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32
УДК 519.6 : 531
Consideration of stress stiffening and material reorientation in modal space based finite element solutions
D. Marinkovic1'2, M. Zehn1
1 Department of Structural Analysis, Technische Universität Berlin, Berlin, 10623, Germany 2 Faculty of Mechanical Engineering, University of Nis, 18000, Serbia
Structural deformations are an important aspect of many engineering tasks. They are typically resolved as "off-line" finite element computations with accuracy set as the primary objective. Though high computational efficiency is always an important aspect, in certain applications its priority is of equal or similar importance as the accuracy itself. This paper tackles the problem of proper extension of linear models with the objective of keeping high numerical efficiency and covering moderate geometric nonlinearities. Modal-space based approach is addressed as one of the standard techniques for robust model reduction. Two extensions are proposed to account for moderate geometric nonlinearities in modal-space based solutions, one accounting for stress stiffening effect and the other for moderate material rigid-body rotations during deformation. Examples are provided to demonstrate the applicability and discuss the aspects of proposed techniques.
Keywords: model reduction, modal space, geometric nonlinearity, finite element method, structural deformation
Учет эффектов упрочнения в напряженно-деформированном состоянии и переориентации материала в решениях на основе модального пространства методом конечных элементов
D. Marinkovic12, M. Zehn1
1 Берлинский технический университет, Берлин, 10623, Германия 2 Нишский университет, Ниш, 18000, Сербия
Структурные деформации являются важным аспектом многих инженерных задач. Как правило, такие деформации рассчитываются в ходе «автономных» вычислений методом конечных элементов, в которых точность является ключевым требованием. В некоторых приложениях точность имеет такой же приоритет, как и высокая вычислительная эффективность. В настоящей работе обсуждается проблема корректного расширения области применения линейных моделей для сохранения высокой численной эффективности и учета геометрических нелинейностей. В качестве стандартного метода уменьшения размерности модели рассмотрен подход на основе модального пространства. Предложены два расширения модели для учета геометрических нелинейностей в решениях на основе модального пространства. Первое расширение позволяет учитывать эффект упрочнения в напряженно-деформированном состоянии, второе — повороты материала в деформируемом твердом теле. На примерах показаны применимость и особенности предлагаемых методов.
Ключевые слова: уменьшение размерности модели, модальное пространство, геометрическая нелинейность, метод конечных элементов, структурная деформация
1. Introduction
A considerable amount of work has been dedicated to the development of formalisms to simulate material deformations. The finite element method (FEM) has established itself as the method of choice for computations in the whole range from micro- to macromodelling, since it offers high accuracy and robustness in computation. In most engineering tasks, the accuracy is set as the highest priority and the finite element computations are performed "off-line" taking hours and occasionally even days to complete. This
is particularly valid for finite element computations that account for nonlinear structural behaviour. On the other hand, in certain fields of applications, such as multibody system (MBS) dynamics or various virtual-reality based simulators, high computational efficiency is an aspect of similar importance as the accuracy itself. Depending on the specific field of application, the requirement for the simulation speed may even be of the highest priority, e.g. in virtual-reality simulators that demand real-time simulation [1]. In some other fields, for instance MBS dynamics,
© Marinkovic D., Zehn M., 2017
it would suffice to speed up the computation significantly, without a strict requirement related to simulation time, whereby the accuracy is kept within the acceptable limits.
The mentioned fields of applications typically involve structures that perform large motions and rotations. Hence, geometric nonlinearities are intrinsic in the considered de-formational behaviour. Apparent savings can be made by neglecting the deformational behaviour, thus reducing the problem to only six degrees of freedom of rigid-body motion per object. The resulting differential equations of motion are still nonlinear and the complexity is relatively high due to a number of constrains defining the links between the parts. The complexity increases dramatically if deforma-tional behaviour must be included for adequate simulation results.
