A digital modeling method for irregular objects based on discrete mathematics and topological geometry

Abstract

The application provides an irregular object digital modeling method based on discrete mathematics and topological geometry. The core innovation is that a node-segment-triangle-polygon-envelope body geometric primitive system following topological relations is constructed by taking a multi-dimensional space discrete node as a minimum unit, and a strict upward increasing and downward compatible hierarchical relation is followed between the primitives. A collection of geometric primitives following specific topological relations is used to construct three-dimensional digital models of different types of irregular objects. This logic is applicable to all physical objects containing irregular primitives, and when applied to rock-soil bodies, it constitutes a technical base for key links such as geological modeling, three-dimensional contour design and interactive mechanical calculation in the whole life cycle of the rock-soil industry. The application is an original innovation at the mathematical theory level, and provides a new idea and reliable base for the research and development of irregular object digital modeling technology.

Description

Technical Field

[0001] This invention belongs to the field of industrial digitalization, and specifically relates to digital modeling of irregular objects, which is most representative in the field of geotechnical engineering based on irregular geological bodies. Background Technology

[0002] The digital technology of service modeling / design is theoretically based on mathematics. Continuous functions are used for regular objects, while topological geometry is used for irregular objects. Given the principle that topological compatibility is continuous, but the converse is not true, topological-continuous coupling is used for situations that simultaneously involve both.

[0003] Digital technologies in industrial design originated in the manufacturing industry. Two-dimensional CAD, which emerged in the late 1960s, and BIM technology, which appeared in the 1980s, have continued to this day. Both 2D and 3D design software have the ability to construct regular and irregular curves. The underlying reason is that these software programs not only build curve function libraries, but also construct the topological relationships contained in spatial curves using points and line segments as basic units, meeting the application requirements for constructing and editing polylines.

[0004] Replacing line segments with triangular meshes and constructing 3D irregular surfaces through sets of triangular meshes has naturally become the technical approach for modeling irregular surfaces. At this stage, modeling only requires the spatial connection of nodes in the triangular mesh as primitives, but the technical difficulty of meeting editing requirements increases exponentially: the surface must first be backward compatible with the complete topological relationships of each primitive, and complex inclusion relationship judgment logic needs to be established to achieve dynamic updates of the topology of each primitive. Because of this, all 3D design software originating from the manufacturing and construction industries, as well as measurement software based on point cloud modeling, have very weak or even no editing capabilities for triangular mesh surfaces. Strengthening the topological relationships of triangular meshes and their sets has become a fundamental problem that must be overcome in the digital modeling of irregular objects, and related research work is mainly driven by the demand for 3D modeling of irregular geological bodies. After unremitting efforts, related technologies appeared in GoCAD (released in 1999) and CnGIM_ma (released in 2016), making them currently the only two products in the world that use triangles as primitives and follow strict topological relationships. Given the complexity of this technical approach, other products such as Leapfrog, EVS, Petrol, Creatar Xmodling, and DepthInside all employ implicit modeling techniques in an attempt to circumvent the complex topological requirements.

[0005] Slope excavation involves the normalization of irregular geological formations, and the contour design process often requires adjustments to the slope ratio based on changes in geological conditions. Therefore, the fundamental logic of slope excavation contour design is to follow strict topological relationships and construct a collection of polygons as basic units. This approach is the correct way to solve the problem, upgrading the modeling logic from triangular meshes to polygons as basic units, where each polygon is a collection of two or more triangles, and backward compatible with and including all lower-order basic units. From this perspective, using triangular topology in GoCAD and CnGIM_ma as a foundation becomes a realistic and feasible implementation method. Dassault Systèmes (France) and Bentley (USA), which have achieved great success in BIM software development for regular objects, attempted to acquire geotechnical engineering software companies to master topology-based modeling technology and extend their product chains to the geotechnical engineering field of designing irregular objects. However, the acquired products did not include the necessary topology modeling technology, resulting in no substantial progress over several years to more than a decade. Dassault retreated from and adhered to function-based parameterized design, while Bentley focused on solving data interaction between different software programs, primarily in file format.

[0006] Stability evaluation through mechanical calculations is a crucial step in geotechnical engineering design. The design process based on 3D geological models provides unprecedented conditions for the automatic construction of computational models, parameter assignment, and even intelligent recommendation of computational assumptions. However, this requires defining the computational scope within the 3D engineering digital model according to the design needs, directly converting the engineering digital model into a computational model. The main task is to convert the surfaces of the engineering digital model, after they are closed together, into envelopes in the 3D computational model. When the engineering surfaces are triangular or polygonal aggregates, advancing to a higher level and constructing a 3D modeling method based on envelopes becomes a natural approach. The common method for constructing the envelope is surface intersection, including topological intersection and Boolean operations after function fitting, which is the mainstream method to date (CN119026223A, CN120449526A / B, CN115359212B, CN118627167B). These methods retain the triangular / quadrilateral mesh in the original digital model and directly generate the tetrahedral / hexahedral mesh in the computational model to ensure topological consistency. This widely used technical route can complete the model conversion, but it lacks technical rationality and practical application. 1) The computational mesh in the computational model depends on the requirements of the engineering digital model and does not follow the principle of computational mesh generation: determining the mesh size based on the stress change gradient, which lacks technical rationality; 2) To reflect the topography and strata undulations, the mesh size of engineering surface models is often small and the number of meshes is large. When directly converted into the computational mesh in the three-dimensional numerical model, the number of computational meshes often reaches tens of millions, far exceeding the number of meshes in general engineering problem computational models (hundreds of thousands to millions), resulting in a huge amount of invalid computation. 3) When generating envelopes between irregular triangular mesh surfaces using the intersection method, it is difficult to control the mesh shape along the intersection line. Possible "illness" meshes may not meet the requirements of mechanical calculations, leading to errors or even collapse.

