Structural optimization, compressive strength analysis, and application exploration of triply periodic minimal surfaces (TPMS) in interior design

Abstract

In modern interior design, the selection of materials and structural optimization are critical to achieving both functionality and aesthetic appeal. Triply periodic minimal surfaces (TPMS), with their outstanding mechanical properties and lightweight characteristics, have increasingly attracted the attention of designers. In this study, Gyroid (G) TPMS structures were fabricated using LCD-based photopolymerization technology, and ten models with periodic parameter T ranging from [1/5, 2] were designed. Finite element simulations and experimental validations were employed to thoroughly analyze the structural optimization and compressive strength of G surfaces, particularly when T = 1/3. The results revealed that at T = 1/3, G surfaces exhibited minimal stress concentration, a moderate number of meshes, and optimal values for compression crush rate and porosity error. Furthermore, with T = 1/3 and a porosity of 50%, additional parameters such as surface offset, cell thickness, offset thickness, and gradient distribution were adjusted to design four distinct G surface structures. Compression tests were conducted to investigate the effects of different loading directions on mechanical properties, deformation behavior, and energy absorption. It was found that under vertical loading, the surface-offset G surface exhibited the highest energy absorption and efficiency, while under parallel loading, the cell-thickness G surface demonstrated the most stable deformation. The application of these structures in interior design is thoroughly discussed, emphasizing how TPMS can serve as effective structural systems and innovative spatial elements. This study provides a solid theoretical foundation and extensive experimental evidence for the application of TPMS structures in interior design.

Introduction

In the fields of architecture and interior design, the development of innovative materials and structural designs is critical to achieving a balance between spatial efficiency, aesthetics, and functionality¹. Triply periodic minimal surfaces (TPMS), as a type of microstructure with unique geometric and mechanical properties, have garnered significant attention in design and engineering in recent years². With their infinite, non-self-intersecting periodic arrangements along three orthogonal axes and zero mean curvature, TPMS exhibit great potential for lightweight design and structural optimization³. In interior design specifically, TPMS applications not only improve material efficiency but also enhance spatial aesthetics and structural stability. TPMS structures possess two notable advantages: first, their entire geometry can be precisely expressed using mathematical functions, allowing parameters such as porosity, volume ratio, and surface area to be directly controlled^4–7. Second, compared to conventional honeycomb and truss lattice structures, TPMS demonstrate uniformly distributed stress under load, balanced load-bearing capacity, and extended fatigue life^8,9. These advantages have made TPMS porous structures highly attractive to researchers across diverse fields, including mechanical engineering, materials science, energy systems, and medicine^10–13. Figure 1 illustrates the applications of TPMS and other minimal surfaces in the fields of architecture and interior design.

Fig. 1.

illustrates the applications of TPMS and other minimal surfaces in the fields of architecture and interior design.

TPMS and related structures are widely found in biological systems in nature. Through continuous evolution, living organisms have optimized their forms and structures across different scales (micro, meso, and macro). The first reported case of such a structure was observed in the skeletal morphology of echinoderms, where minimal surface-like structures were identified¹⁴. Subsequently, G-surface structures were discovered in the wing scales of butterflies^15,16, while D-surface structures were identified in the exoskeletons of beetles and weevils^17,18. Furthermore, G-surface membranes were found in the cone cells of tree shrew retinas¹⁹. Additionally, it has been observed that in the mitochondria of amoeba Chaos carolinensis, at least two minimal surface structures coexist within the same organelle, with G, D, and P surface TPMS structures capable of transitioning into one another under specific conditions²⁰.

