Improved finite difference formulas for boundary value problems

This paper deals with the numerical solution of linear differential equations of fourth order by finite differences. It points out significant (but usually overlooked) errors which result from the conventional method of imposing the boundary conditions in such problems. Revised finite difference formulas are derived which apply near the boundaries and which eliminate the above errors. Three commonly encountered boundary conditions ar econsidered. These correspond, in the terminology of beam analysis, to a clamped end, to a simply supported end and to a free end. The improvement in accuracy that can be achieved with the revised formulas is illustrated by two representative examples. The revised formulas are shown to reduce the overall error of the numerical solution by a factor of about five in a typical case.     

A finite difference approach to acoustic tomography in anisotropic media

In this paper, a finite difference formulation for acoustic wave propagation is used as the basis for tomographic reconstruction. This approach offers some interesting advantages over traditional, ray based methods; particularly for anisotropic media. Since this approach provides information on the full acoustic field (not individual rays), it offers a conventional way to incorporate beam skew and ray bending phenomena directly into the problem formulation. Here, we present a tomographic reconstruction algorithm which is adapted from the algebraic reconstruction technique (ART) to take full advantage of the finite difference formulation of the problem. Results are presented to illustrate the utility of this approach.     

A finite difference model for cMUT devices

A finite difference method was implemented to simulate capacitive micromachined ultrasonic transducers (cMUTs) and compared to models described in the literature such as finite element methods. Similar results were obtained. It was found that one master curve described the clamped capacitance. We introduced normalized capacitance versus normalized bias voltage and metallization rate, independent of layer thickness, gap height, and size membrane, leading to the determination of a coupling factor master curve. We present here calculations and measurements of electrical impedance for cMUTs. An electromechanical equivalent circuit was used to perform simulations. Our experimental measurements confirmed the theoretical results in terms of resonance, anti-resonance frequencies, clamped capacitance, and electromechanical coupling factor. Due to inhomogeneity of the tested element array and strong parasitic capacitance between cells, the maximum coupling coefficient value achieved was 0.27. Good agreement with theory was obtained for all findings.     

A drafting program for isoparametric finite elements

Considerable effort has been invested lately in the application of isoparametric finite elements for numerical solution of a wide range of applied mechanics problems. In fact, several general purpose computer programs are now available which are based upon such finite elements. In the present paper, a new application of the isoparametric finite element concept is introduced which significantly extends its usefulness for many practical structural configurations. In this application, final working or architectural drawings of the structure are made from the same (or similar) finite element model as was utilized in a structural integrity analysis. The hardware necessary to produce such drawing, a computer driven plotter or automated drafting machine, is available commercially or through most data centres, and the software concepts required are described herein.     

A method for calculating meshless finite difference weights

An exact method for calculating the finite difference weights for arbitrary distributed points in multiple dimensions is presented. The method avoids the numerical ill conditioning associated with a small‐scale factor (ε) by delaying the limiting of ε → 0. The Gaussian Radial Basis function is approximated by a Taylor's series expansion in the variable ε, and the resultant matrix equation is solved using the Fraction Free LU decomposition method. Copyright © 2007 John Wiley & Sons, Ltd.     

A finite element solution program for large structures

A straightforward and general computer program for assembling and solving (using Gauss elimination technique) widely sparsed finite element matrix equations with very large bandwidth and capable of handling different degrees-of-freedom and variable bandwidth at different nodes, is described herein. The program assembles any type of finite elements having arbitrary number of nodes and each node may have differnt degrees-of-freedom. It requires only a small core memory in the computer, although a fast random access device is also needed. The two very important features of this program are (i) it does not store any zero submatrices within the band and (ii) during the solution of equations all operations dealing with zero submatrices within the band are automatically skipped and thus the savings of a considerable amount of disc storage space and computer time can be effected in many cases. Another feature is that many right hand sides can be handled simultaneously. Hence the program is very economical for structures having widely sparsed matrix equations. A listing of the computer program written in FORTRAN IV for CDC 6400 computer is readily available from the authors, but unfortunately could not be given here because of lack of space. The program is so general that it can be used to solve a wide class of finite element problems without actually having to understand fully the techniques behind it.     

A nonstandard finite difference scheme for a strain-gradient theory

Strain-gradient theories have proved very useful in describing computational aspects of phase transforming materials such as shape memory alloys. A significant feature implicit to these theories is a relation between the driving force acting on a phase boundary and its velocity. Numerical calculation of the kinetic relation using standard finite difference methods shows significant quantitative and qualitative departures from the analytical kinetic relation. In this paper we derive a nonstandard finite difference scheme and show that the kinetic relation evaluated using this scheme displays the correct qualitative behaviour and matches the analytical solution significantly better quantitatively. In particular, the nonstandard finite difference scheme eliminates spurious lattice trapping in the kinetic relation.     

A generalized finite difference scheme for convection-dominated metal-forming problems

A simple but versatile numerical technique using generalized finite difference discretization has been developed for heat transfer problems involving high convective heat flow, irregular geometry and high local thermal gradients. Upwind differencing is utilized to stabilize the numerical oscillations often induced in convection-dominated heat transfer problems. An arbitrary irregular mesh scheme is adopted to treat the irregular geometry and to achieve high accuracy in zones having high thermal gradients. To demonstrate the validity of the formulation procedure, results predicted from the present scheme are compared with the analytical solution for a problem having a regular boundary. Application to a typical metal-forming process having curved boundaries is then included.     

