LECTURE 17DIRECT TIME INTEGRATION
IN DYNAMICS
Radek KolmanInstitute of Thermomechanics
The Czech Academy of Sciences, Prague
Michal Mracko, Alena Kruisova, Anton Tkachuk
An ECCOMAS Advanced Course on Computational Structural DynamicsJune 4 – 8, 2018
Prague, Czech Republic
Contents
• FEM in dynamics, formulation of dynamic problems
• Introduction into direct time integration
• Basic methods - Newmark method and central differ-ence method
• Solving of nonlinear time-depend problems
• Time step size estimates
•Mass scaling
Basic literature
Belytschko T., Hughes T.J.R. Computational Methods for Transient Analysis. North-Holland:Amsterdam, 1983.
Bathe K.J. Finite Element Procedures, Prantice-Hall, Englewood Cliffs, New York, 1996.
Hughes T.J.R. The Finite Element method: Linear and Dynamic Finite Element Analysis. DoverPublications: New York, 2000.
Har J., Tamma K. Advances in Computational Dynamics of Particles, Materials and Structures.John Wiley: New York, 2011.
Wu S.R., Gu. L. Introduction to the Explicit Finite Element Method for Nonlinear TransientDynamics. John Wiley: New York, 2012.
Felippa C. Introduction to Finite Element Methods, lecture notes, Department of AerospaceEngineering Sciences, University of Colorado at Boulder, 2017.
1. Finite element method in linear dynamics,formulation of dynamic problems
Governing equations for solid mechanics
Strong form:ρ ui = σij,j + bi in Ω× [t0, T ]
ui = gi on ΓD × [t0, T ]
σijnj = hi on ΓN × [t0, T ]
ui(x, t0
)= u0i (x) for x ∈ Ω
ui(x, t0
)= u0i (x) for x ∈ Ω
Hooke’s law: σij = Cijklεkl
Theory of small deformation:εkl =
1
2
(∂uk∂xl
+∂ul∂xk
)ui - the component of displacement vector u(x, t);x ∈ Ω - the position vector;Ω - the domain of interest with the boundary Γσij - the Cauchy stress tensor (symmetric tensor); εkl - the infinitesimal strain tensor;Cijkl - elasticity tensor; ρ - mass density;bi - the component of volume (body) intensity vector b;ni - the component of the outward normal vector n on Γ;gi - the component of prescribed boundary displacement vector g;hi - the component of prescribed traction vector h;u0i and u0i - the components of the initial displacement and velocity fields.
FEM recapitulation
Approximation of displacement field via shape functions N
uh = N q
where q is vector of generalized nodal quantities (displacements/rotations, etc.).
Approximation of velocity and acceleration fields
uh = N q uh = N q
Infinitesimal strain tensor
ε = Duh,
where D is the differential operator. Then
ε = DN q = B q
where B is the strain-displacement matrix. In elasticity problems, stress is given as
σ = D ε
where D is the elasticity matrix.
FEM recapitulation
Energy balance (principle of virtual work):
δEk + δU = δP ,
after definition of kinetic energy Ek, potential (strain) energy U and work of ex-
ternal forces P , it yields∫Ω
δuT %u dΩ +
∫Ω
δεT σ dΩ =
∫Ω
δuT b dΩ +
∫ΓN
δuT h dΓ
Using discretization of kinematic quantities we have
δqT[∫
Ω
%NTNq dΩ +
∫Ω
BT σ dΩ−∫
Ω
NT b dΩ−∫
ΓN
NT h dΓ
]= 0.
