CN114494642A - Stress gradient-based adaptive T spline FCM structure optimization design method - Google Patents

Stress gradient-based adaptive T spline FCM structure optimization design method Download PDF

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CN114494642A
CN114494642A CN202210014504.3A CN202210014504A CN114494642A CN 114494642 A CN114494642 A CN 114494642A CN 202210014504 A CN202210014504 A CN 202210014504A CN 114494642 A CN114494642 A CN 114494642A
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陈亮
张卫红
孟亮
酒丽朋
丰圣起
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Northwestern Polytechnical University
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Abstract

The invention relates to a stress gradient-based adaptive T spline FCM structure optimization design method, which is used for overcoming the problem of calculation redundancy caused by the fact that the existing B spline FCM structure optimization method cannot be locally subdivided. The technical scheme is that firstly, a T spline is introduced into a finite cell method, and the T spline is used as an interpolation shape function of a physical field to be solved; thirdly, deducing analytical expression of the stress gradient by using the high-order continuity of the T spline, calculating to obtain the stress gradient, and determining the cell elements 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-shaped functions are linearly independent; and finally, optimally designing the structure by combining the method with an optimization means. The method can realize self-adaptive local subdivision in the optimization process, can save a large amount of calculated amount while ensuring that the calculation precision meets the requirement, greatly improves the calculation efficiency, and has important practical value for popularization and application of the optimization method.