[0007] In view of this, the present invention proposes a digital modeling method for irregular objects based on discrete mathematics and topological geometry, with the focus on the innovative idea of ​​using polygons and envelopes as basic units. Summary of the Invention

[0008] Industrial digital modeling techniques based on function fitting and low-level geometric primitives have fundamental flaws when dealing with irregular spatial objects. Traditional CAD / BIM technologies, exemplified by manufacturing, rely on continuous functions and parametric design, making it difficult to effectively address the topological consistency and dynamic editing requirements of naturally irregular objects such as geological bodies. While some 3D geological modeling software (such as GoCAD and CnGIM_ma) achieves rigorous topological modeling using triangles as primitives, they lack systematic solutions when higher-order primitives such as polygons and envelopes are required. This has resulted in the following unresolved issues: 1) The design of 3D excavation / fill contours lacks unified topological primitive support, and a geotechnical 3D design technology based on polygon primitives that meets dynamic editing capabilities and multi-scenario applications has not yet emerged; 2) A technology based on primitive reconstruction to convert engineering digital models into computational models has not yet been seen.

[0009] In view of this, the present invention proposes a digital modeling method for irregular objects based on discrete mathematics and topological geometry, including one fundamental innovation and several applied innovations.

[0010] The fundamental innovation refers to the proposed system composed of five levels of primitives, namely nodes, line segments, triangles, polygons, and envelopes. Except for node primitives, any level of primitive is constructed through the topological relationships between lower-level primitives. Thus, there is an upward and downward compatible hierarchical relationship between the primitives, with higher-level primitives inheriting the complete topology of lower-level primitive objects. The system satisfies the judgment of reference relationships and discontinuities between primitives and supports the addition and deletion of primitives through topological updates. The discrete nodes support multi-dimensional attributes, with a default of three dimensions (x, y, z), used to define the spatial geometric position of the nodes. Attributes beyond three dimensions are custom attributes to meet the requirements of describing the uncertainty and spatial variability of soil and rock properties.

[0011] The aforementioned applied innovations refer to the various digital modeling methods proposed for geotechnical engineering based on two higher-order primitives: polygons and envelopes. These methods include 3D contour modeling of planar (slope) and linear (tunnel) geotechnical engineering structures, computational modeling based on 2D engineering drawings, and 3D computational modeling based on 3D digital models. Both computational modeling methods employ a feature point reconstruction technique to ensure that specific requirements for mechanical calculations, such as mesh partitioning and density control, are met.

[0012] Based on the aforementioned fundamental innovation and multiple applied innovations, the proposed digital modeling method for irregular objects based on discrete mathematics and topological geometry includes the following steps: S1: Define multidimensional space discrete nodes as the most basic digital geometric primitives, and construct line segments, triangles, polygons and envelopes in sequence according to complete topological relationships to form a geometric primitive system for digital modeling of irregular objects; S2: Construct two-dimensional and three-dimensional irregular curves based on a collection of line segment primitives that follow complete topological relationships; S3: Based on a set of planar triangular primitives that follow strict topological relationships, an irregular spatial surface is constructed for geological modeling; S4: Based on a set of planar polygon primitives that follow complete topological relationships, construct a two-dimensional computational model or a three-dimensional model of the outline of geotechnical engineering excavation and filling structures; S5: Construct a three-dimensional numerical calculation model based on an envelope primitive set that follows strict topological relationships.

[0013] In a first aspect, this invention discloses a method for constructing a digital geometric primitive system based on discrete mathematics and topological principles. The primitive system consists of nodes, line segments, triangles, polygons, and envelopes, increasing sequentially upwards and being backwards compatible; higher-order primitives contain all lower-order complete topologies. The node supports multi-dimensional attributes, which are three-dimensional (x, y, z) by default and are used to define the node's spatial geometric position. Dimensions above three are custom indicators. The term "discrete" refers to the fact that when constructing a three-dimensional digital model of an irregular object using a collection of geometric primitives, the geometric primitives are independent of each other, do not obey any functional formula, but follow complete topological relationships. The complete topology includes proximity, connection, and containment relationships, and supports topology updates.

[0014] In a second aspect, the present invention discloses a method for creating two-dimensional and three-dimensional irregular curves (polylines) using line segments as basic units and following complete topological relationships.

[0015] Specifically, Connect two adjacent discrete nodes in space with straight lines in a predetermined order to form line segment primitives; associate the line segment primitives in a certain order to construct two-dimensional or three-dimensional irregular curves; When all line segment primitives in the line segment assembly are coplanar, the line segment assembly is used to construct irregular planar curves in the form of two-dimensional polylines; when the line segment primitives are not coplanar, the line segment assembly is used to construct irregular spatial curves. The topological structure of the irregular curve includes the topological relationship between different line segment primitives. Addition and deletion are achieved through topological updates. Moving only changes the coordinate position of the primitives and does not affect the topological relationship. Any line segment in the irregular curve can be replaced by an equation other than the equation of a straight line, thereby realizing curve modeling of topology-function coupling logic; The continuity of a polyline is determined by the number of references to discrete nodes. When the number of references to any discrete node is 1, the discrete node is an endpoint, indicating discontinuity.

[0016] In a third aspect, this invention discloses a method for creating irregular spatial surfaces using planar triangles that follow strict topological relationships as primitives. When used for 3D geological modeling, this method imposes even stricter requirements on the topology of the triangles (half-side structures) and the topology between triangles.

