A numerical method for a magnetostatic inverse problem using the edge element

This paper is concerned with the large‐deflection (bending and free vibration)analysis of thin elastic curved beams by conventional and mixed finite element methods. The conventional finite element method is based on the total potential energy expression, whereas the mixed method is based on a Reissner‐type variational statement and involves the bending moments and deflections as primary dependent variables. Incremental and direct formulations are presented for both methods. The nonlinearity is included in the numerical method via an iterative procedure (i.e. the in‐plane force due to the large deflection is not treated as constant but included under the integral in the energy expression). In the case of vibrations, the transverse shear and rotary inertia effects are also included. A number of numerical examples of beams with various edge conditions are analysed for deflections and natural frequencies, and the results are compared with those reported in the literature.

In this paper an analysis is carried out to study the efficiency of heat distribution through fins using finite element method (FEM) and differential quadrature method ( DQM). Numerical solutions are obtained using both methods for temperature distribution over the fin surface. Analysis of surface solutions in terms of temperature and error distributions have been presented here. Solutions are also obtained using equally spaced (conventional) and non-equally spaced (optimum) mesh (nodal points) distribution techniques. Two-dimensional heat conduction problem has been solved in both equal and non-equal nodal points distribution cases (called here conventional FEM (CFEM) and conventional DQM (CDQM) and optimum FEM (OFEM) and optimum DQM (ODQM)). The obtained results are compared and investigated with exact results. The best results are found in OFEM solutions. It is also found that, ODQM solutions reach very close to OFEM solutions with negligible error, whereas CFEM and CDQM solutions diverges with increasing number of nodal(mesh) points.

Efficient solver for mixed finite element method and control-volume mixed finite element method in 3-D on Hexahedral grids

In this paper we develop a leap-frog-type finite element method for modeling the electromagnetic wave propagation in metamaterials. The metamaterial model equations are represented by integrodifferential Maxwell's equations, which are quite challenging to analyze and solve in that we have to solve a coupled problem with different partial differential equations given in different material regions. Our method is based on a mixed finite element method using edge elements, which can easily handle the tangential continuity of the electric field. Stability analysis and optimal error estimate are carried out for the proposed scheme. The scheme is implemented and confirmed to obey the proved optimal convergence rate by using a smooth analytical solution. Then the scheme is extended to model wave propagation in heterogeneous media composed of metamaterials and free space, and extensive numerical results (using a rectangular edge element, a triangular edge element, and mixed edge elements) demonstrate the effectiveness of our algorithm for modeling the exotic backward wave propagation phenomenon in metamaterials. To the best of our knowledge, this is the first paper with an exhaustive simulation of backward wave propagation in metamaterials using time-domain finite element methods.

The multiscale finite element method (MsFEM) combined with conventional finite element method (FEM) is proposed to solve static magnetic field in the ribbon magnetic core. Firstly, a simple 2-dimensional electrostatic problem is used to introduce the MsFEM part. The results are compared to analytical method, as well as conventional FEM. Then, an example of magneto static problem is considered for a ribbon magnetic core built sheet by sheet. Conventional FEM and MsFEM are used to compute the magneto-static field by adopting scalar magnetic potential. Both magnetic potential and magnetic flux density on a certain path are compared. It is shown that the results obtained by MsFEM agrees well with the one from conventional FEM. Moreover, MsFEM combined with FEM is potentially a general strategy for multiscale modeling of ribbon magnetic cores, which can effectively reduce the computational cost.

Integral representations for the solutions to linear elliptic self adjoint boundary value problems are derived in terms of two functions which are generalisations of the single and double layer potentials used in the theory of harmonic functions. The generalised potentials are constructed in terms of a fundamental solution which is an approximation to the exact kernel of the boundary value problem in question. The representations so obtained are shown to provide a basis from which strict approximations to the solutions of boundary value problems can be developed. In particular the structure of the integral equation representing the given boundary value problem is precisely determined.