In the field of MB S dynamics, a cosimulation involving MBS formalisms and FEM computation offers an all-around solution with very high accuracy and possibility to account for various material effects, all types of nonlinearities, coupled-field problems, etc. [2-4]. However, this irrefutable advantage comes together with the disadvantage of rather large computational effort and extensive data transfer between the two systems (MBS and FEM). Corotational FEM [5] may support highly efficient simulations, but high efficiency would still demand models of moderate size. Therefore, at the present stadium of development, model reduction is a standard approach used to cope with the computational burden. One possible way of performing model reduction is to apply finite element computations in order to provide a set of results that is further used in the training phase of neural networks. Such an approach was applied to various problems of real-time simulation including geometrically nonlinear deformation of a cantilever beam [6] and a shell structure [7], simulation of impact [8], etc. This technique also allows consideration of nonlinear effects since finite element computations in the training phase can be nonlinear as well. But it should be noted that the resulting model strongly depends on the deformation set "density" used and the range of deformation covered in the training phase. The established approach in the MBS is the one that implies usage of modal vectors and coordinates to reduce complex FEM models. It implies that orthogonal mode shapes, computed prior to simulation, represent the degrees of freedom used to describe the structures deformational behaviour. Commercial MBS software packages apply the component mode synthesis (CMS) technique, particularly the Craig-Bampton method [9]. The determined modes and modal parameters (stiffness, mass, damping) are properties of the structure in its original configuration and, thus, the deformational behaviour with respect to the floating reference frame is linear. However, in certain cases, particularly those involving thin-walled structures made of modern composite materials, the range of applicability of a linear model is relatively fast exceeded, as material may exhibit significant stress stiffening effects or local material rotations,
97
which give rise to geometrically nonlinear behaviour. If this is the case, their consideration becomes essential to obtain representative results, which demands appropriate extensions of the modal-based approach. So far the available literature does not offer many attempts with similar objectives [10-13].
In this paper two techniques are proposed to extend the applicability of modal-based solutions from linear to moderate geometrically nonlinear structural deformation. The techniques rely on the consideration of geometric stiffening effects and moderate rigid-body rotations of structural subdomains.
2. Modal reduction and Craig-Bampton method
As already mentioned, FEM enables high fidelity modeling of physical processes on different scales ranging from micro to macro. The same is valid for material deformations with the remark that keeping finite element models in their full extent ("nodal description") is numerically very demanding and, thus, time-consuming. If high computational efficiency is a high priority demand, model reduction techniques are usually addressed. A widely accepted approach to model reduction relies on the modal superposition technique, which implies that the displacement field u is given in the following form:
Nm
u(t) = £ qt(t) Vi, (1)
i=l
where qi are the modal coordinates, Vi the mode shape vectors and Nm is the number of selected mode shapes. This practically means that the solution is obtained as a linear combination of mode shapes, which are determined in a step prior to simulation. The simulation efficiency is, therefore, achieved by performing a great deal of computation prior to actual simulation. The predetermined mode shapes become the degrees of freedom in terms of which the deformational behavior of the structure is described. Furthermore, the mode shapes may be obtained so as to be orthogonal with respect to the finite element stiffness and mass matrices. Not only is the number of degrees of freedom in this manner dramatically reduced, but the resulting equations are also decoupled. In other words, the resulting modal stiffness and mass matrices, K m and Mm are diagonal:
K m = OTK L O = diag(Mi»?), (2)
Mm = Ot MO = diag , (3)
where KL and M are the finite element linear stiffness and mass matrices, O is the matrix of mode shapes, O = = ], M-. the modal mass of mode i and ro. the
corresponding circular eigenfrequency. Typically, normalization is performed so that the modal masses are equal to 1. With the damping matrix in modal space, Cm, also diagonal (modal damping coefficients), the resulting dynamic equations read [14]:
98
M m q + Cmq + K m q = Fm(i), (4)
where q denotes the vector of modal coordinates and Fm are the external forces in modal space:
Fm = ^TFext, (5)
where Fext is the vector of nodal external forces. A static case is resolved by neglecting inertial and damping effects in Eq. (4).
