# 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.

Dynamic response of linear undamped system due to harmonic loading:

## 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 ΓΌ0. 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

## 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