A Structural Design System and Method Based on the Efficient Reconstruction of Bionic Linear Surfaces

Abstract

The present invention discloses a structural design system and method based on efficient reconstruction of bionic line-body-surface, including: establishing an equivalent loading model, calculating the range of bifurcation ratios at all levels of the main leaf veins, constructing a multi-level optimization model, and carrying out optimal structural optimization design. By the above means, the structural design system and method based on efficient reconstruction of bionic line-body-surface of the present invention can not only utilize the multi-objective multidisciplinary optimization design method to achieve the efficient reconstruction of the bionic structure line-body-surface, but also improve the efficiency of structural design and reduce costs by means of the limit distribution model and bionic collaboration.

Description

Technical Field

[0001] The present invention relates to the technical field of bionic mechanical structure design, and particularly to a structure design system and method based on efficient reconstruction of bionic line-surface-body. Background Art

[0002] Mechanical structure design is a complex and crucial link. A good mechanical structure can greatly improve the service life and quality of products. As a reliable, efficient, and accurate analysis method in modern mechanical structure design, finite element analysis predicts mechanical properties such as the strength, stiffness, and stability of a structure by simulating the loads and stress conditions of actual working conditions, and then conducts optimization design of the mechanical structure. Among them, mesh generation and topology optimization occupy a relatively large proportion of the design time, and it is difficult to quickly obtain a better solution, and it cannot guarantee that this solution is close to both the performance optimal solution and the weight optimal solution at the same time. Usually, to obtain both the performance and weight optimal solutions, it is necessary to build a surrogate model on this basis and conduct multi-objective multidisciplinary optimization to select the desired solution set scheme.

[0003] In addition, although there are many lightweight and efficient structures in nature, for different loading directions, the layout differences of their efficient load-bearing structures are very obvious. Although bionic design can effectively improve design efficiency and performance, due to the mismatch between the bionic source and specific application scenarios, and the large differences in shape and size, it is currently difficult for some mechanical structure designs to quickly obtain an ideal load-bearing structure through bionics. Summary of the Invention

[0004] To solve the above technical problems, one technical solution adopted by the present invention is:

[0005] Provide a structure design system and method based on efficient reconstruction of bionic line-surface-body, and its steps include:

[0006] 1) Establish an equivalent loading model for regular units, and construct a limit distribution model of an efficient structure according to a preset load-bearing stability criterion. The load-bearing stability criterion is:

[0007]

[0008] where d i is the diameter of the auxiliary rod, D i is the diameter of the load-bearing rod, i = 1, 2, 3... n, D 2 > D 1 , d i-1 > d i , δ(d i , D 2 ) is the yield strength value of the equivalent loading model when the diameter of the load-bearing rod is D 2 and the diameter of the auxiliary rod is d i , and so on for others; ρ(di , D 2 ) is the density value of the equivalent loading model when the diameter of the load-bearing rod is D 2 and the diameter of the auxiliary rod is d i , and so on for others; among them, the relationship of the three is: D 2 > D 1 > d i-1 ;

[0009] When the load-bearing stability criterion is met, the limit distribution model of its efficient structure is: the diameter of the auxiliary rod is greater than d i-1 , and the diameter of the load-bearing rod is greater than D 2 ;

[0010] 2) Extract the vein distribution of the sample leaves of the structural growth unit, and calculate the average range of the bifurcation ratios of the main veins;

[0011] 2.1) Based on the image recognition algorithm, extract the vein distribution of each leaf, identify all the main veins on the leaf and the secondary vein nodes bifurcated from each main vein, and obtain the vein segment lengths (l 1 , l 2 ,..., l n ) between each secondary vein node of each main vein; among them, l 1 -l n is the average value of the vein segment lengths of several sample leaves extracted;

[0012] 2.2) Calculate the average range of the bifurcation ratios of the main veins:

[0013] (min(l 1 / l 2 , l 2 / l 3 ,..., l n-1 / l n ), max(l 1 / l 2 , l 2 / l 3 ,..., l n-1 / l n ));

[0014] 3) Using the range of the bifurcation ratios of the main veins, combined with the limit distribution model of the efficient structure, construct a multi-level optimization model, and construct a surrogate model unit using the CAE simulation data set and the test set;