The quality of the obtained solution strongly depends on the selection of mode shapes used in the simulation. There are a number of criteria for a proper selection of modes, such as the frequency range of time-dependent external forces, the so-called effective modal masses, etc. In commercial FEM codes a modal analysis is performed with the applied boundary conditions, which comes down to resolving the eigenvalue problem:
(K L - M) = 0. (6)
The obtained normal mode shapes represent patterns of structural vibration at specific frequencies (eigenfrequen-cies) and are further used to perform transient dynamic analysis of the structure by means of the mode superposition technique.
Commercial MBS software packages apply more sophisticated techniques to obtain mode shapes that allow for a more versatile simulation. This is actually a consequence of the demand for relatively high flexibility related to applicable kinematic and dynamic boundary condition in the reduced model. The Craig-Bampton method [9] is a common choice. It requires to partition the deformable structure's degrees of freedom (DOFs) into the boundary (external) DOFs and internal DOFs, the former belonging to the nodes of the finite element model that the user wants to retain in the simulation model for the purpose of defining (kinematic or dynamic) boundary conditions. Those are the nodes at which the interaction between the structure and the rest of the global model can be realized. In the next step, the method requires two sets of modes:
(i) constraint modes, which are static shapes obtained by giving each boundary DOF a unit displacement, while all other boundary DOFs are fixed;
(ii) fixed-boundary normal modes, which are obtained by fixing all boundary DOFs and resolving the eigenvalue problem (Eq. (5)).
The so-obtained Craig-Bampton modes are not an orthogonal set of modes. For better numerical efficiency, they are typically orthonormalized prior to simulation. Consequently, the resulting mode shapes cannot be given a physical interpretation.
3. Consideration of moderate geometric nonlinearities
in modal space
The computed mode shapes and their parameters (modal mass, stiffness, damping), regardless of the method used to determine them, are properties of the structure in its original configuration and are therefore suitable for linear analysis,
i.e. for deformations characterized by small displacements. Though in many cases of engineering structures the linear analysis provides sufficient accuracy, there are yet structures, particularly thin-walled structures made of modern composites, that require consideration of at least moderate geometrically nonlinear effects when simulating their deformations. Two solutions developed to achieve this obj ective are discussed below. For an adequate interpretation of the proposed solutions, the starting point should be the total Lagrangian (TL) formulation of the geometrically nonlinear analysis. The tangent finite element stiffness matrix of the structure in the TL formulation reads:
K T = K L + K u + K „, (7)
where K u is the initial displacement matrix that takes into account the displacements between the current and initial structure configuration and Ka is the initial stress (geometric stiffness) matrix, which takes into account the influence of the stress state in the material on its stiffness.
The modal superposition technique, in its original formulation, uses only the linear stiffness matrix transformed to the modal space. The developed techniques aim at consideration of the remaining two terms in an approximate manner.
3.1. Geometric stiffening effect in modal space
Generally speaking, the presence of stresses in the material (prestress) influences the overall structural stiffness. This influence is described by the geometric stiffness matrix K a. The possibility of including geometric stiffness effects is already available in commercial software package SIMPACK [11]. SIMPACK applies the approach that differentiates between forces, with respect to which the structure is rather flexible (i.e. forces that may cause large deformations) and forces, with respect to which the structure is rather stiff (e.g. forces causing membrane strains in thin-walled structures). The latter forces could induce significant stresses in the structure with small deformations. The induced stresses, depending on their orientation, may further affect the overall structural stiffness. Assuming small deformations, a linear dependence of stresses on forces is adopted. Hence, the stress state is approximated as a linear superposition of the stresses due to each load case (the same assumption used in the linear FEM analysis). The same is done with the geometric stiffness matrix so that it is linearly dependent on acting forces. This approach is implemented in SIMPACK only for beam elements and it is limited to quasistatic forces.
