CN103279645A - Carbon nano tube molecular dynamics simulation method based on GPU parallel computation - Google Patents

Abstract

The invention discloses a carbon nano tube molecular dynamics simulation method based on GPU parallel computation. The method includes the following steps: (1) structuring a carbon nano tube system model composed of carbon nano tubes and C60 molecules through database files; (2) stressing the C60 molecules so that the C60 molecules can oscillate back and forth at an equilibrium position, wherein when the C60 molecules are in an equilibrium state, the positions and speeds of carbon particles on the walls of the carbon nano tubes and the positions and speeds of the C60 molecules are made to change; (3) dividing the carbon nano tubes into a plurality of layers of computing units on a CUDA platform, performing round robin computation by the utilization of a CPU traversal computing unit to obtain a parallel computation queue, and scheduling a stream processing unit of a GPU to perform parallel computation and processing; (4) executing the step (3) repeatedly, finally outputting the motion trajectory of the C60 molecules along with simulation time, drawing a curve chart of temperature change along with energy transmission in the carbon nano tubes, and completing the simulation process. The carbon nano tube molecular dynamics simulation method based on the GPU parallel computation promotes computing efficiency of molecular dynamics simulation.

Description

Carbon nanotube molecule dynamics simulation method based on the GPU concurrent operation

Technical field

The present invention relates to a kind of carbon nano-tube analytic dynamics emulation mode, especially a kind of carbon nanotube molecule dynamics simulation method based on the GPU concurrent operation belongs to simulation technical field.

Background technology

In carbon nanotube molecule dynamics simulation process, owing to relate to the floating-point operation of ultra-large molecular weight and a large amount of complexity in its computing, make the carbon nanotube molecule dynamics simulation need extremely strong computing power.Because carbon nano-tube is made of a large amount of molecules, relate to a large amount of floating-point operations in its analog simulation, along with the expansion of molecular dynamics simulation scale, simulation result visual also extremely important.For these carbon nanotube molecule dynamics simulation systems, the floating-point operation ability and the computation capability that improve simulation algorithm have significant role.

In emerging nanometer engineering field, the macroscopical mechanism that is based upon on the continuous medium basis is difficult to explain some phenomena that occur in the nanometer engineering, thereby the molecular dynamics method becomes one of important research means.The carbon nanotube molecule dynamics simulation, mainly simulate the motion of carbon nano-tube system by Newtonian mechanics, extracted data sample in the data acquisition when its different conditions, thereby calculate the configuration integration of carbon nano-tube system, and serve as thermodynamic quantity and other macroscopic properties that the carbon nano-tube system is further calculated on the basis with the result of configuration integration, thereby efficient emulation carbon nano-tube system.But carbon nanotube molecule dynamics simulation algorithm is limited to greatly by computing power two: comprise the huge amount particle in the first little space, calculated amount is very big; It two is be to guarantee numerical simulation stability, and typical molecular dynamics time step is femtosecond (fs) level, computing power is required high.

Be subject to the development of computing power, for enlarging the simulation algorithm scale, domestic and international many experts and scholars have done big quantity research to the molecular dynamics simulation algorithm.Serial algorithm by initial expansion unit scale, develop into the parallel algorithm by enlarging the simulation scale for a plurality of CPU distribution of computation tasks now, the scale that can simulate is brought up to up to a million, up to ten million by several thousand initial atoms even to the scale of more than one hundred million atoms.Though the computing power of CPU is very powerful, but for the ultra-large simulation system with practical significance, its travelling speed also far can not practical requirement, and for its batch processing carbon, the serial implementation efficiency of single CPU is also lower comparatively speaking.Therefore, large-scale carbon nanotube molecule dynamics simulation algorithm needs further to break through.

Summary of the invention

The objective of the invention is in order to solve the defective of above-mentioned prior art, a kind of carbon nanotube molecule dynamics simulation method based on the GPU concurrent operation is provided, this method can increase substantially the simulation scale of carbon nano-tube, greatly improves the operation efficiency of molecular dynamics emulation.

