Photocuring 3D printing support generation method based on self-adaption of suspension angle

Abstract

The invention relates to the technical field of photocuring 3D printing support generation, in particular to a photocuring 3D printing support generation method based on suspension angle self-adaption, which comprises the following steps of: layering a model and extracting a normal vector and a curvature of a triangular patch; dynamically calculating a critical suspension angle by combining equipment parameters, material characteristics and temperature; identifying a suspension area and marking a risk level; constructing a support density mapping function to generate an initial support point; optimizing a support network topology through gradient descent; and corrected support parameters are fed back based on historical collapse data. According to the invention, adaptive generation and closed-loop optimization of the supporting structure can be realized, and the printing success rate and the supporting efficiency are improved.

Description

Technical Field

[0001] This invention belongs to the field of additive manufacturing technology, specifically a method for generating supports based on adaptive overhang angle photopolymerization 3D printing. Background Technology

[0002] With the widespread application of photopolymer 3D printing technology, the generation strategy of support structures has a decisive impact on print quality, material consumption, and post-processing efficiency. Especially in the printing of complex geometries, if overhanging areas are not properly supported, it can easily lead to interlayer collapse, surface roughness, or even printing failure. Existing technologies mostly use fixed thresholds (such as 45°) to determine whether support needs to be added, lacking the ability to adaptively respond to local geometric features and printing process parameters, and there is room for improvement in balancing support effectiveness and structural lightweighting.

[0003] A search revealed the patent CN115635683B, entitled "Integrated Optimization Design and Manufacturing Method of Structural Layout, Geometry and 3D Printing," published on July 25, 2023. This patent generates a self-supporting structure requiring no additional support by filtering out components that violate overhang angle constraints and iteratively merging components and nodes during layout optimization. This method relies on active reconstruction of the structural topology and is primarily suitable for scenarios where the original design can be modified. However, it has limitations when applied to existing models that cannot be altered. Furthermore, its overhang angle constraint uses a globally fixed threshold, failing to fully consider the impact of local curvature variations, layer thickness differences, or resin rheological properties on the actual printable overhang limit. This may result in insufficient or redundant support in complex curved surfaces or areas with fine features.

[0004] A search revealed the patent CN117235924A, entitled "Optimization Design and Manufacturing Method of Three-Dimensional Self-Supporting Truss Based on Multi-Axis 3D Printing," published on December 15, 2023. This patent reduces the number of overhanging components by integrating and optimizing the structural layout and printing direction, thereby reducing reliance on the supporting structure. This solution primarily targets truss structures in multi-axis motion systems. Its core lies in achieving self-support through coordinated optimization of the printing path and structural form. However, it has limited coverage of the support generation problem for general models in traditional single-axis photopolymerization equipment. Furthermore, its overhang constraints are still based on preset angle thresholds, lacking a dynamic assessment mechanism for local overhang risks and failing to address the adaptive generation and optimization of the supporting structure itself. This limits its adaptability when handling non-truss, high-detail, or organic-shaped photopolymerization printing tasks.

[0005] The above indicates that current photopolymer 3D printing support generation methods mostly use static thresholds for determining the overhang angle, which needs improvement in adaptability to local geometric features and process conditions. Furthermore, achieving efficient support without altering the original model still requires optimization. Therefore, this invention proposes a photopolymer 3D printing support generation method based on adaptive overhang angle. This method aims to dynamically adjust the support generation strategy by combining local geometric features and printing conditions, effectively controlling support usage, improving surface quality, and simplifying post-processing while ensuring printing success rate. Summary of the Invention

[0006] In order to overcome the above-mentioned defects of the prior art, the embodiments of the present invention provide a photopolymerization 3D printing support generation method based on adaptive suspension angle to solve the problems mentioned in the background art. To achieve the above objectives, the present invention provides the following technical solution: a method for generating supports in photopolymerization 3D printing based on adaptive overhang angle, comprising the following steps: S1, inputting a three-dimensional model; S2, performing voxelization layering on the input three-dimensional model and extracting the normal vectors and local curvature information of all surface triangular facets in each layer; S3, calculating the dynamic critical overhang angle threshold corresponding to each local region based on the layer thickness parameters of the current printing equipment, the rheological properties of the resin material, and the ambient temperature; S4, identifying the overhang regions for which supports need to be generated by combining the spatial orientation of the triangular facets and the dynamic critical overhang angle threshold, and marking their risk levels; S5, constructing a support density mapping function based on the risk level, projected area, and distance from the substrate of the overhang region, and generating an initial support point distribution map; S6, adjusting the support point positions and connection topology using a gradient descent optimization algorithm based on the initial support point distribution map, forming a support network with hierarchical density and directional connection structure; S7, performing feedback correction on the connection strength parameters and density weights of the support network based on actual collapse records of similar geometric features in historical printing tasks; S8, outputting an adaptive support network.

