A parameterization method for conformal embedding domain and isogeometric analysis based on THB++ splines

Abstract

The present invention belongs to the field of computational mechanics, and discloses a conformal embedding domain isogeometric analysis parameterization method based on THB++ spline, and the steps are as follows: reading model geometry information, defining background embedding domain, cropping unit identification, THB++ spline conformal embedding domain parameterization representation derivation, model point selection and conformal embedding domain construction. The method proposed in the present invention enables the background embedding domain to be conformal with the internal model at the boundary, facilitates the application of displacement boundary conditions, and solves the numerical oscillation problem caused by boundary non-conformality. It provides a more convenient, efficient and robust analysis algorithm for the numerical simulation of three-dimensional B-Rep models, enriching and expanding the theory and application of isogeometric analysis.

Description

Technical Field

[0001] The invention belongs to the field of computational mechanics and relates to a geometric analysis parameterization method such as conformal embedding domain based on THB++ splines. Background Art

[0002] Most of the current mainstream 3D CAD software uses B-Rep boundary representation CAD models, which describe 3D models through a series of trimmed boundary surfaces, rather than using 3D body splines for parametric representation. B-Rep representation CAD models include two types: one is a B-Rep surface model composed of multiple trimmed surfaces used to represent the shell, and the other is a B-Rep solid CAD model used to describe a 3D entity. In the B-Rep boundary representation method, each boundary surface is constructed independently, and only part of each boundary surface participates in the representation of the entire model after trimming. It should be pointed out that the trimming operation of the boundary surface only changes the shape representation of the surface, but does not change the parametric mathematical description of its underlying spline. In addition, the trimming of the boundary surface is often inaccurate, resulting in gaps or overlaps in the final surface patch when spliced. At the same time, the 3D solid model represented by B-Rep is usually hollow and lacks geometric information.

[0003] Therefore, it is necessary to construct geometry suitable for IGA with the help of geometric parameterization. Traditional geometric reconstruction methods (such as geometric reconstruction based on T-splines) are cumbersome and time-consuming, just like finite element meshing. Geometric analysis methods such as immersion / embedded domain immerse the B-Rep three-dimensional solid model without internal geometric information into a regular three-variable B-spline or NURBS cuboid without geometric reconstruction, and give it three-dimensional geometric information through background B-spline or NURBS. In the recent article "Application of B++ splines in isogeometric analysis" published by Zhu Xuefeng et al. in 2016, the proposed B++ spline satisfies the Kronecker delta property at the model boundary. The background spline embedding domain is constructed using B++ splines, which can effectively solve the problem of non-conformality between the background spline representation and the B-Rep model at the boundary. However, the current mature B++ spline parameterization method is only applicable to two-dimensional models. In order to overcome the limitations of existing methods, a suitable and robust isogeometric analysis method that can handle three-dimensional B-Rep models is needed.

[0004] Based on the above background technology, this paper proposes a parameterization method for conformal embedding domain isogeometric analysis based on THB++ splines. The core idea is to use THB++ splines to construct a spline representation of the background conformal embedding domain of any three-dimensional B-Rep model, and to establish a theory and method for conformal embedding domain isogeometric analysis based on THB++ splines by combining isogeometric analysis and isoparametric ideas. The research results provide a more convenient, efficient and robust analysis algorithm for the numerical simulation of three-dimensional B-Rep models, enriching and expanding the theory and application of isogeometric analysis. Summary of the invention

[0005] The purpose of the present invention is to overcome the shortcomings of existing isogeometric analysis methods and to provide a conformal embedding domain isogeometric analysis method that can process any three-dimensional B-Rep model.

[0006] The technical solution of the present invention:

[0007] A geometric analysis parameterization method based on THB++ splines for conformal embedding domains, including the following steps:

[0008] Step 1: Read model geometry information;

[0009] Read the 3D B-Rep model file and obtain the geometric information of its internal model, such as the coordinate information of the minimum bounding box corner points used to construct the initial NURBS spline background embedding domain, and the geometric information of the underlying uncut surfaces of each boundary surface (node ​​vectors, control points);

[0010] Step 2: Define the background embedding domain;

[0011] The background embedding domain is usually defined by three-parameter B-splines or NURBS (non-uniform rational B-splines), which is a regular hexahedron with a size that can accommodate the model.

