CN114996879B - Interval field geometric uncertainty topology optimization method for flexible clamp mechanism - Google Patents
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Abstract
The invention provides a section field geometric uncertainty topology optimization method aiming at a flexible clamp mechanism, which comprises the steps of firstly, representing a projection threshold value in a Heaviside density filter as a section field, and constructing a robustness topology optimization model based on a worst case; then, a robust objective function and a constraint are efficiently solved based on interval KL expansion and Chebyshev polynomial expansion; and finally, deriving the sensitivity of the robustness objective function and the constraint to the design variable, solving by adopting a gradient-based optimization algorithm, and finally obtaining the robustness topological configuration under the geometric uncertainty. The invention has the advantages that for the topological optimization problem of geometric uncertainty, the higher calculation precision is ensured, the calculation efficiency is greatly improved, the stability and the convergence are better in the optimization iterative calculation process, and the obtained structural configuration has better robustness under the uncertain condition, thereby being more beneficial to production and manufacture.
Description
Technical Field
The invention relates to the field of robustness topology optimization methods, in particular to a section field geometric uncertainty topology optimization method for a flexible clamp mechanism.
Background
Topology optimization techniques, while satisfying structural strength, implement innovative design of structures by re-layout of materials, are often applied to design of complex structures. The development of modern processing methods and technologies such as additive manufacturing technology provides possibility for the production and manufacture of complex structures, and with the continuous progress and perfection of topology optimization technology, the topology optimization method has gradually become an important tool for the design and development of current products, and is increasingly applied to the optimization design of complex equipment structures such as vehicles, ships, bridges, aerospace and the like.
Most of the current structural optimization problems are based on deterministic assumptions, however, there is a large amount of uncertainty in the actual engineering production application. Various uncertainties related to material properties, service environments and manufacturing processes can seriously affect structural properties and even cause structural failure, and a series of researches on consideration of structural material characteristics, geometric characteristics, load sizes, directions and other parameter uncertainties also appear. The uncertainty of the structural geometry boundaries due to manufacturing errors, measurement errors and information imperfections is less studied than the uncertainty of materials, loads. While minor disturbances at the boundaries of the structure in the design production may cause instability and failure of the structure, design ignoring geometric uncertainty may make it difficult for the manufactured part to meet the actual service requirements. Therefore, considering geometric uncertainties is of great importance to the topology optimization design of the structure.
Geometric uncertainty due to manufacturing errors is typically measured in the form of bounded random fields in general studies taking into account the actual physical properties of uncertainty, however, the precise probability distribution of random field parameters requires a large number of experimental samples to construct. In practical engineering, due to cost and technical limitations, the obtained samples are limited, and accurate probability distribution is often difficult to obtain, and the interval uncertainty analysis method which only needs a small amount of samples provides a solution for the dilemma. The general geometric uncertainty is bounded, so that it can be characterized by a bounded field of compartments. The bounded interval model realizes the uncertainty research of the structure under the condition of fewer samples, and improves the calculation efficiency and saves the calculation cost and the experiment cost while meeting the calculation precision.
Disclosure of Invention
The invention aims to provide a section field geometric uncertainty topology optimization method aiming at a flexible clamp mechanism, which can solve the problem of structural geometric uncertainty topology optimization. The invention adopts the interval field model to measure the space-bounded geometric uncertainty of the structure, and combines interval KL (Karhunen-Loeve) expansion and Chebyshev polynomial expansion to provide a Robust Topological Optimization (RTO) efficient solving method based on a variable density method. Firstly, representing a projection threshold value in the Heaviside density filtering as an interval field, and constructing a robust topology optimization model based on worst case; then, a robust objective function and a constraint are efficiently solved based on interval KL expansion and Chebyshev polynomial expansion; and finally, deriving the sensitivity of the robustness objective function and the constraint to the design variable, solving by adopting a gradient-based optimization algorithm, and finally obtaining the robustness topological configuration under the geometric uncertainty. The invention has the advantages that for the topological optimization problem of geometric uncertainty, the higher calculation precision is ensured, the calculation efficiency is greatly improved, the stability and the convergence are better in the optimization iterative calculation process, and the obtained structural configuration has better robustness under the uncertain condition, thereby being more beneficial to production and manufacture.
