Download A New Non-oscillatory Numerical Approach for Structural Dynamics

Proceedings of the IMAC-XXVII
February 9-12, 2009 Orlando, Florida USA
©2009 Society for Experimental Mechanics Inc.
A New Non-Oscillatory Numerical Approach for Structural Dynamics
and Wave Propagation in Solids
A. Idesman1 , M. Schmidt2 , J. R. Foley2 , Y. Tu2 , R. L. Sierakowski2
1
Texas Tech University, Lubbock, Texas 79409-1021, U.S.A.
2
Air Force Research Laboratory, Munitions Directorate, Eglin Air Force Base, FL 32542, U.S.A.
Abstract
One of main issues in the application of the finite element method and other numerical methods to wave propagation
and structural dynamics problems is the appearance of high-frequency spurious oscillations in a numerical solution.
A new numerical approach based on a new solution strategy and new time-integration methods is developed for
elastodynamics. The finite element method is used for the space discretization. The new approach filters spurious
oscillations and retains actual oscillations. 1-D and 2-D impact problems are used for the calibration of numerical
dissipation for the new approach as well as for the validation of the new numerical technique. It is shown that a
simple modification of the mass matrix of the semi-discrete finite element equations leads to the essential reduction
of the space-discretization error. For the first time a reliable, fast, accurate and non-oscillatory solution of acoustic
and elastic wave propagation problems is possible. In contrast to existing approaches, the new technique does not
require any guesswork for the selection of numerical dissipation or artificial viscosity and retains the accuracy of
the basic solution at low modes. New fundamental results have been obtained due to the new approach: e.g., in
contrast to textbooks on finite elements, for long-term integration, the size of time increments for explicit methods
should be much smaller than the stability limit (rather than close to it).
1
Introduction
Most finite element procedures for elastodynamics problems are based upon semi-discrete methods [1, 2, 3]. For
these methods, the application of finite elements in space to linear elastodynamics problems leads to a system of
ordinary differential equations in time
U + KU = R.
U + C U̇
(1)
M Ü
Here M , C , K are the mass, damping, and stiffness matrices, respectively, U is the vector of the nodal displacement,
R is the vector of the nodal load. Eq. (1) can be also obtained by the application of other discretization methods
in space such as the finite difference method, the spectral element method, the boundary element method, the
smoothed particle hydrodynamics (SPH) method and others. Many different numerical methods have been developed for the time integration of Eq. (1). However, for wave propagation problems, the integration of Eq. (1) leads
to the appearance of spurious high-frequency oscillations. Both the spatial discretization used for the derivation of
Eq. (1), and the time integration of Eq. (1) affect spurious oscillations and the accuracy of a numerical solution.
Moreover, the amplitude of spurious oscillations in numerical solutions may even increase at mesh refinement (see
Figs. 3a and 5a). There are no reliable numerical methods that yield an accurate solution without spurious oscillations. Existing numerical approaches that treat this issue are based on the introduction of numerical dissipation or
artificial viscosity from the first time increment for the suppression of spurious high-frequency oscillations. However,
numerical dissipation or artificial viscosity also affects low modes of a numerical solution. Although this effect is
small for a small number of time increments, due to error accumulation during the time integration of Eq. (1), low
modes of numerical solutions become very inaccurate even at a moderate number of time increments. If numerical
dissipation is not used in calculations, then spurious high-frequency oscillations spoil the numerical solution. The
existing methods with numerical dissipation or artificial viscosity do not take into account the effect of the number of
time increments on the accuracy of final numerical results. Therefore, different researchers report different values of
artificial viscosity needed for the suppression of spurious oscillations (e.g., see the recent paper [4] and references
there), and it is not clear what the optimal value of artificial viscosity is or how to select it (there is no understanding that this value depends on the number of time increments). And even if spurious oscillations are suppressed
for a specific number of time increments, the final solution will be inaccurate due to error accumulation at low modes.
This contradiction between the accuracy of low modes and the suppression of spurious high-frequency oscillations
cannot be resolved within the existing approaches.
Recently we suggested a new numerical technique that yields non-oscillatory, accurate and reliable solutions for
wave propagation in solids; see [5, 6]. The paper includes a short description of the new numerical technique based
on the new two-stage solution strategy for elastodynamics and the new first-order time-continuous Galerkin (TCG)
method used for filtering spurious high-frequency oscillations; see [5, 6]. An effective procedure for the reduction of
the space-discretization error is discussed. Finally, we solve 1-D and 2-D impact problems for linear elastodynamics
in order to show the effectiveness of the new numerical technique for wave propagation problems.
