CN111832185A - Precise assembly simulation method and system based on skin model - Google Patents

Abstract

The present disclosure provides an accurate assembly simulation method and system based on a skin model. In order to combine the geometric tolerance of the workpiece in the assembly simulation, a skin model of the part is generated according to the tolerance of shape, position and direction, and the skin model of the part is identified. The mating surfaces and load boundary conditions of Contact deformation, the relative positioning of the part will move further; if a new contact point is generated, another iteration of the secondary optimization will begin with the deformed SMS and reduced load; after all loads are balanced, the iteration stops and the part is fully assembled , to achieve accurate assembly simulation of multiple parts; at the same time, by repeating the above assembly simulation simulation multiple times, effective tolerance analysis can be carried out according to the obtained assembly deviation.

Description

Precise assembly simulation method and system based on skin model

technical field

The present disclosure belongs to the technical field of assembly simulation, and in particular relates to an accurate assembly simulation method and system based on a skin model.

Background technique

The statements in this section merely provide background information related to the present disclosure and do not necessarily constitute prior art.

With regard to manufacturing errors and measurement uncertainty, geometric tolerances inevitably exist on each manufactured workpiece. In order to ensure that parts are still interchangeable in large-scale assemblies on the basis of these, the concept of geometric tolerance is defined, and the limits of geometric tolerance are specified from the perspective of assembly. In addition, the functional requirements of artefacts depend on the effect of tolerance accumulation; Therefore, establishing reasonably accurate tolerances is a critical task that ensures the functionality and quality of the manufacturing process, while optimizing production costs and respecting the manufacturing tooling; Geometric Dimensioning and Tolerancing (GD&T) schemes should be implemented early in the design process , whose purpose is to ensure adequate product quality in terms of manufacturing and inspection costs associated with tolerances, by defining the limits of geometric defects of manufacture.

The inventors found that in most commercial computer-aided tolerancing (CAT) systems, the geometrical tolerances are produced by transforming ideal surfaces, which lacks an understanding of the mechanism and form of geometrical tolerances, and therefore is not consistent with international tolerance standards (i.e. GD&T and ISO geometry). Product Specification (GPS) standards) are inconsistent; in addition, this approximation does not meet the need to accurately predict assembly tolerances, especially for high-precision assemblies; and is not sufficient to cover geometric tolerances throughout the product life cycle; assembly simulation is an integral part of the product design process One of the most important issues is the basis of tolerance analysis; in the design process, these tolerances caused by shape defects can be divided into shape tolerance, orientation tolerance and position tolerance; in order to improve the accuracy of tolerance analysis and synthesis, the above factors must be considered And integrated it into the assembly simulation modeling; the inventors found that the existing assembly simulation method ignores the combined effects of geometric tolerance and contact deformation, due to the geometric tolerance generated in the manufacturing process, contact deformation will occur in the assembly process. , both of which affect the assembly quality; during the assembly process, the two influence each other and affect the final assembly quality.

SUMMARY OF THE INVENTION

In order to solve the above problems, the present disclosure provides an accurate assembly simulation method and system based on a skin model. By considering the influence of contact deformation and geometric tolerance on workpiece assembly, the skin model-based method is used to effectively improve the accuracy of assembly simulation. At the same time, the tolerance analysis is carried out according to the assembly simulation results, and reasonable geometric tolerances are obtained, which provide guidance for the geometric tolerance of workpiece production, effectively avoid the contact deformation during the assembly process of the workpiece, and improve the assembly quality.

According to a first aspect of the embodiments of the present disclosure, there is provided an accurate assembly simulation method based on a skin model, including:

Collect the structural parameters of the workpiece to be assembled;

Generate a skin model based on the structural parameters and geometric tolerances of the workpiece to be assembled;

Define the mating surface and load boundary conditions of the skin model of the workpiece to be assembled;

The assembly of the skin model of the workpiece to be assembled is equivalent to the calculation of the displacement and reaction force of the mating surfaces, and the distance objective function between the mating surfaces and its constraint conditions are defined;

The objective function is iteratively calculated to minimize the distance between the mating surfaces, and the assembly simulation result is obtained.