PreviousNext No AccessSEG Technical Program Expanded Abstracts 1995Two‐dimensional regularized inversion of magnetotelluric dataAuthors: Michael S. ZhdanovPatricia P. de LugaoOleg N. PortniaguineMichael S. ZhdanovUniversity of Utah, Patricia P. de LugaoUniversity of Utah, and Oleg N. PortniaguineUniversity of Utahhttps://doi.org/10.1190/1.1887564 SectionsAboutPDF/ePub ToolsAdd to favoritesDownload CitationsTrack CitationsPermissions ShareFacebookTwitterLinked InRedditEmail Permalink: https://doi.org/10.1190/1.1887564FiguresReferencesRelatedDetailsCited ByA Selection Algorithm of Survey Lines and the Interactive Mode of the Sounding Curve7 January 2015 SEG Technical Program Expanded Abstracts 1995ISSN (print):1052-3812 ISSN (online):1949-4645Copyright: 1995 Pages: 1566 publication data© 1995 Copyright © 1995 Society of Exploration GeophysicistsPublisher:Society of Exploration Geophysicists HistoryPublished: 10 Feb 2005 CITATION INFORMATION Michael S. Zhdanov, Patricia P. de Lugao, and Oleg N. Portniaguine, (1995), "Two‐dimensional regularized inversion of magnetotelluric data," SEG Technical Program Expanded Abstracts : 803-806. https://doi.org/10.1190/1.1887564 Plain-Language Summary PDF DownloadLoading ...

Static and dynamic analysis of two-phase peridynamic one-dimensional systems is thoroughly investigated in this paper for various boundary conditions by exact and numerical approaches. Such a nonlocal study concerns the axial or torsional responses of two-phase peridynamic rods or the behavior of a two-phase peridynamic string. For a normalized exponential kernel, the integro-differential equation of the two-phase peridynamic problem can be converted as a linear higher-order differential equation with constant coefficients. Exact static solutions of fixed–fixed and fixed–free two-phase peridynamic axial rod (TPPAR) under uniform distributed loading and concentrated force are analytically derived. It is shown that the pure peridynamic rod problem can be ill-posed due to a conflict between the natural boundary condition and the peridynamic constitutive boundary condition. The two-phase peridynamic problem may be justified from physical arguments, and is also seen as a regularized peridynamic problem. The exact eigen-frequencies of the two-phase peridynamic rods are also analytically obtained for various boundary conditions, including fixed–free, fixed–fixed or free–free boundary conditions. These theoretical approaches are corroborated by numerical investigations based on Finite Difference Method (FDM), Finite Element Method (FEM), exact FEM and variational iterative methods. Two FDM schemes are presented for the statics and the vibration of two-phase peridynamic rods, with convergence orders of 1 and 2, respectively. Moreover, three different FE procedures are proposed as: (1) FEM for direct solving the integro-differential equations of two-phase peridynamic one-dimensional system, (2) FEM for discretizing the pure differential equations, and (3) Exact FEM to obtain both the exact stiffness matrix and exact dynamic stiffness matrix of two-phase peridynamic one-dimensional system, based on Eisenberger’s methodology. The order of convergence for the first and the second proposed FEM is equal to 2 and 4, according to the type of selected finite elements. Moreover, the third proposed FEM based on the exact shape function of the peridynamic problem gives the exact solution from one single FEM element. In addition, the variational iterative method of Stodola–Vianello is successfully applied to the two-phase peridynamic eigenvalue problems.

It has been well known that exact closed-form solutions are not available for non-Levy-type plates. Thus, more accurate and efficient computational methods have been required for the plates subjected to arbitrary boundary conditions. This paper presents a frequency-domain spectral element model for the rectangular finite plate element. The spectral element model is developed by using two methods in combination: (1) the boundary splitting and (2) the super spectral element method in which the Kantorovich method-based finite strip element method and the frequency-domain waveguide method are utilized. The present spectral element model has nodes on four edges of the finite plate element, but no nodes inside. This can reduce the total number of degrees of freedom a lot to improve the computational efficiency significantly, when compared with the standard finite element method (FEM). The high solution accuracy and computational efficiency of the present spectral element model are evaluated by the comparison with exact solutions and the solutions by the standard FEM.

Numerical approximation of head and flux covariances in three dimensions using mixed finite elements

Acoustic and elastic wave equations are discretized by finite element method with triangles and central difference scheme in space and time, respectively. The generalized eigenvalue problems are constructed to analyze the numerical dispersion of the finite element method. Three types of mass finite element method, which are consistent finite element method (CFEM), lumped finite element method (LFEM) and mixed finite element method (MFEM), are implemented to the numerical scheme, and the numerical dispersion is under a comprehensive study. Then, the MFEM is adopted to investigate the behavior of numerical anisotropy in terms of four typical triangle meshes. Comparing the numerical dispersion and anisotropy behavior when MFEM uses different interpolation orders, one can easily find that numerical error decreases as the interpolation order increases. Specifically, the numerical dispersion and anisotropy are negligible when third‐order interpolation is used. Finally, controlling the other impact factors, the dispersive properties of elastic wave propagation in elastic medium with various velocity ratios are analyzed, and the results of numerical dissipation are presented at the end of this paper.

