Mimetic Methods for Diffusion Problems

DANSIS seminar: High order discretisation 21/3/2018

Kennet OlesenPhD

Aarhus University 2013-2016

A diffusion problem is typically constructed out of 2 parts

• An equilibrium or balance part– E.g. mass balance, energy balance etc.

𝛁 ⋅ 𝒒 = 𝑓

• A constitutive part– E.g. relation between heat flux and a temperature gradient

𝒒 = k 𝛁𝛼

Merge the 2 parts and you get:

𝛁 ⋅ k 𝛁𝛼 = 𝑓

k ∆𝛼 = 𝑓

Structure of a diffusion problem

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Special interpolation polynomials: Edge polynomials

In [1] interpolation polynomials are derived,which are based on integrated line values

𝛽ℎ 𝜉 =

𝑖=1

𝑁

𝐵𝑖𝑒𝑖(𝜉) 𝐵𝑖 = 𝜉𝑖−1

𝜉𝑖

𝛽 𝜉 𝑑𝜉

𝑒𝑖 𝜉 = −

𝑗=0

𝑖−1𝑑ℎ𝑗 𝜉

𝑑𝜉 𝜉𝑘−1

𝜉𝑘

𝑒𝑖 𝜉 = 1 𝑓𝑜𝑟 𝑖 = 𝑘0 𝑓𝑜𝑟 𝑖 ≠ 𝑘

ℎ: Lagrange polynomials

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[1]: Gerritsma M.: Edge Functions for Spectral Element MethodsSelected Papers from the ICOSAHOM '09. Springer, 2011

Special interpolation polynomials: Edge polynomials

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𝛽ℎ 𝜉 =

𝑖=1

𝑁

𝐵𝑖𝑒𝑖(𝜉)

𝐵𝑖 = 𝜉𝑖−1

𝜉𝑖

𝛽 𝜉 𝑑𝜉

𝛼ℎ 𝜉 =

𝑖=0

𝑁

𝛼𝑖ℎ𝑖(𝜉)

Interpolation in 3D -> tensor products

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𝛼ℎ 𝜉1, 𝜉2, 𝜉3 =

𝑖

𝑗

𝑘

𝐴𝑖𝑗𝑘 Ψ𝑖𝑗𝑘 𝜉1, 𝜉2, 𝜉3

Ψ𝑖𝑗𝑘 𝜉1, 𝜉2, 𝜉3 = ℎ𝑖(𝜉1)ℎ𝑗(𝜉2)ℎ𝑘(𝜉3)

Ψ𝑖𝑗𝑘 𝜉1, 𝜉2, 𝜉3 = 𝑒𝑖(𝜉1)ℎ𝑗(𝜉2)ℎ𝑘(𝜉3)

Ψ𝑖𝑗𝑘 𝜉1, 𝜉2, 𝜉3 = ℎ𝑖(𝜉1)𝑒𝑗(𝜉2)ℎ𝑘(𝜉3)

Ψ𝑖𝑗𝑘 𝜉1, 𝜉2, 𝜉3 = ℎ𝑖(𝜉1)ℎ𝑗(𝜉2)𝑒𝑘(𝜉3)Ψ𝑖𝑗𝑘 𝜉1, 𝜉2, 𝜉3 = ℎ𝑖(𝜉1)𝑒𝑗(𝜉2)𝑒𝑘(𝜉3)

Ψ𝑖𝑗𝑘 𝜉1, 𝜉2, 𝜉3 = 𝑒𝑖(𝜉1)ℎ𝑗(𝜉2)𝑒𝑘(𝜉3)

Ψ𝑖𝑗𝑘 𝜉1, 𝜉2, 𝜉3 = 𝑒𝑖(𝜉1)𝑒𝑗(𝜉2)ℎ𝑘(𝜉3)

Ψ𝑖𝑗𝑘 𝜉1, 𝜉2, 𝜉3 = 𝑒𝑖(𝜉1)𝑒𝑗(𝜉2)𝑒𝑘(𝜉3)

𝑄1 𝑄2 𝑄3

𝐹

Differentiation

• Say we have the approximation:

𝛼ℎ 𝜉 =

𝑖=0

𝑁

𝛼𝑖ℎ𝑖(𝜉)

• We choose to approximate the derivative by:

𝑑𝛼ℎ(𝜉)

