# On conservation of energy and kinematic compatibility in dynamics of nonlinear velocity-based three-dimensional beams

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## Abstract

In this paper, we present an original energy-preserving numerical formulation for velocity-based geometrically exact three-dimensional beams. We employ the algebra of quaternions as a suitable tool to express the governing equations and relate rotations with their derivatives, while the finite-element discretization is based on interpolation of velocities in a fixed frame and angular velocities in a moving frame description. The proposed time discretization of governing equations directly relates the energy conservation constraint with the time-discrete kinematic compatibility equations. We show that a suitable choice of primary unknowns together with a convenient choice of the frame of reference for quantities and equations is beneficial for the conservation of energy and enables admissible approximations in a simple manner and without any additional effort. The result of this study is simple and efficient, yet accurate and robust numerical model.

## Keywords

Structural dynamics Nonlinear beams Kinematic compatibility Energy conservation Rotational quaternions Implicit time integration## 1 Introduction

Geometrically nonlinear beam models are popular due to their suitability for modeling frame-like structures undergoing arbitrary large displacements and rotations but small strains with sufficient accuracy and relatively low computational costs. Despite the simplified kinematics, the governing equations remain challenging from the perspective of numerical solution methods. Computationally efficient, accurate, robust and stable numerical formulations demand firm understanding of the model, relationships between the quantities describing the beam and their dependence with respect to both space and time. Modern techniques in numerical analysis allow many possible approaches in handling the problem of such complexity. It is thus not a surprise that after more than three decades of intensive research this field is still a subject of interest for many researchers, as reflected by recent publications, see, e.g., [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12].

Many challenges in beam formulations reported in the literature stem from the use of spatial rotations as members of the configuration space of the beam. Spatial rotations form a multiplicative group with interesting properties that are not easily preserved in numerical solution methods. Since the pioneering work of Simo and Vu-Quoc [13], beam finite elements have often been based on the interpolation of rotational degrees of freedom. In [13], a rotational vector is used for the parametrization of rotations, but this is only one among a variety of choices, see [14]. The chosen parametrization of rotations has an important impact on their discretization and the behavior of solution methods. Thorough discussions and comparisons can be found, e.g., in [15, 16, 17, 18, 19]. Crisfield and Jelenić [20, 21] pointed out that standard additive-type interpolation of rotational vectors could lead to nonobjective strain measures in many rotation-based approaches. Different interpolation strategies for rotational degrees of freedom were studied by Romero [22] and Bauchau and Han [23]. The use of only three parameters for the description of rotations results in lower storage demands for the primary unknown quantities. However, we can not easily avoid the calculation of rotation matrices and computationally more sensitive extraction of rotational parameters from rotation matrices [24]. This concept can be completely avoided by the introduction of rotational quaternions as a four-parameter representation of rotations. Not only that such parametrization is free from singularities, but also it is computationally more efficient regarding the number of calculations needed to rotate a vector. The use of quaternions seems to be standard for the efficient implementation of algorithms involving rotation matrices, see [54, p. 281] and the references therein. In beam formulations, quaternions were first employed by Bottasso [25] and Kehrbaum and Maddocks [26], while McRobie and Lasenby [27] used theoretically equivalent Clifford algebra. The interest in quaternion algebra as a suitable tool for beam formulations has increased in the last decade, see [2, 28, 29, 30, 31, 32, 33]. Another alternative is the interpolation of all nine components of the rotation matrix by Betsch and Steinmann [34] and recently by Sonneville et al. [35].

The description of the relationship between the deformed configuration of the beam and the strain measures is an important aspect of beam theories. In the “geometrically exact” approach, see Reissner [36, 37], strain resultants of the cross section are introduced and related to configuration variables in such a way that the kinematic equations are consistent with the virtual work principle. Several alternatives to this approach can be found, such as co-rotational method [38] and absolute nodal coordinate method [39]. In the present paper, we will stem from geometrically exact model as a suitable basis for the proposed numerical formulation.

In dynamics, a proper treatment of the configuration space with three-dimensional rotations might still result in a loss of numerical stability in the case of long-term calculations [40]. Several energy-preserving and energy-decaying algorithms were proposed as the answer to this phenomenon, e.g., [40, 41, 42, 43, 44, 45, 46]. These algorithms use similar strategy within different frameworks. For a more general approach, the reader is referred to the paper by Bauchau and Bottasso [47]. A common characteristic of these formulations is the special approximation of the kinematic equations imposed by preservation constraints. These approximations can be in contradiction with other properties of continuous systems, i.e., the orthogonality of rotation at the midtime is often violated. Despite this shortcoming, increased stability of long-term calculations using such methods was confirmed numerous times. Another important property of a continuous dynamic system, surprisingly rarely considered in beam formulations, is the direct relation between the strains and the velocities without any configuration variables being present, called the compatibility equations [48] or, alternatively, the intrinsic kinematic equations [49].