[0015] 3.1) Using the range of the bifurcation ratios of the main veins, determine the maximum bifurcation level n - 1 of the main veins, and combined with the limit distribution model of the efficient structure, construct a multi-level optimization model under different bifurcation ratio levels;

[0016] 3.2) Set the threshold of the number k of bifurcation points according to the initial model, calculate the values of the bifurcation ratios at each level, and achieve the positioning of the bifurcation points, so as to CADize the above multi-level optimization model. Among them, the model parameter variables include the number of bifurcation levels, the bifurcation angle, the diameter of the load-bearing rod, and the diameter of the auxiliary rod. The number of bifurcation levels takes values from 1 to n - 1. The bifurcation angle takes values from 0 to 180° respectively according to the number k of bifurcation points. The values of the diameters of the load-bearing rod and the auxiliary rod are random and need to meet the restrictions of step 1. The values of the bifurcation ratios at each level are determined according to the number of bifurcation points, indicating the parameter value differences of each level of bifurcation. The specific values of the bifurcation ratios at each level are as follows:

[0017]

[0018] Among them, a represents min(l 1 / l 2 ,l 2 / l 3 ,...,l n-1 / l n );b represents max(l 1 / l 2 ,l 2 / l 3 ,...,l n-1 / l n );

[0019] 3.3) Divide the finite element mesh, perform CAE finite element analysis, and obtain the simulation data set and test set with different mechanical properties;

[0020] 3.4) According to the simulation data set and the test set, the surrogate model unit obtains the mapping relationship between different structural and shape parameters and the bearing capacity value, and fits this mapping relationship into the following formula:

[0021] Bearing capacity

[0022] Among them, α j is the weighting coefficient, j is a variable (natural number), which can be obtained after the mapping relationship of the radial basis neural network is fitted into this formula. If j is 5 after fitting, there will be α 1 ~α 5 These 5 weighting coefficients; is the basis function; N is the number of phases of the basis function; ε is the error phase; x is the structural and shape parameter variable, including the diameter of the load-bearing rod, the diameter of the auxiliary rod, the number of bifurcation levels, and the kth bifurcation angle, k = 1, 2, 3,....

[0023] 4) The optimization design unit optimizes according to the DOE experimental design and the surrogate model in step 3) using the NSGA-II multi-objective evolutionary algorithm, with the maximization of the bearing capacity and the minimization of the mass as the optimization objectives max(λ 1 F - λ 2(M), obtain the combination of the optimal load-bearing rod diameter, auxiliary rod diameter, bifurcation level, and bifurcation angle (the k-th) of the model structure, that is, the optimal Pareto front solution; where λ 1 , λ 2 are weighting coefficients, and M is the model mass; the structural and shape parameters are constraints (d min < d m < d max , D min < D n < D max ), d min is the minimum auxiliary rod diameter, d max is the maximum auxiliary rod diameter, D min is the minimum load-bearing rod diameter, D max is the maximum load-bearing rod diameter, and the bifurcation level and bifurcation angle take the values in step 3.2.

[0024] In a preferred embodiment of the present invention, the equivalent loading model is an axial compression finite element model.

[0025] In a preferred embodiment of the present invention, the finite element model includes only the load-bearing rod model and the combined model of the load-bearing rod and the auxiliary rod, and their relative densities are the same.

[0026] In a preferred embodiment of the present invention, it includes a structural design system based on the efficient reconstruction of bionic line-surface-bodies, and this system includes regular units, structure-growing units, surrogate model units, and optimization design units.

[0027] The beneficial effects of the present invention are: not only can the efficient reconstruction of bionic structure line-surface-bodies be realized by using the multi-objective multidisciplinary optimization design method, but also the efficiency of structural design can be improved and the cost can be reduced by means of the limit distribution model and bionic cooperation. BRIEF DESCRIPTION OF THE DRAWINGS

[0028] In order to more clearly illustrate the technical solutions in the embodiments of the present invention, the following will briefly introduce the drawings required for the description of the embodiments. Obviously, the following drawings are only some embodiments of the present invention. For those of ordinary skill in the art, without creative efforts, other drawings can be obtained based on these drawings, where:

[0029] Figure 1 is a schematic diagram of the axial compression finite element model of only the load-bearing rod in a preferred embodiment of a structural design system and method based on the efficient reconstruction of bionic line-surface-bodies of the present invention;

