Divide-and-Conquer Based Adaptive Coarse Grained Simulation of RNA

2010 ◽  
Author(s):  
Mohammad Poursina ◽  
Kishor Bhalerao ◽  
Kurt Anderson
Keyword(s):  
Molecular Modeling ◽  
Degrees Of Freedom ◽  
Traditional Approach ◽  
Equations Of Motion ◽  
Critical Role ◽  
Divide And Conquer ◽  
Coarse Grained ◽  
Biomolecular Systems ◽  
Spatial And Temporal Scales ◽  
O 10

Molecular modeling has gained increasing importance in recent years for predicting important structural properties of large biomolecular systems such as RNA which play a critical role in various biological processes. Given the complexity of biopolymers and their interactions within living organisms, efficient and adaptive multi-scale modeling approaches are necessary if one is to reasonably perform computational studies of interest. These studies nominally involve multiple important physical phenomena occurring at different spatial and temporal scales. These systems are typically characterized by large number of degrees of freedom O(103) – O(107). The temporal domains range from sub-femto seconds (O(10−16)) associated with the small high frequency oscillations of individual tightly bonded atoms to milliseconds (O(10−3)) or greater for the larger scale conformational motion. The traditional approach for molecular modeling involved fully atomistic models which results in fully decoupled equations of motion. The problems with this approach are well documented in literature.

scholarly journals "Dividing and Conquering" and "Caching" in Molecular Modeling

2021 ◽  
Author(s):  
Xiaoyong Cao ◽  
Pu Tian
Keyword(s):  
Molecular Modeling ◽  
Degrees Of Freedom ◽  
Materials Science ◽  
Force Fields ◽  
Coarse Graining ◽  
Divide And Conquer ◽  
Collective Variables ◽  
Molecular Systems ◽  
State Models ◽  
Molecular Science

Molecular modeling is widely utilized in subjects including but not limited to physics, chemistry, biology, materials science and engineering. Impressive progress has been made in development of theories, algorithms and software packages. To divide and conquer, and to cache intermediate results have been long standing principles in development of algorithms. Not surprisingly, Most of important methodological advancements in more than half century of molecule modeling are various implementations of these two fundamental principles. In the mainstream classical computational molecular science based on force fields parameterization by coarse graining, tremendous efforts have been invested on two lines of algorithm development. The first is coarse graining, which is to represent multiple basic particles in higher resolution modeling as a single larger and softer particle in lower resolution counterpart, with resulting force fields of partial transferability at the expense of some information loss. The second is enhanced sampling, which realizes "dividing and conquering" and/or "caching" in configurational space with focus either on reaction coordinates and collective variables as in metadynamics and related algorithms, or on the transition matrix and state discretization as in Markov state models. For this line of algorithms, spatial resolution is maintained but no transferability is available. Deep learning has been utilized to realize more efficient and accurate ways of "dividing and conquering" and "caching" along these two lines of algorithmic research. We proposed and demonstrated the local free energy landscape approach, a new framework for classical computational molecular science and a third class of algorithm that facilitates molecular modeling through partially transferable in resolution "caching" of distributions for local clusters of molecular degrees of freedom. Differences, connections and potential interactions among these three algorithmic directions are discussed, with the hope to stimulate development of more elegant, efficient and reliable formulations and algorithms for "dividing and conquering" and "caching" in complex molecular systems.

Fast Far-Field Potential Evaluation in Multibody-Based Simulations of Biopolymers

2013 ◽  
Author(s):  
Mohammad Poursina ◽  
Kurt S. Anderson
Keyword(s):  
Potential Field ◽  
Equations Of Motion ◽  
Charged Particles ◽  
Field Potential ◽  
Divide And Conquer ◽  
Coarse Grained ◽  
Mathematical Framework ◽  
Far Field ◽  
Geometrical Properties ◽  
Divide And Conquer Scheme

A novel algorithm to approximate the long-range potential field in multiscale simulations of biopolymers is presented. These models contain various domains including single particles, as well as regions with coarse-grained clusters in which high frequency modes of motion are suppressed. Herein, coarse-grained regions are formed via treating groups of atoms as rigid and/or flexible bodies/clusters connected together via kinematic joints, and as such, multibody dynamic techniques are used to form and solve the equations of motion. In such simulations with n particles, the evaluation of the potential field with computational complexity of O(n2), if not performed wisely, may become a bottleneck. This paper presents the approximation of the potential field due to the interaction between a charged particle and a body containing charged particles. This approximation is expressed in terms of physical and geometrical properties of the bodies such as the entire charge of the cluster and a pseudo-inertia tensor. Further, a divide-and-conquer scheme is introduced to implement the presented far-field potential evaluations. In this scheme adjacent charged bodies are combined together to form new bodies. The mathematical framework to create these new assemblies is presented. Then the potential of the resulting bodies on the charged particles which are far from them are recursively calculated.