Additive manufacturing technology enables the direct fabrication of CAD-designed models, making it feasible to produce complex TPMS lattice structures. Currently, the primary 3D printing technologies used for fabricating TPMS structures include Selective Laser Sintering (SLS), Selective Laser Melting (SLM), Electron Beam Melting (EBM), and photopolymerization-based methods such as Stereolithography (SLA) and Digital Light Processing (DLP)^21,22. Al-Saedi et al.²³ reported the use of the SLM process to fabricate Al-12Si aluminum alloy functional gradient F2BCC lattice structures, where the mechanical properties and energy absorption capabilities were tested through finite element analysis and compression experiments. Yan et al.^24,25 fabricated D-surface Ti-6Al-4V and G-surface Al-Si10-Mg porous lattice structures using SLM and fitted the compression test results to the Gibson-Ashby model. Maskery et al.²⁶ studied failure modes and energy absorption in Double-Gyroid porous lattice structures made of Al-Si10-Mg, also fabricated using SLM. They found that small unit cells help prevent low-strain structural failure caused by local fractures and crack propagation. Yanez et al.²⁷ employed the EBM process to fabricate G-surface Ti-6Al-4V lattice structures and experimentally analyzed the influence of the angle between the single rod principal axis and the loading direction on mechanical response. Their study concluded that G-surfaces with angles below 35° exhibit high specific strength. Similarly, Ataee et al.²⁸ used EBM to produce G-surface Ti-6Al-4V bone scaffolds and investigated the effects of pore size on the microstructure, mechanical properties, and compressive strength of the scaffolds. Due to its high precision and low cost, ultraviolet (UV) photopolymerization techniques have gained increasing attention^29–31. Zhang et al.³² proposed a Log-sigmoid-based functionally graded TPMS design method and fabricated the structures using SLA technology.

The application of TPMS porous structures in architecture and interior design has significantly increased in recent years. Due to their unique geometric characteristics and superior mechanical properties, TPMS porous structures have been utilized in the optimization of building structures. FSierra et al.³³ reported that employing TPMS porous structures in the supporting frameworks of high-rise buildings effectively reduces structural weight, enhances material utilization, and maintains excellent load-bearing capacity and stability. This optimization approach contributes to reducing construction costs while improving economic efficiency and sustainability. In facade design, TPMS porous structures exhibit distinct advantages. Their complex surface morphology and periodic arrangement provide facades with rich visual appeal and dynamic light and shadow effects, enhancing the aesthetic and artistic qualities of buildings. Additionally, the structures’ excellent breathability and light transmission properties improve indoor comfort and contribute to energy efficiency³⁴. In the context of interior space division, TPMS porous structures can serve as partitions or spatial dividing elements. Their lightweight nature and unique geometric configurations allow such partitions to combine excellent load-bearing capacity with minimal spatial occupation, all while adding decorative value. Furthermore, the porous nature of TPMS structures promotes airflow and sound transmission, helping to create a more comfortable and efficient indoor environment³.

Compared to SLA and DLP, Liquid Crystal Display (LCD) photopolymerization technology^35,36 utilizes a series of UV LCD light sources that project light directly onto the build area in a parallel manner. This type of light projection eliminates pixel distortion and ensures no light scattering, allowing for higher resolution and smaller pixel sizes during printing. As a result, LCD photopolymerization achieves precise model reproduction, making it particularly suitable for fabricating complex TPMS structures. Despite being a relatively recent advancement, there has been limited research on using LCD technology for manufacturing TPMS porous lattice structures.

In this study, Gyroid TPMS structures were fabricated using LCD photopolymerization technology. A parametric modeling method was introduced, with single-parameter controls including periodicity (T), surface offset (C), and cell thickness (O), and dual-parameter controls including offset thickness (CO) and gradient distribution (G). Ten models with periodic parameter T ranging from [1/5, 2] were designed, and the influence of T on G surfaces was explored through a combination of finite element simulation and experimental validation. Keeping T and porosity constant, parameters such as surface offset, cell thickness, offset thickness, and gradient distribution were varied to design four distinct G surface structures. Compression tests were conducted to investigate the effects of different loading directions on mechanical performance, deformation behavior, and energy absorption. This study aims to provide a theoretical foundation and experimental support for the application of TPMS structures in interior design.

Experimental procedures

Parametric design of Gyroid surfaces

This study utilized the nTopology 5.11.2 (https://www.ntop.com) design software^37–39. The line charts in this study were generated using OriginPro 2021 (https://www.originlab.com). By adjusting the parameters controlling implicit functions within the software, properties such as porosity, pore size, distribution, and morphology were manipulated⁴⁰. This parametric modeling approach for TPMS structures enables the creation of superior porous surface designs to meet the evolving demands of advanced technologies. The parametric TPMS model can be represented using the Weierstrass formula⁴¹:

1

where ω is a complex variable, θ is the Bonnet angle, and R(ω) is a surface function. The fundamental part of the contour integral Re represents the Cartesian coordinates of each point on the surface. The type of TPMS that can be expressed by this method is limited, and the implicit sinusoidal periodic function is usually used for approximate expression. The expression is as follows⁴²:

2

where r is the position vector of the Euclidean space, A_k is the amplitude factor, h_k is the kth grid vector in the reciprocal space, λ_k the periodic wavelength, P_k is the phase offset, and C is the offset of the implicit function. A series of TPMS can be obtained by changing the C value, and the value range is [− 1, 1]. The implicit expression of the G surface used in this paper is:

3

where X = 2πx, Y = 2πy, Z = 2πz. Where 2π is the coefficient, the TPMS’s period in the three directions of the spatial coordinate axis is 1. Then, the implicit expression of the G surface can be transformed into:

4

where T represents the period of TPMS, representing the number of cells along a coordinate axis in a cube space of fixed size.

Single-parameter control of G surface structure

The single parameters controlling the G surface include periodicity (T), surface offset (C), and cell thickness (O). Using nTopology software, the G surface structure with T = 1/3 is shown in Fig. 2. The flexibility of parameter control allows designers to adapt structural characteristics to meet specific interior design needs. A smaller periodicity (T) can be used to design complex textured translucent walls or ceiling decorations, while variations in surface offset (C) enhance depth and layering, making it suitable for open-space partitions. Optimizing cell thickness (O) strikes a balance between aesthetics and load-bearing capacity, making it ideal for load-bearing walls or structural supports. Figure 3 illustrates the relationship between T, C, and O under a porosity condition of 50%. As T decreases (as shown in Fig. 4, which depicts the relationship between the G surface and periodic patterns), the structural complexity and cell quantity increase, making it suitable for decorative spatial designs. Precise porosity control also enables functional designs such as ventilated partitions and acoustic walls. By leveraging these parameter control principles, designers can seamlessly integrate functionality and aesthetics, offering innovative possibilities for interior design. The curve-fitting equations for these relationships are as follows:

5

Fig. 2.

Design of G surface structure.

Fig. 3.

Relationship between G surface structure and single-parameter.

Fig. 4.

Relationship between G surface structure and period.

Bi-parameter control of G surface structure

The dual parameters controlling the G surface include offset thickness (CO) and gradient distribution (G). Figure 5 illustrates the relationship between the porosity of the G surface and the parameters CO and G. For offset thickness G surfaces (Fig. 5a), when parameter C remains constant, increasing parameter O leads to a gradual decrease in porosity, while the curve shifts toward higher O values, forming a triangular region. The positive or negative value of C only determines the offset direction without affecting the porosity. This characteristic can be applied in interior design to create partition walls or decorative support structures that balance strength and transparency. For gradient distribution G surfaces (Fig. 5b), gradient changes were achieved using the Ramp command by adjusting Out min and Out max. When the porosity of the G surface exceeds 30%, increasing Out min and Out max results in porosity adjustments and curve shifts. This feature is ideal for designing gradient-style translucent walls or decorative screens, enabling precise control over porosity distribution to meet both aesthetic and functional requirements.

Fig. 5.

Relationship between G surface structure and bi-parameter.

Table 1 lists the porosity variations corresponding to special points for offset thickness and gradient distribution G surfaces under a porosity condition of 50%. These parameters provide designers with clear guidance, enabling flexible applications of TPMS structures in interior design.

Table 1.

G surface structures corresponding to special point.

Finite element method and preparation of G surface structure

Finite element method

When employing the Finite Element (FE) method, the model must undergo mesh generation, material property assignment, and constraint application before being converted into a set of mathematical equations for numerical solving. The nTopology software, which integrates CAD, CAE, and CAM functionalities, was used in this study. As shown in Fig. 6a, the Mesh from Implicit Body command was applied to generate a mesh with a size of 0.5 mm (tetrahedral mesh). The Remesh Surface command was subsequently used to optimize the mesh, increasing density at the model’s edges and beams while keeping flat areas less dense, ultimately producing an FE volume mesh. Figure 6b illustrates the compression model, where sufficient constraints were applied to the bottom face, and the top face was displaced 5 mm in the negative Z-axis direction. In interior design, the FE method provides critical support for optimizing complex structures. By simulating the performance of TPMS structures under various loading conditions, designers can ensure structural safety and functionality. For instance, FE analysis can verify the deformation behavior and mechanical stability of TPMS structures when used as load-bearing partition walls or large decorative elements. This precise computational approach offers a reliable theoretical foundation for innovative and efficient designs.