The previous equation should be valid for an arbitrary δq and, then the discretized
equations of motion have the form
Mq = f ext − f int
FEM recapitulation
Discretized equations of motion:
Mq = f ext − f int
Consistent mass matrix:
M =
∫Ω
%NTN dΩ
Vector of internal forces:
f int =
∫Ω
BT σ dΩ
Vector of external forces:
f ext =
∫Ω
NT b dΩ +
∫ΓN
NT h dΓ + fsingular
In impact-contact problems, equations of motion have the form
Mq = f ext − f int − f contact
FEM for linear problems
The continuous Galerkin-Bubnov approximation method.Finite element approximation of the displacement field u:
uh(x, t) =NDOF∑I=1
NI(x)uI(t), δuh(x, t) =NDOF∑I=1
NI(x)δuI(t)
where uI are unknown nodal displacements.Discrete equations of motion for linear elasticity problems:
Mu + Ku = f ext
+ nodal Dirichlet boundary conditions.Internal forces are given as
f int = Ku
with the stiffness matrix defined as
K =
∫Ω
BTDB dΩ
General linear problems:
Mu(t) + Cdampingu(t) + Ku(t) = f ext(t) (1)
with the damping matrix C. Remark: Rayleigh damping matrix: Cdamping = aM + bK
2. Introduction into direct time integration
nodal displacement vector: u(t)nodal velocity vector : u(t) = v(t)nodal acceleration vector : u(t) = a(t)
Solutions of discretized equations of motion
• modal superposition (LECTURE 15)
• direct time integration
System of second order ordinary differential equations:
Mu(t) + Cu(t) + Ku(t) = f ext(t)− f contact(t) (2)
In direct time integration,
approximation of quantities
at discrete time tn
u(tn) ≈ uh(tn) = un
Temporal discretization:
t = 0, t1, t2, t3, . . . , T
Time step size:
∆ti = ti+1 − tiFor constant time step size ∆t:
tn = n∆t, n = 0, 1, 2, . . . , N
Solutions of discretized equations of motion
Mathematical methods for numerical solution of
the first-order system
y = f (y, t),y = (u, u)T − state space
• The forward Euler method
• The backward Euler method
• The generalized trapeziodal method
• The midpoint method
• Methods of the Runge-Kutta type
• The central difference method
• Linear multi-step methods
• Other methods
the second-order system
u = f (u, u, t)
• The Newmark method
• The Houbolt method
• The Wilson θ method
• The Midpoint method
• The Central difference method
• The HHT method
• The Generalized-α method
• Other methods
A predictor/multi-corrector formof time scheme
The generalized-α method [Chung, Hulbert 1993]
Start
Predictori = 0
Stop
i = i+ 1
Correctorai+1n+1 = ain+1 + ∆a
∥∥Rin+1
∥∥ ≤ ε∥∥R0
n+1
∥∥Test
?Yes
NodRi
dan+1
∆a = −Rin+1
din+1 = dn + ∆tvn +(∆t)2
2
((1− 2β)an + 2βain+1
)ain+1 =(γ − 1)
γan
vin+1 = vn
vi+1n+1 = vin+1 + γ∆t∆a
di+1n+1 = din+1 + β (∆t)2 ∆a
din+αf= dn + αf (d
in+1 − dn)
vin+αf= vn + αf (d
in+1 − dn)
ain+αm= ain+1 + αm(ain+1 − an)
Rin+1 = R(din+αf
,vin+αf, ain+αm
)
Direct time integration methods in FEM
Implicit methods:
• Methods of the Newmark’s family [Newmark 1959]
• The HHT method [Hilber, Hughes, Taylor 1977]
• The midpoint method [Simo 1991]
• The generalized-α method [Chung, Hulbert 1993]
Explicit methods:
• The central difference method [Krieg 1973, Dokainish & Subbaraj 1989, in each FEM book]
• Methods on the next slide.
Implicit-explicit methods:
• Methods of authors: Belytschko, Mullen, Liu, Hughes, Fellipa, K.C. Park, Combescure,
Farhat, Tezaur, and others.
Other methods:
• Asynchronous, symplectic or variational time integrations.
A review of explicit time integration methods in FEM
• the central difference method [Krieg 1973, Dokainish & Subbaraj 1989, in each FEM book]
• the Verlet method [Verlet 1967] (molecular dynamics)
• the Trujillo method [Trujillo 1977]
• the Park variable-step central difference method [K.C. Park & Underwood 1980]
• the Chung and Lee method [Chung & Lee 1994]
• the explicit form of the generalized-α method [Hulbert & Chung 1996]
• the Zhai method [Zhai 1996]
• the Tchamwa–Wielgosz method [Tchamwa & Conway & Wielgosz 1999]
• the explicit predictor/multi-corrector method [Hughes 2000]
• the Tamma et al. method [Tamma et al. 2003]
• the Chang pseudo-dynamic method [Chang 2008]
• the semi-explicit modified mass method [Doyen et al. 2011]
• the Yin method [Yin 2013]
• the two-time step Bathe method [Noh & Bathe 2013]
• the multi-time step Park method [Park et al. 2012, Cho et al. 2013, Kolman et al. 2016]
• survey and comparative papers [Fung 2003, Rio 2005, Nsiampa 2008, Maheo 2013].
