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) Numerical Im- Technische Universit¨ at M¨ unchen, Germany (TUM) plementation Monolithic Staggered Sep 1st, 2016 Comparison of results Conclusion Remarks Future Work 1/20
WIAM 2016 Jorge De Anda Salazar Introduction Incremental VF 1 Introduction Applications Approaches 3 Numerical Implementation Applications Test Case Monolithic Variational Approaches Form Staggered Test Case Postulates Constitutive model Comparison of results 2 Variational Form Kinematics Incremental VF Postulates 4 Conclusion Numerical Im- Constitutive model Remarks plementation Monolithic Kinematics Future Work Staggered Comparison of results Conclusion Remarks Future Work 2/20
Research topic WIAM 2016 Jorge De Anda Salazar Introduction Development of algorithmic strategies for numerical Applications Approaches simulation of coupled mechanical-diffusion problems Test Case Variational Form Applications Postulates Constitutive Reactive flows model Kinematics in solids Incremental VF Numerical Im- Tissue plementation Monolithic diagnosis and Staggered Comparison of reconstruction results Conclusion Remarks Future Work 3/20
Research topic WIAM 2016 Jorge De Anda Salazar Development of algorithmic strategies for numerical simulation of coupled mechanical-diffusion problems Introduction Applications Approaches Test Case Variational Applications Form Postulates Reactive flows Constitutive model Kinematics in solids Incremental VF Numerical Im- Tissue plementation diagnosis and Monolithic Staggered reconstruction Comparison of results 1 Conclusion Remarks Future Work 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 3/20
Research topic WIAM 2016 Jorge De Anda Salazar Development of algorithmic strategies for numerical simulation of coupled mechanical-diffusion problems Introduction Applications Approaches Test Case Variational Applications Form Postulates Reactive flows Constitutive model Kinematics in solids Incremental VF Numerical Im- Tissue plementation diagnosis and Monolithic Staggered reconstruction Comparison of results 1 Conclusion Remarks Future Work 1 By User:Paulnasca - Own work, CC BY 2.0, https://commons.wikimedia.org/w/index.php?curid=7128816 3/20
Research topic WIAM 2016 Jorge De Anda Salazar Development of algorithmic strategies for numerical simulation of coupled mechanical-diffusion problems Introduction Applications Approaches Test Case Variational Applications Form Postulates Reactive flows Constitutive model Kinematics in solids Incremental VF Numerical Im- Tissue plementation diagnosis and Monolithic Staggered reconstruction Comparison of results 1 Conclusion Remarks Future Work 1 By - Carl Fredrik (CFCF)- Own work, CC BY 3.0, https://commons.wikimedia.org/wiki/File:716 Intervertebral Disk.jpg 3/20
Transient Chemical diffusion WIAM 2016 Jorge De Anda Salazar Introduction Applications Approaches Variables Test Case Variational Form � Postulates c : concentration [ mol / m 3 ] j : molar flux [ mol / m 2 s ] Constitutive model D : diffusivity [ m 2 / s ] µ : chemical potential [ J / mol ] Kinematics Incremental VF Numerical Im- plementation Monolithic Staggered Comparison of results Conclusion Remarks Future Work 4/20
Transient Chemical diffusion WIAM 2016 Jorge De Anda Salazar Introduction Applications Approaches Test Case 2 3 Variational Form Postulates Variables Constitutive model Kinematics Incremental VF � c : concentration [ mol / m 3 ] j : molar flux [ mol / m 2 s ] Numerical Im- D : diffusivity [ m 2 / s ] µ : chemical potential [ J / mol ] plementation Monolithic Staggered Comparison of results Conclusion Remarks Future Work 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 4/20
Different approaches WIAM 2016 Jorge De Anda Salazar Introduction Applications Approaches Test Case • Strong Form Variational Form • Weak form Postulates Constitutive • Variational form model Kinematics Incremental VF Numerical Im- plementation Monolithic Staggered Comparison of results Conclusion Remarks Future Work 5/20
