Application of a wavelet-Galerkin method to the forward problem resolution in fluorescence diffuse optical tomography

非线性无力场是天体物理中的重要数学模型，它是一套非线性偏微分方程组，经常用于太阳及恒星磁场的理论研究。在轴对称情形下，该方程组归化为满足特定边界条件的带有未知参数的二阶非线性常微分方程，此即所谓的Low-Lou解法。本文提出一种参数打靶法，作为对Low-Lou解法的探讨和补遗，并给出更多可选的数值无力场。 Nonlinear force-free magnetic field is an important mathematical model in astrophysics, which is a set of nonlinear partial differential equations, often used in the theoretical studies of solar and stellar magnetic fields. In the axisymmetric case, this set of partial differential equations is reduced into a nonlinear ordinary differential equation of second order with an unknown parameter, satisfying certain boundary condition. This is the so-called Low-Lou approach of the problem. In this paper, we propose a parametric shooting method as a technical supplement for the Low- Lou approach, offering more optional numerical force-free magnetic fields.

An enriched finite element method with interpolation cover functions for acoustic analysis in high frequencies

Uniformly convergent finite element methods for singularly perturbed elliptic boundary value problems: convection—diffusion type

The workshop Nonstandard Finite Element Methods , organized by Susanne C. Brenner (Baton Rouge), Carsten Carstensen (Berlin) and Peter Monk (Newark) was held August 10 – 16, 2008. This meeting was well attended with over 40 participants with broad geographic representation. Although Courant is often credited with the discovery of finite elements [2, 3], the first practical use of such methods in engineering seems to date back to the work of Argyris and Clough et. al. (apparently Clough coined the name “finite elements”) in the 1950s [1, 4]. By the 1970s the finite element method was firmly established in engineering practice and the basic theory of conforming elements for elliptic problems was well understood [2]. Engineers and mathematicians have since expanded the use of finite elements to a wide variety of new applications and have improved the theoretical underpinnings of the method. In addition, efficient computational algorithms have been developed, such as the multigrid method, that allow for the application of finite elements to sophisticated three dimensional problems. Within current research, we can distinguish classical or standard finite element methods, and newer nonstandard finite elements The former include conforming elements applied to “standard” Galerkin discretizations of problems. Usually accuracy is obtained by mesh refinement. As the range of applications has increased, and to alleviate perceived shortcomings in standard conforming elements, numerical analysts and engineers have sought more general finite element methods based, for example, on mixed formulations, and non-conforming or even discontinuous Galerkin schemes. These methods may be superior to standard finite elements in particular applications because of enhancements to stability, robustness or conservation properties. We refer to them as non-standard finite element methods. This Oberwolfach workshop was devoted to non-standard finite element methods and their analyses. Inevitability such a snapshot of the subject is biased, but we have tried to include non-standard methods in the broadest sense including mixed, discontinuous Galerkin, generalized, partition of unity and mortar FEM methods. All these schemes have in common that stability and convergence is not obvious and requires mathematical analysis. This is even more true for developing fast solvers, a posteriori error estimation and adaptive mesh design. Through the week there were 29 presentations on various non-standard topics. These provided an overview of current research directions, new developments and open problems.

We propose the nonconforming Finite Element (FE) method based on Nitsche-type mortaring for efficiently solving the inhomogeneous wave equation, where due to the change of material properties the wavelength in the subdomains strongly differs. Therewith, we gain the flexibility to choose for each subdomain an optimal grid. The proposed method fulfills the physical conditions along the nonconforming interfaces, namely the continuity of the acoustic pressure and the normal component of the acoustic particle velocity. We apply the nonconforming grid method to the computation of transmission loss (TL) of an expansion chamber utilizing micro-perforated panels (MPPs), which are modeled by a homogenization approach via a complex fluid. The results clearly demonstrate the superiority of the nonconforming FE method over the standard FE method concerning pre-processing, mesh generation flexibility and computational time.

In this paper, the ‐Galerkin mixed finite element method (MFEM) is used to solve the time‐fraction‐order damped beam vibration equations with simple support at both ends. Compared with the standard finite element method (FEM), the ‐Galerkin MFEM can calculate the deflection, bending moment and other parameters of the beam more directly, so it is more suitable for solving the beam vibration equations, which are high‐order partial differential equations. The Caputo fractional derivative is approximated using the L1 formula, and a fully discrete numerical scheme is established by linear backward Euler method. The stability of the proposed scheme, as well as the existence, uniqueness, and convergence of the numerical solution, are rigorously analyzed. Numerical examples are provided to verify the theoretical results, with real‐world beam data used to investigate how variations in the damping coefficient and the order of the fractional derivative affect the beam's vibrational behavior. Numerical simulations demonstrate that the amplitude of the beam decays more rapidly with larger damping coefficients. Moreover, as the order of the fractional derivative increases, the decay rate of the beam vibration first increases and then decreases, while the peak of the curve gradually shifts to the right.

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.

A simple, efficient finite element method has been presented for the solution of a variety of scalar field problems in two dimensions. It is based on the mapping of the physical problem domain into an ‘image’ domain in the w‐plane. The governing equation(s) and the boundary conditions in the physical plane are also appropriately transformed into the w‐plane. The processes of standard finite element analysis are then implemented to obtain a solution in the w‐plane.The method has been explained in detail, with illustrative examples where appropriate; it has several important advantages over the standard finite element method, particularly for the solution of infinite or semi‐infinite domain problems. The method has been demonstrated to be simple, efficient, economical and potentially capable of dealing with a large repartoire of two‐dimensional problems, including non‐homogeneity, nonlinearity, etc.

