Finite element node area weight solution method, electronic device and storage medium

Abstract

The present invention discloses a method for solving the area weight of a finite element node, an electronic device and a storage medium. The method comprises the following steps: obtaining the finite element nodes of an optical mirror; projecting the finite element nodes along the optical axis direction to the plane composed of the other two axes to obtain a node matrix; triangulating the node matrix according to the Delaunay triangulation condition to obtain a triangular network; dividing the triangle according to the centroid of the triangle and the midpoint of the three sides; obtaining all quadrilaterals that wrap each node to form a polygon; calculating the area of ​​the polygon as the area weight of the wrapped node. The present invention uses the Delaunay triangulation algorithm to triangulate the projected nodes. Due to its empty circle characteristics and maximized minimum angle characteristics, the triangular triangle obtained by triangulation is the closest, most regular and unique, which is the optimal triangulation result; the solution result is used as the node weight to effectively improve the removal of rigid body displacement and the accuracy of polynomial fitting in optical-mechanical integration analysis.

Description

Technical Field

[0001] The present invention belongs to the technical field of optical-mechanical-thermal integration analysis, and specifically relates to a finite element node area weight solution method, electronic equipment and storage medium. Background Art

[0002] Optical-mechanical thermal integration analysis is one of the important branches of optomechanical engineering science, which is used to simulate the optical characteristics and mechanical properties of the system. Under different thermal load conditions, in addition to rigid body displacement, the surface of optical components will also undergo elastic deformation, which will change the wave aberration of the optical system and thus affect the imaging quality of the system.

[0003] In the simulation analysis of optical-mechanical integration, the coordinates of nodes before and after thermal deformation obtained by finite element analysis are usually used to design mathematical algorithms to remove rigid body displacements, and then polynomial fitting is performed on the node coordinates after rigid body displacement to quantitatively analyze the impact of elastic deformation on the system wave aberration. In the current rigid body displacement algorithm and the fitting algorithm for surface deformation after rigid body displacement, the finite element nodes are usually calculated as points with infinitesimal areas, and the impact of the node area on the calculation results has not been considered. In actual calculations, the number of nodes cannot be infinitely large, so each node occupies a certain area and cannot be equivalent to an ideal point. Summary of the invention

[0004] The purpose of the present invention is to provide a method for solving the area weight of a finite element node, an electronic device and a storage medium, so as to solve the problem that the finite element node is equivalent to a point with an infinitely small area for calculation, which affects the calculation result.

[0005] To achieve the above object, the present invention provides a method for solving the finite element node area weight, comprising the following steps:

[0006] Get the finite element nodes of the optical mirror;

[0007] Project the finite element nodes along the optical axis to the plane formed by the other two axes to obtain the node matrix;

[0008] According to the Delaunay triangulation condition, the node matrix is ​​triangulated to obtain a triangular network;

[0009] Get the centroid and the midpoints of the three sides of each triangle in the triangle network;

[0010] Split the triangle according to its centroid and the midpoints of its three sides;

[0011] Get all the quadrilaterals that wrap each node to form a polygon;

[0012] Calculate the area of ​​the polygon as the area weight of the wrapped node.

[0013] In some optional implementation schemes, obtaining the finite element nodes of the optical mirror includes:

[0014] The thermal load analysis of the optical mirror is carried out by using finite element analysis software to obtain the nodes of the optical mirror after finite element analysis.

[0015] In some optional implementations, the node matrix is ​​triangulated according to the Delaunay triangulation condition to obtain a triangular network including:

[0016] S31, constructing a triangle ΔP1P2P3 on the projection plane that includes all the nodes in the node matrix, and making sure that the three vertices of the triangle ΔP1P2P3 are not within the circumscribed circle of any three nodes in the node matrix;

[0017] S32, randomly select a node q in the node matrix, and analyze the positional relationship between the node q and all current triangles; if the node q is located in a certain triangle, then connect the node q and the three vertices of the triangle to form a new triangle; if the node q is located on an edge of a certain triangle, then connect the node q and the vertex corresponding to the edge to form a new triangle;

[0018] S33, judging whether the triangles after the triangulation meet the Delaunay triangulation condition; if so, saving the vertices of the triangles meeting the triangulation condition; if not, removing and flipping the illegal edges of the triangles not meeting the triangulation condition, and updating the triangular network;

[0019] S34, repeating step S32 and step S33 until all nodes in the node matrix are saved;

[0020] S35, deleting the three edges of triangle ΔP1P2P3 and the edges associated therewith, and obtaining a triangle network after Delaunay triangulation.

