Overview of Solution Methods

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1 20 Overview of Solution Methods NFEM Ch 20 Slide 1

2 Nonlinear Structural Analysis is a Multilevel Continuation Process Stages Increments Iterations Individual stage Increments Iterations NFEM Ch 20 Slide 2

3 Study Focus Restriction In this and subsequent Chapters we will stay within an individual stage, which need not be identified. Consequently we will focus on the two innermost solution levels: increments and iterations. These are also called predictor and corrective levels, especially in literature dealing with numerical solution of ODEs NFEM Ch 20 Slide 3

4 Classification (Advancing over an Individual Stage) Nonlinear FEM Purely incremental methods. Also called predictor-only methods. Corrective level is missing. Popular in path dependent problems, e.g., plasticity Incremental-iterative methods. Also called predictor-corrector methods. Corrective phase aims to eliminate drift error. Popular in path independent problems NFEM Ch 20 Slide 4

5 Basic Description and Notation (1) To advance the solution, the stage is broken up into incremental steps, or increments by short. If necessary, increments will be identified by subscript n For example the state vector after the n-th increment is u The state vector before any increment (initial state or stage start) is u Over each incremental step the state vector u and staging parameter λ undergo finite changes denoted by n 0 respectively. u n λ n NFEM Ch 20 Slide 5

6 Basic Description and Notation (2) k n Nonlinear FEM Iteration steps will be usually identified by the superscript k For example { u k k n, λ n } may denote the solution after the kth iteration of the nth step, whereas {u 0 0 n, λ n } is the predicted solution before starting the corrective process. Iterative changes in u and λ are often shortened to d and η, resp. Certain "argument omission" abbreviations will be used throughout the exposition to reduce clutter. If { u n, λ n} is a state-control pair computed after n incremental step we denote r n = r u n,λ n K n = K u n,λ n q n = q u n,λ n etc Similarly if the state control pair is { u,λ } after k iterations carried out at the nth increment, we denote and likewise for other quantities r k n = r uk n,λk n etc k n NFEM Ch 20 Slide 6

7 Decisions Must Be Made in Three Continuation Ingredients Increment control: how far to advance Predictor: advancing from last solution Corrector: eliminating or reducing the drift error NFEM Ch 20 Slide 7

8 Increment Control Nonlinear FEM Performing the nth increment step u n = u n+1 u n λ n = λ n+1 λ n Stepsize constraint c u n, λ n = 0 Rate form of stepsize constraint in which a T u + g λ = 0, a T = c u, g = c λ. NFEM Ch 20 Slide 8

9 Predictor Nonlinear FEM Predictor gives the increments u 0 n, λ0 n, Simplest predictor is Forward Euler applied to the stiffness rate equation ṙ = 0, or K u = q λ. For a prescribed λ increment, it gives u 0 n = K 1 n q n λ 0 n = v n λ 0 n NFEM Ch 20 Slide 9

10 Corrector Inserting the predicted values in the residual gives the drift error r 0 n = r( u n + u 0 n,λ n + λ 0 n) 0. A corrective process generates a series of updates (k = iteration index) u k λ k that hopefully make the total solution converge towards equilibrium thus eliminating the drift error NFEM Ch 20 Slide 10

11 Drift Error in Predictor-Only Continuation Process λ Computed solutions Drift error Actual Equilibrium Path u NFEM Ch 20 Slide 11

12 Stepsize Control Constraint Types P P P C C C S S S c=0 c=0 c=0 Stage parameter control (aka load control if λ is a load factor) State control Arclength control NFEM Ch 20 Slide 12

13 Stepsize Increment Control Types (2) P P P C C C S S S c=0 c=0 c=0 Hyperspherical control Global hyperelliptic control Local hyperelliptic control NFEM Ch 20 Slide 13

14 Predictor With Stage Parameter (aka Load) Control Nonlinear FEM NFEM Ch 20 Slide 14

15 Predictor With Arclength Control Nonlinear FEM NFEM Ch 20 Slide 15

16 Traversing Equilibrium Path in Positive Sense Criterion: positive external work over increment W = q T n u0 n = qt n v n λ n > 0. Generally effective, but fails if q T v = 0 E.g., bifurcation points or places where incremental velocity vector v vanishes NFEM Ch 20 Slide 16

17 Stage Parameter Control Nonlinear FEM Constraint: λ n = l n, Rate form: a = 0, g = 1. NFEM Ch 20 Slide 17

18 State Control (aka Displacement Control) Nonlinear FEM Constraint: c u n u T n u n 2 l n u 2 = 0 Rate form: a T = 2 u n g = 0 NFEM Ch 20 Slide 18

19 Arclength Control Nonlinear FEM Constraint: c( u n, λ n ) = s n l n = 1 f n v T n u n + λ n ln = 0 Rate form: f n =+ 1 + v T n v n. a T = v n /f n g = 1/f n NFEM Ch 20 Slide 19

Residual Force Equations

3 Residual Force Equations NFEM Ch 3 Slide 1 Total Force Residual Equation Vector form r(u,λ) = 0 r = total force residual vector u = state vector with displacement DOF Λ = array of control parameters