Materials Process Design and Control Laboratory

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STOCHASTIC VARIATIONAL MULTISCALE METHOD FOR
ELLIPTIC EQUATIONS WITH MULTISCALE COEFFICIENTS
BADRI VELAMUR ASOKAN and NICHOLAS ZABARAS
Materials Process Design and Control
Laboratory Sibley School of Mechanical and
Aerospace Engineering169 Frank H. T. Rhodes
Hall Cornell University Ithaca, NY
14853-3801 Email zabaras_at_cornell.edu,
bnv2_at_cornell.edu URL http//mpdc.mae.cornell.edu/

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OUTLINE

• Current techniques for multiscale elliptic
  equations
• Variational multiscale VMS method
• Generalized polynomial chaos approach
• Deterministic VMS modeling of multiscale
  elliptic equation
• Issues in extension of approach to stochastic
  elliptic equation
• Presentation of subgrid problems
• Numerical examples
• Extensions to practical systems A brief
  discussion

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CURRENT TECHNIQUES

• Stochastic VMS Zabaras et al. JCP 208(1), 2005
• Residual-Free bubbles Sangalli, SIAM MMS 1(3),
  2003
• Adaptive variational multiscale method Larson,
  Chalmers finite element preprints 2004-18,
  2004-11
• Variational multiscale method Arbogast, SIAM
  J.Num.Anal 42, 2004, Arbogast, SPE J., Dec
  2002
• Multiscale finite elements Hou, JCP 134, 1997,
  Hou, JCP, 2005
• Heterogeneous finite element method Xu, J. Am.
  Math, 2003
• Homogenization and allied techniques

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MODEL MULTISCALE ELLIPTIC EQUATION
Boundary
in
on
Domain

• Multiple scale variations in K
• K is inherently random property predictions are
  at best statistical

Crystal microstructures

• Composites
• Diffusion processes

Permeability of Upper Ness formation
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STOCHASTIC PROCESSES AS FUNCTIONS

• A probability space is a triple comprising of
  collection of basis outcomes , permutation
  of these outcomes and a probability measure

• A real-valued random variable is a function that
  maps the probability space to a real line
  regions in go to intervals in the real line

• Random variable

• A space-time stochastic process is can be
  represented as

other regularity conditions
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SERIES REPRESENTATION CONTD

• Karhunen-Loeve

ON random variables
Mean function
Stochastic process
Deterministic functions

• Generalized polynomial chaos

Stochastic input
Askey polynomials in input
Stochastic process
Deterministic functions
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STOCHASTIC VMS MODELING
in
on

• K is spatially rapidly varying stochastic
  process a multiscale diffusion coefficient

such that, for all
V Find
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VARIATIONAL MULTISCALE METHOD

• Hypothesis Actual solution is a sum of coarse
  scale resolved part and a subgrid scale
  unresolved part Hughes, 95
• This induces a similar decomposition of the
  governing equation into coarse and subgrid parts

• Idea Approximately solve the subgrid equations
  and include the effect on coarse scale equation
• Highly successful with advection-diffusion
  problems, fluid-flow, micromechanics and other

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VMS VARIATIONAL FORMULATION
such that, for all
V Find

• V denotes the full variational formulation
• U and V denote appropriate function spaces for
  the multiscale solution u and test function v
  respectively

• VMS hypothesis
• Induced function space decomposition Hughes
  1995

Exact coarse fine
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VMS COARSE AND SUBGRID SCALES
Using
and the induced function space
decomposition
Find
such that, for all
and
and
Coarse V
Subgrid V

• Solve subgrid V using Greens' functions, PU
  and other
• Substitute the subgrid solution in coarse V

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DEFINITIONS AND DISCRETIZATION

• Since the subgrid solution uF represents rapid
  variations, more terms in GPCE is required
• Let us now split the subgrid solution into two
  parts defined by the following equations

Find
such that, for all
and
and
Subgrid V
Homogeneous V
Affine V
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SOLUTION OF HOMOGENEOUS V
Homogeneous V

• This equation yields an approach similar to
  MsFEM technique for solving multiscale elliptic
  equations Hou et al.
• By examination, is a map of the coarse
  solution on the subgrid scale
• Since, represents subgrid variations, a
  higher order GPCE is used (leading to more terms
  in stochastic series expansion)

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FINITE ELEMENT DISCRETIZATION

• Subgrid mesh
• NelF elements
• Associated with each element sub-domain in the
  coarse mesh

• Coarse mesh
• NelC elements

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DEFINITIONS AND DISCRETIZATION

• Assume a finite element discretization of the
  spatial region D into NelC coarse elements
• Each coarse element is further discretized by a
  subgrid mesh with NelF elements
• In each coarse element, the coarse solution uC
  can be approximated as

nbf number of spatial finite element basis
functions in each coarse element PC number of
terms in the GPCE of coarse solution
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SOLUTION OF HOMOGENEOUS V CONTD

• Considering the following finite element GPCE
  representation for coarse solution uC

