GEO E1050 Finite Element Method Autumn Lecture. 9. Nonlinear Finite Element Method & Summary

Transcription

1 GEO E1050 Finite Element Method Autumn 2016 Lecture. 9. Nonlinear Finite Element Method & Summary

2 To learn today The lecture should give you overview of how non-linear problems in Finite Element Method are solved. 1. Understand the problem: why non-linear problems require special treatment 2. Understand solutions and their limitations: Initial (tangent) stiffness technique Newton-Raphson Modified Newton-Raphson 3. Understand idea behind the arc-length method - full derivation of arc-length method is not required 4. Geometric non-linearity what is that? 2

3 To learn today The lecture should give you also an overview of some other subjects relevant in Finite Element modelling. 1. Understand principles of 3D analysis 2. Understand principles of special elements: - infinite elements - interfaces - contact 3. Understand principles behind dynamic Finite Element analysis 3

4 Initial (tangent) stiffness / incremental solution Basically we divide the load into substeps and treat each as linear but that is usually not accurate enough and inefficient 4

5 Initial (tangent) stiffness / incremental solution Basically we divide the load into substeps and treat each as linear but that is usually not accurate enough and inefficient 5

6 Initial (tangent) stiffness / incremental solution Finite Element Method. W. Sołowski 6

7 Initial (tangent) stiffness / incremental solution Basically we divide the load into substeps and treat each as linear but that is usually not accurate enough and inefficient 7

8 Recalculate stresses in the elements Global marix stiffnes displaceme Vector containing nt increments Vector containing forces and theirincrements of all element nodes at allelement nodes K d = R G G G Depends on stress and strain state Hence it is not constant. In elasto-plasticity most often material softens which means that the predicted displacements are too small (we assume that the material is too stiff ) 8

9 Recalculate stresses in the elements Global stiffnes marix displaceme Vector containing nt increments Vector containing forces and theirincrements of all element nodes at allelement nodes K d = R G For each element: G ε = B Having strain increment, we calculate stress increment accurately (various algorithms exist) σ D ep (...) dε = As material most often softens, the stresses are lower than what we assumed initially. Having stresses, one can compute unbalanced forces in the system and use those to compute next iteration R i d E G 9

10 Newton - Raphson Finite Element Method. W. Sołowski 10

11 Modified Newton Raphson Finite Element Method. W. Sołowski 11

12 Modified Newton Raphson Finite Element Method. W. Sołowski 12

13 Arc-length method Idea: do not solve the exact load step, but solve as much as easily done and than proceed to solve the next bit So we have sort of automatic load stepping (can be achieved differently too) We choose the arc-length l and converge to the loaddisplacement curve Arc-length more and more commonly used (e.g. now it is used in Plaxis) Deals with softening well! 13

14 Arc-length method Finite Element Method. W. Sołowski 14

15 Arc-length method ψ - factor as x & y axes have different scales 15

16 Arc-length method Finite Element Method. W. Sołowski 16

17 Arc-length method Finite Element Method. W. Sołowski 17

18 GEO E1050 Finite Element Method Geometric non-linearity

19 Geometric non-linearity 19

20 Geometric non-linearity 20

21 Geometric non-linearity 21

22 GEO E1050 Finite Element Method 3D FEM

23 3D FEM 23

24 3D FEM 24

25 3D FEM 25

26 3D FEM Global matrix K G can be very big, thus usually iterative solutions are used, not direct inverse; direct inversion still may be quicker for highly non-linear problems 26

27 GEO E1050 Finite Element Method Special elements

28 Infinite elements Zienkiewicz, chapter

29 Infinite elements Zienkiewicz, chapter 7.6, linear interpolation 29

30 Infinite elements Zienkiewicz, chapter 7.6, quadratic interpolation 30

31 Interfaces: to allow for slippage & discontinuity Reduced stiffness element 31

32 Interfaces Springs 32

33 Interfaces Interface elements Always check theory manual!!! 33

34 Interfaces: Plaxis Initial distance between nodal pairs is zero Each node 3 degrees of freedom Allows for creation of gaps, slippage etc. Friction coefficient can be defined, as well as static/dynamic friction coefficients 34

35 Interfaces: Optum in limit analysis 35

36 Contact between domains fake boundary condition, using Lagrange multipliers (Zienkiewicz Chapter 11) 36

37 Contact between domains If we can do that, we also can do slip / Hertzian contact! Algorithms still under development (Zienkiewicz Vol 2 Chapter 8). 37

38 GEO E1050 Finite Element Method Dynamic problems

39 Dynamic problems: require time We add time dependent parts to the equation: 39

40 Dynamic problems: require time We add inertial parts to the equation: viscous forces inertial forces 40

41 Thank you