scholarly journals “Dividing and Conquering” and “Caching” in Molecular Modeling

2021 ◽  
Vol 22 (9) ◽  
pp. 5053
Author(s):  
Xiaoyong Cao ◽  
Pu Tian
Keyword(s):  
Molecular Modeling ◽  
Degrees Of Freedom ◽  
Materials Science ◽  
Coarse Graining ◽  
Divide And Conquer ◽  
Collective Variables ◽  
Molecular Systems ◽  
Materials Science And Engineering ◽  
State Models ◽  
Molecular Science

Molecular modeling is widely utilized in subjects including but not limited to physics, chemistry, biology, materials science and engineering. Impressive progress has been made in development of theories, algorithms and software packages. To divide and conquer, and to cache intermediate results have been long standing principles in development of algorithms. Not surprisingly, most important methodological advancements in more than half century of molecular modeling are various implementations of these two fundamental principles. In the mainstream classical computational molecular science, tremendous efforts have been invested on two lines of algorithm development. The first is coarse graining, which is to represent multiple basic particles in higher resolution modeling as a single larger and softer particle in lower resolution counterpart, with resulting force fields of partial transferability at the expense of some information loss. The second is enhanced sampling, which realizes “dividing and conquering” and/or “caching” in configurational space with focus either on reaction coordinates and collective variables as in metadynamics and related algorithms, or on the transition matrix and state discretization as in Markov state models. For this line of algorithms, spatial resolution is maintained but results are not transferable. Deep learning has been utilized to realize more efficient and accurate ways of “dividing and conquering” and “caching” along these two lines of algorithmic research. We proposed and demonstrated the local free energy landscape approach, a new framework for classical computational molecular science. This framework is based on a third class of algorithm that facilitates molecular modeling through partially transferable in resolution “caching” of distributions for local clusters of molecular degrees of freedom. Differences, connections and potential interactions among these three algorithmic directions are discussed, with the hope to stimulate development of more elegant, efficient and reliable formulations and algorithms for “dividing and conquering” and “caching” in complex molecular systems.

scholarly journals "Dividing and Conquering" and "Caching" in Molecular Modeling

2020 ◽  
Author(s):  
Xiaoyong Cao ◽  
Pu Tian
Keyword(s):  
Machine Learning ◽  
Free Energy ◽  
Molecular Modeling ◽  
Materials Science ◽  
Divide And Conquer ◽  
Machine Learning Techniques ◽  
Collective Variables ◽  
Spatial And Temporal Scales ◽  
Wide Range ◽  
State Models

Molecular modeling is widely utilized in subjects including but not limited to physics, chemistry, biology, materials science and engineering. Impressive progress has been made in development of theories, algorithms and software packages. To divide and conquer, and to cache intermediate results have been long standing principles in development of algorithms. Not surprisingly, Most of important methodological advancements in more than half century of molecule modeling are various implementations of these two fundamental principles. To access interesting behavior of complex molecular systems in a wide range of spatial and temporal scales, the molecular modeling community has invested tremendous efforts on two lines of algorithm development. The first is coarse graining, which is to represent multiple basic particles in higher resolution modeling as a single larger and softer particle in lower resolution counterpart, with resulting force fields of partial transferability at the expense of some information loss. The second is enhanced sampling, which realizes "dividing and conquering" and/or "caching" in configurational space with focus either on reaction coordinates and collective variables as in Metadynamics and related algorithms, or on the transition matrix and state discretization as in Markov state models. For this line of algorithms, spatial resolution is maintained but no transferability is available. With introduction of machine learning techniques, many new developments, particularly those based on deep learning, have been implemented to realize more efficient and accurate ways of "dividing and conquering" and "caching" along these two lines of algorithmic research. We recently developed the generalized solvation free energy theory , which suggests a third class of algorithm that facilitate molecular modeling through partially transferable in resolution "caching" of local free energy landscape. Connections and potential interactions among these three algorithmic directions are discussed. This brief review is on both the traditional development and the application of machine learning in molecular modeling from the perspective of "dividing and conquering" and "caching", with the hope to stimulate development of more elegant, efficient and reliable formulations and algorithms in this regard.

