MPC unit-based mirror rigid body displacement and surface accuracy analysis method

Abstract

The application belongs to the field of space camera mirror analysis, and provides a mirror surface rigid body displacement and surface shape precision analysis method based on MPC unit, comprising: creating a new node in the mirror finite element model; extracting node numbers and spatial coordinates of all nodes on the mirror surface and constructing a rigid body displacement function; creating MPC unit about the new node according to the rigid body displacement function; constructing a rigid body displacement correction function of the mirror surface sag direction according to the node numbers and spatial coordinates of each node; constructing a surface shape error RMS value solving function according to each node on the mirror surface and the rigid body displacement correction function; importing the rigid body displacement function, MPC unit, rigid body displacement correction function and surface shape error RMS value solving function into the mirror finite element model, and calculating the rigid body displacement and surface shape error RMS value of the mirror surface by using a finite element solver. The application can improve the optimization design efficiency of the space camera mirror.

Description

Technical Field

[0001] This invention belongs to the field of space camera mirror analysis technology, and particularly relates to a method for analyzing the rigid body displacement and surface shape accuracy of a mirror based on MPC units. Background Technology

[0002] The rigid body displacement and surface accuracy of a reflector under various operating conditions are crucial indicators for the design of space optical cameras. In the design process of reflector surfaces, simulation analysis is typically performed using the finite element method. However, existing commercial finite element analysis software lacks the capability to analyze the rigid body displacement and surface accuracy of reflector surfaces. This usually requires exporting and calculating the nodal coordinates and displacement data of the reflector surface to complete the rigid body displacement and surface accuracy analysis, which severely impacts the efficiency of iterative optimization in reflector design. Summary of the Invention

[0003] In view of this, the present invention aims to provide a method for analyzing the rigid body displacement and surface shape accuracy of a mirror based on MPC elements, in order to solve the technical problem that existing commercial finite element analysis software does not yet have the function of analyzing the rigid body displacement and surface shape accuracy of a mirror, which seriously affects the efficiency of iterative optimization of mirror design.

[0004] To achieve the above objectives, the technical solution created by this invention is implemented as follows:

[0005] A method for analyzing the rigid body displacement and surface accuracy of a reflector mirror based on MPC units includes the following steps:

[0006] S1: Create a new node in the finite element model of the mirror;

[0007] S2: Extract the node numbers of all nodes located on the mirror surface, and extract the spatial coordinates of the corresponding nodes in the global coordinate system based on the node numbers;

[0008] S3: Construct a rigid body displacement function based on the node number and spatial coordinates of each node on the mirror surface to represent the relationship between the rigid body displacement of the mirror surface and the displacements of all nodes on the mirror surface.

[0009] S4: Create MPC elements for the new nodes to characterize the rigid body displacement of the mirror surface, based on the rigid body displacement function.

[0010] S5: Construct a rigid body displacement correction function in the sagittal direction of the mirror surface based on the node number and spatial coordinates of each node on the mirror surface;

[0011] S6: Construct a function to solve the RMS value of the surface shape error of the mirror based on each node on the mirror surface and the rigid body displacement correction function;

[0012] S7: Import the rigid body displacement function, MPC element, rigid body displacement correction function, and surface error RMS value solution function into the finite element model of the reflector, and use the finite element solver to calculate the rigid body displacement and surface error RMS value of the reflector surface.

[0013] Furthermore, suppose the first... i The spatial coordinates of the nodes are ( x i , y i , z i ), then the node i nodal displacements ( , , ) is represented as:

[0014] ;

[0015] In the formula, T x , T y , T z These represent the edges of the mirror surface in the global coordinate system. x Axial direction, y Axial direction, z Translational displacement of a rigid body in the axial direction. R x , R y , R z These represent the edges of the mirror surface in the global coordinate system. x Axial direction, y Axial direction, z Rigid body rotational displacement in the axial direction;

[0016] Rewriting the above equation in matrix form gives:

[0017] ;

[0018] In the formula, d i Representative node i The displacement vector, t This represents the displacement vector of the rigid body on the mirror surface. , A i The representative coefficient matrix is ​​represented as follows:

[0019] ;

[0020] For all nodes on the mirror surface, the following relationship exists:

[0021] ;

[0022] In the formula, , , ;

[0023] The rigid body displacement of the mirror surface is fitted using the least squares method to obtain the rigid body displacement function, which is expressed as:

[0024] .

