The Finite Element Method for Computational Structural Mechanics

Transcription

1 The Finite Element Method for Computational Structural Mechanics Martin Kronbichler Applied Scientific Computing (Tillämpad beräkningsvetenskap) January 29, 2010 Martin Kronbichler (TDB) FEM for CSM January 29, / 31

2 What is structural mechanics about? A mathematician s viewpoint: Solve the (elliptic) PDE 3X λ 2 u j 3X «2 u i + µ + 2 u j + f x k x j x 2 i = 0, i = 1, 2, 3, j x i x j j,k=1 j=1 for the variable u = [u 1, u 2, u 3 ] on a certain computational domain Ω subject to some boundary conditions. An engineer s viewpoint: Body deformed by loads. Describe deformation and structural failure (body breaks down ). Pictures from open source finite element library deal.ii, Martin Kronbichler (TDB) FEM for CSM January 29, / 31

3 Module scope Get a basic understanding of the mathematical modeling in structural mechanics Perform some easy CSM simulations with the commercial software package Comsol Understand and relate the theoretical model behind simulation results Martin Kronbichler (TDB) FEM for CSM January 29, / 31

4 Schedule Time Room Lecture Jan 29, Lab 1, group 1 (Comsol intro, assignment questions) Feb 2, D Lecture Feb 2, Lab 1, group 2 (Comsol intro, assignment questions) Feb 4, D Lab 2, group 1 (assignment questions) Feb 5, D Lab 2, group 2 (assignment questions) Feb 5, D Seminar (presentation of results) Feb 8, Martin Kronbichler (TDB) FEM for CSM January 29, / 31

5 Module outline and literature 1. General overview over linear elasticity (today) O.C. Zienkiewicz and R.L. Taylor. The Finite Element Method. Volume 1: The Basis. Butterworth-Heinemann, Oxford, Chapters 1, 2, 3, 4. Martin Kronbichler (TDB) FEM for CSM January 29, / 31

6 Module outline and literature 1. General overview over linear elasticity (today) O.C. Zienkiewicz and R.L. Taylor. The Finite Element Method. Volume 1: The Basis. Butterworth-Heinemann, Oxford, Chapters 1, 2, 3, Exemplary elasticity problem and discretization: The Plane Stress Problem (Feb 2) C.A. Felippa s lecture notes on The Plane Stress Problem, Chapter 14. Available online at Martin Kronbichler (TDB) FEM for CSM January 29, / 31

7 Module outline and literature 1. General overview over linear elasticity (today) O.C. Zienkiewicz and R.L. Taylor. The Finite Element Method. Volume 1: The Basis. Butterworth-Heinemann, Oxford, Chapters 1, 2, 3, Exemplary elasticity problem and discretization: The Plane Stress Problem (Feb 2) C.A. Felippa s lecture notes on The Plane Stress Problem, Chapter 14. Available online at 3. Going beyond Plain Stress: A survey on how to pursue general problems in structural mechanics (Feb 8, < 0.5 h) O.C. Zienkiewicz and R.L. Taylor. The Finite Element Method. Volume 2: Solid Mechanics. Butterworth-Heinemann, Oxford, M.A. Crisfield. Non-linear finite element analysis of solids and structures: essentials. Vol. 1. John-Wiley & Sons, Martin Kronbichler (TDB) FEM for CSM January 29, / 31

8 Introduction, history History of structural mechanics Martin Kronbichler (TDB) FEM for CSM January 29, / 31

9 Introduction, history Truss structures [sv. fackverk] E. g. a frame structure Modeled by assembling elementary beam elements [sv. bjälke] (if individual beams not too long) Martin Kronbichler (TDB) FEM for CSM January 29, / 31

10 Introduction, history Truss structures [sv. fackverk] E. g. a frame structure Modeled by assembling elementary beam elements [sv. bjälke] (if individual beams not too long) Beam element: 12 degrees of freedom/element (in the most general 3D case); 3 coordinates + 3 angles per beam end point θ 1 θ 2 u 1 u 2 Martin Kronbichler (TDB) FEM for CSM January 29, / 31