Influence of Frame Stiffness and Rider Position on Bike Dynamics: Analytical Study

2015 ◽  
Author(s):  
Trevor Williams ◽  
Sudhir Kaul ◽  
Anoop Dhingra
Keyword(s):  
Dynamic Characteristics ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Critical Role ◽  
Rear Wheel ◽  
Front Wheel ◽  
Rigid Bodies ◽  
Limited Attention ◽  
Structural Stiffness ◽  
Bicycle Frame

Dynamic characteristics of a bicycle such as handling and stability can be studied during the design phase to comprehend specific aspects associated with the overall layout as well as the frame architecture. Bicycles demonstrate unique properties such as static instability that is overcome by getting them into motion with a minimum velocity threshold. The structural stiffness of a frame plays a critical role in the handling behavior of a bike. However, the influence of structural stiffness has received limited attention in the existing literature. This paper attempts to fill the gap by presenting analytical results from a study that includes the influence of rider positions on three bicycle layouts. The analytical model consists of four rigid bodies: rear frame, front frame (front fork and handle bar assembly), front wheel and rear wheel. The overall model exhibits three degrees-of-freedom: the roll angle of the frame, the steering of the front frame, and the rotation of the rear wheel with respect to the frame. The rear frame is divided into two parts, the rider and the bicycle frame, that are assumed to be rigidly connected. This is done in order to allow the model to account for varying rider positions. The influence of frame flexibility is studied by coupling the structural stiffness of the frame to the governing equations of motion. Layouts from a benchmark bicycle, a commercially manufactured bicycle, and a cargo bicycle are used for this study in conjunction with rider positions ranging from a relaxed position to an extreme prone position. All the results are analyzed and compared with some proven analytical and experimental results in the existing literature. Results indicate that some of the rider positions can play a significant role in influencing the dynamic characteristics of the bike. Structural stiffness is seen to significantly affect the weave mode, only when the stiffness is reduced substantially.

A Logarithmic Complexity Divide-and-Conquer Algorithm for Multi-flexible Articulated Body Dynamics

2006 ◽  
Vol 2 (1) ◽  
pp. 10-21 ◽  
Author(s):  
Rudranarayan M. Mukherjee ◽  
Kurt S. Anderson
Keyword(s):  
Degrees Of Freedom ◽  
Temporal Integration ◽  
Equations Of Motion ◽  
Turnaround Time ◽  
Divide And Conquer ◽  
Dynamics Simulation ◽  
Flexible Body ◽  
Integration Step ◽  
Large Rotations ◽  
Nonlinear Equations Of Motion

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 nb joint variables, providing O(log(nb)) turnaround time per temporal integration step, achieved with O(nb) processors. The actual cost associated with the parallel treatment of the nf 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.

A Logarithmic Complexity Divide and Conquer Algorithm for Flexible Multibody Dynamics

2005 ◽  
Author(s):  
Rudranarayan Mukherjee ◽  
Kurt Anderson
Keyword(s):  
Elastic Deformation ◽  
Degrees Of Freedom ◽  
Parallel Implementation ◽  
Equations Of Motion ◽  
Computational Cost ◽  
Flexible Multibody Dynamics ◽  
Divide And Conquer ◽  
Element Analysis ◽  
Large Rotations ◽  
Nonlinear Equations Of Motion

This paper presents an efficient algorithm for parallel implementation of multi-flexible-body dynamics systems simulation and analysis. The effective overall computational cost of the algorithm is logarithmic when implemented with a processor optimal O(n) processors. This algorithm formulates and solves the nonlinear equations of motion for mechanical systems with interconnected flexible bodies subject to small elastic deformation together with large rotations and translations. The large rotations or translations are modeled as rigid body degree of freedom associated with the interconnecting kinematic joint degrees of freedom. The elastic deformation of the component bodies is modeled through the use of admissible shape functions generated using standard finite element analysis software or otherwise. Apart from the approximation associated with the elastic deformations, this algorithm is exact, non-iterative and applicable to generalized multi-flexible chain and free topologies.

scholarly journals "Dividing and Conquering" and "Caching" in Molecular Modeling

2021 ◽  
Author(s):  
Xiaoyong Cao ◽  
Pu Tian
Keyword(s):  
Molecular Modeling ◽  
Degrees Of Freedom ◽  
Materials Science ◽  
Force Fields ◽  
Coarse Graining ◽  
Divide And Conquer ◽  
Collective Variables ◽  
Molecular Systems ◽  
State Models ◽  
Molecular Science