[0030] Figure 2 is a schematic diagram of the axial compression finite element model of the combination of the load-bearing rod and the auxiliary rod in a preferred embodiment of a structural design system and method based on the efficient reconstruction of bionic line-surface-bodies of the present invention;

[0031] Figure 3 It is a schematic diagram of a multi - level optimization model of a preferred embodiment of a structural design system and method based on the efficient reconstruction of bionic line - surface - body of the present invention;

[0032] Figure 4 It is a schematic diagram of a mapping relationship of a preferred embodiment of a structural design system and method based on the efficient reconstruction of bionic line - surface - body of the present invention;

[0033] Figure 5 It is a schematic diagram of the process of step 4 of a preferred embodiment of a structural design system and method based on the efficient reconstruction of bionic line - surface - body of the present invention;

[0034] Figure 6 It is a schematic diagram of a finite - element model of a preferred embodiment of a structural design system and method based on the efficient reconstruction of bionic line - surface - body of the present invention;

[0035] Figure 7 It is a schematic diagram of the deformation before fracture of the uniform - thickness cross - section of a traditional load - bearing rod and auxiliary rod;

[0036] Figure 8 It is a schematic diagram of the deformation before fracture of the optimal cross - section of a load - bearing rod and an auxiliary rod of a preferred embodiment of a structural design system and method based on the efficient reconstruction of bionic line - surface - body of the present invention;

[0037] Figure 9 It is a schematic diagram of the structure of a preferred embodiment of a high - efficiency reconstruction and structural design system based on bionic line - surface - body of the present invention. Detailed implementation manners

[0038] Next, the technical solutions in the embodiments of the present invention will be described clearly and completely. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts belong to the scope of protection of the present invention.

[0039] Please refer to Figures 1-9 , the embodiments of the present invention include:

[0040] A high - efficiency reconstruction and structural design system based on bionic line - surface - body, which can be applied to the design of all mechanical structures, mainly includes: regular units, structure - growing units, surrogate - model units, and optimal - design units.

[0041] A design method of a structural design system based on the efficient reconstruction of bionic line - surface - body includes the following steps:

[0042] 1) The regular unit establishes an equivalent loading model and constructs the limit distribution model of an efficient structure through the bearing stability criterion; among them, the bearing stability criterion is:

[0043]

[0044] Among them, d i is the diameter of the auxiliary rod, D i is the diameter of the bearing rod, i = 1, 2, 3... n, δ(d i , D 2 ) is the yield strength value of the equivalent loading model (the bearing rod diameter is D 2 , and the auxiliary rod diameter is d i ), and so on for others; ρ(d i , D 2 ) is the density value of the equivalent loading model (the bearing rod diameter is D 2 , and the auxiliary rod diameter is d i ), and so on for others; among them, D 2 > D 1 , d i-1 > d i , D 2 > D 1 > d i-1 .

[0045] Once the bearing stability criterion designed in step 1) is satisfied, the limit distribution model of its efficient structure is: the diameter of the auxiliary rod is greater than d i-1 , and the diameter of the bearing rod is greater than D 2 .

[0046] Further preferably, the equivalent loading model is an axial compression finite element model, and this finite element model includes only the bearing rod model and the combined model of the bearing rod and the auxiliary rod, and their relative densities are all the same.

[0047] 2) The structural growth unit extracts the vein distribution of the sample leaves and calculates the range of the bifurcation ratios of the main veins;

[0048] 2.1) Based on the image recognition algorithm, extract the vein distribution of the leaves, identify all the main veins on the leaves and the secondary vein nodes (bifurcation points) bifurcated from each main vein, and obtain the vein segment lengths (l 1 , l 2 ,..., l n ) between each secondary vein node of each main vein, where l 1 -l n is the average value of the vein segment lengths of several sample leaves extracted, and for the convenience of calculation, the sample leaves can be selected from the sample library with the same number of main veins and secondary vein nodes.

[0049] For example, if there are 5 secondary vein nodes on a main vein, the lengths of the recognized main vein are respectively (l 1 , l 2 , l 3 , l 4 ).

