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The purpose of this paper is to develop an effective numerical algorithm for a gas-melt two-phase flow and use it to simulate a polymer melt filling process. Moreover, the suggested algorithm can deal with the moving interface and discontinuities of unknowns across the interface.

The algebraic sub-grid scales-variational multi-scale (ASGS-VMS) finite element method is used to solve the polymer melt filling process. Meanwhile, the time is discretized using the Crank–Nicolson-based split fractional step algorithm to reduce the computational time. The improved level set method is used to capture the melt front interface, and the related equations are discretized by the second-order Taylor–Galerkin scheme in space and the third-order total variation diminishing Runge–Kutta scheme in time.

The numerical method is validated by the benchmark problem. Moreover, the viscoelastic polymer melt filling process is investigated in a rectangular cavity. The front interface, pressure field and flow-induced stresses of polymer melt during the filling process are predicted. Overall, this paper presents a VMS method for polymer injection molding. The present numerical method is extremely suitable for two free surface problems.

For the first time ever, the ASGS-VMS finite element method is performed for the two-phase flow of polymer melt filling process, and an effective numerical method is designed to catch the moving surface.

The purpose of this paper is an accurate computation of eddy currents in laminated media with minimal computer resources.

Modeling each laminate of the laminated core of electrical devices requires prohibitively many finite elements (FEs). To overcome this restriction a higher order multi-scale FE method with the magnetic vector potential

has been developed to cope with 3D problems considering edge effects.

The multi-scale FE approach facilitates an accurate simulation of the eddy current losses with minimal computer resources. Numerical simulations demonstrate a remarkable accuracy and low computational costs. The effect of regularization on the results is shown.

The eddy current losses are of great interest in the design of electrical devices with laminated cores.

The multi-scale FE approach takes also into account of the edge effects in 3D.

This paper aims to present a computationally efficient finite element model for the simulation of isothermal immiscible two-phase flow in a rigid porous media with a particular application to CO₂ sequestration in underground formations. Focus is placed on developing a numerical procedure, which is effectively mesh-independent and suitable to problems at regional scales.

The averaging theory is utilized to describe the governing equations of the involved unsaturated multiphase flow. The level-set (LS) method and the extended finite element method (XFEM) are utilized to simulate flow of the CO₂ plume. The LS is employed to trace the plume front. A streamline upwind Petrov-Galerkin method is adopted to stabilize possible occurrence of spurious oscillations due to advection. The XFEM is utilized to model the high gradient in the saturation field front, where the LS function is used for enhancing the weighting and the shape functions.

The capability of the proposed model and its features are evaluated by numerical examples, demonstrating its accuracy, stability and convergence, as well as its advantages over standard and upwind techniques. The study showed that a good combination between a mathematical model and a numerical model enables the simulation of complicated processes occurring in complicated and large geometry using minimal computational efforts.

A new computational model for two-phase flow in porous media is introduced with basic requirements for accuracy, stability, and convergence, which are met using relatively coarse meshes.

This study highlights the demand for low-cost and high accuracy products through the design and development of new 3D printing technologies. Besides, significant progress has been made in this field. A comparative study helps to understand the latest development in materials and future prospect of this technology.

Nevertheless, a large amount of progress still remains to be made. While some of the works have focused on the performances of the materials, the rest have focused on the development of new methods and techniques in additive manufacturing.

This paper critically evaluates the current 3D printing technologies, including the development and optimizations made to the printing methods, as well as the printed objects. Meanwhile, previous developments in this area and contributions to the modern trend in manufacturing technology are summarized briefly.

The paper can be summarized in three sections. Firstly, the existing printing methods along with the frequently used printing materials, as well as the processing parameters, and the factors which influence the quality and mechanical performances of the printed objects are discussed. Secondly, the optimization techniques, such as topology, shape, structure and mechanical property, are described. Thirdly, the latest development and applications of additive manufacturing are depicted, and the scope of future research in the relevant area is put forward.

The purpose of this paper is to investigate the feasibility of solving turbulent flows based on smoothed finite element method (S-FEM). Then, the differences between S-FEM and finite element method (FEM) in dealing with turbulent flows are compared.

The stabilization scheme, the streamline-upwind/Petrov-Galerkin stabilization is coupled with stabilized pressure gradient projection in the fractional step framework. The Reynolds-averaged Navier-Stokes equations with standard k-epsilon model are selected to solve turbulent flows based on S-FEM and FEM. Standard wall functions are applied to predict boundary layer profiles.

This paper explores a completely new application of S-FEM on turbulent flows. The adopted stabilization scheme presents a good performance on stabilizing the flows, especially for very high Reynolds numbers flows. An advantage of S-FEM is found in applying wall functions comparing with FEM. The differences between S-FEM and FEM have been investigated.

The research in this work is limited to the two-dimensional incompressible turbulent flow.

The verification and validation of a new combination are conducted by several numerical examples. The new combination could be used to deal with more complicated turbulent flows.

The applications of the new combination to study basic and complex turbulent flow are also presented, which demonstrates its potential to solve more turbulent flows in nature and engineering.