2
Numerical technique
With the help of the new two-stage solution strategy and the new TCG methods suggested in [5, 6, 7, 8, 9], we will
resolve the seemingly irresolvable contradiction of current approaches to filtering spurious oscillations. The main
advantages of the new strategy are as follows. The new numerical technique: a) allows the selection of the best
numerical method for basic computations from simple criteria (the most important one being the accuracy of the
method); b) includes post-processing for filtering spurious high-frequency oscillations, which requires little computation time compared to the stage of basic computations (a small number of time increments with the new implicit
TCG method with large numerical dissipation is used for post-processing); c) yields no error accumulation due to
numerical dissipation (or artificial viscosity) at the stage of basic computations; d) does not require any guesswork
for the selection of numerical dissipation or artificial viscosity as do the existing approaches. Thus, the approach
can be easily incorporated in computer codes and does not require interaction with users for the suppression of
spurious high-frequency oscillations. Below we briefly describe the new solution strategy for elastodynamics and
the new implicit TCG method with large numerical dissipation that, according to the new strategy, is used for postprocessing (see [5, 6, 7, 8, 9] for a detailed description and the application of the new strategy and the new TCG
time-integration methods).
2.1
A new two-stage solution strategy
The idea of the new two-stage strategy is very simple. Because for linear elastodynamics problems there is no interaction between different modes during time integration (they are integrated independently of each other), the most
accurate time-integration method (without numerical dissipation or artificial viscosity) should be used for basic computations (the first stage), especially for a long-term integration. It means that all modes (including high-frequency
modes) are integrated very accurately and the solution will include spurious high-frequency oscillations. Then, for
the damping out of high modes, a method with large numerical dissipation (or with artificial viscosity) is used for
a number of time increments as a post-processor (the second stage). This method can be considered a filter of
high modes. Usually, a small number of time increments is sufficient for the second stage, with negligible error
accumulation at low modes.
In the current paper, for basic computations we will use the explicit central difference method, the implicit trapezoidal rule (second-order time-integration methods with zero numerical dissipation), or the fourth-order explicit TCG
method suggested in [10] in order to obtain an accurate solution of the semi-discrete elastodynamics problem,
Eq. (1) (this solution contains spurious high-frequency oscillations). For long-term integration, high-order timeintegration methods allow much larger time increments than do second-order methods at the same accuracy.
2.2
Implicit TCG method for filtering spurious high-frequency oscillations
Let us describe the implicit TCG method with large numerical dissipation suggested in [6]. The method is based on
the linear approximations of displacements U (t) and velocities V (t) within a time increment ∆t (0 ≤ t ≤ ∆t)
U (t) = U 0 + U 1 t ,
V (t) = V 0 + V 1 t ,
(2)
and has the first order of accuracy. Here U 0 and V 0 are the known initial nodal displacements and velocities, and
the unknown nodal vector V 1 can be expressed in terms of the unknown nodal vector U 1 as follows:
V1 =
1
1
U1 −
V0.
a1
a1
(3)
Finally, the following system of algebraic equations is solved for the determination of U 1
M + a1C + a21K )U
U 1 = − a1K U 0 + M V 0 + R 1 ,
(M
(4)
where
a1 =
m+2
∆t ,
m+3
R1 =
(m + 2)2
(m + 3)∆tm+1
Z
∆t
R (t)tm+1 dt .
(5)
0
The parameter m is responsible for the amount of numerical dissipation. After the calculation of U 1 from Eq. (4), the
values of displacements and velocities at the end of a time increment ∆t are calculated using Eqs. (2) and (3) for
t = ∆t. The maximum numerical dissipation corresponds to m = ∞. For the case m = ∞, the parameter a1 = ∆t,
and R 1 should be calculated analytically; see Eqs. (5). If the analytical calculation of R 1 at m = ∞ is impossible,
then a value m ≥ 15 can be used, because the difference in numerical dissipation for m = ∞ and m ≥ 15 is not very
essential (m = 15 is used in the current computations). The numerical examples show that the first-order accurate
implicit TCG method allows the suppression of spurious high-frequency oscillations for 5 ∼ 10 time steps while
retaining good accuracy of the solution at low modes. C = 0 (no viscosity) is used in the current computations.
