CN114781205B - Sensitivity analysis method and application of structural topology optimization model - Google Patents
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Abstract
The invention discloses a sensitivity analysis method and application of a structural topology optimization model, belonging to the field of structural optimization design; in the sensitivity analysis method provided by the invention, based on the characteristic of B spline function tensor product decomposition, a traditional multi-dimensional implicit filter weight matrix is decomposed into a plurality of single-dimensional sub-weight matrices for calculation. In another sensitivity analysis method provided by the invention, a sub-weight matrix in the direction of each parameter unit is obtained based on the number of grids in each direction of grids divided by an engineering structure and the minimum filter radius. The sub-weight matrix in each direction can obtain the whole filter weight matrix by a Kronecker matrix multiplication method, thereby realizing the equivalent expression of the weight matrix. According to the method, the sensitivity filter weight matrix of each unit is not required to be calculated, only the sub weight matrix in each dimension direction is required to be calculated, the storage efficiency is high, and the calculation efficiency is high.
Description
Technical Field
The invention belongs to the field of structural optimization design, and particularly relates to a sensitivity analysis method and application of a structural topology optimization model.
Background
The traditional engineering structure design often requires engineering designers to have years of practical experience, and the design mode can not meet the requirements of high efficiency, high precision, flexibility and the like of the current industrial products. The structure optimization technology based on the numerical analysis technology can quickly generate a reasonable and effective structural design scheme through a computer, and is one of main implementation modes of intelligent design of engineering structures. The size optimization, the shape optimization and the topology optimization are three main ways for realizing the engineering structure optimization design, and the topology optimization becomes the most potential structure optimization design tool due to the advantages of wider design space, variable internal structure topology form and the like. However, the structural topology optimization model based on the description form of unit variables (or node variables) is essentially a 0-1 integer programming problem, and a series of numerical instability problems such as checkerboard, grid dependence and the like are very easy to occur.
The filter technology can effectively eliminate the numerical instability such as checkerboard, grid dependence and the like of the optimization result, and is the basis for updating the optimization design variables by a series of variable density topology optimization methods such as a solid isotropy punishment model, a progressive structure, discrete variables and the like. Therefore, the variable density topology optimization method based on the filter technology can effectively realize the optimization design of the engineering structure and is gradually applied to the fields of aerospace, automobile engineering, construction, medical appliances and the like. The sensitivity analysis of the structural topological optimization model is used as an important link of structural topological optimization, and reflects the derivative information of the constraint function and the objective function on the optimization design variable, so that the influence degree and the influence rule of the optimization design variable on the output response are represented, and the sensitivity analysis method for researching the structural topological optimization model has important significance;
The sensitivity analysis method based on the filter technology uses the sensitivity weighted average of the area near the unit to replace the initial sensitivity of the corresponding optimization design variable, thereby realizing the regularization treatment of the structural topology optimization model. Whether an explicit filter based on cell center point distance or an implicit filter based on B-spline cross-cell characteristics, the weight matrix of all cells needs to be calculated and stored so as to quickly perform sensitivity analysis of the variable density topology optimization method. However, when the unit size and the radius of the filter are larger, the calculation and storage of the weight values of the filter matrix are performed in a mode of indexing all the units, which results in a series of problems of overlong calculation time, overlarge storage occupation space, slow updating efficiency of the optimized variables and the like of the weight matrix of the filter, and further seriously affects the sensitivity analysis of the structural topology optimization model and the application scene of the variable density topology optimization method. Therefore, a sensitivity analysis method of a non-globally indexed structural topology optimization model is urgently needed to improve the efficiency of updating the filter weight matrix and the optimization variables.
Disclosure of Invention
Aiming at the defects or improvement demands of the prior art, the invention provides a sensitivity analysis method and application of a structural topology optimization model, and aims to solve the technical problems that the calculation time of a structural topology optimization filter weight matrix is long and the storage efficiency of the filter weight matrix is low in the existing sensitivity analysis method.
To achieve the above object, in a first aspect, the present invention provides a sensitivity analysis method of a structural topology optimization model; the structural topology optimization model is constructed by taking the sum of strain energy of all analysis grid cells in the engineering structural analysis grid as an objective function and the sum of material volumes of all analysis grid cells as a volume constraint function, and the optimization design variable is the material density of the B spline control point coordinate position of the analysis grid;
the sensitivity analysis method comprises the following steps:
Arranging and integrating the strain energy of each analysis grid unit of the engineering structure according to the positions of the strain energy in the analysis grid to obtain a strain energy matrix C of the analysis grid; calculating and analyzing the partial conductance of the strain energy matrix C of the grid with respect to the density value of the cell center point to obtain a partial conductance matrix dC 1 of the objective function with respect to the density value of the cell center point;
Arranging and integrating derivative values of the material volumes of all the analysis grid cells to the density of the central points of the analysis grid cells according to the positions of the derivative values in the analysis grid cells to obtain a partial derivative matrix dV 1 of a volume constraint function on the density values of the central points of the cells;
Partial derivative matrix expression of objective function transformed based on tensor integral solution for optimizing design variable Calculating to obtain a partial derivative value of the objective function with respect to each optimization design variable;
partial derivative matrix expression of volume constraint function converted based on tensor integral solution and related to optimization design variable Calculating to obtain the partial derivative value of the volume constraint function with respect to each optimization design variable;
Wherein H m is a sub-weight matrix of the analysis grid in the m-th parameter direction; m=1, 2, …, M; m is the number of parameter directions, and the value is 2 or 3; the number of lines of H m is the number of B spline functions in the mth parameter direction, and the number of columns is the number of analysis grid units in the mth parameter direction; the element H m (r, c) in the r-th row and c-th column of H m is the B-spline function value corresponding to the r-th B-spline function and c-th analysis grid unit in the m-th parameter direction.