[0007] In a preferred embodiment, step S2, which involves voxelizing and layering the input 3D model and extracting the normal vectors and local curvature information of all surface triangular facets in each layer, specifically includes: slicing the 3D model along the Z-axis according to a set layer thickness to generate a discrete layer sequence; performing a neighborhood search on the triangular mesh in each layer to calculate the unit normal vector of each triangular facet in 3D space; using discrete differential geometry to calculate the Gaussian curvature and average curvature at each vertex based on the difference in normal vectors between adjacent triangular facets; using the curvature values ​​and normal vectors together as input variables for subsequent dynamic threshold calculation, and establishing a geometric feature data table indexed by the layer number.

[0008] In a preferred embodiment, step S3, calculating the dynamic critical overhang angle threshold corresponding to each local region based on the layer thickness parameters of the current printing equipment, the rheological properties of the resin material, and the ambient temperature, specifically includes: establishing a process parameter-overhang limit mapping database, which contains the measured maximum printable overhang angle under different combinations of layer thickness (25μm–100μm), resin viscosity (200–2000 mPa·s), and ambient temperature (15–35℃); interpolating and querying the parameters used in the current printing task to obtain the basic critical angle θ0; introducing a local curvature correction factor κ, defining the dynamic critical overhang angle θ = θ0× (1 − α·|κ|), where α is an empirical coefficient with a value range of 0.1–0.3; and assigning an independent θ value to each triangular facet to form a spatially varying overhang judgment threshold field.

[0009] In a preferred embodiment, step S4, combining the spatial orientation of the triangular facets with the dynamic critical overhang angle threshold, identifies the overhang areas requiring support and marks their risk levels. Specifically, this includes: calculating the angle β between the normal vector of each triangular facet and the positive direction of the Z-axis; when β > θ, determining that the facet is in an overhang state and including its region in the candidate support area; classifying the risk level according to the difference between β and θ, Δβ = β − θ: Δβ ∈ (0°, 5°) is low risk, (5°, 15°) is medium risk, and >15° is high risk; merging consecutive adjacent overhang facets of the same level to generate a set of overhang areas with risk labels.

[0010] In a preferred embodiment, step S5, which involves constructing a support density mapping function and generating an initial support point distribution map based on the risk level, projected area, and distance from the substrate of the overhanging region, specifically includes defining a support density function ρ = w1·R + w2·A. p + w3·H, where R is the risk level code value (low=1, medium=2, high=3), A p The normalized value of the projected area of ​​the overhanging region in the XY plane is given by , H is the normalized height of the center point of the region from the printed substrate, and w1, w2, and w3 are preset weighting coefficients. Support points are arranged in each overhanging region according to the Poisson disk sampling strategy, with a point spacing d = d0 / √ρ, where d0 is the baseline spacing. The coordinates of the support points are bound to the risk level of their respective regions, and an initial support point distribution map is output.

[0011] In a preferred embodiment, step S6, based on the initial support point distribution map, uses a gradient descent optimization algorithm to adjust the support point positions and connection topology to form a support network with hierarchical density and directional connection structure. Specifically, this includes: constructing a connection graph between support points, with initial connections using Delaunay triangulation; defining the energy function E = E s+λ·E t E s E is the total length of the supporting structure. t The cumulative value of the angle deviation between the support and the normal vector of the overhanging surface is λ, which is the directional consistency weight. The support point coordinates are used as optimization variables, and the position is updated iteratively through gradient descent to minimize the energy function E. Redundant connections with energy contributions below the threshold are removed, and the main support paths are retained to form a lightweight and directionally adaptable support network.

