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This paper aims to develop a novel numerical method based on bi-cubic B-spline functions and alternating direction (ADI) scheme to study numerical solutions of advection diffusion equation. The method captures important properties in the advection of fluids very efficiently. C.P.U. time has been shown to be very less as compared with other numerical schemes. Problems of great practical importance have been simulated through the proposed numerical scheme to test the efficiency and applicability of method.

A bi-cubic B-spline ADI method has been proposed to capture many complex properties in the advection of fluids.

Bi-cubic B-spline ADI technique to investigate numerical solutions of partial differential equations has been studied. Presented numerical procedure has been applied to important two-dimensional advection diffusion equations. Computed results are efficient and reliable, have been depicted by graphs and several contour forms and confirm the accuracy of the applied technique. Stability analysis has been performed by von Neumann method and the proposed method is shown to satisfy stability criteria unconditionally. In future, the authors aim to extend this study by applying more complex partial differential equations. Though the structure of the method seems to be little complex, the method has the advantage of using small processing time. Consequently, the method may be used to find solutions at higher time levels also.

ADI technique has never been applied with bi-cubic B-spline functions for numerical solutions of partial differential equations.

This paper aims to propose a fast and robust 3D point set registration method for pose estimation of assembly features with few distinctive local features in the manufacturing process.

The distance between the two 3D objects is analytically approximated by the implicit representation of the target model. Specifically, the implicit B-spline surface is adopted as an interface to derive the distance metric. With the distance metric, the point set registration problem is formulated into an unconstrained nonlinear least-squares optimization problem. Simulated annealing nested Gauss-Newton method is designed to solve the non-convex problem. This integration of gradient-based optimization and heuristic searching strategy guarantees both global robustness and sufficient efficiency.

The proposed method improves the registration efficiency while maintaining high accuracy compared with several commonly used approaches. Convergence can be guaranteed even with critical initial poses or in partial overlapping conditions. The multiple flanges pose estimation experiment validates the effectiveness of the proposed method in real-world applications.

The proposed registration method is much more efficient because no feature estimation or point-wise correspondences update are performed. At each iteration of the Gauss–Newton optimization, the poses are updated in a singularity-free format without taking the derivatives of a bunch of scalar trigonometric functions. The advantage of the simulated annealing searching strategy is combined to improve global robustness. The implementation is relatively straightforward, which can be easily integrated to realize automatic pose estimation to guide the assembly process.

The purpose of this paper is to present a new approach of meshless local B-spline based finite difference (FD) method for solving two dimensional transient heat conduction problems.

In the present method, any governing equations are discretized by B-spline approximation which is implemented in the spirit of FD technique using a local B-spline collocation scheme. The key aspect of the method is that any derivative is stated as neighbouring nodal values based on B-spline interpolants. The set of neighbouring nodes are allowed to be randomly distributed thus enhanced flexibility in the numerical simulation can be obtained. The method requires no mesh connectivity at all for either field variable approximation or integration. Time integration is performed by using the Crank-Nicolson implicit time stepping technique.

Several heat conduction problems in complex domains which represent for extended surfaces in industrial applications are examined to demonstrate the effectiveness of the present approach. Comparison of the obtained results with solutions from other numerical method available in literature is given. Excellent agreement with reference numerical method has been found.

The method is presented for 2D problems. Nevertheless, it would be also applicable for 3D problems.

A transient two dimensional heat conduction in complex domains which represent for extended surfaces in industrial applications is presented.

The presented new meshless local method is simple and accurate, while it is also suitable for analysis in domains of arbitrary geometries.

The purpose of this paper is to investigate the numerical solutions of the Burgers' and modified Burgers' equation using sextic B‐spline collocation method.

Crank‐Nicolson central differencing scheme has been used for the time integration and sextic B‐spline functions have been used for the space integration to the modified and time splitted modified Burgers' equation.

It has been found that the proposed method is unconditionally stable and obtained results are consistent with some earlier published studies.

Sextic B‐spline collocation method for the Burgers' and modified Burgers' equation is given.

This paper aims to propose a robust method for non-rigid point set registration, using the Gaussian mixture model and accommodating non-rigid transformations. The posterior probabilities of the mixture model are determined through the proposed integrated feature divergence.

