A parametric migration method for isogeometric analysis based on discrete geometric mapping

Abstract

The present invention discloses a method for isogeometric analysis parametric migration based on discrete geometric mapping, comprising the following steps: step (1) extracting the sequential boundaries of a source model; step (2) inputting the source model and the target model, and making the boundary control points of the two CAD boundary models of the source model and the target model correspond one to one; step (3) constructing a geometric mapping between discrete grids; and step (4) optimizing the flipped facets in the mapping result obtained in step (3). The discrete mapping proposed in the present invention can highly approximate the maintenance of the G1 continuity of the source parameterized model without the need for a complex optimization process of the G1 continuity constraint during the parameterization process. The present invention uses discrete geometric mapping to perform a parametric migration process of similar models, which can effectively reduce the time cost of parameterizing the target model.

Description

Technical Field

[0001] The present invention relates to the technical field of geometric model processing, and in particular to an isogeometric analysis parameterization migration method based on discrete geometric mapping. Background Art

[0002] The results of geometric model parameterization in isogeometric analysis will have a great impact on the calculation accuracy and efficiency of subsequent simulation analysis on the model. At present, there are many mature methods for single-piece spline parameterization of two-dimensional geometric models that meet the requirements of isogeometric analysis. However, if the region is more complex, it is necessary to divide the two-dimensional plane region into several sub-regions, and parameterize these sub-regions separately to obtain the parameterization of the overall boundary model. At present, there are also some relatively mature regional decomposition parameterization methods such as skeleton-based, approximate convex decomposition, polySquare, etc. However, this type of parameterization method usually has a complex optimization process and a high computational cost. Summary of the invention

[0003] In view of the shortcomings of the prior art, the present invention proposes an isogeometric analysis parameterization migration method based on discrete geometric mapping. For two or more complex CAD boundary models with similar shapes, the result of regional decomposition parameterization of one model is known, and an approximate G1 continuous parameterization of the target model can be generated without applying a complex regional decomposition algorithm to the target model.

[0004] In order to solve the above technical problems, the technical solution of the present invention is:

[0005] A parameterized migration method for isogeometric analysis based on discrete geometric mapping includes the following steps:

[0006] Step (1) extracting the sequence boundary of the source model;

[0007] Step (2) inputting a source model and a target model, and making the boundary control points of the two CAD boundary models of the source model and the target model correspond one to one;

[0008] Step (3) Construct geometric mapping between discrete grids

[0009] 3-1. Generate source grid and target grid;

[0010] 3-2. Construct a discrete harmonic mapping that approximately preserves G1 continuity;

[0011] 3-3. Input the generated source mesh and target mesh into discrete harmonic mapping to obtain the geometric mapping result from the source mesh to the target mesh;

[0012] Step (4) optimizes the flipped face in the mapping result obtained in step (3).

[0013] Preferably, the method for extracting the source model sequence boundary in step (1) is:

[0014] Traverse each piece in the source model. If one of its four edges does not have an adjacent face, we consider it to be the outer boundary of the model, mark it and extract it, and then recursively search for boundary segments adjacent to the boundary segment starting from one of the boundary segments, until the sequential boundary of the source model is completely extracted.

[0015] Preferably, the step (2) comprises the following sub-steps:

[0016] 2-1. First, determine the correspondence of some key points on the boundaries of the source model and the target model, sample the boundaries of the source model and the target model to obtain discrete point sets, and count the arc lengths between two adjacent key points on the source model and the target model respectively;

[0017] 2-2. Calculate the proportion of each spline curve on the source model boundary to its arc length, take a small arc length segment of corresponding proportion on the corresponding arc length segment on the target model as the curve segment corresponding to the original model, and fit the target small arc length segment with the representation of the spline curve of the source model according to the sampling points of the small arc length segment on the target model. Process each spline curve segment on the source model boundary in turn, and finally represent the target model with the boundary curve form of the source model to obtain a one-to-one correspondence between the boundary control points of the source model and the target model.

