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Journal ArticleDOI

Improved numerical dissipation for time integration algorithms in structural dynamics

TL;DR: In this article, a new family of unconditionally stable one-step methods for the direct integration of the equations of structural dynamics is introduced and is shown to possess improved algorithmic damping properties which can be continuously controlled.
Abstract: A new family of unconditionally stable one-step methods for the direct integration of the equations of structural dynamics is introduced and is shown to possess improved algorithmic damping properties which can be continuously controlled. The new methods are compared with members of the Newmark family, and the Houbolt and Wilson methods.

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Lawrence Berkeley National Laboratory
Recent Work
Title
IMPROVED NUMERICAL DISSIPATION FOR TIME INTEGRATION ALGORITHMS IN STRUCTURAL
DYNAMICS
Permalink
https://escholarship.org/uc/item/18z548gt
Author
Hilber, Hans M.
Publication Date
1976-04-01
eScholarship.org Powered by the California Digital Library
University of California

-.
u u
Submitted
to
Earthquake
Engineering
and
Structural
Dynamics
LBL-4486
Preprint
C.
IMPROVED
NUMERICAL
DISSIPATION
FOR
TIME
INTEGRATION
ALGORITHMS
IN
STRUCTURAL
DYNAMICS
Hans
M.
Hilber,
Thomas
J.
R.
Hughes,
and
Robert
L.
Taylor
April
1976
Prepared
for
the
U.
S.
Energy
Research
and
Development
Administration
under
Contract
W
-7405-ENG
-48
For
Reference
Not
to
be taken from this
.room

DISCLAIMER
This document was prepared
as
an account
of
work sponsored
by
the United States
Government. While this document is believed to contain correct information, neither the
United States Government nor any agency thereof, nor the Regents
of
the University of
California, nor any
of
their employees, makes any warranty, express or implied, or
assumes any legal responsibility for the accuracy, completeness, or usefulness
of
any
information, apparatus, product, or process disclosed, or represents that its use would not
infringe privately owned rights. Reference herein to any specific commercial product,
process, or service
by
its trade name, trademark, manufacturer, or otherwise, does not
necessarily constitute or imply its endorsement, recommendation, or favoring by the
United
States Government or any agency thereof, or the Regents
of
the University
of
Califomia. The views and opinions
of
authors expressed herein do not necessarily state or
reflect those
of
the United States Government or any agency thereof or the Regents
of
the
University
of
California.

(
*
0 0
0
IMPROVED
NUMERICAL
DISSIPATION
FOR
TIME
INTEGRATION
. *
ALGORITHMS
IN
STRUCTURAL
DYNAMICS
Hans
M.
Hilber
Thomas
J.
R.
Hughes
Robert
L.
Taylor
Division
of
Structural
Engineering
and
Structural
Mechanics
Department
of
Civil
Engineering
and
Lawrence
Berkeley
Laboratory
University
of
California
Berkeley,
California
94720
April
1976
This
work
was
supported
in
part
by
the
U.
S.
Energy
Research
and
Development
Administration

o
u o
~
4 o
ABSTRACT
A new
family
of
unconditionally
stable
one-step
methods
for
the
direct
integration
of
the
equations
of
structural
dynamics
is
introduced
and
is
shown
to
possess
improved
algorithmic
damping
properties
which
can
be
continuously
controlled.
The new
methods
are
compared
with
members
of
the
Newmark
family,
and
the
Houbolt.
and
Wilson
methods.
i.

Citations
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Cites methods from "Improved numerical dissipation for ..."

  • ...Twist also provides a standard implementation of a finite difference time-stepping algorithm that is commonly used in the computational mechanics community: the Hilber–Hughes–Taylor (HHT) method (Hilber et al., 1977)....

    [...]

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TL;DR: It is shown that the combination of the phase-field model and local adaptive refinement provides an effective method for simulating fracture in three dimensions.

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Cites methods from "Improved numerical dissipation for ..."

  • ...A fixed time step of ∆t = 1.25 × 10−8 was chosen, which is slightly less than 0.9∆tcrit, where the critical time step, ∆tcrit, is computed as ∆tcrit = Ωcrit ωmax (92) with ωmax the maximum natural frequency of the momentum equation determined from the undamped eigenproblem and (considering only the undamped case) HHT-α (see Miranda, Ferencz, and Hughes (1989)) Ωcrit = √ 2(γ + 2α(γ − β) γ + 2α(γ − β) (93) Explicit generalized-α (see Hulbert and Chung (1996)) Ωcrit = √ 12(1 + ρb)3(2− ρb) 10 + 15ρb − ρ2b + ρ3b − ρ4b ....

    [...]

  • ...The HHT-α method, parameterized by α, provides a second-order accurate family of algorithms for linear second-order equations if α ∈ [− 13 , 0] and αm = 1, (82) αf = 1 + α, (83) β = (1− α)2 4 , (84) γ = 1− 2α 2 (85) (see Miranda, Ferencz, and Hughes (1989))....

    [...]

  • ...The simulations were performed using the staggered integration scheme described in Section 3.3.2 with the momentum equation being solved explicitly using the HHT-α method with α = −0.3....

    [...]

  • ...For the fully explicit case (M∗ = αmM̃) we use either the HHT-α of Hilber, Hughes, and Tayler (1977) or the explicit generalized-α method of Hulbert and Chung (1996)....

    [...]

  • ...The simulations were performed using the staggered integration scheme described in Section 3.3.2 with the momentum equation being solved explicitly using the HHT-α method with α = −0.1....