Further preferably, the method for obtaining the partial derivative matrix expression of the objective function transformed by the tensor integration method with respect to the optimal design variable and the partial derivative matrix expression of the volume constraint function transformed by the tensor integration method with respect to the optimal design variable includes:
Obtaining a partial derivative matrix expression H multiplied by dC 1 of the objective function on the basis of the chain derivation method and a partial derivative matrix expression H multiplied by dV 1 of the volume constraint function on the basis of the optimal design variable; h is a partial guide matrix of the density value of the cell center point relative to the optimal design variable, namely a cell center point B spline basis function value matrix of the analysis grid;
Decomposing the partial guide matrix H in the partial guide expression H multiplied by dC 1 into sub-weight matrixes in different parameter directions of the analysis grid by adopting a tensor integration method, thereby converting the partial guide expression H multiplied by dC 1 into a form of multiplying the sub-weight matrixes in different parameter directions of the analysis grid by the partial guide matrix dC 1, namely
The tensor integration method is adopted to decompose the partial guide matrix H in the partial guide expression H multiplied by dV 1 into the sub-weight matrix in the directions of different parameters of the analysis grid, so that the partial guide expression H multiplied by dV 1 is converted into the form of multiplying the sub-weight matrix in the directions of different parameters of the analysis grid with the partial guide matrix dV 1, namely
Further preferably, the method for acquiring the sub-weight matrix includes: and analyzing the B spline node vectors in each parameter direction of the grid according to the engineering structure to obtain the B spline function value of the central point of the analysis grid unit in each parameter direction and the index of the B spline function in the parameter direction, thereby obtaining the sub-weight matrix in each parameter direction.
Further preferably, the coordinate position of the B-spline control point of the analysis grid is calculated based on CAD information of the engineering structure by combining with a B-spline node vector insertion method.
In a second aspect, the present invention provides a topology optimization method for an engineering structure, including the following steps:
s11, initializing the Young' S modulus of elasticity of the materials of each analysis grid unit of the engineering structure according to the physical material volume constraint of the topological optimization model of the engineering structure;
S12, obtaining a stiffness matrix of the engineering structure based on the Young' S modulus of elasticity of the material of each analysis grid unit; based on the rigidity matrix of the obtained engineering structure and boundary conditions, calculating displacement vectors of each analysis grid cell, and further calculating strain energy of each analysis grid cell;
S13, constructing a structural topology optimization model by taking the sum of strain energy of each analysis grid unit as an objective function and the sum of material volumes of each analysis grid unit as a volume constraint function; the optimization design variable of the structural topology optimization model is the material density of the coordinate position of the B spline control point of the analysis grid;
s14, performing sensitivity analysis on the structural topology optimization model by adopting the sensitivity analysis method provided by the first aspect of the invention to obtain the partial derivative value of the objective function and the volume constraint function on each optimization design variable;
s15, inputting the partial derivative value into an optimization solution operator to obtain updated values of all the optimization design variables, and updating the Young' S elastic modulus of the material of each analysis grid unit;
S16, repeating the steps S12-S15 until the relative change value of the objective function of the adjacent iteration steps is smaller than a first preset threshold or the maximum change value of the relative material density corresponding to the Young' S elastic modulus of the material of the analysis grid unit is smaller than a second preset threshold.
In a third aspect, the present invention provides a method for analyzing sensitivity of a structural topology optimization model, including: the structural topology optimization model is constructed by taking the sum of strain energy of all analysis grid cells in the engineering structural analysis grid as an objective function and the sum of material volumes of all analysis grid cells as a volume constraint function, and the optimization design variable is the material density of the center point of the analysis grid cells;
the sensitivity analysis method comprises the following steps:
Arranging and integrating the strain energy of each analysis grid unit of the engineering structure according to the positions of the strain energy in the analysis grid to obtain a strain energy matrix C of the analysis grid; calculating and analyzing the partial conductance of the strain energy matrix C of the grid with respect to the optimal design variable to obtain a partial conductance matrix dC 2 of the objective function with respect to the optimal design variable;
Arranging and integrating the derivative values of the material volumes of each analysis grid unit to the density of the central points according to the positions of the material volumes of each analysis grid unit in the analysis grid to obtain a partial derivative matrix dV 2 of the volume constraint function on the optimization design variables, so as to obtain the partial derivative value of the volume constraint function on each optimization design variable; wherein,
Using a sensitivity correction formula based on tensor integration solution after transformationCorrecting the bias guide matrix dC 2 to obtain bias guide values of the corrected objective function with respect to each optimization design variable;
Wherein F n is a sub-weight matrix of the analysis grid in the nth direction; n=1, 2, …, N; n is the number of the geometric dimension directions of the analysis grid, and the value is 2 or 3; the number of rows and columns of F n are the number of units in the nth direction of the analysis grid; element F n (p, q) of the p-th row and q-th column of F n is a distance weight value between the p-th cell and the q-th cell in the n-th direction.
Further preferably, the method for acquiring the sensitivity correction formula includes:
Decomposing the filter weight matrix F in the sensitivity correction formula F multiplied by dC 2 into sub-weight matrices in different directions of the analysis grid by adopting a tensor integration method, thereby converting the sensitivity correction formula F multiplied by dC 2 into a form of multiplying the sub-weight matrices in different directions of the analysis grid by a partial guide matrix dC 2, namely
Further preferably, the filter weight matrix F is a weight matrix between unit center points, and is a Kronecker product of sub-weight matrices of all analysis grids in each direction, namely:
further preferably, the method for acquiring the sub-weight matrix includes:
calculating a sub-weight matrix in each direction according to the distance between the center points of the grid cells in each direction of the analysis grid of the engineering structure and the minimum filter radius;
Wherein, R min is the minimum filter radius of the filter,Is the distance in the direction between the p-th cell and the center point of the q-th cell in the n-th direction.
