Imad M. Khan

Advances in the Application of the Divide-and-Conquer Algorithm to Multibody System Dynamics

This paper summarizes the various recent advancements achieved by utilizing the divide-and-conquer algorithm (DCA) to reduce the computational burden associated with many aspects of modeling, designing, and simulating articulated multibody systems. This basic algorithm provides a framework to realize O(n) computational complexity for serial task scheduling. Furthermore, the framework of this algorithm easily accommodates parallel task scheduling, which results in coarse-grain O(log n) computational complexity. This is a significant increase in efficiency over forming and solving the Newton–Euler equations directly. A survey of the notable previous work accomplished, though not all inclusive, is provided to give a more complete understanding of how this algorithm has been used in this context. These advances include applying the DCA to constrained systems, flexible bodies, sensitivity analysis, contact, and hybridization with other methods. This work reproduces the basic mathematical framework for applying the DCA in each of these applications. The reader is referred to the original work for the details of the discussed methods.

New and Extended Applications of the Divide-and-Conquer Algorithm for Multibody Dynamics

This work presents a survey of the current and ongoing research by the authors who use the divide-and-conquer algorithm (DCA) to reduce the computational burden associated with various aspects of multibody dynamics. This work provides a brief discussion of various topics that are extensions of previous DCA-based algorithms or novel uses of this algorithm in the multibody dynamics context. These topics include constraint error stabilization, spline-based modeling of flexible bodies, model fidelity transitions for flexible-body systems, and large deformations of flexible bodies. It is assumed that the reader is familiar with the “Advances in the Application of the DCA to Multibody System Dynamics” text as the notation used in this work is explained therein and provides a summary of how the DCA has been used previously.

Model transitions and optimization problem in multi-flexible-body systems: Application to modeling molecular systems

Abstract This paper presents an efficient algorithm for the simulation of multi-flexible-body systems undergoing discontinuous changes in model definition. The equations governing the dynamics of the transitions from a higher to a lower fidelity model and vice versa are formulated through imposing/removing certain constraints on/from the system model. The issue of the non-uniqueness of the results associated with the transition from a lower to a higher fidelity model may be handled by solving an optimization problem. This optimization problem is subjected to the satisfaction of the constraint imposed by the generalized impulse–momentum equations. The divide-and-conquer algorithm (DCA) is applied to formulate the jumps in the system states resulting from the model transition. The DCA formulation in its basic form is both time and processor optimal and results in linear and logarithmic complexity when implemented in serial and parallel with O ( n ) processors, respectively. As such, its application can reduce the effective computational cost of formulating and solving the optimization problem in the transitions to the finer models. The principal aspects of the mathematics for the algorithm implementation is developed and numerical examples are provided to validate the method.

Model Transitions and Optimization Problem in Multi-Flexible-Body Modeling of Biopolymers

This paper presents an efficient algorithm for the simulation of multi-flexible-body systems undergoing discontinuous changes in model definition. The equations governing the dynamics of the transitions from a higher to a lower fidelity model and vice versa are formulated through imposing/removing certain constraints on/from the system. Furthermore, the issue of the non-uniqueness of the results associated with the transition from a lower to a higher fidelity model is dealt with as an optimization problem. This optimization problem is subjected to the satisfaction of the impulse-momentum equations. The divide and conquer algorithm (DCA) is applied to formulate the dynamics of the transition. The DCA formulation in its basic form is time optimal and results in linear and logarithmic complexity when implemented in serial and parallel, respectively. As such, it reduces the computational cost of formulating and solving the optimization problem in the transitions to the finer models. Necessary mathematics for the algorithm implementation is developed and a numerical example is given to validate the method.Copyright

Divide-and-Conquer-Based Large Deformation Formulations for Multi-Flexible Body Systems

