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WIAM 2016 Applications Approaches Test Case Variational Form - - PowerPoint PPT Presentation

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WIAM 2016 Jorge De Anda Salazar Introduction WIAM 2016 Applications Approaches Test Case Variational Form Jorge De Anda Salazar Postulates Constitutive model Kinematics Incremental VF Ecole Centrale de Nantes, France (ECN)

slide-1
SLIDE 1

WIAM 2016 Jorge De Anda Salazar Introduction

Applications Approaches Test Case

Variational Form

Postulates Constitutive model Kinematics Incremental VF

Numerical Im- plementation

Monolithic Staggered Comparison of results

Conclusion

Remarks Future Work 1/20

WIAM 2016

Jorge De Anda Salazar

´ Ecole Centrale de Nantes, France (ECN) Technische Universit¨ at M¨ unchen, Germany (TUM)

Sep 1st, 2016

slide-2
SLIDE 2

WIAM 2016 Jorge De Anda Salazar Introduction

Applications Approaches Test Case

Variational Form

Postulates Constitutive model Kinematics Incremental VF

Numerical Im- plementation

Monolithic Staggered Comparison of results

Conclusion

Remarks Future Work 2/20

1 Introduction

Applications Approaches Test Case

2 Variational Form

Postulates Constitutive model Kinematics Incremental VF

3 Numerical Implementation

Monolithic Staggered Comparison of results

4 Conclusion

Remarks Future Work

slide-3
SLIDE 3

WIAM 2016 Jorge De Anda Salazar Introduction

Applications Approaches Test Case

Variational Form

Postulates Constitutive model Kinematics Incremental VF

Numerical Im- plementation

Monolithic Staggered Comparison of results

Conclusion

Remarks Future Work 3/20

Research topic

Development of algorithmic strategies for numerical simulation of coupled mechanical-diffusion problems Applications Reactive flows in solids Tissue diagnosis and reconstruction

slide-4
SLIDE 4

WIAM 2016 Jorge De Anda Salazar Introduction

Applications Approaches Test Case

Variational Form

Postulates Constitutive model Kinematics Incremental VF

Numerical Im- plementation

Monolithic Staggered Comparison of results

Conclusion

Remarks Future Work 3/20

Research topic

Development of algorithmic strategies for numerical simulation of coupled mechanical-diffusion problems Applications Reactive flows in solids Tissue diagnosis and reconstruction

1

1 By - Mpt-matthew - Own work, CC BY 3.0, https://en.wikipedia.org/wiki/File:Expanded lithium-ion polymer battery from an Apple iPhone 3GS.jpg

slide-5
SLIDE 5

WIAM 2016 Jorge De Anda Salazar Introduction

Applications Approaches Test Case

Variational Form

Postulates Constitutive model Kinematics Incremental VF

Numerical Im- plementation

Monolithic Staggered Comparison of results

Conclusion

Remarks Future Work 3/20

Research topic

Development of algorithmic strategies for numerical simulation of coupled mechanical-diffusion problems Applications Reactive flows in solids Tissue diagnosis and reconstruction

1

1 By User:Paulnasca - Own work, CC BY 2.0, https://commons.wikimedia.org/w/index.php?curid=7128816

slide-6
SLIDE 6

WIAM 2016 Jorge De Anda Salazar Introduction

Applications Approaches Test Case

Variational Form

Postulates Constitutive model Kinematics Incremental VF

Numerical Im- plementation

Monolithic Staggered Comparison of results

Conclusion

Remarks Future Work 3/20

Research topic

Development of algorithmic strategies for numerical simulation of coupled mechanical-diffusion problems Applications Reactive flows in solids Tissue diagnosis and reconstruction

1

1 By - Carl Fredrik (CFCF)- Own work, CC BY 3.0, https://commons.wikimedia.org/wiki/File:716 Intervertebral Disk.jpg

slide-7
SLIDE 7

WIAM 2016 Jorge De Anda Salazar Introduction

Applications Approaches Test Case

Variational Form

Postulates Constitutive model Kinematics Incremental VF

Numerical Im- plementation

Monolithic Staggered Comparison of results

Conclusion

Remarks Future Work 4/20

Transient Chemical diffusion

Variables c : concentration [mol/m3] µ : chemical potential [J/mol]

  • j : molar flux [mol/m2s]