According to a second aspect of the embodiments of the present disclosure, a precise assembly simulation system based on a skin model is provided, including:

The model building module is used to collect the structural parameters of the workpiece to be assembled; generate its skin model based on the structural parameters and geometric tolerances of the workpiece to be assembled; define the mating surface and load boundary conditions of the skin model of the workpiece to be assembled;

An objective function building module, which is used to equate the assembly of the skin model of the workpiece to be assembled as the calculation of the displacement and reaction force of the mating surface, and defines the distance objective function between the mating surfaces and its constraints;

An assembly simulation module is used to iteratively calculate the objective function, so as to minimize the distance between the mating surfaces and obtain assembly simulation results.

Compared with the prior art, the beneficial effects of the present disclosure are:

(1) The solution described in the present disclosure avoids using an equivalent ideal surface to replace a paired non-ideal surface by simulating a real part with shape defects by means of a skin model, taking geometrical tolerances and local contact deformations into account in the assembly simulation process, These equivalent surfaces may affect the accuracy of the assembly simulation and prevent application in shaft-hole assembly; the solution described in this disclosure can consider the contact of the mating surfaces to be frictional, which is consistent with the actual situation, while solving multi-part Assembly simulation problems; reasonable geometric tolerances are obtained through assembly simulation, which provides guidance for the geometric tolerances of workpieces produced by manufacturers, effectively avoids contact deformation during workpiece assembly, and improves assembly quality.

(2) The calculation required by the solution described in this disclosure is more efficient than that required by the existing finite element method; currently, the finite element method is reliable for simulating local contact deformations in assemblies; however, for Monte Carlo Simulation, preprocessing of randomly generated skinned models in finite element analysis is ineffective.

Description of drawings

The accompanying drawings, which form a part of the present disclosure, are used to provide a further understanding of the present disclosure, and the exemplary embodiments of the present disclosure and their descriptions are used to explain the present disclosure and do not constitute an improper limitation of the present disclosure.

Fig. 1 is the flow chart of the accurate assembly simulation method based on the shape of the skin model described in the first embodiment of the present disclosure;

Figure 2(a) is a schematic diagram of the initial model of the workpiece described in the first embodiment of the present disclosure;

FIG. 2(b) is a schematic diagram of the workpiece model described in Embodiment 1 of the present disclosure after being segmented;

Fig. 2 (c) is the workpiece model schematic diagram after the geometrical tolerance additional processing described in the present disclosure embodiment one;

FIG. 2(d) is a schematic diagram of the result of the reorganized skin model described in Embodiment 1 of the present disclosure;

3 is a schematic diagram of the relative positional relationship between two mating surfaces described in Embodiment 1 of the present disclosure;

Figure 4(a) is a schematic diagram of the interaction mechanism between contact force and deformation after applying an external load as described in Embodiment 1 of the present disclosure;

Figure 4(b) is a schematic diagram of the interaction mechanism between contact force and deformation after applying an external load to generate a new contact point described in Embodiment 1 of the present disclosure;

Figure 4(c) is a schematic diagram of the interaction mechanism between contact force and deformation after applying another external load as described in the first embodiment of the present disclosure;

4(d) is a schematic diagram of the interaction mechanism between contact force and deformation after applying another external load to generate a new contact point as described in Embodiment 1 of the present disclosure;

5 is a flowchart of the iterative algorithm for contact deformation described in Embodiment 1 of the present disclosure;

FIG. 6 is a schematic diagram of a specific equipment simulation example described in Embodiment 2 of the present disclosure;

FIG. 7 is a schematic diagram showing whether the influence of contact deformation on the X distribution is considered.

Detailed ways

The present disclosure will be further described below with reference to the accompanying drawings and embodiments.

It should be noted that the following detailed description is exemplary and intended to provide further explanation of the present disclosure. Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this disclosure belongs.

It should be noted that the terminology used herein is for the purpose of describing particular embodiments only, and is not intended to limit the exemplary embodiments according to the present disclosure. As used herein, unless the context clearly dictates otherwise, the singular is intended to include the plural as well, furthermore, it is to be understood that when the terms "comprising" and/or "including" are used in this specification, it indicates that There are features, steps, operations, devices, components and/or combinations thereof.

Example 1:

The purpose of this embodiment is to provide an accurate assembly simulation method based on a skin model.