The following empirical formula for the selection of time increments for a time-integration method with large numerical
dissipation is suggested in [6]:
∆t = α(N1 )
∆xΩ0.1 (N )
,
c
(6)
q
where c = Eρ is the wave velocity; ∆x is the size of a finite element; Ω0.1 (N ) is the value of Ω = w∆t at which
the spectral radius has the value 0.1 for the selected number N of time increments; Ω0.1 is used to scale spectral
radii calculated at different numbers of time increments N ; α(N1 ) is the empirical coefficient depending on the timeintegration method, the order of finite elements, and on the number N1 of elements which are passed through by
cT
the wave front (this number can be expressed as N1 = ∆x
). For example, for the first-order implicit TCG method
(a = 0 and m = 15), the following explicit expression is suggested in [8] for the coefficient α(N1 ):
a2
cT
cT
α(
) = a1
,
(7)
∆x
∆x
where a1 = 0.2655 and a2 = 0.3306 for linear elements, and a1 = 0.1785 and a2 = 0.2357 for quadratic elements.
Eq. (7) was obtained by the special calibration procedure for numerical dissipation for a 1-D impact problem (see
[8]). For the implicit first-order TCG method with large numerical dissipation (a = 0 and m = 15), Ω0.1 = 1.31 for 5
time increments and Ω0.1 = 0.81 for 10 time increments.
Remark 1. For the selection of the size of time increments for post-processing 2-D and 3-D problems, the following
modification of Eq. (6) can be used
ci T
∆xj
∆t = max α(
) max
Ω0.1 (N )
i,j
i,j
∆xj
ci

a2
max (∆xj )
max (ci )T
j

= a1  i
Ω0.1 (N ) ,
(8)
min (∆xj )
min (ci )
j
i
where max (ci ) = max (c1 , c2 ) and min (ci ) = min (c1 , c2 ) (i = 1, 2) are the maximum and minimum values between
i
i
q
q
E(1−ν)
E
the velocities of the longitudinal wave c1 =
and
the
transversal
wave
c
=
2
2ρ(1+ν)(1−2ν)
2ρ(1+ν) , ∆xj are the
dimensions of finite elements along the axes xj (i = 1, 2 for the 2-D problems and i = 1, 2, 3 for the 3-D problems).
Eq. (8) is based on Eq. (6) with the selection of the maximum size of a time increment with respect the longitudinal
and transversal waves, and the dimensions of a finite element along the coordinate axes. For uniform meshes with
linear and quadratic quadrilateral elements in the case of multi-dimensional problems, we will use the coefficients
a1 and a2 obtained for the 1-D case; see Eq. (7).
2.3
Reduction of space discretization error
So far we have discussed the numerical technique for the effective time integration of the semi-discrete system, Eq.
(1), for elastodynamics. However, it is known that computation time can be reduced by the reduction of the space-
Figure 1: Impact of an elastic bar of length L = 4 against a rigid wall.
Figure 2: Velocity distribution along the bar for long-term behavior calculated on a uniform mesh with 100 quadratic
3-node finite elements. Curves 1 and 2 correspond to the solutions obtained with different uniform time increments
for basic computations; ∆t1 = 0.0064 for curve 1 and ∆t2 = 2.5∆t1 = 0.016 for curve 2. Curve 3 corresponds to the
analytical solutions at the observation time T = 22.4 (a) and T = 23 (b).
discretization error (i.e., a smaller number of elements can be used in calculations for more accurate elements). The
space discretization error can be indirectly evaluated through the dispersion equation for different finite and spectral
elements; see [11, 12, 13, 14, 15, 16, 17, 18, 19] for acoustics and elastic waves. However, the modification of
finite element formulations for the reduction of space-discretization error for elastic waves requires additional study.
Moreover, finite elements with low dispersion yield spurious oscillations as well, and the comparison of different
elements without a reliable non-oscillatory numerical technique is not realistic. In the proposed study, the reduction
of the space-discretization error is based on the modification of the finite-element mass matrix for implicit time
integration methods. In contrast to the classical lumped M lump or consistent M cons mass matrices, the averaged
M lump + M cons ) is used. The solutions of the 1-D and 2-D impact problems
mass matrix calculated as M av = 0.5(M
with the averaged mass matrix and the new non-oscillatory approach show the essential reduction of the spatial
error; see below.