[0012] In a preferred embodiment, step S7, based on actual collapse records of similar geometric features in historical printing tasks, specifically includes the following steps to correct the connection strength parameters and density weights of the support network: establishing a geometric feature-collapse event database to record the curvature, overhang angle, support density, and actual failure mode of areas that collapsed in past printings; increasing the risk level of overhang areas in the current model that have a geometric similarity of more than 85% with historical collapse areas by one level; increasing the α coefficient of a certain curvature range (e.g., |κ| > 0.8 mm⁻¹) to 0.35 in historical data if the collapse rate is higher than 15% in S2; and simultaneously increasing the w1 weight in S4 to automatically obtain higher support density for high curvature areas, thus completing the closed-loop parameter correction. Attached Figure Description

[0013] Figure 1 This is a flowchart illustrating a photopolymerization 3D printing support generation method based on adaptive overhang angle according to the present invention. It shows the complete processing steps from 3D model input to support network output, including 3D model input, voxelization layering, dynamic critical overhang angle calculation, overhang area identification and risk classification, initial support point distribution generation, support network optimization and feedback correction based on historical data, and adaptive support network output.

[0014] The attached diagrams are labeled as follows: S1, 3D model input; S2, voxelization layering and geometric feature extraction; S3, dynamic critical overhang angle calculation; S4, overhang area identification and risk level marking; S5, initial support point distribution map generation; S6, support network optimization; S7, historical collapse data feedback correction; S8, adaptive support network output. Detailed Implementation

[0015] Reference Figure 1The diagram shown illustrates a flow chart of a photopolymerization 3D printing support generation method based on adaptive overhang angle according to the present invention. The entire method consists of eight main modules connected sequentially: 1. 3D model input; 2. Voxelization layering and geometric feature extraction; 3. Dynamic critical overhang angle calculation; 4. Overhang area identification and risk level marking; 5. Initial support point distribution map generation; 6. Support network optimization; 7. Historical collapse data feedback correction; 8. Adaptive support network output. The following detailed and complete implementation description of each step, in conjunction with the accompanying drawings, ensures that those skilled in the art can fully reproduce the technical solution of the present invention based on this specification.

[0016] First, in step 1, the user imports the 3D model to be printed into the support generation system in STL, OBJ, or 3MF format. This 3D model is represented by a closed mesh composed of several triangular facets, each containing the coordinates of three vertices and their normal vectors. After receiving the model, the system passes it to step 2 for further processing.

[0017] In step 2, the system performs voxelization layering on the 3D model along the Z-axis according to a set layer thickness parameter (e.g., 50 μm), discretizing the continuous 3D geometry into a series of 2D slices parallel to the XY plane, forming a discrete layer sequence. Each layer corresponds to a Z-coordinate value and retains all triangular facets intersecting the slice plane within that layer. Subsequently, for each triangular facet in each layer, the system performs a neighborhood search operation: centered on the current triangular facet, it traverses all adjacent triangular facets sharing vertices or edges, constructing a local neighborhood graph. Based on this, the system calculates the unit normal vector n = (nx, ny, nz) of each triangular facet in 3D space using the vector cross product formula and normalizes it. Then, using discrete differential geometry, based on the angle difference between the normal vectors of adjacent triangular facets, the system calculates the Gaussian curvature K and mean curvature H at each vertex. Specifically, for a vertex v, its Gaussian curvature K(v) is obtained by calculating the difference between the solid angle subtended by the normal vectors of the triangular facets surrounding v and 2π, and then dividing by the Voronoi area of ​​that vertex; the average curvature H(v) is obtained by weighted summation of the dihedral angles of adjacent edges and dividing by the perimeter of the vertex's neighborhood. Finally, the average curvature values ​​of the three vertices associated with each triangular facet are taken as the local curvature κ of that facet, and κ and the normal vector n are stored together in a geometric feature data table indexed by the layer number, completing step 2, and this data table is then passed to step 3.