An accurate assembly simulation method based on a skin model, including:

Collect the structural parameters of the workpiece to be assembled;

Generate a skin model based on the structural parameters and geometric tolerances of the workpiece to be assembled;

Define the mating surface and load boundary conditions of the skin model of the workpiece to be assembled;

The assembly of the skin model of the workpiece to be assembled is equivalent to the calculation of the displacement and reaction force of the mating surfaces, and the distance objective function between the mating surfaces and its constraint conditions are defined;

The objective function is iteratively calculated to minimize the distance between the mating surfaces, and the assembly simulation result is obtained.

Further, the structural parameters include conventional parameters such as workpiece size, curve parameters and lines;

As shown in FIG. 1, the flow chart of the accurate assembly simulation method based on the shape of the skin model described in the present disclosure is shown. In order to combine the geometric tolerance of the workpiece in the assembly simulation, the geometric tolerance is specified according to the product design requirements; according to the shape, Position and orientation tolerances generate SMS (skin model shapes) of the part; utilize GeoSpelling (a formal language for expressing specification semantics) to identify mating surfaces and load boundary conditions for the SMS, which include applied forces and torque, the position and direction of constraints; in terms of geometric tolerance, a quadratic optimization method is used to model SMS-based multibody assemblies, and the relative positions of mating surfaces are determined, as explained by Hertz theory , the contact point is subjected to additional load and deforms further, subsequently, the relative positioning of the part will move further due to the contact deformation; if a new contact point is generated, another step of the quadratic optimization will begin with the deformed SMS and reduced load One iteration; after all loads are balanced, the iteration stops and the parts are all assembled, realizing accurate assembly simulation of multiple parts.

It should be clear to those skilled in the art that geometric tolerance is a combination of systematic and random deviations, and the characteristics of systematic deviations are deterministic, predictable and reproducible; on the contrary, unpredictable fluctuations in the manufacturing process will produce random deviations; design The tolerance can limit the geometric tolerance, the tolerance of the design is specified by the designer according to the product design requirements. to control the quality and function of the assembly; therefore, the present disclosure uses the geometric tolerance of the workpiece to be assembled to generate the SMS, which is specified by the designer according to product design requirements; the SMS based on the geometric tolerance generates a representative workpiece surface profile to close to the actual surface.

Specifically, the generation process of the skin model is as follows:

The SMS for geometric tolerance modeling can be created through mathematical methods (such as second-order shape and random field methods) or through the results of manufacturing process simulations or part prototype measurements; the results of part prototype measurements are used in this example for skinning The creation of the model, and it is necessary to ensure that the surface of the equivalent skin model is consistent with the specified tolerance, and the specified tolerance is specified by the designer according to the product design requirements; the generation of the skin model requires first segmentation of the workpiece structure, To treat each surface independently; each surface is accompanied by geometric tolerances based on design tolerances; different tolerances limit different kinds of geometric features, for example, shape tolerances limit intrinsic features, while orientation and position tolerances limit situations features; in the construction of the skinned model, when geometric tolerances are simulated, they are merged and added to the original nominal model; the components of the skinned model go through three steps: segmentation, geometric tolerance addition, and reorganization; as shown in the figure 2 will generate a skinned model with specified tolerances; the unevenness of the SMS described in Figure 2 has been scaled for easy viewing of geometric tolerances, other modeling issues - non-joining, face-joining, obtuse and sharp The definition of the dihedral angle, and the definition of the mesh size and the deviation size are well known to those skilled in the art, so they will not be repeated here.

Further, described assembly simulation method is equivalent to the assembly of SMS for the calculation of described mating surface displacement and reaction force, and defines the objective function and constraint condition of the distance between described mating surfaces, specifically comprises the steps:

On the mating surface, the contact deformation and the relative position are interdependent, and the contact point is determined by the relative position of the mating surface; however, the relative position of the mating surface changes with the contact deformation. In order to speed up the solution of the relative position, the SMS is assumed to be rigid , Small Displacement Torsion Theory (SDT) is used in rigid body displacement theory to calculate the initial point of contact between two skin surfaces; in SDT theory, the surface displacement expressed as a center point is as follows:

SDT = [rt] ^T = [α β γ uvw] ^T (1)

where r and t are the rotation and translation of the rigid body, respectively, r contains three rotation angles α, β and γ around the x, y and z directions, respectively; u, v and w are the translations along the x, y and z directions, respectively;

