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The purpose of this paper is to present a variational mesh h-adaption approach for strongly coupled thermomechanical problems.

The mesh is adapted by local subdivision controlled by an energy criterion. Thermal and thermomechanical problems are of interest here. In particular, steady and transient purely thermal problems, transient strongly coupled thermoelasticity and thermoplasticity problems are investigated.

Different test cases are performed to test the robustness of the algorithm for the problems listed above. It is found that a better cost-effectiveness can be obtained with that approach compared to a uniform refining procedure. Because the algorithm is based on a set of tolerance parameters, parametric analyses and a study of their respective influence on the mesh adaption are carried out. This detailed analysis is performed on unidimensional problems, and a final example is provided in two dimensions.

This work presents an original approach for independent h-adaption of a mechanical and a thermal mesh in strongly coupled problems, based on an incremental variational formulation. The approach does not rely on (or attempt to provide) error estimation in the classical sense. It could merely be considered to provide an error indicator. Instead, it provides a practical methodology to adapt the mesh on the basis of the variational structure of the underlying mathematical problem.

Using the semi‐inverse method proposed by the present author, a family of variational principle for direct problem of S2‐flow in mixed‐flow turbomachinery is obtained; then, applying the functional variation with variable domain, two families of variational principles are established for the hybrid problems of determining the unknown shape of bladings, where pressure or velocity is over‐specified. The present variational models are well posed for redundant data at boundaries. The theory provides both rational ways for best contouring the hub/casing walls to meet various practical design requirements and a theoretical basis for introducing the finite element method into computational aerodynamics of turbomachinery.

A generalized variational principle of 2D unsteady compressible flow around oscillating airfoils is established directly from the governing equations and boundary/initial conditions via the semi‐inverse method proposed by He. In this method, an energy integral with an unknown F is used as a trial‐functional. The identification of the unknown F is similar to the identification of the Lagrange multiplier. Based on the variational theory with variable domain, a variational principle for the inverse problem (given as the time‐averaged pressure over the airfoil contour, while the corresponding airfoil shape is unknown) is constructed, and all the boundary/initial conditions are converted into natural ones, leading to well‐posedness and the unique solution of the inverse problems.

In an earlier paper (Turkyilmazoglu, 2011a), the author introduced a new optimal variational iteration method. The idea was to insert a parameter into the classical variational iteration formula in an aim to prevent divergence or to accelerate the slow convergence property of the classical approach. The purpose of this paper is to approve the superiority of the proposed method over the traditional one on several physical problems treated before by the classical variational iteration method.

A sufficient condition theorem with an upper bound for the error is also presented to further justify the convergence of the new variational iteration method.

The optimal variational iteration method is found to be useful for heat and fluid flow problems.

The optimal variational iteration method is shown to be convergent under sufficient conditions. A novel approach to obtain the optimal convergence parameter is introduced.

Electrical potentials in a junction field transistor can be calculated using a simplified model based on a complete depletion assumption. This gives rise to a free boundary problem. We show here how we can approximate this problem with a quasi‐variational inequality technique and the shape optimization method. A detailed analysis of these methods is presented. Using some numerical experiments we compare our results with the solution of the discrete drift‐diffusion system, accomplished with a Gummel‐like algorithm. The numerical results suggest that the methods proposed here work successfully and that the shape optimization technique provides a reasonably free boundary without excessive iterations.

The purpose of this paper is to introduce an auxiliary parameter into the well‐known variational iteration algorithm which proves very effective to control the convergence region of approximate solution.

In this paper, an auxiliary parameter is introduced into the well‐known variational iteration algorithm which proves very effective to control the convergence region of approximate solution.

The present technology provides a simple way to adjust and control the convergence region of approximate solution for any values. An optimal auxiliary parameter can be obtained by the error of norm two of the residual function.

It is confirmed that submitted manuscript is original and is not being considered in any other journal.

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.

The main purpose of this paper is to implement the piecewise spectral-variational iteration method (PSVIM) to solve the nonlinear Lane-Emden equations arising in mathematical physics and astrophysics.

This method is based on a combination of Chebyshev interpolation and standard variational iteration method (VIM) and matching it to a sequence of subintervals. Unlike the spectral method and the VIM, the proposed PSVIM does not require the solution of any linear or nonlinear system of equations and analytical integration.

Some well-known classes of Lane-Emden type equations are solved as examples to demonstrate the accuracy and easy implementation of this technique.

In this paper, a new and efficient technique is proposed to solve the nonlinear Lane-Emden equations. The proposed method overcomes the difficulties arising in calculating complicated and time-consuming integrals and terms that are not needed in the standard VIM.

The nonlinear Schrödinger equation plays a vital role in wave mechanics and nonlinear optics. The purpose of this paper is the fractal paradigm of the nonlinear Schrödinger equation for the calculation of novel solitary solutions through the variational principle.

Appropriate traveling wave transform is used to convert a partial differential equation into a dimensionless nonlinear ordinary differential equation that is handled by a semi-inverse variational technique.

This paper sets out the Schrödinger equation fractal model and its variational principle. The results of the solitary solutions have shown that the proposed approach is very accurate and effective and is almost suitable for use in such problems.

Nonlinear Schrödinger equation is an important application of a variety of various situations in nonlinear science and physics, such as photonics, the theory of superfluidity, quantum gravity, quantum mechanics, plasma physics, neutron diffraction, nonlinear optics, fiber-optic communication, capillary fluids, Bose–Einstein condensation, magma transport and open quantum systems.

The variational principle of the Schrödinger equation without the Lagrange multiplier method in the sense of the fractal calculus is developed for the first time in the literature to the best of the author's understanding.

The purpose of this paper is to apply, for the first time, the authors' newly developed perturbation iteration method to heat transfer problems. The effectiveness of the new method in nonlinear heat transfer problems will be tested.

Nonlinear heat transfer problems are solved by perturbation iteration method. They are also solved by the well‐known technique variational iteration method in the literature.

It is found that perturbation iteration solutions converge faster to the numerical solutions. More accurate results can be achieved with this new method for nonlinear heat transfer problems.

A few iterations are actually sufficient. Further iterations need symbolic packages to calculate the solutions.

This new technique can practically be applied to many heat and flow problems.

The new perturbation iteration technique is successfully implemented to nonlinear heat transfer problems. Results show good agreement with the direct numerical simulations and the method performs better than the existing variational iteration method.