[0012] Step 3: cropping unit identification;

[0013] In order to select appropriate matching points, when processing the background embedding domain geometry, it is necessary to judge the cell attribute R of each background embedding domain cell.

[0014] Step 4: Model point selection;

[0015] A series of matching points located on the model boundary are selected to replace the control points of the non-zero basis function of the background embedding domain located on the model boundary;

[0016] Step 5: Derivation of parameterized representation of THB++ spline conformal embedding domain;

[0017] Derive the mathematical expression of the THB++ spline representation of the three-dimensional B-Rep model;

[0018] Step 6: Construct a conformal embedding domain;

[0019] The conformal embedding domain basis function is generated by the selected collocation points and the original embedding domain basis function, and the control points of the conformal embedding domain including the collocation points are deduced in reverse.

[0020] Beneficial effects of the present invention: The present invention provides a conformal embedding domain isogeometric analysis parameterization method based on THB++ splines. The three-variable THB++ splines inherit the characteristic of B++ splines satisfying the Kronecker delta property at the boundary, while retaining the local encryption characteristics of THB splines. It can make the background embedding domain conformal with the internal model at the boundary, facilitate the application of displacement boundary conditions, and solve the numerical oscillation problem caused by boundary non-conformality. It provides a more convenient, efficient and robust analysis algorithm for the numerical simulation of three-dimensional B-Rep models, enriching and expanding the theory and application of isogeometric analysis. BRIEF DESCRIPTION OF THE DRAWINGS

[0021] Figure 1 is a schematic diagram of the initial background embedding domain of the present invention;

[0022] Figure 2 It is a schematic diagram of the embedding domain isogeometric analysis hierarchical encryption of the present invention;

[0023] Figure 3 It is a schematic diagram of the selection of geometric analysis points for embedding domain etc. of the present invention;

[0024] Figure 4 It is a schematic diagram of the THB++ spline conformal embedding domain of the embedding domain isogeometric analysis of the present invention;

[0025] Figure 5 It is a general flow chart of the geometric analysis parameterization method such as conformal embedding domain based on THB++ splines described in the present invention;

[0026] Figure 6 is the initialization flow chart of the present invention;

[0027] Figure 7 is a flow chart of constructing a conformal embedding domain according to the present invention; DETAILED DESCRIPTION

[0028] The specific implementation of the present invention is further described below in conjunction with the accompanying drawings and technical solutions.

[0029] The present invention provides a geometric analysis parameterization method based on THB++ spline conformal embedding domain, etc. The overall flow chart is as follows: Figure 5 As shown, the following steps are included:

[0030] 1) Read model geometry information;

[0031] Perform file I / O operations on the 3D model; use the data exchange module of OCCT to import the B-Rep model data from the external CAD model file into OCCT and save it in the internal data structure of OCCT; obtain the geometric information of the B-Rep model, specifically the coordinate information of the minimum bounding box corner points used to construct the initial NURBS spline background embedding domain and the geometric information of the underlying untrimmed surfaces of each boundary surface;

[0032] 2) Define the background embedding domain;

[0033] The background embedding domain is defined by three-parameter B-spline or NURBS, with a regular hexahedron shape and a size that can accommodate the B-Rep model, such as Figure 1 The figure shows the initial background embedding domain diagram. Figure 2 Schematic diagram of encryption for geometric analysis levels such as embedding domain;

[0034] 3) Cutting unit identification;

[0035] Determine unit attributes : Assume that when the background embedded domain element intersects the model boundary surface, the element attribute ; When the background embedding domain unit is completely inside the model, ; When the background embedding domain unit is completely outside the model, ;

[0036] Collision detection + ray tracing: given a point in space And 3D B-Rep model , through the three-dimensional B-Rep model The relationship between boundary surfaces and surfaces establishes a spatial hierarchical bounding box about the B-Rep model. From any point in space, Starting from, draw a ray in any direction. If the number of intersections between the ray and the spatial level bounding box is an even number, then point If the number of intersection points is odd, then the point Located inside the model;