In order to achieve the aim, the interval field geometric uncertainty topology optimization method for the flexible clamp mechanism is adopted and is characterized in that:
step 1, initializing parameters of a flexible clamp mechanism;
step 2, aiming at the problem that gray units exist in variable density topological optimization, a three-field model based on a projection strategy, namely a design variable field rho, is constructed e Filtering variable fieldsProjection variable field +.>Wherein the filtered variable field is derived from the design variable field by density filtering as follows:
in N e Is the number of units in the filtering neighborhood of the ith unit, wherein the volume of the ith unit is denoted as v i The density is denoted as ρ i The weight function is denoted as H ei ：
H ei ＝max(0,r min -r ei ) (2)
Wherein r is min Represents the filter radius, r ei Physical Density field for center distance of ith cell and ith cellThen is +.>Obtained by filtration through a Heaviside:
where H represents the projection threshold and β is the parameter that approximates the continuous smoothing of the non-steerable Heaviside function;
step 3, to describe the spatial bounded geometric uncertainty of the clamp mechanism structure boundary, the Heaviside projection threshold is expressed as a section field H (z) related to the spatial position; for any position z e D in the design domain space, the value corresponding to the interval field belongs to the following interval:
H(z)＝[H L (z),H U (z)] (4)
wherein H is U (z) and H L (z) represents upper and lower boundary values corresponding to the position z, the upper and lower boundary functions of which are respectively denoted as H for the interval field U (z) and H L (z) then the median function H of the interval field c (z) and radius function H r (z) is expressed as:
step 4, constructing a robust topological optimization model of the flexible clamp mechanism in the worst case:
wherein: l represents a unit vector with a value of 1 at the degree of freedom of the output point and other values of 0, U represents a displacement field, ρ represents a topological design vector composed of topological design variables, K represents a structural overall stiffness matrix, F is a given external load, F (H (z)) represents a structural objective function, g and V 0 Respectively representing the volume of the design structure and the design domain volume, and ζ represents the volume fraction;
step 5, performing interval Karhunen-loeve (KL) expansion on the interval field H (z) to obtain m=6 interval variables η i ，i＝1,2,…,m；
Wherein eta is i ＝[-1,1]Is a standard independent interval variable and meetsλ i Is->The eigenvalues and eigenvectors of the autocorrelation function R (z, z'), respectively, are represented in two dimensions as:
where z' denotes another position in space than z, l x And l y The relative lengths of two directions of the two-dimensional coordinate axes are respectively, and x 'and y' respectively represent another position different from x and y on the coordinate axes;
step 6, solving a robustness objective function, namely max f (eta);
step 7, solving the clamp mechanism robustness objective function and the volume constraint for the design variable ρ e Sensitivity information of (2);
step 8, updating and iterating the design variable by adopting a gradient-based moving asymptote method based on sensitivity information of the design variable;
and 9, judging the convergence, if not, returning to the step 6, and continuing to iterate the loop until the optimal robust topological configuration under the geometric uncertainty of the clamp mechanism is obtained after the calculation is converged.
Further, in step 6, to solve for the maximum value of the interval function f (η), it may be approximately expressed as a p-order chebyshev expansion:
in c χ Representing chebyshev's coefficient, subscript χ (χ) i =0, 1, …, p) represents the natural number spaceIndex set of (a), p is expansion order, ">Is a Chebyshev polynomial corresponding to a single dimensional interval variable>The resulting multidimensional Chebyshev term, wherein +.>Index number, [ theta ] corresponding to the ith interval variable i ]＝arccos(η i )＝[0,π]。
The chebyshev coefficient is calculated using the least squares method as follows:
c χ ＝(A T A) -1 A T f(η) (10)
where a represents a matrix of samples and,η is the element of the r-th row and s-th column in the sample matrix (r) Represents the (th) sample point consisting of the root of the chebyshev polynomial, f (eta) being expressed in allAnd (5) matching the output response matrix of the points.