[0025] Furthermore, the node numbering of the new nodes created in the finite element model of the reflector is... s ,node s The mapping relationship between the displacement of the reflector and the nodal displacements of all nodes on the reflector surface is expressed as:

[0026] ;

[0027] In the formula, u s Representative node s The displacement vector, , u x , u y , u z Representing nodes respectively s Lower edge of global coordinate system x Axial direction, y Axial direction, z Translational displacement in the axial direction. θ x , θ y , θ z Representing nodes respectively s Lower edge of global coordinate system x Axial direction, y Axial direction, z Rotational displacement in the axial direction.

[0028] Furthermore, the rigid body displacement correction function in the mirror surface sagittal direction is expressed as:

[0029] ;

[0030] In the formula, Representative node i The displacement in the sag direction after rigid body displacement correction.

[0031] Furthermore, the constructed function for calculating the RMS value of the surface shape error is expressed as follows:

[0032] ;

[0033] In the formula, n This represents the number of nodes on the mirror surface.

[0034] Compared with the prior art, the present invention can achieve the following beneficial effects:

[0035] This invention creates MPC elements in finite element analysis software to characterize the rigid body displacement of a reflector mirror, and constructs a mirror rigid body correction function and a surface shape error RMS value solution function in the finite element analysis software. It integrates the analysis functions of the rigid body displacement and surface shape accuracy of the reflector mirror into the finite element analysis calculation file, thereby realizing the direct solution of the rigid body displacement and surface shape error RMS value of the reflector mirror, thus improving the optimization design efficiency of space camera reflectors. Attached Figure Description

[0036] The accompanying drawings, which form part of this invention, are used to provide a further understanding of the invention. The illustrative embodiments and descriptions of the invention are used to explain the invention and do not constitute an undue limitation of the invention. In the drawings:

[0037] Figure 1 A flowchart illustrating the method for analyzing the displacement and surface accuracy of a rigid body of a reflective mirror based on an MPC unit, as described in an embodiment of the present invention.

[0038] Figure 2 A schematic diagram of the parameters of the selected MPC unit described in the embodiment of the present invention;

[0039] Figure 3 A schematic diagram of the rigid body displacement correction function in the sagittal direction of the mirror surface, as described in an embodiment of the present invention;

[0040] Figure 4 A schematic diagram of the RMS value calculation function for the surface shape error as described in the embodiments of the present invention;

[0041] Figure 5 A schematic diagram showing the calculation results of the rigid body displacement of the mirror surface described in the embodiment of the present invention;

[0042] Figure 6 A schematic diagram showing the calculation results of the RMS value of the surface shape error of the mirror surface described in the embodiment of the present invention;

[0043] Figure 7 This is a schematic diagram showing the results of the rigid body displacement and surface shape error RMS value of the mirror surface calculated according to traditional analysis methods. Detailed Implementation

[0044] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and specific embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and do not constitute a limitation thereof.

[0045] The invention will now be described in detail with reference to the accompanying drawings and embodiments.

[0046] like Figure 1 As shown, the method for analyzing the rigid body displacement and surface accuracy of a reflector surface based on an MPC unit, provided by an embodiment of the present invention, includes the following steps:

[0047] S1: Create a new node in the finite element model of the mirror.

[0048] Record the node number of the new node when creating it, denoted as . s .

[0049] For example: Create a node at the origin (0,0,0,) with node number 9001000.

[0050] S2: Extract the node numbers of all nodes located on the mirror surface, and extract the spatial coordinates of the corresponding nodes in the global coordinate system based on the node numbers.

[0051] A total of [number] samples were extracted from the surface of the mirror. n The node, the i The node number of each node is i Node number i The corresponding spatial coordinates are ( x i , y i , z i ).

[0052] For example: 312 nodes are extracted from the mirror surface, and the node numbers of the 312 nodes are 101~412.

