PurposeTwo-dimensional (2D) problems are governed by unsteady anisotropic modified-Helmholtz equation of time–space dependent coefficients are considered. The problems are transformed into a boundary-only integral equation which can be solved numerically using a standard boundary element method (BEM). Some examples are solved to show the validity of the analysis and examine the accuracy of the numerical method.Design/methodology/approach The 2D problems which are governed by unsteady anisotropic modified-Helmholtz equation of time–space dependent coefficients are solved using a combined BEM and Laplace transform. The time–space dependent coefficient equation is reduced to a time-dependent coefficient equation using an analytical transformation. Then, the time-dependent coefficient equation is Laplace transformed to get a constant coefficient equation, which can be written as a boundary-only integral equation. By utilizing a BEM, this integral equation is solved to find numerical solutions to the problems in the frame of the Laplace transform. These solutions are then inversely transformed numerically to obtain solutions in the original time–space frame.FindingsThe main finding of this research is the derivation of a boundary-only integral equation for the solutions of initial-boundary value problems governed by a modified-Helmholtz equation of time–space dependent coefficients for anisotropic functionally graded materials with time-dependent properties.Originality/valueThe originality of the research lies on the time dependency of properties of the functionally graded material under consideration.

PurposeThis article proposes a relaxed gradient iterative (RGI) algorithm to solve coupled Sylvester-conjugate transpose matrix equations (CSCTME) with two unknowns.Design/methodology/approachThis article proposes a RGI algorithm to solve CSCTME with two unknowns.FindingsThe introduced (RGI) algorithm is more efficient than the gradient iterative (GI) algorithm presented in Bayoumi (2014), where the author's method exhibits quick convergence behavior.Research limitations/implicationsThe introduced (RGI) algorithm is more efficient than the GI algorithm presented in Bayoumi (2014), where the author's method exhibits quick convergence behavior.Practical implicationsIn systems and control, Lyapunov matrix equations, Sylvester matrix equations and other matrix equations are commonly encountered.Social implicationsIn systems and control, Lyapunov matrix equations, Sylvester matrix equations and other matrix equations are commonly encountered.Originality/valueThis article proposes a relaxed gradient iterative (RGI) algorithm to solve coupled Sylvester conjugate transpose matrix equations (CSCTME) with two unknowns. For any initial matrices, a sufficient condition is derived to determine whether the proposed algorithm converges to the exact solution. To demonstrate the effectiveness of the suggested method and to compare it with the gradient-based iterative algorithm proposed in [6] numerical examples are provided.

PurposeAn accurate prediction of the deformation of retaining structures is critical for ensuring the stability and safety of braced deep excavations, while the high nonlinear and complex nature of the deformation makes the prediction challenging. This paper proposes an explainable boosted combining global and local feature multivariate regression (EB-GLFMR) model with high accuracy, robustness and interpretability to predict the deformation of retaining structures during braced deep excavations.Design/methodology/approachDuring the model development, the time series of deformation data is decomposed using a locally weighted scatterplot smoothing technique into trend and residual terms. The trend terms are analyzed through multiple adaptive spline regressions. The residual terms are reconstructed in phase space to extract both global and local features, which are then fed into a gradient-boosting model for prediction.FindingsThe proposed model outperforms other established approaches in terms of accuracy and robustness, as demonstrated through analyzing two cases of braced deep excavations.Research limitations/implicationsThe model is designed for the prediction of the deformation of deep excavations with stepped, chaotic and fluctuating features. Further research needs to be conducted to expand the model applicability to other time series deformation data.Practical implicationsThe model provides an efficient, robust and transparent approach to predict deformation during braced deep excavations. It serves as an effective decision support tool for engineers to ensure the stability and safety of deep excavations.Originality/valueThe model captures the global and local features of time series deformation of retaining structures and provides explicit expressions and feature importance for deformation trends and residuals, making it an efficient and transparent approach for deformation prediction.