Purpose of the present invention can reach by taking following technical scheme:

Carbon nanotube molecule dynamics simulation method based on the GPU concurrent operation is characterized in that may further comprise the steps:

1) by database file structure carbon nano-tube system model, this model carries out initialization by carbon nano-tube and the C60 molecular composition that is arranged on carbon nano-tube inside to model, reads all particle's velocity and position coordinates in the model;

2) the C60 molecule is subjected to the power that carbon particle produces on the carbon nano-tube tube wall, vibration back and forth on the equilibrium position, when the C60 molecule is in equilibrium state, the condition that different temperatures is set respectively simultaneously at the forward and backward two ends of carbon nano-tube is simulated, and setting simulated time, make that speed and the position of carbon particle and C60 molecule constantly changes on the carbon nano-tube tube wall, and then stressed, the motion state of C60 molecule and energy are constantly changed;

3) on the CUDA platform, carbon nano-tube is cut apart, be divided into the separate computing unit of multilayer, but adopt CPU traversal computing unit to carry out the trocheameter calculation and obtain the concurrent operation formation, dispatch the stream processing unit of GPU again, adopt the Verlet algorithm to carry out concurrent operation and processing, repeat this step and finish until simulated time;

4) adopt database file record related data, export the C60 molecule along with the running orbit of simulated time, and describe the interior temperature changing curve diagram that transmits along with energy of carbon nano-tube, finish simulation process.

As a kind of preferred version, described carbon nano-tube is the stratiform hollow structure, is made up of a plurality of hexagonal carbon ring structures; Described C60 molecule has one or more.

As a kind of preferred version, described step 2) in, adopt Gaussian heating bath simulation that the heating bath condition of different temperatures is set respectively at the forward and backward two ends of carbon nano-tube, and set the heating bath time as simulated time.

As a kind of preferred version, in the described step 3), it is specific as follows that employing CPU traversal computing unit carries out trocheameter:

A) if all particles all as the center calculation particle, jump to step h);

B) find first non-competing particle not by the particle of center calculation in order, but add the concurrent operation formation;

C) all neighbours of this particle of mark are that this parallel queue once can not parallel particle, if the neighbour be two degree can not parallel particle, then promote to once calculating;

D) all time neighbour of this particle of mark be that this parallel queue two degree can not parallel particle, if inferior neighbour does not then revise its number of degrees for once can not parallel particle;

E) this particle of mark is for pressing the center calculation particle;

F) whether traversed the particle tail of the queue? if continue to carry out, jump to step a) if not;

G) but next one parallel queue begin, return step b);

H) finish.

As a kind of preferred version, in the described step 3), it is specific as follows with processing to adopt the Verlet algorithm to carry out concurrent operation:

A) by all particle position of carbon nano-tube system model, calculate key relation and angular relationship between neighbour's particle and inferior neighbour;

B) stream processing unit of scheduling GPU, parallel computation is segmented in the particle in the various computing unit, and the interaction force of each particle and its neighbour's particle on the carbon nano-tube tube wall is calculated in accumulation;

C) according to C60 molecule region, calculate the interaction force of each particle in interior each particle of C60 molecule and its carbon nano-tube zone, place;

D) according to particle suffered power and speed thereof, upgrade particle position, execution in step b again) and c);

E) according to the suffered power of particle, calculate the heat flow value of this speed of particle and carbon nano-tube forward and backward two ends heating bath;

F) if reach the circulation frequency, then calculate and finish; Otherwise, in the data that the CPU end is preserved particle at interval, return step d).

As a kind of preferred version, in the described step 3), Verlet algorithm specific design is as follows:

A) x (t+ Δ t) and x (t-Δ t) being carried out Taylor expansion is shown below:

$\stackrel{\→}{x}(t+\mathrm{\Δt})=\stackrel{\→}{x}\left(t\right)+\stackrel{\→}{v}\left(t\right)\mathrm{\Δt}+\frac{\stackrel{\→}{a}\left(t\right){\mathrm{\Δt}}^2}{2}+\frac{\stackrel{\→}{b}\left(t\right){\mathrm{\Δt}}^3}{2}+O\left(\mathrm{\Δ}{t}^4\right)---\left(1\right)$

