Finite element mesh rapid reconstruction method based on linear elastic mechanics
By using linear elasticity theory and virtual elastic body model, the problems of low computational efficiency and element distortion of finite element mesh under dynamic geometric changes are solved, and efficient and robust mesh reconstruction is achieved, which is suitable for scenarios such as structural shape optimization and topology optimization.
Patent Information
- Application Number
- CN202511032643.9
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-07-25
- Publication Date
- 2026-01-23
AI Technical Summary
Traditional finite element mesh reconstruction methods suffer from low computational efficiency, easy element distortion, and poor adaptability to large deformations in dynamic geometric change scenarios, making it difficult to meet the needs of real-time simulation.
By introducing the theory of linear elasticity and establishing a virtual elastic body model, mesh reconstruction is transformed into a continuous medium deformation process dominated by physical laws. By combining global linear elasticity solution and local adaptive correction technology, rapid mesh reconstruction is achieved.
It improves the efficiency and robustness of mesh reconstruction, enhances the adaptability to large deformations and high curvature geometry, avoids topology inconsistencies, and supports real-time processing of millions of meshes.
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Figure CN121389558A_ABST
Abstract
Description
TECHNICAL FIELD
[0001] The present application relates to the technical field of computer aided engineering (CAE), and is a high-efficiency finite element grid reconstruction method based on linear elasticity, which is suitable for adaptive updating of a grid when a geometric model dynamically changes in engineering simulation analysis. BACKGROUND Finite element simulation analysis, as a core technology of computer aided engineering (CAE), is widely used in simulation calculation in the fields of structural mechanics, engineering mechanics, fluid dynamics, heat and mass transfer, etc. The core idea is to discretize a complex continuum into a finite number of element grids, and to obtain the distribution of physical fields by solving partial differential equations. However, when the simulation involves dynamic changes in geometry (such as shape optimization, topology optimization, phase change process, structural changes in the product iteration process, etc.), the initial grid cannot adapt to the deformed geometry, resulting in problems such as element distortion, negative Jacobian determinant, etc., which seriously affect the calculation accuracy and convergence. Redoing the grid division requires a lot of time, which seriously affects the efficiency of structural strength evaluation. Therefore, the grid needs to be dynamically reconstructed to match the new geometry, while maintaining the element quality and topology rationality.
[0002] The existing dynamic mesh reconstruction method mainly relies on the geometry-driven Delaunay reconstruction method. The Delaunay reconstruction method (Shewchuk, J. R. (1996). Triangle: Engineering a 2D quality mesh generator and Delaunay triangular. In: Lin, M. C., Manocha, D. (eds) Applied Computational Geometry Towards Geometric Engineering. WACG 1996. Lecture Notes in Computer Science, vol 1148. Springer, Berlin, Heidelberg. http: / / doi.org / 10.1007 / BFb0014497) is a widely used mesh generation technique in computational geometry, which is to construct a triangular mesh with the optimal topological structure by satisfying the Delaunay criterion, that is, the circumcircle of any triangle does not contain other nodes. Its core advantage is to maximize the minimum internal angle of the mesh element, thereby avoiding the generation of sharp or degenerate triangular mesh, which significantly improves the mesh quality. In the dynamic reconstruction scene, the Delaunay method adapts to the geometric changes by inserting or deleting nodes: when the target geometry deforms, first detect the deviation area of the new geometric boundary and the original mesh, then incrementally insert new nodes and locally reconstruct the mesh, and delete invalid elements to maintain the Delaunay cavity property. And when extended to three dimensions, the tetrahedralization criterion needs to be introduced, and the adaptability to sharp features or thin-walled features is poor, and large-scale deformation will cause global topological failure, which needs to be frequently reconstructed. However, this method is sensitive to hanging nodes and curvature mutation areas, and needs to be combined with post-processing smoothing technology to repair the element quality, which further increases the computational burden, and the method generates triangular and tetrahedral elements, which is contrary to the requirement of quadrilateral or hexahedral element mesh partitioning in aircraft structure strength analysis. SUMMARY
[0003] The purpose of the present application is: The present application aims to solve the common problems of low computational efficiency, easy distortion of elements, and poor adaptability to large deformation of traditional finite element mesh reconstruction methods in dynamic geometric change scenarios, and to provide a mesh reconstruction scheme with high efficiency, robustness and geometric fidelity for engineering simulation. The traditional geometry-driven reconstruction method relies on local topology operation, although it can ensure the quality of the element, but the computational scale increases sharply with the increase of the grid scale, which is difficult to meet the real-time simulation demand. Although the pure displacement interpolation method is fast, it is easy to lead to element penetration, distortion and other non-physical deformation due to the lack of physical constraints. The present application introduces the theory of linear elasticity to construct a virtual elastic body model, and converts the mesh reconstruction into a continuous medium deformation process dominated by physical laws, which fundamentally ensures the coordination and rationality of the displacement field.