D : diffusivity [m2/s]

slide-8
SLIDE 8

WIAM 2016 Jorge De Anda Salazar Introduction

Applications Approaches Test Case

Variational Form

Postulates Constitutive model Kinematics Incremental VF

Numerical Im- plementation

Monolithic Staggered Comparison of results

Conclusion

Remarks Future Work 4/20

Transient Chemical diffusion

2 3

Variables c : concentration [mol/m3] µ : chemical potential [J/mol]

  • j : molar flux [mol/m2s]

D : diffusivity [m2/s]

2By BruceBlaus - Own work, CC BY 3.0, https://commons.wikimedia.org/w/index.php?curid=29452222 3 Georg Job, Physical chemistry from a different angle

slide-9
SLIDE 9

WIAM 2016 Jorge De Anda Salazar Introduction

Applications Approaches Test Case

Variational Form

Postulates Constitutive model Kinematics Incremental VF

Numerical Im- plementation

Monolithic Staggered Comparison of results

Conclusion

Remarks Future Work 5/20

Different approaches

  • Strong Form
  • Weak form
  • Variational form
slide-10
SLIDE 10

WIAM 2016 Jorge De Anda Salazar Introduction

Applications Approaches Test Case

Variational Form

Postulates Constitutive model Kinematics Incremental VF

Numerical Im- plementation

Monolithic Staggered Comparison of results

Conclusion

Remarks Future Work 5/20

Different approaches

  • Strong Form(Fick’s laws)

1st law: ∂c ∂t = −∇ · j 2nd law: j = −D∇c

  • Weak form
  • Variational form
slide-11
SLIDE 11

WIAM 2016 Jorge De Anda Salazar Introduction

Applications Approaches Test Case

Variational Form

Postulates Constitutive model Kinematics Incremental VF

Numerical Im- plementation

Monolithic Staggered Comparison of results

Conclusion

Remarks Future Work 5/20

Different approaches

  • Strong Form(Fick’s laws)

1st law: ∂c ∂t = −∇ · j 2nd law: j = −D∇c

  • Weak form
  • Ω

∂c ∂t δc + D∇c · ∇δc

  • dx = 0
  • Variational form
slide-12
SLIDE 12

WIAM 2016 Jorge De Anda Salazar Introduction

Applications Approaches Test Case

Variational Form

Postulates Constitutive model Kinematics Incremental VF

Numerical Im- plementation

Monolithic Staggered Comparison of results

Conclusion

Remarks Future Work 5/20

Different approaches

  • Strong Form(Fick’s laws)

1st law: ∂c ∂t = −∇ · j 2nd law: j = −D∇c

  • Weak form
  • Ω

∂c ∂t δc + D∇c · ∇δc

  • dx = 0
  • Variational form

{c} = arg inf

c I(c)

slide-13
SLIDE 13

WIAM 2016 Jorge De Anda Salazar Introduction

Applications Approaches Test Case

Variational Form

Postulates Constitutive model Kinematics Incremental VF

Numerical Im- plementation

Monolithic Staggered Comparison of results

Conclusion

Remarks Future Work 5/20

Different approaches

  • Strong Form(Fick’s laws)

1st law: ∂c ∂t = −∇ · j 2nd law: j = −D∇c

  • Weak form
  • Ω

∂c ∂t δc + D∇c · ∇δc

  • dx = 0
  • Variational form

{c} = arg inf

c I(c)

Variational Form ⇐ ⇒ Weak Form ⇐ ⇒ Strong Form

slide-14
SLIDE 14

WIAM 2016 Jorge De Anda Salazar Introduction

Applications Approaches Test Case

Variational Form

Postulates Constitutive model Kinematics Incremental VF

Numerical Im- plementation

Monolithic Staggered Comparison of results

Conclusion

Remarks Future Work 5/20

Different approaches

  • Strong Form(Fick’s laws)

1st law: ∂c ∂t = −∇ · j 2nd law: j = −D∇c

  • Weak form
  • Ω

∂c ∂t δc + D∇c · ∇δc

  • dx = 0
  • Variational form

{c, µ} = arg inf

c sup µ I(c, µ)

