CN107944189B - Method for optimizing structural parameters based on sparse matrix symbolic operation result - Google Patents
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Abstract
The invention discloses a method for optimizing structural parameters based on sparse matrix symbolic operation results, and belongs to the technical field of airplane structural strength analysis. The method mainly comprises the steps of firstly generating a rigidity matrix through a finite element analysis method; secondly, performing matrix transformation on the stiffness matrix by taking the minimum filling element as a target, and establishing a mapping relation; then, carrying out equal change on the right end item of the load by utilizing the transformation, and then carrying out numerical decomposition to form a triangular decomposition array of the linear equation system sparse array; finite element analysis is carried out by utilizing a matrix decomposition result, optimization calculation is carried out, and numerical item replacement operation is carried out on the updated design variable by utilizing the mapping relation; and finally, repeating the optimization process until the optimization converges. The invention fully utilizes the result of the first symbolic operation to carry out the numerical decomposition operation, only needs to carry out the symbolic operation once in the whole design process, and theoretically adopts the method to shorten the design time by about 50 percent compared with the traditional design process.
Description
Technical Field
The invention belongs to the technical field of airplane structural strength analysis, and particularly relates to a method for optimizing structural parameters based on sparse matrix symbolic operation results.
Background
The research and development design of the structure is finished with high quality under the condition of limited time and expense, the design of an optimization means is necessarily selected, and the optimization design increasingly becomes an effective means for solving the contradiction between safety, performance, time and expense of people. Over decades of effort, the development of optimization techniques has made great progress in theory and application.
For historical reasons, a finite element analyzer is artificially isolated from an optimization process in the traditional optimization design process, a result obtained after the optimizer completes numerical value optimization must be returned to the analyzer to perform complete finite element analysis once, in the process, the finite element analyzer mainly provides information such as structural response, variable sensitivity and the like for constructing an optimization objective function, and then the optimizer selects a proper optimization algorithm to perform optimization processing on the objective function. At present, with the improvement of computer hardware technology, finely divided finite element grids are often adopted for increasingly complex structural design forms, finite element models of millions of nodes are quite common in design, and the design efficiency is relatively low when the finite element models of the scale are repeatedly called for carrying out optimization design. Referring to fig. 2, in the conventional optimization design, an analysis process is used as a black box call, and the optimization efficiency is low because a solver is repeatedly called to perform response acquisition.
Disclosure of Invention
In order to solve the problems, the invention provides an optimization method for reducing the operation amount and improving the problem solving efficiency, the sign operation result in the sparse matrix direct solving method is utilized, the characteristic that the non-zero element distribution of the structural rigidity matrix is unchanged is fully utilized during the structural parameter optimization, and the optimization efficiency is greatly improved by putting the sign operation outside the optimization iteration process. The invention relates to a method for optimizing structural parameters based on sparse matrix symbolic operation results, which mainly comprises the following steps:
the method comprises the steps of firstly, obtaining rigidity parameters of a structure to be designed, and generating a rigidity initial matrix, wherein the rigidity initial matrix corresponds to a load initial right-end item;
secondly, performing matrix transformation on the stiffness initial matrix by taking minimum filling elements as a target to obtain a stiffness transformation matrix, and establishing a corresponding relation between the stiffness initial matrix and the stiffness transformation matrix;
transforming the initial right end item of the load according to the corresponding relation to obtain a right end item of load transformation, and performing numerical decomposition on the right end item of load transformation to form a triangular decomposition array of a sparse matrix of a linear equation set;
performing finite element analysis by using the triangular decomposition array, acquiring response and variable sensitivity values required by structural design, constructing an optimization objective function, and performing optimization calculation to obtain an initial design variable;
fifthly, carrying out numerical item replacement operation on the initial design variables by utilizing the corresponding relation in the second step;
and step six, repeating the step three to the step five until the optimization converges.
Preferably, in the second step, before establishing the correspondence between the stiffness initial matrix and the stiffness transformation matrix, the second step further includes storing the stiffness initial matrix and the stiffness transformation matrix in a CSC format.
Preferably, in the second step, the matrix transforming the stiffness initial matrix includes matrix transforming by means of Metis or MD.
According to the method, the symbol operation process and the numerical decomposition process in the sparse matrix direct solution are effectively combined with the change of the matrix item value caused by variable updating in the parameter optimization design, and the optimization design efficiency is greatly improved after the traditional optimization design process is designed and modified by using the technical method.
Drawings
Fig. 1 is a flowchart of a preferred embodiment of a method for performing structure parameter optimization based on sparse matrix symbolic operation results according to the present invention.
Fig. 2 is a flow chart of an optimization method in the prior art.
FIG. 3 is an initial stiffness matrix and a corresponding right end load matrix for the embodiment of the invention shown in FIG. 1.
