The Finite Element Method for the Analysis of Non-Linear and Dynamic Systems Prof. Dr. Eleni Chatzi Lecture 1-20 September, 2017 Institute of Structural Engineering Method of Finite Elements II 1
Course Information Instructors Prof. Dr. Eleni Chatzi, email: chatzi@ibk.baug.ethz.ch Dr. Giuseppe Abbiati, email: abbiati@ibk.baug.ethz.ch Dr. Konstantinos Agathos, email: agathos@ibk.baug.ethz.ch Assistant Adrian Egger, HIL E19.5, email: egger@ibk.baug.ethz.ch Course Website Lecture Notes and Demos will be posted in: http://www.chatzi.ibk.ethz.ch/education/method-of-finite-elements-ii.html Performance Evaluation - Final Project (100%) Topic Categories announced on 12.10.2017 - You are free to shape your own project! Computer Labs are already be pre-announced and it is advised to bring a laptop along for those sessions Help us Structure the Course! Participate in our online survey: http://goo.gl/forms/ws0asblxiy Institute of Structural Engineering Method of Finite Elements II 2
Course Information Suggested Reading Nonlinear Finite Elements for Continua and Structures, by T. Belytschko, W. K. Liu, and B. Moran, John Wiley and Sons, 2000 The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, by T. J. R. Hughes, Dover Publications, 2000 Nonlinear finite element analysis of solids and structures,crisfield, M.A., Remmers, J.J. and Verhoosel, C.V., John Wiley Sons, 2012 Computational methods for plasticity: theory and applications, De Souza Neto, E.A., Peric, D. and Owen, D.R., 2011 Lecture Notes by Carlos A. Felippa Nonlinear Finite Element Methods (ASEN 6107): http://www. colorado.edu/engineering/cas/courses.d/nfem.d/home.html Institute of Structural Engineering Method of Finite Elements II 3
Learning Goals By the end of the lecture, the goal is to learn: When is a problem nonlinear & what are types of nonlinearity? What types of nonlinearity exist? What are the differences, but also similarities, between linear and nonlinear problems? How to formulate & solve nonlinear problems What algorithms to use & what are typical problems? Institute of Structural Engineering Method of Finite Elements II 4
Learning Goals What we know so far We can recognize linearity As the process following the fundamental rule output = constant input F M=FL When is this linear? M w L θ when M = EI = EI φ EI φ Institute of Structural Engineering Method of Finite Elements II 5
Learning Goals Beyond Linearity F M=FL What if? M plasticity w L θ We have excessive stresses in a given section? Material Nonlinearity EI φ F M=FL What if? w θ We have excessive stresses in a given section? L Fracture Phenomena Institute of Structural Engineering Method of Finite Elements II 6
Learning Goals Beyond Linearity F w F θ M=FL & how does the structure actually deform? M =? L Large Displacements ΔL w F θ M=FL & how does the structure actually deform when heated? L Thermomechanics Problems khanacademy.org Institute of Structural Engineering Method of Finite Elements II 7
Significance of the Lecture The importance of Modeling Models are abstract representations of the real world Simulation models are extensively used in civil engineering practice. Such models allow the user to understand structural system performance predict structural behavior diagnose damage optimize design, etc Institute of Structural Engineering Method of Finite Elements II 8
Significance of the Lecture The importance of Modeling Models are abstract representations of the real world Structures may fail because The models used for their design do not properly represent the complexity of the physics Overly simplified models may not predict response under extreme loads The values of the input parameters have not been selected properly Institute of Structural Engineering Method of Finite Elements II 9