In a fourth aspect, the present invention provides a topology optimization method for an engineering structure, including the following steps:
s21, initializing the Young' S modulus of elasticity of the materials of each analysis grid unit of the engineering structure according to the physical material volume constraint of the topological optimization model of the engineering structure;
S22, obtaining a stiffness matrix of the engineering structure based on the Young' S modulus of elasticity of the material of each analysis grid unit; based on the rigidity matrix of the obtained engineering structure and boundary conditions, calculating displacement vectors of each analysis grid cell, and further calculating strain energy of each analysis grid cell;
S23, constructing a structural topology optimization model by taking the sum of strain energy of each analysis grid unit as an objective function and the sum of material volumes of each analysis grid unit as a volume constraint function; the optimization design variable of the structural topology optimization model is the material density of the central point of the analysis grid unit;
s24, performing sensitivity analysis on the structural topology optimization model by adopting the sensitivity analysis method provided by the third aspect of the invention to obtain the partial derivative value of the objective function and the volume constraint function on each optimization design variable;
S25, inputting the partial derivative value into an optimization solution operator to obtain updated values of all the optimization design variables, and updating the Young' S elastic modulus of the material of each analysis grid unit;
s26, repeating the steps S22-S25 until the maximum change value of the relative material density corresponding to the Young' S modulus of the material of the grid unit is smaller than a third preset threshold value; evenly refining the analysis grid;
s27, repeating the step S26 until the grid refinement times are greater than a fourth preset threshold.
In a fifth aspect, the present invention provides a topology optimization method system for an engineering structure, including: a memory storing a computer program and a processor executing the topology optimization method provided in the second or fourth aspect of the present invention.
In a sixth aspect, the invention also provides a machine-readable storage medium storing machine-executable instructions which, when invoked and executed by a processor, cause the processor to implement one or more of the sensitivity analysis method provided in the first aspect, the topology optimization method provided in the second aspect, the sensitivity analysis method provided in the third aspect, and the topology optimization method provided in the fourth aspect of the invention.
In general, through the above technical solutions conceived by the present invention, the following beneficial effects can be obtained:
1. According to the sensitivity analysis method for the structural topology optimization model provided by the first aspect of the invention, sensitivity analysis is carried out on the structural topology optimization model based on the partial derivative matrix expression of the objective function converted by the tensor integration method with respect to the optimization design variable and the partial derivative matrix expression of the volume constraint function converted by the tensor integration method with respect to the optimization design variable; according to the invention, based on tensor decomposition characteristics of the B spline basis function, a tensor integration method is adopted in advance to decompose the partial guide matrix H in the partial guide expression H multiplied by dC 1 and the partial guide expression H multiplied by dV 1 into the sub-weight matrixes in different parameter directions of the analysis grid, and then the partial guide expression is converted into the sub-weight matrixes in different parameter directions of the analysis grid to be multiplied by the corresponding partial guide matrix dC 1 and the partial guide matrix dV 1 for calculation, so that the equivalent expression of the B spline function value can be realized only by storing the value of the B spline basis function in each direction and not storing the value of the B spline basis function of all units.
2. In the iterative updating process of the topology optimization method of the engineering structure provided by the second aspect of the invention, the method provided by the first aspect of the invention is adopted to perform sensitivity analysis, and the calculation efficiency and the storage efficiency of each sensitivity analysis are obviously improved; when the number of B spline units in each direction of the discrete isogeometric analysis grid of the engineering structure analysis domain is equal and n, the space complexity of the isogeometric analysis weight matrix of the engineering structure can be reduced from O (n 3) to O (n), and more remarkable optimization is obtained.
3. According to the sensitivity analysis method of the structural topology optimization model provided by the third aspect of the invention, after the partial guide matrix dC 2 and the partial guide matrix dV 2 of the objective function and the volume constraint function relative to the optimization design variable are obtained, the partial guide matrix dC 2 of the objective function relative to the optimization design variable is subjected to one-step correction, the explicit filter weight matrix F in the sensitivity correction formula F multiplied by dC 2 is decomposed into the sub weight matrices in different directions of the analysis grid by adopting the tensor integration method in advance based on the decomposition characteristic of the tensor product, so that the sensitivity correction formula F multiplied by dC 2 is converted into the form of multiplying the sub weight matrix in different directions of the analysis grid by the partial guide matrix dC 2, and therefore, the method only needs to store the sub weight matrix in different directions of the analysis grid, and does not need to store all weight values in the explicit filter weight matrix F, and has higher storage efficiency and higher calculation speed.
4. In the iterative updating process of the topology optimization method of the engineering structure provided by the fourth aspect of the invention, the sensitivity analysis is performed by adopting the method provided by the third aspect of the invention, and the calculation efficiency and the storage efficiency of each step of sensitivity analysis are obviously improved.
Drawings
FIG. 1 is a diagram of a geometric analysis grid such as a 2X 2 size B-spline according to example 1 of the present invention;
FIG. 2 is a schematic diagram of IEN array of a geometric analysis grid such as B-spline provided in example 1 of the present invention; wherein, (a) is an IEN array schematic diagram of the geometric analysis grid such as B-spline in the x direction, and (B) is an IEN array schematic diagram of the geometric analysis grid such as B-spline in the y direction;
FIG. 3 is a flowchart of a topology optimization method of an engineering structure according to embodiment 2 of the present invention;
FIG. 4 is a schematic structural diagram of a filter according to embodiment 3 of the present invention, wherein (a) is a conventional filter with a filter radius of 2 based on the Euclidean distance of the center point of the unit, and (b) is an explicit filter with a filter radius of 2 based on the tensor product structure of the axial distance of the center point;
FIG. 5 is a flowchart of a topology optimization method of an engineering structure according to embodiment 4 of the present invention;
FIG. 6 is a schematic diagram of the comparison result of the time required for calculating the weight matrix for performing the isogeometric topological optimization based on the sensitivity analysis method of the implicit filter based on tensor product decomposition provided by the embodiment 7 of the present invention and the sensitivity analysis method based on the conventional implicit filter;
Fig. 7 is a schematic diagram of a comparison result of the memory occupied by the weight matrix when performing topological optimization of an isogeometric structure based on the sensitivity analysis method of the implicit filter based on tensor product decomposition provided by the embodiment 7 of the present invention and the sensitivity analysis method based on the conventional implicit filter;
FIG. 8 is a schematic diagram of the comparison result of the time required for calculating the weight matrix for performing the multi-resolution structure topology optimization based on the sensitivity analysis method of the explicit filter based on tensor product decomposition provided by the embodiment 8 of the present invention and the sensitivity analysis method based on the conventional explicit filter;
Fig. 9 is a schematic diagram of a comparison result between a weight matrix and a memory occupied by the weight matrix when performing topology optimization of a multi-resolution structure based on the sensitivity analysis method of the explicit filter based on tensor product decomposition provided by the embodiment 8 of the present invention and the sensitivity analysis method based on the conventional explicit filter.
