CN114372361B - Coarse grid selection method based on BDDC region decomposition parallel algorithm - Google Patents
Coarse grid selection method based on BDDC region decomposition parallel algorithm Download PDFInfo
- Publication number
- CN114372361B CN114372361B CN202111680165.4A CN202111680165A CN114372361B CN 114372361 B CN114372361 B CN 114372361B CN 202111680165 A CN202111680165 A CN 202111680165A CN 114372361 B CN114372361 B CN 114372361B
- Authority
- CN
- China
- Prior art keywords
- matrix
- flexible multi
- interface
- sub
- bddc
- Prior art date
- Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
- Active
Links
- 238000000354 decomposition reaction Methods 0.000 title claims abstract description 88
- 238000004422 calculation algorithm Methods 0.000 title claims abstract description 86
- 238000010187 selection method Methods 0.000 title claims abstract description 38
- QHHHYLFZGYIBCX-UHFFFAOYSA-N n,n'-bis[(2,2-dimethyl-1,3-dioxolan-4-yl)methyl]methanediimine Chemical group O1C(C)(C)OCC1CN=C=NCC1OC(C)(C)OC1 QHHHYLFZGYIBCX-UHFFFAOYSA-N 0.000 title claims abstract 19
- 239000011159 matrix material Substances 0.000 claims abstract description 308
- 239000000463 material Substances 0.000 claims abstract description 57
- 238000004088 simulation Methods 0.000 claims abstract description 36
- 239000013598 vector Substances 0.000 claims description 89
- 238000000034 method Methods 0.000 claims description 57
- 238000004364 calculation method Methods 0.000 claims description 39
- 230000009466 transformation Effects 0.000 claims description 27
- 238000006243 chemical reaction Methods 0.000 claims description 20
- 230000000295 complement effect Effects 0.000 claims description 17
- 230000003044 adaptive effect Effects 0.000 claims description 16
- 230000007704 transition Effects 0.000 claims description 14
- 108020001568 subdomains Proteins 0.000 claims description 11
- 230000008569 process Effects 0.000 claims description 10
- 230000008878 coupling Effects 0.000 claims description 8
- 238000010168 coupling process Methods 0.000 claims description 8
- 238000005859 coupling reaction Methods 0.000 claims description 8
- 238000000638 solvent extraction Methods 0.000 claims description 5
- 238000010276 construction Methods 0.000 claims description 4
- 238000007781 pre-processing Methods 0.000 claims description 3
- 239000010409 thin film Substances 0.000 claims description 3
- 230000001131 transforming effect Effects 0.000 claims description 3
- 238000012163 sequencing technique Methods 0.000 claims description 2
- 238000005192 partition Methods 0.000 description 11
- 239000010408 film Substances 0.000 description 5
- 230000006854 communication Effects 0.000 description 4
- 238000011161 development Methods 0.000 description 4
- 238000004891 communication Methods 0.000 description 3
- 238000005516 engineering process Methods 0.000 description 3
- 238000005094 computer simulation Methods 0.000 description 2
- 238000010586 diagram Methods 0.000 description 2
- 238000012360 testing method Methods 0.000 description 2
- 238000009827 uniform distribution Methods 0.000 description 2
- 241001229889 Metis Species 0.000 description 1
- 238000004458 analytical method Methods 0.000 description 1
- 238000003491 array Methods 0.000 description 1
- 230000009286 beneficial effect Effects 0.000 description 1
- 230000007547 defect Effects 0.000 description 1
- 230000001419 dependent effect Effects 0.000 description 1
- 238000013461 design Methods 0.000 description 1
- 230000005486 microgravity Effects 0.000 description 1
- APVPOHHVBBYQAV-UHFFFAOYSA-N n-(4-aminophenyl)sulfonyloctadecanamide Chemical compound CCCCCCCCCCCCCCCCCC(=O)NS(=O)(=O)C1=CC=C(N)C=C1 APVPOHHVBBYQAV-UHFFFAOYSA-N 0.000 description 1
- 238000011160 research Methods 0.000 description 1
Classifications
-
- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F30/00—Computer-aided design [CAD]
- G06F30/20—Design optimisation, verification or simulation
Landscapes
- Engineering & Computer Science (AREA)
- Physics & Mathematics (AREA)
- Theoretical Computer Science (AREA)
- Computer Hardware Design (AREA)
- Evolutionary Computation (AREA)
- Geometry (AREA)
- General Engineering & Computer Science (AREA)
- General Physics & Mathematics (AREA)
- Management, Administration, Business Operations System, And Electronic Commerce (AREA)
Abstract
The invention discloses a coarse grid selection method based on a BDDC region decomposition parallel algorithm, and belongs to the field of large-scale flexible multi-body system parallel simulation prediction. The Deluxe scaling matrix is used for constructing the generalized characteristic equation coefficient matrix, so that the situation that the coefficients of the sub-domain interface materials jump in the flexible multi-body system can be effectively aimed at; in addition, the invention calculates the secondary Sul compensation matrix of the shared edge to condense the rest information of the interfaces in the sub-domain of the flexible multi-body system except the shared edge, thereby effectively aiming at the condition that the sub-domain interface material coefficient jump occurs in the flexible multi-body system, ensuring the stability of the BDDC region decomposition parallel algorithm and improving the parallel simulation prediction efficiency of the flexible multi-body system. The invention can solve the problem of low efficiency caused by jump of the material coefficient of the subdomain interface of the flexible multi-body system, improves the parallel simulation prediction efficiency of the flexible multi-body system based on the BDDC region decomposition parallel algorithm, and further solves the related technical problems in the field of parallel simulation prediction of large-scale flexible multi-body systems.
Description
Technical Field
The invention belongs to the field of parallel simulation prediction of a large-scale flexible multi-body system, and relates to a coarse grid selection method based on a BDDC (binary digital direct current) region decomposition parallel algorithm.
Background
With the rapid development of aerospace technology and the increasing demand for aerospace applications, the development of expandable space structures such as antennas and solar sails on spacecraft is towards large and soft development. The development of the large-scale space expandable structure technology has important significance for improving the core competitiveness of the international space technology in the future of China. Because the simulated unfolding test of the large-sized space unfolding structure in space environments such as microgravity and vacuum cannot be implemented on the ground, and the risk and fund of the unfolding test in space are more difficult to bear, accurate dynamic modeling and analysis are urgently needed in the unfolding process of the large-sized space structure, and the effectiveness of design is further ensured.
The large-scale space expandable structure is composed of a large number of beams, rods, films and other flexible rods, and is a typical flexible multi-body system. Studies have shown that a flexible multi-body system dynamics modeling method based on small deformation and small rotation assumption cannot describe an increasingly gentle spatial structure unfolding process, and large deformation and large rotation coupling dynamics characteristics of a flexible member of the spatial unfolding structure can be described by adopting an absolute node coordinate method (ANCF). However, because of increasingly complex structures and functions of the space expandable structure, the method is adopted to research the expansion dynamics process, and has the problems of large calculation scale and high calculation cost. The parallel algorithm based on region decomposition can remarkably improve the dynamic simulation efficiency of the flexible multi-body system by dividing the dynamic model of the flexible multi-body system into a plurality of subdomains, simultaneously calculating the model in each subdomain through multiple cores and communicating a small amount of key information, and effectively solve the problem.
The BDDC (Balancing Domain Decomposition by Constraints) area decomposition parallel algorithm is one of the most commonly used non-overlapping area decomposition methods at present, although the BDDC area decomposition parallel algorithm can calculate the local problem in each sub-domain at the same time by using threads to improve the calculation efficiency, communication between threads to generate a global interface problem is unavoidable, the cost in the communication process increases rapidly with the increase of communication content, the generated global interface problem needs to be solved iteratively by using iterative algorithms such as PCG, and the calculation cost of the process is often intolerable because the calculation efficiency of the iterative algorithms such as PCG is highly dependent on the condition number of coefficient matrixes of the global interface problem, and if the coefficient matrixes of the interface problem tend to be ill, the result meeting the accuracy can be obtained through repeated iterative calculation. Because of the close relation between the inter-subdomain communication and the generation of the global interface problem coefficient matrix and the coarse grid of the BDDC regional decomposition parallel algorithm, how to select the coarse grid of the BDDC regional decomposition parallel algorithm becomes a key for solving the problem. In the traditional coarse grid selection method, angular points with a large number of correlations between adjacent subfields are commonly used as coarse grids, but the coarse grid selection method is difficult to obtain satisfactory calculation efficiency under the condition that material coefficient transitions exist between the subfields, and when the flexible multi-body system contains a plurality of rigid-flexible structure joints and a BDDC region decomposition parallel algorithm is used for carrying out parallel simulation prediction on the flexible multi-body system, the problem that larger material coefficient differences exist between the subfields often occurs, so that the condition number of the global interface equation coefficient matrix is deteriorated, the iteration number of the PCG iteration algorithm for solving the global interface problem is increased, the simulation efficiency of the BDDC region decomposition parallel algorithm is greatly influenced, and therefore, the problem of how to select a proper coarse grid to improve the simulation efficiency of the BDDC region decomposition parallel method is needed to be solved.
