Method for improving bending resistance of TPMS (Tire Pressure Monitor System) sandwich structure by utilizing variable density topological optimization
By using variable density topology optimization design, the relative density function and wall thickness gradient function of the TPMS sandwich structure are constructed to generate variable thickness cells, which solves the problem of insufficient bending resistance of the TPMS sandwich structure and achieves higher bending resistance and lightweight effect.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- TIANJIN POLYTECHNIC UNIV
- Filing Date
- 2025-12-11
- Publication Date
- 2026-04-17
AI Technical Summary
Existing TPMS sandwich structures fail to fully utilize the mechanical properties of the material in terms of bending resistance. Uniform relative density design and conventional relative density gradient design fail to maximize their bending resistance.
A variable density topology optimization method was adopted. The relative density function of the TPMS structure was constructed by polynomial fitting. The relative density field was divided into multiple segments, the wall thickness gradient function matrix was constructed, and variable thickness cells were generated and spliced to realize the multi-segment variable density optimization design. The sandwich structure was printed by combining photopolymerization molding technology, and three-point bending test and numerical verification were carried out.
It significantly improves the bending resistance of TPMS sandwich structures while ensuring lightweight structure and efficient material utilization, with bending stiffness increased by 52.19%~17.18%.
Smart Images
Figure CN121885031A_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of core layer design and performance improvement technology of biomimetic sandwich structures, and in particular relates to a method for improving the bending resistance of a novel lightweight TPMS sandwich structure by using variable density topology optimization design. Background Technology
[0002] Sandwich structures, characterized by their lightweight, high stiffness, and excellent impact resistance, are widely used in aerospace, construction, and transportation. Biomimetic sandwich structures further optimize mechanical properties, such as enhancing energy absorption, improving fatigue life, and increasing versatility. Tri-periodic minimal surface (TPMS) biomimetic structures possess continuous, smooth, and high surface area ratios, exhibiting excellent specific strength, uniform stress distribution, and functional gradient designability. When used as the core layer of a sandwich structure, TPMS structures can achieve both high performance and lightweighting; however, they typically employ uniform relative density designs or conventional relative density gradient designs, resulting in underutilization of the material's mechanical properties. Topology optimization can design material distribution based on the optimal force transmission path of the structure under given loads and constraints. Therefore, this study investigated the effect of variable density topology optimization design on improving the flexural performance of TPMS sandwich structures.
[0003] The diverse topological configurations and functional gradient designability of TPMS structures have made them not only outstanding in compressive performance but also of great interest in flexural performance. In recent years, scholars both domestically and internationally have conducted a series of studies on the flexural performance of TPMS sandwich structures. Inspired by the scales of butterfly wings, Pelanconi et al. optimized the stiffness of Gyroid structures under bending loads by adding carbon fiber reinforcements to the external region. Peng et al. studied the differences in mechanical properties and parametric analysis of monolithic Primitive, Neovius, and IWP sandwich structures, with the Neovius sandwich structure exhibiting excellent flexural performance and energy absorption. Wu et al. conducted parametric analysis on TPMS sandwich structures with a Gyroid core, finding that the relative density of the core and the thickness of the panels are positively correlated with the flexural modulus and peak load of the sandwich structure. Stepinac et al. studied the static bending response of simply supported pure core beams and sandwich beams based on Gyroid structures, with the sandwich beams showing significantly higher load-bearing capacity and stiffness than the pure core beams. Studies by Guo et al. have found that the bending performance and stability of Primitive, Neovius, and Diamond sandwich structures are significantly improved after introducing a relative density gradient design. They also found that Primitive sandwich structures have stronger resistance to deformation and are more sensitive to relative density gradient designs. Wu et al., through finite element analysis and experiments, discovered that the ultimate load and bending stiffness of three-period minimal surface (P-TPMS) sandwich structures are significantly affected by cell length, relative density, and panel thickness. Li et al. compared the mechanical properties of Primitive and Gyroid sandwich structures with those of traditional honeycomb sandwich structures, finding that Primitive sandwich structures have advantages in resisting bending deformation and energy absorption. Mahapatra et al. studied the bending performance of TPMS-based Diamond lattice sandwich panels. They investigated the bending performance of TPMS-based Diamond lattice sandwich panels at different relative densities, finding that the sandwich panel with a relative density of 35% exhibited the best performance in terms of bending strength and energy absorption. Fashanu et al. studied the mechanical properties of TPMS-based Gyroid and Diamond structures as cores in sandwich composites and compared them with traditional honeycomb cores, finding that TPMS structures have advantages in energy absorption and impact resistance. Liu et al. designed a hybrid TPMS core layer by combining Diamond and Schwarz P structures and controlled the width of its transition region, studying its effect on the bending performance of the composite sandwich structure. Lin et al. studied the mechanical properties of different TPMS lattice core layers under three-point bending loads, finding that the SP lattice core layer had the smallest deformation, while the F-RD lattice core bore the largest load.Although the above studies systematically explored the influence of TPMS structure type and relative density as well as panel parameters on the bending performance of sandwich structures, the relative density of the core layer is usually designed uniformly or conventionally with a gradient, which cannot maximize the mechanical properties of the material.
