Urban morphology exhibits fractal characteristics, which can be described by multifractal scaling. Multifractal parameters under positive moment orders primarily capture information about cen-tral areas with relatively stable growth, while those under negative moment orders mainly reflect information about marginal areas with more active growth. However, effectively utilizing mul-tifractal spectrums to uncover the spatio-temporal variations of urban growth remains a challenge. To addresses this issue, this paper proposes a multifractal measurement by combining theoretical principles and empirical analysis. To capture the difference between growth stability in central areas and growth activity in marginal areas, an index based on generalized correlation dimension Dq is defined. This index takes the growth rate of Dq at extreme negative moment order as the numerator, and that at extreme positive moment order as the denominator. During the stable stage of urban growth, the index demonstrates a consistent pattern over time. While during the active stage, the index may exhibit abnormal fluctuations or even jumps. This indicates that the index can reveal spatio-temporal information about urban evolution that cannot be directly observed through multifractal spectrums alone. By integrating this index with multifractal spectrums, we can more comprehensively characterize the evolutionary characteristics of urban spatial structure.

Urban morphology exhibits fractal characteristics, which can be described by multifractal scaling. Multifractal parameters under positive moment orders primarily capture information about central areas characterized by relatively stable growth, while those under negative moment orders mainly reflect information about marginal areas that experience more active growth. However, effectively utilizing multifractal spectra to uncover the spatio-temporal variations of urban growth remains a challenge. To addresses this issue, this paper proposes a multifractal measurement by combining theoretical principles and empirical analysis. To capture the difference between growth stability in central areas and growth activity in marginal areas, an index based on generalized correlation dimension Dq is defined. This index takes the growth rate of Dq at extreme negative moment order as the numerator and that at extreme positive moment order as the denominator. During the stable stage of urban growth, the index demonstrates a consistent pattern over time, while during the active stage, the index may exhibit abnormal fluctuations or even jumps. This indicates that the index can reveal spatio-temporal information about urban evolution that cannot be directly observed through multifractal spectra alone. By integrating this index with multifractal spectra, we can more comprehensively characterize the evolutionary characteristics of urban spatial structure.

In this work, an improved domain decomposition method is developed to address workload imbalance when implementing the parallel computing of a four‐dimensional lattice spring model (4D‐LSM) to solve problems in rock engineering on a large scale. A cubic domain decomposition scheme is adopted and optimized by a simulated annealing algorithm (SAA) to minimize the workload imbalance among subdomains. The improved domain decomposition method is implemented in the parallel computing of the 4D‐LSM. Numerical results indicate that the proposed domain decomposition method can further improve the workload balance among processors, which is helpful to supersede the limit of computational scale when solving large‐scale geotechnical problems and decrease the runtime of the parallel 4D‐LSM by at most 40% compared to the original cubic decomposition method. This shows the practicability of the proposed method in parallel computing. Two types of target functions of SAA are tested, and their influence on the performance of the parallel 4D‐LSM is investigated. Finally, a computational model with one billion particles for one actual engineering application of using 4D‐LSM is realized, and the result shows the advantages of parallel computing.

Parallel computing assigns the computing model to different processors on different devices and implements it simultaneously. Accordingly, it has broad applications in the numerical simulation of geotechnical engineering and underground engineering, of which models are always large-scale. With parallel computing, the computing time or the memory requirements will be reduced by splitting the original domain of the numerical model into many subdomains, which is thus named as the domain decomposition method. In this study, a cubic and equal volume domain decomposition strategy was utilized to realize the parallel computing on the distributed memory system of four-dimensional lattice spring model (4D-LSM) based on the message passing interface. With a more efficient communication strategy introduced, this study aimed at operating an one-billion-particle model on a supercomputer platform. The preprocessing procedure of the parallelized 4D-LSM was restructured and the particle generation strategy suitable for the supercomputer platform was employed to minimize the time consumption in preprocessing and calculation. On this basis, numerical calculations were performed on TianHe-3 prototype E class supercomputer at the National Supercomputer Center in Tianjin. Two fieldscale three-dimensional blasting wave propagation models were carried out, of which the numerical results verify the computing power and the advantage of the parallelized 4D-LSM in the simulation of large-scale three-dimension models. Subsequently, the time complexity and spatial complexity of 4D-LSM and other particle discrete element methods were analyzed.

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