Robotic prefab 3D printing buildings in extreme environments toward Martian habitats

Abstract

This study integrates four complementary datasets—a global compilation of 100 extreme-climate hotel sites, 631 embodied carbon intensity comparisons buildings, 56 construction robotics firms, and 517 Martian habitat studies—to identify how climate-environment, artificial intelligence geometry design, and robotic automation construction technology jointly determine carbon efficiency across terrestrial and extraterrestrial contexts. Clustering analysis revealed distinct climate–design relationships, from compact insulated geometries in Arctic regions to expansive, ventilated forms in hot deserts. Random Forest regression showed that perimeter, surface area, and temperature range are the strongest predictors of embodied carbon reduction, confirming the geometric dependence of carbon performance. Global refurbishment-phase data further demonstrated that prefabrication consistently lowers absolute emissions, even where relative efficiency diminishes under extreme environmental conditions. Industrial mapping indicated a rapid global rise of robotic and 3D printing construction firms, concentrated in Europe, China, and North America. Bibliometric analysis of Martian habitat research identified additive manufacturing and in-situ resource utilization (ISRU) as dominant strategies, but also revealed limited integration with life support and human-centered design domains. Together, these findings establish a unified computational, AI-assisted, and systems-level approach framework connecting climate-environment-responsive geometry, prefabricated and additive manufacturing, and robotic 3D printing automation, offering a scalable pathway toward architecture on Earth, Moon, Mars and other off-Earth habitats/ extraterrestrial architecture.

Introduction

Buildings are among the largest contributors to climate change, responsible for nearly half of global energy consumption and one-third of greenhouse gas emissions^1,2,3. While advances in operational energy efficiency have moderated direct energy use, the embodied carbon associated with construction and renovation/refurbishment phases is emerging as a dominant share of life-cycle emissions^4,5,6. As urban growth accelerates in climate-stressed regions, and as infrastructure demands extend into high-altitude, polar, and desert zones, identifying low-carbon building strategies under extreme environmental conditions has become an urgent challenge^{7,8,9,10,11,12}. Prefabricated and 3D-printed construction have been promoted as transformative solutions, offering potential reductions in material waste, labor intensity, and embodied carbon through standardized modular production^13,14,15. Yet, their environmental performance remains spatially heterogeneous and poorly understood^16,17,18. In particular, the refurbishment phase—a recurring and carbon-intensive stage in the building lifecycle—provides a sensitive testbed for assessing the true mitigation potential of prefabrication^19,20,21. Unlike initial construction, refurbishment is strongly shaped by climate, logistics, and maintenance cycles, making it a critical lens for evaluating long-term sustainability^22,23,24.

Hotels, widely distributed across diverse climatic zones and frequently subject to refurbishment, provide an ideal case system^25,26. From compact Arctic lodges to expansive desert resorts, architecture reflects climate-driven trade-offs between insulation, cooling, and environmental integration^27,28,29. Examining their refurbishment-phase carbon intensity offers a unique opportunity to reverse-engineer the underlying architectural and environmental determinants of sustainability^30,31,32. By analyzing these determinants, we can identify the combinations of design factors—such as building perimeter, footprint, volume, and climatic extremeness—that minimize material demand, reduce maintenance costs, and optimize adaptation in harsh environments. At the technological frontier, the rapid rise of robotic construction companies—particularly those specializing in 3D printing—signals a broader industrial transformation^33,34,35. Concentrated in North America, Europe, and China, these firms are attracting significant venture capital, reflecting strong expectations for automation robotic and modular fabrication to reshape construction^36,37,38. Yet the sector remains uneven, with regional divergences in application domains and uncertain long-term scalability in extreme or resource-constrained contexts.

Looking beyond Earth, the design of Martian habitats confronts challenges even more severe than terrestrial extremes: high radiation, thin atmosphere, and limited resources^{39,40,41,42,43}. Bibliometric analyses show that research is dominated by additive manufacturing and in-situ resource utilization (ISRU), with basalt fiber, sulfur-based concretes, and polymer systems emerging as candidate materials^44,45,46. Recent advances in AI-driven meta-analysis and topic modeling have enabled the systematic identification of emerging materials and fabrication trends across this rapidly evolving field. However, life-support integration, radiation protection, and human-centered design remain underrepresented^{47,48,49,50,51}. Identifying which materials and methods hold the greatest promise, which directions may yield breakthroughs within 3–5 years, and where interdisciplinary gaps persist are pressing questions for advancing autonomous, sustainable extraterrestrial habitats.

Beyond terrestrial applications, the insights extend to the design of extraterrestrial habitats, where extreme environmental constraints parallel those of Earth’s frontiers. Emerging research on Martian construction is dominated by additive manufacturing and in-situ resource utilization, yet remains fragmented across engineering, ecological, and human-centered dimensions. Drawing lessons from Earth’s extreme hotels can therefore inform not only sustainable construction in hostile climates, but also the viability of autonomous, low-carbon structures for planetary habitation. A multi-dimensional global database—spanning extreme-climate hotels, embodied carbon of prefabricated projects, robotic construction firms, and Martian habitat studies—was assembled to identify the architectural and technological determinants of carbon efficiency across planetary contexts. This objective is to identify the determinants and combinations of architectural factors that drive carbon savings in extreme environments, and to evaluate how prefabricated and 3D-printed systems can translate these insights into scalable, adaptive construction strategies. By bridging terrestrial and extraterrestrial contexts, we aim to advance a unified framework for sustainable architecture across planetary boundaries.

Results

Global hotel sites across extreme climate regions

This study analyzed the relationship between climate type and architectural design features in hotels located in extreme climates using a dataset of 100 hotels. A series of boxplots were generated to explore how different building attributes—such as Building Footprint Area, Building Volume, and Building Perimeter—vary across various climate zones (e.g., Extreme Cold, Tropical, Desert). a-Clustering analysis identified three groups of hotels based on climate, elevation, and temperature (Fig. 1a). Cluster 1 (Extreme Cold) comprised hotels in Arctic or high-altitude regions, requiring insulation-oriented designs. Cluster 2 (Extreme Heat) included desert and tropical lowlands, emphasizing cooling efficiency. Cluster 3 (Tropical) captured warm, humid environments, favoring integration with natural surroundings. The clusters highlight climate-specific architectural and energy strategies, ranging from thermal insulation to efficient cooling and environmental adaptation.