PurposeKriging surrogate model has demonstrated a powerful ability to be applied to a variety of engineering challenges by emulating time-consuming simulations. However, when it comes to problems with high-dimensional input variables, it may be difficult to obtain a model with high accuracy and efficiency due to the curse of dimensionality. To meet this challenge, an improved high-dimensional Kriging modeling method based on maximal information coefficient (MIC) is developed in this work.Design/methodology/approachThe hyperparameter domain is first derived and the dataset of hyperparameter and likelihood function is collected by Latin Hypercube Sampling. MIC values are innovatively calculated from the dataset and used as prior knowledge for optimizing hyperparameters. Then, an auxiliary parameter is introduced to establish the relationship between MIC values and hyperparameters. Next, the hyperparameters are obtained by transforming the optimized auxiliary parameter. Finally, to further improve the modeling accuracy, a novel local optimization step is performed to discover more suitable hyperparameters.FindingsThe proposed method is then applied to five representative mathematical functions with dimensions ranging from 20 to 100 and an engineering case with 30 design variables.Originality/valueThe results show that the proposed high-dimensional Kriging modeling method can obtain more accurate results than the other three methods, and it has an acceptable modeling efficiency. Moreover, the proposed method is also suitable for high-dimensional problems with limited sample points.

PurposeThis study aimed to develop a method to calculate the mattress indentation for further estimating spinal alignment.Design/methodology/approachA universal indentation calculation model is derived based on the system theory, and the deformation characteristics of each component are analyzed by the finite element (FE) model of a partial air-spring mattress under the initial air pressure of 0.01–0.025 MPa. Finally, the calculation error of the model is verified.FindingsThe results indicate that the indentation calculation model could describe the stain of a mattress given the load and the constitutive model of each element. In addition, the FE model of a partial air-spring mattress can be used for further simulation analysis with an error of 1.47–3.42 mm. Furthermore, the deformation of the series system is mainly contributed by the air spring and the components directly in contact with it, while the top component is mainly deflection deformation. In addition, the error of the calculation model is 2.17–5.59 mm on the condition of 0.01–0.025 MPa, satisfying the engineering application. Finally, the supine spinal alignment is successfully extracted from the mattress indentation.Research limitations/implicationsThe limitation of this study is that it needs to verify the practicality of the indentation calculation model for the Bonnier spiral spring mattress. The main feature of the Bonnier spring mattress is that all springs are connected, so the mattress deflection and neighborhood effect are more significant than those of the air-spring mattress. Therefore, the applicability of the model needs to be tested. Moreover, it is worth further research to reduce the deformation error of each component.Practical implicationsAs part of the series of studies on the intelligent air-spring mattress, the indentation-based evaluation method of spinal alignment in sleep postures will be studied for hardness and intelligent regulation based on this study.Social implicationsThe results of this research are ultimately used for the intelligent adjustment of air-spring mattresses, which automatically adjusts the hardness according to the user's sleep postures and spinal alignment, thus maintaining optimal spinal biomechanics. The successful application of this result could improve the sleep health of the general public.Originality/valueBased on the series system theory, an indentation calculation model for mattresses with arbitrary structure is proposed, overcoming the dependence of parameters on materials and their combinations when fitting the Burgers model. Further, the spinal alignment in supine posture is extracted from the indentation, laying a theoretical foundation for further recognition and adjustment of the spinal alignment of the intelligent mattress.

PurposeAssembly lines are widely employed in manufacturing processes to produce final products in a flow efficiently. The simple assembly line balancing problem is a basic version of the general problem and has still attracted the attention of researchers. The type-I simple assembly line balancing problems (SALBP-I) aim to minimise the number of workstations on an assembly line by keeping the cycle time constant.Design/methodology/approachThis paper focuses on solving multi-objective SALBP-I problems by utilising an artificial bee colony based-hyper heuristic (ABC-HH) algorithm. The algorithm optimises the efficiency and idleness percentage of the assembly line and concurrently minimises the number of workstations. The proposed ABC-HH algorithm is improved by adding new modifications to each phase of the artificial bee colony framework. Parameter control and calibration are also achieved using the irace method. The proposed model has undergone testing on benchmark problems, and the results obtained have been compared with state-of-the-art algorithms.FindingsThe experimental results of the computational study on the benchmark dataset unequivocally establish the superior performance of the ABC-HH algorithm across 61 problem instances, outperforming the state-of-the-art approach.Originality/valueThis research proposes the ABC-HH algorithm with local search to solve the SALBP-I problems more efficiently.