Variational Form ⇐ ⇒ Weak Form ⇐ ⇒ Strong Form

slide-15
SLIDE 15

WIAM 2016 Jorge De Anda Salazar Introduction

Applications Approaches Test Case

Variational Form

Postulates Constitutive model Kinematics Incremental VF

Numerical Im- plementation

Monolithic Staggered Comparison of results

Conclusion

Remarks Future Work 5/20

Different approaches

  • Strong Form(Fick’s laws)

1st law: ∂c ∂t = −∇ · j 2nd law: j = −D∇c

  • Weak form
  • Ω

∂c ∂t δc + D∇c · ∇δc

  • dx = 0
  • Variational form

{c, µ} = arg inf

c sup µ I(c, µ)

Chemistry Thermodynamics Mechanics Electricity c S ǫ q µ T σ V

slide-16
SLIDE 16

WIAM 2016 Jorge De Anda Salazar Introduction

Applications Approaches Test Case

Variational Form

Postulates Constitutive model Kinematics Incremental VF

Numerical Im- plementation

Monolithic Staggered Comparison of results

Conclusion

Remarks Future Work 6/20

Test Case: Initial discontinuous concentration

Boundary conditions BC :    c = cext ∀x = L, t > 0 ∂c ∂x = 0 ∀x = 0, t > 0 Initial conditions IC : c(x, t = 0) = c0 c0 = cext Exact Solution

0.2 0.4 0.6 0.8 1 + [meters] #10!3

  • 1.84
  • 1.82
  • 1.8
  • 1.78
  • 1.76
  • 1.74

7 #105

Chemical Potential 0.2 0.4 0.6 0.8 1

+ [meters] #10!3 2000 4000 6000 8000 10000 12000 c [mol/m

3] Concentration

0.2 0.4 0.6 0.8 1 + [meters] #10!3

  • 2
  • 1.5
  • 1
  • 0.5

e [J/m ] #109

Internal Energy 0.5 1 1.5 2

c #104

  • 1.85
  • 1.8
  • 1.75

7 #105

Phase Space

slide-17
SLIDE 17

WIAM 2016 Jorge De Anda Salazar Introduction

Applications Approaches Test Case

Variational Form

Postulates Constitutive model Kinematics Incremental VF

Numerical Im- plementation

Monolithic Staggered Comparison of results

Conclusion

Remarks Future Work 6/20

Test Case: Initial discontinuous concentration

Boundary conditions BC :    c = cext ∀x = L, t > 0 ∂c ∂x = 0 ∀x = 0, t > 0 Initial conditions IC : c(x, t = 0) = c0 c0 = cext Exact Solution

0.2 0.4 0.6 0.8 1 + [meters] #10!3

  • 1.9
  • 1.85
  • 1.8
  • 1.75

7 #105

Chemical Potential 0.2 0.4 0.6 0.8 1

+ [meters] #10!3 50 100 150 200 c [mol/m

3] Concentration

0.2 0.4 0.6 0.8 1 + [meters] #10!3

  • 3.5
  • 3
  • 2.5
  • 2
  • 1.5
  • 1
  • 0.5

e [J/m ] #107

Internal Energy 50 100 150 200

c

  • 1.9
  • 1.85
  • 1.8
  • 1.75

7 #105

Phase Space 0.2 0.4 0.6 0.8 1

+ [meters] #10!3

  • 1.84
  • 1.82
  • 1.8
  • 1.78
  • 1.76
  • 1.74

7 #105

Chemical Potential 0.2 0.4 0.6 0.8 1

+ [meters] #10!3 2000 4000 6000 8000 10000 12000 c [mol/m

3] Concentration

0.2 0.4 0.6 0.8 1 + [meters] #10!3

  • 2
  • 1.5
  • 1
  • 0.5

e [J/m ] #109

Internal Energy 0.5 1 1.5 2

c #104

  • 1.85
  • 1.8
  • 1.75

7 #105

Phase Space

slide-18
SLIDE 18

WIAM 2016 Jorge De Anda Salazar Introduction

Applications Approaches Test Case

Variational Form

Postulates Constitutive model Kinematics Incremental VF

Numerical Im- plementation

Monolithic Staggered Comparison of results

Conclusion

Remarks Future Work 6/20

Test Case: Initial discontinuous concentration

Boundary conditions BC :    c = cext ∀x = L, t > 0 ∂c ∂x = 0 ∀x = 0, t > 0 Initial conditions IC : c(x, t = 0) = c0 c0 = cext Spurious oscillations (Stable schemes) Scheme : Crank-Nicolson

slide-19
SLIDE 19

WIAM 2016 Jorge De Anda Salazar Introduction

Applications Approaches Test Case

Variational Form

Postulates Constitutive model Kinematics Incremental VF

Numerical Im- plementation

Monolithic Staggered Comparison of results

Conclusion

Remarks Future Work 7/20

Variational formulation

Advantages

  • A suitable framework for coupled problems.
  • Versatility for numerical implementation.
  • Use of optimization algorithms.
  • Symmetric mathematical structure → cost reduction.
  • Mathematical analysis of solutions.
slide-20
SLIDE 20