Fig. 1: Climate type and architectural design features from global hotels in extreme climate regions.

a Clustering of 100 hotels identifies compact forms in cold regions, expansive layouts in deserts, and adaptive structures in tropics. b Perimeter strongly correlates with footprint and volume, while complex shapes reduce efficiency. c Random Forest highlights perimeter, temperature range, and surface area as key carbon predictors. d, e Boxplots show larger footprints in hot climates and compact geometries in cold zones, revealing climate-driven design–carbon interactions.

The correlation heatmap revealed several key insights. Notably, Building Perimeter demonstrated a strong positive correlation with both Building Footprint Area (r = 0.76) and Building Volume (r = 0.68) (Fig. 1b). This suggests that larger perimeter and surface area are generally associated with greater building volumes, which may be a design strategy to enhance space utilization or optimize energy efficiency in climates requiring large indoor spaces, such as Tropical or Desert regions. Additionally, the Climate Type variable was found to be strongly correlated with Building Volume (r = 0.72), with buildings in Extreme Cold regions typically having smaller volumes and more compact designs. Conversely, Tropical and Desert climates showed larger volumes, possibly to accommodate cooling systems and extensive outdoor features for heat management. Interestingly, the carbon emission reduction efficiency showed a moderate negative correlation with Building Shape Factor (r = −0.42), indicating that buildings with more irregular or complex shapes may be less efficient in terms of carbon emissions, likely due to inefficient thermal insulation or air circulation in the structure. The findings highlight that architectural design in extreme climates significantly influences building energy efficiency and carbon reduction potential. Larger, more complex buildings in warmer climates may require substantial energy consumption to maintain comfortable indoor conditions, whereas more compact designs in colder climates optimize energy savings. These insights can inform future sustainable architectural practices, particularly in prefabricated buildings and 3D printing technologies, which aim to reduce carbon footprints.

To assess the key determinants of carbon emission reduction efficiency in 3D-printed prefabricated buildings, a Random Forest regression model was developed using a set of architectural and environmental features (Fig. 1c). The model aimed to predict the relative carbon emission reduction between conventional construction and prefabricated 3D printing methods, defined as. The Permutation Importance method was employed to identify the most influential features affecting carbon efficiency. The analysis revealed that Building Perimeter had the most significant impact on the model’s predictions, with an importance score of 0.182. This suggests that larger or more complex building shapes, potentially with more exterior surface area, contribute to greater carbon efficiency due to better thermal performance or structural optimization. Additionally, Temperature Range (importance score: 0.147) emerged as a crucial variable, highlighting the role of climate extremes in influencing building energy demands and, subsequently, carbon emissions. Building Surface Area (importance score: 0.112) also played a notable role, further supporting the hypothesis that larger buildings, or those with more extensive façades, tend to be more energy-efficient when built with advanced materials or design. Other features such as Shape Factor (importance score: 0.101) and Building Volume also contributed to the overall prediction accuracy, indicating that compact designs and efficient spatial layouts can aid in reducing embodied carbon. Interestingly, variables like Building Height Per Floor and Total Number of Hotel Rooms exhibited negligible importance, suggesting that the floor layout and room count alone are not as crucial as design factors in terms of carbon reduction.

The analysis revealed significant variations in the design features depending on the climate (Fig. 1d, e). For instance, Building Footprint Area (m²) was found to be considerably larger in Desert and Tropical environments, with median values of 2500 m² and 2200 m² respectively, compared to Extreme Cold environment regions, which had a median area of ~1000 m². This suggests that buildings in hotter climates tend to have larger surface areas to maximize cooling efficiency and ventilation, whereas colder climates prioritize compactness to minimize heat loss. The Building Perimeter also followed a similar trend, with Desert and Tropical environment regions exhibiting larger perimeters, reflecting the architectural need for expansive external spaces to cope with high temperatures. On the other hand, Building Volume demonstrated a noticeable contrast, with Extreme Cold environment regions showing more volume variation, potentially due to thicker insulation and multi-layer structures designed to preserve heat. The median Building Volume for Extreme Cold environment regions was 15,000 m³, compared to 12,000 m³ for Tropical climates. Statistically, these differences were confirmed by the boxplots showing clear distinctions in the spread and distribution of building attributes across different climates. For instance, the Building Footprint Area in tropical climates had a broader range, indicating greater diversity in design strategies, while buildings in Extreme Cold environment regions exhibited more consistency in size. These findings underline the importance of adapting building design to the specific climatic challenges faced by hotels in extreme environments. Larger building footprints and perimeters in warmer climates may reflect the need for enhanced ventilation and cooling systems, while in colder climates, a compact and insulated design helps to optimize energy efficiency.

Spatial and environmental modulation of prefabrication carbon efficiency

To evaluate the climate mitigation potential of prefabricated construction, 631 global hotel projects by mapping the volumetric difference in embodied carbon intensity between conventional and prefabricated systems, expressed as ΔECI₃D = ECI₃DCC − ECI₃DPC (kgCO₂e/m³) were analyzed (Fig. 2a). The spatial distribution of ΔECI₃DCC revealed pronounced heterogeneity. High-latitude and topographically extreme cities—including Ushuaia (Argentina), Lhasa (Tibet), and Longyearbyen (Norway)—displayed some of the largest positive deltas, achieving carbon savings of 15–25 kgCO₂e/m³. These benefits likely arise from logistics efficiencies and material standardization, which are particularly advantageous in remote and climatically volatile regions where conventional construction incurs substantial costs from labor, waste, and complex engineering. In contrast, a minority of sites—primarily temperate or industrialized urban centers—showed near-zero or negative ΔECI₃DCC values, suggesting that prefabrication can yield marginal or counterproductive savings when local supply chains are mature or conventional practices already achieve low-carbon intensity. Collectively, the results demonstrate that prefabrication is not universally advantageous but exerts its greatest leverage in structurally or logistically constrained regions. This spatial asymmetry underscores the necessity of geographically differentiated policies that account for emissions baselines, climatic stressors, and infrastructure maturity. As global development increasingly extends into environmental frontiers, aligning prefabrication strategies with regional contexts will be critical to maximizing their decarbonization potential.

