A high-order discontinuous hybrid Control-volume/Finite ... › documentos_public › archivos › ... · On a discontinuous hybrid CV & FE algorithm The idea: motivations and features

A high-order discontinuous hybrid

Control-volume/Finite-element method onmulti-dimensional domains

BCAM Workshop on Computational Mathematics, October 17-18, 2013

G. Stipcich1 M. Piller2

1BCAM -Basque Center for Applied Mathematics, Bilbao, Spain

2Universita degli Studi di Trieste, Trieste, Italy

[email protected]

Outline

1. On a discontinuous hybrid CV & FE algorithmThe idea: motivations and featuresBackground

2. Space discretization

3. Numerical results

4. Past, Present & Future

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On a discontinuous hybrid CV & FE algorithm

Outline





2 / 22

On a discontinuous hybrid CV & FE algorithm The idea: motivations and features

Outline





2 / 22


Motivations and features

1 Hybrid CV and FE merges the advantages of both

Control Volume: integral method conservative at CV levelGeometric flexibility

2 Discontinuous methods:

local P-adaptivity for high-order accuracyextremely local data structure: high-efficiency parallelizationmainly used for hyperbolic problems (discontinuities) oradvection-dominated problems (high-Reynolds numbers)

3 We focused on diffusion problems

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On a discontinuous hybrid CV & FE algorithm Background

Outline





3 / 22

On a discontinuous hybrid CV & FE algorithm Background

Background I: shape function-based FV

Low order High-order or Spectral

Baliga & Patankar (1980) Wang (2002)

Banaszek (1989) Iskandarani (2004)

Martinez (2006) Piller & Stalio (2011)

DCVFEM

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Space discretization

Outline





4 / 22


Discretization I: Mapping of generic curvilinear elements

Elemental mapping:e = Me (e) ∀e ∈ Th

The blending method is used:Me = l (η) + c (η)

The reference element ise ≡ [−1, 1]d ∀e ∈ Th

Advantage: single assemblyprocedure

collapsed

linear mapping +

blending

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Discretization I: Mapping of generic curvilinear elements

Elemental mapping:e = Me (e) ∀e ∈ Th

The blending method is used:Me = l (η) + c (η)

The reference element ise ≡ [−1, 1]d ∀e ∈ Th

Advantage: single assemblyprocedure

collapsed

linear mapping +

blending

5 / 22


Discretization II: Reference element and Control Volumes

The CVs (dashed lines) areinterpreted as a dual mesh

Location of the nodes(Discontinuous approach)

Reference element for Pe = 2

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Discretization III: Natural subdivision into CV for any Pe

(a) Pe = 1 (b) Pe = 2 (c) Pe = 3

(d) Pe = 4 (e) Pe = 5 (f) Pe = 6

7 / 22


Discretization IV: Elements in the Physical space

A B

C

D

Quadrilateral element, Pe = 5

A

B

C

Triangular element, Pe = 5

8 / 22


Discretization V: Analysis of the Condition Number

Condition Number ≡ ‖Me‖ ‖M−1e ‖

1 2 3 4 5 6 7 8 910

101

102

103

104

P e

(P e)2

(P e)3

ConditionNumber

Quadrilateral elements

1 2 3 4 5 6 7 8 910

101

102

103

104

P e

(P e)4

(P e)4

ConditionNumber

Triangular elements

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Numerical results

Outline





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Numerical results

Numerical results

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Numerical results

Governing equations

The diffusion equation is solved

Fundamental formulation:

−∇ · (D∇T ) = Q

The mixed formulation is used: constitutive and conservation equations

q+D · ∇T = 0

∇ · q = Q

11 / 22

Numerical results

Governing equations

The diffusion equation is solved

Fundamental formulation:

−∇ · (D∇T ) = Q

The mixed formulation is used: constitutive and conservation equations

q+D · ∇T = 0

∇ · q = Q

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Numerical results

Numerical tests I: 2D hollow cylinder

2D section of hollow cylinder

Dirichlet and Neumann b.c.

Q = 0

Analytical solution available

12 / 22

Numerical results


Hybrid mesh

High-order approximation ofthe (non-polynomial) curvedboundaries

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Numerical results


L2 Norm

10−0.6

10−0.4

10−0.2

100

100.2

10−12

10−10

10−8

10−6

10−4

10−2

100

h-convergence isoparametric (P e = Q)

log(h)

log(eL2)

P e = 1

P e = 2

P e = 3

P e = 4

P e = 5

P e + 2

P e + 1

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Numerical results


L∞ Norm

10−0.6

10−0.4

10−0.2

100

100.2

10−12

10−10

10−8

10−6

10−4

10−2

100

h-convergence isoparametric (P e = Q)

log(h)

log(eL∞)

P e = 1

P e = 2

P e = 3

P e = 4

P e = 5

P e + 2

P e + 1

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Numerical results

Numerical tests II: Non-homogeneous diffusivity

Non-homogeneous diffusivity D

Zero Dirichlet b.c.

