CN109325260A - A kind of kinetic model shows hidden mixed asynchronous long stagger scheme calculation method - Google Patents
A kind of kinetic model shows hidden mixed asynchronous long stagger scheme calculation method Download PDFInfo
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Abstract
Hidden mixed asynchronous long stagger scheme calculation method is shown the present invention provides a kind of kinetic model, which comprises the steps of: step 1: generating the data file that explicit partition calculates and implicit subregion calculating is required；Step 2: according to the data file, obtaining the internal node data of explicit partition and the boundary node data of explicit partition, and the boundary node data with explicit partition are sent to implicit subregion；Step 3: implicit subregion receives explicit partition data, solves implicit subregion internal node data and boundary node data, and send explicit partition for boundary node data；Step 4: regeneration block node data exports subregion output file.The present invention during structural dynamical model according to zoning unit characteristic and external load frequency select explicitly with implicit integration format and time step.It realizes that asynchronous long subarea management calculates using multiple boundary facility strategy, effectively increases dynamics model analysis precision and efficiency.
Description
Technical field
The present invention relates to structural mechanics fields, and in particular, to kinetic model shows hidden mixed asynchronous long stagger scheme
Calculation method.Especially, it is related to a kind of finite element dynamics and shows hidden mixed implementation method.It is especially to be related to one
Kind stagger scheme shows hidden mixed asynchronous long calculation method.
Background technique
Direct integation method is the common method of finite element dynamic analysis, including explicitly with Implicit Method two major classes.It is aobvious
Formula and implicit two methods respectively have advantage and disadvantage: the essence of explicit integral is under inquiring into from the joint movements equation at current time
The movement of one moment node, its advantage is that not needing to carry out rigidity, quality, the general assembly for damping battle array, the formation of right-hand vector need only
It is formed on unit Primary plateaus according to contribution of each unit to useful load vector is cumulative, entire calculate substantially exists in this way
It is carried out on unit Primary plateaus, only needs the high speed storage portions of very little, computational efficiency is higher, wave of this method in open system
There is more application in dynamic response analysis.But explicit integral is a kind of integration method of conditional stability, calculates and uses
Time step limited by Integration Scheme stability.And implicit integration method has centainly superior in terms of numerical stability
Property, still, implicit integration format is Space Coupling, simultaneous equations must be all solved in the calculating of each time step, thus, with
The number of unit of FEM calculation and the increase of interstitial content, the calculation amount of this Integration Scheme and to computer data storage
It is required that also will all sharply increase.
There are local fine grid dividings, and finite element dynamic analysis model is existed obviously in large-scale structure analysis
Time and space different scale.On the one hand, the grid of local fine makes finite element dynamic analysis explicit solution process
In can only use lesser critical time step.Single time of integration step-length, which can greatly improve, calculates the time.On the other hand, exist
The different phase or different zones of structural finite element model dynamic analysis, high frequency response always occur respectively with low frequency response,
It usually needs to combine in numerical value calculating and explicitly be solved with implicit algorithm, to play the advantage of algorithms of different.
The implicit algorithm of unconditional file can use biggish time step, and the explicit algorithm of conditional stability is using small
Time step.It is this to show hidden mixed method and have many applications in large-scale complex fine-structure distribution, such as multiple physical field, stream
Response of labyrinth caused by solid coupling analysis, Multibody System Dynamic Analysis, vehicle collision and security partition, impact etc.
Deng.These project analysis need the time in wider time scale, local acknowledgement's high precision computation and overall calculation too much
Efficiency is in acceptable time range.
Find by prior art documents: Chen Shaolin etc. is in Chinese science: technological sciences, 2017,47 (12):
Publishing an article on 1321-1330, " asynchronous parallel of the Three-dimensional Time Domain interaction of soil and structure analysis based on aobvious-implied format is calculated
Method ", this article use explicit centered Finite Difference Methods and implicit Newmark method analyzing three-dimensional time domain soil-structure power phase interaction
With.Using viscous-spring artificial boundary simulate infinite field ground, based in MPI protocol realization soil-structure system soil body region it is different
Step-length parallel computation.But program is asked by the artificial self-compiling program that is divided into Ansys program is called during this method is realized
Solution preocess, this reduces computational efficiency to a certain extent, expands demand of the program to memory, is unfavorable for the further of program
Extension.
Du Xiuli etc. is in geotechnical engineering journal, and 2016,38 (3): publishing an article on 395-403, " u-p format is saturated two-phase and is situated between
Aobvious-implicit alternative algorithm of matter dynamical problem ".This article is proposed using solid phase displacement u and pore water pressure p as the u- of basic variable
The kinetic equation of p form.The power balance equation of solid-liquid two-phase is solved using central difference method, and the kinetic equation of liquid phase utilizes essence
Thin step-by-step integral method calculates.The method of the different coupling terms for solving liquid phase compatible equations is provided simultaneously.But this method is directed to
Central difference method and Precise Time-integration method use identical step-length, this is used in calculation amount and memory to Matrix Calculating
It is a huge challenge for inverse Precise integration method.
It is existing to show hidden mixing calculation method and mostly use non-overlap Border joint form partition method.According to external applied load
Time scale selects explicit or Implicit Method method.It is continuous to show hidden partition boundaries constrained motion variable (such as displacement, acceleration)
Or partition boundaries energy balance is coupled using Lagrange multiplier, use subcycle method to realize step-length between asynchronous long subregion
Matching.Showing has Data Matching between hidden subregion difference step-length, mostly uses linear interpolation method, this is to a certain extent
The stability of algorithm is reduced, while also limiting the selection of maximum step-length ratio.
Summary of the invention
For the defects in the prior art, hidden mixed asynchronous length is shown the object of the present invention is to provide a kind of kinetic model
Stagger scheme calculation method.
A kind of kinetic model provided according to the present invention shows hidden mixed asynchronous long stagger scheme calculation method, including such as
Lower step:
Step 1: generating the data file that explicit partition calculates and implicit subregion calculating is required；
Step 2: according to the data file, obtaining the internal node data of explicit partition and the boundary node of explicit partition
Data, and the boundary node data with explicit partition are sent to implicit subregion；
Step 3: implicit subregion receives the boundary node data of the explicit partition, by the boundary node of the explicit partition
External node data of the data as implicit subregion, solve the internal node data of implicit subregion and the boundary node of implicit subregion
Data, and the boundary node data of implicit subregion are sent to explicit partition；
Step 4: regeneration block node data exports subregion output file；Judge whether main time step terminates, if so,
Terminate process, otherwise, then return step 2 continues to execute.
