LIGGGHTS – Open Source Discrete Element Simulations of Granular Materials Based on Lammps

Investigation of thermal properties of DNA structure with precise atomic arrangement via equilibrium and non-equilibrium molecular dynamics approaches.

Biomechanical systems share many properties with mechanically engineered systems, and researchers have successfully employed mechanical engineering simulation software to investigate the mechanical behavior of diverse biological mechanisms, ranging from biomolecules to human joints. Unlike their man-made counterparts, however, biomechanisms rarely exhibit the simple, uncoupled, pure-axial motion that is engineered into mechanical joints such as sliders, pins, and ball-and-socket joints. Current mechanical modeling software based on internal-coordinate multibody dynamics can formulate engineered joints directly in minimal coordinates, but requires additional coordinates restricted by constraints to model more complex motions. This approach can be inefficient, inaccurate, and difficult for biomechanists to customize. Since complex motion is the rule rather than the exception in biomechanisms, the benefits of minimal coordinate modeling are not fully realized in biomedical research. Here we introduce a practical implementation for empirically-defined internal-coordinate joints, which we call “mobilizers.” A mobilizer encapsulates the observations, measurement frame, and modeling requirements into a hinge specification of the permissible-motion manifold for a minimal set of internal coordinates. Mobilizers support nonlinear mappings that are mathematically equivalent to constraint manifolds but have the advantages of fewer coordinates, no constraints, and exact representation of the biomechanical motion-space—the benefits long enjoyed for internal-coordinate models of mechanical joints. Hinge matrices within the mobilizer are easily specified by user-supplied functions, and provide a direct means of mapping permissible motion derived from empirical data. We present computational results showing substantial performance and accuracy gains for mobilizers versus equivalent joints implemented with constraints. Examples of mobilizers for joints from human biomechanics and molecular dynamics are given. All methods and examples were implemented in Simbody™—an open source multibody-dynamics solver available at https://Simtk.org.

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Minimal formulation of joint motion for biomechanisms

Thin elastic rods such as cables, phone coils, tree branches, or hair, are common objects in the real world but computing their dynamics accurately remains challenging. The recent Super-Helix model, based on the discrete equations of Kirchhoff for a piecewise helical rod, is one of the most promising models for simulating non-stretchable rods that can bend and twist. However, this model suffers from a quadratic complexity in the number of discrete elements, which, in the context of interactive applications, makes it limited to a few number of degrees of freedom - or equivalently to a low number of variations in curvature along the mean curve. This paper proposes a new, recursive scheme for the dynamics of a Super-Helix, inspired by the popular algorithm of Featherstone for serial multibody chains. Similarly to Featherstone's algorithm, we exploit the recursive kinematics of a Super-Helix to propagate elements inertias from the free end to the clamped end of the rod, while the dynamics is solved within a second pass traversing the rod in the reverse way. Besides the gain in linear complexity, which allows us to simulate a rod of complex shape much faster than the original approach, our algorithm makes it straightforward to simulate tree-like structures of Super-Helices, which turns out to be particularly useful for animating trees and plants realistically, under large displacements.

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Linear Time Super-Helices

Complex variable methods for shape sensitivity of finite element models

A new algorithm, Orthogonal Complement based Divide-and-Conquer Algorithm (O-DCA), is presented in this paper for calculating the forward dynamics of constrained multi-rigid bodies including topologies involving single or coupled closed kinematic loops. The algorithm is exact and noniterative. The constraints are imposed at the acceleration level by utilizing a kinematic relation between the joint motion subspace (or partial velocities) and its orthogonal complement. Sample test cases indicate excellent constraint satisfaction and robust handling of singular configurations. Since the present algorithm does not use either a reduction or augmentation approach in the traditional sense for imposing the constraints, it does not suffer from the associated problems for systems passing through singular configurations. The computational complexity of the algorithm is expected to be O(n+m) and O(log(n+m)) for serial and parallel implementation, respectively, where n is the number of generalized coordinates and m is the number of independent algebraic constraints.