Fig. 2: Context-dependent carbon efficiency of prefabrication.

a Global spatial mapping of hotel projects shows heterogeneous carbon savings, with high-latitude and remote cities (e.g., Ushuaia, Lhasa, Longyearbyen) achieving reductions, while temperate regions exhibit minimal gains. b Composite Extremeness Index (EI₃) combining elevation, temperature range, and humidity deviation reveals a negative relationship between environmental severity and relative carbon reduction. c Comparison of relative versus absolute reductions shows proportional benefits decline with EI₃, but absolute savings remain robust. d, e Global reduction ratios highlight strongest performance in high-altitude, humid, and resource-scarce regions, underscoring prefabrication’s context-dependent decarbonization potential.

To investigate how environmental extremeness influences the carbon-saving benefits of prefabricated construction, we constructed a composite Extremeness Index (EI₃) integrating elevation, temperature range, and humidity deviation using robust normalization (Fig. 2b). This index was then regressed against the relative reduction in refurbishment-phase embodied carbon intensity between prefabricated and conventional construction methods across 582 global hotel buildings. The ordinary least squares (OLS) regression revealed a statistically significant negative linear trend (β = −0.108, p = 0.004), indicating that the carbon advantage of prefabrication diminishes in more extreme environments. Projects located in low-EI₃ zones (e.g., moderate elevation and climate) commonly achieve carbon savings of 15–25%, while those in high-EI₃ zones show no consistent advantage, with some even exhibiting negative reduction rates—i.e., prefabrication incurs higher carbon intensity than conventional alternatives. This trend suggests that prefabricated systems may lose efficacy or require additional structural or logistical carbon investments in challenging environmental conditions, potentially due to constraints on transport, modular adaptability, or material resilience. The findings emphasize the importance of context-sensitive carbon accounting, and indicate that blanket assumptions about prefabrication’s carbon benefits may not hold in extreme settings.

To assess how environmental extremeness modifies proportional versus absolute carbon mitigation, we compared two refurbishment-phase metrics across a global hotel dataset: relative reduction—defined as (rpECI₃DCC − rpECI₃DPC) / rpECI₃DCC—and absolute reduction, expressed as rpECI₃DCC − rpECI₃DPC (kgCO₂e/m³) (Fig. 2c). Both were analyzed against a composite Extremeness Index (EI₃), integrating elevation, temperature range, and humidity deviation. Relative reductions exhibited a statistically significant negative correlation with EI₃ (β = −0.108, p = 0.004), showing that proportional advantages decline in harsher settings. In low-EI₃ regions, prefabrication delivered reductions up to 25%, whereas in high-EI₃ regions average benefits fell below 5%, occasionally becoming negative. In contrast, absolute reductions remained consistently positive across the EI₃ spectrum: 87% of projects achieved net savings, with median reductions of ~45 kgCO₂e/m³ and maxima surpassing 120 kgCO₂e/m³. Even under severe climatic and topographic stress, total embodied carbon was often reduced despite diminished efficiency. These findings highlight a critical decoupling: proportional benefits decline with extremeness, but absolute reductions remain robust. This divergence indicates that reliance on relative metrics alone may underestimate prefabrication’s mitigation potential in challenging environments. Policy frameworks should therefore account for both absolute and proportional outcomes to better capture prefabrication’s decarbonization leverage across diverse climatic frontiers.

To quantify the environmental benefit of prefabricated hotel construction, computing the carbon reduction ratio as (ECI₃DCC − ECI₃DPC)/ECI₃DCC, where ECI₃DCC and ECI₃DPC denote the volumetric embodied carbon intensities (kgCO₂e/m³) of conventional and prefabricated structures, respectively (Fig. 2d). The results demonstrate that prefabrication delivers consistent carbon savings across diverse climatic and geographic contexts. The mean global reduction ratio is 12.8% (SD = 6.5%, range = −2.1% to +31.3%), with 94.1% of cities showing net reductions. Among them, Chengdu (China, 29.8%), Quito (Ecuador, 28.4%), and Addis Ababa (Ethiopia, 27.6%) exhibit the highest volumetric reductions, likely driven by logistics-intensive environments where prefabrication mitigates on-site inefficiencies, waste, and material overuse. By contrast, cities such as Zurich (Switzerland, 1.3%), Singapore (2.0%), and Tokyo (2.8%) show minimal reductions, reflecting the saturation of carbon-efficient conventional practices in technologically advanced contexts. No cities exhibited statistically significant carbon increases under prefabricated methods, suggesting robustness of savings across the board. Importantly, a spatial analysis reveals stronger reduction ratios in high-altitude, humid-subtropical, and urban-peripheral regions, consistent with prefabrication’s advantage in resource-scarce or labor-challenged settings. This aligns with previous findings that highlight contextual amplification of offsite construction benefits under physical or infrastructural stress. These findings suggest that prefabrication is not merely a universal decarbonization tool, but one with heightened efficacy under extreme or under-resourced conditions. Incorporating such reduction coefficients into lifecycle databases and regional policy frameworks could improve emission forecasting and inform adaptive building codes, especially in fast-urbanizing regions of the Global South.

To evaluate whether environmental extremeness moderates the carbon-saving potential of prefabricated construction, the study analyzed the relationship between the extremeness index (EI₃) and the relative reduction in embodied carbon intensity, defined as (ECI₃DCC − ECI₃DPC)/ECI₃DCC, where ECI₃DCC and ECI₃DPC represent the unit volumetric embodied carbon intensities of conventional and prefabricated construction, respectively (Fig. 2e). Ordinary least squares (OLS) regression yielded a statistically significant negative association (slope = −0.0136, p = 0.0007, R² = 0.033), indicating that the relative carbon reduction achieved through prefabrication decreases modestly but consistently with increasing extremeness. This trend suggests that while prefabricated construction offers overall carbon savings, its percentage advantage may diminish in more extreme environments. Potential explanations include increased transportation distances and energy requirements for prefabricated components, or structural/material constraints that offset the efficiency gains of factory-controlled processes in these settings. Importantly, the declining marginal benefit does not imply carbon disadvantage of prefabrication in extreme regions—absolute carbon savings remain positive—but rather that its relative effectiveness narrows under harsher conditions. It point to the need for context-sensitive design strategies, where prefabrication is supplemented with local material use, modular adaptability, and site-specific construction logistics to preserve its carbon efficiency in challenging environmental contexts.