$\stackrel{\→}{x}(t-\mathrm{\Δt})=\stackrel{\→}{x}\left(t\right)-\stackrel{\→}{v}\left(t\right)\mathrm{\Δt}+\frac{\stackrel{\→}{a}\left(t\right){\mathrm{\Δt}}^2}{2}-\frac{\stackrel{\→}{b}\left(t\right){\mathrm{\Δt}}^3}{2}+O\left(\mathrm{\Δ}{t}^4\right)---\left(2\right)$

Wherein, x (t+ Δ t) is expressed as the position of previous moment, and x (t-Δ t) is expressed as position constantly, back one;

B) with formula (1) and formula (2) addition, it is as follows to obtain L-expression:

$\stackrel{\→}{x}(t+\mathrm{\Δt})=2\stackrel{\→}{x}\left(t\right)-\stackrel{\→}{x}(t-\mathrm{\Δt})+\stackrel{\→}{a}\left(t\right){\mathrm{\Δt}}^2+O\left(\mathrm{\Δ}{t}^4\right)---\left(3\right)$

At position and the acceleration of known carbon nano-tube system model particle current time t, and under the situation of the position of previous moment t-Δ, extrapolate next position of t+ Δ t constantly;

C) formula (1) and formula (2) are subtracted each other, both sides are simultaneously divided by 2 Δ t again, and the expression formula that obtains speed is as follows:

$\stackrel{\→}{v}\left(t\right)=\frac{\stackrel{\→}{x}(t+\mathrm{\Δt})-\stackrel{\→}{x}(t-\mathrm{\Δt})}{2\mathrm{\Δt}}+O\left({\mathrm{\Δt}}^2\right)---\left(4\right)$

Constantly under the situation of the position of t+ Δ t, extrapolate the speed v (t) of current time t in the position of known carbon nano-tube system model particle previous moment t-Δ t and back one;

D) by formula (1)～(4), under the situation of known particle t-2 Δ t position, the position in the t-Δ t moment and t-Δ t acceleration constantly constantly, start the Verlet algorithm and carry out integration: according to t-2 Δ t position, t-Δ t position and t-Δ t acceleration constantly constantly constantly, with t=t-Δ t substitution formula (3), obtain the position of current time t; According to the position of current time t, upgrade the acceleration of current time t based on certain potential function; Simultaneously, according to position and the t-2 Δ t position constantly of current time t, with t=t-Δ t substitution formula (4), upgrade t-Δ t speed constantly; Namely obtain position, t-Δ t speed and the t acceleration constantly constantly of particle current time t, repeat this step.

The present invention has following beneficial effect with respect to prior art:

1, the present invention adopts database file structure Nano carbon balls C60 model and carbon nano tube structure model, thereby carbon nanotube molecule dynamics is carried out emulation, not only convenient calculating, also being convenient to construct multiple different model compares, portable higher, and utilize algorithm can simulate the running orbit of C60 molecule in carbon nano-tube of football shaped structure.

2, the present invention adopts the concurrent operation platform CUDA based on GPU by the issue of NVIDIA company to carry out the image processing, the CUDA platform be one based on the architecture of C language, possessing has a large amount of high-performance calculation instructions and good DLL (dynamic link library), can greatly improve the efficient of carbon nanotube molecule dynamics simulation method.

3, the present invention can be cut apart large-scale carbon nano-tube system model by the CUDA platform, be divided into multilayer suitable size, separate and computing unit that GPU can bear, make the parallel running in CUDA of each algorithm, greatly improved the speed of carbon nanotube molecule dynamics simulation method.

Description of drawings

Fig. 1 is C60 molecular schematic diagram of the present invention.

Fig. 2 is carbon nano-tube synoptic diagram of the present invention.

Fig. 3 is the structural representation of the carbon nano-tube system model of the present invention's structure.

Fig. 4 is the data stream block diagram of the present invention's carbon nanotube molecule dynamics simulation in CPU and GPU.

Fig. 5 carries out the schematic flow sheet of carbon nanotube molecule dynamics simulation under the CUDA environment for the present invention.

