Jeremy Laflin

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.

Enhancing the Performance of the DCA When Forming and Solving the Equations of Motion for Multibody Systems

This chapter provides an initial investigation into using the Graphics Processing Unit (GPU) (or similar hardware) to execute the Divide-and-Conquer Algorithm (DCA), which forms and solves the equations-of-motion for articulated multibody systems. The computational time required to form and solve the equations-of-motion of a simple n-length pendulum using the GPU is compared with a standard serial CPU implementation, a rudimentary parallelization on the CPU using OpenMP, and some combinations of the CPU and the GPU. The hybrid version uses the GPU for a select number of levels in the recursive sweeps and uses an OpenMP parallelization on a multi-core CPU for the remaining levels of recursion. The results demonstrate a significant performance increase when the GPU is used despite recursive algorithms being ill-suited to hardware designed for Single Instruction Multi-Data (SIMD). This is largely due to the tree-type structure of recursive processes, with half of the required operations being contained in the first level of recursion for a binary tree.

Fast Electrostatic Force and Moment Calculations in Multibody-Based Simulations of Coarse-Grained Biopolymers

In molecular simulations, the dominant portion of the computational cost is associated with force field calculations. Herein, we extend the approach used to approximate long range gravitational force and the associated moment in spacecraft dynamics to the coulomb forces present in coarse grained biopolymer simulations. We approximate the resultant force and moment for long-range particle-body and body-body interactions due to the electrostatic force field. The resultant moment approximated here is due to the fact that the net force does not necessarily act through the center of mass of the body (pseudoatom). This moment is considered in multibody-based coarse grain simulations while neglected in bead models which use particle dynamics to address the dynamics of the system. A novel binary divide and conquer algorithm (BDCA) is presented to implement the force field approximation. The proposed algorithm is implemented by considering each rigid/flexible domain as a node of the leaf level of the binary tree. This substructuring strategy is well suited to coarse grain simulations of chain biopolymers using an articulated multibody approach.Copyright

Investigation of GPU Use in Conjunction With DCA-Based Articulated Multibody Systems Simulation

Since computational performance is critically important for simulations to be used as an effective tool to study and design dynamic systems, the computing performance gains offered by Graphics Processing Units (GPUs) cannot be ignored. Since the GPU is designed to execute a very large number of simultaneous tasks (nominally Single Instruction Multi-Data (SIMD)), recursive algorithms in general, such as the DCA, are not well suited to be executed on GPU-type architecture. This is because each level of recursion is dependent on the previous level. However, there are some ways that the GPU can be leveraged to increase computational performance when using the DCA to form and solve the equations of motion for articulated multibody systems with a very large number of degrees-of-freedom.Computational performance of dynamic simulations is highly dependent on the nature of the underlying formulation and the number of generalized coordinates used to characterize the system. Therefore, algorithms that scale in a more desirable (lower order) fashion with the number of degrees-of-freedom are generally preferred when dealing with large (N > 10) systems. However, the utility of using simulations as a scientific tool is directly related to actual compute time. The DCA, and other top performing methods, have demonstrated the desirable property of the required compute time scaling linearly with (O(n)) with the number of degrees-of-freedom (n) and sublinearly (O(logn) performance when implemented in parallel. However for the DCA, total compute time could be further reduced by exploiting the large number of independent operations involved in the first few levels of recursion.A simple chain-type pendulum example is used to explore the feasibility of using the GPU to execute the assembly and disassembly operations for the levels of recursion that contain enough bodies for this process to be computationally advantageous. A multi-core CPU is used to perform the operations in parallel using Open MP for the remaining levels. The number of levels of recursion that utilizes the GPU is varied from zero to all levels. The data corresponding to zero utilization of the GPU provides the reference compute-time in which the assembly and disassembly operations necessary at each level are performed in parallel using Open MP. The computational time required to simulate the system for one time-step where the GPU is utilized for various levels of recursion is compared to the reference compute time also varying the number of bodies in the system.A decrease in the compute-time when using the GPU is demonstrated relative to the reference compute-time even for systems of moderate size n < 1000 for arrangements using the GPU. This is a lower number of bodies than was expected for this test case and confirms that the GPU can bring significant increases in computational efficiency for large systems, while preserving the attractive sub-linear scalability (w.r.t. compute time) of the DCA.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.