PurposeThermal fractures initiated under cooling at the surfaces of a 2-D or 3-D structure propagate, arrest and coalesce, leading to its structural failure and material-property changes, while the same processes can happen in the rock mass between parallel hydraulic fractures filled with cold fluid, leading to enhanced fracture connectivity and permeability.Design/methodology/approachThis study used a 2-D plane strain fracture model for mixed-mode thermal fractures from two parallel cooling surfaces. Fracture propagation was governed by the theory of linear elastic fracture mechanics, while the displacement and temperature fields were discretized using the adaptive finite element method. This model was validated using two numerical benchmarks with strong fracture curvature and then used to simulate the propagation and coalescence of thermal fractures in a long rock mass.FindingsModeling results show two regimes: (1) thermal fractures from a cooling surface propagate and arrest by following the theoretical solutions of half-plane fractures before the unfractured portion decreases to 20% rock-mass width and (2) some pairs of fractures from the opposite cooling surfaces tend to eventually coalesce. The fracture coalescence time is in a power law with rock-mass width.Originality/valueThese findings are relevant to both subsurface engineering and material engineering: structure failure is a key concern in the latter, while fracture coalescence can enhance the connectivity of thermal and hydraulic fractures and thus reservoir permeability in the former.

PurposeFeature extraction from 3D datasets is a current problem. Machine learning is an important tool for classification of complex 3D datasets. Machine learning classification techniques are widely used in various fields, such as text classification, pattern recognition, medical disease analysis, etc. The aim of this study is to apply the most popular classification and regression methods to determine the best classification and regression method based on the geodesics.Design/methodology/approachThe feature vector is determined by the unit normal vector and the unit principal vector at each point of the 3D surface along with the point coordinates themselves. Moreover, different examples are compared according to the classification methods in terms of accuracy and the regression algorithms in terms of R-squared value.FindingsSeveral surface examples are analyzed for the feature vector using classification (31 methods) and regression (23 methods) machine learning algorithms. In addition, two ensemble methods XGBoost and LightGBM are used for classification and regression. Also, the scores for each surface example are compared.Originality/valueTo the best of the author’s knowledge, this is the first study to analyze datasets based on geodesics using machine learning algorithms for classification and regression.

PurposeThe curve construction on surfaces is becoming more and more important in computer-aided design (CAD), computer graphics (CG) and the other related fields. This problem is often encountered in NC machining, tool path generation, automated fiber placement and so on. However, designing curves on curved surfaces is quite different from constructing a curve in Euclidean space. Therefore, the traditional methods of curve design are not suitable for constructing a continuous curve on surface. The authors need to perform interpolation directly on surface so that the final target curve is embedded into the given surface and also meets the continuous conditions.Design/methodology/approachFirstly, adopting a series of Hermite blending functions, the authors design a space curve passing the given knots on the point-cloud surface. Then, the authors construct a class of directrixes that are adopted to determine vector fields for projection. Finally, a complete G2 continuous curve embedded in point-cloud surfaces is constructed by solving the first-order ordinary differential equations (ODEs).Findings The authors’ main contribution is to overcome the problem of constructing G1 and G2 continuous curves on point-cloud surfaces and the authors’ schemes are based on the projection moving least square (MLS) surfaces and traditional differential geometric.Originality/valueBased on the framework of projection MLS surfaces, a novel method to overcome the problem of constructing G2 continuous curves on point-cloud surfaces is proposed.

Purpose3D steady heat conduction analysis considering heat source is conducted on the fundamental of the fast multipole method (FMM)-accelerated line integration boundary element method (LIBEM).Design/methodology/approachDue to considering the heat source, domain integral is generated in the traditional heat conduction boundary integral equation (BIE), which will counteract the well-known merit of the BEM, namely, boundary-only discretization. To avoid volume discretization, the enhanced BEM, the LIBEM with dimension reduction property is introduced to transfer the domain integral into line integrals. Besides, owing to the unsatisfactory performance of the LIBEM when it comes to large-scale structures requiring massive computation, the FMM-accelerated LIBEM (FM-LIBEM) is proposed to improve the computation efficiency further.FindingsAssuming N and M are the numbers of nodes and integral lines, respectively, the FM-LIBEM can reduce the time complexity from O(NM) to about O(N+ M), and a full discussion and verification of the advantage are done based on numerical examples under heat conduction.Originality/value(1) The LIBEM is applied to 3D heat conduction analysis with heat source. (2) The domain integrals can be transformed into boundary integrals with straight line integrals by the LIM. (3) A FM-LIBEM is proposed and can reduce the time complexity from O(NM) to O(N+ M). (4) The FM-LIBEM with high computational efficiency is exerted to solve 3D heat conduction analysis with heat source in massive computation successfully.