[0004] The technical scheme of the present application is: A finite element mesh rapid reconstruction method based on linear elasticity is provided, comprising the following steps: Step 1: Establishing a seeker antenna cover geometric model, performing finite element mesh division on the geometric model, comparing the geometric structure difference before and after deformation, identifying the undeformed region and the deformed region; the undeformed region and the deformed region are spatial regions; Step 2: Assigning virtual material properties to the finite element mesh, wherein the Young's modulus determines the anti-deformation stiffness of the medium, and the Poisson's ratio adjusts the volume compression characteristics; a first elastic modulus is applied in the undeformed region to maintain geometric integrity; a second elastic modulus is used in the deformed region; the first elastic modulus is greater than the second elastic modulus; Step 3: Classifying the grid nodes of the geometric model after the finite element mesh division before deformation, the grid nodes located in the spatial range of the undeformed region are fixed boundary nodes, the grid nodes located in the spatial interior of the deformed region are free deformation nodes, and the grid nodes located on the spatial surface of the deformed region are forced displacement nodes; The displacement vector of the fixed boundary node is set to 0, and the calculation formula of the displacement of the forced displacement node after the deformation processing is:
[0005] Wherein, is the displacement vector of the forced displacement node, is the coordinate of the forced displacement node before deformation, is the coordinate of the forced displacement node after deformation; Step 4, establishing linear elastic equilibrium equation according to finite element model
[0006] Wherein, is the displacement vector of all nodes of the finite element model, written as Wherein, The displacement vector of the fixed boundary node is The displacement vector of the free deformation node is The degree of freedom of the free deformation node is The degree of freedom of the fixed boundary node is The degree of freedom of the forced deformation node is The stiffness matrix of the finite element model is
[0007] Wherein, The stiffness matrix corresponding to the degree of freedom of the free deformation node is The stiffness matrix corresponding to the degree of freedom of the fixed boundary node is The stiffness matrix corresponding to the degree of freedom of the forced deformation node is And The coupling stiffness matrix corresponding to the degree of freedom of the free deformation node and the degree of freedom of the fixed boundary node is transposed to each other, And The coupling stiffness matrix corresponding to the degree of freedom of the free deformation node and the degree of freedom of the forced deformation node is transposed to each other, And The coupling stiffness matrix corresponding to the degree of freedom of the fixed boundary node and the degree of freedom of the forced deformation node is transposed to each other. The node load vector of the finite element model is set to 0. Step 5, establish linear elastic equilibrium equation under displacement constraint
[0008] Solve by applying Gaussian elimination method, get
[0009] Step 6, get all node displacement vectors Then, all node displacement vectors are superimposed to all corresponding node coordinates before deformation to obtain all node coordinates after deformation:
[0010] In the formula, The node coordinates after deformation are The node coordinates before deformation are kept unchanged in the corresponding relationship between the unit and the node number, and the node coordinates in the finite element solving file are modified to complete the finite element grid reconstruction. Step 7, the Jacobian determinant, aspect ratio, warping factor index of the reconstructed grid are calculated, and it is judged whether all the unit Jacobian determinants of the reconstruction are greater than 0.5, the aspect ratio is less than 5, and the warping factor is less than 6.5, if yes, then the finite element grid meets the use requirement, if not, the unit with too large distortion is modified, and steps 4 to 6 are repeated.
[0011] Further, the method for modifying the unit with too large distortion is to increase the second elastic modulus.
[0012] Further, the undeformed region includes structural edges and sharp corners.
[0013] Further, the Poisson's ratio is set to 0 to avoid the influence of coupling deformation in different directions.
[0014] Further, the first elastic modulus is at least 1000 times of the second elastic modulus, and further, the first elastic modulus is E=1000 and the second elastic modulus is E=1. The local flexibility is enhanced to adapt to the target deformation requirement.
[0015] Further, in step 4, the linear elastic equilibrium equation under displacement constraint can be processed by using the Lagrange multiplier method or the penalty function method, and the Gauss-Seidel iteration, the conjugate gradient method or the generalized least residual method is used for solving.