[0017] Strict topology refers to the complete topology and inheritance followed when constructing triangular primitives and sets. The triangles are Delaunay meshes using a half-edge data structure, backward compatible and containing line segments, nodes, and their topology. Continuity is determined by the proximity of nodes and edges between triangles, meaning adjacent triangles are either connected or disconnected. The topological structure of the constructed spatial surface includes topological relationships within and between triangular meshes. Additions and deletions are achieved through topology updates, and moving primitives updates their coordinate positions without affecting the topology.

[0018] When using strictly topological triangular primitives for 3D geological modeling, it is preferable to employ adaptive generation technology to construct triangular mesh assemblies, i.e., irregular spatial surfaces, to meet the requirements of geometric operations (surface topological intersection, attribute data operations) for model applications.

[0019] In a fourth aspect, the present invention discloses a method for creating a two-dimensional computational model using planar polygons that follow strict topological relationships as primitives, when all primitives in the assembly are coplanar.

[0020] A polygon is formed by connecting nodes sequentially to create a closed outline. The closed outline consists of two or more triangles and is backward compatible with and includes lower-order primitives and their topology. A collection of polygonal primitives formed according to certain rules. When all primitives obey the plane equation, that is, when all primitives are coplanar, it is used to construct different types of closed partitions in a two-dimensional calculation model and to convert two-dimensional engineering drawings into two-dimensional calculation models. A collection of polygonal primitives formed according to certain rules. When all primitives obey the plane equation, that is, when all primitives are coplanar, control points on the polygon boundary are extracted and connected in sequence to reconstruct closed polylines. This is used to construct different types of closed partitions in a two-dimensional calculation model and to convert two-dimensional engineering drawings into two-dimensional calculation models.

[0021] The polygons in the computational model are constructed using a control point reconstruction method. The control points include the intersection points generated by the intersection of selected polylines in the two-dimensional engineering drawing, representative inflection points reflecting the topography and stratum undulations, and nodes in the two-dimensional drawing describing the structural outline morphology. The minimum interval between adjacent control points should not be less than the minimum side length of the grid cell / soil strip in the calculation model; After reconstruction, the geological units or engineering units (excavation or filling zones) in the engineering drawings correspond to the corresponding zones (material zones, excavation zones, etc.) in the calculation model, and the relevant parameter attributes are automatically migrated to the calculation model according to the topological relationship.

[0022] In a fifth aspect, this invention discloses a three-dimensional modeling (design) method for various excavation and filling contours in geotechnical engineering when polygons are used as basic units and the aggregate does not conform to the plane equation. This method is applicable to both linear structures (dams, tunnels, ditches, etc.) and planar structures (excavation, embankment slopes, etc.).

[0023] The topology of the planar structure is directly constructed from polygon primitives; its topology includes the topological relationships between polygons and between the low-order primitives contained in the polygons. The geometric operation requirements for adding and deleting are realized through topology updates, and the movement operation only changes the node coordinates.

[0024] The linear structure 3D contour model is generated through the "axis + cross-section" method, where the axis and cross-section are collections of line segments, that is, polygons (such as tunnel sidewalls, arches, and bottom slabs) are generated using lower-order primitives. Its topology includes the topological relationships between polygons and between cross-sections and axes. The topology between polygons meets the requirements of operations based on excavation part type (such as sidewalls, arches, and bottom slabs) (such as geological logging and support optimization), filling zone interface area and zone volume calculation. The topology between cross-sections and axes enables efficient design through 2D-3D linkage editing.

[0025] The line segment primitives in the interrupted surface or axis of the linear structure can be replaced by equations other than the linear equation (circle, ellipse, etc.); the polygon primitives in the planar structure can be replaced by equations other than the planar equation (sphere, ellipsoid, etc.); thus realizing the contour design of topology-function coupling, supporting regular transition changes in local contour morphology.

[0026] In a sixth aspect, the present invention discloses a method for constructing an envelope volume based on a three-dimensional digital model of an engineering project using a feature point reconstruction method. The method first generates feature points and then reconstructs and generates the envelope volume according to the technical route of point ~ line segment ~ polygon ~ envelope volume.

[0027] Feature points are divided into initial points and mapped points. Initial points mainly come from the control points (inflection points) of the excavation / filling contour in the engineering digital model. Secondly, auxiliary points are added manually as needed (such as control points of the grid partition boundary surface of the calculation model). Mapping points are the intersection points generated by the lines or rays generated from the starting point in the preferred direction and intersecting with the corresponding surfaces in the engineering digital model. The correspondence between the initial point and the mapping points is recorded to construct the corresponding topology. The preferred direction refers to the following: when a straight line or ray is drawn from the starting point, if it intersects a surface in the engineering digital model at an angle that is as close to a right angle as possible, then the direction of the straight line or ray is the preferred direction. When feature points are used to generate material partitions, the straight line or ray intersects the ground plane as close to a right angle as possible; when used to reconstruct discontinuous contact surfaces corresponding to faults and joints, it intersects the corresponding fault and dominant joint groups as close to a right angle as possible.

[0028] When reconstructing the envelope based on feature points, first construct closed polylines for the initial points in the order of point to line segment. Then, triangulate the nodes of the closed polylines to form polygons, completing the progressive process from point to line segment to triangle to polygon, following strict topology.

[0029] Based on the topology of the mapping points and the initial point, mapping points are obtained sequentially at various levels and on the model boundary surface. Corresponding polygons are constructed accordingly and then combined into an envelope to complete the envelope reconstruction. The reconstructed envelope consists of only a small number of sparse meshes. The mesh can be subdivided according to different accuracy requirements for a specified envelope based on computational needs, so as to obtain the best balance between computational load and computational accuracy.

[0030] In a seventh aspect, this invention discloses a method for constructing an envelope volume primitive set, a three-dimensional numerical calculation model, and a method for transferring parameter attributes. The method is based on the aforementioned feature point-based envelope reconstruction method, and establishes a mapping relationship between secondary polygon primitives and the faces of the engineering model during the reconstruction process.