Distance Estimation between Coarsened Regions of Biopolymers for Far Field Force Approximation

Current strategies to simulate dynamic behavior of large molecular systems involve computationally expensive fully atomistic models, or lower resolution models that have been coarsened. Coarsening is accomplished by grouping tightly bonded atoms, with little relative motion, in two main ways: spherical beads, and rigid bodies. The latter method, which preserves system geometry, has been shown to better capture the system physics by including rotational equations of motion of the coarsened region. This can have a significant effect on the system dynamics. The most advanced of these methods adaptively determine the regions that should be coarsened, and approach O(log(n)) computational performance (n is the number of coarsened regions in the system). Low computational order methods are limited by the pairwise force computation at each time step, which is required for biochemical systems. Thus, an approximation has been proposed for use with these methods that reduce the computational complexity of the force computation to O(nlog(n)).This approximation method (constructed similarly to the Fast Multipole Method) requires that the minimum distance between coarsened regions be computed. Intuitively obvious strategies, such as tracking the exact system geometry, are often so expensive that they negate the benefits of using a reduced order method. To this end, pseudo-radius of gyration is proposed that is computed from a tensor quantity similar to the inertia tensor, but describes the charge distribution of the coarsened region. The mechanisms for manipulating this quantity during the assembly and disassembly of coarsened regions would be similar to what is done for the actual inertia tensor in the current dynamics model. This quantity will be computationally inexpensive to store and manipulate, therefore will preserve the overall low computational cost of the force approximation, while allowing for a more accurate coarsening strategy.

A Recursive Body-Body Formulation for Reducing the Computational Cost of Pairwise Coulomb Force Computation for Modeling and Simulation of Biopolymers, Using a Multibody Approach to Model Reduction

This work presents a method for recursively assembling tensor-like quantities that parameterize the charge distribution of rigid bodies, which result from model reduction of biopolymeric systems using an articulated multibody approach. This is done with the goal of reducing the computational cost associated with the pairwise force determination encountered in molecular dynamics simulations. To achieve a linear computational cost complexity of the force determination, with respect to the number of bodies in the system (N), a recursive assembly and disassembly (evaluation) sweep is proposed. This work proposes assembling these tensor quantities (pseudo-inertia tensors), which are associated with the body’s charge distribution, with a method that uses the standard parallel axis theorem to shift these tensors to a common point so they may be summed.This work presents a preliminary numerical example that examines the accuracy of the force and moment computation using a pseudo-inertia tensor resulting after one level of recursive assembly. The Coulomb force and associated moment on a target body due to the assembled body is computed. The test problem approximates a system that is highly negatively or positively charged. The orientation of the bodies that are assembled is varied, along with the distance between the assembly and the target body. The preliminary results presented herein suggest that this is a viable method of efficiently representing the charge distribution of an assembly. The numerical example presented determines the Coulomb force and the associated moment, as a function of distance and the pseudo-inertia tensor. However, the approximation can be used for any force that is of the form 1/rs, where s is any power.Copyright

Parallel Algorithm for Modeling Constrained Multi-Flexible Body System Dynamics

This paper presents an algorithm for modeling the dynamics of multi-flexible body systems in closed kinematic loop configurations where the component bodies are modeled using the large displacement small deformation formulation. The algorithm uses a hierarchic assembly disassembly process in parallel implementation and a recursive assembly disassembly process in serial implementation to achieve highly efficient simulation turn-around times. The operational inertias arising from the rigid body modes of motion at the joint locations on a component body are modified to account for the nonlinear inertial effects and body forces arising from the body based deformations. Traditional issues, such as motion induced stiffness and temporal invariance of deformation field related inertia terms, are robustly addressed in this algorithm. The algorithm uses a mixed set of coordinates viz. (i) absolute coordinates for expressing the equations of motion of a body fixed reference frame, (ii) relative or internal coordinates to express the kinematic joint constraints and (iii) body fixed coordinates to account for the body’s deformation field. The kinematic joint constraints and the closed loop constraints are treated alike through the formalism of relative coordinates, joint motion spaces and their orthogonal complements. Verification of the algorithm is demonstrated using the planar fourbar mechanism problem that has been traditionally used in literature.Copyright

Implementation Issues With Far Field Approximations With Multi-Resolution Biopolymer Simulations

Current strategies to simulate the dynamic behavior of large molecular systems involve either computationally expensive fully atomistic models, or lower resolution models that have been coarsened in some manner. Coarsening is nominally accomplished by grouping tightly bonded atoms that have little relative motion. Traditionally this accomplished by treating a region as pseudo-atom and connecting it to other psuedo-atoms to reproduce the system. Alternatively this can be done with a multibody-based approach characterizing the regions as rigid or flexible bodies [1]–[6].Copyright