Fig. 6 is comparison diagram working time that the present invention is based on GPU concurrent operation and traditional serial arithmetic.

Embodiment

Embodiment 1:

As Fig. 1～shown in Figure 5, present embodiment is as follows based on the carbon nanotube molecule dynamics simulation method of GPU concurrent operation:

1) in database file, reads the parameter condition of carbon nano-tube system, as initial temperature, population, the density time etc., structure carbon nano-tube system model MOD1 and parameter model MOD2, model M OD2 can arrange various parameters and use at model M OD1, model M OD1 such as Fig. 1～shown in Figure 3, by carbon nano-tube be arranged on a football shaped C60 molecule (molecule that is constituted by 60 carbon atoms of carbon nano-tube inside, the likeness in form football, have another name called football alkene) form, described carbon nano-tube is the stratiform hollow structure, pipe shaft is the director circle tubular construction, be made up of a plurality of hexagonal carbon ring structures unit (being made of carbon atom), generally between one to tens nanometer, length then is far longer than its diameter to its diameter, this model is carried out initialization, read all particle's velocity and position coordinates in the model;

2) the C60 molecule is subjected to the power that carbon particle produces on the carbon nano-tube tube wall, vibration back and forth on the equilibrium position, when the C60 molecule is in equilibrium state, adopt Gaussian hot bath method (constraint temperature control method), the heating bath condition of different temperatures is set respectively simultaneously at the carbon nano-tube two ends, and the setting heating bath time, make the variation that carbon particle and C60 molecule are broken occurrence positions and speed on the carbon nano-tube tube wall because of equilibrium state, and these variations are interactional, has transitivity, be accompanied by the variation of these particle positions and speed, energy transmits in the carbon nano-tube system of elongated tubular, because the C60 molecule is positioned at carbon nano-tube, its stressed meeting is along with the position of carbon nano-tube constituent particle changes and changes, therefore follow the process of this energy transmission, the motion state of C60 molecule and energy also can change;

The ultimate principle that adopts the Gaussian hot bath method is to add friction force f in the equation of motion _i, and with itself and particle rapidity v _iConnect, its force bearing formulae is:

f _i＝ma _i+ξmv _i

When equilibrium state, system temperature is constant, and therefore dEk/dt=0 is arranged

Namely have: $\underset{i}{\mathrm{\Σ}}{v}_i{a}_i=0$

Can get thus: $\mathrm{\ξ}=\frac{\underset{i}{\mathrm{\Σ}}{f}_i{v}_i}{m\underset{i}{\mathrm{\Σ}}{v}_i^2}$

3) on the CUDA platform, carbon nano-tube is cut apart, be divided into multilayer suitable size, separate computing unit, partition principle is to improve degree of parallelism in the prerequisite of avoiding too much double counting, the computing unit of namely cutting apart is thicknessization rationally, because computing unit is excessive, then parallel not obvious, computing unit is too small, can cause the calculating of too many unnecessary repetition;

As shown in Figure 4, obtain n concurrent operation formation but adopt CPU traversal computing unit to carry out the trocheameter calculation, as follows:

A) if all particles all as the center calculation particle, jump to step h);

B) find first non-competing particle not by the particle of center calculation in order, but add the concurrent operation formation;

C) all neighbours of this particle of mark are that this parallel queue once can not parallel particle, if the neighbour be two degree can not parallel particle, then promote to once calculating;

D) all time neighbour of this particle of mark be that this parallel queue two degree can not parallel particle, if inferior neighbour does not then revise its number of degrees for once can not parallel particle;

E) this particle of mark is for pressing the center calculation particle;

F) whether traversed the particle tail of the queue? if continue to carry out, jump to step a) if not;

G) but next one parallel queue begin, return step b);

H) finish;

As shown in Figure 4 and Figure 5, the stream processing unit of scheduling GPU adopts the Verlet algorithm to carry out concurrent operation and processing, and is as follows:

A) by all particle position of carbon nano-tube system model, calculate key relation and angular relationship between neighbour's particle and inferior neighbour;

B) stream processing unit of scheduling GPU, parallel computation is segmented in the particle in the various computing unit, and the interaction force of each particle and its neighbour's particle on the carbon nano-tube tube wall is calculated in accumulation;

C) according to C60 molecule region, calculate the interaction force (Van der Waals force) of each particle in interior each particle of C60 molecule and its carbon nano-tube zone, place;

D) according to particle suffered power and speed thereof, upgrade particle position, execution in step b again) and c);

E) according to the suffered power of particle, calculate the heat flow value of this speed of particle and carbon nano-tube forward and backward two ends heating bath;

F) if reach the circulation frequency, then calculate and finish; Otherwise, in the data that the CPU end is preserved particle at interval, return step d).