In summary, the BDDC region decomposition parallel algorithm is widely applied to parallel simulation prediction of a large-scale flexible multi-body system, however, the flexible multi-body system has a plurality of rigid-flexible structure connected parts, so that region decomposition frequently encounters material coefficient jump conditions between sub-domain interfaces, the iteration number of solving a global interface equation by the BDDC region decomposition parallel algorithm in the traditional coarse grid selection method is increased, and further the dynamic simulation efficiency of the BDDC region decomposition parallel algorithm on the flexible multi-body system is reduced, so that a new coarse grid selection method is required to be researched to improve the parallel simulation prediction efficiency of the BDDC region decomposition parallel algorithm on the flexible multi-body system.
Disclosure of Invention
In order to solve the defects in the prior art, the invention discloses a coarse grid selection method based on a BDDC region decomposition parallel algorithm, which aims to solve the technical problems that: by calculating a Deluxe scaling matrix of the shared edge, material coefficient information of a sub-domain interface of the flexible multi-body system is recorded, and the situation of material coefficient jump of the sub-domain interface of the flexible multi-body system can be effectively aimed; by calculating the secondary Sulbumin matrix of the shared edge, the rest part of information of the interfaces in the sub-domains of the flexible multi-system except the shared edge is condensed, and more sub-domain boundary information of the flexible multi-system can be provided; by constructing a generalized characteristic equation coefficient matrix with a shared edge and containing the material coefficient characteristics of the sub-domain interface of the flexible multi-body system, corresponding material coefficient transition information can be provided for the condition of material coefficient transition of the sub-domain interface of the flexible multi-body system effectively; by unitizing the generalized eigenvector matrix with the shared edge with respect to the generalized eigenvector coefficient matrix, a set of new bases of the flexible multi-body system is constructed, so that the rigidity matrix of the flexible multi-body system under the representation of the new bases can have a better condition number; rearranging the unit generalized eigenvector matrix according to the generalized eigenvalue order; setting a characteristic value parameter, selecting a specific unit characteristic vector, constructing an adaptive coarse grid, obtaining a rigidity matrix and a force vector corresponding to the adaptive coarse grid, and applying the rigidity matrix and the force vector corresponding to the adaptive coarse grid to a BDDC region decomposition parallel algorithm. The method can effectively solve the problem of low calculation efficiency caused by jump of the material coefficients of the subdomain interface of the flexible multi-body system, further improve the parallel simulation prediction efficiency of the flexible multi-body system based on the BDDC region decomposition parallel algorithm, and further solve the technical problem of related engineering in the field of parallel simulation prediction of the large-scale flexible multi-body system.
Related engineering technical problems in the field of parallel simulation prediction of large-scale flexible multi-body systems include on-orbit assembly of space solar power stations, on-orbit unfolding of large truss antenna structures, on-orbit unfolding of thin film solar wings and the like.
The invention aims at realizing the following technical scheme:
The invention discloses a coarse grid selection method based on BDDC region decomposition parallel algorithm, which is used for partitioning the whole calculation domain of a flexible multi-body system; calculating a Sul complement matrix of the internal interface; calculating a Deluxe scaling matrix of the shared edge; calculating a quadratic Sul compensation matrix of the shared edge; calculating a generalized characteristic equation coefficient matrix of the shared edge; calculating generalized characteristic equation of the shared edge; unitizing the generalized characteristic vector with respect to the generalized characteristic equation coefficient matrix to construct a group of new bases of the flexible multi-body system; re-ordering the unit eigenvector matrix from small to large according to the eigenvalue size; generating a conversion matrix according to the rearranged unit feature vector matrix and performing base transformation on the rigidity matrix and the force vector of the flexible multi-body system; setting a characteristic value parameter, selecting a specific unit characteristic vector, constructing a self-adaptive coarse grid, and obtaining a rigidity matrix and a force vector corresponding to the self-adaptive coarse grid; and applying the rigidity matrix and the force vector corresponding to the self-adaptive coarse grid to a BDDC region decomposition parallel algorithm. The invention can solve the problem of low simulation efficiency caused by jump of the material coefficient of the subdomain interface when the BDDC region decomposition parallel algorithm is adopted to carry out parallel simulation prediction on the flexible multi-body system; by calculating a Deluxe scaling matrix of the shared edge, material coefficient information of a sub-domain interface of the flexible multi-body system is recorded, and the situation of material coefficient jump of the sub-domain interface of the flexible multi-body system can be effectively aimed; by calculating the secondary Sul compensation matrix of the shared edge, the rest part of information of the interfaces in the sub-domain of the flexible multi-body system except the shared edge is condensed, and more information of the sub-domain interfaces of the flexible multi-body system can be provided; by constructing a generalized characteristic equation coefficient matrix with a shared edge and containing the material coefficient characteristics of the sub-domain interface of the flexible multi-body system, corresponding material coefficient transition information can be provided for the condition of material coefficient transition of the sub-domain interface of the flexible multi-body system effectively; by unitizing the generalized eigenvector matrix with the shared edge with respect to the generalized eigenvector coefficient matrix, a set of new bases of the flexible multi-body system is constructed, so that the rigidity matrix of the flexible multi-body system under the representation of the new bases can have a better condition number; the unit generalized eigenvector matrix is rearranged according to the generalized eigenvalue order, eigenvalue parameters are set, specific unit eigenvectors are selected, an adaptive coarse grid is constructed, a stiffness matrix and a force vector corresponding to the adaptive coarse grid are obtained, the stiffness matrix and the force vector corresponding to the adaptive coarse grid are applied to a BDDC region decomposition parallel algorithm, calculation efficiency of material coefficient jump of a subdomain interface can be improved, and simulation prediction efficiency of the BDDC region decomposition parallel algorithm is guaranteed.
The invention discloses a coarse grid selection method based on BDDC (Balancing Domain Decomposition by Constraints) region decomposition parallel algorithm, which comprises the following steps:
Step one, partitioning the whole calculation domain of the flexible multi-body system to realize the division of an inner interface Γ and an inner area I.
Dividing the overall computational domain Ω of the flexible multi-body system into n sub-domains Ω i (i=1, 2,3, …, n), there isThe boundary of the overall computational domain constitutes an external interface as/>The interfaces Γ i between all adjacent subfields constitute the internal interface of the overall computational domain as Γ, i.e./>External interface/>The inner region outside the inner interface Γ is I, which is also constituted by the inner regions I i of the subfields, i.e./>
Step two, calculating the sulf matrix of the interface Γ i of all subfields Ω i (i=1, 2,3, …, n) in step oneThe Sul complement matrix/>And (3) Deluxe scaling matrix calculation in the third step.
The stiffness matrix corresponding to the inner region I i and the interface Γ i of the sub-region Ω i is represented asAnd/>Wherein the method comprises the steps ofAnd/>Representing the coupling term of the inner region I i of the subdomain Ω i with the interface Γ i.
Sul compensation of interface Γ i for stiffness matrixThe method comprises the following steps:
The subfield Ω i interface Γ i is formed by a plurality of common edges common to the subfields and surrounding adjacent subfields, if the subfield Ω i interface Γ i is formed by m common edges, where the k (k=1, …, m) common edge is denoted as E i,k, then there is The corresponding Sulbumin matrix/>, of the interface Γ i of the corresponding subdomain Ω i The corresponding amount of the common edge E i,k can be expressed as:
Wherein the diagonal member Representing the Sulbumin matrix corresponding to common edge E i,k in subdomain Ω i, off-diagonal elements/>Then the coupling term between common edge E i,1 and common edge E i,k is represented.
Step three, calculating Deluxe scaling matrices of all the common edges E i,k (k=1, …, m) in step twoDeluxe scaling matrix/>, of the common edgeAnd the method is used for calculating the coefficient matrix of the generalized characteristic equation in the step five.
