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The purpose of this paper is to demonstrate the eligibility of the Weighted Residual Methods (WRMs) applied to magneto hydro dynamic (MHD) nanofluid flow in divergent and convergent channels. Selecting the most appropriate method among the WRMs and discussing about Jeffery-Hamel flow's treatment in divergent and convergent channels are the other important purposes of the present research.

Three analytical methods (Collocation, Galerkin and Least Square Method) and numerical method have been applied to solve the governing equations. The reliability of the methods is also approved by a comparison made between the fourth-order Runge-Kutta numerical method.

The obtained solutions revealed that WRMs can be simple, powerful and efficient techniques for finding analytical solutions in science and engineering non-linear differential equations.

It could be considered as a first endeavor to use the solution of the MHD nanofluid flow in divergent and convergent channels using these kinds of analytical methods along with the numerical approach.

This work is concerned with computational modelling of viscoplastic fluids. The flows considered are assumed to be incompressible, while the viscoplastic laws are obtained by incorporating a yield stress below which the fluid is assumed to remain non‐deformable. The Bingham fluid is chosen as a model problem and is considered in detail in the text. The finite element formulation adopted in this work is based on a version of the stabilised finite element method, known as the Galerkin/least‐squares method, originally developed by Hughes and co‐workers. This methodology allows use of low and equal order interpolation of the pressure and velocity fields, thus providing an efficient finite element framework. The Newton‐Raphson method has been chosen for solution of the incremental non‐linear problem arising through the temporal discretisation of the evolution problem. Numerical examples are provided to illustrate the main features of the described methodology.

This article aims to present a direct solution to handle linear constraints in finite element (FE) analysis without penalties or the Lagrange multipliers introduced.

First, the system of linear equations corresponding to the linear constraints is solved for the leading variables in terms of the free variables and the constants. Then, the reduced system of equilibrium equations with respect to the free variables is derived from the finite-dimensional virtual work equation. Finally, the algorithm is designed.

The proposed procedure is promising in three typical cases: (1) to enforce displacement constraints in any direction; (2) to implement local refinements by allowing hanging nodes from element subdivision and (3) to treat non-matching grids of distinct parts of the problem domain. The procedure is general and suitable for 3D non-linear analyses.

The algorithm is fitted only to the Galerkin-based numerical methods.

The proposed procedure does not need Lagrange multipliers or penalties. The tangential stiffness matrix of the reduced system of equilibrium equations reserves positive definiteness and symmetry. Besides, many contemporary Galerkin-based numerical methods need to tackle the enforcement of the essential conditions, whose weak forms reduce to linear constraints. As a result, the proposed procedure is quite promising.

The purpose of this paper is to investigate a novel stabilization scheme to handle convection and pressure oscillation in the process of solving incompressible laminar flows by finite element method (FEM).

The semi-implicit stabilization scheme, characteristic-based polynomial pressure projection (CBP3) consists of the Characteristic-Galerkin method and polynomial pressure projection. Theoretically, the proposed scheme works for any type of element using equal-order approximation for velocity and pressure. In this work, linear 3-node triangular and 4-node tetrahedral elements are the focus, which are the simplest but most difficult elements for pressure stabilizations.

The present paper proposes a new scheme, which can stabilize FEM solution for flows of both low and relatively high Reynolds numbers. And the influence of stabilization parameters of the CBP3 scheme has also been investigated.

The research in this work is limited to the laminar incompressible flow.

The verification and validation of the CBP3 scheme are conducted by several 2 D and 3 D numerical examples. The scheme could be used to deal with more practical fluid problems.

The application of scheme to study complex hemodynamics of patient-specific abdominal aortic aneurysm is also presented, which demonstrates its potential to solve bio-flows.

The paper simulated 2 D and 3 D numerical examples with superior results compared to existing results and experiments. The novel CBP3 scheme is verified to be very effective in handling convection and pressure oscillation.

The purpose of this paper is to assess the performance of the stabilised moving least squares (MLS) scheme in the meshless local Petrov–Galerkin (MLPG) method for heat conduction method.

In the current work, the authors extend the stabilised MLS approach to the MLPG method for heat conduction problem. Its performance has been compared with the MLPG method based on the standard MLS and local coordinate MLS. The patch tests of MLS and modified MLS schemes have been presented along with the one- and two-dimensional examples for MLPG method of the heat conduction problem.

In the stabilised MLS, the condition number of moment matrix is independent of the nodal spacing and it is nearly constant in the global domain for all grid sizes. The shifted polynomials based MLS and stabilised MLS approaches are more robust than the standard MLS scheme in the MLPG method analysis of heat conduction problems.

The MLPG method based on the stabilised MLS scheme.

Using the Heaviside operator, a single partial differential equation is obtained for the space‐time variation of the pore pressure in two adjacent soil layers undergoing simultaneous consolidation. A closed form expression for the solution to the problem is given as a generalized Fourier series. The coordinate functions of the series are the eigenfunctions of the composite medium obtained computationally through the application of the extended Galerkin method.