[0018] Preferably, in step 3-1, the method for generating the source mesh and the target mesh is: based on the Triangle library, the boundary control points of the source model and the boundary control points of the target model are used as constraints to construct the Delaunay triangulation results, thereby generating the source mesh and the target mesh.

[0019] Preferably, in step 3-2, in order to obtain a discrete harmonic mapping φ that approximately preserves the continuity of G1 12 The energy optimization function is defined as follows:

[0020] E=E[φ 12 ]+λ1E boundary

[0021] stBφ 12 (P)=0

[0022] in,

[0023]

[0024]

[0025] Where E[φ 12 ] represents the discrete harmonic energy, which is used to measure the distortion of the mapping, Eboundary Describes the constraints on the correspondence between the source and target grid boundary marker points, λ1 measures the degree of dependence on the boundary markers; Bφ 12 (P) = 0 indicates that the target model maintains the continuity of the source model G1 after mapping. The initial mapping distortion is large, and λ1 is as small as possible. When the distortion is continuously reduced, λ1 is continuously increased to satisfy the constraint of the corresponding boundary marker points of the source mesh and the target mesh. uv Represents an edge in the source mesh<u,v> The cotangent weight value of<p,q> i=1,…,r represents any pair of corresponding marked points in the source mesh and the target mesh, P represents the internal vertices of the source mesh, and B represents the constraint matrix that maintains the continuity of G1.

[0026] Preferably, the optimization method of step (4) is:

[0027] According to the definition of non-flipping bounded conformal distortion mentioned in the Foldover-free method, plus the constraint of preserving G1 continuity defined in step (3) of this paper, the optimization energy function is defined as follows:

[0028]

[0029]

[0030] A u =b,B u =0

[0031] Among them, E d Represents each facet t of the flipped face in the geometric mapping result i ,i=1,…,N Jacobian matrix J i Projection into bounded conformal warp space On, τ(H i ) describes the projected back piece t i The conformal distortion, A u =b indicates that the mapping needs to satisfy the correspondence on the boundary, B represents the constraint matrix that maintains the continuity of G1, and the unknown quantity is solved and the position u of the mesh vertex after flipping optimization is obtained through an alternating algorithm.

[0032] Preferably, the method further includes step (5) parameterizing migration results and quality evaluation.

[0033] Preferably, in step (5), the evaluation method of parameterized migration is:

[0034] The two common indicators of the normalized Jacobian matrix and the condition number of the Jacobian matrix are used to measure whether the parameterization result of the target boundary after parameterization migration meets the requirements of isogeometric analysis. For a known parameterization B(u,v), let J be the Jacobian matrix of the mapping B,

[0035] Assumption: B(u,v)=(x(u,v),y(u,v)) T ,x(u,v),y(u,v)∈R

[0036] Then the canonical Jacobian is calculated as follows:

[0037]

[0038] Another important evaluation metric is the condition number of the Jacobian matrix calculated as follows

[0039] K(J)=||J||||J -1 ||,||J||=(tr(J T J)) 1 / 2

[0040] in

[0041]

[0042] Experiments have shown that the closer the parameterized normed Jacobian matrix is ​​to 1.0, the better the parameterization quality is. Similarly, the closer the condition number of the parameterized Jacobian matrix is ​​to 2.0, the better the parameterization quality is.

[0043] The present invention has the following characteristics and beneficial effects:

[0044] The present invention utilizes discrete geometric mapping to perform the parametric migration process of similar models, which can effectively reduce the time cost of parameterizing the target model. When processing complex CAD models, the traditional isogeometric analysis parameterization method will first divide the model, and then perform a separate parameterization operation on each sub-block of the division. The whole process involves a very complex optimization process. If the two models are very similar, that is, the shape of the target model and the shape of the source model after rigid transformation or articulated transformation are very similar, the mapping can be used to migrate the parameterization of the source model to the target model to avoid the process of segmentation optimization. Since the target model only has boundary splines and the source model facets are in the form of high-order polynomials, it is very difficult to directly perform the parameterization migration operation, but the discrete geometric mapping between grids is very convenient. Therefore, this paper performs a triangulation operation to generate a background grid to transfer the migration process between grids, which can reduce the time cost while ensuring that the parameterization results meet the subsequent analysis.