[0018] In step 3, the system queries a pre-established process parameter-sag limit mapping database based on the process parameters configured for the current printing task—including layer thickness h (e.g., 50 μm), viscosity η of the photosensitive resin used (e.g., 800 mPa·s), and ambient temperature T (e.g., 25℃). This database, obtained through extensive experimental calibration, records the maximum sag angle θ0 that can be stably printed under different combinations of (h, η, T). For example, when h=50 μm, η=800 mPa·s, and T=25℃, θ0=45° is found. Subsequently, the system introduces a local curvature correction factor κ, defining the dynamic critical sag angle θ = θ0 × (1 − α·|κ|), where the empirical coefficient α is preset to 0.2 based on material properties. For each triangular facet, the |κ| calculated in step 2 is substituted into the above formula to calculate an independent θ value, thus forming a spatially varying sag judgment threshold field across the entire model surface. This threshold field serves as the judgment criterion for subsequent identification of sag regions.

[0019] In step 4, the system traverses all triangular faces and calculates the angle β = arccos(nz) between its normal vector n and the positive Z-axis unit vector (0,0,1). If β > θ, the triangular face is determined to be in a suspended state and marked as a candidate support face. Subsequently, Δβ = β − θ is calculated, and risk levels are divided according to preset intervals: when Δβ ∈ (0°, 5°], it is marked as low risk (R=1); when Δβ ∈ (5°, 15°], it is marked as medium risk (R=2); when Δβ > 15°, it is marked as high risk (R=3). Next, the system performs a region merging operation on spatially continuous suspended faces with the same risk level: through graph connectivity analysis, faces of the same level that share edges or vertices are clustered into single suspended regions, and each region is assigned a unique region ID and a unified risk level label, finally generating a set of suspended regions with risk labels, which serves as the input for step 5.

[0020] In step 5, the system constructs a support density mapping function for each overhanging region. First, it calculates the projected area A of that region on the XY plane. p The coordinates are then normalized to the [0,1] interval; simultaneously, the Z-coordinate of the geometric center point of this region is calculated and normalized relative to the total printing height to obtain H ∈ [0,1]. Subsequently, the support density function ρ = w1·R + w2·A is defined. p+ w3·H, where the weighting coefficients w1=0.6, w2=0.25, and w3=0.15 are preset constants. Based on the ρ value, the system uses a Poisson disk sampling strategy to deploy support points in this area: the baseline spacing d0=2.0 mm is set, and the actual point spacing d = d0 / √ρ. During the sampling process, it is ensured that the Euclidean distance between any two support points is not less than d, and priority is given to covering high curvature or high-risk sub-regions. Each support point records its three-dimensional coordinates (x, y, z), the ID of the overhanging region to which it belongs, and the risk level R. Finally, the initial support point distribution map is output and passed to step 6.

[0021] In step 6, the system constructs a support network using the initial support point distribution map as input. First, Delaunay triangulation is performed on all support points to generate an initial connection diagram, where each edge represents a potential support rod between two points. Then, the energy function E = E0 is defined. s + λ·E t E s E is the sum of the lengths of all support rods. t This is the cumulative value of the angular deviation between the direction of each support rod and the normal vector of the corresponding overhanging surface. Specifically, for the support rod connecting points p_i and p_j, its direction vector is v_ij = p_j − p_i; if the rod serves a certain overhanging surface f, its normal vector is n_f, then the angular deviation is 1 − |v_ij·n_f| / (||v_ij||·||n_f||). λ is the directional consistency weight, set to 0.4. The system uses the (x,y) coordinates of all support points as optimization variables (the z coordinate is fixed to the corresponding floor height), and uses the gradient descent algorithm to iteratively update the point positions: in each iteration, the partial derivative of E with respect to each variable is calculated, and the point coordinates are adjusted along the negative gradient direction with a learning rate η=0.01 until the energy change is less than 1e-5 or the maximum number of iterations of 100 is reached. After optimization, the system iterates through all support rods and calculates their contribution to the total energy E. If a rod's contribution is below a set threshold (e.g., 0.1% of the total E), it is removed from the connection diagram. The final retained connection structure forms a lightweight support network with hierarchical density and directional connection characteristics.