This work carries out a great extension of S-FEM in simulations of fluid dynamics. The new combination is verified to be very effective in handling turbulent flows. The performances of S-FEM and FEM on turbulent flows were analyzed by several numerical examples. Superior results were found compared with existing results and experiments. Meanwhile, S-FEM has an advantage of accuracy in predicting boundary layer profile.

The variational principle views a complex problem in an energy way, it gives good physical understanding of an iteration method, and the variational-based numerical methods always have a conservation scheme with a fast convergent rate. The purpose of this paper is to establish a variational principle for a fractal nano/microelectromechanical (N/MEMS) system.

This paper begins with an approximate variational principle in literature for the studied problem, and a genuine variational principle is obtained by the semi-inverse method.

The semi-inverse method is a good mathematical tool to the search for a genuine fractal variational formulation for the N/MEMS system.

The established variational principle can be used for both analytical and numerical analyses of the N/MEMS systems, and it can be extended to some more complex cases.

The variational principle can be used for variational-based finite element methods and energy-based analytical methods.

The new and genuine variational principle is obtained. This paper discovers the missing piece of the puzzle for the establishment of a variational principle from governing equations for a complex problem by the semi-inverse method. The new variational theory opens a new direction in fractal MEMS systems.

The purpose of this paper is to present an efficient return mapping algorithm for elastoplastic constitutive problems of ductile metals with an exact closed form solution of the local constitutive problem in the small strain regime. A Newton Raphson iterative method is adopted for the solution of the boundary value problem.

An efficient return mapping algorithm is illustrated which is based on an elastic predictor and a plastic corrector scheme resulting in an implicit and accurate numerical integration method. Nonlinear kinematic hardening rules and linear isotropic hardening rules are used to describe the components of the hardening variables. In the adopted algorithmic approach the solution of the local constitutive equations reduces to only one straightforward nonlinear scalar equation.

The presented algorithmic scheme naturally leads to a particularly simple form of the nonlinear scalar equation which ultimately scales down to an algebraic (polynomial) equation with a single variable. The straightforwardness of the present approach allows to find the analytical solution of the algebraic equation in a closed form. Further, the consistent tangent operator is derived as associated with the proposed algorithmic scheme and it is shown that the proposed computational procedure ensures a quadratic rate of asymptotic convergence when used with a Newton Raphson iterative method for the global solution procedure.

In the present approach the solution of the algebraic nonlinear equation is found in a closed form and accordingly no iterative method is required to solve the problem of the local constitutive equations. The computational procedure ensures a quadratic rate of asymptotic convergence for the global solution procedure typical of computationally efficient solution schemes. In the paper it is shown that the proposed algorithmic scheme provides an efficient and robust computational solution procedure for elastoplasticity boundary value problems. Numerical examples and computational results are reported which illustrate the effectiveness and robustness of the adopted integration algorithm for the finite element analysis of elastoplastic structures also under elaborate loading conditions.

This paper aims to review some effective methods for fully fourth-order nonlinear integral boundary value problems with fractal derivatives.

Boundary value problems arise everywhere in engineering, hence two-scale thermodynamics and fractal calculus have been introduced. Some analytical methods are reviewed, mainly including the variational iteration method, the Ritz method, the homotopy perturbation method, the variational principle and the Taylor series method. An example is given to show the simple solution process and the high accuracy of the solution.

An elemental and heuristic explanation of fractal calculus is given, and the main solution process and merits of each reviewed method are elucidated. The fractal boundary value problem in a fractal space can be approximately converted into a classical one by the two-scale transform.

This paper can be served as a paradigm for various practical applications.

The purpose of this paper is to suggest a general numerical algorithm for nonlinear problems by the variational iteration method (VIM).

Firstly, the Laplace transform technique is used to reconstruct the variational iteration algorithm-II. Secondly, its convergence is strictly proved. Thirdly, the numerical steps for the algorithm is given. Finally, some examples are given to show the solution process and the effectiveness of the method.

No variational theory is needed to construct the numerical algorithm, and the incorporation of the Laplace method into the VIM makes the solution process much simpler.

A universal iteration formulation is suggested for nonlinear problems. The VIM cleans up the numerical road to differential equations.

The purpose of this paper is to apply the variational multi-scale framework to the finite element approximation of the compressible Navier-Stokes equations written in conservation form. Even though this formulation is relatively well known, some particular features that have been applied with great success in other flow problems are incorporated.

The orthogonal subgrid scales, the non-linear tracking of these subscales, and their time evolution are applied. Moreover, a systematic way to design the matrix of algorithmic parameters from the perspective of a Fourier analysis is given, and the adjoint of the non-linear operator including the volumetric part of the convective term is defined. Because the subgrid stabilization method works in the streamline direction, an anisotropic shock capturing method that keeps the diffusion unaltered in the direction of the streamlines, but modifies the crosswind diffusion is implemented. The artificial shock capturing diffusivity is calculated by using the orthogonal projection onto the finite element space of the gradient of the solution, instead of the common residual definition. Temporal derivatives are integrated in an explicit fashion.

Subsonic and supersonic numerical experiments show that including the orthogonal, dynamic, and the non-linear subscales improve the accuracy of the compressible formulation. The non-linearity introduced by the anisotropic shock capturing method has less effect in the convergence behavior to the steady state.

A complete investigation of the stabilized formulation of the compressible problem is addressed.