For the common edge E i,k in the subdomain Ω i, its Deluxe scaling matrixThe calculation is as follows:
Where I (E i,k) represents the set of all subfields that share E i,k, the Deluxe scaling matrix of common edge E i,k if E i,k is shared only by two subfields, Ω i and Ω j The expression is simplified as:
calculate the Deluxe scaling matrix for all the common edges E i,k (k=1, …, m) The Deluxe scaling matrix/>Recording the material coefficient information of all the common subfields E i,k can effectively provide material coefficient jump information for the situation of the material coefficient jump of the subfield interface of the flexible multi-body system.
Step four, calculating the quadratic sull matrix of all the shared edges E i,k (k=1, …, m) in step twoThe quadratic Sul complement matrix/>And the method is used for calculating the coefficient matrix of the generalized characteristic equation in the step five.
For the common edge E i,k in the subdomain omega i interface gamma i, the Sul complement matrix isLet the rest of the interface Γ i except the common edge E i,k be C, i.e. c=Γ i\{Ei,k, then the common edge E i,k solependence matrix/>Sulbumin/>The method comprises the following steps:
Wherein, Diagonal matrix representing the remainder C,/>And/>Representing the coupling term of the common edge E i,k to the remaining portion C.
Calculate the quadratic solependent matrix of all the common edges E i,k (k=1, …, m)The quadratic Sul complement matrix/>The soft multi-body system subdomain interface gamma i Sulbumin/>The part except the shared edge E i,k is condensed into the shared edge E i,k, contains more soft multi-body system subdomain boundary information and is used for constructing a more effective generalized characteristic equation coefficient matrix in the step five.
Step five, constructing a generalized characteristic equation coefficient matrix of all the common edges E i,k (k=1, …, m)AndThe generalized characteristic equation coefficient matrix/>And/>And (3) calculating the generalized characteristic equation in the step six.
Step 5.1: constructing a generalized eigen equation coefficient matrix of all the common edges E i,k (k=1, …, m)The generalized characteristic equation coefficient matrix/>From the common edge Sul complement matrix/>, in the step twoDeluxe scaling matrix/>, in step threeThe structure comprises the material coefficient characteristics of the sub-domain interface of the flexible multi-body system, and can effectively provide corresponding material coefficient jump information for the situation of material coefficient jump of the sub-domain interface of the flexible multi-body system.
For the common edge E i,k of the subdomain omega i, the common edge E i,k corresponds to the Sulzer matrix and Deluxe scaling matrix belonging to the subdomain omega i, respectivelyAnd/>If the sub-field Ω j and the sub-field Ω i share E i,k, the shared edge E i,k corresponds to the Sulzer matrix and Deluxe scaling matrix belonging to the sub-field Ω j, respectivelyAnd/>Generalized eigen equation coefficient matrix for the common edge E i,k The method comprises the following steps:
Wherein I (E i,k) represents the sub-field set of all shared E i,k, and if E i,k is shared only by two sub-fields of Ω i and Ω j, the generalized characteristic equation coefficient matrix of the shared edge E i,k The expression is reduced to a two-subdomain case:
Constructing a generalized eigenequation coefficient matrix of all the common edges E i,k (k=1, …, m)
Step 5.2: constructing a coefficient matrix of a generalized characteristic equation of a common edge E i,k The generalized characteristic equation coefficient matrix/>From the quadratic Sul complement matrix/>, in the fourth stepThe parallel and constitution, the information of the rest parts except the shared edge E i,k of the sub-domain of the flexible multi-system is condensed, and the material coefficient characteristics of the sub-domain interface of the flexible multi-system are included, so that corresponding material coefficient jump information can be provided for the condition of material coefficient jump of the sub-domain interface of the flexible multi-system effectively.
For the common edge E i,k of the sub-field Ω i, if the common edge E i,k is shared by m sub-fields Ω i (i=1, …, m), the generalized characteristic equation coefficient matrix of the common edge E i,k The method comprises the following steps:
If E i,k is shared only by two sub-domains of Ω i and Ω j, then the generalized characteristic equation coefficient matrix of the shared edge E i,k The expression is simplified as:
Wherein, For/>And/>The parallel sum expression of (2) is as follows:
Wherein, For/>Is the generalized inverse of (a).
Constructing a generalized eigen equation coefficient matrix of all the common edges E i,k (k=1, …, m)
Step six, calculating generalized characteristic equations of all the common edges E i,k (k=1, …, m) to obtain corresponding characteristic vector matrixesEigenvalue matrix/>The eigenvalue vector matrix/>The coefficient calculation of the generalized characteristic equation in the fifth step is adopted to obtain the material coefficient jump characteristic of the sub-domain interface of the flexible multi-body system, and the unitized characteristic vector matrix/>Constructing a new set of bases allows the stiffness matrix of the flexible multi-body system to have better condition numbers under the representation of the new bases.
According to the two generalized characteristic equation coefficient matrixes calculated in the fifth step, solving the following generalized characteristic equation:
Wherein, For a matrix of feature vectors, each column of the matrix represents a feature vector; /(I)For a matrix of eigenvalues, each diagonal element of the matrix corresponds to an eigenvalue.
Generalized characteristic equations of all the common edges E i,k (k=1, …, m) are calculated.
Step seven, the feature vector matrix of all the shared edges E i,k (k=1, …, m) in the step six is obtainedProceeding with respect to coefficient matrix/>A new set of bases for the flexible multi-body system is constructed for use in step nine to generate a transformation matrix Q (i) for each sub-field Ω i (i=1, 2,3, …, n), thereby constructing an adaptive coarse grid.
For the common edge E i,k of the subdomain omega i, the generalized eigenvector isWherein the kth vector is denoted/>Feature vector/>With respect to coefficient matrix/>Is a unitized feature vector/>The method comprises the following steps:
with respect to the coefficient matrix for all eigenvectors Is unitized to obtain a unit eigenvector matrixThe unit eigenvector matrix/>As a new set of bases for the flexible multi-body system, the stiffness matrix of the flexible multi-body system in the new base representation can be given a better condition number.
Step eight, the eigenvalue matrix of all the shared edges E i,k (k=1, …, m) in the step sixThe corresponding eigenvalues are arranged from small to large, and the unit eigenvector matrix/>, is arranged according to the same sequenceRearranging to ensure the one-to-one correspondence between the unit eigenvectors and eigenvalues, and sequencing the unit eigenvector matrix/>For constructing the transformation matrix in step nine.
Step nine, constructing a conversion matrix Q (i) of each sub-field Ω i (i=1, 2,3, …, n), and performing a basis transformation on the stiffness matrix K (i) and the force vector v (i) of each sub-field for step ten construction of the adaptive coarse grid.
Step 9.1: a conversion matrix Q (i) of the subfields Ω i is constructed.
The stiffness matrix K (i) of the subdomain Ω i and the force vector v (i) are represented in order of the inner region I followed by the inner interface Γ as follows:
The conversion matrix Q (i) of the subfields Ω i is constructed as follows:
Wherein the method comprises the steps of Is a unit array with the same degree of freedom as the internal region I i of the subdomain omega i,/>A unit eigenvector matrix/>, consisting of all the common edges E i,k (k=1, …, m) of the subfield Ω i in step sevenThe composition is as follows:
Step 9.2: the rigidity matrix K (i) of the subdomain omega i and the force vector v (i) are subjected to base transformation, and the rigidity matrix after the base transformation has a better condition number.
For stiffness matrix K (i), a transpose of the left-hand conversion matrix is requiredRight multiplying the conversion matrix Q (i) to obtain a rigidity matrix/>, after the base conversion
Wherein,Represented by the portion corresponding to the common edge E i,k (k=1, …, m):
For force vectors, a transpose of the left-hand conversion matrix is required Obtain force vector/>, after basis transformation
Wherein,Represented by the corresponding portion of the common edge E ij:
Step ten, setting a characteristic value parameter lambda Tol to be more than or equal to 1 for constructing a self-adaptive coarse grid, and carrying out size sorting on the characteristic value matrix of all the common edges E i,k (k=1, …, m) in the step eight The stiffness matrix and the force vector after the basis transformation obtained by multiplying the unit eigenvector corresponding to the eigenvalue larger than lambda Tol by the stiffness matrix K (i) and the force vector v (i) represent the stiffness matrix/>, corresponding to the adaptive coarse gridSum force vector/>The node corresponding to the rigidity matrix and the force vector after the base transformation is the self-adaptive coarse grid pi Adapt. The self-adaptive coarse mesh pi Adapt has higher stability aiming at the condition of transition of the material coefficients of the subdomain interface of the flexible multi-body system, and ensures the high efficiency and stability of BDDC region decomposition parallel algorithm.