Course Outline FEM II - Lecture Schedule Introduction - From Linear to Nonlinear Considerations Block #1 Material Nonlinearity I - Constitutive Modeling Material Nonlinearity II - FE Implementation Computer Lab I Block #2 Large Deformations I Block #3 Dynamics I Large Deformations II Nonlinear Dynamics II Institute of Structural Engineering Method of Finite Elements II 10
Course Outline FEM II - Lecture Schedule Computer Lab II Block #4 Fracture I - General principles Fracture II - The extended Finite Element Method Block #5 Thermomechanics I Special Topics - Coupling/Penalty Methods (Lagrange Multipliers), Contact, Imposition of Essential BCs Computer Lab III Institute of Structural Engineering Method of Finite Elements II 11
Review of the Finite Element Method (FEM) FEM: Numerical Technique for approximating the solution of continuous systems. The MFE is based on the physical discretization of the observed domain, thus reducing the number of the degrees of freedom; and shifting from an analytical to a numerical formulation. Continuous Discrete q y+dy h1 q x dy q x+dx dx q y h2 L Flow of water Permeable Soil ( ) k 2 φ 2 x + 2 φ 2 y Impermeable Rock = 0 Laplace Equation F = KX Direct Stiffness Method Institute of Structural Engineering Method of Finite Elements II 12
Review of the Linear Finite Element Method (FEM) How is the Physical Problem formulated? A linear elastic system under specific loads and constraints at its boundaries can be described by the following fundamental equations: Linear balance (Equilibrium) equation. Linear constitutive model. Kinematic Equation, or Static balance equation or, if we consider modal analysis as an equivalent static analysis, at least stationary. Example of a bridge FE model. Institute of Structural Engineering Method of Finite Elements II 13
Review of the Linear Finite Element Method (FEM) How is the Physical Problem formulated? A linear elastic system under specific loads and constraints at its boundaries can be described by the following fundamental equations: Linear balance (Equilibrium) equation. Linear constitutive model. Kinematic Equation, or Static balance equation or, if we consider modal analysis as an equivalent static analysis, at least stationary. Example of a bridge FE model. Institute of Structural Engineering Method of Finite Elements II 13
Review of the Linear Finite Element Method (FEM) How is the Physical Problem formulated? A linear elastic system under specific loads and constraints at its boundaries can be described by the following fundamental equations: Linear balance (Equilibrium) equation. Linear constitutive model. Kinematic Equation, or Static balance equation or, if we consider modal analysis as an equivalent static analysis, at least stationary. Example of a bridge FE model. Institute of Structural Engineering Method of Finite Elements II 13
Review of the Linear Finite Element Method (FEM) The Strong Form of a Linear Problem On the basis of the previous fundamental euations, a governing differential equation, also known as the strong form of the problem may be extracted: 1 Equilibrium Equations ex. f (x) = σa = q(l x) dσ dx + q/a = 0 2 Constitutive Model ex. σ = Eɛ 3 Kinematics Relationship ex. ɛ = du dx Axial bar Example q x q f(x) Institute of Structural Engineering Method of Finite Elements II 14 L-x
Examples of Linear Balance Equations Mechanical Axial bar Example q Thermomechanical θ1 θ(x) x φ(x)>0 θ2 Force balance: dσ dx + P = 0 u : displacement [m] σ : stress [Pa] P : body load [ ] N m 3 Heat flux balance: dφ dx + R = 0 θ : temperature [K] φ : heat flux [ W m 2 ] R : heat source [ W m 3 ] Institute of Structural Engineering Method of Finite Elements II 15
Examples of Linear Constitutive Models Mechanical Thermomechanical Hooke s law: σ = E du dx u : displacement [m] x : space coordinate [m] du dx : strain [ ] m m σ : stress [Pa] E : Young modulus [Pa] Fourier s law: φ = k dθ dx θ : temperature [K] x : space coordinate [m] dθ dx : temperature grad. [ ] K m φ : heat flux [ ] W m 2 k : thermal cond. [ ] W mk Institute of Structural Engineering Method of Finite Elements II 16