Molecular modeling is widely utilized in subjects including but not limited to physics, chemistry, biology, materials science and engineering. Impressive progress has been made in development of theories, algorithms and software packages. To divide and conquer, and to cache intermediate results have been long standing principles in development of algorithms. Not surprisingly, Most of important methodological advancements in more than half century of molecule modeling are various implementations of these two fundamental principles. In the mainstream classical computational molecular science based on force fields parameterization by coarse graining, tremendous efforts have been invested on two lines of algorithm development. The first is coarse graining, which is to represent multiple basic particles in higher resolution modeling as a single larger and softer particle in lower resolution counterpart, with resulting force fields of partial transferability at the expense of some information loss. The second is enhanced sampling, which realizes "dividing and conquering" and/or "caching" in configurational space with focus either on reaction coordinates and collective variables as in metadynamics and related algorithms, or on the transition matrix and state discretization as in Markov state models. For this line of algorithms, spatial resolution is maintained but no transferability is available. Deep learning has been utilized to realize more efficient and accurate ways of "dividing and conquering" and "caching" along these two lines of algorithmic research. We proposed and demonstrated the local free energy landscape approach, a new framework for classical computational molecular science and a third class of algorithm that facilitates molecular modeling through partially transferable in resolution "caching" of distributions for local clusters of molecular degrees of freedom. Differences, connections and potential interactions among these three algorithmic directions are discussed, with the hope to stimulate development of more elegant, efficient and reliable formulations and algorithms for "dividing and conquering" and "caching" in complex molecular systems.

The motion of a lunar orbiter

1966 ◽  
Vol 25 ◽  
pp. 373
Author(s):  
Y. Kozai
Keyword(s):  
Degrees Of Freedom ◽  
Dimensional Space ◽  
Artificial Satellite ◽  
Equations Of Motion ◽  
Triaxial Ellipsoid ◽  
The Moon ◽  
Secular Motion ◽  
The Earth ◽  
Close Satellite ◽  
The Mean

The motion of an artificial satellite around the Moon is much more complicated than that around the Earth, since the shape of the Moon is a triaxial ellipsoid and the effect of the Earth on the motion is very important even for a very close satellite.The differential equations of motion of the satellite are written in canonical form of three degrees of freedom with time depending Hamiltonian. By eliminating short-periodic terms depending on the mean longitude of the satellite and by assuming that the Earth is moving on the lunar equator, however, the equations are reduced to those of two degrees of freedom with an energy integral.Since the mean motion of the Earth around the Moon is more rapid than the secular motion of the argument of pericentre of the satellite by a factor of one order, the terms depending on the longitude of the Earth can be eliminated, and the degree of freedom is reduced to one.Then the motion can be discussed by drawing equi-energy curves in two-dimensional space. According to these figures satellites with high inclination have large possibilities of falling down to the lunar surface even if the initial eccentricities are very small.The principal properties of the motion are not changed even if plausible values ofJ3andJ4of the Moon are included.This paper has been published in Publ. astr. Soc.Japan15, 301, 1963.

The Influence of Loading Position in A Priori High Stress Detection using Mode Superposition

2020 ◽  
Vol 1 (1) ◽  
pp. 93-102
Author(s):  
Carsten Strzalka ◽  
◽  
Manfred Zehn ◽  
Keyword(s):  
Finite Element ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
A Priori ◽  
High Stress ◽  
Von Mises Stress ◽  
Detailed Knowledge ◽  
Variable Loading ◽  
Numerical Effort ◽  
Von Mises

For the analysis of structural components, the finite element method (FEM) has become the most widely applied tool for numerical stress- and subsequent durability analyses. In industrial application advanced FE-models result in high numbers of degrees of freedom, making dynamic analyses time-consuming and expensive. As detailed finite element models are necessary for accurate stress results, the resulting data and connected numerical effort from dynamic stress analysis can be high. For the reduction of that effort, sophisticated methods have been developed to limit numerical calculations and processing of data to only small fractions of the global model. Therefore, detailed knowledge of the position of a component’s highly stressed areas is of great advantage for any present or subsequent analysis steps. In this paper an efficient method for the a priori detection of highly stressed areas of force-excited components is presented, based on modal stress superposition. As the component’s dynamic response and corresponding stress is always a function of its excitation, special attention is paid to the influence of the loading position. Based on the frequency domain solution of the modally decoupled equations of motion, a coefficient for a priori weighted superposition of modal von Mises stress fields is developed and validated on a simply supported cantilever beam structure with variable loading positions. The proposed approach is then applied to a simplified industrial model of a twist beam rear axle.

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