Locally conservative Galerkin and finite volume methods for two-phase flow in porous media

The modelling of a flexible robot arm is concerned with its rigid body motion as well as its oscillation due to the elasticity of the arm [1-4]. As the elastic deformation is distributed along the arm, it is obvious that this system is a distributed parameter system. Although a set of partial differential equations is presumed to be the best model for this system, it is limited to a relatively low frequency oscillation of a homogeneous, simply shaped arm. In the case of a more complicated flexible arm, the partial differential equation model is not necessarily advantageous, and another method such as the finite element method will be used. First, this paper presents the detailed development of the partial differential equations and boundary conditions for two models of a flexible robot arm. One is a rigorous model obtained through the use of Timoshenko beam theory, which takes into account the effect of transverse shear deformation and rotary inertia; the other is a simpler and more frequently used model ...

Purpose – The purpose of this paper is to focus on the numerical modelling of transient natural convection flow of an incompressible viscous nanofluid past an impulsively started semi-infinite vertical plate with variable surface temperature. Design/methodology/approach – The problem is governed by the coupled non-linear partial differential equations with appropriate boundary conditions. A robust, well-tested, Crank-Nicolson type of implicit finite-difference method, which is unconditionally stable and convergent, is used to solve the governing non-linear set of partial differential equations. Findings – The local and average values of the skin-friction coefficient (viscous drag) and the average Nusselt number (the rate of heat transfer) decreased, while the local Nusselt number increased for all nanofluids, namely, aluminium oxide-water, copper-water, titanium oxide-water and silver-water with an increase in the temperature exponent m. Selecting aluminium oxide as the dispersing nanoparticles leads to the maximum average Nusselt number (the rate of heat transfer), while choosing silver as the dispersing nanoparticles leads to the minimum local Nusselt number compared to the other nanofluids for all values of the temperature exponent m. Also, choosing silver as the dispersing nanoparticles leads to the minimum skin-friction coefficient (viscous drag), while selecting aluminium oxide as the dispersing nanoparticles leads to the maximum skin-friction coefficient (viscous drag) for all values of the temperature exponent m. Research limitations/implications – The Brinkman model for dynamic viscosity and Maxwell-Garnett model for thermal conductivity are employed. The governing boundary layer equations are written according to The Tiwari-Das nanofluid model. A range of nanofluids containing nanoparticles of aluminium oxide, copper, titanium oxide and silver with nanoparticle volume fraction range less than or equal to 0.04 are considered. Practical implications – The present simulations are relevant to nanomaterials thermal flow processing in the chemical engineering and metallurgy industries. This study also provides an important benchmark for further simulations of nanofluid dynamic transport phenomena of relevance to materials processing, with alternative computational algorithms (e.g. finite element methods). Originality/value – This paper is relatively original and illustrates the influence of variable surface temperature on transient natural convection flow of a viscous incompressible nanofluid and heat transfer from an impulsively started semi-infinite vertical plate.

XFEM analysis for effectively modeling the singularity of the capacitor edge

A topology optimization, based on a coupling method of finite element and meshless method, is proposed for continuum structure. A reasonable arrangement of the meshless domain can guarantee the accuracy of the meshless methods and meanwhile keep the computational efficiency of the finite element method. Besides, as the coupling method is adopted, the displacement boundary conditions are applied to finite elements nodes by standard finite element method. A dual-level density approximant is carried out to approximate and interpolate a unified and continuous density field. An unstable nodes phenomenon is observed in the interface domain, leading to nonconvergence of the equilibrium iterations. A smooth blending function and an energy convergence criterion are used to circumvent the convergent difficulty. Three benchmark problems of topology optimization are given to demonstrate the effectiveness of the proposed approach. Neither “islanding” phenomenon nor other common numerical instabilities of the SIMP method occur in this approach.

Analysis of a continuous Galerkin method with mesh modification for two-dimensional telegraph equation

A weakly singular symmetric Galerkin boundary element method (SGBEM) is coupled with the standard finite element method (FEM) in order to establish an accurate and efficient numerical technique for analysis of fractures in three‐dimensional, anisotropic, linearly elastic media. In the strategy, the weakly singular SGBEM developed by Rungamornrat and Mear (textitInt. J. Solids Struct. 2008; 45:1283–1301; Comput. Methods Appl. Mech. Engrg 2008; 197:4319–4332) is utilized to model a small‐scale region containing the crack while the (possibly large and complex) compliment region is treated by the FEM. The coupled technique exploits the positive features of both methods; the SGBEM proves to be a convenient and highly accurate method for obtaining mixed‐mode stress intensity factors along the crack front, whereas the FEM is very efficient for modeling large‐scale problems in the absence of cracks. An important aspect of the formulation and implementation of the technique is that continuity of displacement and traction across the interface between the SGBEM and FEM regions is enforced in a weak sense. This allows the two regions (one modeled by the SGBEM and the other by the FEM) to be discretized independently without the need for the resulting meshes to conform on the interface separating the regions, and this flexibility in the discretization process leads to a significant reduction in the modeling effort. To demonstrate the utility and accuracy of the technique, several boundary value problems involving both embedded and surface breaking cracks are treated, and it is shown that the coupled technique yields highly accurate stress intensity factors that exhibit only a slight dependence upon mesh refinement. Copyright © 2010 John Wiley & Sons, Ltd.