CN114494642B - Stress gradient-based adaptive T-spline FCM structure optimization design method - Google Patents
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Abstract
The invention relates to a self-adaptive T-spline FCM structure optimization design method based on stress gradient, which is used for solving the problem of computational redundancy caused by incapability of local subdivision in the existing B-spline FCM structure optimization method. Firstly, introducing a T spline into a finite cell method, and using the T spline as an interpolation shape function of a physical field to be solved; then, deducing analysis expression of the stress gradient by using high-order continuity of the T spline, calculating to obtain the stress gradient, and determining cells needing to be subdivided according to a set threshold value; then, generating a T grid by adopting a quadtree subdivision strategy, and ensuring that the obtained T spline function is independent of linearity; and finally, combining the method with an optimization means to optimally design the structure. The method can realize self-adaptive local subdivision in the optimization process, can ensure that the calculation accuracy meets the requirement, saves a large amount of calculation amount, greatly improves the calculation efficiency, and has important practical value for popularization and application of the optimization method.
Description
Technical Field
The invention relates to a structural optimization design method, in particular to a structural optimization design method based on a self-adaptive T spline FCM.
Background
The literature Cai S, zhang W, zhu J, et al, stress constrained shape and topology optimization with fixed mesh, AB-spline finite cell method combined with level set function, J, computer Methods in Applied Mechanics and Engineering,2014,278, 361-387 discloses a structural optimization design method based on a B-spline FCM, the method is based on a finite cell method combined with a level set method, and meanwhile, a high-order continuous B-spline is used as a shape function of an interpolation displacement field, so that grid updating is avoided continuously in an optimization process, a high-order continuous displacement field can be obtained, the precision of the stress field obtained through calculation is ensured, stress constraint can be met, and a final obtained optimization result can play a guiding role in actual engineering design.
Although the method can obtain high-precision analysis results and clear optimization results, the method is limited by the tensor product construction form of the B-spline, local subdivision cannot be realized, and in order to ensure the calculation precision in the optimization process, the initial cell division must be ensured to be fine enough, so that the situation of calculation redundancy can be inevitably generated in the optimization process, the calculation amount is increased, and the calculation efficiency is reduced. This is especially true when dealing with practical complex engineering problems, which is not conducive to popularization and use in engineering practice.
Disclosure of Invention
Technical problem to be solved
In order to solve the problem of computational redundancy caused by the fact that the existing B-spline FCM structure optimization method cannot be partially subdivided, the invention provides a structure optimization design method based on a self-adaptive T-spline FCM.
Technical proposal
A self-adaptive T spline FCM structure optimization design method based on stress gradient is characterized in that: establishing a T spline finite cell analysis method, introducing the T spline into the finite cell method, and weighting the T spline shape function by combining the existing shape function weighting method to realize accurate application of homogeneous Dirichlet boundary conditions; the method comprises the following steps:
step 1: deducing an analysis expression of the stress gradient by utilizing the high-order continuity of the T spline function, and judging cells needing to be subdivided according to a set threshold value;
modeling Von-Mises stress gradientsIs used as a basis for determination; due to sigma von By stress vectors->Calculated, thus, nextEmphasis is placed on the derivative calculation of σ:
wherein D represents an elastic matrix,the strain matrix after the weight correction is adopted, and U refers to a node displacement vector;
is not difficult to obtain according to the definition of the strain matrixIs the derivative of:
wherein w is a weight function, and x y is a physical domain coordinate;
the specific elements in the above formula are calculated as follows:
due to the shape function T i Is defined in the parameter space, and the derivatives in the equation are for x and y in the physical domain, so a second order Jacobian matrix for both spaces needs to be introduced:
wherein, zeta eta is the coordinate of the parameter domain;
in addition, since the method is based on regular grid division of rectangular areas, the second-order Jacobian matrix of the above formula can be further simplified into a diagonal matrix:wherein the method comprises the steps ofL of (2) x ,l y The dimensions of the cells in the x and y directions are shown, respectively;
according to the above formulas, the higher-order continuity of the cubic T spline function used by the method can be obtained by analysisAnd->In the case of obtaining sigma von After the derivative of (a), a threshold is first givenAs a criterion for determining whether the internal stress variation of the cell is severe; once it is determined that the maximum value of the stress gradient inside certain cells is greater than a given threshold +.>Then the cells are recorded and determined to be cells to be subdivided;
step 2: after determining the cells to be subdivided, implementing subdivision of the cells by adopting a quadtree subdivision strategy so as to generate a T grid;
step 3: and according to the optimization problem defined in advance, adopting an MMA optimization algorithm to solve the optimization problem, and obtaining a final optimization result.