[0022] Then, in step 7, the system invokes a historical collapse data feedback correction mechanism. This mechanism relies on a pre-built geometric feature-collapse event database, which records information about areas where collapses occurred in past printing tasks, including local curvature κ, actual overhang angle β, support density ρ used, and failure modes (such as fracture, peeling, deformation). For each overhang region in the current model, the system extracts its geometric feature vector (κ, β, R) and compares it with all collapse records in the database: using cosine similarity calculation, if the similarity exceeds 85%, the risk level R of that region is increased by one level (e.g., from medium risk to high risk). Furthermore, if the database shows a historical collapse rate exceeding 15% within a curvature range of |κ| > 0.8 mm⁻¹, the system temporarily adjusts the α coefficient corresponding to that region from 0.2 to 0.35 in step 3 to reduce the dynamic critical angle θ and enhance the sensitivity of overhang determination. Simultaneously, in step 5, the system increases the weight of w1 from 0.6 to 0.75, giving higher weight to high-curvature regions in the support density calculation, thereby automatically increasing the support point density. This feedback correction process achieves closed-loop parameter correction, ensuring that the support strategy dynamically evolves with historical experience.

[0023] Finally, in step 8, the system outputs the optimized and corrected support network in a standard format (such as support-specific SLT or internal binary format) for use by the slicing software. This support network contains the three-dimensional coordinates of all support points, connection topology, local density levels, and connection strength parameters, and can be directly used for support structure generation and printing path planning in photopolymer 3D printers.

[0024] Throughout the process, the modules are tightly coupled through data structures: the geometric feature data table output in step 2 serves as the input for step 3; the dynamic threshold field generated in step 3 drives the overhang determination in step 4; the set of labeled overhang regions output in step 4 is used for density calculation in step 5; the initial support point distribution map in step 5 forms the basis for optimization in step 6; the support network in step 6 is corrected using historical data in step 7; and the final correction result is output in step 8. All parameters (such as α, w1, w2, w3, λ, d0, etc.) can be configured through the user interface or automatically loaded by the system according to the material / equipment type, ensuring the method's universality and configurability. Through the above detailed steps, this invention achieves adaptive, refined, and data-driven generation of overhang structure supports in photopolymerization 3D printing, solving the problem of support redundancy or insufficiency caused by traditional fixed threshold methods. To better enable those skilled in the art to fully understand and implement this invention, the specific implementation principle of this invention is further supplemented below with a specific application scenario.

[0025] In a practical photopolymerization 3D printing task, taking the printing of a dental model with a complex organic curved surface as an example, this model contains multiple high-curvature overhanging regions (such as the gingival margin and pit and fissure structures) and large-area low-curvature overhanging surfaces (such as the buccal outer contour). The system first receives the STL file of the model in step 1, and then performs voxelization layering with a layer thickness of 50 μm in step 2. For each triangular facet in each layer, the system constructs a local geometry through neighborhood search and calculates the normal vector n and local curvature κ for each facet. For example, at the cusp and pit fissure, due to the drastic abrupt change in the normal vector direction of adjacent triangular facets, the calculated |κ| is as high as 1.2 mm⁻¹, while in the gentler buccal region, |κ| is only 0.1 mm⁻¹.

[0026] In step 3, based on the resin viscosity η = 800 mPa·s, layer thickness h = 50 μm, and ambient temperature T = 25℃, the system retrieves the basic critical angle θ0 = 45° from the process parameter-sag limit mapping database. Subsequently, for the high-curvature pit region, substituting α = 0.2, θ = 45° × (1 − 0.2 × 1.2) = 34.2° is calculated; while for the gentler region, θ = 45° × (1 − 0.2 × 0.1) ≈ 44.1°. This creates a spatially varying dynamic threshold field, making the sag determination in high-curvature regions more stringent, thereby avoiding localized collapse due to insufficient resin flowability.

[0027] In step 4, the system calculates the angle β between the normal vector of each patch and the Z-axis. For the pit and fissure region, the measured β is 48°, which is greater than its dynamic threshold of 34.2°, Δβ=13.8°, and it is marked as medium risk (R=2). However, for a certain buccal region, β=46°, which slightly exceeds the traditional 45° threshold, but because its dynamic threshold is 44.1°, Δβ=1.9°, it is only marked as low risk (R=1). Through graph connectivity analysis, the system clusters adjacent medium-risk patches into an independent overhang region (ID: A07) and assigns a unified risk label to ensure regional consistency in subsequent support generation.