Preferably, the data file includes: explicit partition model file, explicit partition boundary information file, implicit point
Section model file, implicit partition boundaries message file；
Explicit partition model file includes: the unit, node, load, boundary constraint information of explicit partition；
Implicit partition model file includes: the unit, node, load, boundary constraint information of implicit subregion；
Explicit partition boundary information file includes: boundary node number, node serial number and the node coordinate of explicit partition；
Implicit partition boundaries message file includes: boundary node number, node serial number and the node coordinate of implicit subregion；
The partitioned nodes data include: the node data of the node data of explicit partition, implicit subregion；
The node of explicit partition include: the internal node of explicit partition, the boundary node of explicit partition, explicit partition it is outer
Portion's node；
The node of implicit subregion include: the internal node of implicit subregion, the boundary node of implicit subregion, implicit subregion it is outer
Portion's node；
The node data of explicit partition includes: the modal displacement of explicit partition, the node speed of explicit partition and explicitly divides
The node acceleration in area；
The node data of implicit subregion includes: the modal displacement of implicit subregion, the node speed of implicit subregion and implicitly divides
The node acceleration in area；
The subregion output file includes: explicit partition output file, implicit subregion output file.
Preferably, the step 1 includes:
Structure finite element grid is divided according to multiple boundary nodal method: structure finite element grid is subjected to node
Segmentation, the finite element grid are explicit partition and implicit subregion by node allocation；
The boundary node of explicit partition, external node subregion respectively with the boundary node of implicit partition, external node
It is overlapped, constitutes the overlapping partition of explicit partition and implicit subregion；
Explicit partition model file and explicit partition boundary information file are generated according to the explicit partition；
Implicit partition model file and implicit partition boundaries message file are generated according to the implicit subregion.
Preferably, in the step 2, the acquisition explicit partition internal node data and explicit partition boundary node data
The step of, comprising:
Wherein,
Indicate modal displacement predicted value；
Indicate node speed predicted value；
Indicate that main time step number is n, subcycle time step is the explicit partition modal displacement predicted value of j+1；
Indicate that main time step number is n, subcycle time step is the explicit partition node speed predicted value of j+1；
N indicates main time step number；
J indicates subcycle time step number；
M indicates that implicit subregion and explicit partition integration step ratio, m are the positive integer more than or equal to 1；
Δ T indicates implicit subregion integration step；
Δ t indicates that explicit partition time step, explicit partition time step are all explicit partition unit minimum times step
It is long；
The Z unit is indicated in the critical time step of n-th of main time step, Z=1,2 ..., N, N are positive whole
Number；
ΔtcrIndicate unit critical time step；
μ indicates proportionality coefficient；
L indicates element characteristic length；
The density of ρ expression unit material therefor；
The Poisson's ratio of ν expression unit material therefor；
The elasticity modulus of E expression unit material therefor；
Indicate that main time step number is n, subcycle time step is the explicit partition modal displacement of j；
Indicate that main time step number is n, subcycle time step is the explicit partition node speed of j；
Indicate that main time step number is n, subcycle time step is the explicit partition node acceleration of j；
βEWith γEIndicate explicit time-integration parameter；
It indicatesWithIt is multiplied；
Indicate that subregion mass matrix presses internal node freedom degree BIMatrix in block form；
Indicate that the corresponding piecemeal vector of internal node, subscript T are the transposition of vector；
MEIndicate the mass matrix of explicit partition；
BIIndicate internal node freedom degree vector；
üIIndicate explicit partition internal node acceleration；
Indicate that main time step number is n, subcycle time step is the explicit partition internal node acceleration of j+1；
fext,iIndicate the corresponding outer force vector applied of internal node；
Indicate main time step number be n, subcycle time step be j+1 when explicit partition correspond to internal node external force to
Amount；
Indicate that subregion stiffness matrix presses internal node freedom degree BIMatrix in block form；
KEIndicate the stiffness matrix of explicit partition；
Indicate internal node displacement prediction value；
Indicate that main time step is n, subcycle time step is the explicit partition internal node displacement prediction of j+1 time step
Value；
Indicate boundary node displacement prediction value；
Indicate that main time step is n, subcycle time step is the explicit partition boundary node displacement prediction of j+1 time step
Value；
MEIndicate explicit partition mass matrix；
KEIndicate explicit partition stiffness matrix；
BcIndicate boundary node freedom degree BcVector；
It indicatesWithIt is multiplied；
Indicate boundary node freedom degree BcCorresponding piecemeal mass matrix；
Indicate the corresponding piecemeal vector of boundary node；
ucIndicate boundary node acceleration；
Indicate that main time step number is n, explicit partition boundary node acceleration when subcycle time step is j+1；
fext,cIndicate outer force vector corresponding with boundary node freedom degree；
Indicate that main time step number is n, explicit partition corresponds to the external force of boundary node when subcycle time step is j+1
Vector；
Indicate boundary node freedom degree BcCorresponding piecemeal stiffness matrix；
∑ indicates summation symbol；
Indicate that main time step number is n, subregion modal displacement when subcycle time step is j+1；
Indicate that main time step number is n, subregion node speed when subcycle time step is j+1；
U indicates explicit partition node acceleration；
Indicate that main time step number is n, subregion node acceleration when subcycle time step is j+1；
Preferably, described to solve implicit subregion internal node data and implicit partition boundaries node data in the step 3
The step of, comprising:
Wherein,
Indicate modal displacement predicted value when the main time step number of implicit subregion is n+1；
Indicate node speed predicted value when the main time step number of implicit subregion is n+1；
unIndicate modal displacement when the main time step of implicit subregion is n；
Indicate node speed when the main time step of implicit subregion is n；
unIndicate node acceleration when the main time step of implicit subregion is n；
(MI+γIΔTC+βIΔT2KI)ün+1Indicate MI+γIΔTC+βIΔT2KIWith ün+1It is multiplied
MI+γIΔTC+βIΔT2KIIndicate the effective stiffness matrix K implicitly calculatedeff；
MIIndicate implicit subregion quality；
KIIndicate implicit subregion rigidity；
CIIndicate implicit subregion damping matrix；
γI, βIFor implicit time integration parameter；
ün+1Indicate node acceleration when the main time step of implicit subregion is n+1；
Subscript I indicates the internal node of implicit partitioned nodes；
Subscript B indicates the boundary node of implicit partitioned nodes；
Subscript E indicates the external node of implicit partitioned nodes；
Indicate outer force vector when the main time step of implicit subregion is n+1；
fextIndicate the outer force vector of implicit subregion；