In the field of tolerance design, the displacements are usually small and usually linearized, for any normal vector n _j,i of the discrete surface S _j , after rigid body displacement, its orientation will change as follows:

n′ _j,i =n _j,i -n _j,i ×r _j (2) The distance between mating surfaces is the basic constraint for solving the initial contact point; for the vertex A _j,i on the surface S _j , it is along the Projection to the closest projection A _j+1,i on S _j+1 along the normal direction (n _j+1,i ) of the line segment, as shown in Figure 3, S _j,i and A _j+1,i The distance d _j,i between can be expressed as:

The distance d' _{j, i} after displacement is:

After ignoring the second-order term, the distance d' _j,i of the displacement is linearized, and the linearization result is as follows:

where C _j and C _j+1 are the center points of S _j and S _j+1 , respectively;

Therefore, for the entire workpiece, the distance constraints can be reorganized in matrix form as follows:

d′=A·SDTs+d (6)

Since the rigid body displacement theory is adopted, it is ensured that no interpenetration occurs between the mating surfaces. Therefore, d' _j,i should be greater than or equal to zero.

Further, the determination of the balance constraint specifically includes:

For each SMS, the contact point is responsible for bearing internal and external loads; during assembly, the external force F, external torque T and internal force or support reaction force R should be balanced, and the balance constraint can be expressed as:

Among them, j represents different surfaces to be assembled, i is different discrete points on surface j, k is the serial number of the applied force on surface j, q is the serial number of applied torque on surface j, and L _j,i is the difference between the force point and the center of the object The _distance between the ) and the value of the tangent direction of the contact point, which is modeled using equation (7); as another constraint, the value of d' _j,i should be greater than or equal to zero, and if the distance d' _j,i is greater than zero, then The value of the reaction force R _j,i is correspondingly zero; if R _j,i is greater than zero, then d′ _j,i =0; therefore, the combined constraint of the distance constraint and the equilibrium constraint can be expressed as:

R ^T d′=0 (8)

After that, the assembly simulation of the SMS is equivalent to finding the displacement SDT and the reaction force R that minimize the distance between the mating surfaces and satisfy the above constraints, as follows:

Furthermore, the problem is restructured with quadratic objectives and quadratic constraints in matrix form, i.e.

The above problem can be solved by a quadratic optimization algorithm. In order to speed up the solution speed, the present embodiment adopts the Hessian function of calculating the secondary objective lens to solve.

Further, since the influence of local contact deformation on the lamination deviation is not negligible with the increase of assembly load, and the contact deformation caused by external load on the mating surface shows topological changes, therefore, it is determined that the effect of local contact deformation on the assembly deviation is impact is crucial;

With the applied external load, as shown in Fig. 4a, the deformations Δ ¹ _j,i and Δ ¹ _j+1, i generated by the reaction force R ¹ _j,i between the two contact points may generate as shown in Fig. 4b another point of contact shown. Since new contact points are prevented, only much smaller deviations of Δ ¹ ' _j,i are allowed. Then the allowable sum of Δ ¹ ' _j,i and Δ ¹ ' _j+1,i ^shall be d ¹ _min , _which is also the minimum distance from the mating surface, except for the zero distance on the contact point, i.e. when an external load is applied , as shown in Fig. 4a, the deformations Δ ¹ _j,i and Δ ¹ _j+1,i caused by the reaction force R ¹ _j,i between the contact points may generate other contact points, as shown in Fig. 4b. Due to the resistance from the newly created contact point, only a much smaller Δ ¹ ' _j,i deformation is allowed. The allowable sum of Δ ¹ ' _j,i and Δ ¹ ' _j+1,i should be d ¹ _min , d ¹ _min is also the minimum distance of the contact surface, and the distance of the contact point is zero, namely

减小到R¹'_j,i，这不满足载 荷边界约束；反作用力与接触变形之间的下一轮相互作用是从不平衡的外力

开始的，如图4c所示。新发现的接触点将在第二次迭代中变形(图4d中所示)。The contact deformation depends on the reaction force. Correspondingly, the reaction force

reduced to R ¹ ' _j,i , which does not satisfy the load boundary constraint; the next round of interaction between the reaction force and the contact deformation is the external force from the unbalanced

started, as shown in Figure 4c. The newly discovered contact points will be deformed in the second iteration (shown in Fig. 4d).

