Variational Formulation Method Research Articles

Existence and Uniqueness in the Linearised One and Two-dimensional Problem of Partial Differential Equations With Variational Method

The classical solution and the strong solution of a partial differential equation problem are continuously differentiable solutions. This solution has a derivative for a continuous infinity level. However, not all problems of partial differential equations can be easily obtained by strong solutions. Even the existence of a solution requires in-depth investigation. The variational formulation method can qualitatively analyze a single solution to a partial differential equation problem. This study provides an alternative method in analyzing the problem model of partial differential equations analytically. In this research, we will examine the partial differential equation modelling built from fluid dynamics modelling.

Variational formulation of a hybrid perspective shape from shading model

Over a long period of time, it is challenging to solve the completed shape from shading (SFS) problem which has no limitations on light position and surface roughness. Many existing researches developed Hamilton–Jacobi equation (HJE)-based method of such SFS problem. However, HJE-based methods have one major weakness of boundary condition requirement. Finding a proper algorithm for calculating direct depth in a completed SFS model is desirable. In this paper, a completed hybrid perspective shape from shading model based on Cook–Torrance BRDF reflectance model is established. The depth is expressed and calculated directly on the Cartesian coordinates rather than spherical coordinates, which prevents exponential explosion risks caused by substitution of natural exponential. A direct variational formulation method is employed to solve the complex partial differential equation (PDE) of the image irradiance model. The major contribution of the method is that depth is considered as the only variable to deduce iterative equation. After discretizing the partial derivative terms of the iterative equation, the variational method could be programed conveniently and intuitively. Simulations and experiments demonstrate the accuracy and robustness of the method.

Limit Elastic Analysis of FG Ceramic Rotating Disk on the Basis of Effective Mechanical Properties

Functionally graded materials are prevalent among industries for its low weight compared to alloys. Depending upon the applications ceramic-ceramic materials can be graded to obtain novel functional combination. In a disk, yielding mainly occurs at the root, hence to overcome the yielding problem, Al2O3 material was used at the root and graded with different ceramics; as ceramics possess high yield strength. In composites, many methods are available to calculate effective mechanical properties and modified rule of mixture (MROM) is one of these. In MROM only single parameter, stress to strain transfer ratio needs to be identified which in present case was calculated using inverse approach i.e. by comparing Halpin-Tsai model with MROM. Variational formulation method was employed taking radial displacement field as unknown variable. The effective Young’s modulus calculated was then verified with experimental data and good agreement was seen. Further effective yield stress distribution was quantified and plotted to compare with von-Mises stress. Limit elastic speed then obtained was used to rank different ceramic-ceramic graded combinations. Limit speed defines the performance of disk and thus it was chosen as a selection parameter.

Proučavanje rotirajućeg diska od funkcionalno gradiranog materijala sa metalnom matricom bazirano na prenosnom odnosu napona i deformacije primenom helpin-cajovog i MROM modela za analizu granice elastičnosti

Limit elastic analysis of a functionally graded (FG) rotating disk with material grading based on modified rule of mixture (MROM) is reported in the present study. In MROM, stress to strain transfer ratio (q) is an unknown parameter which restricts the application of MROM as yield strength estimation of a material depends on this ratio. Till now the determination of stress to strain transfer ratio, which varies with size, shape, manufacturing processes and composition, is reported by means of experiments. In current work, the effective Young's modulus is calculated by two means i.e. MROM and Halpin-Tsai. Later stress to strain transfer ratio is quantified using MROM using inverse approach and then effective yield stress variation of FGM has been calculated. Different combinations of metal matrix ceramic reinforced FG material were selected. Metals having high strength to weight ratio were combined with non-oxide ceramics of low density. Variational formulation method has been employed to solve the elasticity problem of the rotating disk taking radial displacement field as unknown variable and best material combination is proposed on the basis of maximum limit elastic speed obtained.

Limit angular speed of L-FGM rotating disk for both temperature dependent and temperature independent mechanical properties

Abstract The necessity of weight saving, and high strength at elevated temperatures in a rotating disk motivates researchers to study FGM, to overcome such restrains effectively metal-ceramic graded disk is found superior among other classifications. The present study was carried out for uniform thickness FG rotating disk by employing variational formulation method considering radial displacement field as unknown variable. Temperature variation was considered along radius, Halpin–Tsai method was used to calculate effective Young’s modulus for two cases namely; temperature dependent (TD) and temperature independent (TID). Effective yield stress variation was calculated to obtain accurate limit angular speed for the above mentioned cases. Yield stress variation was plotted along with other stresses to identify yield location. From results it was found that for L-FGM disk, limit speed decreases as aspect ratio increase, a comparison of limit speed with TD and TID properties is also reported.

Stress and deformation of functionally graded rotating disk based on modified rule of mixture

Rotation of disks at high speed induces centrifugal loading and builds up stresses and deformations. In current work, gradient wise FGM disk of Al/SiC is considered, bearing linear volume fraction variation. For uniform thickness disk, stresses, deformation and variation in limit elastic speed and yield stress were computed for different aspect ratios. Variational formulation method is employed taking radial displacement field as unknown variable. Since, investigations reveal that modified Young’s Modulus formulation gives results similar to experimental investigations, hence modified rule of mixture (MROM) was used for calculation of effective Young’s Modulus. Stress and deformation results are reported at limit elastic speed of the disks for different aspect ratio

Robust point-to-point trajectory planning for nonlinear underactuated systems: Theory and experimental assessment