Examples of Problems Equations Mechanical Thermomechanical Force balance: E d 2 u dx 2 + P = 0 u : displacement [m] E : Young modulus [Pa] P : body load [ N m 3 ] Heat flux balance: k d 2 θ dx 2 + R = 0 θ : temperature [K] k : thermal cond. [ ] W mk R : heat source [ ] W m 3 Institute of Structural Engineering Method of Finite Elements II 17
Boundary Value Problems (BVPs) The Strong Form Mechanical BVP Thermal BVP E d 2 u dx 2 + P = 0 { u 0,L = u 0,L du dx 0,L = du 0,L dx u : displacement [m] E : Young modulus [Pa] P : body load [ N m 3 ] k d 2 θ dx 2 + R = 0 { θ 0,L = θ 0,L dθ dx 0,L = dθ 0,L dx θ : temperature [K] k : thermal cond. [ ] W mk R : heat source [ ] W m 3 Institute of Structural Engineering Method of Finite Elements II 18
Boundary Value Problems (BVPs) The Strong Form Mechanical BVP Thermal BVP E d 2 u dx 2 + P = 0 { u 0,L = u 0,L du dx 0,L = du 0,L dx FEM Formulation: K u n u n u u = F u n u 1 n u 1 k d 2 θ dx 2 + R = 0 { θ 0,L = θ 0,L dθ dx 0,L = dθ 0,L dx FEM Formulation: K θ n θ n θ θ = F θ n θ 1 n θ 1 Institute of Structural Engineering Method of Finite Elements II 18
Boundary Value Problems (BVPs) The Strong Form Mechanical BVP Thermal BVP E d 2 u dx 2 + P = 0 { u 0,L = u 0,L du dx 0,L = du 0,L dx FEM Formulation: K u n u n u u = F u n u 1 n u 1 k d 2 θ dx 2 + R = 0 { θ 0,L = θ 0,L dθ dx 0,L = dθ 0,L dx FEM Formulation: K θ n θ n θ θ = F θ n θ 1 n θ 1 Institute of Structural Engineering Method of Finite Elements II 18
Weak Form - 1D FEM So how to solve the Strong Form The strong form requires strong continuity on the dependent field variables (usually displacements). Whatever functions define these variables have to be differentiable up to the order of the PDE that exist in the strong form of the system equations. Obtaining the exact solution for a strong form of the system equation is a quite difficult task for practical engineering problems. The finite difference method can be used to solve the system equations of the strong form and obtain an approximate solution. However, this method usually works well for problems with simple and regular geometry and boundary conditions. Alternatively we can use the finite element method on a weak form of the system. This weak form is usually obtained through energy principles which is why it is also known as variational form. Institute of Structural Engineering Method of Finite Elements II 19
Weak Form - 1D FEM From Strong Form to Weak form Three are the approaches commonly used to go from strong to weak form: Principle of Virtual Work Principle of Minimum Potential Energy Methods of weighted residuals (Galerkin, Collocation, Least Squares methods, etc) Visit our Course page on Linear FEM for further details: http://www.chatzi.ibk.ethz.ch/education/ method-of-finite-elements-i.html Institute of Structural Engineering Method of Finite Elements II 20
Weak Form - 1D FEM From Strong Form to Weak form - Approach #1 Principle of Virtual Work For any set of compatible small virtual displacements imposed on the body in its state of equilibrium, the total internal virtual work is equal to the total external virtual work. The principle of virtual displacements facilitates the derivation of discretized equations: where Ω R u (u) = F u δɛ T σdω = Ω δu T P vol u dω + δu T P sur u dγ Γ σ : stress state generated by volume P vol u surface P sur u loads. δu and δɛ : compatible variations of displacement u and strain ɛ fields. and Institute of Structural Engineering Method of Finite Elements II 21