Fig. 6.

G surface meshing process and FE model.

Preparation of G surface structure

The samples were fabricated using a JG MAKER G5 photopolymerization printer (Aurora Co., Shenzhen, China), which employs LCD photopolymerization technology with a 405 nm laser light source and a resolution of 3840 × 2400 (4 K). As shown in Fig. 7, the models were first sliced using CHITUBOX software with a slicing thickness of 0.05 mm, and the resulting files were imported into the printer. To ensure the integrity of the complex G surface structures, high-flow aqueous photopolymer resin was used. After printing, the samples were ultrasonically cleaned in an alcohol solution to remove excess resin and then cured in a UV curing chamber for 12 h, resulting in finalized G surface samples. In interior design, this high-resolution 3D printing technology enables the creation of intricate and refined TPMS structures, such as translucent decorative partitions or precision-engineered load-bearing components. LCD photopolymerization technology not only meets aesthetic requirements but also enhances spatial functionality through lightweight and modular design, offering expanded possibilities for modern interior design.

Fig. 7.

G surface structure preparation process.

Results and discussion

Effect of the period

The tensile properties of photosensitive resin samples were tested by an electronic universal testing machine (WDW-50, China) at a 1 mm/min loading rate using ISO 527 standard in uniaxial compression test. The density of the sample was tested by Archimedes’ drainage method. The weighing process measures the average porosity of the G surface, and the formula is⁴³:

6

where is the average porosity; m₁ is the mass of the solid structure of the same size as the G surface; m₂ is the mass of the G surface.

The results are shown in Table 2, and Fig. 8a illustrates the relationship between the periodic parameter T, maximum equivalent stress (MEF), and the number of meshes. The study found that as T decreases, MEF initially decreases and then fluctuates, reaching its minimum value of 1140.8 MPa at T = 1/3. This behavior is attributed to the increased structural complexity and higher cell and beam density at smaller T values, which allow the load to be distributed across more supports, reducing stress concentration. However, when T is further reduced, the beams become too thin, weakening load resistance, increasing stress concentration, and making the structure unstable. Figure 9 shows the stress distribution at different T values, with the least stress concentration observed at T = 1/3, indicating optimal performance. Additionally, as T decreases, the number of meshes grows exponentially, reaching 3,042,729 at T = 1/5, significantly increasing computational cost and printing difficulty. In interior design, selecting T = 1/3 for the G surface optimizes load-bearing performance while balancing material efficiency and manufacturability. This parametric design is suitable for high-strength, lightweight interior structures, such as load-bearing partition walls or large decorative ceilings, fulfilling both mechanical performance and practical manufacturing requirements.

Table 2.

Photosensitive resin property.

Property	Value	Test method	Unit
Density	1.21	Archimedes’ law	g/cm3
Modulus of elasticity	2300	ISO 527	MPa
Poisson’s ratio	0.32	ISO 527	–

Fig. 8.

G-surfaces with different periodic parameter T: (a)MEF and number of meshes; (b) printing error and compression crush rate.

Fig. 9.

Stress distribution of G surface structures with different period.

The uniaxial compression experiment was conducted according to the ISO 604 standard using a universal testing machine at a loading rate of 1 mm/min to evaluate the compressive performance of G surfaces with different periodic T values. Table 3 summarizes the sample mass, 3D printing porosity error, and compression fracture rate for various T values. Figure 8b illustrates the trends in porosity error and fracture rate. The study revealed that as T decreases, the structural complexity and load-bearing capacity of the samples improve, leading to a gradual reduction in fracture rate, which reaches its minimum of 1.32% at T = 1/3. However, further reductions in T result in thinner beams, weakening the load resistance and causing the fracture rate to rise again.

Table 3.

Printing and compression of data with different period G surface structure.