[0037] Using 3D B-Rep Model If the background embedded domain unit collides with the model, it means that the background embedded domain unit intersects with the model. ; If no collision occurs and any corner point of the background embedded domain unit is inside the model, then the background embedded domain unit is completely inside the model ; Otherwise, the background embedding domain unit has no contact with the model ;

[0038] 4) Model point selection;

[0039] Based on THB++ spline, the model is subjected to geometric analysis such as conformal embedding domain, and points on the model boundary are selected as the matching points of THB++ spline to obtain a suitable matching point set. ; The matching points are mainly points at the model boundary and at the key geometric features of the model, such as Figure 3 The figure shows the schematic diagram of the geometric analysis point selection of the embedded domain of the model;

[0040] 5) Derivation of parameterized representation of THB++ spline conformal embedding domain;

[0041] First, the B-Rep model is given The THB spline representation is:

[0042]

[0043] in, is the control point, is a three-dimensional truncated hierarchical NURBS spline basis function, which is defined as:

[0044]

[0045] in, For the Layer basis function The truncation operation, For the definition in NURBS spline basis functions on layer-nested hierarchical clipping domains, is a hierarchical NURBS spline basis function defined on a clipped nested hierarchical domain;

[0046] According to whether the three-dimensional truncation level NURBS spline unit is clipped, Written as:

[0047]

[0048] in, For the A complete basis function, the local support satisfies the following conditions:

[0049]

[0050] For the The local support of the pruned basis function satisfies the following conditions:

[0051]

[0052] in, is a complete basis function The corresponding control points, is the clipped basis function The corresponding control points, The set of all basis functions for the three-dimensional truncated hierarchical NURBS spline of the B-Rep model;

[0053] The goal is to Transformed into THB++ spline conformal embedding domain parameterized representation; the key to the transformation method is to use a reversible matrix , the active control points of the background embedded domain units on the boundary of the three-dimensional B-Rep model are converted into a set of matching points and a set of enrichment points located on the model surface;

[0054] matrix Should meet: is a reversible matrix, is a right Markov matrix;

[0055] exist matrix ;

[0056]

[0057] Make

[0058]

[0059] Among them, the matrix yes The identity matrix, is a zero matrix;

[0060] Depend on and For a reversible matrix, we can deduce the matrix is a row-full-rank matrix, i.e., the collocation point The selection of is not arbitrary, and needs to be selected so that the matrix is the collocation point of the full row rank matrix;

[0061] Matrix in three-dimensional space Depend on and Two parts; for the collocation set points in , because the collocation points are located on the model boundary, we have:

[0062]

[0063] The vector form is expressed as:

[0064]

[0065] Using two orthogonal vectors and Rewrite Dimensional vector ,in For the matrix A vector in row space, For the matrix vectors in the null space;

[0066]

[0067] in, , Both dimensional vector, and satisfy ;

[0068] have:

[0069]

[0070] matrix is a full row rank matrix, so is an invertible matrix; therefore:

[0071]

[0072] get:

[0073]

[0074] in, is the enhanced control point, and the matching point located on the boundary Replace the control points of the boundary non-zero basis function ;

[0075] Further, we get:

[0076]

[0077] Known The matrix is ​​a full row rank matrix that satisfies ,because It is not a square array, so Not unique; use singular value decomposition to obtain A set of standard orthogonal basis vectors in the matrix null space, forming the matrix ,satisfy , ;make , and finally get:

[0078]

[0079] Then we have:

[0080]

[0081] Among them, the matrix Is Composition phalanx, yes The matrix of yes The matrix of For matching The number of Enrichment point The number of

[0082] Finally, we get the vector representation of THB++ spline of the three-dimensional B-Rep model. for:

[0083]

[0084] in, , and Defined as "3D THB++ spline basis function";

[0085] The expanded form is:

[0086]

[0087] in, , , ;

[0088] 6) Constructing conformal embedding domain;