[0004] Domestic and international scholars have made some progress in improving the mechanical properties of TPMS structures through topology optimization. Parlayan et al. proposed a novel mapping method based on adjoint sensitivity, which efficiently transforms topology optimization results into manufacturable gradient TPMS structures, and verified the effectiveness of this method in improving mechanical properties using bending tests. Alkebsi et al. combined topology optimization and TPMS lattice structure gradient design to optimize the internal solid structure of gas turbine blades, reducing the weight, stress, and deformation of the blades. Panesar et al. found that combining topology optimization results with TPMS cells not only optimized the load transfer efficiency of the lattice structure, but also enhanced the mechanical properties of the structure under different loading directions through gradient design. Zeng Yuanhui et al. proposed a density gradient hybridization optimization design method based on the anisotropy of TPMS lattice structures, comprehensively considering the influence of load direction and relative density changes on mechanical properties. Simsek et al. proposed a topology optimization design method integrating homogenization theory and radial basis function mapping strategy, realizing the efficient design and performance optimization of functionally graded lattice structures based on Gyroid structures. Li et al. proposed a functionally graded lattice structure optimization method based on a three-period horizontal plane. By combining topology optimization algorithms with the geometric properties of TPLS structures, they significantly improved the mechanical properties of the structures. The above research indicates that while topology optimization can enhance the mechanical properties of TPMS structures, it did not consider applying topology optimization to improve the bending resistance of TPMS sandwich structures. Summary of the Invention
[0005] The problem this invention aims to solve is to provide a method for improving the flexural resistance of TPMS sandwich structures using variable density topology optimization. This method utilizes polynomial fitting to construct the isosurface threshold t and relative density ρ of the TPMS structure. TPMS function ρ TPMS (t); Variable density topology optimization is performed with the relative density of topological elements as the design variable and minimizing structural flexibility as the design objective. The relative density field is divided into multiple segments, and a function matrix ρ(x) is constructed using polynomial fitting; ρ(x) and ρ TPMS(t) mapping is used to construct the TPMS structure wall thickness gradient function matrix t(x), generating TPMS cells with varying thicknesses and splicing them together to achieve multi-segment variable density optimization design of the TPMS core layer; photopolymerization molding technology is used to print photosensitive resin panels and core layers respectively, and three-point bending tests of the TPMS sandwich structure before and after variable density topology optimization are carried out and compared and analyzed with numerical results. A method for improving the bending resistance of TPMS sandwich structure by using variable density topology optimization is proposed, so that the TPMS sandwich structure can achieve higher bending resistance while ensuring the lightweight of the structure and realizing efficient use of materials.