[0045] In addition, the discrete mapping proposed in the present invention can highly approximately maintain the G1 continuity of the source parameterized model. The discrete mapping itself can only make the mapping result satisfy G0 continuity, but by applying the constraints of this article, the obtained parameterized migration result can approximately satisfy the G1 continuity constraint without the need for a complex optimization process of the G1 continuity constraint during the parameterization process. BRIEF DESCRIPTION OF THE DRAWINGS

[0046] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings required for use in the embodiments or the description of the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative labor.

[0047] Figure 1 It is a flowchart of the steps of an embodiment of the present invention.

[0048] Figure 2 It is the parameterization result of the source model and the target boundary model diagram in the embodiment of the present invention.

[0049] Figure 3 It is a schematic diagram of the result of using the boundary curve of the source model to fit the target boundary model in an embodiment of the present invention.

[0050] Figure 4 It is a schematic diagram of the background mesh of the source model and the background mesh of the target model after triangulation in an embodiment of the present invention.

[0051] Figure 5 It is a result diagram of the background grid of the source model in the embodiment of the present invention being subjected to discrete mapping and flip optimization to the target grid.

[0052] Figure 6 It is a schematic diagram of the final result of parameterized migration according to an embodiment of the present invention. DETAILED DESCRIPTION

[0053] It should be noted that, in the absence of conflict, the embodiments of the present invention and the features in the embodiments may be combined with each other.

[0054] In the description of the present invention, it should be understood that the terms "center", "longitudinal", "lateral", "up", "down", "front", "back", "left", "right", "vertical", "horizontal", "top", "bottom", "inside", "outside" and the like indicate positions or positional relationships based on the positions or positional relationships shown in the accompanying drawings, and are only for the convenience of describing the present invention and simplifying the description, rather than indicating or implying that the device or element referred to must have a specific orientation, be constructed and operated in a specific orientation, and therefore cannot be understood as limiting the present invention. In addition, the terms "first", "second", and the like are only used for descriptive purposes, and cannot be understood as indicating or implying relative importance or implicitly indicating the number of technical features indicated. Thus, features defined as "first", "second", and the like may explicitly or implicitly include one or more of the features. In the description of the present invention, unless otherwise specified, "multiple" means two or more.

[0055] In the description of the present invention, it should be noted that, unless otherwise clearly specified and limited, the terms "installed", "connected", and "connected" should be understood in a broad sense, for example, it can be a fixed connection, a detachable connection, or an integral connection; it can be a mechanical connection or an electrical connection; it can be a direct connection, or it can be indirectly connected through an intermediate medium, or it can be the internal communication of two components. For ordinary technicians in this field, the specific meanings of the above terms in the present invention can be understood by specific circumstances.

[0056] The present invention provides a parameterized migration method of isogeometric analysis based on discrete geometric mapping, such as Figure 1 As shown, the following steps are included:

[0057] Step (1) extracts the sequential boundary of the source model. Traverse each piece in the source model. If one of its four edges does not have an adjacent face, we consider it to be the outer boundary of the model. We mark it and extract it. Then, we start from one of the boundary segments and recursively search downward for the boundary segments adjacent to the boundary segment until the sequential boundary of the source model is completely extracted. Figure 2 shown.

[0058] Specifically,

[0059] 1-1. Extract the sequential boundaries of the source model. First, mark the boundaries of each face of the source model. If there is no adjacent edge, the edge is a boundary edge, and the edgemark is marked as -1. After marking all faces, traverse all faces and extract boundary edges to obtain the overall unordered boundary model of the original model.