[0050] 2.2) Calculate the range of bifurcation ratios at all levels of the main vein:

[0051] (min(l 1 / l 2 , l 2 / l 3 ,..., l n-1 / l n ), max(l 1 / l 2 , l 2 / l 3 ,..., l n-1 / l n ). Among them, the length ratio of two adjacent vein segments of each main vein is the bifurcation ratio of the main vein, and the range of bifurcation ratios at all levels is to take the minimum value and the maximum value among the length ratios of all adjacent two vein segments.

[0052] Through the above algorithm, the specific series range and this bifurcation ratio can be obtained, so as to design new main vein bifurcation points for guiding the construction of an optimization model, so that different bifurcations will form different model structures and shapes.

[0053] 3) Utilize the range of bifurcation ratios at all levels of the main vein, and then combine with the limit distribution model of the efficient structure to construct a multi-level optimization model, and construct a surrogate model unit by using the CAE simulation data set and the test set;

[0054] 3.1) Refer to Figure 3 , 1 is the load-bearing rod, 2 / 3 is the auxiliary rod. Utilize the range of bifurcation ratios at all levels of the main vein to determine the maximum bifurcation level n - 1 of the main vein, and combine with the limit distribution model of the efficient structure to construct multi-level optimization models at different bifurcation ratio levels (0 level, 1 level, 2 level respectively).

[0055] 3.2) According to the threshold of the number k of bifurcation points preset by analyzing the initial model, calculate the values of bifurcation ratios at all levels to realize the positioning of bifurcation points, so as to CAD-ize the above multi-level optimization model, thereby establishing and optimizing the CAD model. Because once the bifurcation points are determined, the number of bifurcation angles is determined and the values of bifurcation ratios are fixed, so that the CAD model can be generated completely.

[0056] Among them, the model parameter variables include bifurcation level, bifurcation angle, load-bearing rod diameter, and auxiliary rod diameter.

[0057] The bifurcation series takes values from 1 to n - 1; the bifurcation angles are respectively taken as 0 to 180° according to the number of bifurcation points k. Each bifurcation point has a random zone angle. For example, for 2 bifurcation points, the angle of the first bifurcation point is 0 - 180°, and the angle of the second bifurcation point is also 0 - 180°; the diameters of the load-bearing rod and the auxiliary rod are randomly taken and need to meet the restrictions of step 1; the above-mentioned values of the bifurcation ratios at all levels are determined according to the number of bifurcation points, indicating the parameter value differences of each level of bifurcation.

[0058] The specific values of the bifurcation ratios at all levels are as follows:

[0059]

[0060] Among them, a represents min(l 1 / l 2 ,l 2 / l 3 ,...,l n-1 / l n ); b represents max(l 1 / l 2 ,l 2 / l 3 ,...,l n-1 / l n ).

[0061] 3.3) Divide the finite element mesh and conduct CAE finite element analysis to obtain simulation data sets and test sets with different mechanical properties; among them, for any model, meshing, conducting CAE analysis, and summarizing multiple simulation results can form a data set, and artificial division can be made into a re-simulation set and a test set, which is common knowledge in this field.

[0062] 3.4) According to the simulation data set and the test set, the surrogate model unit obtains the mapping relationship between different structural parameters and bearing capacity values based on artificial intelligence and its learning algorithms (mainly using radial basis neural networks), and fits this mapping relationship into the following formula:

[0063] Bearing capacity

[0064] Among them, α j is the weighting coefficient, j is a variable (natural number), which can be obtained after the mapping relationship of the radial basis neural network is fitted into this formula. If j is 5 after fitting, there will be α 1 ~α 5 These 5 weighting coefficients; is the basis function; N is the number of phases of the basis function; ε is the error phase; x is the structural and shape parameter variable, specifically including the diameter of the load-bearing rod, the diameter of the auxiliary rod, the bifurcation level, and the k-th bifurcation angle, where k = 1, 2, 3,.... When calculating the bearing capacity, these 4 parameters need to be input simultaneously for x each time. For example, when k = 2, x includes the diameter of the load-bearing rod, the diameter of the auxiliary rod, the bifurcation level, the 1st bifurcation angle, and the 2nd bifurcation angle. The diameter of the load-bearing rod and the diameter of the auxiliary rod are directly obtained from step 1, and the bifurcation level and bifurcation angle are obtained from step 3.2.

[0065] 4) Establish an optimization design unit and carry out the optimal structural optimization design.