Description

Stress gradient-based adaptive T spline FCM structure optimization design method
Technical Field
The invention relates to a structure optimization design method, in particular to a structure optimization design method based on self-adaptive T spline FCM.
Background
A structural optimization design method based on B-spline FCM is disclosed in the documents ' Cai S, Zhang W, Zhu J, et al, stress constrained shape and optimization with fixed mesh ' AB-spline fine cell coordinated with level set function [ J ]. Computer Methods in Applied Mechanics and Engineering,2014,278:361-387 ', and is based on a finite cell method combined with a level set method, and simultaneously 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 the optimization process, a high-order continuous displacement field can be obtained, and the precision of the stress field obtained by calculation is ensured, so that stress constraint can be met, and the final optimization result can play a guiding role in the actual Engineering design.
Although the method described in the document can obtain a high-precision analysis result and a clear optimization result, the method is limited by a tensor product structural form of a B-spline, local subdivision cannot be achieved, 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 inevitably occurs in the optimization process, the calculation amount is increased, and the calculation efficiency is reduced. This is especially prominent when dealing with the actual complex engineering problem, and is not conducive to the 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 locally subdivided, the invention provides a structure optimization design method based on an adaptive T spline FCM.
Technical scheme
A stress gradient-based adaptive T spline FCM structure optimization design method is characterized by comprising the following steps: establishing a T spline finite cell element analysis method, introducing a T spline into the finite cell element method, and weighting a T spline shape function by combining an existing shape function weighting method to realize accurate application of homogeneous Dirichlet boundary conditions; the method comprises the following steps:
step 1: deducing an analytical expression of the stress gradient by using the high-order continuity of the T-spline function, and judging the cell elements needing to be subdivided according to a set threshold;
modeling the Von-Mises stress gradient
Figure BDA0003458110840000021
Is used as a basis for determination; due to sigmavonBy stress vectors
Figure BDA0003458110840000022
Calculated, so the following focus is on the derivative calculation of σ:
Figure BDA0003458110840000023
in the formula, D represents an elastic matrix,
Figure BDA0003458110840000024
is the strain matrix after weighting correction, and U refers to the node displacement vector;
is not difficult to obtain according to the definition of the strain matrix
Figure BDA0003458110840000025
Derivative of (a):
Figure BDA0003458110840000026
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:
Figure BDA0003458110840000031
due to the shape function TiIs defined in the parameter space and the derivatives in the formula are with respect to x and y in the physical domain, so it is necessary to introduce a second order Jacobian matrix with respect to two spaces:
Figure BDA0003458110840000032
wherein ξ η is the coordinate of the parameter domain;
in addition, because the method carries out regular grid division based on the rectangular region, the second-order Jacobian matrix of the above formula can be further simplified into a diagonal matrix:
Figure BDA0003458110840000033
wherein lx,lyThe dimensions in the x and y directions of the cell are shown;
according to the formulas, the high-order continuity of the cubic T-spline function used by the method can be obtained analytically
Figure BDA0003458110840000034
And
Figure BDA0003458110840000035
in addition, σ is obtainedvonAfter the derivative, a threshold value is first specified
Figure BDA0003458110840000036
To be used as a standard for judging whether the internal stress of the cell element changes violently; upon determining that the maximum value of the stress gradient within some cells is greater than a given threshold
Figure BDA0003458110840000037
Then these cells are recorded and determined as the cells to be subdivided;
step 2: after determining the cell to be subdivided, adopting a quadtree subdivision strategy to realize subdivision of the cell so as to generate a T grid;
and 3, step 3: and solving the optimization problem by adopting an MMA optimization algorithm according to the predefined optimization problem to obtain a final optimization result.
Advantageous effects
The invention provides a stress gradient-based adaptive T spline FCM structure optimization design method, which can realize local subdivision of an analysis grid by introducing a T spline into an FCM, deduces an analytical expression of a stress gradient and provides a grid division criterion capable of ensuring the linear independence of a generated T spline-shaped function. The method can realize self-adaptive local subdivision in the optimization process, can greatly improve the calculation efficiency and ensure the calculation precision. Compared with the design method in the background art, the T spline interpolation shape function can realize local subdivision, so that the method can use a thicker grid for calculation in the initial stage of calculation, and the adaptive local subdivision in the optimization iteration process can effectively ensure the calculation precision, and simultaneously 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 finite cell method calculation and the clearness of the optimization result boundary of the level set method, and has strong engineering practical value.
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The drawings are only for purposes of illustrating particular embodiments and are not to be construed as limiting the invention, wherein like reference numerals are used to designate like parts throughout.
FIG. 1 is a diagram illustrating the geometry and boundary conditions of a model according to an embodiment of the present invention.
Fig. 2 is a schematic diagram of a T-grid obtained through local subdivision in the embodiment of the present invention.
Fig. 3 is a comparison of the calculation accuracy in the embodiment of the present invention with the background art.
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
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention. In addition, the technical features involved in the respective embodiments of the present invention described below may be combined with each other as long as they do not conflict with each other.
Referring to fig. 1, the shape optimization of the triangular frame structure is taken as an example, and the model is set to have a young's modulus of 207.4MPa and a poisson's ratio of 0.3. The optimization aims at the minimum volume of the structure, the constraint is that the maximum Von Mises stress does not exceed 800MPa, and the optimization design variables are given in figure 1. The specific method comprises the following steps:
1. first, based on B-spline FCM of the background art, mechanical property analysis is performed on an original structure by using an initial uniform coarse grid of 20 × 35 units, and the maximum Von Mises stress at this time is obtained to be 374.82MPa, as shown in fig. 3 (a). The maximum Von Mises stress was calculated to be 413.33MPa using a 40 x 70 cell dense grid as shown in fig. 3(b) for stress analysis. Comparing these two results makes it easy to see that: when the grid is too thick, an analysis result with enough precision cannot be obtained, which is not favorable for obtaining a reasonable optimization result; due to the tensor product property of B-splines, local encryption of the grid is not possible, and obtaining a result with sufficient accuracy increases a large amount of computation (the number of computation units is four times of the initial grid). The invention establishes the T spline finite cell analysis method, introduces the T spline into the finite cell analysis method, combines the existing shape function weighting method, and weights the T spline shape function to realize the accurate application of the homogeneous Dirichlet boundary condition, and can realize the improvement of the calculation precision of the key part in the structural analysis by virtue of the local subdivision characteristic of the T spline.
2. The analytic expression of the stress gradient can be deduced by utilizing the high-order continuity of the T-spline function, and the cell cells needing to be subdivided are judged according to the set threshold value. Von-Mises stress gradient mode in the method
Figure BDA0003458110840000051
Is used as the basis for the determination. In addition, since σvonIs by means of stress vectors
Figure BDA0003458110840000052
Calculated, so the following focus is on the derivative calculation of σ:
Figure BDA0003458110840000053
in the formula, D represents an elastic matrix,
Figure BDA0003458110840000054
is the strain matrix after weighted correction, and U refers to the node displacement vector.
Is not difficult to obtain according to the definition of the strain matrix
Figure BDA0003458110840000061
Derivative of (a):
Figure BDA0003458110840000062
wherein w is a weight function, and x y is a physical domain coordinate;
the specific element calculation form in the above formula is not difficult to obtain by the chain derivation method, as follows:
Figure BDA0003458110840000063
due to the shape function TiIs defined in the parameter space and the derivatives in the formula are with respect to x and y in the physical domain, so it is necessary to introduce a second order Jacobian matrix with respect to two spaces:
Figure BDA0003458110840000071
wherein ξ η is the coordinate of the parameter domain;
in addition, because the method carries out regular grid division based on the rectangular region, the second-order Jacobian matrix of the above formula can be further simplified into a diagonal matrix:
Figure BDA0003458110840000072
wherein lx,lyRespectively representing the x and y directions of the cellThe dimension of the direction.
According to the formulas, the high-order continuity of the cubic T-spline function used by the method can be obtained analytically
Figure BDA0003458110840000073
And
Figure BDA0003458110840000074
in addition, σ is obtainedvonAfter the derivative, we first give a threshold
Figure BDA0003458110840000075
As a criterion for judging whether the internal stress of the cell changes violently. Upon determining that the maximum value of the stress gradient within some cells is greater than a given threshold
Figure BDA0003458110840000076
Then the cells are recorded and determined to be the cells that need to be subdivided.
3. After determining the cells to be subdivided, a simple and direct quadtree subdivision strategy is adopted to realize subdivision of the cells so as to generate a T-grid, which can ensure that the corresponding T-spline-shaped 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 point, it can be obtained that the number of cells of the T-grid is increased by only 10% with respect to the initial grid, while the corresponding maximum Von Mises stress results are only 0.5% out of tolerance with respect to the reference value.
4. In the shape optimization process, along with the structure evolution, the T grid needs to be updated during each iteration, so that the locally refined position is consistent with the currently calculated larger position of the stress gradient module. The generation of the T grid depends on a pre-analysis calculation, and in order to avoid the pre-analysis calculation in each iteration, the invention adopts the T grid generated in the previous iteration to calculate the stress gradient. And updating the design parameters by using a gradient-based MMA optimization algorithm, inputting the updated variables into the next iteration, and repeating the steps until the optimization process converges.
In the method, the final optimization design result is obtained through 40 steps of iteration and convergence of the optimization process. The initial structure volume is 35000mm3The optimized structure volume is 19000mm3The maximum stress at this time just satisfies the set upper limit. Compared with the design method in the background art, the design result of the invention is very similar, however, the limited cell number 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 obvious improvement on the calculation efficiency on the premise of ensuring the precision, and has a strong engineering application prospect.
While the invention has been described with reference to specific embodiments, the invention is not limited thereto, and various equivalent modifications or substitutions can be easily made by those skilled in the art within the technical scope of the present disclosure.