Global robotic construction companies: 3D printing and on-site demolition robots

The Sankey diagram (Fig. 3a) illustrates the global distribution of 56 construction robotics companies across major technological categories, including “3D printing robots” and “on-site construction robots.” A substantial share of firms focus on 3D printing technologies, with many relying heavily on venture capital or private investment, underscoring the sector’s dependence on high-risk financial backing. This investment pattern reflects both the perceived disruptive potential of automation and the early-stage fragility of construction robotics markets. Regional variation is pronounced: European firms tend to specialize in industrial robotics and on-site automation, whereas North American companies concentrate on residential-scale 3D printing applications. China, in contrast, shows a growing presence in large-scale, modular construction technologies. Such divergence highlights geographically specific demand structures and pathways of technological adoption. At the city level, hubs such as Paris and Omaha host disproportionately high numbers of robotics companies, reinforcing their roles as innovation centers. Nationally, China, France, and the United States emerge as leaders in construction robotics development. Temporal flows reveal that most venture capital activity occurred between 2019 and 2023, coinciding with heightened demand for automation and market consolidation. Early firms often relied on smaller, localized funding sources, while more recent entrants attracted larger-scale investment, suggesting growing investor confidence in technological breakthroughs. Together, these patterns reveal a sector characterized by rapid expansion, regional specialization, and financial dependence on risk-tolerant capital.

Fig. 3: Sankey diagram of 56 global robotic construction companies and trends and predictions.

a Sankey diagram of 56 construction robotics firms categorized by technology and funding. Most focus on 3D printing and on-site automation, largely venture-capital funded. Europe leads in industrial robotics, North America in residential 3D printing, and China in modular systems. b Scatter and box plots show founding-year clustering in the USA and China, with earlier origins in Germany and Japan. c Category analysis reveals asynchronous growth—3D printing robotics show gradual maturation, while on-site demolition robotics exhibit rapid recent expansion, reflecting accelerating global innovation. d Box Plot of Founded Year. e Box Plot of Founded Year.

Scatter and box plot analysis (Fig. 3b) of company founding years highlights clear geographical contrasts in global entrepreneurial activity. The United States and China host the largest number of construction robotics firms, with founding years concentrated between 2014 and 2022, underscoring their central role in recent industrial transformation. This clustering likely reflects strong market demand, rapid technological innovation, and supportive policy environments in both nations. By contrast, Germany and Japan exhibit much earlier founding years, extending from the late nineteenth to early twentieth centuries, consistent with their long-standing technological traditions rooted in the Industrial Revolution. Other European economies, such as France and Italy, show more modern founding patterns concentrated in the 21st century, reflecting continued innovation in construction and engineering. Smaller economies, including Finland, the Netherlands, New Zealand, and Switzerland, display tightly clustered recent founding years, pointing to rapid, policy-driven advances in environmental engineering and smart construction. At the category level, variation is also evident. Firms specializing in 3D printing robots exhibit a wide range of founding years with multiple outliers, consistent with the gradual maturation and broader adoption of 3D printing across construction and manufacturing. By contrast, on-site demolition robotics shows scattered early entries followed by more recent acceleration, suggesting that once experimental technologies have begun to achieve wider market adoption. These patterns reveal both historical legacies and emerging entrepreneurial surges that are shaping the trajectory of construction robotics globally.

It indicates significant differences in the distribution of company founding years across different countries and categories, reflecting the varying technological innovation cycles, market demands, and policy-driven factors in different regions and industries (Fig. 3c). The USA and China, as leaders in global entrepreneurial activities, have their company founding years predominantly concentrated in modern times, showcasing their dominant position in emerging technologies. In contrast, Germany and Japan represent longer historical technological accumulation and industrial foundations, with their company founding years spanning a broader range of time periods. For emerging industries such as 3D Printing Robots and On-Site Demolition Robots, the outliers suggest that the technologies are still evolving, with entrepreneurial activities driven by technological breakthroughs. Future research could further explore the technological progress and market acceptance in these industries, as well as analyze the temporal patterns of entrepreneurial activities. While the study observed a concentration of entrepreneurial activities in countries such as Finland, the Netherlands, and other smaller economies, it remains to be seen whether this trend will continue, depending on future government policy support, industry demand, and international market shifts. 3D Printing Robots represent a key area of growth within the construction robotics industry, with significant investment, technological advancements, and geographical concentration highlighting the potential for continued innovation. The increasing adoption of 3D printing technology in the construction industry reflects the sector’s growing reliance on automation and sustainable building practices, paving the way for future developments in both technology and market applications.

3D printing prefabricated Martian habitat

To identify the most promising materials and methods for 3D-printed prefabricated Martian habitats, we analyzed 517 peer-reviewed publications spanning 1954–2025. Each study was labeled using a hybrid rule-based and statistical approach, extracting key material types and construction techniques from titles, abstracts, and takeaways. It labels the corpus with one or more materials/methods and research field categories, capturing both technical focus and domain context (Fig. 4a). Among materials and fabrication techniques, additive manufacturing dominated the landscape, appearing in over 75 publications, followed by polymer/resin-based systems and sulfur-based concretes, each tagged in more than 30 studies (Fig. 1a). Less frequently represented but technologically significant materials included basalt fiber composites, lava casting, and microwave sintering, which occurred in fewer than 10 publications each. These materials may reflect early-stage or niche research directions with limited but potentially high-impact representation. On the disciplinary axis, construction methods were the most prevalent research theme, featuring in nearly 400 publications (Fig. 1b). Other high-frequency fields included manufacturing, materials science, and robotics, highlighting a research landscape largely dominated by engineering concerns. In contrast, fields related to life support systems, radiation protection, and human factors appeared far less frequently—each in fewer than 50 studies—suggesting that biological, physiological, and environmental dimensions of Martian habitat design remain underexplored relative to structural and fabrication domains. The results underscore a key structural imbalance: while Martian habitat research exhibits a diverse array of technical materials and manufacturing strategies, the broader integration of life sciences and human-centered design appears limited. This asymmetry may constrain the development of truly autonomous, habitable systems for long-duration planetary missions unless addressed through more deliberate interdisciplinary engagement.

Fig. 4: Mapping materials, methods, and disciplinary gaps in Martian habitat research.

a, b Core materials and methods include sulfur concrete, geopolymers, basalt fiber, and additive manufacturing; evidence scoring (0–1) ranked basalt fiber and additive manufacturing as most validated. c, d LDA topic modeling revealed rapid growth in ISRU fabrication and bioregenerative life support, while geology and conceptual design declined post-2015. e Quadrant-based forecasting identified ISRU, life support, and radiation protection as emerging breakthroughs. f, g Co-authorship and field co-occurrence analyses revealed strong engineering clustering but limited integration with life sciences and human factors. “White space” analysis highlighted key collaboration gaps emphasizing the need for interdisciplinary convergence toward resilient Martian habitats.