11 Introduction, history Truss structures [sv. fackverk] E. g. a frame structure Modeled by assembling elementary beam elements [sv. bjälke] (if individual beams not too long) Beam element: 12 degrees of freedom/element (in the most general 3D 2 θ θ case); 3 coordinates + 3 angles per beam 1 u u 2 1 end point Shape of the deformed beam described by a cubic (so-called Hermite) polynomial (4 parameters) in each of the 3 space coordinates Calculate displacement of beam parameters: A discrete model balance forces and moments at joints [sv. länk] solve sparse linear system Martin Kronbichler (TDB) FEM for CSM January 29, / 31

12 Introduction, history Beams Shape of long beams not accurately described by above model Idea: line up several elementary beam elements Enforce continuity for positions and angles at interfaces between elements (i.e. couple parameters) A mathematical continuum model: the discrete elements are not physical Martin Kronbichler (TDB) FEM for CSM January 29, / 31

13 Introduction, history Above ideas very old ( 1860) 1950 s: triangular and rectangular elements developed for analysis of e. g. airplane wings Idea by engineers in structural mechanics (at the same time when computers became available) Berkeley professor R. W. Clough coined the term finite element Martin Kronbichler (TDB) FEM for CSM January 29, / 31

14 Introduction, history Above ideas very old ( 1860) 1950 s: triangular and rectangular elements developed for analysis of e. g. airplane wings Idea by engineers in structural mechanics (at the same time when computers became available) Berkeley professor R. W. Clough coined the term finite element Later recognized as a variant of an old technique ( Ritz method ) FEM as a general method for discretization of PDEs developed in the 60 70s (mathematics, scientific computing). Mathematical foundation already in a paper by Courant (1942), unknown by Clough et al.! Martin Kronbichler (TDB) FEM for CSM January 29, / 31

15 Linear elasticity The basics of linear elasticity Martin Kronbichler (TDB) FEM for CSM January 29, / 31

16 Linear elasticity The general model for 3D linear static elasticity Ω: reference configuration f: body force g: surface traction Γ 0 : fixed (clamped) surface Γ 1 : surface subject to load; a free surface has zero load Hand-drawn picture goes here Martin Kronbichler (TDB) FEM for CSM January 29, / 31

17 Linear elasticity The general model for 3D linear static elasticity Ω: reference configuration f: body force g: surface traction Γ 0 : fixed (clamped) surface Γ 1 : surface subject to load; a free surface has zero load Problem Under the steady loads f and g, determine Hand-drawn picture goes here (i) the equilibrium displacement field u(x) with components [u(x)] = ( u 1 (x), u 2 (x), u 3 (x) ) T with respect to the given reference configuration and (ii) the stresses in the body. Martin Kronbichler (TDB) FEM for CSM January 29, / 31

18 Linear elasticity Forces and stresses Forces Two types of forces: body and contact forces Body force: e. g. gravity, magnetic forces. If f is a force density (in N/m 3 ), the force (in N) on a given volume V is: F = f dω V Martin Kronbichler (TDB) FEM for CSM January 29, / 31

19 Linear elasticity Forces and stresses Forces Two types of forces: body and contact forces Body force: e. g. gravity, magnetic forces. If f is a force density (in N/m 3 ), the force (in N) on a given volume V is: F = f dω Contact force: force between parts of material (inside or on the surface of the body) S: a surface with a normal n Force on S from outside depends both on position x S and on normal n: F S = t(n, x) ds = σ(x) n ds S t(n, x): the Cauchy stress vector σ(x): the stress tensor S V t(n,x) S n(x) Martin Kronbichler (TDB) FEM for CSM January 29, / 31

20 Linear elasticity Forces and stresses The stress tensor components: σ 11 σ 12 σ 13 [σ] = σ 21 σ 22 σ 23, σ 31 σ 32 σ 33 σ T = σ Symmetric matrix. Spectral theorem: 3 real eigenvalues, there is an orthogonal set of eigenvectors The eigenvalues: principal stresses The orthogonal eigenvectors: the principal directions of stress Choose coordinate system aligned with the principal directions of stress Stress tensor becomes diagonal Forces are directed orthogonal to the coordinate planes Martin Kronbichler (TDB) FEM for CSM January 29, / 31

