CN112560312B - Topology optimization method for identifying structural defects - Google Patents
Topology optimization method for identifying structural defects Download PDFInfo
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- CN112560312B CN112560312B CN202011491689.4A CN202011491689A CN112560312B CN 112560312 B CN112560312 B CN 112560312B CN 202011491689 A CN202011491689 A CN 202011491689A CN 112560312 B CN112560312 B CN 112560312B
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Abstract
The invention discloses a topological optimization method for identifying structural defects, which takes the pseudo density of a finite element model unit of a structure to be detected as a design variable, takes the square of the displacement response error of the structure to be detected and the structure containing the defects as a topological optimization objective function, carries out the finite element analysis of the structure to obtain the real-time structural displacement response, carries out the topological optimization iterative solution of the defect identification by deducing the derivative of the objective function relative to the design variable, and can finish the defect identification when the objective function converges to the minimum value. The invention adopts a topological optimization method to realize the defect identification of the structure, converts the defect identification problem into an inverse problem and solves the problem, and has a rigorous mathematical theory. The invention can position and shape the structural defect, without prior information, without influence of defect quantity, and the structural defect display is visual.
Description
Technical Field
The invention belongs to the field of structural defect identification, and particularly relates to a topological optimization method for structural defect identification.
Background
The structural defect identification is a technical guarantee for safe operation of important key mechanical equipment, and the principle is to invert the position and the size of the structural defect according to the physical response of the structure. The basic idea of structural defect identification is to develop an identification algorithm according to the physical response of a structure and perform qualitative and quantitative identification on structural defects. The technology for identifying the structural defects without damage is more, such as ultrasonic detection, ray imaging, vibration analysis and the like. Ultrasonic detection is one of the important technical means of structural defect identification, and the ultrasonic detection requires that users have richer engineering experience, and is difficult when the structure is complex and the inside of the structure contains more holes.
The optimization theory and the computational mechanics numerical method which are rapidly developed in recent years provide a solution for identifying the structural defects, and the structural defects can be identified as an inverse problem to be solved. For the structure to be identified, firstly, a test method is adopted to obtain a structure response, and then defect information is obtained based on the structure response and a reverse algorithm. At present, the level set method is adopted to describe the structural defect boundary in the existing research, and the structural defect identification research is carried out. Due to the numerical problem of the level set method, certain prior information is needed during the identification of the structural defects, the algorithm steps are complex, and more preset parameters exist. The topological optimization method is a conceptual design method in the field of structural optimization, and can obtain a topological structure meeting performance requirements through the change of structural topology. The topology optimization and the structural defect identification are the same in nature and belong to the optimization problem. Thus, topological optimization methods can be used to find topological configurations that satisfy responsive structural defects.
Disclosure of Invention
Aiming at the defects or improvement requirements in the prior art, the invention provides a topological optimization method for identifying the structure defects, which adopts the topological optimization method to realize the defect identification of the structure, converts the defect identification problem into an inverse problem solution and has a strict mathematical theory. The invention can position and shape the structural defect, without prior information, without influence of defect quantity, and the structural defect display is visual.
In order to achieve the above object, the present invention provides a topology optimization method for identifying structural defects, which comprises the following steps:
s1: measuring a displacement response of the defect-containing structure at the designated location;
s2: establishing a two-dimensional/three-dimensional model of a defect-free structure, and establishing a finite element model by considering the measurement position points;
s3: defining design variables on all units of the finite element model with the defect-free structure, and defining artificial material properties on the units;
s4: solving a finite element model of a defect-free structure containing artificial materials to obtain a structure displacement response;
s5: solving a topological optimization objective function;
s6: solving the derivative of the objective function with respect to the design variable;
s7: updating the design variables;
s8: judging whether a convergence condition is met;
s9: if the convergence is satisfied, outputting the geometry of the structural defect, and if the convergence is not satisfied, repeating the above steps S4-S8.
In the step S1, the two-dimensional structure measurement position point may be arbitrarily selected, and the three-dimensional structure measurement position point is located on the outer surface, but the structure measurement position point needs to avoid the structure constraint position and the excitation load application position.
The nodes of the finite element model with no defect structure in the step S2 need to pass through the measurement position points.
In the step S3, the value range of the design variable on the unit is more than or equal to 0.001 and less than or equal to 1, and the value of the initial design variable is 1; the artificial material properties are defined as follows:
E(x)＝xpE0
wherein x is a design variable representing the presence or absence of a cell, and p is a penalty coefficient (1)<p<5) The effect of this is to make the design variables as close to 0 or 1 as possible, E0The elastic modulus when the unit design variable is 1, and the elastic modulus when the unit design variable is less than 1.