WIAM 2016 Jorge De Anda Salazar Introduction

Applications Approaches Test Case

Variational Form

Postulates Constitutive model Kinematics Incremental VF

Numerical Im- plementation

Monolithic Staggered Comparison of results

Conclusion

Remarks Future Work 7/20

Variational formulation

Advantages

  • A suitable framework for coupled problems.
  • Versatility for numerical implementation.
  • Use of optimization algorithms.
  • Symmetric mathematical structure → cost reduction.
  • Mathematical analysis of solutions.

Starting Point Framework : Generalized Standard Materials. Postulates : ∃ Internal energy U(c) ∃ Dissipation potential χ(∇µ, c)

slide-21
SLIDE 21

WIAM 2016 Jorge De Anda Salazar Introduction

Applications Approaches Test Case

Variational Form

Postulates Constitutive model Kinematics Incremental VF

Numerical Im- plementation

Monolithic Staggered Comparison of results

Conclusion

Remarks Future Work 7/20

Variational formulation

Advantages

  • A suitable framework for coupled problems.
  • Versatility for numerical implementation.
  • Use of optimization algorithms.
  • Symmetric mathematical structure → cost reduction.
  • Mathematical analysis of solutions.

Starting Point Framework : Generalized Standard Materials. Postulates : ∃ Internal energy U(c) → µ = dU dc ∃ Dissipation potential χ(∇µ, c) → j = − dχ d(∇µ) Variables are energetically conjugated.

slide-22
SLIDE 22

WIAM 2016 Jorge De Anda Salazar Introduction

Applications Approaches Test Case

Variational Form

Postulates Constitutive model Kinematics Incremental VF

Numerical Im- plementation

Monolithic Staggered Comparison of results

Conclusion

Remarks Future Work 8/20

Constitutive model

Fickean Model U(c) = µ0c + RT

  • cln

c c0

  • − c + c0
  • g := −

∇µ χ( g, c) = cM 2 g · g M = D RT Linear Model U(c) = µ0c + H 2 (c − c0)2 H = RT c0 χ( g) = Dµ 2 g · g Dµ := D H M, Dµ : mobility coefficients H : Chemical capacity R : Boltzman constant T: Temperature

slide-23
SLIDE 23

WIAM 2016 Jorge De Anda Salazar Introduction

Applications Approaches Test Case

Variational Form

Postulates Constitutive model Kinematics Incremental VF

Numerical Im- plementation

Monolithic Staggered Comparison of results

Conclusion

Remarks Future Work 9/20

X-Field formulations

  • 3-Field form

I3(c, j, µ) =

  • τ

Ω

   ˙ U − µ˙ c + χ∗(c, j) − g · j

  • χ(

g)

+µr    dV −

  • ∂jΩ

µjdS

  • dt

{c, j, µ} = arg inf

c,

  • j

sup

µ I3(c,

j, µ)

  • 2-Field form

I2(c, µ) =

  • τ

Ω

   ˙ U − µ˙ c

˙ U∗(µ)− ˙ µc

−χ( g, c) + µr    dV −

  • ∂jΩ

µjdS

  • dt

{c, µ} = arg inf

c sup µ I2(c, µ)

  • 1-Field form

I1(µ) =

  • τ

Ω

  • − ˙

U∗(µ) − c ˙ µ − χ( g, dU∗ dµ ) + µr

  • dV −
  • ∂jΩ

µjdS

  • dt

{µ} = arg sup

µ I1(µ)

slide-24
SLIDE 24

WIAM 2016 Jorge De Anda Salazar Introduction

Applications Approaches Test Case

Variational Form

Postulates Constitutive model Kinematics Incremental VF

Numerical Im- plementation

Monolithic Staggered Comparison of results

Conclusion

Remarks Future Work 10/20

Incremental variational formulation (IVF)