[0028] In step 5, for region A07, the system calculates its XY projected area A. p =12.5 mm², normalized to 0.38; its center point Z coordinate is 8.2 mm above the substrate, and the total model height is 20 mm, so H=0.41. Substituting into the support density function ρ= 0.6×2 + 0.25×0.38 + 0.15×0.41 ≈ 1.35. The reference spacing d0=2.0 mm, then the actual point spacing d =2.0 / √1.35 ≈ 1.72 mm. The system uses Poisson disk sampling to place support points in this area, prioritizing denser placement near the peak curvature to ensure sufficient support reaction force in high-stress areas.

[0029] In step 6, all support points are triangulated using Delaunay triangulation to generate an initial connection diagram. The system defines the energy function E = E s + 0.4·E t E t The consistency between the direction of the support rod and the normal vector of the suspended surface is measured. For example, if a support rod connects two points p_i and p_j, and its direction vector v_ij makes a 30° angle with the surface normal vector n_f, then the angle deviation term is 1 − cos(30°) ≈ 0.134. Through gradient descent optimization, the system fine-tunes the position of the support point (x, y) to make the support rod more perpendicular to the suspended surface, thereby improving the support stiffness. After optimization, several redundant connections are eliminated because their energy contribution is less than 0.1%, ultimately forming a lightweight network with a clear backbone and reasonable branches.

[0030] In step 7, the system queries the historical collapse database and finds that in areas where |κ|>0.8 mm⁻¹ and Δβ>10°, the collapse rate reached 18%. The geometric feature vector of the current region A07 (κ=1.2, β=48°, R=2) has a cosine similarity of 89% with a record in the database (κ=1.15, β=47.5°, failure mode: peeling), exceeding the 85% threshold. Therefore, the system automatically increases the risk level of A07 from R=2 to R=3, and in step 5, temporarily adjusts w1 from 0.6 to 0.75, recalculates ρ≈1.52, and further reduces the point spacing to 1.62 mm. Simultaneously, in step 3, the α coefficient for this region is increased from 0.2 to 0.35, reducing θ to 31.5° and enhancing overhang sensitivity.

[0031] Finally, in step 8, the system outputs a support network containing precise coordinates, topological connections, and local density levels, which the slicing software uses to generate support entities. This support effectively suppresses resin flow deformation in the pit and fissure areas during printing, while avoiding over-support in low-risk areas. During post-processing, the support is easy to peel off, and the model surface roughness Ra < 15 μm, which is significantly better than the traditional fixed 45° threshold method (Ra > 35 μm, and there is local collapse).

[0032] All content not described in detail in this specification belongs to existing technology known to those skilled in the art, and the implementation of each algorithm module can be based on open-source geometry processing libraries (such as CGAL, Open3D) or commercial slicing engines. Software control logic not mentioned in this technical solution is not shown in the figures due to conventional programming implementation, and will not be elaborated here. The above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A method for generating supports using photopolymerization 3D printing based on adaptive overhang angle, characterized in that, The process includes the following steps: S1, inputting a 3D model; S2, performing voxelization and layering on the input 3D model, and extracting the normal vectors and local curvature information of all surface triangular facets in each layer; S3, calculating the dynamic critical overhang angle threshold for each local area based on the layer thickness parameters of the current printing equipment, the rheological properties of the resin material, and the ambient temperature; S4, combining the spatial orientation of the triangular facets and the dynamic critical overhang angle threshold, identifying the overhang areas that need to be supported, and marking their risk levels. S5. Based on the risk level, projected area, and distance from the substrate of the overhanging area, construct a support density mapping function to generate an initial support point distribution map; S6. Based on the initial support point distribution map, use a gradient descent optimization algorithm to adjust the support point positions and connection topology to form a support network with hierarchical density and directional connection structure; S7. Based on the actual collapse records of similar geometric features in historical printing tasks, the connection strength parameters and density weights of the support network are corrected. S8. The corrected support network is output as adaptive support structure data.

2. The method for generating a photopolymer 3D printing support based on adaptive overhang angle according to claim 1, characterized in that, In S2, the input three-dimensional model is voxelized and layered, and the normal vectors and local curvature information of all surface triangular facets in each layer are extracted (2). Specifically, the three-dimensional model is sliced ​​along the Z-axis according to a set layer thickness to generate a discrete layer sequence; a neighborhood search is performed on the triangular mesh in each layer to calculate the unit normal vector of each triangular facet in the three-dimensional space; the discrete differential geometry method is used to calculate the Gaussian curvature and average curvature at each vertex based on the difference in normal vectors of adjacent triangular facets; the curvature value and normal vector are used together as input variables for subsequent dynamic threshold calculation, and a geometric feature data table is established with the layer number as the index.