可以计算如下：Based on the study of the correlation between the reaction force and the contact deformation, we propose an iterative method to evaluate the relative position of the assembled SMS with contact deformation, as shown in Fig. 5. It can be used to determine the local surface deformation on non-ideal surfaces caused by the displacement of each point. Usually, the basic theory for evaluating elastic contact problems is Hertzian theory. Elastic deformation of a single contact point in the rth iteration based on Hertz theory

It can be calculated as follows:

where R ^r _j,i and ρ ^r _j,i are the reaction force and average curvature at the surface contact points, respectively, and E _j is the elastic modulus of S _j . Since R ^r _j+1,i is equal to R ^r _j,i , Δ ^r ' _j+1,i can be replaced by Δ ^r ' _j,i using equation (12), then

If the mating surfaces have the same material properties, equation (13) can be simplified as follows:

where Δ ^r ' _j,i is the temporary displacement of the contact point in the rth iteration. After one iteration, the SMS morphs with it. In this case, only a part of the external force F ^r _j,k is balanced by the reduced reaction force R' _j,i . For deformed SMS, unbalanced forces and torques are imported into the next iteration as load boundary constraints. The graph of this step is highlighted in grey in Figure 5, after which the next round of secondary optimization begins, and the iterative process continues until all external forces and torques are balanced.

Embodiment 2:

The purpose of this embodiment is to provide an accurate assembly simulation system based on a skin model.

An accurate assembly simulation system based on a skin model, including:

The model building module is used to collect the structural parameters of the workpiece to be assembled; generate its skin model based on the structural parameters and geometric tolerances of the workpiece to be assembled; define the mating surface and load boundary conditions of the skin model of the workpiece to be assembled;

An objective function building module, which is used to equate the assembly of the skin model of the workpiece to be assembled as the calculation of the displacement and reaction force of the mating surface, and defines the distance objective function between the mating surfaces and its constraints;

An assembly simulation module is used to iteratively calculate the objective function, so as to minimize the distance between the mating surfaces and obtain assembly simulation results.

Embodiment three:

The purpose of this embodiment is to provide a skin model-based tolerance analysis method.

A skin model-based tolerance analysis method, the tolerance analysis method adopts the above-mentioned skin model-based precise assembly simulation method, and the tolerance analysis is performed according to the assembly deviation distribution by repeating the precise assembly simulation method many times, The tolerance analysis of the workpiece is carried out by using the assembly deviation distribution, thereby providing accurate geometric tolerance guidance for the workpiece production process and improving the assembly quality of the workpiece.

Specifically, Monte Carlo simulation is used in this embodiment to show how geometric tolerances and local contact deformation affect the value of the functional requirement X. The ultimate goal is to compare the cumulative assembly deviation of the deformed mating surface and the undeformed surface; Tolerance analysis based on Monte Carlo simulation to show the effect of geometric tolerance and local contact deformation on assembly deviation; one simulation only considers the effect of geometric tolerance, and the other simulation considers the effect of both deviations; Figure 6 shows a An assembly simulation example of a component named RGB, the RGB component consists of three plane parts, and the tolerance of the mating surface is 0.3 mm; the functional requirements (X=100±1.50mm) are defined between the surfaces Gr and B8, And 1000 simulations were carried out; wherein, the functional requirement was assembly deviation.

As shown in Figure 7, the distribution of X from 1000 simulations is shown, and a tolerance analysis using initial tolerance values (tp=0.3mm and tf=0.3mm) and taking part deformation into account shows that X meets the functional requirements; in order to show the contact deformation , the distribution of X was calculated in the same 1000 simulation runs without contact deformation, as shown in Fig. 7, the contact deformation exhibited a considerable effect on X, indicating that the influence of contact deformation is not negligible in this example ; However, the mating surface is flattened due to the assembly force, so the tolerance value can be defined to a wider range than that defined by the traditional method, which also ensures that X meets the functional requirements; moreover, the larger tolerance can greatly reduce the manufacturing cost ; In particular, for assemblies composed of a large number of components, assembly deviation analysis based on assembly simulation will undoubtedly be more important. In addition, in 1000 Monte Carlo simulations, the average time consumption of each run considering contact deformation is 90s, and the scheme described in the present disclosure has higher execution efficiency compared to FEM.