[0060] 1-2. To facilitate the subsequent boundary correspondence, the disordered boundary model is subjected to sequential operations. That is, starting from a boundary curve segment, the adjacent boundary curve segments are searched in sequence, and the sequential boundary of the source model can be obtained by recursively performing the above operations.

[0061] Step (2) inputs the source model and the target model, and makes the boundary control points of the two CAD boundary models of the source model and the target model correspond one to one.

[0062] The specific implementation method is as follows:

[0063] 2-1. First, determine the correspondence of some key points on the boundaries of the source model and the target model, sample the boundaries of the source model and the target model to obtain discrete point sets, and count the arc lengths between two adjacent key points on the source model and the target model respectively;

[0064] 2-2. Calculate the proportion of each spline curve on the source model boundary to the arc length where it is located, take a small arc length segment of the corresponding proportion on the corresponding arc length segment on the target model as the curve segment corresponding to the original model, and fit the target small arc length segment with the representation of the spline curve of the source model according to the sampling points of the small arc length segment on the target model. Process each spline curve segment on the boundary of the source model in turn, and finally represent the target model with the boundary curve form of the source model, such as Figure 3 As shown, the boundary control points of the source model and the target model correspond one to one.

[0065] Step (3) Construct geometric mapping between discrete grids

[0066] 3-1. Generate source grid and target grid.

[0067] Specifically, Figure 4 As shown, the method for generating the source mesh and the target mesh is: based on the Triangle library, the boundary control points of the source model and the boundary control points of the target model are used as constraints to construct the Delaunay triangulation results, thereby generating the source mesh and the target mesh.

[0068] 3-2. Construct a discrete harmonic mapping that approximately preserves G1 continuity. Construct a mapping that maps the source grid to the target grid. The input of the mapping is the source grid and the target grid, as well as the one-to-one correspondence constraints of the grid boundary points.

[0069] Specifically, in order to obtain a discrete harmonic mapping φ that approximately preserves the continuity of G1 12 The energy optimization function is defined as follows:

[0070] E=E[φ 12 ]+λ1E boundary

[0071] stBφ 12 (P)=0

[0072] in,

[0073]

[0074]

[0075] Where E[φ 12 ] represents the discrete harmonic energy, which is used to measure the distortion of the mapping, E boundary Describes the constraints on the correspondence between the source and target grid boundary marker points, λ1 measures the degree of dependence on the boundary markers; Bφ 12(P) = 0 indicates that the target model maintains the continuity of the source model G1 after mapping. The initial mapping distortion is large, and λ1 is as small as possible. When the distortion is continuously reduced, λ1 is continuously increased to satisfy the constraint of the corresponding boundary marker points of the source mesh and the target mesh. uv Represents an edge in the source mesh<u,v> The cotangent weight value of<p,q> i=1,…,r represents any pair of corresponding marked points in the source mesh and the target mesh, P represents the internal vertices of the source mesh, and B represents the constraint matrix that maintains the continuity of G1.

[0076] Further explanation, such as Figure 5 As shown, in this embodiment, it is assumed that the mapping of each point in the source grid to a certain surface of the target grid is φ 12 , usually, the mapping φ 12 The quality of φ can be considered from the following two aspects: 1. The distortion of the mapping. 2. Whether the mapping is bijective, that is, whether a point in the source grid corresponds to a point in the target grid surface. 12 The quality directly determines the location of the internal control points of the source model mapped to the internal part of the target model, that is, the quality of the parameterized migration result. The present invention constructs the mapping from the source mesh to the target mesh based on discrete harmonic mapping. In discrete differential geometry, the degree of distortion of the mapping can be measured by discrete Dirichlet energy. For discrete triangular meshes, Dirichlet energy is defined as follows:

[0077]

[0078] where dφ 12 (f)∈R 2*2 Any patch f∈F1 in the source mesh and its mapped 12 The description of the linear transformation between the obtained patches f, a f Describes the area of ​​the patch f, the above energy can be equivalently described as:

[0079]

[0080] w uv Represents an edge in the source mesh<u,v> The cotangent weight value of .