[0031] The mapping relationship includes the mapping between material partitions in the calculation model and strata, excavation partitions and excavation contour surfaces in the engineering digital model; it also includes the mapping between discontinuous contact and fracture. Based on the mapping, the required input parameters for calculation are further transferred from the engineering model, including the physical and mechanical parameters of soil and rock materials, and the mechanical parameters of discontinuous contact surfaces. When the engineering digital model includes reinforcement design, the geometric and physical and mechanical parameters of the corresponding reinforcement structure are obtained from the knowledge base according to the data layer association in the application software and transferred to the calculation model.

[0032] Beneficial effects Compared with the prior art, the present invention has the following significant advantages: A complete primitive system has been constructed: For the first time, a complete hierarchical system of geometric primitives has been established, from nodes, line segments, triangles, polygons to envelopes. Each primitive strictly follows the principle of upward increment and downward compatibility, providing a complete and unified technical foundation for digital modeling of irregular objects in the industrial field.

[0033] A new modeling method is proposed: a new method for constructing two-dimensional computational models, geotechnical engineering excavation and filling contour models, and three-dimensional computational models based on high-order primitives (polygons, envelopes); and a feature point reconstruction method is adopted when constructing the computational model, which effectively solves the incompatibility problem between the computational model and the engineering model caused by the difference in requirements.

[0034] Topological consistency guarantee: By strictly constraining the proximity, connection and containment relationships between primitives, the model is ensured to maintain topological consistency throughout the editing, updating and transformation process, which fundamentally solves the problems of heterogeneous data formats, cumbersome model transformation and even failure in traditional methods.

[0035] Dynamic editing capability: Based on the topology update mechanism, it supports adding, deleting and moving primitives at any level without rebuilding the entire model, which greatly improves the efficiency of design iteration.

[0036] Geological to computational model conversion: By adopting a feature point-based reconstruction method, the automatic conversion from two-dimensional engineering drawings to two-dimensional computational models and from three-dimensional engineering digital models to three-dimensional numerical computational models is realized, which solves the practical problems of insufficient technical rationality and lack of engineering application value of existing methods.

[0037] Topology-function coupling design: It supports the replacement of primitives such as line segments and polygons with equations other than linear equations. While maintaining the topological structure, it introduces the ability to describe functions, taking into account the flexibility of irregular objects and the accuracy of regular objects, and is suitable for complex industrial scenarios.

[0038] This invention is not limited to the field of geotechnical engineering, but can be extended to all industrial fields involving digital modeling of irregular objects, including but not limited to geological exploration, water conservancy and hydropower, transportation engineering, mining, machinery manufacturing, aerospace, etc., and is especially suitable for coupled scenarios that contain both regular and irregular objects. Attached Figure Description

[0039] Figure 1 This invention provides five geometric primitive systems and corresponding models that are progressively upgraded. Figure 2 This invention uses line segments as basic units to construct irregular curves; Figure 3 This invention uses line segments as basic units and includes irregular curves with arc segments; Figure 4 The geological model constructed in this invention follows strict topology and is based on triangles. Figure 5 This invention provides a two-dimensional computational model constructed using polygons as primitives and when they are coplanar. Figure 6 This invention provides an excavation slope profile model constructed using polygons as basic units. Figure 7 This invention presents a tunnel excavation outline model constructed using polygons as basic units. Figure 8 This invention presents a tailings dam model constructed using polygons as basic units. Figure 9 This invention presents a three-dimensional numerical calculation model constructed using the envelope as a basic unit. Detailed Implementation

[0040] The technical solutions of the embodiments of this application will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of this application, and not all embodiments. Based on the embodiments of this application, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of this application.

[0041] In view of the shortcomings of the existing technology, the first aspect of the present invention proposes a digital modeling method for irregular objects based on discrete mathematics and topological geometry. Its basic logic constructs a five-level geometric primitive system that is upwardly incremental, downward compatible and maintains topological consistency. The collection of different geometric primitives constructs a digital model that meets the application requirements of different scenarios.

[0042] S100: Figure 1 It represents the 5-level geometric primitives and the corresponding partial digital model.

[0043] Figure 1'a' represents a multidimensional discrete node, and the solid circle represents a collection of four discrete nodes, where each node is a geometric primitive. The four nodes in the collection lack any correlation; deleting or moving one node has no effect on the other three. This is discreteness. Each node is described by default using three parameters (x, y, z) to represent its geometric position. In addition, additional parameters such as dip, tilt, wave velocity, and RMR can be added according to specific requirements, thus enabling multidimensionality. The other four primitives all contain the lowest-level node primitives. Based on downward compatibility, the nodes in the remaining primitives also exhibit multidimensionality, with a non-uniform spatial distribution of recorded characteristic indicators.

[0044] Figure 1 b and Figure 1 e represents the polyline constructed from line segments, triangles, polygons, envelope primitives, geological interfaces, excavated slope outlines, and three-dimensional numerical models of slopes, respectively. Examples and descriptions will follow.

[0045] S200: Figure 2 It represents a polyline model constructed from line segment primitive geometry, and is backward compatible with 7 node primitives.

[0046] The polyline consists of seven nodes numbered 1 to 7. On the left, these seven nodes are connected sequentially by straight lines, forming a collection of six line segments. Nodes 1 and 7 are connected to only one line segment each, serving as endpoints, totaling two, indicating that this collection of segments constitutes a continuous polyline. In contrast, nodes 3 and 4 on the right are also referenced by only one line segment each, serving as endpoints (breaks), making the total number of endpoints in the right-side collection exceed two, thus constituting a non-continuous polyline.

[0047] Figure 2 As shown, it demonstrates that line segment primitives and their sets, constructed according to a complete topology, can construct continuous and discontinuous polylines.

