Time Integration: Fundamentals © Thomas J.R. Hughes

Outline 1)Semi-discrete Methods  Heat equation  Structural dynamics  Nonlinear systems 1)Space-time Methods

Semi-discrete Heat Equation

Generalized Trapezoidal Method

Explicit algorithms economical per step but “stability” limits size of time step. Implicit algorithms more expensive per step, but more stable, so larger time steps may generally be taken. Remarks

Commutative Diagram

Stability

Therefore, the stability condition is satisfied if either of the following is true: Numerical Stability

Significance of Stability Concept n A × × × × × ×10 41

Consistency and Convergence Rearrange algorithm: Arrange the exact solution in a similar fashion:

Theorem: Stability + Consistency Convergence In fact,

Semi-Discrete Equations of Motion Newmark Algorithm Nathan Newmark (Courtesy of the University of Illinois archives)

Various forms are useful:

Predictor/multicorrector Newmark algorithm (H. et al, 1979, H. 2000)

1.Average Acceleration Method Implicit Unconditionally Stable Second-order accurate Examples 

Average acceleration method is equivalent to the trapezoidal rule applied to first-order form of the equation of motion:

4.Central Difference Method Conditionally stable Second-order accurate 

Stability for Newmark Unconditional stability: Conditional stability:

Survey of Structural Dynamics Algorithms Implicit, unconditionally stable, (usually) second-order accurate, linear multi-step methods. Two-step displacement-difference equation algorithms: Newmark + Simple - High-frequency dissipation requires implying first-order accuracy

Three-step displacement-difference equation algorithms Houbolt + Very strong high-frequency dissipation - “Asymptotic annihilation” - “Poor” second-order accuracy Collocation/Wilson + Fair combination of low-frequency accuracy and high-frequency dissipation - “Overshoot” pathology Hilber-Hughes-Taylor (HHT α-method) + Good combination of low-frequency accuracy and high- frequency dissipation Chung-Hulbert (Generalized α-method) + Similar to HHT α-method + “Asymptotic annihilation”

Generalized α-method

Hilber-Hughes-Taylor (HHT α-method) David Hilbert (not Hilber!) Hans-Martin Hilber

Nonlinear Systems Outline 1.Semi-discrete equations of nonlinear mechanics 2.A simple class of nonlinear problems 3.Newmark algorithms 4.Predictor-corrector algorithms 5.Implicit-explicit finite element algorithms (“mesh partitions”)

Semi-Discrete Equations of Nonlinear Mechanics

A Simple Class of Nonlinear Problems - Includes nonlinear elasticity and some nonlinear “rate- type” viscoelastic materials Assume:

Step 1: Implicit Algorithm Newmark Algorithms

Implementation by Newton-Raphson: Displacement Form

Step 2: Explicit Algorithm Predictor-Corrector Algorithm Same as Newmark, except predictors

Implementation: same, except

Step 3: Synthesis Implicit-Explicit FE Algorithms Elements are divided into two groups: implicit group and explicit group Notation:

Implementation: same, except Note

Convergence and Accuracy

Explicit Predictor/Multicorrector Algorithms Implementation: same, except

Implementation by Newton-Raphson: Displacement Form Only changes compared with implicit algorithms

Space-time Formulations Example: Initial-Value Problem of Elastodynamics

Space-time Formulations

Discontinuous Galerkin Method in Time

Remarks: 1. 2.Continuity of the solution across time slabs is weakly enforced. 3.A complete mathematical convergence theory exists. 4.The issue of time integrators is eliminated by the choice of space-time interpolation. 5.Unconditional stability in all cases. 6.A system of linear algebraic equations on each time slab. 7.See Hughes-Hulbert: Vol. 36, pp (1988) Computer Methods in Applied Mechanics and Engineering.

Features of space-time discontinuous Galerkin finite element methods Inter-element discontinuous basis functions –Weak enforcement of balance/conservation conditions in space-time (e.g., Rankine- Hugoniot conditions in for conservation laws) –Enables exact conservation per element and O(N) complexity for hyperbolic problems

Features of space-time discontinuous Galerkin finite element methods Inter-element discontinuous basis functions –Weak enforcement of balance/conservation conditions in space-time (e.g., Rankine- Hugoniot conditions for conservation laws) –Enables exact conservation per element and O(N) complexity for hyperbolic problems

Causal space-time mesh and O(N) advancing-front solution strategy

Tent Pitcher: Causal space-time meshing causality constraint tent–pitching sequence Given a space mesh, Tent Pitcher constructs a space-time mesh such that every facet on sequence of advancing fronts is space-like (patch height bounded by causality constraint) Similar to CFL condition, except entirely local and not related to stability (required for O(N) solution)

Patch–by–patch meshing and solution Patches (‘tents’) of tetrahedra; solve immediately for O(N) method with rich parallel structure Maintain “space mesh” as advancing, space-like front with non-uniform time coordinates

Space-time Discontinuous Galerkin Methods for the Dynamics of Solids Robert B. Haber University of Illinois at Urbana–Champaign NSF: ITR/AP DMR ITR/AP DMR Center for Process Simulation & Design Materials Computation Center Structured Integration Workshop California Institute of Technology 7-8 May 2009

Crack-tip Wave Scattering