[0006] To solve the above-mentioned technical problems, the technical solution adopted by the present invention is: a method for improving the bending resistance of TPMS sandwich structures using variable density topology optimization, characterized by including the following steps:
[0007] Step (1), Construction of the function of TPMS structure isosurface threshold and relative density: Taking the typical Diamond structure in TPMS structure as an example, the Diamond structure at relative density ρ TPMS When the coefficient is [0.2, 0.6], there is a clear linear relationship between the coefficient and the isosurface threshold t, meaning that the coefficient of determination R is within this range. 2 >0.999, therefore, t is taken as the independent variable and ρ TPMS Polynomial fitting was performed on the dependent variable to obtain the relative density function ρ of the Diamond structure. TPMS (t):
[0008] ρ TPMS (t)=0.0575t 2 +1.14101t+8.6×10 -4 (1)
[0009] Step (2), Variable density topology optimization of the TPMS core layer: Taking the typical Diamond structure in TPMS as an example, firstly, with the relative density of topological units as the design variable and minimizing structural flexibility as the design objective, a variable density topology optimization model is established based on the SIMP interpolation model; secondly, the relative density field after variable density topology optimization is divided into multiple equal-length segments, and the length of the Diamond cell is the same as that of a single segment, and a function matrix ρ(x) is constructed by polynomial fitting; subsequently, ρ(x) and ρ TPMS (t) Mapping to obtain the Diamond structure wall thickness gradient function matrix t(x), and inputting it into MSLattice software to generate the variable thickness Diamond cell corresponding to each segment; finally, all variable thickness Diamond cells are spliced together in sequence to obtain the multi-segment variable density optimized Diamond core layer.
[0010] The expressions for ρ(x) and t(x) are as follows:
[0011]
[0012] In the formula: l c Here, x represents the length of the Diamond cell and the length of a single segment, and x is the position of the topological element along the axis of the design domain, with a value ranging from [-l]. c / 2,l c / 2];ρ lst (x), ρ 2nd (x), ρ 3rd (x), ρ 4th (x), ρ 5th (x) represent the relative density gradient functions of segments 1, 2, 3, 4, and 5 under a 40% volume constraint, respectively;
[0013] Step (3), Experimental and numerical verification of the improved bending performance of TPMS sandwich structure: Taking the typical Diamond structure in TPMS structure as an example, the Diamond core layer model before and after variable density topology optimization was generated using MSLattice software. Photopolymerization molding technology was used to print the photosensitive resin panel and core layer respectively. Three-point bending test of Diamond sandwich structure before and after variable density topology optimization was carried out, and the numerical and experimental results were compared and analyzed.
[0014] Step (4) is achieved by using variable density topology optimization to improve the bending performance of TPMS sandwich structure through steps (1) to (3).
[0015] Compared with existing methods, this invention utilizes variable-density topology optimization to improve the bending performance of TPMS sandwich structures. By employing the above technical solution, a relative density function of the TPMS structure is constructed using polynomial fitting, clarifying the mapping relationship between the isosurface threshold and relative density. Variable-density topology optimization is performed with the relative density of topological elements as the design variable and minimizing structural flexibility as the design objective. The relative density field is divided into multiple segments, and a function matrix is constructed using polynomial fitting, clarifying the changing trend of the relative density of topological elements within each segment. Mapping the relative density of topological elements to the relative density of the TPMS structure is performed to construct the TPMS structure wall thickness gradient function matrix, generating TPMS cells of varying thickness and splicing them together, enabling multi-segment variable-density optimization design of the TPMS core layer. Three-point bending tests are conducted on the TPMS sandwich structure before and after variable-density topology optimization, and the numerical results are compared and analyzed. This demonstrates that while achieving higher bending performance in the TPMS sandwich structure, it also ensures lightweight construction and efficient material utilization. This invention solves the aforementioned problems and provides technical support and important reference for improving the bending performance of TPMS sandwich structures using variable-density topology optimization. Attached Figure Description
[0016] The present invention will be described in detail below with reference to the accompanying drawings and examples. The advantages and implementation methods of the present invention will become more apparent from this description. The accompanying drawings are for illustrative purposes only and do not constitute any limitation on the present invention. In the accompanying drawings:
[0017] Figure 1 This is a force-displacement test curve of the Diamond sandwich structure before and after the variable density topology optimization of this invention.