Numerical errors, properties of time integrators
Numerical errors:
• dispersion (distortion of pulse), anisotropy and diffraction, polarization errors
• spurious oscillations, parasitic modes
• numerical dissipation and attenuation
• period elongation and amplification
Requirements and properties of explicit methods:
• diagonal mass and damping matrices
• second-order accuracy
• symplectic and energy and momentum conserving
• unconditionally/conditionally stability, time step size estimator
• numerical dissipation controlled by a parameter
• the numerical dissipation should affect higher modes; lower modes should not be affected
• an effective evaluator of RHS, underintegration of linear FEs or Hourglass controlling.
Numerical methods for wave problems
• finite difference method (FDM) in time and space
• finite element method (FEM)
• boundary element method (BEM)
• finite volume method (FVM)
• spectral methods
• pseudo-spectral method
• collocation methods
• wavelet-based methods
• mass-spring approximations
• smooth particle hydrodynamics method
• discontinuous Galerkin (DG) method and more others.
3. Basic methods - The Newmark method
Nathan M. Newmark, 1910 – 1981University of Illinois at Urbana-Champaign
The Newmark method
• Newmark, N.M.: A method of computation for structural dynamic. Journal of
the Engineering Mechanics Division, 85, pp 67–94, 1959.
Kinematic quantities:
ut+∆t = ut + ∆t ut +∆t2
2
((1− 2β) ut + 2β ut+∆t
)ut+∆t = ut + ∆t
((1− γ) ut + γ ut+∆t
)Newmark’s parameters γ and β.
Equations of motion at time t + ∆t:
Mut+∆t + Kut+∆t = f t+∆text
Discrete operator (displacement form)
Keffut+∆t = f t+∆t
eff
where
Keff = K + a0M, f t+∆teff = f t+∆t
ext + M(a0u
t + a1ut + a2u
t)
a0 = 1/(β∆t2
), a1 = 1/ (β∆t) , a2 = 1/ (2β)− 1
Newmark method - predictor/corrector form
Kinematic quantities:
ut+∆t = ut + ∆t ut +∆t2
2
((1− 2β) ut + 2β ut+∆t
)ut+∆t = ut + ∆t
((1− γ) ut + γ ut+∆t
)Predictor phase:
ut+∆t = ut + ∆tvt +∆t2
2(1− 2β) at
vt+∆t = vt + ∆t (1− γ) at
Equations of motion at time t + ∆t for update acceleration:(M + β∆t2K
)at+∆t = f t+∆t
ext −Kut+∆t
Corrector phase:
ut+∆t = ut+∆t + β∆t2at+∆t
vt+∆t = vt+∆t + γ∆t at+∆t
Advantage: in memory only ut+∆t, vt+∆t, at+∆t
The Newmark method
Special cases:
• The average acceleration method: β = 14, γ = 1
2
• The linear acceleration method: β = 16, γ = 1
2
• The Fox-Goodwin method: β = 112, γ = 1
2
• The central difference method: β = 0, γ = 12
Unconditional stability (time step can be chosen arbitrary)
2β ≥ γ ≥ 1
2
The Newmark parameters for conditional stability must satisfy:
γ ≥ 1
2, β <
1
2γ, Ω ≤ Ωcrit
where Ω = ωh∆t
Ωcrit = (γ/2− β)−12
The Newmark method with β = 14, γ = 1
2
Requirements:
• Consistent mass matrix
• An efficient and performance linear solver
Properties:
• implicit method, so-called the constant-average-acceleration method
• unconditionally stable (time step can be chosen arbitrary)
• second order accuracy
• conserves of total energy
• no amplitude decay
• period elongation
• for 5% relative period error is ∆t.= 0.125Tmin, where Tmin = 2π/ωmax is
minimal period vibration.
One-dimensional stress wave in a bar
Linear (classical) wave equation
∂2u
∂t2= c2
0
∂2u
∂x2
u - displacement, x - position, t - time, c0 =√E/ρ - wave speed
L
xF(t)A,E,r
Scheme of a free-fixed bar under an impact loading.
Loading
σ(0, t) = −σ0H(t)
σ is the stress, H is the Heaviside step function.