Weak Form - 1D FEM From Strong Form to Weak form - Approach #1 Principle of Virtual Work For any set of compatible small virtual displacements imposed on the body in its state of equilibrium, the total internal virtual work is equal to the total external virtual work. The principle of virtual displacements facilitates the derivation of discretized equations: where Ω R u (u) = F u δɛ T σdω = Ω δu T P vol u dω + δu T P sur u dγ Γ σ : stress state generated by volume P vol u surface P sur u loads. δu and δɛ : compatible variations of displacement u and strain ɛ fields. and Institute of Structural Engineering Method of Finite Elements II 21
Weak Form - 1D FEM From Strong Form to Weak form - Approach #1 Principle of Virtual Work For any set of compatible small virtual displacements imposed on the body in its state of equilibrium, the total internal virtual work is equal to the total external virtual work. The principle of virtual displacements facilitates the derivation of discretized equations: where Ω R u (u) = F u δɛ T σdω = Ω δu T P vol u dω + δu T P sur u dγ Γ σ : stress state generated by volume P vol u surface P sur u loads. δu and δɛ : compatible variations of displacement u and strain ɛ fields. and Institute of Structural Engineering Method of Finite Elements II 21
Weak Form - 1D FEM From Strong Form to Weak form - Approach #3 Consider the following (1D) generic strong form: ( d c(x) du ) + f(x) = 0 dx dx c(0) d dx u(0) = C 1 (Neumann BC) & u(l) = 0 (Dirichlet BC) Galerkin s Method Given an arbitrary weight function w, where S = {u u C 0, u(l) = 0}, S 0 = {w w C 0, w(l) = 0} C 0 is the collection of all continuous functions. Multiplying by w and l integrating over Ω w(x)[(c(x)u (x)) + f (x)]dx = 0 0 [w(0)(c(0)u (0) + C 1 ] = 0 Institute of Structural Engineering Method of Finite Elements II 22
Weak Form - 1D FEM Using the divergence theorem (integration by parts) we reduce the order of the differential: l 0 l wg dx = [wg] l 0 gw dx 0 The weak form is then reduced to the following problem. Also, in what follows we assume constant properties c(x) = c = const. Find u(x) S such that: l 0 w cu dx = l 0 wfdx + w(0)c 1 S = {u u C 0, u(l) = 0} S 0 = {w w C 0, w(l) = 0} Institute of Structural Engineering Method of Finite Elements II 23
Weak Form Notes: 1 Natural (Neumann) boundary conditions, are imposed on the secondary variables like forces and tractions. For example, u y (x 0, y 0 ) = u 0. 2 Essential (Dirichlet) or geometric boundary conditions, are imposed on the primary variables like displacements. For example, u(x 0, y 0 ) = u 0. 3 A solution to the strong form will also satisfy the weak form, but not vice versa. Since the weak form uses a lower order of derivatives it can be satisfied by a larger set of functions. 4 For the derivation of the weak form we can choose any weighting function w, since it is arbitrary, so we usually choose one that satisfies homogeneous boundary conditions wherever the actual solution satisfies essential boundary conditions. Note that this does not hold for natural boundary conditions! Institute of Structural Engineering Method of Finite Elements II 24
FE formulation: Reminder Let s recall the steps to the Galerking approach 1 Step #1: - Discretization 2 Step #2: - Aprroximate u(x) 3 Step #3: - Aprroximate w(x) 4 Step #4: - Substitute in the weak form w(x) 5 Step #5: - Obtain the discrete system matrices Institute of Structural Engineering Method of Finite Elements II 25
FE formulation: Discretization How to derive a solution to the weak form? Step #1:Follow the FE approach: Divide the body into finite elements, e, connected to each other through nodes. e x 1 e x 2 e Then break the overall integral into a summation over the finite elements: [ x e ] 2 x e w cu 2 dx wfdx w(0)c 1 = 0 x1 e x1 e e Institute of Structural Engineering Method of Finite Elements II 26