[0018] Figure 2 This is a force-displacement numerical curve of the Diamond sandwich structure before and after the variable density topology optimization of this invention.
[0019] Figure 3 This is a comparison chart of experimental and numerical results of the bending stiffness of the Diamond sandwich structure before and after the variable density topology optimization of this invention under different core layer relative densities. Detailed Implementation
[0020] The present invention will be further described below with reference to the embodiments and accompanying drawings:
[0021] This invention utilizes variable density topology optimization to improve the flexural performance of TPMS sandwich structures based on the following design principles:
[0022] 1. A gradient function matrix for the wall thickness of the TPMS structure is constructed using polynomial fitting and relative density mapping to account for the influence of variable density topology optimization design on the wall thickness at different locations of the TPMS core layer.
[0023] 2. By splicing TPMS cells with varying thickness, a multi-segment variable density optimized TPMS core layer was obtained. The results of three-point bending test and numerical simulation of the photosensitive resin TPMS sandwich structure before and after variable density optimization were analyzed to verify the effectiveness of variable density topology optimization in improving the bending resistance of TPMS sandwich structure.
[0024] To address the aforementioned technical problems, this invention employs innovative designs in areas such as the construction of the TPMS structure wall thickness gradient function matrix and the multi-segment variable density optimization of the TPMS core layer.
[0025] 1. Construction of the wall thickness gradient function matrix of TPMS structure
[0026] To account for the influence of variable density topology optimization on the wall thickness at different locations of the TPMS core, a TPMS structure wall thickness gradient function matrix was constructed using polynomial fitting and relative density mapping.
[0027] 2. Multi-segment variable density optimization of TPMS core layer
[0028] To verify the effectiveness of variable density topology optimization in improving the bending resistance of TPMS sandwich structures, TPMS cells with varying thicknesses were spliced together to obtain a multi-segment variable density optimized TPMS core layer. The results of three-point bending tests and numerical simulations of the photosensitive resin TPMS sandwich structures before and after variable density optimization were analyzed.
[0029] This invention utilizes variable density topology optimization to improve the flexural performance of TPMS sandwich structures, comprising the following steps:
[0030] Step (1), Construction of the function of TPMS structure isosurface threshold and relative density: Taking the typical Diamond structure in TPMS structure as an example, the Diamond structure at relative density ρ TPMS When the coefficient is [0.2, 0.6], there is a clear linear relationship between the coefficient and the isosurface threshold t, meaning that the coefficient of determination R is within this range. 2 >0.999, therefore, t is taken as the independent variable and ρ TPMS Polynomial fitting was performed on the dependent variable to obtain the relative density function ρ of the Diamond structure. TPMS (t):
[0031] ρ TPMS (t)=0.0575t 2 +1.14101t+8.6×10 -4 (1)
[0032] Step (2), Variable density topology optimization of the TPMS core layer: Taking the typical Diamond structure in TPMS as an example, firstly, with the relative density of topological units as the design variable and minimizing structural flexibility as the design objective, a variable density topology optimization model is established based on the SIMP interpolation model; secondly, the relative density field after variable density topology optimization is divided into multiple equal-length segments, and the length of the Diamond cell is the same as that of a single segment, and a function matrix ρ(x) is constructed by polynomial fitting; subsequently, ρ(x) and ρ TPMS (t) Mapping to obtain the Diamond structure wall thickness gradient function matrix t(x), and inputting it into MSLattice software to generate the variable thickness Diamond cell corresponding to each segment; finally, all variable thickness Diamond cells are spliced together in sequence to obtain the multi-segment variable density optimized Diamond core layer.
[0033] The expressions for ρ(x) and t(x) are as follows:
[0034]
[0035] In the formula: l c Here, x represents the length of the Diamond cell and the length of a single segment, and x is the position of the topological element along the axis of the design domain, with a value ranging from [-l].c / 2,l c / 2];ρ lst (x), ρ 2nd (x), ρ 3rd (x), ρ 4th (x), ρ 5th (x) represent the relative density gradient functions of segments 1, 2, 3, 4, and 5 under a 40% volume constraint, respectively;
[0036] Step (3), Experimental and numerical verification of the improved bending performance of TPMS sandwich structure: Taking the typical Diamond structure in TPMS structure as an example, the Diamond core layer model before and after variable density topology optimization was generated using MSLattice software. Photopolymerization molding technology was used to print the photosensitive resin panel and core layer respectively. Three-point bending test of Diamond sandwich structure before and after variable density topology optimization was carried out, and the numerical and experimental results were compared and analyzed.