[0081] Furthermore, if the Dirichlet energy is directly used as the optimization objective function and the boundary control point set is used as the constraint for optimization, the result of mapping the source grid to the target grid will be flipped and degraded, resulting in flipping and degradation of the parameterized migration result, which does not meet the requirements of subsequent geometric analysis. In addition, the discrete mapping is a piecewise linear mapping and cannot maintain the G1 continuity of the source segmentation parameterization result. Based on the RHM method, according to the G1 continuity of the source model and the constraints corresponding to the boundary, this paper defines the total optimization energy function as follows:

[0082] E=E[φ 12 ]+λ1E boundary

[0083] Where E[φ 12 ] represents the discrete harmonic energy, which is used to measure the distortion of the mapping. In this embodiment, it is hoped that it will be as small as possible during the optimization process; E boundary The constraints corresponding to the boundary marker points of the source grid and the target grid are described. It is hoped that the boundary points after mapping are as close as possible to the boundary points of the target grid. Therefore, in the iterative process, the size of λ1 is continuously increased; B is the constraint matrix for maintaining the continuity of the source model G1. Experiments show that the parameterized migration result after mapping in the present invention is highly close to G1 continuity.

[0084]

[0085]

[0086] By combining semi-quadratic splitting and block coordinates to optimize the above energy E, the final mapping result is obtained, as shown in Figure 5 shown.

[0087] 3-3. Input the generated source mesh and target mesh into discrete harmonic mapping to obtain the geometric mapping result from the source mesh to the target mesh;

[0088] Step (4) optimizes the flipped face in the mapping result obtained in step (3).

[0089] Specifically, according to the Foldover-free method, in order to ensure that the discrete mapping result does not have flipped triangles, it is proposed to limit the distortion of the patch to the range of bounded conformal distortion, so that the patch does not flip. i , i=1,…,N represents any patch t i ,i=1,…,N, the Jacobian matrix for J i Perform singular value decomposition SVD to get J i =U i S i V i T , where Ui and V i is an orthogonal matrix, S i =diag(σ i,1 ,σ i,2 ), assuming that σ i,1 ≥σ i,2 In order to define a bounded conformal distortion without flipping, this embodiment uses the SSVD to decompose the Jacobian matrix. When detJ i ≥0, SSVD is equivalent to SVD, otherwise, U i and V i Change to a rotation matrix with a minimum singular value less than zero, where the conformal distortion is defined as:

[0090]

[0091] When the smallest singular value σ i,1 ≥σ i,2 When τ(J i )≥1, i=1,…,N, patch t i Without flipping, the bounded conformal mapping that preserves G1 continuity needs to satisfy:

[0092] 1≤τ(J i )≤k i ,i=1,…,N

[0093] Au=b,keep G1 continuity

[0094] Where 1≤τ(J i )≤k i , i=1,…,N is equivalent to adding the patch t i The Jacobian matrix J i Projection into bounded conformal warp space In the above, Au=b indicates that the mapping needs to satisfy the correspondence on the boundary. The unknown quantities in the above formula, namely the upper bound k of the bounded conformal distortion and the coordinates u of the optimized mesh vertices, are solved by the alternating algorithm.

[0095] Solution for k. If all flips cannot be eliminated after projection, the upper bound k of the bounded vacuum interval may be very small, and a larger k should be assigned, as shown in the following formula, where β increases with the number of iterations.