Periods	Pre-compression (g)	Post-compression (g)	Theoretical porosity	Actual porosity	Inaccuracy	Crush rate
Non-porous	31.825	–	–	–	–	–
T = 2	16.021	10.598	50.00%	49.66%	0.68%	33.85%
T = 1	16.068	12.615	50.00%	49.51%	0.98%	21.58%
T =	16.091	13.413	50.00%	49.44%	1.12%	16.88%
T =	16.119	14.836	50.00%	49.35%	1.30%	8.33%
T =	16.142	15.537	50.00%	49.28%	1.44%	4.30%
T =	16.170	16.065	50.00%	49.19%	1.62%	1.32%
T =	16.487	16.225	50.00%	48.19%	3.62%	1.59%
T =	16.683	16.335	50.00%	47.23%	5.54%	2.10%
T =	17.067	16.595	50.00%	46.38%	7.24%	2.77%
T =	17.237	16.862	50.00%	45.84%	8.32%	2.17%

The characteristics of 3D printing contribute to an increase in slicing layers and extended printing times for complex structures. From a microscopic perspective (Fig. 10), the printed surface forms a gear-like structure (Fig. 10a), increasing the actual volume and reducing porosity. This phenomenon is exacerbated by the layered accumulation of complex structures, particularly when T < 1/3, where porosity error significantly increases. Additionally, residual photopolymer resin (Fig. 10b, c) slightly enlarges the actual volume.

Fig. 10.

Microscopic morphology of G surface structure.

Based on finite element analysis and experimental results, the G surface at T = 1/3 achieves an optimal balance between porosity error and mechanical performance. In interior design, this finding provides valuable guidance for creating high-strength, lightweight structures. Selecting T = 1/3 for G surfaces not only optimizes mechanical properties but also makes them ideal for designing translucent partition walls or load-bearing components, offering innovative solutions that combine practicality and aesthetics.

Compression performance analysis

The study revealed that the G surface achieves optimal overall performance when the periodic parameter T = 1/3. Under the conditions of T = 1/3 and 50% porosity, four G surface structures were designed by varying surface offset, cell thickness, offset thickness, and gradient distribution (Fig. 11). Figure 12 illustrates the two loading modes in the uniaxial compression experiment, while Fig. 13 shows the stress–strain curves under different loading directions. The experiment demonstrated that both vertical and parallel loading directions exhibit three deformation stages: the elastic deformation stage (< 0.1), the plastic yield stage (0.1–0.3), and the densification stage (> 0.3). Notably, under parallel loading, the cell-thickness G surface displayed a smooth transition from the plastic yield stage to the densification stage, indicating higher stability during loading^21,23–26. This stability and direction dependence provide critical design guidance in interior applications. Gradient-distribution G surfaces are suitable for partition walls with high load-bearing requirements, while cell-thickness G surfaces are more appropriate for decorative columns or cushioning structures requiring stable deformation. Such mechanics-based optimization offers scientific support for improving the performance and aesthetics of interior structures.

Fig. 11.

3D printed G surface structure: (a) surface offset, C = 0; (b) wall thickness, O = 1.2779 mm; (c) offset thickness, C = − 0.25, O = 1.2359; (d) gradient arrangement, out min = − 0.5 mm, out max = 0.5 mm.

Fig. 12.

Load direction: (a) vertical print direction; (b) parallel print direction.

Fig. 13.

Stress–strain in different loading directions: (a) vertical print direction; (b) parallel print direction.

Table 4 summarizes the mechanical properties of the four G surface structures under different compression directions. The findings revealed that under vertical loading, the cell-thickness G surface exhibited the highest Young’s modulus, while the other three structures performed better under parallel loading. For all four G surfaces, the yield strength was higher under parallel loading than under vertical loading. Notably, the cell-thickness G surface exhibited the largest differences in Young’s modulus and yield strength between the two loading directions, at 21.67% and 8.27%, respectively. Specifically, the surface-offset G surface, characterized by thicker beams, demonstrated the best mechanical properties; the cell-thickness G surface, with thinner and denser beams, showed good mechanical performance; the offset-thickness G surface, due to pore distribution, exhibited the weakest performance. The gradient-distribution G surface, under parallel loading, showed an elastic modulus and yield strength 2.3 and 2.6 times higher, respectively, than those under vertical loading. This is because, under vertical loading, deformation begins in the weaker high-porosity regions, while under parallel loading, both high- and low-porosity regions deform simultaneously, with the low-porosity areas providing greater strength.

Table 4.

Mechanical properties of four G surface structures in different compression directions.

