Proceedings in Applied Mathematics and Mechanics, 29 May 2008
Efficient Finite Element Analysis of Inelastic Structures with Iterative
Solvers
Karsten J. Quint∗1 , Stefan Hartmann1 , Jurjen Duintjer Tebbens2 , and Andreas Meister3
1
2
3
Institute of Mechanics, University of Kassel, Mönchebergstrasse 7, 34109 Kassel, Germany
Institute of Computer Science Academy of Sciences of the Czech Republic, Prague, Czech Republic
Department of Mathematics, University of Kassel, Heinrich-Plett Str. 40, 34125 Kassel, Germany
This article treats the efficient solution of the linear systems of equations which arise during the iterative process within the
finite element analysis of inelastic structures. Up to 80% of the total computation time is spend by the linear solver which
suggests investigating this process. To this end high order time integration methods, diagonally implicit Runge-Kutta methods
(DIRK), in combination with an inexact Multilevel-Newton algorithm (MLNA) are applied.
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1 Constitutive Model
In many engineering applications inelastic material properties such as plastic or viscous effects play a crucial role. These
applications lead to nonlinear initial boundary value problems. In the case of isothermal quasi-static processes these consist
of the local balance of linear momentum, suitable initial conditions for the displacement field and the velocity field as well
as geometric and dynamic boundary conditions. The material can be described by constitutive models of evolutionary type.
There the materials state is given by internal variables q, which develop according to either ordinary differential equations or
differential-algebraic equations (DAE) Aq̇ = r(q, t).
In the present paper the material model describes small strain viscoplasticity of the polymer polyoxmethylene. The model
is based on an additive decomposition of the linearized Green strain tensor E = Ee + Ev into an elastic Ee and viscous Ev
part. This ansatz and further assumptions yield the Cauchy stress tensor T = Teeq + Theq + Tov = h(E, Theq , Ev ) as a sum of
an equilibrium part with Theq and without Teeq = Teeq (E) hysteresis and an overstress part Tov = Tov (Ee ). Here the internal
p
variables develop according to Ėv = η1 Tov and Ṫheq = cĖD − bTheq ĖD · ĖD , see [1] for details.
2 FEM and DIRK Approach
The local balance of linear momentum is transformed into an equivalent weak form, which is in the following discretized
using finite elements and nodal displacements u. The numerical integration over space leads to the DAE-system, see [2],
g(t, u, q) = 0
and
q̇(t) − r(t, u, u̇, q) = 0.
The algebraic part is the discretized balance equation and the differential part is due to the process dependence of the material.
The DAE system is solved by applying diagonally implicit Runge-Kutta methods, which form a class of implicit integration
methods. The most popular and most simple method of this class is the implicit Euler method, but higher order methods can
be implemented easily and allow a cheap but effective step size control, [2]. The application yields a non linear system of
equations G(Un+1 , Qn+1 ) = 0 and L(Un+1 , Qn+1 ) = 0. The second equation is the time integration step, which is here given
for the implicit Euler-method for simplicity L(Un+1 , Qn+1 ) = (Qn+1 − Qn )/∆tn − r(tn+1 , Un+1 , (Un+1 − Un )/∆t, Qn+1 ).
In current non-linear finite element codes this equation system is solved iteratively on two levels by the MLNA, [2]. The
L-part is decomposed naturally into small subsystems of the size of local internal variables (in the case of the material model
described above nq = 12) and solved for given displacements U on Gauss-point level. This is often referred to as the stressalgorithm in the FE-literature. The remaining G-part is in the next step solved on the global level with the internal variables
Q and dQ/dU, which is related to the consistent tangent, from the local (element) level
i
h
i
h
dG ∆U = ∂G + ∂G dQ ∆U = −G(z), with z := (U, Q) and U ← U + ∆U.
(1)
dU
∂U z
∂Q z dU z
For each time step the linear system of equations (1) of dimension nu ≫ nq has to be solved several times until convergence
is achieved. The property of solving a sequence of similar systems can be exploited. Another property which can be taken
advantage of is the structure of the stiffness matrix [dG/dU]. This matrix is large, structurally symmetric and sparse. Iterative
solvers are well suited for this kind of matrices. They can be divided into two classes: splitting and projection methods.