In the dynamic modeling and simulation of multi-flexible-body systems, large deformations and rotations has been a focus of keen interest. The reason is a wide variety of application area where highly elastic components play important role. Model complexity and high computational cost of simulations are the factors that contribute to the difficulty associated with these systems. As such, an efficient algorithm for modeling and simulation of systems undergoing large rotations and large deflections may be of great importance. We investigate the use of absolute nodal coordinate formulation (ANCF) for modeling articulated flexible bodies in a divide-and-conquer (DCA) framework. It is demonstrated that the equations of motion for individual finite elements or elastic bodies, as obtained by the ANCF, may be assembled and solved using a DCA type method. The current discussion is limited to planar problems but may easily be extended to spatial applications. Using numerical examples, we show that the present algorithm provides an efficient and robust method to model multibody systems employing highly elastic bodies.Copyright

Dynamics and control of multi-flexible-body systems in a divide-and-conquer scheme

In this paper, we present an extension of the generalized divide-and-conquer algorithm (GDCA) for modeling constrained multi-flexible-body systems. The constraints of interest in this case are not the motion constraints or the presence of closed kinematic loops but those that arise due to inverse dynamics or control laws. The introductory GDCA paper introduced an efficient methodology to include generalized constraint forces in the handle equations of motion of the original divide-and-conquer algorithm for rigid multibody systems. Here, the methodology is applied to flexible bodies connected by kinematic joints. We develop necessary equations that define the algorithm and present a well known numerical example to validate the method.Copyright

Strategies for Adaptive Model Reduction with DCA-Based Multibody Modeling of Biopolymers

This contribution discusses the need for adaptive model reduction when simulating biopolymeric systems and the issues surrounding the execution of these model changes in a computationally efficient manner. These systems include nucleic acids, proteins, and traditional polymers such as polyethylene. Two distinct general strategies of reducing selected degrees-of-freedom from the model are presented and the appropriateness of use is discussed. The strategies discussed herein are a momentum based approach and a velocity based approach. The momentum-based approach is derived from modeling discontinuous changes in model definition as instantaneous application (or removal) of constraints. The velocity-based approach is based on removing a degree-of-freedom when the associated generalized speed is zero. A Numerical example is included that demonstrates that both methods similarly characterize long-time conformational motion of a system.

Efficient Large Deformation Simulations of Multi-Flexible-Body Systems Using Absolute Nodal Coordinate Beam and Plate Elements

In this paper, we investigate the absolute nodal coordinate finite element (FE) formulations for modeling multi-flexible-body systems in a divide-and-conquer framework. Large elastic deformations in the individual components (beams and plates) are modeled using the absolute nodal coordinate formulation (ANCF). The divide-and-conquer algorithm (DCA) is utilized to model the constraints arising due to kinematic joints between the flexible components. We develop necessary equations of the new algorithm and present numerical examples to test and validate the method.Copyright

Orthogonal Complement-Based Divide and Conquer Algorithm: A Detailed Investigation of Its Performance

In this paper, we characterize the orthogonal complement-based divide-and-conquer (ODCA) [1] algorithm in terms of the constraint violation error growth rate and singularity handling capabilities. In addition, we present a new constraint stabilization method for the ODCA architecture. The proposed stabilization method is applicable to general multibody systems with arbitrary number of closed kinematic loops. We compare the performance of the ODCA with augmented [2] and reduction [3] methods. The results indicate that the performance of the ODCA falls between these two traditional techniques. Furthermore, using a numerical example, we demonstrate the effectiveness of the new stabilization scheme.Copyright

Multi-Flexible-Body Simulations Using Interpolating Splines in a Divide-and-Conquer Scheme

A new method for modeling multi-flexible-body systems is presented that incorporates interpolating splines in a divide-and-conquer scheme. This algorithm uses the floating frame of reference formulation and piece-wise interpolation spline functions to construct and solve the non-linear equations of motion of the multi-flexible-body systems undergoing large rotations and translations. We compare the new algorithm with the flexible divide-and-conquer algorithm (FDCA) that uses the assumed modes method and may resort to sub-structuring in many cases [1]. We demonstrate, through numerical examples, that in such cases the interpolating spline-based approach is comparable in accuracy and superior in efficiency to the FDCA. The algorithm retains the theoretical logarithmic complexity inherent to the divide-and-conquer algorithm when implemented in parallel.Copyright