Loading direction	Modulus of elasticity (MPa)		Yield strength (MPa)		Efficiency (%)
	Vertical	Parallel	Vertical	Parallel	Vertical	Parallel
Surface offset	208.33	219.49	10.21	10.82	38.10	34.89
Wall thickness	195.75	172.51	9.79	10.28	34.17	35.32
Offset thickness	140.93	179.93	8.76	9.55	31.95	36.01
Gradient arrangement	98.16	227.34	4.38	11.23	21.42	41.28

Figure 14 illustrates the deformation behavior of the four G surfaces under vertical loading. During compression, deformation was primarily characterized by the sequential folding of cells, a result of the orderly arrangement and smooth connectivity of TPMS structures. The reaction force at the fixed support caused cells to fail from the bottom, as shown in Fig. 13a–c. For the gradient-distribution G surface, however, failure initiated from the top due to the higher porosity and thinner beams in this region (Fig. 13d). At a strain of 40% (ε = 0.4), the bottom layers densified into a nearly solid structure, while the top layers became increasingly compact, resulting in lateral expansion. At a strain of 60% (ε = 0.6), a 45° fracture band formed from the upper left to the lower right, caused by interlayer leftward sliding induced by parallel-to-printing loading. After compression, the surface-offset and cell-thickness G surfaces formed structures resembling upright trapezoids, while the gradient-distribution G surface, due to its gradually decreasing porosity from top to bottom, exhibited a rapid stress increase at ε > 0.3ε > 0.3ε > 0.3, leading to significant densification and a final structure resembling an inverted trapezoid⁴⁴. In interior design, these deformation characteristics hold practical significance. For example, gradient-distribution G surfaces are well-suited for decorative partitions or walls requiring gradient strength, while surface-offset and cell-thickness G surfaces are better suited for load-bearing structures or columns, providing a balance of structural safety and aesthetic appeal.

Fig. 14.

Deformation behavior of 4 types of G Surfaces with loads applied in the vertical printing direction: (a) surface offset; (b) wall thickness; (c) offset thickness; (d) gradient arrangement.

Figure 15 illustrates the deformation behavior of four G surfaces under parallel-to-3D-printing-direction loading. The study revealed that at the initial loading stage, all four structures began to deform and fail from the bottom, with cells breaking layer by layer from the bottom upward. This deformation behavior differed from that under vertical loading, as gradient-distribution G surfaces also failed from the bottom during parallel loading due to the uniformity of porosity along the top and bottom layers.

Fig. 15.

Deformation behavior of 4 types of G Surfaces with loads applied in the parallel printing direction: (a) surface offset; (b) wall thickness; (c) offset thickness; (d) gradient arrangement.

As loading continued, the bottom layers gradually densified into a solid-like structure, while the top layers became increasingly compact, reaching the plastic yield stage. At a strain of 40% (ε = 0.4), all structures entered the densification stage, characterized by lateral expansion. At this stage, the gradient-distribution G surface exhibited interlayer rightward sliding due to lower porosity and corresponding strength differences on the right side. At a strain of 60% (ε = 0.6), all four structures developed a 45° fracture band extending from the upper right to the lower left, a result of interlayer rightward sliding induced by parallel loading.

In interior design, this directional deformation characteristic provides valuable insights for material selection. For instance, gradient-distribution G surfaces are well-suited for partition walls or dividers that need to withstand dynamic loads, as their gradient properties help balance load distribution. On the other hand, other G surface structures may be more advantageous in applications requiring balanced compression and expansion, such as support columns or load-bearing walls. These structures provide innovative solutions that integrate functionality and aesthetics in interior design.

Energy absorption analysis

The research on the energy absorption characteristics of TPMS porous structure in the compression process is of great significance. It can help to design lightweight metamaterials, reduce energy waste, improve energy efficiency, and determine how to absorb better and disperse energy when stressed. It is also one of the important indicators for evaluating porous structures. The formula for calculating the total energy absorption per unit volume is given in⁴⁵.

7

where W_ε is the sum of energy absorption per unit volume; σ(ε) is the stress at the strain ε; ε for strain. The energy efficiency (η) is the ratio of the absorbed energy to the given strain (ε) divided by the corresponding stress (σ(ε))¹, which can be expressed as:

8

The equation can be used to determine where the slope of the energy efficiency (η) curve is 0, which indicates that the lattice structure begins to densify during compression deformation. The energy efficiency at a certain point of strain (ε) is the largest.