The precise assembly simulation method and system based on the skin model provided by the above embodiments are completely achievable and have broad application prospects.

The above descriptions are only preferred embodiments of the present disclosure, and are not intended to limit the present disclosure. For those skilled in the art, the present disclosure may have various modifications and changes. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present disclosure should be included within the protection scope of the present disclosure.

Although the specific embodiments of the present disclosure are described above in conjunction with the accompanying drawings, they do not limit the protection scope of the present disclosure. Those skilled in the art should understand that on the basis of the technical solutions of the present disclosure, those skilled in the art do not need to pay creative efforts. Various modifications or variations that can be made are still within the protection scope of the present disclosure.

Claims (10)

1. an accurate assembly simulation method based on skin model, is characterized in that, comprises: Collect the structural parameters of the workpiece to be assembled; Generate a skin model based on the structural parameters and geometric tolerances of the workpiece to be assembled; Define the mating surface and load boundary conditions of the skin model of the workpiece to be assembled; The assembly of the skin model of the workpiece to be assembled is equivalent to the calculation of the displacement and reaction force of the mating surfaces, and the distance objective function between the mating surfaces and its constraint conditions are defined; The objective function is iteratively calculated to minimize the distance between the mating surfaces, and an assembly simulation result is obtained. 2 . The method for accurate assembly simulation based on a skin model according to claim 1 , wherein the skin model is generated through segmentation, geometric tolerance addition and reorganization. 3 . 3. An accurate assembly simulation method based on a skin model according to claim 2, characterized in that, the segmentation is to divide the created workpiece model into independent surfaces for separate processing, and the geometric tolerance is attached. It is the geometric tolerance generated by adding a specified tolerance to each surface, and the reorganization is to recombine the divided surfaces according to the positional relationship of the original model. 4. An accurate assembly simulation method based on a skin model according to claim 1, wherein the constraints include distance constraints and balance constraints. 5. An accurate assembly simulation method based on a skin model according to claim 4, characterized in that, the distance constraint needs to use a contact point, and the contact point is determined by the matching surface of the skin model of the workpiece to be assembled. The relative position is determined, and the contact point is the basic condition for solving the distance between the mating surfaces. 6. An accurate assembly simulation method based on a skin model according to claim 1, wherein the relative position of the mating surface changes with the contact deformation. In order to speed up the solution speed of the relative position, it is necessary to assume that the The skin model is a rigid deformation, and the contact points are calculated by the small displacement torsion theory SDT. 7. A kind of accurate assembly simulation method based on skin model as claimed in claim 1 is characterized in that, for each to-be-assembled workpiece skin model, its contact point is responsible for bearing internal and external loads, so, in the assembly process , the external force F, external torque T and support reaction force R should be balanced, and the balance constraint is expressed as:

Among them, j represents different surfaces to be assembled, i is different discrete points on surface j, k is the serial number of the applied force on surface j, q is the serial number of applied torque on surface j, and L _j,i is the difference between the force point and the center of the object distance between. 8. The skin model-based precise assembly simulation method according to any one of claims 1 to 7, wherein the skin model-based precise assembly simulation method is used to perform several assembly simulations to obtain assembly deviation distribution According to the assembly deviation distribution, the tolerance analysis of the workpiece is carried out, so as to provide accurate geometric tolerance guidance for the workpiece production process and improve the assembly quality of the workpiece. 9. An accurate assembly simulation system based on a skin model, characterized in that, comprising: The model building module is used to collect the structural parameters of the workpiece to be assembled; generate its skin model based on the structural parameters and geometric tolerances of the workpiece to be assembled; define the mating surface and load boundary conditions of the skin model of the workpiece to be assembled; an objective function building module, which is used to equate the assembly of the skin model of the workpiece to be assembled as the calculation of the displacement and reaction force of the mating surfaces, and define the distance objective function between the mating surfaces and its constraints; The assembly simulation module is used for iterative calculation of the objective function, so as to minimize the distance between the mating surfaces and obtain the assembly simulation result. 10 . The precise assembly simulation system based on a skin model according to claim 9 , wherein the skin model is generated through segmentation, geometric tolerance addition and reorganization. 11 .

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