[0053] S3: Construct a rigid body displacement function based on the node number and spatial coordinates of each node on the mirror surface to represent the relationship between the rigid body displacement of the mirror surface and the displacements of all nodes on the mirror surface.

[0054] For example: Construct a rigid body displacement function between the rigid body displacement of the mirror surface and the node positions of the 312 mirror surface nodes.

[0055] Mirror Node i nodal displacements ( , , ) is represented as:

[0056] ;

[0057] In the formula, T x , T y , T z These represent the edges of the mirror surface in the global coordinate system. x Axial direction, y Axial direction, z Translational displacement of a rigid body in the axial direction. R x , R y , R z These represent the edges of the mirror surface in the global coordinate system. x Axial direction, y Axial direction, z Rigid body rotational displacement in the axial direction;

[0058] Rewriting the above equation in matrix form gives:

[0059] ;

[0060] In the formula, d i Representative node i The displacement vector, t This represents the displacement vector of the rigid body on the mirror surface. , A i The representative coefficient matrix is ​​represented as follows:

[0061] ;

[0062] For all nodes on the mirror surface, the following relationship exists:

[0063] ;

[0064] In the formula, , , By fitting the rigid body displacement of the mirror surface using the least squares method, the rigid body displacement function representing the relationship between the rigid body displacement of the mirror surface and the displacements of all nodes on the mirror surface is:

[0065] .

[0066] S4: Create MPC elements for the new nodes to characterize the rigid body displacement of the mirror surface, based on the rigid body displacement function.

[0067] Multi-point constrained (MPC) elements define the coupling relationships between nodal degrees of freedom. Specifically, they use certain degrees of freedom of one node as standard values, and then establish mathematical relationships between certain degrees of freedom of other specified nodes and these standard values. MPC elements are widely used in finite element analysis due to their ability to simulate complex physical phenomena and improve computational efficiency.

[0068] For example: Creating about nodes s The MPC elements used to characterize the rigid body displacement of the mirror surface, such as... Figure 2 As shown.

[0069] Specifically, nodes s The mapping relationship between the displacement of the reflector and the nodal displacements of all nodes on the reflector surface is expressed as:

[0070] ;

[0071] In the formula, u s Representative node s The displacement vector, , u x , u y , u z Representing nodes respectively s Lower edge of global coordinate system x Axial direction, y Axial direction, z Translational displacement in the axial direction. θ x , θ y , θ z Representing nodes respectively s Lower edge of global coordinate system x Axial direction, y Axial direction, z Rotational displacement in the axial direction.

[0072] S5: Construct a rigid body displacement correction function in the sagittal direction of the mirror surface based on the node number and spatial coordinates of each node on the mirror surface.

[0073] The rigid body displacement correction function in the sag direction of the mirror surface is constructed as follows: Figure 3 As shown, specifically:

[0074] ;

[0075] In the formula, Representative node iThe displacement in the sag direction after rigid body displacement correction.

[0076] S6: Construct a function to solve the RMS value of the surface shape error of the mirror surface based on each node on the mirror surface and the rigid body displacement correction function.

[0077] The constructed function for solving the RMS value of the surface shape error is as follows: Figure 4 As shown, specifically:

[0078] ;

[0079] In the formula, n This represents the number of nodes on the mirror surface.

[0080] S7: Import the rigid body displacement function, MPC element, rigid body displacement correction function, and surface error RMS value solution function into the finite element model of the reflector, and use the finite element solver to calculate the rigid body displacement and surface error RMS value of the reflector surface.

[0081] The calculation results of the rigid body displacement of the mirror surface are as follows: Figure 5 As shown, the rigid body displacement values ​​of the mirror surface are obtained as follows:

[0082] .

[0083] The calculation results of the surface shape error RMS value are as follows Figure 6 As shown, the surface shape error RMS value is 147.57 nm.

[0084] The results of the rigid body displacement and surface shape error RMS values ​​of the reflector surface calculated using traditional analysis methods are as follows: Figure 7 As shown, the rigid body displacement values ​​of the mirror surface calculated by the traditional analysis method are:

[0085] .

[0086] The RMS value of the surface shape error calculated by the traditional analysis method is 147.6 nm.