In the step S4, finite element solution can be performed on a defect-free structure containing an artificial material in a plurality of physical fields, when static analysis is adopted, the structural response is displacement, and the finite element solution is represented by the following equation:
K(x)U＝F
wherein K is a structural stiffness matrix related to a design variable x, and U is a displacement response of a defect-free structure containing an artificial material; and F is the excitation load.
The mathematical model of the topology optimization objective function in step S5 is defined as follows:
J(x)＝(U(x)-U0)2
where J is an objective function with respect to a design variable x, U is a displacement response of a defect-free structure containing an artificial material, and U is0Is the displacement response of the defect-containing structure.
The solution of the objective function with respect to the design variable derivatives in step S6 is divided into two steps:
K(x)λ＝-2(U(x)-U0)
in the formula, lambda is an accompanying displacement vector;
in the formula (I), the compound is shown in the specification,
to optimize the sensitivity of the model, T represents the matrix transpose.
The update strategy of the design variables in step S7 is as follows:
in the formula, n represents the number of iteration steps, xi is the step size, max represents the maximum value in brackets, and min represents the minimum value in brackets.
The convergence criterion in step S8 is defined as follows:
In which ε represents a very small positive number, nmaxRepresenting the maximum number of iterations.
Generally, compared with the prior art, the above technical solution conceived by the present invention mainly has the following technical advantages:
1. the invention adopts a topological optimization method to realize the defect identification of the structure, converts the defect identification problem into an inverse problem and solves the problem, and has a rigorous mathematical theory.
2. Meanwhile, the method can be used for positioning and shaping identification of the structural defects, does not need prior information, is not influenced by the number of the defects, and is visual in structural defect display.
Drawings
The invention is further illustrated by the following figures and examples.
FIG. 1 is a flowchart of a topology optimization method for structural defect identification according to the present invention.
FIG. 2 is a schematic diagram of the two-dimensional defect-free topology optimization design variable definition of the present invention.
FIG. 3 is a schematic diagram of two-dimensional structural defect identification constructed in a preferred embodiment of the present invention.
Fig. 4 is a schematic diagram of the two-dimensional structure defect identification result in the preferred embodiment of the present invention.
Detailed Description
Embodiments of the present invention will be further described with reference to the accompanying drawings.
Example 1:
referring to fig. 1, the present invention provides a topology optimization method for structural defect identification, as shown in fig. 1, the method includes the following steps:
s1: and measuring the displacement response of the defect-containing structure at a specified position, wherein a two-dimensional structure measuring position point can be selected at will, and a three-dimensional structure measuring position point is positioned on the outer surface, but the structure measuring position point needs to avoid a structure constraint position and an excitation load application position.
S2: establishing a two-dimensional/three-dimensional model of a defect-free structure, and establishing a finite element model by considering the measurement position points, wherein the nodes of the finite element model of the defect-free structure need to pass through the measurement position points.
S3: as shown in fig. 2, design variables are defined on all elements of the finite element model of the defect-free structure, and artificial material properties are defined on the elements;
more specifically, the value range of the design variable on the unit is 0.001-1, and the value of the initial design variable is 1. The artificial material properties are defined as follows:
E(x)＝xpE0
wherein x is a design variable representing the presence or absence of a cell, and p is a penalty coefficient (1)<p<5) The effect of this is to make the design variables as close to 0 or 1 as possible, E0The elastic modulus when the unit design variable is 1, and the elastic modulus when the unit design variable is less than 1.
S4: solving a finite element model of a defect-free structure containing artificial materials to obtain a structure displacement response;
more specifically, the finite element solution of defect-free structures containing artificial materials in various physical fields may be performed in step S4. When static analysis is used, the structural response is displacement. Finite element solution can be represented by the following equation:
K(x)U＝F
where K is the structural stiffness matrix for the design variable x, U is the displacement response of a defect-free structure containing an artificial material, and F is the excitation load.
S5: solving a topological optimization objective function;
more specifically, the mathematical model of the topology optimization objective function in step S5 is defined as follows:
J(x)＝(U(x)-U0)2
where J is an objective function with respect to a design variable x, U is a displacement response of a defect-free structure containing an artificial material, and U is0Is the displacement response of the defect-containing structure.
S6: solving the derivative of the objective function with respect to the design variable;
more specifically, the derivative of the objective function with respect to the design variable in step S6 is derived as follows.
Constructing a Lagrangian function:
L(x)＝(U(x)-U0)2+λT(K(x)U(x)-F)
derivative of the lagrangian function:
order bag
The coefficients of (a) are zero:
K(x)λ＝-2(U(x)-U0)
in the formula, λ is an associated displacement vector.
In the formula (I), the compound is shown in the specification,
to optimize the sensitivity of the model, T represents the matrix transpose.
S7: updating the design variables;
more specifically, the update strategy for the design variables is as follows:
in the formula, n represents the number of iteration steps, xi is the step size, max represents the maximum value in brackets, and min represents the minimum value in brackets.