• The subgrid solution can be represented as
  follows

• Since represents subgrid variations, a
  nonlinear coarse scale mapping, a higher order
  GPCE is used implies more terms in GPCE of

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SOLUTION OF HOMOGENEOUS V CONTD

• Now, we have

• Thus, we end up with Nmax homogeneous subgrid
  problems in each coarse element D(e)
• Following representation is used for
  approximation

nbf number of spatial finite element basis
functions in each element defined on the subgrid
mesh PF number of terms in
the GPCE. Also, PF gtPC
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HOMOGENEOUS V BOUNDARY CONDITIONS
Coarse element D(e)

• Also, reduced problems are solved on element
  sides for obtaining oscillatory boundary
  conditions

Subgrid mesh
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SOLVING REDUCED PROBLEM

• Each coarse element edge is mapped to a line
  grid
• Line grid yields coordinates (s along the line
  grid, n normal to line grid)
• The reduced problem specified below is solved on
  the line grid

Mapping element edge
Subgrid mesh
s
n
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FEM FOR HOMOGENEOUS V

• Thus in each coarse element EC, we can solve for
  the subgrid basis functions as follows

• Note that we solve for the sum of coarse
  subgrid basis functions
• The boundary conditions for this equation are
  obtained as the solution of the reduced problem
  on coarse element edges
• DOF for the problem (Nno-subgrid)(PF1)

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SOLUTION OF AFFINE V
Affine V

• This affine correction is unique to the VMS
  formulation Arbogast et al. and is not obtained
  in MsFEM type of formulations
• Crucial in case of localized sources and sinks
• Again, similar to the homogeneous V, we have

• This affine correction solution has no
  dependence on coarse scale behavior
• We solve this equation on each coarse element
  with zero boundary conditions

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DESCRIPTION OF NUMERICAL PROBLEMS
Coarse element D(e)

• Based on boundary conditions used and subgrid
  problems solved, we have three studies

Subgrid mesh
VMS-Os
MsFEM-L
MsFEM-Os

• Reduced solution as BC
• No affine correction

• Reduced solution as BC
• Affine correction explicitly solved

• Linear boundary conditions are used
• No affine correction

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NUMERICAL EXAMPLES

• Deterministic studies
• Case 1 Periodic media single fast scale
  separation
• Case 2 non-periodic media multiple scales
• Stochastic studies
• Case 1 Pseudo-periodic media
• Effect of PF -PC difference
• Future studies and research directions

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DETERMINISTIC CASE I PERIODIC MEDIA

• A 128x128 mesh for complete resolution
• 4-noded bilinear quads for FEM

K periodic
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DETERMINISTIC CASE I RESULTS
Resolved FEM
MsFEM-L

• coarse 8x8 subgrid 16x16 mesh
• VMS yields consistent low error values

MsFEM-Os
VMS-Os
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DETERMINISTIC CASE II NON-PERIODIC MEDIA

• Presence of multiple spatial scales
• non-periodic spatial variation
• vertices v1(0,0) v2(1,0) v3(0,1) v4(1,1)

K non-periodic
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DETERMINISTIC CASE II NON-PERIODIC MEDIA

• A 512x512 mesh for complete resolution
• 4-noded bilinear quads for FEM

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DETERMINISTIC CASE II RESULTS
Resolved FEM
MsFEM-L

• coarse 16x16 subgrid 32x32 mesh
• VMS yields consistent low error values

MsFEM-Os
VMS-Os
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STOCHASTIC CASE I PSEUDO-PERIODIC MEDIA

• uniformly distributed diffusion coefficient

K0
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DETERMINISTIC CASE II NON-PERIODIC MEDIA

• A 512x512 mesh for complete resolution
• 4-noded bilinear quads for FEM
• MsFEM-L is not attempted owing to superior
  performance of MsFEM-Os and VMS-Os for
  deterministic studies

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STOCHASTIC CASE I RESULTS
VMS-Os
FEM
MsFEM-Os
U0
U1
U2
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STOCHASTIC CASE I ERROR MEASURES

• L-inf error was calculated on the mean value
• Again, VMS is consistently better than MsFEM-Os.

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STOCHASTIC CASE I EFFECT OF PC TERMS

• We have assumed that the fine scale solution has
  more PC terms in its expansion
• While reconstructing the fully resolved solution
  from the fine scale solution, we can only
  reconstruct up to the PCC terms. Beyond those
  terms, the fine scale solution is no longer a
  one-to-one map, hence, we see abnormalities
  (still equal in L-2)

L2 error
Linf error
PF-PC
PF-PC
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PRESENT AND FUTURE

• Are currently applying VMS to transient
  diffusion problems in heterogeneous media
• Applying VMS for solving convection-diffusion in
  random media
• Implementation of over-sampling method in the
  context of VMS
• Implementation of support-space techniques with
  VMS hypothesis applied in sample space spatial
  and stochastic VMS
• Adaptive generation and solution of subgrid
  problems, specifically in convection-diffusion
  applications