Fig. 1. Schematic diagram of the overhanging structure, where the red areas is the overhanging structure areas

Fig. 2. SLM printing at seven different overhang angles^[21]

Fig. 3. Schematic diagram of self-supporting. The orange area represents the print layer, and the green areas represent the finish print layers. When the orange area can be printed, the structure is considered to be self-supporting

Fig. 4. Definition of supporting region S(i,j) for element (i,j)^[30]

Fig. 5. Definition of 3D AM filter, where the blue region S(i,j,k) denotes the supporting region of an element at position (i,j,k) in a mesh^[31]

Fig. 6. Flowchart of the topology optimization process with overhang constraint^[37] (After selecting the appropriate reference domain, define the underlying data for the structural analysis. Then, the optimization loop is performed, which includes defining the topology optimization parameters, obtaining the physical density field, profile evaluation, finite element method calculations, and sensitivity analysis)

Fig. 7. Two-dimensional test results of MBB and cantilever beams. (a) MBB beams obtained by Langelaar^[30]; (b) MBB beams obtained by Garaigordobil et al.^[37]; (c)–(f) optimal topology structures with different critical angles of 45°, 60°, 80°, and 90° obtained by Garaigordobil et al.^[37]

Fig. 8. Smooth topology structure and density of grid points within elements^[40] ( Divide the element points into N grid points. When ρe,g=1, grid points are solid points; when ρe,g=ρmin, the grid point is a void point)

Fig. 9. Cubic univariate B-spline basis functions with uniform and open knot vectors^[42]

Fig. 10. Optimal structural configuration of the cantilever beam without constraints^[45]

Fig. 11. Graph of convergence history and calculation time of the cantilever beam re-optimization step^[45]

Fig. 12. Structural skeleton deposition path under level set and overhang constraint modeling based on multi-layer level set^[47]. (a)‒(b) Two examples of a horizontal set-based deposition path; (c) modeling of overhang constraint base on multi-layer level set, where d represents the maximum overhang length constraint and the critical overhang angle is set to 45°

Fig. 13. Downward cusp^[48]

Fig. 14. Skeleton segmentation^[49]. (a) Structural topology; (b) identified skeleton; (c) identified intersection and end points; (d) segmented structural skeletons; (e) segmented structural areas

Fig. 15. Threshold self-supporting conditions^[49]. (a) Threshold self-supporting condition when the building direction aligns with the y-axis; (b) threshold self-supporting condition when the building direction deviates from the y-axis

Fig. 16. Overhang recognition^[52]. (a) Five yellow elements in the supporting region can support the target element; (b) the target element can support five green elements in the supported region

Fig. 17. Optimization results of the self-supporting cantilever beams. (a) Self-supporting cantilever beam proposed by Bi et al.^[52]; (b) self-supporting cantilever beam proposed by Langelaar^[31]

Fig. 18. Basic idea of the MMC-based and MMV-based topology optimization approaches^[53]. (a) Topology optimization evolution process based on MMC; (b) topology optimization evolution process based on MMV

Fig. 19. Definition of polygons with odd and even sides^[54]. (a) An 11-side polygon; (b) a 12-side polygon

Fig. 20. Illustration of proposed approach for self-supporting design of cantilever beam proposed based on solid polygon features^[55]

Fig. 21. Three-dimensional cantilever beam with virtual skeleton^[56]

Fig. 22. Hemp cantilever: comparison of solutions obtained through layout optimization and continuous topology optimization^[57]

Fig. 23. Test results of the structural layout, geometric optimization and 3D printing of the bidirectional center-stressed vertical truss considering different self-supporting critical angles^[22]

Fig. 24. A direction-driven shape optimization program that significantly reduces the use of support materials^[58]. (a) Initial design; (b) adjusted new scheme, with the support structure indicated in blue

Fig. 25. Filling optimization process using a rhombic structure: by performing a so-called carving operation on each leaf node of the rhombic tree, the rhombic element is converted into a rhombic shell with a given wall thickness^[61]

Fig. 26. Flow diagram of the self-supporting hollow filling structure algorithm^[62]. (a) A triangular mesh is first entered, and the self-supporting infill structure is generated in its offset version; (b) randomly generate a large number of inner pillars and nodes, and initialize the radius of all pillars; (c) elimination of all redundant struts and non-self-supporting struts; (d) further optimization of inner pillars and joints to reduce material usage; (e) the model is generated by three forces: one at the top and the other two on the sides

Fig. 27. Microscopic infill structures. (a) Multi-faceted Voronoi structure^[63]; (b) CrossFill-filled structure similar to foam^[64]

Fig. 28. Unit cells of the layer-based infill structure are reconstructed by cross-sections, where p_i represents the starting point, the blue polygon is the hexagonal cross-section, the red and green polygons are two equilateral triangular cross-sections, and the orange and purple polygons are two adjacent cross-sections^[65]

Fig. 29. Printing effect of a kitten model. (a) A porous structure similar to a skeleton^[66]; (b) elliptical hollow structure^[67]

Fig. 30. Filling structural elements based on layers and the printed result^[68]. (a) Filling elements; (b) print result

Fig. 31. Overhang constraints based on AM filter^[69]

Fig. 32. Improved parametrization scheme based on the two fields formulation (green: the objective function; blue: global volume constraint; red: local volume constraint; brown: overhang constraint)^[70]

Fig. 33. Sensitivity parallel solving flowchart^[71]

Fig. 34. Optimization effect of MBB beam^[73]. (a) Unconstrained; (b) overhang angle constraints; (c) overhang angle and height constraints

Fig. 35. Optimized MBB beam results under self-support constraint, considering the printing direction and the overhang angle: the change of the substrate represents different printing directions^[74]

Fig. 36. Printing of beam parts^[75]. (a) Original model; (b) printed model

Fig. 37. Simulation results of residual stress distribution of MBB beam^[77]. (a) Self-supporting but without residual stress constraints; (b) self-supporting and residual stress constraints