I2(c, µ) =

  • τ

Ω

  • ˙

U − µ˙ c − χ(c, g) + µr

  • dV −
  • ∂jΩ

µjdS

  • dt

⇓ I2(c, µ) ≈

N−1

  • n=0

I n

2 (µn+1, cn+1; µn, cn)

I n

2 (µ, c) =

  • Ω
  • ∆U(c) − µ∆c − ∆tχ(

gn, cn+α)

  • dV

{cn+1, µn+1} = arg inf

c sup µ I n 2 (µ, c; µn, cn)

graphically...

slide-25
SLIDE 25

WIAM 2016 Jorge De Anda Salazar Introduction

Applications Approaches Test Case

Variational Form

Postulates Constitutive model Kinematics Incremental VF

Numerical Im- plementation

Monolithic Staggered Comparison of results

Conclusion

Remarks Future Work 11/20

Monolithic approach (Implicit-Explicit scheme)

Xi+1 = xi + δxi

Initial guess Initial guess (i=1) (i=1)

X Xi

i =x

=xn

n

Solve Ki δxi = Ri

Ri < Tol

Xn+1 = xi+1

i←i+1

Monolithic Approach

NO NO YES

1-Field form µ = µ(a)N(X) R = δI1 δµ K = δR δµ

REMARK: If U∗(µ) and χ(µ) are convex the tangent matrix is positive definite

slide-26
SLIDE 26

WIAM 2016 Jorge De Anda Salazar Introduction

Applications Approaches Test Case

Variational Form

Postulates Constitutive model Kinematics Incremental VF

Numerical Im- plementation

Monolithic Staggered Comparison of results

Conclusion

Remarks Future Work 12/20

Monolithic approach (Implicit-Explicit scheme)

Linear model

10 0 10 2 10 4 Time Step " t 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 Relative Error 7 Temporal Convergence Plot 512 256 128 64 32 16 8 10 -6 10 -4 10 -2 Spacing " x 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 Relative Error 7 Spatial Convergence Plot 1025 513 257 129 65 33 17 9 10 0 10 2 10 4 Time Step " t 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 Relative Error 7 Temporal Convergence Plot 8 16 32 64 128 256 512 10 -6 10 -4 10 -2 Spacing " x 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 Relative Error 7 Spatial Convergence Plot 9 17 33 65 129 257 513 1025

1st Order Element 2nd Order element

slide-27
SLIDE 27

WIAM 2016 Jorge De Anda Salazar Introduction

Applications Approaches Test Case

Variational Form

Postulates Constitutive model Kinematics Incremental VF

Numerical Im- plementation

Monolithic Staggered Comparison of results

Conclusion

Remarks Future Work 12/20

Monolithic approach (Implicit-Explicit scheme)

Fickean model

10 0 10 2 10 4 Time Step " t 10 -6 10 -4 10 -2 10 0 10 2 10 4 10 6 Relative Error 7 Temporal Convergence Plot 512 256 128 64 32 16 8 10 -6 10 -4 10 -2 Spacing " x 10 -6 10 -4 10 -2 10 0 10 2 10 4 10 6 Relative Error 7 Spatial Convergence Plot 1025 513 257 129 65 33 17 9 10 0 10 2 10 4 Time Step " t 10 -6 10 -5 10 -4 10 -3 Relative Error 7 Temporal Convergence Plot 8 16 32 64 128 256 512 10 -6 10 -4 10 -2 Spacing " x 10 -6 10 -5 10 -4 10 -3 Relative Error 7 Spatial Convergence Plot 9 17 33 65 129 257 513 1025

1st Order Element 2nd Order element

slide-28
SLIDE 28

WIAM 2016 Jorge De Anda Salazar Introduction

Applications Approaches Test Case

Variational Form

Postulates Constitutive model Kinematics Incremental VF

Numerical Im- plementation

Monolithic Staggered Comparison of results

Conclusion

Remarks Future Work 13/20

Monolithic approach (Implicit-Explicit scheme)

Xi+1 = xi + δxi Initial guess Initial guess (i=1) (i=1)