FIG. 4 is a lower triangular matrix after decomposition of the initial stiffness matrix of the embodiment of the invention shown in FIG. 3.
FIG. 5 is a stiffness transformation matrix after least padding transform according to the embodiment of the invention shown in FIG. 1.
Fig. 6 is a decomposed lower triangular matrix of the stiffness transformation matrix of the embodiment of the invention shown in fig. 5.
FIG. 7 is a transformation mapping matrix according to the embodiment of the invention shown in FIG. 1.
Fig. 8 is a sparse matrix deposited in CSC format according to the embodiment of the present invention shown in fig. 1.
Detailed Description
In order to make the implementation objects, technical solutions and advantages of the present invention clearer, the technical solutions in the embodiments of the present invention will be described in more detail below with reference to the accompanying drawings in the embodiments of the present invention. In the drawings, the same or similar reference numerals denote the same or similar elements or elements having the same or similar functions throughout. The described embodiments are only some, but not all embodiments of the invention. The embodiments described below with reference to the drawings are illustrative and intended to be illustrative of the invention and are not to be construed as limiting the invention. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention. Embodiments of the present invention will be described in detail below with reference to the accompanying drawings.
In the description of the present invention, it is to be understood that the terms "center", "longitudinal", "lateral", "front", "rear", "left", "right", "vertical", "horizontal", "top", "bottom", "inner", "outer", etc., indicate orientations or positional relationships based on those shown in the drawings, and are used merely for convenience in describing the present invention and for simplifying the description, but do not indicate or imply that the device or element being referred to must have a particular orientation, be constructed and operated in a particular orientation, and thus, should not be construed as limiting the scope of the present invention.
The invention provides a novel design method for carrying out parameter optimization design by utilizing symbolic operation results in a sparse matrix direct solving method, which fully utilizes the characteristic that non-zero elements of a structural rigidity matrix are not changed in distribution when structural parameters are optimized, directly replaces coefficient values in a linear equation set by design variable current values subjected to optimal parameter adjustment through an optimization algorithm, and directly carries out numerical decomposition operation by utilizing the first symbolic operation results. Referring to fig. 1, a schematic flow chart of the optimization method of the present invention, and fig. 2 is a flow chart of a conventional optimization method, in combination with the two diagrams, the present invention greatly improves the optimization efficiency by putting symbolic operations outside the optimization iteration process and performing symbolic operations only once in the whole parameter optimization design process. Compared with the traditional design process, the design time is shortened by about 50% by adopting the method theoretically.
The invention relates to a method for optimizing structural parameters based on sparse matrix symbolic operation results, which mainly comprises the following steps:
the method comprises the steps of firstly, obtaining rigidity parameters of a structure to be designed, and generating a rigidity initial matrix, wherein the rigidity initial matrix corresponds to a load initial right-end item;
secondly, performing matrix transformation on the stiffness initial matrix by taking minimum filling elements as a target to obtain a stiffness transformation matrix, and establishing a corresponding relation between the stiffness initial matrix and the stiffness transformation matrix;
transforming the initial right end item of the load according to the corresponding relation to obtain a right end item of load transformation, and performing numerical decomposition on the right end item of load transformation to form a triangular decomposition array of a sparse matrix of a linear equation set;
performing finite element analysis by using the triangular decomposition array, acquiring response and variable sensitivity values required by structural design, constructing an optimization objective function, and performing optimization calculation to obtain an initial design variable;
fifthly, carrying out numerical item replacement operation on the initial design variables by utilizing the corresponding relation in the second step;
and step six, repeating the step three to the step five until the optimization converges.
The invention is further described below with reference to the accompanying drawings.
Step 1: carrying out structural finite element analysis, and assembling a structural overall rigidity matrix according to a conventional mode; for effective explanation, a simple stiffness matrix a and a corresponding right-end term B of the structural stiffness are assumed, referring to fig. 3, where a is an initial stiffness matrix and B is an initial right-end term of a load corresponding to the initial stiffness matrix a, and when equation solution is performed by a direct method, a most basic method is to decompose the matrix into an upper triangular matrix and a lower triangular matrix. That is, for a linear equation: ax ═ B requires finding a lower triangular matrix L, as shown in fig. 4, with an upper triangular matrix U (a symmetric matrix with U being L for structural issues), so that a ═ LU- > LU ═ B.
Step 2: in the above process, although the matrix a is a sparse matrix, the lower triangular matrix L has more non-zero elements after decomposition. Thus, if the equations are solved directly with L, we will actually get a similar computational complexity and data storage as the dense matrix. After decomposing an element with an initial value of zero in the matrix a, if a non-zero element is generated at a corresponding position of the matrix L, it is generally called a non-zero element Fill (Fill-in). From the calculation point of view, an effective method is that the non-zero elements in the matrix A are traversed, and the number of the non-zero elements in the matrix L after decomposition is reduced as much as possible, so that only a small number of the non-zero elements in the L need to be considered during solving.