[0037] Step (4) is achieved by using variable density topology optimization to improve the bending performance of TPMS sandwich structure through steps (1) to (3).
[0038] Any aspects not covered in this invention are applicable to existing technologies.
[0039] Example:
[0040] 1. Variable density topology optimization design of the Diamond core layer
[0041] First, the ρ structure of the Diamond structure is obtained using polynomial fitting. TPMS (t); secondly, a two-dimensional design domain is established and discretized into 36×1 topological elements. Fixed constraints and moving constraints are applied at positions 5 topological elements from each end of the design domain, and a vertically downward point load F is applied at the middle position of the design domain; thirdly, based on the SIMP interpolation model, variable density topology optimization is performed with ρ as the design variable and minimizing structural flexibility as the design objective. The relative density field is divided into multiple equal-length segments, and the length of the Diamond cell is the same as that of a single segment. ρ(x) is constructed through polynomial fitting; subsequently, ρ(x) and ρ TPMS (t) is mapped to obtain t(x), and t(x) is input into MSLattice software to generate the variable thickness Diamond cell corresponding to each segment; finally, the variable thickness Diamond cells are spliced sequentially along the axis of the design domain to obtain a single beam, and then two single beams are spliced along the Y direction to obtain the Diamond core layer of the multi-segment variable density optimization design.
[0042] 2. Preparation of the photosensitive resin Diamond core layer and panel
[0043] The Diamond core model and the panel model generated using ABAQUS software were input into Magics software for printing orientation setting and slicing. The sliced model was then imported into the ZERO software of the Zhongrui Technology 3D printer (model: Isla880) and printing parameters were set. Photopolymerization printing was performed using photosensitive resin to obtain the panel and Diamond core. Instant adhesive was used to bond the panel and Diamond core to obtain the Diamond sandwich structure. The geometric parameters of the Diamond sandwich structure were selected as follows: panel thickness h... panel The thickness is 1.5mm, and the core layer thickness is h. core The core layer and panel are 20mm thick, with a length l of 180mm and a width w of 40mm.
[0044] 3. Three-point bending test and numerical simulation
[0045] A three-point bending test platform for the Diamond sandwich structure was built using a precision electronic universal testing machine (model: AGS-X-50kND) and a computer to obtain the force-displacement curves at the loading point of the upper panel of the Diamond sandwich structure. The parameters of the precision electronic universal testing machine were set as follows: limit load 10000N, loading speed 2mm / min, 100 samples per cycle, and sampling rate 50 samples per second.
[0046] Input the nodal equations and dimensional parameters of the Diamond structure into the MSLattice software to generate a Diamond cell STL model with a relatively coarse surface mesh, and then import it into the HyperMesh software. Perform mesh quality checks, mesh repairs, and 3D mesh generation on the Diamond cell STL model sequentially, and then stitch them together along the Z-axis to obtain the INP model of the Diamond core layer. Use ABAQUS software to create geometric models of the panel, indenter, and support, and import the INP model of the Diamond core layer. Assign photosensitive resin material properties to the panel and core layer, and define the indenter and support components as rigid bodies. Create a reference point on the indenter, apply displacement boundary conditions along the negative Z-axis at this reference point, and set an output set to record the indenter displacement and reaction force to obtain the force-displacement curve of the Diamond sandwich structure.