[0089] After the point selection is completed, an initial size of The initial point set , resulting from the row vector The size of the composition is The matrix ; Use Gauss-Jordan elimination or QR decomposition to find the matrix A maximal linearly independent group ;pass The row index in derives the construction row full rank A set of suitable collocation points required by the matrix , and The difference between them represents the redundant matching points that need to be eliminated;

[0090] The basis function matrix B of the background embedding domain basis function at the set of collocation points is obtained by the selected collocation points;

[0091] Furthermore, the THB++ spline transformation matrix T is obtained from the matrix B:

[0092]

[0093] Where R is a special null space of B, satisfying BR=0;

[0094] The conformal embedding domain basis function is obtained from the matrix B and the matrix T;

[0095] Furthermore, the control points of the original embedded domain background geometry are replaced by the control points of the original boundary non-zero basis function multiplied by the inverse of the matrix T, and the new control points are deduced. Then, the new control points replace the control points of the original boundary non-zero basis function, thereby obtaining the background embedded domain geometry that is conformal to the model at the boundary, as shown in Figure 4 The figure shows a schematic diagram of the THB++ spline conformal embedding domain of the model's embedded domain isogeometric analysis. The THB++ spline conformal embedding domain isogeometric analysis parameterization method obtained by the above method satisfies the Kronecker delta property at the boundary, which enables the background embedded domain to be conformal with the internal model at the boundary, making it easier to apply displacement boundary conditions and solve the numerical oscillation problem caused by boundary non-conformity.

[0096] The present invention provides a parameterization method for conformal embedding domain isogeometric analysis based on THB++ splines. The three-variable THB++ splines inherit the characteristic of B++ splines satisfying the Kronecker delta property at the boundary, while retaining the local encryption characteristic of THB splines. It can make the background embedding domain conformal with the internal model at the boundary, facilitate the application of displacement boundary conditions, and solve the numerical oscillation problem caused by boundary non-conformality. It provides a more convenient, efficient and robust analysis algorithm for the numerical simulation of three-dimensional B-Rep models, enriching and expanding the theory and application of isogeometric analysis.

[0097] Although the embodiments of the present invention have been described for the purpose of illustration, it will be understood by those skilled in the art that various modifications, additions and substitutions may be made in form and detail without departing from the scope and spirit of the invention disclosed in the appended claims, and all these changes shall fall within the scope of protection of the appended claims of the present invention, and the various steps in the various departments and methods of the products claimed by the present invention may be combined together in any combination. Therefore, the description of the embodiments disclosed in the present invention is not intended to limit the scope of the present invention, but is used to describe the present invention. Accordingly, the scope of the present invention is not limited by the above embodiments, but is defined by the claims or their equivalents.