[0021] In some optional implementations, whether the triangulated triangles meet the Delaunay triangulation condition is determined according to the following formula:

[0022]

[0023] Where (xi, yi), (xj, yj) and (xk, yk) represent the coordinates of the three vertices of the triangle after segmentation; (xr, yr) represents the coordinates of node r in the node matrix;

[0024] If there is a node r in the node matrix such that s ijkr >0, the triangle after segmentation does not meet the segmentation condition; if there is no node r in the node matrix, s ijkr >0, the triangles after segmentation meet the segmentation conditions.

[0025] In some optional implementation schemes, removing and flipping illegal edges of triangles that do not meet the splitting conditions include:

[0026] The edge closest to the node r in the triangle that does not meet the splitting conditions is an illegal edge. The illegal edge is removed, and the node r is connected to the three vertices of the triangle that does not meet the splitting conditions.

[0027] In some alternative embodiments, the centroid c of the triangle n The calculation formula for (x,y) is as follows:

[0028]

[0029] In the formula, (x n1 ,y n1 )、(x n2 ,y n2 ) and (x n3 ,y n3 ) represents the coordinates of the three vertices of the nth triangle.

[0030] In some optional implementations, the area calculation formula of the polygon is as follows:

[0031]

[0032] Where Wn represents the area of ​​the nth polygon, that is, the area weight; M represents the number of all quadrilaterals constituting the polygon; Snk' represents the area of ​​the kth quadrilateral.

[0033] In some optional implementations, the area of ​​the quadrilateral is 1 / 3 of the area of ​​the segmented triangle; the area of ​​the segmented triangle is calculated as follows:

[0034]

[0035] In the formula, (x n1 ,y n1 )、(x n2 ,y n2 ) and (x n3 ,y n3 ) represents the coordinates of the three vertices of the nth triangle.

[0036] An electronic device includes one or more processors and a memory; one or more programs are stored in the memory and are configured to be executed by the one or more processors, and the one or more programs are configured to execute the above-mentioned finite element node area weight solution method.

[0037] The present invention also provides a computer-readable storage medium, in which a program code is stored, wherein the above-mentioned finite element node area weight solution method is executed when the program code is run.

[0038] Compared with the prior art, the present invention has the following advantages and beneficial effects:

[0039] The present invention uses the Delaunay triangulation algorithm to triangulate the projected nodes. Due to its empty circle characteristics and maximized minimum angle characteristics, the triangulated triangles are the closest, most regular and unique, which is the optimal triangulation result. The solution result can be directly used as the node weight of the rigid body displacement solution algorithm or the surface deformation fitting algorithm after removing the rigid body displacement, which is very important to improve the accuracy of removing the rigid body displacement and the accuracy of polynomial fitting in the optical-mechanical integration analysis. BRIEF DESCRIPTION OF THE DRAWINGS

[0040] Figure 1 A flow chart showing the general finite element node area weight solution method.

[0041] Figure 2 Represents the three-dimensional node data obtained by finite element analysis.

[0042] Figure 3 Represents the projection diagram of the finite element node along the optical axis.

[0043] Figure 4 Schematic diagram representing the projected node triangulation.

[0044] Figure 5 A schematic diagram showing the flipping of an illegal edge.

[0045] Figure 6 Represents the projection node triangulation result.

[0046] Figure 7 Represents the method of splitting and combining node area weighted polygons.

[0047] Figure 8 Represents the calculation results of finite element node weights.

[0048] Fig. 9 It represents the surface deformation fitting result after removing the rigid body displacement without considering the area weight.