Unit feature vector matrix of common edge E i,k In total m columns of vectors, then/>Can be expressed as:
if the characteristic value matrix of the shared edge E i,k The first (l < m) eigenvalue/>Greater than the parameter lambda Tol, the eigenvalue matrix/>, according to step eightAccording to the order from small to large, the characteristic values after the first characteristic value is larger than the parameter lambda Tol, and the characteristic vector corresponding to the characteristic value is multiplied by the rigidity matrix and the force vector to obtain the rigidity matrix and the force vector corresponding to the self-adaptive coarse grid.
Unit feature vector matrix for common edge E i,k The method comprises the following steps:
Performing base transformation on the rigidity matrix:
Wherein the rigidity matrix with the row number larger than l is a rigidity matrix corresponding to the self-adaptive coarse grid
Performing base transformation on the force vector:
Wherein the force vector with the row number larger than l is the force vector corresponding to the self-adaptive coarse grid
Thereby completing the selection of the self-adaptive coarse mesh pi Adapt.
Step eleven: applying the self-adaptive coarse mesh pi Adapt obtained in the step ten to BDDC region decomposition parallel algorithm, namely, transforming the base into a rigidity matrixSum force vector/>Preprocessing operator construction for BDDC region decomposition parallel algorithms. The adaptive coarse grid pi Adapt contains information of the sub-domain interface material coefficients of the flexible multi-domain system, and in the parallel simulation prediction process of the flexible multi-domain system with the sub-domain interface material coefficient jump by adopting the BDDC region decomposition parallel algorithm, the condition number and the iteration number of the interface equation can be effectively reduced, and the high efficiency and the stability of the BDDC region decomposition parallel algorithm are ensured, so that the problem of low calculation efficiency caused by the sub-domain interface material coefficient jump of the flexible multi-domain system is solved.
The method also comprises the steps of: and carrying out parallel simulation prediction on the large-scale flexible multi-body system according to the eleventh step, thereby solving the technical problem of engineering related to the parallel simulation prediction of the large-scale flexible multi-body system.
The related engineering technical problems in the field of large-scale flexible multi-body system parallel simulation prediction comprise efficient parallel simulation prediction of space solar power station on-orbit assembly, efficient parallel simulation prediction of on-orbit expansion of a large truss antenna structure and efficient parallel simulation prediction of on-orbit expansion of a thin film solar wing.
The beneficial effects are that:
1. According to the coarse grid selection method based on the BDDC regional decomposition parallel algorithm, the Deluxe scaling matrix with the shared edge is calculated, material coefficient information of a sub-domain interface of the flexible multi-body system is recorded, the Deluxe scaling matrix is used for constructing a generalized characteristic equation coefficient matrix, the situation that the sub-domain interface material coefficient transitions occur in the flexible multi-body system can be effectively aimed at, the calculated characteristic vector matrix and characteristic value matrix can provide the flexible multi-body system sub-domain interface material coefficient transition information, the stability of the BDDC regional decomposition parallel algorithm for parallel simulation prediction under the situation that the BDDC regional decomposition parallel algorithm transitions the flexible multi-body system sub-domain interface material coefficient is guaranteed, and the parallel simulation prediction efficiency of the BDDC regional decomposition parallel algorithm for the flexible multi-body system is improved.
2. According to the coarse grid selection method based on the BDDC regional decomposition parallel algorithm, the secondary Sul compensation matrix of the shared edge is calculated, the rest information of the interfaces in the sub-domains of the flexible multi-body system except the shared edge is aggregated, and the secondary Sul compensation matrix is applied to the generalized characteristic equation coefficient matrix structure, so that the material coefficient jump information of the sub-domain interfaces of the flexible multi-body system can be provided when the generalized characteristic equation is calculated, the condition that the material coefficient jump of the sub-domain interfaces occurs to the flexible multi-body system is effectively aimed, the stability of the BDDC regional decomposition parallel algorithm is ensured, and the parallel simulation prediction efficiency of the BDDC regional decomposition parallel algorithm to the flexible multi-body system is improved.
3. According to the coarse grid selection method based on the BDDC regional decomposition parallel algorithm, the generalized eigenvector of the shared edge is calculated, the eigenvector matrix is subjected to unitization of the coefficient matrix related to the generalized eigenvector equation to obtain the unit eigenvector matrix, a group of new basis and conversion matrix aiming at the sub-domain interface of the flexible multi-body system are constructed, the condition number of the rigidity matrix of the flexible multi-body system under the new basis representation can be improved, the stability of the BDDC regional decomposition parallel algorithm is ensured, and the parallel simulation prediction efficiency of the BDDC regional decomposition parallel algorithm on the flexible multi-body system is improved.
4. According to the coarse grid selection method based on the BDDC regional decomposition parallel algorithm, a generalized eigenvalue matrix of a shared edge is calculated, a unit eigenvector matrix is rearranged according to the order of the eigenvalue, a specific unit eigenvector is selected by setting eigenvalue parameters, a self-adaptive coarse grid is constructed, a stiffness matrix and a force vector corresponding to the self-adaptive coarse grid are obtained, the stiffness matrix and the force vector are applied to the BDDC regional decomposition parallel algorithm, the condition of transition of material coefficients of a sub-domain interface of a flexible multi-body system can be effectively aimed, the condition number of the coefficient matrix of the interface problem and the iteration number of the iteration process can be effectively reduced in the parallel simulation prediction process of the flexible multi-body system, and the high efficiency and the stability of the BDDC regional decomposition parallel algorithm are ensured.
5. The coarse grid selection method based on the BDDC regional decomposition parallel algorithm can effectively improve the calculation efficiency of the BDDC regional decomposition parallel algorithm when the material coefficient of the sub-domain interface of the flexible multi-body system is in transition, ensure the stability of the BDDC regional decomposition parallel algorithm, and further solve the technical problems of related engineering in the field of large-scale flexible multi-body system parallel simulation prediction.
Drawings
FIG. 1 is a schematic view of a vibration model of a sheet;
FIG. 2 is a schematic diagram of division of sub-domains with different division numbers for a thin plate overall calculation domain, wherein (a) is a division situation of dividing each sub-domain into 256 units for an overall calculation domain division 4 sub-domain, (b) is a division situation of dividing each sub-domain into 64 units for an overall calculation domain division 16 sub-domain, (c) is a division situation of dividing each sub-domain into 16 units for an overall calculation domain division 64 sub-domain, and (d) is a division situation of dividing each sub-domain into 4 units for an overall calculation domain division 256 sub-domain;
FIG. 3 is a schematic diagram of a partitioning of each sub-domain of the thin-plate partition 4 sub-domain into 256 units, wherein Ω 1、Ω2、Ω3、Ω4 corresponds to the number of the 4 sub-domains; e 1,1 and E 1,2 are the common edges of the subdomains Ω 1, E 1,1 and E 1,2 make up the inner boundary Γ 1, Γ 1＝E1,1∪E1,2, of the subdomain Ω 1; similarly, with E 2,1 and E 2,2 being the common edge of subfield OMEGA 2, E 2,1 and E 2,2 constituting the inner boundary Γ 2 of subfield OMEGA 2, i.e., Γ 2＝E2,1∪E2,2;E3,1 and E 3,2 being the common edge of subfield OMEGA 3, E 3,1 and E 3,2 constituting the inner boundary Γ 3 of subfield OMEGA 3, i.e., Γ 3＝E3,1∪E3,2;E4,1 and E 4,2 being the common edge of subfield OMEGA 4, e 4,1 and E 4,2 form the inner boundary Γ 4 of the sub-field Ω 4, Γ 4＝E4,1∪E4,2; the common edges E 1,1 and E 2,1 are the same edge; likewise, E 1,2 and E 3,1、E2,2 and E 4,1, and E 3,2 and E 4,2 are the same side;
FIG. 4 is a comparison of calculation results of BDDC region decomposition parallel algorithm of the coarse grid selection method based on BDDC region decomposition parallel algorithm disclosed in the invention under different region division conditions, wherein, the graph (a) is the comparison of the result of each sub-field 256 unit of the whole calculation region division 4 sub-field, the graph (b) is the comparison of the result of each sub-field 64 unit of the whole calculation region division 16 sub-field, the graph (c) is the comparison of the result of each sub-field 16 unit of the whole calculation region division 64 sub-field, and the graph (d) is the comparison of the result of each sub-field 4 unit of the whole calculation region division 256 sub-field;
FIG. 5 is a schematic view of a triangular prism type deployable film structure;
Fig. 6 shows the result of dividing the triangular prism-shaped expandable film structure into regions with different partition numbers, wherein fig. 6 (a) shows the partition situation when the partition number p=6, fig. 6 (b) shows the partition situation when the partition number p=12, fig. 6 (c) shows the partition situation when the partition number p=24, and fig. 6 (d) shows the partition situation when the partition number p=36;
Fig. 7 is a flowchart of a coarse mesh selection method based on BDDC region decomposition parallel algorithm disclosed in the present invention.