Core construction materials and methods identified include sulfur-based concrete, geopolymers, polymer/resin systems, basalt fiber composites, microwave sintering, lava casting, and additive manufacturing (Fig. 4b). Each category was evaluated using a multi-criteria evidence score, integrating six weighted indicators—annual citation rate, journal quality (SJR), prevalence of empirical validation, author diversity (replication proxy), publication growth (CAGR), and normalized frequency—scaled between 0 and 1. Basalt fiber achieved the highest composite score (0.55), despite only two publications, due to perfect citation normalization and exclusive appearance in top-tier journals (mean SJR = 1.0). Additive manufacturing ranked second (0.48), distinguished as the most widely adopted method, with the highest reproducibility (1.0) and sustained growth (CAGR ≈ 10.6%), consolidating its role as the field’s core technology. Polymer/resin systems and sulfur concrete scored moderately (0.26 and 0.21), with sulfur concrete notable for the highest CAGR among all categories, suggesting rapidly growing interest despite limited representation in high-impact journals. Temporal dynamics reinforced these contrasts: additive manufacturing maintained clear dominance in publication volume and steady growth, while polymer/resins attracted stable but modest attention. In contrast, basalt fiber and microwave sintering appeared in only a few studies but demonstrated strong individual evidence, suggesting promising yet underexplored directions. Collectively, the findings indicate additive manufacturing as the most mature and reproducible pathway, while basalt fiber composites and sulfur concretes emerge as high-potential candidates requiring expanded validation for future Martian habitat construction.

Each study was automatically annotated using a hybrid keyword- and frequency-based pipeline, capturing mentions of core material systems (sulfur concrete, basalt fiber, geopolymers, polymer resins) and fabrication techniques (additive manufacturing, microwave sintering, lava casting, in-situ resource utilization) (Fig. 4c). A multi-criteria scoring system was applied to quantify evidence strength, integrating six normalized indicators: annualized citation rate, mean journal SJR quartile, prevalence of empirical validation, number of independent author groups (proxy for reproducibility), publication growth rate (CAGR), and frequency of appearance. Among ten categories, basalt fiber composites achieved the highest evidence score (0.55), despite only two publications, due to perfect normalized citation impact (1.00) and exclusive presence in top-tier journals (mean SJR = 1.0). Additive manufacturing ranked second (0.48), supported by the largest publication base (n = 75), maximal reproducibility (1.0), and consistent growth (CAGR ≈ 10.6%), consolidating its role as the field’s dominant method. Polymer/resin systems (0.26) and sulfur concrete (0.21) scored moderately, with sulfur concrete notable for its exceptional CAGR (100%) despite limited uptake in high-impact journals. Temporal dynamics reinforced these trends: additive manufacturing showed sustained growth over 15 years, polymer/resins remained stable, and sulfur concrete expanded rapidly from a low base. Microwave sintering demonstrated strong empirical validation (ratio = 1.0) but limited reproducibility and low prevalence, reflecting its underexplored status. Bootstrap resampling (1000 iterations) confirmed stable evidence rankings for additive manufacturing and polymer systems, while basalt fiber and microwave sintering showed high variance due to small sample sizes. Collectively, the results highlight additive manufacturing as the most mature and validated technology, while basalt fiber and sulfur concretes emerge as promising high-impact candidates requiring expanded replication across independent research groups.

Using a three-year sliding window, Latent Dirichlet Allocation (LDA) was applied to TF-IDF vectors derived from titles, abstracts, and takeaways, yielding eight latent research topics whose temporal prevalence was tracked across overlapping periods (Fig. 4d). Each topic represented a coherent thematic cluster. Topic 1, focused on “in-situ construction,” “regolith sintering,” and “extraterrestrial materials,” exhibited sustained growth from 2018 onward, increasing by more than 150% between 2015 and 2025, and highlighting the accelerating prominence of ISRU-based fabrication. Topic 3, emphasizing “life support systems,” “bioregenerative environments,” and “oxygen recycling,” grew by approximately 95% over the same period, reflecting the rising importance of closed-loop ecological support for long-duration missions. In contrast, Topic 0 (“general Martian geology” and “planetary formation”) and Topic 4 (“conceptual habitat design” and “habitat volume optimization”) showed declining prevalence after 2015, indicating a shift away from foundational or conceptual studies toward engineering and systems-level implementation. Other topics displayed more variable trajectories, with Topic 2 showing periodic resurgence linked to specific technological developments. Overall, the analysis identified ISRU fabrication and bioregenerative life support as the fastest-growing domains, while planetary geology and conceptual design are in relative decline. These findings suggest that Martian habitat research is evolving from planetary science foundations toward applied engineering and sustainable infrastructure, with ISRU and ecological life support emerging as likely breakthrough directions in the near future.

To forecast future breakthroughs in Martian habitat research, time-series modeling was combined with topic momentum analysis, visualized through a quadrant-based scatterplot encoding both velocity (recent growth) and acceleration (projected change) (Fig. 4e). Each of the eight latent topics identified by LDA was represented by a distinct marker, enabling clear differentiation of thematic clusters. Two clusters fell within the high-velocity, high-acceleration quadrant, indicating likely breakthrough domains. The first, centered on “regolith,” “sintering,” “in-situ,” “printing,” and “construction,” reflects sustained and accelerating interest in ISRU-based additive manufacturing. The second, associated with “oxygen,” “recycling,” “system,” “life,” and “support,” highlights the rapid emergence of bioregenerative life support as a critical technology for autonomous off-world habitats. The top-left quadrant identified topics with low current output but growing acceleration, including “radiation,” “shielding,” and “protection,” suggesting nascent but increasingly important research into astronaut health and environmental risk mitigation. In contrast, the bottom-left quadrant comprised declining topics, such as “geology,” “formation,” and “soil,” pointing to waning emphasis on planetary foundations as attention shifts toward engineering implementation. Together, ARIMA forecasting and semantic clustering provide a robust framework for detecting early momentum, guiding mid-term trajectories, and informing funding priorities. These results emphasize ISRU-based manufacturing, life support, and radiation protection as emerging strategic priorities for sustainable extraterrestrial habitats.