Detailed Description
The present invention will be described in further detail with reference to the drawings and examples, in order to make the objects, technical solutions and advantages of the present invention more apparent. It should be understood that the specific embodiments described herein are for purposes of illustration only and are not intended to limit the scope of the invention. In addition, the technical features of the embodiments of the present invention described below may be combined with each other as long as they do not collide with each other.
Example 1,
The embodiment provides a sensitivity analysis method of a structural topology optimization model, which is based on an implicit filter technology; the structural topology optimization model is constructed by taking the sum of strain energy of all analysis grid cells in the engineering structural analysis grid as an objective function and the sum of material volumes of all analysis grid cells as a volume constraint function, and the optimization design variable is the material density of the B spline control point coordinate position of the analysis grid; the coordinate position of the B spline control point of the analysis grid is calculated based on CAD information of the engineering structure by combining with a B spline node vector insertion method.
Specifically, the sensitivity analysis method provided in the present embodiment includes:
Arranging and integrating the strain energy of each analysis grid unit of the engineering structure according to the positions of the strain energy in the analysis grid to obtain a strain energy matrix C of the analysis grid; calculating and analyzing the partial conductance of the strain energy matrix C of the grid with respect to the density value of the cell center point to obtain a partial conductance matrix dC 1 of the objective function with respect to the density value of the cell center point;
Arranging and integrating derivative values of the material volumes of all the analysis grid cells to the density of the central points of the analysis grid cells according to the positions of the derivative values in the analysis grid cells to obtain a partial derivative matrix dV 1 of a volume constraint function on the density values of the central points of the cells;
Partial derivative matrix expression of objective function transformed based on tensor integral solution for optimizing design variable Calculating to obtain a partial derivative value of the objective function with respect to each optimization design variable;
partial derivative matrix expression of volume constraint function converted based on tensor integral solution and related to optimization design variable Calculating to obtain the partial derivative value of the volume constraint function with respect to each optimization design variable;
Wherein H m is a sub-weight matrix of the analysis grid in the m-th parameter direction; m=1, 2, …, M; m is the number of parameter directions, and the value is 2 or 3; the number of lines of H m is the number of B spline functions in the mth parameter direction, and the number of columns is the number of analysis grid units in the mth parameter direction; the element H m (r, c) in the r-th row and c-th column of H m is the B-spline function value corresponding to the r-th B-spline function and c-th analysis grid unit in the m-th parameter direction.
Specifically, the method for acquiring the sub-weight matrix comprises the following steps: and analyzing the B spline node vectors in each parameter direction of the grid according to the engineering structure to obtain the B spline function value of the central point of the analysis grid unit in each parameter direction and the index of the B spline function in the parameter direction, thereby obtaining the sub-weight matrix in each parameter direction.
In order to further explain the sensitivity analysis method provided in this embodiment, the following details are given by taking a geometric analysis grid such as a2×2-sized B-spline as an example:
FIG. 1 is a schematic diagram of a geometric analysis grid such as a B-spline with a size of 2×2, wherein the geometric analysis grid such as the B-spline is generated based on CAD model data information of an engineering structure; the IEN array of the geometric grids such as the B-spline in each parameter direction (geometric parameters, namely the x direction and the y direction) shown in figure 2 is obtained by carrying out combined sequencing on the local index of the basis function of the geometric analysis grid such as the B-spline of the engineering structure and the unit numbers of all geometric analysis units in the geometric analysis grid such as the B-spline; wherein, (a) is IEN array schematic diagram of geometric analysis grid such as B spline in x direction; (b) An IEN array schematic diagram of a geometric analysis grid such as a B spline in the y direction; the number of geometric analysis units in the x direction, the y direction and the like is 2, and each B spline unit corresponds to 3B spline basis functions in each parameter direction, namely, corresponds to the local index of the 3B spline basis functions. Finally, based on IEN arrays in each direction, assembling the isogeometric unit data of the engineering structure by a sparse matrix assembly method; in this example, the row index of the final assembled sub-weight matrix corresponds to the position of the relevant B-spline function and the column index corresponds to the position of the cell. The final results obtained are as follows:
wherein H 1 is a sub-weight matrix of the analysis grid in the x-direction, and H 2 is a sub-weight matrix of the analysis grid in the y-direction.
After the sub-weight matrix is obtained, a partial derivative matrix of the objective function about the density value of the central point of the unit is further obtainedAnd then, a partial derivative matrix expression of the objective function converted based on the tensor integration method with respect to the optimization design variable is adopted, and the partial derivative matrix of the objective function with respect to the optimization design variable is calculated as follows:
it should be noted that, the method for obtaining the partial derivative matrix expression of the objective function transformed by the tensor integration method with respect to the optimization design variable and the partial derivative matrix expression of the volume constraint function transformed by the tensor integration method with respect to the optimization design variable includes:
Obtaining a partial derivative matrix expression H multiplied by dC 1 of the objective function on the basis of the chain derivation method and a partial derivative matrix expression H multiplied by dV 1 of the volume constraint function on the basis of the optimal design variable; h is a partial guide matrix of the density value of the cell center point relative to the optimal design variable, namely a cell center point B spline basis function value matrix of the analysis grid;
Decomposing the partial guide matrix H in the partial guide expression H multiplied by dC 1 into sub-weight matrixes in different parameter directions of the analysis grid by adopting a tensor integration method, thereby converting the partial guide expression H multiplied by dC 1 into a form of multiplying the sub-weight matrixes in different parameter directions of the analysis grid by the partial guide matrix dC 1, namely
The tensor integration method is adopted to decompose the partial guide matrix H in the partial guide expression H multiplied by dV 1 into the sub-weight matrix in the directions of different parameters of the analysis grid, so that the partial guide expression H multiplied by dV 1 is converted into the form of multiplying the sub-weight matrix in the directions of different parameters of the analysis grid with the partial guide matrix dV 1, namely
The present embodiment is based on tensor decomposition characteristics of B-spline basis functions, that is, there is redundancy in the B-spline basis function value matrix H, and knowing the values of the B-spline basis functions in each parameter direction, the matrix H can be restored, so that only the values of the B-spline basis functions in each direction need to be stored, and the values of the B-spline basis functions of all units in the storage space are not needed. The invention can realize the equivalent expression of the B spline function value, does not need to store the function value and the index of each unit, only needs to store the function value and the index of all units in each direction, has higher storage efficiency and higher calculation speed.