21 Linear elasticity Forces and stresses Stress state examples (i) Simple extension (uniaxial tension) Cylindrical specimen, arbitrary cross section with area A, Pulling at each side with evenly distributed force f f x 2 f Stress tensor constant with components f/a 0 0 [σ] = x 1 Martin Kronbichler (TDB) FEM for CSM January 29, / 31

22 Linear elasticity Forces and stresses (ii) Simple shear Plate specimen Pulling along top and bottom surface (area A) with evenly distributed force f Stress tensor constant with components [σ] = Hand-drawn picture goes here Martin Kronbichler (TDB) FEM for CSM January 29, / 31

23 Linear elasticity Forces and stresses (iii) Hydrostatic stress Spherical ball specimen Body at the exposure of gas at pressure p Stress tensor constant with components [σ] = p x 2 p x 1 Martin Kronbichler (TDB) FEM for CSM January 29, / 31

24 Linear elasticity Forces and stresses Relevance of stress Stress is the variable that enters force balance, see slide 22. Stress is a model to predict structural failure due to excessive loads, which is a key task of mechanical engineering: choose the size of component parts such that the maximum stress for a given material (e.g. structural steel) is not exceeded anywhere in the body (often add safety factors) Martin Kronbichler (TDB) FEM for CSM January 29, / 31

25 Linear elasticity Forces and stresses Stress analysis and visualization Stress tensor: 6 independent components ([σ] symmetric) How to analyze & visualize stress field? Concept 1: look at the Frobenius norm of the tensor σ 2 F = 3 3 i=1 j=1 σ 2 ij Can show: σ F independent of choice of coordinate system for components σ ij However, σ F not useful as a criterium for predicting structural failure (specimen breakdown) Martin Kronbichler (TDB) FEM for CSM January 29, / 31

26 Linear elasticity Forces and stresses Stress analysis II Usually: material very resistant to pure pressure loading (compression, expansion) Decompose stress tensor into an isotropic (pressure-like, volume-changing) σ I and a deviatoric part σ D : σ = 1 tr σi + (σ 13 ) } 3 tr σi {{}}{{} σ I σ D Martin Kronbichler (TDB) FEM for CSM January 29, / 31

27 Linear elasticity Forces and stresses Stress analysis II Usually: material very resistant to pure pressure loading (compression, expansion) Decompose stress tensor into an isotropic (pressure-like, volume-changing) σ I and a deviatoric part σ D : σ = 1 tr σi + (σ 13 ) } 3 tr σi {{}}{{} σ I σ D The von Mises stress is defined as σ v = σ D F = σ 1 F 3 tr σi, (Frobenius norm of the deviatoric part of the stress tensor) Martin Kronbichler (TDB) FEM for CSM January 29, / 31

28 Linear elasticity Forces and stresses Examples: Simple extension: σ 0 0 [σ] = 0 0 0, [σ I ] = σ , [σ D ] = σ so σ v = σ 2/3 0.82σ. Martin Kronbichler (TDB) FEM for CSM January 29, / 31

29 Linear elasticity Forces and stresses Examples: Simple extension: σ 0 0 [σ] = 0 0 0, [σ I ] = σ , [σ D ] = σ so σ v = σ 2/3 0.82σ. Simple shear: 0 σ σ 0 [σ] = σ 0 0, [σ I ] = 0 0 0, [σ D ] = σ so σ v = σ. Martin Kronbichler (TDB) FEM for CSM January 29, / 31

30 Linear elasticity Forces and stresses Examples: Simple extension: σ 0 0 [σ] = 0 0 0, [σ I ] = σ , [σ D ] = σ so σ v = σ 2/3 0.82σ. Simple shear: 0 σ σ 0 [σ] = σ 0 0, [σ I ] = 0 0 0, [σ D ] = σ so σ v = σ. Hydrostatic stress: [σ] = σ 0 1 0, [σ I ] = σ 0 1 0, [σ D ] = so σ v = 0. Martin Kronbichler (TDB) FEM for CSM January 29, / 31

