Time-Marching Solution Methods

Time-Marching Solution Methods

1. Time Integration

   • Overview of Methods
     • The same form of approximation is used in both techniques to define the variation of acceleration over the time interval.
     • Implicit
       • Solve EXACT equations at t2, but variation of the response is APPROXIMATED over the time interval t1 to t2.
     • Explicit
       • Solve APPROXIMATE form of equations at t2, but variation of the response is APPROXIMATED over the time interval t1 to t2.
   • Basic Requirements
     • Accuracy & Convergence
       • Intrinsic error expressed in powers of \(\Delta\)t.
     • Stability
       • Stable if error remains bounded by a small number epsilon whilst marching in time.
     • Consistency
       • For an infinitesimally small time step the integration must equal the boundary condition. 

2. 1-DoF Systems

   • Acceleration Variation
     • Constant Acceleration
     • Linear Acceleration
   • Newmark-Beta
     • \(\gamma=a/2\) & \(\beta=b/6\)
     • Modified Velocity Assumption
       • Proportion \(\alpha\) of velocity response of constant acceleration and proportion a of velocity response of linear acceleration.
     • Modified Displacement Assumption
       • Proportion \(\beta\) of velocity response of constant acceleration and proportion b of velocity response of linear acceleration.
   • Central Difference Method
     • Assume constant acceleration over a time interval.

3. Multi-DoF Systems

   • Use single-DoF methods with vectors of displacements, velocities and accelerations.

4. Explicit Solution Methods

   • If \(\Delta\)t tends to 0 the equation tends to the exact one.
   • Damping & Stiffness forces assumed constant over the time step.
     • Newmark-Beta Explicit Form
     • Central Difference Explicit Form
       • EoM
   • Highly nonlinear problems, where the response is only required over a short time.

5. Implicit Solution Methods

   • Satisfy the complete solution at \(t_{2}\) knowing the response at \(t_{1}\). This has to be done by some iterative technique (e.g. Newton-Raphson). 
   • Computationally very much less efficient due to inversion of non-diagonal matrix.
   • Newton-Raphson
     • Accurate solution to nonlinear equations.
     • Requires more inversions.
   • Linear, or mildly nonlinear problems, where response duration is long.

6. Stability

   • Recurrence Matrix
     • Stable is largest eigenvalue of H has a modulus less than or equal to 1.

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