Users:General FEM Analysis/Analyses Reference/Dynamic Analysis

Contents 1 General Description 1.1 Discretization in Space 1.2 Discretization in Time 1.3 Linear Equation System 1.4 Time Integration Methods 1.5 Geometric Linearity and Nonlinearity 1.6 Rayleigh Damping 1.7 Loading and Load Curves 1.8 Initial Load 1.9 Restart Functionaliy 1.10 Path Following Methods 2 Input Parameters 2.1 Parameter Description 2.2 Example of a Complete Input Block

General Description

The dynamic analysis is used to analyze the time-dependent or transient behavior of structures. There exist various analysis types with different time integration algorithms and geometric linear and nonlinear formulations to cover small and large structural displacements.

Discretization in Space

The unknown variables used to describe the transient behavior of a structure are the displacement d, the velocity v and the acceleration a. The discretization in space by the Finite Element Method (FEM) leads to the general form of the linear equation of motion:

M a(t)+C v(t)+K d(t)=f(t)

with M as the mass matrix, C as the damping matrix, K as the linear stiffness matrix and f as the external and time-dependent force vector. It has to be noted, that the unknown variables are time-dependent.

Discretization in Time

For the discretization in time a time integration algorithm is used. The discretization is done in two steps. First, the examined time period t=[t₀, t_tot] is subdivided into discrete intervalls [t_n, t_n+1] and corresponding time steps Δt_n=t_n+1-t_n. Seccond, for the progress of the variables within a timestep certain assumptions are made, depending on the choosen time integration algorithm. The solution is then only computed at discrete times. In Carat++ implicit time integration algorithms are used, which satisfy dynamic equilibrium at time n+1. Now, the system has the form:

M a_n+1+C v_n+1+K d_n+1=f_n+1

This is an equation system with ndof equations, but 3 x ndof unknowns.

Linear Equation System

The chosen time integration algorithm allows a reduction to ndof equations, so it makes the system solvable. The acceleration and velocity can be formulated depending on the displacements only, so the main variable is the displacement. Using e.g. the Newmark-beta method leads to the following linear equation system:

(1/βΔt² M + γ/2Δt C + K) d_n+1=f_n+1 + M(1/βΔt²d_n + 1/βΔt v_n + ...) +C(-v_n- (1-γ)Δt a_n+...)

This system is solved in every timestep. All terms on the right hand side are already known and the system can be solved for the unknown displacements d_n+1. After this, the corresponding velocity and acceleration are determined by backsubstitution.

Time Integration Methods

There are currently two time integration methods implemented:

• Newmark-beta method,
• Generalized-alpha method.

The Newmark-beta method uses two control parameters, β and γ. The choice of these parameters influences the accuracy and stability of the algorithm. Accuracy of 2nd order is given for β=0.25 and γ=0.5. Dissipation of higher frequencies is reached for γ≥0.5, but the accuracy is diminished to 1st order. Unconditional stability is guaranteed for a combination that ensures 2β≥γ≥0.5.

The Generalized-alpha method is based on the Newmark-beta method, but the equilibrium condition is formulated between two timesteps by linear blending. It uses four control parameters, β and γ from the Newmark-beta method and additionally the two linear shift-parameters α_m and α_f. The advantage of the Generalized-alpha method is to ensure 2nd order accuracy with controllable damping of higher frequency modes. The control parameters depend on the wanted damping of the higher frequencies, which is described by the spectral radius ρ_∞. For ρ_∞=1.0 no damping occurs, for ρ_∞<1.0 higher frequencies are damped. An overview on parameter combinations is given in the following table. Other values can be found in the corresponding literature.

Parameter Combination for Generalized-Alpha Method
Spectral radius ρ∞	β	γ	αm	αf
1.0	0.250	0.500	0.500	0.500
0.9	0.277	0.553	0.421	0.474
0.8	0.309	0.611	0.333	0.444

Geometric Linearity and Nonlinearity

Rayleigh Damping

Loading and Load Curves

Initial Load

Restart Functionaliy

Path Following Methods

The iterative solution of the nonlinear problem requires path following methods. The most simple path following method is a pure Load Control. Here the parameter lambda is increased according to the specified load curve. This allows nonlinear analysis for relatively robust nonlinear problems. But as soon as the structure shows instabilities the load control method usually does not not converge anymore. In this case the Arc Length method can be applied to compute the equilibrium path. This method controls the so called arc length which is defined as a combination from displacement increment and load increment. In general it is also possible to directly control a specified displacement. But actually this path control method is not available in Carat++. More information about path following methods can be found in the thesis of Reiner Reitinger^[1] and Amphon Jarusjarungkiat.

Example:  Actual iteration step:                                   i
          Number of desired equilibrium iterations:                J_d = 8.
          Number of equilibrium iterations in the last time step:  J_i-1 = 12.
          Number of restarts:                                      Nmb_Restarts = 0.
          Exponent:                                                p=1.0.
          Step length of last step:                                StepLength_i-1 = 0.2.

          alpha = ( 8 / (12*(0+1)))^(1.0) = (8 / 12)^(1.0) = 2/3.
          StepLength_i = 2/3 * 0.2 = 0.1333. 

Input Parameters

Parameter Description

Example of a Complete Input Block

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