31 Linear elasticity Forces and stresses Why stresses are relevant, revisited Model to predict structural failure due to excessive load: base criterion on von Mises stress works well for steel, aluminium, copper, gold ( ductile [sv. töjbar, duktil] materials) Von Mises criteria not appropriate to predict failure by crack propagation ( fatigue [sv. materialutmattning]) For cracks, better to use criteria based on maximum principal stress Martin Kronbichler (TDB) FEM for CSM January 29, / 31

32 Linear elasticity Elasticity equations Elasticity equations: integral and differential form Fundamental axiom in mechanics: total forces on each volume V Ω are in balance σ n ds + V V f dω = 0 V Ω, (Integral form) Martin Kronbichler (TDB) FEM for CSM January 29, / 31

33 Linear elasticity Elasticity equations Elasticity equations: integral and differential form Fundamental axiom in mechanics: total forces on each volume V Ω are in balance σ n ds + V V f dω = 0 V Ω, (Integral form) Divergence theorem (Gauss) ( σ + f) dω = 0 V Ω, thus, where V σ = f in Ω, u = 0 on Γ 0, σ n = g on Γ 1, [ σ] i = 3 j=1 x j σ ij. (Differential form) Martin Kronbichler (TDB) FEM for CSM January 29, / 31

34 Linear elasticity Elasticity equations Constitutive law Above problem statements incomplete (3 equations + b.c., 6 unknown stresses and 3 unknown displacements) Need a constitutive law, a relation between stress and displacement, σ = σ(u) Depends on material properties (rubber and steel at the same load behave very differently!) Assume: Homogeneous material (same property at each point) Isotropic material (same property in each direction) Small deformations (geometric linear problems) Martin Kronbichler (TDB) FEM for CSM January 29, / 31

35 Linear elasticity Elasticity equations Then σ = λi tr(ɛ(u)) + 2µɛ(u), where ɛ (= ɛ T ) is the strain tensor [sv. töjning]: (constitutive relation) ɛ(u) = 2( 1 u + ( u) T ), (kinematic relation) ɛ ij (u) = 1 ( ui + u ) j. 2 x j x i Martin Kronbichler (TDB) FEM for CSM January 29, / 31

36 Linear elasticity Elasticity equations Then σ = λi tr(ɛ(u)) + 2µɛ(u), where ɛ (= ɛ T ) is the strain tensor [sv. töjning]: (constitutive relation) ɛ(u) = 2( 1 u + ( u) T ), (kinematic relation) ɛ ij (u) = 1 ( ui + u ) j. 2 x j x i Parameters λ, µ: Lamé coefficients, usually expressed as µ = E 2(1 + ν), λ = Eν (1 + ν)(1 2ν) E: Young modulus [sv. elasticitetsmodul]. Ex. around 200 GPa for steel ν: Poisson ratio; 0 < ν < 1/2. Ex. around 0.3 for steel, almost 0.5 for rubber (indicating incompressible material) Martin Kronbichler (TDB) FEM for CSM January 29, / 31

37 Linear elasticity Elasticity equations Visualization of quantities and equations in structural mechanics: Tonti diagram Displacement (Dirichlet) BCs Prescribed displacements û u = û on Γ 0 Displacements u Body forces f Kinematic relation: ɛ(u) = 1 2 ` u + ( u) T Equilibrium (balance eq.): σ + f = 0 quantity quantity Unknown fields Known fields Strains ɛ Constitutive relation: ɛ = Cσ or σ = λi tr(ɛ) + 2µɛ Stresses σ Force (Neumann) BCs: σ n = g = t on Γ 1 Prescribed tractions t Martin Kronbichler (TDB) FEM for CSM January 29, / 31

38 Energy and FEM Linear elasticity from an energy point of view the weak form as a basis for FEM Martin Kronbichler (TDB) FEM for CSM January 29, / 31

39 Energy and FEM Equations: energy viewpoint Alternative derivation of the equations: energy principle (basis for FEM) Elastic energy stored in the body, the elastic strain energy: S(u) = 1 2 Ω ɛ(u) : σ(u) dω := i,j=1 Ω ɛ ij σ ij dω cf. the energy stored in a spring: 1 2ɛ f (force f, stretched by ɛ) Martin Kronbichler (TDB) FEM for CSM January 29, / 31