[0066] The optimization design unit is optimized using the NSGA-II multi-objective evolutionary algorithm according to the DOE experimental design and the surrogate model in step 3), with the maximization of bearing capacity and the minimization of mass as the optimization objectives max(λ 1 F - λ 2 M), to obtain the combination of the optimal diameter of the load-bearing rod, the diameter of the auxiliary rod, the bifurcation level, and the bifurcation angle (the k-th) of the model structure, that is, the optimal Pareto front solution. Among them, λ 1 and λ 2 are the weighting coefficients, M is the mass of the model; the structural and shape parameters are constraints (d min <d m <d max , D min <D n <D max ), d min is the minimum diameter of the auxiliary rod, d max is the maximum diameter of the auxiliary rod, D min is the minimum diameter of the load-bearing rod, D max is the maximum diameter of the load-bearing rod, and the bifurcation level and bifurcation angle directly take the values in step 3.2.

[0067] Before obtaining the optimal Pareto front solution, step 4 needs to calculate the output result according to the mapping relationship obtained in step 3, that is, the internal iteration update of step 4, and it is not necessary to run the CAE analysis for each model. Specific Example 1

[0069] Taking the optimization design of the extruded aluminum section as an example (considering the extrusion process limitation, the minimum wall thickness is 1.8 mm, so the minimum diameter of the auxiliary rod is 1.8 mm, and the minimum cavity limitation, the bifurcation level is 1), the number of bifurcation points is 1 - 3.

[0070] Under the extrusion bending condition, compared with the optimal cross-section optimization design scheme of the traditional uniform size, the optimal cross-section structure designed by the structural design system and method based on the efficient reconstruction of bionic line, surface, and body of the present invention has a higher breaking force before bending fracture, as shown in the following table.

[0071]

[0072] The beneficial effects of a structural design system and method based on efficient reconstruction of bionic line-surface bodies according to the present invention are as follows:

[0073] 1. Based on the equivalent loading model, combined with load-bearing stability, a limit distribution model of an efficient structure is constructed, solving problems such as the dependence on topology optimization and low design efficiency in the initial layout of an efficient structure. With the aid of the limit distribution model and bionic coordination, the structural design process is a growth design from 0 to 1, with high design efficiency and low cost;

[0074] 2. Utilize the golden ratio characteristics presented by the vein distribution of leaves to quickly form the structural vein distribution under a specific shape and load direction, realizing the efficient reconstruction of bionic structural line-surface bodies;

[0075] 3. Based on CAE simulation analysis, by establishing a data set and a test set between the bionic line-surface body structure and performance, constructing a surrogate model, and using a multi-objective multidisciplinary optimization design method, an efficient bionic line-surface body structure is obtained.

[0076] The above are only embodiments of the present invention, and do not limit the patent scope of the present invention accordingly. Any equivalent structural or equivalent process transformation made using the content of the specification of the present invention, or directly or indirectly applied in other related technical fields, shall be similarly included in the patent protection scope of the present invention.