Finally, based on the bounded nature of the trigonometric function, it can be seen thatThe upper and lower bounds of the structural performance response f (η) can thus be approximated as:
in c 0 Representing the first term chebyshev coefficient, a robust objective function of the flexible clamping mechanism is then obtained:
the beneficial effects of the invention are as follows:
1. the invention can realize the topology optimization design of geometric uncertainty under a small number of sample points.
2. The method solves the problem of geometric uncertainty based on interval KL expansion and Chebyshev polynomial expansion, and improves the calculation efficiency while guaranteeing the precision.
3. The structural configuration obtained by the invention eliminates the fine branch structure in optimization and improves the phenomenon of 'hinging'. Compared with the traditional topological optimization method based on deterministic hypothesis, the structural configuration obtained by the method has better robustness and is more convenient to produce and manufacture.
Drawings
FIG. 1 is a flow chart of the present invention.
FIG. 2 is a schematic diagram of structural design domains and boundary conditions.
FIG. 3 is a schematic diagram of deterministic and robust topology optimization results.
FIG. 4 is a schematic diagram of a deterministic and robust topology optimization iteration curve.
Detailed Description
The following description of the embodiments of the present invention will be made clearly and completely with reference to the accompanying drawings, in which it is apparent that the embodiments described are only some embodiments of the present invention, but not all embodiments. All other embodiments, which can be made by those skilled in the art based on the embodiments of the invention without making any inventive effort, are intended to be within the scope of the invention.
The embodiment provides a geometric uncertainty topology optimization method based on a flexible clamp mechanism interval field, which is characterized in that boundary disturbance of a structure is represented through interval field description of a projection threshold value in a large scale density filter, the interval field is approximately discretized into a limited interval variable based on interval KL expansion, a robust objective function and constraint are solved by combining a Chebyshev polynomial expansion method, sensitivity of the objective function is solved, a gradient algorithm is used for optimizing a model, and finally a structure topology optimization configuration is obtained.
The invention will be described in further detail with respect to a flexible clamping mechanism. In this embodiment, the side length of the flexible clamping mechanism is l=2, and as shown in fig. 1, the optimization target is u out1 And u out2 Is provided.
The specific implementation steps are as follows:
and step 1, initializing parameters of the flexible clamp mechanism.
The elastic modulus of the entity and the hole is set as E 0 =1 and e= -9 min The volume fraction is set to 0.3, and the parameter f is optimized in ＝1，k in ＝1，k out =0.005, penalty factor p is set to 3, and the structure finite element is divided into 200×200;
step 2, aiming at the problem that gray units exist in variable density topological optimization, a three-field model based on a projection strategy, namely a design variable field rho, is constructed e Filtering variable fieldsProjection variable field +.>Wherein the filtering variable field is set up byThe calculated variable field was obtained by density filtration as follows:
in N e Is the number of units in the filtering neighborhood of the ith unit, wherein the volume of the ith unit is denoted as v i The density is denoted as ρ i The weight function is denoted as H ei ：
H ei ＝max(0,r min -r ei ) (14)
Wherein r is min Represents the filter radius, r in this example min Taken as 6.5, r ei Physical Density field for center distance of ith cell and ith cellThen is +.>Obtained by filtration through a Heaviside:
where H represents the projection threshold, β is a parameter that approximates the non-guided Heaiside function to a continuous smooth, in iterative solution, β increases from 1 to 32 by doubling its value every 25 iterations;
step 3, to describe the spatial bounded geometric uncertainty of the clamp mechanism structure boundary, the Heaviside projection threshold is expressed as a section field H (z) related to the spatial position. For any position z e D in the design domain space, the value corresponding to the interval field belongs to the following interval:
wherein H is U (z) and H L (z) represents upper and lower boundary values corresponding to the position zFor the interval field, the upper boundary function and the lower boundary function are respectively marked as H U (z) and H L (z) then the median function H of the interval field c (z) and radius function H r (z) is expressed as:
in this embodiment, the upper bound H U (z) =0.6, lower bound H L (z) =0.4, median H of interval field c (z) taken as 0.5, radius H r (z) is taken as 0.1;
step 4, constructing a robust topological optimization model of the flexible clamp mechanism in the worst case:
wherein L represents a unit vector with a value of 1 at the degree of freedom of the output point and other values of 0, U represents a displacement field, ρ represents a topological design vector composed of topological design variables, K represents a structural overall stiffness matrix, F is a given external load, F (H (z)) represents a structural objective function, g and V 0 Respectively representing the volume of the design structure and the design domain volume, and ζ represents the volume fraction;
step 5, performing interval Karhunen-loeve (KL) expansion on the interval field H (z) to obtain m=6 interval variables η i (i＝1,2,…,m)：
Wherein eta is i ＝[-1,1]Is a standard independent interval variable and meetsλ i Is->Respectively, is an autocorrelation functionThe eigenvalues and eigenvectors of the number R (z, z '), the autocorrelation function R (z, z'), is expressed in two dimensions as:
where z' denotes another position in space than z, l x And l y Respectively taking l as the correlation length of two directions of two-dimensional coordinate axes x ＝l y =4, x 'and y' represent another position on the coordinate axis different from x and y, respectively; since R (z, z ') is positive with bounded symmetry, the correlation function can be characterized as follows according to the Mercer's theorem:
wherein the eigenvalues and eigenvectors can be found by solving the following friedel integration equation:
the eigenvalue lambda= [2.9123,0.2839,0.2839,0.0818,0.0818,0.0375] is calculated, and the eigenvalue lambda= [2.9123,0.2839,0.2839,0.0818,0.0818,0.0375] is:
thus, the interval field characterizing the boundary uncertainty of the flexible mechanism is approximately expressed as 6 interval variables, so the structural performance response can be further expressed as f (η), where η represents an interval vector consisting of 6 interval variables;
and 6, solving a robustness objective function, namely max f (eta). To solve for the maxima of the interval function f (η), it can be approximated as a p-order chebyshev expansion:
in c χ Representing chebyshev's coefficient, subscript χ (χ) i =0, 1, …, p) represents the natural number spaceIndex set of (a), p is expansion order, ">Is a Chebyshev polynomial corresponding to a single dimensional interval variable>The resulting multidimensional Chebyshev term, wherein +.>Index number, [ theta ] corresponding to the ith interval variable i ]＝arccos(η i )＝[0,π]；
In this embodiment, the number of elements in the index set can be obtained by taking the expansion order p as 2 and the number of interval variables as m=6According to χ ε N m Determining a matrix with index set χ of 6×56:
the chebyshev coefficient is calculated using the least squares method as follows:
c χ ＝(A T A) -1 A T f(η) (24)
where a represents a matrix of samples and,η is the element of the r-th row and s-th column in the sample matrix (r) Representing the (th) composed of chebyshev polynomial rootsThe sample points, f (η), represent the output response matrix at all the distribution points. In the implementation process, the fitting point is formed by combining high-dimension Chebyshev polynomial zero points, and is expressed as a matrix of 6×56 as follows:
based on the above-mentioned fitting, the sample matrix a is a 56×28 matrix:
then, according to the formula (12), the Chebyshev coefficient can be obtained through the sample point matrix value and the function value corresponding to the distribution point;
finally, based on the bounded nature of the trigonometric function, it can be seen thatThe upper and lower bounds of the structural performance response f (η) can thus be approximated as:
in c 0 Representing the first term chebyshev coefficient, a robust objective function of the flexible clamping mechanism is then obtained:
step 7, solving the clamp mechanism robustness objective function and the volume constraint for the design variable ρ e Sensitivity information of (a) is provided. The robustness objective function of equation (14) is rewritten as:
where sign (·) represents the sign function, since the sign function is not conductive at x=0, then the tanh function approximation is used:
sign(x)＝tanh(lx) (28)
wherein l is a larger positive number, taken here as 100;
then, equation (27) is applied to the design variable ρ e The sensitivity of (2) is as follows:
the variable ρ is related to the left and right sides of equation (23) e And (3) deriving:
when considering interval uncertainty, the sensitivity of the deterministic objective function with respect to the design variable in equation (30) can be considered as an interval function, expressed by chebyshev's expansion:
wherein g χ Is the expansion coefficient. By comparing the formula (30) with the formula (31), it is possible to obtain:
the sensitivity of the flex clamp mechanism robustness objective function with respect to the design variable can then be expressed as:
the solution of the flexible clamping mechanism volume constraint is similar to the solution of the robust objective function, also first approximately denoted as the p-order chebyshev expansion, where the sensitivity analysis is performed. But for volume constraints not onlyThe worst-case maxima need to be solved and the corresponding projection threshold η needs to be found max Final filtering of the structure is achieved. Thus here we select a series of sample points to chebyshev expansion which are further analyzed by scanning to obtain the maximum value of the volumeAnd obtain the corresponding projection threshold eta max And then the sensitivity of the robustness volume constraint on the design variable can be obtained by substituting the sensitivity into a function equation of the deterministic volume sensitivity.
Step 8, updating and iterating the design variable by adopting a gradient-based moving asymptote method based on sensitivity information of the design variable;
and 9, judging the convergence if the iteration step is more than 200 steps or the relative change of the objective function in the final iteration is less than 0.01, and returning to the step 6 if the objective function is not converged, and continuing to iterate until the optimal robust topological configuration under the geometric uncertainty of the clamp mechanism is obtained after the calculation convergence.
The deterministic and robust topological optimization design results for the flexible clamping mechanism of fig. 2 are given in fig. 3, where (a) in fig. 3 is a deterministic topological optimization design configuration and (b) is a robust topological optimization design configuration that takes into account geometric uncertainty. Obviously, the design taking into account uncertainty is different from the design under deterministic assumption, and there is no hinging phenomenon in fig. 3 (b). The dashed and solid lines in fig. 4 are objective function iteration curves for deterministic and robust topological optimization, respectively. From the iteration history, the proposed robust topology optimization method has good stability and convergence.
To verify the validity of the proposed method, a robustness objective function of the deterministic design and the robust design under geometrical uncertainty conditions is calculated. As can be seen from table 1, the maximum displacement value for the robust design structure is-1.656, while the maximum displacement value for the deterministic structure is-1.212. That is, robust design results that take into account geometric uncertainties are faced with several issuesWhich boundary fluctuates with better robustness. In addition, to verify the calculation accuracy of the method, 10 are also shown in the table 5 Subsampled MCS (Monte Carlo Scanning) results. For the robustness design, the robustness objective function obtained by the proposed method is-1.656, while the comparison result of the MCS method is-1.700, and the error is 2.59%. Meanwhile, the method provided by the invention only needs to call the function 56 times, thereby ensuring good precision and simultaneously having higher calculation efficiency.
Table 1 robustness objective function of structural topology optimization design
The steps in the present application may be sequentially adjusted, combined, and pruned according to actual requirements.
Although the present application is disclosed in detail with reference to the accompanying drawings, it is to be understood that such descriptions are merely illustrative and are not intended to limit the application of the present application. The scope of the present application is defined by the appended claims and may include various modifications, alterations, and equivalents to the invention without departing from the scope and spirit of the application.