4) repeated execution of steps 3), finish until the heating bath time, adopt database file record related data, export the C60 molecule along with the running orbit of simulated time, and describe the interior temperature changing curve diagram that transmits along with energy of carbon nano-tube, finish simulation process.

In the described step 3), because interparticle acting force spacing influence is bigger, must set one apart from critical value, to judge its position relation, when interparticle spacing less than the distance critical value, be neighbour's particle, when interparticle spacing greater than the distance critical value, be inferior neighbour, think that then its mutual molecular force can ignore.

In the described step 3), Verlet algorithm specific design is as follows:

A) x (t+ Δ t) and x (t-Δ t) being carried out Taylor expansion is shown below:

$\stackrel{\→}{x}(t+\mathrm{\Δt})=\stackrel{\→}{x}\left(t\right)+\stackrel{\→}{v}\left(t\right)\mathrm{\Δt}+\frac{\stackrel{\→}{a}\left(t\right){\mathrm{\Δt}}^2}{2}+\frac{\stackrel{\→}{b}\left(t\right){\mathrm{\Δt}}^3}{2}+O\left(\mathrm{\Δ}{t}^4\right)---\left(1\right)$

$\stackrel{\→}{x}(t-\mathrm{\Δt})=\stackrel{\→}{x}\left(t\right)-\stackrel{\→}{v}\left(t\right)\mathrm{\Δt}+\frac{\stackrel{\→}{a}\left(t\right){\mathrm{\Δt}}^2}{2}-\frac{\stackrel{\→}{b}\left(t\right){\mathrm{\Δt}}^3}{2}+O\left(\mathrm{\Δ}{t}^4\right)---\left(2\right)$

Wherein, x (t+ Δ t) is expressed as the position of previous moment, and x (t-Δ t) is expressed as position constantly, back one;

B) with formula (1) and formula (2) addition, it is as follows to obtain L-expression:

$\stackrel{\→}{x}(t+\mathrm{\Δt})=2\stackrel{\→}{x}\left(t\right)-\stackrel{\→}{x}(t-\mathrm{\Δt})+\stackrel{\→}{a}\left(t\right){\mathrm{\Δt}}^2+O\left(\mathrm{\Δ}{t}^4\right)---\left(3\right)$

At position and the acceleration of known carbon nano-tube system model particle current time t, and under the situation of the position of previous moment t-Δ, extrapolate next position of t+ Δ t constantly;

C) formula (1) and formula (2) are subtracted each other, both sides are simultaneously divided by 2 Δ t again, and the expression formula that obtains speed is as follows:

$\stackrel{\→}{v}\left(t\right)=\frac{\stackrel{\→}{x}(t+\mathrm{\Δt})-\stackrel{\→}{x}(t-\mathrm{\Δt})}{2\mathrm{\Δt}}+O\left({\mathrm{\Δt}}^2\right)---\left(4\right)$

Constantly under the situation of the position of t+ Δ t, extrapolate the speed v (t) of current time t in the position of known carbon nano-tube system model particle previous moment t-Δ t and back one;

D) by formula (1)～(4), under the situation of known particle t-2 Δ t position, the position in the t-Δ t moment and t-Δ t acceleration constantly constantly, start the Verlet algorithm and carry out integration: according to t-2 Δ t position, t-Δ t position and t-Δ t acceleration constantly constantly constantly, with t=t-Δ t substitution formula (3), obtain the position of current time t; According to the position of current time t, upgrade the acceleration of current time t based on certain potential function, thereby obtain that it is stressed; Simultaneously, according to position and the t-2 Δ t position constantly of current time t, with t=t-Δ t substitution formula (4), upgrade t-Δ t speed constantly; Namely obtain position, t-Δ t speed and the t acceleration (stressed) constantly constantly of particle current time t, repeat this step.