Figure 16 presents the energy absorption versus strain relationship and energy efficiency curves for the four G surfaces under different loading directions. Figure 16a, c depict the vertical loading scenario. The results show that when the strain ≤ 0.6, the surface-offset G surface exhibits the highest energy absorption, indicating its strong energy absorption capacity per unit volume, making it highly resistant to external loads without breaking. The offset-thickness G surface surpasses the total energy absorption of the cell-thickness G surface at ε ≥ 0.32, demonstrating superior energy absorption during the densification stage. For the gradient-distribution G surface, energy absorption increases rapidly after the high-porosity regions are compressed, with the curve slope rising sharply. At ε ≥ 0.56, its energy absorption exceeds that of the cell-thickness G surface and continues to increase. In the energy efficiency (η) curves, at ε = 0.3, all four G surfaces begin densification. The surface-offset, wall-thickness, offset-thickness, and gradient-distribution G surfaces achieve their peak energy efficiency at strains of 0.47, 0.55, 0.53, and 0.6, with maximum values of 38.10%, 34.17%, 31.95%, and 21.42%, respectively. Among them, the surface-offset G surface has the highest overall energy efficiency, indicating that it can absorb more energy per unit volume. In interior design, these characteristics provide valuable guidelines for functional applications. The surface-offset G surface is ideal for load-bearing walls or seismic support structures requiring high energy absorption capacity, while the gradient-distribution G surface is suitable for energy-absorbing buffer zones or vibration-damping modules, offering enhanced safety and functional optimization for architectural spaces.

Fig. 16.

Analysis of energy absorption and efficiency for different loading directions: (a, c) vertical print direction; (b, d) parallel print direction.

Figure 16b, d illustrate the energy absorption and energy efficiency curves under parallel loading. The study found that while the energy absorption curves of the four structures are similar, the cell-thickness G surface has the lowest total energy absorption per unit volume due to its thinner beams, which limit its energy absorption capacity and make it more prone to failure under load. The gradient-distribution G surface surpasses the surface-offset G surface in total energy absorption at ε ≥ 0.39 and maintains a small difference. Similar to vertical loading, the energy efficiency (η) curves of the surface-offset, offset-thickness, and gradient-distribution G surfaces have a slope of zero at ε = 0.3, indicating the onset of densification. However, the cell-thickness G surface shows a non-zero slope due to its structural characteristics, reflecting a smoother deformation process. The surface-offset, wall-thickness, offset-thickness, and gradient-distribution G surfaces reach their energy efficiency peaks at strains of 0.48, 0.53, 0.55, and 0.48, with corresponding values of 34.89%, 35.32%, 36.01%, and 41.28%, respectively. The gradient-distribution G surface exhibits the fastest growth in energy efficiency during the densification stage, indicating its superior ability to absorb energy per unit volume at this stage. In interior design, these findings offer significant insights for material selection. The gradient-distribution G surface is well-suited for dynamic buffering walls or seismic support structures requiring high energy absorption capabilities, while the cell-thickness G surface is ideal for functional decorative components requiring stable deformation. These features provide innovative options for integrating functionality and safety into interior design.

Exploration of TPMS applications in interior design

Based on the aforementioned research on the structural characteristics, optimization, and compressive strength of Triply Periodic Minimal Surfaces (TPMS), it was observed that the periodic nature of TPMS allows for regular repetition and arrangement in specific directions, making them suitable for modular applications. The inherent complexity and fluidity of TPMS forms also help to break the rigidity of traditional modular units, enabling spaces to exhibit both order and variation. This feature enhances spatial richness while maintaining a structured layout. From two distinct perspectives, this study explores how TPMS can be utilized as a method for spatial design and organization in interior design, effectively integrating structural systems with innovative decorative design elements.

Figure 17 illustrates a conceptual design of interior walls using TPMS structures. Modern interior design emphasizes spatial efficiency and circulation patterns, often requiring large-span spaces. In floor plans, the aim is to maximize usable space by reducing the number of columns or even eliminating them entirely. For smaller-span spaces in older buildings, secondary structural adjustments are often needed. This involves redesigning the interior layout by reducing or optimizing existing walls, structural columns, and other architectural elements. Non-load-bearing columns and walls that interfere with spatial division can be removed without affecting the main load-bearing structure. Through mechanical calculations and spatial planning, new load-bearing structures can be strategically designed in appropriate locations. The parametric design capabilities of TPMS, combined with their excellent load-bearing performance and the convenience and lightweight nature of 3D printing technology, provide a viable alternative to traditional steel columns and partition walls. This approach effectively addresses the challenges of prolonged construction cycles and high material consumption, offering a more efficient and sustainable solution for interior design.

