −0.4

kΩ =0

−0.6 −0.8

kΓ =1

1

−0.6 −0.8 1

1 0.5 0

Nicolas Fourrier (U.V.A.)

1 0.5

0.5

0.5 0

0

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0

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