[0048] Figure 3 This involves editing a polyline containing a complete topological relationship in two ways. The polyline is formed by connecting three nodes p0, p1, and p2 in sequence, and is a collection of two line segments e0 and e1. Inserting nodes p3 and p4 into the two line segments within this collection results in: Based on the proximity relationship between the insertion point and the original point, that is, the distance between P3 and e0 is the smallest and the distance between P4 and e1 is the smallest, it can be determined that P3 is located at e0 and P4 is located at e1. Topology update: After inserting 2 nodes, the number of nodes in the polyline increases to 5, and the connection order between them is P0~P3~P1~P4~P0; the number of segments increases to 4, and the nodes referenced by each segment are updated accordingly.

[0049] The above describes how, after adding a new node, the number of nodes, the number of line segments, and the line segment reference relationships to nodes are updated.

[0050] Furthermore, Figure 3 Following the principle of tangency at the endpoints, e1 is replaced with an arc. At this time, the topology of the polyline remains unchanged, but the arc function is coupled with the parameter as an additional attribute of e1, thus realizing the coupling between the topology and the function.

[0051] Figure 2 and Figure 3 The related operations are applicable to models constructed using geometric primitives of triangles, polygons, and envelopes, but are difficult to represent graphically in space and involve tedious and complex processes. Therefore, [the following is omitted as the text is incomplete and cannot be translated]. Figure 2 and Figure 3 Taking this as an example, we will introduce the case study's effects and the topological principles it embodies in detail.

[0052] S300: Figure 4 It represents the topology of triangles and the spatial discontinuous surface constructed from a collection of triangles.

[0053] Figure 4 The left side consists of two triangles that support the half-edge data structure. The left triangle (1) is formed by connecting nodes 1, 2, and 3 in a counterclockwise direction. It contains two low-level primitives: three line segments and three discrete nodes, thus achieving backward compatibility.

[0054] exist Figure 4 In the set formed by the two triangles on the left, when any one triangle chooses to connect three adjacent nodes in a counter-clockwise direction, all other triangles must follow this rule. When the line segment (edge) formed by connecting nodes 1 and 2 is defined as the initial edge, if these two nodes form an edge of another triangle (triangle ②), its connection order must be 2 to 1, thus forming a mirror image of the initial edge of triangle ①. Therefore, the triangular geometric primitives that follow strict topology have the following characteristics / capabilities: When any triangle has a mirror side, the triangle is connected to another triangle through the initial side, thus constructing a continuous model; If any triangle has no mirror side (which can be represented as empty in programming), then the initial side is composed of discontinuous boundary lines, thereby constructing a discontinuous model and identifying the model boundaries (inner and outer boundaries).

[0055] Figure 4 The left side represents the operation of moving node 4. At this time, the topology, such as the connection relationship between nodes, is not changed. Only the coordinates of the node are updated to achieve movement and corresponding contour editing.

[0056] Figure 4The right side shows a discontinuous surface constructed using triangular primitives that follow strict topological relationships, in the form of a triangular assembly. This surface not only includes the topology of both triangles and line segments as primitives, as well as the multidimensional properties of discrete nodes, but also... Figure 4 As shown on the left, the topological relationship between two adjacent triangles constructs a discontinuous spatial surface.

[0057] Figure 4 Each triangle on the right satisfies the Delaunay criterion.

[0058] S400: Figure 5 This represents a two-dimensional computational model constructed from two-dimensional cross-sectional views, using polygons as primitives.

[0059] Figure 5 The left side shows the main contents of the slope engineering profile, which is an irregular collection of polylines. The discrete curves representing topography, strong weathering, weak upper weathering, and weak lower weathering describe the changes in rock mass properties on this profile, corresponding to the material partitions in the calculation model. Figure 5 The "slope" on the left refers to the generalized outline of the slope excavation that reflects the overall slope ratio. Figure 5 The rectangle on the left represents the boundary of the computational model to be constructed.

[0060] Figure 5 After the topography, slopes, weathering boundaries, and model boundaries on the left intersect, they form several coplanar polygons. Each polygon represents the difference in the degree of rock weathering, i.e., the material partitioning in the calculation model, thus constructing a two-dimensional calculation model in the form of a polygonal assembly. This process adopts a reconstruction method, automatically eliminating overly dense nodes in the engineering drawing lines and merging overly short line segments into adjacent segments to meet the minimum size requirements of soil strips (limit equilibrium method) and mesh (numerical method) in the calculation model.

[0061] Based on the computational model constructed from the two-dimensional engineering drawings, the features are extracted and automatically reconstructed using default settings. Geometric editing is generally not performed, but the necessary information for the calculation, such as stratigraphic names, fault codes, and their corresponding physical and mechanical parameters, should be obtained through "migration." Therefore, the inheritance of low-order primitive geometric topology is not emphasized at this stage; instead, the focus is on object mapping and parameter migration, which will be implemented during software functional design and will not be elaborated upon here.

[0062] S500: Figure 6 This illustrates the outline model of the excavated slope constructed using polygons as basic units. Given the complexity of the principles and processes, only a brief overview of the implementation effects and the consistency of the technical features with the instruction manual is provided here.

[0063] Figure 6 The excavated slope profile shown includes the following types of polygonal primitives: Coplanar quadrilateral: The outer boundary is formed by connecting four nodes sequentially, and the interior includes at least two triangles, reflecting downward compatibility. In this case, the slope ratio is an important parameter to meet parametric design requirements; Triangle: A triangle, a degenerated form of a polygon, exhibiting backward compatibility. Twisted quadrilateral: The four nodes of the outer boundary are not coplanar and do not conform to a given function. It belongs to the primitive of irregular polygons and is the basic manifestation of irregularity. Arc-shaped surfaces: The result of replacing certain polygons with given arc / surface equations, thereby achieving topological-functional coupling. That is, the slope profile as a whole is regarded as a collection of polygons (topology), and the local profile shape supports the use of high-level function substitution (relative to plane equations).