The above process is usually implemented by means of Metis or MD method in engineering, and this process is called symbol operation. For the matrix A, B exemplified herein, the sign operation process is essentially to obtain the corresponding matrix a ' and the corresponding matrix B ' with the smallest padding elements, as shown in fig. 5, a ' is a stiffness transformation matrix, B ' is a right-end term of load transformation, and the corresponding decomposition matrix L ' refers to fig. 6, it can be seen that the non-zero elements contained in the matrix are much smaller than the original lower triangular matrix L, and in practical application, the gain is quite obvious for the matrix scale of millions.
It is understood that, before performing matrix transformation, the present embodiment may further include constructing a directed graph of the initial stiffness matrix and a corresponding wavefront tree.
And step 3: the symbol operation process is essentially a series of row-column transformation to the initial matrix, and a switching matrix P can be used to represent the switching of the rows and columns of the matrix. As shown in FIG. 7, examples are illustrated herein but are, for example, trueEssentially, the first and fifth rows and the first and fifth columns of matrix a are exchanged, i.e. matrix a' ═ PAPTAnd the matrix transformation process corresponding to the symbol operation is a transformation matrix P.
And 4, step 4: in order to reduce the computation and I \ O operation, the CSC format may be used to store the corresponding sparse matrix and the transformed minimum matrix of padding elements, as shown in fig. 8, in the case of using finite elements to solve the equilibrium equation in the engineering practice. For the CSC format to store the values of the two matrices before and after, which are only arranged in different orders, we can see that an explicit correspondence between the values of the matrices is established.
And 5: by utilizing the corresponding relation, after the optimizer returns the current value of the design variable subjected to numerical optimization, the transformed matrix item can be automatically modified through a program, matrix decomposition operation is carried out on the transformed matrix item, the structural response is obtained, and the structural response is submitted to the optimizer for next optimization iteration until optimization convergence, the optimization process is a conventional means, and details are not repeated.
Step 6: according to the above steps, the symbolic operation process is set outside the design iteration loop shown in fig. 1, and referring to fig. 1, the time of the symbolic operation process of the matrix operation is basically equivalent to the time of the numerical decomposition process with respect to the time consumption of the structural analysis, so that the structural analysis time can be reduced by half each time according to the process, and the program organization mode has significant benefit with respect to the structural optimization design which consumes a long time for the structural analysis.
According to the method, the symbol operation process and the numerical decomposition process in the sparse matrix direct solution are effectively combined with the change of the matrix item value caused by variable updating in the parameter optimization design, and the optimization design efficiency is greatly improved after the traditional optimization design process is designed and modified by using the technical method.
Finally, it should be pointed out that: the above examples are only for illustrating the technical solutions of the present invention, and are not limited thereto. Although the present invention has been described in detail with reference to the foregoing embodiments, it will be understood by those of ordinary skill in the art that: the technical solutions described in the foregoing embodiments may still be modified, or some technical features may be equivalently replaced; and such modifications or substitutions do not depart from the spirit and scope of the corresponding technical solutions of the embodiments of the present invention.
Claims (3)
1. A method for optimizing structural parameters based on sparse matrix symbolic operation results is characterized by comprising the following steps:
the method comprises the steps of firstly, obtaining rigidity parameters of a structure to be designed, and generating a rigidity initial matrix, wherein the rigidity initial matrix corresponds to a load initial right-end item;
secondly, performing matrix transformation on the stiffness initial matrix by taking minimum filling elements as a target to obtain a stiffness transformation matrix, and establishing a corresponding relation between the stiffness initial matrix and the stiffness transformation matrix;
transforming the initial right end item of the load according to the corresponding relation to obtain a right end item of load transformation, and performing numerical decomposition on the right end item of load transformation to form a triangular decomposition array of a sparse matrix of a linear equation set;
performing finite element analysis by using the triangular decomposition array, acquiring response and variable sensitivity values required by structural design, constructing an optimization objective function, and performing optimization calculation to obtain an initial design variable;
fifthly, carrying out numerical item replacement operation on the initial design variables by utilizing the corresponding relation in the second step;
and step six, repeating the step three to the step five until the optimization converges.
2. The method according to claim 1, wherein in the second step, before establishing the correspondence between the stiffness initial matrix and the stiffness transformation matrix, the method further comprises storing the stiffness initial matrix and the stiffness transformation matrix in a CSC format.
3. The method of claim 1, wherein in the second step, the matrix transformation of the stiffness initial matrix comprises matrix transformation using Metis or MD.
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