𝑑𝜉=

𝑖=1

𝑁

𝑖−𝑖

𝑖𝑑𝛼ℎ(𝜉)𝑑𝜉

𝑑𝜉 𝑒𝑖(𝜉)

• This reduces to (first fundamental theorem of calculus):

𝑑𝛼ℎ(𝜉)

𝑑𝜉=

𝑖=1

𝑁

𝛼𝑖 − 𝛼𝑖−1 𝑒𝑖(𝜉)

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𝛼𝑖

𝛼𝑖−1

Divergence

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𝛁 ⋅ 𝒒ℎ =𝜕𝑞1

ℎ 𝜉1, 𝜉2, 𝜉3𝜕𝜉1

+𝜕𝑞2

ℎ 𝜉1, 𝜉2, 𝜉3𝜕𝜉2

+𝜕𝑞3

ℎ 𝜉1, 𝜉2, 𝜉3𝜕𝜉3

𝑞1ℎ 𝜉1, 𝜉2, 𝜉3 =

𝑖=0

𝑁

𝑗=1

𝑁

𝑘=1

𝑁

𝑄𝑖,𝑗,𝑘1 ℎ𝑖(𝜉1)𝑒𝑗(𝜉2)𝑒𝑘(𝜉3)

𝑞2ℎ 𝜉1, 𝜉2, 𝜉3 =

𝑖=1

𝑁

𝑗=0

𝑁

𝑘=1

𝑁

𝑄𝑖,𝑗,𝑘2 𝑒𝑖(𝜉1)ℎ𝑗(𝜉2)𝑒𝑘(𝜉3)

𝑞3ℎ 𝜉1, 𝜉2, 𝜉3 =

𝑖=1

𝑁

𝑗=1

𝑁

𝑘=0

𝑁

𝑄𝑖,𝑗,𝑘3 𝑒𝑖(𝜉1)𝑒𝑗(𝜉2)ℎ𝑘(𝜉3)

𝛁 ⋅ 𝒒ℎ =

𝑖=1

𝑁

𝑗=1

𝑁

𝑘=1

𝑁

𝑄𝑖,𝑗,𝑘1 − 𝑄𝑖−1,𝑗,𝑘

1 + 𝑄𝑖,𝑗,𝑘2 − 𝑄𝑖,𝑗−1,𝑘

2 + 𝑄𝑖,𝑗,𝑘3 − 𝑄𝑖,𝑗,𝑘−1

3 𝑒𝑖(𝜉1)𝑒𝑗(𝜉2)𝑒𝑘(𝜉3)

𝑑iv

𝑬(3,2)

Divergence – Incidence matrix

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The discrete balance equation

Mimetic Methods for Diffusion Problems 9

𝛁 ⋅ 𝒒ℎ = 𝑓h

This is written by:

𝑖=1

𝑁

𝑗=1

𝑁

𝑘=1

𝑁

𝑄𝑖,𝑗,𝑘1 − 𝑄𝑖−1,𝑗,𝑘

1 + 𝑄𝑖,𝑗,𝑘2 − 𝑄𝑖,𝑗−1,𝑘

2 + 𝑄𝑖,𝑗,𝑘3 − 𝑄𝑖,𝑗,𝑘−1

3 𝑒𝑖(𝜉1)𝑒𝑗(𝜉2)𝑒𝑘(𝜉3) =

𝑖=1

𝑁

𝑗=1

𝑁

𝑘=1

𝑁

𝐹𝑖,𝑗,𝑘 𝑒𝑖(𝜉1)𝑒𝑗(𝜉2)𝑒𝑘(𝜉3)

They share common interpolation polynomials:

𝑖=1

𝑁

𝑗=1

𝑁

𝑘=1

𝑁

𝑄𝑖,𝑗,𝑘1 − 𝑄𝑖−1,𝑗,𝑘

1 + 𝑄𝑖,𝑗,𝑘2 − 𝑄𝑖,𝑗−1,𝑘

2 + 𝑄𝑖,𝑗,𝑘3 − 𝑄𝑖,𝑗,𝑘−1

3 − 𝐹𝑖,𝑗,𝑘 𝑒𝑖(𝜉1)𝑒𝑗(𝜉2)𝑒𝑘(𝜉3) = 0

This implies:

𝑄𝑖,𝑗,𝑘1 − 𝑄𝑖−1,𝑗,𝑘

1 + 𝑄𝑖,𝑗,𝑘2 − 𝑄𝑖,𝑗−1,𝑘

2 + 𝑄𝑖,𝑗,𝑘3 − 𝑄𝑖,𝑗,𝑘−1

3 − 𝐹𝑖,𝑗,𝑘 = 0

Exact balance equation for our mesh

Putting together the system of equations

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• Problem: More unknowns than equations