S8: judging whether a convergence condition is met;
more specifically, the convergence criterion is defined as follows:
In which ε represents a very small positive number, nmaxRepresenting the maximum number of iterations.
S9: if the convergence is satisfied, outputting the geometry of the structural defect, and if the convergence is not satisfied, repeating the above steps S4-S8.
Example 2:
fig. 3 shows a two-dimensional beam structure with an aspect ratio of 2:1, considering only the static displacement response of the structure, totally constrained by the degrees of freedom of the left boundary, applying a vertically downward concentrated load at the midpoint of the right boundary, setting displacement response measuring points on the upper and lower boundaries, respectively, setting two rectangular defect regions in the middle, and dividing the structure domain into 60 × 30 four-node units according to a preferred embodiment of the present invention. The method provided by the invention is adopted to carry out structural defect identification topology optimization, and the result shown in figure 4 is obtained after convergence, wherein a dark color area is a defect-free structural domain, and a light color area is an identified defect.
The invention provides a topological optimization method for structure defect identification, which is a systematic defect identification method, and needs to consider key factors such as boundary conditions, mechanical response, defect positions, measurement positions, objective functions and the like of a structure.
Claims (5)
1. A topological optimization method for identifying structural defects is characterized by comprising the following steps:
s1: measuring a displacement response of the defect-containing structure at the designated location;
s2: establishing a two-dimensional/three-dimensional model of a defect-free structure, and establishing a finite element model by considering the measurement position points;
s3: defining design variables on all units of the finite element model with the defect-free structure, and defining artificial material properties on the units;
s4: solving a finite element model of a defect-free structure containing artificial materials to obtain a structure displacement response;
s5: solving a topological optimization objective function;
s6: solving the derivative of the objective function with respect to the design variable;
s7: updating the design variables;
s8: judging whether a convergence condition is met;
s9: if the convergence is satisfied, outputting the geometric shape of the structural defect, and if the convergence is not satisfied, repeating the steps S4-S8;
in the step S4, finite element solution can be performed on a defect-free structure containing an artificial material in a plurality of physical fields, when static analysis is adopted, the structural response is displacement, and the finite element solution is represented by the following equation:
K(x)U＝F
wherein K is a structural stiffness matrix related to a design variable x, and U is a displacement response of a defect-free structure containing an artificial material; f is an excitation load;
the mathematical model of the topology optimization objective function in step S5 is defined as follows:
J(x)＝(U(x)-U0)2
wherein J is an objective function with respect to a design variable x, and U is a displacement response of a defect-free structure containing an artificial material，U0Is a displacement response containing a defective structure;
the solution of the objective function with respect to the design variable derivatives in step S6 is divided into two steps:
K(x)λ＝-2(U(x)-U0)
in the formula, lambda is an accompanying displacement vector;
in the formula (I), the compound is shown in the specification,
in order to optimize the sensitivity of the model, T represents matrix transposition;
the update strategy of the design variables in step S7 is as follows:
in the formula, n represents the number of iteration steps, xi is the step size, max represents the maximum value in brackets, and min represents the minimum value in brackets.
2. The topological optimization method for structural defect identification according to claim 1, characterized in that: in the step S1, the two-dimensional structure measurement position point may be arbitrarily selected, and the three-dimensional structure measurement position point is located on the outer surface, but the structure measurement position point needs to avoid the structure constraint position and the excitation load application position.
3. The topological optimization method for structural defect identification according to claim 1, characterized in that: the nodes of the finite element model with no defect structure in the step S2 need to pass through the measurement position points.
4. The topological optimization method for structural defect identification according to claim 1, characterized in that: in the step S3, the value range of the design variable on the unit is more than or equal to 0.001 and less than or equal to 1, and the value of the initial design variable is 1; the artificial material properties are defined as follows:
E(x)＝xpE0
wherein x is a design variable representing the presence or absence of a cell, p is a penalty coefficient, 1<p<5, the effect of which is to bring the design variables as close as possible to 0 or 1, E0The elastic modulus when the unit design variable is 1, and the elastic modulus when the unit design variable is less than 1.
5. The topological optimization method for structural defect identification according to claim 1, characterized in that: the convergence criterion in step S8 is defined as follows:
In which ε represents a very small positive number, nmaxRepresenting the maximum number of iterations.
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| US20100324870A1 (en) * | 2009-06-19 | 2010-12-23 | Microsoft Corporation | Solver-based visualization framework |
| CN101697176A (en) * | 2009-10-29 | 2010-04-21 | 西北工业大学 | Method for layout optimal design of multi-assembly structure system |
| CN102902845A (en) * | 2012-09-12 | 2013-01-30 | 北京航空航天大学 | Method for designing blade sections of helicopter rotors |
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