The method involves an alternating two-step framework, comprising correspondence estimation and subsequent transformation updating. For correspondence estimation, integrated feature divergences including both global and local features, are coupled with deterministic annealing to address the non-convexity problem of registration. For transformation updating, the expectation-maximization iteration scheme is introduced to iteratively refine correspondence and transformation estimation until convergence.

The experiments confirm that the proposed registration approach exhibits remarkable robustness on deformation, noise, outliers and occlusion for both 2D and 3D point clouds. Furthermore, the proposed method outperforms existing analogous algorithms in terms of time complexity. Application of stabilizing and securing intermodal containers loaded on ships is performed. The results demonstrate that the proposed registration framework exhibits excellent adaptability for real-scan point clouds, and achieves comparatively superior alignments in a shorter time.

The integrated feature divergence, involving both global and local information of points, is proven to be an effective indicator for measuring the reliability of point correspondences. This inclusion prevents premature convergence, resulting in more robust registration results for our proposed method. Simultaneously, the total operating time is reduced due to a lower number of iterations.

This paper aims to propose a new boundary condition and a web-spline basis of finite element space approximation to remedy the problems of constraints due to homogeneous and non-homogeneous; Dirichlet boundary conditions. This paper considered the two-dimensional linear elasticity equation of Navier–Lamé with the condition CAB. The latter allows to have a total insertion of the essential boundary condition in the linear system obtained; without using a numerical method as Lagrange multiplier. This study have developed mixed finite element; method using the B-splines Web-spline space. These provide an exact implementation of the homogeneous; Dirichlet boundary conditions, which removes the constraints caused by the standard; conditions. This paper showed the existence and the uniqueness of the weak solution, as well as the convergence of the numerical solution for the quadratic case are proved. The weighted extended B-spline; approach have become a much more workmanlike solution.

In this paper, this study used the implementation of weighted finite element methods to solve the Navier–Lamé system with a new boundary condition CA, B (Koubaiti et al., 2020), that generalises the well-known basis, especially the Dirichlet and the Neumann conditions. The novel proposed boundary condition permits to use a single Matlab code, which summarises all kind of boundary conditions encountered in the system. By using this model is possible to save time and programming recourses while reap several programs in a single directory.

The results have shown that the Web-spline-based quadratic-linear finite elements satisfy the inf–sup condition, which is necessary for existence and uniqueness of the solution. It was demonstrated by the existence of the discrete solution. A full convergence was established using the numerical solution for the quadratic case. Due to limited regularity of the Navier–Lamé problem, it will not change by increasing the degree of the Web-spline. The computed relative errors and their rates indicate that they are of order 1/H. Thus, it was provided their theoretical validity for the numerical solution stability. The advantage of this problem that uses the CA, B boundary condition is associated to reduce Matlab programming complexity.

The mixed finite element method is a robust technique to solve difficult challenges from engineering and physical sciences using the partial differential equations. Some of the important applications include structural mechanics, fluid flow, thermodynamics and electromagnetic fields (Zienkiewicz and Taylor, 2000) that are mainly based on the approximation of Lagrange. However, this type of approximation has experienced a great restriction in the level of domain modelling, especially in the case of complicated boundaries such as that in the form of curvilinear graphs. Recently, the research community tried to develop a new way of approximation based on the so-called B-spline that seems to have superior results in solving the engineering problems.

The purpose of this paper is to present a new method, namely, “Re-modified quintic B-spline collocation method” to solve the Kuramoto–Sivashinsky (KS) type equations. In this method, re-modified quintic B-spline functions and the Crank–Nicolson formulation is used for space and time integration, respectively. Five examples are considered to test out the efficiency and accuracy of the method. The main objective is to develop a method which gives more accurate results and reduces the computational cost so that the authors require less memory storage.

A new collocation technique is developed to solve the KS type equations. In this technique, quintic B-spline basis functions are re-modified and used to integrate the space derivatives while time derivative is discretized by using Crank–Nicolson formulation. The discretization yields systems of linear equations, which are solved by using Gauss elimination method with partial pivoting.

Five examples are considered to test out the efficiency and accuracy of the method. Finally, the present study summarizes the following outcomes: first, the computational cost of the proposed method is the less than quintic B-spline collocation method. Second, the present method produces better results than those obtained by Lattice Boltzmann method (Lai and Ma, 2009), quintic B-spline collocation method (Mittal and Arora, 2010), quintic B-spline differential quadrature method (DQM) (Mittal and Dahiya, 2017), extended modified cubic B-spline DQM (Tamsir et al., 2016) and modified cubic B-splines collocation method (Mittal and Jain, 2012).