We modify this approach by relating the geometric stiffness matrix directly to the deformation, i.e. modal coefficients, rather than to the acting forces. In our approach, for each mode shape the corresponding material stress state < and geometric stiffness matrix Kare calculated for the modal coefficient value equal to 1 (ql = 1). We refer to the so-obtained Kas a "unit geometric stiffness matrix" for the ith mode. Now, assuming that a structure deforms
according to a mode shape, the actual geometric stiffness matrix is simply approximated by scaling Kwith the modal coefficient q.:
K m- = qi K m0. (8)
At this point, it should be emphasized that the mode shapes are not orthogonal with respect to the geometric stiffness matrix for any mode shape. However, having in mind the objective of covering moderate geometric nonli-nearities and providing high numerical efficiency, we introduce a simplification that the geometric stiffness matrix related to a mode shape influences only the size of deformation caused by that mode shape. This implies a diagonal form of the modal geometric stiffness matrix. For each mode shape, the unit geometric stiffness matrix is transformed to the modal space in a straight-forward manner:
kai = vK Vi, (9)
where k^i is the unit modal geometric stiffness coefficient for the ith mode. Hence, the unit modal geometric stiffness matrix has the following diagonal form:
kal
(10)
For a deformation state described by the vector of modal coefficients q, the modal geometric stiffness matrix is given as:
ViKi
k m -
qNm K
'cNL
(11)
Hence, the introduced assumptions lead to the modal geometric stiffness matrix that is linearly dependent on q. The internal structural forces in modal space are then:
- (Km + Km)q.
(12)
Obviously, according to Eq. (12), the internal forces exhibit a quadratic dependence on the modal coefficients through the modal geometric stiffness matrix, while the linear dependence remains through the matrix K m.
Two relatively simple examples of academic nature are chosen to demonstrate the applicability of the geometric stiffness matrix based approach (Fig. 1). Both structures are thin-walled structures as such structures are well-known
for their high susceptibility to geometrically nonlinear effects. Figure 1, a depicts a plate structure with the in-plane dimensions 1x0.8 m and the thickness of 8 mm. All four corner points are clamped and the plate is exposed to the force of F = 7.5 kN acting at the midpoint of the structure (point A). Figure 1, b shows a cylindrical shell of similar dimensions (see Fig. 1, b for dimensions), the same thickness of 8 mm, and with the two parallel straight edges clamped over the whole length. The shell is exposed to the line distributed force acting at the midspan along the whole width. The resulting overall force has the magnitude of 100 kN. Both structures are considered to be made of steel (E =2.1 x 1011 Pa, v = 0.3) and the loads are chosen so as to cause deformations characterized by moderate geometrically nonlinear effects. Obviously, in the case of the curved structure, the required force is significantly larger, which is the consequence of larger stiffness of the shell (curved geometry), boundary conditions distributed along whole edges and the line distributed force.
The displacements are observed since the proposed technique aims at improvements in accuracy of predicting the deformed structural configuration. This is a very important aspect in systems consisting of several linked parts as the system configuration strongly depends on deformations of single parts. As a representative result for the above presented cases, the midpoint deflections of both structures are considered (point A of the plate and point B of the cylindrical shell). The deflections are computed using various approaches and their development with increasing force is shown in diagrams in Fig. 2. The results obtained with the full FEM models in ABAQUS are used as the reference solutions. The results by the presented approach (denoted by "geometric stiffness") are computed in originally developed program and in ADAMS, which is denoted by MODAL and ADAMS in diagrams, respectively.