[0049] Fig.10 It represents the surface deformation fitting result after the area weight is introduced to remove the rigid body displacement. DETAILED DESCRIPTION

[0050] In order to make the purpose, technical solutions and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not intended to limit the present invention. In addition, the technical features involved in the various embodiments of the present invention described below can be combined with each other as long as they do not conflict with each other.

[0051] The present invention discloses a general finite element node area indexing, partitioning and weight solution method, which is suitable for calculating the finite element node area weight in the optical-mechanical thermal integration analysis. The present invention first performs thermal load analysis on the optical mirror through finite element analysis software to obtain the node data after finite element analysis; the finite element node coordinates are projected to the plane along the optical axis direction, and the projected nodes are triangulated according to the Delaunary triangulation condition; according to the centroid coordinates of the triangle after partitioning and the midpoint positions of the three sides, the triangular network is partitioned and combined to obtain the polygon that wraps the node; according to the geometric relationship between the triangulation and the coordinates after partitioning, the area of ​​the polygon is obtained and used as the weight of the node. The calculation result of the present invention can be directly used as the node weight of the rigid body displacement solution algorithm and the surface fitting algorithm after removing the rigid body displacement for analysis and calculation, avoiding the problem of insufficient precision in the solution of the rigid body displacement or the surface deformation fitting after removing the rigid body displacement due to uneven finite element network partitioning or different density.

[0052] In the analysis example of the universal finite element node area indexing, partitioning and weight solving method of the present invention, a hyperbolic mirror with a diameter of 53mm, a quadratic surface coefficient of -2.106 and a curvature radius of -100.6025mm was subjected to a +60°C thermal analysis. The present invention is used to calculate the finite element node area weight and is brought into the rigid body displacement algorithm and the rigid body displacement surface deformation fitting algorithm for verification.

[0053] The specific implementation process is as follows: Figure 1 As shown, the specific steps include:

[0054] (1) Use finite element analysis software to perform thermal load analysis on the mirror surface and obtain the coordinate position P0 (x, y, z) of the node before deformation and the change in the node coordinate ΔP (x, y, z), as shown in Figure 2 shown.

[0055] (2) Project the finite element node coordinates P0 (x, y, z) along the optical axis to the plane formed by the other two axes to obtain the projected node matrix Q (x, y), as follows: Figure 3 shown.

[0056] (3) Take any three points P1(x1, y1), P2(x2, y2), and P3(x3, y3) on the projection plane, so that the triangle ΔP1P2P3 they form can contain all the projected nodes Q(x, y), and at the same time ensure that the three points P1, P2, and P3 are not in the circumscribed circle of any three points in Q(x, y). In this embodiment, the coordinates of the three points are P1(22, -60), P2(-50, 10), and P3(30, 40), respectively.

[0057] (4) Take any point q(x q ,y q )∈Q(x,y), and analyze its relationship with all current ΔP i P j P k The position relationship of i P j P k Indicates the triangle that has been obtained so far: in the first iteration, it represents the triangle ΔP1P2P3; in subsequent iterations, it refers to the newly formed triangle.

[0058] If the point q(x q ,y q ) is located at ΔP i P j P k If q(x q ,y q ) and ΔP i P j P k Three vertices form a new triangle, such as Figure 4 -a; if point q(x q ,y q ) is located at ΔP i P j P k If the point q is on an edge of , then the connection point q and the vertex corresponding to the edge form a new triangle, such as Figure 4 -b as shown.

[0059] (5) According to formula (1), determine whether the triangle after triangulation satisfies the Delaunay triangulation condition. In the formula, (xi, yi), (xj, yj) and (xk, yk) represent the coordinates of the three vertices of the triangle after triangulation; (xr, yr) represents the coordinates of the node r in the node matrix; if there is a node r in the node matrix such that s ijkr >0, it indicates that there is a node r located at ΔP i P j P k If the node matrix does not contain a node r, then s ijkr>0, the triangles after segmentation meet the segmentation conditions.

[0060]

[0061] If ijkr >0 does not meet the segmentation conditions, it is necessary to remove and flip the illegal edges of the triangle to update the current triangle network. Figure 5 As shown, PjPk is an illegal edge. And save the coordinates of the triangle vertices that meet the conditions in real time ni (x,y).