    [...]

Journal ArticleDOI
TL;DR: In this paper, the concept of k-refinement is explored and shown to produce more accurate and robust results than corresponding finite elements, including rods, thin beams, membranes, and thin plates.

1,008 citations

Journal ArticleDOI
Juan C. Simo1
TL;DR: In this paper, a formulation and algorithmic treatment of static and dynamic plasticity at finite strains based on the multiplicative decomposition is presented which inherits all the features of the classical models of infinitesimal plasticity.
Abstract: A formulation and algorithmic treatment of static and dynamic plasticity at finite strains based on the multiplicative decomposition is presented which inherits all the features of the classical models of infinitesimal plasticity. The key computational implication is this: the closest-point-projection algorithm of any classical simple-surface or multi-surface model of infinitesimal plasticity carries over to the present finite deformation context without modification. In particular, the algorithmic elastoplastic tangent moduli of the infinitesimal theory remain unchanged. For the static problem, the proposed class of algorithms preserve exactly plastic volume changes if the yield criterion is pressure insensitive. For the dynamic problem, a class of time-stepping algorithms is presented which inherits exactly the conservation laws of total linear and angular momentum. The actual performance of the methodology is illustrated in a number of representative large scale static and dynamic simulations.

734 citations

References
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Journal ArticleDOI
TL;DR: A systematic procedure is developed for the calculation of the structural response of an airplane subject to dynamic loads, with particular attention given to determining the stresses developed due to flight through gusts.
Abstract: A systematic procedure is developed for the calculation of the structural response of an airplane subject to dynamic loads. Particular attention is given the problem of determining the stresses developed due to flight through gusts. Difference equivalents for derivatives and matrix notation are used to develop a recurrence relation that permits step-by-step calculation of the response and of the loads that occur on the structure. The chief feature of this recurrence approach is that the generality and physical aspects of the basic equilibrium relations of the problem are preserved without loss of ease in application. The use of difference equivalents for derivatives in the solution of dynamic problems is first illustrated by means of a simple damped oscillator example, and the application to the flexible aircraft structure is then made. For brevity, the case of wing bending and vertical motion of the airplane is treated, although the method may be readily extended to take into account also wing torsional deformations, pitching motion of the airplane, fuselage deflections, and tail forces of known character. Either a sharp-edge gust or a gust of arbitrary shape in the spanwise or flight directions may be treated. Some results obtained by application of the recurrence matrix relation are presented, and the advantages of this method over other methods of evaluating the dynamic response of an aircraft are discussed.

567 citations


"Improved numerical dissipation for ..." refers background or methods in this paper

  • ...Elaboration on these points and further properties of the Newmark family of algorithms may be found in [3L To analyze systems such as (1), or equivalently (3a), it is convenient to invoke the property of orthogonality of....

    [...]

  • ...Mu + Ku = F (1) where M is the mass matrix, K is the....

    [...]

  • ...for (1) consists of finding a function u = u(t), where t£[0,T], T > 0, satisfying (1) and the initial conditions: u(O)_ = (2) u<o> = where d and v are the given vectors of initial data....

    [...]

  • ...We are interested in obtaining approximate solutions of (1) by one-step difference methods....

    [...]

  • ...Employing the obvious notations, the singledegree-of-freedom analogs of (1) and (3a) - (3c) are: Mi....

    [...]

Journal ArticleDOI
TL;DR: In this article, a systematic procedure for the stability and accuracy analysis of direct integration methods in structural dynamics is presented, and the specific methods studied are the Newmark generalized acceleration scheme, the Houbolt method and the Wilson θ-method.
Abstract: A systematic procedure is presented for the stability and accuracy analysis of direct integration methods in structural dynamics. Amplitude decay and period elongation are used as the basic parameters in order to compare various integration methods. The specific methods studied are the Newmark generalized acceleration scheme, the Houbolt method and the Wilson θ-method. The advantages of each of these methods are discussed. In addition, it is shown how the direct integration of the equations of motion is related to the mode superposition analysis.

434 citations


"Improved numerical dissipation for ..." refers background in this paper

  • ...for (1) consists of finding a function u = u(t), where t£[0,T], T > 0, satisfying (1) and the initial conditions: u(O)_ = (2) u<o> = where d and v are the given vectors of initial data....

    [...]

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TL;DR: In this paper, the Newmark family of second-order difference approximations is compared with the original or extended Wilson and Houboult methods for the direct time integration of the spatially discretized equations of linear elastodynamics.

260 citations

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TL;DR: In this paper, the authors show that the conditionally stable explicit schemes and the unconditionally stable implicit schemes can be divided into two classes: the conditionably stable explicit and implicit schemes.
Abstract: In using the finite element method to compute a transient response, two choices must be made. First, some form of mass matrix must be decided upon. Either the consistent mass matrix prescribed by the finite element method can be employed or some form of diagonal mass matrix may be introduced. Secondly, some particular time integration procedure must be adopted. The procedures available divide themselves into two classes: the conditionally stable explicit schemes and the unconditionally or conditionally stable implicit schemes. The choices should be guided by both economy and accuracy. Using exact discrete solutions compared to the exact solutions of the differential equations, the results of these choices are displayed. Concrete examples of well-matched methods, as well as ill-matched methods, are identified and demonstrated. In particular, the diagonal mass matrix and the explicit central difference time integration method are shown to be a good combination in terms of accuracy and economy.

221 citations