[0064] Figure 6 The left and right sides together represent editing operations such as inserting slope segments, moving nodes, and optimizing the outline shape, based on the premise that each primitive follows strict topological relationships: Figure 6 The interpolation shown on the left refers to inserting a node on a given horse trail, thereby activating it. Figure 3 The polyline (bridle path) shown is updated in topology; and a new polygon is added simultaneously, triggering the topology update of the polygon (at this time, the triangle primitives are contained and compatible with the polygon), thus realizing the editing operation of adding a new stair segment. Figure 6 As shown on the left, all the upper catwalk lines are inserted synchronously and the corresponding topology is updated. This is an option designed according to application requirements, which supports "parametric" design with linkage characteristics (or non-parametric design that is only effective for the current catwalk). Moving nodes does not affect the topology, but it changes the shape of adjacent polygons after moving node coordinates, resulting in non-coplanar "twisted" polygons. This satisfies the design of "irregular transition sections" and solves the problems that are common to any other technical approach.

[0065] S700: Figure 7 This represents a tunnel excavation outline model constructed using polygons as primitives. The model is created using an "axis + cross-section" approach, where both axes and cross-sections are polylines. Adjacent cross-sections are connected based on topological relationships to create polygons, and the tunnel model is a collection of polygons. The entire process follows complete topological relationships and is backward compatible.

[0066] The axis is a polyline supporting topological-functional coupling, allowing for arbitrary node insertion. Each node contains four-dimensional data (x, y, z, station number) by default, meaning the axis inherits the multidimensional attributes of discrete points. In subsequent tunnel stability analysis, parameters can be freely defined to record changes in tunnel depth, ground stress, and surrounding rock quality, providing sensory data for "intelligent evaluation" and "intelligent design," thus meeting the requirements for upgrading from data to intelligence. The cross-section is a closed polyline following a complete topology. Each polyline consists of at least four control nodes connected sequentially and comprises four segments. These segments define the arch, sidewalls, and base of the cross-section. Furthermore, this polyline supports the substitution of segments with higher-order functions (circular arc equations), achieving topological-functional coupling (gateway shape). The circular cross-section is divided into four segments according to the defined control points, each segment being substituted by a circular arc equation, thus constructing a topologically consistent circular cross-section. After the cross-sections are associated with the specified station number on the axis, they are connected according to the principle of "same type control points" to construct several (4 or more) polygons representing the top arch, side walls, and bottom slab. Therefore, connections between different types of cross-sections are supported, such as... Figure 7 The irregular transition tunnel segment is constructed by connecting the circular and arch-shaped cross sections. Through the topology of the downward compatible cross section, the irregular tunnel segment can still be "disassembled" into polygonal components such as side walls, arches, and bottom slabs. This meets the requirements of geological logging and support optimization by location during the construction period.

[0067] S800: Figure 8 This paper presents a 3D model of a tailings dam constructed using polygons as primitives, demonstrating that this logic applies to structures formed by filling methods. The modeling process follows complete topological relationships while satisfying the design requirements of both the external contour and internal partitioning.

[0068] Figure 8 The above is a dam model constructed using the "axis + cross-section" method. Similar to the tunnel mentioned above, it can distinguish the dam crest, upstream and downstream slopes, and foundation surface, and supports modification of design parameters (dam crest width, slope ratio, and slope height, etc.) for any part of it. When modifying one of the cross-sections, based on the dam's topology's backward compatibility with cross-sections, a three-dimensional update is simultaneously achieved, i.e., two-dimensional and three-dimensional simultaneous editing.

[0069] Unlike the tunnel outline, the dam design process also constructs the topology of each internal section of the dam. The three-dimensional internal sections of the dam and their outer boundaries are also a collection of polygons, which allows the volume and boundary area of ​​each section to be obtained automatically.

[0070] The dam body contains internal partitions, each of which is actually an envelope formed by a closed polygon. There is a coplanar connection between adjacent envelopes. Essentially, it is a case where a set of "polygons" is closed and then forms a set of "envelopes", indicating an upward increasing relationship between the two.

[0071] S900: Figure 9This represents a three-dimensional numerical computation model constructed using envelopes as primitives. The input data here consists of spatially irregular curved surfaces with topological relationships, intersecting to form envelopes. Similar to the principle of constructing a two-dimensional computation model, the difference lies in the increased complexity of the topological relationships involved when upgrading from two-dimensional to three-dimensional.

[0072] Figure 9 It is a three-dimensional numerical calculation model constructed using a spatial irregular surface intersection method, based on an engineering digital model that includes a stratigraphic interface model and a tunnel model. The material partitions and excavation partitions in the calculation model are sets of closed envelopes. Each envelope is composed of multiple polygons, and each polygon is a collection of multiple triangles, thus achieving backward compatibility.

[0073] Figure 9 The envelope drawn by the intersection of three sets of surfaces (the model surface is represented by lines) is clearly shown. It is reconstructed based on feature points. Its typical feature is that it is relatively neat and regular. In the near-horizontal direction, after extracting feature points along the ground plane, it is only ensured that the feature points match the ground plane. The line segments between the feature points are not necessarily located on the ground plane. This reflects the simplification of the geological model according to the calculation requirements, meets the requirements of real-world operation standards, and solves many practical problems that exist when directly converting geological grids into computational grids.

[0074] The material partition boundaries in the computational model correspond to the stratigraphic interfaces in the engineering digital model, while the excavation partitions correspond to the tunnel outline and working face. These are used as a basis to transfer the relevant parameters required by the computational model.