As shown in Figure 6, can see the GPU concurrent operation that the present invention adopts, its execution time will be far smaller than traditional serial arithmetic, and therefore, its simulation efficiency improves a lot than traditional serial arithmetic.

Embodiment 2:

The principal feature of present embodiment is: in the described step 1), the C60 molecule that is arranged on carbon nano-tube inside can be for a plurality of, and all the other are with embodiment 1.

The above; it only is the preferred embodiment of the invention; but protection scope of the present invention is not limited thereto; anyly be familiar with those skilled in the art in scope disclosed in this invention; be equal to replacement or change according to technical scheme of the present invention and inventive concept thereof, all belonged to protection scope of the present invention.

Claims (6)

1. based on the carbon nanotube molecule dynamics simulation method of GPU concurrent operation, it is characterized in that may further comprise the steps:

1) by database file structure carbon nano-tube system model, this model carries out initialization by carbon nano-tube and the C60 molecular composition that is arranged on carbon nano-tube inside to model, reads all particle's velocity and position coordinates in the model;

2) the C60 molecule is subjected to the power that carbon particle produces on the carbon nano-tube tube wall, vibration back and forth on the equilibrium position, when the C60 molecule is in equilibrium state, the condition that different temperatures is set respectively simultaneously at the forward and backward two ends of carbon nano-tube is simulated, and setting simulated time, make that speed and the position of carbon particle and C60 molecule constantly changes on the carbon nano-tube tube wall, and then stressed, the motion state of C60 molecule and energy are constantly changed;

3) on the CUDA platform, carbon nano-tube is cut apart, be divided into the separate computing unit of multilayer, but adopt CPU traversal computing unit to carry out the trocheameter calculation and obtain the concurrent operation formation, dispatch the stream processing unit of GPU again, adopt the Verlet algorithm to carry out concurrent operation and processing;

4) repeated execution of steps 3), finish until simulated time, adopt database file record related data, export the C60 molecule along with the running orbit of simulated time, and describe the interior temperature changing curve diagram that transmits along with energy of carbon nano-tube, finish simulation process.

2. the carbon nanotube molecule dynamics simulation method based on the GPU concurrent operation according to claim 1, it is characterized in that: described carbon nano-tube is the stratiform hollow structure, is made up of a plurality of hexagonal carbon ring structures; Described C60 molecule has one or more.

3. the carbon nanotube molecule dynamics simulation method based on the GPU concurrent operation according to claim 1 and 2, it is characterized in that: described step 2), adopt Gaussian heating bath simulation that the heating bath condition of different temperatures is set respectively at the forward and backward two ends of carbon nano-tube, and set the heating bath time as simulated time.

4. the carbon nanotube molecule dynamics simulation method based on the GPU concurrent operation according to claim 3 is characterized in that: in the described step 3), it is specific as follows to adopt CPU traversal computing unit to carry out trocheameter:

A) if all particles all as the center calculation particle, jump to step h);

B) find first non-competing particle not by the particle of center calculation in order, but add the concurrent operation formation;

C) all neighbours of this particle of mark are that this parallel queue once can not parallel particle, if the neighbour be two degree can not parallel particle, then promote to once calculating;

D) all time neighbour of this particle of mark be that this parallel queue two degree can not parallel particle, if inferior neighbour does not then revise its number of degrees for once can not parallel particle;

E) this particle of mark is for pressing the center calculation particle;

F) whether traversed the particle tail of the queue? if continue to carry out, jump to step a) if not;

G) but next one parallel queue begin, return step b);

H) finish.