The purpose of this paper is to study the resonance failure sensitivity analysis of straight-tapered assembled pipe conveying nonuniform axial fluid by an active learning Kriging (ALK) method.

In this study, first, the motion equation of straight-tapered assembled pipe conveying nonuniform fluid is built. Second, the Galerkin method is used for calculating the natural frequency of assembled pipe conveying nonuniform fluid. Third, the ALK method based on expected risk function (ERF) is used to calculate the resonance failure probability and moment independent global sensitivity analysis.

The findings of this paper highlight that the eigenfrequency and critical velocity of uniform fluid-conveying pipe are less than the reality and the error is biggest in first-order natural frequency. The importance ranking of input variables affecting the resonance failure can be obtained. The importance ranking is different for a different velocity and mode number. By reducing the uncertainty of variables with a high index, the resonance failure probability can be reduced maximally.

There are no experiments on the eigenfrequency and critical velocity. There is no experiments about natural frequency and critical velocity of straight tapered assembled pipe to verify the theory in this paper.

The originality of this paper lies as follows: the motion equation of straight-tapered pipe conveying nonuniform fluid is first obtained. The eigenfrequency of nonuniform fluid and uniform fluid inside the assembled pipe are compared. The resonance reliability analysis of straight-tapered assembled pipe is first proposed. From the results, it is observed that the resonance failure probability can be reduced efficiently.

Gives a bibliographical review of the error estimates and adaptive finite element methods from the theoretical as well as the application point of view. The bibliography at the end contains 2,177 references to papers, conference proceedings and theses/dissertations dealing with the subjects that were published in 1990‐2000.

This paper aims to provide a comprehensive review of uncertainty quantification methods supported by evidence-based comparison studies. Uncertainties are widely encountered in engineering practice, arising from such diverse sources as heterogeneity of materials, variability in measurement, lack of data and ambiguity in knowledge. Academia and industries have long been researching for uncertainty quantification (UQ) methods to quantitatively account for the effects of various input uncertainties on the system response. Despite the rich literature of relevant research, UQ is not an easy subject for novice researchers/practitioners, where many different methods and techniques coexist with inconsistent input/output requirements and analysis schemes.

This confusing status significantly hampers the research progress and practical application of UQ methods in engineering. In the context of engineering analysis, the research efforts of UQ are most focused in two largely separate research fields: structural reliability analysis (SRA) and stochastic finite element method (SFEM). This paper provides a state-of-the-art review of SRA and SFEM, covering both technology and application aspects. Moreover, unlike standard survey papers that focus primarily on description and explanation, a thorough and rigorous comparative study is performed to test all UQ methods reviewed in the paper on a common set of reprehensive examples.

Over 20 uncertainty quantification methods in the fields of structural reliability analysis and stochastic finite element methods are reviewed and rigorously tested on carefully designed numerical examples. They include FORM/SORM, importance sampling, subset simulation, response surface method, surrogate methods, polynomial chaos expansion, perturbation method, stochastic collocation method, etc. The review and comparison tests comment and conclude not only on accuracy and efficiency of each method but also their applicability in different types of uncertainty propagation problems.

The research fields of structural reliability analysis and stochastic finite element methods have largely been developed separately, although both tackle uncertainty quantification in engineering problems. For the first time, all major uncertainty quantification methods in both fields are reviewed and rigorously tested on a common set of examples. Critical opinions and concluding remarks are drawn from the rigorous comparative study, providing objective evidence-based information for further research and practical applications.

To provide sufficient conditions for existence, uniqueness and finite element approximability of the solution of time‐harmonic electromagnetic boundary value problems involving metamaterials.

The objectives are achieved by analysing the most simple conditions under which radiation, scattering and cavity problems are well posed and can be reliably solved by the finite element method. The above “most simple conditions” refer to the hypotheses allowing the exploitation of the simplest mathematical tools dealing with the well posedness of variationally formulated problems, i.e. Lax‐Milgram and first Strang lemmas.

The results of interest are found to hold true whenever the effective dielectric permittivity is uniformly positive definite on the regions where no losses are modelled in it and, moreover, the effective magnetic permeability is uniformly negative definite on the regions where no losses are modelled in it. The same good features hold true if “positive” is replaced by “negative” and vice versa in the previous sentence.

It is a priori known that more sophisticated mathematical tools, like Fredholm alternative and compactness results, can provide more general results. However this would require a more complicated analysis and could be considered in a future research.

The design of practical devices involving metamaterials requires the use of reliable electromagnetic simulators. The finite element method is shown to be reliable even when metamaterials are involved, provided some simple conditions are satisfied.

For the first time to the best of authors' knowledge a numerical method is shown to be reliable in problems involving metamaterials.