LEAST-SQUARES FINITE ELEMENT MODELS

LEAST-SQUARES FINITE ELEMENTMODELS General idea of the least-squares formulationapplied to an abstract boundary-value problem Works of our group Application to Poisson’s equation Application to flows of viscousincompressible fluids Numerical ExamplesJ N ReddyCoupled Problems : 1

Least-Squares Variational Formulation:Abstract Nonlinear Formulation Abstract nonlinear boundary value problem (u) fin Ω g (u) hon ΓΩ: domain of boundary value problemΓ: boundary of Ω : first-order nonlinear partialdifferential operatorg: linear boundary condition operatorf, h: dataCoupled Problems : 2

Least-Squares Variational Formulation:Abstract Nonlinear FormulationAbstract least-squares variational principle ( u; f , h)(1 (u) f22Ω ,0 g (u) h2Γ ,0)– Find u such that (u; f, h) (ũ; f, h) for all ũ ,1(Ω)where is an appropriate vector space,suchasHδ ( u , u ) 0Necessary condition for minimization: ,u) (u( ( u ) u , ( u ) f )Ω,0 ( g ( u ) , g ( u ) h )Γ,0Coupled Problems : 3

Least-Squares Finite Element Models:Formulations for Nonlinear Problems Two approaches may be adopted when formulatingleast-squares finite element models of nonlinearproblems– (1) Linearize PDE prior to construction andminimization of least-squares functional Element matrices will always be symmetricSimplest possible form of the element matrices– (2) Linearize finite element equations followingconstruction and minimization of least-squaresfunctional Approach is consistent with variational settingFinite element matrices are more complicatedResulting coefficient matrix may not be symmetricCoupled Problems : 4

Finite Element ImplementationSpectral/hp Finite ElementsOne-dimensional high-order Lagrange interpolation functions i ( ) 1( 1)( 1) L′p ( )p ( p 1) Lp ( i )( i )0.5ψi 0-0.5-1-0.50ξ0.51Multi-dimensional interpolation functions constructed from tensorproducts of the one-dimensional functionsWe employ full Gauss Legendre quadrature rules in evaluation ofthe integralsCoupled Problems : 5

APPLICATIONS OF LSFEM TO DATEby JNReddy and his coauthors Fluid Dynamics (2-D)––––– Viscous incompressible fluidsViscous compressible fluids (with shocks)Non-Newtonian (polymer and power-law) fluidsCoupled fluid flow and heat transferFluid-solid interactionSolid Mechanics (static and free vibration analysis)–––––BeamsPlatesShellsFracture mechanicsHelmholtz equationCoupled Problems : 6

Finite Element FormulationsOf the Poisson Equation(Primal ) Problem : 2u fin Ω( )u uˆ on Γu2 u gˆ on Γ g n(Mixed) Problem :v u 0 in Ω v fin Ωu uˆon Γunˆ v gˆ on ΓgJN ReddyLSFEM- 7

Least-Squares Formulation - Primal1.I1 (u ) u f 2.Minimize I1 (u )220,Ω u gˆ 02,Γg nB1 (u , v) l 1(v)() v u B1 (u , v) v, u 0,Ω , n n 0,Γg v 2l1 (v) v, f 0,Ω , g n 0,Γg(22)JN ReddyLSFEM- 8

Least Squares Formulation - Mixedˆ v ĝ 02,ΓI m ( u ) v u 02,Ω v f 02,Ω ngM inimize I m : δI m 0 givesBm ((u , v), (δu , δv)) l m ((δu , δv))JN ReddyLSFEM- 9

Least-squares Mixed Fe ModelFinite element approximationmu(x) uh (x) n u (x), v(x) v (x) v (x)jjhjj 1j 111Finite element model K K12 T K ij11 K12 u F1 v 2 K22 F i j dx, K ij12 K ij22 j i j dx K ji21 ( ij i j ) dx Fi1 0, Fi2 nˆ nˆ dsij f dx nˆ gˆ dsi i LSFEM- 10

Example (Using LSFEM Mixed Model):Differential Equationy 2u f in -1 x, y 1x 1Boundary Conditions u v 0 on y 1 y u w q * ( y) 0 on x 1 xy 1u u * ( y ) 8 cos πy on x 1y 1x 1xAnalytical solution:u ( x, y ) (7 x x ) cos πy7JN ReddyLSFEM- 11

L2 Norm of ErrorsPlots of the L2-Error norms as a function of pPolynomial order, pCoupled Problems : 12

LEAST-SQUARES FORMULATIONOF VISCOUS INCOMPRESSIBLE FLUIDSGoverning equations (Navier-Stokes equations)1(u )u p [( u) ( u)T ] f in ΩRe u 0in Ωu uˆ on Γunˆ σ tˆon ΓσFluid Flow (LSFEM) 13

VELOCITY-PRESSURE-VORTICITYFORMULATION OF N-S EQUATIONS FORVISCOUS INCOMPRESSIBLE FLUIDS1(u )u p ω fReω u 0 u 0 ω 0 in ˆ on uu uˆω ωon 1T U fReU ( u)T 0 u 0(u U)T p U 0 (tr U) 0ˆ on uu uˆ on U U

NUMERICAL EXAMPLES

Lid-driven Cavity-1

Lid-driven Cavity-2Fluid Flow (LSFEM) 17

Lid-driven Cavity-3Finite element mesh(20x20)Stream function (Re 5,000,p 6)Fluid Flow (LSFEM) 18

StreamlinesRe 104Pressure contoursDilatation contoursFluidMech LSFEM - 19Fluid Flow (LSFEM) 19

Oscillatory flow of a viscous incompressiblefluid in a lid-driven cavityFluid Flow (LSFEM) 20

Flow of a viscous fluid in a narrowchannel (backward facing step)Fluid Flow (LSFEM) 21

Flow of a Viscous Incompressible Fluidaround a Cylinder-1Close-up of mesh aroundMesh (501 elements; p 4)the cylinderFluid Flow (LSFEM) 22

Fluid Flow (LSFEM) 23

2D Flows Past a Circular Cylinder-2Robust at moderately high Reynolds numbers: Re 100 – 104High p-level solution: p 4, 6, 8, 10No filters or stabilization are neededFluid Flow (LSFEM) 24

Flow of a viscous fluid past acircular cylinder-3Fluid Flow (LSFEM) 25

Steady Flow Past a Circular Cylinder-4Fluid Flow (LSFEM) 26

Flow Past Two Circular CylindersFluid Flow (LSFEM) 27

Motion of a Cylinder in a Square CavityInitial boundary value problemProblem parametersΩtΓ cylvcyl 1.0Γ wallJN Reddy– ρ 1,Re 100– vwalls 0 vcyl 1.0– t (0, 0.70]Finite element discretization–––––NE 400, p-level 431,360 degrees of freedomTime step: t 0.005α 0.5 (α-family)ε 10 6 (nonlinear iteration)

Motion of a Cylinder in a Square Cavity

Motion of a Cylinder in a Square CavityFinite element mesh at t 0 and t 0.70 Pseudo-elasticity technique used to update mesh at each time step Non-uniform Young’s modulus specified for each element

Fluid-Solid Interaction(movement of a rigid solid circularcylinder in a viscous fluid)2-D Problems: 31JN Reddy2-D Problems: 31

least-squares finite element models of nonlinear problems – (1) Linearize PDE prior to construction and minimization of least-squares functional Element matrices will always be symmetric Simplest possible form of the element matrices – (2) Linearize finite element equations following construction and minimization of least-squares. functional