Analytical solution
σ(x, t) = −σ0H(c0t− x)
KF Graff. Wave motion in elastic solids. Oxford University Press, 1975
One dimensional wave propagation test
The time step size defined by the Courant number C0 = ∆tc0H , H is the finite
element length.
0 0.2 0.4 0.6 0.7 0.8 1−1.5
−1
−0.5
0
0.5
x/L
σ/σ 0
Newmark’s method, linear FEM, CMM, Co = 0.1
0 0.2 0.4 0.6 0.7 0.8 1−1.5
−1
−0.5
0
0.5
x/L
σ/σ 0
Newmark’s method, linear FEM, CMM, Co = 0.5
0 0.2 0.4 0.6 0.7 0.8 1−1.5
−1
−0.5
0
0.5
x/L
σ/σ 0
Newmark’s method, linear FEM, CMM, Co = 1.0
One dimensional wave propagation test
0 0.2 0.4 0.6 0.7 0.8 1−1.5
−1
−0.5
0
0.5
x/L
σ/σ 0
Newmark’s method, quadratic FEM, CMM, Co = 0.1
0 0.2 0.4 0.6 0.7 0.8 1−1.5
−1
−0.5
0
0.5
x/L
σ/σ 0
Newmark’s method, quadratic FEM, CMM, Co = 0.5
0 0.2 0.4 0.6 0.7 0.8 1−1.5
−1
−0.5
0
0.5
x/L
σ/σ 0
Newmark’s method, quadratic FEM, CMM, Co = 1.0
4. Basic methods - Central difference method
The central difference method
Dokainish M.A., Subbaraj K. A survey of direct time-integration methods in com-
putational structural dynamics - I. Explicit methods. Comput. & Struct., 32(6),
1371–1386, 1989.
Equations of motion at the time t:
Mut = f ext − f int − f cont
Approximation of time derivatives - Central difference scheme in time:
ut ≈ ut+∆t − ut−∆t
2∆tut ≈ ut+∆t − 2ut + ut−∆t
∆t2
The central difference method
The Newmark method with β = 0, γ = 1/2.
Kinematic quantities:
ut+∆t = ut + ∆t ut +∆t2
2ut
ut+∆t = ut +∆t
2
(ut + ut+∆t
)Equations of motion at the time t:
Mut + Kut = f text
Approximation of velocity and acceleration by the central differencies:
ut ≈ 1
2∆t
(ut+∆t − ut−∆t
)ut ≈ 1
∆t2(ut+∆t − 2ut + ut−∆t
)
Implementation I
F teff = F t
ext −[K − 2
∆t2M]ut − 1
∆t2Mut−∆t
Meff =1
∆t2M
ut+∆t = M−1eff F
teff
In memory: displacements ut+∆t, ut, ut−∆t
The rest of quantities are computed if they are needed.
Implementation II -
Solve time t = 0:
Evaluate force residual: r0 = fext(t = 0)−Ku0
Compute acceleration: u0 = M−1r0
for n = 1...N (time steps)
Evaluate force residual: rn = fnext −KunCompute nodal accelerations: un = M−1rn
Update nodal velocities: un+1/2 = un−1/2 + ∆tun
Update nodal displacements: un+1 = un + ∆tun+1/2
end for
In memory: displacements ut+∆t, velocities ut+∆/2, accelerations ut
Implementation III -Predictor-corrector form
Predictorun+1 = un + ∆tun +
∆t2
2un
˙un+1
= un +∆t
2un
¨un+1
= 0
Solve equations of motion at the time tn+1 = tn + ∆t
MƬun+1
= fext(tn+1)− fint(t
n+1, un+1, ˙un+1
)− fcont(tn+1, un+1, ˙u
n+1)
Correctorun+1 = un+1
un+1 = ˙un+1
+∆t
2Ƭu
n+1
un+1 = Ƭun+1
Advantage: in memory only ut+∆t, vt+∆t, at+∆t
Central difference method
Requirement:
• for efficient computations, it is needed the inversion of M
• lumped (diagonal) mass matrix - no required a linear solver
Properties:
• explicit method
• conditionally stable (time step can not be chosen arbitrary)
• second order accuracy
• conserving of total energy in the limit ∆t→ 0, energy oscillations in sense of
the shadow Hamiltonian
• no amplitude decay
• period shortening
• the best choice for the time step size is the critical time step, so ∆t = α∆tcr,
where α = 0.9
One dimensional wave test
0 0.2 0.4 0.6 0.7 0.8 1−1.5
−1
−0.5
0
0.5
x/L
σ/σ 0
CDM, linear FEM, LMM, Co = 0.1
0 0.2 0.4 0.6 0.7 0.8 1−1.5
−1
−0.5
0
0.5
x/L
σ/σ 0
CDM, linear FEM, LMM, Co = 0.5
0 0.2 0.4 0.6 0.7 0.8 1−1.5
−1
−0.5
0
0.5
x/L
σ/σ 0
CDM, linear FEM, LMM, Co = 1.0
5. Lumping techniques for mass matrices
See Lecture 8.