Q 6= 0

D discontinuous in a sharp andsmall portion of Ω

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Numerical results


Triangular coarse mesh

In relation to the source Q,we can set any Pe

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Numerical results


The discontinuity interface isaccurately captured

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Numerical results

Numerical tests III: Anisotropic, non-hom. diffusivity

Anisotropic andNon-homogeneous diffusivity D

Piecewise constant Dirichlet b.c.

Q = 0

D discontinuous in a rectangularportion of Ω

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Numerical results

Numerical tests III: Anisotropic, non-hom. diffusivity

The streamlines of the vectorquantity q are properlydeviated in a ”z” shape

0.324

7.85

1.00 2.00 3.00 4.00 5.00 6.00 7.00

Temperature

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Past, Present & Future

Outline





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Background II: previous work on 1D DCVFEM

1 Formal derivation of weakformulation

2 Fourier analysis

3 P-convergence T and q4 Numerical results:

confirm Fourier analysisDCVFEM comparable toDGFEM

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2 Fourier analysis



φV (x) =

1 if x ∈ V

0 otherwise

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2 Fourier analysis



0 1 2 30

1

2

3

k

ke

Modified wavenumber: diffusion

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2 Fourier analysis



0.5 1 1.5 2 2.5 3 3.50

1

2

3

4

5

6

log (h0/h)

log

(e/e 0

)

log

(h/h

0)

DCVFEM: P-convergence

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2 Fourier analysis



−1 −0.5 0 0.5 1−2

−1

0

1

2

x

T

The Hemker problem

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2 Fourier analysis



−1 −0.5 0 0.5 1−2

−1

0

1

2

x

T

The Hemker problem

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2 Fourier analysis



−1 −0.5 0 0.5 1−2

−1

0

1

2

x

T

The Hemker problem

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Conclusions, ongoing work & issues

Conclusions:

We propose a discontinuous CVdiscretization on genericelements, with straightforward“spectral” accuracy

Ability of handling curvedgeometries, sharp gradients,anisotropic media

Ongoing work & issues:

3D extension tohexahedral/tetrahedral elementsof arbitrary order

Bottleneck: computational cost

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Conclusions:






C

A

B

Curved elements, arbitrary order

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Conclusions:





Bottleneck: computational costHigh gradients, non-homog. media

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Conclusions:






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Conclusions:





Bottleneck: computational cost Hexahedron, arbitrary order

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Conclusions:





Bottleneck: computational costTetrahedron, arbitrary order

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Conclusions:





Bottleneck: computational costComputational cost

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Thank you

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Weak formulation: formal derivation I

Consider the volume indicator function

φV (x) =

1 if x ∈ V

0 otherwise

(1)

And take the constitutive equation qd + D∇T = 0. After

1 multiplication with (1)

2 integration over the whole domain Ω

3 application of the divergence theorem

we have∫

V

D−1qd dx+

∫

∂V

T−n ds = 0 (2)

where T− denotes the trace on ∂V of the restriction of the function T tothe control volume V .

24 / 22



φV (x) =

1 if x ∈ V

0 otherwise

(1)





we have∫

V

D−1qd dx+

∫

∂V

T−n ds = 0 (2)


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φV (x) =

1 if x ∈ V

0 otherwise

(1)





we have∫

V

D−1qd dx+

∫

∂V

T−n ds = 0 (2)


24 / 22



φV (x) =

1 if x ∈ V

0 otherwise

(1)





we have∫

V

D−1qd dx+

∫

∂V

T−n ds = 0 (2)


24 / 22



φV (x) =

1 if x ∈ V

0 otherwise

(1)





we have∫

V

D−1qd dx+

∫

∂V

T−n ds = 0 (2)


24 / 22



φV (x) =

1 if x ∈ V

0 otherwise

(1)





we have∫

V

D−1qd dx+

∫

∂V

T−n ds = 0 (2)


24 / 22



φV (x) =

1 if x ∈ V

0 otherwise

(1)





we have∫

V

D−1qd dx+

∫

∂V

T−n ds = 0 (2)


24 / 22



φV (x) =

1 if x ∈ V

0 otherwise

(1)





we have∫

V

D−1qd dx+

∫

∂V

T−n ds = 0 (2)


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Weak formulation: formal derivation II

Regularity conditions:

T ∈ H1(

V ;R2)

→ theory of the mollifiers [Salsa, 2007].

