Materials Process Design and Control Laboratory Sibley School of Mechanical and Aerospace Engineering 169 Frank H. T. Rhodes Hall Cornell University Ithaca, NY URL: BADRI VELAMUR ASOKAN and NICHOLAS ZABARAS STOCHASTIC VARIATIONAL MULTISCALE METHOD FOR ELLIPTIC EQUATIONS WITH MULTISCALE COEFFICIENTS

Materials Process Design and Control Laboratory 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

Materials Process Design and Control Laboratory 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 , ]  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

Materials Process Design and Control Laboratory MODEL MULTISCALE ELLIPTIC EQUATION Permeability of Upper Ness formation Domain Boundary in on  Multiple scale variations in K  K is inherently random [property predictions are at best statistical] Crystal microstructures  Composites  Diffusion processes

Materials Process Design and Control Laboratory 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

Materials Process Design and Control Laboratory SERIES REPRESENTATION [CONTD]  Karhunen-Loeve Stochastic process Mean function ON random variables Deterministic functions  Generalized polynomial chaos Stochastic process Askey polynomials in input Deterministic functions Stochastic input

Materials Process Design and Control Laboratory STOCHASTIC VMS MODELING in on  K is spatially rapidly varying stochastic process [a multiscale diffusion coefficient] [V] Find such that, for all

Materials Process Design and Control Laboratory VARIATIONAL MULTISCALE METHOD h Subgrid scale solution Coarse scale solution Actual solution  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

Materials Process Design and Control Laboratory VMS [VARIATIONAL FORMULATION] [V] Find such that, for all  [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

Materials Process Design and Control Laboratory VMS [COARSE AND SUBGRID SCALES] Using and the induced function space decomposition Find such that, for all and Coarse [V] Subgrid [V]  Solve subgrid [V] using Greens' functions, PU and other  Substitute the subgrid solution in coarse [V]

Materials Process Design and Control Laboratory DEFINITIONS AND DISCRETIZATION  Since the subgrid solution u F 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 Subgrid [V] Homogeneous [V] Affine [V]

Materials Process Design and Control Laboratory 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)

Materials Process Design and Control Laboratory FINITE ELEMENT DISCRETIZATION Coarse mesh  Nel C elements Subgrid mesh  Nel F elements  Associated with each element sub-domain in the coarse mesh

Materials Process Design and Control Laboratory DEFINITIONS AND DISCRETIZATION  Assume a finite element discretization of the spatial region D into Nel C coarse elements  Each coarse element is further discretized by a subgrid mesh with Nel F elements  In each coarse element, the coarse solution u C can be approximated as nbf = number of spatial finite element basis functions in each coarse element P C = number of terms in the GPCE of coarse solution

Materials Process Design and Control Laboratory SOLUTION OF HOMOGENEOUS [V] CONTD  Considering the following finite element – GPCE representation for coarse solution u C  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 ]

Materials Process Design and Control Laboratory 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 P F = number of terms in the GPCE. Also, P F >P C

Materials Process Design and Control Laboratory HOMOGENEOUS [V] BOUNDARY CONDITIONS Coarse element D (e) Subgrid mesh  Also, reduced problems are solved on element sides for obtaining oscillatory boundary conditions

Materials Process Design and Control Laboratory SOLVING REDUCED PROBLEM Subgrid mesh Mapping element edge s n  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

Materials Process Design and Control Laboratory FEM FOR HOMOGENEOUS [V]  Thus in each coarse element E C, 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)(P F +1)

Materials Process Design and Control Laboratory 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

Materials Process Design and Control Laboratory DESCRIPTION OF NUMERICAL PROBLEMS Coarse element D (e) Subgrid mesh  Based on boundary conditions used and subgrid problems solved, we have three studies MsFEM-LMsFEM-Os VMS-Os  Linear boundary conditions are used  No affine correction  Reduced solution as BC  No affine correction  Reduced solution as BC  Affine correction explicitly solved

Materials Process Design and Control Laboratory 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 P F -P C difference  Future studies and research directions

Materials Process Design and Control Laboratory DETERMINISTIC [CASE I] PERIODIC MEDIA  A 128x128 mesh for complete resolution  4-noded bilinear quads for FEM K [periodic] Coarse/ subgrid MsFEM-L [L2,Linf]x10 3 MsFEM-Os [L2,Linf] x10 3 VMS-Os [L2,Linf] x10 3 [4,32]1.09, , ,1.34 [8,16]0.59, , ,0.57 [16,8]0.79, , ,0.57

Materials Process Design and Control Laboratory DETERMINISTIC [CASE I] RESULTS Resolved FEM MsFEM-L MsFEM-Os VMS-Os  coarse 8x8 subgrid 16x16 mesh  VMS yields consistent low error values

Materials Process Design and Control Laboratory DETERMINISTIC [CASE II] NON-PERIODIC MEDIA K [non-periodic]  Presence of multiple spatial scales  non-periodic spatial variation  vertices v 1 (0,0) v 2 (1,0) v 3 (0,1) v 4 (1,1)

Materials Process Design and Control Laboratory DETERMINISTIC [CASE II] NON-PERIODIC MEDIA  A 512x512 mesh for complete resolution  4-noded bilinear quads for FEM Coarse/ subgrid MsFEM-L [L2,Linf]x10 5 MsFEM-Os [L2,Linf] x10 5 VMS-Os [L2,Linf] x10 5 [32,16]2.18, ,5.50 [16,32]1.99, , ,7.46 [8,64]3.10, , ,9.97

Materials Process Design and Control Laboratory DETERMINISTIC [CASE II] RESULTS Resolved FEM MsFEM-L MsFEM-Os VMS-Os  coarse 16x16 subgrid 32x32 mesh  VMS yields consistent low error values

Materials Process Design and Control Laboratory STOCHASTIC [CASE I] PSEUDO-PERIODIC MEDIA  uniformly distributed diffusion coefficient K0K0

Materials Process Design and Control Laboratory 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] Coarse/ subgrid MsFEM-Os [L2,Linf] x10 5 VMS-Os [L2,Linf] x10 5 [32,8]0.77, ,12.6 [16,16]1.97, ,28.6 [8,32]4.75, ,32.9

Materials Process Design and Control Laboratory STOCHASTIC [CASE I] RESULTS FEM MsFEM-Os VMS-Os U0 U1 U2

Materials Process Design and Control Laboratory STOCHASTIC [CASE I] ERROR MEASURES  L-inf error was calculated on the mean value  Again, VMS is consistently better than MsFEM-Os.

Materials Process Design and Control Laboratory 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 PC C 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 P F -P C

Materials Process Design and Control Laboratory 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