Advantageous effects
According to the self-adaptive T-spline FCM structure optimization design method based on the stress gradient, the T-spline is introduced into the FCM, so that local subdivision of an analysis grid can be realized, an analytical expression of the stress gradient is deduced, and a grid division criterion capable of guaranteeing linearity independence of a generated T-spline function is provided. The method can realize self-adaptive local subdivision in the optimization process, can greatly improve the calculation efficiency and ensures the calculation precision. Compared with the design method in the background art, the T spline interpolation function can realize local subdivision, so that the method can use coarser grids to calculate in the initial stage of calculation, the self-adaptive local subdivision in the optimization iteration process can effectively ensure the calculation accuracy, and meanwhile, the method can overcome the calculation redundancy, save a large amount of calculation amount and greatly improve the calculation efficiency. In addition, the method provided by the invention can be simultaneously suitable for shape and topology optimization, can also keep the high accuracy of the calculation of the finite cell method and the definition of the boundary of the optimization result of the level set method, and has stronger engineering practical value.
Drawings
The drawings are only for purposes of illustrating particular embodiments and are not to be construed as limiting the invention, like reference numerals being used to refer to like parts throughout the several views.
FIG. 1 is a schematic diagram of model geometry and boundary conditions in an embodiment of the present invention.
FIG. 2 is a schematic diagram of a T-grid obtained by partial subdivision in an embodiment of the present invention.
FIG. 3 is a comparison of the calculation accuracy with the background art in the embodiment of the present invention.
Fig. 4 is the final design result in an embodiment of the present invention.
Fig. 5 is a flow chart of the present invention.
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, 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.
Referring to fig. 1, the shape optimization of the tripod structure is taken as an example, and the young's modulus of the model is set to 207.4MPa and poisson's ratio to 0.3. The objective of this optimization is that the volume of the structure is minimum, the constraint is that the maximum Von Mises stress does not exceed 800MPa, and the optimal design variables are given in fig. 1. The specific method comprises the following steps:
1. first, based on the B-spline FCM of the background art, mechanical property analysis is performed on the original structure by using an initial uniform coarse mesh with 20×35 units, and the maximum Von Mises stress at this time is 374.82MPa, as shown in fig. 3 (a). The maximum Von Mises stress was calculated to be 413.33MPa by force analysis using the 40 x 70 unit encrypted grid in the background literature, as shown in fig. 3 (b). Comparing these two results is not difficult to see: when the grid is too thick, an analysis result with enough precision cannot be obtained, which is unfavorable for obtaining a reasonable optimization result; because of the tensor product nature of B-splines, a partially encrypted mesh cannot be achieved, and a large number of computations (four times the number of computation units as the initial mesh) are added to obtain a result of sufficient accuracy. The invention establishes a T spline finite cell analysis method, introduces the T spline into the finite cell analysis method, combines the existing shape function weighting method to weight the T spline shape function to realize accurate application of homogeneous Dirichlet boundary conditions, and can realize the improvement of the calculation accuracy of key parts in structural analysis by virtue of the local subdivision characteristic of the T spline.
2. The high-order continuity of the T spline function can be utilized to deduce an analytical expression of the stress gradient, and the cell needing to be subdivided is judged according to a set threshold value. Von-Mises stress gradient mode in the present methodIs used as a basis for determination. In addition, due to sigma von By stress vectors->Calculated, and therefore focused next on the derivative calculation of σ:
wherein D represents an elastic matrix,is the strain matrix after the weight correction, and U refers to the node displacement vector.
Is not difficult to obtain according to the definition of the strain matrixIs the derivative of:
wherein w is a weight function, and x y is a physical domain coordinate;
the calculation form of the specific elements in the above formula is not difficult to obtain by a chain derivation rule, and is as follows:
due to the shape function T i Is defined in the parameter space, and the derivatives in the equation are for x and y in the physical domain, so a second order Jacobian matrix for both spaces needs to be introduced:
wherein, zeta eta is the coordinate of the parameter domain;
in addition, since the method is based on regular grid division of rectangular areas, the second-order Jacobian matrix of the above formula can be further simplified into a diagonal matrix:wherein l is x ,l y The dimensions of the cells in the x and y directions are shown, respectively.
According to the above formulas, the higher-order continuity of the cubic T spline function used by the method can be obtained by analysisAnd->In the case of obtaining sigma von After the derivative of (a) we first have to give a threshold +.>As a criterion for determining whether the internal stress of the cell is severely changed. Once it is determined that the maximum value of the stress gradient inside certain cells is greater than a given threshold +.>Then the cells are recorded and determined to be cells that need subdivision.