In the present paper, we will employ the rotational quaternions for the description of three-dimensional rotations. However, in contrast to our quaternion-based approach presented in [32], we will avoid the interpolation of rotational quaternions due to the above-mentioned problems. The crucial idea exploited here assumes the spatial and temporal derivatives of configuration variables to be the natural quantities for the description of total mechanical energy of the system. The previously developed strain-based beam formulation by Zupan and Saje [50] for static analysis of three-dimensional beams was motivated by this idea. In dynamics, angular velocities have a similar role as the rotational strains in statics, which was exploited in the paper by Zupan and Zupan [51], where velocities and angular velocities were taken as the primary unknowns of both spatial and temporal discretization. The time discretization employed in [51] can be interpreted as a modification of the implicit Newmark scheme. Unfortunately, a more suitable choice of the primary variables does not automatically assure the energy preservation. For sparse meshes and larger time steps, we can observe in some cases a loss of convergence accompanied with unrealistic increase in the total mechanical energy [51]. To avoid this phenomenon, we here propose a novel energy-conserving method based on a more suitable form of discrete governing equations while preserving the advantages of the velocity-based approach. Three important computational benefits of our approach are: (i) the additivity of the primary unknowns enables the use of standard interpolation techniques in space and simplifies the iteration procedure; (ii) the kinematic compatibility equations are satisfied with the same level of accuracy as the governing equations; and (iii) the energy of conservative systems is preserved without the need to introduce any special approximation of rotation or strain field. The computational advantages of the quaternion representation of rotations are preserved; moreover, proper treatment of the primary unknowns and the governing equations provides us with a relatively simple, numerically stable and robust method. The accuracy and excellent numerical performance of the present approach will be demonstrated by several benchmark examples.

## 2 Governing equations

*t*denotes the time and \(x\in \left[ 0,L\right] \) the arc-length parameter of the reference curve at the initial state of the beam with

*L*being its initial length. An arbitrary configuration of such a beam can be described by the position vectors \(\overset{\rightharpoonup }{r}\left( x,t\right) \) and the rotations between fixed and local bases. We will use the rotational quaternions as a suitable representation of rotations. Taking \(\vartheta \) to be the angle of rotation and \(\overset{\rightharpoonup }{e}\) denoting the unit vector on the axis of rotation, a rotational quaternion \(\widehat{q}\) is expressed as

It should be noted that all vector quantities in the sequel take values on the current configuration of the beam. The initial length of the line of centroids is used only as a suitable parameter for computational purposes. The transformation between the spatial and material form of equations is straightforward since the pull-back and push-forward are performed by the quaternion rotation (3). For further details, see the discussion in [37, 53].

### 2.1 Kinematic equations

*x*and the coordinate transformation is carried out using rotational quaternions. Thus, translational strain describes the rate of change of the position vector along the length of the beam. The rotational strain, \({\mathbf {K}}\), consists of a torsional and two bending strain components. In terms of quaternions, it is determined by

*x*and is expressed with respect to the local basis. The analogy of the above formula to the standard definition using rotation matrices is evident, while the factor 2 that additionally appears arises from the definitions (1)–(2).

*compatibility equations*and will play an important role in our numerical formulation.

### 2.2 Equations of motion

*A*is the area of the cross section; \(\mathbf {J}_{\rho }\) is the mass-inertia matrix of the cross section.

### 2.3 Constitutive equations

## 3 Numerical formulation

### 3.1 Time discretization

### 3.2 Spatial discretization

*t*replaced by a set of their space-discrete values \({\mathbf {v}}^{p}\left( t\right) \) and \({\varvec{\Omega }}^{p}\left( t\right) \) and interpolated by a set of

*N*interpolation functions \(I_{p}\left( x\right) \)

*N*, are chosen from the interval \( \left[ 0,L\right] \) with \(x_{1}=0\) and \(x_{N}=L\). Thus, the sets of discrete vectors \({\mathbf {v}}^{p}\) and \({\varvec{\Omega }}^{p}\) become the unknowns of the problem. In accord with the time discretization presented above, the only unknowns of the present formulation at each time step are the average velocities and angular velocities \(\overline{{\mathbf {v}}}^{p}\) and \(\overline{{\varvec{\Omega }}} ^{p}\) at interpolation points.