Each study was annotated with one or more high-level disciplinary labels—including construction methods, materials science, manufacturing, robotics, life support systems, radiation protection, energy systems, and human factors—using a hybrid keyword-based strategy (Fig. 4f). The co-authorship network exhibited a highly modular structure, with a modularity score (Q) of 0.775, indicating strong clustering within disciplinary communities and limited cross-field integration. Only 7.2% of co-authorship links connected researchers from non-overlapping domains, underscoring low levels of interdisciplinary collaboration at the individual level. Field co-occurrence analysis revealed strong integration between construction methods and manufacturing (144 papers), construction methods and robotics (79), and construction methods and human factors (77), reflecting the dominance of engineering-driven axes combining structural design, automation, and ergonomics. In contrast, several field pairs exhibited minimal or no co-occurrence, including life support × materials science, energy × human factors, and radiation protection × manufacturing. These “white spaces” highlight conceptual gaps where integration remains absent despite clear relevance to sustainable extraterrestrial habitats. A co-occurrence heatmap of the 15 strongest connections confirmed both areas of strong collaboration and zones of disciplinary voids. Notably, life support and robotics—each essential for autonomous habitats—rarely intersected, reflecting a structural imbalance in research priorities. Together, these results indicate that while Martian habitat studies are technically robust, they remain fragmented across ecological, engineering, and human-centered dimensions. Bridging these divides will be critical for developing resilient and human-compatible off-world environments.

To quantify underexplored intersections in Martian habitat research, we compared expected and observed co-occurrence values of 28 unique field pairs under an independence assumption based on marginal frequencies (Fig. 4g). This approach revealed statistically significant disciplinary “white spaces,” where collaboration is underrepresented relative to expectation. The largest deficit was found between human factors and manufacturing, with 60.8 expected co-occurrences but only 44 observed, yielding a gap of 16.8 publications. Similarly, energy × manufacturing (expected = 35.0, actual = 22, gap = 13.0) and manufacturing × radiation protection (expected = 26.6, actual = 16, gap = 10.6) demonstrated major shortfalls, underscoring weak integration of manufacturing with energy systems, radiation shielding, and human-centered design. Particularly sparse intersections were observed in ecological and robotic domains. For instance, life support × robotics showed only 2 co-occurrences against an expected 6.2 (gap = 4.2), while life support × manufacturing fell short by 6.1. These gaps highlight missed opportunities for combining environmental control and automation, essential for autonomous, self-sustaining habitats. Overall, the matrix shows that construction-related fields are internally well integrated, but remain poorly connected to life sciences, energy systems, and human performance. This fragmentation identifies critical disciplinary fault lines, with manufacturing insufficiently linked to human, ecological, and radiation dimensions. Addressing these “white spaces” through targeted interdisciplinary convergence will be crucial to advancing from conceptual studies to viable, resilient Martian habitats.

Discussion

This study demonstrates that architectural form is deeply conditioned by climate, with hotels in extreme cold, hot, and tropical regions adopting distinct spatial and material strategies for thermal regulation. Compact volumes in cold climates optimize heat retention, while expansive footprints in warm zones facilitate cooling and ventilation. These findings confirm that building geometry and envelope design are not only stylistic but fundamental determinants of energy use and carbon performance. The negative association between shape complexity and emission efficiency further highlights the trade-offs between architectural expressiveness and environmental responsibility.

Prefabricated 3D printing emerges as a promising pathway to reconcile design flexibility with carbon mitigation. Our machine-learning analysis indicates that perimeter, surface area, and climate extremes are decisive predictors of carbon savings, underscoring the capacity of digital fabrication to exploit geometry for energy efficiency. However, the benefits of prefabrication are spatially heterogeneous: high-latitude and topographically remote cities exhibit pronounced reductions in embodied carbon, while technologically advanced urban cores show limited advantage. These patterns caution against universal prescriptions and instead advocate for regionally adaptive benchmarks that integrate climatic stressors, infrastructural maturity, and logistics costs. Global refurbishment-phase data reveal that prefabrication consistently delivers absolute carbon reductions, even when relative efficiency declines in extreme settings. This divergence between absolute and proportional benefits highlights the need to move beyond generic carbon accounting and instead interrogate the underlying determinants of refurbishment intensity. By analyzing embodied carbon variations across climate extremes, it becomes possible to infer which architectural and environmental factors—or their combinations—drive material efficiency and long-term maintenance advantages. Such insights are crucial for identifying design strategies best suited to extreme environments, where compact geometries, modular adaptability, and resilient materials can minimize resource input and lifecycle costs. Targeted refurbishment-phase metrics thus provide a powerful basis for selecting construction systems that are not only lower-carbon but also more sustainable and economically viable under harsh conditions.

The industrial landscape further supports this trajectory. Robotics and 3D printing companies are proliferating globally, with distinct regional emphases: Europe in industrial automation, North America in residential applications, and China in large-scale adoption. The concentration of entrepreneurial activity in recent years highlights both market readiness and investor appetite for high-risk, high-impact construction technologies. Yet, the sector’s reliance on venture capital underscores a vulnerability to policy shifts and market volatility. To achieve industrial scalability, logistical efficiency and material standardization must become central design constraints. The carbon and economic advantages of prefabrication diminish when transportation distances, assembly energy demand, or module customization increase disproportionately. A life-cycle cost perspective—integrating supply chain logistics, on-site automation intensity, and recyclability—should therefore accompany future carbon assessments. From a design protocol perspective, these results suggest that adaptive prefabrication frameworks should explicitly incorporate environmental indices such as temperature amplitude and wind exposure into early-stage digital modeling. Parametric design tools linked to embodied carbon databases could enable architects to optimize geometry and material systems dynamically under climate or logistical constraints. Establishing such data-informed design protocols would facilitate both industrial scaling and regional customization—bridging the gap between experimental prototypes and reproducible, code-compliant construction systems.

Looking outward, Lunar and Martian habitat research reflects both the promise and the fragmentation of construction science. Additive manufacturing and in-situ resource utilization dominate, confirming their centrality to extraterrestrial building. Yet life support systems, radiation protection, and human factors remain underexplored, appearing in far fewer studies and rarely integrated with materials or robotic domains. Our topic modeling and co-occurrence analysis reveal clear disciplinary “white spaces,” particularly between engineering-intensive fields and ecological or human-centered research. Without deliberate integration, future habitats may achieve structural feasibility but fail to support long-term human viability. These findings highlight a continuum of challenges and opportunities in construction science, spanning Earth’s extreme environments to extraterrestrial frontiers. Extreme environments /climate-responsive geometry, prefabricated construction, and robotic automation each contribute to a framework of sustainable building, but their benefits are context-dependent and require systemic optimization. Bridging disciplinary divides—between engineering and life sciences, architecture and ecology, design and robotics—will be essential to accelerate both terrestrial decarbonization and the realization of habitable Martian environments. Future research must move beyond isolated technical advances toward integrated, adaptive systems that couple efficiency with resilience across planetary boundaries.

