## Most Relevant Recent Publications

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T. Lauß, S. Oberpeilsteiner, W. Steiner and K. Nachbagauer. The Discrete Adjoint Gradient Computation for Optimization Problems in Multibody Dynamics. Journal for Computational and Nonlinear Dynamics, DOI 10.1115/1.4035197, 2016.
[see Abstract]
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The adjoint method is a very efficient way to compute the gradient of a cost functional associated to a dynamical system depending on a set of input signals. However, the numerical solution of the adjoint differential equations raises several questions with respect to stability and accuracy. An alternative and maybe more natural approach is the discrete adjoint method (DAM), which constructs a finite difference scheme for the adjoint system directly from the numerical solution procedure, which is used for the solution of the equations of motion. The method delivers the exact gradient of the discretized cost functional subjected to the discretized equations of motion. For the application of the discrete adjoint method to the forward solver, several matrices are necessary. In this contribution, the matrices are derived for the simple Euler explicit method and for the classical implicit Hilber–Hughes–Taylor (HHT) solver.

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S. Oberpeilsteiner, T. Lauß, K. Nachbagauer and W. Steiner. Optimal input design for multibody systems by using

an extended adjoint approach. Journal for Multibody System Dynamics, DOI 10.1007/s11044-016-9541-8, 2016.
[see Abstract]
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an extended adjoint approach. Journal for Multibody System Dynamics, DOI 10.1007/s11044-016-9541-8, 2016.

We present a method for optimizing inputs of multibody systems for a subsequently performed parameter identification. Herein, optimality with respect to identifiability is attained by maximizing the information content in measurements described by the Fisher information matrix. For solving the resulting optimization problem, the adjoint system of the sensitivity differential equation system is employed. The proposed approach combines these two well-established methods and can be applied to multibody systems in a systematic, automated manner. Furthermore, additional optimization goals can be added and used to find inputs satisfying, for example, end conditions or state constraints.

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K. Nachbagauer, S. Oberpeilsteiner and W. Steiner. Enhancement of the Adjoint Method by Error Control of Accelerations for Parameter Identification in Multibody Dynamics. Universal Journal of Control and Automation, Vol. 3(3), pp.47-52, DOI: 10.13189/ujca.2015.030302, 2015.
[see Abstract]
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The present paper shows the embedding of the adjoint method in multibody dynamics and its broad applicability for examples for both, parameter identification and optimal control. Especially, in case of parameter identifications in engineering multibody applications, a theoretical enhancement of the proposed adjoint method by an error control of accelerations is inevitable in order to meet the circumstances of experimental studies using acceleration sensors in general.

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K. Sherif, K. Nachbagauer and W. Steiner. On the rotational equations of motion in rigid body dynamics when using Euler parameters. Journal for Nonlinear Dynamics, Springer Netherlands, doi: 10.1007/s11071-015-1995-3, pp. 1-10, 2015.
[see Abstract]
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Many models of three-dimensional rigid body dynamics employ Euler parameters as rotational coordinates. Since the four Euler parameters are not independent, one has to consider the quaternion constraint in the equations of motion. This is usually done by the Lagrange multiplier technique. In the present paper, various forms of the rotational equations of motion will be derived, and it will be shown that they can be transformed into each other. Special attention is hereby given to the value of the Lagrange multiplier and the complexity of terms representing the inertia forces. Particular attention is also paid to the rotational generalized external force vector, which is not unique when using Euler parameters as rotational coordinates.

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K. Nachbagauer, S. Oberpeilsteiner, K. Sherif and W. Steiner. The Use of the Adjoint Method for Solving Typical Optimization Problems in Multibody Dynamics. Journal for Computational and Nonlinear Dynamics, doi:10.1115/1.4028417 (10 pages), 2014.
[see Abstract]
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The present paper illustrates the potential of the adjoint method for a wide range of optimization problems in multibody dynamics such as inverse dynamics and parameter identification.

Although the equations and matrices included show a complicated structure, the additional effort when combining the standard forward solver to the adjoint backward solver, is kept in limits.

Therefore, the adjoint method shows an efficient way to incorporate inverse dynamics to engineering multibody applications, e.g., trajectory tracking or parameter identification in the field of robotics.

The present paper studies examples for both, parameter identification and optimal control, and shows the potential of the adjoint method in solving classical optimization problems in multibody dynamics.

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K.Sherif and K.Nachbagauer. A detailed derivation of the velocity-dependent inertia forces in the floating frame of reference formulation. Journal for Computational and Nonlinear Dynamics, doi:10.1115/1.4026083 (8 pages), 2014.
[see Abstract]
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In case of complex multibody systems, an efficient and time-saving computation of the equations of motion is essential, in particular concerning the inertia forces. When using the floating frame of reference formulation for modeling a multibody system, the inertia forces, which include velocity-dependent forces, depend non-linearly on the system state and therefore have to be updated in each time step of the dynamic simulation. Since the emphasis of the present investigation is on the efficient computation of the velocity-dependent inertia forces as well as a fast simulation of multibody systems, a detailed derivation of the latter forces for the case of a general rotational parametrization is given. It has to be emphasized that the present investigations revealed a simpler representation of the velocity-dependent inertia forces compared to results presented in literature. In contrast to the formulas presented in literature, the presented formulas do not depend on the type of utilized rotational parametrization or on any associated assumptions.

