A Finite Volume Approach to Multiscale Elasticity Paul Delgado NSF Fellow (HRD-1139929) Doctoral Candidate - Computational Science University of Texas at El Paso November 1, 2014 A Finite Volume Approach to Multiscale Elasticity
Inspiration William Kamkwamba, South Africa A Finite Volume Approach to Multiscale Elasticity
Definition Poroelasticity A Finite Volume Approach to Multiscale Elasticity
Applications Fluid flow affects solid deformation! A Finite Volume Approach to Multiscale Elasticity
The Challenge Large variations in material parameters over small spatial scales. A Finite Volume Approach to Multiscale Elasticity
The Goldilocks Problem Assume ǫ << | Ω | ◮ If h > ǫ , then simulation is fast, but highly inaccurate. ◮ If h < ǫ , then simulation is accurate, but extremely slow. A Finite Volume Approach to Multiscale Elasticity
The Curse of Dimensionality Assuming 10 3 nodes per µ m , a Petascale computer solves the equations in ◮ In 2D ⇒≈ 3 , 000 yrs ◮ In 3D ⇒≈ 31 quadrillion yrs Moral: Parallelization, alone, will not solve this problem!!! A Finite Volume Approach to Multiscale Elasticity
Conventional Methods How can we balance the need for accuracy with the need for efficiency? A Finite Volume Approach to Multiscale Elasticity
Multiscale Method A Finite Volume Approach to Multiscale Elasticity
Our approach Toward a multiscale method for poroelasticity ◮ Decouple solid & fluid equations ◮ Develop multiscale methods for each equation A Finite Volume Approach to Multiscale Elasticity
Progress ◮ Developed & Verified Operator Splitting Method ◮ Developed 1D Multiscale Flow & Deformation Methods ◮ Improved methods for neumann conditions & source terms ◮ Higher Dimensional method for Fluid Flow Today, we demonstrate our multiscale method for the solid equation in higher dimensions A Finite Volume Approach to Multiscale Elasticity
Solid Equation −∇ · σ = � F in Ω σ = σ ( ǫ ) ǫ = ǫ ( ∇ � u ) � � u x u y ∇ � u = v x v y u = d ( x , y ) on ∂ Ω d σ · n = t ( x , y ) on ∂ Ω t Momentum balance relates stress to displacement A Finite Volume Approach to Multiscale Elasticity
Methodology Heterogeneous Multiscale Framework (E & Engquist 2003). A Finite Volume Approach to Multiscale Elasticity
Key Idea Microgrid Macrogrid ◮ A fully coupled microscopic model on the entire computational domain Ω ◮ Seek a solution at a small subset of the microgrid. ◮ Key to Efficiency: Use less info than what is available! A Finite Volume Approach to Multiscale Elasticity
Macro Model Incomplete Finite Volume Method � � � σ · � − n = F ∂ CV CV � � � � � − CV E σ x + CV W σ x − CV N τ xy + CV S τ xy = f (1) CV � � � � � − CV N σ y + CV S σ y − CV E τ xy + CV W τ xy = g (2) CV � � σ x τ x y ◮ σ = τ xy σ y ◮ No explicit constitutive relation σ = σ ( ǫ ( ∇ � u )) A Finite Volume Approach to Multiscale Elasticity
Micro Model Linear Heterogeneous Isotropic Model ∇ · σ + � F = 0 � � σ x τ xy σ ( ǫ ) = = 2 µ ( � x ) ǫ + λ ( � x ) tr ( ǫ ) I τ xy σ y u ) = 1 � u T � ǫ ( ∇ � ∇ � u + ∇ � 2 � u x � u y ∇ � u = v x v y Other models are also possible (molecular dynamics, lattice structures, etc...) A Finite Volume Approach to Multiscale Elasticity
Step 1: Initial Guess Old field variables ( u ij , v ij ) K A Finite Volume Approach to Multiscale Elasticity
Step 2: Loop For each control volume boundary D A Finite Volume Approach to Multiscale Elasticity
Step 3: Sample Micro data near CV D boundary midpoint A Finite Volume Approach to Multiscale Elasticity
Step 4: Constraint Projection Interpolate BC’s from local macro field ( u ij , v ij ) K A Finite Volume Approach to Multiscale Elasticity
Step 5: Solve Micromodel Obtain local micro field in B D δ A Finite Volume Approach to Multiscale Elasticity
Step 6: Data Estimation (1) Calculate total normal & shear force along mid cross-section. A Finite Volume Approach to Multiscale Elasticity
Step 6: Data Estimation (2) Rescale total forces to entire control volume boundary D A Finite Volume Approach to Multiscale Elasticity
Step 7: Solve Macro Model Obtain updated field variables ( u ij , v ij ) K + 1 A Finite Volume Approach to Multiscale Elasticity
Key to Micro-Macro Iterations ◮ Assume σ = 0 when ∇ � u = 0. ◮ Assume u x , u y , v x and v y are independent variables . ◮ Taylor series expansion of σ y , σ x , and τ xy ◮ Fixed Point Iteration over K Then ∂ CV D ν D , K � � G D , K � 4 � i ∂ CV D ν K + 1 � G D , K + 1 · e i (3) = i G D , K · e i i = 1 i ◮ Stress Component: ν = σ y , σ x , and τ xy ◮ Boundary: D = N , S , E , W ◮ Subgradient: 1 G D , K u D , K � � ∇ � ≡ vec ◦ e i (i=1,...,4) i 1 ( ◦ ) denotes the Hadamard Product and e i denotes standard basis in R 4 A Finite Volume Approach to Multiscale Elasticity
Numerical Experiments Unit Square Domain Ω = [ 0 , 1 ] 2 Cases w/ Analytical Solutions ◮ Prescribed displacement u , v functions ◮ Smooth material functions λ, µ ◮ Derived source terms f , g Cases w/o Analytical Solutions ◮ Random material parameters λ, µ ◮ Prescribed source terms f , g ◮ Reference Solution obtained numerically Analyze convergence as the total sampling area → | Ω | A Finite Volume Approach to Multiscale Elasticity
Results - Analytical Case Displacement 2 ) sin ( π y u = v = sin ( π x 2 ) , λ = µ = 11 + sin ( 2 π x ) sin ( 2 π y ) A Finite Volume Approach to Multiscale Elasticity
Results - Analytical Case Normal Stress A Finite Volume Approach to Multiscale Elasticity
Results - Analytical Case Shear Stress A Finite Volume Approach to Multiscale Elasticity
Results - Analytical Case Convergence A Finite Volume Approach to Multiscale Elasticity
Results - Random Case Displacement A Finite Volume Approach to Multiscale Elasticity
Results - Random Case Normal Stress A Finite Volume Approach to Multiscale Elasticity
Results - Random Case Shear Stress A Finite Volume Approach to Multiscale Elasticity
Results - Random Case Convergence A Finite Volume Approach to Multiscale Elasticity
Conclusions ◮ Our method fails as a general purpose PDE solver ◮ Works best in the worse case scenario: random heterogeneity ◮ Displacement is well approximated, but not stress. ◮ Algorithm is highly parallelizable ◮ Results are consistent with other implementations of HMM. A Finite Volume Approach to Multiscale Elasticity
Future Work ◮ Multiphysics Simulation ◮ Parallelization ◮ Improve stress estimation ◮ Test with other micromodels A Finite Volume Approach to Multiscale Elasticity
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