EXAMPLE 2,
The invention provides a topology optimization method of an engineering structure, which is a structure topology optimization method based on a decomposable implicit filter, as shown in figure 3, and comprises the following steps:
s11, initializing the Young' S modulus of elasticity of the materials of each analysis grid unit of the engineering structure according to the physical material volume constraint of the topological optimization model of the engineering structure;
S12, obtaining a stiffness matrix of the engineering structure based on the Young' S modulus of elasticity of the material of each analysis grid unit; based on the rigidity matrix of the obtained engineering structure and boundary conditions, calculating displacement vectors of each analysis grid cell, and further calculating strain energy of each analysis grid cell;
S13, constructing a structural topology optimization model by taking the sum of strain energy of each analysis grid unit as an objective function and the sum of material volumes of each analysis grid unit as a volume constraint function; the optimization design variable of the structural topology optimization model is the material density of the coordinate position of the B spline control point of the analysis grid;
S14, performing sensitivity analysis on the structural topology optimization model by adopting the sensitivity analysis method provided by the embodiment 1 of the invention to obtain the partial derivative value of the objective function and the volume constraint function on each optimization design variable;
S15, inputting the partial derivative value into an optimization solution operator to obtain updated values of all the optimization design variables, and updating the Young' S elastic modulus of the material of each analysis grid unit; the adopted optimization solver can be an optimization criterion method, a moving gradual approach line and the like;
S16, repeating the steps S12-S15 until the relative change value of the objective function of the adjacent iteration steps is smaller than a first preset threshold (the value is 10 -6 in the embodiment) or the maximum change value of the relative material density corresponding to the Young' S modulus of the material of the analysis grid unit is smaller than a second preset threshold (the value is 10 -2 in the embodiment).
The related technical solutions are the same as embodiment 1, and are not described here in detail.
EXAMPLE 3,
The embodiment provides a sensitivity analysis method of a structural topology optimization model, which is based on an explicit filter technology; comprising the following steps: the structural topology optimization model is constructed by taking the sum of strain energy of all analysis grid cells in the engineering structural analysis grid as an objective function and the sum of material volumes of all analysis grid cells as a volume constraint function, and the optimization design variable is the material density of the center point of the analysis grid cells;
the sensitivity analysis method comprises the following steps:
Arranging and integrating the strain energy of each analysis grid unit of the engineering structure according to the positions of the strain energy in the analysis grid to obtain a strain energy matrix C of the analysis grid; calculating and analyzing the partial conductance of the strain energy matrix C of the grid with respect to the optimal design variable to obtain a partial conductance matrix dC 2 of the objective function with respect to the optimal design variable;
Arranging and integrating the derivative values of the material volumes of each analysis grid unit to the density of the central points according to the positions of the material volumes of each analysis grid unit in the analysis grid to obtain a partial derivative matrix dV 2 of the volume constraint function on the optimization design variables, so as to obtain the partial derivative value of the volume constraint function on each optimization design variable; wherein, The partial derivative value of the volume constraint function on each optimization design variable is 1;
Using a sensitivity correction formula based on tensor integration solution after transformation Correcting the bias guide matrix dC 2 to obtain bias guide values of the corrected objective function with respect to each optimization design variable; updating the partial derivative value of the objective function with respect to each optimization design variable to the partial derivative value of the corrected objective function with respect to each optimization design variable;
Wherein F n is a sub-weight matrix of the analysis grid in the nth direction; n=1, 2, …, N; n is the number of the geometric dimension directions of the analysis grid, and the value is 2 or 3; the number of rows and columns of F n are the number of units in the nth direction of the analysis grid; element F n (p, q) of the p-th row and q-th column of F n is a distance weight value between the p-th cell and the q-th cell in the n-th direction.
Specifically, the method for acquiring the sub-weight matrix comprises the following steps:
calculating a sub-weight matrix in each direction according to the distance between the center points of the grid cells in each direction of the analysis grid of the engineering structure and the minimum filter radius;
Wherein, R min is the minimum filter radius of the filter,Is the distance in the direction between the p-th cell and the center point of the q-th cell in the n-th direction.
Taking N equal to 2 as an example, the center point position coordinate of the p-th cell is (i p,jp), and the center point position coordinate of the q-th cell is (i q,jq), thenThe distance in the x direction between the center points of the p-th cell and the q-th cell, i.e. |i p-iq |,Is the distance in the y-direction between the center points of the p-th cell and the q-th cell, i.e., |j p-jq |. The case where N is equal to 3 is similar to the case where N is equal to 2, and a description thereof will be omitted.
It should be noted that, the method for obtaining the sensitivity correction formula includes:
Decomposing the filter weight matrix F in the sensitivity correction formula F multiplied by dC 2 into sub-weight matrices in different directions of the analysis grid by adopting a tensor integration method, thereby converting the sensitivity correction formula F multiplied by dC 2 into a form of multiplying the sub-weight matrices in different directions of the analysis grid by a partial guide matrix dC 2, namely
Further, the filter weight matrix F is a weight matrix between unit center points, and is a Kronecker product of sub-weight matrices of all analysis grids in all directions, namely:
Taking a two-dimensional filter as an example, where N is 2, as shown in fig. 4, a schematic structure of the filter is shown, where (a) is a conventional filter with a filtering radius of 2 and based on the euclidean distance of the center point of the unit; (b) For an explicit filter based on a center point axial distance tensor product structure with a filter radius of 2, the filter is constructed as follows:
Wherein, The weight influence value representing the p-th design variable (i.e., the p-th unit) on the q-th design variable (i.e., the q-th unit) may be composed of sub-weight values in both directions, and thus may constitute an explicit filter based on a tensor product structure.
