LEAST-SQUARES FINITE ELEMENT MODELS

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1 LEAST-SQUARES FINITE ELEMENT MODELS General idea of the least-squares formulation applied to an abstract boundary-value problem Works of our group Application to Poisson s equation Application to flows of viscous incompressible fluids Numerical Examples J N Reddy Coupled Problems : 1

2 Least-Squares Variational Formulation: Abstract Nonlinear Formulation Abstract nonlinear boundary value problem g ( u) ( u) = f in Ω = h on Γ Ω: domain of boundary value problem Γ: boundary of Ω : first-order nonlinear partial differential operator g: linear boundary condition operator f, h: data Coupled Problems : 2

3 Least-Squares Variational Formulation: Abstract Nonlinear Formulation Abstract least-squares variational principle 1 u f h = u f + u h 2 ( 2 2 ) Ω,0 Γ,0 ( ;, ) ( ) g ( ) Find u such that (u; f, h) (ũ; f, h) for all ũ, where is an appropriate vector space, δ such, as= H0 1 (Ω) Necessary condition for minimization: ( ) ( ) ( ) ( uu) ( ) ( g( ) g( ) ) u, u = u u, u f + u, u h Ω,0 Γ,0 Coupled Problems : 3

4 Least-Squares Finite Element Models: Formulations for Nonlinear Problems Two approaches may be adopted when formulating 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 Approach is consistent with variational setting Finite element matrices are more complicated Resulting coefficient matrix may not be symmetric Coupled Problems : 4

5 Finite Element Implementation Spectral/hp Finite Elements One-dimensional high-order Lagrange interpolation functions 1 i ( ) = ( 1)( + 1) L p ( ) ( + 1) L ( )( ) p p p i i ψ i ξ Multi-dimensional interpolation functions constructed from tensor products of the one-dimensional functions We employ full Gauss Legendre quadrature rules in evaluation of the integrals Coupled Problems : 5

6 APPLICATIONS OF LSFEM TO DATE by JNReddy and his coauthors Fluid Dynamics (2-D) Viscous incompressible fluids Viscous compressible fluids (with shocks) Non-Newtonian (polymer and power-law) fluids Coupled fluid flow and heat transfer Fluid-solid interaction Solid Mechanics (static and free vibration analysis) Beams Plates Shells Fracture mechanics Helmholtz equation Coupled Problems : 6

7 Finite Element Formulations Of the Poisson Equation (Primal ) Problem : (Mixed) Problem : 2 u = f in Ω v u = 0 in Ω 2 ( = ) u = uˆ on u = gˆ n on Γ Γ u g v = f in Ω u = uˆ on nˆ v = gˆ on Γ Γ u g JN Reddy LSFEM- 7

8 ( ) ( ) g g g g n v f v v l n u n v u v v u B v l v u B u I g n u f u u I Γ Ω Γ Ω Γ Ω + = + = = + = 0, 0, 2 1 0, 0, , 2 0, 2 1,, ) (,, ), ( ) ( ), ( ) ( ˆ ) ( 1. Minimize 2. Least-Squares Formulation - Primal JN Reddy LSFEM- 8

9 Least Squares Formulation - Mixed I m ( u ) = v u Ω + v f Ω + nˆ v ĝ Γ 0, 0, 0, g Minimize I m : δi m = 0 gives B m (( u, v),( δu, δv)) = l m (( δu, δv)) JN Reddy LSFEM- 9

10 Least-squares Mixed Fe Model Finite element approximation u( x) uh( x) uj j( x), vx ( ) vh( x) vj j( x) K K u 1 F 12 T 22 v 2 K K F ij i j ij i j ji K dx, K dx K ˆ ˆ 22 ij i j i j i j K ( ) dx n n ds m Finite element model j 1 j F 0, F f dx nˆ gˆ ds i i i i n LSFEM- 10

11 Example (Using LSFEM Mixed Model): Differential Equation 2 u = f in -1 x, y 1 Boundary Conditions u v = 0 on y = ± 1 y y y = +1 x = 1 x = +1 x u w = q * ( y ) = 0 on x = 1 x u = u* ( y) = 8cos π y on x = 1 Analytical solution: y = 1 u( x, y) = (7x + x 7 )cos πy JN Reddy LSFEM- 11

12 Plots of the L 2 -Error norms as a function of p L 2 Norm of Errors Polynomial order, p Coupled Problems : 12

13 LEAST-SQUARES FORMULATION OF VISCOUS INCOMPRESSIBLE FLUIDS Governing equations (Navier-Stokes equations) ( u ) u + p 1 Re [( u) + ( u) T ] = f in Ω u = 0 in Ω u = uˆ on Γ u nˆ σ = tˆ on Γ σ Fluid Flow (LSFEM) 13

14 VELOCITY-PRESSURE-VORTICITY FORMULATION OF N-S EQUATIONS FOR VISCOUS INCOMPRESSIBLE FLUIDS 1 (u )u p ω f Re ω u 0 u 0 ω 0 in u uˆ on u ω ωˆ on T 1 T (u U) p U f Re T U ( u) 0 u 0 U 0 (tr U) 0 u uˆ on U Uˆ on u

15 NUMERICAL EXAMPLES

16 Lid-driven Cavity-1

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

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

19 Streamlines 4 Re = 10 Pressure contours Dilatation contours FluidMech LSFEM - 19 Fluid Flow (LSFEM) 19

20 Oscillatory flow of a viscous incompressible fluid in a lid-driven cavity Fluid Flow (LSFEM) 20

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

22 Flow of a Viscous Incompressible Fluid around a Cylinder-1 Mesh (501 elements; p=4) Close-up of mesh around the cylinder Fluid Flow (LSFEM) 22

23 Fluid Flow (LSFEM) 23

2D Flows Past a Circular Cylinder-2 Robust at moderately high Reynolds numbers: Re = 100 10 4

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

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

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

27 Flow Past Two Circular Cylinders Fluid Flow (LSFEM) 27

28 Motion of a Cylinder in a Square Cavity Initial boundary value problem Γ cyl Ω t v cyl = 1.0 Problem parameters ρ = 1, Re = 100 v walls = 0 v cyl = 1.0 t (0, 0.70] Finite element discretization Γ wall NE = 400, p-level = 4 31,360 degrees of freedom Time step: t=0.005 α = 0.5 (α-family) ε =10 6 (nonlinear iteration) JN Reddy

29 Motion of a Cylinder in a Square Cavity

30 Motion of a Cylinder in a Square Cavity Finite 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

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

Chapter 9: Differential Analysis

Chapter 9: Differential Analysis 9-1 Introduction 9-2 Conservation of Mass 9-3 The Stream Function 9-4 Conservation of Linear Momentum 9-5 Navier Stokes Equation 9-6 Differential Analysis Problems Recall 9-1 Introduction (1) Chap 5: Control