# Time integration

## Equation of Motion

Single degree of freedom system Forces acting on mass m Equilibrium – d Alembert s principle :  Equation of motion for linear elasticity : linear ordinary d.e.

Nonlinear case : nonlinear ordinary d.e.

Analytical solutions of linear ordinary di erential equations are available.  ## Direct Integration of Equation of Motion

• For nonlinear problems only numerical solutions are possible.
• Focus is on explicit methods, in particular Central Difference method.
• LS-DYNA uses a modification of the central di erence time integration.
• Central difference scheme is an explicit method.
• For explicit schemes the equation of motion is evaluated at the old time step tn, whereas implicit methods use the equation of motion at the new time step tn+1.

## Central Difference

• discretization • difference formula : Substitute equations (??) and (??) into (??) : • For lumped mass and damping the matrices M are diagonal. • Inversion of diagonal matrices M and C is trivial. • At timer t = 0 we have initial conditions u0 and &u-odot0. From equilibrium we find ü0. From equation (??) and (??) : • The central di erence scheme is conditionally stable, i.e. the size of the time step is limited.

## Stability of the Central Difference Scheme

Uncouple the system of linear equations of motion into the modal equations. Φ … modal matrix with M-orthonormalized eigenvectors stored in columns N uncoupled equations of motion with generalized displacements χ : central differences : Substitute xn and xn into equation of motion () at time t^n* : In matrix form A… time integration operator for discrete

For m-time steps and L = 0 Spectral decomposition of A : P … orthonormal matrix; contains eigenvectors of A

J … Jordan form; eigenvalues λi of A are stored on diagonal

spectral radius = ρ(A) = largest eigenvalue of A = max ((diag(J)) Eigenvalues of A for the undamped equation of motion For the damped equation of motion : Damping reduces the critical time step.

For varying time step sizes : • The time integration is stable, if the time step size decreases.
• The time step is bounded by the largest natural frequency of the structure.
• For shells: bending and membrane modes are present the frequency of the membrane mode usually limits the critical time step, since membrane stiffness is much larger than bending stiffness

## Critical time step of a rog ## Time Integration in LS-DYNA • discretization In LS-DYNA actual geometry x is used instead of displacements. Thus x replaces u.
• difference formula : equation of motion at time t^n for the nonlinear case:  (asynchronous damping)

assumption : update of accelerations

update formulas for velocities and displacements:

From (2): From (1): Remarks :

• starting procedure for first time step with • Standard central di erence method approximates time step limit for LS-DYNA time integration scheme.
• No stability proofs are available for time integration of nonlinear problems. Default in LS-DYNA : • critical time step for varying time increments 