X Xi

i =x

=xn

n

Solve Ki δxi = Ri

Ri < Tol

Xn+1 = xi+1 i←i+1 Monolithic Approach NO NO YES

2-Field form µ = µ(a)N(X) c = c(a)N(X) Rµ = δI2 δµ Kµµ = δRµ δµ Rc = δI2 δc Kcc = δRc δc Kcµ = δRµ δc Kµc = Kcµ Kµµ Kµc Kcµ Kcc δµ δc

  • = −

Rµ Rc

slide-29
SLIDE 29

WIAM 2016 Jorge De Anda Salazar Introduction

Applications Approaches Test Case

Variational Form

Postulates Constitutive model Kinematics Incremental VF

Numerical Im- plementation

Monolithic Staggered Comparison of results

Conclusion

Remarks Future Work 14/20

Monolithic approach (Implicit-Explicit scheme)

Linear model

10 0 10 2 10 4 Time Step " t 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 Relative Error 7 Temporal Convergence Plot 512 256 128 64 32 16 8 10 -6 10 -4 10 -2 Spacing " x 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 Relative Error 7 Spatial Convergence Plot 1025 513 257 129 65 33 17 9 10 0 10 2 10 4 Time Step " t 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 Relative Error 7 Temporal Convergence Plot 8 16 32 64 128 256 512 10 -6 10 -4 10 -2 Spacing " x 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 Relative Error 7 Spatial Convergence Plot 9 17 33 65 129 257 513 1025

1st Order Element 2nd Order element

slide-30
SLIDE 30

WIAM 2016 Jorge De Anda Salazar Introduction

Applications Approaches Test Case

Variational Form

Postulates Constitutive model Kinematics Incremental VF

Numerical Im- plementation

Monolithic Staggered Comparison of results

Conclusion

Remarks Future Work 14/20

Monolithic approach (Implicit-Explicit scheme)

Fickean model

10 0 10 2 10 4 Time Step " t 10 -6 10 -5 10 -4 10 -3 Relative Error 7 Temporal Convergence Plot 512 256 128 64 32 16 8 10 -6 10 -4 10 -2 Spacing " x 10 -6 10 -5 10 -4 10 -3 Relative Error 7 Spatial Convergence Plot 1025 513 257 129 65 33 17 9 10 0 10 2 10 4 Time Step " t 10 -6 10 -5 10 -4 10 -3 Relative Error 7 Temporal Convergence Plot 8 16 32 64 128 256 512 10 -6 10 -4 10 -2 Spacing " x 10 -6 10 -5 10 -4 10 -3 Relative Error 7 Spatial Convergence Plot 9 17 33 65 129 257 513 1025

1st Order Element 2nd Order element

slide-31
SLIDE 31

WIAM 2016 Jorge De Anda Salazar Introduction

Applications Approaches Test Case

Variational Form

Postulates Constitutive model Kinematics Incremental VF

Numerical Im- plementation

Monolithic Staggered Comparison of results

Conclusion

Remarks Future Work 15/20

Global/Local approach (Implicit-Explicit scheme)

Xi+1 = xi + δxi Initial guess Initial guess (i=1) (i=1)

X Xi

i =x

=xn

n

y yi

i =y

=yn

n

Solve Ki δxi = Ri

Ri < Tol

Xn+1 = xi+1 i←i+1 Global/Local Approach NO NO YES

Update yi (Monolithic Approach) x : Global var y : Local var

2-Field form µ = µ(a)N(X) c = c(q)δ(X − Xp) Global : {µn+1} = arg sup

µ I n 2 (µ; ˇ

c) Local : {ˇ c} = arg inf

c I n 2 (µ, c)

slide-32
SLIDE 32

WIAM 2016 Jorge De Anda Salazar Introduction

Applications Approaches Test Case

Variational Form

Postulates Constitutive model Kinematics Incremental VF

Numerical Im- plementation

Monolithic Staggered Comparison of results

Conclusion

Remarks Future Work 16/20

Global/Local approach (Implicit-Explicit scheme)

Linear model

10 0 10 2 10 4 Time Step " t 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 Relative Error 7 Temporal Convergence Plot 512 256 128 64 32 16 8 10 -6 10 -4 10 -2 Spacing " x 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 Relative Error 7 Spatial Convergence Plot 1025 513 257 129 65 33 17 9 10 0 10 2 10 4 Time Step " t 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 Relative Error 7 Temporal Convergence Plot 8 16 32 64 128 256 512 10 -6 10 -4 10 -2 Spacing " x 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 Relative Error 7 Spatial Convergence Plot 9 17 33 65 129 257 513 1025