[0016] Further, in step 6, in addition to dynamically adjusting the local area material elastic modulus, a Laplace smoothing operator can be introduced to iteratively relax the distorted unit, so that the node position is adaptively migrated to the neighborhood set center, thereby restoring the unit quality.
[0017] Further, in step 5, when the deformation amount of the forced displacement node is too large An incremental loading strategy is used to decompose the forced displacement node displacement vector into a small deformation sequence of multiple stages: , wherein After each step is solved, the grid node coordinates are updated:
[0018] In the formula, is the coordinates of all grid nodes before deformation in the kth incremental step, is the coordinates of all grid nodes after deformation in the kth incremental step, is the node displacement vector in the kth incremental step.
[0019] Further, in step 5, the finite element software is used for solving, and the node displacement vector is directly extracted in the result file.
[0020] The advantages of the present application are: In comparison with the prior art, the technical effects of the product and device of the present application are improved, expanded and extended in combination with the technical effects mentioned in the core and secondary invention points, so as to improve the technical advantages and creativity of the present application.
[0021] (1) A physical and geometric dual-driven mesh deformation mechanism is established, the global displacement distribution is constrained by the linear elasticity constitutive equation, unreasonable unit deformation is inhibited, and the topological unreasonable problem caused by the traditional geometric interpolation method is avoided.
[0022] (2) The global linear elasticity solution is combined with the local adaptive correction technology, the efficiency and quality are considered, the adaptive ability to large deformation and high curvature geometry is enhanced, and the waste of computing resources caused by the traditional method due to layer-by-layer subdivision is avoided.
[0023] (3) A lightweight solving strategy is designed, the incremental step-by-step deformation loading strategy can handle large deformation conditions, the sparse matrix iterative algorithm and parallel computing technology are used to break through the real-time processing bottleneck of large-scale mesh, support the reconstruction of million-level mesh quickly, and expand the use scenarios of the method, such as structure shape optimization and topology optimization. BRIEF DESCRIPTION OF DRAWINGS
[0024] Figure 1 It is a schematic diagram of the original finite element mesh of the seeker antenna cover; Figure 2 It is a schematic diagram of the forced displacement boundary constraint list; Figure 3 It is a schematic diagram of the solution file before and after mesh deformation; Figure 4a It is a schematic diagram of the node coordinates of the solution file before deformation; Figure 4b It is a schematic diagram of the node coordinates of the solution file after deformation; Figure 5 It is a schematic diagram of the finite element mesh of the reconstructed seeker antenna cover. DETAILED DESCRIPTION
[0025] The disclosed examples will be described more fully with reference to the accompanying drawings, in which some, but not all, of the disclosed examples are shown. Indeed, a great variety of examples can be described and should be understood as not being necessarily mutually exclusive, but rather the various examples can be selectively combined into other examples. These examples are described in sufficient detail to facilitate the understanding of the disclosure and of exemplary, but not alternative, embodiments thereof. The examples described herein should be understood as being examples merely.
[0026] A finite element mesh rapid reconstruction method based on linear elasticity is provided, comprising the following steps: Step 1: Establish the seeker radome geometric model, perform finite element meshing on the geometric model, deform the geometric model, compare the geometric structure difference before and after deformation, and identify the undeformed region and the deformed region; the undeformed region and the deformed region are spatial regions; Step 2: Assign virtual material properties to the finite element mesh, wherein the Young's modulus determines the medium's anti-deformation stiffness, and the Poisson's ratio adjusts the volume compression characteristics; a first elastic modulus is applied in the undeformed region to maintain geometric integrity; a second elastic modulus is used in the deformed region; the first elastic modulus is greater than the second elastic modulus; Step 3: Classify the grid nodes of the deformed geometric model after finite element meshing, the grid nodes located in the spatial range of the undeformed region are fixed boundary nodes, the grid nodes located in the spatial interior of the deformed region are free deformation nodes, and the grid nodes located on the spatial surface of the deformed region are forced displacement nodes; The displacement vector of the fixed boundary node is set to 0, and the calculation formula of the displacement of the forced displacement node after deformation is:
[0027] Wherein, is the displacement vector of the forced displacement node, is the coordinate of the forced displacement node before deformation, is the coordinate of the forced displacement node after deformation; Step 4, establish linear elastic equilibrium equation according to finite element model
[0028] Wherein, is the displacement vector of all nodes of the finite element model, written as , wherein, is the displacement vector of the fixed boundary node, is the displacement vector of the free deformation node, is the degree of freedom of the free deformation node, is the degree of freedom of the fixed boundary node, is the degree of freedom of the forced deformation node; The stiffness matrix of the finite element model is:
[0029] Wherein, is the stiffness matrix corresponding to the degree of freedom of the free deformation node, is the stiffness matrix corresponding to the degree of freedom of the fixed boundary node, is the stiffness matrix corresponding to the degree of freedom of the forced deformation node, and is the coupling stiffness matrix corresponding to the freedom of the free deformation node and the freedom of the fixed boundary node, and is the transpose of each other, and is the coupling stiffness matrix corresponding to the freedom of the free deformation node and the freedom of the forced displacement node, and is the transpose of each other, and is the coupling stiffness matrix corresponding to the freedom of the fixed boundary node and the freedom of the forced displacement node, and is the transpose of each other. is the node load vector of the finite element model, and is set to 0. Step 5, establish the linear elastic equilibrium equation under displacement constraint
[0030] Solve by applying Gaussian elimination method, get
[0031] Step 6, get all node displacement vectors After that, all node displacement vectors are superimposed to all corresponding node coordinates before deformation to get all node coordinates after deformation:
[0032] In the formula, is the node coordinate after deformation, is the node coordinate before deformation, keep the corresponding relationship between the unit and the node number unchanged, modify the node coordinate in the finite element solving file to complete the finite element mesh reconstruction; Step 7, calculate the Jacobian determinant, aspect ratio and warping factor index of the reconstructed mesh, judge whether all the elements of the reconstructed mesh meet the requirements of Jacobian determinant > 0.5, aspect ratio < 5 and warping factor < 6.5, if yes, the finite element mesh meets the use requirements, if not, correct the elements with too large distortion and repeat steps 4-6.
[0033] The method for correcting the elements with too large distortion is to increase the second elastic modulus.
[0034] The undeformed area includes the structure edges and sharp corners.
[0035] The Poisson's ratio is set to 0 to avoid the influence of coupling deformation in different directions.
[0036] The first elastic modulus is at least 1000 times of the second elastic modulus. Further, the first elastic modulus is E=1000 and the second elastic modulus is E=1. The local flexibility is enhanced to adapt to the target deformation requirement.
[0037] In step 4, the linear elastic equilibrium equation under displacement constraint can be handled by Lagrange multiplier method or penalty function method, and solved by Gauss-Seidel iteration, conjugate gradient method, generalized minimum residual method.
[0038] In step 6, in addition to dynamically adjusting the local area material elastic modulus, a Laplacian smoothing operator can also be introduced to iteratively relax the distorted elements, so that the node position is adaptively migrated to the center of the neighborhood set, thereby restoring the element quality.
[0039] In step 5, when the deformation of the forced displacement node is too large, an incremental loading strategy is adopted to decompose the forced displacement node displacement vector into a small deformation sequence of multiple stages: , where After each step, update the grid node coordinates:
[0040] In the formula, is the coordinates of all grid nodes before deformation in the kth incremental step, is the coordinates of all grid nodes after deformation in the kth incremental step, is the node displacement vector in the kth incremental step.
[0041] Step 5, solve by finite element software, and directly extract the node displacement vector in the result file.
[0042] The descriptions of the different advantageous arrangements have been presented for purposes of illustration and description. They are not intended to be exhaustive or to limit the example to the forms disclosed. Many modifications and variations will be apparent to those of ordinary skill in the art. Different advantageous examples can provide different advantages than the other examples. The embodiments were chosen and described in order to best explain the principles of the example, the practical application, and to enable others skilled in the art to understand the example with various modifications being adapted to the particular use being contemplated.