Fig. 17.

Schematic diagram of interior wall design form of TPMS structure. This image were generated using AutoCAD 2024 (https://www.autodesk.com).

Figures 18 and 19 illustrate the application of TPMS structures in interior spatial planning, including a floor plan and a rendered design of an office space. The case study focuses on an office with a total area of approximately 140 m². The building’s column grid is distributed along the periphery as well as within the interior space, with a span size of 4200 × 4200 mm. Three structural columns are located in the center of the space, significantly hindering the plasticity and diversity of the layout. On one hand, the columns obstruct sightlines and circulation paths; on the other hand, the dense arrangement of central columns reduces the overall spatial efficiency. The design proposal aims to open up the space, eliminating restrictions imposed by the original structural columns and enabling flexible spatial reconfiguration. First, the central row of structural columns is removed. Below the structural beams, and based on mechanical calculations, partition walls made of Triply Periodic Minimal Surface (TPMS) structures are strategically positioned to provide both support and spatial division. These partition walls can be designed in any desired shape. For this specific office design, curved walls of varying lengths were created to meet functional requirements. These walls not only solve the load-bearing issue but also emphasize the modern aesthetics of TPMS structures through their sequence, rhythm, and transparency. The TPMS partition walls are fabricated as single integrated units, characterized by their lightweight nature and ease of assembly and disassembly. Furthermore, these walls are reusable, making them a sustainable and flexible solution for modern interior spaces. By combining functionality and aesthetic appeal, the design highlights the potential of TPMS structures to transform interior environments effectively.

Fig. 18.

Floor plan of TPMS structure for indoor space application. This image were generated using AutoCAD 2024 (https://www.autodesk.com).

Fig. 19.

Effect of TPMS structure for indoor space application. This image were generated using SketchUp 2024 (https://www.sketchup.com).

Conclusion

This study focused on the structural optimization, compressive strength analysis, and application exploration of Triply Periodic Minimal Surfaces (TPMS) in interior design. Gyroid (G) TPMS models were successfully fabricated using LCD photopolymerization technology, and a parametric modeling approach was employed to investigate the effect of the periodic parameter T on the structural performance of G surfaces. The findings revealed that when T = 1/3, the G surface demonstrated optimal structural performance, characterized by minimal stress concentration, a moderate number of meshes, and the lowest compression fracture rate, while maintaining porosity error at a reasonable level. Additionally, the study highlighted the influence of microstructural features on the actual volume and porosity of G surfaces, which holds significant implications for material selection and application in interior design.

Under the conditions of T = 1/3 and 50% porosity, four types of G surface structures were designed and comprehensively analyzed in terms of mechanical performance, deformation behavior, and energy absorption under different loading directions. The results showed significant directional differences in mechanical performance during vertical and parallel compression, with compressive strength exhibiting clear directional dependence. Notably, the surface-offset G surface demonstrated the highest energy absorption under vertical loading, while the gradient-distribution G surface exhibited the fastest increase in energy efficiency during the densification stage. These characteristics are critical for ensuring the safety and functionality of structural components such as columns and partition walls in interior design applications.

In conclusion, this study not only provides scientific evidence for the structural optimization of TPMS in interior design but also offers an in-depth analysis of their compressive performance. By applying TPMS as effective structural systems and innovative design elements, this research provides valuable insights for future interior design applications. The findings enable designers and engineers to better understand and utilize the unique properties of TPMS, paving the way for more efficient, safe, and aesthetically pleasing interior design solutions.

Author contributions

Y.H..: Project administration, Writing—review & editing, Writing—original draft, Investigation. L.Z.: Validation, Conceptualization, Visualization, Funding acquisition, Project administration, Supervision, Writing—review & editing, Resources. All authors reviewed the manuscript.

Data availability

All the raw data are available upon request to the corresponding author.

Declarations

Competing interests

The authors declare no competing interests.