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Orthogonal Complement Based Divide-and-Conquer Algorithm for constrained multibody systems

This paper presents an efficient algorithm for the dynamics simulation and analysis of multi-flexible-body systems. This algorithm formulates and solves the nonlinear equations of motion for mechanical systems with interconnected flexible bodies subject to the limitations of modal superposition, and body substructuring, with arbitrarily large rotations and translations. The large rotations or translations are modelled as rigid body degrees of freedom associated with the interconnecting kinematic joint degrees of freedom. The elastic deformation of the component bodies is modelled through the use of modal coordinates and associated admissible shape functions. Apart from the approximation associated with the elastic deformations, this algorithm is exact, non-iterative, and applicable to generalized multi-flexible chain and tree topologies. In its basic form, the algorithm is both time and processor optimal in its treatment of the n b joint variables, providing O(log(n b )) turnaround time per temporal integration step, achieved with O(n b ) processors. The actual cost associated with the parallel treatment of the n f flexible degrees of freedom depends on the specific parallel method chosen for dealing with the individual coefficient matrices which are associated locally with each flexible body.

/pdf/a-logarithmic-complexity-divide-and-conquer-algorithm-for-4t5fgff712.pdf

A logarithmic complexity divide-and-conquer algorithm for multi-flexible articulated body dynamics

A new method is presented for accurately and efficiently simulating multi-scale multibody systems with discontinuous changes in system definitions as encountered in adaptive switching between models with different resolutions as well as models with different system topologies. An example of model resolution change is a transition of a system from a discrete particle model to a reduced order articulated multi-rigid body model. The discontinuous changes in system definition may be viewed as an instantaneous change (release or impulsive application of) the system constraints. The method uses a spatial impulse–momentum formulation in a divide and conquer scheme. The approach utilizes a hierarchic assembly–disassembly process by traversing the system topology in a binary tree map to solve for the jumps in the system generalized speeds and the constraint impulsive loads in linear and logarithmic cost in serial and parallel implementations, respectively. The method is applicable for systems in serial chain as well as kinematical loop topologies. The coupling between the unilateral and bilateral constraints is handled efficiently through the use of kinematic joint definitions. The equations of motion for the system are produced in a hierarchic sub-structured form. This has the advantage that changes in sub-structure definitions/models results in a change to the system equations only within the associated sub-structure. This allows for significant changes in model types and definitions without having to reformulate the equations for the whole system.

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Efficient methodology for multibody simulations with discontinuous changes in system definition

In the design and analysis of multibody dynamics systems, sensitivity analysis is a critical tool for good design decisions. Unless efficient algorithms are used, sensitivity analysis can be computationally expensive, and hence, limited in its efficacy. Traditional direct differentiation methods can be computationally expensive with complexity as large as O(n4+n2m2+nm3), where n is the number of generalized coordinates in the system and m is the number of algebraic constraints. In this paper, a direct differentiation divide-and-conquer approach is presented for efficient sensitivity analysis of multibody systems with general topologies. This approach uses a binary tree structure to traverse the topology of the system and recursively generate the sensitivity data in linear and logarithmic complexities for serial and parallel implementations, respectively. This method works concurrently with the forward dynamics problem, and hence, requires minimal data storage. The differentiation required in this algorithm is minimum as compared to traditional methods, and is generated locally on each body as a preprocessing step. The method provides sensitivity values accurately up to integration tolerance and is insensitive to perturbations in design parameter values. This approach is a good alternative to existing methodologies, as it is fairly simple to implement for general topologies and is computationally efficient.

A divide-and-conquer direct differentiation approach for multibody system sensitivity analysis

This paper reports a new research effort aimed at using efficient multibody dynamics methods to simulate coarse-grained molecular systems. Various molecular systems are studied and the results of nanosecond-long simulations are analyzed to validate the method. The systems studied include bulk water, alkane chains, alanine dipeptide and carboxyl terminal fragments of calmodulin, ribosomal L7/L12 and rhodopsin proteins. The stability and validity of the simulations are studied through conservation of energy, thermodynamics properties and conformational analysis. In these simulations, a speed up of an order of magnitude is realized for conservative error bounds with a fixed timestep integration scheme. A discussion is presented on the open-source software developed to facilitate future research using multibody dynamics with molecular dynamics.

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Rudranarayan M. Mukherjee

Papers

Orthogonal Complement Based Divide-and-Conquer Algorithm for constrained multibody systems

A logarithmic complexity divide-and-conquer algorithm for multi-flexible articulated body dynamics

Efficient methodology for multibody simulations with discontinuous changes in system definition

A divide-and-conquer direct differentiation approach for multibody system sensitivity analysis

Substructured molecular dynamics using multibody dynamics algorithms