1st Order Element 2nd Order element

slide-33
SLIDE 33

WIAM 2016 Jorge De Anda Salazar Introduction

Applications Approaches Test Case

Variational Form

Postulates Constitutive model Kinematics Incremental VF

Numerical Im- plementation

Monolithic Staggered Comparison of results

Conclusion

Remarks Future Work 16/20

Global/Local approach (Implicit-Explicit scheme)

Fickean model

10 0 10 2 10 4 Time Step " t 10 -8 10 -7 10 -6 10 -5 Relative Error 7 Temporal Convergence Plot 512 256 128 64 32 16 8 10 -6 10 -4 10 -2 Spacing " x 10 -8 10 -7 10 -6 10 -5 Relative Error 7 Spatial Convergence Plot 1025 513 257 129 65 33 17 9 10 0 10 2 10 4 Time Step " t 10 -8 10 -7 10 -6 10 -5 Relative Error 7 Temporal Convergence Plot 8 16 32 64 128 256 512 10 -6 10 -4 10 -2 Spacing " x 10 -8 10 -7 10 -6 10 -5 Relative Error 7 Spatial Convergence Plot 9 17 33 65 129 257 513 1025

1st Order Element 2nd Order element

slide-34
SLIDE 34

WIAM 2016 Jorge De Anda Salazar Introduction

Applications Approaches Test Case

Variational Form

Postulates Constitutive model Kinematics Incremental VF

Numerical Im- plementation

Monolithic Staggered Comparison of results

Conclusion

Remarks Future Work 17/20

Comparison of results

Linear model Variables FE Order WF VF1s VFM VFGL slope 1 1.4981 1.4982 1.4978 1.4988 2 1.5330 1.5329 1.6485 1.5329 y- 1

  • 18.756
  • 18.754
  • 18.756
  • 18.756

intercept 2

  • 18.882
  • 18.882
  • 19.327
  • 18.882

Fickean model Variables FE Order WF VF1s VFM VFGL slope 1 x 0.8387 0.9550 1.3053 2 x 1.1411 1.1252 1.3186 y- 1 x

  • 11.6990
  • 12.0939
  • 16.5942

intercept 2 x

  • 12.7585
  • 13.1300
  • 16.9043

WF : Implicit scheme (Weak formulation) VF1s : Implicit-Explicit scheme - Monolithic approach (1-Field VF) VF1M : Implicit-Explicit scheme - Monolithic approach (2-Field VF) VF1GL : Implicit-Explicit scheme - Global/Local approach (2-Field VF)

slide-35
SLIDE 35

WIAM 2016 Jorge De Anda Salazar Introduction

Applications Approaches Test Case

Variational Form

Postulates Constitutive model Kinematics Incremental VF

Numerical Im- plementation

Monolithic Staggered Comparison of results

Conclusion

Remarks Future Work 18/20

Conclusions

  • Validation of VF for transient chemical diffusion.
  • VF is equivalent to SF and WF.
  • No post-process relation between variables.
  • Definition of Incremental VF.
  • Solution → Optimization process.
  • Variety of algorithms to implement.
  • Different perspective for mathematical analysis
  • Easier to coupled with different physics.
slide-36
SLIDE 36

WIAM 2016 Jorge De Anda Salazar Introduction

Applications Approaches Test Case

Variational Form

Postulates Constitutive model Kinematics Incremental VF

Numerical Im- plementation

Monolithic Staggered Comparison of results

Conclusion

Remarks Future Work 19/20

Work in progress

Chemo-Mechanical coupling { u, µ, c} = arg inf

  • u sup

µ inf c I(

u, µ, c)