KEIndicate that implicit partitioned external node corresponds to effective stiffness matrix matrix in block form；
K(B+I)EIndicate that implicit partitioned nodes unit of reporting to the leadship after accomplishing a task corresponds to effective stiffness matrix matrix in block form；
KE(B+I)Indicate that implicit partitioned nodes unit of reporting to the leadship after accomplishing a task corresponds to effective stiffness matrix matrix in block form, wherein
KB+IIndicate effective stiffness matrix matrix in block form corresponding with boundary node inside implicit subregion；
Indicate effective stiffness matrix inversion of partitioned matrix corresponding with boundary node inside implicit subregion；
B+I indicates boundary and internal node；
üEIndicate implicit partitioned external node acceleration；
Indicate implicit partitioned external node acceleration when main time step number is n+1；
üB+IIndicate implicit partition boundaries and internal node acceleration；
Indicate implicit partition boundaries and internal node acceleration when main time step number is n+1；
fEIndicate implicit partitioned external panel load；
Indicate external node load when main time step number is n+1；
fB+IIndicate boundary and internal node load；
Indicate boundary and internal node load when main time step number is n+1；
Indicate implicit subregion internal node displacement when main time step is n+1；
Indicate implicit subregion internal node speed when main time step is n+1；
Indicate implicit subregion internal node displacement prediction value when main time step is n+1；
Indicate implicit subregion internal node acceleration when main time step is n+1；
Indicate implicit subregion internal node rate predictions when main time step is n+1；
Indicate implicit partition boundaries modal displacement when main time step is n+1；
Indicate implicit partition boundaries node bit rate when main time step is n+1；
Indicate that implicit partition boundaries point is displaced predicted value when main time step is n+1；
Indicate implicit partition boundaries node acceleration when main time step is n+1；
Indicate implicit partition boundaries node speed predicted value when main time step is n+1.
Preferably, the step 4 includes:
Implicit partition boundaries node data updates explicit partition node data, implicit subregion root to explicit partition based on the received
Implicit partitioned nodes data are updated according to received explicit partition boundary node data；
Export explicit partition output file and implicit subregion output file；
Main time step, which counts, increases by a step；
If main time step, which reaches, calculates total time, main time the end of the step is determined, terminate process；
Otherwise, then return step 2 continues to execute.
Preferably, the explicit partition output file includes: the node speed of the modal displacement of explicit partition, explicit partition
And the node acceleration of explicit partition；
Implicit subregion output file includes: the node speed and implicit subregion of the modal displacement of implicit subregion, implicit subregion
Node acceleration.
Compared with prior art, the present invention have it is following the utility model has the advantages that
1, explicit partition of the invention is according to multiblock technique attribute, and for implicit subregion according to decomposed domain precision, selection is respective
Zone time integration step has biggish flexibility under the premise of guaranteeing partitioned nodes computational accuracy.It is different to show hidden subregion
Guarantee that calculating process is unified by subcycle process between step-length, solves subregion step-length caused by dividing because of grid local fine
Inconsistent problem.
2, the present invention uses the subarea management strategy of multiple knot, and the partial region of adjacent sectors is set as to couple subregion.
Hidden subregion step-length is shown than different, the unit number of layers of overlapping is different.During explicit subcycle, boundary node data are according to explicit
It calculates information transmission characteristic to solve, is not related to Interpolation Process, solving precision is higher.
3, explicit partition of the present invention solves format using the explicit Newmark of prediction correction, and implicit subregion use is simultaneous therewith
The Implicit Method format of appearance.It shows hidden subregion and all uses loose line compression storage, used inside implicit subregion with boundary node dilute
Direct solution is dredged to solve.Parallel schema is expanded to convenient for program.This is conducive to the rule of further expansion structural dynamical model
Mould.
4, the present invention selects during structural dynamical model according to zoning unit characteristic and external load frequency aobvious
Formula and implicit integration format and time step.Realize that asynchronous long subarea management calculates using multiple boundary facility strategy, the present invention
Effectively increase dynamics model analysis precision and efficiency.
Detailed description of the invention
Upon reading the detailed description of non-limiting embodiments with reference to the following drawings, other feature of the invention,
Objects and advantages will become more apparent upon:
Fig. 1 is the multiple knot partitioning strategies schematic diagram in preference of the invention.
Fig. 2 is the explicit subcycle flow diagram in preference of the invention.
Fig. 3 is implicit sparse storage, solution flow diagram in preference of the invention.
Fig. 4 is that the hidden formatted data that staggeredly solves that shows in preference of the invention transmits schematic diagram.
Fig. 5 is the process step for showing hidden mixed asynchronous long stagger scheme calculation method of kinetic model according to the present invention
Schematic diagram.
Specific embodiment
The present invention is described in detail combined with specific embodiments below.Following embodiment will be helpful to the technology of this field
Personnel further understand the present invention, but the invention is not limited in any way.It should be pointed out that the ordinary skill of this field
For personnel, without departing from the inventive concept of the premise, several changes and improvements can also be made.These belong to the present invention
Protection scope.
As shown in figure 5, a kind of hidden mixed asynchronous long stagger scheme that shows of the kinetic model provided according to the present invention calculates
Method includes the following steps:
Step S101: the data file that explicit partition calculates and implicit subregion calculating is required is generated；
Step S102: according to the data file, internal node data and the boundary of explicit partition of explicit partition are obtained
Node data, and the boundary node data with explicit partition are sent to implicit subregion；
Further, explicit partition is successively successively decreased using prediction syndrome round-robin method seeks the internal node of explicit partition
The boundary node data of data and explicit partition, and explicit partition boundary node data are sent by MPI data-transfer protocols
To implicit subregion as implicit partitioned external node data；
Further, the prediction syndrome cycle calculations are by the explicit Newmark integral based on prediction correction
Format realizes that explicit Newmark is divided into prediction step and correction two steps of step, and prediction step is for calculating partitioned nodes speed and position
The discreet value of shifting, correction step obtain node speed according to calculated node acceleration true value correction node speed and displacement
With displacement true value.