40 Energy and FEM Equations: energy viewpoint Alternative derivation of the equations: energy principle (basis for FEM) Elastic energy stored in the body, the elastic strain energy: S(u) = 1 2 Ω ɛ(u) : σ(u) dω := i,j=1 Ω ɛ ij σ ij dω cf. the energy stored in a spring: 1 2ɛ f (force f, stretched by ɛ) Work exerted on the body by external forces: W (u) = u f dω + u g ds Ω Γ 1 Martin Kronbichler (TDB) FEM for CSM January 29, / 31

41 Energy and FEM Equations: energy viewpoint Alternative derivation of the equations: energy principle (basis for FEM) Elastic energy stored in the body, the elastic strain energy: S(u) = 1 2 Ω ɛ(u) : σ(u) dω := i,j=1 Ω ɛ ij σ ij dω cf. the energy stored in a spring: 1 2ɛ f (force f, stretched by ɛ) Work exerted on the body by external forces: W (u) = u f dω + u g ds Ω Γ 1 Total potential energy (fundamental axiom in mechanics) T (u) = S(u) W (u) Martin Kronbichler (TDB) FEM for CSM January 29, / 31

42 Energy and FEM The principle of minimal potential energy Equilibrium: displacement u minimizes total potential energy T (u) = 1 ɛ(u) : σ(u) dω u f dω u g ds 2 Ω Ω Γ }{{}}{{ 1 } S(u) W (u) Martin Kronbichler (TDB) FEM for CSM January 29, / 31

43 Energy and FEM The principle of minimal potential energy Equilibrium: displacement u minimizes total potential energy T (u) = 1 ɛ(u) : σ(u) dω u f dω u g ds 2 Ω Ω Γ }{{}}{{ 1 } S(u) W (u) W (u) is linear in u For linear elastic materials, σ(u) is linear in u and thus S(u) quadratic in u For linear, homogeneous, and isotropic material: T (u) = 1 [λ ( tr(ɛ(u)) ) ] 2 + 2µɛ(u) : ɛ(u) dω 2 Ω = 1 a(u, u) l(u) 2 u f dω u g ds Γ 1 where a is linear in both arguments and l is a linear form in u. Martin Kronbichler (TDB) FEM for CSM January 29, / 31 Ω

44 Energy and FEM Mathematics: Variational Principle Variational Reformulation Now assume that u minimizes T. Let v be an arbitrary admissible displacement (satisfying v = 0 on Γ 0 ). Perturb the minimum and define, for t R, φ(t) = T (u + tv) = 1 a(u + tv, u + tv) l(u + tv) 2 = 1 2 a(u, u) + l(u) + t( a(v, u) l(v) ) + t2 a(v, v) 2 φ is convex quadratic in t since a is positive definite (a(v, v) > 0 when v 0) Thus, φ is minimized iff φ (0) = 0, that is, a(v, u) = l(v) for each admissible v (1) Martin Kronbichler (TDB) FEM for CSM January 29, / 31

45 Energy and FEM Mathematics: Variational Principle Relation to linear algebra A quadratic function in R n has the same structure as T (u): J(u) = 1 2 ut Au b T u 1 a(u, u) l(u) 2 Here, the variational principle reads for A symmetric and positive definite: J minimized by u J(u) = 0 and J(u) pos. def. Au = b, A pos. def. v T (Au b) = 0 for all v R n, A pos. def. Martin Kronbichler (TDB) FEM for CSM January 29, / 31

46 Energy and FEM Mathematics: Variational Principle Equation (1) in explicit form: the equilibrium displacement u satisfies [ ( ) ( ) ] λ tr ɛ(v) tr ɛ(u) + µɛ(v) : ɛ(u) dω = v f dω + v g ds Ω Ω Γ 1 for each admissible v. Above relation is a variational expression, called the principle of virtual work in mechanics, and is the basis for the discretization with the Finite Element Method (FEM) In FEM, we want to fulfill (1) not for all admissible functions v, but only for those described by a finite-dimensional subset V h of admissible displacements. Martin Kronbichler (TDB) FEM for CSM January 29, / 31