3
Numerical modeling
The new technique based on the new two-stage solution strategy and the new TCG method is implemented into the
finite element code FEAP [3]. For the post-processing stage of the two-stage strategy, the implicit TCG method with
large numerical dissipation is used for all considered problems with 10 time increments selected according to Eqs.
(6)-(8).
3.1
Impact of an elastic bar against a rigid wall
The first 1-D problem is related to the impact of an elastic bar of the length L = 4 against a rigid wall (see Fig. 1).
Young’s modulus is chosen to be E = 1 and the density to be ρ = 1. The following boundary conditions are applied:
a)
b)
Figure 3: Velocity distribution along the bar at time T = 10 calculated on uniform meshes with 100 (curve 2) and
500 (curve 3) quadratic 3-node finite elements after basic computations (a) and after post-processing (b). Curve 1
corresponds to the analytical solutions.
the displacement u(0, t) = t (which corresponds to the velocity v(0, t) = v0 = 1) and u(4, t) = 0 (which corresponds
to the velocity v(4, t) = 0). Initial displacements and velocities are zero; i.e., u(x, 0) = v(x, 0) = 0. The problem has
the continuous solution for displacements ua (x, t) = t − x for t ≥ x and ua (x, t) = 0 for t ≤ x, and the discontinuous
solution for velocities and stresses va (x, t) = −σ a (x, t) = 1 for t ≥ x and va (x, t) = σ a (x, t) = 0 for t ≤ x (at the
interface x = t, jumps in stresses and velocities occur).
First, we solved the problem on a uniform mesh with 100 quadratic finite element at the observation time T = 22.4
using the explicit central-difference method for basic computations. A lumped mass matrix and two different uniform
time increments were selected for basic computations ∆t1 = 0.0064 and ∆t2 = 2.5∆t1 = 0.016 (the value ∆t2 =
0.016 is close to the stability limit). Velocity distributions along the bar at time T = 22.4 are shown in Fig. 2a for the
central difference method. As expected, the oscillatory numerical solutions were obtained; see curves 1 and 2 in
Fig. 2a. However, due to error accumulation during long-term integration, curve 2 is more diffusive than curve 1 in
Fig. 2a. Therefore, post-processing these results with the TCG method yielded the different non-oscillatory results;
see Fig. 2b. As we can see from Fig. 2, in contrast to textbooks on finite elements, for long-term integration, the
size of time increments for explicit methods should be much smaller than the stability limit (rather than close to it).
Next, we show that mesh refinement leads to locally divergent results if very small time increments are used with
existing approaches (in this case all time-integration methods yield an accurate solution of semi-discrete system, Eq.
(1) corresponding to the results after basic computations of the suggested approach). Fig. 3a shows results after
basic computations with the trapezoidal rule for meshes with 100 and 500 quadratic elements for the observation
time T = 10. Similar results are also obtained with the explicit central difference scheme or the new fourth-order
explicit TCG method suggested in [10]. As can be seen from Fig. 3a, the local error for the velocity in the vicinity of
the wave front (x = 2) grows at mesh refinement. However, non-oscillatory convergent results were obtained with
the new approach including the post-processing stage with the new TCG method; see Fig. 3b.
3.2
Impact of an elastic cylinder against a rigid wall
A cylinder of length L = 2.5 and radius R = 1 is considered; see Fig. 4. The z-axis is the axis of revolution. Young’s
modulus is chosen to be E = 1, Poisson’s ratio ν = 0.3, and the density to be ρ = 1. The following boundary
conditions are applied: along boundary AB un = t (which corresponds to the instantaneous application of velocity
vn = v0 = 1) and τn = 0; along boundaries CD and AD σn = 0 and τn = 0; along boundary BC un = 0 and
τn = 0, where un , vn , and σn are the normal displacements, velocities and the tractive forces, respectively; τn are
the tangential tractive forces. Initial displacements and velocities are zero; i.e., ui (r, z, 0) = vi (r, z, 0) = 0.
Similar to the 1-D impact problem, mesh refinement for the 2-D impact problem leads to locally divergent results
if very small time increments are used with existing approaches (in this case all time-integration methods yield
accurate solution of semi-discrete system, Eq. (1) corresponding to the results after basic computations of the
suggested approach). Fig. 5a shows results after basic computations with the trapezoidal rule for meshes with
Figure 4: Axisymmetric impact of an elastic cylinder of length L and radius R against a rigid wall. z is the axis of
revolution.