Step S103: implicit subregion receives the boundary node data of the explicit partition, by the boundary of the explicit partition
External node data of the node data as implicit subregion, solve the internal node data of implicit subregion and the boundary of implicit subregion
Node data, and the boundary node data of implicit subregion are sent to explicit partition；Further, explicit partition and implicit subregion
It is all made of the row compression storage format CRS storing data of sparse matrix, it is described to solve implicit subregion internal node data and boundary
Node data is realized by sparse direct solver.Implicit subregion stiffness matrix, mass matrix and damping matrix are according to dilute
Dredge the row compression storage format CRS storing data of symmetrical matrix.Stiffness matrix, mass matrix and damping matrix and external applied load data
It is then corresponding to be assigned to sparse direct solver solution according to implicit partitioned nodes number piecemeal；
Further, the boundary node data of implicit subregion are sent to explicit partition by MPI data-transfer protocols, show
The boundary node data of formula subregion and the boundary node data of implicit subregion include node serial number.
Step S104: regeneration block node data exports subregion output file；Judge whether main time step terminates, if so,
Then terminate process, otherwise, then return step 2 continues to execute.It further, can also be according to updated partitioned nodes data meter
Strain, stress are calculated, and output strain, stress as needed.
Specifically, the data file includes: explicit partition model file, explicit partition boundary information file, implicit point
Section model file, implicit partition boundaries message file；
Explicit partition model file includes: the unit, node, load, boundary constraint information of explicit partition；Further,
Explicit partition internal node number information is stored in explicit partition model file.
Implicit partition model file includes: the unit, node, load, boundary constraint information of implicit subregion；Further,
Implicit subregion internal node number information is stored in implicit partition model file.
Explicit partition boundary information file includes: boundary node number, node serial number and the node coordinate of explicit partition；
Further, the explicit partition boundary information file is for the data communication in asynchronous long calculating process.
Implicit partition boundaries message file includes: boundary node number, node serial number and the node coordinate of implicit subregion；
Further, the implicit partition boundaries message file is for the data communication in asynchronous long calculating process.
The partitioned nodes data include: the node data of the node data of explicit partition, implicit subregion；
The node of explicit partition include: the internal node of explicit partition, the boundary node of explicit partition, explicit partition it is outer
Portion's node；
The node of implicit subregion include: the internal node of implicit subregion, the boundary node of implicit subregion, implicit subregion it is outer
Portion's node；
The node data of explicit partition includes: the modal displacement of explicit partition, the node speed of explicit partition and explicitly divides
The node acceleration in area；
The node data of implicit subregion includes: the modal displacement of implicit subregion, the node speed of implicit subregion and implicitly divides
The node acceleration in area；
The subregion output file includes: explicit partition output file, implicit subregion output file.Further, subregion
The content that output file includes includes any one of following or appoints multinomial: element stress, strain, modal displacement, speed, acceleration.
Further, the result that Structural Dynamics calculates includes: displacement, speed, acceleration, stress, strain, is for model analysis
It is no whether to reach the strength of materials, displacement more than preset displacement value, determine a need for damping.
Specifically, the step 1 includes:
Structure finite element grid is divided according to multiple boundary nodal method: structure finite element grid is subjected to node
Segmentation, the finite element grid are explicit partition and implicit subregion by node allocation；
The boundary node of explicit partition, external node subregion respectively with the boundary node of implicit partition, external node
It is overlapped, constitutes the overlapping partition of explicit partition and implicit subregion；
Explicit partition model file and explicit partition boundary information file are generated according to the explicit partition；
Implicit partition model file and implicit partition boundaries message file are generated according to the implicit subregion.
Specifically, in the step 2, the acquisition explicit partition internal node data and explicit partition boundary node data
The step of, comprising:
Wherein,
Indicate modal displacement predicted value；
Indicate node speed predicted value；
Indicate that main time step number is n, subcycle time step is the explicit partition modal displacement predicted value of j+1；
Indicate that main time step number is n, subcycle time step is the explicit partition node speed predicted value of j+1；
N indicates main time step number；
J indicates subcycle time step number；
M indicates that implicit subregion and explicit partition integration step ratio, m are the positive integer more than or equal to 1；
Δ T indicates implicit subregion integration step；
Δ t indicates that explicit partition time step, explicit partition time step are all explicit partition unit minimum times step
It is long；
The Z unit is indicated in the critical time step of n-th of main time step, Z=1,2 ..., N, N are positive whole
Number；Further,Pass through Δ tcrCalculation formula calculate obtain, each unit is when each time step will calculate critical
Between step-length；
ΔtcrIndicate unit critical time step；
μ indicates proportionality coefficient；
L indicates element characteristic length；
The density of ρ expression unit material therefor；
The Poisson's ratio of ν expression unit material therefor；
The elasticity modulus of E expression unit material therefor；
Indicate that main time step number is n, subcycle time step is the explicit partition modal displacement of j；
Indicate that main time step number is n, subcycle time step is the explicit partition node speed of j；
Indicate that main time step number is n, subcycle time step is the explicit partition node acceleration of j；
βEWith γEIndicate explicit time-integration parameter；
It indicatesWithIt is multiplied；
Indicate that subregion mass matrix presses internal node freedom degree BIMatrix in block form；
Indicate that the corresponding piecemeal vector of internal node, subscript T are the transposition of vector；
MEIndicate the mass matrix of explicit partition；Further, explicit partition mass matrix uses lumped mass matrix, son
Round-robin method refers to that a time step includes that the i.e. main time step of two kinds of different time step-lengths and the dynamics of subcycle time step are asked
Solution preocess.