Mass lumping (diagonalization)
Row sum method: meii =
n∑j=1
meij
The row sum method produces negative diagonal terms for higher-order FEM.The HRZ (Hinton-Rock-Zienkiewicz) method:A scaling method for conserving of total element mass. Procedure is as follows.
1. For each coordinate direction, select the DOFs that contribute to motion in that direction.From this set, separate translational DOF and rotational DOF subsets.
2. Add up the CMM diagonal entries pertaining to the translational DOF subset only. Callthe sum S.
3. Apportion Me to DLMM entries of both subsets on dividing the CMM diagonal entriesby S.
4. Repeat for all coordinate directions.
The HRZ method can be used for higher-order FEM or FEM with rotation DOFs (beams, plates,shells).
• Hinton, E., Rock, T.A. & Zienkiewicz, O.C.: A note on mass lumping and related processesin the finite element method. Int. J. Earthquake Eng. Struct. Dyn., 4, pp 245–249, 1976.
Direct inversion of mass matrix of consistent type
In explicit time integration, we need to solve
u = M-1(f ext − f int − f contact
)The aim is to take the direct inversion of the mass matrix M-1 from the consistent mass Mwithout a lumping so that M-1 satisfies following properties
• It should accurately keep both low and intermediate-frequency response components;
• Except for discontinuous wave propagation problems, its numerically stable explicit inte-gration step size should be much larger than employing the standard mass matrix.
• Its inverse should be inexpensive to generate, preferably without factorization computations.
This issue is still an open problem.
6. Solving of nonlinear time-depend problems
Solving of nonlinear time-depend problems
Vector of internal forces:
fint =
∫Ω
BT σ(deformation tensor, strain-rate, time, temperature, internal variables) dΩ
Often, vector local internal forces are evaluated by one-point Gauss integration (only one inte-
gration point taken place in the centroid of a finite element) and the tress tensor σ are kept as
an internal state variable.
Algorithm:Initial conditions and initialization, time t = 0.Set initial velocity u0, σ0 and initial values of other internal material variablesSet initial displacement u0, n = 0, compute M or M-1
Evaluate internal force fnint, Evaluate external force fnext, Evaluate contact force fncontEvaluate force residual: rn = fnext − fnint − fncontCompute accelerations un = M−1rn
Time update: tn+1 = tn + ∆tn+1/2, tn+1/2 = 12(tn + tn+1)
Update nodal velocities un+1/2 = un + (tn+1/2 − tn)un
Enforce velocity boundary conditionsUpdate nodal displacements un+1 = un + ∆tn+1/2un+1/2
Evaluate internal force fn+1int for un+1
Compute force residual rn+1 at tn+1 and accelerations un+1
Update nodal velocities un+1 = un+1/2 + (tn+1 − tn+1/2)un+1
Check energy balance at the time step n+ 1Update counter n = n+ 1Goto to STEP TIME UPDATE
7. Stability of time schemes
Stability theory - modal transformation
Eigen-value problem:KΦ = ω2MΦ
Modal transformation:u = ΦX,
with propertiesΦTKΦ = I, ΦTMΦ = Ω2
Ω is the diagonal matrix.Equation of motion after modal transformation:
X(t) + Ω2X(t) = ΦTR(t),
We have an independent equation
Xi(t) + ω2iXi(t) = (ΦTR)i(t),
For analysis of stability, it is sufficient to study
X(t) + ω2maxX(t) = r(t),
Stability theory
In direct time integration, the recursive relationship in time stepping process has a form[ut+∆t
ut+∆t
]= A
[ut
ut
]+ Lt+ν(r), (3)
where A marks the amplification operator, which dictates stability behaviour of the method.We define the spectral radius of A
ρ (A) = maxi=1,2,...,n
|λi| , (4)
where λi denotes the i-the eigen value of the operator AStability criterion yields:
1. if all eigenvalues are distinct, it must be satisfied ρ (A) ≤ 1 whereas
2. If A contains multiple eigenvalues, we require that all such eigenvalues |λi| < 1.
Stability theory - The Newmark method
The Newmark method with β = 1/2 and γ = 1/4
A = (1 + ω2∆t2/4)
[1− ω2∆t2
4 , ∆t
−ω2∆t, 1− ω2∆t2
4
](5)
eigen-values
λ1,2 =−(
1− ω2∆t2
4
)± iω∆t
1 + ω2∆t2
4
(6)
with propertiesρ (A) = 1. (7)
It means that the Newmark method with β = 1/2 and γ = 1/4 is unconditionally stable.
Stability theory - Central difference method
A =
[2− ω2∆t2, −1
1, 0
], (8)
with eigen-values
λ1,2 =2− ω2∆t2
2±√
(2− ω2∆t2)2
4− 1. (9)
0 0.5 1 1.5 2 2.5 30
1
2
3
4
5
ω ∆ t
ρ(A)
central difference methodNewmark method
The central difference method is conditionally stable.Stability limit for the central difference method
∆tω ≤ 2 (10)
It yields the stability formula for the time step size ∆t as
∆t ≤ 2
ωmax(11)
where ωmax is the maximum eigen value of the discretized system.
8. Time step size estimations for FEM
Belytschko, T., Liu, W. K., Moran, B., Elkhodary, K. (2013). Nonlinear finite elements forcontinua and structures. II edition. John Wiley & Sons.
Wu, S. R., Gu, L. (2012). Introduction to the explicit finite element method for nonlineartransient dynamics. John Wiley & Sons.
Benson (1998). Stable time step estimation for multi-material Eulerianhydrocodes. CMAME
Belytschko, Smolinski, Liu (1985). Stability of multi-time step partitioned integrators for first-order finite element systems. CMAME
Kulak (1989). Critical time step estimation for three-dimensional explicit impact analysis: Struc-tures under Shock and Impact, ed, Bulson: 155-163, Elsvier, Amsterdam
Time step size estimations for FEM
We define the Courant number:
Co =∆tc1
H
∆t - time step size, c1 is wave speed of longitudinal wave, H - characteristic length
(length of finite element edge)
The non-dimensional angular velocity:
ω =ωH
c1
The critical time step size for the cetral difference method
∆tcr =2
ωmax
Then, the critical time step size is given
Cocr =∆tcrc1
H=
2
ωmax
Time step size estimations for FEM
Wave speeds in elastic solids under small deformation theory
3D longitudinal wave: c1 =√
(Λ + 2G)/ρ
3D shear wave: c2 =√G/ρ
2D longitudinal wave under plane strain state: c1 =√
(Λ + 2G)/ρ
2D shear wave under plane strain state: c2 =√G/ρ
2D longitudinal wave under plane stress state: c1 =
√E
(1− ν2) ρ
2D shear wave under plane stress state: c2 =√G/ρ
1D longitudinal wave under uniaxial strain state: c =
√(1− ν)E
(1 + ν)(1− 2ν) ρ
1D longitudinal wave under uniaxial stress state: c =√E/ρ
Time step size estimations for FEM
Stability limit for the central difference method ∆t ≤ ∆tcr = 2ωmax
Methods of time step size estimations
• global methods (computation or estimation of ωmax, KΦ = ω2MΦ )
- ωmax can be computed or estimated using global mass and stiffness matrices.
• element based methods (computation or estimation of ωemax on elemental
level,
KeΦe = ω2eM
eΦe
- respecting the element eigenvalue inequality ωmax ≤ maxi ωei over all finite
elements.
The highest eigenvalue of dissassembled system is higher than the highest
eigenvalue of the assembled system.