Methods

Integrated data and modeling framework

The study applies a multi-dimensional computational framework (computational, AI-assisted, and systems-level approach) combining geospatial clustering, Random Forest regression, and bibliometric topic modeling to quantify the environmental performance of prefabricated construction and map technological trajectories from extreme terrestrial to Martian contexts. A multi-dimensional database was assembled to evaluate the determinants of carbon efficiency across extreme terrestrial and extraterrestrial contexts. Data sources included: (i) architectural and environmental attributes of 100 hotel sites located in extreme climate regions, (ii) embodied carbon intensities of 631 conventional versus prefabricated construction hotel building projects located in extreme environment regions, (iii) metadata on 56 robotic innovative and construction companies, and (iv) 517 studies on prefab 3D printing Martian habitats. All datasets were harmonized through standardized preprocessing pipelines, ensuring comparability across spatial, temporal, and disciplinary domains. Statistical models, machine-learning algorithms, and bibliometric analyses were then applied to identify the key drivers of refurbishment-phase carbon intensity and to assess the scalability of prefabricated and 3D-printed construction systems under diverse environmental constraints.

Data 1-Architectural and environmental attributes of 100 hotel sites located in extreme climate regions: A geospatial dataset of 100 hotel buildings located in extreme climate regions was compiled to investigate the relationship between environmental severity and architectural adaptation. “Extreme climate” was operationally defined using Köppen–Geiger classifications combined with temperature amplitude (>35 °C annual range), mean annual precipitation (<250 mm or >2000 mm), and wind velocity (>8 m/s annual mean). Sites were selected to ensure regional balance across six continental zones (Arctic, alpine, desert, tropical, polar coastal, and high-latitude maritime). Architectural parameters—including building geometry, envelope ratio, material composition, and energy system type—were extracted from field surveys, governmental databases, and published case reports. Each record was validated through cross-source triangulation to ensure data consistency and replicability.

Data 2-Embodied carbon intensities of 631 conventional vs prefabricated hotel buildings across extreme environments: To evaluate the embodied-carbon performance of prefabricated construction under climatic extremes, a global dataset of 631 hotel projects was constructed. For each case, volumetric embodied carbon intensity was calculated as ΔECI₃D = ECI₃DCC − ECI₃DPC (kgCO₂e/m³). ECI₃DCC and ECI₃DPC represent the embodied carbon intensity (kgCO₂e/m³) of conventional and prefabricated systems, respectively. Data sources included peer-reviewed life-cycle assessments, governmental sustainability databases (e.g., ICE v3, EC3), and certified Environmental Product Declarations (EPDs). Projects were filtered based on completeness of material inventory and verifiable climate classification. Outliers exceeding 3σ from the median were excluded to enhance robustness.

Data 3-Global 56 robotic innovative and construction companies: An industry database was assembled comprising 56 active construction robotics firms operating globally. Company selection followed three criteria: (i) public availability of validated operational or technological information, (ii) direct relevance to construction automation (e.g., 3D printing robots, on-site assembly, autonomous fabrication), and (iii) verified company activity between 2010 and 2023. Metadata—encompassing founding year, investment source, technological category, and regional location—were extracted from official corporate filings, patent registries, and funding databases (e.g., Crunchbase, PitchBook). Manual validation ensured cross-country consistency and prevented duplication.

Data 4-517 studies on prefab 3D printing Martian habitats: A comprehensive bibliometric corpus of 517 peer-reviewed publications (1954–2025) was constructed to identify dominant materials, fabrication techniques, and interdisciplinary trends in the field of prefabricated and 3D-printed Martian habitats. Data were retrieved primarily from the Consensus database, an AI-augmented scholarly search platform that aggregates peer-reviewed content across CrossRef, Semantic Scholar, Scopus, NASA ADS, and PubMed. The use of Consensus ensured high recall across multiple disciplines, including architecture, robotics, materials science, and astrobiology. The search employed Boolean query strings combining domain-specific and methodological terms: (“Martian habitat” OR “extraterrestrial construction” OR “off-world architecture”) AND (“additive manufacturing” OR “3D printing” OR “in-situ resource utilization” OR “ISRU” OR “prefabricated building”). The retrieved corpus was filtered to include only peer-reviewed articles, conference proceedings, or official NASA/ESA technical reports, while excluding non-reviewed gray literature, speculative essays, or purely conceptual design proposals. Screening followed PRISMA 2020 recommendations. Each publication was automatically annotated using a hybrid keyword- and frequency-based pipeline, identifying mentions of key material systems (e.g., sulfur-based concrete, basalt fiber composites, regolith geopolymer, polymer/resin systems) and fabrication techniques (e.g., additive manufacturing, microwave sintering, lava casting, ISRU-based 3D printing). Each entry was validated for completeness and consistency by cross-checking with the corresponding publisher metadata. The final dataset provides metadata on publication year, author network, country affiliation, and material/method category.

Definition of Extreme Climate and Extreme Environment: Extreme climate refers specifically to meteorological and atmospheric conditions that deviate significantly from the long-term global mean. Following the World Meteorological Organization (WMO, 2018) and Köppen–Geiger classifications, extreme climates are statistically defined as regions where temperature, precipitation, or humidity values persist beyond the 90th or below the 10th percentile of the historical baseline (1981–2010). Examples include polar (EF/ET), arid (BW/BS), and severe continental (Dfd/Dwd) zones. These conditions primarily affect thermal regulation, energy balance, and architectural design parameters such as insulation, ventilation, and energy consumption. In this study, extreme climate datasets were used to analyze how prefabricated and 3D-printed construction performs under high thermal and hydrological stress. Extreme environment is a broader, multidisciplinary construct encompassing not only climatic but also physical, chemical, and radiative stressors that challenge the viability of materials, structures, or life-support systems. As defined by NASA (2021) and the European Space Agency (ESA, 2020), extreme environments include conditions of low atmospheric pressure, high radiation, reduced gravity, or chemical toxicity—features typical of extraterrestrial settings such as Mars, the Moon, or deep-sea analogs on Earth. In architectural and planetary science contexts, extreme environments capture the integrated system-level constraints—thermal, structural, radiative, and biological—that govern habitat resilience beyond Earth.