BIIndicate internal node freedom degree vector；
üIIndicate explicit partition internal node acceleration；
Indicate that main time step number is n, subcycle time step is the explicit partition internal node acceleration of j+1；
fext,iIndicate the corresponding outer force vector applied of internal node；
Indicate main time step number be n, subcycle time step be j+1 when explicit partition correspond to internal node external force to
Amount；
Indicate that subregion stiffness matrix presses internal node freedom degree BIMatrix in block form；
KEIndicate the stiffness matrix of explicit partition；
Indicate internal node displacement prediction value；
Indicate that main time step is n, subcycle time step is that the explicit partition internal node displacement of j+1 time step is pre-
Measured value；
Indicate boundary node displacement prediction value；
Indicate that main time step is n, subcycle time step is the explicit partition boundary node displacement prediction of j+1 time step
Value；
MEIndicate explicit partition mass matrix；
KEIndicate explicit partition stiffness matrix；
BcIndicate boundary node freedom degree BcVector；
It indicatesWithIt is multiplied；
Indicate boundary node freedom degree BcCorresponding piecemeal mass matrix；
Indicate the corresponding piecemeal vector of boundary node；
ücIndicate boundary node acceleration；
Indicate that main time step number is n, explicit partition boundary node acceleration when subcycle time step is j+1；
fext,cIndicate outer force vector corresponding with boundary node freedom degree；
Indicate that main time step number is n, explicit partition corresponds to the external force of boundary node when subcycle time step is j+1
Vector；
Indicate boundary node freedom degree BcCorresponding piecemeal stiffness matrix；
∑ indicates summation symbol；
Indicate that main time step number is n, subregion modal displacement when subcycle time step is j+1；
Indicate that main time step number is n, subregion node speed when subcycle time step is j+1；
ü indicates explicit partition node acceleration；
Indicate that main time step number is n, subregion node acceleration when subcycle time step is j+1；
Specifically, described to solve implicit subregion internal node data and implicit partition boundaries node data in the step 3
The step of, comprising:
Wherein,
Indicate modal displacement predicted value when the main time step number of implicit subregion is n+1；
Indicate node speed predicted value when the main time step number of implicit subregion is n+1；
unIndicate modal displacement when the main time step of implicit subregion is n；
Indicate node speed when the main time step of implicit subregion is n；
unIndicate node acceleration when the main time step of implicit subregion is n；
(MI+γIΔTC+βIΔT2KI)ün+1Indicate MI+γIΔTC+βIΔT2KIWith ün+1It is multiplied
MI+γIΔTC+βIΔT2KIIndicate the effective stiffness matrix K implicitly calculatedeff；
MIIndicate implicit subregion quality；
KIIndicate implicit subregion rigidity；
CIIndicate implicit subregion damping matrix；
γI, βIFor implicit time integration parameter；
ün+1Indicate node acceleration when the main time step of implicit subregion is n+1；
Subscript I indicates the internal node of implicit partitioned nodes；
Subscript B indicates the boundary node of implicit partitioned nodes；
Subscript E indicates the external node of implicit partitioned nodes；
Indicate outer force vector when the main time step of implicit subregion is n+1；
fextIndicate the outer force vector of implicit subregion；
KEIndicate that implicit partitioned external node corresponds to effective stiffness matrix matrix in block form；
K(B+I)EIndicate that implicit partitioned nodes unit of reporting to the leadship after accomplishing a task corresponds to effective stiffness matrix matrix in block form；
KE(B+I)Indicate that implicit partitioned nodes unit of reporting to the leadship after accomplishing a task corresponds to effective stiffness matrix matrix in block form, wherein
KB+IIndicate effective stiffness matrix matrix in block form corresponding with boundary node inside implicit subregion；
Indicate effective stiffness matrix inversion of partitioned matrix corresponding with boundary node inside implicit subregion；
B+I indicates boundary and internal node；
üEIndicate implicit partitioned external node acceleration；
Indicate implicit partitioned external node acceleration when main time step number is n+1；Further, by will implicitly divide
The received explicit partition boundary node acceleration in area is obtained as implicit partitioned external node acceleration
üB+IIndicate implicit partition boundaries and internal node acceleration；
Indicate implicit partition boundaries and internal node acceleration when main time step number is n+1；
fEIndicate implicit partitioned external panel load；
Indicate external node load when main time step number is n+1；
fB+IIndicate boundary and internal node load；
Indicate boundary and internal node load when main time step number is n+1；
Indicate implicit subregion internal node displacement when main time step is n+1；
Indicate implicit subregion internal node speed when main time step is n+1；
Indicate implicit subregion internal node displacement prediction value when main time step is n+1；
Indicate implicit subregion internal node acceleration when main time step is n+1；
Indicate implicit subregion internal node rate predictions when main time step is n+1；
Indicate implicit partition boundaries modal displacement when main time step is n+1；
Indicate implicit partition boundaries node bit rate when main time step is n+1；
Indicate that implicit partition boundaries point is displaced predicted value when main time step is n+1；
Indicate implicit partition boundaries node acceleration when main time step is n+1；
Indicate implicit partition boundaries node speed predicted value when main time step is n+1.
Specifically, the step 4 includes:
Implicit partition boundaries node data updates explicit partition node data, implicit subregion root to explicit partition based on the received
Implicit partitioned nodes data are updated according to received explicit partition boundary node data；
Export explicit partition output file and implicit subregion output file；
Main time step, which counts, increases by a step；
If main time step, which reaches, calculates total time, main time the end of the step is determined, terminate process；
Otherwise, then return step 2 continues to execute.
Specifically, the explicit partition output file includes: the node speed of the modal displacement of explicit partition, explicit partition
And the node acceleration of explicit partition；
Implicit subregion output file includes: the node speed and implicit subregion of the modal displacement of implicit subregion, implicit subregion
Node acceleration.
Below by preference, more specific detail is carried out to the present invention:
Embodiment 1:
A kind of kinetic model show hidden mixed asynchronous long stagger scheme calculation method overall flow include: first by point
Area's multiple knot strategy generating shows data file required for hidden subregion calculates；Then, explicit partition is selected according to zoning unit attribute
Time of integration step-length is selected, implicit subregion selects zone time step-length according to solving precision, and each partitioned process forms alone subregion
Stiffness matrix and mass matrix；Then, explicit partition is successively successively decreased according to time step using explicit subcycle method and is sought in subregion
Portion's node and boundary node data, and partition boundaries node data is sent to implicit subregion by MPI data-transfer protocols and is made
For implicit partitioned external node data；External node data known to implicit subregion solve the process of inner boundary data as domain decomposition
Back substitution process in method Domain decomposition method method.And implicit boundary node data are transmitted to explicitly
Subregion；The boundary node information that last each subregion comes according to transmitting, regeneration block information, and strain, answer according to requiring to calculate
Power, the displacement of output partitioned nodes, velocity and acceleration data.Otherwise the program determination if time the end of the step re-starts explicit
The prediction of subregion, which walks, to be calculated.
It is successively described in detail below as the sequencing of calculation process:
The first step generates the data file shown needed for hidden subregion calculates by multiple boundary nodal method.