Detailed Description
The following examples are illustrative of the invention and are not intended to limit the scope of the invention.
For a better description of the objects and advantages of the present invention, the following detailed description of the invention refers to the accompanying drawings and examples.
A coarse grid selection method based on BDDC region decomposition parallel algorithm comprises the following specific steps:
example 1 sheet vibration model
As shown in fig. 1, the single sheet is vibrated when the uniform distribution force acts on one side of the single sheet, the side length of the square sheet is 1m, the left boundary of the square sheet is fixedly supported, the uniform distribution force f y acts as 200MPa, the direction is along the negative y-axis direction, the density ρ of the sheet is=1600 kg/m 3, and the poisson ratio μ is=0.3. Dividing the whole calculation domain of the thin plate into 4, 16, 64 and 256 sub-domains, respectively dividing each sub-domain into 256, 64, 16 and 4 units, ensuring that the total unit number of the whole calculation domain is 1024, the material coefficients between the sub-domains are E 1＝1×1010 Pa and E 2＝1×1013 Pa are distributed at intervals, and the characteristic value parameter is lambda Tol =1. Four zone division cases are shown in fig. 2.
Taking 256 units of each sub-field of 4 sub-fields as an example, a coarse grid selection method based on BDDC region decomposition parallel algorithm is applied, and the specific steps are as follows:
step one: partitioning is carried out on the whole calculation domain of the thin plate, and the division of the inner interface gamma and the inner area I is realized.
As shown in fig. 3, the thin plate is divided into 4 subfields Ω i (i=1, 2,3, 4) of the same size, and each subfield is divided into 16×16=256 units, so that the thin plate overall calculation field is composed of 1024 units in total. The outermost boundary of the sheet is called the outer boundaryThe boundary shared between subfields is called an inner interface Γ, which consists of 4 boundaries Γ i (i=1, 2,3, 4), i.e.Except for the external interface/>The remainder of the interface Γ with the inner surface is referred to as the inner region I.
Step two: calculating the solependent matrix of the subdomain Ω i (i=1, 2,3, 4) interface Γ i in step one
The stiffness matrix K (i) for the subdomain Ω i (i=1, 2,3, 4) is:
matrix corresponding to condensed internal region I i Obtaining a Sul complement matrix of the interface Γ i:
Interface solependent matrix of 4 subfields Ω i (i=1, 2,3, 4) The corresponding quantities with the common edges are expressed as:
Wherein the diagonal member Representing the Sulbumin matrix corresponding to common edge E 1,1 in subdomain Ω 1, off-diagonal elements/>Then the coupling term between common edge E 1,1 and common edge E 1,2 in subdomain Ω 1 is represented; the same applies to the rest of the subfields.
Step three: calculating Deluxe scaling matrices of all the common edges E i,k (i, =1, 2,3,4; k=1, 2) in step two
For one common edge E 1,1,E1,1 in subfield OMEGA 1 to be shared by subfields OMEGA 1 and OMEGA 2, then the Deluxe matrix of subfield OMEGA 1 common edge E 1,1 The method comprises the following steps:
Wherein, The solependent matrix corresponding to the subfield Ω 2 representing the common E 1,1 is known to be:
Therefore, the subzone Ω 1 shares the Deluxe matrix of edge E 1,1 The final representation is:
Similarly, the Deluxe matrix of the remaining common edges can be obtained:
Step four: calculating the quadratic sull matrix of all the common edges E i,k (i, =1, 2,3,4; k=1, 2) in step two
For one common edge E 1,1 in the subdomain Ω 1, since Γ 1＝E1,1∪E1,2, then c=Γ 1\{E1,1}＝E1,2, the quadratic sulr matrix of the common edge E 1,1 The method comprises the following steps:
similarly, the quadratic suler matrix of the remaining common edges can be obtained:
Step five: constructing a generalized characteristic equation coefficient matrix of all the common edges E i,k (i, =1, 2,3,4; k=1, 2) And/>
Step 5.1: constructing a generalized characteristic equation coefficient matrix of all the common edges E i,k (i, =1, 2,3,4; k=1, 2)
A common edge E 1,1,E1,1 for one of subfields Ω 1 is shared by subfields Ω 1 and Ω 2. The SulDu matrix and Deluxe scaling matrix of the common edge E 1,1 in the subfield Omega 1 are respectivelyThe SulBu matrix and Deluxe scaling matrix of the common edge E 1,1 in the subfield Omega 2 are/>, respectivelyThen the common edge E 1,1 generalized characteristic equation coefficient matrix/>The method comprises the following steps: /(I)
And similarly, the coefficient matrix of the generalized characteristic equation of the other shared edges can be obtained:
it can be seen that the generalized characteristic equation coefficient matrix of the same common edge in the different subfields containing the same Are identical.
Step 5.2: constructing a generalized characteristic equation coefficient matrix of a common edge E i,k (i, =1, 2,3,4; k=1, 2)
A common edge E 1,1,E1,1 for one of subfields Ω 1 is shared by subfields Ω 1 and Ω 2. The quadratic Sulbumin matrix of the common edge E 1,1 in the subfield omega 1 isThe quadratic Sul complement matrix of the common edge E 1,1 in the subdomain Ω 2 is/>Then the common edge E 1,1 generalized characteristic equation coefficient matrix/>The method comprises the following steps:
Wherein, For/>Is the generalized inverse of (a).
Similarly, the generalized characteristic equation coefficient matrix of the other shared edges can be obtained:
it can be seen that the generalized characteristic equation coefficient matrix of the same common edge in the different subfields containing the same Are identical.
Step six: calculating generalized eigenvalues of all the common edges E i,k (i, =1, 2,3,4; k=1, 2) to obtain corresponding eigenvector matrixEigenvalue matrix/>
For one common edge E 1,1 in the subdomain omega 1, the generalized characteristic equation coefficient matrix isAnd/>The common edge generalized characteristic equation is:
Since the common edge E 1,1 is shared by the subfields Ω 1 and Ω 2, and there is />Namely, the generalized characteristic equation of the common edge E 2,1 in the subdomain omega 2 is completely consistent with the generalized characteristic equation of the common edge E 1,1 in the subdomain omega 1, and the obtained generalized characteristic vector matrix/> issolvedGeneralized eigenvalue matrix/>The method is applicable to the subdomains omega 1 and omega 2 of the common E 1,1.
Similarly, the generalized characteristic equation of the remaining common edges is:
/>
and solving generalized characteristic equations of the rest shared edges to obtain a generalized characteristic vector matrix and a generalized characteristic value matrix.
Step seven: the eigenvector matrix of all the common edges E i,k (i, =1, 2,3,4; k=1, 2) in step six is appliedWith respect to coefficient matrix/>Unitizing.
For one common edge E 1,1 in the subdomain omega 1, the generalized eigenvector matrix and the generalized eigenvalue matrix are respectivelyAnd/>Let/>Representing feature vector matrix/>For the k-th feature vector of (2), then for the feature vector/>Performing a matrix/>, with respect to generalized eigen equation coefficientsTo obtain the k-th single-bit feature vector/>
Matrix of generalized eigenvectorsAll eigenvectors in (a) are subjected to coefficient matrix related to generalized eigenvalueTo obtain a unit eigenvector matrix/>
Similarly, unitizing the generalized eigenvector matrix of the rest shared edges to obtain a unit eigenvector matrix />Since the generalized eigen equations in the sub-domains containing the same common edge are identical, then there are:
step eight: matrix of unit eigenvectors Reordered from small to large according to eigenvalue size. For one common edge E 1,1 in the subdomain omega 1, the unit eigenvector matrix and the generalized eigenvalue matrix are respectively/>And/>Due to the bit eigenvector matrix/>Each column vector and generalized eigenvalue matrix/>The eigenvalues of each column diagonal element are in one-to-one correspondence and will/>Rearranging the eigenvalues in the order from small to large to obtain a rearranged generalized eigenvalue matrix, and arranging the unit eigenvector matrix/>And rearranging according to the same sequence, and maintaining a one-to-one correspondence with the generalized eigenvalue matrix.
