This survey reviews spherical designs from a numerical-analysis perspective while retaining their connections with algebraic combinatorics and discrete geometry. A finite set X-N subset of S-d is a spherical t-design when equal-weight integration over X-N is exact for all spherical polynomials of degree at most t. We organize the subject around three themes: cardinality bounds, existence and nonexistence; certified and large-scale numerical construction; and approximation and computational applications. Particular attention is paid to the relations among polynomial exactness, covering and separation geometry, conditioning, and numerical stability. We also discuss applications to numerical integration, interpolation, hyper-interpolation, differential and integral equations, spherical signal and image processing. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
After long-term efforts, the Hamilton path (cycle) problem for connected vertex-transitive graphs of order pq (where p and q are primes) was finally resolved in 2021, see [10]. Fifteen years ago, mathematicians began addressing this problem for graphs of order 2pq. Among these studies, it was proved in 2012 (see [22]) that every connected vertex-transitive graph of order 10p (where p≠7 is a prime) contains a Hamilton path, with the exception of a family of graphs that was recently confirmed in [11]. In this paper, we achieve a further result: every connected vertex-transitive graph of order 10p (where p is a prime) contains a Hamilton cycle, except for the truncation of the Petersen graph.
Brittle failure of fractured rock materials under quasi-static loading is characterized by strong suddenness and severe consequences, making accurate prediction of crack evolution and failure load critically important. However, conventional continuum mechanics-based methods encounter singularities when addressing discontinuities, while the widely adopted dynamic relaxation (DR) method within peridynamics is applicable solely to the continuous deformation stage. To address this limitation, this paper proposes a two-stage joint algorithm combining dynamic relaxation and conventional explicit analysis within the peridynamics framework. In this method, the DR method is employed exclusively during the continuous deformation stage before the onset of loading-induced bond failure, while the conventional explicit algorithm is activated to simulate the subsequent crack propagation process. The feasibility and stability of the proposed method are validated through the failure simulation of an L-shaped concrete plate. Subsequently, the quasi-static failure behavior of Brazilian discs with pre-existing central cracks and slope models containing steep-gently dipping pre-existing fissures is systematically investigated. For Brazilian discs, the influences of crack inclination angle and relative crack length on failure modes and bearing capacity are revealed. The numerical results are consistent with experimental observations. For the two specific slope models considered, the geometric configuration characterized by theta = 90 degrees exhibits tensile coalescence failure, whereas that characterized by theta = 56 degrees exhibits tensile-shear coalescence failure; both numerical results agree with the corresponding experimental observations. The findings demonstrate that the proposed joint algorithm combines the computational efficiency of dynamic relaxation during the continuous deformation stage with an explicit approximation of the quasi-static fracture process during damage evolution. This provides a simple yet effective numerical solution for quasi-static failure analysis of fractured rock materials.
To investigate the earth pressure acting on shield tunnel linings in longitudinally non-uniform ground under nonstationary hydraulic conditions, this study develops a nonlinear analytical framework considering the mobilized shear strength of unsaturated soils. Closed-form displacement fields induced by the ground loss are first derived using the complex variable method. The soil domain is then discretized. Deviatoric strain, mobilized shear strength, and strain energy contribution of each soil unit are determined through a nonlinear stress-strain relationship. An energy balance equation is established by equating accumulated strain energy to external work. The circumferential earth pressure acting on the lining and the corresponding stress release coefficient are accordingly obtained. The proposed method agrees well with centrifuge test data and numerical simulations. It can reasonably capture the continuous transition of earth pressures from the full overburden state to the loosening pressure state during construction. The stress release coefficient increases nonlinearly with the ground loss ratio. It rises rapidly at small deformation and then gradually approaches a stable level. A higher groundwater level significantly weakens the stress-release capacity of the ground. For a ground loss ratio of 0.5 %, the stress release coefficient decreases by about 25.7 % when the groundwater table changes from a depth of 0.8 h (h = 12 m) to the ground surface. The hydraulic parameters of unsaturated soils and soil type also exert strong control on stress-release behavior. Among the soils considered, clay shows the highest sensitivity, followed by silt, whereas loess and sand are much less sensitive. The proposed framework provides a rational basis for support pressure design of shield tunnels in complex hydraulic environments.