The present embodiment can decompose the explicit filter weight matrix into sub-weight matrices along each direction based on the decomposition characteristics of the tensor product. Compared with the traditional explicit filtering matrix, the method has the advantages that the storage efficiency and the calculation speed are improved remarkably.
EXAMPLE 4,
The topology optimization method of the engineering structure is a structure topology optimization method based on a decomposable explicit filter, and is also a multi-resolution structure topology optimization method, as shown in fig. 5, and comprises the following steps:
s21, initializing the Young' S modulus of elasticity of the materials of each analysis grid unit of the engineering structure according to the physical material volume constraint of the topological optimization model of the engineering structure;
S22, obtaining a stiffness matrix of the engineering structure based on the Young' S modulus of elasticity of the material of each analysis grid unit; based on the rigidity matrix of the obtained engineering structure and boundary conditions, calculating displacement vectors of each analysis grid cell, and further calculating strain energy of each analysis grid cell;
S23, constructing a structural topology optimization model by taking the sum of strain energy of each analysis grid unit as an objective function and the sum of material volumes of each analysis grid unit as a volume constraint function; the optimization design variable of the structural topology optimization model is the material density of the central point of the analysis grid unit;
S24, performing sensitivity analysis on the structural topology optimization model by adopting the sensitivity analysis method provided by the embodiment 3 of the invention to obtain the partial derivative value of the objective function and the volume constraint function on each optimization design variable;
S25, inputting the partial derivative value into an optimization solution operator to obtain updated values of all the optimization design variables, and updating the Young' S elastic modulus of the material of each analysis grid unit;
S26, repeating the steps S22-S25 until the maximum change value of the relative material density corresponding to the Young' S modulus of the material of the grid unit is smaller than a third preset threshold (the value is 10 -1 in the embodiment); evenly refining the analysis grid;
After the analysis grid is uniformly refined, the sub-weight matrix adopted in the next iteration process for sensitivity analysis is changed, and the sub-weight matrix is updated according to the center point distance of the refined grid and the number of units in each dimension direction; preferably, after the analysis grid is uniformly refined, the minimum filter radius r min in the next sensitivity analysis can be updated to k times of the current value of the minimum filter radius r min in combination with the refinement degree; preferably, k.epsilon.1.5, 2.
S27, repeating the step S26 until the grid refinement times are greater than a fourth preset threshold (the value is 3 in the embodiment).
The related technical solution is the same as embodiment 3, and will not be described here.
EXAMPLE 5,
A topology optimization method system of an engineering structure, comprising: a memory storing a computer program and a processor that when executing the computer program performs the topology optimization method provided in embodiment 2 or embodiment 4 of the present invention.
The related technical solutions are the same as embodiment 2 or embodiment 4, and are not described here again.
EXAMPLE 6,
A machine-readable storage medium storing machine-executable instructions that, when invoked and executed by a processor, cause the processor to implement one or more of the sensitivity analysis method provided by embodiment 1, the topology optimization method provided by embodiment 2, the sensitivity analysis method provided by embodiment 3, and the topology optimization method provided by embodiment 4 of the present invention.
The related technical solutions are the same as those of embodiment 1 to embodiment 4, and are not described here in detail.
EXAMPLE 7,
To further illustrate the effectiveness of the sensitivity analysis method based on the decomposable implicit filter provided in example 1 of the present invention, a numerical experiment was performed as follows:
The validity of the implicit filter sensitivity analysis method based on tensor product decomposition provided in embodiment 1 of the present invention was verified in a computer with hardware conditions Xeon (R) Gold 5120CPU@2.20GHz 2.19GHz intel core and 64GB RAM and software environment consisting of Windows 10 operating system and MATLAB R2021 a. The sensitivity analysis method provided by the invention and the traditional sensitivity analysis method based on the implicit filter are respectively adopted for comparison in the isogeometric analysis, wherein the B spline function order used for expressing the density distribution of the design domain material is 10. The experimental results are shown in fig. 6 and 7: fig. 6 is a schematic diagram of a comparison result of time required for calculating a weight matrix based on a tensor product decomposition-based implicit filter sensitivity analysis method provided by the present invention and a traditional implicit filter-based sensitivity analysis method to perform isogeometric topological optimization (the abscissa is resolution of an isogeometric analysis grid, and the ordinate is time required for performing sensitivity analysis to calculate the weight matrix); fig. 7 is a schematic diagram of a comparison result of a memory occupied by a weight matrix when performing topological optimization of an isogeometric structure based on the sensitivity analysis method of the implicit filter based on tensor product decomposition provided by the invention and the conventional sensitivity analysis method based on the implicit filter (the abscissa is the resolution of an isogeometric analysis grid, and the ordinate is the memory occupied by the weight matrix when performing sensitivity analysis); as can be seen from fig. 6, when the decomposable implicit filter provided by the present invention is substituted for the conventional implicit filter, the ratio of the time required for calculating the weight matrix by the conventional method to the time required for calculating the weight matrix by embodiment 1 of the present invention can reach 5300 at the maximum. Meanwhile, as can be seen from fig. 7, the maximum value of the ratio of the weight matrix of the conventional implicit filter to the weight matrix of the decomposable implicit filter provided by the present invention is 4976. Therefore, the memory efficiency and the preprocessing efficiency of the sensitivity analysis method based on the decomposable implicit filter provided by the invention are obviously improved.