The originally developed program uses the updated "tangent" modal stiffness matrix for the computation and the first 10 fixed-boundary normal modes determined in ABAQUS. The implementation in ADAMS is done by using the subroutine MFOSUB to add the part of the internal force due to the stress stiffening effects as an external force with the opposite presign. And it should be highlighted again that ADAMS uses the orthonormalized set of Craig-Bamp-tion modes. Hence, for the originally developed program
- ABAQUS linear
■ ABAQUS nonlinear
- MODAL linear
- MODAL geometric stiffness
■ ADAMS geometric^ stiffness
| a
Load level F/F^
a 4 -
Qh
CO
s
0.0
-ABAQUS linear^
■ ABAQUS nonlinear
- MODAL linear
- MODAL geometric stiffness
■ ADAMS geometric stiffness
Load level F/F-„
Fig. 2. Plate-point A (a) and shell-point B (b) deflection computed with different approaches
the modal geometric stiffness coefficients are determined for the fixed-boundary normal modes, whereas for ADAMS they are determined for the orthonormalized Craig-Bampton modes. The fact that the originally developed program and ADAMS use different set of mode shapes explains slightly different results by the two computations. However, it is obvious that both results are close to the strict nonlinear results by the full FEM models obtained in ABAQUS and therewith expand the applicability of the modal-space based solution from the realm of linear deformations into the realm of moderate geometrically nonlinear deformations. At the same time the number of degrees of freedom is dramatically reduced.
It can also be noticed that in the case of plate, the actual (nonlinear) response of the structure is stiffer than predicted by the linear approach, whereas in the case of cylindrical shell it is the other way around. In its original deformation, the plate resists the load using only the bending stiffness, but upon deformation, the plate becomes essentially a shell that uses its membrane stiffness (the induced tensile membrane stresses) to resists the load more efficiently. In the case of cylindrical shell, the load gives rise to negative membrane stresses which reduce the structural stiffness. With the increasing load this structure would experience the loss of structural stability and a snap-through effect would be observed. Surely, such a large deformation and the associated geometrically nonlinear effects are beyond the applicability of the proposed technique.
3.2. Modal displacement warping
Geometrically nonlinear effects may be caused by material rigid-body rotations of the considered structure or its subdomains. If the motion of a structure can be adequately described by a single (averaged) rigid-body rotation and small deformation, then the approach used by the MBS programs yields good results. The MBS approach resolves such a case by determining the rigid-body motion and then superposes the small deformation described in modal space onto the rotated structure. However, with certain structures, subdomains can perform relatively large rigid-body rotations with respect to each other. Such a case poses difficulties for the standard MBS approach.
We propose an approach to improve the accuracy of predicting the deformed structure configuration by using the modal-space based solution for the above described type of deformation. The idea behind the approach is relatively simple and consists in two steps. In the first step, the structural deformation is determined by using directly the modal superposition technique. This motion incorporates the material rigid-body motion as well. Strictly speaking, the amount of rigid-body motion is different for different material particles of the structure. Therefore, in the second step it is necessary to determine a kind of average rigid-body rotation for the substructure (or the whole structure) under consideration. The obtained rotation is then used to rotate the displacement field yielded in the first step.
The idea can be conveniently illustrated and some aspects discussed on an academic example of a clamped beam structure (Fig. 3). Let us assume that the beam deforms according to the first mode shape. As mode shapes are properties of the undeformed structure, the predicted beam tip displacement would be such that the tip remains on the line perpendicular to the initial configuration (Fig. 3, a). In reality, however, the pattern of the beam tip would be similar to the dotted line in Fig. 3, b.
Hence, the displacements are first computed using the modal superposition. Then, an average rigid-body rotation
Fig. 3. Clamped beam: linear deformation according to the first mode (a) and actual beam tip pattern (b)
Fig. 4. Clamped beam: averaged beam rotation (a) and rotation of displacements (b)
Fig. 5. Clamped beam: force acting upon the deformed configuration (a) and equivalent load (force + moment) acting upon the initial configuration (b)
is determined. This can be done by placing a local coordinate system conveniently onto the structure or a substructure (e.g. at the center of mass) and using its original orientation and the orientation in the deformed configuration. In Fig. 4, a this rotation is represented symbolically by angle 0. Finally, the displacement field obtained in the first step is rotated by the determined rotation (Fig. 4, b) hence the term "displacement warping".