### 3.3 Energy conservation

It is obvious that the use of above formulas for the evaluation of midtime stress resultants does not restrict the applicability of our approach for nonconservative problems. Let us stress that no special approximation of rotations or strains was needed in the present approach. Due to a suitable choice of primary variables and governing equations, we get the desired properties of the formulation, while the kinematic compatibility is satisfied with the same accuracy as the governing equations.

## 4 Computational aspects

At each discrete time, the nonlinear discrete governing equations (36)–(37) are solved for the nodal values of average velocities and angular velocities using the Newton–Raphson method. This requires the construction of the Jacobian matrix, which is in our case expressed analytically. The analytical Jacobian is advantageous compared to the approximative one due to the presence of rotational degrees of freedom. Its derivation is relatively simple as the primary variables were conveniently chosen.

### 4.1 Linearization of equations

*i*, and \(\delta \mathbf {y}\) a vector of corrections of all nodal unknowns

### 4.2 Prediction of initial values

### 4.3 Compatibility of boundary conditions

### 4.4 Energy dissipation

## 5 Numerical simulations

*E*and

*G*are the elastic and shear moduli of material. The remaining quantities represent the geometric properties of the cross section expressed with respect to its centroid: \(A_{1}\) is the area; \(J_{1}\) is the torsional inertial moment; \(A_{2}\) and \(A_{3}\) are the effective shear areas; \(J_{2}\) and \(J_{3}\) are the bending inertial moments. All problems involve deformable beam members. In first three examples, a part of the motion is conservative, while the last one is a nonconservative problem.

The interpolation points in all of the examples were taken to be equidistant and standard Lagrange polynomials were used as shape functions. Integrals were evaluated numerically using the Gaussian quadrature rule. The number of integration points was taken to be equal to the number of interpolation points, *N*, for full integration and \(N-1\) for the reduced integration. Newton–Raphson iteration was terminated when the Euclidean norm of the vector of all unknowns at the structural level, \( \left\| \delta \mathbf {y}\right\| _{2}\), was less than \(10^{-8}\). A quadratic convergence of iteration procedure has been observed in all test problems at all time steps.

### 5.1 Free flight of a beam

In our simulation, the finite-element mesh consisted of 10 quadratic elements, \(N=3\). To demonstrate the performance of the proposed approach, we study the long-term behavior of the beam. We show the results for constant time step \(h=0.1\) until the time \(t=1000\), but need to stress that our solver experienced no difficulties and we could continue with calculation.

*XY*and

*XZ*, respectively, is depicted at the beginning and at the end of the calculation period. The problem is nonlinear and nonperiodic. Still, we can observe a paddling-like pattern, which is preserved during the whole analysis. The energy remains constant after the load is removed as illustrated in Fig. 6.

### 5.2 Right-angle cantilever

This example, also presented by Simo and Vu-Quoc [13], studies a right-angle cantilever beam under triangular pulse load in the direction out of the plane of the beam, see Fig. 7.

### 5.3 Circular beam

*A*and

*B*, see Fig. 10. Other data are as follows:

*A*, see Fig. 11. Figure 11 also reveals large magnitudes of displacements and a complicated response of structure without any evident pattern of movement.

### 5.4 Four-bar mechanism

*A*,

*B*and

*D*are orthogonal to the plane of mechanism. The revolute joint at point

*C*is inclined with respect to the normal axis by the angle \(\varphi =5^{\circ }\) simulating initial imperfection. Bars

*AB*and

*BC*share equal properties:

*CD*are:

*A*. Initial imperfection results in three-dimensional motion of the system.

*BC*and

*CD*at point

*C*and \(\widehat{\mathbf {q}}_{e}\) is the rotational quaternion describing the initial imperfection of the joint. The simulation was carried out until \(t=12\). Following [57], we obtained the results presented here using numerical dumping and constant time step \(h=0.004\).

## 6 Conclusions

We have presented a novel energy-preserving scheme based on velocities and angular velocities, where we satisfy the kinematic compatibility equations with the same accuracy as the governing equations. The discrete kinematic compatibility equations enforce the admissible update of strain vectors. This update is in our approach completely harmonized with the energy conservation demands, which additionally provides the consistent approximation of stress resultants. The finite element proposed is based on the interpolation of velocities in a fixed basis and angular velocities in the local basis. The use of standard shape functions is therefore completely consistent with properties of the configuration space. The main advantage of the proposed solution method is in its simplicity, robustness and long-term stability. Its favorable behavior was demonstrated by several numerical examples.

## Notes

### Acknowledgements

This work was supported by the Slovenian Research Agency through the research programme P2-0260 and the research project J2-8170. The support is gratefully acknowledged.

### Compliance with ethical standards

### Conflicts of interest

The authors declare that they have no conflict of interest.

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