PurposeThis paper aims to discuss inverse problems that arise in a variety of practical thermal processes and systems. It presents some of the approaches that may be used to obtain results that lie within a small region of uncertainty. Therefore, the non-uniqueness of the solution is reduced so that the final design and boundary conditions may be determined. Optimization methods that may be used to reduce the uncertainty and to select locations for experimental data and for minimizing the error are presented. A few examples of thermal systems are given to illustrate the applicability of these methods and the challenges that must be addressed in solving inverse problems.Design/methodology/approachIn most analytical and numerical solutions, the basic equations that describe the process, as well as the relevant and appropriate boundary conditions, are known. The interest lies in obtaining a unique solution that satisfies the equations and boundary conditions. This may be termed as a direct or forward solution. However, there are many problems, particularly in practical systems, where the desired result is known but the conditions needed for achieving it are not known. These are generally known as inverse problems. In manufacturing, for instance, the temperature variation to which a component must be subjected to obtain desired characteristics is prescribed, but the means to achieve this variation are not known. An example of this circumstance is the annealing, tempering or hardening of steel. In such cases, the boundary and initial conditions are not known and must be determined by solving the inverse problem to obtain the desired temperature variation in the component. The solutions, thus, obtained are generally not unique. This is a review paper, which discusses inverse problems that arise in a variety of practical thermal processes and systems. It presents some of the approaches or strategies that may be used to obtain results that lie within a small region of uncertainty. It is important to realize that the solution is not unique, and this non-uniqueness must be reduced so that the final design and boundary conditions may be determined with acceptable accuracy and repeatability. Optimization techniques are often used for minimizing the error. This review presents several methods that may be applied to reduce the uncertainty and to select locations for experimental data for the best results. A few examples of thermal systems are given to illustrate the applicability of these methods and the challenges that must be addressed in solving inverse problems. By considering a variety of systems, the paper also shows the importance of solving inverse problems to obtain results that may be used to model and design thermal processes and systems.FindingsThe solution of inverse problems, which frequently arise in thermal processes, is discussed. Different strategies to obtain the conditions that lead to the desired result are given. The goal of these approaches is to reduce uncertainty and obtain essentially unique solutions for different circumstances. The error of the method can be checked against known conditions to see if it is acceptable for the given problem. Several examples are given to illustrate the use of these methods.Originality/valueThe basic strategies presented here for solving inverse problems that arise in thermal processes and systems, as well as the optimization techniques used to reduce the domain of uncertainty, are fairly original. They are used for certain challenging problems that have not been considered in detail earlier. Several methods are outlined for considering different types of problems.

A two-grid mixed finite element method for a nonlinear fourth-order reaction–diffusion problem with time-fractional derivative

In this paper, a meshless local maximum-entropy finite element method (LME-FEM) is proposed to solve 1D Poisson equation and steady state convection---diffusion problems at various Peclet numbers in both 1D and 2D. By using local maximum-entropy (LME) approximation scheme to construct the element shape functions in the formulation of finite element method (FEM), additional nodes can be introduced within element without any mesh refinement to increase the accuracy of numerical approximation of unknown function, which procedure is similar to conventional p-refinement but without increasing the element connectivity to avoid the high conditioning matrix. The resulted LME-FEM preserves several significant characteristics of conventional FEM such as Kronecker-delta property on element vertices, partition of unity of shape function and exact reproduction of constant and linear functions. Furthermore, according to the essential properties of LME approximation scheme, nodes can be introduced in an arbitrary way and the $$C^0$$ C 0 continuity of the shape function along element edge is kept at the same time. No transition element is needed to connect elements of different orders. The property of arbitrary local refinement makes LME-FEM be a numerical method that can adaptively solve the numerical solutions of various problems where troublesome local mesh refinement is in general necessary to obtain reasonable solutions. Several numerical examples with dramatically varying solutions are presented to test the capability of the current method. The numerical results show that LME-FEM can obtain much better and stable solutions than conventional FEM with linear element.

A new mixed finite-element method is proposed for computing the anisotropic, lossy, and open waveguide modes. By incorporating Gauss' law into the vectorial wave equation, the variational formulation is completely free of spurious modes. It utilizes the curl-conforming linear tangential and quadratic normal (LT/QN) edge elements to expand the tangential component of the electric field, and the modified nodal-based scalar basis functions to expand the longitudinal component. A smaller scale generalized eigenvalue problem is obtained by employing the mass lumping technique. Numerical examples verify that the mixed finite-element method with mass lumping is free of any spurious eigenmode and has high efficiency.