Abstract This paper proposes the theory and the experimental assessment of a robust model-based trajectory planning algorithm for underactuated nonlinear systems in point-to-point motion. The method has been developed to increase the insensitivity of the resulting trajectory to parametric uncertainties of the plant. The proposed method is based on an augmented model that considers an approximate dynamics of the servo-controlled axis driving the actuated degrees of freedom. Trajectory planning is accomplished by computing the motion reference for the actuated degrees of freedom to reduce the effects of the uncertainty on the dynamic model. By exploiting an indirect variational formulation method, the necessary optimality conditions deriving from the Pontryagin’s minimum principle are imposed, thus leading to a differential Two-Point Boundary Value Problem (TPBVP). Numerical solution of the latter is accomplished by means of collocation techniques to handle model nonlinearities. Robustness is achieved by including additional conditions on the sensitivity functions for the initial and final points of the trajectory. The experimental evaluation of the effectiveness of the proposed method is performed on a double-pendulum crane, by comparing the transient and residual vibration. A comparison is provided with three well-established input-shaping methods, and robustness against unmodeled parametric perturbations and tracking errors is evaluated. The experimental evidence indicates that the inclusion of the additional constrains results in an effective reduction of the residual vibration, and that the proposed method is well suited to perform high speed motion.

Nodal Analysis of Urban Water Distribution Networks

There are three methods for analysing the flow and pressure distribution in looped water supply networks (the loop method, the node method, the pipe method), accounting for the chosen unknown hydraulic parameters. For all of these methods, the nonlinear system of equations can be solved using iterative procedures (Hardy–Cross, Newton–Raphson, linear theory). In the cases of the extension or the rehabilitation of distribution networks, the unknown parameters are the hydraulic heads at nodes, and the nodal method for network analysis is preferred. In this paper, a generalised classic model is developed for the nodal analysis of complex looped systems with non-standard network components and the solvability of new problems, along with the determination of the pressure state in the system. In addition, this paper exhibits a different approach to this problem by using the variational formulation method for the development of a new analysis model based on unconditioned optimisation techniques. This model has the advantage of using a specialised optimisation algorithm, which directly minimises an objective multivariable function without constraints, implemented in a computer program. The two proposed models are compared with the classic Hardy–Cross method, and the results indicated a good performance of these models. Finally, a study is performed regarding the implications of the long-term operation of the pipe network on energy consumption using these models. The new models can serve as guidelines to supplement existing procedures of network analysis.

A Plane-Wave Least-Squares Method for Time-Harmonic Maxwell's Equations in Absorbing Media

In this paper we discuss numerical methods for solving time-harmonic Maxwell equations in three-dimensional absorbing media. We propose a new variational problem on the space spanned by the plane-wave basis functions for the discretization of these types of equations. The variational problem is derived mathematically from a minimization problem, and the resulting stiffness matrix is Hermitian positive definite. A simple domain decomposition preconditioner is introduced for the system generated by the proposed variational formula. Numerical results indicate that the approximate solution generated by the proposed discretization method is clearly more accurate than that generated by the ultra weak variational formulation method, and the proposed preconditioner is effective.

Simulations of head-related transfer functions in wideband acoustics

Head-related transfer functions (HRTFs) have been simulated in three dimensions for a head-and-torso model for the entire audio frequency range, from 20 Hz to 20 kHz. As opposed to data acquired through measurements, the results derived from computer simulations are free from the effects of noise and imperfections in the electro-acoustic chain. In addition, the spatial resolution can easily be made better than in any practical experiment. The simulations have been performed using an ultraweak variational formulation method for solving Helmholtz equation. The numerical method with its parallel computing capability is efficient, enabling numerical calculations of large physical systems, e.g., of size 0.4×0.5×0.8 m, at high frequencies. The method uses plane-wave basis functions instead of polynomial basis as in standard finite-element methods (FEM), resulting in the need of much sparser volume mesh than in FEM. For HRTF calculations, a so-called perfectly matched layer has been utilized and the solutions are derived in the far field. Thus, exterior problems often solved using boundary-element method are dealt with. The specific HRTF simulations have been performed using fully reflecting, sound hard, boundaries on the head and torso, and also with complex impedance boundaries on the torso such as clothing.

Optimal thrust program for deviating an Earth-pointing asteroid

The purpose of this study is to investigate the optimal thrust program for deviating an asteroid when it is flying directly toward or crossing the Earth. Under some assumptions, the problem can be considered as a two-body problem of Earth-asteroid system. The initial relative speed and distance are specified to be 10 km/s and 150 times of Earth's radius, respectively. We have about 1 day (or, 93681 seconds exactly) to take action. If a single impulse is applied to the asteroid at the specified initial point, the required impulse to obtain a miss distance of 2 times of Earth's radius is 169.5 m/s per kg mass. For an asteroid of 10 m in diameter, the total impulse required is 3.02 × 10 8 m/s. It needs a typical large launching rocket to provide the total impulse. When the asteroid is larger or the initial distance is shorter, the number of launching rockets required increases rapidly. For further analysis with physical and engineering constraints imposed, we shall have to use the variational formulation method.

Diffusional release of a dispersed solute from a spherical polymer matrix

The exact analytical solution and an approximate solution based on the variational formulation method have been obtained for the release kinetics of a solute from a spherical polymeric matrix in which the initial solute loading A is above the solubility limit C s. Comparisons between these new solutions and previously reported approximate solutions based on either pseudo-steady state assumptions or the refined integral method indicate that the approximate solutions are very accurate when A C s is much larger than unity. However, as drug loading decreases, deviations between the approximate solutions and the exact solution increase so that the approximate solutions are least accurate in the limit A C s → 1 .