EXAMPLE 8,
To further illustrate the effectiveness of the sensitivity analysis method based on the decomposable explicit filter provided in embodiment 3 of the present invention. The following numerical experiments were performed in the same hardware and software environment as in example 7:
The explicit filter based on tensor product decomposition provided by the present invention and the conventional explicit filter are used for comparison, respectively, wherein the initial filter radius value in each filter is 8. The experimental results are shown in fig. 8 and 9: FIG. 8 is a schematic diagram of a comparison result of time required for calculating a weight matrix based on a tensor product decomposition-based explicit filter sensitivity analysis method provided by the present invention and a conventional explicit filter-based sensitivity analysis method for performing multi-resolution structure topology optimization (the abscissa is the resolution of a grid, and the ordinate is the time required for performing sensitivity analysis) of calculating the weight matrix; fig. 9 is a schematic diagram of a comparison result of a memory occupied by a weight matrix when performing topology optimization of a multi-resolution structure based on a sensitivity analysis method based on tensor product decomposition provided by the present invention and a conventional sensitivity analysis method based on an explicit filter (an abscissa is a resolution of a grid, and an ordinate is a memory occupied by the weight matrix when performing sensitivity analysis); as can be seen from fig. 8, when the explicit filter based on tensor product decomposition provided by the present invention is replaced with the conventional explicit filter, the maximum value of the ratio of the time required for calculating the weight matrix by the conventional method to the time required for calculating the weight matrix by embodiment 3 of the present invention is 2217.1. Meanwhile, as can be seen from fig. 9, the maximum value of the ratio of the weight matrix of the conventional explicit filter to the weight matrix of the decomposable explicit filter provided by the present invention is 727.5. Therefore, the preprocessing efficiency and the memory efficiency of the explicit filter based on tensor product decomposition provided by the invention are improved remarkably.
In summary, embodiment 1 of the present invention decomposes the conventional multi-dimensional implicit filter weight matrix into several single-dimensional sub-weight matrices based on the characteristics of B-spline function tensor product decomposition, thereby greatly reducing the storage space and computation time of the weight matrix. In the embodiment 3 of the invention, the sub-weight matrix in the direction of each parameter unit is obtained based on the number of grids in each direction of grids divided by the engineering structure and the minimum filter radius. The sub-weight matrix in each direction can obtain an integral matrix by a Kronecker matrix product method, so that the equivalent expression of the weight matrix is realized. Based on the method, the sensitivity filtering weight matrix of each unit is not required to be calculated, only the sub weight matrix in each dimension direction is required to be calculated, the storage efficiency is high, and the calculation efficiency is high.
It will be readily appreciated by those skilled in the art that the foregoing description is merely a preferred embodiment of the invention and is not intended to limit the invention, but any modifications, equivalents, improvements or alternatives falling within the spirit and principles of the invention are intended to be included within the scope of the invention.
Claims (10)
1. The sensitivity analysis method of the structural topological optimization model is characterized in that the structural topological optimization model is constructed by taking the sum of strain energy of all analysis grid cells in an engineering structural analysis grid as an objective function and the sum of material volumes of all analysis grid cells as a volume constraint function, and the optimal design variable is the material density of the B spline control point coordinate position of the analysis grid;
The sensitivity analysis method comprises the following steps:
Arranging and integrating the strain energy of each analysis grid unit of the engineering structure according to the positions of the strain energy in the analysis grid to obtain a strain energy matrix C of the analysis grid; calculating and analyzing the partial conductance of the strain energy matrix C of the grid with respect to the density value of the cell center point to obtain a partial conductance matrix dC 1 of the objective function with respect to the density value of the cell center point;
Arranging and integrating derivative values of the material volumes of all the analysis grid cells to the density of the central points of the analysis grid cells according to the positions of the derivative values in the analysis grid cells to obtain a partial derivative matrix dV 1 of a volume constraint function on the density values of the central points of the cells;
Partial derivative matrix expression of objective function transformed based on tensor integral solution for optimizing design variable Calculating to obtain a partial derivative value of the objective function with respect to each optimization design variable;
partial derivative matrix expression of volume constraint function converted based on tensor integral solution and related to optimization design variable Calculating to obtain the partial derivative value of the volume constraint function with respect to each optimization design variable;
Wherein H m is a sub-weight matrix of the analysis grid in the m-th parameter direction; m=1, 2, …, M; m is the number of parameter directions, and the value is 2 or 3; the number of lines of H m is the number of B spline functions in the mth parameter direction, and the number of columns is the number of analysis grid units in the mth parameter direction; the element H m (r, c) in the r-th row and c-th column of H m is the B-spline function value corresponding to the r-th B-spline function and c-th analysis grid unit in the m-th parameter direction.
2. The sensitivity analysis method according to claim 1, wherein the method for obtaining the partial conductance matrix expression of the objective function transformed by the tensor integration method with respect to the optimal design variable and the partial conductance matrix expression of the volume constraint function transformed by the tensor integration method with respect to the optimal design variable comprises:
Obtaining a partial derivative matrix expression H multiplied by dC 1 of the objective function on the basis of the chain derivation method and a partial derivative matrix expression H multiplied by dV 1 of the volume constraint function on the basis of the optimal design variable; h is a partial guide matrix of the density value of the cell center point relative to the optimal design variable, namely a cell center point B spline basis function value matrix of the analysis grid;
Decomposing a partial guide matrix H in the partial guide expression H multiplied by dC 1 into sub-weight matrixes in different parameter directions of an analysis grid by adopting a tensor integration method, so as to convert the partial guide expression H multiplied by dC 1 into a form of multiplying the sub-weight matrixes in different parameter directions of the analysis grid by the partial guide matrix dC 1, namely
Decomposing the partial guide matrix H in the partial guide expression H multiplied by dV 1 into sub-weight matrixes in different parameter directions of the analysis grid by adopting a tensor integration method, so as to convert the partial guide expression H multiplied by the partial guide matrix dV 1 into the form that the sub-weight matrixes in different parameter directions of the analysis grid are multiplied by the partial guide matrix dV 1, namely
3. The sensitivity analysis method according to claim 1 or 2, wherein the method of acquiring the sub-weight matrix includes: and analyzing the B spline node vectors in each parameter direction of the grid according to the engineering structure to obtain the B spline function value of the central point of the analysis grid unit in each parameter direction and the index of the B spline function in the parameter direction, thereby obtaining the sub-weight matrix in each parameter direction.