Similarly, the unit feature vector matrix of the rest common edges is rearranged in the same way.
Step nine: a transformation matrix Q (i) of the subfields Ω i (i=1, 2,3, 4) is constructed, and the stiffness matrix K (i) of the respective subfields and the force vector v (i) are transformed based.
Step 9.1: a conversion matrix Q (i) of the subfields Ω i (i=1, 2,3, 4) is constructed.
For subdomain Ω 1, its stiffness matrix K (1) and force vector v (1) are expressed in order of inner region I followed by inner interface Γ as follows:
the conversion matrix Q (1) of the subfields Ω 1 is constructed as follows:
Wherein, From the unit eigenvector matrix/>, of all the common edges E 1,1 and E 1,2 of the subdomain Ω 1 in step sevenThe composition is as follows:
Similarly, the transformation matrix of the rest subfields is:
Step 9.2: the stiffness matrix K (i) of the subdomain Ω i (i=1, 2,3, 4) and the force vector v (i) are base transformed.
For subdomain omega 1, performing base transformation on a rigidity matrix K (1) to obtain:
Wherein, Represented by the corresponding portion of the common edge E 1,1、E1,2:
And similarly, performing basic transformation on the rigidity arrays of the rest subdomains to obtain:
for subdomain omega 1, the rigidity matrix v (1) is subjected to base transformation to obtain:
Wherein, Represented by the corresponding portion of the common edge E 1,1、E1,2:
similarly, the force vectors of the rest subdomains are subjected to base transformation to obtain:
Step ten: and setting a characteristic value parameter lambda Tol =1, and constructing an adaptive coarse grid.
For one common edge E 1,1 in the subdomain omega 1, the unit feature vector matrix after reordering isLet the unit eigenvector matrix/>In total m columns of eigenvectors, then/>Can be expressed as:
If the characteristic value matrix of the shared edge E 1,1 The first (l < m) eigenvalue/>Greater than the parameter lambda Tol, the eigenvalue matrix/>, according to step eightThe feature values after the first feature value are all larger than the parameter lambda Tol in the order from small to large.
Unit feature vector matrix of common edge E 1,1 The method comprises the following steps:
and performing base transformation on the rigidity matrix corresponding to the common edge E 1,1:
Wherein the rigidity matrix with the row number larger than l is the rigidity matrix corresponding to the self-adaptive coarse grid in the common side E 1,1 of the subdomain omega 1 />
And similarly, the rigidity matrix corresponding to the self-adaptive coarse grid in the rest shared edges can be obtained.
The force vector corresponding to the common edge E 1,1 is subjected to base transformation:
Wherein the force vector with the row number larger than l is the force vector corresponding to the self-adaptive coarse grid in the common edge E 1,1 of the subdomain omega 1
And similarly, the force vectors corresponding to the adaptive coarse meshes in the rest common edges can be obtained.
And selecting the self-adaptive coarse meshes of all the shared edges to finish the selection of the self-adaptive coarse meshes pi Adapt of the whole calculation domain of the sheet.
Step eleven: applying the self-adaptive coarse mesh pi Adapt obtained in the step ten to BDDC region decomposition parallel algorithm, namely, transforming the base into a rigidity matrixSum force vector/>Preprocessing operator construction for BDDC region decomposition parallel algorithms.
Step twelve: and carrying out parallel simulation prediction on the sheet vibration model according to the step eleven.
The BDDC region decomposition parallel calculation is carried out by comparing the coarse grids generated by the coarse grid selection method based on the BDDC region decomposition parallel algorithm disclosed by the invention with the coarse grids generated by the coarse grid selection method based on the BDDC region decomposition parallel algorithm, the simulation time is 1 multiplied by 10 -4 s, the time step is 5 multiplied by 10 -6 s, the iteration number in the simulation process is taken as the judgment basis of the calculation efficiency, and the comparison result is shown in figure 4. The algorithm iteration number of the self-adaptive coarse grid obtained by the coarse grid selection method based on the BDDC region decomposition parallel algorithm is obviously reduced compared with the algorithm iteration number of the self-adaptive coarse grid which is not used in the prior art after the self-adaptive coarse grid is applied to the BDDC region decomposition parallel algorithm, and the superiority of the self-adaptive coarse grid selection method is reflected.
Example 2 triangular prism type Expandable film Structure
As shown in fig. 5, a triangular prism type expandable film structure with the length of 100m is established, the degree of freedom of the model is 154848, and the self-adaptive coarse mesh obtained by the coarse mesh selection method based on the BDDC region decomposition parallel algorithm disclosed by the invention is applied to the iteration result of the BDDC region decomposition parallel algorithm and the original coarse mesh selection method under the condition that the division region number p=6, 12, 24 and 36, and is compared as shown in table 1:
TABLE 1 METIS regional decomposition results
The condition number and the iteration number of the BDDC region decomposition parallel algorithm iteration result after the original coarse grid selection method are obviously increased along with the increase of the partition number p, and the condition number and the iteration number of the BDDC region decomposition parallel algorithm iteration result after the original coarse grid selection method are adopted are obviously increased. The self-adaptive coarse grid is obtained by adopting the coarse grid selection method based on the BDDC regional decomposition parallel algorithm, and after the self-adaptive coarse grid is applied to the BDDC regional decomposition parallel algorithm, the condition of jump of the material coefficient of the subdomain interface of the flexible multi-body system can be effectively aimed, so that the iteration number and condition number in the solution process of the BDDC regional decomposition parallel algorithm to the flexible multi-body system are ensured to be in a smaller range, the calculation efficiency of the BDDC regional decomposition parallel algorithm is effectively improved, and the superiority of the coarse grid selection method is embodied.
While the foregoing is directed to embodiments of the present invention, other and further embodiments of the invention may be devised without departing from the basic scope thereof, and the scope thereof is determined by the claims that follow.
Claims (10)
1. A coarse grid selection method based on BDDC region decomposition parallel algorithm is characterized in that: comprises the following steps of the method,
Partitioning the whole calculation domain of the flexible multi-body system to realize the division of an inner interface gamma and an inner area I;
Step two, calculating the Sulzer matrix of all the subdomain omega i interfaces Γ i in the step one The Sul complement matrix/>For Deluxe scaling matrix calculation in step three, i=1, 2,3, …, n;
Step three, calculating Deluxe scaling matrix of all the shared edges E i,k in step two Deluxe scaling matrix/>, of the common edgeThe method is used for calculating a coefficient matrix of the generalized characteristic equation in the step five, wherein k is k=1, … and m;
step four, calculating the quadratic Sul compensation matrix of all the shared edges E i,k in the step two The quadratic Sul complement matrix/>The method is used for calculating a coefficient matrix of the generalized characteristic equation in the step five, wherein k=1, … and m;
Step five, constructing a generalized characteristic equation coefficient matrix of all the shared edges E i,k And/>The generalized characteristic equation coefficient matrix/>And/>The method is used for calculating a step six generalized characteristic equation, wherein k is k=1, … and m;
Step six, calculating generalized characteristic equations of all the shared edges E i,k to obtain corresponding characteristic vector matrixes Eigenvalue matrix/>The eigenvalue vector matrix/>The coefficient calculation of the generalized characteristic equation in the fifth step is adopted to obtain the material coefficient jump characteristic of the sub-domain interface of the flexible multi-body system, and the unitized characteristic vector matrix is utilizedConstructing a group of new bases, so that the rigidity matrix of the flexible multi-body system under the representation of the new bases has better condition number;
Step seven, the feature vector matrix of all the shared edges E i,k in the step six is obtained Proceeding with respect to coefficient matrix/>A new set of bases of the flexible multi-body system is constructed for use in step nine to generate a conversion matrix Q (i), i=1, 2,3, …, n for each sub-field Ω i, thereby constructing an adaptive coarse grid;
Step eight, the eigenvalue matrix of all the shared edges E i,k in the step six is obtained The corresponding eigenvalues are arranged from small to large, k=1, …, m, and the unit eigenvector matrix/>, according to the same sequenceRearranging to ensure the one-to-one correspondence between the unit eigenvectors and eigenvalues, and sequencing the unit eigenvector matrix/>Constructing a conversion matrix for step nine;
Step nine, constructing a conversion matrix Q (i), i=1, 2,3, … and n of each subdomain omega i, and performing base transformation on a rigidity matrix K (i) and a force vector v (i) of each subdomain for constructing an adaptive coarse grid in step ten;
Step ten, setting a characteristic value parameter lambda Tol to be more than or equal to 1 for constructing a self-adaptive coarse grid, and sorting the characteristic value matrix of all the common edges E i,k subjected to size sorting in step eight The stiffness matrix and the force vector after the basis transformation obtained by multiplying the unit eigenvector corresponding to the eigenvalue larger than lambda Tol by the stiffness matrix K (i) and the force vector v (i) represent the stiffness matrix/>, corresponding to the adaptive coarse gridSum force vector/>The node corresponding to the rigidity matrix and the force vector after the base transformation is the self-adaptive coarse grid pi Adapt; the self-adaptive coarse mesh pi Adapt has higher stability aiming at the condition of transition of the material coefficients of the subdomain interface of the flexible multi-body system, and ensures the high efficiency and stability of BDDC region decomposition parallel algorithm;
step eleven: applying the self-adaptive coarse mesh pi Adapt obtained in the step ten to BDDC region decomposition parallel algorithm, namely, transforming the base into a rigidity matrix Sum force vector/>Preprocessing operator construction for BDDC region decomposition parallel algorithm; the adaptive coarse grid pi Adapt contains information of the sub-domain interface material coefficients of the flexible multi-domain system, and in the parallel simulation prediction process of the flexible multi-domain system with the sub-domain interface material coefficient jump by adopting the BDDC region decomposition parallel algorithm, the condition number and the iteration number of the interface equation can be effectively reduced, and the high efficiency and the stability of the BDDC region decomposition parallel algorithm are ensured, so that the problem of low calculation efficiency caused by the sub-domain interface material coefficient jump of the flexible multi-domain system is solved.