[0062] (6) Repeat steps (4) and (5) until all projection node coordinates are inserted and the triangulation of all nodes is completed.

[0063] (7) Delete the three edges ΔP1P2P3 and their associated edges to obtain the triangle set T after Delaunay triangulation n The result after mirror triangulation in this embodiment is as follows Figure 6 shown.

[0064] (8) Calculate the area of ​​each triangle after segmentation. n The coordinates of the three vertices of the nth triangle are t n1 (x n1 ,y n1 ), t n2 (x n2 ,y n2 ), t n3 (x n3 ,y n3 ), t n1 ,t n2 ,t n3 ∈T ni , the area of ​​the triangle is:

[0065]

[0066] After calculation, the area set of the triangles after segmentation is S n , that is, s n ∈S n .

[0067] (9) Calculate the centroid coordinate position c of each triangle after segmentation n (x,y), the calculation method is as follows

[0068]

[0069] After calculation, the centroid coordinate set of the triangle after segmentation is C n , that is, c n ∈Cn .

[0070] (10) Calculate the midpoint coordinates R of the three sides of each triangle nj (x,y)=[r n1 r n2 r n2 ] T , j∈{1,2,3}, representing the three edge indices of the triangle.

[0071]

[0072] (11) Connect lines from the centroid of the triangle to the midpoints of the three sides, and divide the triangle t n (t n ∈T n ) is divided into three sub-quadrilaterals. The specific division method is as follows Figure 7 As shown. From the mathematical relationship, we can see that the areas of the three sub-quadrilaterals are equal, namely:

[0073]

[0074] After calculation, the area set of the segmented quadrilaterals is S n ',s n '∈S n '.

[0075] (12) Traverse all node position coordinates and search for all quadrilaterals surrounding them to form a polygonal area. Let the number of quadrilaterals around the nth node be M, and the area of ​​a single quadrilateral be s nk ', then the area weight of the node is W n for:

[0076]

[0077] The area weight of all nodes is calculated as W n , W n ∈W n In this example, the face weight of the mirror node is Figure 8 shown.

[0078] (13) The calculation results of the node area weights are brought into the rigid body displacement solution algorithm and the surface deformation fitting algorithm after the rigid body displacement is removed, and the calculation results of the rigid body displacement are shown in Table 1. Under the conditions of considering the node weights and not considering the node area weights, the results of the mirror after the rigid body displacement is removed are subjected to the Zernike fitting, and the results are obtained. Fig. 9 and Fig.10 result.

[0079] Table 1 Comparison of rigid body displacement calculation results

[0080] Calculation results of rigid body displacement Node Weight Without node weight Rx 1.5799E-05 2.1846E-05 R 6.4525E-06 5.5829E-06 R 2.4282E-04 2.6788E-07 Tx 4.4532E-04 -1.2004E-04 Ty 6.6188E-04 3.1587E-04 Tz 0.0249E-00 0.0248E-00

[0081] An electronic device includes one or more processors and a memory; one or more programs are stored in the memory and are configured to be executed by the one or more processors, and the one or more programs are configured to execute the above-mentioned finite element node area weight solution method.

[0082] The present invention also provides a computer-readable storage medium, in which a program code is stored, wherein the above-mentioned finite element node area weight solution method is executed when the program code is run.

[0083] It should be pointed out that, according to the needs of implementation, the various steps / components described in this application can be split into more steps / components, and two or more steps / components or partial operations of steps / components can be combined into new steps / components to achieve the purpose of the present invention.

[0084] It will be easily understood by those skilled in the art that the above are only preferred embodiments of the present invention and are not intended to limit the present invention. Any modifications, equivalent substitutions and improvements made within the spirit and principles of the present invention should be included in the protection scope of the present invention.