4. A topology optimization method for an engineering structure, comprising the steps of:
s11, initializing the Young' S modulus of elasticity of the materials of each analysis grid unit of the engineering structure according to the physical material volume constraint of the topological optimization model of the engineering structure;
S12, obtaining a stiffness matrix of the engineering structure based on the Young' S modulus of elasticity of the material of each analysis grid unit; based on the stiffness matrix of the engineering structure and boundary conditions, calculating displacement vectors of each analysis grid cell, and further calculating strain energy of each analysis grid cell;
S13, constructing a structural topology optimization model by taking the sum of strain energy of each analysis grid unit as an objective function and the sum of material volumes of each analysis grid unit as a volume constraint function; the optimization design variable of the structural topology optimization model is the material density of the coordinate position of the B spline control point of the analysis grid;
s14, performing sensitivity analysis on the structural topology optimization model by adopting the sensitivity analysis method of any one of claims 1-3 to obtain the partial derivative value of the objective function and the volume constraint function on each optimization design variable;
S15, inputting the partial derivative value into an optimization solving operator to obtain updated values of all the optimization design variables, and updating the Young' S elastic modulus of the material of each analysis grid unit;
S16, repeating the steps S12-S15 until the relative change value of the objective function of the adjacent iteration steps is smaller than a first preset threshold or the maximum change value of the relative material density corresponding to the Young' S elastic modulus of the material of the analysis grid unit is smaller than a second preset threshold.
5. The sensitivity analysis method of the structural topological optimization model is characterized in that the structural topological optimization model is constructed by taking the sum of strain energy of all analysis grid cells in an engineering structural analysis grid as an objective function and the sum of material volumes of all analysis grid cells as a volume constraint function, and the optimal design variable is the material density of the center point of the analysis grid cells;
The sensitivity analysis method comprises the following steps:
Arranging and integrating the strain energy of each analysis grid unit of the engineering structure according to the positions of the strain energy in the analysis grid to obtain a strain energy matrix C of the analysis grid; calculating and analyzing the partial conductance of the strain energy matrix C of the grid with respect to the optimal design variable to obtain a partial conductance matrix dC 2 of the objective function with respect to the optimal design variable;
Arranging and integrating the derivative values of the material volumes of each analysis grid unit to the density of the central points according to the positions of the material volumes of each analysis grid unit in the analysis grid to obtain a partial derivative matrix dV 2 of the volume constraint function on the optimization design variables, so as to obtain the partial derivative value of the volume constraint function on each optimization design variable; wherein,
Using a sensitivity correction formula based on tensor integration solution after transformationCorrecting the partial guide matrix dC 2 to obtain the partial guide value of the corrected objective function with respect to each optimal design variable;
Wherein F n is a sub-weight matrix of the analysis grid in the nth direction; n=1, 2, …, N; n is the number of the geometric dimension directions of the analysis grid, and the value is 2 or 3; the number of rows and columns of F n are the number of units in the nth direction of the analysis grid; element F n (p, q) of the p-th row and q-th column of F n is a distance weight value between the p-th cell and the q-th cell in the n-th direction.
6. The sensitivity analysis method according to claim 5, wherein the method for obtaining the sensitivity correction formula includes:
Decomposing the filter weight matrix F in the sensitivity correction formula F x dC 2 into sub-weight matrices in different directions of the analysis grid by adopting a tensor integration method, thereby converting the sensitivity correction formula F x dC 2 into a form of multiplying the sub-weight matrices in different directions of the analysis grid by a partial conductance matrix dC 2, namely
The filter weight matrix F is a weight matrix between unit center points, specifically, a Kronecker product of sub-weight matrices of all analysis grids in each direction, namely:
7. The sensitivity analysis method according to claim 5 or 6, wherein the method for acquiring the sub-weight matrix comprises:
calculating a sub-weight matrix in each direction according to the distance between the center points of the grid cells in each direction of the analysis grid of the engineering structure and the minimum filter radius;
Wherein, R min is the minimum filter radius of the filter,Is the distance in the direction between the p-th cell and the center point of the q-th cell in the n-th direction.
8. A topology optimization method for an engineering structure, comprising the steps of:
s21, initializing the Young' S modulus of elasticity of the materials of each analysis grid unit of the engineering structure according to the physical material volume constraint of the topological optimization model of the engineering structure;
S22, obtaining a stiffness matrix of the engineering structure based on the Young' S modulus of elasticity of the material of each analysis grid unit; based on the rigidity matrix of the obtained engineering structure and boundary conditions, calculating displacement vectors of each analysis grid cell, and further calculating strain energy of each analysis grid cell;
S23, constructing a structural topology optimization model by taking the sum of strain energy of each analysis grid unit as an objective function and the sum of material volumes of each analysis grid unit as a volume constraint function; the optimization design variable of the structural topology optimization model is the material density of the center point of the analysis grid unit;
S24, performing sensitivity analysis on the structural topology optimization model by adopting the sensitivity analysis method of any one of claims 5-7 to obtain the partial derivative value of the objective function and the volume constraint function on each optimization design variable;
S25, inputting the partial derivative value into an optimization solution operator to obtain updated values of all the optimization design variables, and updating the Young' S elastic modulus of the material of each analysis grid unit;
s26, repeating the steps S22-S25 until the maximum change value of the relative material density corresponding to the Young' S modulus of the material of the grid unit is smaller than a third preset threshold value; evenly refining the analysis grid;
s27, repeating the step S26 until the grid refinement times are greater than a fourth preset threshold.
9. A topology optimization system for an engineering structure, comprising: a memory storing a computer program and a processor executing the topology optimization method of claim 4 or 8.
10. A machine-readable storage medium storing machine-executable instructions that, when invoked and executed by a processor, cause the processor to implement one or more of the sensitivity analysis method of any one of claims 1-3, the topology optimization method of claim 4, the sensitivity analysis method of any one of claims 5-7, and the topology optimization method of claim 8.
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