Claims (2)
1. The interval field geometric uncertainty topology optimization method for the flexible clamp mechanism is characterized by comprising the following steps of:
step 1, initializing parameters of a flexible clamp mechanism;
step 2, aiming at the problem that gray units exist in variable density topological optimization, a three-field model based on a projection strategy, namely a design variable field rho, is constructed e Filtering variable fieldsProjection variable field +.>Wherein the filtered variable field is filtered from the design variable field by the following densityThe method comprises the following steps:
in N e Is the number of units in the filtering neighborhood of the ith unit, wherein the volume of the ith unit is denoted as v i The density is denoted as ρ i The weight function is denoted as H ei ：
H ei ＝max(0,r min -r ei ) (2)
Wherein r is min Represents the filter radius, r ei Physical Density field for center distance of ith cell and ith cellThen is +.>Obtained by filtration through a Heaviside:
where H represents the projection threshold and β is the parameter that approximates the continuous smoothing of the non-steerable Heaviside function;
step 3, to describe the spatial bounded geometric uncertainty of the clamp mechanism structure boundary, the Heaviside projection threshold is expressed as a section field H (z) related to the spatial position; for any position z e d in the design domain space, the value corresponding to the interval field belongs to the following interval:
H(z)＝[H L (z),H U (z)] (4)
wherein H is U (z) and H L (z) represents upper and lower boundary values corresponding to the position z, the upper and lower boundary functions of which are respectively denoted as H for the interval field U (z) and H L (z) then the median function H of the interval field c (z) and radius function H r (z) is expressed as:
step 4, constructing a robust topological optimization model of the flexible clamp mechanism in the worst case:
wherein: l represents a unit vector with a value of 1 at the degree of freedom of the output point and other values of 0, U represents a displacement field, ρ represents a topological design vector composed of topological design variables, K represents a structural overall stiffness matrix, F is a given external load, F (H (z)) represents a structural objective function, g and V 0 Respectively representing the volume of the design structure and the design domain volume, and ζ represents the volume fraction;
step 5, performing interval Karhunen-loeve expansion on the interval field H (z) to obtain m=6 interval variables eta i ，i＝1,2,…,m；
Wherein eta is i ＝[-1,1]Is a standard independent interval variable and meetsλ i Is->The eigenvalues and eigenvectors of the autocorrelation function R (z, z'), respectively, are represented in two dimensions as:
where z' denotes another position in space than z, l x And l y The relative lengths of two directions of the two-dimensional coordinate axes are respectively, and x 'and y' respectively represent another position different from x and y on the coordinate axes;
step 6, solving a robustness objective function, namely max f (eta);
step 7, solving the clamp mechanism robustness objective function and the volume constraint for the design variable ρ e Sensitivity information of (2);
step 8, updating and iterating the design variable by adopting a gradient-based moving asymptote method based on sensitivity information of the design variable;
and 9, judging the convergence, if not, returning to the step 6, and continuing to iterate the loop until the optimal robust topological configuration under the geometric uncertainty of the clamp mechanism is obtained after the calculation is converged.
2. A method of interval field geometric uncertainty topology optimization for a flexible clamping mechanism as recited in claim 1, wherein: in step 6, to find the maximum value of the solution interval function f (η), it may be approximately expressed as a p-order chebyshev expansion:
in c χ Representing Chebyshev's coefficient, subscript χ, χ i =0, 1, …, p represents the natural number spaceIndex set of (a), p is expansion order, ">Is a Chebyshev polynomial corresponding to a single dimensional interval variable>The resulting multi-dimensional chebyshev term,wherein->Index number, [ theta ] corresponding to the ith interval variable i ]＝arccos(η i )＝[0,π]；
The chebyshev coefficient is calculated using the least squares method as follows:
c χ ＝(A T A) -1 A T f(η) (10)
where a represents a matrix of samples and,η is the element of the r-th row and s-th column in the sample matrix (r) Representing the r-th sample point composed of chebyshev polynomial root, f (η) representing the output response matrix at all the fitting points;
finally, based on the bounded nature of the trigonometric function, it can be seen thatThe upper and lower bounds of the structural performance response f (η) can thus be approximated as:
in c 0 Representing the first term chebyshev coefficient, a robust objective function of the flexible clamping mechanism is then obtained:
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