2. The coarse grid selection method based on BDDC region decomposition parallel algorithm as set forth in claim 1, wherein: and step twelve, carrying out parallel simulation prediction on the large-scale flexible multi-body system according to step eleven, thereby solving the technical problem of related engineering of the parallel simulation prediction of the large-scale flexible multi-body system.
3. The coarse grid selection method based on BDDC region decomposition parallel algorithm as set forth in claim 2, wherein: related engineering technical problems in the field of parallel simulation prediction of large-scale flexible multi-body systems include on-orbit assembly of space solar power stations, on-orbit unfolding of large truss antenna structures and on-orbit unfolding of thin film solar wings.
4. A coarse mesh selection method based on BDDC region decomposition parallel algorithm as claimed in claim 3, wherein: the first implementation method of the step is that,
Dividing the overall computational domain Ω of the flexible multi-body system into n sub-domains Ω i, i=1, 2,3, …, n, then there areThe boundary of the overall computational domain constitutes an external interface as/>The interfaces Γ i between all adjacent subfields constitute the internal interface of the overall computational domain as Γ, i.e./>External interface/>The inner area outside the inner interface Γ is i, which is also composed of the inner areas of the subfields i i, i.e./>
The implementation method of the second step is that,
The stiffness matrix corresponding to the inner region I i and the interface Γ i of the subdomain omega i is respectively expressed asAnd/>Wherein/>AndThe coupling term of the inner region i i of the subdomain Ω i and the interface Γ i;
sul complement matrix of interface Γ i corresponding to stiffness matrix The method comprises the following steps:
The subfield Ω i interface Γ i is formed by a plurality of common edges common to the subfield and surrounding adjacent subfields, if the subfield Ω i interface Γ i is formed by m common edges, wherein the kth common edge is denoted as E i,k, k=1, …, m, then there is The corresponding Sulbumin matrix/>, of the interface Γ i of the corresponding subdomain Ω i The corresponding amount of the common edge E i,k is expressed by:
Wherein the diagonal member Representing the Sulbumin matrix corresponding to common edge E i,k in subdomain Ω i, off-diagonal elements/>Then the coupling term between common edge E i,1 and common edge E i,k is represented;
The implementation method of the third step is that,
For the common edge E i,k in the subdomain Ω i, its Deluxe scaling matrixThe calculation is as follows:
Where I (E i,k) represents the set of all subfields that share E i,k, the Deluxe scaling matrix of common edge E i,k if E i,k is shared only by two subfields, Ω i and Ω j The expression is simplified as:
Calculating Deluxe scaling matrix of all the shared edges E i,k The Deluxe scaling matrix/>Recording the material coefficient information of all the common subfields E i,k, wherein k is k=1, …, m, and the material coefficient jump information can be effectively provided for the situation of the material coefficient jump of the subfield interface of the flexible multi-body system.
5. The method for selecting the coarse grid based on the BDDC region decomposition parallel algorithm as recited in claim 4, wherein the method comprises the following steps: the realization method of the fourth step is that,
For the common edge E i,k in the subdomain omega i interface gamma i, the Sul complement matrix isLet the rest of the interface Γ i except the common edge E i,k be C, i.e. c=Γ i\{Ei,k, then the common edge E i,k solependence matrix/>Sul complement matrix/>The method comprises the following steps:
Wherein, Diagonal matrix representing the remainder C,/>And/>A coupling term representing the common edge E i,k with the remaining portion C;
calculating the quadratic Sul compensation matrix of all the common edges E i,k The quadratic Sul complement matrix/>The soft multi-body system subdomain interface gamma i Sulbumin/>The part except the shared edge E i,k is condensed into the shared edge E i,k, contains more soft multi-body system subdomain boundary information and is used for constructing a more effective generalized characteristic equation coefficient matrix in the step five.
6. The method for selecting the coarse grid based on the BDDC region decomposition parallel algorithm as recited in claim 5, wherein the method comprises the following steps: the fifth implementation method is that,
Step 5.1: constructing a generalized characteristic equation coefficient matrix of all the shared edges E i,k The generalized characteristic equation coefficient matrix/>From the common edge Sul complement matrix/>, in the step twoDeluxe scaling matrix in step threeThe structure comprises the material coefficient characteristics of the sub-domain interface of the flexible multi-body system, and can effectively provide corresponding material coefficient transition information for the situation of material coefficient transition of the sub-domain interface of the flexible multi-body system;
For the common edge E i,k of the subdomain omega i, the common edge E i,k corresponds to the Sulzer matrix and Deluxe scaling matrix belonging to the subdomain omega i, respectively And/>If the sub-field Ω j and the sub-field Ω i share E i,k, the shared edge E i,k corresponds to the Sulzer matrix and Deluxe scaling matrix belonging to the sub-field Ω j, respectivelyAnd/>Generalized eigenequation coefficient matrix/>, of the common edge E i,k The method comprises the following steps:
Where I (E i,k) represents the set of subfields of all common E i,k, if E i,k is shared only by two subfields Ω i and Ω j, the generalized eigenequation coefficient matrix a Ei,k expression of common edge E i,k is reduced to a two subfield case:
Constructing a generalized characteristic equation coefficient matrix of all the shared edges E i,k
Step 5.2: constructing a coefficient matrix of a generalized characteristic equation of a common edge E i,k The generalized characteristic equation coefficient matrixFrom the quadratic Sul complement matrix/>, in the fourth stepThe parallel and constitution, the information of the rest parts except the shared edge E i,k of the sub-domain of the flexible multi-system is condensed, the material coefficient characteristics of the sub-domain interface of the flexible multi-system are included, and corresponding material coefficient jump information can be provided for the condition of material coefficient jump of the sub-domain interface of the flexible multi-system effectively;
For the common edge E i,k of the sub-field Ω i, if the common edge E i,k is shared by m sub-fields Ω i, i=1, 2,3, …, m, the generalized eigen equation coefficient matrix of the common edge E i,k The method comprises the following steps:
If E i,k is shared only by two sub-domains of Ω i and Ω j, then the generalized characteristic equation coefficient matrix of the shared edge E i,k The expression is simplified as:
Wherein, For/>And/>The parallel sum expression of (2) is as follows:
Wherein, For/>Is the generalized inverse of (2);
Constructing a generalized characteristic equation coefficient matrix of all the shared edges E i,k
7. The coarse grid selection method based on BDDC region decomposition parallel algorithm as set forth in claim 6, wherein: the implementation method of the step six is that,
According to the two generalized characteristic equation coefficient matrixes calculated in the fifth step, solving the following generalized characteristic equation:
Wherein, For a matrix of feature vectors, each column of the matrix represents a feature vector; /(I)As a characteristic value matrix, each diagonal element of the matrix corresponds to a characteristic value;
the generalized characteristic equation for all the common edges E i,k is calculated, k=1, …, m.