The method presented in this paper is new to best of the authors’ knowledge. This work is the original work of authors and the manuscript is not submitted anywhere else for publication.

The purpose of this study is to construct quartic trigonometric tension (QTT) B-spline collocation algorithms for the numerical solutions of the Coupled Burgers’ equation.

The finite elements method (FEM) is a numerical method for obtaining an approximate solution of partial differential equations (PDEs). The development of high-speed computers enables to development FEM to solve PDEs on both complex domain and complicated boundary conditions. It also provides higher-order approximation which consists of a vector of coefficients multiplied by a set of basis functions. FEM with the B-splines is efficient due both to giving a smaller system of algebraic equations that has lower computational complexity and providing higher-order continuous approximation depending on using the B-splines of high degree.

The result of the test problems indicates the reliability of the method to get solutions to the CBE. QTT B-spline collocation approach has convergence order 3 in space and order 1 in time. So that nonpolynomial splines provide smooth solutions during the run of the program.

There are few numerical methods build-up using the trigonometric tension spline for solving differential equations. The tension B-spline collocation method is used for finding the solution of Coupled Burgers’ equation.

This paper aims to present a nonlocal gradient plasticity damage model to demonstrate the crack pattern of a body, in an elastic and plastic state, in terms of damage law. The main objective of this paper is to reconsider the nonlocal theory by including the material in-homogeneity caused by damage and plasticity. The nonlocal nature of the strain field provides a regularization to overcome the analytical and computational problems induced by softening constitutive laws. Such an approach requires C1 continuous approximation. This is achieved by using an isogeometric approximation (IGA). Numerical examples in one and two dimensions are presented.

In this work, the authors propose a nonlocal elastic plastic damage model. The nonlocal nature of the strain field provides a regularization to overcome the analytical and computational problems induced by softening constitutive laws. An additive decomposition of strains in to elastic and inelastic or plastic part is considered. To obtain stable damage, a higher gradient order is considered for an integral equation, which is obtained by the Taylor series expansion of the local inelastic strain around the point under consideration. The higher-order continuity of nonuniform rational B-splines (NURBS) functions used in isogeometric analysis are adopted here to implement in a numerical scheme. To demonstrate the validity of the proposed model, numerical examples in one and two dimensions are presented.

The proposed nonlocal elastic plastic damage model is able to predict the damage in an accurate manner. The numerical results are mesh independent. The nonlocal terms add a regularization to the model especially for strain softening type of materials. The consideration of nonlocality in inelastic strains is more meaningful to the physics of damage. The use of IGA framework and NURBS basis functions add to the nonlocal nature in approximations of the field variables.

The method can be extended to 3D. The model does not consider the effect of temperature and the dissipation of energy due to temperature. The method needs to be implemented for more real practical problems and compare with experimental work. This is an ongoing work.

The nonlocal models are suitable for predicting damage in quasi brittle materials. The use of elastic plastic theories allows to capture the inelastic deformations more accurately.

The nonlocal models are suitable for predicting damage in quasi brittle materials. The use of elastic plastic theories allows to capture the inelastic deformations more accurately.

The present work includes the formulation and implementation of a nonlocal damage plasticity model using an isogeometric discretization, which is the novel contribution of this paper. An implicit gradient enhancement is considered to the inelastic strain. During inelastic deformations, the proposed strain tensor partitioning allows the use of a distinct potential surface and distinct failure criterion for both damage and plasticity models. The use of NURBS basis functions adds to more nonlocality in the approximation.

The purpose of this paper is to illustrate how the numerical solution of the Burgers' equation is obtained using the methods of cubic B‐spline collocation and quadratic B‐spline Galerkin over the geometrically graded mesh.

The spatial domain is partitioned into geometrically graded mesh. The finite element methods are constructed within the Galerkin and collocation methods using an expansion of the quadratic and cubic B‐splines as an approximate function, respectively, over this mesh.

When the higher errors are observed at near boundaries for shock‐like and travelling wave solutions of the Burgers' equation, accuracy of the defined methods increase by using finer mesh at near this boundary.

Over the geometrically graded mesh definitions of the quadratic B‐spline Galerkin and cubic B‐spline collocation are given.