A Finite Volume Approach to Multiscale Elasticity Paul Delgado NSF - - PowerPoint PPT Presentation

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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

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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

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Inspiration

William Kamkwamba, South Africa

A Finite Volume Approach to Multiscale Elasticity

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Definition

Poroelasticity

A Finite Volume Approach to Multiscale Elasticity

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Applications

Fluid flow affects solid deformation!

A Finite Volume Approach to Multiscale Elasticity

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The Challenge

Large variations in material parameters over small spatial scales.

A Finite Volume Approach to Multiscale Elasticity

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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

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The Curse of Dimensionality

Assuming 103 nodes per µm, 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

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Conventional Methods

How can we balance the need for accuracy with the need for efficiency?

A Finite Volume Approach to Multiscale Elasticity

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Multiscale Method

A Finite Volume Approach to Multiscale Elasticity

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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

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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

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Solid Equation

−∇ · σ = F in Ω σ = σ(ǫ) ǫ = ǫ(∇ u) ∇ u =

  • ux

uy vx vy

  • u = d(x, y) on ∂Ωd

σ · n = t(x, y) on ∂Ωt Momentum balance relates stress to displacement

A Finite Volume Approach to Multiscale Elasticity

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Methodology

Heterogeneous Multiscale Framework (E & Engquist 2003).

A Finite Volume Approach to Multiscale Elasticity

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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

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Macro Model

Incomplete Finite Volume Method −

  • ∂CV

σ · n =

  • CV
  • F

−

  • CV E σx +
  • CV W σx −
  • CV N τxy +
  • CV S τxy =
  • CV

f (1) −

  • CV N σy +
  • CV S σy −
  • CV E τxy +
  • CV W τxy =
  • CV

g (2)

◮ σ =

  • σx

τxy τxy σy

  • ◮ No explicit constitutive relation σ = σ(ǫ(∇

u))

A Finite Volume Approach to Multiscale Elasticity

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Micro Model

Linear Heterogeneous Isotropic Model ∇ · σ + F = 0 σ(ǫ) =

  • σx

τxy τxy σy

  • = 2µ(

x)ǫ + λ( x)tr(ǫ)I ǫ(∇ u) = 1 2

  • ∇

u + ∇ uT ∇ u = ux uy vx vy

  • Other models are also possible (molecular dynamics, lattice

structures, etc...)

A Finite Volume Approach to Multiscale Elasticity

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Step 1: Initial Guess

Old field variables (uij, vij)K

A Finite Volume Approach to Multiscale Elasticity

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Step 2: Loop

For each control volume boundary D

A Finite Volume Approach to Multiscale Elasticity

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Step 3: Sample

Micro data near CV D boundary midpoint

A Finite Volume Approach to Multiscale Elasticity

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Step 4: Constraint Projection

Interpolate BC’s from local macro field (uij, vij)K

A Finite Volume Approach to Multiscale Elasticity

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Step 5: Solve Micromodel

Obtain local micro field in BD

δ

A Finite Volume Approach to Multiscale Elasticity

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Step 6: Data Estimation (1)

Calculate total normal & shear force along mid cross-section.

A Finite Volume Approach to Multiscale Elasticity

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Step 6: Data Estimation (2)

Rescale total forces to entire control volume boundary D

A Finite Volume Approach to Multiscale Elasticity

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Step 7: Solve Macro Model

Obtain updated field variables (uij, vij)K+1

A Finite Volume Approach to Multiscale Elasticity

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Key to Micro-Macro Iterations

◮ Assume σ = 0 when ∇

u = 0.

◮ Assume ux, uy, vx and vy are independent variables. ◮ Taylor series expansion of σy, σx, and τxy ◮ Fixed Point Iteration over K

Then

  • ∂CV D νK+1

=

4

  • i=1
  • ∂CV D νD,K

GD,K

i

  • GD,K

i

· ei GD,K+1

i

· ei (3)

◮ Stress Component: ν = σy, σx, and τxy ◮ Boundary: D = N, S, E, W ◮ Subgradient:1 GD,K

i

≡ vec

  • ∇

uD,K

  • ei (i=1,...,4)

1(◦)denotes the Hadamard Product and ei denotes standard basis in R4

A Finite Volume Approach to Multiscale Elasticity

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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

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Results - Analytical Case

Displacement u = v = sin( πx

2 )sin( πy 2 ), λ = µ = 11 + sin(2πx)sin(2πy)

A Finite Volume Approach to Multiscale Elasticity

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Results - Analytical Case

Normal Stress

A Finite Volume Approach to Multiscale Elasticity

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Results - Analytical Case

Shear Stress

A Finite Volume Approach to Multiscale Elasticity

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Results - Analytical Case

Convergence

A Finite Volume Approach to Multiscale Elasticity

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Results - Random Case

Displacement

A Finite Volume Approach to Multiscale Elasticity

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Results - Random Case

Normal Stress

A Finite Volume Approach to Multiscale Elasticity

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Results - Random Case

Shear Stress

A Finite Volume Approach to Multiscale Elasticity

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Results - Random Case

Convergence

A Finite Volume Approach to Multiscale Elasticity

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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

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Future Work

◮ Multiphysics Simulation ◮ Parallelization ◮ Improve stress estimation ◮ Test with other micromodels

A Finite Volume Approach to Multiscale Elasticity