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K. Nachbagauer. State of the Art of ANCF Elements Regarding Geometric Description, Interpolation Strategies, Definition of Elastic Forces, Validation and the Locking Phenomenon in Comparison with Proposed Beam Finite Elements. Archives of Computational Methods in Engineering, doi: 10.1007/s11831-014-9117-9 (27 pages), 2014.
[see Abstract]
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The focus of the present article lies on new enhanced beam finite element formulations in the absolute nodal coordinate formulation (ANCF) and its embedding in the available formulations in the literature. The ANCF has been developed in the past for the modeling of large deformations in multibody dynamics problems. In contrast to the classical nonlinear beam finite elements in literature, the ANCF does not use rotational degrees of freedom, but slope vectors for the parameterization of the orientation of the cross section. This leads to several advantages compared to the classical formulations, e.g. ANCF elements do not necessarily suffer from singularities emerging from the parameterization of rotations. In the classical large rotation vector formulation, the mass matrix is not constant with respect to the generalized coordinates. In the case of ANCF elements, a constant mass matrix follows, which is advantageous in dynamic analysis.

Within the present paper the state of the art of ANCF elements in literature is reviewed including the basic description of the kinematics, interpolation strategies and definition of elastic forces, but also problems and known disadvantages arising in the existing elements, as e.g. the locking phenomenon.

It has to be mentioned here that most of the studies on nonlinear elements based on the ANCF in literature use linear constitutive laws. Regarding many applications in which geometric and material nonlinearities arise, elastic material models are not sufficient to represent the real problem accurately. For this reason, an extension of the material model is necessary in order to fulfill the requirements of current but also of future materials arising in engineering or research. An overview of existing nonlinear material models in literature can be found in the “Appendix”.

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K. Sherif, W. Witteveen. Deformation mode selection and orthonormalization for an efficient simulation of the rolling contact problem, In: Proceedings of the 32th IMAC 2014, A Conference on Structural Dynamics, Orlando, FL, USA.
[see Abstract]
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The dynamic simulation of contact in rolling processes is very time-consuming. This is mainly based on the fine resolution of the surface domain of each roll, which, however, is essential in order to capture the effect of concentrated contact forces. Existing model order reduction techniques cannot be readily applied due to the nonlinear nature of the contact dynamics. In order to improve the speed of contact analysis, the present paper proposes a sophisticated combination of so-called characteristic static correction modes and vibration normal modes for describing the deformation of each roll. While the characteristic static correction modes are required to capture the concentrated nonlinear contact forces, the vibration normal modes describe the global deformation behavior of the rolls. For the computation of the characteristic static correction modes, first attachment modes are computed for a longitudinal sub-area of each roll. Then an eigenanalysis is performed on the component mode synthesis mass and stiffness matrices that correspond to the attachment modes. The resultant eigenvectors have been truncated and applied to the entire surface domain of the rolls. In order to obtain well-conditioned equations, all modes are finally orthonormalized. An example from the metal forming industry is used to demonstrate the results.

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K. Nachbagauer, K. Sherif, W. Witteveen. FreeDyn – A Multibody Simulation Research Code. In: Proceedings of the 11th World Congress on Computational Mechanics (WCCM) and the 5th European Conference on Computational Mechanics (ECCM), July 20-25, 2014, Barcelona, Spain.
[see Abstract]
[http://www.wccm-eccm-ecfd2014.org/]

The present paper introduces the non-commercial academic MBS-software FREEDYN, which is currently developed at the University of Applied Sciences Upper Austria. Special attention is turned on the efficient computation of highly nonlinear multibody dynamics systems including flexible components arising in nearly every industrial brunch. A fast and accurate simulation code and its open-source release are the main goals. Several examples have been performed in order to show its user-friendly applicability and its accuracy and efficiency according to the analytically derived components of the Jacobian matrix as well as their description using inertia shape integrals.

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W. Steiner and S. Reichl. The optimal control approach to dynamical inverse problems. J.Dyn. Sys.,Meas.,Control, 134(2), doi 10.1115/1.4005365, Nr.021010 (13 pages), 2012.
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This paper considers solution strategies for “dynamical inverse problems”, where the main goal is to determine the excitation of a dynamical system, such that some output variables, which are derived from the system’s state variables, coincide with desired time functions. The paper demonstrates how such problems can be restated as optimal control problems and presents a numerical solution approach based on the method of steepest descent. First, a Performance measure is introduced, which characterizes the deviation of the output variables from the desired values and which is minimized by the solution of the inverse problem. Second, we show, how the gradient of this error functional can be computed efficiently by applying the theory of optimal control, in particular by following an idea of H. J. Kelley and A. E. Bryson. As the major contribution of this paper we present a modification of this method which allows the application to the case where the state equations are given by a set of differential algebraic equations. This situation has great practical importance since multibody systems are mostly described in this way. For comparison we also discuss an approach which bases an a direct transcription of the optimal control problem. Moreover, other methods to solve dynamical inverse problems are summarized.