Claims

1. A geometric analysis parameterization method based on THB++ splines for conformal embedding domains, characterized in that: The following steps are involved: 1) Read model geometry information; Perform file I / O operations on the B-Rep model; use the data exchange module of OCCT to import the B-Rep model data from the external CAD model file into OCCT and save it in the internal data structure of OCCT; obtain the geometric information of the B-Rep model, specifically the coordinate information of the minimum bounding box corner points used to construct the initial NURBS spline background embedding domain and the geometric information of the underlying untrimmed surfaces of each boundary surface; 2) Define the background embedding domain; The background embedding domain unit is defined by three-parameter B-spline or NURBS, with the shape of a regular hexahedron and a size that can accommodate the B-Rep model; 3) Cutting unit identification; Determine unit attributes :Assume that when the background embedded domain element intersects the B-Rep model boundary surface, the background embedded domain element attribute ; When the background embedding domain unit is completely inside the B-Rep model, ; When the background embedding domain unit is completely outside the B-Rep model, ; Collision detection and ray tracing: Given a point in space And the B-Rep model , through the B-Rep model The relationship between boundary surfaces and surfaces establishes a spatial hierarchical bounding box about the B-Rep model. From any point in space, Starting from, draw a ray in any direction. If the number of intersections between the ray and the spatial level bounding box is an even number, then point If the number of intersection points is odd, then the point Located inside the B-Rep model; Using the B-Rep Model If the background embedded domain unit collides with the B-Rep model, it means that the background embedded domain unit intersects with the B-Rep model. ; If no collision occurs and any corner point of the background embedding domain unit is inside the B-Rep model, then the background embedding domain unit is completely inside the B-Rep model ; Otherwise, the background embedding domain unit has no contact with the B-Rep model ; 4) Model point selection; Based on THB++ spline, the B-Rep model is subjected to geometric analysis such as conformal embedding domain, and the points on the boundary of the B-Rep model are selected as the matching points of THB++ spline to obtain a suitable matching point set. ; The matching points are mainly points at the boundaries of the B-Rep model and at the key geometric feature positions of the B-Rep model; 5) Derivation of parameterized representation of THB++ spline conformal embedding domain; First, the B-Rep model is given The THB spline representation is: in, is the control point, is a three-dimensional truncated hierarchical NURBS spline basis function, which is defined as: in, For the Layer basis function The truncation operation, For the definition in NURBS spline basis functions on layer-nested hierarchical clipping domains, is a hierarchical NURBS spline basis function defined on a clipped nested hierarchical domain; According to whether the three-dimensional truncation level NURBS spline unit is clipped, Written as: in, For the A complete basis function, the local support satisfies the following conditions: For the The local support of the pruned basis function satisfies the following conditions: in, is a complete basis function The corresponding control points, is the clipped basis function The corresponding control points, The set of all basis functions for the three-dimensional truncated hierarchical NURBS spline of the B-Rep model; The goal is to Transformed into THB++ spline conformal embedding domain parameterized representation; the key to the transformation method is to use a reversible matrix , the control points of the background embedded domain units that are active on the boundary of the B-Rep model are converted into a set of matching points and a set of enrichment points located on the model surface; matrix Should meet: is a reversible matrix, is a right Markov matrix; exist matrix ; Make Among them, the matrix yes The identity matrix, is a zero matrix; Depend on and For a reversible matrix, we can deduce the matrix is a row-full-rank matrix, i.e., the collocation point The selection of is not arbitrary, and needs to be selected so that the matrix is the collocation point of the full row rank matrix; Matrix in three-dimensional space Depend on and Two parts; for the collocation set points in , because the collocation points are located on the model boundary, we have: The vector form is expressed as: Using two orthogonal vectors and Rewrite Dimensional vector ,in For the matrix A vector in row space, For the matrix vectors in the null space; in, , Both dimensional vector, and satisfy ; have: matrix is a full row rank matrix, so is an invertible matrix; therefore: get: in, is the enhanced control point, and the matching point located on the boundary Replace the control points of the boundary non-zero basis function ; Further, we get: Known The matrix is ​​a full row rank matrix that satisfies ,because It is not a square array, so Not unique; use singular value decomposition to obtain A set of standard orthogonal basis vectors in the matrix null space, forming the matrix ,satisfy , ;make , and finally get: Then we have: Among them, the matrix Is Composition phalanx, yes The matrix of yes The matrix of For matching The number of Enrichment point The number of Finally, we get the vector representation of THB++ spline of the three-dimensional B-Rep model. for: in, , and Defined as "3D THB++ spline basis function"; The expanded form is: in, , , ; 6) Constructing conformal embedding domain; After the point selection is completed, an initial size of The initial point set , resulting from the row vector The size of the composition is The matrix ; Use Gauss-Jordan elimination or QR decomposition to find the matrix A maximal linearly independent group ;pass The row index in derives the construction row full rank A set of suitable collocation points required by the matrix , and The difference between them represents the redundant matching points that need to be eliminated; The basis function matrix B of the background embedding domain basis function at the set of collocation points is obtained by the selected collocation points; Furthermore, the THB++ spline transformation matrix T is obtained from the matrix B: Where R is a special null space of B, satisfying BR=0; The conformal embedding domain basis function is obtained from the matrix B and the matrix T; Furthermore, the control points of the original embedded domain background geometry are replaced. The new control points are deduced by multiplying the original boundary non-zero basis function control points by the inverse of the matrix T. The new control points then replace the control points of the original boundary non-zero basis function, thereby obtaining the background embedded domain geometry that is conformal to the B-Rep model at the boundary.

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