3. After determining the cells to be subdivided, a simple and direct quadtree subdivision strategy is adopted to implement subdivision of the cells so as to generate a T-grid, which can ensure that the corresponding T-spline functions are linearly independent, as shown in fig. 3, and the corresponding structure Von Mises stress distribution is shown in fig. 2 (c). At this time, it can be obtained that the number of cells of the T-grid is increased by only 10% with respect to the initial grid, and the corresponding maximum Von Mises stress result has an error of only 0.5% with respect to the reference value.
4. In the shape optimization process, along with the structure evolution, the T grid needs to be updated in each iteration, so that the locally refined position is consistent with the larger part of the currently calculated stress gradient module. The generation of the T-grid depends on a prior analysis calculation, in order to avoid such a pre-analysis calculation in each iteration, the present invention uses the T-grid generated in the previous iteration to calculate the stress gradient. And updating design parameters by using a MMA optimization algorithm based on gradient, inputting the updated variables into the next iteration, and repeating the steps until the optimization process converges.
The method of the embodiment is iterated in 40 steps, and the optimization process converges to obtain a final optimization design result. Initial structural volume of 35000mm 3 The optimized structure volume is 19000mm 3 The maximum stress at this time just meets the set upper limit. Compared with the design method in the background art, the design result of the invention is very similar, however, the number of the limited cells is reduced by more than 70% in the whole optimization process.The result shows that the T-spline FCM shape optimization method provided by the invention has very remarkable improvement on the calculation efficiency on the premise of ensuring the accuracy, and has stronger engineering application prospect.
While the invention has been described with reference to certain preferred embodiments, it will be understood by those skilled in the art that various changes and substitutions of equivalents may be made without departing from the spirit and scope of the invention.
Claims (1)
1. A self-adaptive T spline FCM structure optimization design method based on stress gradient is characterized in that: establishing a T spline finite cell analysis method, introducing the T spline into the finite cell method, and weighting the T spline shape function by combining the existing shape function weighting method to realize accurate application of homogeneous Dirichlet boundary conditions; the method comprises the following steps:
step 1: deducing an analysis expression of the stress gradient by utilizing the high-order continuity of the T spline function, and judging cells needing to be subdivided according to a set threshold value;
modeling Von-Mises stress gradientsIs used as a basis for determination; due to sigma von By stress vectors->Calculated, and therefore focused next on the derivative calculation of σ:
wherein D represents an elastic matrix,is the strain moment after the weighted correctionAn array, U refers to a node displacement vector;
is not difficult to obtain according to the definition of the strain matrixIs the derivative of:
wherein w is a weight function, and x y is a physical domain coordinate;
the specific elements in the above formula are calculated as follows:
due to the shape function T i Is defined in the parameter space, and the derivatives in the equation are for x and y in the physical domain, so a second order Jacobian matrix for both spaces needs to be introduced:
wherein, zeta eta is the coordinate of the parameter domain;
in addition, since the method is based on regular grid division of rectangular areas, the second-order Jacobian matrix of the above formula can be further simplified into a diagonal matrix:wherein l is x ,l y The dimensions of the cells in the x and y directions are shown, respectively;
according to the above formulas, the higher-order continuity of the cubic T spline function used by the method can be obtained by analysisAnd->In the case of obtaining sigma von After the derivative of (a) a threshold value is first given>As a criterion for determining whether the internal stress variation of the cell is severe; once it is determined that the maximum value of the stress gradient inside certain cells is greater than a given threshold +.>Then the cells are recorded and determined to be cells to be subdivided;
step 2: after determining the cells to be subdivided, implementing subdivision of the cells by adopting a quadtree subdivision strategy so as to generate a T grid;
step 3: and according to the optimization problem defined in advance, adopting an MMA optimization algorithm to solve the optimization problem, and obtaining a final optimization result.
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| CN111832116A (en) * | 2020-06-05 | 2020-10-27 | 中国科学院力学研究所 | Lattice sandwich plate damage identification method based on dynamic characteristics and deep learning |
| CN113434921A (en) * | 2021-07-05 | 2021-09-24 | 西安交通大学 | Structure equal-geometry topological optimization method considering mesoscale effect |
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| CN111832116A (en) * | 2020-06-05 | 2020-10-27 | 中国科学院力学研究所 | Lattice sandwich plate damage identification method based on dynamic characteristics and deep learning |
| CN113434921A (en) * | 2021-07-05 | 2021-09-24 | 西安交通大学 | Structure equal-geometry topological optimization method considering mesoscale effect |
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| Stress constrained shape and topology optimization with fixed mesh: A B-spline finite cell method combined with level set function;Shouyu Cai, Weihong Zhang， Jihong Zhu，Tong Gao；;《 Computer Methods in Applied Mechanics and Engineering》;20140815;第361-387页 * |
| 基于适合分析T样条的高阶数值流形方法;刘登学;张友良;刘高敏;;力学学报;20170131(第01期);第212-222 页 * |
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