5. the carbon nanotube molecule dynamics simulation method based on the GPU concurrent operation according to claim 4 is characterized in that: in the described step 3), adopt the Verlet algorithm to carry out concurrent operation and handle specific as follows:

A) by all particle position of carbon nano-tube system model, calculate key relation and angular relationship between neighbour's particle and inferior neighbour;

B) stream processing unit of scheduling GPU, parallel computation is segmented in the particle in the various computing unit, and accumulation calculate on the carbon nano-tube tube wall in the interaction force of each particle and its neighbour's particle;

C) according to C60 molecule region, calculate the interaction force of each particle in interior each particle of C60 molecule and its carbon nano-tube zone, place;

D) according to particle suffered power and speed thereof, upgrade particle position, execution in step b again) and c);

E) according to the suffered power of particle, calculate the heat flow value of this speed of particle and carbon nano-tube forward and backward two ends heating bath;

F) if reach the circulation frequency, then calculate and finish; Otherwise, in the data that the CPU end is preserved particle at interval, return step d).

6. the carbon nanotube molecule dynamics simulation method based on the GPU concurrent operation according to claim 5, it is characterized in that: in the described step 3), Verlet algorithm specific design is as follows:

A) x (t+ Δ t) and x (t-Δ t) being carried out Taylor expansion is shown below:

$\stackrel{\→}{x}(t+\mathrm{\Δt})=\stackrel{\→}{x}\left(t\right)+\stackrel{\→}{v}\left(t\right)\mathrm{\Δt}+\frac{\stackrel{\→}{a}\left(t\right){\mathrm{\Δt}}^2}{2}+\frac{\stackrel{\→}{b}\left(t\right){\mathrm{\Δt}}^3}{2}+O\left(\mathrm{\Δ}{t}^4\right)---\left(1\right)$

$\stackrel{\→}{x}(t-\mathrm{\Δt})=\stackrel{\→}{x}\left(t\right)-\stackrel{\→}{v}\left(t\right)\mathrm{\Δt}+\frac{\stackrel{\→}{a}\left(t\right){\mathrm{\Δt}}^2}{2}-\frac{\stackrel{\→}{b}\left(t\right){\mathrm{\Δt}}^3}{2}+O\left(\mathrm{\Δ}{t}^4\right)---\left(2\right)$

Wherein, x (t+ Δ t) is expressed as the position of previous moment, and x (t-Δ t) is expressed as position constantly, back one;

B) with formula (1) and formula (2) addition, it is as follows to obtain L-expression:

$\stackrel{\→}{x}(t+\mathrm{\Δt})=2\stackrel{\→}{x}\left(t\right)-\stackrel{\→}{x}(t-\mathrm{\Δt})+\stackrel{\→}{a}\left(t\right){\mathrm{\Δt}}^2+O\left(\mathrm{\Δ}{t}^4\right)---\left(3\right)$

At position and the acceleration of known carbon nano-tube system model particle current time t, and under the situation of the position of previous moment t-Δ, extrapolate next position of t+ Δ t constantly;

C) formula (1) and formula (2) are subtracted each other, both sides are simultaneously divided by 2 Δ t again, and the expression formula that obtains speed is as follows:

$\stackrel{\→}{v}\left(t\right)=\frac{\stackrel{\→}{x}(t+\mathrm{\Δt})-\stackrel{\→}{x}(t-\mathrm{\Δt})}{2\mathrm{\Δt}}+O\left({\mathrm{\Δt}}^2\right)---\left(4\right)$

Constantly under the situation of the position of t+ Δ t, extrapolate the speed v (t) of current time t in the position of known carbon nano-tube system model particle previous moment t-Δ t and back one;

D) by formula (1)～(4), under the situation of known particle t-2 Δ t position, the position in the t-Δ t moment and t-Δ t acceleration constantly constantly, start the Verlet algorithm and carry out integration: according to t-2 Δ t position, t-Δ t position and t-Δ t acceleration constantly constantly constantly, with t=t-Δ t substitution formula (3), obtain the position of current time t; According to the position of current time t, upgrade the acceleration of current time t based on certain potential function; Simultaneously, according to position and the t-2 Δ t position constantly of current time t, with t=t-Δ t substitution formula (4), upgrade t-Δ t speed constantly; Namely obtain position, t-Δ t speed and the t acceleration constantly constantly of particle current time t, repeat this step.

Images