0.2 0.4 0.6 0.8 1 + [mts] #10!3

  • 1.76
  • 1.758
  • 1.756
  • 1.754
  • 1.752
  • 1.75

7 #105

Chemical Potential

0.2 0.4 0.6 0.8 1 + [mts] #10!3 7000 8000 9000 10000 11000 c [mol/m

3] Concentration

0.2 0.4 0.6 0.8 1 + [mts] #10!3 0.2 0.4 0.6 0.8 1 u [mts] #10!4

Displacement

0.2 0.4 0.6 0.8 1 + [meters] #10!3

  • 1.9
  • 1.8
  • 1.7
  • 1.6
  • 1.5
  • 1.4
  • 1.3

e [J/m 3] #109

Free Energy

LIMITING CASE: Diffusion → Mechanics

slide-37
SLIDE 37

WIAM 2016 Jorge De Anda Salazar Introduction

Applications Approaches Test Case

Variational Form

Postulates Constitutive model Kinematics Incremental VF

Numerical Im- plementation

Monolithic Staggered Comparison of results

Conclusion

Remarks Future Work 19/20

Work in progress

Chemo-Mechanical coupling { u, µ, c} = arg inf

  • u sup

µ inf c I(

u, µ, c)

0.5 1 1.5 c #104

  • 1.76
  • 1.758
  • 1.756
  • 1.754
  • 1.752
  • 1.75

7 #105

Ch-Phase Space

0.5 1 1.5 c #104 2 4 6 8 u #10!5

Mec-Phase Space

  • 1.76
  • 1.758
  • 1.756
  • 1.754
  • 1.752
  • 1.75

7 #105 2 4 6 8 u #10!5

Cp-Phase Space

  • 1.75

0.2 0.4 1.06 u #10 -4

  • 1.7505

0.6 1.05 #10 5 7 0.8 1.04 #10 c 1

  • 1.751

1.03 1.02

  • 1.7515

1.01

LIMITING CASE: Diffusion → Mechanics

slide-38
SLIDE 38

WIAM 2016 Jorge De Anda Salazar Introduction

Applications Approaches Test Case

Variational Form

Postulates Constitutive model Kinematics Incremental VF

Numerical Im- plementation

Monolithic Staggered Comparison of results

Conclusion

Remarks Future Work 19/20

Work in progress

Chemo-Mechanical coupling { u, µ, c} = arg inf

  • u sup

µ inf c I(

u, µ, c)

0.5 1 + [mts] #10!3

  • 1.75
  • 1.74999999999
  • 1.74999999998
  • 1.74999999997
  • 1.74999999996

7 #105

Chemical Potential 0.5 1

+ [mts] #10!3 1.1249 1.12495 1.125 1.12505 1.1251 1.12515 c [mol/m

3]

#104 Concentration 0.5 1 + [mts] #10!3

  • 5
  • 4
  • 3
  • 2
  • 1

u [mts] #10!6

Displacement 0.5 1

+ [meters] #10!3

  • 1.9687606793785
  • 1.968760679378
  • 1.9687606793775
  • 1.968760679377
  • 1.9687606793765
  • 1.968760679376

e [J/m 3] #109

Free Energy

LIMITING CASE: Mechanics → Diffusion

slide-39
SLIDE 39

WIAM 2016 Jorge De Anda Salazar Introduction

Applications Approaches Test Case

Variational Form

Postulates Constitutive model Kinematics Incremental VF

Numerical Im- plementation

Monolithic Staggered Comparison of results

Conclusion

Remarks Future Work 19/20

Work in progress

Chemo-Mechanical coupling { u, µ, c} = arg inf

  • u sup

µ inf c I(

u, µ, c)

1.1248 1.1249 1.125 1.1251 1.1252 c #104

  • 1.75
  • 1.74999999999
  • 1.74999999998
  • 1.74999999997
  • 1.74999999996

7 #105

Ch-Phase Space 1.1248 1.1249 1.125 1.1251 1.1252

c #104

  • 6
  • 5
  • 4
  • 3
  • 2
  • 1

u #10!6

Mec-Phase Space

  • 1.75
  • 1.74999999998 -1.74999999996

7 #105

  • 6
  • 5
  • 4
  • 3
  • 2
  • 1

u #10!6

Cp-Phase Space

1.1249 1.125 1.1251 1.1252 c #10

  • 1.749999999999
  • 1.749999999998
  • 1.749999999997
  • 1.749999999996
  • 1.749999999995

7 #10 5

  • 5
  • 4
  • 3
  • 2
  • 1

u #10 -6

LIMITING CASE: Mechanics → Diffusion

slide-40
SLIDE 40

WIAM 2016 Jorge De Anda Salazar Introduction

Applications Approaches Test Case

Variational Form

Postulates Constitutive model Kinematics Incremental VF

Numerical Im- plementation

Monolithic Staggered Comparison of results

Conclusion

Remarks Future Work 20/20

Questions