A frontal solution program for finite element analysis

The program given here assembles and solves symmetric positive–definite equations as met in finite element applications. The technique is more involved than the standard band–matrix algorithms, but it is more efficient in the important case when two-dimensional or three-dimensional elements have other than corner nodes. Artifices are included to improve efficiency when there are many right hand sides, as in automated design. The organization of the program is described with reference to diagrams, full notation, specimen input data and supplementary comments on the ASA FORTRAN print-out.     

A new finite element concurrent computer program architecture

A new computer program architecture for the solution of finite elemet systems using concurrent processing is presented. The basic approach involves the automatic splitting of an arbitrary spatial domain. Processors are dynamically re-assigned during the several phases of an analysis. Direct and iterative solution strategies are considered. Computational algorithms for finite element dynamic analysis of large-scale structural problems that exploit concurrent features of MIMD computers are implemented in modules around the basic architecture. Also, problems with localized non-linearities are treated. A first implementation on a 32-processor hypercube achieves efficiency rates of up to 90 per cent.     

On the Galerkin finite difference method: A multi-bases approach

At present there are many general methods of approximating a possibly nonlinear operator equation by a finite equation system. The most commonly applied methods are the finite element and the finite difference. Numerous papers (e.g. References 9 and 10) have dealt with a comparison of these methods. In Reference 12 the Galerkin finite difference method (GFDM) is developed. The GFDM is a special finite element method designed t o solve nonlinear and possibly coupled partial differential equations numerically. It consists of a finite difference scheme derived from a Galerkin finite element method through the use of special local basis functions and a special grid. In this paper, we are concerned with extending the GFDM to derive ‘normal’ difference schemes. e.g. five-point schemes on two-dimensional domains for a general class of operators, even in nonlinear cases. Using GFDM or the finite element method on two-dimensional regions generally leads to at least 7- or 9-point schemes as well as expensive approximations of the nonlinear terms. Often, numerical integration is necessary. These computational costs are due to the non-orthogonality of the continuous and differentiable local basis functions that are needed in this case. The basic idea of the multi-bases approaches, which are the major concern in this paper¹¹, is to reduce the smoothnessproperties ofthelocal basis functions in favour oftheir orthogonality. Thiscan beachieved with the help of so-called transfer operators which map the non-orthogonal and differentiable basis onto an only bounded but orthogonal basis.     

Electrolyte formulas of aqueous zinc ion battery: A physical difference with chemical consequences

Researchers have recently redirected their attention toward aqueous batteries in pursuit of safety and affordability. The water-based electrolyte promises many appealing merits such as non-flammability and environmental friendliness but is also cursed with low energy density. In an attempt of lifting the curse, many efforts have been made to re-formulate the aqueous electrolyte, which seems unlikely to succeed without understanding the interplay between electrolytes’ physical properties and electrochemical performance. Starting from the composition of electrolytes, we discuss how various formulas of zinc ion battery electrolytes lead to diverse electrochemical performances. By evaluating the electrochemical performance of batteries in five metrics, it provides a la carte electrolyte designing strategies with different battery purposes.     

A finite difference calculation procedure for three-dimensional turbulent separated flows

A general finite difference scheme has been proposed along with a three-dimensional co-ordinate transformation procedure for the prediction of three-dimensional fully elliptic flows. This numerical scheme has been successfully employed for the calculations of the three-dimensional turbulent separated flow in a rectangular diffuser. The complexity of the phenomena is seen to increase tremendously for the three-dimensional flows of this class.     

A finite difference perturbation procedure for non-linear elastic stability problems

A finite difference perturbation scheme is developed which allows a simple solution to a wide class of non-linear difurcation problems. The analysis shows that in order to determine the inital post bucking behaviour accurately, it is not necessary to solve more than the linear eigenvalue difference equation with similar accuracy.     

Multivariable approach to the finite difference solution of elasto-plastic problems

The solutions of plane elasto-plastic problems usually use one or two field variables—namely a stress functions or the displacements. The use of three, four or five field variables is investigated and it is concluded that the three stresses from the best basis for a multivariable approach. Attempts to solve the governing equations with an initial value technique were exhaustively tried and discarded in preference to a boundary value or elliptical technique. The problem solved to check the method is that of a hole in plane strain uniaxial tension. Effective plastic strain distributions are plotted, along with stress and strain concentration data and distributions of the mean to effective stress ratio. Comparison is made between solutions produced using incremental and deformation theories.     

Fortran computer program for inextensional bending of a doubly curved shell triangular element

A Fortran computer program is described which calculates physical quantities for a class of shell triangular elements undergoing inextensional bending. These elements are in quadratic parametric representation and may have positive, zero or negative Gaussian curvature. The exact inextensional bending solutions for the rectangular displacement components are cubic in the surface co-ordinates and the curvature changes are relatively slowly varying.     

A method for constructing explicit solutions to integral equations with difference kernels on finite segments

A method for constructing explicit solutions of integral equations with difference kernels on finite segments is proposed. Equations with kernels of sufficiently general type are considered. The method opens a promising direction for the investigation and practical application of solutions for this class of equations.