[0047] 4. Comparison of three-point bending test and numerical results of Diamond sandwich structure before and after variable density topology optimization
[0048] Based on the above three-point bending tests and numerical simulations, the force-displacement curves and experimental and numerical results of the bending stiffness of the Diamond sandwich structure before and after variable-density topology optimization were obtained, as shown in the figures below. Figure 1 , Figure 2 and Figure 3 As shown. By Figure 1 and Figure 2 It can be seen that the force-displacement curves from the experiments and numerical simulations agree well, demonstrating the accuracy of the numerical model of the Diamond sandwich structure. Figure 3 It can be seen that when the relative density of the Diamond core layer is 40%, the variable density topology optimization improves the bending stiffness of the Diamond sandwich structure by 52.19%; when the relative density of the Diamond core layer is 55%, the variable density topology optimization improves the bending stiffness of the Diamond sandwich structure by 17.18%.
[0049] The embodiments of the present invention have been described in detail above, but the content described is only a preferred embodiment of the present invention and should not be considered as limiting the scope of the present invention. All equivalent changes and improvements made within the scope of the present invention should still fall within the scope of this patent.
Claims
1. A method for improving the bending resistance of a TPMS sandwich structure using variable density topology optimization, characterized in that: Includes the following steps: Step (1), Construction of the function of TPMS structure isosurface threshold and relative density: Taking the typical Diamond structure in TPMS structure as an example, the Diamond structure at relative density ρ TPMS When the coefficient is [0.2, 0.6], there is a clear linear relationship between the coefficient and the isosurface threshold t, meaning that the coefficient of determination R is within this range. 2 >0.999, therefore t is taken as the independent variable and ρ TPMS Polynomial fitting was performed on the dependent variable to obtain the relative density function ρ of the Diamond structure. TPMS (t): p TPMS (t) = 0.0575t 2 + 1.14101t + 8.6 x 10 -4 (1) Step (2), Variable density topology optimization of the TPMS core layer: Taking the typical Diamond structure in TPMS as an example, firstly, with the relative density of the topological units as the design variable and minimizing the structural flexibility as the design objective, a variable density topology optimization model is established based on the SIMP interpolation model; secondly, the relative density field after variable density topology optimization is divided into multiple equal-length segments, and the length of the Diamond cell is the same as that of a single segment, and a function matrix ρ(x) is constructed by polynomial fitting; subsequently, ρ(x) and ρ TPMS (t) Mapping to obtain the Diamond structure wall thickness gradient function matrix t(x), and inputting it into MSLattice software to generate the variable thickness Diamond cell corresponding to each segment; finally, all variable thickness Diamond cells are spliced together in sequence to obtain the multi-segment variable density optimized Diamond core layer. The expressions for ρ(x) and t(x) are as follows: In the formula: l c Here, represents the length of the Diamond cell and the length of a single segment, and x is the position of the topological element along the design domain axis, ranging from [-l]. c / 2,l c / 2];ρ 1st (x), ρ 2nd (x), ρ 3rd (x), ρ 4th (x), ρ 5th (x) represent the relative density gradient functions of segments 1, 2, 3, 4, and 5 under a 40% volume constraint, respectively; Step (3), Experimental and numerical verification of the improved bending performance of TPMS sandwich structure: Taking the typical Diamond structure in TPMS structure as an example, the Diamond core layer model before and after variable density topology optimization was generated using MSLattice software. Photopolymerization molding technology was used to print the photosensitive resin panel and core layer respectively. Three-point bending test of Diamond sandwich structure before and after variable density topology optimization was carried out, and the numerical and experimental results were compared and analyzed. Step (4) is achieved by using variable density topology optimization to improve the bending performance of TPMS sandwich structure through steps (1) to (3).
2. The method of claim 1, wherein the method is characterized by: In step (1), a polynomial fitting method is used to construct the relative density function ρ of the TPMS structure. TPMS (t) and step (2) using ρ(x) and ρ TPMS After mapping t(x) to achieve the multi-segment variable density optimization design of the TPMS core layer, the variable density topology optimization of the TPMS sandwich structure is analyzed by combining three-point bending test and numerical simulation.
3. The method for improving the flexural performance of TPMS sandwich structures using variable density topology optimization according to claim 1 or 2, characterized in that: The flexural performance of the TPMS sandwich structure was significantly improved after variable density topology optimization, and the numerical results were consistent with those of the three-point bending test.