Claims (1)

1. A stress gradient-based adaptive T spline FCM structure optimization design method is characterized by comprising the following steps: establishing a T spline finite cell element analysis method, introducing a T spline into the finite cell element method, and weighting a T spline shape function by combining an existing shape function weighting method to realize accurate application of homogeneous Dirichlet boundary conditions; the method comprises the following steps:
step 1: deducing an analytical expression of the stress gradient by using the high-order continuity of the T-spline function, and judging the cell elements needing to be subdivided according to a set threshold;
modeling the Von-Mises stress gradient
Figure FDA0003458110830000011
Is used as a basis for determination; due to sigmavonIs by means of stress vectors
Figure FDA0003458110830000012
Calculated, so the following focus is on the derivative calculation of σ:
Figure FDA0003458110830000013
in the formula, D represents an elastic matrix,
Figure FDA0003458110830000014
is the strain matrix after weighting correction, and U refers to the node displacement vector;
is not difficult to obtain according to the definition of the strain matrix
Figure FDA0003458110830000015
Derivative of (a):
Figure FDA0003458110830000016
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:
Figure FDA0003458110830000021
due to the shape function TiIs defined in the parameter space, and the derivatives in the equation are with respect to x and y in the physical domain, so it is necessary to introduce a second order Jacobian matrix with respect to two spaces:
Figure FDA0003458110830000022
wherein ξ η is the coordinate of the parameter domain;
in addition, because the method carries out regular grid division based on the rectangular region, the second-order Jacobian matrix of the above formula can be further simplified into a diagonal matrix:
Figure FDA0003458110830000023
wherein lx,lyThe dimensions in the x and y directions of the cell are shown;
according to the formulas, the high-order continuity of the cubic T-spline function used by the method can be obtained analytically
Figure FDA0003458110830000024
And
Figure FDA0003458110830000025
in addition, σ is obtainedvonAfter the derivative, a threshold value is first specified
Figure FDA0003458110830000026
To be used as a standard for judging whether the internal stress of the cell element changes violently; upon determining that the maximum value of the stress gradient within some cells is greater than a given threshold
Figure FDA0003458110830000027
Then these cells are recorded and determined as the cells that need to be subdivided;
step 2: after determining the cell to be subdivided, adopting a quadtree subdivision strategy to realize subdivision of the cell so as to generate a T grid;
and step 3: and solving the optimization problem by adopting an MMA optimization algorithm according to the predefined optimization problem to obtain a final optimization result.
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Citations (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US20200134918A1 (en) * 2018-10-31 2020-04-30 The Hong Kong University Of Science And Technology Methods of high-definition cellular level set in b-splines for modeling and topology optimization of three-dimensional cellular structures
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

Patent Citations (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US20200134918A1 (en) * 2018-10-31 2020-04-30 The Hong Kong University Of Science And Technology Methods of high-definition cellular level set in b-splines for modeling and topology optimization of three-dimensional cellular structures
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

Non-Patent Citations (2)

* Cited by examiner, † Cited by third party
Title
SHOUYU CAI, WEIHONG ZHANG， JIHONG ZHU，TONG GAO；: "Stress constrained shape and topology optimization with fixed mesh: A B-spline finite cell method combined with level set function", 《 COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING》, 15 August 2014 (2014-08-15), pages 361 - 387 *
刘登学;张友良;刘高敏;: "基于适合分析T样条的高阶数值流形方法", 力学学报, no. 01, 31 January 2017 (2017-01-31), pages 212 - 222 *

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