[0075] The reconstruction scheme of this invention can effectively control the envelope and mesh shape by selecting the initial point and optimizing the mapping direction. Based on the triangulation of the control point, it generates only the initial sparse mesh that controls the shape of the envelope, supports selective subdivision of the mesh according to the expected stress change gradient, and balances the relationship between the number of meshes and the calculation accuracy, thus solving all the above problems well.

[0076] The evolution of geometric primitives described in this invention, from nodes to line segments to triangles to polygons to envelopes, covers the development history of geometric techniques for irregular objects, thus yielding numerous results. Among these, the topologies of node, line segment, and triangle primitives are already very well-developed and widely used (BG Baumgart 1975, CN 113610983 A, etc.), yet they are still included in this invention with the intention of constructing a complete, upwardly incremental, and downwardly compatible system. This invention asserts: 1) Node-line-triangle-polygon-envelope primitive system: This indicates the development direction of digital modeling technology based on the relationships between them and serving irregular primitive objects, which is different from the existing technical route adopted by Dassault, Bentley, etc. 2) Technical approach for constructing the outline of geotechnical engineering objects based on polygon primitives and its modeling methods for three scenarios: excavated slopes, excavated outlines of unfilled underground caverns, and filled dams. The innovative effects achieved by this technology in these three scenarios include: allowing arbitrary editing during excavated slope design, enabling the outline to adapt to random changes in terrain and geological conditions; supporting integrated 2D and 3D operations for underground caverns, such as supporting geological logging and reinforcement design based on 2D visualizations in a 3D environment, avoiding cumbersome 2D / 3D coordinate transformations; and automatically obtaining engineering quantity data such as the volume of filling zones and the area of ​​interfaces during dam design, eliminating the need for recalculation. 3) Two-dimensional and three-dimensional numerical modeling methods based on feature point reconstruction. The reconstruction scheme is derived from the technical system of point-line-triangle-polygon-envelope volume of this invention, and is also a component of this system. Reconstruction solves the defects in the technical principles of other methods and the resulting practical feasibility problems: the accuracy of engineering two-dimensional drawings and three-dimensional digital models (line segment length, mesh size and density variation) depends on the degree of geometric undulation of the engineering object (geological body), and the variation of two-dimensional and three-dimensional mesh size in the calculation model is determined by the stress change gradient. In reality, it is impossible to consider the application requirements under unknown conditions (stress changes caused by excavation) during the engineering modeling stage.

[0077] This invention not only constructs the aforementioned technical system but also completes the transformation of technological achievements. The developed 3D geological modeling software CnGIM_ma is a product that advances from primitives to triangles. Starting from this point, the primitives are further advanced to polygons and envelopes, completing the corresponding digital modeling methods, developing the corresponding software products CnGIM_sd and CnGIM_hw, and putting them into production applications, ensuring the practical feasibility and value of the innovative technology.

Claims

1. A digital modeling method for irregular objects based on discrete mathematics and topological geometry, characterized in that, Multidimensional spatial discrete nodes are defined as the most basic digital geometric primitives. Following complete topological relationships, geometric primitives of different levels for digital modeling are constructed. The geometric primitives include line segments, triangles, polygons and envelopes, and the levels of line segments, triangles, polygons and envelopes are from low to high. There is an upward and downward compatible logical relationship between geometric primitives at different levels, with higher-order geometric primitives inheriting the complete topology of lower-order geometric primitives; The digital modeling method for irregular objects based on the aforementioned geometric primitives includes: S1: Define multidimensional space discrete nodes as the most basic digital geometric primitives, and construct line segments, triangles, polygons and envelopes in sequence according to complete topological relationships to form a geometric primitive system for digital modeling of irregular objects; S2: Construct two-dimensional and three-dimensional irregular curves based on a collection of line segment primitives that follow complete topological relationships; S3: Based on a set of planar triangular primitives that follow strict topological relationships, an irregular spatial surface is constructed for geological modeling; S4: Based on a set of planar polygon primitives that follow complete topological relationships, construct a two-dimensional computational model or a three-dimensional model of the outline of geotechnical engineering excavation and filling structures; S5: Construct a three-dimensional numerical calculation model based on an envelope primitive set that follows strict topological relationships.

2. The method as described in claim 1, characterized in that, The primitive system based on discrete mathematics and topological geometry includes: The node supports multi-dimensional attributes, which are three-dimensional (x, y, z) by default and are used to define the spatial geometric position of the node. Dimensions above three are custom indicators. The geometric primitive system consists of nodes, line segments, triangles, polygons, and envelopes, which are sequentially increasing upwards and compatible downwards. Higher-level primitives contain all the complete topologies of lower-level primitives. The term "discrete" refers to the fact that when constructing a three-dimensional digital model of an irregular object using a collection of geometric primitives, the geometric primitives are independent of each other, do not obey any functional formula, but follow complete topological relationships. The complete topology includes proximity, connection, and containment relationships, and supports topology updates.

3. The method as described in claim 1, characterized in that, The aforementioned collection of line segments based on complete topological relationships, constructing two-dimensional and three-dimensional irregular curves, includes: Connect two adjacent discrete nodes in space with straight lines in a predetermined order to form line segment primitives; connect the line segment primitives in a certain order. When all nodes are coplanar, a two-dimensional irregular curve is constructed; when they are not coplanar, a three-dimensional irregular curve is constructed.

