Users:General FEM Analysis/Analyses Reference/Dynamic Analysis

Revision as of 11:49, 6 September 2010

• Newmark-Beta
• Generalized Alpha
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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.

Structure of Equation System

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. The chosen time integration algorithm allows a redcution 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+...)

d bekannt: rückrechnung für a und v

dämpfung rayleigh

einteilung in zeitschritte,n+1,

exp + imp, hier impl-> gg an n+1

3*ndof variable aber nur ndof gleichung pro zeitschritt -> time itnegration algorithm,l beziehungen zw. a,v und d -> reduktion auf die variable Verschiebungen, lösbar, d bekannt: rückrechnung für a und v

dämpfung rayleigh

A nonlinear problem is formulated by the equation r = f_int(u) - λ * f_ext(u) where r specifies the residual vector and f_int and f_ext define the internal and external forces respectively. In general, the internal forces as well as the external forces depend on the actual displacement field u. Thus, the equation is nonlinear with respect to the a priori unknown equilibrium displacements. The parameter λ specifies a scaling parameter for the external load. Usually this parameter is controlled by a load curve which specifies λ as a function of the pseudo time t. The slope of this load curve must be small enough such that convergence is possible in each step. Whenever a nonlinear analysis has convergence problems it is appropriate to reduce the slope of this load curve and calculate more time steps.

At the equilibrium point the internal forces are equal to the external forces and the residual vector is equal to zero. The above specified nonlinear problem is linearized for the actual displacement state and solved by a Newton-Raphson scheme where the residual vector is used to compute incremental displacements by K_t u_inc = r. Here the tangential stiffness matrix is specified by K_t whereas u_inc denotes the incremental displacements.

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.

Step Length Control

Actually there exist two methods for step length control in Carat++. The most simple method is the fixed step length which computes the whole load displacement path with a constant step length. This procedure is robust but inefficient because a large number of load steps has to be computed. Usually it is more efficient to apply adaptive step length methods. A very robust method was presented by Crisfield and Ramm, c.f. ^[1]. This method specifies the actual step length for the time step i by the operation StepLength_i = alpha * StepLength_i-1. The parameter alpha is computed by (J_d / (J_i-1 * (Nmb_Restarts+1)))^p with: J_d = desired number of equilibrium iterations, J_i-1 = equilibrium iterations of last time step and p equals the exponent.

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. 

Thus, if the number of equilibrium iterations is larger than a desired value the step length is decreased for the next time step. If the number of equilibrium iterations is smaller than a desired value the step length is increased for the next time step. The exponent p can be used to manipulate the update speed. It should be used with care.

Simultaneous Eigenvalue Analysis

Reaching critical points of the load displacement path (limit points, bifurcation points, ...) requires special investigations. A simultaneous eigenvalue analysis solving the problem (K_t - eigenvalue * I) phi = 0 allows for approximation of the load factor at the next critical point lambda_k. Whenever a critical point is reached the above equations yields to a zero eigenvalue. Extrapolating the current load factor according to the eigenvalue allows for good approximations of the critical load.

Imperfect Designs

Many structures subjected to large compression loading are sensitive to geometrical imperfections. Unfortunately the shape of the imperfection is a priori unknown but mostly some information about tolerances exist. In the nonlinear analysis the imperfection shapes can be computed as sum of several normalized eigenmodes. The final imperfection mode is then scaled by the specified tolerance. This operation can be expressed by dx_imp = tol * (sum_i phi_i). After computation of the imperfection mode, the geometry of the structure is modified and the nonlinear analysis starts with the imperfect geometry.

Input Parameters

Parameter Description

Example of a Complete Input Block