[0087] pass Figure 5 , Figure 6 and Figure 7 The comparison shows that the results of the two analysis methods are consistent, indicating that the method for analyzing the rigid body displacement and surface accuracy of the mirror surface based on MPC units is practical and effective.

[0088] Although embodiments of the present invention have been shown and described above, it is to be understood that the above embodiments are exemplary and should not be construed as limiting the present invention. Those skilled in the art can make changes, modifications, substitutions, and variations to the above embodiments within the scope of the present invention.

[0089] The specific embodiments of the present invention described above do not constitute a limitation on the scope of protection of the present invention. Any other corresponding changes and modifications made in accordance with the technical concept of the present invention should be included within the scope of protection of the claims of the present invention.

Claims

1. A method for analyzing the rigid body displacement and surface shape accuracy of a reflector surface based on MPC units, characterized in that, Includes the following steps: S1: Create a new node in the finite element model of the mirror; S2: Extract the node numbers of all nodes located on the mirror surface, and extract the spatial coordinates of the corresponding nodes in the global coordinate system based on the node numbers; S3: Construct a rigid body displacement function based on the node number and spatial coordinates of each node on the mirror surface to represent the relationship between the rigid body displacement of the mirror surface and the displacements of all nodes on the mirror surface. S4: Create MPC elements for the new nodes to characterize the rigid body displacement of the mirror surface, based on the rigid body displacement function. S5: Construct a rigid body displacement correction function in the sagittal direction of the mirror surface based on the node number and spatial coordinates of each node on the mirror surface; S6: Construct a function to solve the RMS value of the surface shape error of the mirror based on each node on the mirror surface and the rigid body displacement correction function; S7: Import the rigid body displacement function, MPC element, rigid body displacement correction function, and surface error RMS value solution function into the finite element model of the reflector, and use the finite element solver to calculate the rigid body displacement and surface error RMS value of the reflector surface.

2. The method for analyzing the rigid body displacement and surface accuracy of a reflector surface based on an MPC unit according to claim 1, characterized in that, Let the first mirror on the surface of the mirror be... i The spatial coordinates of the nodes are ( x i , y i , z i ), then the node i nodal displacements ( , , ) is represented as: ； In the formula, T x , T y , T z These represent the edges of the mirror surface in the global coordinate system. x Axial direction, y Axial direction, z Translational displacement of a rigid body in the axial direction. R x , R y , R z These represent the edges of the mirror surface in the global coordinate system. x Axial direction, y Axial direction, z Rigid body rotational displacement in the axial direction; Rewriting the above equation in matrix form gives: ； In the formula, d i Representative node i The displacement vector, t This represents the displacement vector of the rigid body on the mirror surface. , A i The representative coefficient matrix is ​​represented as follows: ； For all nodes on the mirror surface, the following relationship exists: ； In the formula, , ; The rigid body displacement of the mirror surface is fitted using the least squares method to obtain the rigid body displacement function, which is expressed as: 。 3. The method for analyzing the rigid body displacement and surface accuracy of a reflector surface based on an MPC unit according to claim 2, characterized in that, The node number of the new node created in the finite element model of the reflector is s ,node s The mapping relationship between the displacement of the reflector and the nodal displacements of all nodes on the reflector surface is expressed as: ； In the formula, u s Representative node s The displacement vector, , u x , u y , u z Representing nodes respectively s Lower edge of global coordinate system x Axial direction, y Axial direction, z Translational displacement in the axial direction. θ x , θ y , θ z Representing nodes respectively s Lower edge of global coordinate system x Axial direction, y Axial direction, z Rotational displacement in the axial direction.

4. The method for analyzing the rigid body displacement and surface accuracy of a reflector surface based on an MPC unit according to claim 3, characterized in that, The rigid body displacement correction function in the sagittal direction of the mirror surface is expressed as: ； In the formula, Representative node i The displacement in the sag direction after rigid body displacement correction.

5. The method for analyzing the rigid body displacement and surface accuracy of a reflector surface based on an MPC unit according to claim 4, characterized in that, The constructed function for solving the RMS value of the surface shape error is expressed as follows: ； In the formula, n This represents the number of nodes on the mirror surface.

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