METIS is the serial subregion software of internationally famous open source, is suitable for large-scale structure grid division.This example uses
METIS realizes sectoring function.Kinetic model finite element grid is split into explicit and implicit two regions by METIS.
METIS divide after the result is that structure finite element grid node belongs to explicit or implicit region, show hidden each subregion corresponding one
A partition model file and boundary information file.
During grid dividing, is numbered first, in accordance with grid and read in finite element model node serial number.According to node allocation number
According to the lookup affiliated subregion of corresponding node judges whether the affiliated node of grid fully belongs to a subregion, if then numbering grid
And its corresponding partitioned file of the node serial number write-in affiliated partition number of node.It, will if the affiliated different subregions of grid internal node
All partition numbers are written in grid number, and node is written corresponding subregion and needs to judge whether node has existed the subregion, do not deposit
Then be written.External force and boundary constraint are added according to node serial number.Multiple knot partitioning strategies are as shown in Figure 1.
It needs that structured grid information is written to different message file and boundary file by subregion multiple knot strategy.Structure
FEM meshing completes output according to the file of grid number order and node serial number sequence.Grid data is compiled comprising grid
Number and grid inside node serial number.Node data includes the coordinate value of node serial number and node.
Second step, explicit partition is successively successively decreased using prediction syndrome round-robin method seeks subregion inside and boundary node number
According to, and partition boundaries node data is transmitted to implicit subregion.
Explicit partition reads in grid data from partition model file according to grid number order, is superimposed according to the degree of freedom on a node basis
Subregion stiffness matrix and mass matrix are generated, mass matrix uses lumped mass form.Explicit time-integration format is conditional equilibrium constant
Fixed algorithm, minimum time step-length are limited by CFL condition
In formula: Δ tcrFor unit critical time step, μ is proportionality coefficient, and l is element characteristic length, and ρ, ν and E represent institute
With the density, Poisson's ratio and elasticity modulus of material.The time step that can be used for explicit partition is minimum for all units
Time step
In formula: Δ t is explicit partition time step,Indicate n-th unit in the critical step-length of the n-th time step.It is aobvious
Formula subregion uses the Newmark Integration Scheme based on prediction correction form, is divided into prediction step and the step greatly of correction step two.Prediction step can
To be expressed as
In formula: subscript n is system time step-length,WithRepresent jth subcycle time step explicit partition section
Point displacement, velocity and acceleration.WithModal displacement and rate predictions, subscriptIndicate n-th of main+1 son of time step jth
Circulation time walks the moment to dependent variable.βEWith γEIt is two adjustable parameters of explicit Newmark Integration Scheme.It adjusts β and γ can
To change algorithm for the filtering characteristic of different frequency waveform.Parameter m is implicit subregion and explicit partition integration step ratio
In formula: Δ T is implicit subregion integration step, and Δ t is explicit partition time step.M is just whole more than or equal to 1
Number.Successively successively decrease with the increase of subcycle time step according to explicit integration calculating rule and seek internal node and boundary node
Acceleration, internal node acceleration can be expressed as
In formula: MEWith KEIndicate explicit partition mass matrix and stiffness matrix,WithIt indicates to divide
Area's mass matrix and stiffness matrix press internal node freedom degree BIMatrix in block form.Indicate boundary node displacement prediction value.It indicates
Internal node displacement prediction value.fext,iIndicate the corresponding outer force vector applied of internal node.Boundary node freedom degree BcVector.
The acceleration formula of boundary node can be expressed as
In formula:WithIndicate boundary node freedom degree BcCorresponding piecemeal mass matrix, just
Spend matrix.fext,cOuter force vector corresponding with boundary node freedom degree, ücIndicate boundary node acceleration.∑ indicates summation symbol
Number.
Node acceleration is calculated, displacement can be walked by Newmark integral correction with speed and be calculated
In formula:WithIndicate n-th of main step moment time step jth+1 sub- circulation time, the displacement of node with
Speed.Equation (8) and equation (9) are the correction equation that explicit Newmark format is corrected based on prediction, with equation (3), (4)
Used time step is consistent.Displacement, the velocity and acceleration number of explicit partition internal node can be calculated by this step
According to.Explicit partition subcycle operation is as shown in Figure 2.
Third step, implicit subregion receive data as partitioned external node data, solve implicit subregion internal node and side
Boundary's node, and partition boundaries node data is sent to explicit partition.
Implicit subregion carries out prediction step using main time step Δ T and calculates, and speed is calculated with displacement prediction step and be can be written as
In formula: unIndicate modal displacement when the main time step of implicit subregion is n.When indicating that the main time step of implicit subregion is n
Node speed.ünIndicate node acceleration when the main time step of implicit subregion is n.γI, βIFor implicit Newmark integral parameter.
The acceleration value of implicit partitioned nodes can be rewritten into
In formula: subscript n+1 indicates main time step n+1 to dependent variable.MI, KI, CIRespectively indicate implicit subregion quality, rigidity
With damping matrix.Implicit partitioned nodes are divided into internal node, boundary node and external node, are indicated respectively with subscript I, B and E.
MI+γIΔTC+βIΔT2KIAlso further it is represented as the effective stiffness matrix K implicitly calculatedeff.Implicit subregion is according to node
Piecemeal effective stiffness matrix can be write as
In formula: KESubscript E indicates that external node corresponds to effective stiffness matrix, and B+I is boundary and internal node.fEFor correspondence
External node load item.Known external node is asked internal and is operated with boundary node for the back substitution in the decomposition method of domain
Implicit subregion internal node and boundary node acceleration are calculated, the speed of internal node and displacement use
Newmark correction format calculates
In formula:WithFor the implicit subregion internal node displacement of the (n+1)th main time step and speed.
Implicitly partition boundaries node speed is with displacement correction value
In formula:WithFor the implicit partition boundaries modal displacement of the (n+1)th main time step and speed.
Inside and the boundary node data for calculating implicit subregion need for boundary node data to be sent to explicit partition work
For explicit partition external node information, to participate in the calculating of next main time step.
Mass matrix, effective stiffness matrix, stiffness matrix and damping matrix are all made of the row compression storage lattice of sparse storage
The piecemeal stiffness matrix of formula CRS, corresponding formula (14) solve internal node and boundary node acceleration, and call directly sparse
Direct solution is realized.Implicit subregion sparse storage, solution process are as shown in Figure 3.
4th step shows hidden subregion regeneration block nodal information, and according to calculating strain, stress is required, exports partitioned nodes
Displacement, velocity and acceleration data.The program determination if main time the end of the step, otherwise again since second step.