There is another aspect that demands attention. Namely, assuming the loads are conservative, during the deformation the external forces might change their lines of action, keeping the same direction (Fig. 5). The linear analysis does not recognize this effect, since the reference configuration is always the initial one, but the effect is accounted for in the geometrically nonlinear analysis. Since modal superposition technique is intrinsically a linear method, the reference configuration is the initial one. This requires adequate correction. Once the deformed configuration is computed using the above described approach, the new lines of action of external forces are also determined, see Fig. 5, a.
F= 10 kN
S {
y I <
<
10m
y i
% 1
z ,, <
0.05 m
Fig. 6. Clamped beam with dimensions and excitation force
With this information, one can compute the corrective moments MF that compensate for the change of the line of action caused by the displacements uF of the point at which the force F acts (Fig. 5), when computation is performed using the initial configuration as a reference configuation.
To get the impression on achievable improvements, the above example is considered with specific parameters— dimensions and force magnitude as given in Fig. 6 and the beam is assumed to be made of steel (E = 2 x 1011 N/m2, v = 0.3).
The computation in modal space is done using the first 10 mode shapes, although it is obvious that the first mode shape plays the major role in this case. Diagrams in Fig. 7 depict the beam tip displacements in the global x- and y-directions, respectively. Again, in the linear analysis, the displacement in the global x direction is equal to zero. One may notice a good agreement of the results obtained by the presented approach and the strict geometrically nonlinear
0.0 f
2-0.2-
$-0.6-
Î-0.8
■ ABAQUS nonlinear
■ ABAQUS linear
■ MODAL rotation
4000 Force, N
8000
4000 Force, N
Fig. 7. Displacement of the beam tip in the global x and y direction
Fig. 8. Rear car axle: CAD model (a); sub-domains for modal displacement warping (b)
Fig. 9. Rear car axle with kinematic boundary conditions and load case 1 (a) and 2 (b)
FEM results computed in ABAQUS. In other words, regarding the accuracy of determining deformed shape, the results by the presented method offer a significant improvement compared to the linear result.
Verification of the presented idea on a complex structure that can be encountered in engineering practice would be worthwhile. MBS dynamics is frequently used in car industry to assess solutions related to suspension kinematics, handling performance, ride comfort, durability, etc. The design assessment requires a large number of simulations to check all the aspects, which puts emphasis onto numerical efficiency, besides the classical requirement for high level of accuracy.
The rear car axle depicted in Fig. 8 is considered below. A large portion of the considered structure can be classified as a thin-walled structure. The full FEM model (courtesy of Volkswagen AG) contains approximately 44000 linear shell elements and 5000 solids. Altogether the model has over 300 000 degrees of freedom (DOF). The deformation of the axle during the car drive in a curve is a very important aspect for an accurate prediction of the trajec-
tory. The linear analysis does not provide in this case acceptable accuracy because the crank arms may perform an average rotation of up to 15° during the deformation, whereby the deformation of the crank arms remains relatively small. For that reason, this structure is a viable candidate to apply the proposed solution.
The crank arms are chosen as structural subdomains (Fig. 8, b), for which the modal displacement warping is to be performed in order to improve the accuracy of predicting the geometry of deformed structure. To determine the average crank arm's rotation, the positions of the axle bushing and flange are used together with the axle flange orientation, i.e. a line perpendicular to the flange. The turning curve of a car is significantly influenced by the vertical wheel displacement (suspension travel) and toe angle. Therefore, in the simulation of car dynamics in an MBS program, it is important to account for the flexible behavior of the rear car axle including the discussed geometrically nonlinear effects.
Two particularly selected representative load cases are computed to test the applicability of the proposed solution.
100£
8013 % 60-
§ 40-
S 20-
Oh
ISI
GO 0-50 -40 -30 -20 -10 0 Toe-in angle, min
20
£
£ 0 13
I-20
■i -40 a
<D Oh
§ -60 -80
0 10 20 30 40 50 Toe-in angle, min
Fig. 10. Suspension travel versus toe-in angle for load case 1 (a) and 2 kN (b)
Load case 1 is a vertical force of 1 kN acting upon the wheel, while load case 2 is a side force of 7 kN, both of which are depicted in Fig. 9 together with the applied boundary conditions. The deformed configuration of the rear car axle is determined using linear and geometrically nonlinear computation with the full FEM model in ABAQUS and, additionally, using the modal superposition method with the displacement warping technique applied only to crank arms as substructures.