[0096]

[0097] After determining k, try to convert the patch t i , the Jacobian matrix J of i=1,…,N i , i = 1, ..., N projected onto bounded conformal warp space i=1,…,N to update the coordinates u of the grid points. The energy optimization function of this process is defined as follows:

[0098]

[0099]

[0100] A u =b,B u =0

[0101] The unknown quantities in the above formula are u and H i , determine H by local-global method i and point u, and in the process of determining u, the boundary correspondence Au=b is satisfied, maintaining the continuity of G1.

[0102] Among them, E d Represents each facet t of the flipped face in the geometric mapping result i ,i=1,…,N Jacobian matrix J i Projection into bounded conformal warp space On, τ(H i ) describes the projected back piece t i The conformal distortion, A u =b indicates that the mapping needs to satisfy the correspondence on the boundary, B represents the constraint matrix that maintains the continuity of G1, and the unknown quantity is solved and the position u of the mesh vertex after flipping optimization is obtained through an alternating algorithm.

[0103] Furthermore, the method also includes step (5) parameterizing the migration results and quality evaluation.

[0104] After obtaining the mapping result based on the background grid, we can get the position of the internal control points of the source model after being mapped to the internal part of the target model. Since the boundary control points are known, and the order and node vectors are also known, we can further get the result after parametric migration. The method in this paper was used to test the parametric migration results of some complex CAD boundary models with similar shapes. One of the sample results is shown in the attached figure. Figure 6 As shown. And through data analysis, it is observed that the results of parameterized migration are highly close to G1 continuity. In addition, the quality of parameterized migration results is measured by indicators such as the normalized Jacobian and the condition number of the Jacobian matrix, verifying that it is suitable for the requirements of isogeometric analysis.

[0105] Specifically, the evaluation method of parameterized migration is:

[0106] The two common indicators of the normalized Jacobian matrix and the condition number of the Jacobian matrix are used to measure whether the parameterization result of the target boundary after parameterization migration meets the requirements of isogeometric analysis. For a known parameterization B(u,v), let J be the Jacobian matrix of the mapping B,

[0107] Assumption: B(u,v)=(x(u,v),y(u,v)) T ,x(u,v),y(u,v)∈R

[0108] Then the canonical Jacobian is calculated as follows:

[0109]

[0110] Another important evaluation metric is the condition number of the Jacobian matrix calculated as follows

[0111] K(J)=||J||||J -1 ||,||J||=(tr(J T J)) 1 / 2

[0112] in

[0113]

[0114] like Figure 6 As shown, experiments have shown that the closer the parameterized normed Jacobian matrix is ​​to 1.0, the better the parameterization quality is. Similarly, the closer the condition number of the parameterized Jacobian matrix is ​​to 2.0, the better the parameterization quality is.

[0115] The embodiments of the present invention are described in detail above with reference to the accompanying drawings, but the present invention is not limited to the described embodiments. For those skilled in the art, various changes, modifications, substitutions and variations of these embodiments including components are made without departing from the principles and spirit of the present invention, and still fall within the scope of protection of the present invention.