The transmitting of the data of explicit partition and implicit by stages carries out data using MPI data-transfer protocols and sends and receives,
In the data transfer of stagger scheme, explicit partition carries out explicit subcycle time step first and calculates, by explicit partition boundary
Node data is sent to implicit subregion.Implicit subregion receives explicit partition boundary node data as implicit partitioned external number of nodes
According to.Implicit partition data is then calculated, transmitting partition boundaries node data to explicit partition so staggeredly calculates.Show hidden update
Complete partitioned nodes data can choose the displacement of output partitioned nodes, speed or acceleration value, computing unit as needed and answer
Change and stress value, can export as needed.
Main time step, which counts, increases by a step, calculates total time, program stopped if reaching, no person calculates from algorithm second step
Start to continue to calculate.It is as shown in Figure 4 to show hidden intercrossed data transmitting schematic diagram.
In the description of the present application, it is to be understood that term " on ", "front", "rear", "left", "right", " is erected at "lower"
Directly ", the orientation or positional relationship of the instructions such as "horizontal", "top", "bottom", "inner", "outside" is orientation based on the figure or position
Relationship is set, description the application is merely for convenience of and simplifies description, rather than the device or element of indication or suggestion meaning are necessary
It with specific orientation, is constructed and operated in a specific orientation, therefore should not be understood as the limitation to the application.
One skilled in the art will appreciate that in addition to realizing system provided by the invention in a manner of pure computer readable program code
It, completely can be by the way that method and step be carried out programming in logic come so that provided by the invention other than system, device and its modules
System, device and its modules are declined with logic gate, switch, specific integrated circuit, programmable logic controller (PLC) and insertion
The form of controller etc. realizes identical program.So system provided by the invention, device and its modules may be considered that
It is a kind of hardware component, and the knot that the module for realizing various programs for including in it can also be considered as in hardware component
Structure；It can also will be considered as realizing the module of various functions either the software program of implementation method can be Hardware Subdivision again
Structure in part.
Specific embodiments of the present invention are described above.It is to be appreciated that the invention is not limited to above-mentioned
Particular implementation, those skilled in the art can make a variety of changes or modify within the scope of the claims, this not shadow
Ring substantive content of the invention.In the absence of conflict, the feature in embodiments herein and embodiment can any phase
Mutually combination.
Claims (7)
1. a kind of kinetic model shows hidden mixed asynchronous long stagger scheme calculation method, which comprises the steps of:
Step 1: generating the data file that explicit partition calculates and implicit subregion calculating is required；
Step 2: according to the data file, obtaining the internal node data of explicit partition and the boundary node number of explicit partition
According to, and the boundary node data with explicit partition are sent to implicit subregion；
Step 3: implicit subregion receives the boundary node data of the explicit partition, by the boundary node data of the explicit partition
As the external node data of implicit subregion, the internal node data of implicit subregion and the boundary node number of implicit subregion are solved
According to, and the boundary node data of implicit subregion are sent to explicit partition；
Step 4: regeneration block node data exports subregion output file；Judge whether main time step terminates, if so, terminating
Process, otherwise, then return step 2 continues to execute.
2. kinetic model according to claim 1 shows hidden mixed asynchronous long stagger scheme calculation method, feature exists
In, the data file include: explicit partition model file, it is explicit partition boundary information file, implicit partition model file, hidden
Formula partition boundaries message file；
Explicit partition model file includes: the unit, node, load, boundary constraint information of explicit partition；
Implicit partition model file includes: the unit, node, load, boundary constraint information of implicit subregion；
Explicit partition boundary information file includes: boundary node number, node serial number and the node coordinate of explicit partition；
Implicit partition boundaries message file includes: boundary node number, node serial number and the node coordinate of implicit subregion；
The partitioned nodes data include: the node data of the node data of explicit partition, implicit subregion；
The node of explicit partition includes: the outside segments of the internal node of explicit partition, the boundary node of explicit partition, explicit partition
Point；
The node of implicit subregion includes: the outside segments of the internal node of implicit subregion, the boundary node of implicit subregion, implicit subregion
Point；
The node data of explicit partition includes: the node speed and explicit partition of the modal displacement of explicit partition, explicit partition
Node acceleration；
The node data of implicit subregion includes: the node speed and implicit subregion of the modal displacement of implicit subregion, implicit subregion
Node acceleration；
The subregion output file includes: explicit partition output file, implicit subregion output file.
3. kinetic model according to claim 2 shows hidden mixed asynchronous long stagger scheme calculation method, feature exists
In the step 1 includes:
Structure finite element grid is divided according to multiple boundary nodal method: structure finite element grid is subjected to node point
It cuts, the finite element grid is explicit partition and implicit subregion by node allocation；
Boundary node, the external node subregion of explicit partition are overlapped with the boundary node of implicit partition, external node respectively,
Constitute the overlapping partition of explicit partition and implicit subregion；
Explicit partition model file and explicit partition boundary information file are generated according to the explicit partition；
Implicit partition model file and implicit partition boundaries message file are generated according to the implicit subregion.