• I fix this when I discretize the constitutive equation𝒒ℎ = k 𝛁𝛼ℎ

• Weigh with 𝒒ℎ and integrate over domain1

𝑘𝒒ℎ, 𝒒ℎ

Ω= 𝒒ℎ, 𝛁𝛼ℎ

Ω= − 𝛁 ⋅ 𝒒ℎ, 𝛼ℎ

Ω+ 𝒒ℎ, 𝛼ℎ

𝜕Ω

𝐼𝑛𝑡𝑒𝑔𝑟𝑎𝑡𝑖𝑜𝑛 𝑏𝑦 𝑝𝑎𝑟𝑡𝑠

• Choose same amount of discrete points for 𝛼ℎ as number of elements

• Square system of equations

Transpose of the divergence in thebalance equations

Putting together the system of equations

Mimetic Methods for Diffusion Problems 11

• Balance equation + constitutive equation → Square system of equations

𝟎 𝑬(3,2)

𝑬(3,2)𝑽ℎ 𝑇 1

𝑘𝑯ℎ

𝚫𝛼

𝚫𝑄=

𝚫𝐹

𝑩ℎ𝚫𝛼𝐵

Potential boundary conditions weakly enforced

Flux boundary conditions strongly enforced

• Multiply balance equation with 𝑽ℎ→ Symmetric system of equations

𝟎 𝑽ℎ𝑬(3,2)

𝑬(3,2)𝑽ℎ 𝑇 1

𝑘𝑯ℎ

𝚫𝛼

𝚫𝑄=

𝑽ℎ𝚫𝐹

𝑩ℎ𝚫𝛼𝐵

Complex geometry

Mimetic Methods for Diffusion Problems 12

• Mapping will only affect the matrices with expansion polynomials and not the incidence matrices

• I.e. mapping will not affect the mimicking property of the balance equation

𝑬(3,2)𝚫𝑄 = 𝚫𝐹 𝑬(3,2)𝚫𝑄 = 𝚫𝐹

Test case

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• Choose potential field• Calculate 𝒒 and 𝑓• Apply as boundary conditions

• Solve on different meshes• Refine mesh

• Mesh size ℎ𝑒𝑙 ↓

• Evaluate in 100 × 100 points in each element

• Plot the maximum value in the entire domain

Error of the Potential (𝛼 = 𝜙)

Residual of balance equation (𝑅 = 𝛁 ⋅ 𝒒ℎ − 𝑓ℎ)

Mimicking vector valued fields

Mimetic Methods for Diffusion Problems 14

• So far we have considered Poisson’s equation of a scalar field• But what about Poisson’s equation of a vector field?

• For example the equilibrium of forces in Continuum mechanics

𝛁 ⋅ 𝝈 = −𝒇

Mimicking vector valued fields

Mimetic Methods for Diffusion Problems 15

• The procedure is the same just in 3 directions𝑫𝚫𝑇 = −𝚫𝐹

• With

𝑫 =

𝑬(3,2) 𝟎 𝟎

𝟎 𝑬(3,2) 𝟎

𝟎 𝟎 𝑬(3,2)

• Consider the structural problem• Constitutive equations

• 𝜀𝑖𝑗 =𝜕

𝜕𝑥𝑖𝑢𝑗 − 𝜔𝑖𝑗 = 𝐶𝑖𝑗𝑘𝑙 𝜎𝑘𝑙

• Equilibrium of forces

•𝜕

𝜕𝑥𝑖𝜎𝑖𝑗 + 𝑓𝑗 = 0

• Symmetry of the stress tensor• 𝜎𝑖𝑗 − 𝜎𝑗𝑖 = 0

𝑯ℎ 𝑫 𝑽ℎ 𝑇− 𝑹ℎ 𝑇

𝑽ℎ 𝑫 𝟎 𝟎−𝑹ℎ 𝟎 𝟎

𝚫𝑇

𝚫𝑢

𝚫𝜔

=𝑩ℎ𝚫𝑢

𝐵𝐶

−𝑽ℎ𝚫𝐹

𝟎

Test case

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Test case

Mimetic Methods for Diffusion Problems 17

FEMimetic

Thank you for your time!

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