Finite element method: the basics

Transcription

1 Finite element method: the basics Prof. José E. Andrade Department of Civil & Environmental Engineering July, 2007

2 Outline Applications of FEM Fundamentals Examples

3 Applications of FEM

4 patient-specific model bypass alternatives Biomechanics

5 !a P FOOTING GRAIN SHEAR BAND COMPACTIVE ZONE!r FAILURE SURFACE 'HOMOGENEOUS' SOIL DILATIVE ZONE VOID FIELD SCALE LOG (m) >1 SPECIMEN SCALE 0-1 MESO SCALE -2 Geomechanics GRAIN SCALE -3

6 shear DEVIATORIC strain STRAIN consolidation FE SOLN ANAL SOLN VERTICAL COORDINATE, m PRESSURE, kpa Solid-fluid interactions

7 Simulation engineering

8 Fundamentals of FEM

9 ) '(1 '(0 '(/ '(. '(, '(+ '(* '() ' & ' '() '(* '(+ '(, '(- '(. '(/ '(0 '(1 )!!"!#$ & ) '(1 '(0 '(/ '(. '(- '(- '(, '(+ '(* '() ' FEM Designed to approximately solve PDE s PDE s model physical phenomena Three types of PDE s: Parabolic: fluid flow Hyperbolic: wave eqn Elliptic: elastostatics %!"!#$

10 FEM recipe Strong from Weak form Galerkin form Matrix form

11 Elastostatics: strong PDE B.C. s u, xx +f = 0 u(1) = g u, x (0) = h strong form 0 derived from continuum mechanics strong form = PDE + B.C. s usually, there is no exact sln for strong form 1 x

12 Elastostatics: weak Use principle of virtual work Introduce virtual displacement Use strong form 1 w; w(1) = 0 0 w(u, xx +f) dx = w, x u, x dx = 1 0 wf dx + w(0)h

13 Elastostatics: Galerkin Construct approximate solution like g u h = v h + g h like w Construct functions based on shape functions w h = n N A c A v h = n A=1 A=1 g h = gn n+1 N A d A

14 Piecewise linear FE 1 N 1 N A N n+1 x 1 x 2 x A-1 x A x A+1 x n x n+1 x 0 1 node finite element

15 Elastostatics: matrix Use weak form, plug-in Galerkin approximation n B=1 F A = K AB d B = F A K AB = 1 0 N A f dx + N A (0)h it all boils down to... K d = F 0 N A,x N b,x dx N A,x N n+1,x dxg stiffness matrix force vector

16 Properties of K d = F Stiffness matrix is symmetric banded positive-definite may not always apply Displacement vector = unknowns only Can use any linear algebra solver to find solution

17 Multi-D deformation σ + f = 0 in Ω u = g on Γ g σ n = h on Γ h equilibrium e.g., clamp e.g., confinement Γ g Ω Γ h Constitutive relation given u get σ e.g., elasticity, plasticity

18 FEM program TIME STEP LOOP ITERATION LOOP ASSEMBLE FORCE VECTOR AND STIFFNESS MATRIX ELEMENT LOOP: N=1, NUMEL GAUSS INTEGRATION LOOP: L=1, NINT CALL MATERIAL SUBROUTINE CONTINUE CONTINUE CONTINUE T = T +!T

19 Element technology: 2D Serendipity family of quads Lagrange family of quads Gauss integration point displacement node Standard triangular elements

20 Modeling ingredients 1. Set geometry 2. Discretize domain 3. Set matl parameters 4. Set B.C. s 5. Solve H B

21 Modeling ingredients 1. Set geometry 2. Discretize domain 3. Set matl parameters 4. Set B.C. s 5. Solve

22 Modeling ingredients 1. Set geometry 2. Discretize domain 3. Set matl parameters 4. Set B.C. s 5. Solve

23 Modeling ingredients! a 1. Set geometry 2. Discretize domain 3. Set matl parameters 4. Set B.C. s 5. Solve! r

24 Modeling ingredients! a 1. Set geometry 2. Discretize domain 3. Set matl parameters 4. Set B.C. s 5. Solve! r

25 Examples

26 Hyperbolic: LSST array output input

27 Geometry and B.C.s output input

28 Material parameters shear modulus degradation & damping

29 Input base acceleration

30 Acceleration output

31 Soil-fluid interaction Model ingredients Nonlinear continuum mechanics s Robust constitutive theory X x f Computational inelasticity x 2 x 1 f Nonlinear finite elements

32 Soil-fluid interaction Model ingredients Nonlinear continuum mechanics = = 2 3 Robust constitutive theory Computational inelasticity Nonlinear finite elements

33 Soil-fluid interaction Model ingredients Nonlinear continuum mechanics Robust constitutive theory F n F n+1 n tr n+1 n+1 Computational inelasticity Nonlinear finite elements

34 Soil-fluid interaction Model ingredients Nonlinear continuum mechanics Robust constitutive theory Computational inelasticity Nonlinear finite elements Displacement node Pressure node

35 Plane-strain compress specific volume CT scan FE model

36 Plane-strain compress specific volume shear strain and flow fluid pressure

37 Plane-strain compress

38 Questions?