4. The method as described in claim 1, characterized in that, The strict topological relationship of the irregular spatial surface refers to the fact that, based on the complete topological relationship, the triangular mesh follows a half-side data structure and obeys the Delaunay criterion. When the irregular spatial surface is used for geological modeling, it must simultaneously satisfy the following conditions: The triangle primitive is a Delaunay mesh with a half-side data structure, which is backward compatible and includes line segments, discrete nodes and their topological relationships. Based on this, the criteria for determining whether triangles are connected or disconnected are established. An adaptive generation technique is used to construct a planar triangular assembly. The irregular curved surface model constructed according to this requirement meets the requirements of subsequent applications, including surface topological intersection and attribute data calculation. Topology updates enable geometric operations such as adding or deleting arbitrary primitives, while moving primitives does not affect the topology and only changes the node coordinates.

5. The method as described in claim 1, characterized in that, In the planar polygon assembly, any polygon forms a closed contour line by sequentially connecting discrete nodes. The closed contour line consists of at least two triangles, is backward compatible, and includes low-order geometric primitives and their topology. Polygonal primitives form a planar polygonal assembly according to certain rules. When all primitives in the planar polygonal assembly obey the plane equation, control points on the polygon boundary are extracted, and closed polylines are reconstructed by connecting the control points in sequence. This is used to construct different types of closed regions in the two-dimensional calculation model and convert the two-dimensional engineering drawing into a two-dimensional calculation model. The control points include the intersection points generated by the intersection of selected polylines in the two-dimensional engineering drawing, the representative inflection points reflecting the topography and stratum undulations, and the nodes describing the structural outline in the two-dimensional engineering drawing. The minimum interval between adjacent control points shall not be less than the minimum side length of the calculation unit / soil strip in the calculation model; After reconstruction, the geological or engineering units in the engineering drawing correspond to the regions with the same meaning in the calculation model, and the relevant parameter attributes are automatically migrated to the calculation model according to the topological relationship.

6. The method as described in claim 4, characterized in that, When the primitives in the planar polygon assembly do not obey the plane equation, a three-dimensional model of the outline of various excavation and filling structures in geotechnical engineering is constructed based on the planar polygon assembly, that is, three-dimensional design of geotechnical engineering. Based on the differences in modeling and implementation methods, the various structures in geotechnical engineering are divided into linear structures and planar structures; The topology of the planar structure is directly constructed from polygonal primitives; the linear structure is generated by the "axis + cross-section" method, where the axis and cross-section are a collection of line segments, that is, lower-order geometric primitives are used to generate polygons. The irregular curved surface of the planar structure has a topological structure that includes the topological relationships between different polygonal primitives and among the lower-order geometric primitives within the polygonal primitives. The geometric operation requirements for adding and deleting are realized through topological updates, and the movement operation only changes the node coordinates. The irregular curved surface of the linear structure has a topological structure that includes the topological relationships between different polygonal primitives and between cross sections and axes. The topological relationships between polygonal primitives satisfy the requirements for operation according to the type of excavation location, calculation of interface area and volume of filling zones, and the topological relationships between cross sections and axes are used to achieve efficient design for two-dimensional-three-dimensional linkage editing. The line segment primitives in the interrupted surface or axis of the linear structure can be replaced by equations other than the equation of a straight line (circle, ellipse, etc.); the polygon primitives in the planar structure can be replaced by equations other than the equation of a plane (sphere, ellipsoid, etc.).

7. The method as described in claim 1, characterized in that, A three-dimensional numerical computation model is constructed based on a collection of envelopes that follow strict and complete topological relationships. This includes: using a feature-point-based reconstruction method to generate envelopes, constructing spatially closed partitions and the topological relationships between different partitions to form the three-dimensional numerical computation model. Specifically, the following steps are included: Generate feature points; the feature points are divided into initial points and mapping points; wherein, the initial points include control points of the excavation / filling contour shape in the engineering digital model, as well as auxiliary points added manually as needed; the mapping points are the intersection points generated by the intersection of a straight line or ray generated according to the preferred direction based on the starting point and the corresponding curved surface in the engineering digital model, and the corresponding topology is constructed by recording the correspondence between the initial points and the mapping points; The envelope is reconstructed and generated using the technical route of node ~ line segment ~ polygon ~ envelope. First, for the initial point, a closed polyline is constructed in the order of node ~ line segment. Based on the triangulation of the closed polyline nodes, a polygon is formed, completing the progressive process of node ~ line segment ~ triangle ~ polygon, following strict topology. Based on the topology of the mapping points and the initial point, mapping points are obtained sequentially at various levels and on the model boundary surface. Corresponding polygons are then constructed and combined to form an envelope, thus completing the envelope reconstruction.

8. The method as described in claim 7, characterized in that, The preferred direction refers to: drawing a straight line or ray through the starting point such that the straight line or ray intersects a surface in the engineering digital model at a large acute angle that is as close to a right angle as possible; the direction of the straight line or ray is then the preferred direction.

9. The method as described in claim 7, characterized in that, The straight line or ray intersects a surface in the engineering digital model at an angle that is as close to a right angle as possible, including: When feature points are used to generate material partitions, the straight lines or rays intersect the ground plane at a large acute angle that is as close to a right angle as possible; When feature points are used to reconstruct discontinuous contact surfaces corresponding to faults and joints, the straight lines or rays intersect the corresponding fault and dominant joint groups at large acute angles that are as close to right angles as possible.

10. The method as described in claim 7, characterized in that, The envelope also includes unique associations and indices between the envelope and each discrete surface in the engineering digital model, thereby sequentially calculating the mapping between each feature partition in the model and the engineering model, including the mapping between material partitions and strata, excavation partitions and excavation contours, and discontinuous contact and fractures. Based on the mapping, the required input parameters for calculation are further transferred from the engineering model, including the physical and mechanical parameters of soil and rock materials and the mechanical parameters of discontinuous contact surfaces. When the engineering digital model includes reinforcement design, the geometric and physical mechanical parameters of the corresponding reinforcement structure are obtained from the knowledge base according to the data layer association relationship during software implementation and then transferred to the computational model.

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