• nodal based methods - ωmax can be estimated from nodal stiffness and
mass properties based on the Gershgorin’s theorem
- estimation of maximum eigenvalue of system (M-1K− ω2I)Φ = 0
Time step size estimations for FEM
Global methodsPower iteration:
AΦ = λΦ A = M−1K
Algorithm:
1. Initialize eigenvector Φ0, e.g. randon in range [-1,1], i = 0
2. i=i+1
3. Compute Ψi+1 = KΦi or as internal force Ψi+1 = fint(Φi)
4. Compute χi+1 = M-1Ψi
5. Compute estimate of eigenvalue λmaxi+1 = ‖χi+1‖
6. Update eigenvector Φi+1 = χi+1/λmaxi+1
7. If |λmaxi+1 /λmaxi − 1| > ε or i < N iter go to STEP 2.
8. Finish
Time step size estimations for FEM
Element based methodsUsing the eigenvalue inequality ωmax ≤ maxi ω
ei
1. Power iteration of elemental level
2. Upper bound for eigenfrequency for 2D four-noded quadrilateral or 3D eight-
noded brick solid elements
ωmax ≤ maxiωei ≤
√∑k,j
BkjBkj
with longitudinal wave speed c1 and strain-displacement matrix B
Flanagan, Belytschko (1981). A uniform strain hexahedron and quadrilateral with
orthogonal hourglass control. International Journal for Numerical Methods in En-
gineering
Time step size estimations for FEM
Element based methods
3. Upper bound for eigenfrequency
ωmax ≤ maxiωei ≤
c1
lewith longitudinal wave speed c and characteristic length of element le.
How to choose le?1
l2De =Aelement
lmaxl3De =
VelementAmax
4. CFL (Courant-Friedrichs-Lewy 2) condition ∆t ≤ α lec1
, α depends on element
type, integration type, order, shape, mass matrix, mass scaling, etc.1LS-DYNA manual2Courant, R., Friedrichs, K., Lewy, H., 1967, On the partial difference equations of mathematical physics
Critical Courant number -bilinear elements
Linear 1D FEM with the lumped mass matrix
ωhmax =2c0
H, ωhmax = 2, Cocr =
∆tcr c0
H=
2
ωhmax= 1
Linear 1D FEM with the consistent mass matrix
ωhmax =
√12c0
H, ω =
√12, Cocr =
∆tcr c0
H=
2
ωhmax= 1/√
3 ≈ 0.577
Square linear 2D and 3D FEM the with diagonal mass matrix
Cocrit =∆tcrit c1
H= 1
Serendipity quadratic (eight-noded) 2D and 3D FEM with the lumped mass by the
HRZ method
Cocrit =∆tcrit c1
H≈ 0.2
Critical Courant number -bilinear elements
Numerical experiment with bilinear FEM, diagonal mass matrix, free boundary
conditions, plain strain problem
mesh density ωmax Cocrit1× 1 2.3905 0.83662× 2 2.0723 0.96514× 4 2.0327 0.98398× 8 2.0199 0.9901
16× 16 2.0179 0.991132× 32 2.0176 0.991264× 64 2.0176 0.9912
infinity mesh 2.0 1.0
ω =ωH
c1, Co =
∆t c1
H, Cocrit =
2
ωmax
Time step size estimations for FEM
Nodal based methods
Gershgorin circle theorem3 based method: For a given square matrix A (complex
n× n matrix) the Gershgorin’s circle which belongs to the i-th diagonal entry Aii
is defined as
Si(Aii,Ri =n∑
j=1,i 6=j| Aij |), i = 1, ..., n
where Si defines a circle with radius Ri and position around x-axis at the posi-
tion Aii.
Gershgorin’s circle: D(Aii, Ri) Ri =n∑
j=1,i 6=j| Aij |
Example: A =
3 −0.5 0.4−0.75 4 −0.5
0 −0.7 1
row-wise column-wiseD(3, 0.9) D(3, 0.75)D(4, 1.25) D(4, 1.2)D(1, 0.7) D(1, 0.9)
3Gerschgorin, S., 1931, Uber die Abgrenzung der Eigenwerte einer Matrix
Time step size estimations for FEM
Nodal based methods
Application for FEM with lumped mass matrix4:
ω2max ≤ max
i
n∑j=1
| Kij |
Mii
This method respects Dirichlet boundary conditions.
Application for FEM with lumped mass matrix in contact-impact problems using
penalty formulation
ω2max ≤ max
i
n∑j=1
| Kij | +Kpi
Mii
where Kp is the corresponding penalized stiffness matrix.
4Kulak, R., F., 1989, Critical Time Step Estimation for Three-Dimensional Explicit Impact Analysis
Time step size estimations for FEM
Example: fix-free bar.