1D FE formulation: Galerkin Method Step #2: Approximate the continuous displacement field u(x) using a discrete equivalent: Galerkin s method assumes that the approximate (or trial) solution, u, can be expressed as a linear combination of the nodal point displacements u i, where i refers to the corresponding node number. u(x) u h (x) = i N i (x)u i = N(x)u where bold notation signifies a vector and N i (x) are the shape functions. In fact, the shape function can be any mathematical formula that helps us infer (interpolate) what happens at points that lie within the nodes of the mesh. In the 1-D case that we are using as a reference, N i (x) are defined as 1st degree polynomials indicating a linear interpolation. Institute of Structural Engineering Method of Finite Elements II 27
1D FE formulation: Galerkin s Method Shape function Properties: Bounded and Continuous One for each node Ni e(x j e) = δ ij, where { 1 if i = j δ ij = 0 if i j The shape functions can be written as piecewise functions of the x coordinate: This is not a convenient notation. x x i 1 Instead of using the global coordinate, x i 1 x < x i x i x i 1 x, things become simplified when N i (x) = xi + 1 x, x i x < x using coordinate ξ referring to the i+1 x i+1 x i local system of the element (see page 0, otherwise 25). Institute of Structural Engineering Method of Finite Elements II 28
1D FE formulation: Galerkin s Method Step #3: Approximate w(x) using a discrete equivalent: The weighting function, w is usually (although not necessarily) chosen to be of the same form as u w(x) w h (x) = i N i (x)w i = N(x)w i.e. for 2 nodes: N = [N 1 N 2 ] u = [u 1 u 2 ] T w = [w 1 w 2 ] T Alternatively we could have a Petrov-Galerkin formulation, where w(x) is obtained through the following relationships: w(x) = i (N i + δ he σ δ = coth( Pee 2 ) 2 Pe e dn i dx )w i coth = ex + e x e x e x Institute of Structural Engineering Method of Finite Elements II 29
1D FE formulation: Galerkin s Method Step #4: Substituting into the weak formulation and rearranging terms we obtain the following in matrix notation: l 0 l 0 w cu dx l (w T N T ) c(nu) dx 0 wfdx w(0)c 1 = 0 l 0 w T N T fdx w T N(0) T C 1 = 0 Since w, u are vectors, each one containing a set of discrete values corresponding at the nodes i, it follows that the above set of equations can be rewritten in the following form, i.e. as a summation over the w i, u i components (Einstein notation): l 0 ( l f 0 j i ) dn i (x) u i c dx j w j N j (x)dx j dn j (x) w j dx dx w j N j (x)c 1 = 0 x=0 Institute of Structural Engineering Method of Finite Elements II 30
1D FE formulation: Galerkin s Method This is rewritten as, [ ( l w j 0 j i 0 i cu i dn i (x) dx ) ] dn j (x) fn j (x)dx + (N j (x)c 1 ) x=0 = 0 dx The above equation has to hold w j since the weighting function w(x) is an arbitrary one. Therefore the following system of equations has to hold: ( ) l dn i (x) dn j (x) cu i fn j (x)dx + (N j (x)c 1 ) dx dx x=0 = 0 j = 1,..., n After reorganizing and moving the summation outside the integral, this becomes: [ ] l c dn i(x) dn j (x) l u i = fn j (x)dx + (N j (x)c 1 ) dx dx x=0 = 0 j = 1,..., n 0 i 0 Institute of Structural Engineering Method of Finite Elements II 31
1D FE formulation: Galerkin s Method We finally obtain the following discrete system in matrix notation: Ku = f where writing the integral from 0 to l as a summation over the subelements we obtain: K = A e K e K e = x e 2 x1 e [ ] dn(x) T c dn(x) dx = dx dx x e 2 x e 1 B T cbdx f = A e f e f e = x e 2 x e 1 N T fdx + N T h x=0 where A is not a sum but an assembly, and, x denotes differentiation with respect to x. In addition, B = dn(x) is known as the strain-displacement matrix. dx Institute of Structural Engineering Method of Finite Elements II 32
Transitioning to the Nonlinear Case How to treat the Nonlinear Case? Linear Problems K d= F or Pd ( ) = F Stiffness matrix K is constant Pd ( + d) = Pd ( ) + Pd ( ) 1 2 1 2 P( α d) = α P( d) = αf If the load is doubled, displacement is doubled, too Superposition is possible 2F F d K 2d Nonlinear Problems F Pd ( ) = F, P(2 d) 2F K T How to find d for a given F? d i source: Nam-Ho Kim, EGM6352, Lecture Notes Institute of Structural Engineering Method of Finite Elements II 33 d