Figure 5: The distribution of the dimensionless velocity v̄z = vz /v0 along the axis of revolution at dimensionless time
T̄ = c0 t/R = 1.724 on uniform meshes with 40 × 100 = 4000 (curve 2) and 160 × 400 = 64000 (curve 3) linear 4-node
finite elements after basic computations (a) and after post-processing (b). Curve 1 corresponds to the analytical and
reference solutions.
40 × 100 = 4000
q and 160 × 400 = 64000 linear elements for the dimensionless observation time T̄ = c0 t/R = 1.724
E
where c0 =
ρ is the longitudinal wave rate in the 1-D case. Similar results were also obtained with the explicit
central difference scheme or the new fourth-order explicit TCG method suggested in [10]. As can be seen from
Fig. 5a, the local error for the velocity distribution along the axis of revolution grows at mesh refinement. However,
non-oscillatory convergent results are obtained with the new approach including post-processing stage with the new
TCG method; see Fig. 5b. As a reference solution, we used the solution obtained by the new approach on a fine
mesh. The reference solution is close to the approximate analytical solution for the same problem derived in [20] by
the Laplace transform.
3.3
Numerical examples with the reduced space-discretization error
Here, we solve the 1-D and 2-D impact problems considered above using linear finite elements and the consistent
and averaged mass matrices (see section 2.3). The new non-oscillatory approach described above and based
on the two-stage solution strategy and the new TCG method for post-processing was used. The trapezoidal rule
with uniform time increments ∆t̄ = 0.0004 for a mesh with 100 linear elements for the 1-D impact problem, and
∆t̄ = 0.0001 for a mesh with 40 × 100 = 4000 linear elements for the 2-D impact problem was applied for basic
computations. At these small time increments, the error in time was small and could be neglected. The velocity distribution after basic computations contained high-frequency spurious oscillations. After the post-processing stage,
the spurious oscillations were removed; see Fig. 6. In contrast to the classical lumped M lump or consistent M cons
M lump + M cons ) allows the reduction of the
mass matrices, the averaged mass matrix calculated as M av = 0.5(M
number of nodes by a factor of 2 − 4 for the 1-D problem (see Fig. 6a) and by a factor 4 − 16 for the 2-D problem (see
Fig. 6b) without change in accuracy. For example, the results on the mesh with 160 × 400 = 64000 linear elements
and the consistent mass matrix (see curve 3 in Fig. 4b) and on the mesh with 40 × 100 = 4000 linear elements and
the averaged mass matrix (see curve 2 in Fig. 6b) are close to each other. If standard solvers, for which computation
time is proportional to N 3 (N is the number of degrees of freedom) are used, then the averaged mass matrix will
lead to a great reduction in computation time for elastic waves. These results are very important, because many
commercial finite element codes use linear elements for elastodynamics. A detailed study of the space discretization
error for low-order finite elements with implicit and explicit time-integration methods will be conducted.
a)
b)
Figure 6: The velocity distribution along the bar4 at time T = 198 for the 1-D impact problem (a) and along the
the axis of revolution at time T = 2 for the 2-D impact problem (b) after post-processing with the TCG method.
Curves 1 correspond to the analytical and reference solutions. Curves 2 and 3 correspond to the results obtained
with the averaged and consistent mass matrices.
4
Concluding remarks
The new numerical approach based on the new solution strategy and the new-time integration methods for the first
time yields accurate and non-oscillatory solutions for wave propagation in elastic materials. In contrast to existing
approaches, the new technique does not require any guesswork for the selection of numerical dissipation (or artificial
viscosity) and does not require interaction between users and computer codes for the suppression of spurious highfrequency oscillations. A simple modification of the mass matrix of the semi-discrete finite element equations leads
to the essential reduction of the space-discretization error. A detailed analytical study of the reduction of the space
discretization error for low-order finite elements with implicit and explicit time-integration methods will be considered
in the future.
Acknowledgments
The research of AVI has been supported in part by the Texas Higher Education Coordinating Board under grant
003644 0008 2006, DOE under grant DE-FC03-03NA00144 as well as the support of the Air Force Summer Faculty
Fellowship Program 2008.
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