4. kinetic model according to claim 3 shows hidden mixed asynchronous long stagger scheme calculation method, feature exists
In, in the step 2, acquisition explicit partition internal node data and the step of explicit partition boundary node data, comprising:
Wherein,
Indicate modal displacement predicted value；
Indicate node speed predicted value；
Indicate that main time step number is n, subcycle time step is the explicit partition modal displacement predicted value of j+1；
Indicate that main time step number is n, subcycle time step is the explicit partition node speed predicted value of j+1；
N indicates main time step number；
J indicates subcycle time step number；
M indicates that implicit subregion and explicit partition integration step ratio, m are the positive integer more than or equal to 1；
Δ T indicates implicit subregion integration step；
Δ t indicates that explicit partition time step, explicit partition time step are all explicit partition unit minimum time step-lengths；
The Z unit is indicated in the critical time step of n-th of main time step, Z=1,2 ..., N, N are positive integer；
ΔtcrIndicate unit critical time step；
μ indicates proportionality coefficient；
L indicates element characteristic length；
The density of ρ expression unit material therefor；
The Poisson's ratio of ν expression unit material therefor；
The elasticity modulus of E expression unit material therefor；
Indicate that main time step number is n, subcycle time step is the explicit partition modal displacement of j；
Indicate that main time step number is n, subcycle time step is the explicit partition node speed of j；
Indicate that main time step number is n, subcycle time step is the explicit partition node acceleration of j；
βEWith γEIndicate explicit time-integration parameter；
It indicatesWithIt is multiplied；
Indicate that subregion mass matrix presses internal node freedom degree BIMatrix in block form；
Indicate that the corresponding piecemeal vector of internal node, subscript T are the transposition of vector；
MEIndicate the mass matrix of explicit partition；
BIIndicate internal node freedom degree vector；
Indicate explicit partition internal node acceleration；
Indicate that main time step number is n, subcycle time step is the explicit partition internal node acceleration of j+1；
fext,iIndicate the corresponding outer force vector applied of internal node；
Indicate that main time step number is n, explicit partition corresponds to the outer force vector of internal node when subcycle time step is j+1；
Indicate that subregion stiffness matrix presses internal node freedom degree BIMatrix in block form；
KEIndicate the stiffness matrix of explicit partition；
Indicate internal node displacement prediction value；
Indicate that main time step is n, subcycle time step is the explicit partition internal node displacement prediction value of j+1 time step；
Indicate boundary node displacement prediction value；
Indicate that main time step is n, subcycle time step is the explicit partition boundary node displacement prediction value of j+1 time step；
MEIndicate explicit partition mass matrix；
KEIndicate explicit partition stiffness matrix；
BcIndicate boundary node freedom degree BcVector；
It indicatesWithIt is multiplied；
Indicate boundary node freedom degree BcCorresponding piecemeal mass matrix；
Indicate the corresponding piecemeal vector of boundary node；
Indicate boundary node acceleration；
Indicate that main time step number is n, explicit partition boundary node acceleration when subcycle time step is j+1；
fext,cIndicate outer force vector corresponding with boundary node freedom degree；
Indicate that main time step number is n, explicit partition corresponds to the outer force vector of boundary node when subcycle time step is j+1；
Indicate boundary node freedom degree BcCorresponding piecemeal stiffness matrix；
∑ indicates summation symbol；
Indicate that main time step number is n, subregion modal displacement when subcycle time step is j+1；
Indicate that main time step number is n, subregion node speed when subcycle time step is j+1；
Indicate explicit partition node acceleration；
Indicate that main time step number is n, subregion node acceleration when subcycle time step is j+1.
5. kinetic model according to claim 4 shows hidden mixed asynchronous long stagger scheme calculation method, feature exists
In, in the step 3, described the step of solving implicit subregion internal node data and implicit partition boundaries node data, comprising:
Wherein,
Indicate modal displacement predicted value when the main time step number of implicit subregion is n+1；
Indicate node speed predicted value when the main time step number of implicit subregion is n+1；
unIndicate modal displacement when the main time step of implicit subregion is n；
Indicate node speed when the main time step of implicit subregion is n；
Indicate node acceleration when the main time step of implicit subregion is n；
(MI+γIΔTC+βIΔT2KI)Indicate MI+γIΔTC+βIΔT2KIWithIt is multiplied
MI+γIΔTC+βIΔT2KIIndicate the effective stiffness matrix K implicitly calculatedeff；
MIIndicate implicit subregion quality；
KIIndicate implicit subregion rigidity；
CIIndicate implicit subregion damping matrix；
γI, βIFor implicit time integration parameter；
Indicate node acceleration when the main time step of implicit subregion is n+1；
Subscript I indicates the internal node of implicit partitioned nodes；
Subscript B indicates the boundary node of implicit partitioned nodes；
Subscript E indicates the external node of implicit partitioned nodes；
Indicate outer force vector when the main time step of implicit subregion is n+1；
fextIndicate the outer force vector of implicit subregion；
KEIndicate that implicit partitioned external node corresponds to effective stiffness matrix matrix in block form；
K(B+I)EIndicate that implicit partitioned nodes unit of reporting to the leadship after accomplishing a task corresponds to effective stiffness matrix matrix in block form；
KE(B+I)Indicate that implicit partitioned nodes unit of reporting to the leadship after accomplishing a task corresponds to effective stiffness matrix matrix in block form, wherein
KB+IIndicate effective stiffness matrix matrix in block form corresponding with boundary node inside implicit subregion；
Indicate effective stiffness matrix inversion of partitioned matrix corresponding with boundary node inside implicit subregion；
B+I indicates boundary and internal node；
Indicate implicit partitioned external node acceleration；
Indicate implicit partitioned external node acceleration when main time step number is n+1；
Indicate implicit partition boundaries and internal node acceleration；
Indicate implicit partition boundaries and internal node acceleration when main time step number is n+1；
fEIndicate implicit partitioned external panel load；
Indicate external node load when main time step number is n+1；
fB+IIndicate boundary and internal node load；
Indicate boundary and internal node load when main time step number is n+1；
Indicate implicit subregion internal node displacement when main time step is n+1；
Indicate implicit subregion internal node speed when main time step is n+1；
Indicate implicit subregion internal node displacement prediction value when main time step is n+1；
Indicate implicit subregion internal node acceleration when main time step is n+1；
Indicate implicit subregion internal node rate predictions when main time step is n+1；
Indicate implicit partition boundaries modal displacement when main time step is n+1；
Indicate implicit partition boundaries node bit rate when main time step is n+1；
Indicate that implicit partition boundaries point is displaced predicted value when main time step is n+1；
Indicate implicit partition boundaries node acceleration when main time step is n+1；
Indicate implicit partition boundaries node speed predicted value when main time step is n+1.
6. kinetic model according to claim 2 shows hidden mixed asynchronous long stagger scheme calculation method, feature exists
In the step 4 includes:
Implicit partition boundaries node data updates explicit partition node data to explicit partition based on the received, and implicit subregion is according to connecing
The explicit partition boundary node data of receipts update implicit partitioned nodes data；
Export explicit partition output file and implicit subregion output file；
Main time step, which counts, increases by a step；
If main time step, which reaches, calculates total time, main time the end of the step is determined, terminate process；
Otherwise, then return step 2 continues to execute.
7. kinetic model according to claim 6 shows hidden mixed asynchronous long stagger scheme calculation method, feature exists
In the explicit partition output file includes: the node speed and explicit partition of the modal displacement of explicit partition, explicit partition
Node acceleration；
Implicit subregion output file includes: the section of the modal displacement of implicit subregion, the node speed of implicit subregion and implicit subregion
Point acceleration.
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