Claims

1. A method for solving finite element node area weights, characterized in that: The following steps are involved: Get the finite element nodes of the optical mirror; Project the finite element nodes along the optical axis to the plane formed by the other two axes to obtain the node matrix; According to the Delaunay triangulation condition, the node matrix is ​​triangulated to obtain a triangular network; Get the centroid and the midpoints of the three sides of each triangle in the triangle network; Split the triangle according to its centroid and the midpoints of its three sides; Get all the quadrilaterals that wrap each node to form a polygon; Calculate the area of ​​the polygon as the area weight of the wrapped node.

2. The finite element node area weight solution method according to claim 1, characterized in that: The finite element nodes for obtaining optical mirrors include: The thermal load analysis of the optical mirror is carried out by using finite element analysis software to obtain the nodes of the optical mirror after finite element analysis.

3. The finite element node area weight solution method according to claim 1, characterized in that: According to the Delaunay triangulation condition, the node matrix is ​​triangulated to obtain a triangular network including: S31, constructing a triangle ΔP1P2P3 on the projection plane that includes all the nodes in the node matrix, and making sure that the three vertices of the triangle ΔP1P2P3 are not within the circumscribed circle of any three nodes in the node matrix; S32, randomly select a node q in the node matrix, and analyze the positional relationship between the node q and all current triangles; if the node q is located in a certain triangle, then connect the node q and the three vertices of the triangle to form a new triangle; if the node q is located on an edge of a certain triangle, then connect the node q and the vertex corresponding to the edge to form a new triangle; S33, judging whether the triangles after the triangulation meet the Delaunay triangulation condition; if so, saving the vertices of the triangles meeting the triangulation condition; if not, removing and flipping the illegal edges of the triangles not meeting the triangulation condition, and updating the triangular network; S34, repeating step S32 and step S33 until all nodes in the node matrix are saved; S35, deleting the three edges of triangle ΔP1P2P3 and the edges associated therewith, and obtaining a triangle network after Delaunay triangulation.

4. The method for solving the finite element node area weight according to claim 3, characterized in that: Determine whether the triangle after triangulation satisfies the Delaunay triangulation condition according to the following formula: Where (xi, yi), (xj, yj) and (xk, yk) represent the coordinates of the three vertices of the triangle after segmentation; (xr, yr) represents the coordinates of node r in the node matrix; If there is a node r in the node matrix such that s ijkr >0, the triangle after segmentation does not meet the segmentation conditions; If there is no node r in the node matrix, s ijkr >0, the triangles after segmentation meet the segmentation conditions.

5. The method for solving the finite element node area weight according to claim 4, characterized in that: Eliminating and flipping illegal edges of triangles that do not meet the splitting conditions include: The edge closest to the node r in the triangle that does not meet the splitting conditions is an illegal edge. The illegal edge is removed, and the node r is connected to the three vertices of the triangle that does not meet the splitting conditions.

6. The finite element node area weight solution method according to claim 1, characterized in that: The centroid of the triangle c n The calculation formula for (x,y) is as follows: In the formula, (x n1 ,y n1 )、(x n2 ,y n2 ) and (x n3 ,y n3 ) represents the coordinates of the three vertices of the nth triangle.

7. The finite element node area weight solution method according to claim 1, characterized in that: The area of ​​a polygon is calculated as follows: Where Wn represents the area of ​​the nth polygon, that is, the area weight; M represents the number of all quadrilaterals constituting the polygon; Snk' represents the area of ​​the kth quadrilateral.

8. The method for solving the finite element node area weight according to claim 7, characterized in that: The area of ​​the quadrilateral is 1 / 3 of the area of ​​the divided triangle; the area of ​​the divided triangle is calculated as follows: In the formula, (x n1 ,y n1 )、(x n2 ,y n2 ) and (x n3 ,y n3 ) represents the coordinates of the three vertices of the nth triangle.

9. An electronic device, characterized in that: including one or more processors and memory; One or more programs are stored in a memory and configured to be executed by one or more processors, and the one or more programs are configured to execute the finite element node area weight solution method described in any one of claims 1-8.

10. A computer-readable storage medium, characterized in that: A computer-readable storage medium stores program code, wherein when the program code is run, the finite element node area weight solution method described in any one of claims 1 to 8 is executed.

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