A family of mollifiers ρǫ ∈ C∞

0 (R2), ǫ > 0 is defined as follows

ρǫ(x) =

1

ǫ2exp

−

1

1−∣

∣

∣

x

ǫ

∣

∣

∣

2

for |x | < 1

0 for |x | ≥ 1

(3)

A sequence of smooth functions Tǫ is defined using the convolution

Tǫ(x) =

∫

R2

ρǫ(x − y)E (T (y)) dy

where E (·) is a linear extension operator

E : H1 (V ) −→ H1(

R2)

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T ∈ H1(

V ;R2)




ρǫ(x) =

1

ǫ2exp

−

1

1−∣

∣

∣

x

ǫ

∣

∣

∣

2

for |x | < 1

0 for |x | ≥ 1

(3)


Tǫ(x) =

∫

R2



E : H1 (V ) −→ H1(

R2)

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T ∈ H1(

V ;R2)




ρǫ(x) =

1

ǫ2exp

−

1

1−∣

∣

∣

x

ǫ

∣

∣

∣

2

for |x | < 1

0 for |x | ≥ 1

(3)


Tǫ(x) =

∫

R2



E : H1 (V ) −→ H1(

R2)

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T ∈ H1(

V ;R2)




ρǫ(x) =

1

ǫ2exp

−

1

1−∣

∣

∣

x

ǫ

∣

∣

∣

2

for |x | < 1

0 for |x | ≥ 1

(3)


Tǫ(x) =

∫

R2



E : H1 (V ) −→ H1(

R2)

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Weak formulation: formal derivation III

It can be proven that Tǫ converges to T

limǫ→0

‖T − Tǫ‖H1(V ) = 0 (4)

The second integral in the constitutive equation can be rearranged as:

∫

Ω∇T ·Φx dx =

∫

V

∂T

∂xdx = lim

ǫ→0

∫

V

∂Tǫ

∂xdx =

limǫ→0

∫

∂V

Tǫnx ds =

∫

∂V

T−nx ds

(5)

Analogously for the conservation equation. Choosing T , q ∈ H1 (V )

guarantees the existence of the integrals on the control volume V .

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limǫ→0

‖T − Tǫ‖H1(V ) = 0 (4)


∫

Ω∇T ·Φx dx =

∫

V

∂T

∂xdx = lim

ǫ→0

∫

V

∂Tǫ

∂xdx =

limǫ→0

∫

∂V

Tǫnx ds =

∫

∂V

T−nx ds

(5)



26 / 22



limǫ→0

‖T − Tǫ‖H1(V ) = 0 (4)


∫

Ω∇T ·Φx dx =

∫

V

∂T

∂xdx = lim

ǫ→0

∫

V

∂Tǫ

∂xdx =

limǫ→0

∫

∂V

Tǫnx ds =

∫

∂V

T−nx ds

(5)



26 / 22



limǫ→0

‖T − Tǫ‖H1(V ) = 0 (4)


∫

Ω∇T ·Φx dx =

∫

V

∂T

∂xdx = lim

ǫ→0

∫

V

∂Tǫ

∂xdx =

limǫ→0

∫

∂V

Tǫnx ds =

∫

∂V

T−nx ds

(5)



26 / 22



limǫ→0

‖T − Tǫ‖H1(V ) = 0 (4)


∫

Ω∇T ·Φx dx =

∫

V

∂T

∂xdx = lim

ǫ→0

∫

V

∂Tǫ

∂xdx =

limǫ→0

∫

∂V

Tǫnx ds =

∫

∂V

T−nx ds

(5)



26 / 22



limǫ→0

‖T − Tǫ‖H1(V ) = 0 (4)


∫

Ω∇T ·Φx dx =

∫

V

∂T

∂xdx = lim

ǫ→0

∫

V

∂Tǫ

∂xdx =

limǫ→0

∫

∂V

Tǫnx ds =

∫

∂V

T−nx ds

(5)



26 / 22



limǫ→0

‖T − Tǫ‖H1(V ) = 0 (4)


∫

Ω∇T ·Φx dx =

∫

V

∂T

∂xdx = lim

ǫ→0

∫

V

∂Tǫ

∂xdx =

limǫ→0

∫

∂V

Tǫnx ds =

∫

∂V

T−nx ds

(5)



26 / 22

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A high-order discontinuous hybrid Control-volume/Finite ... › documentos_public › archivos › ... · On a discontinuous hybrid CV & FE algorithm The idea: motivations and features