Claims

1. A structural design method based on the efficient reconstruction of bionic linear-surface-solids, characterized in that the steps include: 1) The regular unit establishes an equivalent loading model, and constructs the limit distribution model of the efficient structure according to the preset bearing stability criterion. The bearing stability criterion is: where d i is the diameter of the auxiliary rod, D i is the diameter of the load-bearing rod, i = 1, 2, 3... n, D 2 > D 1 , d i-1 > d i , δ(d i , D 2 ) is the yield strength value of the equivalent loading model when the diameter of the load-bearing rod is D 2 and the diameter of the auxiliary rod is d i , and so on for the others; ρ(d i , D 2 ) is the density value of the equivalent loading model when the diameter of the load-bearing rod is D 2 and the diameter of the auxiliary rod is d i , and so on for the others; where the relationship among the three is: D 2 > D 1 > d i-1 ; When the bearing stability criterion is met, the limit distribution model of its efficient structure is: the diameter of the auxiliary rod is greater than d i-1 , the diameter of the bearing rod is greater than D 2 ; 2) The structural growth unit extracts the vein distribution of the sample leaves and calculates the average range of the bifurcation ratios at all levels of the main vein. 2.1) Based on the image recognition algorithm, extract the vein distribution of each leaf, identify all the main veins on the leaf and the secondary vein nodes branched out from each main vein, and obtain the length of the vein segment body (l 1 , l 2 ,..., l n ) between each secondary vein node of each main vein; where l 1 -l n is the average value of the lengths of the vein segment bodies of several sample leaves extracted. 2.2) Calculate the average range of the bifurcation ratios at all levels of the main vein: (min(l 1 / l 2 ,l 2 / l 3 ,...,l n-1 / l n ), max(l 1 / l 2 ,l 2 / l 3 ,...,l n-1 / l n )); 3) Using the range of the bifurcation ratios at all levels of the main vein, combined with the limit distribution model of the efficient structure, construct a multi-level optimization model, and construct the surrogate model unit using the CAE simulation dataset and the test set. 3.1) Using the range of the bifurcation ratios at all levels of the main vein, determine the maximum bifurcation level n - 1 of the main vein, and combined with the limit distribution model of the efficient structure, construct the multi-level optimization model under different bifurcation ratio levels. 3.2) According to the initial model, set the threshold value of the number k of bifurcation points, calculate the values of the bifurcation ratios at all levels, and realize the positioning of the bifurcation points, so as to CAD-ize the above multi-level optimization model. Among them, the model parameter variables include the bifurcation level, the bifurcation angle, the diameter of the bearing rod, and the diameter of the auxiliary rod. The bifurcation level takes values from 1 to n - 1, the bifurcation angle takes values from 0 to 180° respectively according to the number k of bifurcation points, the values of the diameters of the bearing rod and the auxiliary rod are random and need to meet the restrictions in step 1). The values of the bifurcation ratios at all levels are determined according to the number of bifurcation points, indicating the parameter value differences of each level of bifurcation. The specific values of the bifurcation ratios at all levels are: where a represents min(l 1 / l 2 ,l 2 / l 3 ,...,l n-1 / l n ); b represents max(l 1 / l 2 ,l 2 / l 3 ,...,l n-1 / l n ); 3.3) Divide the finite element mesh and perform CAE finite element analysis to obtain the simulation dataset and the test set with different mechanical properties. 3.4) According to the simulation dataset and the test set, the surrogate model unit obtains the mapping relationship between different structure and shape parameters and the bearing capacity value, and fits this mapping relationship into the following formula: Bearing capacity Among them, α j is the weighting coefficient, j is a variable and a natural number. After the mapping relationship of the radial basis neural network is fitted to this formula, if j is 5 after fitting, there will be α 1 ~α 5 These 5 weighting coefficients; is the basis function; N is the number of phases of the basis function; ε is the error phase; x is the structure and shape parameter variable, including the diameter of the load-bearing rod, the diameter of the auxiliary rod, the bifurcation level, the k-th bifurcation angle, k = 1, 2, 3,...; 4) The optimization design unit optimizes using the NSGA-II multi-objective evolutionary algorithm according to the DOE experimental design and the surrogate model in step 3), with the maximization of bearing capacity and the minimization of mass as the optimization objectives max(λ 1 F - λ 2 M), to obtain the combination of the optimal bearing rod diameter, auxiliary rod diameter, bifurcation level, and k-th bifurcation angle of the model structure, that is, the optimal Pareto front solution; where λ 1 and λ 2 are weighting coefficients, and M is the model mass; the structure and shape parameters are constraints d min < d m < d max , D min < D n < D max , d min is the minimum auxiliary rod diameter, d max is the maximum auxiliary rod diameter, D min is the minimum bearing rod diameter, D max is the maximum bearing rod diameter, and the bifurcation level and bifurcation angle take the values in step 3.2).

2. The structural design method based on the efficient reconstruction of bionic linear-surface-solids according to claim 1, characterized in that the equivalent loading model is an axial compression finite element model.

3. The structural design method based on the efficient reconstruction of bionic linear-surface-solids according to claim 2, characterized in that the finite element model includes only the bearing rod model and the combined model of the bearing rod and the auxiliary rod, and their relative densities are the same.

4. The structural design method based on the efficient reconstruction of bionic linear-surface-solids according to claim 1, characterized in that it includes a structural design system based on the efficient reconstruction of bionic linear-surface-solids, and this system includes a regular unit, a structural growth unit, a surrogate model unit, and an optimal design unit.

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