L = 10m A = 0.1m2
ρ = 7850 kg ·m−3 E = 210GPanumber of elements = 6 le = see table below
Different meshes with corresponding element lengths le [m]
Regular 1.667 1.667 1.667 1.667 1.667 1.667Irregular 1 1.68 1.65 1.67 1.65 1.67 1.68Irregular 2 1 0.5 3 1.5 2 2Irregular 3 0.5 1 1.5 2 2 3Irregular 4 3 2 2 1.5 1 0.5
Graphical representation of FE meshes:
Time step size estimations for FEM
Comparison of the dimensionless estimated eigen-frequencies for free-free bar:
ωdimless =ωestωnom
Local GlobalMesh max 2c
leGer. Eig Ger. Eig Nominal value [Hz]
Regular 1.000 1.000 1.000 1.000 1.000 987.8Irregular 1 1.010 1.010 1.010 1.038 1.000 988.0Irregular 2 1.706 1.706 1.706 1.207 1.000 1929.8Irregular 3 1.202 1.202 1.202 1.202 1.000 2738.4Irregular 4 1.202 1.202 1.202 1.202 1.000 2738.4
Comparison of the dimensionless estimated eigen-frequencies for fix-free bar:
Local GlobalMesh max 2c
leGer. Eig Ger. Eig Nominal value [Hz]
Regular 1.009 1.009 1.009 1.009 1.000 979.4Irregular 1 1.019 1.019 1.019 1.013 1.000 978.8Irregular 2 1.814 1.814 1.814 1.171 1.000 1815.6Irregular 3 1.916 1.916 1.916 1.106 1.000 1718.7Irregular 4 1.202 1.202 1.202 1.202 1.000 2738.4
9. Mass scaling
Motivation: change frequency spectrum of FEM model via modification of mass
matrix, affect maximum eigen-frequency of FE system so that the critical time step
is larger and computations is efficient.
Smaller maximum eigen-value ⇒ larger time step size
Modification of mass matrix as
Mo = M + λo
where λo is the artificial added mass matrix.
Mass scaling
Methods of mass scaling in FEM
• convential mass scaling - adding artificial mass in diagonal terms of mass
matrix
mλe =
ρAle2
[1 0
0 1
]mo
e = me + αmλe
- preserving the diagonal structure of mass matrix
- increasing element inertia - applied only to a small number of element -
applied to structural finite element (beam, shell, solid-like shell, applied only
on rotation degrees of freedom)
Frequency spectrum:
• selective mass matrix5 - adding artificial mass so so that translation inertia
is preserving.
mλe = β
ρAle2
[1 −1
−1 1
]mo
e = me + βmλe
- only selected modes are affected
- off-diagonal mass matrix structure ⇒ using the reciprocal mass matrix
Frequency spectrum:
5Olovsson Etal. (2005) Selective Mass Scaling for explicit Finite Element Analyses, IJNME 63
General form for preserving of translation inertia6
moe =
∆m
n− 1(I−
n∑i=1
oioT
i )
For example for 2D, rigid body modes for a four-noded element are chosen as
o1 =[
1 0 1 0 1 0 1 0]T
o2 =[
0 1 0 1 0 1 0 1]T
General form for elimination of selected eigen-modes with corresponding modal
vectors Φl7
moe = αPemeP
T
e
where
Pe = I−Φl[ΦT
lΦl]-1Φl
6Olovsson, et al. (2005) Selective Mass Scaling for explicit Finite Element Analyses, IJNME7J. Gonzalez, et al. (2018) Inverse Mass Matrix via the Method of Localized Lagrange Multipliers IJNME.
The Taylor test
x
y
z
ØD0
L
v0
rigid wall
Geometry: bar radius R = 3.2 [mm]length L = 32.4 [mm]
Impact velocity L = 227.0 [m/s]3D problemElastic and mass parameters of copper:
E = 117 [GPa]
ν = 0.35 [-]
ρ = 8.93 [kg/m3]
Simo J2 finite plasticity theory
Bilinear stress-strain curve
Isotropic hardening
Yield strength σY = 400 [MPa]
Plastic modulus E ′ = 100 [MPa]
The proposed method with θ = 0.5.
Taylor GI. The use of flat ended projectiles for determining yield stress. I. Theoretical consider-
ations. Proceedings of the Royal Society A , 194, 289–299, 1948.
The Taylor test
Distributions of σekv at the time t = 80 µs.
Thank you for you attention!