Claims

1. A parameterized migration method for isogeometric analysis based on discrete geometric mapping, characterized in that: The steps include: Step (1) extracts the sequence boundary of the source model. The method for extracting the sequence boundary of the source model is: Traverse each piece in the source model. If one of its four edges does not have an adjacent face, it is considered to be the outer boundary of the model. It is marked and extracted. Then, starting from one of the boundary segments, recursively search downward for the boundary segments adjacent to the boundary segment until the sequential boundary of the source model is completely extracted. Step (2) inputting a source model and a target model, and making the boundary control points of the two CAD boundary models of the source model and the target model correspond one to one; 2-1. First, determine the correspondence of some key points on the boundaries of the source model and the target model, sample the boundaries of the source model and the target model to obtain discrete point sets, and count the arc lengths between two adjacent key points on the source model and the target model respectively; 2-2. Calculate the proportion of each spline curve on the source model boundary to the arc length where it is located, take a small arc length segment of the corresponding proportion on the corresponding arc length segment on the target model as the curve segment corresponding to the original model, and fit the target small arc length segment with the representation of the spline curve of the source model according to the sampling points of the small arc length segment on the target model. Process each spline curve segment on the source model boundary in turn, and finally represent the target model with the boundary curve form of the source model, so as to obtain a one-to-one correspondence result between the boundary control points of the source model and the target model. Step (3) Construct geometric mapping between discrete grids 3-1. Generate source grid and target grid; The method of generating the source mesh and the target mesh is as follows: based on the Triangle library, the boundary control points of the source model and the boundary control points of the target model are used as constraints to construct the Delaunay triangulation results, thereby generating the source mesh and the target mesh; 3-2. Construct a discrete harmonic mapping that approximately preserves G1 continuity; In order to obtain a discrete harmonic map φ that approximately preserves the continuity of G1 12 The energy optimization function is defined as follows: E=E[φ 12 ]+λ1E boundary s.t.Bφ 12 (P)=0 in, Where E[φ 12 ] As shown above, the discrete harmonic energy is used to measure the distortion of the mapping, E boundary Describes the constraints on the correspondence between the source and target grid boundary marker points, λ1 measures the degree of dependence on the boundary markers; Bφ 12 (P) = 0 indicates that the target model maintains the continuity of the source model G1 after mapping. The initial mapping distortion is large, and λ1 is as small as possible. When the distortion is continuously reduced, λ1 is continuously increased to satisfy the constraint of the corresponding boundary marker points of the source mesh and the target mesh. uv Represents an edge in the source mesh<u,v> The cotangent weight value of<p,q> i=1,…,r represents any pair of corresponding marked points in the source mesh and the target mesh, P represents the internal vertices of the source mesh, and B represents the constraint matrix that maintains the continuity of G1; 3-3. Input the generated source mesh and target mesh into discrete harmonic mapping to obtain the geometric mapping result from the source mesh to the target mesh; Step (4) optimizes the flipped face in the mapping result obtained in step (3).

2. The isogeometric analysis parameterization migration method based on discrete geometric mapping according to claim 1 is characterized in that: The optimization method of step (4) is: According to the definition of non-flipping bounded conformal distortion mentioned in the Foldover-free method, plus the constraint of preserving G1 continuity defined in step (3), the optimization energy function is defined as follows: A u =b,B u =0 Among them, E d Represents each facet t of the flipped face in the geometric mapping result i ,i=1,…,N Jacobian matrix J i Projection into bounded conformal warp space On, τ(H i ) describes the projected back piece t i The conformal distortion, A u =b indicates that the mapping needs to satisfy the correspondence on the boundary, B represents the constraint matrix that maintains the continuity of G1, and the unknown quantity is solved and the position u of the mesh vertex after flipping optimization is obtained through an alternating algorithm.

3. The isogeometric analysis parameterization migration method based on discrete geometric mapping according to any one of claims 1 to 2, characterized in that: It also includes step (5) parameterizing migration results and quality evaluation.

4. The isogeometric analysis parameterization migration method based on discrete geometric mapping according to claim 3 is characterized in that: In step (5), the evaluation method of parameterized migration is: The two common indicators of the normalized Jacobian matrix and the condition number of the Jacobian matrix are used to measure whether the parameterization result of the target boundary after parameterization migration meets the requirements of isogeometric analysis. For a known parameterization B(u,v), let J be the Jacobian matrix of the mapping B, Example: B(u,v)=(x(u,v),y(u,v)) T ,x(u,v),y(u,v)∈R Then the canonical Jacobian is calculated as follows: Another important evaluation metric is the condition number of the Jacobian matrix calculated as follows K(J)=||J||||J -1 ||,||J||=(tr(J T J)) 1 / 2 in Experiments have shown that the closer the parameterized normed Jacobian matrix is ​​to 1.0, the better the parameterization quality is. Similarly, the closer the condition number of the parameterized Jacobian matrix is ​​to 2.0, the better the parameterization quality is.

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