Different approaches WIAM 2016 Jorge De Anda Salazar Introduction Applications • Strong Form (Fick’s laws) Approaches Test Case Variational Form 1st law: ∂ c ∂ t = −∇ · � 2nd law: � Postulates j = − D ∇ c j Constitutive model Kinematics Incremental VF • Weak form Numerical Im- • Variational form plementation Monolithic Staggered Comparison of results Conclusion Remarks Future Work 5/20
Different approaches WIAM 2016 Jorge De Anda Salazar • Strong Form (Fick’s laws) Introduction Applications Approaches 1st law: ∂ c Test Case ∂ t = −∇ · � 2nd law: � j = − D ∇ c j Variational Form Postulates • Weak form Constitutive model Kinematics Incremental VF Numerical Im- � � ∂ c � plementation ∂ t δ c + D ∇ c · ∇ δ c dx = 0 Monolithic Staggered Ω Comparison of results • Variational form Conclusion Remarks Future Work 5/20
Different approaches WIAM 2016 • Strong Form (Fick’s laws) Jorge De Anda Salazar 1st law: ∂ c Introduction ∂ t = −∇ · � 2nd law: � j = − D ∇ c j Applications Approaches Test Case • Weak form Variational Form Postulates Constitutive model � � ∂ c � Kinematics ∂ t δ c + D ∇ c · ∇ δ c dx = 0 Incremental VF Ω Numerical Im- plementation Monolithic • Variational form Staggered Comparison of results Conclusion Remarks { c } = arg inf c I ( c ) Future Work 5/20
Different approaches • Strong Form (Fick’s laws) WIAM 2016 Jorge De Anda Salazar 1st law: ∂ c ∂ t = −∇ · � 2nd law: � j j = − D ∇ c Introduction Applications Approaches Test Case • Weak form Variational Form Postulates � � ∂ c � Constitutive model ∂ t δ c + D ∇ c · ∇ δ c dx = 0 Kinematics Incremental VF Ω Numerical Im- plementation • Variational form Monolithic Staggered Comparison of results Conclusion { c } = arg inf c I ( c ) Remarks Future Work Variational Form ⇐ ⇒ Weak Form ⇐ ⇒ Strong Form 5/20
Different approaches • Strong Form (Fick’s laws) WIAM 2016 Jorge De Anda Salazar 1st law: ∂ c ∂ t = −∇ · � 2nd law: � j = − D ∇ c j Introduction Applications Approaches Test Case • Weak form Variational Form Postulates � ∂ c � � Constitutive model ∂ t δ c + D ∇ c · ∇ δ c dx = 0 Kinematics Incremental VF Ω Numerical Im- plementation • Variational form Monolithic Staggered Comparison of results Conclusion { c , µ } = arg inf c sup µ I ( c , µ ) Remarks Future Work Variational Form ⇐ ⇒ Weak Form ⇐ ⇒ Strong Form 5/20
Different approaches • Strong Form (Fick’s laws) WIAM 2016 Jorge De Anda Salazar 1st law: ∂ c ∂ t = −∇ · � 2nd law: � j j = − D ∇ c Introduction Applications Approaches • Weak form Test Case Variational Form � ∂ c � � Postulates ∂ t δ c + D ∇ c · ∇ δ c dx = 0 Constitutive model Ω Kinematics Incremental VF • Variational form Numerical Im- plementation Monolithic Staggered { c , µ } = arg inf c sup µ I ( c , µ ) Comparison of results Conclusion Chemistry Thermodynamics Mechanics Electricity Remarks Future Work c S ǫ q µ σ T V 5/20
Test Case: Initial discontinuous concentration WIAM 2016 Boundary conditions Initial conditions Jorge De Anda Salazar c = c ext ∀ x = L , t > 0 Introduction IC : c ( x , t = 0) = c 0 BC : ∂ c Applications Approaches ∂ x = 0 ∀ x = 0 , t > 0 c 0 � = c ext Test Case Variational Exact Solution Form Postulates # 10 5 Chemical Potential Concentration -1.74 12000 Constitutive 10000 -1.76 model 3 ] 8000 Kinematics -1.78 c [mol/m 6000 Incremental VF -1.8 7 4000 Numerical Im- -1.82 2000 plementation -1.84 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 # 10 ! 3 # 10 ! 3 Monolithic + [meters] + [meters] Staggered # 10 9 Internal Energy # 10 5 Phase Space 0 -1.75 Comparison of results -0.5 e [J/m ] Conclusion -1 7 -1.8 Remarks -1.5 Future Work -2 -1.85 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 # 10 ! 3 # 10 4 + [meters] c 6/20
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