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This study aims to introduce the hybrid finite element (FE) – meshfree method and multiscale variational principle into the traditional mixed FE formulation, leading to a stable mixed formulation for incompressible linear elasticity which circumvents the need to satisfy inf-sup condition.

Using the hybrid FE–meshfree method, the displacement and pressure are interpolated conveniently with the same order so that a continuous pressure field can be obtained with low-order elements. The multiscale variational principle is then introduced into the Galerkin form to obtain stable and convergent results.

The present method is capable of overcoming volume locking and does not exhibit unphysical oscillations near the incompressible limit. Moreover, there are no extra unknowns introduced in the present method because the fine-scale unknowns are eliminated using the static condensation technique, and there is no need to evaluate any user-defined stability parameter as the classical stabilization methods do. The shape functions constructed in the present model possess continuous derivatives at nodes, which gives a continuous and more precise stress field with no need of an additional smooth process. The shape functions in the present model also possess the Kronecker delta property, so that it is convenient to impose essential boundary conditions.

The proposed model can be implemented easily. Its convergence rates and accuracy in displacement, energy and pressure are even comparable to those of second-order mixed elements.

To develop an overview of generalized scales based on pansystems‐relative quantification.

This is a discussion paper exploring the key issues surrounding generalized measures.

The concrete contents of the study include generalized measure views, dimension theory, concepts, logic, theories, Einstein's relativity, quality‐quantity‐degree, methodology of physics, theorems in pansystems mathematics and physics explained within the framework of pan‐scale transformations.

Provides an overview of generalized scales based on pansystems‐relative quantification.

In this paper, the influence of fractal interface geometry to the evolution of the friction mechanism is studied. The paper is based on fractal approaches for the modeling of the multiscale self‐affine topography of these interfaces. More specifically, these approaches are based on scale‐independent parameters such as the fractal dimension. Here, friction between rough surfaces is assumed to be the result of the gradual plastification of the fractal interface asperities. In order to study the resulting highly nonlinear problem a variational formulation is used in order to describe contact between the interfaces. The numerical method used here leads to the successive solution of quadratic optimization problems. Finally, structures with different fractal interfaces are analyzed in order to obtain results for the relation between the fractal dimension and the overall response of the structures.

This paper aims to introduce a multiscale computational method for structural failure analysis with inheriting simulation of moving trans-scale boundary (MTB). This method is motivated from the error in domain bridging caused by cross-scale damage evolution, which is common in structural failure induced by damage accumulation.

Within the method, vulnerable regions with high stress level are described by continuum damage mechanics, while elastic structural theory is sufficient for the rest, dividing the structural model into two scale domains. The two domains are bridged to generate mixed dimensional finite element equation of the whole system. Inheriting simulation is developed to make the computation of MTB sustainable.

Numerical tests of a notched three-point bending beam and a steel frame show that this MTB method can improve efficiency and ensure accuracy while capturing the effect of material damage on deterioration of components and structure.

The proposed MTB method with inheriting simulation is an extension of multiscale simulation to structural failure analysis. Most importantly, it can deal with cross-scale damage evolution and improve computation efficiency significantly.

The purpose of this paper is to describe a variational multiscale finite element approximation for the incompressible Navier‐Stokes equations using the Boussinesq approximation to model thermal coupling.

The main feature of the formulation, in contrast to other stabilized methods, is that the subscales are considered as transient and orthogonal to the finite element space. These subscales are solution of a differential equation in time that needs to be integrated. Likewise, the effect of the subscales is kept, both in the nonlinear convective terms of the momentum and temperature equations and, if required, in the thermal coupling term of the momentum equation.

This strategy allows the approaching of the problem of dealing with thermal turbulence from a strictly numerical point of view and discussion important issues, such as the relationship between the turbulent mechanical dissipation and the turbulent thermal dissipation.

The treatment of thermal turbulence from a strictly numerical point of view is the main originality of the work.

The purpose of this paper is to present a finite element approximation of the low Mach number equations coupled with radiative equations to account for radiative heat transfer. For high-temperature flows this coupling can have strong effects on the temperature and velocity fields.

The basic numerical formulation has been proposed in previous works. It is based on the variational multiscale (VMS) concept in which the unknowns of the problem are divided into resolved and subgrid parts which are modeled to consider their effect into the former. The aim of the present paper is to extend this modeling to the case in which the low Mach number equations are coupled with radiation, also introducing the concept of subgrid scales for the radiation equations.

As in the non-radiative case, an important improvement in the accuracy of the numerical scheme is observed when the nonlinear effects of the subgrid scales are taken into account. Besides it is possible to show global conservation of thermal energy.

The original contribution of the work is the proposal of keeping the VMS splitting into the nonlinear coupling between the low Mach number and the radiative transport equations, its numerical evaluation and the description of its properties.

This paper aims to develop a robust set of advanced numerical tools to simulate multiphase flows under the superimposition of external uniform magnetic fields.

The flow has been simulated in a fully Eulerian framework by a {\it variational multi-scale} method, which allows to take into account the small-scale turbulence without explicitly model it. The multi-fluid problem has been solved through the convectively re-initialized level-set method to robustly deal with high density and viscosity ratio between the phases and the surface tension has been modelled implicitly in the level-set framework. The interaction with the magnetic field has been modelled through the classic induction equation for 2D problems and the time step computation is based on the electromagnetic interaction to guarantee convergence of the method. Anisotropic mesh adaptation is then used to adapt the mesh to the main problem’s variables and to reach good accuracy with a small number of degrees of freedom. Finally, the variational multiscale method leads to a natural stabilization of the finite elements algorithm, preventing numerical spurious oscillations in the solution of Navier–Stokes equations (fluid mechanics) and the transport equation (level-set convection).

The methodology has been validated, and it is shown to produce accurate results also with a low number of degrees of freedom. The physical effect of the external magnetic field on the multiphase flow has been analysed.

The dam-break benchmark case has been extended to include magnetically constrained flows.

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.

This paper aims to solve the problem that strong noise interference seriously affects the direction of arrival (DOA) estimation in complex underwater acoustic environment. In this paper, a combined noise reduction algorithm and micro-electro-mechanical system (MEMS) vector hydrophone DOA estimation algorithm based on singular value decomposition (SVD), variational mode decomposition (VMD) and wavelet threshold denoising (WTD) is proposed.

Firstly, the parameters of VMD are determined by SVD, and the VMD method can decompose the signal into multiple intrinsic mode functions (IMFs). Secondly, the effective IMF component is determined according to the correlation coefficient criterion and the IMF less than the threshold is processed by WTD. Then, reconstruction is carried out to achieve the purpose of denoising and calibration baseline drift. Finally, DOA estimation is achieved by the combined directional algorithm of preprocessed signal.

Simulation and field experiments results show that the algorithm has good noise reduction and baseline drift correction effects for nonstationary underwater signals, and high-precision azimuth estimation is realized.

This research provides the basis for MEMS hydrophone detection and positioning and has great engineering significance in underwater detection system.