8. The coarse grid selection method based on BDDC region decomposition parallel algorithm as set forth in claim 7, wherein: the seventh implementation method is that,
For the common edge E i,k of the subdomain omega i, the generalized eigenvector is v Ei,k, where the kth vector is denoted asFeature vector/>With respect to coefficient matrix/>Is a unitized feature vector/>The method comprises the following steps:
The unit eigenvector matrix is obtained by unitizing all eigenvectors with respect to the coefficient matrix A Ei,k The unit eigenvector matrix/>As a new set of bases for the flexible multi-body system, the stiffness matrix of the flexible multi-body system in the new base representation can be given a better condition number.
9. The coarse grid selection method based on BDDC region decomposition parallel algorithm as set forth in claim 8, wherein: the implementation method of the step nine is that,
Step 9.1: constructing a conversion matrix Q (i) of the subdomains omega i;
The stiffness matrix K (i) and the force vector v (i) of the subfields Ω i are represented in the order of the inner region i and then the inner interface Γ as follows:
The conversion matrix Q (i) of the subfields Ω i is constructed as follows:
Wherein the method comprises the steps of Is a unit array with the same degree of freedom as the inner area I i of the subdomain omega i,/>The unit eigenvector matrix/>, of all the common edges E i,k in the subdomain Ω i in step sevenComposition, k=1, …, m:
Step 9.2: the rigidity matrix K (i) of the subdomain omega i and the force vector v (i) are subjected to base transformation, and the rigidity matrix after the base transformation has a better condition number;
for stiffness matrix K (i), a transpose of the left-hand conversion matrix is required Right multiplying the conversion matrix Q (i) to obtain a rigidity matrix/>, after the base conversion
Wherein,Denoted by the corresponding portion of the common edge E i,k, k=1, …, m:
For force vectors, a transpose of the left-hand conversion matrix is required Obtain force vector/>, after basis transformation
Wherein,Represented by the corresponding portion of the common edge E ij:
10. The coarse grid selection method based on BDDC region decomposition parallel algorithm as set forth in claim 9, wherein: the implementation method of the step ten is that,
Unit feature vector matrix of common edge E i,k In total m columns of vectors, then/>Expressed as:
if the characteristic value matrix of the shared edge E i,k The first (l < m) eigenvalue/>Greater than the parameter lambda Tol, the eigenvalue matrix/>, according to step eightAccording to the order from small to large, the characteristic values after the first characteristic value is larger than the parameter lambda Tol, and the characteristic vector corresponding to the characteristic value is multiplied by the rigidity matrix and the force vector to obtain the rigidity matrix and the force vector corresponding to the self-adaptive coarse grid;
unit feature vector matrix for common edge E i,k The method comprises the following steps:
Performing base transformation on the rigidity matrix:
Wherein the rigidity matrix with the row number larger than l is a rigidity matrix corresponding to the self-adaptive coarse grid
Performing base transformation on the force vector:
Wherein the force vector with the row number larger than l is the force vector corresponding to the self-adaptive coarse grid
Thereby completing the selection of the self-adaptive coarse mesh pi Adapt.
Priority Applications (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN202111680165.4A CN114372361B (en) | 2021-12-23 | 2021-12-23 | Coarse grid selection method based on BDDC region decomposition parallel algorithm |
Applications Claiming Priority (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN202111680165.4A CN114372361B (en) | 2021-12-23 | 2021-12-23 | Coarse grid selection method based on BDDC region decomposition parallel algorithm |
Publications (2)
| Publication Number | Publication Date |
|---|---|
| CN114372361A CN114372361A (en) | 2022-04-19 |
| CN114372361B true CN114372361B (en) | 2024-05-28 |
Family
ID=81141980
Family Applications (1)
| Application Number | Title | Priority Date | Filing Date |
|---|---|---|---|
| CN202111680165.4A Active CN114372361B (en) | 2021-12-23 | 2021-12-23 | Coarse grid selection method based on BDDC region decomposition parallel algorithm |
Country Status (1)
| Country | Link |
|---|---|
| CN (1) | CN114372361B (en) |
Citations (5)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN106096132A (en) * | 2016-06-12 | 2016-11-09 | 北京航空航天大学 | A kind of emulation mode of the different materials clothing fold based on differential domain |
| CN107220421A (en) * | 2017-05-18 | 2017-09-29 | 北京理工大学 | A kind of spatial complex flexible structure dynamics of multibody systems modeling and computational methods |
| EP3754521A1 (en) * | 2019-04-26 | 2020-12-23 | Dalian University Of Technology | Structural topology optimization method based on material-field reduction series expansion |
| CN112290601A (en) * | 2020-10-27 | 2021-01-29 | 国网山东省电力公司电力科学研究院 | Optimized scheduling method and system for flexible interconnection alternating current-direct current power distribution system |
| CN113779831A (en) * | 2021-09-02 | 2021-12-10 | 西北工业大学 | Polycondensation FETI engineering numerical method based on regional decomposition |
-
2021
- 2021-12-23 CN CN202111680165.4A patent/CN114372361B/en active Active
Patent Citations (5)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN106096132A (en) * | 2016-06-12 | 2016-11-09 | 北京航空航天大学 | A kind of emulation mode of the different materials clothing fold based on differential domain |
| CN107220421A (en) * | 2017-05-18 | 2017-09-29 | 北京理工大学 | A kind of spatial complex flexible structure dynamics of multibody systems modeling and computational methods |
| EP3754521A1 (en) * | 2019-04-26 | 2020-12-23 | Dalian University Of Technology | Structural topology optimization method based on material-field reduction series expansion |
| CN112290601A (en) * | 2020-10-27 | 2021-01-29 | 国网山东省电力公司电力科学研究院 | Optimized scheduling method and system for flexible interconnection alternating current-direct current power distribution system |
| CN113779831A (en) * | 2021-09-02 | 2021-12-10 | 西北工业大学 | Polycondensation FETI engineering numerical method based on regional decomposition |
Also Published As
| Publication number | Publication date |
|---|---|
| CN114372361A (en) | 2022-04-19 |
Similar Documents
| Publication | Publication Date | Title |
|---|---|---|
| CN110110413B (en) | Structural topology optimization method based on material field reduction progression expansion | |
| Kammer et al. | Optimal placement of triaxial accelerometers for modal vibration tests | |
| Adeli et al. | Algorithms for nonlinear structural dynamics | |
| CN109918712B (en) | Calculation method for solving composite stress intensity factor based on p-type finite element method and contour integral method | |
| Dodds Jr et al. | Substructuring in linear and nonlinear analysis | |
| Bogomolny | Topology optimization for free vibrations using combined approximations | |
| CN107491585B (en) | Structural topology optimization design method taking random displacement response variance as target | |
| CN110826132B (en) | Design method of structure-dispersed vibration control system | |
| CN114372361B (en) | Coarse grid selection method based on BDDC region decomposition parallel algorithm | |
| CN114347029A (en) | Model order reduction method for rapid simulation of pneumatic soft robot | |
| CN113779831B (en) | Polycondensation FETI engineering numerical method based on regional decomposition | |
| CN108875132B (en) | Topological optimization method of stress constraint type truss continuum structure under multiple working conditions | |
| Beckers | Dual methods for discrete structural optimization problems | |
| Khot et al. | Optimum design of composite structures with stress and displacement constraints | |
| CN116644634A (en) | Structural topology optimization method based on SDOFR and pretreatment recovery conjugate gradient method | |
| CN113836766B (en) | METIS-based spherical hinge-containing multi-body system diagram dividing method | |
| Prasad | Approximation, adaptation and automation concepts for large-scale structural optimization | |
| CN114237055A (en) | Multi-order modal active vibration suppression method for extremely-large space rigid-flexible coupling system | |
| Karpel | Modal-based enhancement of integrated design optimization schemes | |
| CN113722993A (en) | Modular spatial structure multi-constraint sequence planning method oriented to stable assembly | |
| CN111259589A (en) | Continuum frequency constraint topology optimization design method considering damage-safety | |
| CN113032910A (en) | Layout optimization method for actuator of flexible spacecraft | |
| CN115963859B (en) | Spacecraft cluster clustering cooperative control method based on full-drive system theory | |
| Padovan et al. | Hierarchically parallelized constrained nonlinear solvers with automated substructuring | |
| CN115392081A (en) | High-precision flow-induced vibration simulation method based on FETI |
Legal Events
| Date | Code | Title | Description |
|---|---|---|---|
| PB01 | Publication | ||
| PB01 | Publication | ||
| SE01 | Entry into force of request for substantive examination | ||
| SE01 | Entry into force of request for substantive examination | ||
| GR01 | Patent grant | ||
| GR01 | Patent grant |