∗
Corresponding author: e-mail: [email protected], Phone: +49 561 804 3606, Fax: +49 561 804 2720
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2
Currently the projection methods (Krylov subspace methods) as GMRES and BiCGSTAB are considered most important, [3].
Iterative solvers allow to specify to which accuracy the linear system is solved. This can be exploited by solving the linear
system (1) only to limited accuracy at the beginning of the Newton process and considering instead the Newton-condition
k[dG/dU]∆U + G(z)k ≤ η kG(z)k .
(2)
The forcing term η < 1 is reduced as the iteration progresses towards the solution. Here we follow an approach from
2 2
(m)
[4] with the forcing term at the m-th iteration ηa = γ G z(m) / G z(m−1) . For the beginning of the iteration
the forcing term is set to η (0) = ηmax . To avoid oversolving on the final iterate the forcing term is bounded from below
η (m) = max(ηa , ηmin ).
Both efficiency and robustness of iterative methods are improved by applying preconditioners, which transform the original
linear system into one with the same solution, but that is likely to be easier to solve. Examples are the incomplete factorizations
ILU(0) and ILUT. Depending on the number of unknowns the cost of the preconditioner can be significant. The aforementioned solution of a sequence of linear systems can be taken advantage of by keeping the preconditioner constant for a
number of linear solutions.
3 Numerical Examples
80
iterations per lin. system
iterations per lin. system
80
constant η
adaptive η
75
70
65
60
55
50
45
40
35
30
25
10
every
freeze
75
70
65
60
55
50
45
0
10
20
30
40
50
60
70
0
10
(a) Geometry and mesh
(b) Adaptive stopping tolerance
20
30
40
50
60
70
global iteration
global iteration
(c) Recomputation of the preconditioner
Fig. 1 Tensile test of a plate with hole (The 2nd order method of Ellsiepen is used for the time integration)
For all computations a BiCGSTAB solver in combination with an ILUT preconditioner (lfil=6, droptol=1.e-4) is used. A
homogenous loading of a plate with hole Fig. 1(a) is considered. Due to the symmetries only one eighth of the plate is meshed
with ne =30888 elements leading to a total of nu =100520 unknown displacements and nQ =2965248 internal variables. The
plate is stretched by 2% and the performance of the adaptive forcing term η with ηmax =1.d-4 and γ=0.9 is examined. In
Fig. 1(b) the number of iterations to solve (2) is plotted over the global iteration number. Using the adaptive forcing term
reduces the number of iterations to solve the linear systems at the beginning of each time step but leads to an additional
iteration on the global level. Comparing the computation time still about 7% of the total computation time is saved. In the
next investigation three preconditioner update strategies are compared, see Fig. 1(c): recomputing every iteration, every 10th
iteration and freezing (in this case the preconditioner is computed only once at the beginning of the simulation). Recomputing
the preconditioner periodically results in approximately the same number of iterations per linear solve. With the frozen
preconditioner the number of iterations increases with progress in time and the algorithm might even break down. But the
gain in computational time is significant: using the periodic recomputation saves 53% and the frozen preconditioner even saves
56% of the computation time. Keeping the preconditioner constant as long as possible has the greatest effect on computation
time and can be easily implemented.
Acknowledgements This paper is based on investigations of the collaborative research center SFB/TR TRR 30, which is kindly supported
by the German Research Foundation DFG.
References
[1]
[2]
[3]
[4]
S. Hartmann, Archive of Applied Mechanics 76, 349–366 (2006).
S. Hartmann, K. J. Quint, and A. W. Hamkar, Journal of Applied Mathematics and Mechanics 88(5), 342–364 (2008).
Y. Saad, Iterative methods for sparse linear systems, 2nd. Edition (SIAM, Philadelphia, 2003).
S. C. Eisenstat and H. F. Walker, SIAM Journal on Scientific Computing 17(1), 16–32 (1996).
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