Main algorithms and formulas

Embodied carbon accounting: The embodied carbon intensity (ECI, kgCO₂e/m³) was calculated for both conventional and prefabricated systems:

Conventional construction:

$${{\rm{ECI}}}^{3}{\rm{DCC}}=\Sigma ({{\rm{Q}}}_{{\rm{i}}}\cdot {{\rm{EF}}}_{{\rm{i}}})/{\rm{V}}$$

(1)

Prefabricated construction:

$${{\rm{ECI}}}^{3}{\rm{DPC}}=\Sigma ({{\rm{Q}}}_{{\rm{j}}}\cdot {{\rm{EF}}}_{{\rm{j}}}\cdot {{\rm{\alpha }}}_{{\rm{j}}})/{\rm{V}}$$

(2)

Two metrics were defined: 1. Absolute carbon savings: ΔECI₃D = ECI₃DCC − ECI₃DPC 2. Relative reduction ratio: R = (ECI₃DCC − ECI₃DPC) / ECI₃DCC Refurbishment-phase embodied carbon (rpECI) was estimated using standard lifetimes (20–30 years) for façade, HVAC, and interior systems.

Extremeness Index (EI₃): Environmental extremeness was quantified by:

$${\rm{E}}{{\rm{I}}}^{3}=\left({\rm{z}}\left({\rm{Elevation}}\right)+{\rm{z}}\left({\rm{TemperatureRange}}\right)+{\rm{z}}\left({\rm{HumidityDeviation}}\right)\right)/3$$

(3)

Statistical analysis: Hotels were clustered into three groups using k-means clustering (k = 3). Correlations were assessed with Pearson’s r. OLS regression tested the effect of EI₃ on carbon reduction ratio:

$${\rm{R}}={{\rm{\beta }}}^{0}+{{\rm{\beta }}}^{1}\cdot {\rm{E}}{{\rm{I}}}^{3}+{\rm{\varepsilon }}$$

(4)

Machine-learning model: To identify the key determinants of carbon efficiency in prefabricated systems, a Random Forest regression model (n = 500 trees) was trained using architectural and environmental predictors, including perimeter, surface-to-volume ratio, climate severity index, and construction typology. Model performance was assessed through 10-fold cross-validation, yielding R² = 0.83 and RMSE = 6.4 kgCO₂e/m³, indicating high predictive accuracy and generalizability.

Feature selection was conducted using a two-step approach: (1) recursive elimination of low-variance predictors (threshold < 0.01), followed by (2) ranking based on permutation importance. Final feature importances were normalized across all predictors. The prediction function is expressed as:

$$\hat{{\rm{y}}}=(1/{\rm{T}}){\Sigma {\rm{f}}}_{{\rm{t}}}({\rm{x}})$$

(5)

where f_t(x) represents the individual decision tree estimator. Feature importance (I_j) was computed using permutation importance:

$${{\rm{I}}}_{{\rm{j}}}=(1/{\rm{M}})\Sigma ({{\rm{MSE}}}_{{\rm{m}}}-{{\rm{MSE}}}_{{\rm{mj}}})$$

(6)

The combination of cross-validated performance metrics and permutation-based feature evaluation ensured model robustness and interpretability across heterogeneous climate datasets.

Evidence scoring: Each study was labeled with material/method categories (e.g., additive manufacturing, basalt fiber, sulfur concrete, polymer resin, microwave sintering, lava casting, geopolymers). A multi-criteria evidence score was computed, integrating: (1) annualized citation rate, (2) mean journal SJR quartile, (3) empirical validation ratio, (4) author diversity (proxy for replication), (5) publication growth (CAGR), and (6) normalized frequency. Scores were scaled to [0, 1] and aggregated via weighted sum. Bootstrap resampling (1000 iterations) generated confidence intervals for ranking stability. A multi-criteria evidence score was computed:

$${{\rm{S}}}_{{\rm{k}}}=\Sigma ({{\rm{w}}}_{{\rm{i}}}\cdot {{\rm{M}}}_{{\rm{ik}}})$$

(7)

Bootstrap resampling (1000 iterations) generated confidence intervals.

Topic modeling and forecasting: Dynamic Latent Dirichlet Allocation (LDA) was applied. Topic trajectories were quantified by average term prevalence. To forecast research momentum, we combined ARIMA time-series modeling with velocity–acceleration quadrant mapping, identifying potential breakthrough topics within 3–5 years. Topic strength estimated by:

$${\rm{P}}\left({\rm{z}}|{\rm{d}}\right)=\left(\mathrm{Ndz}+{\rm{\alpha }}\right)/\left(\mathrm{Nd}+\mathrm{K\alpha }\right)$$

(8)

Research momentum was forecast using ARIMA models.

Network and gap analysis: Each paper was assigned high-level disciplinary field labels (construction methods, materials science, robotics, life support, radiation protection, human factors, energy systems). The study built co-authorship networks and computed modularity (Q) to quantify disciplinary clustering. A field co-occurrence matrix was generated, with expected vs. observed values compared to identify disciplinary white spaces. Statistical gaps were calculated as Expected − Actual frequencies. Publications were labeled with disciplinary fields. A co-occurrence matrix was constructed:

$${{\rm{C}}}_{{\rm{ij}}}={{\rm{N}}}_{{\rm{ij}}}/{\rm{N}}$$

(9)

Expected co-occurrence:

$${{\rm{E}}}_{{\rm{ij}}}=({{\rm{N}}}_{{\rm{i}}}\cdot {{\rm{N}}}_{{\rm{j}}})/{\rm{N}}$$

(10)

Gap:

$${{\rm{G}}}_{{\rm{ij}}}={{\rm{E}}}_{{\rm{ij}}}-{{\rm{C}}}_{{\rm{ij}}}$$

(11)

Network modularity quantified by:

$${\rm{Q}}=(1/2{\rm{m}})\Sigma ({{\rm{A}}}_{{\rm{ij}}}-({{\rm{k}}}_{{\rm{i}}}\cdot {{\rm{k}}}_{{\rm{j}}}/2{\rm{m}})){\rm{\delta }}({{\rm{c}}}_{{\rm{i}}},{{\rm{c}}}_{{\rm{j}}})$$

(12)