The results by modal superposition are obtained using 10 and 20 mode shapes in both considered cases. The results are presented in the form of diagrams in Fig. 10, where suspension travel is given versus toe-angle. Hence, two geometric quantities are given one vs. the other, one of which is a displacement directly obtained as a computational result, while the other one is an angle derived from displacements. For that reason, achieving high accuracy is rather demanding. Nevertheless, a compelling improvement of the prediction offered by the modal displacement warping technique compared to the linear results is easily observable in both cases. This improvement is obtained using the model with 10 and 20 DOFs (mode shapes), which is a massive reduction in model size. As a reminder, the full FEM model contains over 300000 DOFs. The computational time is reduced by several orders of magnitude. On an average personal computer configuration (CPU: Intel i3-2120) the geometrically nonlinear computation with the full FEM model takes approximately 40 min, but the time
for the solution in modal space is measured in milliseconds.
4. Conclusions
Certain fields of engineering call for numerically highly efficient simulations involving deformational structural behaviour that is beyond the applicability of linear models. This is particularly the case with the MBS dynamics of modern mechanisms and other structures with some parts belonging to the group of thin-walled structures. The requests for numerical efficiency and accuracy are opposite in their nature and quite difficult to conciliate. This paper offers solutions to tackle this problem.
Modal superposition technique was revisited in this paper as a standard model reduction technique in MBS and FEM computations that enables massive numerical savings but covers only small deformations, i.e. linear analysis, in its original formulation. The possibilities of expanding its application to moderate geometrically nonlinear deformations, with the numerical effort kept low, were investigated. Two approaches were proposed to achieve this objective. The first one aims at the stress stiffening effects as a cause of geometric nonlinearities and is based on inclusion of the geometric stiffness matrix in the computation, whereas the second approach handles moderate rigid-body rotations of structural subdomains by performing the procedure referred to as "displacement warping".
The examples have demonstrated that the proposed methods may indeed expand the span of applicability of the modal space based solution with the accuracy kept in acceptable limits (depending on requirements). The success of the methods strongly depends on the character of deformation. Obviously, in the cases used to illustrate the approach based on the geometric stiffness matrix, the considered structures exhibit no average rigid-body rotation and even the local material rotations are rather small. So, the stress stiffening effect plays a significant role in the geometric nonlinearities, at least in a small vicinity of the range in which the linear model is applicable. On the other hand, the beam and rear car axle examples are characterized by significant material rotations and hence the success of the proposed technique based on displacement warping. Therefore, one needs to analyze the deformational behaviour and understand it in details in order to make an appropriate choice of the method, prepare the model properly and apply it with care. The effort invested in the model preparation is paid-off by fast simulation, which may even enable realtime simulation. Since the proposed two methods are complementary regarding the effects they account for, in the further work the possibility of their combined application should be examined.
The future work should also consider the possibility of including the effects originating from contact problems in modal space finite element solutions. This could require
| a
\
-o- ABAQUS nonlinear^v
— ABAQUS linear ^
Warped modal (10 modes)
-•- Warped modal (20 modes)
substructuring, with separate sub-structures modelled by either full finite element or modal space reduced finite element models. The method of dimensionality reduction [15, 16] offers a solid basis for such an extension as it enables fast solution to contact problems, which is an important prerequisite for high numerical efficiency.
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Dragan Marinkovic, Dr.-Ing., Technische Universität Berlin, Germany, dragan.marinkovic@tu-berlin.de Manfred Zehn, Prof. habil. Dr.-Ing., Technische Universität Berlin, Germany, manfred. zehn@tu-berlin.de