3. The method for generating a support based on adaptive overhang angle photopolymerization 3D printing according to claim 1, characterized in that, In S3, the dynamic critical overhang angle threshold (3) corresponding to each local area is calculated based on the layer thickness parameters of the current printing equipment, the rheological properties of the resin material and the ambient temperature. Specifically, this includes: establishing a process parameter-overhang limit mapping database, which contains the measured maximum printable overhang angle under different combinations of layer thickness (25μm–100μm), resin viscosity (200–2000 mPa·s) and ambient temperature (15–35℃); interpolating and querying the parameters used in the current printing task to obtain the basic critical angle θ0; introducing a local curvature correction factor κ, defining the dynamic critical overhang angle θ = θ0 × (1 − α·|κ|), where α is an empirical coefficient with a value range of 0.1–0.3; assigning an independent θ value to each triangular facet to form a spatially varying overhang judgment threshold field.

4. The method for generating a support based on adaptive overhang angle photopolymerization 3D printing according to claim 1, characterized in that, In S4, the spatial orientation of the triangular facets and the dynamic critical overhang angle threshold are combined to identify the overhang areas that need to be supported and mark their risk levels. (4) Specifically, this includes: calculating the angle β between the normal vector of each triangular facet and the positive direction of the Z-axis; when β > θ, the facet is determined to be in an overhang state and its area is included in the candidate support area; the risk level is divided according to the difference between β and θ, Δβ = β − θ: Δβ ∈ (0°, 5°) is low risk, (5°, 15°) is medium risk, and >15° is high risk; the regions of consecutive adjacent overhang facets of the same level are merged to generate a set of overhang areas with risk labels.

5. The method for generating a photopolymer 3D printing support based on adaptive overhang angle according to claim 1, characterized in that, In step S5, based on the risk level, projected area, and distance from the substrate of the overhanging region, a support density mapping function is constructed to generate an initial support point distribution map (5). Specifically, this includes defining the support density function ρ = w1·R + w2·A. p +w3·H, where R is the risk level code value (low=1, medium=2, high=3), A p The normalized value of the projected area of ​​the overhanging region in the XY plane is given by , H is the normalized height of the center point of the region from the printed substrate, and w1, w2, and w3 are preset weighting coefficients. Support points are arranged in each overhanging region according to the Poisson disk sampling strategy, with a point spacing d = d0 / √ρ, where d0 is the baseline spacing. The coordinates of the support points are bound to the risk level of their respective regions, and an initial support point distribution map is output.

6. The method for generating a photopolymer 3D printing support based on adaptive overhang angle according to claim 1, characterized in that, In S6, based on the initial support point distribution map, the gradient descent optimization algorithm is used to adjust the support point positions and connection topology to form a support network with hierarchical density and directional connection structure (6). Specifically, this includes: constructing a connection map between support points, with the initial connection using Delaunay triangulation; defining the energy function E = E s + λ·E t E s E is the total length of the supporting structure. t The cumulative value of the angle deviation between the support and the normal vector of the overhanging surface is λ, which is the directional consistency weight. The support point coordinates are used as optimization variables, and the position is updated iteratively through gradient descent to minimize the energy function E. Redundant connections with energy contributions below the threshold are removed, and the main support paths are retained to form a lightweight and directionally adaptable support network.

7. The method for generating a photopolymerization 3D printing support based on adaptive overhang angle according to claim 1, characterized in that, In S7, the connection strength parameters and density weights of the support network are corrected based on the actual collapse records of similar geometric features in historical printing tasks (7). Specifically, this includes: establishing a geometric feature-collapse event database to record the curvature, overhang angle, support density and actual failure mode of the area that collapsed in the past printing; increasing the risk level of overhang areas in the current model that have a geometric similarity of more than 85% with historical collapse areas by one level; increasing the α coefficient of the area to 0.35 in S2 if the collapse rate of a certain curvature interval (such as |κ| > 0.8 mm⁻¹) is higher than 15% in historical data; and simultaneously increasing the w1 weight in S4 so that the high curvature area automatically obtains a higher support density and completes the closed-loop parameter correction.

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