Claims
1. A fast finite element mesh reconstruction method based on linear elasticity, characterized in that, The method comprises the following steps: Step 1: a seeker antenna cover geometric model is established, finite element mesh is divided on the geometric model, the geometric model is subjected to deformation processing, the geometric structure difference before and after deformation is compared, and the undeformed region and the deformed region are identified; The undeformed region and the deformed region are spatial regions; Step 2: virtual material properties are assigned to the finite element mesh, wherein the Young's modulus determines the anti-deformation stiffness of the medium, the Poisson's ratio adjusts the volume compression characteristics, the first elastic modulus is applied in the undeformed region to maintain geometric integrity; The second elastic modulus is used in the deformed region; the first elastic modulus is greater than the second elastic modulus; Step 3: the grid nodes of the geometric model after the finite element mesh division before deformation are classified, the grid nodes located in the spatial range of the undeformed region are fixed boundary nodes, and the grid nodes located in the spatial interior of the deformed region are free deformation nodes; The grid nodes located on the spatial surface of the deformed region are forced displacement nodes; The displacement vector of the fixed boundary node is set to 0, and the calculation formula of the displacement of the forced displacement node after the deformation processing is: wherein, is the displacement vector of the forced displacement node, is the coordinate of the forced displacement node before deformation, is the coordinate of the forced displacement node after deformation; Step 4, linear elastic equilibrium equation is established according to the finite element model wherein, is the displacement vector of all nodes of the finite element model, written as wherein, is the displacement vector of the fixed boundary nodes, is the displacement vector of the free-form nodes, is the degree of freedom of the free-form nodes, is the degree of freedom of the fixed boundary nodes, is the degree of freedom of the forced deformation nodes; is the stiffness matrix of the finite element model: wherein, Kff is the stiffness matrix corresponding to the freedom of the free deformation node, Kbb is the stiffness matrix corresponding to the freedom of the fixed boundary node, Kdd is the stiffness matrix corresponding to the freedom of the forced displacement node, and Kfb is the coupling stiffness matrix corresponding to the freedom of the free deformation node and the fixed boundary node, and are transposes of each other, and Kfd is the coupling stiffness matrix corresponding to the freedom of the free deformation node and the forced displacement node, and are transposes of each other, and Kbd is the coupling stiffness matrix corresponding to the freedom of the fixed boundary node and the forced displacement node, and are transposes of each other. is the nodal load vector for the finite element model, set to 0; Step 5, linear elastic equilibrium equation under displacement constraint is established Gaussian elimination method is applied to solve, and Step 6, get all node displacement vectors After that, all node displacement vectors are superimposed to all corresponding node coordinates before deformation to get all node coordinates after deformation: In the formula, are the deformed node coordinates, are the node coordinates before deformation, the correspondence between the holding unit and the node number is unchanged, and the node coordinates in the finite element solution file are modified to complete the finite element mesh reconstruction; Step 7, Jacobian determinant, aspect ratio and warping factor indexes of the reconstructed grid are calculated, whether all the unit Jacobian determinants are greater than 0.5, the aspect ratio is less than 5, and the warping factor is less than 6.5, if yes, the finite element mesh meets the use requirement, if not, the unit with excessive distortion is corrected, and steps 4-6 are repeatedly performed.
2. The fast reconstruction method of a linear elastic finite element mesh based on the linear elasticity mechanics according to claim 1, characterized in that: The method for correcting the unit with excessive distortion is to increase the second elastic modulus.
3. The fast reconfiguration method of a linear-elasticity-based finite element mesh according to claim 1, wherein: The undeformed region comprises structural edges and sharp corners.
4. The fast reconfiguration method of a linear-elasticity-based finite element mesh according to claim 1, wherein: The Poisson's ratio is set to 0.
5. The fast reconfiguration method of a linear-elasticity-based finite element mesh according to claim 1, wherein: The first elastic modulus is at least 1000 times of the second elastic modulus.
6. The fast reconfiguration method of a linear-elasticity-based finite element mesh according to claim 5, characterized in that: Further, the first elastic modulus is E=1000, and the second elastic modulus is E=1.
7. The fast reconfiguration method of a linear-elasticity-based finite element mesh according to claim 1, wherein: In step 4, the linear elastic equilibrium equation under displacement constraint can be processed by using the Lagrange multiplier method or the penalty function method, and is solved by using the Gauss-Seidel iteration, the conjugate gradient method or the generalized least residual method.
8. The fast reconfiguration method of a linear-elasticity-based finite element mesh according to claim 1, wherein: In step 6, in addition to dynamically adjusting the local region material elastic modulus, a Laplace smoothing operator can be introduced to iteratively relax the distorted unit, so that the node position is adaptively migrated to the neighborhood center, thereby restoring the unit quality.
9. The fast reconfiguration method of a linear-elasticity-based finite element mesh according to claim 1, wherein: In Step 5, when the forced displacement node deformation is too large, an incremental loading strategy is adopted, and the forced displacement node displacement vector is decomposed into a multi-stage small deformation sequence: where The mesh node coordinates are updated after each step of solving: wherein is the coordinates of all mesh nodes before deformation at the kth increment step, is the coordinates of all mesh nodes after deformation at the kth increment step, is the node displacement vector at the kth increment step.
10. The fast reconfiguration method of a linear-elasticity-based finite element mesh according to claim 1, wherein: In step 5, the finite element software is used for solving, and the node displacement vector is directly extracted in the result file.